Observations and Dynamics of Circulations in the North Indian Ocean 9811958637, 9789811958632

Citation preview

Atmosphere, Earth, Ocean & Space

Julian P. McCreary Satish R. Shetye

Observations and Dynamics of Circulations in the North Indian Ocean

Atmosphere, Earth, Ocean & Space Editor-in-Chief Wing-Huen Ip, Institute of Astronomy, National Central University, Zhongli, Taoyuan, Taiwan

The series Atmosphere, Earth, Ocean & Space (AEONS) publishes state-of-art studies spanning all areas of Earth and Space Sciences. It aims to provide the academic communities with new theories, observations, analytical and experimental methods, and technical advances in related fields. The series includes monographs, edited volumes, lecture notes and professional books with high quality. The key topics in AEONS include but are not limited to: Aeronomy and ionospheric physics, Atmospheric sciences, Biogeosciences, Cryosphere sciences, Geochemistry, Geodesy, Geomagnetism, Environmental informatics, Hydrological sciences, Magnetospheric physics, Mineral physics, Natural hazards, Nonlinear geophysics, Ocean sciences, Seismology, Solar-terrestrial sciences, Tectonics and Volcanology.

Julian P. McCreary · Satish R. Shetye

Observations and Dynamics of Circulations in the North Indian Ocean

Julian P. McCreary Emeritus Professor of Oceanography University of Hawaii Honolulu, HI, USA

Satish R. Shetye Scientist (Retired) CSIR-National Institute of Oceanography Dona Paula, Goa, India

ISSN 2524-440X ISSN 2524-4418 (electronic) Atmosphere, Earth, Ocean & Space ISBN 978-981-19-5863-2 ISBN 978-981-19-5864-9 (eBook) https://doi.org/10.1007/978-981-19-5864-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The North Indian Ocean (NIO) is the part of the Indian Ocean where the ocean comes under the influence of monsoon winds. The region is important both scientifically and societally. Scientifically, it is interesting because its circulations are very different from those in the other tropical oceans. One reason for the difference is simply basin geometry: The NIO is small in comparison and divided into two sub-basins, the Arabian Sea and Bay of Bengal. Another is the forcing by monsoon winds, which are dominated by annual variability with a weak mean component and strong subseasonal (intraseasonal) oscillations. Societally, people in the countries that surround the NIO have used the ocean for trade, fisheries, and other maritime activities throughout recorded history, and the importance of these activities has recently risen exponentially, owing to the increase in the region’s population (currently about a sixth of the world’s population) and its economic development. Since early 1960, when the global community of oceanographers and meteorologists conducted the International Indian Ocean Expedition (IIOE), there has been a rapid increase in the observations that reveal characteristics of the large-scale circulation of the NIO. Simultaneously, ideas on the dynamics underlying NIO circulation evolved extensively. These advances have led to the need for a comprehensive source of information about NIO circulation and dynamics, particularly for new entrants to the field, both students and researchers. This book aims at addressing this need. Because similar processes occur in other oceans, the book is written so that its theoretical chapters (Parts II–IV) can be read separately from its observational ones (Part I). Thus, we expect that readers interested in wind-driven ocean dynamics in other regions will also find our book useful. The theoretical chapters are built around finding analytic solutions to a linearized version of the fluid equations, namely, the linear, continuously stratified (LCS) model (Sect. 5.2). Readers must therefore have an appropriate level of mathematical expertise. We address this issue by including almost all steps in derivations and by reviewing the mathematics required to obtain forced solutions (Chap. 9). As such, the ocean-dynamics chapters are appropriate for readers with only a basic knowledge of solutions to partial differential equations. The origins of the book go back well over a decade, when some physical oceanographers associated with the Council of Scientific and Industrial Research’s (CSIR’s) v

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National Institute of Oceanography, Goa (NIO-Goa), India recognized that such a book would be particularly helpful for the doctoral students at the institute. An important event that shaped its evolution was a summer school (NIOSS), entitled “Dynamics of the North Indian Ocean,” held at NIO-Goa during 2010. It not only inspired the book’s contents and organization, but also defined its most unique aspect: the inclusion of videos that illustrate each dynamical process. During morning sessions at the summer school, solutions to the LCS model were derived analytically, and in afternoon sessions participants obtained corresponding numerical solutions that could be viewed as videos. Feedback concerning the afternoon activity was all positive, indicating that the videos were critical in helping participants understand the more abstract, analytic solutions. We expect our readers will have a similar reaction. The set of videos in the book is an update and expansion of the set prepared at the summer school. A number of people, institutions, and agencies contributed to the preparation of various aspects of the book. Scientists who were helpful in discussing NIO observations and dynamics include: P. Amol, H. Annamalai, G. S. Bhat, John Boyd, Ted Burkhardt, Abhisek Chatterjee, Ted Durland, Tom Farrar, Eric Firing, Ryo Furue, Sulochana Gadgil, Lei Han, Ke Huang, Ashok Karamuri, Xiaopei Lin, Roger Lukas, Mike McPhaden, Dennis Moore, Ricardo Matano, Arnab Mukherjee, Kelvin Richards, Fabian Schloesser, D. Shankar, S. S. C. Shenoi, Stefan Smith, Alexander Soloviev, V. Vijith, P. N. Vinaychandran, Robert Weller, and William Young. Julian McCreary thanks Nova Southeastern University, the International Pacific Research Center (IPRC) at the University of Hawaii at Manoa, and US funding agencies (National Science Foundation, NSF; National Aeronautics and Space Administration, NASA; and National Oceanic and Atmospheric Administration, NOAA) for supporting his research that is reported in this book. He also acknowledges NIOGoa for supporting his visits to the institute under its Adjunct Scientist programme. Satish Shetye thanks NIO-Goa, and its parent organization CSIR, for their support during the three decades of his tenure at NIO-Goa (1982–2012). He remains grateful to the Ministry of Earth Sciences (earlier the Department of Ocean Development), Government of India, for research grants, and thanks Goa University for making it possible for him to continue his research during 2012–16. We also acknowledge the people who helped write the codes we used to integrate the LCS model and to make videos. The codes are all based on programs originally written for the NIOSS by Abhisek Chatterjee, Aparna Gandhi, Arnab Mukherjee, and D. Shankar. Special thanks go to Ryo Furue, for showing us how to update the NIOSS codes to a user-friendly form. Indeed, it is hard to imagine how the book’s videos could have been made without Ryo’s assistance, in particular, his guidance was invaluable to Julian McCreary for setting up the routines needed to run the model and video codes on his laptop. In making the videos we used PyFerret, a graphics

Preface

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and analysis product of NOAA’s Pacific Marine Environmental Laboratory (http:// ferret.pmel.noaa.gov/Ferret/). We thank Ryo Furue, Billy Kessler, Arnab Mukherjee, and V. Vijith for their help in writing PyFerret code. Finally, we much appreciate the assistance of V. A. Abhishek, P. Amol, N. Anup, G. S. Michael, R. Prasanth, V. Vijith, and P. N. Vinayachandran in making videos and figures of observational data. Honolulu, USA Goa, India

Julian P. McCreary Satish R. Shetye

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Observational Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Book Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1 1 4 5 7

Observations: Atmospheric Forcing and Ocean Response

2

Atmospheric Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Monsoons and the ITCZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 South Asian ITCZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 South-Asian Summer Monsoon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Precipitation and Orography . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Monsoon Winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 SST and Ocean Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Non-orographic Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Interannual Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 ENSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 IOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Sub-annual Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Madden-Julian Oscillations . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Quasi-biweekly Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Sub-weekly Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Diurnal Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 12 12 14 18 18 19 21 22 24 25 25 28 31 32 35 36 37 38

3

Ocean Forcing and the Surface Mixed Layer . . . . . . . . . . . . . . . . . . . . . 3.1 Ocean Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Climatological Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 40 40 48 ix

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3.2

4

Surface Mixed Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Mixed-layer Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Mixed-layer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Basin-wide Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Arabian Sea and Bay of Bengal . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 54 56 58 63 69

Ocean Circulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Thermohaline Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Bottom Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Deep Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Intermediate Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Upper Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Climatological Surface Currents and Sea Level . . . . . . . . . . . . . . . 4.2.1 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Sea Level and Geostrophic Velocity . . . . . . . . . . . . . . . . . 4.3 Southern-Hemisphere Circulations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 South Equatorial Thermocline Ridge . . . . . . . . . . . . . . . . 4.3.2 South Equatorial Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 South Equatorial Countercurrent . . . . . . . . . . . . . . . . . . . . 4.3.4 East Africa Coastal Current . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Equatorial Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Wyrtki Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Upwelling Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Equatorial Undercurrents and Deeper Flows . . . . . . . . . . 4.4.4 Intraseasonal Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Sumatra/Java Coast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Mean Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Semiannual and Annual Cycles . . . . . . . . . . . . . . . . . . . . . 4.5.3 Intraseasonal Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Andaman Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Background Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Annual Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Intraseasonal Variability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Bay of Bengal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Interior Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Intraseasonal Variability and Eddies . . . . . . . . . . . . . . . . . 4.7.3 East India Coastal Current . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Western-Boundary Gyres . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.5 Salt Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Monsoon Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Summer Monsoon Current . . . . . . . . . . . . . . . . . . . . . . . . .

71 72 72 74 77 79 80 83 84 86 90 90 92 94 95 95 96 100 101 104 105 105 106 107 107 107 108 109 110 112 114 114 115 117 124 125 126 126

Contents

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4.8.2 Winter Monsoon Current . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Arabian Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Lakshadweep High and Low . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 West India Coastal Current . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.3 Pakistan Coastal Current . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.4 Interior Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.5 East Arabia Coastal Current . . . . . . . . . . . . . . . . . . . . . . . . 4.9.6 Somali Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.7 Salt Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Marginal Seas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.1 Persian Gulf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.2 Gulf of Oman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.3 Red Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.4 Gulf of Aden . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II 5

127 127 128 128 129 137 138 140 142 147 148 148 151 153 157 160

Models

Ocean Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Ocean General Circulation Models . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linear, Continuously Stratified Model . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Vertical Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Barotropic and Baroclinic Modes . . . . . . . . . . . . . . . . . . . 5.2.5 Modal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Layer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Nonlinear 1 21 -layer Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Linear 1 21 -layer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Multi-layer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Potential Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

163 164 164 165 165 166 167 167 170 171 173 173 176 177 177 178 180 181 181

Part III Free Waves 6

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Wave Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Plane Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 General Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 186 186 186

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6.1.3 Other Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wave Groups in a Uniform Medium . . . . . . . . . . . . . . . . . . . . . . . . Impact of a Slowly-varying Medium . . . . . . . . . . . . . . . . . . . . . . . . Impact of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188 189 189 192 193

7

Midlatitude Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Gravity and Rossby Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Phase and Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Extension to Variable f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Critical Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Kelvin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Zonal Coasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Meridional Coasts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Boundary-Generated Rossby Waves . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Meridional Energy Propagation . . . . . . . . . . . . . . . . . . . . . 7.4 Waves Along a Slanted Coast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Kelvin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Rossby-Wave Pathways . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Critical Latitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Meridional Energy Propagation . . . . . . . . . . . . . . . . . . . . . 7.5.3 Zero-Group-Velocity Resonance . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 196 196 201 202 203 205 207 207 210 214 214 216 217 221 222 223 225 225 225 227 227

8

Equatorial Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Hermite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Equatorial Gravity and Rossby Waves . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Phase and Group Velocities . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Mixed Rossby-Gravity Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Equatorial Kelvin Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Relationship to Midlatitude Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231 232 235 236 240 242 243 244 246 247 248

6.2 6.3 6.4 6.5

Part IV Forced Solutions 9

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 9.1 Common Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 9.1.1 Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

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9.1.2 Basin and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 9.1.3 Radiation and Initial Conditions . . . . . . . . . . . . . . . . . . . . 9.1.4 Wind-stress Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dirac δ-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transform Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Switched-on Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Periodic Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 256 257 258 258 260 261 261 267

10 Ekman Drift and Inertial Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Midlatitude Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 Constant f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Variable f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Equatorial Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Single-mode Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Solutions with z-dependence . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271 272 272 280 283 284 284 291 295 296 297

11 Sverdrup Flow and Boundary Currents . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Interior Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 Pycnocline Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Boundary Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Western Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Other Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 300 301 302 303 303 303 308 310 312

12 Interior Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Simplified Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Constant- f Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Switched-On Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 Periodic Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Variable- f Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Switched-On Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Periodic Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313 313 315 316 320 321 322 322 326 330 331

9.2 9.3

9.4

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13 Coastal Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Simplified Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Switched-On Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Two-Dimensional Response . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Three-Dimensional Response, β = 0 . . . . . . . . . . . . . . . . 13.2.3 Three-Dimensional Response, β = 0 . . . . . . . . . . . . . . . . 13.3 Periodic Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.4 Solutions for a Slanted Boundary . . . . . . . . . . . . . . . . . . . 13.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 River-Driven Circulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

333 334 335 336 338 341 351 351 352 354 356 356 357 358

14 Equatorial Ocean: Switched-On Forcing . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Simplified Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Interior Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Small-Time Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Long-Time Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Reflections from a Single Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 East-Coast Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 West-coast Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Reflections from Both Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

361 362 363 364 367 375 375 379 381 381 382

15 Equatorial Ocean: Periodic Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Interior Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 v Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2 u Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 p Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.5 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Reflections from a Single Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Eastern Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Western Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Meridional Energy Propagation . . . . . . . . . . . . . . . . . . . . . 15.2.4 Reflections from a Slanted Boundary . . . . . . . . . . . . . . . . 15.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Equatorial Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.1 Equatorial Basin Resonance . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Zero-Group-Velocity Resonance . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

385 386 386 389 392 393 394 395 396 397 398 400 403 405 405 407 409

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16 Beams and Undercurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.2 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Undercurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.1 Conceptual Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.3 Videos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

413 413 414 419 428 430 430 431 432 436 436

17 Cross-Equatorial and Subtropical Cells . . . . . . . . . . . . . . . . . . . . . . . . . 17.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Two-Dimensional Overturning . . . . . . . . . . . . . . . . . . . . . . 17.2.2 Three-dimensional Pathways . . . . . . . . . . . . . . . . . . . . . . . 17.2.3 Transports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.4 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Model Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 STC Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3 CEC Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.4 Cross-Equatorial Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.5 Equatorial Roll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Video Captions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

441 442 443 443 447 452 455 455 456 459 462 465 469 471

Appendix A: List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 Appendix B: Simplified LCS Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Appendix C: Video Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

Chapter 1

Introduction

Abstract The goal of this book is twofold: to summarize observations of large-scale, climatological circulations in the region of the Indian Ocean north of 10◦S (the North Indian Ocean); and to describe the basic processes that determine them. The book is divided into four parts, which separately consider observations (Part I) and processes (Parts II–IV). Because the basic processes are linear, we illustrate them with analytic and numerical solutions to a linear model, namely, the linear continuously stratified (LCS) model. An essential part of the book are videos of these solutions, which are available on the web. Keywords North Indian Ocean · Process solutions · Videos of observations and solutions · Online address of videos · Walt Whitman · Bertrand Russell The North Indian Ocean (NIO) is the part of the Indian Ocean north of about 10◦S. Because the Asian landmass restricts the NIO to latitudes south of about 25◦N, the NIO is tropically confined and small in comparison to other oceans, as is evident in Fig. 1.1. In addition, while the other tropical oceans experience quasi-steady, easterly trade winds, the NIO comes under the influence of seasonally-reversing monsoon1 winds. All these differences (its small size, tropical location, and influence of monsoon winds) give the NIO circulations a unique character.

1.1 Observational Background Given its small size, the NIO is strongly influenced by the geometry and topography of the surrounding land (Fig. 1.2). The East African Mountains and Ethiopian Highlands are an important factor in generating the cross-equatorial winds that impact the western NIO. The Himalayas insulate the moist warm winds that support deep 1 The

word “monsoon” is derived from the Arabic word “mausam” for “season”.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_1. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_1

1

2

1 Introduction

Fig. 1.1 Map of the world ocean with bottom topography and land elevation in Lambert Cylindrical Equal Area projection. In this projection, the areas of the rectangles (φ = 30◦ , θ = 10◦ ) have the same relative size that they do on the surface of the earth. The NIO occupies about 5% of the total area of the world ocean

atmospheric convection from the dry cold winds farther north on the Tibetan Plateau (Boos and Kuang 2010), profoundly influencing the structure of the NIO wind and precipitation fields. The Western Ghats and the Arakan Range/Tenasserim Hills, lead to local precipitation maxima that freshen the Arabian Sea and Bay of Bengal, respectively. The downward slope of the Deccan Plateau towards the Bay of Bengal, ensures that the bay receives far more river runoff than the Arabian Sea, one of the reasons sea-surface salinity (SSS) is much fresher in the bay than in the Arabian Sea. Collectively, these topographic features determine the structure of the South-Asian monsoon winds, which in turn define the NIO circulation. Perhaps the most distinguishing feature of NIO circulations is their temporal variation, which, like the winds that drive them, reverse seasonally in many regions. One striking example is the Somali Current (SC), a western-boundary current as strong as the Gulf Stream in the Atlantic and Kuroshio in the Pacific, which flows northeastward during the Southwest Monsoon and southwestward during the Northeast monsoon. Furthermore, oceanic variability occurs on a wide range of time scales, again a reflection of the atmospheric forcing. Three prominent time scales are the annual cycle (the annual period plus its harmonics), intraseasonal variability (periods less than 100 days or so) and interannual variability (periods greater than one year). Although these time scales differ considerably, they tend to have similar amplitudes. This similarity complicates the analysis of observed phenomena, for example, making it difficult to isolate the cause and impact of each time scale. In the atmosphere, for example, a strong monsoon year often occurs because there are more active periods in the monsoon active-break cycle (Gadgil 2003): This increased, intraseasonal activity strengthens the overall forcing during the summer, thereby directly impacting the annual cycle.

1.1 Observational Background

3

Fig. 1.2 Topography of South Asia and NIO with rivers. Labels indicate the East African Mountains (EM), the Ethiopean Highlands (EH), Western Ghats (WG), Deccan Plateau (DP), Himalayan Mountains (HM), Tibetan Plateau (TP), Arakan Mountains (AM), Tenasserim Hills (TH); rivers Ganges (GN), Brahmaputra (BR), and Irrawaddy (IR); Red Sea (RS), Persian Gulf (PG), Gulf of Oman (GO), and Gulf of Aden (GA)

Another NIO uniqueness is the ubiquitous presence of basin-scale waves (Rossby, Kelvin, and coastally-trapped shelf waves), which are more apparent than in other oceans. One reason for their prominence is the tropical location of the NIO, where the propagation speeds of Rossby waves are high. A second is the high variability of the wind forcing; as a result, forcing in one region acts as a generator of waves that then propagate freely throughout the basin. A third is the presence of India, which enhances the visibility and importance of coastally-trapped waves in NIO dynamics. Video 1.1 illustrates the surface expressions of phenomena noted in the previous paragraph and discussed later in this book, showing the climatological annual cycle of sea-level anomalies determined from satellite observations during 1993–2019. Being based on a 27-year climatology, the video averages out intraseasonal oscillations and eddies, which typically don’t have a fixed time and location for their onset, and weakens the impact of interannual variability. The remaining signals, while still complex, exhibit remarkable regular features. The overall response in the video is a complex mixture of locally-forced responses and free waves. One indication of wave propagation is the prevalence of westwardpropagating signals (Rossby waves), most of which radiate away from either the eastern boundary of the basin or the west coast of India. During April/May, there is an eastward-propagating signal of high (red/yellow) sea level that crosses the

4

1 Introduction

basin along the equator (equatorial Kelvin wave); at the eastern boundary, it reflects as a wave packet that propagates north and south along the boundary (coastally trapped waves) and westward into the interior ocean (Rossby wave). During October/November, there is a similar, eastward-propagating wave and reflection. Also during October/November, a coastally-trapped signal of high sea level spreads first southward along the east coast of India, then around the tip of India, and finally northward along the west coast, a clear example of how east-coast processes are linked to west-coast currents. Finally, a notable locally-forced response occurs in the western Arabian Sea during the Southwest Monsoon: Beginning in June, sea level drops markedly there (becomes blue), first along the Somali and Arabian coasts and then farther offshore, and the low sea level persists until November.

1.2 Theoretical Background Historically, a hierarchy of models has been used to deal with this complexity, the models varying from systems that are simple enough to solve analytically (with paper and pencil) to state-of-the-art, oceanic general circulation models (OGCMs). Each model type has its own advantages and limitations. To produce the most realistic solutions, OGCMs are ideal as they contain parameterizations of most processes believed to be important in the ocean. For identifying, isolating, and analyzing basic processes, however, OGCMs are limited by their complexity and computational expense. For this purpose, simpler models are more useful, and at the same time their solutions are still sufficiently detailed to be validated against observations. To illustrate the ability of models to represent NIO observations, Videos 1.2 and 1.3 show sea-level anomalies (SLAs) from solutions to an OGCM and to a linear, continuously stratified (LCS) model (Sect. 5.2), respectively, which are comparable to Video 1.1. The two solutions are forced by observed wind products (see the video captions for details), and show the daily-climatological, sea-level anomalies generated by their respective forcings. The agreement between observed and modeled videos is striking: Both solutions are able to simulate the large-scale, observed features remarkably well. Further, the LCS solution simulates them as well as the OGCM one does, indicating that the simpler dynamics in the LCS model capture the essential physics of the observations. Notable differences among the videos are that the amplitudes of their large-scale, SLAs differ, with that of the OGCM (LCS) variability being somewhat weaker (stronger) than that for observed SLA. One reason for the different amplitudes is that the time periods used to prepare the climatologies are not the same, and so the climatologies do not include the same interannual signals. Another is that the wind products used to force the models differ from the actual winds; for example, the National Center for Environmental Prediction (NCEP) winds that force the OGCM are known to be weak in the Indian Ocean (e.g., Goswami and Sengupta 2003). Such model/data differences are expected (cannot be avoided), and do not detract from the overall agreement. Finally, note that because the LCS solution is obtained in a closed

1.3 Book Overview

5

basin, the sea-level signal propagates across the eastern boundary gap (from about 10–20◦S) and along the southern boundary; however, the closed boundaries do not significantly impact the climatological signals in the NIO of interest here.

1.3 Book Overview The purpose of this book is twofold: (1) to provide a comprehensive description of the large-scale, wind-forced circulations in the NIO, and (2) to understand the dynamical processes that account for them. Because these circulations are reproducible with simpler models (as evidenced by the comparison of Videos 1.3 and 1.1), we base our theoretical discussion on solutions to the LCS model. Specifically, we obtain a suite of LCS solutions under idealized forcings and in idealized basins, which are designed to isolate specific processes and simple enough to obtain analytically. The usefulness of the analytic solutions cannot be overstated: They allow the identification and interpretation of the underlying processes to a degree that would not otherwise be possible. With this set of process solutions in hand, it is then possible to see that similar responses occur in observations and realistic solutions (Videos 1.1–1.3). In this way, the process solutions provide pieces of the NIO “puzzle,” which can then be put together to provide a complete dynamical picture. To put it another way, the process solutions provide a “language” that facilitates (makes possible) discussion of more complex, ocean dynamics. Historically, the approach outlined in the previous paragraph has provided a powerful methodology for understanding ocean dynamics, not only in the NIO but other oceans as well. Indeed, it is hard to imagine how the more complex dynamics at work in OGCMs could ever have been understood without the theoretical foundation represented by the LCS solutions discussed in this book. At the same time, it is important to keep in mind their dynamical limitations, two of which are noted here. First and most importantly, because the LCS model lacks nonlinearities, its solutions cannot represent current instabilities that generate eddies; as a result, we can expect LCS solutions that develop narrow (likely unstable) currents are not realistic. Second, the coasts in the LCS model are represented by vertical walls without continental shelves, and consequently the internal waves that propagate along coasts are Kelvin waves rather than shelf waves; however, this simplification is not as severe as one might think because the coastal adjustments caused by Kelvin and shelf waves are so similar (for our purposes essentially the same). The book is organized into four parts. Part I reviews observed NIO phenomena, discussing: properties of the wind and precipitation fields over the NIO (Chap. 2), how this atmospheric forcing enters the ocean across the air-sea interface (Chap. 3), and the oceanic circulations generated by that forcing (Chap. 4). Part II provides an overview of ocean models, among other things highlighting how the governing equations of the LCS model are obtained from those for a typical OGCM (Chap. 5). Part III discusses free-wave (unforced) LCS solutions, first discussing general wave properties (Chap. 6), and then deriving specific characteristics of midlatitude (Chap. 7) and

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equatorial (Chap. 8) waves. Part IV, the rest of the book, reports forced solutions to the LCS model and, in Sect. 17.3, to a 2 21 -layer model. Topics covered are: common features among all the forced solutions, and the mathematical background needed for their derivation (Chap. 9); Ekman drift and inertial oscillations both at midlatitudes and near the equator (Chap. 10); Sverdrup balance and steady boundary currents (Chap. 11); adjustment processes in the midlatitude-interior, coastal, and equatorial regions of the ocean (Chaps. 12–15); vertically-propagating waves and undercurrents (Chap. 16); and the Indian Ocean’s shallow overturning cells (Chap. 17). The above organization separates observational and theoretical parts. That separation explicitly recognizes that the ocean dynamics and process solutions discussed in Parts II–IV are applicable to several NIO phenomena (as well as to phenomena in other oceans). A result of the separation is that Part I has many forward references to theoretical concepts discussed later. Conversely, Parts III and IV include links back to relevant observations discussed in Part I. (There is, however, no logical need for readers to follow those links to understand the text; they can do so at their own discretion.) A unique aspect of the book are videos that illustrate mathematical solutions and observed fields. They can be downloaded from (or viewed on) the web at the link to supplementary material provided at the bottom of the first page of each chapter. Captions for the videos are included at the end of each chapter. In addition, the videos can be downloaded from the site at https://drive.google.com/drive/folders/178RmQgw-wu3ry_Gv64at61D9zF640RK?usp=share_link. There, they are kept in subfolders “Videos” within folder “NIOBook.” From this site, readers can download video files as they prefer: one at a time, from a particular chapter, or throughout the entire book (the latter about 6.5 GB). Appendix C provides an overview of common aspects of the solutions. We recommend that readers go over Appendix C carefully before viewing the videos in Parts III and IV. In writing the book, we found it best (most logical) to discuss the videos after the derivation of each solution. On the other hand, although a complete understanding of a process requires knowledge of its underlying mathematics, it is also essential to have a visual impression of its impacts, which the videos provide. Therefore, in their initial reading of a chapter, we expect that many readers may find it useful first to read the chapter introduction and view the videos, before delving into mathematical derivations. We are interested in hearing from our readers any thoughts and questions they have about the book, particularly concerning errors they may find. To provide this feedback, readers can send an email to [email protected]. If useful for the entire readership, we will include such information in the subfolder “Feedback” in “NIOBook.” In conclusion, we have sought to write a “beautiful” book, one that is both informative and enjoyable to read. That task has not been easy, in part because beauty lies in the eyes of the beholder. For sure, all of our readers love the ocean, but some might not feel as highly about rigorous data analysis and theory. It is our hope that we have spared our readers the predicament narrated so elegantly by Walt Whitman about an astronomy lecture he attended · · ·

Video Captions

7

When I heard the learn’d astronomer, When the proofs, the figures, were ranged in columns before me, When I was shown the charts and diagrams, to add, divide, and measure them, When I sitting heard the astronomer where he lectured with much applause in the lectureroom, How soon unaccountable I became tired and sick, Till rising and gliding out I wander’d off by myself, In the mystical moist night-air, and from time to time, Look’d up in perfect silence at the stars.

Instead, we hope they come to feel the way Bertrand Russell does about mathematics when he wrote · · · Mathematics, rightly viewed, possesses not only truth, but supreme beauty–a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

We certainly feel this way about the remarkable structures in the NIO that data analysis and theory have revealed, and hope our readers will as well.

Video Captions Video 1.1 Daily-climatological SLA, prepared from Archiving, Validation, and Interpretation of Satellite Oceanographic (AVISO) data for the period 1993–2019 (Sect. 4.2.2). The data resolution is 25◦ , the unit of the color bar is cm, and blackdashed contours indicate 200-m isobaths. Video 1.2 Daily-climatological SLA from an OGCM solution forced by daily NCEP winds for the period 2000–2017. The OGCM is the Ocean model For the Earth Simulator (OFES), developed at the Japan Marine Science and Technology Center (JAMSTEC). The resolution of the model grid is 0.1◦ , which is sufficiently high to resolve eddies. To prepare the video climatology, daily sea-level data from the OFES solution for 2000–2017 was downloaded from the Asian Pacific Data Research Center (APDRC) and then daily averaged. The resolution of the model grid is 0.1◦ , which is sufficiently high to resolve eddies. The unit of the color bar is cm, and black-dashed contours indicate 200-m isobaths.

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Video 1.3 Daily-climatological SLA from a solution to the LCS model (Sect. 5.2) forced by daily winds from the daily Scatterometer Climatology of Ocean Winds (SCOW) data set for the period 1999–2007. The solution is a superposition of the responses to the n = 1–5 baroclinic modes. The resolution of the model grid is 0.25◦ , the unit of the color bar is cm, and black-dashed contours indicate 200-m isobaths.

Part I

Observations: Atmospheric Forcing and Ocean Response

Chapter 2

Atmospheric Circulation

Abstract The South Asian Monsoon dominates the atmospheric circulation over the North Indian Ocean. It results from the annual, meridional migration of the Intertropical Convergence Zone (ITCZ), the global-scale tropical rain belt. During boreal summer, the ITCZ lies in the northern hemisphere, ensuring that atmospheric fields associated with the summer monsoon (e.g., winds and precipitation) have significantly higher magnitudes than they do during the rest of the year. The spatial and temporal structures of the ITCZ result from complex interactions among the atmosphere, orography, and ocean. One consequence of these interactions is that during summer the Bay of Bengal becomes the most active region of the global ITCZ. Superimposed on the climatological-mean annual cycle is variability at interannual and sub-annual time scales, with amplitudes as large as that of the annual cycle itself. Keywords Intertropical Convergence Zone · Annual migration · South-Asian summer monsoon · Climatological precipitation and winds · Interannual and sub-annual variability In this chapter, we review properties of the atmospheric circulation over the NIO, focussing on its near-surface features that impact the ocean. Much of the chapter discusses their climatological annual cycle, commonly referred to as the South-Asian monsoon, Indian monsoon, or (when the location is clear) simply the monsoon. We first note the fundamental connection between all tropical monsoons and the tropical rain-bearing belt, the Intertropical Convergence Zone (ITCZ), and comment on the special characteristics of the South-Asian ITCZ (Sect. 2.1). Then, because the SouthAsian summer monsoon is much stronger than its winter counterpart, and of direct relevance to many topics covered in this book, we discuss its properties in detail (Sect. 2.2). Finally, we provide an overview of prominent atmospheric variations over the NIO at interannual (Sect. 2.3) and sub-annual (Sect. 2.4) time scales, which are often strong enough to impact the climatological variability significantly.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_2. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_2

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2.1 Monsoons and the ITCZ 2.1.1 General Properties Figure 2.1 shows global, climatological-mean precipitation and near-surface wind fields over the ocean during July and January. The most conspicuous feature in both panels is a belt of high precipitation girdling the equatorial region, typically with convergent, near-surface winds to either side. These features (high precipitation and convergent winds) define the ITCZ. Similar figures for other months show that the ITCZ is present throughout the year. In the Atlantic and Pacific its width is roughly 10◦ , although it varies considerably in both location and time.1 In the Western Pacific, the core of the belt is split into northern and southern parts most of the year. In the Indian Ocean during winter (bottom panel), the ITCZ extends across the basin at about 8◦S. In contrast, during summer (top panel) it is located from about 10◦S– 20◦N, and is broken into several parts whose precipitation varies from intense to negligible. In a global sense, the ITCZ is the location of the rising branch of the Hadley circulation. In this atmospheric overturning cell, air rises in the tropics, flows poleward at the top of the troposphere, sinks in the subtropics, and returns to the tropics at low levels. In the Atlantic and Pacific, the low-level branch is apparent in Fig. 2.1 during both seasons, with the ITCZ positioned between convergent trade winds. In the Indian Ocean, a similar pattern is present during winter. During summer the wind field is more complicated, but the Hadley circulation is still apparent when the wind field is averaged over the zonal extent of the ocean (see the discussion of Fig. 2.3 below). The ITCZ is of special interest to the NIO, as it determines both where rainfall occurs and the distribution of winds. Globally, the annual-mean ITCZ position occurs near 6◦N. That northern-hemisphere location has been linked to the Atlantic Ocean’s transport of energy northward across the equator, making the northern hemisphere warmer than the southern hemisphere (Schneider et al. 2014). The ITCZ migrates seasonally around its mean position, and the extent of migration differs with longitude. Over oceans, the migration tends to be small ( 10◦ ), whereas over continents, or both continents and oceans, it is larger (∼20◦ ) and interhemispheric (Adam et al. 2016a, b). Wherever the migration is large, the underlying regions experience wet and dry seasons during a year: a wet season when the ITCZ is overhead, and a dry season when it is not. The monsoons over South and East Asia, Indonesia-Australia, North and South Africa, and North and South America are of this sort (Wang and Ding 2008). 1

The Atlantic and Pacific ITCZs have been known to mariners for as long as those oceans have been used for trade and travel. Traditionally, they were referred to as the “doldrums,” because they are regions of weak mean winds, in striking contrast to the strong and steady trade winds just to their north and south. At the same time, they are regions of high storm activity. For both reasons, ITCZs were dreaded by sailors in the pre-steam-engine period.

2.1 Monsoons and the ITCZ

13

Fig. 2.1 Monthly climatological 1000-mb winds (m/s) and rain (mm/day) during July (upper panel) and January (lower panel). The wind data are from the NCEP-National Center for Atmospheric Research (NCAR) Reanalysis for 1981–2010 (downloaded from https://iridl.ldeo.columbia.edu) and precipitation data from the GPCP V2.2 climatology (Adler et al. 2018)

Although the causes of ITCZ migration are not completely understood, it is clear that moist processes are crucially important. They are an integral part of the dynamics of the Hadley circulation, indeed its driving force. In the tropics, moist surface air is warmed until it becomes convectively unstable and begins to rise. As it rises, water vapor condenses to generate rainfall, and the resulting release of latent heat lowers air density, thereby strengthening the upward flow. A consequence of the precipitation is that the poleward and sinking branches of the Hadley cell are composed of dry air, with moisture replaced in the equatorward branch by evaporation across the sea surface. A key dynamical question is: What processes determine the location of the rising branch and, hence, tropical precipitation? In the tropical oceans, convective instability occurs if sea-surface temperature (SST) is higher than about 28◦C (Gadgil et al. 1984; Graham and Barnett 1987; Sud and Walker 1999a, b). As a result, the ITCZ shifts to the summer hemisphere with the warmest SST. On the other hand, the SST condition is necessary but not sufficient, that is, there can be areas with SST higher than 28◦C without precipitation. Factors that prevent precipitation in regions where SST >28◦C are complex, involving remotely-generated signals and land interactions.

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2.1.2 South Asian ITCZ 2.1.2.1

Characteristics

During boreal winter the ITCZ lies over the Indian Ocean at about 8◦S (Fig. 2.1, bottom panel), and it begins to migrate northward in boreal spring. Schneider et al. (2014) have discussed the nature of this migration, comparing it with those over the Pacific and Atlantic Oceans. Over these oceans, the average location of the maximum precipitation in the rain belt varies smoothly (roughly sinusoidally) in time from 2◦N during boreal winter to 9◦N during summer (Fig. 2.2, left panel). In striking contrast, the NIO migration is more abrupt during both the onset and withdrawal of the summer monsoon, and its location shifts from 8◦S to 20◦N, four times the amplitude in the Pacific and Atlantic (Fig. 2.2, right panel). Indeed, the latitudinal range of summertime precipitation in the South-Asian region is the largest of the entire tropical rain belt (Figs. 2.1 and 2.2). Another characteristic of the summer ITCZ over the Indian Ocean is that it has two branches, one near 20◦N and the other about 5◦S. The southern branch has weaker precipitation and occurs throughout the year. The northern branch forms rapidly during the monsoon onset and persists throughout the season; during the withdrawal of the summer monsoon, it moves southward and merges with the southern branch. One difference between the two branches and ITCZs elsewhere is the nature of the seasonal-mean winds. Over the Indian Ocean, both branches are associated with prominent winds (Sect. 2.2.2): for the northern branch, southwesterlies or westerlies over the Arabian Sea and southwesterlies over the Bay of Bengal; and southeasterlies for the southern branch. In contrast, the seasonal-mean winds in the latitude bands of the Pacific and Atlantic ITCZs are weak (doldrums).

Fig. 2.2 Seasonal migration of the ITCZ over a the Pacific (160◦ E–100◦W) and b the South-Asian monsoon region (65◦ E–95◦E). Zonally averaged mean precipitation (colour scale) and surface winds (vectors) are shown as a function of time of year. The location of maximum precipitation within the rain belt is marked by red lines. The precipitation data are the daily TMPA data (from Tropical Rainfall Measuring Mission, TRMM; see Liu et al. 2012). The data were smoothed temporally and meridionally by robust local linear regression, spanning 11 days in time and 1◦ in latitude. The wind data are the 10 metre winds from the ECMWF interim reanalysis for the same years (Dee et al. 2011). The longest wind vector (in South Asian monsoon region at 18◦S in September) corresponds to a wind speed of 9.1 m s−1 . From Schneider et al. (2014).

2.1 Monsoons and the ITCZ

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Fig. 2.3 Climatological Hadley circulation averaged over 70–90◦E during January (left panel) and July (right panel). The figure is based on the dataset ERA-40 (Uppala et al. 2005), which is a reanalysis of meteorological observations from September 1957 to August 2002 produced by the European Centre for Medium Range Weather Forecasts (ECMWF). The vertical axis shows atmospheric pressure in millibars. The red arrows mark approximate positions of maximum precipitation. Taken from Gadgil (2018). © Indian Academy of Sciences. Used with permission

In the northern hemisphere of the NIO, the summertime winds are generally southwesterly, leading to the monsoon at this time being referred to as the “southwest” or “summer” monsoon. Similarly, the wintertime winds are northeasterly, leading to the names “northeast” and “winter” monsoon. The Northeast-Monsoon winds, which originate in the mountainous terrain of South Asia, are cool and dry. In contrast, the Southwest Monsoon winds come from the tropical ocean, and are warm, moist, and stronger than the winter winds.

2.1.2.2

Hadley Circulation

As expected, the ascending and descending branches of the Hadley circulation shift their locations as the ITCZ migrates. This shift is illustrated in Fig. 2.3, which shows climatological meridional and vertical velocities averaged over the width of the NIO (from 70–90◦E) during January and July, at which times the ascents take place approximately from 10◦S–4◦N and 5◦ –20◦N, respectively. The wider region of ascent during July reflects the broadening of the ITCZ during the summer monsoon. Also note that the upward velocity has a weak relative maximum during July centered near 5◦S, consistent with the band of precipitation there (Fig. 2.2, right panel). The locations of maximum precipitation (red arrows) are located at about 5◦S (22◦N) in January (July).

2.1.2.3

SST Impact

Although the processes that determine the above characteristics of the South-Asian ITCZ (large migration, abrupt onset and withdrawal, and large meridional extent) are complex, a major factor appears to be the seasonal evolution of SST (Fig. 2.4;

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Video 2.1). Regions of tropical oceans where SST is in excess of 28◦C, thereby favoring precipitation, are referred to as “warm pools.” Consistent with the location of strongest solar heating and the rising branches of the Hadley circulations, the NIO warm pool during summer (June/July/August, JJA) is confined primarily to the northern hemisphere and vice versa during winter (December/January/February, DJF). The impact of solar heating on summertime SST is in fact more evident earlier during April/May, before ocean processes begin to cool the northern NIO (Sect. 2.2.3) when the warm pool extends across the entire basin; at this time, its area peaks at 24×106 km2 , a value close to that of the springtime warm pool in the Pacific (26×106 km2 ; Vinayachandran and Shetye 1991). During JJA, SST drops in the western Arabian Sea due to ocean processes (Sect. 2.2.3), and the area of the warm pool decreases. Finally, note in Fig. 2.4 and Video 2.1 that SST rises rapidly above 30◦C in the northeastern Bay of Bengal during April, a consequence of the thinness of the oceanic mixed layer there (Sect. 3.2). Jiang and Li (2011) noted that deep atmospheric convection over the northeastern bay starts about 10 days after the April warming, marking the ITCZ’s northward shift (Fig. 2.2). In addition, they observed that the warming favors the northeastward movement of the monsoon trough and the formation of a region of high convective instability, like the monsoon onset over the South China Sea and western North Pacific (Wu and Wang 2001).

2.1.2.4

History

It is noteworthy that the linkage of the South-Asian monsoon to ITCZ migration is a relatively new idea. In the 17th century, Halley (1686) hypothesized that heating of the Asian landmass by the sun, particularly over the Tibetan Plateau, was the driving force for the South-Asian monsoon, essentially viewing it as a large-scale, sea- and land-breeze (Sect. 2.4.4). That idea prevailed unchallenged for about two centuries. As meteorological observations of the South Asian monsoon improved, however, doubts about Halley’s theory arose (e.g., Blandford 1886; Simpson 1921), with Blandford (1886) first proposing the monsoon/ITCZ linkage. As global meteorological observations increased, the idea gained wider acceptance (Riehl 1954, 1979; Charney 1969), particularly with the advent of cloud-cover observations from satellites (Sikka and Gadgil 1980). Most recently, support for the idea has come from numerical models that are able to simulate the South-Asian monsoon realistically (e.g., Sabeerali et al. 2013; Sperber et al. 2013; Boos and Kuang 2010, 2013). While today the monsoon/ITCZ linkage is largely accepted (e.g., Gadgil 2003; Schneider et al. 2014), it is also recognized that ITCZ structure is strongly impacted by factors other than solar heating, such as the processes that determine moist convection and atmospheric interactions with the ocean and land topography.

Fig. 2.4 Annual cycle of climatological SST (◦C) in the NIO from the World Ocean Atlas 2018. See Locarnini et al. (2019; Section 2.1) for details on data collection and analysis

2.1 Monsoons and the ITCZ 17

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2.2 South-Asian Summer Monsoon The structure of the South-Asian summer monsoon is far more complex than suggested by the zonally-averaged fields in Figs. 2.2 and 2.3. This complexity results from orography, wind-driven ocean circulations, and zonal SST variations. We consider each of these factors in Sects. 2.2.1, 2.2.2, and 2.2.3, respectively. To conclude, we discuss summertime storm activity and other non-orographic processes, which also contribute to the overall rainfall (Sect. 2.2.4).

2.2.1 Precipitation and Orography Figure 2.5 plots precipitation over South Asia and East Africa during summer (June/July/August/September, JJAS; see Video 2.2 for its time development.) Rainfall occurs throughout most of the domain, with distinct regions of elevated rainfall. Four regions of very high rainfall occur: along the coasts of western India, Myanmar, and Thailand/Malaysia; and against the Himalaya foothills. Convection tends to spread westward or west-north-westward from these regions, its intensity decreasing with distance. The spread is largest in the Bay of Bengal, where it extends to the Indian subcontinent and is often linked to the arrival of rain-bearing, low-pressure systems (Sect. 2.2.4). Regions of moderate precipitation are located: centered along 5◦S between 50◦E and Sumatra, and over India and East Africa.

Fig. 2.5 Rainfall (m) during June–September, based on TRMM (Tropical Rainfall Measuring Mission) climatology for the period 1998–2015. The green curves show contours for 500-m land elevation

2.2 South-Asian Summer Monsoon

19

The regions of very high rainfall are clearly linked to neighboring orography (Xie et al. 2006), namely, the Sahyadri (also called the Western Ghats, India), Arakan (Myanmar), and Tenasserim (Thailand/Malaysia) mountain ranges (Fig. 1.2). Given the prevailing westerly winds during summer (Sect. 2.2.2), these mountains provide the uplift needed to trigger atmospheric convection: Even though their height is less than a kilometer, they are still able to anchor centers of precipitation on their windward sides. Uplift, associated with low-pressure systems (LPSs; Sect. 2.2.4) that originate over the Bay of Bengal, causes the precipitation along the Himalayan foothills and over the Deccan plateau of India. Uplift, due to the prevailing southerly winds in the presence of Ethiopian/Kenyan mountains (Fig. 1.2; Sect. 2.2.2) provides the precipitation over East Africa. The weaker rainfall maximum off the Sumatran west coast results from the southern branch of the South Asian ITCZ. Mountain ranges not only generate high precipitation on their windward side, but also form rain shallows on their leeward side. Prominent examples are the regions of very low rainfall east of the southern tip of India and Sri Lanka (Arushi et al. 2017). Another example is caused by mountains of the Andaman archipelago (near 12◦N, 90◦E), with stronger (weaker) rainfall on its westward (eastward) side. One expects a shadow to form east of the Western Ghats; however, it is weak or absent because the region is filled with precipitation from westward-propagating, low-pressure systems (Sect. 2.2.4).

2.2.2 Monsoon Winds In late April and May, rainfall starts over the eastern part of the NIO north of the equator and, to provide a source for the rising air, surface winds begin to converge onto precipitating regions. Figure 2.6 shows the NIO winds during July at the height of the summer monsoon. (See Video 2.2 for their climatological annual cycle.) Within the latitudinal band of the ITCZ (north of the equator), the convergent winds have a westerly component. This structure is in marked contrast to the Atlantic and Pacific, where winds within the ITCZ band are weak and have a strong easterly component to either side (Fig. 2.1). Several factors determine the predominance of northern-hemisphere westerlies. Most important is that the strong precipitation that drives the winds is zonally confined by orography (Fig. 2.5 and Video 2.2), rather than being distributed relatively uniformly along a latitude band as in the Pacific and Atlantic (Fig. 2.1). The atmosphere responds to this zonally-confined forcing both locally and remotely by radiating waves, the former generating rising air and the latter providing convergent, low-level, moist winds needed to maintain the convection. A prominent part of the remote response is the excitation of long-wavelength Rossby waves with westward group velocity (similar to wind-generated, oceanic Rossby waves discussed in Chap. 12 and elsewhere in the book). As a result, the convergent winds extend

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Fig. 2.6 July 1000-mb monthly climatological (1981–2010 base period) winds from the NCEPNCAR Reanalysis (downloaded from https://iridl.ldeo.columbia.edu)

primarily westward from the forcing region, that is, atmospheric dynamics prevents convergent easterlies from developing east of the forcing region. Another factor is the Himalayas, which tend to block any convergent flow from the north. A third is that it is difficult for air to cross the equator: In the absence of mixing, flows must conserve potential vorticity (PV, Sect. 5.4), which provides a strong constraint against cross-equatorial flow owing to the Coriolis parameter changing sign. To this point, it is noteworthy that cross-equatorial flow does exist in the western ocean (Fig. 2.6), indicating that the importance of mixing in the planetary boundary layer (PBL) there. Westerly flow across the northern NIO is eventually blocked by the East African Mountains (Fig. 1.2), and as a result cross-equatorial flow must develop to supply air for the westerlies. At the peak of the summer monsoon, it becomes a strong jet (Findlater 1969; Hart et al. 1978), and Fig. 2.7 illustrates its structure on July 4, 1977. Anderson (1976) proposed that the mountains provide the needed dissipation to overcome the potential-vorticity constraint: The core of the jet (Fig. 2.7) hugs the East African Mountains, thereby dissipating potential vorticity in a frictional boundary layer (similar to oceanic western-boundary currents, as discussed in Sect. 11.2.1 and elsewhere). Thus, the source of air for northern-NIO convection ultimately arises from regions of sinking air (subsidence) south of the equator, a remarkable example of interhemispheric exchange.

2.2 South-Asian Summer Monsoon

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Fig. 2.7 Contours of northward wind component (m/s) in a cross-section at the equator. The data were collected on July 4, 1977. The contours clearly show the low-level jet with its core hugging the eastern slopes of the East African Mountains. After Hart et al. (1978). © American Meteorological Society. Used with permission

2.2.3 SST and Ocean Processes Figure 2.8 plots climatological SST during summer (JJAS). The SST pattern is striking, with cool SST throughout most of the Arabian Sea, south and east of India/Sri Lanka, and extending into the southern Bay of Bengal. Even the interior of the bay is cooler than it was during April/May (Fig. 2.4; Video 2.1). This pattern cannot be caused by solar heating alone; rather, it results from the ocean’s response to forcing by the monsoon winds. Near the coasts of Somalia and Oman, alongshore winds of the Findlater Jet drive offshore (southeastward) Ekman drift (Fig. 2.6 and Video 2.2; Chap. 10), which upwells cool subsurface water to the ocean surface and then advects the upwelled water offshore. In the interior of the Arabian Sea, the strong winds cool SST by surface evaporation and by entrainment of subsurface water into the mixed layer (Sect. 3.2). In the southeastern Arabian Sea, the westerlies develop a northerly component near India (Fig. 2.6 and Video 2.2), which again leads to offshore (southeastward) Ekman drift, coastal upwelling, and SST cooling. In the region surrounding Sri Lanka, open-ocean Ekman pumping drives additional upwelling (Sects. 4.7.4, 12.2, and elsewhere). Cool surface water from both of these sources is advected into the southern Bay of Bengal by the swift Summer Monsoon Current (Sect. 4.8.1). Elsewhere in the bay, SST cooling is weak despite the winds having a similar structure to those in the Arabian Sea. One reason for this difference is that the southwesterlies in the bay are weaker than in the Arabian Sea (Fig. 2.6). Another is that the near-surface, oceanic stratification is stronger, owing to the large freshwater flux into the bay from rainfall and river runoff that makes near-surface waters less dense. A third is the summertime deepening of the thermocline along the eastern boundary of the bay due to the reflection of the spring Wyrtki Jet (WJ; Sect. 4.7.3.4): Part of this signal propagates around the perimeter of the bay where it tends to deepen the thermocline along the north and east coasts of India (visible as a sea-level rise in Fig. 4.7 below and Video 1.1). These factors ensure that coastal upwelling along

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Fig. 2.8 SST (◦C) averaged over summer months (JJAS), based on World Ocean Atlas 2018 (Locarnini et al. 2019). The dashed countours show 200-m isobaths

the continental boundaries of the bay is not usually strong enough to bring cool subsurface water to the surface (Sect. 12.2.1), and so SST remains warm there; in the interior of the bay, the stronger stratification weakens mixed-layer entrainment (Sect. 3.2), again inhibiting surface cooling. The SST pattern in Fig. 2.8 has a profound impact on precipitation (Fig. 2.5 and Video 2.2). Overall, the two distributions have a similar structure, consistent with the constraint that SST >28◦C for deep convection to occur. In the Arabian Sea, convection is weak or absent in regions where SST 28◦C there. Near and just south of the equator, the bands of elevated SST and rainfall nearly overlap. There are also instances where the patterns differ, indicating that factors other than SST (orography, signals from remote precipitating regions, ocean processes, etc.) also impact rainfall. For example, rainfall is absent east of Sri Lanka even though SST is high there (Sect. 2.2.1), a consequence of the Sahyadri and SriLankan rain shadow; likewise, rainfall is weak off Pakistan, despite SST being high.

2.2.4 Non-orographic Rainfall Although the most intense, summer-monsoon precipitation occurs along mountain ranges, there is significant rainfall over areas where orography plays no role. Five such areas can be identified in Fig. 2.5 and Video 2.2: the interior of the Bay of

2.2 South-Asian Summer Monsoon

23

Bengal, much of and India, the eastern Arabian Sea, off the Sumatran coast, and a rainfall band along 5◦S. In the first three regions, rain-bearing, convective systems account for much of the summer precipitation. Such systems are associated with cyclonic winds and low surface pressure, and are referred to as low-pressure systems (LPSs). They form in almost all monsoon regions of the world throughout the year, with their frequency of genesis typically peaking during summer, and they account for a large fraction of summer precipitation (Hurley and Boos 2015). Ranked in order of strength, LPSs are classified as a low, depression, storm, or severe storm (Mooley and Shukla 1989). The area of an LPS within which winds are significantly high varies inversely with LPS strength, being larger for lows and depressions than for storms and severe storms. Therefore, although storms and severe storms have more intense precipitation, lows and depressions are the main contributors to seasonal rainfall because they cover a larger area (Sikka 1977; Gadgil 2003). These three regions have received special attention, because they are linked to seasonal rainfall over the Indian subcontinent. Sikka (1980) pointed out that, during a summer monsoon with good rainfall over India, the number of days when LPSs are active over the three areas is significantly higher than during a monsoon with deficient rainfall. Based on data in Indian Daily Weather Reports from 1888 onward, Mooley and Shukla (1989), Sikka (2006), Krishnamurthy and Ajaymohan (2010) and Praveen et al. (2015) identified other prominent characteristics of summertime LPSs. First, they are either generated within the three areas by various instability processes, mostly north of 15◦N, or they are remnants of systems that cross over from the Pacific. Figure 2.9 illustrates locations of summertime (JJAS) LPS genesis within the three areas, and it indicates that on average about a dozen LPSs form each year, two thirds in the Bay of Bengal. Second, the number of days when LPSs are active is approximately half of the summer monsoon, with about 70% (25%) having a life span less than 5 days (6–10 days). Third, although their preferred propagation direction is west-northwest, LPSs that move northward or westward are not uncommon. Finally, their propagation speeds vary from 2–12◦ per day, and their mean central pressure anomaly is 5.7±3.6 mb (Mooley and Shukla 1989). Praveen et al. (2015) estimated that about 60% of the summer-monsoon precipitation over eastern and central India is associated with LPSs. The western coastal region of Sumatra experiences precipitation throughout the year due to diurnal processes. The diurnal cycle of precipitation peaks in the afternoon over land and at night or early morning offshore (Kamimera et al. 2012). Yokoi et al. (2017) proposed that gravity waves emanating from the convective systems over land play a significant role in the offshore migration of this precipitation zone. Further, the amplitude of the diurnal cycle is related to Madden-Julian Oscillations (MJOs; Sect. 2.4.1), being more intense when the active MJO phase is present over the eastern Indian Ocean. There is a belt of precipitation located from 10◦S to the equator and from east of about 50◦E to just west of Sumatra (Fig. 2.5 and Video 2.2) throughout the year. The processes that generate this precipitation have not received specific attention; however, they may be similar to those that cause the Pacific and Atlantic ITCZs.

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Fig. 2.9 Tracks (blue lines) of 1520 low pressure systems covering 6841 days formed during the JJAS in the period 1888–2003. The red (black) dot shows the location of genesis (termination). From Krishnamurthy and Ajayamohan (2010). © American Meteorological Society. Used with permission

2.2.5 Summary From the above, it is clear that the South-Asian summer monsoon is a remarkable example of coupled interactions between the atmosphere, orography, and the ocean. To summarize the interactions, consider the development of the summer monsoon. When heating strengthens in the northern hemisphere during spring, SST becomes higher than 28◦C there. As a result, convective instability begins in the northeastern Bay of Bengal where the SST criterion is satisfied, likely due to the mixed-layer being thin there. As convection over the bay strengthens, westerly winds form to supply the necessary low-level air, and the convection becomes orographically locked to the Arakan and Tenasserim mountain ranges. The westerly winds quickly extend westward across the bay and the Arabian Sea due to Rossby-wave propagation. They are forced to rise as they cross the Western Ghats, and this uplift generates rainfall along the Indian west coast. The Rossby waves are blocked by the East African Mountains, thereby generating the Findlater Jet. At this point, oceanic processes begin to impact the atmosphere. In the Arabian Sea, the strong monsoon winds force coastal upwelling along the western boundary of the basin and lead to evaporative cooling and mixed-layer entrainment in its interior. These processes cool SST below 28◦C throughout much of the basin, inhibiting convection there. In the Bay of Bengal, similar processes occur but more weakly since the winds are weaker than in Arabian Sea. In addition, the layer of low-salinity surface water in the northern bay resists entrainment. As a result, SST remains higher than 28◦C in the bay, allowing convection to occur there. Given that SST remains warm, the bay is capable of supporting deep convective processes in the atmosphere, making it the most active region of the ITCZ associated with South Asian summer monsoon. An essential component of ITCZ activity there is the formation and movement of LPSs. After forming mostly over the northern

2.3 Interannual Variability

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and eastern bay, they typically move northwestward but often either north or west, bringing precipitation to the regions over which they pass. While topographic rainfall is intense, it is localized over a small area. In contrast, precipitation associated with LPSs is locally less intense, it is spread over a larger area. As such, LPSs play a significant role in making South Asia livable.

2.3 Interannual Variability The primary example of interannual variability in the global ocean and atmosphere is El Nino and the Southern Oscillation (ENSO). Analyses of long-term records of SLP and SST data, band-passed on ENSO time scales (3–6 years), reveal a signal of global extent that is associated with eastward propagation (e.g., Allan 2000; White and Cayan 2000; White and Tourre 2003). These studies highlight common properties of ENSO signals that exist throughout the entire data records. During the past two decades, research has focused on properties of individual interannual events and on the coupled, ocean-land-atmosphere processes that determine them (e.g., Cai et al. 2019). One finding is that individual ENSO events have very different characteristics. Another is that a second climate event, the Indian Ocean Dipole (IOD), appears to be generated by coupled processes within in the Indian Ocean and can occur independently from ENSO. As is common among climate scientists today, our discussion of ENSO and IOD views them as distinct events. On the other hand, they often occur together and both involve similar coupled processes. Consistent with the aforementioned data analyses, then, an alternate interpretation of ENSO and IOD is that they describe a single ENSO/IOD event, with the two parts varying in their relative strengths from one event to another.

2.3.1 ENSO ENSO is a dominant mode of Earth’s natural climate variability, generated by ocean-atmosphere interactions in the tropical Pacific. Prominent characteristics of the “warm” phase of ENSO include: warming in the eastern, equatorial Pacific Ocean (EEPO) and off Peru, the El Nino part of ENSO; and weakening of the zonal gradient of sea-level pressure (SLP) across the basin, the Southern-Oscillation part. Typically, warm events are phase-locked to the annual cycle, with eastern-Pacific SST anomalies developing during boreal summer, peaking in winter, and decaying in the following spring. During the 20th century there were 9 strong and 16 moderate El Niño events, occurring roughly every 4–6 years with strongest events at intervals of ∼20 years.

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2.3.1.1

2 Atmospheric Circulation

History

The discovery of the strong connection of El Nino to the Southern Oscillation was a remarkable scientific achievement. As part of his long-term (1904–24) investigation of the cause of droughts in the Indian summer monsoon, Sir Gilbert Walker discovered that SLP anomalies in Darwin, Australia, and Tahiti vary out-of-phase—when SLP is high over one region, it is low over the other—and he named the phenomenon the Southern Oscillation (Walker 1924). Decades later, scientists at the University of California at Los Angeles (UCLA) were requested by the Inter-American Tropical Tuna Commission (IATTC) to investigate the cause of El Nino events, which were accompanied by a collapse of the Peruvian fishing industry. One of those scientists, Jacob Bjerknes, noticed that El Nino events occur only when SLP at Tahiti minus that at Darwin (a Southern Oscillation index, SOI) peaks at negative values. Indeed, given the high correlation between time series of the SOI and eastern-Pacific SST anomalies (an El Nino index), he concluded that the two phenomena, El Niño and the Southern Oscillation, are different aspects of a single dynamical system (Bjerknes 1966, 1969).

2.3.1.2

Properties

During a typical (moderate) El Nino (Fig. 2.10, right panel), convection shifts to the central/eastern Pacific, bringing rainfall to normally-dry Pacific islands; in addition, SLP in the WEPO increases, weakening the zonal SLP gradient, and the near-surface, equatorial easterlies collapse. The Walker circulation is split into two parts: a normal (clockwise) cell confined to the eastern basin, and an anomalous (counterclockwise) cell in the western ocean with subsidence over Indonesia. In the ocean, the thermocline flattens enough that it no longer outcrops in the eastern ocean and off Peru, warming SST in those regions. The resulting lack of upwelling drastically impacts the ecosystem there, raising ocean temperatures beyond tolerable levels for some fish

Fig. 2.10 Schematic pictures illustrating oceanic and atmospheric conditions in the Pacific during normal (left panel) and moderate El Nino (right panel) years. Adapted from diagrams prepared by Emily Eng for PMEL, NOAA (www.climate.gov/news-features/blogs/enso/rise-el-nino-andla-nina)

2.3 Interannual Variability

27

species and cutting off the nutrient supply needed for phytoplankton growth. It is this episodic ecological catastrophe that caused the IATTC to contact UCLA scientists for help. In addition to the normal and moderate ENSO states shown in Fig. 2.10, a continuum of others exists. During very strong El Ninos, for example, all convection moves across the Pacific to Peru, bringing torrential rains to that normally-arid region. The structure of this event is similar to that in the right panel of Fig. 2.10, except with all the cloud symbols shifted to far-eastern Pacific, a counterclockwise Walker circulation with low-level westerlies spanning the basin, and the thermocline tilting down to the east. During weaker El Ninos, the eastward shift of precipitation stalls in the central Pacific; this state, referred to as a “Modoki” or “central-Pacific” El Nino, is similar to the one in the right panel of Fig. 2.10 except with convection shifted somewhat farther west (Larkin and Harrison 2005; Ashok et al. 2007; Kug et al. 2009; Kao and Yu 2009). Conversely, the “cold” ENSO phase, “La Nina,” has a structure like that in the left panel of Fig. 2.10, except that the thermocline tilt is larger, regions of cold SST are expanded, and the normal Walker circulation is strengthened.

2.3.1.3

Dynamics

Bjerknes (1969) proposed an ocean-atmosphere, positive-feedback loop to be a key aspect of the evolution of ENSO events. To illustrate a modern version of his loop, consider the generation of an El Nino event. Suppose some process (trigger) initially shifts precipitation in the WEPO eastward. Even if the total rainfall doesn’t change as the convection shifts eastward, the equatorial winds do: Easterlies are generally confined east of the convecting region, with westerly winds to its west; thus, the shift causes basin-wide equatorial easterlies to weaken, thereby flattening the thermocline tilt, increasing SST in the EEPO, and shrinking the cold tongue. The latter change expands the region where SST >28◦C (the warm pool), allowing convection to shift even farther eastward, closing the positive-feedback loop. Continuation of this loop eventually leads to a moderate (Fig. 2.10, right panel) or strong El Nino. All the above steps work in reverse, and so provide a positive feedback loop that leads from an El Nino state to a normal one or to La Nina. One view of ENSO is that it is an oscillatory phenomenon that regularly shifts from El Nino to La Nina and back. As such, it requires negative, as well as positive, feedback in order to force the system from either of its extremes toward normality. To illustrate the primary negative-feedback process for ENSO, consider the ocean’s adjustment to anomalous westerly winds during an El Nino event. Part of the response is a pair of upwelling-favorable Rossby waves in each hemisphere. These waves propagate to the western boundary, where they reflect as an upwellingfavorable equatorial Kelvin wave that propagates eastward across the basin; it raises the equatorial thermocline, stops the positive-feedback loop that leads to El Nino, and initiates another that leads to La Nina. A corollary of this theory is that it explains the 4–5 year time scale of ENSO: roughly twice the time it takes the Rossby waves to propagate across the basin. Another view of ENSO is that it is a “damped” oscillator; in this view, processes similar to the above occur but they are weak enough to

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require external triggers to initiate them. For further discussions of ENSO dynamics, see Philander (1990), McCreary and Anderson (1991), Neelin et al. (1998), Wang and Picaut (2004), Clarke (2008), Schott et al. (2009), Sarachik and Cane (2010), Kohyama et al. (2018), and Xie et al. (2018).

2.3.1.4

Impacts

Although ENSO is generated in the tropical Pacific, it remotely impacts the atmosphere in other regions through the radiation of atmospheric waves (e.g., Liu and Alexander 2007; Xie et al. 2009). For example, Fig. 2.13 (bottom panel) illustrates ENSO’s impact on Indian-Ocean winds. Another major ENSO impact, discovered by Sir Gilbert Walker in the early 20th century, is a statistical linkage between the Southern Oscillation and summer monsoon rainfall over India (e.g., Sikka 1980; Pant and Parthasarathy 1981; Rasmusson and Carpenter 1983). The relationship between the two, however, is not one to one. Although historical rainfall records show that severe droughts in India are accompanied by ENSO events, not every ENSO event leads to a drought; further, ENSO events with warmest SST anomalies in the central equatorial Pacific are more effective in producing Indian droughts than those with warmest SSTs in the eastern basin (Krishna Kumar et al. 2006).

2.3.2 IOD The IOD is an interannual climate mode that originates in the Indian Ocean during fall (Hastenrath et al. 1993; Saji et al. 1999; Webster et al. 1999; Murtugudde et al. 2000). A typical “cold” (positive) event involves strong SST cooling in the eastern, equatorial Indian Ocean (EEIO) and off Sumatra, and weaker warming in the western, equatorial Indian Ocean (WEIO), forming an SST dipole. Analyzing a time series of the SST difference between the western and southeastern equatorial Indian Ocean (EIO), an IOD index, Saji and Yamagata (2003) concluded that 19 moderate-to-strong IOD events occurred from 1958–1997.

2.3.2.1

Properties

Figure 2.11 schematically illustrates normal and IOD states. Normally (left panel), there is rising air and low SLP over the EEIO and Southeast Asia, subsidence and high SLP in the WEIO, and low-level westerlies; in the ocean, the thermocline is flat or even slopes down a bit to the east, and SST is warm across the basin. During a cold IOD event (right panel), there is convection and low SLP over the WEIO and equatorial West Africa, subsidence and high SLP over the EEIO and Indonesia, and low-level equatorial easterlies; in response to the easterlies, the thermocline tilts upward to the east across the basin, and it outcrops in the EEIO and along Sumatra/Java, cooling SST in both regions.

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Significant IOD anomalies begin during the fall, although signatures of the event can appear as early as late spring or early summer (Vinayachandran et al. 2010). The anomalies typically reach their peak during November and dissipate by January under the influence of winter monsoon. Figure 2.12 shows anomalies of SST, depth of the 20◦C isotherm, cloudiness, and winds associated with a cold IOD during September and November. Prominent features are anomalous southeasterlies along the southeast coast of the tropical Indian Ocean and easterlies in the EEIO. They lead to a shallower thermocline, colder SST, and reduced rainfall (increased outgoing longwave radiation, OLR) in the region. In contrast, the WEIO experiences a deeper thermocline, increased SST, and decreased OLR.

2.3.2.2

Dynamics

As for ENSO, atmosphere-ocean positive feedback is involved in the establishment of an IOD cold event. During September/October, climatological SST is greater than 28◦C in the central/western EIO (CWEIO; Fig. 2.4 and Video 2.1) and the EEIO thermocline is shallow. At this time, then, the system is primed for the onset of positive feedback. In a normal year, convection in the CWEIO is prevented by subsidence associated with the (counterclockwise) Walker circulation (Fig. 2.11, left panel). Suppose some process (trigger) initially weakens convection over Southeast Asia. Part of the remote response to that anomalous forcing is westward radiation of atmospheric Rossby waves, which weakens the normal Walker circulation and hence CWEIO subsidence. If the subsidence is sufficiently reduced, convection can develop in the CWEIO, and this anomalous forcing generates eastward-propagating, atmospheric Kelvin waves that further weaken the normal Walker circulation and its low-level westerlies. In response to the weakened (or even reversed) westerlies, the thermocline develops an anomalous upwards tilt toward the east. If the tilt is large enough, the thermocline outcrops and cools SST in the EEIO, further suppressing precipitation over Southeast Asia and strengthening the IOD event (Fig. 2.11, right

Fig. 2.11 Schematic pictures illustrating oceanic and atmospheric conditions in the Indian Ocean during normal (left panel) and cold IOD or positive IOD (right panel) years. Also shown is the nature of oceanic thermocline. Normally flat, it tilts upward towards the east during cold IOD. From www. bom.gov.au

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Fig. 2.12 Upper panels: Monthly-mean anomalies of SST (colour) and depth of the 20◦C isotherm (contours) for cold IOD years during September (left) and November (right). Lower panels: Monthly-mean anomalies of OLR (W/m2 , colour) and surface wind (m/s, vectors). Based on Figures 5 and 6 of Vinayachandran et al. (2010). © Indian Academy of Sciences. Used with permission

panel). Note that this sequence of events is analogous to Bjerknes positive feedback in the Pacific that generates La Nina. For this feedback loop to occur, there must be an external trigger and the equatorial thermocline must be sufficiently shallow (preconditioned) to allow anomalous upwelling to impact SST. A prominent IOD trigger is ENSO, which shifts convection away from Indonesia (Annamalai et al. 2005a). Other triggers that have been proposed to impact Indonesian convection are: anomalous Hadley circulations (Kajikawa et al. 2001; Fischer et al. 2005); severe cyclones in the Bay of Bengal (Francis et al. 2007); and MJOs in the Indian Ocean (Rao et al. 2009). Regarding preconditioning, the thermocline in the EEIO, h, must be shallow enough to allow upwelling to be effective in cooling the SST (Vinayachandran et al. 1999, 2002, 2007; Murtugudde et al. 2000; Li et al. 2003; Annamalai et al. 2005b). One process that influences h is Pacific decadal variability, by modifying either the EEIO zonal wind stress or transport of the Indonesian Throughflow (Annamalai and Murtugudde 2004; Annamalai et al. 2005). Wind anomalies associated with MJOs can also impact h (Rao et al. 2009).

2.4 Sub-annual Variability

2.3.2.3

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Connection to ENSO

The fact that ENSO and IOD are often linked has led some researchers to conclude that IOD is simply an aspect of ENSO. On the other hand, the property that IOD and ENSO result from feedback processes in different oceans suggests that they can occur independently. In this regard, Saji and Yamagata (2003) concluded that 11 out of the 19 moderate-to-strong IOD events from 1958–1997 occurred independently of ENSO, and Meyers et al. (2007) found that about half of the IOD events from 1876 to 1999 were independent. There are also distinct differences in the anomalies associated with the two events. For example, Fig. 2.13 compares Indian-Ocean wind anomalies associated with IOD (top) and ENSO (bottom). Both panels have lowlevel anticyclones from 80–90◦E, but they are located from 10–20◦S during ENSO and from the equator to 20◦S during an IOD. The wind patterns seen in the figure were obtained by a zero-lag correlation with indices that mark the state of ENSO and IOD, and hence they represent wind anomalies during the peaks of events. As noted above, ENSO is known to be a prominent trigger for IOD events. In the damped-oscillator view of its dynamics, ENSO also needs a trigger. Interestingly, Luo et al. (2010) reported a particularly large IOD event when that seems to have been the case.

2.3.2.4

Impacts

Note that the anomalous equatorial easterlies during an IOD (Fig. 2.13, top panel) are much stronger than they are during ENSO (bottom panel). This difference is significant, given the large impact these winds have on equatorial currents and SST (Sects. 4.4, 10.2 and 14.2). Given this difference, one might expect that IOD impacts Indian rainfall more strongly than ENSO, but that does not seem to be the case, as correlations between the two variables are weak. In contrast, IOD events tend to increase rainfall over tropical eastern Africa and reduce it over the Maritime Continent and northwest Australia (Ashok et al. 2001; Vinayachandran et al. 2010).

2.4 Sub-annual Variability There are a number of atmospheric oscillations over the NIO and South Asia at periods less than the annual. Some are simply harmonics of the climatological annual cycle (180, 120, and 90 days), which are generated primarily because the summer monsoon is much stronger than the winter one. Others occur at periods less than a season and more than a week (intraseasonal variability), and they are generated by processes very different from those that cause the climatological monsoon. The two prominent examples of intraseasonal variability in the NIO are: Madden-Julian

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Fig. 2.13 a Partial correlation of 1000 HPa winds (vectors) and wind curl (colours) with Indian Ocean Dipole (IOD) index. b Same as (a) but for Niño-3 index. Only correlations significant at 99% level are shown. The figure is based on Figure 1 of Yu et al. (2005)

Oscillations (30–60 days, MJOs; Sect. 2.4.1) and the quasi-biweekly mode (10–20 days, QBM; Sect. 2.4.2). At even shorter periods, atmospheric variability occurs at sub-weekly (2–6 days; Sect. 2.4.3) and diurnal (Sect. 2.4.4) time scales.

2.4.1 Madden-Julian Oscillations Based on analyses of tropical-island observations, Madden and Julian (1971) first reported eastward-propagating patterns of wind, rainfall, cloud, and pressure anomalies at periods of 30–60 days. They are now known to be the dominant mode of tropical intraseasonal variability, accounting for about 66% of all subseasonal monsoon variability (Annamalai and Slingo 2001). In honor of their discoverers, they are referred to as MJOs.

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Fig. 2.14 The pattern of precipitation and winds associated with an idealized Madden-Julian Oscillation. From https://www.climate.gov/news-features/blogs/enso/what-mjo-and-why-do-we-care

Figure 2.14 schematically depicts a typical MJO. It consists of regions of enhanced and suppressed precipitation (a rainfall dipole) and an overturning circulation that connects them (Rui and Wang 1990). Typically, MJOs form by convective instability in the WEIO, and then propagate eastward into the western Pacific at speeds of about 5 m/s (Zhang 2005). Their convection decays in the central/eastern Pacific, when the MJO propagates over the cool SSTs in the Pacific cold tongue. Nevertheless, a pressure signal still remains that can propagate around the globe as an atmospheric Kelvin wave to return to the Indian Ocean; this property has led some researchers to suggest that a previous MJO can act as a trigger to generate the next one (Matthews 2008).

2.4.1.1

Properties

The structure of MJOs varies seasonally, with their convection centered somewhat off the equator in the summer hemisphere. During the boreal summer, they also interact with the monsoon, developing a northward-propagating component (e.g., Lawrence and Webster 2002, their Figure 8) often referred to as “northward-propagating intraseasonal variabilities” (NPISVs). The northward propagation has a profound impact on monsoon precipitation: It accounts for active and break cycles of rainfall over India that last from a week to a month, and also impacts the overall seasonal rainfall (Yasunari 1980; Lawrence and Webster 2002; Annamalai and Sperber 2005).

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Fig. 2.15 Composite of active minus break days of winds at 850 hPa for the NPISV. The winds (either the zonal or meridional component) significant at 5% level are plotted. The vector in the lower left corner has magnitude of 3 m/s. The figure is based on Figure 3 in Annamalai and Slingo (2001)

Figure 2.15 shows the wind field at 850 hPa associated with a typical NPISV at a time when it most strongly impacts Indian rainfall. It is noteworthy that the circulation anomalies associated with NPISVs are of global extent, an indication of the far-reaching impact of MJOs.

2.4.1.2

Dynamics

The causes of MJOs are still not well understood. It is generally accepted that they result from an atmospheric instability, but the specific processes that set their frequency, zonal scale, and eastward propagation speed are not clear (e.g., Wang 2005; Zhang 2005; Waliser 2012; Zhang 2013). There are also indications that MJOs are secondarily influenced by the ocean state. For example, in response to MJO forcing SST is cooler (warmer) in its precipitating (dry) parts. A comparison of MJO solutions in atmosphere-only (AGCMs) and coupled (CGCMs) general circulation models suggests that this SST pattern feeds back to affect MJO structure, with MJOs being “better formed” in coupled solutions (Fu et al. 2003; Inness et al. 2003; Zheng et al. 2004). In addition, the predictive skill of monsoon rainfall is better in CGCM solutions than in AGCM solutions that are forced by prescribed SST anomalies (Fu et al. 2002; Zhu and Shukla 2013).

2.4.1.3

Impacts

MJOs profoundly impact atmospheric, oceanic, and coupled phenomena in the NIO and Pacific Ocean. As noted above, in the northern hemisphere NPISVs cause active and break periods of Indian rainfall. In the southern hemisphere, MJO-related con-

2.4 Sub-annual Variability

35

vection interrupts the ITCZ south of the equator (Yoneyama et al. 2013). Surface winds associated with the MJO generate intraseasonal fluctuations of the Wyrtki Jets (Sect. 4.4.1; Iskander and McPhaden 2011; Jensen et al. 2015; Prerna et al. 2019), the SC (Sect. 4.9.6; Mysak and Mertz 1984) and the South Equatorial Current (Sect. 4.3.2; Zhou et al. 2008). In many areas of the Indian Ocean, intraseasonal sea-level fluctuations are associated with MJO-generated, equatorial Kelvin, Rossby waves, and coastal waves (Sects. 4.6.3 and 4.7.2; Oliver and Thompson 2010). Regarding IOD, events can be initiated by MJOs if their convectively suppressed phase occurs over the Indian Ocean during May/June (Rao et al. 2009); additionally, an IOD mature phase can occur only in the absence of strong MJO events (Rao and Yamagata 2004). Regarding ENSO, El Nino events are often preceded by MJOs in the western Pacific, suggesting the latter trigger the former (McPhaden 1999, 2008; McPhaden et al. 2006).

2.4.2 Quasi-biweekly Mode Using data from Daily Weather Report 1962 and upper-air observations from Indian stations, Murakami (1976) identified oscillations in wind, precipitation, and other fields with periods near 15 days. Based on analyses of the cross-equatorial low-level jet, monsoon cloud cover, and monsoon rainfall, Krishnamurti and Balme (1976) identified similar oscillations. Subsequent studies showed that the periods of the oscillations vary from 10 to 20 days, and they are now known as 10–20 day oscillations, quasi-biweekly oscillations, or the quasi-biweekly mode (QBM). They occur throughout the monsoonal region (Kikuchi and Wang 2009) year round (Chatterjee and Goswami 2004), and contribute about 25% of all subseasonal variability there (Annamalai and Slingo 2001). Annamalai and Slingo (2001) found that QBMs originate in the western Pacific and then propagate westward as an atmospheric Rossby wave, and Chatterjee and Goswami (2004) estimated the Rossby-wave wavelength and phase speed to be about 6000 km and 4.5 m/s. Figure 2.16 shows the wind field associated with a QBM after it has moved to be over South Asia and the NIO. The field consists of two cyclonic cells, one located over the bay and the other just south of the equator. In contrast to the NPISV composites in Fig. 2.15, the QBM composites are much more local, being confined largely to the western Pacific and Indian Oceans. In their analyses of observations and an AGCM solution, Wang et al. (2017) confirmed that QBM variability originates primarily in the equatorial western Pacific, and thereafter propagates northwestward across the Bay of Bengal into northern India and then northward onto the Tibetan Plateau. We note that the above description of QBM movement and structure is consistent with, and extends, earlier results reported by Chen and Chen (1993). The QBM significantly impacts the equatorial Indian Ocean, primarily through the excitation of Yanai waves (Sects. 4.4.4, 8.3, and 16.1.2.2). Sengupta et al. (2001) forced an OGCM with realistic winds, finding that at QBM times scales: the solution’s

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Fig. 2.16 Composite of active minus break days of winds at 850 hPa for the QBO. The winds (either the zonal or meridional component) significant at 5% level are plotted. The vector in the lower left corner has magnitude of 1 m/s. The figure is based on Figure 4 in Annamalai and Slingo (2001)

across-equatorial velocity, v, correlated remarkably well (in both amplitude and phase) with current-meter observations from the central equatorial ocean (Schott et al. 1994); and that v was associated with Yanai waves. In a theoretical study, Miyama et al. (2006) examined how QBM winds excite Yanai waves in detail, among other things showing that they are generated by both near-equatorial, meridional winds and zonal-wind curl. Chatterjee et al. (2013) demonstrated that much of the QBM variability in the WEIO was also due to wind-forced Yanai waves.

2.4.3 Sub-weekly Oscillations In addition to QBMs, Murakami (1976) and Krishnamurti and Balme (1976) also identified oscillations at periods of 2–6 days. Murakami (1976), whose data was limited to a dozen stations from the Indian subcontinent and one from Port Blair in the Andaman Islands, concluded that the summertime oscillations originated in the northern Bay of Bengal and propagated westward with a wavelength of about 3000 km. Krishnamurty and Balme (1976), using data from a wider geographical area, found 2–6-day oscillations in all the fields they examined, attributing them to “local instabilities and local disturbance passages.” More recently, Yasunaga et al. (2010) observed 3–4-day oscillations in the central equatorial Indian Ocean. Since their discovery four decades ago, there has been little progress in understanding the geographical distribution and underlying dynamics of sub-weekly oscillations. There is a need to do so, as they appear in ocean observations. For example, 2–6-day signals have been reported in Indian coastal regions (Amol et al. 2012, 2014; Mukherjee et al. 2013, 2014), and there are indications they are wind-driven. In par-

2.4 Sub-annual Variability

37

ticular, inertial periods (Chap. 10) along the coasts of India south of about 15◦N fall in the 2–6-day range. Therefore, we can expect that variability in this period band can be preferentially excited by the wind, and the coastal observations are consistent with this expectation.

2.4.4 Diurnal Variability Dai and Deser (1999) studied diurnal variability in global (50◦S–70◦N) surface wind and divergence fields using 3–hourly wind observations from approximately 10,000 stations and available marine reports during 1976–1997. Over the oceans, they found that the amplitude of the diurnal cycle was generally about 0.3–0.4 m/s, peaks around 1200–1400 hours local time, and has little seasonal variation. The diurnal cycle has been observed in other atmospheric variables as well. Using a global archive of high-resolution (3-hourly, 0.58 latitude-longitude grid) data, Yang and Slingo (2001) constructed a climatology of the diurnal cycle for convection, cloudiness, and surface temperature in all tropical regions. They found that the strong signal over land spreads several hundred kilometers offshore, likely by the offshore radiation of gravity waves. Offshore from the perimeter of the Bay of Bengal and from Sumatra, they lead to substantial diurnal variations in convection and precipitation, the latter tending to reach its maximum during the early morning. Dai and Deser (1999) also reported a semidiurnal signal in all seasons with a weaker wind-speed amplitude of 0.2–0.3 m/s; it explained about 15–25% of the daily variance over the ocean (compared to 30–40% for the diurnal cycle). Generally, during the afternoon and early evening diurnal variability is a circulation in which air rises over the continents and sinks over the nearby ocean, with connecting near-surface onshore and elevated offshore flows that join two regions; an opposite circulation occurs during the early morning (Dai and Deser 1999). Near coasts, the onshore flow of marine air is referred to as “sea breeze,” and it is observed at locations from the tropics to polar regions. It develops when solar radiation under relatively cloud-free conditions heats the land surface faster than the sea surface. The resulting land/sea temperature difference creates a pressure gradient that drives a shallow layer of cool marine air toward the land as a gravity current. On reaching the land, the air is warmed and rises. Thereafter, it often flows offshore at higher altitudes, and weakly descends over a distance that can vary from tens to several hundred kilometers offshore (e.g., Miller et al. 2003). Coastal areas of the NIO are expected to have a sea breeze, but data to confirm this idea are unavailable from many parts of the coast. There are data to show that sea breeze is ubiquitous along the west coast of India during the dry season. During the summer monsoon, however, it is usually absent because cloudy skies and rainfall inhibit its formation, and because at this time there are usually westerlies with speeds high enough to overwhelm the sea breeze circulation. Whenever there is a break in

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the summer monsoon leading to a cloud-free day, the sea-breeze cycle sets in. Aparna et al. (2005) used NASA quick scatterometer (QuikSCAT) winds to conclude that the west-coast sea breeze extends 160–210 km offshore, depending on the time of year and location along the coast.

Video Captions Video 2.1 Daily-climatological SST using the NOAA OI SST V2 High Resolution Dataset for the period 1982–2020. The resolution of the data is 0.25◦ . The color bar gives SST in ◦C. Black-dashed contours indicate 200-m isobaths. Video 2.2 Daily-climatological precipitation (color shading) and winds (vectors). The winds are determined based on NCEP Reanalysis data from the Physical Sciences Laboratory, NOAA, for the period 1998–2014. The precipitation is determined from TRMM daily data during 1998–2015. The resolution of both datasets is 0.25◦ . The unit of the color bar is mm/day and vector key (lower-left corner) is m/s. The green contour indicates 500-m elevation. Precipitation in the video is still not smooth despite being determined from climatological data, an indication of just how sporadic individual rainfall events are.

Chapter 3

Ocean Forcing and the Surface Mixed Layer

Abstract The ocean is forced by radiation from and to the atmosphere, fluxes across the air-sea interface, and by freshwater input from precipitation and rivers. They drive circulations in the surface mixed layer (ML) of the ocean, which in turn force deeper circulations. We review each of these forcings, and discuss their impacts on ML properties in both the real ocean and ML models. One impact of evaporation and freshwater input is that the ML thickness differs markedly in the northern areas of the Bay of Bengal and Arabian Sea. In the northern Bay, high freshwater input decreases near-surface salinity and density, and the resulting increase in near-surface stratification ensures that the ML remains relatively thin. Conversely, in the northern Arabian Sea high evaporation increases near-surface salinity and density, decreasing the near-surface stratification and allowing the ML to thicken to larger values. During the summer monsoon, the thinner ML in the northern Bay leads to sea-surface temperature being warm enough to support atmospheric convection, making the northern Bay one of the rainiest regions in the global tropics. Keywords Downward and backward radiation · Air-sea and freshwater fluxes · Mixed-layer processes and models · Mixed-layer properties in Bay of Bengal versus Arabian Sea Almost all forcings enter the ocean as fluxes across, or very near, the air-sea interface. (The only exceptions are the tidal forcings by the sun and moon, not considered in this book, which act throughout the water column.) One part of those fluxes is radiative, in which incoming solar (shortwave) radiation heats the upper ocean and outgoing (longwave) radiation cools it. Another is freshwater flux from rainfall and river/land runoff, which affects the near-surface ocean by lowering its salinity and density. The other fluxes involve complex interactions on either side of the air-sea interface. Typically, the mismatch between atmospheric and oceanic properties at the interface generates turbulence. It leads to the formation of well-mixed layers to either side: the PBL in the atmosphere, which varies in thickness from tens of meters to 1–2 km Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_3. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_3

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(Peixoto and Oort 1992; Wallace and Hobbs 2006); and the surface mixed layer (ML) in the ocean with thicknesses of 10–100 m or even larger (Sect. 3.2). The two layers are associated with fluxes that exchange momentum (stress), mass (evaporation), and heat (sensible and latent heat) between the two fluids, so that each forces the other. As the place where most forcing enters the ocean, the ML plays an important role in the generation of ocean circulations, but its properties also affect other phenomena. For example, its thickness h m impacts biological activity: phytoplankton growth is inhibited in regions where h m is thicker than the depth of the euphotic zone, through its impact on the light intensity experienced by the phytoplankton (e.g., Wiggert et al. 2002). Its temperature Tm impacts climate, particularly in regions where Tm is greater than 28◦C, the necessary condition for deep convection (end of Sect. 2.1.1). Its salinity Sm is important, because large errors in h m occur in ocean models when it is improperly simulated (Nagura et al. 2018). Given these (and other) impacts, it is important to understand the processes that determine ML properties and to be able to simulate those properties accurately in ocean models. The goals of this chapter are twofold: to review the atmospheric fluxes that force the ocean, and to describe how they impact the ML. Toward these ends, we first define each of the atmospheric fluxes and describe their climatological variations over the NIO (Sect. 3.1). Then, we discuss the processes by which these forcings determine ML properties, comment on the models that are commonly used to simulate those processes, and describe the resulting patterns of h m , Tm , and Sm both throughout the basin and at specific locations in the Arabian Sea and Bay of Bengal (Sect. 3.2).

3.1 Ocean Forcing 3.1.1 Definitions As noted above, the external fluxes that impact the ocean divide into three distinct categories: radiative, air-sea, and freshwater fluxes. Many of their definitions are complex, requiring approximations for practical use. On the positive side, most approximations use variables that can be measured by satellites, allowing estimates of each flux to be determined globally. Talley et al. (2011) is a useful reference that describes the climatological distribution of the fluxes and practical methods to estimate them. Recently, Cronin et al. (2019) reviewed existing measurement methods and commented on how they could improve in the future.

3.1.1.1

Downward Radiation

Radiation from the sun radiates downward through the atmosphere and into the ocean. Its characteristics are modified as it does by a variety of processes (reflection, scattering, and absorption).

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Top of the Atmosphere: The sun has a surface temperature of about T = 6000◦K and, consistent with Stefan’s law, radiates energy at a rate proportional to T 4 . The radiation has a sharp peak near the wavelength of 500 nanometers (nm), a value predicted by Wien’s law. The energy is spread over the electromagnetic spectrum with wavelengths that include the extreme ultraviolet (10–100 nm), ultraviolet (100– 400 nm), visible (400–800 nm), and infrared (800–100,000 nm) (Haigh 2011). In atmospheric and ocean sciences, solar energy with wavelengths less (greater) than 4,000 nm is called shortwave (longwave) radiation. Shortwave (longwave) radiation covers the visible and near-infrared (infrared) range (Wallace and Hobbs 2006). The rate at which shortwave energy reaches the top of the atmosphere, S∞ varies in time. That rate changes seasonally owing to the earth’s distance from the sun in its elliptic orbit. It also changes on climatic time scales due to internal stellar processes (solar activity) and long-term changes in the earth’s orbit (Haigh 2011). The yearly average of S∞ is referred to as the “solar constant,” and in recent years its value is typically 1365–1372 W/m2 (Talley et al. 2011). Top of the Atmosphere to Ocean Surface: As solar energy travels into the atmosphere, it is partly scattered and absorbed, the primary absorption due to molecules of ozone, water, oxygen, and carbon dioxide with an additional contribution from aerosols. On a global and annual average, of the 100 units of energy propagating towards the ocean surface, 19 units are absorbed in the atmospheric column, and 29 scattered back to space. Of the 52 units that reach the earth’s surface, 4 are reflected there. As a result, only 48 units enter the oceans (Fig. 5.5 in Talley et al. 2011), generating the first term (Q sw ) on the right-hand side of (3.8) below. In essence, incoming solar radiation is decreased by an average factor of about φ¯ sw = 50% from S∞ to its entry into the ocean. Ocean Surface: The unaveraged factor, φsw , is not spatially and temporally uniform. It depends on the length of the air column in a direct line between the sun and the observation point, which is determined by the angle at which the solar radiation reaches the earth’s surface (see Peixoto and Oort 1992, their Eq. 6.18; Wallace and Hobbs 2006, their Fig. 10.5). Thus, φsw (y, t) varies seasonally and with latitude y, as well as diurnally, with its maximum value at the latitude where the sun is directly above the earth’s surface. Radiation S∞ is further reduced by reflection from the earth’s surface and clouds. An overall expression for the shortwave radiation at the sea surface is then (3.1) Q sw = S∞ φsw (1 − αs ) (1 − 0.7n c ) , where αs and n c are defined next, an equation similar to Eq. (2.6.1) in Gill (1982). Reflectivity of solar radiation from the ocean and land is measured by albedo αs . Albedo depends strongly on the angle at which solar radiation reaches the earth’s surface, with more reflection occurring at higher solar-zenith angles, such as during sunset and sunrise or during winter at high latitudes; for example, at low angles, αs = 0.03–0.06, whereas at high angles αs can approach 0.3. A fixed albedo of 0.055 has often been used in tropical areas (Cronin et al. 2019). Reflectivity, to some extent, is also dependent on suspended matter and sea state (hence on wind field): The smoother the sea state, the higher the reflection.

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Parameter n c is the fraction of the sky covered by clouds. Part of the incoming radiation is reflected, absorbed, or scattered by clouds, and the factor of 0.7 multiplying n c in (3.1) is determined empirically from observations. A major source of error in estimating Q sw is determining n c . Before the 1980s, its values were mainly estimated subjectively; since that time, satellite observations and land-based radar observations have increasingly been used to provide objective estimates. Two methods are used to measure shortwave radiation entering the sea at a given location. One is to use a pyranometer, which is suitable for direct measurements over a small area. The other, which allows global coverage, is to use a bulk formula such as (3.1) or its equivalent (e.g., Eq. 5.10 in Talley et al. 2011). Data needed to evaluate the terms in (3.1) are based on both in situ and satellite observations. Relevant information from satellites includes: measurements of incident solar radiation at the top of the atmosphere, composition of the atmosphere including water vapor content and clouds, and surface conditions including atmospheric reflectivity. Below the Ocean Surface: Each frequency σ of Q sw decays exponentially in the water column with a form like exp [z/λ (σ )]. In ocean models, the contributions are often split into two bins: one in which λ is only a few centimeters (non-penetrating radiation); and another where it is tens of meters (penetrating radiation). The latter impacts the way the mixed layer restratifies after being mixed by wind or cooling. In addition, the radiation can sometimes penetrate below the mixed layer, the penetration depth depending not only λ but also the optical properties of water. Those properties depend on the concentration of particles in sea water, the primary contributors being plankton and sediments with the latter found mostly in coastal areas. The former can lead to absorption of Q sw close to the surface when blooms occur, a process that impacts SST.

3.1.1.2

Back Radiation

The ocean loses heat from its surface in the form of longwave radiation. Like the energy that leaves the sun’s surface, the longwave radiation is describable as blackbody radiation from the earth’s surface that follows Stefan’s and Wien’s laws. Its wavelengths vary from 4,000–100,000 nm, the thermal infrared range. A part of this outgoing radiation is reflected back to land/ocean by the atmosphere. In addition, the atmosphere can also radiate longwave radiation downward according to Stefan’s and Wien’s laws. The ocean’s net radiative heat loss Q lw (called “back radiation”) is the longwave energy radiated from the ocean surface minus that received from the atmosphere. It contributes about 40% of the heat lost from the ocean surface annually (Talley et al. 2011; their Fig. 5.5). Over small areas, Q lw can be measured directly using a radiometer, but over large regions the use of a semi-empirical bulk formula is the only practical measurement method. One example is   √  Q lw = −σ S B Ts4 0.39 − 0.05 e 1 − kn 2c − 4σ S B Ts3 (Ts − Ta ) ,

(3.2)

3.1 Ocean Forcing

43

which has been in use since the 1950s (Josey et al. 1999). In (3.2),  is the emittance of the sea surface, usually taken to be 0.98, a value less than 1 because sea surface is not a perfect black body; σ S B is the Stefan-Boltzmann constant 5.67×10−8 W/m2 /◦K4 ; Ts is the surface-water temperature expressed in ◦K; Ta is air temperature in ◦K at a height of about 10 m from the sea surface; e is the water vapor pressure, also at a height of about 10 m; and k is a cloud cover coefficient that is determined empirically and increases with latitude. The first term on the right-hand side of (3.2) represents Stefan’s Law √ corrected for downward radiation by the atmosphere (the term proportional to e) and cloud effects. The second term is a correction that is significant only in areas where the air-sea temperature is large. An example is over the western-boundary currents of the North Atlantic and Pacific where cold continental air can flow over warm SST (Talley et al. 2011). Minus signs are included on both terms to indicate that Q lw is a heat loss from the ocean.

3.1.1.3

Air-Sea Fluxes

Two approaches are used to determine the fluxes across the air-sea interface: the eddy-correlation and bulk-aerodynamic methods (Peixoto and Oort 1992). The former is labor intensive, requiring three-dimensional (3-d) measurements of velocity, temperature, and humidity fields within the PBL at high temporal resolution: Timeaveraged correlations among fluctuations of these variables then lead to accurate determinations of fluxes (e.g., Krishnamurti et al. 2013). Because the measurements require sophisticated equipment, the method is mainly used during research campaigns. The simpler bulk-aerodynamic method estimates fluxes from variables that are easier to measure. Specifically, a measure of atmospheric turbulence is taken to be the mean wind speed va at a prescribed height above the ocean surface, typically 10 m. The surface flux for variable q is then assumed to be represented by Fq = Cq ρa va q,

(3.3)

that is, Fq is proportional to the turbulence magnitude va times a difference q = qa − qo between relevant variables in the atmosphere qa and surface ocean qo . The constant of proportionality Cq is determined by comparing estimates of Fq from (3.3) with observed values, and ρa = 0.0012 gm/cm3 is a typical air density that is included so that Cq is dimensionless. Wind Stress The bulk-aerodynamic formula used to compute the wind-stress vector, τ = (τ x , τ y ) is (3.4) τ = Cm ρa va (v a − v) , where v a is the wind-velocity vector at 10 m and v is the surface ocean current. Although Cm is taken to be constant with typical value of 1.1×10−3 , research spread over decades has shown that its value varies primarily with wind speed, gustiness,

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and rainfall (Smith 1988; Fairall et al. 1996 and 2003; Yu and Jin 2014). According to (3.4), the ocean will always be stressed as long as its surface velocity differs from that of the wind. Commonly, v is ignored in (3.4), in which case the ocean always feels a stress in the direction of the wind. Recent studies, however, have suggested that the impact of v is not always negligible (Dawe and Thompson 2006; Kara et al. 2007; Shi and Bourassa 2019). Evaporation Evaporation from the ocean is estimated by E = Ce ρa va (qs − qa ) ,

(3.5)

where: qs (Ts ) is the saturation (highest possible) specific humidity at the sea-surface temperature Ts ; qa is the specific humidity near the ocean surface, usually measured at a height of 10 m; and a common value for Ce is 1.6×10−3 . According to (3.5), evaporation is a diffusive process driven by air turbulence measured by va , and its magnitude depends on (qs − qa ), that is, on how much qa differs from qs . The saturation specific humidity just above the air-sea interface is qs (Ta ). Over most of the oceans Ta < Ts and, because qs increases with temperature, it follows that qa ≤ qs (Ta ) < qs (Ts ). In these regions, then, E > 0 and there is a loss of fresh water from the ocean due to evaporation. Interestingly, there are a few exceptional regions where Ta > Ts so that qa > qs and E < 0, examples being the Grand Banks off Newfoundland and coastal areas of Northern California (Talley et al. 2011). Here, air moisture condenses into fog droplets in response to the cool Ts , and the droplets enter the ocean to cause “reverse evaporation.” Latent and Sensible Heat The latent heat flux is expressed in terms of E by Q l = −LE = −Ce Lρa va (qs − qa ) ,

(3.6)

where L = 2.44×106 J/kg is the latent heat of evaporation, and the minus sign is included to indicate that the ocean is cooled by positive evaporation. The sensible heat flux is given similarly by Q s = −Cs ρa va C p (Ts − Ta ),

(3.7)

where C p = 1004 (J/kg)/◦C is the specific heat of air and C S is typically 1.6×10−3 . Since Ts − Ta > 0 over most of the ocean, Q s is usually negative and so cools the ocean. Unlike the radiative fluxes, Q l and Q s cannot be measured by satellites because there are no satellite-based sensors that measure qa or Ta . The modern approach is to use (3.6) and (3.7) together with a blend of satellite-based measurements and estimates of qa and Ta from reanalysis products (e.g., Yu et al. 2008; Yu and Jin 2014).

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Improvements Bulk-aerodynamic formulae provide the only practical method for estimating surface fluxes globally, mainly because of two developments in the last few decades. First is that satellite-based sensors now provide most of the variables needed to evaluate bulk formulae. Second, those variables not measured by satellites can be estimated from global, atmospheric reanalyses. Hence, today the importance of bulk formulae cannot be underestimated. (They are, for example, used in all the coupled models that predict climate variability and change.) It is, therefore, essential that the formulae represent the fluxes as accurately as possible. Given their simplicity, it is remarkable that they work as well as they do. As might be expected, however, they break down under extreme conditions, such as under very high and very low wind speeds. There has been considerable work devoted to improving their accuracy under these conditions, typically by allowing drag coefficients Cq to vary with state variables. A comprehensive description of these efforts is given in Soloviev and Lukas (2014).

3.1.1.4

Freshwater Fluxes

Freshwater is added to the ocean primarily through precipitation and runoff from rivers and land. Globally, the freshwater that enters the ocean must balance the net loss due to evaporation. Precipitation Rainfall adds freshwater to the ocean surface, reducing surface density, and creating a density stratified, surface layer. Rain is measured by dividing the volume of freshwater added per unit area per unit time. Units used to measure rain include mm/hr, cm/month, or similar units depending on the context. Before the 1990s, rainfall rates on land and at sea were almost exclusively measured directly using gauges that determine rainwater accumulation. Because rain is sporadic both temporally and spatially, obtaining accurate direct measurements is difficult. Over land, this difficulty could be handled simply by increasing the number of rain gauges. Over the oceans, however, only a few ships, usually research vessels, carried rain gauges, and as a consequence the ocean remained poorly sampled. During the 1990s, satellite-based sensors began to be used to estimate rainfall. Their advantage is that they cover large areas, including oceans. On the other hand, because satellite sensors measure rainfall indirectly, the relationship between the satellite measurement and actual rainfall must be determined. The accuracy of that relationship requires careful comparison of the satellite measurement with direct estimates from rain gauges (see below). One of the early satellite rainfall sensors measured electromagnetic radiation from the earth’s surface in the infrared (IR) range. It detected OLR, from which cloud-top temperature and cloud height could be determined. Rainfall was then estimated by correlating OLR with rain gauge observations (Xie and Arkins 1996). An advan-

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tage of IR sensors is that they usually are located on geosynchronous geostationary satellites and provide data at a high sampling frequency. Sensors in the microwave range have also been used for rainfall estimation (Wentz 1997). They can be either passive or active: the former observing natural radiation from earth’s surface, and the latter detecting the scatter of a microwave beam emitted by the sensor. Active sensors can measure raindrop properties and provide better estimates of rainfall. They are generally flown on low earth-orbiting satellites, and hence have a lower sampling frequency. The well-known Tropical Rain Measuring Mission (TRMM) Microwave Imager (TMI) was an active sensor flown from 1997–2015. Its rain radar significantly improved the capability to measure precipitation over tropics and subtropics. A number of studies have investigated the accuracy of TRMM data, indicating that it is good but not perfect. For example, Prakash and Gairola (2013) compared daily rain rates over the tropical Indian Ocean during 2004–2011 obtained from the TRMM Multi-satellite Precipitation Analysis (TMPA) product over the tropical Indian Ocean with rates from a contemporaneous buoy array. Their results show a statistically significant linear correlation between the two precipitation estimates ranging from 0.40–0.89, with the highest correlation observed during the Southwest Monsoon (June–September). The root-mean-square error varied from about 1–22 mm/day. While the TMPA product overestimated precipitation over most buoy locations, it underestimated precipitation during periods of high (100 mm/day) and light (0.5 mm/day) precipitation. While there has been significant improvement in measuring global rainfall, strategies being pursued now bear the promise of further improvement. One is the use of a combination of sensors and specialized algorithms (Maggioni et al. 2016). Another is to assimilate satellite and gauge-based observations into global atmospheric models to create a reanalysis (combined observation/model) product. Such reanalyses appear to provide the best estimates of overall precipitation (Pena-Arancibia et al. 2013). Popular products are the European Center for Medium-range Weather Forecast (ECMWF) reanalysis (ERA-Interim), Japanese 25-yr reanalysis (JRA-25), and NCEP Global Reanalysis 2. Another promising development is the maturing of technologies that can produce higher-quality surface-precipitation measurements through optical disdrometers, which can measure the sea-drop size distribution and velocity of falling rain (e.g., Klepp 2015). River and Land Runoff Most runoff comes from rivers, but some also occurs along the land (seashore) between them. River runoff R (volume/time) is measured by multiplying the crosssectional area of the river and velocity of the river flow. Such measurements are usually made with a gauge in the main channel of a river upstream of the location where the river enters the sea. Because there are often rivulets (tributaries) that join the river between the gauge and the ocean, river gauge data typically have to be corrected to estimate the freshwater flow into the sea. Land runoff L is even more difficult to measure. It enters the sea through rivers, rivulets, and streams that are much too small and far too many to gauge. However, together they can make a significant

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contribution. An example is the west coast of India, a region of high precipitation (Sect. 2.2.1). About 50 small rivers carry the rainwater to the sea. As we see below, their combined contribution is now estimated to exceed that of the three major rivers along the eastern Arabian Sea (Table 3.1). A similar situation occurs in the high precipitation areas bordering the Bay of Bengal. Dai and Trenberth (2002) listed the 200 rivers with the largest R, and Table 3.1 lists the 12 rivers in the NIO from their list. There are 9 in the Bay of Bengal and 3 in the Arabian Sea, which discharge 1733 and 164 km3 /yr into the two regions, respectively. Smaller rivers also contribute to the overall runoff but, because they are often not gauged, their contributions remain unquantified. For example, about 50 small (a few tens of kilometers long) rivers originate on the slopes of the Western Ghats, a region of high precipitation. Their contribution to runoff along the eastern boundary of the Arabian Sea is unknown. Likewise, the contribution of small rivers and streams along the eastern boundary of the bay remains uncertain. In the last several decades, two approaches have been followed to reduce uncertainties: combining runoff observations with the output of continental-scale land surface hydrology models (e.g., Dai et al. 2009); and estimating the runoff required to balance the global water cycle (Rodell et al. 2015). Clark et al. (2015) used the former approach to estimate runoff into the world oceans on a 0.5◦ grid. Their global runoff rate is 44200 km3 /year, similar to the value of 45900 km3 /year required to close the global water cycle (Rodell et al. 2015). We note that (Schneider et al. 2017) estimated that about 1000 km3 /year enter the ocean as groundwater at a subsurface level through porous soil. The new estimates highlight the importance of land runoff to the NIO. According to a model estimate in Clark et al. (2015), the average runoff into the eastern Arabian

Table 3.1 List of the 12 strongest rivers in the NIO, showing from left to right their name, country of origin, estimated outflow transport at the river mouth (km3 /yr), and the region where the river enters the ocean. Adapted from Dai and Trenberth (2002) River Country Runoff Region Brahamaputra Ganges Irrawady Indus Godavari Mahanadi Krishna Brahmani Narmada Tista Tapi Cauvery

Bangladesh India Myanmar Pakistan India India India India India Bangladesh India India

628 404 393 104 97 73 55 48 44 27 16 7.7

Northern Bay of Bengal Northern Bay of Bengal Northeastern Bay of Bengal Northeastern Arabian Sea Western Bay of Bengal Northwestern Bay of Bengal Western Bay of Bengal Western Bay of Bengal Eastern Arabian Sea Northern Bay of Bengal Eastern Arabian Sea Western Bay of Bengal

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Sea is 380 km3 /year, and into the Bay of Bengal north of 6◦N is 3400 km3 /year. Both estimates include land runoff from ungauged areas and are significantly higher than runoff from the major rivers listed in Table 3.1. In another approach, Yaremchuk et al. (2005) used an inverse method to estimate the seasonal cycle of river plus land runoff (R+L) into the Bay of Bengal by assimilating climatological salinity/temperature (S/T ) and ocean-precipitation (P) data into an ocean model. Their estimate of ∼2500 km3 /year is intermediate between the total runoff of the rivers listed in Table 3.1 (about 1500 km3 /year) and the Clark et al. (2015) estimate. Inverse methods’ reliability depends on the accuracy of fields used: in this case, S/T and P fields. As measurements of these fields improve, we can expect that estimates of R+L using inverse methods to converge with those using hydrological models (e.g., Clark et al. 2015) and the water balance method (e.g., Rodell et al. 2015).

3.1.2 Climatological Fluxes Figures 3.1 and 3.2 illustrate the climatological annual cycle of the fluxes discussed in Sect. 3.1.1 for the NIO, showing bimonthly maps of wind stress τ , wind-stress curl, evaporation minus precipitation E − P, and net heat flux Q (Fig. 3.1) and the four components of Q (Fig. 3.2). The datasets used to make the figures are listed in their captions. Though these datasets are currently considered the most reliable, it is important to keep in mind that they are approximations of the actual fluxes, which are under continual improvement (Cronin et al. 2019).

3.1.2.1

Wind Stress

Figure 3.1 (top-left panel) plots wind-stress vectors τ (also see Video 3.1). The most prominent features are the seasonally reversing monsoon winds, which are directed southwesterly (northeasterly) during the summer (winter) monsoon. The winds are much stronger during summer, so much so that the annual-mean wind field (not shown) is essentially a weaker version of the summer one. Along the equator, the winds are directed primarily meridionally during the monsoons, and only during the intermonsoons (April/May and October/November) do they have a significant zonal (westerly) component (Fig. 4.12, top-left panel). Although not nearly as strong as the winds during the monsoons, the westerlies drive swift equatorial currents, the spring and fall Wyrtki Jets (Sect. 4.4.1). One reason for the strong response is the vanishing of the Coriolis force f at the equator (Sects. 10.2.1 and 14.2). Another is “equatorial basin resonance,” in which boundary-reflected, equatorial Rossby and Kelvin waves interfere constructively with the directly winddriven response (Jensen, 1993; Han et al. 2011; Sect. 15.4.1).

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49

Fig. 3.1 Annual cycle of wind stress (τ , top-left), wind stress curl (curl τ , top-right), evaporation minus precipitation (E –P , bottom-left), and net heat flux Q (bottom-right). Each panel shows monthly-mean values for the month shown in the white box. Positive (negative) Q implies gain (loss) by the ocean. The τ and curl τ fields are based on winds from the SCOW data set described in Risien and Chelton (2008). The units of their color bars are N/m2 and 10−8 N/m3 , respectively, and the unit of the vector key (lower left-hand corner of the τ panel) is N/m2 . The E –P and Q panels plot daily, climatological data for 2000–2008 daily from NCEP (evaporation, shortwave and longwave radiation, sensible and latent heat fluxes) and TRMM (rainfall), as described in Behara and Vinayachandran (2016), and the units of their color bars are mm/day and W/m2 , respectively

The equatorial winds in the Indian Ocean are strikingly different from those in the Pacific and Atlantic Oceans, where quasi-steady easterlies prevail. Not surprisingly, the circulations are very different as well. In particular, in the latter two oceans easterlies force upwelling in the eastern equatorial ocean, which lowers SST below 28◦C and prevents precipitation from occurring there. In contrast, the absence of Indian-Ocean easterlies ensures that the eastern ocean remains warm enough to support precipitation.

50

3.1.2.2

3 Ocean Forcing and the Surface Mixed Layer

Wind-Stress Curl y

An important forcing derived from τ is wind-stress curl, curl τ ≡ τx − τ yx , where τ x (τ y ) is the eastward (northward) component of τ and subscripts indicate partial derivatives.1 Figure 3.1 (top-right panel) and Video 3.1 (shading) show the annual cycle of curl τ . In the northern hemisphere, regions of positive (negative) curl tend to shallow (deepen) isopycnals, and vice versa in the southern hemisphere, a process known as Ekman pumping (Sect. 12.2.1). The most prominent wind-curl signals occur during the summer (May–September panels, particularly July), when there are regions of strong, positive and negative curl in the Arabian Sea northwest and southeast of the axis of the Findlater Jet, respectively, and there are similar features (albeit weaker) in the Bay of Bengal. There is also a region of positive curl surrounding the southern tip of India and Sri Lanka, a result of part of the near-surface winds being forced to flow around those landmasses. During winter (November and January panels), the curl patterns in the Arabian Sea and Bay of Bengal tend to be reversed from and weaker than the summertime patterns, a reflection of the weaker northeasterly winds. In the southern hemisphere, there is negative wind curl on the northern flank of the Southeast Trades, which intensifies during the boreal winter. Other noteworthy features are narrow patches of strong curl wherever wind blows offshore from land. They are caused by wind shadows, generated at locations where the wind is blocked by orography. We discuss the impacts of several of these forcings in Chap. 4: just east of Sri Lanka (Sect. 4.7.4), and in the Gulfs of Oman (Sect. 4.10.2) and Aden (Sect. 4.10.4).

3.1.2.3

Evaporation Minus Precipitation

Figure 3.1 (bottom-left panel) shows the annual cycle of net mass flux (E–P) across the air-sea interface. During the summer monsoon (May–September plots), E–P is negative in the Bay of Bengal, eastern Arabian Sea, and EEIO (0–10◦S), the three regions of highest precipitation in Fig. 2.5 and Video 2.2, and is positive elsewhere. During the winter monsoon (November–March plots), E–P is positive throughout the Bay of Bengal and Arabian Sea, a consequence of high evaporation in response to cold northeasterly winds from the Asian continent. The difference is particularly positive in the northeastern Arabian Sea, where the winds originate in snow-clad mountain ranges. In contrast, E–P remains negative in the EEIO throughout the year: It is only during episodes of ENSO or IOD (Sect. 2.3) that E–P > 0 there (Saji et al. 1999; Webster et al. 1999). To visualize the relationship of curl τ to τ , imagine putting a small paddle wheel at a location in any of the τ plots. Then, if τ is stronger on one side of the paddle wheel than another (true almost everywhere), it will spin either clockwise or counterclockwise indicating negative or positive curl, respectively. Alternately, imagine curling the fingers of your right hand in the direction of the spin: If your thumb points down (up), the curl is negative (positive).

1

3.1 Ocean Forcing

51

Annual-mean E–P (not shown) is negative over the central and eastern EIO and Bay of Bengal, except in its western part in the vicinity of Sri Lanka, reaching a minimum of about –2 m/year in northeastern Bay of Bengal. It is positive in the WEIO and entire Arabian Sea except its southeastern corner, with a maximum greater than 2 m/year in the Red Sea. As we shall see, this east/west difference profoundly impacts properties of the surface mixed layer in the two regions (Sects. 3.2.3 and 3.2.4).

3.1.2.4

Heat Fluxes

Figure 3.1 (bottom-right panel) illustrates the climatological annual cycle of the net heat flux Q into the ocean. It is defined by Q = Q sw + Q lw + Q l + Q s ,

(3.8)

the sum of the four components defined earlier. Its structure is complex, owing to the different components that contribute to it (see below). A noteworthy property is that its annual-mean value integrated over the NIO (not shown) is positive, with annualmean estimates ranging from 0.2–0.8 pW (e.g., Hastenrath and Greischar 1993; McCreary et al. 1993; Schott and McCreary 2001, their Fig. 67). This heat must be removed each year, since otherwise NIO temperatures would increase indefinitely, and balance is achieved by southward advection of warm surface water by the Cross Equatorial Cell (CEC, Chap. 17). Figure 3.2 shows the four components of Q: shortwave radiation (Q sw , top-left), latent heat flux (Q l , top-right), longwave radiation (Q lw , bottom-left), and sensible heat flux (Q s , bottom-right). Note that the color-scale ranges differ markedly among the components: They are positive for Q sw , negative for Q l and Q lw , and are smaller for Q s by about an order of magnitude. Shortwave Radiation The annual cycle of monthly-mean solar radiation Q sw (Fig. 3.2, top-left panel) is the driving force behind all the other fluxes. As noted above, without clouds (n c = 0 in Eq. 3.1) Q sw is zonally independent with its maximum value directly under the sun: Its maximum is located at 23.5◦S on the summer solstice (June 20–22); it shifts northward during the latter half of the year, reaching 23.5◦N on the winter solstice (December 21–22); and the opposite migration happens during the first half of the year. Consistent with this property, Q sw increases in the northern hemisphere from January through March. During April-May, however, clouds form in the northeastern NIO and Q sw begins to decrease there. During June–August, Q sw decreases throughout most of the NIO in response to the rainy (cloudy) summer monsoon. During boreal winter, the Sun shifts to the southern hemisphere, intensifying Q sw there. This intensification is visible during November and January from 10◦S to 10◦N, but only west of 80◦E; it does not extend farther east owing to the rainband that is present there throughout the year (Fig. 2.2 and Video 2.2).

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3 Ocean Forcing and the Surface Mixed Layer

Fig. 3.2 Annual cycle of short-wave radiation (Q sw , top-left), latent heat flux (Q l , top-right), longwave radiation (Q lw , bottom-left), and sensible heat flux (Q s , bottom-right). Each panel shows monthly-mean values in W/m2 for the month shown in the white box. Positive (negative) heat flux implies gain (loss) by the ocean. The fluxes are based on 2000–2008 daily data taken from NCEP, as described in Behara and Vinayachandran (2016)

Latent Heat Flux The latent heat flux Q l (Fig. 3.2, top-right panel) is proportional to evaporation E (Eq. 3.6), and hence its magnitude |Q l | is determined by va and q = qs (Ts ) − qa . Consider the impact of these variables on prominent features of Q l seen in the figure. During the winter monsoon (November and January panels), |Q l | is large in the northern NIO, a result of dry and cold, northeasterly winds from the Asian continent that intensify va , q, and hence |Q l |. South of about 10◦N, Q l is determined largely by va , as is apparent from a comparison of the November and January panels of Q l with those of wind amplitude (shading) in Fig. 3.1 (top-left). During the monsoontransition seasons (March and September/October) when the winds and va are weak, |Q l | is reduced throughout much of the NIO. During summer (May and July) in the Arabian Sea and Bay of Bengal, va and q impact |Q l | oppositely: The strong monsoon winds ensure that va is high in both regions, tending to strengthen |Q l |;

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53

in the northwestern Arabian Sea qs decreases owing to the decrease in Ts , whereas in the northern bay qa increases due to high rainfall, and both tendencies tend to decrease q and reduce |Q l |. As a result, |Q l | increases during the summer in the interior of both basins, but remains weak in the north. Longwave Radiation Over most of the ocean, heat loss due to longwave radiation Q lw (lower-left panels of Fig. 3.2) is determined by the first term on the right-hand side of (3.2). That term depends on three factors: Ts (◦K), water-vapor pressure of air e, and cloud cover n c (higher cover implies weaker Q lw ). The first factor has little spatial or temporal variation over the NIO, so that Q lw is modulated by the other two, more so by cloud cover. During winter (November, January, and March panels), the northern NIO is largely cloud-free (n c ≈ 0) and e is low because dry and cold northeasterlies blow over the Arabian Sea and Bay of Bengal, and the northern NIO experiences a large heat loss (∼120 W/m2 ). During this season, though, cloud cover remains high over the equatorial region, and the heat loss is much less (∼40 W/m2 ). During summer (July and September panels), the distribution of Q lw is largely controlled by cloud cover, which is high in the Bay of Bengal and the eastern Arabian Sea, and Q lw varies from about −30 W/m2 in the northern bay to ∼45 W/m2 in equatorial areas. Sensible Heat Flux According to (3.7), the sensible heat flux Q s (Fig. 3.2, bottom-right panel) mirrors the air-sea temperature difference Ta − Ts . Away from ocean boundaries, the tropical ocean and atmosphere interact so that Ta  Ts (see, for example, Fig. 3.7) and Q s  0. Near continental boundaries, however, the interior-ocean balance is broken. In upwelling regions, where Ts is cooled by ocean dynamics, Ts < Ta and Q s > 0; this process is evident in the western Arabian Sea during the summer monsoon. Note also that during winter (January plot) the amplitude of Q s attains a minimum in the northern Arabian Sea and Bay of Bengal, a consequence of Ta being decreased by the cool, northeasterly winds from Asia.

3.2 Surface Mixed Layer Three variables characterize the ML: its thickness h m (MLT), temperature Tm , and salinity Sm . (During times of weak instability, the latter variables differ from SST and SSS in a thin “skin layer”; see, for example, Castro et al. 2003. This difference is important in the interpretation of satellite observations that measure SST and SSS, but for our purposes is negligible.) In this section, we first provide a general overview of the processes that determine ML properties (Sect. 3.2.1) and the models that are used to simulate them (Sect. 3.2.2). Then, we describe ML properties throughout the NIO during a typical year (Sect. 3.2.3) and in the Arabian Sea and Bay of Bengal during specific experiments (Sect. 3.2.4), linking their prominent features to the processes discussed in Sects. 3.2.1 and 3.2.2.

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3.2.1 Mixed-layer Processes The surface ML is a region of strong turbulence. That turbulence can be strong enough to entrain water from below the layer, causing h m to thicken. Conversely, when turbulence weakens water leaves (detrains from) the ML, thinning h m . The ML thickness is also impacted isopycnally by dynamical processes.

3.2.1.1

ML Turbulence

Much of the turbulence within the ML originates from τ . Wind stress τ can be viewed as being composed of two parts (τ = τ + τ¯ ): a turbulent part τ ; and a time-averaged part τ¯ , where the averaging time is larger than the time scale of τ . Turbulence is generated directly by τ , but also indirectly by τ¯ . In that case, τ¯ generates Ekman drift and inertial oscillations (Chap. 10), which cause strong bottom velocities v and hence shear across h m . In response to this shear, the interface becomes unstable due to Kelvin-Helmholtz instability (Smyth and Moum 2012), developing smallamplitude undulations; they rapidly grow to form large-amplitude “billows” that eventually break (rather like the breaking of surface waves), an efficient source of ML turbulence. Another turbulence source arises when the surface density of the ML is altered by Q, P, and E. A useful quantity, which measures the combined impact of all these forcings on ML density, is D = −αt

Q ρo − ρo αs (P − E)Sm ≡ B, cp g

(3.9)

where αt ≡ − (∂ρ/∂ T ) /ρ and αs ≡ (∂ρ/∂ S) /ρ are coefficients of temperature expansion and salinity contraction, c p is the heat capacity of sea water, and ρo is a typical sea-water density. Essentially, D defines a surface “density” flux that determines the rate at which ML density decreases in response to Q, P, and E fluxes. A related, and more commonly used, quantity is “buoyancy flux,” B = Dg/ρo , where g is the acceleration of gravity. Negative B (or D), due to surface cooling (Q < 0) or a deficit of precipitation over evaporation (P–E < 0), increases the density of surface water, thereby inducing convective overturning in the ML. Under idealized conditions, the initial transient response to this forcing is describable by Rayleigh-Bernard instability, in which overturning occurs in well-organized (hexagonal) cells with a width of the order of h m , sinking in their middle, and upwelling on their edges (Bodenschatz et al. 2000). Very quickly, this initial response intensifies to become a fully turbulent one, filled with overturning cells with a width scale of h m but of indefinite shape.

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Turbulence, generated by either wind or negative B, consists of a hierarchy of eddies over a wide range of length scales. Their kinetic energy “cascades” from larger to smaller eddies, generating ever smaller eddies. Eventually, the eddies are small enough for molecular diffusion to become important, in which case eddy energy is converted into heat. This process, “dissipation,” is an integral property of ML turbulence (see, for example, Chap. 3 in Soloviev and Lukas 2014). Because of dissipation, a constant energy source is required to sustain turbulence.

3.2.1.2

Entrainment and Detrainment

Because ML density is vertically uniform, turbulence quickly spreads throughout the layer. Deeper spreading, however, is inhibited by the stratification below the ML. Nevertheless, ML turbulence can still mix water from just below the ML into the layer so that h m gradually thickens at the rate we = h mt , a process known as “entrainment.” The opposite process occurs when positive B (Q > 0 and E–P < 0) lowers the surface density in an existing ML of thickness h m and density ρm . In response, a new mixed layer forms within the original one with a thickness h m < h m and density ρm < ρm , leaving behind a layer in the region from −h m < z < −h m where there is no longer active instability (a “relict” or “fossil” mixed layer). This thinning of the ML is referred to as “detrainment.” Detrainment can also occur when wind stress, a source of turbulence for the ML, decreases. Entrainment impacts Tm and Sm by mixing water from just below the ML into the layer. These changes are in addition to those brought about by air-sea fluxes of heat and mass. Typically, temperature decreases with depth so that entrainment cools Tm ; however, a notable exception occurs in the Bay of Bengal where temperature inversions can lead to entrainment warming (Sect. 3.2.4). In contrast, detrainment doesn’t affect Tm and Sm , since sub-mixed-layer water is never mixed into the newlyformed (thinner) ML.

3.2.1.3

Dynamical Processes

Even without entrainment/detrainment, h m can change isopycnically due to dynamically-induced upwelling and downwelling. For example, the divergence (or convergence) of Ekman flows (Ekman pumping) in the open ocean (Sect. 12.2.1) and from coasts (Sect. 13.2.1) alters h m simply by mass conservation. Horizontal advection can also impact h m , a process that is strikingly evident off Somalia during the summer monsoon when the SC is very strong (Sect. 4.9.6). Similar to detrainment, dynamical upwelling and downwelling don’t impact Tm and Sm because they occur largely isopycnally. On the other hand, if the upwelling is so strong that the ML bottom rises close to the surface, entrainment is required to maintain h m at some minimum value; this dynamically-induced entrainment accounts for the cooling seen in upwelling regions, such as off Somalia and Arabia during the summer monsoon (Sects. 4.9.5 and 4.9.6).

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3.2.2 Mixed-layer Models For the idealized solutions obtained in this book, accurate representation of ML properties is not necessary, and we simply represent the ML as a “body force” region with a vertical structure Z (z) that is spatially uniform and constant in time (see Step 7 in Sect. 5.2.1). Therefore, here we first briefly comment on the variety of ML models commonly used today. Then, we discuss one of them in a bit more detail, the Kraus-Turner (KT) model, because it links ML properties to surface forcings in a straightforward manner, and contrast it to another popular ML model in which the link is indirect, the Price-Weller-Pinkel (PWP) model.

3.2.2.1

Overview

Given the inherent complexity of turbulence, it is not possible to represent MLs in ocean models precisely. As a result, several ML models have been developed that do so approximately, each differing in the processes and parameterizations used to represent turbulence and its impacts. In this regard, Nagura et al. (2018) noted that six different types were used in 31 coupled ocean-atmosphere models used to study climate change. Consistent with observations, some ocean models include a distinct surface ML, in which variables are well mixed and h m is determined by either surface fluxes (Kraus and Turner 1967; Turner and Kraus 1967; Niiler and Kraus, 1977, KT model) or shear instability across its bottom (Price et al. 1986, PWP model). These models are often referred to as bulk models. Other models alter the values of the vertical mixing coefficients, κ and ν, a simpler approach for simulating the ML in a level (non-layer) model. In this latter approach, some models add a set of equations that explicitly determine turbulence and mixing coefficients (Mellor and Yamada 1982, MY; Kantha and Clayson 1994), whereas others parameterize the coefficients in terms of the large-scale current shear and stratification (Pacanowski and Philander 1981, PP; Large et al. 1994, KPP). Regardless of their type, the essential physics of ML models is expressed in a one-dimensional (z-coordinate) framework. They are embedded into 3-d models (like those discussed in Chapter 5), by allowing mixed-layer variables to be advected and mixed horizontally, as is the case in the real ocean (e.g., Mellor 2001).

3.2.2.2

KT Model

A slightly simplified form of the KT model is summarized by the equation 1 1 Pr = mu 3∗ − nh m B = we gρh m , 2 2

(3.10)

where ρ = ρm − ρd is the density jump across the ML bottom of the layer. Equation (3.10) is an approximate version of the general equation for turbulent kinetic energy

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57

in a well-mixed layer, in which turbulence quantities are parameterized in terms of time-averaged (easily-observed) variables. (It is Eq. 10.30 in Niiler and Kraus 1977, without the terms labelled C and F.) The equation states that the rate of production of turbulent kinetic energy Pr is generated by wind stirring (mu 3∗ ) and negative buoyancy forcing ( 21 nh m B, B < 0), and is always balanced by the rate of potentialenergy increase due to entrainment of sub-ML water into the layer, 21 we gρh m . In the wind-stirring term, mu 3∗ , u ∗ is a characteristic ocean velocity (“friction velocity”) wind-stress amplitude τ by τ = ρa Cd va2 = ρo Cd u 2∗ , so that √ related to the√ u ∗ = τ/ (ρo Cd ) = va ρa /ρo . There is no physically-based reason why windgenerated, oceanic turbulence should have the form mu 3∗ ; rather, the expression is useful because it relates turbulence to measurable, time-averaged quantities (either τ or va ). The positive factor m is then adjusted to ensure that ML thicknesses predicted by the model best fits observations. Regarding the buoyancy term, − 21 nh m B, as noted above it is reasonable to assume that turbulence production is proportional to negative B, a necessary condition for convective overturning, but there is no reason for assuming that all of that term leads to entrainment. Similar to m, the positive factor n allows for this possibility, with its value adjusted to best fit observations. Best-fit values for n differ depending on the sign of B: n < 1 when B < 0 and n = 1 when B ≥ 0 (Niiler and Kraus 1977). Further, estimated values of n values for negative B are quite small, of the order of 0.1 or less; that is, the fraction of energy available from convective overturning for entrainment is quite small, typically less than 10%. Solutions to (3.10) differ depending on whether turbulence is (Pr > 0) or is not (Pr = 0) being generated. When Pr > 0, (3.10) can be solved for we to get we =

Pr , 1 gh m ρ 2

Pr > 0.

(3.11a)

When Pr = 0, (3.10) implies that we = 0. This situation can happen only when B > 0, in which case (3.10) with n = 1 yields hm =

mu 3∗ ≡ h mo 1 B 2

Pr = 0,

(3.11b)

a generalization of the Monin-Obukhov depth to include forcing by P–E as well as Q. According to Eqs. (3.11), then, the mixed layer entrains (we > 0) when there is production of turbulence (Pr > 0), and detrains instantly to h mo when there is none (Pr = 0). Note that negative B impacts we in two ways: directly by increasing Pr , and indirectly by increasing ρm and hence by decreasing ρ. As noted above, the impact of the first process tends to be small (since typically n 1), but that of the second can be influential. In the real world, ρ in (3.11a) represents the vertical gradient of density immediately below the ML; thus, when B < 0, ρ is decreased and, hence by (3.11a), we is intensified (e.g., Nagura et al. 2018).

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3.2.2.3

3 Ocean Forcing and the Surface Mixed Layer

PWP Model

Like the KT model, the PWP model consists of a distinct, well-mixed, surface layer. Yet, the underlying physics of the PWP and KT models seem very different. While in the KT model turbulence is generated by u ∗ and negative B, in the PWP model it results entirely from an instability of the velocity gradient across the base of the layer (Kelvin-Helmholtz instability). Nevertheless, when implemented in ocean models the two ML models produce similar ML properties in response to the same atmospheric forcing. The similarity must happen because bottom shear is correlated with surface fluxes, so that both models tend to increase h m under the same forcings.

3.2.3 Basin-wide Properties In this section, we discuss climatological properties of ML variables (h m , Tm , and Sm ) in the NIO. Figures that illustrate basin-wide properties are based on monthly data from the World Ocean Atlas 2018 (Locarnini et al. 2019; Zweng et al. 2019). To show the temporal evolution with greater resolution, other figures are based on data from fixed locations in the Arabian Sea and Bay of Bengal.

3.2.3.1

Mixed-layer Thickness

The ML thickness is the depth below which strong turbulence ceases. Turbulence, however, is difficult to measure directly. Since density is well-mixed in the ML, a practical proxy for MLT is the depth at which density first increases by a small amount δρ from its surface value (e.g., Soloviev and Lukas 2014). Figure 3.3 illustrates the MLT climatological annual cycle throughout the NIO, when MLT is determined by a density change equivalent to a temperature decrease of 0.5◦C. The thinnest MLs occur throughout the year in the Bay of Bengal, particularly over its northern part. This property is due to the large freshwater input (P + R) into the bay, which establishes a new thin ML at the end of the summer monsoon (see the discussion of Fig. 3.9); subsequently, the new ML thickens slowly because the fresh water increases ρ in (3.11a), thereby weakening we . The thickest MLs occur in the Arabian Sea twice a year, during winter and summer (January and July panels in Fig. 3.3). During winter, the thick layer results from both northeasterly winds and evaporative cooling, the latter primarily by decreasing ρ and, hence, intensifying we . In contrast, the summertime thickening is induced only by wind-generated turbulence (see the discussion of Fig. 3.7 below). Interestingly, summertime MLs remain thin along the western, northern, and eastern boundaries of the Arabian Sea, due to coastal upwelling (Sects. 4.9 and 13.2).

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Fig. 3.3 Monthly-mean MLT in the NIO based on World Oceanographic Atlas 2018. The MLT is defined as the depth at which density changes from the surface value by equivalent of 0.5◦C

3.2.3.2

Mixed-layer Temperature

Figure 2.4 shows monthly maps of climatological Tm (SST). To review, during the summer the northern NIO is warmed by surface Q everywhere except in the western Arabian Sea, where it is cooled by offshore advection of waters upwelled at the coast and by entrainment (Sects. 2.2.2, 4.9.5, and 4.9.6). During the winter, northeasterly winds bring cooler continental winds over the Arabian Sea and Bay of Bengal, cooling Tm in both basins. The cooling continues throughout the winter monsoon, followed by rapid warming leading to expansion of the warm pool in the Indian Ocean.  The impact of Q on Tm is described by the balance, Tmt = Q/ c p ρo h m . To j estimate how well this balance holds, we estimated Tmt at month j from the maps in j j j+1 j−1 Fig. 2.4 by Tm / (2tm ), where Tm = Tm − Tm and tm = 30 days. Figure 3.4 j j j j then plots bimonthly maps of Q m = c p ρo h m Tm / (2tm ), where h m is taken from the maps in Fig. 3.3.

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j

Fig. 3.4 Bi-monthly plots of Q m , as defined in the text. The 6 plots are comparable to those in the bottom-right panel of Fig. 3.1, with significant differences between corresponding pairs indicating the importance of processes other than Q in determining Tm

The similarity of Fig. 3.4 to the bottom-right panel of Fig. 3.1 indicates that much of Tmt results from Q forcing. Perhaps more interesting are places where the two fields differ significantly, an indication of the importance of other processes. For example, in the northern Bay of Bengal during winter (December and January panels) the j decrease of Tm is weak (Q m  0), despite the strong cooling (Q < 0) caused by the northeasterly winds from Asia. This weak decrease is likely a consequence of temperature inversions in the region, in which warmer waters lie below the mixed layer (see the discussion of Fig. 3.9 below); as a result, entrainment warms Tm , counteracting the surface cooling by Q. In the western and central Arabian Sea j during summer (May and July panels), Tm decreases (Q m < 0) despite the presence of surface warming (Q > 0). In the western Arabian Sea, the decrease is caused by the offshore advection of cold upwelled coastal waters off Arabia and Somalia (Sects. 4.9.5, and 4.9.6), whereas in the mid-basin it results from cooling by winddriven entrainment (Sects. 3.2.1 and 3.2.4).

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Fig. 3.5 Monthly-mean SSS during January, March, May, July, September and November based on World Oceanographic Atlas 2018 (Zweng et al. (2019)

3.2.3.3

Mixed-layer Salinity

Figure 3.5 provides bimonthly maps of Sm (SSS). During summer, Sm is lower (higher) in the eastern (western) ocean, a result of climatological E–P being mostly negative (positive) there (Fig. 3.1, bottom-left panel). The difference is a maximum between the Bay of Bengal and Arabian Sea at the end of the summer monsoon (Fig. 3.5, September panel), by which time the bay has received most of its precipitation and runoff (Table 3.1). In this regard, Behara and Vinayachandran (2016) concluded that river runoff R plays a major role in keeping the salinities low in northern and western bay, with advection spreading the runoff offshore into the interior of the northern bay and, later in the season, southward along the east coast of India (Fig. 3.5, November and January panels; Sect. 4.9.7). During winter, Sm increases in both the Arabian Sea and Bay of Bengal, a consequence of E–P being positive in both basins (Fig. 3.1, November and January panels) and R being negligible compared to its summertime values. As a result, the Sm contrast between the two basins, generated during the summer, persists throughout the year.

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Fig. 3.6 Bimonthly plots of (E –P ) j , as defined in the text. The 6 plots are comparable to those in the bottom-left panel of Fig. 3.1, with significant differences between corresponding pairs indicating the importance of processes other than E –P in determining Sm

The impact of only E–P on Sm is described by the equation, Smt = (E − P) Sm / h m . Following the approach for the analogous Tm equation in the prej j vious subsection, we estimated its terms using monthly values Sm and h m from the j j maps in Figs. 3.5 and 3.3 to obtain the balance (E–P) j = h m Sm / (2Sm tm ), where j j+1 j−1 Sm = Sm − Sm . Figure 3.6 plots bimonthly (E–P) j . The large differences between corresponding panels in Fig. 3.6 and the bottomleft panel of Fig. 3.1 demonstrate that processes other than E–P (advection, river and land runoff, and entrainment) significantly impact month-to-month variability of Sm . For example, during November freshwater flows southward and westward within the East Indian Coastal Current (EICC) and Winter Monsoon Current (WMC) to impact Sm south of India and Sri Lanka (Fig. 3.5, November panel); subsequently, it is transported northward and westward by the West Indian Coastal Current (WICC) and the westward extension of the WMC to freshen Sm throughout the southeastern Arabian Sea (January panel). During summer, this process is reversed when the eastward extension of the Summer Monsoon Current (SMC) transports Arabian Sea High Salinity Water (ASHSW) eastward south of Sri Lanka and then northeastward

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into the Bay of Bengal (Fig. 3.5, July and September panels). Throughout the summer monsoon, freshening in the northern and northeastern bay by rainfall is enhanced by river runoff, and this coastal source of freshwater spreads into the the open sea primarily by Ekman drift (July panel). A similar, but weaker, impact due to freshwater input is also seen along the southwest coast of India, where near-coastal salinity remains relatively fresh, despite the southward movement of ASHSW within the SMC extension (Fig. 3.5, July panel).

3.2.4 Arabian Sea and Bay of Bengal While the plots in Sect. 3.2.3 provide a basin-wide picture of the seasonal cycle of ML properties in the NIO, they miss important aspects of that evolution (e.g., diurnal and intraseasonal variability). Campaigns to observe ML properties in detail are rare, because of the high effort required to mount them. Here, we discuss two of them: one in the Arabian Sea that covered an entire annual cycle and allowed the atmospheric forcings to be quantified; and the other in the Bay of Bengal, which, although lasting for only one week, illustrates salinity effects not seen in the Arabian Sea.

3.2.4.1

Arabian Sea

To identify the role of air-sea forcing on ML evolution in the Arabian Sea throughout the year, the Woods Hole Oceanographic Institution (WHOI) deployed a mooring in the central Arabian Sea from October 1994 to October 1995 (Weller et al. 2002). Figure 3.7 summarizes the salient features of the atmospheric forcing and MLT properties in the upper 250 m. Consistent with the basin-wide maps of h m and Tm (Figs. 3.3 and 3.5), the mixed layer thickens and cools during both monsoons (bottom three panels of Fig. 3.7). The air-sea fluxes measured at the mooring (top two panels) were able to distinguish between the processes that led to the thickening. Annual Cycle During the experiment, strong winds, moist air, and cloudy skies characterized the summer monsoon (JJA). There was a net heat flux into the ocean at that time, with an average value of 89.5 W/m2 . Despite this heat gain, Tm cooled and by the end of the season had dropped by 5.5◦C. Simultaneously, h m thickened to 80 m. An obvious inference is that wind-stress-induced entrainment was responsible for both the cooling and thickening. In support of this idea, Weller et al. (2002) noted that there was strong velocity shear across the base of the mixed layer, concluding that wind-driven shear instabilities caused the entrainment as in the PWP model (also see the discussion of Fig. 3.8 below).

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3 Ocean Forcing and the Surface Mixed Layer

Fig. 3.7 An overview of the surface forcing and response at the WHOI mooring in central Arabian Sea at 15.5◦N, 61.5◦E. From top to bottom, hourly values of wind-stress magnitude (τ ), daily average net heat flux (Q net ), mixed-layer thickness (based on the depth at which temperature is 0.1◦C cooler than the surface), SST and air temperature (AirT), and a contour plot of 36-houraveraged temperature in the upper 250 m. In the bottom panel, the daily-mean MLT is indicated as a black line. The gap in the time-series during April comes from the recovery and redeployment of the mooring. Dots by the depth scale in the bottom panel mark the depths of temperature observations. Based on Weller et al. (2002)

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Another possible cause of the summertime h m thickening is forcing by wind curl. As the Findlater Jet flows eastward over the Arabian Sea, it induces upwelling (downwelling) on its northern (southern) flank through open-ocean Ekman pumping (Bauer et al. 1991; Sect. 12.2), which impacts h m dynamically. Weller et al. (2002), however, found the contribution to h m thickening was insignificant compared to the wind-stress-induced entrainment, a consequence of the mooring being located near the axis of the jet where the curl is weak. During the winter monsoon (November–February), the winds were moderate with a prominent 5–7 day variability (Fig. 3.7). In addition, the sky was clear and air dry, so that Q cooled Tm at an average rate of –19.7 W/m2 . Further, Sm increased due to the loss of freshwater by surface evaporation. By the end of the monsoon, h m thickened to about 100 m, Tm dropped by 3◦C, and Sm increase by 0.4 psu. The moderate winds suggest that wind-driven turbulence was not the most important cause of the increase in h m ; likely more important is the decrease of ρ due to the increase in ρm by the negative buoyancy forcing B, which intensifies we (see the discussion of Fig. 3.8). During intermonsoon periods (usually March–May and October/November), the winds are weak and Q > 0. As a result, h m thins to form a new layer within the thicker mixed layers generated during the monsoons. As the season progresses, the newly formed ML thickens and warms, leading to a shallow thermocline (red shading in Fig. 3.7, bottom panel), which is referred to as the “seasonal thermocline.” Other Variability A noteworthy aspect of h m throughout the year is the existence of diurnal oscillations, which account for the jitteriness of h m in Fig. 3.7. They arise from the diurnal cycle of Q sw : There is net heat flux into the sea during the day when Q s exceeds the loss terms in (3.8), whereas at night when Q sw = 0 there is net heat loss. Large-amplitude, diurnal variations of h m occur whenever the winds are weak enough for h m to be determined primarily by Q, in which case h m can thin to almost zero thickness during the day. This condition (weak winds) is satisfied in the mooring record except during the summer monsoon. The oscillations are most prominent during the winter when their peak-to-peak amplitude can be 100 m. During the spring intermonsoon, diurnal variability includes oscillations in both h m and Tm . The Tm oscillations happen when winds are weak and h m is thin throughout the day. As a result, Q can warm the ML rapidly it acts on a thinner layer with less mass. After sunset, heat loss at the surface leads to a rapid reduction in Tm due to the thin ML. The resulting diurnal oscillations in h m and Tm should not be characterized merely as noise superimposed on the climatological signal. Instead, they have important dynamical implications. For example, nonlinear effects can enhance the vertical exchange between ML and sub-ML waters, with significant consequences to biology (McCreary et al. 2001).

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Also evident in Fig. 3.7 is subseasonal variability at periods of a week to a month. Note that the τ and Q fields associated with this variability are tightly linked: when wind speed increases, Q reduces in response to increased Q l . These forcings have the expected impact on h m with h m increasing during times of stronger τ and more negative Q. As for diurnal variability, the subseasonal changes in h m significantly impact biological activity (McCreary et al. 2001), by altering the time variability of the spring and fall blooms. Processes To illustrate the ability of the KT model to simulate h m at the WHOI mooring site, Fig. 3.8 compares h m at the site (dashed curve) with modeled h m at a nearby location. The three modeled curves are from a 4 21 -layer model, the upper layer of which is a KT mixed layer (Sect. 3.2.2). To avoid the jitteriness of h m in Fig. 3.7, the curves all plot daily-maximum values of MLT, hˆ m . (See Figs. 1 and 7b of McCreary et al. 2001, for a comparison of observed and modeled diurnal variability.) All three modeled curves reproduce the observed hˆ m reasonably well. Given the wide range of m and n values, the hˆ m curves are surprisingly close, the largest difference among them occurring during the winter when Solution 2 is considerably thicker than the others, likely because n is too large. The authors selected Solution 3 for their main run, as it reproduced h m most faithfully, not only at the mooring site but elsewhere in the Arabian Sea as well. It is striking that hˆ m is thicker during the winter rather than the summer. Given that the winter winds are much weaker than the summer winds, this property cannot be caused by τ forcing (mu 3∗ term in Eq. 3.10). Further, the property holds even in Solution 1 with when n = 0, so it isn’t caused by increased turbulence production Pr in response to negative B. Thus, it must be that the large wintertime increase in hˆ m happens because we increases due to the decrease in ρ caused by negative B (Eq. 3.11a), the only remaining possibility.

Fig. 3.8 Depth-time plots, showing observed daily-maximum MLT, hˆ m , at the WHOI mooring site, and three curves of modeled hˆ m at a nearby location. Consistent with (3.11a), the curve for Solution 2 is always deeper than that for Solution 3. After Hood et al. (2003)

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The good agreement of observed and modeled hˆ m curves in Fig. 3.8 indicates that most of the variability at the mooring site can be explained using 1-d ML concepts. The thinning of hˆ m during November/December and August, however, cannot. Note in the bottom panel of Fig. 3.7 that the thinnings are associated with shallowing of isotherms throughout the water column, indicating that they had a dynamical cause. Analysis of satellite altimetry confirmed this idea, showing that the shallowing resulted from the passage of two cyclonic eddies past the mooring site.

3.2.4.2

Bay of Bengal

Mixed-layer processes, similar to those in the Arabian Sea, are also active in the Bay of Bengal. There, however, MLs have additional features owing to the large freshwater flux. Specifically, precipitation P lowers surface salinity, establishing a thin (10–20 m thick) layer of fresh water. runoff R does so as well, and the thin mixed layer is then advected alongshore by coastal currents and offshore in plumes and lenses. As seen in Fig. 3.3, the shallowest mixed layers in the NIO (∼10 m) form in the northern Bay of Bengal, where the contribution from R is highest. The presence of the freshwater layer leads to the formation of two notable features: “barrier layers” and “temperature inversions.” The barrier layer is the region beneath the bottom of the thin freshwater layer and the top of the thermocline (Lukas and Lindstrom, 1991). Typically, it is a relict ML that results from detrainment, when a newly-formed, freshwater ML cuts off mixing at deeper levels. Barrier-layer temperatures are sometimes higher than in the freshwater layer, forming a temperature inversion. In such cases, the ocean is still stably stratified because density is so low in the ML owing to the freshwater there. Vinayachandran et al. (2002) observed the formation of freshwater and barrier layers at a location in the northern bay 17.5◦N, 89.0◦E). Figure 3.9 summarizes their observations, showing vertical profiles of temperature and salinity every 3 h from 27 July to 6 August 1999. During the first two days of the observations, the ML was 30 m deep, and Tm and Sm were almost uniform in depth with values of 32.9 ppt and 28.5◦C (Fig. 3.9, lower-left panel). On 29 July there was an abrupt change, when the salinity of the surface water dropped by 4 ppt. Thereafter, there was a new, significantly thinner (10–15 m) mixed layer, the base of which was marked by a sharp increase in salinity with depth (lower-right panel of Fig. 3.9). The barrier layer that formed at this time was located from about 15–30 m and persisted through the entire observation period. Within the barrier layer, there was a weak temperature inversion of approximately 0.5◦C magnitude. As noted by the authors, although some precipitation occurred during the period of the observations, it was too little to explain the magnitude of the freshening. They attributed the large salinity changes to a freshwater plume that formed near the coast and was subsequently advected towards the observation site by Ekman drift. Archived hydrographic data have been used to document the distribution of barrier layers and temperature inversions. Both occur in northern and western coastal areas of the bay, usually near mouths of major rivers. Barrier layers are found from June–February (Pankajakshan et al. 2007), their thicknesses varying from 40–60 m

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Fig. 3.9 Evolution of the thermohaline structure near the surface in the Bay of Bengal at 17.5◦N, 89.0◦E during 27 July to 6 August 1999. Upper-left (upper-right) panel panel shows the evolution of temperature (salinity) field. Lower-left (lower-right) panel gives the thermohaline structure at the point of observation on 28 July 1999 (6 August 1999), that is before (after) the arrival of freshwater plume at the location of observations. The barrier layer is marked as BL. Dashed contours are 28.5◦C (29 and 33 psu) in the upper-left (upper-right) panel. A temperature inversion of about 0.5◦C is present in the barrier layer. Vinaychandran (priv. comm.)

Video Captions

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with larger values more common during winter. Pankajakshan et al. (2002) showed that temperature inversions are usually observed during November–February, implying the importance of surface cooling in generating them. In the largest inversions, subsurface temperatures increase from SST by 1.6–2.4◦C and extend over a depth change of 10–20 m. During the winter, the EICC transports barrier layers and temperature inversions from the northern bay southward. These inversions have even been observed off the southwest coast of India (Shenoi et al. 2004; Durand et al. 2004) having been advected there by the extension of the southward-flowing EICC into the northward-flowing WICC.

Video Captions Video 3.1 Daily-climatological wind stress (vectors) and wind curl (color shading) based on SCOW data. The resolution of the dataset is 0.25◦ . The units of the vector key (lower-left corner) and color bar are N/m2 and 10−8 N/m3 , respectively.

Chapter 4

Ocean Circulations

Abstract This chapter provides a comprehensive overview of prominent NIO circulations, as determined from historical observations (hydrography, current-meter records, daily sea level, monthly drifter data, etc.). We begin by reviewing the NIO’s water masses, which provide a picture of its mean flow and overturning circulations. Then, we discuss the annual cycle of upper-ocean circulations in geographically distinct locations: the southern hemisphere, equatorial region, Sumatra/Java coast, Andaman Sea, Bay of Bengal, south of Sri Lanka, Arabian Sea (including the East Arabian and Somali Currents), and marginal seas (Persian Gulf and Gulf of Oman, Red Sea and Gulf of Aden). In each location, major currents are identified, as well as upwelling regions (if there are any). Key processes that drive the currents are also noted, together with references to later chapters where those processes are discussed in detail. One notable process, which dynamically links circulations in many locations, is the reflection of equatorial Kelvin waves from the NIO eastern boundary as Rossby waves and coastally-trapped waves. The reflected waves propagate into the Bay of Bengal and Arabian Sea, where they impact the East India Coastal Current, the monsoon currents south of Sri Lanka, the West India Coastal Current, and circulations in the Gulf of Oman Keywords Thermohaline structure · Water masses · Overturning cells · Near-surface drifters · Climatological sea level and circulations In this chapter, we review the large-scale circulations in the NIO that are driven by the surface fluxes discussed in Chap. 3. Our intent is to provide primarily a description of observed NIO phenomena. At the same time, it is often useful (necessary) to comment on the processes that determine them. When doing that, we provide references to later chapters where those dynamics are discussed in detail. Much of the chapter discusses wind-driven, near-surface currents. In addition, we review circulations associated with overturning cells, which involve sinking and water-mass formation in one region and upwelling in another. Two of those cells, the CEC and Subtropical Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_4. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_4

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Cell (STC), have major impacts on NIO climate and biological activity, and we consider them in detail in Chap. 17. We begin with a description of the thermohaline structure throughout the Indian Ocean, which, among other things, reveals a complex set of overturning circulations and helps to trace movement of waters (Sect. 4.1). Next, we introduce the NIO’s major surface currents and sea-level anomalies that we discuss in later sections (Sect. 4.2). These sections are organized geographically, from south to north and east to west: the southern hemisphere; equatorial region; the Sumatra/Java coastal region; Andaman Sea; Bay of Bengal, south of the tip of Sri Lanka; Arabian Sea; and the NIO’s marginal seas (Sects. 4.3–4.10). This organization is dynamically sensible because it follows prominent pathways by which information propagates through the system by wave radiation (Part III): westward across the South Indian Ocean (Rossby waves) and then northward to the equator along the western boundary (coastal waves); eastward along the equator (equatorial Kelvin waves) and then poleward along the eastern boundary (coastal waves); and westward across the Bay of Bengal and Arabian Sea into the marginal seas (Rossby waves and coastal waves).

4.1 Thermohaline Structure The thermohaline structure of the Indian Ocean is characterized by prominent water masses, each associated with an overturning circulation in which water sinks in one region and then flows horizontally a considerable distance before rising in another. Here, we first provide a general overview of water-mass formation and overturning cells, and list the Indian-Ocean water masses (Sect. 4.1.1). Then, we discuss each water mass in greater detail, organizing them into four layers depending on their depth range (Sects. 4.1.2–4.1.5).

4.1.1 Overview Overturning cells can be divided into two types, depending on the processes that determine their sinking and subsurface branches. In one type (deep-convection), the cells are driven primarily by thermohaline processes. In the other (subduction), wind forcing is an integral part of their dynamics. In deep convection, fluxes of heat and mass across the air-sea interface determine the temperature and salinity, and hence density, of the surface water. When they increase its density from ρ to ρ ∗ , say, it sinks to the depth where the density of the surrounding water is also ρ ∗ . Mass conservation, of course, requires that if water sinks in one region, it must rise in another. In the open ocean, sinking and rising occur over areas of deep turbulent convective mixing where fluid overturns in numerous plumes, with a horizontal scale of about a kilometer and vertical velocities of up to 10 cm s−1 (Marshall and Schott 1999). If the density increase extends to basin boundaries, however, the sinking generates a large-scale circulation, which advects the freshly-

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sunk water away from its generation region, requiring that it rises elsewhere in the ocean and generating an overturning cell (Spall 2010; Schloesser et al. 2012). Convectively-driven cells of this sort are typically generated in marginal seas, where wintertime cooling is most intense due to cold winds from the surrounding land. The most famous of these cells involves the sinking in the North Atlantic and Southern Ocean that forms North Atlantic Deep Water (NADW), and Antarctic Bottom Water (AABW), respectively. The pathways by which these waters rise to the surface and return to their respective sinking regions is complex and global in extent, and hence the complete cell is referred to as the “global overturning circulation.” In the Indian Ocean, convective overturning is associated with the formation of Persian Gulf Water (PGW) and Red Sea Water (RSW); their circulations are relatively shallow, confined to the depths less than about 200 m and 1000 m, respectively (Sects. 4.10.1 and 4.10.3). The subduction process differs from deep convection in that it requires wind forcing and can occur in the open ocean. Like deep convection, it happens during winter when the density of surface water is increased to ρ ∗ . The density increase causes convective mixing, which, together with intensified wind mixing, thickens the surface mixed layer (Sect. 3.2). When the mixed layer thins during the following spring, it leaves behind a subsurface water mass with the characteristics of wintertime surface water. If the newly formed water is subsequently advected by the background circulation to a region where the wintertime mixed layer is thinner, a subsurface water mass is created. This process occurs in the poleward parts of the subtropical gyres of all oceans, the background circulation being the cyclonic, subtropical gyres that then carry the subducted water into the tropics. In the Indian Ocean, the water masses formed in this way are South Indian Central Water (SICW; Sect. 4.1.5) and North Arabian Sea High Salinity Water (NASHSW; Sect. 4.9.4.3), despite almost all of the latter being located in the tropics. SICW is a major source of water for the CEC and STC (Chap. 17). Ocean tracers (salinity, oxygen, and nutrients, etc.) are used to identify particular water masses. For example, Fig. 4.1 shows the distribution of salinity along Section I08I09 from the World Ocean Circulation Experiment (WOCE) Hydrography Program (WHP), which stretches from the Southern Ocean to the northern Bay of Bengal along approximately 90◦E longitude. The various water masses are defined by regions of high and low salinity, and where the potential temperature-salinity (θ S) diagram tends to be linear. The figure has signatures of 12 distinct water masses, and illustrates the depths along which their cores spread. Table 4.1 and Fig. 4.2 summarize properties of the significant water masses that are either located in the NIO or that impact them. Table 4.1 lists the 12 water masses visible in Fig. 4.2. It also includes Persian Gulf Water (PGW), ASHSW, and North Arabian Sea High-Salinity Water (NASHSW), which are formed in the North Arabian Sea in volumes too small to be visible in the eastern-ocean section in Fig. 4.1. For each water mass, the table provides its approximate depth range, defining characteristics (primarily salinity variation in the vertical), and the processes that lead to its formation. Table 4.2 lists the range of potential temperature, salinity, and density (σθ ) of each water mass in Table 4.1. Figure 4.2 illustrates the ranges in a potential

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Fig. 4.1 Distribution of salinity along WOCE Section I08I09. Major water masses found in the section are identified by acronyms defined in Table 4.1 (red labels). (From the WOCE Indian Ocean Atlas; Talley 2013)

temperature–salinity diagram. The figure also shows observed (θ –S) profiles at six locations in the NIO. The preceding tables and figures highlight the characteristics of the water masses discussed in the rest of this section. For convenience, the water masses are organized into four “ layers,” corresponding to bottom, deep, intermediate, and upper-ocean waters.

4.1.2 Bottom Layer Talley et al. (2011), among others, have noted that in the Indian Ocean there are no surface sources of water to feed the bottom layer (3,000 m to bottom). Rather, the bottom layer is supplied by the sinking of dense water in the Atlantic and Southern Oceans. This water is referred to as Lower Circumpolar Deep Water (LCDW) or Antarctic Bottom Water. In the Indian Ocean, it is identified by a deep salinity minimum, as evident in Fig. 4.1. Figure 4.3 depicts the circulation of LCDW at 3500 m, inferred from steric-height gradients. The flow is strongly influenced by the topography and all of it is directed northward. The most conspicuous current flows along the Madagascar coast, passes

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Table 4.1 List of important water masses in the NIO, showing: their name and acronym in parentheses (column 1), their defining characteristics (column 2), the depth range where they are observed (column 3), and processes involved in their formation. The list is separated into four layers by thicker horizontal lines, indicating upper-ocean (top), intermediate (upper middle), deep (lower middle), and bottom (bottom) waters. The labels IO, AO, PO, and SO are acronyms for the Indian, Atlantic, Pacific, and Southern Oceans, respectively. Based on lists from Talley et al. (2011) and Schott and McCreary (2001) Water Mass Characteristic Depth range Process South Indian Central Water (SICW) Indonesian Throughflow Water (ITFW) Indian Equatorial Water (IEW) North Indian Central Water (NICW) Bay of Bengal Surface Water (BBSW) Arabian Sea High Salinity Water (ASHSW) North Arabian Sea High-salinity Water (NASHSW) Persian Gulf Water (PGW)

Subtropical thermocline Low S in S. Eq. Current (SEC)

0–1000 m

Quasi-linear θ-S relation Linear θ-S relation, 150–1000 m Warm and fresh

0–1000 m

0–100 m

Warm and salty

0–200 m

Cool and salty

50–150 m

Subduction in north Arabian Sea

Salty

200–350 m

E > 0, Q < 0 in Persian Gulf

800–1200 m

Throughflow from Pacific

400–1200 m

Indonesian Low S in SEC Intermediate Water (IIW) Red Sea Water (RSW) S maximum

0–500 m

0–1000 m

Subduction in southern IO Flow from PO to IO

Mix of ITFW with surrounding waters Mix of SICW with surrounding waters P + R > E in Bay of Bengal P + R < E in Arabian Sea

Antarctic Intermediate S minimum Water (AAIW)

500–1200 m

E > 0, Q < 0 in Red Sea Advection from SO

Indian Deep Water (IDW) North Atlantic Deep Water (NADW) Upper Circumpolar Deep Water (UCDW)

O2 minimum, N maximum S maximum

2000–3500 m

Mix of IO deep waters

2200–3500 m

Advection from AO

High S in SO

1000–3000 m

Mixing of deep waters in SO

Lower Circumpolar Deep Water (LCDW)

S maximum in SO

1000–3000 m

E > 0, Q < 0 in SO

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Fig. 4.2 Potential temperature and salinity diagram showing their ranges for the water masses listed in Table 4.2. The six black theta-S curves are based on 1◦ annual data from World Ocean Atlas 2018 (Locarnini et al. 2019; Zweng et al. 2019) taken from regions of well-known currents: East Indian Coastal Current (EICC; 85.5◦E, 18.5◦N), South Equatorial Current (SEC; 79.5◦E, 9.5◦S), Monsoon Currents (MC; 77.5◦E, 4.5◦N), East African Coastal Current (EACC; 41.5◦E, 4.5◦S), Somali Current (SC; 49.5◦E, 4.5◦N), and West Indian Coastal Current (WICC; 68.5◦E, 19.5◦N)

through Amirante Passage, and finally enters the Somali and Arabian basins. Given the lack of any southward flow, LCDW must rise in the water column as its spreads northward. This rise is accomplished by diffusion, primarily generated by flow over rough bottom topography (Waterhouse et al. 2014; Sect. 4.1.3), which mixes LCDW with the lower-density deep water above it. Thus, as it flows northward the density of LCDW decreases.

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Table 4.2 Ranges of potential temperature T (◦C), salinity S (psu) and potential density σθ (kg/m3 ) for each of the water masses listed in Table 4.1. Also listed are reference to the literature on which the ranges are based, using the abbreviations: BP: Banse and Postel (2009); M: Marakim et al. (2019); SM: Schott and McCreary (2001); T: Talley (2013), WOCE Indian Ocean Atlas Water mass T (◦C) S (psu) σθ (kg m−3 ) Reference SICW ITFW IEW NICW BBSW ASHSW NASHSW PGW IIW RSW AAIW IDW NADW UCDW LCDW

8–25 8–23 8–23 8–25 25–29 22–30 19–22 5–14 3.5–5.5 5–14 4–8 1.8–2.2 1.2–1.8 1.2–1.8 0.8–1.0

34.6–35.8 34.4–35.0 34.6–35.0 34.6–35.8 28.0–35.0 36.0–36.8 36.0–36.5 34.8–35.4 34.6–34.7 34.8–35.4 34.4–34.7 34.6–34.8 34.72 34.72 34.71–34.72

23.5–27.0 23.0–27.0 23.5–27.0 23.5–27.0 21.0–23.0 ∼24.0 ∼25.0 26.0–27.0 27.0–27.5 27.0–27.5 27.1–27.3 27.4–27.7 27.7–27.8 27.7–27.8 27.82

M M M M M BP BP M M SM SM SM, T SM, T SM, T SM, T

4.1.3 Deep Layer As for bottom water, there is no source of deep water in the Indian Ocean. Instead, lower deep waters (below approximately 2000 m, a neutral density of 27.96 kg/m3 ) move northward into the IO from the Antarctic Circumpolar Current (ACC). These waters include NADW and Upper Circumpolar Deep Water (UCDW). A salinity maximum marks the former, whereas UCDW is a mixture of three deep waters— NADW, Indian Deep Water (IDW) and Pacific Deep Water (PDW)—that forms in the ACC. Most of the water enters the Indian Ocean across the width of the southern boundary, but some NADW enters directly from the South Atlantic. UCDW and NADW and are identifiable in Fig. 4.1 as a single tongue of saltier water that deepens to the north. As they flow northward, both bottom and lower-deep waters mix with overlying water masses, and in so doing lead to the formation of a less-dense water mass, IDW. Thus, IDW is not formed by sinking of dense water (subduction), which is the norm for deep-water formation in the Southern Ocean and North Atlantic. After its formation, IDW flows southward, eventually joining a deep part of the Agulhas current to exit the basin. Recent work, however, suggests that part of the outflow may occur within a deep eastern-boundary current off the Australia coast (Tamsitt et al. 2019).

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Fig. 4.3 Adjusted steric height at 3500 db (10 m2 /s2 or 10 J/kg). Depths less than 3500 m are shaded. From Reid (2003)

The northward movement of bottom and lower-deep waters from the Southern Ocean into the Indian Ocean, their warming due to mixing with overlying waters, and exit from the basin as IDW form the Indian Ocean’s Deep Overturning Circulation (DOC). Estimates of its strength (i.e., the transport of its northward branch, which is equal to that of the shallower southward branch) are of the order of 10–15 Sv, the variability arising from both differences in analysis methods and natural temporal changes in the abyssal oceanic circulation. The DOC is important because it refreshes the NIO’s deep and bottom waters, including in the Arabian Sea and Bay of Bengal. Further, the DOC is a significant part of the global thermohaline circulation (Talley et al. 2011). Dynamics: The theoretical basis of the overturning of abyssal waters is provided by the classical work of Stommel and Arons (1960). In this study, deep water is thought of as being in a uniform deep layer, out of which a horizontally uniform upwelling velocity w continually draws water into a shallower part of the water column. In an ocean with realistic stratification, w is assumed to be related to vertical diffusion through the steady-state balance, wρz = κρzz . A simple representation of the area-averaged density stratification in the Indian Ocean is ρ (z) = ρo exp (z/z), where z ≈ 1000 m. It follows that κ/w = z. With a mean deep upwelling of w = 4×10−7 m/s (Talley et al. 2011) and z = 1000 m, the resulting diffusion coefficient is κ = 4×10−4 m2 /s.

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Waterhouse et al. (2014), using a global set of direct microstructure measurements and indirect estimates based on parameterizations, concluded that the globalaveraged diapycnal diffusivity below 1000 m depth is of the order of 10−4 m2 /s. They further found that the turbulent dissipation rate is enhanced over rough bottom topography, perhaps due to internal-wave generation. This diffusivity value is of the same order as those determined in classical studies (see Munk 1966); in addition, it agrees with the value of κ = 2–10×10−4 m2 /s needed to sustain a steady-state DOC (Talley et al. 2011). While this agreement is encouraging, there are many aspects of the Indian Ocean’s DOC that remain to be explored. For example, empirical studies (see Fig. 4.3) suggest that the northward flow of the deep ocean (∼3500 m) behaves like a western boundary current along the slopes of bottom topography. Further, the pathways followed by the southward DOC branch are at present not clear, nor are there theories that suggest what they might be.

4.1.4 Intermediate Layer There are three major water masses in the intermediate layer (400–1500 m) of the Indian Ocean: Antarctic Intermediate Water (AAIW), Indonesian Intermediate Water (IIW), and Red Sea Water (RSW). Antarctic Intermediate Water (AAIW): AAIW is a water mass characterized by a salinity minimum in the vertical at densities of σθ = 27.0–27.3 kg/m3 and depths of 500–1000 m. It is formed by subduction just north of the ACC near South America, and thereafter is advected into all the ocean basins. Within the Indian Ocean, AAIW spreads northward within the deep part of the subtropical Indian Ocean, where it is readily recognizable as a distinct layer of minimum salinity south of about 12◦S (Fig. 4.1). The minimum, located at about 1100 m near the northern edge of the ACC, shoals to about 500 m at about 15◦S in the western IO. Although in an eroded form, it can also be seen near the equator along the western boundary in the East Africa Coastal Current (EACC) and SC and is found in the western Arabian Sea (Talley 1996; Talley et al. 2011). AAIW leaves the IO through the western boundary current of the South Indian Ocean subtropical gyre, the Agulhas current. As AAIW moves northward in the deep part of the gyre, its volume increases due to mixing with water in the upper branch of the DOC. As a result, the transport of AAIW that exits the Indian Ocean at about 33◦S is higher by about 5 Sv than the transport that enters the basin near the southern edge of the subtropical gyre (Talley et al. 2011). Indonesian Intermediate Water (IIW): The Indonesian archipelago consists of thousands of islands (estimated to be 13,000–18,000) and the passages between them. Sea level, typically being higher on the Pacific side, results in a unidirectional, annualmean flow from the Pacific to the Indian Ocean, the Indonesian Throughflow (ITF). The throughflow waters exit the Indonesian Archipelago through three principal

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passages: Lombok Strait, Ombai Strait, and Leti Strait, the latter being the deepest with a sill depth of 1250 m. The ITF has two relative maxima: a shallower one (400 m), which contains most of the ITF transport (Sect. 4.1.5); and a deeper one (500– 1500 m). When this deeper water enters the Indian Ocean, it flows westward within the South Equatorial Current with a transport of 3–7 Sv (Talley and Sprintall 2005). Along the way, it mixes with the other two intermediate waters of the region, AAIW to the south and RSW to the north, to form IIW. The IIW core occurs as a salinity minimum at about the same depth as AAIW (Fig. 4.1), and is also recognizable as a band of high silicate (Talley and Sprintall 2005). IIW exits the Indian Ocean via the Agulhas current after passing through the Mozambique channel (Makarim et al. 2019). Note in Fig. 4.1 that a pocket of lower salinity occurs at the depth of approximately 1200 m and 7.5◦S. Similar features are often observed in the vicinity of the front associated with the South Equatorial Current (Sect. 4.4), as that front separates lower salinity waters of the SIO from the higher salinities of the NIO. Red Sea Water (RSW): The core of RSW is marked by a density of σθ = 27.2–27.4 kg/m3 in the western Arabian Sea. This water results from a deep overflow of about 0.4 Sv of highly saline water from the Red Sea with a density of σθ = 27.6 kg/m3 (Sect. 4.10.3). It then spreads eastward and southward, the latter pathway along the western boundary towards the Agulhas Current (Beal et al. 2000). The high-salinity signature of RSW is seen distinctly in Fig. 4.4, which plots data from WOCE Section I1: It is the cause of the relative maximum from 400–800 m near the western edge of section (top panel), and the bump in salinity near 10◦C in the westernmost (blue) θ -S diagram (bottom-right panel). In general, the presence of RSW impacts the Arabian Sea at depths of 400–1400 m. As noted by Talley et al. (2011), however, because “the saline overflow water mainly results in a salty ‘dye’ for the deep northern Indian waters,” the signature of RSW progressively weakens towards the east as the salinity maximum erodes (Rochford 1964; Shenoi et al. 1993; Bower et al. 2000); for example, note the eastward weakening of the aforementioned salinity bump in the θ -S diagrams of Fig. 4.4. Despite this weakening, the core of RSW is still detectable within the bay in high-vertical-resolution, conductivity-temperature-depth (CTD) data (Jain et al. 2017).

4.1.5 Upper Layer There are a number of water masses in the upper Indian Ocean. Collectively, they determine the near-surface thermohaline structure of the NIO. South Indian Central Water (SICW): SICW is a water mass in the South Indian Ocean that lies above AAIW (1000 m). It is characterized by high salinity, which in Fig. 4.1 appears as a high-salinity tongue extending from the Subantarctic Front near 44◦S to 12◦S. Note that surface isohalines descend directly into the subsurface ocean

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81

Fig. 4.4 The upper panel shows the distribution of salinity in WOCE Section I1 along 9◦N. The lower panel on the left shows geographical stretch of the section. On the right is θ–S diagram for the stations identified on the left. (From the WOCE Indian Ocean Atlas; Talley 2013.)

from about 44–30◦S, an indication that subduction occurs there (Sect. 4.1.1). North of 30◦S, salinity decreases toward the surface, indicating the presence of a shallow layer (100–200 m) of South Indian Subtropical Underwater (Talley et al. 2011). The newly subducted water in the SICW flows westward within the South Equatorial Current, north in the East Africa Coastal Current, and then either joins the South Equatorial Countercurrent or crosses the equator (Sect. 4.3). Eventually it upwells, either along the South Equatorial Thermocline Ridge or in the northern-hemisphere upwelling regions, and then returns to the southern hemisphere near the surface. As such, SICW is the primary part of the subsurface branch of Indian Ocean’s shallow overturning cells (Chap. 17). It is noteworthy that no comparable subducted water mass forms in the northern hemisphere, a consequence of the Asian landmass and the absence of trade winds. (An exception is a weak overturning cell associated with the formation of NASHSW in the northern Arabian Sea, see below, which Banse and Postel, 2009, estimate to have a transport of only 0.5–0.7 Sv.) One result is a striking contrast in concentrations of dissolved oxygen (and other tracers) between the NIO and the subtropical gyre of

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the South Indian Ocean. As noted by Wyrtki (1973b), there is a front near 12◦S (the latitude of Indonesian Throughflow Water discussed next) that separates NIO and subtropical waters, with low (high) dissolved oxygen north (south) of this latitude. Indonesian Throughflow Water (ITFW): The ITFW is an upper-ocean water mass that enters the Indian Ocean at depths less than 500 m through the Indonesian passages. Owing to high precipitation and mixing in the Indonesian seas (Ffield and Gordon 1992), it is characterized by low salinity. After entering the Indian Ocean, it flows westward within the South Equatorial Current near 12◦S. As it does, it mixes with higher salinity waters to the north (ASHSW and RSW) and the south (SICW), but still maintains its low-salinity signature (Fig. 4.1). Estimates of the total annual-mean transport of the Indonesian Throughflow (IIW + ITFW) range from 2 to more than 20 Sv, the large variability a consequence of the lack of direct observations during pre-2000 years and its large seasonal-tointerannual variability (Wyrtki 1961; Gordon 1986; Godfrey 1996; Meyers et al. 1995; Schiller et al. 1998; Potemra 1999; Lebedev and Yaremchuk 2000; Song and Gordon 2004). More recent estimates, however, favor a value in the upper half of this range (Gordon et al. 1999; Vranes et al. 2002; Susanto and Gordon 2005; Gordon et al. 2010). For example, using data from current meters deployed in major passages of the archipelago, Gordon et al. (2010) reported a mean value of 15 Sv for the throughflow during 2004–2006. A transport of this order is the same as that of the global, overturning transport itself, making the throughflow a significant component of that overturning. Indian Equatorial Water (IEW): IEW is a thermocline water mass located approximately between 5 and 10◦S at depths of 200–800 m in the western Indian Ocean (see Fig. 4.1). A low vertical salinity gradient (Fig. 4.1) and a quasi-linear, θ -S relation mark this water. The mechanism that produces this water is not known. Sharma (1976) and You and Tomczak (1993) suggest that it forms in the western Indian Ocean from mixing of ITW coming from the east, and central-Indian Ocean waters to the south (SICW, see Table 4.2) and the north (North Indian Central Water, discussed next). Being a mixture of other water masses, it forms without a contribution of air-sea fluxes. North Indian Central Water (NICW): SICW spreads westward within the South Equatorial Current and then northward across the equator in the SC. It supplies upwelling water off Somalia and Arabia. In the Northern Hemisphere, it is referred to as NICW, an aged (lower oxygen and higher nutrients) version of SICW that is found everywhere in the NIO at depths from 150 to 1000 m (Fig. 4.1). It is characterized by a linear θ -S curve (Fig. 4.4, lower right-hand panel), which, depending on location, is perturbed by one or the other of RSW or PGW. Persian Gulf Water (PGW): PGW is a high-salinity, near-surface water mass in the NIO, formed in the Persian Gulf. Its interior is, on average, about 60-m deep. The gulf experiences high evaporation, leading to a dense, saline, water mass that

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83

flows through the Straits of Hormuz into the Arabian Sea at depths of 200–250 m (Sect. 4.10.1). PGW is present throughout the northern Arabian Sea, identifiable by its core salinity maximum (see Fig. 4.2 and Table 4.2), but loses its identity farther to the south (Rochford 1964; Bower et al. 2000; Prasad et al. 2001). However, Jain et al. (2017) showed that careful study of CTD profiles from the Bay of Bengal can reveal traces of PGW. Arabian Sea High Salinity Water (ASHSW): ASHSW is found over most of the Arabian Sea (darker orange shading in Fig. 4.4). It is a high-salinity, surface water mass, generated by the excess of evaporation over precipitation in the region, in which salinity values often exceed 36.5 psu. North Arabian Sea High Salinity Water (NASHSW): NASHSW, a cousin of ASHSW, is a shallow (50–150 m) subsurface water mass that forms annually along the northern boundary of the Arabian Sea during the northeast monsoon, when cold and dry, northeasterly winds blowing off the continent cause high evaporation (Banse and Postel 2009). The newly formed NASHSW subducts and interleaves below the ASHSW. In other seasons, it spreads southward as a salinity maximum just underneath the surface-mixed layer (Morrison 1997; Schott and Fischer 2000). While the difference in salinities of NASHSW (36.0–36.5 ppt) and ASHSW (36.0–36.8 ppt) is small, the former is cooler (19–22◦C vs. 22–30◦C) and denser (σt about 25 kg/m3 versus 24 kg/m3 ). In the north, a salinity minimum often separates the two water masses (Banse and Postel 2009). In the south, distinction between them is lost, and the two together are referred to as ASHSW (Prasanna Kumar and Prasad 1999). Bay of Bengal Surface Water (BBSW): BBSW is a surface water mass found in the Bay of Bengal north of about 5◦N. Its salinity decreases from about 34 psu at 5◦N to 31 psu or less near the northern end of the bay. High runoff from the rivers bordering the bay and high precipitation over the bay contribute to the formation of this water. The near-surface waters of the θ -S diagram in the lower panel of Fig. 4.4 give the signature of BBSW. The upper panel of the figure reveals the striking contrast of near-surface waters (shallower than ∼150 m) between the Bay of Bengal and Arabian Sea.

4.2 Climatological Surface Currents and Sea Level This short section serves to introduce the major surface currents and sea-level signals in the NIO, which are then discussed in detail in Sects. 4.3–4.10. The list of currents is based on earlier reviews of NIO circulation, notably those of Wyrtki (1973a) and Schott and McCreary (2001). The listed sea-level anomalies are key indicators of specific wind-driven processes.

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Table 4.3 List of NIO currents (top portion) and sea-level features (bottom portion), showing their name and acronym in parentheses (column 1) and their annual cycle (other columns). The two portions are separated by a thicker horizontal line. Arrows indicate the direction of a current during a particular month: predominantly northward (↑), southward (↓), eastward (→), and westward (←). Letters indicate whether sea level during a month is high (H) or low (L) Current

J

South Equatorial Current (SEC)

← ← ← ← ← ← ← ← ← ← ← ← ← ← ←

F

M A

M J

South Equatorial Countercurrent (SECC)

→ → → → →

East Africa Coastal Current (EACC)

↑

↑

↑

↑

A

S

O

→ → → ↑

East Indian Coastal Current (EICC)

↑

Summer/Winter Monsoon Currents (SMC/WMC)

← ← ←

West Indian Coastal Current (WICC)

↑

East Arabia Coastal Current (EArCC)

↓

Somali Current (SC)

↓

↓

↓

South Equatorial Thermocline Ridge (SETR)

L

L

L

Andaman Sea (AndS)

L

L

L

↑

↑

↑

↑

H

H

↑

↑

↑

↓

↓

J

F

M

↑

↑

↑

↑

→ → → → ↓

↑

↑

↑

← ← ← ←

↓

↓

↓

↓

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

↑

L

L

L

L

L

H

H

H

H

H

L

L

L

L

L

L

L

L

Socotra Eddy (SE)

H

H

H

H

Great Whirl (GW)

H

H

H

H

Southern Gyre (SG)

H

H

H

H

Lakshadweep High (LH) and Low (LL)

↑

→ → → →

L

D

→ →

← ← ← ← ← ← ↑ ↓

Sri Lanka Dome (SLD)

N

→ → → → → → ↑

→ →

Wyrtki Jets (WJs) South Java Current (SJC)

↑

J

↑

↑

↑

↑

↓

↓

↓

↓

↓

↓

L

L

L

L

L

L

H

H

H

L

L

L

H

H

H

↓

4.2.1 Currents Figure 4.5 shows the monthly climatology of near-surface currents based on data from the Global Drifter Program (GDP). Lumpkin and Johnson (2013) used a dataset of 15-m drogued drifters to construct a monthly climatology. Subsequently, Laurindo et al. (2017) enlarged the dataset by including undrogued drifters that were corrected for slip bias, thus recovering about half of the GDP dataset. Figure 4.5 uses monthly climatology based on the enlarged dataset. The top portion of Table 4.3 lists the major currents visible in the figure, and the schematic in Fig. 4.6 shows their locations. The table also indicates the months during which a particular current occurs and its flow direction. Of the 10 currents listed, only the South Equatorial Current and the East Africa Coastal Current, located near the southern boundary of the NIO, flow in the same direction and exist throughout the year. The others come under the influence of monsoon winds and, hence, have a distinct annual cycle in intensity and direction, a distinguishing character of NIO circulations.

Fig. 4.5 Climatological monthly-mean annual cycle of surface currents based on surface drifter data (Laurindo et al. 2017). Because the range of velocity    1/4 1 amplitudes is so large, plotted arrows show vˆ = (u/a, v/a), a = u 2 + v 2 , which has the same direction as v but an amplitude of vˆ  = a = |v| 2

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Fig. 4.6 Schematic diagram showing geographical locations of the climatological currents and sea-level anomalies listed in Table 4.3. Black-dashed contours indicate 200-m isobaths. Open lines with arrowheads mark currents and their direction. Closed lines show locations of prominent sealevel anomalies and their arrowheads the direction of rotation around them. Colors indicate the time of occurrence of a particular feature as follows: throughout the year (magenta); throughout the year except for summer (dashed green); spring and fall (green); summer (red); summer and winter with a reversal (blue); and extensions of the SMC/WMC into the Arabian Sea and Bay of Bengal (dashed-blue)

4.2.2 Sea Level and Geostrophic Velocity We use five sea-level-related products from the Archiving, Validation, and Interpretation of Satellite Oceanographic (AVISO) dataset (Pujol et al. 2016): SLA, mean dynamic topography (MDT), absolute dynamic topography (ADT), zonal geostrophic velocity (UGOS), and meridional geostrophic velocity (VGOS). Technical information about these products is available in (Taburet and Pujol 2020). The products have a horizontal resolution of 0.25◦×0.25◦ , a temporal resolution of a day, and are available from 1993 onward. We have used daily data during 1993–2019 to construct daily climatologies for each of the 5 products for 365 days (29 February data were removed). The climatologies are used in many of the figures and videos throughout the book. SLA: To highlight major features of the annual cycle, Fig. 4.7 plots the monthlymean climatology of AVISO sea-level anomalies, which is essentially a version of Video 1.1 in graphical form. The bottom block of Table 4.3 lists sea-level features visible in the figure that are discussed later in this chapter, and Fig. 4.6 indicates where they occur.

Fig. 4.7 Monthly-mean SLA using daily quarter-degree data from AVISO during 1993-2019. See Taburet and Pujol (2020) for details on data collection and analysis

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Fig. 4.8 MDT (dyn. cm) over the Indian Ocean based on quarter-degree data from AVISO (Taburet et al. 2020)

MDT: Sea-level features that persist throughout the year (absent in SLA) are provided by MDT. MDT is more difficult to determine than SLA, because it has to take into account the shape of the geoid, which, owing to regional variations in the gravitational acceleration g, is not a perfect sphere. It is determined from a combination of observations and models: a geoid model based on Gravity Recovery and Climate Experiment (GRACE) data; drifting buoy velocities; hydrographic profiles from the Array for Real-time Geostrophic Oceanography (Argo) array; and an Ekman-drift model to extract the geostrophic velocity component from velocity measurements (Pujol et al. 2016). Figure 4.8 shows MDT for the Indian Ocean. The plot has a number of prominent features. There is a region of low sea level 5–10◦S in the central/western ocean, associated with the underlying thermocline ridge (Sect. 4.3.1), and a high region to its south; together, they indicate the presence of a Sverdrup circulation, driven by the annual-mean component of the Southeast Trades (Sect. 11.1). Along the equator the MDT slope is positive, a consequence of the annual-mean, equatorial winds being primarily westerlies (Fig. 4.12, top-right panel). (This slope is opposite to those in the Atlantic and Pacific Oceans, where the mean equatorial winds are easterlies.) In the northern Bay of Bengal and Arabian Sea, MDT is relatively high and low, respectively, largely a result of salinity forcing (Sects. 4.7.5 and 4.9.7). Finally, there are small-scale, eddy-like features along the western boundary of the Arabian Sea and within the Gulfs of Oman and Aden.

Fig. 4.9 Monthly-mean ADT using daily quarter-degree data from AVISO during 1993-2019. See Taburet and Pujol (2020) for details on data collection and analysis

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ADT: ADT is the sum of MDT and SLA, and Fig. 4.9 shows its annual cycle. A comparison of Figs. 4.7 and 4.9 highlights the important contribution of MDT to overall sea level. As discussed later in this chapter, off-equatorial patterns of ADT (and SLA) are linked by geostrophy to the annual variations of the currents listed in Table 4.3 and depicted in Fig. 4.6. Geostrophic Velocity: In off-equatorial regions, geostrophic velocity (UGOS, VGOS) supplied by AVISO is computed from gradients of ADT (Taburet and Pujol 2020). We frequently use this field in figures and videos to describe the annual cycle of circulations depicted in Fig. 4.6.

4.3 Southern-Hemisphere Circulations South of the equator, the South Equatorial Current (SEC) and South Equatorial Countercurrent (SECC) are the major currents in the interior ocean. They are located on either side of the South Equatorial Thermocline Ridge (SETR), and are linked to it through geostrophy. Another important southern-hemisphere current is the EACC along the East African coast.

4.3.1 South Equatorial Thermocline Ridge Indian-Ocean hydrographic data show that isotherms rise south of the equator to form a “ridge,” the SETR. Figure 4.10 plots a WOCE section of potential temperature along approximately 54◦ E (Section I7), which shows the ridge peaking from 5–10◦N, and WOCE sections along 80◦E (Section I8N) and 95◦E (Section I8I9) show similar rises. The rise of isotherms (and isopycnals) persists to a depth of about 1000 m, the bottom of the thermocline, and for this reason it is referred to as a “thermocline ridge” (Schott et al. 2009). Reid’s (2003) synthesis of historical hydrographic data shows the ridge’s impact on adjusted surface steric height (their Fig. 5a), with the ridge appearing as a steric-height “valley,” which, although better developed in the west, extends over much of the basin. Figs. 4.8 and 4.9 and Video 4.1 provide an improved description of the valley based on ADT. They show the valley as being centered near 5–10◦S, stretching across the basin, and best developed in the longitude range 45–60◦E. In Fig. 4.9 and Video 4.1, the valley has a distinct annual variation with a maximum (minimum) in March (October). The valley is deepest (bluest shading), and hence the underlying thermocline is shallowest, in a smaller area from 45–65◦E, making the region sensitive to air-sea coupling (see below). Interestingly, the size of this smaller region has a prominent semiannual cycle, with the area where sea level is less than ∼60 cm reaching maxima in November/December and June/July (Masumoto and Meyers 1998; Rao and Sivakumar 2000; Wang et al. 2001; Hermes and Reason 2008; Yokoi et al. 2008).

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91

Fig. 4.10 Potential temperature in WOCE section I7. The cruise track followed while collecting data for the section is shown on the right. (From the WOCE Indian Ocean Atlas; Talley 2013)

Dynamics: The existence of the ridge is a response to the weakening of wind stress along the northern edge of the Southeast Trades. There is a divergence of surface Ekman drift there that, by mass conservation, requires the thermocline to rise (see Chaps. 10 and 12, and Sect. 17.3.2). Yokoi et al. (2008) showed that the Ekmanpumping velocity was mostly positive (upwelling favorable) throughout the year, with prominent annual and semiannual cycles (see their Figs. 9 and 13) and the amplitude of the latter about twice that of the former (their Table 2). As a result, Ekman pumping peaks twice a year, during March/April and September/October, forcing a response that lags the Ekman pumping by a quarter cycle (Sect. 12.2.2). Hermes and Reason (2008) noted that remote forcing through the arrival of upwelling- and downwellingfavorable Rossby waves from the eastern Indian Ocean also impacts the ridge. As noted below, this process significantly impacts air-sea coupling at interannual time scales. Impacts: The Ekman pumping is important, not only because it raises the thermocline, but also because it upwells subsurface water to the ocean surface. This upwelling is important in two aspects. First, it establishes the upwelling branch of one of the Indian-Ocean’s shallow overturning cells, the STC (Chap. 17). Second, because the upwelled water is cool, it lowers SST somewhat (McCreary et al. 1993, noted this ridge cooling, in their Indian-Ocean solution; however, because there was little observational evidence for the cooling when the paper was published, the authors speculated it might be a model artifact rather than a real feature). Xie et al. (2002) suggested that the cooling is usually not noticed because it is weak enough for the equatorward SST gradient to mask it. Nevertheless, the upwelling is influential because SST over the ridge is near the threshold temperature for atmospheric convection (∼28◦C; Sect. 2.1), making the region sensitive to air-sea coupling: Vialard et al. (2008) noted that SST over the ridge typically remains above 27◦C most of the year and is 28.5–30◦C during austral summer. In this temperature range, the magnitude of upwelling and cooling, which in turn depends on the depth of the ridge, determines whether ocean-to-atmosphere coupling occurs (Schott et al. 2009).

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During IOD and ENSO events, the coupling is particularly apparent. A downwelling-favorable Rossby wave, generated in the eastern ocean during fall and winter, arrives in the western portion of the ridge several months later during spring, deepening the ridge there, weakening upwelling, and warming SST. The warming impacts the atmosphere is several ways. One impact is that the pathway of tropical cyclones shifts northward (Xie et al. 2002), increasing the number of cyclones that occur in the region (Jury et al. 1999; Xie et al. 2002; Reason and Keibel 2004; Washington and Preston 2006). Another is that the warming leads to increased rainfall along the Indian Western Ghats during the following monsoon (Vecchi and Harrison 2004; Izumo et al. 2014). Finally, atmospheric model experiments suggest that SST anomalies over the ridge force a substantial amount of interannual precipitation anomalies over the Maritime Continent and western Pacific (Annamalai et al. 2005a), and that they are linked to extratropical circulation in the northern hemisphere during boreal winter (Annamalai et al. 2007). There are also indications that ridge variability impacts SST at intraseasonal time scales. The model studies of Murtugudde and Busalacchi (1999), Murtugudde et al. (2000), and Behera et al. (2000) support this possibility. Jayakumar et al. (2011), however, concluded that ridge variations were only of moderate importance: Using a combination of empirical and modelling studies, they concluded that forcing by air-sea fluxes plays a dominant role (about 70%) in determining intraseasonal SST variability. Finally, additional support for the existence of ridge upwelling is provided by satellite imagery of ocean color, which sometimes indicates the presence of phytoplankton blooms there. For example, Murtugudde et al. (1999) reported enhanced biological activity during 1997-98 in ocean-color data from NASA’s Sea-viewing Wide Field of view Sensor (SeaWiFS), and linked them to Ekman pumping and ridge shoaling.

4.3.2 South Equatorial Current The SEC is a broad region of westward flow across the basin from 5–20◦S, the northern edge of which lies somewhat north of the NIO southern boundary. It is apparent in surface drifter data (Fig. 4.5; Shenoi et al. 1999), ship-drift currents (e.g., Cutler and Swallow 1984; Schmitz 1996), and geostrophic currents (e.g., the region in Fig. 4.8 where mean sea level slopes downward to the north in the southern Indian Ocean). The SEC is present throughout the year. It has a weak seasonal variability (an amplitude only 10% of the mean transport), is strongest during June–August, and weakest during December–February (e.g., Schott et al. 1988; Swallow et al. 1988; and Yamagami and Tozuka 2015). Wijffels et al. (2008) estimated the annual-mean transport of the SEC near its eastern boundary (∼110◦E) to be 12.5 Sv, where the transport is largely due to the ITF. Using the Levitus (1982) annual-mean climatology, Schott et al. (1988) estimated the geostrophic SEC transport above 1100 dB to be 39.4 Sv at 54◦E, with

4.3 Southern-Hemisphere Circulations

93

Fig. 4.11 Schematic of the mean transports in the near surface and thermocline layers past northern Madagascar and in the Somali Current at the equator, derived by Swallow et al. (1991) from moored and shipboard observations. There is a loss of 7 Sv out of the thermocline layer, entering the Mozambique Channel. Taken from Figure 55 of Schott and McCreary (2001)

a maximum (minimum) during July (February). Farther west, the southern half of the SEC (12–17◦S) is impacted by Madagascar. There, it splits into northward- and southward-flowing branches, forming the Northeast Madagascar Current (NEMC) and Southeast Madagascar Current (SEMC), respectively. Based on hydrographic and a year-long current-meter record, Swallow et al. (1988) estimated the annualmean transport of the NEMC (SEMC) to be 29.6±8 (20.6±6) Sv; their sum (50.2 Sv) estimates the SEC transport at 50◦E south of the northern tip of Madagascar, a somewhat larger value than that of Schott et al. (1988) likely due to the different data sets used. Figure 4.11 illustrates pathways of the annual-mean NEMC after it leaves the northern tip of Madagascar: Almost all of its near-surface (0–300 m) transport (16.1 of 18.4 Sv) flows westward to join the northward-flowing EACC (Sect. 4.3.4). About 7 Sv of the 10.8 Sv of the deeper (300–900 m) flow, however, turns southward into the Mozambique Channel. In support of the southward flow, de Ruiter et al. (2005) reported that a train of large (diameters greater than 300 km), southwardpropagating, anticyclonic eddies dominate the channel flow, and that they transport water southward with them at rates as high as 15 Sv. Further, Chapman et al. (2003) found that some of the Autonomous Lagrangian Circulation Explorers (ALACE) floats released at a nominal depth of 900 m moved southward into the channel past the northern tip of Madagascar.

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Dynamics: The annual-mean SEC is the westward-flowing branch of the Sverdrup gyre driven by the Southeast Trades (Chaps. 11 and 12, and Sect. 17.3.2). Because its westward component is primarily geostrophic (entirely so for a zonal wind stress), the SEC must occur in a region where the meridional pressure gradient, p y , is negative. As such, its northern edge occurs at the latitude where p y vanishes, that is, at the latitude of the minimum depth of the thermocline ridge. From this linkage, it follows that the SEC transport follows that of the ridge depth, being strongest (weakest) during austral summer (winter).

4.3.3 South Equatorial Countercurrent In contrast to the SEC, the SECC is not present throughout the year. When it exists, it is an eastward current centered from 3–5◦S and approximately 2◦ wide that extends across the basin. It is strongest during January–March, and at this time is easy to identify in Fig. 4.5 because current speeds are low to its north and south. In April/May, the SECC is blurred by the presence of a strong equatorial jet (the spring Wyrtki Jet discussed in Sect. 4.4.1), but it reappears as a distinct flow in June with the waning of the jet. The SECC is absent from July to September, and begins to reappear during October, as evidenced by the eastward turn of the EACC near 5◦S (Fig. 4.5). It is difficult to identify during October/November, owing to the presence of an equatorial jet (the fall Wyrtki Jet; Sect. 4.4.1), and only in December is it seen as a distinct zonal flow. The SECC turns southward to join the SEC during all seasons except boreal winter, when part of the SECC turns northward to merge with the westward-flowing, equatorial current. This wintertime northward flow was noted by Shenoi et al. (1999) in monthly-mean currents determined from surface drifter data, and is visible during January–March in Fig. 4.5. It also occurs in solutions reported in this book (Videos 17.1a and 17.1b; Chap. 17). Interestingly, northward flow is less clear in geostrophic currents, suggesting that it is primarily ageostrophic Ekman drift (Video 4.1; Sect. 17.3.4). In Fig. 4.11, which summarizes the annual-mean circulation in the western tropical Indian Ocean, the strength of the SECC at its western end is approximately 10 Sv, and it is confined above about 300 m. On the other hand, its transport undergoes significant seasonal variation. As noted above, the current vanishes during July– September, whereas during boreal winter Swallow et al. (1991) estimated its transport to be 24±6 Sv. Dynamics: The SECC is also part of the Sverdrup gyre driven by the Southeast Trades. A major success of dynamical oceanography was the realization that winds, which weaken substantially in a direction perpendicular to the air flow (like the NIO Southeast Trades), can drive a countercurrent that opposes the wind (Sverdrup 1947; Chaps. 11 and 12, and Sect. 17.3.2). The theory was first applied to explain the North Equatorial Countercurrent (NECC) in the Pacific Ocean as a response to the weakened trades within the ITCZ (Sverdrup 1947). The SECC is another example

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of such a countercurrent. Specifically, it is an eastward, geostrophic flow that exists where p y is positive, which occurs north of the minimum depth of the thermocline ridge. It is weaker (less apparent) than the SEC because of its proximity to the equator, where its northern edge is also impacted by equatorial winds.

4.3.4 East Africa Coastal Current The EACC is a narrow (∼120 km) current that flows northward along the east coast of Africa throughout the year. It is fed by the branch of the SEC that flows westward north of Madagascar, and hence its southernmost latitude is located near 10–11◦S. It flows northward to 3–5◦S, where it meets the SC. During boreal winter, the SC flows southward, and the two currents bend eastward at 2–3◦S to supply water for the SECC (Table 4.3; Duing and Schott 1978). During boreal summer, the EACC merges into the northward-flowing SC, which crosses the equator and extends northward along the western boundary of the NIO. At this time, the EACC carries subsurface, thermocline water from the subduction regions in the southern hemisphere to upwelling regions in the northern NIO, making the EACC an important branch of the Cross Equatorial Cell (CEC; Chap. 17; Schott and McCreary 2001). The EACC transport strengthens from south to north in response to inflow from the SEC. Near its northern end, its annual-mean transport above 500 dbar is about 20 Sv, whereas at greater depths it is weak (∼1 Sv) and variable (Swallow et al. 1991). About half of the upper part of this transport turns eastward to join the SECC, with the deeper part flowing into the northern hemisphere (Fig. 4.11). Dynamics: Western boundary currents are driven both locally, primarily by alongshore winds, and remotely by the reflection of Rossby waves from the interior ocean (Chaps. 12, 14, and 15). Over the EACC, the alongshore winds are weak enough for the current to be almost entirely remotely driven. Essentially, then, the EACC reacts to the extension of interior currents to the western boundary by Rossby-wave radiation, providing a source (sink) for eastward (westward) flows. Videos 11.3 show the development of a western-boundary current driven in this way. (We note that in other coastal regions of the NIO, notably along the coasts of Somalia and Arabia, the alongshore winds are so strong that they dominate the forcing of coastal currents; see Sects. 4.9.5 and 4.9.6.).

4.4 Equatorial Region In the equatorial Indian Ocean (2.5◦S–2.5◦N), the annual-mean winds are weak and westerly, in marked contrast to the sustained easterlies that occur over the Pacific and the Atlantic Oceans. Figure 4.12 shows various parts of the climatological annual cycle of equatorial τ x , providing longitude-time plots of anomalous τ x (top-left panel), and its annual-mean (top-right), annual (bottom-left), and semian-

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Fig. 4.12 Longitude-time sections of various parts of climatological τ x (N/m2 ) averaged from 1◦S to 1◦N, showing annual-mean τ x , τ x (top-right panel), anomalous τ x , τ x − τ x (top-left), and the annual (bottom-left) and semiannual (bottom-right) components of τ x . Data are from the SCOW data set (Risien and Chelton 2008)

nual (bottom-right) components. Annual-mean τ x is positive (westerly) everywhere except in the far-western ocean (50◦E), the annual component is concentrated in the western Indian Ocean (40–60◦E), the semiannual one in the central ocean from 50–80◦E (Han et al. 1999). Superimposed on these parts is variability at periods that range from intraseasonal to interannual. Given this different forcing, the equatorial currents in the Indian Ocean have a different character than they do in the other oceans, in particular being much more variable. Here, we review four of their prominent features: intense eastward jets during spring and fall (Sect. 4.4.1), upwelling events (Sect. 4.4.2), equatorial undercurrents and deeper flows (Sect. 4.4.3), and intraseasonal oscillations (Sect. 4.4.4).

4.4.1 Wyrtki Jets Based on monthly surface-current maps published by hydrographic offices of the U.S.A., Netherlands, and Germany, Wyrtki (1973b) first showed that swift jets flow

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eastward along the equator across much of the basin during spring (April/May) and fall (October/November), the transition times between the summer and winter monsoons. In honor of their discoverer, today, they are referred to as the Wyrtki Jets (WJs). Ship-drift Data: During the three decades following their discovery, the WJs were studied primarily using monthly-mean, ship-drift data (e.g., Han et al. 1999). Figure 4.5 illustrates the climatological jets seen in the data. The spring WJ appears in April and continues throughout May. During both months, its southern edge is difficult to identify because it merges with the SECC, but it is still identifiable as an approximately 500-km broad flow symmetric about the equator from 60–90◦E. The jet disappears in June. The fall WJ appears in October, is most energetic in November (when, as in fall, it occurs north of SECC), and starts to disintegrate in December. As seen in Fig. 4.5, the fall jet is the longer-lived and stronger of the two. Figure 4.13 gives a more detailed look at the WJs from ship-drift data, plotting various components of equatorial u from ship-drift data: anomalous u (top-left panel), and its annual-mean (top-right), annual (bottom-left), and semiannual (bottom-right) parts. Note that, in addition to the eastward-flowing WJs, there is across-basin westward flow during the summer and winter between the two WJs (Sect. 4.4.2). The weaker summertime event (July-August) is less clear in Fig. 4.5, owing to many of the velocity vectors also having a southward component that can be as large as the eastward one. RAMA Data: With the advent of the Research Moored Array for African/Asian/ Australian Monsoon Analysis and Prediction (RAMA) program (McPhaden et al. 1981), it has been possible to study the WJs with current-meter data. Iskandar and McPhaden (2011) used RAMA data, in combination with other in situ and satellite data sets, to describe the variability of equatorial currents at 80.5◦E. Figure 4.14, taken from their paper, shows a time series of zonal currents from 40 to 200 m determined from Acoustic Doppler Current Profiler (ADCP) data at that longitude. The near-surface flow field is more complicated than the climatologies of ship-drift or drifter data suggest, containing variability at both intraseasonal and interannual time scales blurring the climatological signal. Despite this complexity, seasonal variations associated with the spring and fall eastward WJs are still evident, and spectral analysis of the record has a dominant peak at the semiannual period. Based on RAMA data collected during 2008–2013, Fig. 4.15 plots the climatological annual cycle of the zonal transport from 2.5◦S–2.5◦N at 80.5◦E (McPhaden et al. 2015). The figure shows the dominance of the semiannual response, with two eastward (the WJs) and two westward events. Consistent with the ship-drift data (Fig. 4.13), the fall jet is the stronger of the WJs, with a peak transport of 19.7±2.4 Sv in November compared to 14.9±2.9 Sv in May; in addition, it lasts longer and penetrates 20 m deeper than the spring jet. McPhaden et al. (2015) cautioned, however, that the jets exhibit spatial and temporal variability, so where and over what time period comparisons are made can affect their relative strengths. The eastward transport of the WJs results in a shift of near-surface water from the western to the eastern ocean, and Fig. 4.7 and Video 1.1 illustrate its annual cycle. During March, sea level is high in the western, equatorial ocean, in April the high sea level extends eastward as part of the spring WJ, and by May it has shifted to the

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Fig. 4.13 Longitude-time sections of various parts of climatological u (cm/s) averaged from 1◦S to 1◦N, showing annual-mean u, u (top-right panel), anomalous u, u − u (top-left), and the annual (bottom-left) and semiannual (bottom-right) components of u. Data are climatological monthlymean annual cycle of surface currents based on drifter observations (Laurindo et al. 2017)

eastern ocean. A similar response occurs during October/November in response to the fall jet. Averaged over the year, the zonal transport in Fig. 4.15 is dominated by the eastward flow, resulting in an annual-mean, eastward transport of 5.4±0.8 Sv. As a result, there is a redistribution of 1.5–2×105 km3 of water from the westernto-eastern basin, causing the upper layer to thicken and sea level to rise in the west, and vice versa in the east (Rao et al. 1989; Quadfasel 1982). Dynamics: The WJs coincide with times of westerly winds along the equator, suggesting that they are directly wind-forced (Figs. 4.12 and 4.13). O’Brien and Hurlburt (1974) confirmed this idea in their numerical solution forced by equatorial westerlies. We obtain similar solutions analytically, which are forced by both xindependent (Sect. 10.2) and zonally bounded (Chap. 14) winds. In the latter case, an important response is the excitation of an equatorial Kelvin wave (Sect. 8.4). In Video 1.1, two such Kelvin waves are visible as eastward-propagating signals during the WJ seasons. Also visible in the video is their reflection from the eastern boundary (Sect. 14.3), an extremely important process that not only influences dynamics of WJs but also links equatorial processes to phenomena in the Bay of Bengal and Arabian Sea (Sects. 4.5 and 4.6).

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Fig. 4.14 Time-depth plot of zonal current at 0◦S, 90◦E taken from Fig. 3 of Iskander and McPhaden (2011). The data were smoothed by a 7-day running mean filter. The reddish (bluish) color indicates eastward (westward) currents, and their zero value is marked by the black contours. The green (blue) curve shows the 28◦C (20◦C) isotherm as a proxy for the top (bottom) of the thermocline

The amplitude of the semiannual WJs is noteworthy, being larger than might be expected from a purely wind-driven response. Based on their analysis of winds and ship-drift observations, Han et al. (1999) reported that “the semiannual response is more than twice as large as the annual one, even though the corresponding wind components have comparable amplitudes,” and these properties are also evident from a comparison of the bottom panels of Figs. 4.12 and 4.13. The large response has been explained as an example of “equatorial basin resonance,” in which second-baroclinicmode equatorial waves reflected from basin boundaries interfere constructively with the directly forced response in the interior ocean (Jensen 1993; Han et al. 1999; 2011;

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Fig. 4.15 Comparison of mean seasonal transport variations based on 5 years of data (August 2008 to July 2013) relative to a fixed depth of 80 m (grey line) and relative to the 26◦C isotherm (black line). One standard error for each time series is plotted at 2-month intervals. Annual mean transports ± one standard error are indicated by the squares on the abscissas. From McPhaden et al. (2015)

Iskandar and McPhaden 2011; Sect. 15.4.1). Consistent with this theory, Nagura and McPhaden (2010) noted that the second baroclinic mode dominated zonal-velocity variability in the RAMA data at 80.5◦E.

4.4.2 Upwelling Events The two periods of westward flow between the WJs (Figs. 4.13 and 4.15) are associated with a redistribution of water from east to west along the equator, which causes the thermocline to rise (upwell) in the eastern, equatorial Indian Ocean. Of the two upwelling events, the winter one (January–March) is much stronger than the summer one (July–August). With the onset of winter monsoon, the zonal component of the winds turns easterly in the western equatorial region and strengthens throughout January (Fig. 4.12). In response, the thermocline rises and sea level drops in the western ocean. These changes propagate eastward, and by February negative sea-level anomalies are present everywhere along the equator; in addition, they spread poleward, both north into the Andaman Sea and south along the coasts of Sumatra and Java (Fig. 4.7 and Video 1.1). The easterly winds decrease during February/March and turn westerly by April. In response, sea-level anomalies in the eastern, equatorial ocean turn positive, signaling the end of the event and the onset of the spring WJ. Consistent with the summertime, westward flow in Figs. 4.13 and 4.15, Rao et al. (2010) reported the existence of an upwelling event in the eastern, equatorial ocean during August/September. Based on a climatology of equatorial SLA for the period 1993–2006, they noted that the summer upwelling event was much weaker than the winter one and confined east of 70◦E. Similar features are visible in Fig. 4.7 and

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Video 1.1, except that positive SLAs only weaken in the eastern, equatorial Indian Ocean with negative SLAs appearing only in a small region off Sumatra/Java. Rao et al. (2010) note that the easterly winds, which can drive eastern-ocean upwelling, were weak and restricted to 50–80◦E during 1993–2006, likely accounting the event’s weakness. Furthermore, in many years summertime easterlies don’t even occur. Dynamics: The basic dynamics of the upwelling events are similar to those for the WJs. The wintertime upwelling event is generated by strengthened easterly winds in the western and central tropical Indian Ocean (Fig. 4.12), and the propagation of upwelling-favorable, equatorial Kelvin waves into the eastern ocean. Subsequently, reflection of the Kelvin waves from the eastern boundary produces westward-propagating Rossby waves and coastal waves that lower sea level off Sumatra/Java (Sect. 4.5), in the Andaman Sea (Sect. 4.6), and increasingly offshore into the interior ocean (Sect. 14.3). The summertime event is weaker because the impact of easterly winds in the central ocean is weakened by westerlies in the far-western ocean (Fig. 4.12).

4.4.3 Equatorial Undercurrents and Deeper Flows In the Atlantic and Pacific Oceans where the prevailing tradewinds are quasi-steady, vertical velocity sections almost always show the presence of an eastward subsurface current (Equatorial Undercurrent, EUC) just beneath westward surface flow (Sect. 16.2). In the NIO, where the winds are highly variable, the situation is very different. There, the near-surface equatorial currents are not steady, but rather vary seasonally; as a result, they radiate vertically and so are linked to flows in the deeper ocean. EUCs: Taft (1967) documented eastward EUCs in the Indian Ocean at 53◦E and ◦ 91 E during March and April, 1963, but not during the Southwest Monsoon of 1962. Subsequent observations (e.g., Swallow 1967; Knox 1976; Leetmaa and Stommel 1980; Schott et al. 1997; Reppin et al. 1999), however, did not reveal any spatial or temporal regularity for the appearance of eastward EUCs. This lack is consistent with the RAMA time series in Fig. 4.14, which shows that an eastward EUC beneath westward near-surface flow happens only sporadically. Given this variability, the EUC is defined more broadly in the Indian Ocean. One definition is that an EUC is any current within the upper thermocline that lies beneath opposite-flowing surface currents or, more generally, beneath weaker surface currents in the same direction. Note that, with this definition, the EUC can be either eastward or westward. Another definition, used by Chen et al. (2015), restricts the previous definition to eastward flows that last for at least one month. Using this definition and data from the RAMA moorings, they described the behavior of the EUC at two locations on the equator, 80.5◦E and 90◦E. Figure 4.16 from their study plots equatorial currents versus time at 80.5◦E. It shows that an EUC occurs regularly during boreal winter/spring, particularly during February and March, with a core depth near the 20◦C isotherm; further, similar undercurrents can be present along much of the equator. An eastward EUC reappears during summer/fall of most years

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Fig. 4.16 Daily zonal current observed by the RAMA mooring at (0◦ , 80.5◦E) from 2005–2012. The data has been smoothed by a 31-day running-mean filter. White (thin black) lines indicate the 0 (0.2) m/s contours, and red (thick black) lines show depths of the 23◦C (20◦C) isotherms obtained from RAMA observations (ORAS4, a reanalysis model output). Gray contours show temperature from the RAMA mooring. After Chen et al. (2015). © American Meteorological Society. Used with permission

mostly in the western basin, with its maximum speed located at different longitudes and depths. In the eastern basin, it exhibits interannual variability linked to occurrence of the IOD (see Sect. 2.3.2). Deeper Flows: Luyten and Swallow (1976) first reported profiles of horizontal velocity in the equatorial Indian Ocean, which extended from the ocean surface to the bottom for a period of a month. Figure 4.17 shows the zonal velocity component from their equatorial profiles at 53◦E. Remarkably, they show a series of alternating jets throughout the water column. The jets were equatorially trapped and had a vertical scale that increased with depth. The records were too short to determine a precise time scale but, given that the jets changed little throughout the observation period, it was several months or longer. In a follow-on experiment, Luyten and Roemmich (1982) deployed a suite of current meters near the equator from 47–59◦E in the western Indian Ocean during April 1979 to June 1980. Spectra of zonal velocity from the records showed a predominant semiannual cycle; moreover, they showed that the phase of deeper currents led that of shallower ones, suggesting that the signals were vertically-propagating waves propagating downward from the ocean surface. A similar phase change is visible in Fig. 4.14, in that bands of positive (red) and negative (blue) flow tilt eastward from depth to the surface.

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Fig. 4.17 Zonal velocity in vertical profiles on the equator at 53◦E. The observations were made in 1976, and the date and month when each profile was obtained is shown at its top. The velocity scale for the profiles is shown above the left-most profile, and positive velocities are eastward. From Luyten and Swallow (1976)

Dynamics: A simple explanation for the Atlantic and Pacific EUCs is that the quasi-steady tradewinds (easterlies) pile up near-surface water in the western ocean, thereby creating an eastward, pressure-gradient force that drives eastward flow beneath the wind-driven surface layer (e.g., Sect. 16.2; Stommel 1960; Charney 1960; McCreary 1981a; Wacongne 1989). This idea clearly is insufficient for the variable, Indian Ocean EUCs. To explore the response of the equatorial ocean to variable winds, Wunsch (1977) obtained a solution forced by a westward-propagating, surface vertical-velocity field (presumably established by the wind) of infinite extent and with a period of one year. In qualitative agreement with the Luyten and Swallow (1976) observations, the solution had a rich vertical structure narrowly confined to the equator. McCreary (1984) extended Wunsch’s (1977) solution to more realistic forcing, considering the response to an oscillating wind of finite zonal extent. In his solutions, wind-driven energy descended into the deep ocean along “beams,” in which the beam path is determined by wave-group theory and phase propagates upwards across them (Sect. 16.1). The videos in group 16.5 (Sect. 16.1.2.2) are numerical versions of several of McCreary’s (1984) solutions, showing the equatorial ocean’s response to forcing by an idealized version of the semiannual zonal wind in the Indian Ocean. Similar to the currents in Fig. 4.17, when the basin has an eastern boundary the response is complex with features having short vertical scales, a result of interference among a number of vertically-propagating waves. Recent analyses of deep current-meter

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records from the equatorial Indian Ocean have shown that properties of observed signals are consistent with this interpretation (Sect. 16.1.3).1

4.4.4 Intraseasonal Oscillations In addition to seasonal variability, winds over the equatorial NIO also have prominent intraseasonal oscillations (ISOs), including Madden-Julian Oscillations (MJOs) at periods of 30–60 days and the Quasi-Biweekly Mode (QBM) at periods of 10–20 days (Sect. 2.4). As might be expected, this atmospheric variability drives oceanic variability in the same period bands. In the MJO band, Iskandar and McPhaden (2011) reported considerable power at 30–70 days in the u field from equatorial RAMA moorings, and the existence of this variability is apparent in Figs. 4.14 and 4.16. The authors noted that in the mixed layer the oscillations were highly correlated with the local winds, and that they extended into the thermocline where their phase led that in the surface layer. These results are consistent with those determined from prior current observations. For example, Luyten and Roemmich (1982) reported variability of near-surface u that peaked near 50 days in data from their current-meter array in the western, equatorial ocean. McPhaden (1982) reported 30–60 day oscillations of u in current-meter data from (70.16◦E, 0.75◦S), noting that they were highly coherent with the local windstress above 100 m. Reppin et al. (1999) reported that near-surface u had significant power at 40–60 days in their array south of Sri Lanka along 80.5◦E from 0.75◦S–5◦N during July 1993 to September 1994. A spectral analysis of the record of u in Fig. 4.14 has little energy in the QBM band. In striking contrast, the spectrum of the v field at the same site is dominated by QBM variability, with little power in the MJO band (Iskandar and McPhaden 2011). Similar results were reported earlier by Reppin et al. (1999), who noted variability of equatorial v with a period of about two weeks; in addition, wind variability measured simultaneously at a nearly surface buoy was also at a maximum in the quasi-biweekly band. Dynamics: The good correspondence between near-surface oceanic variability and local winds at intraseasonal time scales suggests the ocean responds directly to wind forcing at those periods. The property that surface and subsurface variability are correlated, with the latter leading in phase, suggests that beams of equatorial waves 1

Zonal jets, similar to those in Fig. 4.17, are also present in the Pacific and Atlantic Oceans where they are referred to as equatorial deep jets (EDJs). Proprties of EDJs include: vertical wavelengths of the order of 300–700 m; a depth range extending from 3000–3500 m to just below the EUC; and very slow vertical propagation, for example, with a period of 4–5 years in the Atlantic (Youngs and Johnson 2015; Ménesguen et al. 2019). Given their long period, they cannot be remotely-forced waves of the sort illustrated in video group 16.5, and their underlying dynamics remain unclear. In the Indian Ocean, observations are insufficient to determine whether any jet-like features in Fig. 4.17 are EDJs (i.e., have periods significantly longer than a year). One reason is the prominence of shorter-period (one year and less) signals, which do appear to be remotely-forced waves.

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form at intraseasonal periods, just as they do at longer periods (Sect. 16.1; McCreary 1984; Miyama et al. 2006). Interestingly, different equatorial waves dominate in the two period bands, namely, equatorial Kelvin and Yanai waves in the MJO and QBM bands, respectively (Sects. 8.4 and 8.3). The two wave types have different symmetries about the equator: For the equatorial Kelvin wave, u is symmetric and v is identically zero; for Yanai waves, v is symmetric and u is antisymmetric, so that u vanishes at the equator. These different symmetries account for the different spectral properties in the two bands. Videos in groups 16.2 and 16.3 show solutions forced by idealized winds that oscillate at several intraseasonal periods. Signals descend from the forcing region along “beams,” and there is clear upward phase propagation across them. The above discussion focusses on wind-driven, intraseasonal variability. On the other hand, the first observations of Indian-Ocean intraseasonal variability appeared to have another cause. Luyten and Roemmich (1982) reported a sharp spectral peak in v near 27 days in their current-meter data, and Tsai et al. (1992) identified a similar 26-day peak in satellite SST observations in the western ocean. In both studies, the authors noted that features of the oscillations are in good agreement with Yanai-wave characteristics. In their numerical solution, Kindle and Thompson (1989) noted that Yanai waves in the western equatorial Indian Ocean were produced with a distinct period of 26 days, and that they were generated by an instability of the SC as it crossed the equator during the Southwest Monsoon, rather than by wind forcing. Recently, Chatterjee et al. (2013) re-examined the cause of intraseasonal variability in the western, equatorial Indian Ocean, concluding that it was generated by both wind forcing and SC instability.

4.5 Sumatra/Java Coast The Sumatra/Java coast consists of the western and southern sides of Sumatra, Java, Nusa, Tenggara, and other smaller islands to their east. It is dynamically important, because it is where the strong equatorial currents first encounter the Indian-Ocean eastern boundary. From there, signals propagate poleward to impact circulations in both hemispheres. As a result, the Sumatra/Java current system, located almost entirely in the southern hemisphere, is strongly influenced by equatorial processes. It is also impacted by alongshore monsoon winds, which are directed northwesterly (southeasterly) during austral summer (winter).

4.5.1 Mean Currents The coastal flow is about 100 km wide, comparable to the local Rossby radius of deformation, and has two cores with a well-defined velocity minimum between the two: an upper core in the upper 100 m and a deeper one from 200–1000 m (Wijffels

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et al. 2008). The shallower current is referred to as the South Java Current (SJC; also known as the Java-Sumatra Coastal Current), and has been observed to extend from Sumatra to the northern part of the Ombai Strait on the Savu Sea (Sprintall et al. 2010). The deeper current is referred to as the South Java Undercurrent (SJUC), even though it can have the same direction as the near-surface flow. Annual-mean transport estimates and water-mass properties for both currents have been reported in several studies. Using twenty years of monthly, or more frequent, repeat expendable bathythermograph data, Wijffels et al. (2008) estimated the SJC mean transport to be 1.2±0.5 Sv to the east, and noted that the current carried warm (27.8◦C) and fresh (33.70 ppt) water characteristic of the high-rainfall, warm-pool region of the eastern, equatorial Indian Ocean (Fieux et al. 1994, 1996; Sprintall et al. 1999; Wijffels et al. 2002; Wijffels and Meyers 2004; Sprintall et al. 2010). Wijffels et al. (2008) estimated the SJUC mean transport to be 1.1±0.1 Sv to the east, with the current carrying cooler (10.3◦C), more-saline (34.81 ppt) and lower-oxygen water characteristic of RSW (Bray et al. 1997; Fieux et al. 1994, 2005; Wijffels et al. 2002). Thus, water-mass properties point toward the equatorial origin of both currents, suggesting that they are generated by the reflection of equatorial currents from the eastern boundary.

4.5.2 Semiannual and Annual Cycles In their investigation of tide-gauge data in the region, Clarke and Liu (1993) found a large difference between the semiannual and annual components, the former being in-phase for stations between 8◦S and 8◦N with an amplitude of ∼5 cm and the latter out-of-phase across the equator with a maximum amplitude of ∼10 cm. They concluded that the semiannual signal was largely a response to the reflection of the semiannual WJs: As noted previously, downwelling-favorable equatorial Kelvin waves associated with the WJs reach the eastern boundary approximately in April and October, where they reflect as wedge-shaped packets of Rossby and coastal waves that expand in time, raising sea level throughout region (Fig. 4.7 and Video 1.1). In contrast, the annual cycle was driven mostly by the local monsoon winds: The largely, alongshore winds excite coastally trapped waves that radiate poleward from the equator; after their passage, sea level develops a coastal tilt (a surface pressure gradient) that balances (counteracts) the wind, thereby accounting for the equatorial antisymmetry. Based on ship-drift climatology, Quadfasel and Cresswell (1992) showed that similar properties occur in the SJC annual cycle. They also highlighted the competition between equatorial and local forcings. In November, a southward and eastward coastal current is apparent, generated by the reflection of the fall WJ, and it is enhanced by the prevailing northeasterly alongshore winds; conversely, during May the southward WJ-generated current is weakened by southeasterly winds.

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4.5.3 Intraseasonal Variability Similar processes also happen at intraseasonal time scales. Iskandar et al. (2005) studied intraseasonal variability along Sumatra/Java using three datasets that covered the period from January 1995 to April 1997: hourly sea-level data recorded at four tide gauges along the coast of Sumatra/Java and two in the Indonesian seas; sea-level anomalies from TOPEX/Poseidon; and wind stress from the ECMWF reanalysis. During July/August, they found 20–40-day intraseasonal variability that was linked to forcing from the equatorial Indian Ocean. During December–February, they found 60–90 days oscillations, which appeared to be generated both remotely from the equator and locally by alongshore winds.

4.5.4 Dynamics Analytic solutions and videos that are relevant to the dynamics of these currents are presented at several places in the book. Solutions for the reflection of periodic, equatorial Kelvin waves from a meridionally-oriented, eastern-ocean boundary are discussed in Sects. 15.1 and 15.2.1 and illustrated in Videos 15.2a–15.2d. Solutions for the wind-forced response along a meridionally-oriented, eastern boundary are obtained, both when the forcing is confined off the equator and when it crosses the equator: the former derived in Sect. 13.3 and illustrated by the videos in group 13.4; and the latter found in Sects. 15.1 and 15.2.1 and illustrated by Videos 15.2f–15.2h. Effects of an inclined eastern boundary like Sumatra are discussed in Sect. 13.3.4.

4.6 Andaman Sea The eastern boundary of the Andaman Sea follows the western coasts of Myanmar, Thailand, and the Malay Peninsula (Fig. 4.18). It crosses the mouth of the Malacca Strait from Phuket Island in Thailand to the northernmost point of Sumatra Island. The sea’s western boundary extends southward from Cape Negrais in Mayanmar to follow the Andaman and Nicobar Archipelago (572 islands), and then from the southernmost island in the chain to the northernmost point of Sumatra. The sea has an area of 670,000 km2 , roughly a fourth of that of the adjoining Bay of Bengal. It is a critical region for circulations in the northern-hemisphere NIO because, given its gappy western boundary, it connects events that originate in the equatorial region to those occurring in the Bay of Bengal.

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Fig. 4.18 Map of the Andaman Sea. The colour palette on the right shows ocean depth (negative) and land heights (positive) in meters. Lines show coastline (black), international boundaries (red), rivers (blue), and 200 m depth countour (dashed light red). Letters identify locations referred to in the text: Great Channel (A), Ten Degree Channel (B), Preparis Channel (C), Bay of Bengal (D), Cape Negrais (E), Irrawaddy Delta (F), Mayanmar (G), South China Sea (H), Thailand (I), Phuket Island (J), Malay Peninsula (K), Malacca Strait (L), and Sumatra Island (M)

4.6.1 Background Properties Figure 4.18 shows the bottom topography of the sea. The continental shelf off the eastern boundary is wide, with the mid-depth of the continental slope (1000 m) lying more than 200 km from the coastline. Offshore from the shelf, the basin has an uneven bottom with deep terraces and seamounts. The Malacca Strait provides an eastern-boundary gap, one that connects the Andaman Sea to the South China Sea; however, because the strait is very narrow and shallow at its southeastern end (52 km and 25 m), the annual-mean northwestward transport that occurs through

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the inlet is so small (e.g., Wyrtki 1961) that, for our purposes, it is essentially a closed barrier. There are many gaps in the western boundary of the sea, but only three of them are deep and wide enough to allow a significant exchange of water with the Bay of Bengal: the Preparis Channel, which lies between Cape Negrais on the Irrawaddy Delta and the northern edge of the Andaman/Nicobar Archipelago (about 200-km wide and shallower than 250 m); the Ten Degree Channel, which separates the Andaman and Nicobar island groups (145 km wide and a maximum depth of about 800 m; Rodolfo 1969); and the Great Channel, which lies between the southernmost of Nicobar group of islands and the northern tip of the Sumatra island (maximum depth of 1,400 m). The Andaman Sea receives a massive influx of freshwater, both due to high precipitation and runoff from the Irrawaddy River and its tributaries. As a result, SSS in the sea is very low (Fig. 3.5) increasing to 31.8–33.4 psu in the western basin (Sarma and Narvekar 2001). They also found that vertical distributions of temperature and salinity in the western Andaman Sea are similar to those in the eastern Bay of Bengal from the surface to about 1200 m, pointing toward a significant exchange between the two regions. Further, they found that characteristics of Andaman-Sea water below 1300 correspond to those found at about 1,250 m water in the Bay of Bengal. Specifically, salinity in the Andaman Sea below 1200 m remains constant at 34.9 psu, whereas its temperature is warmer than the temperature at corresponding depths in the bay (e.g., about 2◦C warmer at a depth of 2000 m). These properties are consistent with those of a “silled basin,” with the maximum depths of the Ten Degree Channel and Great Channel defining the sill depth. Waters below that depth are well mixed, with renewal times of 5–6 years (Dutta et al. 2007; Okubo et al. 2004).

4.6.2 Annual Cycle Because direct observations in the Andaman Sea are rare, its circulations have been studied mostly using satellite altimetry and numerical model simulations. These studies have shown that the annual cycle arises from two sources: remote forcing by climatological winds along the equator (Potemra et al. 1991; Yu et al. 1991; McCreary et al. 1993), and local winds over the sea (McCreary et al. 1993, 1996; Chatterjee et al. 2017). The equatorial winds generate two downwelling-favorable and two upwelling-favorable packets of equatorial Kelvin waves (Sects. 4.4.1 and 4.4.2; Fig. 4.15), which reflect from the Sumatran coast: About half of the reflection propagates northward into the Andaman Sea, the rest extending southward to impact the Sumatra/Java current system (Sect. 4.5). The local winds are southwesterly during May–September, northeasterly during November–March, and are variable during the transition months, April and October (Fig. 3.1 and Video 3.1). The impact of equatorial forcing is clearly seen in sea level. Note in Fig. 4.7 and in Videos 1.1 and 4.2, that the Sumatran waves cross the mouth of the Malacca Strait to the Malay Peninsula with little difficulty, likely propagating along the continental slope. In so doing, they affect sea level in the Malacca Strait from its mouth to its

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narrowest point: SLAs tend to change sign beyond this point, suggesting it acts as a wall that blocks further wave propagation. After crossing the strait, coastal signals spread northward along the eastern boundary of the basin, and then offshore into the interior of the Andaman Sea as westward-propagating Rossby waves. The sea has three prominent SLA episodes: two with positive anomalies related to the fall and spring WJs; and one with negative anomalies due to the wintertime, equatorial upwelling event. Based on numerical experiments, Chatterjee et al. (2017) concluded that equatorial forcing dominates the sea-level response everywhere in the sea, with local wind forcing having only a minor impact. Figure 4.19 and Video 4.2 are regional versions of Fig. 4.9 and Video 4.1, which show surface geostrophic currents associated with sea-level gradients. During May and October/November, for example, sea-level anomalies increase to the east across the sea, indicating the presence of a northward geostrophic current; conversely, during late January/early February they decrease eastward and there is southward geostrophic flow. On a larger scale, these features are clearly linked to the arrival of wave packets from Sumatra. At other times of the year, the sea-level-gradient field within the Andaman Sea does not have a simple pattern, and the geostrophic currents are not dominated by a single direction. Monthly-mean currents derived from surface drifters (not shown) compare well with geostrophic currents when the currents are strong and local winds are weak, during November, for example. Conversely, when the winds are strong, Ekman flow dominates drifter currents.

4.6.3 Intraseasonal Variability The Andaman Sea is also impacted by the reflection of equatorial waves at intraseasonal periods. Cheng et al. (2013) used altimeter data to examine the propagation of intraseasonal (30–120 day) waves from the equator and along the eastern boundary of the NIO. Panel b) of Fig. 4.20 shows lagged regressions of sea-surface height (SSH) at Stations 1–60 along the red line in panel (a). The regressions show clear evidence of propagation from Stations 1–45: eastward along the equator (Stations 1–20); along Sumatra and across the Malacca Strait (Stations 21–33); and northward along the Malay Peninsula (Stations 34–45). Interestingly, the propagation speed varies along the track, being fastest in the middle region where the signal almost appears to jump across the mouth of the strait. It is likely that, because the strait is so shallow, swiftly-propagating, barotropic waves determine the response there (Sect. 5.2.4). Similar to the seasonal cycle, intraseasonal variability is not confined to the eastern boundary. For example, Kiran (2015) also reported intraseasonal signals in ADCP data from a mooring deployed from May 2011 to April 2012 in the interior of the Andaman Sea at 94◦E, 10.5◦N. Cheng et al. (2013; their Fig. 3a) show that sealevel standard deviations are highest along the coast (4.5–9 cm) with lower values in the interior of the basin (3–4 cm). It is tempting to conclude that this interior variability is linked to the more prominent coastal variability through Rossby-wave

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Fig. 4.19 Bimonthly maps of sea level and geostrophic surface currents in the Andaman Sea. The fields are for the 15th of the month shown in the upper-right corner of each panel

propagation. In support of this idea, Fig. 4.21 shows maps of lagged regressions of SSH in the region (color shading): Beginning at –21 days, a downwelling-favorable (high-sea-level) equatorial Kelvin wave begins to reflect from Sumatra, and by +7 days the coastal response is well developed, both in the Andaman Sea and around the perimeter of the Bay of Bengal; subsequently, the coastal signal propagates offshore, and is replaced by a low-sea-level coastal signal. The weakening of the signal is likely due to randomness in the intraseasonal variability: the greater the lag, the weaker the regression. Although the dominant intraseasonal forcing for the Andaman Sea is from the equator, circulations there are also impacted by local winds. Figure 4.21 also plots lagged regressions of QuikScat winds (arrows), and they are linked to the AndamanSea signal. At zero lag, for example, there are southwesterly winds over the Andaman

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Fig. 4.20 Time-station plot (panel b), showing lagged regressions of intraseasonal SSH anomalies (cm) for stations along the equator and around the perimeter of the Bay of Bengal and Andaman Sea (panel a). The data is averaged in 1◦ bins at Stations 1–70 laong the red line, and the regressions are taken with respect to the normalized SSH anomaly at Station 10. From Fig. 7 of Cheng et al. (2013)

Sea, and they generate Ekman drift (southwesterly surface flow) that tends to enhance the coastal signal. Nevertheless, in a series of numerical experiments Chatterjee et al. (2017) examined the relative impacts of equatorial and local wind forcing, concluding that the latter was an order of magnitude weaker than the former.

4.6.4 Dynamics Altimeter data and surface currents (Fig. 4.19 and Video 4.2) demonstrate that the dominant driving mechanism of circulations in the Andaman Sea is the reflection of wind-forced equatorial Kelvin waves from the eastern boundary of the Indian Ocean, which subsequently radiate off the boundary as a packet of Rossby waves (e.g., Potemra et al. 1991; and McCreary et al. 1993). In this book, we obtain analytic solutions and videos that illustrate this reflection when the eastern boundary is a meridionally-oriented wall (Chaps. 14 and 15). In the real ocean, of course, the Rossby waves encounter the gappy western boundary of the Andaman Sea before entering the Bay of Bengal. Chatterjee et al. (2017) explored the impact of a realistic western boundary in a suite of numerical solutions, finding that its impact is surprisingly small. As an equatorial signal first enters the Andaman Sea, it propagates northward along the eastern-boundary slope as a coastally-trapped wave. When the wave encounters the Preparis Channel, its part shallower than the sill depth (250 m) easily passes through the channel, despite the channel being narrow. (This numerical result is consistent with the analytic solutions of Durland and Qiu, 2003, which show that such signals can pass through narrow passages even with a width of the order of 10 km.) Its

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Fig. 4.21 Regression of the 30–120 day observed sea surface height (cm) and QuikSCAT wind stress to the normalized 30–120 day sea surface height within the black box (93–95◦E, 3–5◦N). From Cheng et al. (2013)

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deeper part, however, must detour around the Andaman Islands before continuing to propagate northward. The coastal wave subsequently propagates offshore as a Rossby wave, which intersects the Andaman Islands south of Preparis Channel. In this situation, the Rossby wave first reflects as a coastal wave that circulates around the islands, and only then does the signal propagate westward into the Bay of Bengal as another Rossby wave. The overall effect of the western boundary, then, is to delay somewhat Rossby-wave propagation from the eastern boundary of the Andaman Sea into the bay.

4.7 Bay of Bengal The Bay of Bengal is bounded to the south approximately by a line joining the southern extreme of Sri Lanka to northernmost point of Sumatra, and to its east by the western boundary of the Andaman Sea. We begin with descriptions of the climatological circulation in the interior of the bay (Sect. 4.7.1) and the prominent intraseasonal variability and eddy activity that occurs there (Sect. 4.7.2). Then, we consider its western-boundary current, the EICC (Sect. 4.7.3), including the formation of smaller-scale seasonal gyres just offshore (Sect. 4.7.4). Finally, we discuss the circulations that maintain the salt balance in the bay (Sect. 4.7.5).

4.7.1 Interior Circulation Prominent features in the Bay of Bengal are basin-scale gyres: an anticyclonic gyre from February–April and a cyclonic one during October–December. There are indications of the gyres in drifter currents (Fig. 4.5), but they are much clearer in sea level (Fig. 4.9, and Videos 4.1 and 4.2). The southward-flowing, eastern branch of the anticyclonic gyre is located where sea level drops eastward from high (red) to low (blue) across the bay, and vice versa for the cyclonic gyre. The western branch of both gyres is the EICC (Sect. 4.7.3). Both videos link the formation of the two gyres to equatorial processes, namely, the reflection of upwelling- and downwelling-favorable, equatorial Kelvin waves from Sumatra, respectively, and the subsequent propagation of the reflected waves through the Andaman Sea and around the perimeter of the bay (Sects. 4.4 and 4.6). In addition, both circulations are enhanced by wind forcing. During the time of the anticyclonic gyre (February–April), wind curl is negative throughout much of the bay (Fig. 3.1, top-left panel; Video 3.1). The curl is associated with negative Ekman pumping that tends to deepen the thermocline in the interior of the bay (e.g., see the discussion in Sect. 12.2.1), thereby intensifying high sea level at the center of the anticyclone. For the cyclonic gyre (October–December), the Ekman pumping has the opposite effect. In addition, the winds along the east coast of India turn northeasterly during October/November, driving a southward coastal flow that strengthens the

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Fig. 4.22 Observed annual-mean standard deviation of SSH (cm, color shading) in the 30–120-day band for the period 2000–2008. Adapted from a figure prepared by Xuhua Cheng

western flank of the gyre (Sect. 4.7.3.4; Chap. 13). In their companion modeling studies, Shankar et al. (1996) and McCreary et al. (1996) separated the responses due to each of the forcing mechanisms noted above (see Fig. 5a–d in the latter paper), confirming the above impacts. A notable feature in Fig. 4.5 is the northward bending of the summertime, eastward current south of the Sri Lanka, the Southwest Monsoon Current (Sect. 4.8), into the bay, and it is also visible in Fig. 4.9 and Video 4.2 as a sea-level drop to the northwest across the current (see Sect. 4.8.1 for further discussion). Two other organized currents in Fig. 4.5 are eastward flow across the northern bay from 15– 18◦N during June–August and northwestward flow across the southern bay during December/January. There are no clear pressure gradients associated with these currents (Fig. 4.9 and Video 4.2), so they are not geostrophic; rather, they must be Ekman drift (Chap. 10) and, consistent with this idea, both are directed to the right of the prevailing Southwest Monsoon and Northeast Monsoon winds.

4.7.2 Intraseasonal Variability and Eddies A number of studies have demonstrated the presence of intraseasonal variability and eddy activity in the Bay of Bengal (Gopalan et al. 2000; Girishkumar et al. 2011, 2013; Durand et al. 2009; Cheng et al. 2013, 2017, 2018; Mukherjee et al. 2019). Cheng et al. (2013, 2017) used satellite sea-level observations to examine their basinwide properties. The data sets were constructed on a 0.33◦ grid from weekly AVISO SLA data to which the mean, surface dynamic topography of Rio et al. (2011) was added (Fig. 4.8). In their first (second) study, the data sets covered the periods 2000– 2008 (1993–2012) and, to focus on intraseasonal variability, were filtered to contain periods from 30–120 (20–190) days.

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Fig. 4.23 Tracks of a anticyclonic and b cyclonic eddies that last more than 30 days, overlying the annual-mean standard deviation of intraseasonal sea level (grayscale, cm) derived from altimetry data during 2000–2008. From Cheng et al. (2013)

Figure 4.22, from Cheng et al. (2013), shows the mean standard deviation of the first data set at each grid point. Interestingly, it is not spread uniformly over the bay, but rather is concentrated in five regions: a zonal band along about 5◦N (Region B); an area just east of Sri Lanka (Region C); a band that extends northeastward across the central bay (Region D); the far-western bay offshore from the EICC (Region E); and four regions of very high standard deviation along the eastern boundaries of the Andaman Sea and Bay of Bengal (Regions R). A key finding is that properties of the variability differ in each region. Regions B and C are dominated by 30–60 day variability. Sensitivity experiments, using a nonlinear version of a reduced-gravity model (Sect. 5.3) forced by realistic winds, were able to reproduce the observed patterns of intraseasonal variability in both regions. The solutions showed that eddies (nonlinear Rossby waves) propagating from the east, rather than local wind forcing, accounted for most of their variance. Further, they demonstrated that the variance in Region C is significantly enhanced by the nonlinear transfer of energy with periods of 90–120 days into the intraseasonal (30–60 day) band. In Regions D and E, the dominant periods are 90–120 days and are associated with eddies. Cheng et al. (2013) defined an eddy to be any closed feature of SSH, in which its center differed by 4 cm from the edge value. Figure 4.23 shows the resulting tracks of eddies that lasted more than a month. As indicated in the figure, there are more cyclonic than anticyclonic eddies, and both types tend to move toward the southwest. As might be expected, both pathways occur in areas that are local maxima in SSH variance. Dynamics: To explore the causes of the regions of high standard deviation, Cheng et al. (2013) carried out a set of numerical experiments using a global, eddy-resolving OGCM with a horizontal resolution of 0.1◦ and 54 vertical levels. To force the OGCM, the authors used four different wind fields, including daily-mean surface wind from satellite-based QuickSCAT measurements. From a comparison of solutions, they concluded that remote forcing from the equator was the primary cause of variability

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in the Andaman Sea and Region B, with some contribution from local winds. (See the discussion of Video 15.2b in Sect. 15.2.1, which provides an idealized representation of the remote response when wind oscillates with a period of 45 days.) In contrast, wind forcing was not able to account for the high variability in Regions D and E: Based on energetics analyses of their solutions, the authors proposed that baroclinic and barotropic instabilities are the principal triggers for eddy generation there. To explain the four areas of high standard deviation in Regions R, the authors invoked high runoff from the large Irrawady, Brahmaputra, Ganga and Mahanadi Rivers, noting that their solutions, which did not include forcing by runoff, did not produce that variability.

4.7.3 East India Coastal Current Considerable effort has gone into observing and understanding the annual cycle of the EICC, which flows from about 8–20◦N. Information about the current comes from drifter and sea-level data, hydrography, and coastal moorings.

4.7.3.1

Drifter and Sea-Level Data

The climatological, monthly-mean drifter data reveal a complex structure, with the current flowing in one direction along the entire coast only during February–April and November/December (Fig. 4.5), and its northern and southern parts having opposite directions at other times of the year. This annual variability can also be seen in sea level (Fig. 4.9 and Video 4.2). During February–April, sea level rises away from the coast everywhere except near its northern end, suggesting a northward geostrophic current. Conversely, during November/December sea level drops away from the coast, suggesting a downwelling situation and a southward geostrophic current. During the rest of the year, the offshore slope differs between northern and southern regions. Because the current is narrow, the offshore sea-level slopes are difficult to see in Fig. 4.9; however, Video 4.2 clearly shows them during February– April and November/December. Shankar et al. (2010) pointed out that sea level in the EICC region has high variability at interannual time scales, which are primarily associated with ENSO and IOD. Sea-level anomalies associated with these events are generated primarily along the equator: Equatorial Kelvin waves reflect from the eastern boundary as coastally-trapped and Rossby waves, which propagate around the perimeter, and into the interior, of the Bay of Bengal (Aparna et al. 2012). During a positive IOD event, negative anomalies occur during April–December, with a peak in September– November. During El Niño, negative anomalies are observed twice, during April– December and November–July. Anomalies during negative IOD and La Niña events are much weaker.

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Durand et al. (2009) used satellite altimeter data for the period 1992–2002, which was specially processed to study EICC variability. The data set allowed the authors to study both seasonal and non-seasonal variability at periods from a few weeks (intraseasonal) to a few years (interannual). They found that non-seasonal variability in the EICC region is significant but not coherent along the coast. They linked the lack of coherence for interannual and interseasonal variability to Ekman pumping over the bay and local alongshore winds, respectively.

4.7.3.2

Hydrography

Historically, observations of the EICC began with coastal hydrography. In particular, during the late 1980s and early 1990s scientists at the National Institute of Oceanography, Goa, India (NIO-Goa) undertook a series of expeditions designed to sample the Indian coastal currents at a representative set of locations and seasons. Horizontal Structure and Transports During 1989–1991, cruises were undertaken to observe the EICC annual evolution. Figure 4.24 shows the dynamic topography of the sea surface relative to 1000 db during the three seasons covered by the expedition: pre-summer monsoon (March/April), summer monsoon (July/August) and winter monsoon (December). During March/April, 1991 (left panel), the topography shows the EICC flowing northward from 11–20◦N along the coast, consistent with the climatology of drifter currents (Fig. 4.5). During the cruise, the alongshore poleward flow was disturbed by two cyclonic eddies, the smaller one at 82◦E, 16◦N, and the larger at 87◦E, 19◦N. In addition, the topography revealed two anticyclonic recirculation zones, centered at 82◦E, 13◦N and 87◦E, 16◦N. Shetye et al. (1993) estimated the EICC poleward transport during March/April to be 10 Sv. The July/August topography (middle panel) did not show any consistent poleward flow along the coast. Instead, three gyres hugged the coastline from 11–19◦N and there was southward flow farther north. The southernmost of the gyres was anticyclonic with upwelling and northward flow at the coast with a transport of 1 Sv (Shetye et al. 1991a). During December (right panel), a well-defined equatorward coastal current with a transport of 8 Sv is seen (Shetye et al. 1993), consistent with the drifter data (Fig. 4.5). Estimates of the EICC transport based on hydrography in other years indicates that there is strong interannual variability. Transport estimates vary from 3–17 Sv for the March/April poleward transport (Murty et al. 1993; Shetye et al. 1993; Sanilkumar et al. 1997; Babu et al. 2003), 1–3 Sv during the summer monsoon (Shetye et al. 1991a; Murty et al. 1993) and 7–8 Sv during the winter monsoon (Murty et al. 1993; Shetye et al. 1996). Vertical Structure Figure 4.25 shows across-shore sections of hydrographic data collected during July/August 1989 at the locations noted in the figure caption (Shetye et al. 1991a). In Legs A, C, and F, there are signatures of coastal upwelling in the upper 50 m

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Fig. 4.24 Dynamic topography of the surface with respect to 1000 db, determined from hydrographic observations taken during 1989–1991: March 14–April 7, 1991 (left panel, Shetye et al. 1993), July 24–August 15, 1989 (middle panel, Shetye et al. 1991a), and December 1–25, 1991 (right panel, Shetye et al. 1996). The contour interval is 5 dyne/cm2 . Dots represent points at which the data were sampled

or so, with isolines of T , S, and ρ all rising toward the coast, consistent with the local winds being upwelling favorable at all three stations. As noted by the authors, these features are similar to those found in a typical, wind-driven, eastern-boundary current, even though the EICC exists along a western boundary (Chap. 13). Interestingly, in Leg F subsurface isolines tilt downward toward the coast. This feature is also often observed along eastern boundaries, where downward-sloping isolines are associated with a geostrophic current that flows in a direction opposite to the surface flow, a Coastal Undercurrent (Sect. 16.2), as seen in the σθ field of Leg F. In contrast, along Leg H, the northernmost of the sections, isolines slope downward uniformly, indicating a southward geostrophic current throughout the water column. Hydrographic data collected during March/April 1991 revealed a current structure in marked contrast with that seen in July/August 1989 (Shetye et al. 1993). They showed isolines of T , S, and ρ rising toward the coast from the bottom of each section (150 m), indicating the presence of a northward, geostrophic current everywhere north of about 10◦N. This strong current was not wind-driven as the winds were weak during this intermonsoon period. The current carried warmer waters of southern origin, and its inshore side was marked by cooler, more saline waters, owing to the uplifting of isolines by the geostrophic flow. The authors estimated the transport of the current to be about 10 Sv. They further noted that the hydrography was suggestive of features associated with the western-boundary currents of subtropical gyres (a recirculation zone, eddies, etc.), although these features were not fully resolved by the hydrographic data (Sect. 4.7.4). Salinity The Bay of Bengal is known for its low near-surface salinities, which result from the high freshwater flux by both precipitation and river runoff. One consequence is that low-salinity, near-surface water is typically present in the EICC from the

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Fig. 4.25 Across-shore sections of hydrographic data collected along the east coast of India during July-August 1989, showing T (◦C, top row), S (ppt, middle row), and σθ (g/cm3 , bottom row) at 11◦N (Leg A), 14◦N (Leg C), 19◦N (Leg F), and 21◦N (Leg H). Each section is approximately 55 km long. At Leg H, there is an approximately 100-km wide shelf, and the section starts at the location where the ocean depth is 50 m. Contour intervals for T , S, and are 1◦C, 0.2 ppt, and 0.5 gm/cm3 . Dots represent points at which the data were sampled. See Shetye et al. (1991a)

coastline to about 55 km offshore. That influence increases to the north, suggesting that an important source is outflow of fresh water from the Ganges-Brahmaputra River. In support of this idea, the near-surface salinity on Leg H has a minimum of 19.79 psu, and minima salinities on the other legs steadily increase to the south. From drifters and sea-level data (Figs. 4.5 and 4.9, and Video 4.2), it can be inferred that the southward near-surface flow seen in Leg H extends southward during the next couple of months leading to a southward EICC along the entire coast during October/November, and this current leads to reduction of salinity along the entire coast (Shetye et al. 1996; Behara and Vinaychandran 2016). Recording this reduction has been a challenge. In a novel observation program, Chaitanya et al. (2014) used local fishing communities’ services on the east coast of India to record variations in near-coast salinity as the EICC transported freshwater to the south. They found a drop of more than 10 ppt in the northern BoB during summer monsoon. The signal of the salinity reduction propagated southward as a

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Fig. 4.26 Black filled circles show the locations of Indian coastal moorings (Amol et al. 2014; Mukherjee et al. 2014). Each is named after a nearby prominent place. Data from moorings Paradip, Gopalpur, Kakinada, and Cuddalore are shown in Fig. 4.27, and from Mumbai, Goa, Kollam, and Kanyakumari in Fig. 4.31

narrow (∼100 km wide) strip along the east coast of India, reaching the southern tip of India after about 2.5 months.

4.7.3.3

Coastal Moorings

In a major boost to understanding the temporal variability of the EICC, ADCPs were deployed in moorings on the shelf and slope off the coasts of India from 2009–2013 at the locations shown in Fig. 4.26. Mukherjee et al. (2014) used the data collected at the Paradip, Gopalpur, Kakinada, and Cuddalore mooring sites to study the EICC. At each site, Fig. 4.27 plots the observed alongshore currents, each record with tides removed and low-pass-filtered at 2.5 days. The seasonal cycle (annual, semiannual, and 120-day periods) is very clear in Fig. 4.27, with the 120-day peak strong at Gopalpur and Kakinada, dominating the variability at Paradip. The annual period is coherent along the entire coast, albeit with significant phase differences between moorings, but alongshore coherence is much weaker at the semiannual and 120-day periods. There is considerable variation of the flow with depth, with the currents almost always being surface concentrated

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Fig. 4.27 De-tided and 2.5-day low-pass-filtered alongshore currents (cm/s) at moorings Paradip, Gopalpur, Kakinada, and Cuddalore (label in top-left corner), with northeastward (eastward at Paradip) flow being positive. Solid vertical lines mark years, and color bars and vertical dashed lines denote seasons. In the depth/time plots, dashed horizontal lines indicate depths of 40 m (black), 100 m (red), and 200 m (blue), and in the current/time plots curves at those depths are plotted using that color code. From Mukherjee et al. (2014). © Indian Academy of Sciences. Used with permission

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and tending to be unidirectional from the surface to 300 m, the maximum depth to which observations were made. A striking aspect of the records is significant interannual variability, which is strong enough to blur the climatological seasonal cycle at some locations. In contrast, intraseasonal variability is typically weaker than the seasonal cycle, particularly at Cuddalore and Paradip. The strongest intraseasonal variability occurs during spring (February–April). Peaks near 12 and 20–22 days are also seen at Gopalpur, Kakinada, and Cuddalore. Other features sometimes seen in the records are instances of vertically propagating phase, indicated by color bands tilting upward (downward) to the right. They result from vertically-propagating waves (Sect. 16.1) that are carry wave energy downward (upward). At other locations, particularly Cuddalore, there is little or no vertical propagation, and the flow structure consists of an undercurrent beneath an opposite-flowing surface current (Sect. 16.2). The across-shore component of the EICC (not shown) is much weaker than the alongshore component at Cuddalore and, except for a few bursts during spring, also at Kakinada and Gopalpur. It is only at Paradip, on the slope of the northern boundary, that significant across-shore flows are seen during spring and the summer monsoon (June–August). In altimeter data, these strong flows are associated with eddy-like circulations.

4.7.3.4

Dynamics

The processes that force the EICC remained a long-lasting puzzle. One reason for the slow progress was the complicated flow structure, which is not always unidirectional along the coast. Another was that several forcing mechanisms had been identified as being important. Potemra et al. (1991) and Yu et al. (1991) noted that equatorial winds impact the EICC through the reflection of equatorial Kelvin waves at the eastern boundary (see Sect. 14.3). Shetye et al. (1993) showed that Ekman pumping (suction) over the interior of the Bay of Bengal during winter produced an anticyclonic gyre, for which the EICC is its northward-flowing western boundary (Sect. 12.2.1). McCreary et al. (1993) argued that alongshore winds along India’s east coast drive a coastal current in the direction of the wind (Sects. 13.2.2 and 17.3.3.1). To resolve the issues, Shankar et al. (1996) and McCreary et al. (1996) examined the separate contributions of all three forcings (equatorial winds, interior Ekman pumping, and alongshore winds) in a suite of solutions: They concluded that each forcing was an important EICC driver at different locations along the coast and times of the year. To illustrate, consider the EICC during the winter and summer monsoons as seen in Video 4.2. During late fall and winter (October–December), the EICC is a well-defined, southward flow everywhere along the coast. One driver of the current, evident in the video, is equatorial forcing: The reflection of a downwelling-favorable, equatorial Kelvin wave leads to a coastally-trapped wave that propagates around the perimeter of the bay to create a southward-flowing geostrophic current along the Indian east coast (see Sects. 15.2.1). Equally important, but not clear in the video,

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is forcing by northeasterly alongshore winds along the east coast: It drives onshore Ekman drift that deepens the coastal thermocline and raises sea level there, and this signal propagates southward as a coastally-trapped wave (Sect. 13.2.2). The EICC reverses abruptly in January, in response to a coastally-trapped wave generated by the reflection of an upwelling-favorable, equatorial Kelvin wave. During the summer monsoon (May–September), the EICC is not as well-defined. During June and July, southwesterly alongshore winds drive offshore Ekman drift, which shallows the thermocline and lowers sea level near the coast, generating a northward geostrophic current there. In contrast to the winter monsoon, however, equatorial forcing does not enhance northward flow: In response to the reflection of the spring WJ during April and May, sea level remains high in the northern bay, and that high extends southward to weaken the wind-driven response; when the winds weaken at the end of the summer monsoon (late September), the high sea level spreads southward and the EICC is then southward along the entire coast. Other complicating factors are regions of Ekman pumping in the neighboring interior ocean: They excite Rossby waves that propagate to the coast (Sect. 17.3.3.1) and generate westernboundary gyres (discussed next) that disrupt unidirectional flow.

4.7.4 Western-Boundary Gyres In addition to allowing for the EICC, the western boundary of the Bay of Bengal also supports the formation of seasonal gyres with a spatial extent on the order of a few hundred kilometers and a lifetime of several months. The best known among them is the “Sri Lankan Dome” (SLD), which occurs off the east coast of Sri Lanka during the summer monsoon (Vinayachandran and Yamagata, 1998). In the climatology of Levitus and Boyer (1994), upper-ocean isotherms rise sharply toward the middle of the SLD during July, leading to the label “dome” (Vinayachandran and Yamagata 1998). Figure 4.9 and Video 4.2 capture the life cycle of the SLD: It begins as a region of low sea level in June, intensifies during July and August, decays in September, and high sea levels appear in the region by October. Figure 4.5 also has signatures of the SLD circulation. The most obvious indicator is the circulation on its southern flank, which is provided by eastward flow within the SMC (Sect. 4.8.1). Further, currents with a southward component are present off the Sri Lankan east coast during summer, which flow against the local alongshore winds; that component must result from the western flank of the dome circulation. Its eastern and northern flanks, however, are not clearly visible in Fig. 4.5, likely due to large Ekman drift in the drifter data. Another seasonal gyre, the “Bay of Bengal Dome” (BBD), forms to the north of the SLD as it decays (Vinaychandran and Yamagata 1998). It is visible in Fig. 4.9 and Video 4.2 as a patch of low sea level that forms in September just north of SLD. By November, the new gyre occupies the region west of approximately 90◦E and south of about 20◦S. It decays by January. Other similar gyres are also visible in Fig. 4.9 and Video 4.2. Note, for example, the patch of low sea level seen during May–July

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in the region 15–20◦N, 80–85◦E; drifter-based surface currents show that a cyclonic gyre forms around the anomaly. Dynamics: Several modelling studies have discussed the dynamics of these domes: McCreary et al. (1993), Shankar et al. (1996), and McCreary et al. (1996) using simpler models, and Vinaychandran and Yamagata (1998) using an OGCM. Through analyses of their model output, Vinaychandran and Yamagata (1998) identified the cause and time evolution of the SLD dome to be primarily the intense, cyclonic, wind curl east of Sri Lanka during May–September (top-right panel of Fig. 3.1 and Video 3.1): Ekman pumping associated with the curl lifts the thermocline and lowers sea level within the dome (Sect. 12.2.1). The decay of the dome is heralded by the arrival of downwelling-favorable Rossby waves off Sri Lanka, which were generated by the reflection of the springtime WJ (Sect. 14.3.1). As the season progresses, the region of positive Ekman pumping shifts northward, and so does the SLD to form the BBD (Vinayachandran and Yamagata 1998; McCreary et al. 1993, their Fig. 2b–d).

4.7.5 Salt Balance The upper panel in Fig. 4.4 illustrates the extreme freshness of the near-surface waters in the Bay of Bengal, contrasting its salinities with those in the Arabian Sea. The freshness happens because each year the bay receives more freshwater from precipitation and river runoff than it loses to evaporation across the sea surface. To achieve an equilibrium state in which salinity doesn’t continually decrease, the annual-mean ocean circulation must remove freshwater from the bay and replace it with salty water. The freshwater removal occurs near both the eastern and western boundaries of the bay. In the eastern pathway, equatorward flow occurs during the summer, with part passing through the Andaman Sea (Han and McCreary 2001; Jensen 2001, 2003; Sengupta et al. 2006; Behara and Vinaychandran 2016; Sect. 4.6). The western pathway occurs during the winter by the EICC and Winter Monsoon Current (Shetye et al. 1996; Behara and Vinaychandran 2016; Sect. 4.8.2). In both pathways, the exported near-surface water exported has salinities in the range of ∼33–35 ppt. The compensating inflow of salty water occurs during the summer when highsalinity water from the Arabian Sea (∼34.5–35.0 ppt) is carried into the Bay of Bengal by the shallow (∼100 m deep) Summer Monsoon Current (Sect. 4.8.1) and the northward flow along the eastern flank of the SLD (Murty et al. 1992; Vinaychandran et al. 1999; Schott and McCreary 2001, Vinaychandran et al. 2013; Sect. 4.7.4). Meandering of this current farther to the north disperses the salty water throughout much of the interior of the bay.

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4.8 Monsoon Currents The seasonal monsoon currents are zonal flows between the Arabian Sea and Bay of Bengal during the monsoon seasons, flowing eastward during summer (June– September) and westward during winter (November–February). We define the SMC and WMC to be the part of the seasonal flow near and south of Sri Lanka, since that location is critical for (determines) the exchange of water between the Arabian Sea and Bay of Bengal (Wyrtki 1971a; Murty et al. 1992; Gopalakrishna et al. 1996; Han and McCreary 2001). The broader seasonal flows within the Arabian Sea and Bay of Bengal (dashed curves in Figure 4.6) are dynamically distinct from the currents south of Sri Lanka, as they form at different times and experience different forcings (Shankar et al. 2002). We refer to them as “extensions” of the SMC and WMC into each basin.

4.8.1 Summer Monsoon Current In climatological drifter data (Fig. 4.5), eastward flow of the SMC first appears south of Sri Lanka in April, reaches a peak during July/August, and continues through October. Consistent with this life cycle, sea level (Fig. 4.9 and Video 4.3) begins to decrease in an area surrounding Sri Lanka in May, and remain low until November. At the same time, sea level is high near the equator, so that there is a poleward pressure gradient and eastward geostrophic flow south of Sri Lanka. At its peak (July/August), the SMC is linked to flows in the Bay of Bengal and Arabian Sea. In the bay, it feeds the northeastward flow in the western basin that flows around the cyclonic Sri Lanka Dome (Sect. 4.7.4). In the Arabian Sea, it is supplied by a southeastward, geostrophic current off the west coast of India and by eastward Ekman drift across the interior of the Arabian Sea (Sect. 4.9.4). Direct current measurements have demonstrated that the SMC is highly surface trapped. Schott and McCreary (2001) reported earlier shipboard observations that showed the SMC vertical structure along 80.5◦E during August 1993. The section (their Figure 48a) shows an eastward SMC located from about 2–5◦N in the upper 100 m. Interestingly, a westward undercurrent was present below the near-surface flow. Schott et al. (1994) used an array of ADCPs and current meters deployed along 80.5◦E from January 1991 to February 1992 south of Sri Lanka to estimate the SMC transport. Consistent with the shipboard measurements, the current decayed so quickly with depth that its transport had to be estimated either by extrapolation of subsurface current observations to the surface or by using seasonal-mean, ship-drift currents to estimate surface values. Based on the former method, the authors estimated the eastward SMC transport to be about 8 Sv north of 3.75◦E; however, they cautioned that the current might have extended farther south to 2◦E, which would have increased its total transport to 15 Sv. Transport estimates based on dynamic

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computations vary from 6–14 Sv (Shankar et al. 2002). The reliability of these estimates, however, is questionable on three counts: choice of the depth of no motion; Ekman flow is not accounted for; and the presence of strong intraseasonal oscillations in the upper 300 m (Schott et al. 1994).

4.8.2 Winter Monsoon Current In the drifter climatology (Fig. 4.5), the first indications of the WMC appear in November, when the EICC extends southward along the east coasts of India and Sri Lanka to turn westward along the southern tip of Sri Lanka. The current reaches its peak strength in January, and is hardly noticeable in March. In their analysis, however, Hacker et al. (1998) suggested that the WMC persists through March into April. The sea-level climatology in Fig. 4.9 and Video 4.3 reflects this WMC lifecycle. In November, sea level along the east coast of India, around Sri Lanka, and along the west coast begins to rise. At this time, sea level is low south of Sri Lanka, and the resulting positive pressure gradient drives the westward-flowing WMC. The positive gradient weakens in March and is gone by April, owing to the increase in sea level near the equator due to the spring WJ. Note that at its peak (December/January), extensions of the WMC are present well into the Bay of Bengal and across most of the Arabian Sea (Figs. 4.5 and 4.9; and Videos 4.2, 4.3, and 4.4). During January, when the WMC reaches its peak, it is fed from the north by the EICC, and from the east by WMC extension in the Bay of Bengal. In the Arabian Sea, the WMC supplies water for the WMC extension, a westward flow, and the WICC that flows northward. The current-meter observations reported in Schott et al. (1994) estimated the westward transport of the WMC to be 12 Sv in early 1991 and 10 Sv in early 1992. These measurements showed that the flow was mostly confined to the upper 100 m, similar to that of SMC during summer 1991. In contrast to the SMC, however, there were no signatures of an undercurrent (Schott and McCreary 2001).

4.8.3 Dynamics Given that the monsoon currents are largely geostrophic, they are identifiable in Fig. 4.9 and Video 4.3 by the sign of the sea-level gradient p y south of Sri Lanka. As noted above, it is positive (negative) from November–March (April–October) indicating the presence of the eastward WMC (westward SMC). Positive p y begins in late October and November, through the extension of the wintertime EICC around Sri Lanka via coastal-wave propagation. At this time, because coastally trapped waves are narrow, the resulting WMC hugs the Sri Lanka coastline. In January, positive p y is maintained by two events: the arrival of a positive Rossbywave packet north of about 5◦N, which was generated by the reflection of the fall WJ; and lower sea level near the equator forced by easterly winds. As a result, by

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February the WMC widens considerably, extending from Sri Lanka to the equator. In addition, the region of positive p y is zonally broad owing to Rossby-wave propagation, thereby establishing WMC extensions into the Bay of Bengal and Arabian Sea. The WMC ends in April when p y is reversed due to the springtime WJ. Negative p y begins in late May and June, with the arrival of a positive, reflected Rossby wave south of 5◦N. At the same time, sea level decreases along the east coast of India and surrounding Sri Lanka, owing to offshore Ekman drift (Sect. 13.2) and local Ekman pumping (Sect. 12.2 and elsewhere), thereby enhancing p y south of Sri Lanka. The SMC weakens during September and October with the withdrawal of the monsoon winds.

4.9 Arabian Sea We define the Arabian Sea to be the part of the NIO north of the equatorial region (2◦N), west of the southern tip of Sri Lanka, and east of the Gulfs of Aden and Oman. We begin with a description of the circulations along and near the eastern boundary of the sea, namely, the seasonally varying gyres that develop off the southwest tip of India (Sect. 4.9.1). We then discuss the Arabian Sea’s eastern- and northern-boundary currents: the WICC (Sect. 4.9.2) and the Pakistan Coastal Current (PCC; Sect. 4.9.3), the latter an extension of the former. Next, we consider circulations in the interior of the Arabian Sea (Sect. 4.9.4), followed by descriptions of its western boundary currents, the East Arabia Coastal Current (EArCC; Sect. 4.9.5), and SC (Sect. 4.9.6), which are separated from the northern boundary by the Gulf of Oman and from each other by the Gulf of Aden. We conclude with a discussion of the salt balance of the Arabian Sea (Sect. 4.9.7).

4.9.1 Lakshadweep High and Low Figure 4.9, and Videos 4.1 and 4.4, identify regions of high (low) sea level during the winter (summer) centered southwest of the tip of India near 75◦E and 8◦N, the two anomalies referred to as the Lakshadweep High and Low (Bruce et al. 1994; Shankar and Shetye 1997; Brandt et al. 2002). Both were also noticed in the annual cycle of solutions discussed in McCreary et al. (1993). They are associated with a thicker (Rao and Sivakumar 2000) and thinner upper layer, respectively, and clearly linked to the monsoon currents south of Sri Lanka and the WICC. The Lakshadweep High (LH) appears in December and persists through February (Fig. 4.9 and Video 4.4). It is associated with an anticyclonic, geostrophic, surface circulation, in which the WMC turns northward to join the WICC near 10◦N (Fig. 4.5). The LH was first described by Bruce et al. (1994) in an analysis of Geosat altimetry, hydrographic observations, and model results. They found velocities of ∼30 cm/s and estimated the total LH transport to be ∼15 Sv. In addition, they concluded that

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the decay of the LH was related to the westward propagation of Rossby waves, first noted in phase plots along 7◦N by Perigaud and Delecluse (1992). This propagation is evident in Fig. 4.9 and Video 4.4 by the continual westward migration of the region of high sea level during March–May. As a result of the migration, the LH loses its regional character, evolving into a band of high sea level across the southern Arabian Sea. The Lakshadweep Low (LL) is present from June–September (Fig. 4.9 and Video 4.4). In contrast to the LH, its connection to the drifter currents is less clear (Fig. 4.5): There is little indication of a cyclonic circulation about the LL, but rather a southeastward or southward flow across it. The likely reason for this difference from the LH is that the upper layer associated with the LL is thin, which strengthens the Ekman drift enough to overwhelm the cyclonic geostrophic flow associated with LL pressure field. Dynamics: Models indicate that the LH and LL are both remotely forced by a continuation of the processes that form the monsoon currents. In an initial stage, coastal waves radiate around Sri Lanka and then poleward along the Indian coast, as is seen by the narrow regions of lower sea level along the Indian west coast during during May and November (Fig. 4.9 and Videos 4.3 and 4.4). Subsequently, these coastal signals propagate westward as a packet of Rossby waves, much faster nearer the equator since the propagation speed of off-equatorial Rossby waves is inversely proportional to the square of the Coriolis parameter (see the discussion of Eq. 7.7). It is this latitudinal difference in propagation speed that creates the distinctive sea-level structures associated with the LH and LL. McCreary et al. (1993) first reported these processes in their numerical solution. Using both observations and a model, Bruce et al. (1994) also presented evidence for the strong Bay of Bengal influence on the LH. Shankar and Shetye (1997) showed the existence of the LL and LH in a linear model, highlighting the linear nature of their basic dynamics.

4.9.2 West India Coastal Current The WICC flows from about 8–22◦N along the west coast of India. As for the EICC, information about the current has come from drifter and sea-level data, hydrography, and coastal moorings.

4.9.2.1

Drifter Currents and Sea-Level Data

The WICC reverses annually, flowing equatorward during spring and summer and poleward during fall and winter (Table 4.3 and Fig. 4.6). This reversal is visible in the climatological drifter data (Fig. 4.5), which shows equatorward flow near the coast from May–September and poleward flow during November–January. It is also apparent in climatological sea-level data (Fig. 4.9 and Video 4.4). Beginning in May (with a hint in April), sea level is lower along the coast than it is offshore from

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Fig. 4.28 Horizontal sections of dynamic topography (dyn cm) relative to 1000 dB, showing the surface dynamic topography during summer (June–July, 1987, left panel) and winter (December 1987 to January 1988, right panel). Dots represent points at which the data were sampled. See Shetye et al. (1990, 1991b)

Pakistan to Sri Lanka, indicating the presence of equatorward geostrophic flow. During June–September, low sea level spreads from the coast, owing to Rossbywave radiation (see the dynamics discussion below). The opposite changes occur from November–February. Beginning in October, high sea level appears along the west coast of India, establishing a zonal sea-level gradient that supports a poleward geostrophic WICC, and from November–February the signal strengthens at the coast and spreads westward.

4.9.2.2

Hydrography

As they did for the EICC, NIO-Goa scientists undertook expeditions during the late 1980s and early 1990s to investigate the poleward and equatorward phases of the WICC. Horizontal Structure: Figure 4.28 summarizes results from their expeditions, showing the dynamic topography off the Indian west coast during the summer (June/July 1987) and winter (December 1987–January 1988) monsoon (Shetye et al. 1990, 1991b) in the left and right panels, respectively. A striking feature of the figure is the seasonal difference in flow structures: well organized and poleward along the entire coast during winter; equatorward and much weaker during summer.

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Fig. 4.29 Upper 3 panels: Vertical sections taken normal to the coast at about 22◦N during summer 1987, showing temperature (◦C, left panel); salinity (pp, middle panel); and σθ (g/cm3 , right panel). Dots represent points at which the data were sampled. Lower 3 panels: Same as the upper panels, except for a section normal to the coast at about 8◦N. From Figs. 2 and 8 of Shetye et al. (1990)

Vertical Structure and Transports: In the summer section (Fig. 4.28, left panel), there is an eastward current near the southern tip of India (∼9◦N) with a transport of ∼4 Sv, which appears to be part of the large-scale circulation rather than the WICC. Farther north, there is a weak, equatorward, coastal current; its width is about 150 km, and it weakens from about 1 Sv along the southwest coast to about 0.5 Sv near 20◦N. Figure 4.29, which plots offshore sections of temperature, salinity, and density at 22◦N (top panels) and 8◦N (bottom panels), illustrates the weakening. At 8◦N, isolines tilt towards the coast near the surface, a vertical structure indicative of an equatorward geostrophic surface current; in addition, deeper isolines tilt downward, indicating the presence of a poleward undercurrent with its core near 150 m. The existence of a poleward undercurrent has also been reported by Rama Sastry and Myrland (1959), Antony (1990), and Muraleedharan et al. (1995). At 22◦N, the upward tilt towards the coast was hardly noticeable in comparison to that in the south.

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During the winter of 1987-88 near the southern end of the coast, the WICC near 10◦N flowed northward, was approximately 400 km wide and 200 m deep, and carried low-salinity Bay of Bengal Surface Water (BBSW; end of Sect. 4.1.5 and Fig. 4.2). Figure 4.30 (bottom panels) shows the vertical structures of temperature, salinity, and density during this season. Isopycnals of temperature and density tilt down at both locations, indicative of the northward flow. At 22◦N (upper panels), the current was restricted mainly to the vicinity of the continental slope. At this location, the current was narrow (100 km), present mostly over the continental slope, extended to a depth of 400 m, and had a transport of about 7 Sv. Narrowing of the current to the north while maintaining the same transport can also be seen from Fig. 4.28, in which the distance between the 155 and 170 dynamic-cm contours decreased from about 400 to 100 km from 10 to 20◦N. The map of 300/1000 dB dynamic topography in Shetye et al. (1991b) shows the existence of a southward undercurrent below the surface current along most of the coast.

4.9.2.3

Coastal Moorings

Amol et al. (2014) analyzed data collected at the Mumbai, Goa, Kollam, and Kanyakumari moorings in Fig. 4.26, located on the continental slope off the west coast of India, and Fig. 4.31 plots the alongshore currents at the four sites, arranged from south (bottom panels) to north (top panels). It is useful to compare the curves with those along the east coast in Fig. 4.27. Consistent with the drifter climatology (Fig. 4.5), the mooring records have a prominent seasonal cycle, with near-surface flow at 40 m being equatorward (poleward) WICC during February–September (November/December). As for the east coast, however, there are significant intraseasonal oscillations at periods of 30–90 days that blur the annual signal. A comparison of current-time curves in Figs. 4.27 and 4.31 shows that the oscillations are relatively stronger than along the east coast, in part because the seasonal cycle (100–400 day band) is weaker (note the different scales of the vertical axes of the two figures). In addition, there is a tendency for the variability to be stronger during and following the winter monsoon. As for the eastern coast (Fig. 4.27), a striking feature of the annual period in the depth-time plots in Fig. 4.31 is the upward tilting to the right of red and blue bands, which indicates that the signal is a vertically-propagating coastal wave with upward phase propagation and downward energy propagation (Sect. 16.1). The phase propagation leads to the formation of coastal undercurrents as the season progresses (blue currents overlying red and vice versa), and it explains why undercurrents were often observed at different depths in earlier hydrographic observations (Shetye et al. 1990, 1991b; Hareeshkumar and Mohankumar 1996; Stramma et al. 1996). Currents in the intraseasonal band also show upward phase propagation, but not as prominently as in the seasonal band. Further, they can have downward phase propagation, and there are occasions when no vertical propagation is seen (e.g., Fig. 9 in Amol et al. 2014).

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Fig. 4.30 Upper 3 panels: Vertical sections taken normal to the coast at about 22◦N during winter 1987-88, showing temperature (◦C, left panel); salinity (pp, middle panel); and σθ (g/cm3 , right panel). Dots represent points at which the data were sampled. Lower 3 panels: Same as the upper panels, except for a section normal to the coast at about 8◦N. Note that the zonal width of the upper panels is less than in the lower panels. See Figs. 2 and 3 of Shetye et al. (1991b)

Amol et al. (2014) also reported correlations of ADCP data along the coast. At the annual cycle, they found correlations between the Goa and Mumbai moorings to be high, whereas those between Kollam and either Kanyakumari or Mumbai to be weak. Video 4.5, which plots the annual signals at moorings 1–4, provides a possible explanation for these differences. The signals at Kanyakumari (7◦N), Goa (15◦N), and Mumbai (20◦N) are all similar, exhibiting vertical phase propagation with the moorings farther north extending to greater depths. These properties are consistent with those of a coastal beam that carries energy downward at a shallow angle (Sect. 16.1). For reasons that are not clear, the Kollam (9◦N) signal did not follow this pattern. A possible cause is interannual variability: Based on longer data records at the four sites, Choudhuri et al. (2020) reported that correlations of the Kollam record with data at other sites was higher during other years. In contrast, at intraseasonal periods alongshore correlations are weak, even for moorings separated

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Fig. 4.31 Panels (a), (c), (e), and (g) show the 5-day low-passed alongshore currents as a function of depth. Blue shade implies equatorward (eastward) flow and red shade poleward (westward) flow at Kollam, Goa, and Mumbai (Kanyakumari); this sign convention is used in all figures for the alongshore currents. The dotted and dashed vertical lines are drawn to delineate summer and winter monsoons, respectively. The dashed horizontal lines mark the 48 m (black), 150 m (red), and 250 m (blue) water depth. Panels (b), (c), (f), and (h) show corresponding line plots for currents at 48 m (black), 150 m (red), and 250 m (blue). From Amol et al. (2014). © Indian Academy of Sciences. Used with permission

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by only 200–250 km. The authors attributed the weakness to the propagation angle of coastal beams being larger at intraseasonal periods: intraseasonal signals are not present at sites located farther north simply because they have already propagated into the deeper ocean (Sect. 16.1).

4.9.2.4

Dynamics

As for the EICC, the WICC forcing mechanisms of its annual cycle and intraseasonal variability were difficult to determine, owing to contributions from both locally- and remotely-generated signals. Annual Cycle Shetye and Shenoi (1988) explored the possibility that the WICC annual cycle could be forced by local winds. They showed that the climatological wind stress along the west Indian coast has a northerly (upwelling favorable) component throughout the year, strengthening in April, reaching a peak strength of only 0.5 dyn/cm2 in August, and weakening in September (Video 3.1); this time evolution suggested that local winds might drive the equatorward (summertime) phase of the WICC (Shetye et al. 1990). In contrast, based on their numerical experiments McCreary et al. (1993) suggested that the WICC is almost entirely remotely driven by winds along the east coast of India during both monsoons: In this scenario, coastally trapped waves excited along the east coast propagate around Sri Lanka to generate the geostrophicallybalanced WICC. Suresh et al. (2016) considered WICC seasonal variability in greater detail, separating out the contributions from winds on the equator, Bay of Bengal, southern tip of Sri Lanka, and west coast of India; they concluded that the dominant WICC forcing arose from winds along the eastern and southern coasts of Sri Lanka, rather than the east coast of India. As discussed next, the highly-resolved wind and sea-level data in Videos 3.1 and 4.3 provide a useful means for visualizing and interpreting these processes. During winter, sea-level data (Video 4.3) support the idea the WICC is almost entirely driven by remote forcing from the Bay of Bengal. As summer ends, the collapse of southwesterly winds and the onset of northeasterly winds along the east coast of India excite downwelling-favorable, coastally-trapped waves. In addition, downwelling-favorable coastal waves, generated by the reflection of fall Wyrtki Jet, reach the east coast where they add constructively to the locally-forced response. Both signals are associated with a southward coastal current that extends equatorward along the east coast during September. By November, having propagated around Sri Lanka, the current continues as a northward WICC, which then spreads westward as a packet of Rossby waves. WICC dynamics are more complicated during the spring and summer. During the spring (April through May), the WICC appears to respond separately from the EICC (Video 4.3). As upwelling-favorable winds intensify along the west coast, negative SLAs appear there, and by the end of May the WICC is southward. At the same

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time, there is little change to SLA along the east coast, indicating that the impact of remote forcing on the WICC is weak. During the summer (June through midAugust), winds strengthen along both coasts of India and around Sri Lanka. As a result, although local winds still force the WICC, remote forcing also contributes: In response to southwesterly winds, SLAs turn negative during June around Sri Lanka and along the east coast to about 18◦N, and these signals are carried by coastal-wave propagation to add to the locally-forced signal on the west coast. It is noteworthy that the east-coast signal during summer is much weaker than it is during winter. The reason is the different contributions of equatorial forcing during the two seasons. During summer, sea level is high around the perimeter of the bay to about 18◦N along the east coast, owing to the reflection of the springtime Wyrtki Jet; this high signal is carried southward by coastal waves, where it weakens the lowsea-level signal driven by alongshore winds. In contrast, during winter the equatorial and locally-forced signals have the same sign, and hence superpose constructively. The impact on the WICC is striking, with its maximum strengths during winter (7 Sv) and summer (1 Sv) differing by almost an order of magnitude. Intraseasonal Variability Except during the summer monsoon, intraseasonal winds are also weak along the Indian west coast, suggesting that the intraseasonal signals there are remotely forced. In this regard, Vialard et al. (2009) argued that MJOs (Sect. 2.4.1) force the intraseasonal variability along the west coast. The MJOs are most active near the equator, their strength peaks during winter, and they excite coastal waves that propagate northward along the coast. Sensitivity experiments by Suresh et al. (2013) showed that about 60–70% of the west-coast, sea-level variations are driven remotely from the equator. During the summer monsoon, intraseasonal wind variations are stronger over the west coast, and the contribution of local wind forcing to sea-level variability rises to about 60% there. As noted above, wind-generated, remotely-forced waves are known to propagate horizontally and vertically along midlatitude coasts and the equator, forming beams that descend into the ocean along well-defined pathways (Sect. 16.1). Nethery and Shankar (2007) considered an alternate mechanism unique to the WICC, arguing that beams along the Indian west coast can also be generated by variations in the the monsoon currents south of Sri Lanka. In their modeling study, India was represented by a latitudinal barrier, and the monsoon currents specified to be a zonal flow through a narrow gap in the barrier centered on 5◦N. For realistic parameters choices, they found that at 20◦N, the beams extended downward to 350 m for a 30-day wave and 40 m for the annual cycle, consistent with their theoretical ray paths (see the discussion of Eqs. 16.9).

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4.9.3 Pakistan Coastal Current The PCC is the northern boundary current of the Arabian Sea. Meridional sea-level gradients near the coast (Fig. 4.9 and Video 4.4) show that it flows westward during winter (November–February) and eastward during late spring/early summer (May– July), as shown in the bottom-right panel of Fig. 4.32. The similarity of sea-level signal along the Pakistani and west-Indian coasts (Fig. 4.9 and Video 4.4) indicates that the PCC can be viewed as a continuation of the WICC, that is, it is predominantly remotely forced by the radiation of coastal waves from the west coast of India. In support of this conclusion, the time dependence of the local winds off Pakistan is very different from that of the PCC (compare the bottom-left and bottom-right panels of Fig. 4.32, and Videos 3.1 and 4.4), suggesting the impact of local forcing is secondary. Wind stress along Pakistan is eastward almost year-round (Fig. 4.32, bottom-left panel), peaking during summer with a mean amplitude of 0.4 dyn/cm2 , consistent with values quoted in Shetye et al. (1985). Despite the upwelling-favorable winds, there is little evidence that cool subsurface water upwells to the surface (Banse 1994), likely a consequence of strong, near-surface surface stratification (see the discussion of Fig. 13.2 in Sect. 13.2.2).

Fig. 4.32 (top panel) Bottom topography of the Arabian Sea near the PCC. The color key gives heights/depths (m), and depth contours are in m. (bottom-left panel) Annual cycle of wind stress (N/m2 ) based on SCOW (Risien and Chelton 2008) along the blue line (24.75◦N) shown in the top panel. (lower-right panel) SLA (cm) and geostrophic velocity (cm/s) along the blue line is based on daily climatology constructed from 1993–2019 daily 0.25◦ data from AVISO (Taburet and Pujol 2020)

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4.9.4 Interior Circulation As for the Bay of Bengal, the large-scale, horizontal circulation in the Arabian Sea is a combination of remote-generated and locally-forced responses. The sea also contains mesoscale features, with eddies in the interior ocean and jets that extend eastward from the Arabian coast. It also develops an overturning circulation associated with the generation of ASHSW.

4.9.4.1

Surface Circulation

As expected, much of the near-surface flow field is geostrophically balanced, and so related to sea-level variability. Given the strength of the monsoon winds, however, drifters reveal that it also has a significant contribution from ageostrophic flow. Sea-level Variability The remotely-forced component of the large-scale, horizontal circulation is visible in Fig. 4.9 and Video 4.4 as the westward spreading of sea-level signals from the WICC, LH, and LL as packets of Rossby waves. The packets have opposite signs during the summer (high, orange) and winter (low, green), but very similar structures. Consider, for example, the offshore spreading of high sea level that appears along the Indian west coast in November. The spreading is faster at lower latitudes, owing to the increase in Rossby-wave propagation speed. As a result, by March the signal reaches the Somali coast in the southern half of the basin, at which time it appears as a broad band of high sea level that extends across the basin and bends to the northwest in the eastern sea. As time passes, this structure continues to propagate westward, gradually dissipating during the summer. A similar spreading occurs for the low-sea-level signal along India and Sri Lanka that appears in May, and part of the relative minimum of sea level centered near 60◦E, 10◦N, in January is a remnant of the original coastal signal. Local forcing by monsoon winds also impacts sea level in the interior Arabian Sea through Ekman pumping wek (see the discussion of Fig. 12.1). The structure of wek is very similar to that of the wind-stress-curl field in Fig. 3.1, and Video 3.1 so that the thermocline tends to rise in regions where the curl is positive (orange/red shading) and to deepen in regions where it is negative (green/blue shading). Ekman pumping is most intense during the Southwest Monsoon, when it is generated on the flanks of the intense Findlater Jet (July plot in Fig. 3.1, top-right panel; Video 3.1). This wek field acts to raise (deepen) the thermocline, and to lower (raise) sea level, north (south) of the jet axis. Its impact is evident in Fig. 4.9 and Video 4.4 in that the August and September plots show a similar pattern, except with the color shading reversed. The net result is a bowl-shaped thermocline in the central sea (Bauer et al. 1991; Schott and McCreary 2001) by the end of the Southwest Monsoon. During the Northeast Monsoon, wek is much weaker than during the summer and it tends to have the opposite pattern, being negative (positive) in the northern (southern)

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Arabian Sea (November and January plots in Fig. 3.1, top-right panel). Given its weakness, its ocean impact is less clear. One impact visible in Fig. 4.9 and Video 4.4 is the establishment of a region of low ADT from 5–10◦N off the Somali coast in November/December. Drifter Currents Some of the features present in the sea-level field are also visible in drifter data (Figs. 4.5 and Movie 4.4), indicating their geostrophic nature. Others are not and so must be ageostrophic Ekman flows. During the Northeast Monsoon, drifter currents tend to flow along the edges of the high-sea level region that extends offshore from the Indian coast. During November, the flow is northward just offshore from India. In December and January, as the Rossby-wave packet of high sea level propagates westward, the northward flow moves increasingly farther offshore, and an anticyclonic circulation forms in the southeastern Arabian Sea. In February, the band of high sea level extends over much of the Arabian Sea. At this time, although the westward flow on its southern flank is clear, return flow on its northern flank is not; it is possibly due to westward Ekman drift forced by the prevailing northeasterlies (Fig. 3.1, top-left panel; Video 3.1). During the Southwest Monsoon, sea level drops near the Indian coast from May– September, and drifter currents tend to flow southeastward along its edges. Elsewhere in the basin, the connection of the drifter currents to sea-level patterns is less clear. In particular, there is eastward flow across much of the Arabian Sea, which appears to have little relation to sea level. In this region, then, the dominant contribution to the drifter currents must be eastward-flowing Ekman drift driven by the strong southwesterly winds. In this regard, Chereskin et al. (1997) have estimated the southward Ekman transport across 8◦N to be nearly 20 Sv.

4.9.4.2

Mesoscale Features

Several types of mesoscale features are present in the Arabian Sea. As in the Bay of Bengal, the Arabian Sea has eddies, but they are typically fewer in number, smaller in diameter, and lower in amplitude than in the bay (e.g., see Fig. 3 in Chelton et al. 2007). Nevertheless, they can impact ocean properties significantly. For example, during the Winter Monsoon, eddy advection of heat can dominate the near-surface temperature trend in the central Arabian Sea (Fischer et al. 2002). Trott et al. (2019), who studied eddies in the NIO west of 80◦E using altimeter SLAs during 1993–2014, found an average of over 100 cyclonic and anticyclonic eddies in the region, with diameters of 60–90 km, life spans from 40–170 days, and peak SLA amplitudes of 1.5–2.5 cm. Their number peaked during the summer monsoon in response to forcing by the intense Findlater jet formation (Flagg and Kim 1998; McCreary et al. 1989), and their principal regions of formation were offshore from the Arabian peninsula and Somalia.

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During the summer monsoon, other mesoscale features are narrow filaments (jets) of water that extend as much as 600 km offshore from the Arabian coast (Sect. 4.9.5) and the Ras Al Hadd Jet. The latter begins at the northeast corner of Oman, flowing into the northern Arabian between two gyres, cyclonic (anticyclonic) to its northwest (southeast), and from there meanders eastward (Böhm et al. 1999; July/August panel of Fig. 4.35 below); it is 150–400 m thick, has a transport of 2–8 Sv and is visible at least 400 km offshore.

4.9.4.3

NASHSW Formation

During the winter, northeasterly winds carry cold and dry air from the snow-covered Hindukush mountain range over the northern Arabian Sea. They increase the density of the surface water (Morrison 1997), leading to intensified vertical mixing through convective overturning that thickens the mixed layer (Sects. 4.1.1 and 3.2). Interestingly, support for this mixed-layer thickening (entrainment) is provided by biological activity: In response to the entrained, subsurface nutrients, the northern Arabian Sea has a wintertime phytoplankton bloom (Banse and McClain 1986; Andruleit et al. 2000). When the mixed layer thins during the spring, a newly formed water mass, NASHSW (Sect. 4.1.5), is left behind at depth (Banse and Postel 2009; Kumar and Prasad 1999; Prasad and Ikeda 2002a). Schott and Fischer (2000) reported evidence that NASHSW forms in this manner over much of the region north of Socotra. Subsequently, it spreads throughout the interior of the northern Arabian Sea at a depth of about 75 m (Rochford 1964; Shenoi et al. 1993; Morrison 1997; Prasad and Ikeda 2002b), mixes with overlying Arabian Sea High Salinity Water (ASHSW), and some possibly upwells off Oman to close the NASHSW overturning cell. Dynamics: This subduction process is similar to the one that occurs in the interior of subtropical oceans (Sect. 4.1.1 and Chap. 17): In both situations, wintertime thickening of the mixed layer occurs, followed by the advection of the new water mass away from its formation region by wind-driven currents. In the northern Arabian Sea, however, some of the subduction also involves coastal processes, like those discussed in Sect. 13.2.2: It is referred to as “coastal subduction” to differentiate it from its subtropical counterpart (Kumar and Prasad 1999; Prasad and Ikeda 2002a; Nagura et al. 2018; Nagura and McPhaden 2018).

4.9.5 East Arabia Coastal Current The eastern boundary of the Arabian Peninsula, the coasts of Oman and Yemen, stretches southward from the northeastern tip of the peninsula to the mouth of the Gulf of Aden (Fig. 1.2). The continental shelf along the boundary is typically about 50 km wide, except offshore from a series of capes where it narrows to ∼10 km. The East Arabia Coastal Current (EArCC) flows along the coast. The northern part of the EArCC is often referred to as the Omani Current.

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Beginning in April, signatures of a weak poleward current appear along the Arabian coast, with lower sea level present in a narrow band near the coast (Fig. 4.9 and more clearly in Fig. 4.37 below, and Video 4.4). The poleward current strengthens during May/June, reaches its peak strength in July/August, and weakens during September. During May/June, a noteworthy feature is the region of high sea level off the coast that contains three anticyclonic gyres. From November–February, the current reverses to flow equatorward, with sea level increasing toward the coast. During the summer, circular features of higher or lower sea level (circulation cells) exist adjacent to the coastline. Between these cells, upwelled waters are transported in narrow (∼10 km scale) filaments (also referred to as squirts or jets) as far as 600 km offshore, and they can be detected in satellite SST and color imagery (Arnone et al. 1996; Manghnani et al. 1998; Fischer et al. 2002). (The climatological gyres visible near the Arabian coast during July/August in Figs. 4.9 and 4.37 are not likely representative of these cells.) The offshore currents in the filaments can exceed 20 cm/s, with transports of 1–2 Sv. Such filaments were first reported in the California Coastal Current, and have also been observed in other eastern-boundary upwelling regions (e.g., Strub et al. 1991; Pelegri et al. 2005.) Dynamics: The EArCC is primarily a response to forcing by monsoon winds (e.g., Prell and Streeter 1982; Shetye and Shenoi 1988; Brock and McClain 1992). In response to the summertime southwesterlies, offshore Ekman drift lowers sea level at the coast, generating a northeastward, geostrophic current there, and opposite effects happen in response to the wintertime northeasterlies (see Chap. 13 and Sect. 16.2.3.1). Offshore wind curl also impacts the coastal region. For example, the steady fall of sea level off the Arabian coast from mid-February through May (Figs. 4.9 and 4.37; Video 4.4) is driven by open-ocean Ekman pumping due to negative wind curl (see Sect. 12.2 and elsewhere): Initially, the negative curl is caused by the offshore strengthening of the wintertime northeasterlies, and beginning in April by the offshore weakening of southwesterlies (Fig. 3.1, top panels; Video 3.1). There are indications that the EArCC is also remotely forced. For example, during November/December high coastal sea level extends all around the perimeter of the Arabian Sea to Arabia, and during July/August low coastal sea level does as well (Fig. 4.9 and Video 4.4). These features suggest that at least part of the sea-level response along Arabia arises from coastally trapped waves that have propagated around the perimeter of the Arabian Sea from the west coast of India. Finally, we note that the small-scale filaments are not directly wind-generated. Rather, they result from instabilities of wind-driven coastal currents. As such, they can only be simulated in nonlinear models with high horizontal and vertical resolution (e.g., Lee et al. 1994). Based on high-resolution, SST and scatterometer winds, however, it is now recognized that offshore Ekman pumping does impact the filaments offshore (Vecchi et al. 2004).

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Fig. 4.33 Schematic diagram of the Somali Current upper-layer flow patterns over the course of the year. Also marked are undercurrents as presently known. Socotra is the island located near 54◦E, 12◦N. After Schott and McCreary (2001)

4.9.6 Somali Current The SC system has long fascinated oceanographers because it is a western boundary current, as strong as the Gulf Stream in the Atlantic and the Kuroshio in the Pacific, which reverses direction annually. This reversal is apparent in Fig. 4.5 and Video 4.4, with currents in the western equatorial region flowing northward (southward) in July (January). Historically, describing and understanding the annual cycle of the SC system, with its remarkable reversal, has been a great challenge for the IndianOcean oceanographic community. Studies of the system have also revealed its broader implications to climate and ocean biology: Upwelling along the Somali coast cools the western Arabian Sea, suppressing atmospheric convection there (Sect. 2.2.3), and also injects nutrients into near-surface waters to support biological productivity.

4.9.6.1

Surface Circulation

The surface circulation of the Somali Current undergoes a remarkable seasonal cycle, which varies significantly interannually. Seasonal Cycle Schott et al. (1990) described the seasonal evolution of the SC system, and Schott and McCreary (2001) updated that description. Figure 4.33 reproduces a schematic from the latter paper, which provides an overview of the events associated with the current during a typical year. The sea-level cycle in Video 4.4 follows this time development very well. March–May: Beginning in May before the onset of the monsoon, the southern SC is the extension of the EACC that flows northward across the equator to about 3–4◦N (Video 4.4). There, it turns offshore, and a cold wedge (low sea level) develops along its shoreward shoulder. Farther north, alongshore winds cause an upwelling

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regime with a shallow northward coastal flow above a southward undercurrent. Its width scale is of the order of 50–100 km. June/July: With the monsoon onset in June, a gyre – the “Great Whirl”—develops from 4–10◦N, with a second cold wedge at the latitude where it turns offshore (10– 12◦N). A cut through the Great Whirl for the mean currents of the summer monsoon determined from a line of moored stations south of Socotra, showed that it extends to almost 1000 m with speeds of 10 cm/s and that its structure remains visible even at greater depths. Cross-equatorial flow continues, now carrying a transport of about 20 Sv in the upper 500 m. It bends offshore south of 4◦N and then southward to form the “Southern Gyre” as shown in Fig. 4.33. Water-mass signatures of both gyres confirm that there is very little exchange between the Great Whirl and the Southern Gyre at this time. (The southward branch of the Southern gyre is not clear in Video 4.4, likely due to the limitations of geostrophy near the equator.) Coastal upwelling is largely confined to the cold wedges associated with the two gyres. For the northern wedge, the upwelled water can come from depths of 200–300 m, with temperatures colder than 15◦C and densities above 26.5 kg/m3 . These densities correspond to surface densities in the southern subtropics near 40◦S. Typically, though, the upwelling is more moderate. In 1993 and 1995, for example, temperatures were in the 25–27◦C range in the northern wedge, indicating that the upwelled water originated from the upper 100 m. Fischer et al. (1996) identified the relatively low salinities of the upwelled water as being of southern-hemisphere origin from the EACC just south of the equator. Finally, there is a net upper-layer outflow from the SC system into the Gulf of Aden through the passage between Socotra and the Horn of Africa. From moored current-meter observations in the passage, Schott et al. (1997) estimated the mean northward outflow throughout the summer monsoon to be about 5 Sv, and the upperlayer water masses have characteristics of upwelled subsurface water (Fischer et al. 1996). August/September: During the late summer monsoon, transports of the SC can exceed 70 Sv (Fischer et al. 1996; Schott et al. 1997). Strong upwelling exists in a wedge just north of where the current turns offshore. Typical upwelling temperatures in the wedge are 19–23◦C, although upwelled waters colder than 17◦C have been observed. At this time, the Great Whirl becomes an almost-closed circulation cell with very little exchange between its offshore recirculation branch and the interior of the Arabian Sea, as evidenced by differences in surface salinities between the Great Whirl and the region east of it. Instead, Schott et al. (1997) and Fischer et al. (1996) reported that a northward-flowing current west of the Great Whirl, which extended from 4–12◦N, supplied water to the central Arabian Sea, and that current is also present in Video 4.4. October/November:When the Southwest Monsoon dies down, the crossequatorial SC turns offshore again at 3◦N, while the Great Whirl continues to spin in its original position (Fig. 4.33 and Video 4.4). The Great Whirl is even discernible underneath the developing Northeast Monsoon circulation well toward the end of the year (Bruce et al. 1981).

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December–February: During the Northeast Monsoon, with the winds blowing away from the Indian subcontinent, the surface SC reverses to flow southward. After crossing the equator, it encounters the northward-flowing EACC, resulting in a confluence and eastward turnoff at 2–4◦S (Duing and Schott 1978; Swallow et al. 1991) that supplies the South Equatorial Countercurrent (Sect. 4.3.3). At the equator, the southward SC is quite shallow, carrying 5 Sv in the upper 150 m, because there is a northward undercurrent at this time (Sect. 4.9.6.2). Observations from a current-meter array made during WOCE 1995–96, and shipboard sections during winter 1997/98 of Schott and Fischer (2000) south of Socotra and in the Socotra Passage, showed that inflow from the east characterized the northern SC during this time. In their analysis of an expendable bathythermograph (XBT) line from Perth, Australia, to the Red Sea, Donguy and Meyers (1995) confirmed the presence of wintertime, westward inflow into the SC system north of 7◦N, and estimated its transport to be 11 Sv above 400 m. In Video 4.4 the inflow begins in November, as a westward flow along the southern flank of an anticyclonic eddy. Interestingly, this eddy is a remnant of the deep thermocline (high sea level) region in the central Arabian Sea, formed during the previous summer by Ekman pumping east of the Findlater Jet: After the summer monsoon, the region propagates westward as a Rossby wave and splits into smaller circulations, one of which is the anticyclone. During December, the inflow is also supplied by a wind-driven cyclonic gyre from 2–10 10◦N. Socotra Eddy: In addition to the circulations depicted in Fig. 4.33, an anticyclonic eddy (Socotra Eddy) is usually present east of Socotra Island during the summer monsoon (Fig. 4.6) and, although well offshore, it is considered to be part of the Somali Current system. Bruce and Beatty (1985) used XBT data from 1975–79 to describe its structure. They found that, although its properties vary from year to year, it usually is present from 10–14◦N with a transport of 9–15 Sv. Because of its interannual variability and low sea-level signature, it is not clearly seen in the climatologies in Figs. 4.7 and 4.9 and Videos 1.1 and 4.4. Interannual Variations The above climatological annual cycle is known to vary significantly interannually, particularly concerning the location of the gyres, wedges, and their movement along the coast. For example, moored and shipboard observations during 1993–96 showed considerable interannual variation in the Great Whirl (Fischer et al. 1996; Schott et al. 1997): In 1993, the northern boundary of the Great Whirl was located about 200 km south of the banks of Socotra; in 1995, it was banked against the slope south of Socotra, and remained as a single, well-organized circulation until midOctober; in 1996, it was again located much more to the south similar to the 1993 situation, the gyre transports were weaker than in 1995, and it became disorganized in August. Further, there have also been observations suggesting that the two-gyre system may at times collapse. In these instances, satellite SST images indicated a rapid (about 1 m/s) northward propagation of the southern cold wedge, which in some cases merged with the northern wedge, suggesting that the Southern Gyre and

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Great Whirl had coalesced (Evans and Brown 1981). Ship-survey measurements supported the northward movement of the southern cold wedge, indicating that lowsalinity southern waters were present in its wake (Schott 1983; Swallow et al. 1983).

4.9.6.2

Undercurrents

Three Somali undercurrents have been observed at different locations and at various times during the year. Figure 4.33 indicates their locations. From April to early June, a southward undercurrent with a depth range of 100–300 m develops beneath northward surface flow from about 4–10◦N. It has a maximum instantaneous speed as large as 60 cm/s (Leetmaa et al. 1982), although the monthly average speed has a maximum of only about 20 cm/s (Quadfasel and Schott 1983; Schott and Quadfasel 1982). It is terminated by the establishment of the deep-reaching Great Whirl. During fall and winter, a southward undercurrent occurs from 8–12◦N (Quadfasel and Schott 1983; Schott and Fischer 2000). Schott and Fischer (2000) measured southward velocities of about 30 cm/s below 100 m in the passage between Socotra and the mainland during January 1998. Based on water-mass properties, they showed the lower part of this undercurrent to be the main supplier of Red Sea Water out of the Gulf of Aden into the Indian Ocean, with another branch entering the region from the northeast around Socotra. Finally, during winter there is a northward, cross-equatorial undercurrent in a depth range from 150–400 m. Its strength is significant in that it almost compensates for the southward, surface transport (Schott 1986; Schott et al. 1990). Observations are sparse, however, and it is not known how it connects to other regions along the boundary or with the equatorial interior ocean. A reasonable supposition is that it is a continuation of the northward EACC. On the other hand, in moored observations that show the winter confluence of the EACC and SC (Düing and Schott 1978), the currents were northward at 4◦S at the undercurrent level but were fluctuating at 2–3◦S, and did not suggest a continuation of the lower part of the EACC into the Somali Undercurrent.

4.9.6.3

Dynamics

The dynamics of the SC have been studied for decades. In a seminal study, Lighthill (1969) proposed that the summertime SC is driven by the arrival of offshore-generated Rossby waves (such as those that radiate from the Indian coast; see Sect. 4.9.4). Today, however, it is well established that its primary driver throughout the year is forcing by the local alongshore wind stress, with other forcings, including Lighthill’s (1969) process, contributing secondarily. McCreary and Kundu (1988) provide a useful review of various forcings, and examine each of their impacts on solutions to an ocean model.

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Alongshore Winds To explore the development of summertime coastal gyres, Cox (1979) forced an OGCM with alongshore winds that tapered offshore, in an ocean basin with its western boundary tilted at 45◦ to the equator. His solution was able to simulate a coastal current with two gyres and cold wedges. Interestingly, the boundary tilt is essential for the establishment of gyres: In solutions with a meridionally-oriented western boundary, although gyres develop during the spin up, they continue to propagate northward along the boundary and are not present in the equilibrium response (Hurlburt and Thompson 1976; Cox 1979; McCreary and Kundu 1988). The complex response with gyres and wedges is also highly nonlinear, and does not appear in linear models (Chap. 13). In their 2-layer model, McCreary and Kundu (1988) concluded it resulted from the advection of thick upper layer (h 1 large) into the region from the south, which restricted the regions where upwelling can occur (h 1 → 0). In this regard, it is noteworthy that the upwelling regime off the Arabian coast lacks gyres and wedges, a consequence of the alongshore winds and coastal currents being weaker there. Southeast Trades Anderson and Moore (1979) and Knox and Anderson (1985) explored the idea that the Southeast trades might also drive the SC, at least in its southern part. These winds drive a Sverdrup gyre in the interior ocean, which is closed by a northwardflowing western-boundary current (Chap. 11). In a nonlinear model, the westernboundary current “overshoots,” extending farther to the north than expected from linear theory before retroflecting offshore to return southward. In the Anderson and Moore (1979) model with a meridionally-oriented boundary, the overshoot extended to 8◦N resulting in a pattern that resembled the Southern Gyre (June/July panel of Fig. 4.33). In contrast, in the McCreary and Kundu (1988) model the overshoot hardly crossed the equator at all. They attributed the lack of a significant overshoot to the boundary of their model having a realistic tilt, and to their forcing being located (more realistically) 2◦ farther south than in the Anderson and Moore (1979) solution. They concluded that inertial overshoot was not likely a realistic forcing mechanism for the SC or Southern Gyre. Offshore Wind Curl Undercurrents are well-known features of coastal circulations along eastern-ocean boundaries. McCreary and Kundu (1985) explored the dynamics of of undercurrents along western boundaries like the Somali coast using the LCS model (Sect. 5.2; see Vic et al. 2017, for a similar study using an OGCM). A surprising result is that steady coastal undercurrents cannot be generated along a western boundary by alongshore winds, a consequence of Rossby waves propagating onshore there (Sect. 16.2). They also showed that undercurrents can still be generated by offshore wind curl. Specifically, they showed that wind stress curl associated with the offshore weakening of the Findlater Jet excite Rossby waves that, upon arriving at the Somali coast, produce an undercurrent. Subsequently, McCreary et al. (1993) noted that

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147

an undercurrent could develop in the Somali coastal area in spring (that is before the onset of summer winds) due to the arrival of mode-2 baroclinic Rossby waves radiated from the west coast of India during winter. In their solution, a northward undercurrent formed during winter, but they were unable to diagnose its cause. Recently, Chatterjee et al. (2019) noted an even more surprising result about forcing by offshore wind curl: The negative wind curl on the eastern flank of the Findlater Jet is strong enough to severely limit the amount of Somali coastal upwelling. Specifically, the curl generates a downwelling-favorable Rossby wave that, when it arrives at the coast, thickens the upper layer enough to prevent the pycnocline from surfacing. See Sect. 17.3.3.1 for a detailed discussion of his process.

4.9.7 Salt Balance The upper panel in Fig. 4.4 illustrates the relative saltiness of near-surface water in the Arabian Sea (ASHSW) over that in the Bay of Bengal (BBSW), the saltier water resulting from the excess of evaporative over freshwater flux there (Sect. 3.1.2 and Fig. 3.1, bottom-left panel). To ensure an equilibrium salt balance in the Arabian Sea, ocean currents must remove this salty water, replacing it with fresher water. This exchanges occurs along several different pathways. The primary pathway for the export of ASHSW occurs during the summer and fall. During that time, southeastward flow carries ASHSW to the southeast corner of the basin where it joins the SMC to flow into the Bay of Bengal (Fig. 4.5). A secondary pathway occurs during winter when ASHSW spreads southward within the SC (Prasad and Ikeda 2002a). Subsequently, it spreads eastward along the equator or within the SECC (Nagura and McPhaden 2018). Three pathways carry freshwater into the Arabian Sea. The primary one is provided by the summertime SC, which carries fresher water from the southern hemisphere northward along the coasts of Somalia and Arabia, and offshore into the interior basin (Fig. 4.6). Another occurs along the west coast of India during the winter, when freshwater from the bay is carried into the northern Arabian Sea by the northward-flowing WICC. (Nagura et al. 2018, highlighted this pathway in their study of MLT errors that develop in the northern Arabian Sea in many IPPC models. In these solutions, surface water there is too salty because the wintertime WICC that supplies it is also too salty; as a result, the near-surface stratification is too weak, and mixed layers can become unrealistically thick.) A third pathway is the WMC that flows west-northwestward, carrying waters of southern Bay of Bengal into the interior of the Arabian Sea. Finally, a local source of freshwater occurs in the eastern Arabian Sea, a high-precipitation region that also receives heavy sizable runoff from the slopes of the Western Ghats (Fig. 3.1, bottom-left panel).

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4.10 Marginal Seas The Persian Gulf and Red Sea are the marginal seas of the NIO, joined to the Arabian Sea by the Gulfs of Oman and Aden, respectively. Being spatially confined, surface cooling and evaporation have a large impact on their density, leading to the formation of overturning circulations and distinct water masses: Persian Gulf Water (PGW) and Red Sea Water (RSW).

4.10.1 Persian Gulf The Persian Gulf is about 1000 km long and 350 km wide, lying between Iran to its north and the Arabian Peninsula to its south (Fig. 1.2). The gulf is shallow, being only about 60–80 m deep in the interior of the basin with a mean depth of only 36 m. It deepens to the southeast to about 100 m in a narrow passage Strait of Hormuz, which is only 56-km wide at its narrowest point.

4.10.1.1

Forcing

Located between the monsoon climate system to its southeast and synoptic weather systems of midlatitudes, the Persian Gulf experiences winds known as “Shamal,” a northwest wind that occurs year round. The wind usually occurs first in the northwest and then spreads south. During the Mt. Mitchell oceanographic expedition of February–June 1993 (Reynolds 1993), the Shamal outbreaks generally lasted from two to several days. The winter Shamal is related to the synoptic systems to the north, lasts several days, and usually has a peak speed of about 10 m/s (the expedition encountered speeds as high as 40 m/s). Winter Shamals are generally stronger than the summer ones and produce the roughest seas in the Gulf. The summer Shamal is usually continuous during June/July, and results from the influence of weather conditions over both India and Arabia. The gulf also experiences southeasterly winds that precede the onset of the Shamal and a diurnal sea breeze. The latter occurs along all the coasts of the gulf and is particularly strong on the Arabian coast. Evaporation in the gulf is much greater than the inflow from rivers and precipitation. The resulting net loss of water creates the conditions found in a negative (reverse) estuary, with surface inflow across its mouth and bottom outflow. Meshal and Hassan (1986) estimated that average evaporation across the surface of the gulf is about 200 cm/yr, most of which occurs during winter due to higher wind speeds, and Al-Hajri (1990) found the total river runoff and precipitation into the gulf to be 16 cm/yr and 7 cm/yr, respectively. Thus, every year the volume of water lost to evaporation in the gulf is an order of magnitude larger than that gained by freshwater input, and this net loss must be compensated by a surface inflow through the Strait of Hormuz.

4.10 Marginal Seas

4.10.1.2

149

Circulation

Figure 4.34 shows isolines of T , salinity, S, and σθ along the length of the gulf during January. Their structure suggests that the gulf behaves like a two-layer system, consisting of: near-surface inflow of lower-salinity water through the Straits of Hormuz, and outflow of denser, higher-salinity water (PGW) near the bottom. After entering the gulf, the inflow flows to the northern portions of the gulf along the Iran coast, its density increasing along the way due to evaporation. At the northern end, the inflow water becomes dense enough to sink, providing the source of the outflow (Swift and Bower 2003). Johns et al. (2003) studied the exchange through the Strait of Hormuz, using hydrographic and moored ADCP data from instruments deployed from December 1996 to March 1998. They estimated the deep outflow of PGW to be relatively steady throughout the year with a mean transport of 0.15±0.03 Sv and a layer-averaged bulk salinity of 39.5 ppt. They also noted that the salinity of the outflow varied considerably on intraseasonal time scales, with the most substantial fluctuations (39.5–40.8 ppt) occurring at the bottom of the outflow layer during boreal winter.

4.10.1.3

Dynamics

Based on solutions to a numerical model, Chao et al. (1992) concluded that the inflow is driven by evaporation, which, by lowering sea level toward the head of the gulf, causes a northward, pressure-gradient force. Further, the authors showed that during winter this thermohaline-driven inflow was opposed and weakened by northwesterly winds (Shamals), whereas during summer the inflow was enhanced by a shallower thermocline and weaker winds. Consistent with this result, Swift and Bower (2003) reported that the density front, which separates denser water in the northern gulf from the lighter inflow water, is located farthest into the gulf during late spring. They also suggested that the density difference between the gulf’s deep water in the interior of the basin and water at comparable depths outside the gulf drive the overturning. To study the dynamics of the sinking (downwelling) branch of the Atlantic Meridional Overturning circulation (AMOC), Schloesser et al. (2012) obtained solutions to both an analytic 2-layer model and an Persian Gulf. They were found in a northernhemisphere, rectangular basin, and by a surface heat flux that increased surface density polewards. For the 2-layer model, they derived a formula (their Eq. 23) for the AMOC overturning transport, M=

  h max 1 −1 ρ 2 , f g He 1 − 2 ρo D

(4.1)

where f −1 is the average value of f −1 over a latitude band near the sinking region, ρ is the upper-layer density difference between the sinking region and near the equator, ρo = 1 gm/cm3 is a background density, He and h max are upper-layer thicknesses near

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Fig. 4.34 Vertical sections of monthly-mean T , S, and σθ during January along the Persian Gulf section (red line) shown in the top panel. Data for T and S are from World Ocean Atlas 2018 (Locarnini et al. 2019; Zweng et al. 2019), and σθ is computed from T and S. Data are plotted at 14 locations (red circles), with locations 1, 7, and 14 identified

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the equator and sinking region, respectively, and D is the ocean depth. The authors also defined upper and lower layers for their OGCM solutions, and verified that (4.1) also accurately estimated the strengths of their overturning circulations. The processes that lead to (4.1) are general and straightforward (involving adjustment processes like those discussed later in this book). Further, (4.1) does not depend on either the width or breadth of the basin. So, it is reasonable to expect that (4.1) also applies to the smaller Persian Gulf. To check, we set f −1 = f (28◦ )−1 , ρ = 0.004 gm/cm3 , D = 80 m, and He = 40 m, and h max = 60 m, the latter three values estimated from Fig. 4.34. For these parameter values, M = 0.13 Sv, consistent with the estimate of Johns et al. (2003).

4.10.2 Gulf of Oman The Gulf of Oman stretches approximately 350 km, from its mouth (the line separating Iran and Pakistan to the cape of Ras Al Hadd, Oman) to the Straits of Hormuz (Figs. 1.2 and 4.35). The gulf and straits link the estuarine-like environment of the Persian Gulf to the open-ocean regime of the northern Arabian Sea.

4.10.2.1

Surface Circulation

To illustrate the near-surface circulation in the gulf, Fig. 4.35 and Video 4.6 show climatological sea level and surface geostrophic velocities using AVISO data during 1993–2015, the figure plotting bimonthly averages. Many of the prominent features in the figure were noted in previous studies based on hydrographic and other in situ data, albeit not as comprehensively depicted owing to data scarcity (e.g., Reynolds 1993; Pous et al. 2004). A striking feature is an intense cyclonic circulation (gyre) that forms near the mouth of the gulf during summer/fall. It spins up from June–August, peaks in September, and decays from October–December. Thereafter part of it appears to shift westward, and is present in the central gulf in May. Another prominent feature is a cyclonic circulation around the perimeter of the entire gulf (northern Arabian Sea) during November/December. The northern branch of this circulation is the Iran Coastal Current (ICC), and at this time the ICC carries salty ASHSW to the Persian Gulf (Sect. 4.10.1). At other times of the year, there is no uniform perimeter circulation, and the ICC is either weak or impacted by the northern flank of eddies within the gulf.

4.10.2.2

Dynamics

The likely cause of the prominent cyclonic gyre noted above is Ekman pumping by local wind curl. Beginning in April, a region of strong positive curl is established

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Fig. 4.35 Bimonthly mean ADT (color, cm) and geostrophic velocity (vectors; cm/s) in Gulf of Oman and Northern Arabian Sea from daily climatology of AVISO data 1993–2019 (Taburet and Pujol 2020)

north of the southeastern corner of the gulf due to the blockage of southwesterly winds by Arabia, and it lasts throughout the summer monsoon (Fig. 3.1, top-right panel; Video 4.6). The positive curl raises the thermocline and lowers sea level in the region through Ekman pumping (e.g., Sect. 12.2), intensifying the gyre until mid-September. When there is high sea level and anticyclonic flow around the perimeter of the gulf (October–December), the winds are weak over most of the basin, indicating that these signals are remotely forced by downwelling-favorable, coastally-trapped waves propagating from the west coast of India. In contrast, a perimeter cyclonic flow is not clearly present within the gulf during the summer (July–September), even though it exists around the perimeter of most of the Arabian Sea. This seasonal difference is likely due to stronger, local forcing within gulf by the summertime southwesterlies.

4.10 Marginal Seas

4.10.2.3

153

PGW Flow

Based on historical hydrographic data, Bower et al. (2000) traced PGW as it moved from the Straits of Hormuz to the Arabian Sea. They found it is carried in a narrow, southern-boundary current that: varies seasonally, is most continuous in winter, is patchy in spring and fall, and absent in summer. L’Hegaret et al. (2015) reported that most PGW flowed southeastward along the southern boundary at a depth of 200–250 m, with some pathways impacted by eddies. As it moves through the gulf, PGW mixes with (entrains) surrounding waters. A measure of the strength of the entrainment is the “dilution factor,” that is, the ratio of the PGW transport at the head of the Gulf of Oman divided by the transport at its mouth, and Bower et al. (2000) determined a dilution factor of about 4. The mixing also lowers the salinity and density of PGW, so that their values near the mouth lie in the ranges 37.50–38.5 psu and σθ = 26.6. PGW follows two pathways in the northern Arabian Sea, both suggested by the currents in Fig. 4.35. During winter, it bends southward to flow along the East Arabian coast, whereas in summer it flows eastward within the deep part of the Ras Al Hadd Jet (Böhm et al. 1999; Sect. 4.9.4). Throughout much of the northern Arabian Sea, PGW is identifiable by its salinity maximum; however, it loses this signature as it flows south owing to further mixing with surrounding waters, and it is barely detectable in the Bay of Bengal (Jain et al. 2017).

4.10.3 Red Sea The Red Sea is contained in a 1930 km long and 270 km wide rift valley (Fig. 1.2). Its average depth is 490 m, and it deepens to 3040 m in a central trough, the Suakin Trough (Johns et al. 1999). At its northern end, it opens into two gulfs, the shallower (maximum depth 70 m) Gulf of Suez on the west and the deeper (maximum depth 1850 m) Gulf of Aqaba on the east. At its southern end, the sea opens to the Gulf of Aden through the strait of Bab al Mandeb. The strait is about 50 km long, 20 km wide, and has an average depth of about 190 m. A region of sills lies near the southern end of the Red Sea, the shallowest being the 137-m deep Hanish Sill.

4.10.3.1

Forcing

During October–May, the Red Sea comes under the influence of the continental winds of Arabia and northeast Africa in the north, and monsoon winds in the south. Mountains surrounding the region play a role in defining the wind field throughout the year. The Ethiopian highlands (Fig. 1.2) steer the northeasterly winter monsoon winds over the NIO into the southern Red Sea. As a result, the winds are southeasterly over the southern part of the sea, northerly over its northern part, and the two converge near 18◦N, the boundary between the two wind systems. The northern winds are

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dominated by events (frontal passages), which propagate southward from Egypt and the southern Mediterranean Sea to decay near the convergence latitude (Johns et al. 1999). During summer (June–September), the convergent wind pattern is replaced by weaker northwesterly winds over the entire Red Sea. The highly arid nature of the lands bordering the Red Sea leads to high rates of evaporation from the sea surface. Estimates of the annual rate of evaporation is 1.61– 1.74 m/yr whereas precipitation is only 0.07–0.15 m/yr, an average loss of water from the sea of 1.46–1.67 m/yr (Smeed 2004). The evaporation varies seasonally, being higher when the winds are stronger.

4.10.3.2

Circulation

Figure 4.36 shows the thermohaline structure in a section along the axis of the Red Sea. The structure can be divided into two distinct regions separated by a pycnocline in the depth range of 80–120 m, which is close to the depth of the Hanish Sill. The deeper region is nearly homogeneous with a potential temperature range of 21.5– 22◦C and salinity range of about 40.4–40.6 ppt. Woelk and Quadfasel (1996) have shown that oxygen increases with depth below 500 m (their Fig. 2), indicating that bottom waters are better ventilated than intermediate waters. In the upper region (z  −100 m), the circulation divides into two layers: nearsurface inflow (layer 1) that carries lighter, lower-salinity water from the Gulf of Aden into the Red Sea; and an upper-thermocline outflow (layer 2), carrying denser, higher-salinity water from the Red Sea into the gulf. The inflow waters become denser as they flow north due to the net evaporation (Phillips 1966), and eventually downwell in the northern basin to close the circulation in the upper region. In the lower region (z  −100 m), Cember (1988) used tracer (carbon-14 and tritium-helium) distributions to argue that the circulation has a complex structure consisting of three distinct layers: a southward-flowing bottom layer with very high salinity greater than about 40 ppt (layer 5); another southward-flowing layer of subthermocline water (layer 3); and a northward-flowing, intermediate mass between the other two (layer 4). Woelk and Quadfasel (1996) concluded that the bottom water forms in the Gulf of Suez, its formation is not seasonal but instead occurs episodically every 4–7 years during particularly severe winters, and its renewal time is 40–90 years. The Red Sea is separated from the Gulf of Aden by a narrow sill at Bab el Mandeb that is only 160 m deep. Exchanges that take place across this sill measure the overall strength of the overturning within the sea. Studies of the exchange (e.g., Morcos 1970; Siedler 1968; Maillard and Soliman 1986; Murray and Johns 1997) indicate that the outflow has an annual-average transport of 0.3 Sv, with a winter maximum of 0.7 Sv. During winter, when the buoyancy forcing is much higher than in summer, the winds are southeasterly over the strait and the Red Sea. As a result, both the surface inflow into the Red Sea and the bottom outflow increase. During summer, however, the winds over the strait typically have a northerly component, opposing the buoyancy-forced circulation. Consistent with this wind forcing, Murray

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Fig. 4.36 Vertical sections of monthly-mean T , S, and σθ during January along the Red Sea section (red line) shown in the top panel. Data for T and S are from World Ocean Atlas 2018 (Locarnini et al. 2019; Zweng et al. 2019), and σθ is computed from T and S. Data are plotted at 15 locations (red circles), with locations 1, 8, and 15 identified. The white area marks the shallowest part of the section with data missing below about 75 m

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and Johns (1997) reported that the flow through the strait in summer, 1995, had a 3-layer structure, with flow out of the Red Sea occurring in the top (presumably wind-driven) layer and bottom layers and inflow confined to the middle layer.

4.10.3.3

Dynamics

Eshel et al. (1994) explored the dynamics of the complex Red Sea circulations using a linear inverse model, that is, one that solves for the flow field required in order to balance observed distributions of heat, mass, salt, and tritium. They concluded that the flow is predominantly driven by thermohaline forcing, with the surface flow being directed opposite to the wind during winter. Further, they found that two different processes were active during winter in the northern Red Sea: one involving the sinking of near-surface (layer 1) water into the pycnocline (layer 2), and the other the sinking of very dense water from outside the model domain (the Gulf of Suez) into the deep ocean (layer 5). Consistent with Cember (1988), their solution developed northwardflowing intermediate (layer 4) and southward-flowing subthermocline (layer 3) flows. At Bab el Mandeb, the pycnocline (layer 2) and some subthermocline (layer 3) waters merged to provide the outflow into the Gulf of Aden. Eshel and Naik (1997) used an OGCM to investigate Red-Sea circulations, particularly concerning the dynamics of intermediate-water formation. Their solution was able to form northward-flowing intermediate water, similar to the flow deduced from observations by Cember (1988). The circulation was driven by the northward density gradient that results from northern surface cooling. This density gradient forced an across-channel geostrophic flow, with divergence and upwelling on the western side and vice versa on the eastern side. The western upwelling was supplied by two western-boundary currents: a southward-flowing current along the northern portion of the coast; and a northward-flowing one along its southern part. Where they met, the denser southward current subducted under the northward one, thereby generating the intermediate water. The transport of the intermediate flow was 0.11 Sv, close to Cember’s (1988) observational estimation. As discussed above, the overturning circulation associated with RSW in winter is confined primarily above 200 m, where it consists of surface-inflow and subsurfaceoutflow layers. Thus, it is sensible to use (4.1) to estimate the strength of this overturning. To estimate the transport of the wintertime overturning, we set f −1 = f (27◦ )−1 , ρ = ρ2 − ρ1 where ρ2 = 1.0281 gm/cm3 and ρ1 = 1.0254 gm/cm3 are estimates of near-surface density at the sinking region and sea entrance, respectively, D = 300 m, He = 100 m, and h max = 200 m. Values of ρ1 , ρ2 , He , and h max are estimated from the hydrographic data used in Fig. 4.36. Equation (4.1) then gives M = 0.65 Sv, consistent with winter observations. We also applied (4.1) during summer when He reduces to 40 m, h max to 80 m, and ρ = ρ2 − ρ1 is 0.005 gm/cm3 (Locarnini et al. 2019; Zweng et al. 2019). With these choices, M reduces to 0.12 Sv somewhat larger than the observed 0.05 Sv reported by Murray and Johns (1997). They also

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noted that the circulation has a three-layer, rather than two-layer, structure during this season, suggesting that it resulted from wind forcing. Hence, we caution that use of (4.1) is questionable during summer.

4.10.4 Gulf of Aden The Gulf of Aden is the region through which the outflow of high salinity water from the Red Sea (RSW) joins the Arabian Sea (Fig. 1.2). The basin is approximately rectangular, with an average depth of 500 m and a maximum depth of ∼2,700 m. The north coast of Somalia forms its 1,000-km-long southern boundary along approximately 11◦N. Its northern boundary is along the south coast of Yemen oriented east-northeast. Its open-eastern (the mouth of the gulf) and western boundaries are roughly 350 km and 250 km wide, respectively.

4.10.4.1

Surface Circulation

Figure 4.37 and Video 4.7 illustrate the annual cycle of surface circulations within the gulf, showing bimonthly averages of ADT and the associated, surface, geostrophic velocity field in the Gulf of Aden. The figure also includes the region of the EArCC, which significantly impacts the gulf’s circulation. During the springtime intermonsoon (March/April), the winds are weak over the region (Fig. 3.1, top-left panel; Video 3.1), and so is the circulation within the gulf and along the Arabian coast (Fig. 4.37, March/April panel). One noticeable feature at this time is a weak anticyclonic gyre centered at near 53◦E and 13◦N; it is the early stage of the “Gulf-of-Aden Eddy” noted by Prasad and Ikeda (2001). During April/May, the eddy strengthens and is one of three anticyclonic eddies found off the Arabian coast. Simultaneously, sea level drops at the coast, and a northeastward EArCC forms under the influence of southwesterly winds. They trigger upwellingfavorable Kelvin waves that propagate anticlockwise around the perimeter of the gulf, lowering sea level there in June. As the summer monsoon progresses (July/August), negative sea-level anomalies cover the western-boundary region of the Arabian Sea, including the Gulf of Aden. At this time, a noteworthy feature within the gulf is the presence of an anticyclonic eddy near 48◦E, 12◦N, marked by high sea level (Fig. 4.37, July/August panel; Video 4.7), named the “Summer Eddy” by Bower and Furey (2012). Toward the end of the summer monsoon (September/October panel), circulations are again weak in the gulf and along the Arabian coast. In October, northeasterly winds begin over the Gulf, establishing a southwestward current along its northern boundary. During November/December, the northeasterlies spread along the Arabian coast, increasing sea level locally; they also generate downwelling-favorable Kelvin waves that propagate cyclonically around the perimeter of the gulf, increasing sea level there as well. Note that at this time, and extending into January, the circulation is oppo-

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Fig. 4.37 Bimonthly mean ADT (color, cm) and geostrophic velocity (vectors; cm/s) in the Gulf of Aden from daily climatology of AVISO data 1993–2019 (Taburet and Pujol 2020)

site to that during July/August, with a “Winter Eddy” in the central gulf (Fig. 4.37, November–February panels; Video 4.7). In addition to the two eddies noted above, other eddies are present in the gulf at specific times and locations (e.g., Bower and Furey 2012). Since they all appear in the SLA climatology, they are present every year; consequently, they must be either: directly wind forced; or indirectly generated by unstable currents, in which the locations of the instabilities are fixed to topographic features. Another set of eddies, which don’t appear in Fig. 4.37, occur randomly. They are initiated in the open Arabian Sea (e.g., Trotte et al. 2018), and sometimes enter the gulf as they propagate westward (Prasad and Ikeda 2001; Bower et al. 2002; Frantatoni et al. 2006; Al Saafani et al. 2007; Bower and Furey 2012). The largest of these eddies have horizontal scales approaching the width of the gulf (∼300 km), amplitudes of ∼30 cm, and can be easily tracked. Although they are surface intensified with azimuthal velocities as high as 50–60 cm/s, they extend to the depth range of RSW outflow (400–800 m) where their velocities are still 20–30 cm/s (Bower and Furey 2012).

4.10 Marginal Seas

4.10.4.2

159

Dynamics

The climatological surface currents in the gulf respond locally to wind forcing and remotely to coastal waves propagating from Arabia. To illustrate, consider their impact on the prominent eddies noted above. The Gulf-of-Aden Eddy spins up during the spring in response to negative Ekman pumping along Arabia (Sect. 4.9.5). Beginning in June and continuing throughout the summer, there is strong positive Ekman pumping across the mouth of the gulf, due to the blockage of the southeasterly winds by the Horn of Africa. That pumping quickly lowers sea level, weakening and eventually eliminating the Gulf-of-Aden eddy. The anticyclonic Summer Eddy appears to be a remnant of the high sea level throughout the gulf established during the winter. Beginning in June, sea level drops in the eastern and western gulf by two different processes: in the east by the positive Ekman pumping near the mouth; and in the west by an upwelling-favorable coastal wave propagating from the EArCC (Fig. 4.37 and Video 4.7). The two processes leave behind a region of high sea level in the central gulf, the Summer Eddy, which gradually weakens as the two downwelling processes intensify. Similar processes in the opposite sense generate the Winter Eddy.

4.10.4.3

RSW Flow

The flow of RSW through the gulf lies well underneath the near-surface circulations in Fig. 4.37. Just after passing the narrows at Bab el Mandeb, RSW outflow appears to separate into two pathways: a northern route through a narrow channel, and a southern one following the topography along the African continent (Siedler 1968). Somewhat farther into the gulf, most of it flows along the southern boundary at a core depth of about 700 m (Bower et al. 2000; their plate 4). Even farther east, RSW pathways are impacted by the deeply-penetrating eddies in the gulf (Bower and Furey 2012). As for PGW, surrounding waters are entrained into the outflow as it moves through the gulf. Along the length of the gulf, Bower et al. (2000) estimated a dilution factor (defined above for PGW) of about 2.5, a lower value than for PGW due to the larger transport of RSW and smaller initial density difference. Near the mouth of the Gulf of Aden, the mixed RSW has a density of 27.00–27.48 σθ , lies in the depth range 400–800 m, and has a salinity maximum of 38.8–39.2 ppt, (Bower and Furey 2012). Beyond the Horn of Africa, the main pathway of RSW into the Arabian Sea is the passage between Socotra and the African continent, where Schott and Fischer (2000) found maximum southward flows during the winter monsoon of 1995-96, in agreement with the seasonal cycle of Bab el Mandeb outflow. This transport cycle confirms the earlier findings of Schott et al. (1990; their Fig. 7), who reported the seasonal salinity maximum at the core density of RSW (σθ = 27.2) to occur from February to April off Somalia (8–12◦N). Using WOCE data, Beal et al. (2000; their plate 3) also found an annual variability of RSW transport in the Gulf of Aden

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4 Ocean Circulations

and northern SC. At farther distances, the presence of RSW becomes less clear as its characteristic high salinity is eroded by mixing with surrounding waters (Sect. 4.1.4).

Video Captions Video 4.1 Daily-climatological ADT and surface velocity (UGOS, VGOS) in the entire Indian Ocean. The video is prepared from AVISO data for the years 1993– 2019. Velocities outside the ±5◦ latitude band are geostrophic and computed from the ADT gradient, whereas they are determined by the method of Lagerloef et al. (1999) inside the band (Pujol et al. 2016). The resolution of the dataset is 0.25◦ . The units of the velocity-vector key (lower left-hand corner) and color bar are cm/s and cm, respectively. Black-dashed contours indicate 200-m isobaths. Video 4.2 As in Video 4.1, except in the Bay of Bengal and Andaman Sea. Video 4.3 As in Video 4.1, except showing the regions of the monsoon currents south of Sri Lanka, their extensions into the Bay of Bengal and Arabian Sea, the EICC, and the WICC. Video 4.4 As in Video 4.1, except in the Arabian Sea. Video 4.5 Alongshore currents from moorings 1–4 in Fig. 4.26. The data has been filtered to highlight the annual cycle. Courtesy of Amol and Mukherjee (2015; priv. comm.). Video 4.6 As in Video 4.1, except in the vicinity of the Gulf of Oman. Video 4.7 As in Video 4.1, except in the vicinity of the Gulf of Aden and western Arabian Sea.

Part II

Models

Chapter 5

Ocean Models

Abstract Models commonly used to study the ocean are reviewed. Most of the chapter derives the equations of motion for the linear, continuously stratified (LCS) model, which are used to obtain most of the solutions in this book. They are obtained by a step-by-step simplification of the equations for a typical, ocean general circulation model. Solutions to the LCS model are represented as expansions in the vertical normal modes of the system: one barotropic mode and an infinite set of baroclinic ones. Layer models are also discussed, and their close relationship to the LCS model noted. Keywords Ocean general circulation models (OGCMs) · Linear continously stratified (LCS) model · Layer models · Barotropic and baroclinic modes · Characteristic speed Three types of 3-d models are commonly used to study circulations and processes in the NIO and elsewhere: ocean general circulation models (OCGMs), the linear, continuously stratified (LCS) model, and layer models. In this chapter, we focus on deriving the governing equations for the LCS and layer models, from which the solutions discussed in later chapters are obtained. We begin with an overview of OGCMs, the most dynamically complete systems that represent the fluid equations most faithfully (Sect. 5.1), and then show how the LCS equations result from simplifications to the OGCM equation set (Sect. 5.2). Next, we discuss layer models, among other things noting that the linear version of a one-layer, reduced-gravity (1 21 -layer) model corresponds to one of the vertical modes of the LCS model (Sect. 5.3). We conclude with a derivation of potential-vorticity conservation for a layer model, a concept we utilize at several places in the text (Sect. 5.4). The importance of the simpler (LCS and layer) models to our understanding of ocean dynamics cannot be overstated. They allow for a suite of analytic (paper and pencil) solutions, in which the impacts of individual processes are clear. As such, they provide a “vocabulary” of idealized solutions (dynamical building blocks), which has proven to be invaluable (essential) for discussing both observations and Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_5. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_5

163

164

5 Ocean Models

OGCM solutions with realistic forcings and basin geometries. Moreover, solutions to simpler models are often quite similar to observations, indicating that, despite their simplifications, the dynamics contained in them capture fundamental processes at work in the ocean. The simpler solutions are useful even where they differ from observations, as they highlight the importance of neglected processes in these regions. Almost all the analytic solutions obtained in the book are solutions to the LCS model. Likewise, almost all the videos in later chapters show numerical solutions to the LCS model (Appendix C), with groups of videos designed to illustrate a specific analytic solution. The only exceptions are the solutions to a 2 21 -layer model discussed in Sect. 17.3.

5.1 Ocean General Circulation Models 5.1.1 Equations A typical equation set for an OGCM can be written u t + uu x + vu y + wu z − f v +

1 px = (νu z )z + ∇· (νh ∇u) , ρ¯

(5.1a)

vt + uvx + vv y + wvz + f u +

1 p y = (νvz )z + ∇· (νh ∇v) , ρ¯

(5.1b)

wt + uwx + vw y + wwz +

1 ρ pz = − g + (νwz )z + ∇· (νh ∇w) , ρ¯ ρ¯

(5.1c)

Tt + uTx + vTy + wTz = (κT Tz )z + ∇· (κhT ∇T ) ,

(5.1d)

St + u Sx + vS y + wSz = (κ S Sz )z + ∇· (κh S ∇S) ,

(5.1e)

u x + v y + wz = 0,

(5.1f)

where ∇ = i∂x + j∂ y is the horizontal gradient operator. For mathematical convenience, Eqs. (5.1), and all other equations in this book, are written in Cartesian (x and y) and level (z) coordinates, but other coordinate systems are also common (see the end of this section). Equations (5.1) are a set of 6 equations in 7 unknowns: zonal, meridional, and vertical velocities, u, v, and w, respectively; pressure p, density ρ, temperature T , and salinity S. The seventh equation, needed to close the system, is ρ = ρ(T, S, p),

(5.1g)

the “equation of state” that defines ρ in terms of temperature, salinity and pressure.

5.1 Ocean General Circulation Models

165

Equations (5.1a)–(5.1e) are statements of the conservation of momentum, heat, and salt (e.g., Kundu et al. 2016). They have the form of advective-diffusive equa¯ p y /ρ, ¯ and tions, with the momentum equations also impacted by pressure ( px /ρ, pz /ρ¯ terms), the u and v equations responding to the Coriolis force (− f v and f u terms), and the w equation affected by gravity (−ρg/ρ¯ term). Note that, because ρ varies so little in the ocean, it is replaced by a constant background value ρ¯ in the momentum equations (the Boussinesq approximation). Usually, the w terms in (5.1c) are neglected (the hydrostatic approximation), because for typical ocean problems (large time and space scales) the pressure and gravity terms are overwhelmingly dominant (see Step 2 in Sect. 5.2.1 below). Finally, (5.1f) follows from mass conservation, under the restriction that sea water is incompressible; this constraint filters sound waves out of the system, but has an insignificant impact on the solutions of interest in this book.

5.1.2 Mixing The horizontal and vertical mixing in Eqs. (5.1) involve second-order derivatives. As such, they are scale selective, preferentially smoothing small-scale numerical noise while leaving large-scale variations relatively undisturbed. For horizontal mixing, the scale selection of second-order mixing is often not adequate: If mixing coefficients are large enough to control noise, the large-scale signals of interest can be too strongly smoothed. One solution to this problem is to replace second-order mixing with fourthorder mixing (proportional to ∇ 4 q for variable q), which is much more scale selective. Another is to allow the horizontal-mixing coefficients in (5.1) to increase in regions of high shear such as western boundary currents (e.g., Smagorinsky 1963, 1993). For vertical mixing, mixing is very large within the surface mixed layer and much weaker at depth. This property is simulated in models either by parameterizing ν and κ so that they are large in regions of high shear and low stratification (low Richardson number) or by explicitly including a constant-density mixed layer in the OGCM (see Chap. 3).

5.1.3 Boundary Conditions Boundary conditions are required at the surface and bottom of the ocean, and typical conditions are τx τy pt , νvz = , κT Tz = Q, κ S Sz = E − P, w = @ z = 0, ρ¯ ρ¯ ρg ¯ (5.2a) u = v = 0, κT Tz = κ S Sz = 0, w = −u Dx − v D y = 0 @ z = −D(x, y). (5.2b) νu z =

166

5 Ocean Models

In (5.2a), wind-stress and buoyancy forcings enter the ocean as fluxes across the ocean surface. The surface condition on w is derived from the constraint that seasurface pressure is equal to atmospheric pressure, the latter usually set to zero. The bottom condition on w in (5.2b) ensures that there is no flow across a sloping bottom. The condition on w in (5.2a) allows sea level, which is expressible in terms of the other model variables, to evolve freely (a “free-surface” condition). In many OGCMs and the LCS model, however, this condition is replaced by w = 0 (the “rigid-lid” approximation; see Step 10 below), which prevents sea level from changing at all. On the other hand, rigid-lid solutions do develop a surface pressure field, p (x, y, 0, t), and p (x, y, 0, t) , (5.3) d= ρg ¯ the height of a water column that can cause p (x, y, 0, t), provides a useful (and accurate) proxy for sea level. Use of the rigid-lid approximation, rather than free-surface conditions, has essentially no impact on internal (baroclinic) variability considered in this book, because it involves displacements of isopycnals (10–100 m) much larger than d (∼10 cm). It does, however, modify the external (barotropic) response; see comments at the end of Sect. 5.2.4. Boundary conditions are also required at the horizontal edges of the model domain. Typically, u = v = 0 (closed, no-slip conditions) are applied adjacent to continents. In some models, the condition on the velocity component tangential to the boundary is relaxed (a slip condition); for example, a common slip condition is to set the normal derivative of the tangential velocity to zero (i.e., for a north-south-oriented boundary, v = 0 is replaced by vx = 0), so that stress vanishes at the side wall. In regional (nonglobal) models, one or more of the domain boundaries is open ocean; in that case, boundary conditions are commonly specified by relaxing model variables near the boundary to observed values (open conditions).

5.1.4 Coordinate Systems A variety of coordinate systems are used to express the equations of motion, each with its own advantages. Here, we list a few commonly-used models of each type. Details for each of them are readily available on the web. In most OGCMs, x and y are replaced by spherical coordinates (longitude φ and latitude θ), so that they apply more readily to all regions of the earth. Popular models of this sort are the Modular Ocean Model (MOM), Parallel Ocean Program (POP) model, and Ocean Circulation and Climate Advanced Modeling project (OCCAM). The Ocean Model For the Earth Simulator (OFES), based on the third version of MOM (MOM3), was used to prepare the high-resolution SLA climatology shown in Video 1.2.

5.2 Linear, Continuously Stratified Model

167

Other OGCMs replace z with another vertical coordinate σ = z/D(x, y), where D is the bottom depth (a σ-coordinate model). In this system, σ “follows” the bottom topography, varying from zero at the ocean surface to −1 at the ocean bottom. It has the advantage that the vertical resolution of the model’s grid increases in regions of shallow topography. The Regional Ocean Modeling System (ROMS) and Princeton Ocean Model (POM) are well-known examples of σ-coordinate models. A third, vertical coordinate system replaces z with ρ (an isopycnal model). Its advantage is that it eliminates spurious (numerical) vertical diffusion that can occur in level models, and the Miami Isopycnal Coordinate Ocean Model (MICOM), Ocean IsoPYCnal Model (OPYC), and Generalized Ocean Layered model (GOLD) are popular isopycnal models. Models can also combine coordinate systems (“hybrid models”), for example, with HYbrid Coordinate Ocean Model (HYCOM) having a z-coordinate, surface mixed layer overlying the deeper isopycnal layers.

5.2 Linear, Continuously Stratified Model The LCS equations and its boundary conditions are simplified versions of Eqs. (5.1) and (5.2). Following McCreary (1980), we highlight the simplifications imposed in a series of steps (Sects. 5.2.1 and 5.2.2), and then show how solutions can be represented as expansions in the vertical normal modes of the simpler system (Sects. 5.2.3–5.2.5). Moore and Philander (1978) provide a more formal development that first writes the equations in non-dimensional form (Appendix B illustrates this approach in another context).

5.2.1 Equations of Motion To simplify the OGCM equations of motion, we carry out the following steps. For each step, we note the processes that are neglected or distorted by the approximation, and rate its overall validity. Step 1: Drop the advection terms from (5.1a) and (5.1b). Many studies have demonstrated their importance in a variety of ocean phenomena; for example, they are necessary for currents to become unstable and to generate eddies. On the other hand, the remaining linear terms are always important, often dominant. Overall, then, we rate this assumption as QUESTIONABLE, reserving judgment as to its usefulness until the LCS solutions are compared to observations (e.g., compare Videos 1.1 and 1.3). Step 2: Impose the hydrostatic relation by neglecting the w terms in (5.1c). The neglect of wt only affects problems where frequencies of the order of the Vaisala frequency or above are involved (see the paragraph after Eq. 7.5). The neglect of the

168

5 Ocean Models

advection terms, uwx + vw y + wwz , is valid everywhere except in regions of very high w, such as density-driven overflows (e.g., Gordon 2005. The neglect of (νwz )z filters out a very thin boundary layer near the ocean surface that is dynamically unimportant for the rest of the flow field. For our purposes, then, this assumption is VERY GOOD. Step 3: Linearize equation of state (5.1g) to      ρ = ρ¯ 1 + αT T − T¯ + α S S − S¯ ,

(5.4)

where ρ, ¯ T¯ , and S¯ are typical density, temperature, and salinity values in the upper NIO, and the coefficients of thermal and salinity expansion, αT and α S , are assumed constant. One issue with (5.4) is that αT and α S vary considerably with temperature, salinity, and pressure (Gill 1982). Another is that the linearization eliminates cabbeling in which waters, initially with the same density but different temperatures and salinities, mix to form a water mass with higher density (Witte 1902; Talley and Yun 2001). Approximation (5.4) does impact solutions, but only qualitatively. So, we rate this assumption OKAY. Step 4: Set κT = κs = κ and κhT = κhs = κh , and then combine Eqs. (5.1d) and (5.1e) into a single equation for density, ρt + uρx + vρ y + wρz = (κρz )z + ∇· (κh ∇ρ) .

(5.5)

Requiring the mixing coefficients for salinity and temperature to be equal eliminates the possibility of mixing by double diffusion, a small-scale process not of interest to us. So, we rate this assumption GOOD. Step 5: Drop uρx and vρ y from (Sect. 5.5), but retain and linearize wρz . The wρz term plays a critical role in ocean dynamics, by vertically advecting isopycnals in the water column. As a result, it directly affects the subsurface pressure field, for example, allowing for the existence of internal waves. To retain this important effect of wρz in linear form, we replace it with wρbz , where ρb (z) represents the background stratification of the ocean and is a function only of z. In effect, these changes linearize the system about a background state of rest with density ρb (z) and pressure pb (z) = −gρbz . The approach is commonly used in analytic models, for example, first being used by Fjeldstad (1933). Horizontal density advection is essential for baroclinic instability and the spreading of tracers, and ρb (z) is not spatially uniform in the real ocean. So, we rate these modifications QUESTIONABLE, assessing their worth only after a comparison of solutions to observations.

5.2 Linear, Continuously Stratified Model

169

With these modifications, Eqs. (5.1) simplify to ut − f v +

1 px = (νu z )z + ∇· (νh ∇u) , ρ¯

(5.6a)

vt + f u +

1 p y = (νvz )z + ∇· (νh ∇v) , ρ¯

(5.6b)

pz = −ρg,

(5.6c)

u x + v y + wz = 0,

(5.6d)

ρt + wρbz = (κρz )z + ∇· (κh ∇ρ) ,

(5.6e)

where ρ and p now represent deviations from their background values, ρb and pb . Continuing, we simplify this set a bit more with two more modifications: one relating to assumptions on the forms of mixing, and the other to the introduction of wind forcing into the model. Step 6: Since precise expressions of mixing of momentum and density in the ocean are not known, we choose mathematically convenient forms. First, restrict νh and κh to be independent of z. Second, replace (κρz )z with (κρ)zz . Finally, choose depthdependent, vertical mixing coefficients of the form ν = A/Nb2 ,

κ = A /Nb2 .

(5.7)

where Nb2 (z) = −gρbz /ρ¯ is the Vaisala frequency of the background state and A and A are constants. Previous studies have recognized the mathematical convenience of (5.7); for example, Fjeldstad (1963) and Mork (1972) used them in their studies of internal waves. Further, (5.7) states that ν and κ decrease wherever background stratification is large, a sensible dependency since stratification does inhibit vertical mixing in the ocean (Turner 1973). These choices are necessary in order to allow second-order mixing to be included in the modal equations. We rate them NECESSARY and OKAY. Step 7: It is possible to introduce wind forcing into the LCS model as stress conditions at the ocean surface (the first two expressions in Eq. 5.2a). In OGCMs and the real ocean, however, wind stress is quickly spread throughout a surface mixed layer (Chap. 3). To represent this process, it is common to introduce τ into the LCS model as a “body force,” (τ /ρ) ¯ Z (z), rather than through a surface stress condition. In this formulation, structure function Z (z) represents spreading by an idealized mixed layer with a time-independent and horizontally-uniform, vertical structure. The choice of 0 Z (z) must satisfy the constraint that −D Z dz = 1, since the total amount of stress that enters the ocean is τ , but is otherwise arbitrary. As such, its unit is inverse length. We rate this change as USEFUL.

170

5 Ocean Models

With the modifications of Steps 6 and 7, Eqs. (5.6) can be written   1 1 τx Z (z) + A ∂z 2 ∂z u + ∇· (νh ∇u) , u t − f v + px = ρ¯ ρ¯ Nb

(5.8a)

  1 1 τy py = Z (z) + A ∂z 2 ∂z v + ∇· (νh ∇v) , ρ¯ ρ¯ Nb

(5.8b)

vt + f u +

  2      pt 1 1 p 1 p  , − ∂z 2 ∂z + u x + v y = −A ∂z 2 ∂z − ∂z 2 ∂z ∇· κh ∇ ρ¯ ρ¯ ρ¯ Nb Nb Nb (5.8c)    

p 1 pzt 1 p z z − A ∂z2 2 − ∇· κh ∇ , (5.8d) w=− 2 ρ¯ Nb ρ¯ Nb ρ¯ 1 ρ = − pz . g

(5.8e)

To obtain the last three equations of (5.8), we solve (5.6c) and (5.6e) for ρ and w in terms of p, and then use the expression for w to rewrite  (5.6d). Note that all the z operators in Eqs. (5.8a)–(5.8c) have the same form, ∂z Nb−2 ∂z , an essential property in order to represent solutions as expansions of the vertical normal modes of the system.

5.2.2 Boundary Conditions To expand solutions into vertical modes also requires that boundary conditions (5.2) are simplified. Step 8: Require the bottom to be flat, that is, D is constant. In the interior ocean, this restriction is reasonable since our focus is on upper-ocean circulations. Along coasts, however, it eliminates continental shelves from the domain, replacing them with vertical walls. As a result, wind-forced coastal circulations involve adjustments by coastal Kelvin waves (Sect. 7.2 and Chap. 13) rather than by a set of shelf-dependent, coastally trapped waves. Because the adjustments are similar in both cases, coastal processes are still well represented under this simplification. So, we rate this assumption as NECESSARY and OKAY. Step 9: Replace the flux conditions on density (the S and T conditions in Eqs. 5.2) with the requirement that ρ = ρ∗ (x, y, t) and, for convenience, set ρ∗ = 0. (See Rothstein 1984, for a solution when ρ∗ = 0.) At the surface, this condition assumes that the atmosphere is a constant-temperature source of heat, and that there is no rain to modify salinity. This choice is unpleasant because it prohibits the model from developing SST and SSS variations. At the ocean bottom, the choice is less problematic,

5.2 Linear, Continuously Stratified Model

171

since we focus on upper-ocean flows. We rate this assumption UNFORTUNATE, but NECESSARY. Step 10: Set w = 0 at the surface, thereby imposing a rigid lid. Because sea level d is so small, this restriction has a negligible impact on the internal-ocean solutions obtained in this book (see comments at the end of Sect. 5.2.4). We therefore rate this assumption VERY GOOD. Step 11: Set u z = vz = 0 at the ocean surface and bottom. At the surface, this restriction presumes that wind stress enters Eqs. (5.8a) and (5.8b) through the body-force terms, τ x Z (z) and τ y Z (z), not by the surface stress conditions in (5.2a). At the bottom, the zero-stress condition is problematic only in regions that develop large bottom velocities, and it has little or no effect on upper-ocean currents. We rate the surface conditions GOOD and the bottom conditions NECESSARY and OKAY. With these restrictions, boundary conditions (5.2) simplify to νu z = νvz = 0, ρ = 0, w = 0,

@

z = 0, −D,

(5.9)

a set consistent with the boundary conditions (5.10b) for the normal modes.

5.2.3 Vertical Normal Modes Given the special forms of Eqs. (5.8) and boundary conditions (5.9), it is possible to represent solutions as expansions in the set of solutions ψn (eigenfunctions, normal modes) defined by   1 1 ψ = −λn ψn ≡ − 2 ψn , (5.10a) nz c Nb2 n z n = 0, 1, 2 . . ., subject to the boundary conditions ψnz (−D) = ψnz (0) = 0.

(5.10b)

The set of constants√λn (eigenvalues) are determined by imposing (5.10b). The related constants, cn = 1/ λn , have the unit of speed, they are referred to as the “characteristic speed of mode n,” and their values for two choices of Nb (z) are listed in Table 5.1. Note that, because (5.10a) is homogeneous, the amplitudes of the eigenfunctions are arbitrary. They are specified by an additional “normalization” condition, and ψn (0) = 1 is a convenient choice (see the comment after Eqs. 5.15a).

(5.10c)

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5 Ocean Models

Table 5.1 Lists of mode parameters Parameter

Modenumber 0

1

2

3

4

5

15

25

50

75

100

cn (cm/s)

∞

285

142

94.9

71.2

56.9

19.0

11.4

5.69

3.80

2.85

xpsi2 (m)

4000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

xZpsi

1.000

1.000

0.999

0.998

0.996

0.994

0.943

0.847

0.470

0.0662

−0.180

H−1 n (km−1 )

0.250

0.500

0.499

0.499

0.498

0.497

0.472

0.423

0.235

0.033

−0.09

rn

0.500

1.000

0.999

0.998

0.996

0.994

0.943

0.847

0.470

0.066

−0.18

cn (cm/s)

∞

265

167

105

74.8

59.7

19.3

11.6

5.76

3.84

2.88 15960

xpsi2 (m)

4000

275

174

804

930

1880

4080

5411

9774

15567

xZpsi

1.000

0.986

0.964

0.910

0.831

0.747

0.064

0.084

0.031

0.016

0.14

H−1 n (km−1 )

0.250

3.59

5.57

1.13

0.894

0.397

0.016

0.0015

0.0032

0.0010

8.6e−4

rn

0.696

1.00

1.55

0.312

0.249

0.111

0.0043

0.0043

8.9e−4

2.8e−4

2.4e−4

Parameters (left column) and their values for various modes (other columns), when Nb (z) is constant (upper block) and has a realistic profile (lower block). The two blocks are separated by a thicker horizontal line. The calculation of all quantities assumes the ocean surface is a rigid lid and its depth is D = 4000 m. If the surface is free, all values in the table are almost unchanged except for c0 , √ which decreases to c0 = g D = 1.98 × 104 cm/s

Equations (5.10) satisfy the requirements of a Sturm-Liouville problem, and so solution properties are well known. Perhaps the most important property is that the eigenfunctions ψn form a complete set, ensuring that model variables can be represented as expansions in ψn , its derivative, or its integral (as in Eqs. 5.15 below). Another important property is orthogonality. To prove orthogonality, multiply (5.10a) by ψm and integrate from −D to 0, obtain a second version by switching the dummy indices m and n, and subtract the two expressions to get 

0 −D

 ψm

1 ψnz Nb2



 dz −

0 −D

z

 ψn

1 ψmz Nb2



 dz = z

1 1 − 2 2 cm cn



0

−D

ψm ψn dz.

(5.11) By integrating the first (or second) term in (5.11) by parts twice and with the aid of (5.10b), it is straightforward to show that the left-hand side of (5.11) vanishes. It follows that, if m = n so that cm = cn , then 

0

−D

ψm ψn dz = 0,

m = n,

(5.12)

the mathematical statement of orthogonality. A final noteworthy property is that, when ordered sequentially, the eigenvalues increase like λn ∼ n 2 for large n; thus, the characteristic speeds cn decrease like n −1 .

5.2 Linear, Continuously Stratified Model

173

5.2.4 Barotropic and Baroclinic Modes Integrating (5.10a) over the water column, and applying boundary conditions (5.10b), gives the integral constraint  1 0 ψn dz = 0, (5.13) cn2 −D Equation (5.13) can be satisfied in two ways. Either c0 = ∞ (the single barotropic mode) or cn is finite and the integral of ψn vanishes (an infinite set of baroclinic modes for n ≥ 1). With c0 = ∞, the solution to (5.10a) for the barotropic (n = 0) mode is ψ0 (z) = 1. In contrast, the structure of the baroclinic modes depends on Nb . For a constant-Nb background stratification, the general solution to (5.10a) is ψn (z) = A cos (Nb z/cn ) + B sin (Nb z/cn ). Then, the imposition of (5.10b) requires that B = 0 and that Nb D/cn = nπ, n = 1, 2, 3, . . ., so that with normalization (5.10c) the baroclinicmode structures and characteristic speeds are ψn (z) = cos

nπz D

,

cn =

Nb D . nπ

(5.14)

When Nb is not constant, ψn (z) and cn are modified versions of (5.14). Figure 5.1 plots several baroclinic modes for constant (middle panel) and realistic (right panel) Nb profiles, and Table 5.1 lists the cn values for each profile. Note that for the realistic stratification the vertical wavelength of the modes varies with depth, being shorter in the upper ocean where Nb is larger; as a result, the first zero crossing of the n > 1 modes (red, blue, and green curves) now occurs near the surface (near the depth of the middle of the pycnocline). When free-surface conditions are applied, it is still possible to represent solutions to (5.8) as modal expansions (5.15). Given the smallness of sea level, ψn (z) and cn are almost unchanged for all the modes. The only exception is for the characteristic speed of the barotropic mode √ c0 , which is reduced from its unrealistic, rigid-lid value (c0 = ∞) to be close to g D, its value in a homogeneous (Nb2 = 0) ocean.

5.2.5 Modal Equations Because functions ψn are a complete set, solutions to (5.8) can be expressed as the expansions q(x, y, z, t) =

N  n=0

 1 p(x, y, z, t) = pn (x, y, t)ψn (z), ρ¯ n=0 (5.15a) N

qn (x, y, t)ψn (z),

174

5 Ocean Models

Fig. 5.1 Constant and realistic Nb (z) profiles (left panel) and mode structures ψn (z) determined from them (middle and right panels, respectively). The n = 1–4 modes are shown, indicated consecutively by the colors black, red, blue, and green. For the constant profile (vertical blue line), the value of Nb is the depth average of the realistic one

where q is u or v, and  z coefficients are functions of only x, y, and t.  the expansion  Given that wz = − u x + v y and −D ρdz  = − p/g, appropriate expansions for w and ρ are w(x, y, z, t) =

N  n=0

 wn (x, y, t)

z −D

ψn (z  ) dz  ,

ρ(x, y, z, t) =

N 

ρn (x, y, t)ψnz .

n=0

(5.15b) In principle, the sums in Eqs. (5.15) should extend to infinity. For realistic problems, however, they can be truncated at a finite value N because their expansion coefficients tend to zero for large n (see below). Note that, since the modes have the normalization ψn (0) = 1, qn and pn are the contributions of mode n to q and p/ρ¯ at the ocean surface, a useful property for interpreting the impact of individual modes. Note also that, although the dimensions of u n and vn remain velocity, the others are changed. Specifically, the dimension of pn is velocity squared, wn is velocity/depth, and ρn is density times depth. The

5.2 Linear, Continuously Stratified Model

175

absorption of a factor of ρ¯−1 into the definition of pn is common and notationally convenient; as a result, in later chapters factors of ρ¯ only appear in the forcing term of modal equations (like Eqs. 5.16) and their solutions. To determine equations for the expansion coefficients, u n , vn , and pn , first multiply the first three equations of (5.8) by ψm and integrate them from −D to 0. Then, integrate the terms involving the operator (∂z Nb−2 ∂z ) by parts twice, using boundary conditions (5.9) and (5.10b) to eliminate unwanted parts integrals. Finally, after introducing expansions (5.15a) into each term and using (5.12), the equations are (∂t + γn ) u n − f vn + pnx =

τx + ∇· (νh ∇u n ) , ρH ¯ n

(5.16a)

(∂t + γn ) vn + f u n + pny =

τy + ∇· (νh ∇vn ) , ρH ¯ n

(5.16b)

    pn pn ∂t + γn 2 + u nx + vny = ∇· κh ∇ 2 , cn cn

(5.16c)

where γn = A/cn2 and γn = A /cn2 are damping coefficients determined by vertical mixing. The coefficient 0 Hn−1

=

Z (z)ψn dz = Zn 0 2 dz ψ n −D

−D

(5.17)

is the expansion coefficient of Z (z), and it measures how well each mode couples to the wind (see Steps 7 and 11). To determine wn , first take the z-derivative of (5.8d) and then follow the steps in the previous paragraph, to get     pn  pn wn = ∂t + γn 2 − ∇· κh ∇ 2 , cn cn

(5.18a)

so that wn is known in terms of pn . Similarly, multiply (5.8e) by the operator ∂z Nb−2 and follow the above steps to obtain 1 ρn = − pn , g

(5.18b)

which expresses ρn in terms of pn . Finally, from (5.3) it follows that dn =

pn ρg ¯

gives the contribution to sea level from mode n.

(5.18c)

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5 Ocean Models

Equations (5.16) are a set of 3 equations in 3 unknowns, which can be solved for u n , vn , and pn , and with pn known Eqs. (5.18) allow wn and ρn to be determined. With all the coupling coefficients known, Eqs. (5.15) then allow the complete 3-d flow field to be constructed. Thus, the LCS model reduces the 3-d OGCM Eqs. (5.1) to the two-dimensional (2-d) set (5.16). In so doing, there is an advantage in computer cost, as N solutions to (5.16) are still much cheaper to obtain than a direct solution to (5.1). More importantly, the LCS model has an enormous conceptual advantage: It allows basic ocean dynamics to be discussed in terms of the solutions to (5.16), that is, separately for each vertical mode, an approach we follow in subsequent chapters. Despite these advantages, it is important to keep in mind the simplifications that led to the LCS model, primarily elimination of all nonlinearities; as a result, the model is incapable of simulating some ocean phenomena known to be important, notably eddies generated by unstable currents.

5.2.6 Convergence Expansions (5.15), however, won’t be useful (practical) unless the summations converge rapidly with n. Rapid convergence is ensured when vertical mixing by γn and γn is included, since damping by those terms increases rapidly with n (cn−2 ∼ n 2 for large n). Even without damping, there is convergence provided that Hn−1 (the factor that determines the forcing strength for a particular mode) decreases with n. Table 5.1 lists values of Hn−1 for the constant (upper block) and realistic (lower block) Nb profiles in Fig. 5.1 (left panel). For both profiles, Z (z) is given by (C.6) in Appendix C, the body-force structure used in all the videos of LCS solutions. Other 0 0 parameters listed are: xpsi2 = −D ψn2 dz and xZpsi = −D Z (z)ψn dz, factors that determine Hn−1 ; and rn = Hn−1 /H1−1 . Parameter rn , the ratio between Hn−1 with its value for n = 1, measures how well a particular mode couples with the wind relative to the first mode. For the baroclinic modes with constant Nb , xpis2 has the constant value D/2, and xZpsi weakens slowly with n because ψn (z) ≈ 1 for z ≥ −H for most of the modes. As a result, Hn−1 and rn weaken slowly with n, so that the wind couples well to all the modes. In contrast, for modes with realistic Nb (z), xpsi2 tends to increase with modenumber whereas xZpsi decreases, so that Hn−1 and rn decrease rapidly with n. The only exception to this tendency is that the second mode couples to the wind better than mode 1 (r2 > r1 ). This interesting property results from the particular structure of the realistic profile, and is one of the reasons the n = 2 mode is often detected in NIO observations. For realistic stratifications, then, the summations converge even without damping. See Chap. 16 for further discussions of convergence. For the barotropic mode, xpsi2 = D and xZpsi = 1 since ψ0 (z) = 1. Note that r0 1 for realistic stratification, a consequence of xpsi2 for n = 0 being much larger than it is for n = 1. One implication is that wind-driven baroclinic currents dominate barotropic flows. For this reason, it is common to ignore the barotropic response

5.3 Layer Models

177

by setting the lower limit of the sums in Eqs. (5.15) to n = 1; for example, all the videos of multi-mode solutions reported in Chap. 16 do so. A notable exception to baroclinic-flow dominance occurs when the circulation integrated over the entire water column is of interest: The depth-integrated currents associated with all baroclinic modes vanishes (Eq. 5.13), so that only flows associated with the barotropic mode contribute. A well-known example of a depth-integrated flow is the steady-state Sverdrup circulation (Chap. 11).

5.3 Layer Models 5.3.1 Overview Layer models represent the ocean as a discrete set of layers. (As such, they are akin to σ-coordinate OGCMs.) A general layer model consists of n layers that extend from the ocean surface to the bottom. Alternately, the n layers can be confined to the upper ocean, with the deep ocean assumed to be in a state of rest. Such models are referred to as n 21 -layer models, the “ 21 ” referring to the deep quiescent layer. In traditional layer models, the density of each layer is uniformly constant. In many situations, however, it is useful to allow layer densities to vary (a variable-density layer model). For example, it is useful to allow at least the layer-1 temperature to vary in order to simulate SST cooling in the intense upwelling regions off Somalia and Oman during the Southwest Monsoon (e.g., McCreary et al. 1993); likewise, layer-1 salinity must vary in order to represent SSS freshening in the Bay of Bengal due to the strong freshwater input there during the summer and fall (Han and McCreary 2001). The conceptual advantage of layer models is that each layer can be designed to represent a particular region or water-mass type in the ocean, thereby representing them more efficiently than they are in OGCMs. For example, McCreary et al. (1993) used a 2 21 -layer model in order to investigate the dynamics of both surface currents and undercurrents (thermocline flows), and Jensen (1991, 2001) used 3 21 - and 4 21 -layer models to consider deeper flows. In this book, we use a 2 21 -layer model to investigate the dynamics of the Indian Ocean’s shallow overturning cells (Sect. 17.3.1). To study coupled physical/biological processes in the Arabian Sea, Hood et al. (2003) used a 4 21 -layer model with an additional ecosystem component, interpreting layers 1–4 to be: a KT mixed layer; a diurnal thermocline (the region left behind when the mixed layer thins during the day); the seasonal thermocline; and the main thermocline. To study the dynamics of oxygen minimum zones (OMZs) in the north Indian Ocean, McCreary et al. (2013) extended the Hood et al. (2003) model to include 6 active layers and an oxygen component; in their model, layer 4 represents both the main thermocline and the upper part of the OMZ, layer 5 corresponds to upper intermediate water and the lower OMZ, and layer 6 simulates deep intermediate water below the OMZ (Fig. 5.2).

178

5 Ocean Models

Fig. 5.2 Schematic diagram of the 6 21 -layer model used by McCreary et al. (2013). Velocities vi , temperatures Ti , and salinities Si can vary within each layer. Water can transfer between each layer by entrainment/detrainment velocities wi . Layers are not allowed to become thinner than minimum thicknesses h i min

5.3.2 Nonlinear 1 21 -layer Model To illustrate the concept of the “ 21 ” layer, we consider only the simplest version, a constant-density, 1 21 -layer model. (Section 17.3.1 provides a similar overview of a 2 21 -layer model.) It consists of a surface layer of thickness h, density ρ1 , and depthindependent currents, u and v, overlying a deep ocean of density ρ2 and no currents. Without forcing, it is in a background state of rest with h = H . As discussed below, pressure gradients are assumed to vanish in the deeper ocean (the “ 21 ” layer). Typical equations for a nonlinear, 1 21 -layer model are u t + uu x + vu y − f v +

1 τx − γu + ∇· (νh ∇u) , px = ρ1 ρ1 h

(5.19a)

vt + uvx + vv y + f u +

1 τy − γv + ∇· (νh ∇v) , py = ρ1 ρ1 h

(5.19b)

h t + (hu)x + (hv) y = −γ  (h − H ) + ∇· (κh ∇h) ,

(5.19c)

where the first two equations arise from momentum conservation and the third from mass continuity. Wind stress enters the ocean as a body force spread throughout the top layer. The γ terms parameterize mixing that occurs from the exchange of water particles between the active layer and deep ocean. As written, the γ  term

5.3 Layer Models

179

parameterizes diffusive processes that tend to return h to its background state, that is, entrainment into, or detrainment from, the layer at the rate we = −γ  (h − H ). Alternate forms for we are also common. In later chapters, for example, we consider an entrainment rate of the form we = −

h − hm θ (h m − h) , δt

(5.20)

which simulates entrainment into an idealized, surface mixed layer of constant thickness h m < H . According to (5.20), entrainment happens wherever wind-driven upwelling attempts to thin h to be less than h m . Typically, δt is assumed small enough so that (5.20) ensures h can never much less than h m .   become An expression for ∇ p = px , p y in terms of h is obtained by requiring the pressure-gradient field to vanish in the deep ocean, thereby ensuring that currents also vanish there. Consider the pressures on level surfaces below (z = −z 2 ) and within the layer (z = −z 1 ). Assuming the hydrostatic relation and that atmospheric pressure is zero, they are given by the weight of water above the surfaces, so that p2 at z = −z 2 and p1 at z = −z 1 are p2 = gρ1 h + gρ2 (d + z 2 − h) ,

p1 = gρ1 (d + z 1 ) ,

(5.21)

where d is sea level to be determined. It follows that ∇ p2 = −g  ρ2 ∇h + gρ2 ∇d,

∇ p1 = gρ1 ∇d,

(5.22)

where g  = g,  = ρ/ρ2 , and ρ = ρ2 − ρ1 . Note that neither z 1 nor z 2 appear in the equations, a statement that both pressure gradients are independent of z in their respective regions. One implication of this independence is that ∇ p1 = ∇ p, and so specifies the pressure-gradient terms in Eqs. (5.19a) and (5.19b). To ensure that the deep ocean is at rest, we set ∇ p2 = 0, in which case Eqs. (5.22) with ∇ p1 → ∇ p can be rewritten ∇d = ∇h,

1 ∇ p = g  ∇h. ρ1

(5.23)

Integrating (5.23) yields d = h,

1 p = g  h = gd, ρ1

(5.24)

where h = h − H and the constants of integration are chosen so that d = p = 0 when h = H . It is noteworthy that, in defining p in this way, we have essentially taken out the pressure associated with the background state, so that p represents deviations from that state, just as pn does in the LCS model.

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5 Ocean Models

5.3.3 Linear 1 21 -layer Model A linear version of Eqs. (5.19) is obtained by dropping the advection terms from the momentum equations, and by replacing h with H in the wind-forcing terms and the nonlinear terms of the continuity equation. The resulting equations can be expressed in terms of either h or p using the second of Eqs. (5.24). Expressed in terms of p, Eqs. (5.19) are (∂t + γ) u t − f v +

px τx + ∇· (νh ∇u) , = ρ1 ρ1 H

(5.25a)

(∂t + γ) vt + f u +

py τy + ∇· (νh ∇v) , = ρ1 ρ1 H

(5.25b)

 p  p . + u + v = ∇· κ ∇ ∂t + γ  x y h ρ1 c2 c2

(5.25c)

where c2 = g  H . Equations (5.25) have the same form as (5.16). Indeed, with the replacements ρ¯ → ρ1 ,

cn2 → c2 ,

qn →

H q, Hn

pn →

H p , Hn ρ1

(5.26)

where q = u or v, the latter are transformed into the former. This property allows the response of any baroclinic mode of the LCS model to be interpreted as a solution to the ocean, typical parameter choices are 1 21 -layer model. For a surface layer in the tropical √ g  = 3 cm/s2 and H = 200 m so that c = g  H = 245 cm/s, a characteristic speed close to that of the n = 1 mode. Given this correspondence, it is most reasonable to interpret only the response of the n = 1 mode as that for a 1 21 -layer model, and it is common to do so in oceanographic literature. Many of the videos presented in later chapters show the response of the n = 1 mode, and it is often helpful (more intuitive) to discuss them within the framework of a 1 21 -layer model. To do that, we must convert variables from the LCS solution (q1 and p1 ) to those for a 1 21 -layer model (q, p, d, and h). We first assign a value for H (typically 150–200 m). Then, (5.24) and (5.26) give q=

H1 q1 , H

p H1 p1 , = ρ1 H

d=

H1 p H1 p1 = d1 . = gρ1 H g H

(5.27a)

5.4 Potential Vorticity

181

For the 1 21 -layer solution to be comparable to the video solution, it must have the √ same wave speeds, so that c = g  H must equal c1 . From (5.24) and (5.27a), it follows that g  H = c12

⇒

h =

gp gH1 =  p1 = φ p1 = φgd1 ,  g ρ1 gH

(5.27b)

where φ = gH1 /c12 .

5.3.4 Multi-layer Models A similar development and linearization can be carried out for a 2-layer model. The linearized set has two vertical modes: one corresponding to the barotropic (n = 0) mode of the LCS model, and the other a close approximation to the baroclinic response of the 1 21 -model or the n = 1 mode of the LCS model. Thus, the 1 21 -layer model behaves like a 2-layer model, except without the barotropic response. These results can be extended to an arbitrary number of layers. Specifically, a linearized (n + 1)-layer model has a barotropic mode and n baroclinic ones, the latter analogous to the n baroclinic modes in a n 21 -layer model and the first n baroclinic modes in the LCS model. The isopycnal OGCMs mentioned at the end of Sect. 5.1 can be viewed as nonlinear models of this sort.

5.4 Potential Vorticity Equations of motion can be combined in various ways to define new variables (e.g., momentum and energy) that, in the absence of forcing and mixing, are “conserved” for a particular water parcel, that is, their values are unchanged as the parcel flows about the domain. One of those variables is potential vorticity (PV). Here, we derive an expression for PV in a layer model like (5.19). For simplicity, we set γ = γ  = 0, although a similar PV equation results when those mixing terms are included. (A version of PV can also be derived from OGCM equations like Eqs. 5.1, and is referred to as “Ertel” PV.) As is commonly done, it is convenient to define the operator Dq = qt + uqx + vq y , Dt

(5.28)

which defines the rate of change of variable q following a water parcel. Two useful corollaries of (5.28) are Dq Dr D + = (q + r ) , Dt Dt Dt

r

Dq Dr D +q = (qr ) . Dt Dt Dt

(5.29)

182

5 Ocean Models

Another useful identity is D Dt

   1 1 = u x + vy , h h

(5.30)

an expansion of continuity Eq. (5.19c) with γ  = 0. Setting γ = 0, derivatives of (5.19a) and (5.19b) with respect to y and x, respectively, are τ yx   px y = + νh ∇ 2 u y , (5.31) u yt + uu x y + vu yy + u y u x + v y − f v y − f y v + ρ¯ ρh ¯ y   p yx τx = + νh ∇ 2 v x , vxt + uvx x + vv yx + vx u x + v y + f u x + ρ¯ ρh ¯

(5.32)

and subtracting (5.31) from (5.32), gives     τ Dζ + (ζ + f ) u x + v y + v f y = curl + νh ∇ 2 ζ, Dt ρh ¯

(5.33)

where ζ ≡ vx − u y is “relative vorticity.” Since f is a function only of y, we can write v f y = f t + u f x + v f y = D f /Dt. Using this relation and the first of Eqs. (5.29), (5.33) becomes     D τ + νh ∇ 2 ζ. (ζ + f ) + (ζ + f ) u x + v y = curl Dt ρh ¯

(5.34)

Dividing by h, and using (5.30) and the second of Eqs. (5.29), then gives D Dt



ζ+ f h



  1 νh τ = curl + ∇ 2 ζ. h ρh ¯ h

(5.35)

The quantity PV = (ζ + f ) / h defines the potential vorticity of a water parcel. Note that D(P V )/Dt = 0 in the absence of forcing (τ = 0) and mixing (νh = 0), a statement that PV does not change its value (is conserved) following a water parcel. PV conservation has proven to be a useful concept in a variety of topics of ocean and atmospheric dynamics. In this book, we use it to discuss the properties of acrossequatorial flows. In the interior ocean, currents typically have a large spatial scale so that |ζ| | f |. In that case, PV ≈ f / h and it cannot be conserved as a water parcel crosses the equator since f changes sign; therefore, across-equatorial flow is possible only in regions where either forcing or mixing allows PV to change sign. In contrast, in a swift western-boundary current (like the EACC) where |ζ| ∼ | f |, across-equatorial flow is possible while conserving PV = (ζ + f ) / h because the change in f can be balanced by an opposite change in ζ (Anderson and Moore 1979).

Part III

Free Waves

Chapter 6

Overview

Abstract A general property of wind-forced solutions is that the ocean adjusts locally within the forcing region and remotely by radiating waves, the latter process being very prominent in the NIO owing to the variability of monsoon winds. In this overview chapter, we review general properties of all waves: their spatial structure, phase, and phase velocity; dispersion relation; group velocities in uniform and slowlyvarying media; and the impact of damping. The following two chapters then apply these concepts to waves that exist in the NIO, both at midlatitudes and near the equator. Keywords Plane versus general waves · Dispersion relation · Phase and group velocities · Propagation through uniform and variable media Although the wave concepts discussed here are generally applicable, waves considered in this book are of a specific kind: They are almost always the complete set of solutions to the unforced and inviscid (τ x = τ y = νh = κh = γn = γn = 0) version of the LCS modal Eqs. (5.16), u t − f v + px = 0,

(6.1a)

vt + f u + p y = 0,

(6.1b)

pt + u x + v y = 0, c2

(6.1c)

the exceptions being a few cases where some mixing terms are retained (e.g., Sect. 6.5). (For notational convenience, we delete subscripts n in Eqs. 6.1. We do so as well in all subsequent modal equations in the book, replacing them only if confusion results from their absence.) We discuss the waves Eqs. (6.1) allow (gravity, Rossby, and Kelvin waves) in detail in Chaps. 7 and 8.

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_6. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_6

185

186

6 Overview

6.1 Wave Structure All wave solutions are assumed to oscillate in time with a single frequency σ . There is, however, no loss of generality in this restriction, since an arbitrary time dependence can be constructed of packets of such oscillations. In addition, the basin is assumed to be unbounded, which allows waves to have a purely sinusoidal structure. The impact of boundaries on waves is considered in later chapters.

6.1.1 Plane Wave A familiar, idealized, wave solution is q(x, y, t) = Qo exp(ikx + iy − iσ t),

(6.2)

a “plane” wave, where q = u, v, or p. In (6.2), Qo is a constant amplitude, θ = kx + y − σ t is the wave’s phase, and σ , k, and  are constant values of frequency and zonal and meridional wavenumbers.1 Solution (6.2), however, is an exact solution only when parameters in the governing equations, which define the medium through which the wave propagates, are constant. (In Eqs. 6.1, for example, f varies with y so the medium is not “uniform,” and solution (6.2) is not exact.) Nevertheless, the concept of a plane wave is still useful for understanding general wave properties, as it provides a reasonable “local” approximation to the exact signal when the medium is “slowly varying” (Sects. 6.4, 7.3.3, and 15.2.1). Figure 6.1 illustrates a plane wave (left panel) and a more general wave with constant frequency σ but variable k and  (right panel).

6.1.2 General Wave A general representation of a wave solution is q(x, y, t) = Qo (x, y, t) exp [iθ (x, y, t)] ,

(6.3)

where θ is the wave’s phase, Qo is its amplitude, and both vary smoothly in space and time.

1

Throughout the book, we express waves (and forced solutions) in complex form. It is understood that the physically relevant response is either their real or imaginary part.

6.1 Wave Structure

187

Fig. 6.1 Schematic diagrams of waves with constant σ , showing a plane wave (left panel) and a more general wave with variable k and  (right panel). In both panels, phase increases to the northwest in the direction of warmer colors, with wave crests (θn = 2π n) and troughs indicated by solid and dashed curves, respectively. After a time equal to the wave period, P = 2π/σ , the plots are unchanged except that phase is everywhere increased by 2π (so that θn → θn+1 and so on). Arrows are centered on a particular wave crest, are oriented perpendicular to the wave crest in the direction of wave propagation (κ/κ), and their length is λ = 2π/κ. For the plane wave, arrows are the same throughout the basin, and their heads and tails extend to neighboring troughs. For the general wave, the direction and wavelength of the arrows depend on location and, hence, the heads and tails of arrows do not intersect troughs

6.1.2.1

Frequency and Wavenumber

The frequency and wavenumbers of the wave are then defined by σ = −θt ,

κ = ∇θ,

(6.4)

where κ = ki + j is the “wavevector” and i and j are unit vectors in the zonal and meridional directions (Whitham 1960; 1961). For plane wave (6.2), note that Eqs. (6.4) reproduce the constant values of κ and σ defined by its linear phase function, θ = kx + y − σ t. For a general wave with a non-planar θ , however, κ and σ are not constant but rather define “local” values that can change smoothly in space and time since θ does (right panel of Fig. 6.1). One consequence is that the heads and tails of arrows do not intersect troughs (right panel) as they do for a plane wave (left panel). Two useful properties follow from definitions (6.4). Taking the gradient of the first of Eqs. (6.4), the time derivative of the second, and combining the two expressions gives (6.5a) κ t + ∇σ = 0;

188

6 Overview

it describes a basic linkage between κ and σ , referred to as the “conservation of wavenumber” or “conservation of wave crests.” Taking the curl of the second of Eqs. (6.4) with the gradient operator ∇, and using the identity ∇ × ∇θ ≡ 0, gives ∇×κ =0

⇒

k y = x .

(6.5b)

In Sect. 6.4, we use both of these relations to derive expressions that determine how κ and σ change as a wave packet propagates through a variable medium.

6.1.2.2

Period and Wavelength

Common definitions of the period and wavelength of a wave, P = 2π/σ and λ = 2π/ |κ|, are the time it takes one wave crest (or trough) to be replaced by another and the minimum distance between crests, the latter property indicated by the arrows extending to adjacent troughs in the left panel of Fig. 6.1. For a general wave in which P and λ vary spatially and in time, the visualizations in the previous sentence are no longer correct, as indicated by arrows not extending between troughs in the right panel. Thus, although P and λ are still useful concepts, for a general wave they are accurate only where P and λ vary slowly between crests.

6.1.2.3

Phase Velocity

The phase velocity of (6.3) is the speed at which θ propagates in the direction perpendicular to θ isolines that points toward increasing θ , that is, in the direction of ∇θ . Suppose isoline θ shifts a distance ξ in that direction in time t. With the aid of (6.4), the general change in θ caused by arbitrary ξ and t is θ = θx x + θ y y + θt t = κ·ξ − σ t = κξ − σ t, where ξ = x i + y j , κ = |κ|, and ξ = |ξ |. Because we are following isoline θ , however, θ = 0 so that the preceding equation implies that κξ = σ t. Then, the instantaneous propagation speed of isoline θ is given by limt→0 ξ/t = σ/κ, and the phase velocity is cp =

σ ki + j σκ = , κκ κ κ

(6.6)

where κ/κ is the unit vector pointing in the direction of κ.

6.1.3 Other Waves We also consider waves with different forms than (6.2). For example, as written above both k and  are real numbers (a trigonometric wave), but one or the other can be imaginary or complex (an evanescent wave). The coastal Kelvin wave, which

6.3 Wave Groups in a Uniform Medium

189

propagates alongshore and decays offshore, is an important example of an evanescent wave (Sect. 7.2). We also consider one-dimensional (1-d) waves, in which either k or  is zero in (6.2) or, for equatorial waves, the factor exp (iy) is replaced by a Hermite function (Chap. 8). Finally, we allow for vertically propagating waves, in which exp(ikx + iy) → exp(ikx + imz) in (6.2), where m is a vertical wavenumber (Sect. 16.1).

6.2 Dispersion Relation The frequency and wavenumber in (6.4) are dynamically linked by the governing equations of motion. The wave dispersion relation, σ = σ (κ, x, t)

(6.7)

with x = xi + yj, describes that linkage.2 The dispersion relation provides a “biography” of all the waves in the system, describing their basic properties; in particular, it determines their phase and “group” velocities, the latter discussed next. Note that σ can be a function of x and t as well as κ, dependencies that arise when waves propagate through a medium that is spatially non-uniform and varies in time. In Eqs. (6.1), the medium varies spatially since f is a function of y but is independent of t, so that the dispersion relation has the form σ (κ, y). In later chapters, we obtain σ (κ, y) by first solving Eqs. (6.1), or approximations to them, for a single equation in v or p, and then inserting a wave form like (6.2) into that equation. If the medium is uniform ( f is constant), this procedure gives an exact expression for σ (κ). If it is non-uniform, an exact expression for σ (κ, y) is generally not possible; however, useful approximations can be found either with the restriction that the medium is “slowly varying” (Sects. 6.3, 7.3.3, and 15.2.1) or by replacing Eqs. (6.1) with an approximate set (Sects. 7.2.2 and Chaps. 12, 13, and 14).

6.3 Wave Groups in a Uniform Medium To introduce the concept of group velocity, consider a spatially confined patch of radiation in an unbounded ocean and when the medium is uniform ( f is constant in Eqs. 6.1). Under these conditions, plane waves are theoretically possible, and the patch can be viewed as a packet (superposition) of many plane waves with different wavenumbers and frequencies, which interfere destructively in the far field. For such a packet, a sensible question to ask is: To where does the overall packet propagate 2

For notational convenience, here and elsewhere we sometimes summarize the arguments of variables, and other quantities, in vector form, that is, (6.7) σ (κ, x) is shorthand for σ (k, , x, y), and so on.

190

6 Overview

as time progresses? (See Sects. 6.2 and 6.3 of Vallis 2017, for similar discussions of this question, both for a uniform medium and the slowly-varying one that we discuss in the next subsection.) There is no insightful answer for a packet composed of a random set of waves, as the packet disperses randomly in time. On the other hand, there is one when the packet is dominated by waves centered on a single frequency σo and wavevector κ o = ko i + o j. More precisely, there is an insightful answer when the initial packet has the form (6.8) q(x, 0) = Q(x) exp (iκ o ·x) , provided the width scale of amplitude Q(x), L, is large with respect to a wavelength of the wave (i.e., L  λo = 2π/κo ) so that many wave cycles are contained in the packet. To introduce time dependence, we first take the Fourier transform of (6.8) to get  ∞ ∞ q(x, 0) exp (−iκ · x) d x q(κ, ˜ 0) =  ∞  ∞ −∞ −∞   ˜  ), Q(x) exp −iκ  ·x d x ≡ Q(κ =

(6.9)

−∞ −∞

where d x = d xd y and κ  = κ − κ o . (Equation 6.9 is a 2-d version of the 1-d trans˜  ) is the transform of the plane form discussed in Sect. 9.3.1.) Variable q(κ, ˜ 0) = Q(κ wave in the packet with wavenumber κ at time t = 0. From the dispersion relation, we know that the wave oscillates at frequency σ (κ). Therefore, ˜  ) exp [−iσ (κ)t] , q(κ, ˜ t) = Q(κ

(6.10)

is its transform at other times.3 The inverse transform of (6.10), 1 q(x, t) = 4π 2



∞ −∞



∞ −∞

˜  ) exp [iκ · x − iσ (κ)t] dκ, Q(κ

(6.11)

where dκ = dkd, then describes the response of the packet at all times. Because σ (κ) is usually a complicated function of κ, it is generally not possible to evaluate the integral in (6.11) exactly.  On the  other hand, with the restriction that   ˜   −1   L  λo ( κ  L ) it follows that Q(κ ) is appreciable only when κ is close

3 A caveat with this step is that σ (κ) often allows more than one σ for a given κ. In such cases, information other than just q(x, 0) is needed to specify the value of σ in (6.10). For example, this property holds for the gravity/Rossby dispersion relation plotted in Fig. 7.1. Imagine drawing a line of constant κ (a vertical line) anywhere in the plot. It intersects both the gravity-wave (upper) and Rossby-wave (lower) curves, and those intersections provided two possible σ values for κ. To apply the above procedure, then, requires knowing whether q(x, 0) is composed of gravity or Rossby waves.

6.3 Wave Groups in a Uniform Medium

191

to κ o ,4 , that is, only waves with wavenumbers close to κ o contribute significantly to q(x, t). With this restriction, we can expand σ in a Taylor series about κ o to get σ = σ (κ o ) + σk (κ o )k  + σ (κ o ) + · · · = σo + κ  · cgo + · · · ,

(6.12)

where σo = σ (κ o ), cgo = σk (κ o ) i + σ (κ o ) j, and higher-order terms are negligible. Keeping only the indicated terms in (6.12), (6.11) becomes  ∞ ∞   1 ˜  ) exp iκ · x − iσo t − iκ  ·cgo t dκ Q(κ q(x, t) = 2  ∞4π  ∞ −∞ −∞    1 ˜  ) exp iκ  · x − cgo t dκ  exp i (κ o ·x −σo t) Q(κ = 4π 2 −∞ −∞ = Q(x − cgo t) exp i (κ o ·x −σo t) ,

(6.13)

where the integration variable is changed from κ to κ  = κ − κ o in the second line. According to (6.13), the phase speed of the waves in the packet is still c p = σo /κo , whereas its amplitude, |q(x, t)| = Q(x − cgo t), (6.14) propagates at the velocity cgo unchanged in shape. The latter property happens because we dropped the extra terms in (6.12): If those terms are retained, |q(x, t)| gradually distorts from its original shape as the packet propagates, with the rate of distortion depending on how well the inequality, L  2π/κo , is satisfied. Given the form of (6.14), it is common to say that the amplitude (energy) of the packet propagates along a “ray path,” in this case defined by any of the lines, x r = x o + cgo t,

(6.15)

where x o is any location within the packet at t = 0. In summary, the amplitude of a group of waves with dominant wavevector κ and frequency σ (neglecting the arbitrary subscript “o”) propagates at the velocity cg = σk i + σ j = ∇k σ,

(6.16)

defining its “group velocity.” Its direction is the same as that of the gradient of σ in k,  space, that is, it is perpendicular to lines of constant σ and points toward increasing σ . Comparing (6.6) and (6.16) it is clear that generally cg = c p . Indeed, only in the special case that σ = aκ, where a is a constant, does cg = c p . Waves for   That Q˜ (κ  ) = q(k, ˜ 0) is narrow when q(x, 0) is broad (κ    L −1 ) is an inherent property of a function and its Fourier transform. To illustrate, let q(x) = exp(iko x) exp(−x 2 /L 2 ), so that the envelope of the wave   √  is a simple  Gaussian. It is straightforward to show that its Fourier transform is q˜ k  = π L exp − 41 k 2 L 2 where k  = k − ko . The inverse relationship is clear from the form   of   the  two−1expressions: q broadens and q˜ narrows as L increases, and hence q˜ k is narrow when k   L . 4

192

6 Overview

which cg = c p (cg = c p ) are referred to as dispersive (non-dispersive) waves. Note that for non-dispersive waves, there are no extra terms in (6.12), so that the wave packet never distorts.

6.4 Impact of a Slowly-varying Medium Now, consider the propagation of radiation through a medium that changes spatially so that σ = σ (κ, x). Following Whitham (1960, 1961), we introduce group velocity in a more general way than in Sect. 6.3. Rather than focusing on the entire packet, we ask the questions: To where does an individual wave in the packet with wavenumber κ and frequency σ (κ, x) propagate, and how are its properties (κ and σ ) modified as it does? One advantage of focussing on these “local” questions is that the groupvelocity concept can be shown to apply in situations other than an isolated radiation patch, as was assumed in Sect. 6.3. More importantly, another is that it is not necessary to impose the restriction that κ remains close to κ o . Let L m be the space scale over which the medium varies and λˆ be a typical wavelength of a wave in the neighborhood of any point Pˆ in the packet. Then, provided that λˆ  L  L m the medium is approximately uniform (varies slowly) ˆ they for individual waves within the packet. Therefore, in the neighborhood of P, can be represented as plane waves of the form (6.3) with κ and σ defined by (6.4) for which relations (6.5) hold. Expanding ∇σ (κ, x) in (6.5a) into ∇σ = σk ∇k + σ ∇ + ∇σ |k, , writing each component of the resulting expression separately, and using (6.5b) to modify some terms, leads to kt + cg ·∇ k = −

 ∂σ  , ∂ x k,

t + cg ·∇  = −

 ∂σ  . ∂ y k,

(6.17a)

Taking the dot product of (6.16) with (6.5a) and using the identity cg ·κ t = σk kt + σ t = σt , gives (6.17b) σt + cg ·∇ σ = 0.   Expressions of the form, ∂t + cg ·∇ q, q = k, , or σ , define the rate of change of q in the direction of cg , that is, along ray paths x r defined by the time integral of x r t = cg .

(6.18)

In this case, because cg isn’t constant (i.e., it isn’t evaluated only for κ = κ o as in Eq. 6.15), ray paths x r are curves rather than straight lines. ˆ Equations (6.17) describe how its σ and Consider the wave that originates at P. κ change along x r . According to (6.17b), σ remains constant along x r . Further, in all the examples considered in this book σ (κ, x) does not depend on x so that −∂σ/∂ x|k, = 0 in the first of Eqs. (6.17a), and k is also constant along x r . On the

6.5 Impact of Mixing

193

other hand, we do consider solutions in which σ (κ, x) depends on y (when f varies), so that  and hence cg change as the wave propagates meridionally (Sects. 7.3.3 and 15.2.3). Finally, we note that in some realistic situations, the medium changes rapidly enough to violate the inequality λo  L m . (For example, as noted in Sect. 16.1 this situation can occur for vertically-propagating waves as they propagate through a strong pycnocline.) In that case, the impact of the sharply varying medium is that only some of the packet propagates through the sharp variation, with the rest reflecting from it.

6.5 Impact of Mixing Mixing broadens and weakens (damps) wave packets. Let the time scale of the damping be td . If the mixing is strong enough for td to be of the order of or less than the period of the waves in the packet (td  2π/σo ), then all properties of the inviscid waves, and hence wave groups, are lost. On the other hand, if mixing is weak (td  2π/σo ), as is usually the case in the open ocean, inviscid wave properties are still apparent and the wave simply decays gradually in the direction of cg . To illustrate the impact of mixing in solution (6.13), we retain the vertical-mixing terms of (5.16) in Eqs. (6.1) by the replacement qt → qt + iγ q, where q = u, v, or p and set γ = γn = γn . Then, the dispersion relation is altered only by an additional imaginary term iγ , that is, σ = σ (κ, x) + iγ . With this replacement, (6.12) becomes σ = σo + iγ + κ  · cgo + · · · ,

(6.19)

and the resulting wave amplitude is |q(x, t)| = Q(x − cgo t) exp(−γ t).

(6.20)

According to (6.20), for small damping (γ  σo ) the packet simply decays slowly in time as it propagates. For large damping (γ  σo ), however, the packet decays so rapidly that it loses its wave properties. This situation always occurs for high-order vertical modes, because γn = A/cn2 increases with n owing to the decrease in cn .

Chapter 7

Midlatitude Waves

Abstract Free-wave solutions for a mode of the LCS model are obtained that are valid at midlatitudes (away from the equator). The dispersion relation for Rossby waves is derived when the Coriolis parameter f is constant, and under a realistic restriction it is shown to be valid even when f varies. The concepts of critical frequency σcr and critical latitude θcr are introduced. Kelvin waves are found along both zonal and meridional coasts. Along a meridional coast and when f varies, Kelvin waves exist only poleward of θcr (β-plane Kelvin waves); as for f -plane Kelvin waves, β-plane Kelvin waves decay offshore, but they also have a weak westward propagation. Equatorward of θcr , Rossby waves radiate offshore along ray paths that are directed meridionally, as well as westward. Similar properties exist even if the coast if the coast is slanted (i.e., not directed precisely north-south), the major difference being that the value of θcr is decreased. Keywords Rossby/gravity waves · Constant- f and β-plane Kelvin waves · Phase and group velocities · Critical frequency and latitude · Meridional energy propagation · Slanted coasts Solutions to (6.1) are difficult to find analytically because f is a sinusoidal function of y ( f = 2 sin θ, θ = y/Re ,  = 2π day−1 , and Re is the radius of the earth). Here, we find approximate solutions for “midlatitude” waves, that is, for waves obtained using approximations that break down near the equator. We first consider gravity and Rossby waves in the open ocean, which are found by either simplifying f to be constant or applying the Wentzel-Kramers-Brillouin (WKB) approximation (Sect. 7.1). Next, we discuss coastal Kelvin waves along zonal and meridional boundaries (Sect. 7.2), Rossby waves generated at meridional boundaries (Sect. 7.3), and then extend the discussions to allow for coasts slanted at an arbitrary angle (Sect. 7.4). To conclude, we comment on observations of these waves and their properties in the NIO (Sect. 7.5).

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_7. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_7

195

196

7 Midlatitude Waves

7.1 Gravity and Rossby Waves A useful starting point for deriving gravity- and Rossby-wave solutions, both at midlatitudes and near the equator, is to rewrite Eqs. (6.1) in an alternate form. First, solve (6.1a) and (6.1c) for equations that express u and p in terms of v, to get u tt − c2 u x x = f vt + c2 v yx ,

(7.1a)

ptt − c2 px x = −c2 f vx − c2 v yt .

(7.1b)

Then, multiplying (6.1b) by the operator ∂tt − c2 ∂x x and using (7.1a) and (7.1b) gives f2 1 (7.1c) − 2 vttt + vx xt + v yyt − 2 vt + βvx = 0, c c where β = f y . The advantage of Eqs. (7.1) is that (7.1c) can be solved for v, and then u and p are also known in terms of v. Since no approximations are made in deriving Eqs. (7.1), they are almost always an equivalent set to (6.1). The exception is for a Kelvin wave along a zonal boundary (Sect. 7.2.1) for which v ≡ 0; in this case, (7.1c) vanishes, and equation set (7.1) is degenerate. In most of this section, we avoid the difficulty of f being variable in Eqs. (7.1) ˆ (This by replacing f and β with their constant values at a particular midlatitude θ. specification is commonly referred to as the “mid-latitude β-plane approximation.” In this book, however, we use that label to apply to Eq. C.2, an approximation in which f varies linearly about θˆ = yo .) The advantage of this approach is that, because f and β are constants, all the coefficients in (7.1c) are constant; consequently, general wave solutions for u, v, and p are plane waves of the form (6.2), allowing key properties of gravity and Rossby waves to be derived in a simple mathematical framework. In the low-latitude NIO, however, the magnitude of f varies significantly. Later in the section, then, we show that under reasonable restrictions wave properties derived for constant f and β are valid even when they vary (Sect. 7.1.3). To conclude, we introduce the important concept of the “critical latitude” of gravity and Rossby waves (Sect. 7.1.4).

7.1.1 Dispersion Relation Inserting (6.2) into (7.1c) gives   σ2 f2 σ k 2 + 2 − 2 + 2 + kβ = 0, c c

(7.2a)

7.1 Gravity and Rossby Waves

197

Fig. 7.1 A 3-d rendering of the mid-latitude dispersion relation at θˆ = ±15◦ , showing σ(k, ) as functions of k/α and α, where α = | f | /c is the inverse of the Rossby radius of deformation. The curves form circular “bowls,” with the upper (high-frequency) bowl (top panel) depicting the dispersion relation for gravity waves and the lower (low-frequency) one that for Rossby waves (bottom panel)

the dispersion relation for the system. To distinguish gravity and Rossby waves, it is useful to write (7.2a) in the alternate form   β 2 k+ + 2 = r 2 , 2σ

 r = 2

β2 σ2 + 4σ 2 c2

 −

f2 . c2

(7.2b)

According to (7.2b), for each σ the dispersion relation consists of a circle of radius r in the k- plane with its center at k = kc = −β/ (2σ),  = 0. Our focus here is on trigonometric waves (both k and  are real). They are possible only when r 2 > 0, which is satisfied in two situations: for sufficiently large σ (gravity waves) when the term σ 2 /c2 dominates the term in parentheses in r 2 , and for sufficiently small σ (Rossby waves) when β 2 /4σ 2 dominates. Equation (7.2a) also allows evanescent waves (one or both of k and  are complex), and we comment on them at the end of this subsection. Figure 7.1 plots a 3-d rendering of the resulting dispersion curves, σ(k, ), when θˆ = ±15◦ .1 The curves form circular “bowls,” with the upper (high-frequency) bowl depicting the dispersion relation for gravity waves and the lower (low-frequency) inverted bowl that for Rossby waves. To help with the visualization of σ (k, l), Note that Eqs. (7.2a) and (7.2b) are unchanged with the replacements σ → −σ, k → −k, and  → −. As a result, dispersion curves for σ < 0 are the mirror images of the ones for σ > 0, and hence provide no new information. Following common usage, then, in Fig. 7.1 and all other dispersion-curve figures in this book, σ is plotted only for its positive values. 1

198

7 Midlatitude Waves

Fig. 7.2 (top panels) Horizontal sections across Fig. 7.1 for two σ values, one each in the gravity (left) and Rossby (right) bowls. The curves are circles and the black dots designate their centers. Arrow pairs for each color indicate directions of the phase and group velocities for each wave (Sect. 7.1.2). The gravity-wave frequency (σ  = 1.01) is close to the bottom of the bowl, since otherwise the two arrows for each wave are not visually distinct. (bottom panels) Vertical sections across Fig. 7.1 along  = 0, showing curves for the gravity (left) and Rossby (right) bowls. Line pairs for each color indicate the direction and amplitude of the phase and group velocities for the wave at point (σ, k) indicated by a black dot: the former by the lines that extend from (0, 0) to (σ, k), and the latter by lines that are tangents to the curves (Sect. 7.1.2). In the right panel, the horizontal, dashed line intersects the dispersion curve at two locations, indicating a long- and shortwavelength, Rossby-wave pair. The slanted dashed line plots the Rossby-wave dispersion curve in the long-wavelength limit (Eq. 7.7)

Fig. 7.2 plots sections across Fig. 7.1 for fixed σ (top panels) and along  = 0 (bottom panels), for the gravity (left panels) and Rossby (right panels) bowls. One striking difference between the gravity and Rossby bowls is that the former exists at frequencies greater than f whereas the latter have much lower frequencies. Another is that the gravity bowl is centered near the origin whereas the Rossby bowl is contained entirely in the region where k < 0. These properties are apparent in (7.2b) when  = 0. Since the center of each circular dispersion curve is at kc = −β/ (2σ), it fol-

7.1 Gravity and Rossby Waves

199

lows that: r > |kc | when σ > | f | so that the circular dispersion curve necessarily extends into the k > 0 region; conversely, r < |kc | when σ < | f | and hence k < 0 always.

7.1.1.1

Critical Frequencies

The two extremum (critical) frequencies are the values of σ (σG and σ R ) at the center of the bowls where the radii of the circles shrink to zero. Setting r 2 = 0 in (7.2b) yields a quadratic equation for σ 2 , which has the roots ⎡  1 ⎤ 2 2  2 2 1 2 R ⎦, σG , σ R = f ⎣1 ± 1 − 2 Rˆ e2

(7.3)

where Rˆ e = | f | /β = Re tan θˆ is the distance to the earth’s axis at latitude θˆ and (replacing neglected subscripts n) R → Rn = cn / | f | is referred to as the “Rossby radius of deformation” of the nth vertical mode. At midlatitudes, Rn / Rˆ e is a small number for all the baroclinic modes; for example, at θˆ = 30◦N and with cn ≈ 250/n cm/s (a rough but reasonable approximation), Rˆ e ≈ 3700 km, Rn ≈ 34/n km, and Rn / Rˆ e ≈ 0.009/n. We can then simplify the radical in (7.3) using the approximation √ 1 + ξ ≈ 1 + 21 ξ, ξ  1, to get  σG2

= f

2

R2 1− 4 Rˆ e2

≈ f 2,

σ 2Rn ≈

R2 2 β 2 cn2 f = . 4f2 4 Rˆ e2

(7.4)

Thus, the two frequencies are well separated, that is, σ Rn  σG . In contrast to σG , σ Rn varies with n through its dependence on the characteristic velocity cn ; however, for notational convenience we usually drop the subscript (σ Rn → σ R ) unless there is a specific need to keep it.

7.1.1.2

Gravity-Wave and Rossby-Wave Limits

Because the gravity- and Rossby-wave dispersion curves are so well separated in frequency, it is possible to split the complete dispersion relation into two parts, one for each wave type.   For gravity waves, in the second of Eqs. (7.2b) the term β 2 / 4σ 2  σ 2 /c2 because σ > σG and σ Rn  σG ; in the first equation, with the additional restriction that |k| Rˆ e−1 , it follows that |k| β/ (2σ) since σ > σG . Neglecting both terms in (7.2b), the dispersion relation simplifies to f2 σ2 2 2 = k +  + , c2 c2

|k| Rˆ e−1 ,

(7.5)

200

7 Midlatitude Waves

the gravity-wave dispersion relation on the f -plane (β = 0). Equation (7.5) describes the upper bowl in Fig. 7.1 accurately everywhere except very near its bottom, where |k| ≈ Rˆ e−1 . The resulting error is minor: Rather than its center being located at σ =   σG = | f | 1 − 1 R 2 / Rˆ e2 ≈ | f | 1 − 1 R 2 / Rˆ e2 and k = −β/ (2σG ) ≈ − Rˆ e−1 /2 in 4

8

(7.3), it is shifted in (7.5) to σ = | f | and k = 0. Note that both (7.2a) and (7.5) allow |σ| to increase indefinitely with |k| and ||. That property is a consequence of the imposition of the hydrostatic approximation in Step 2 of Sect. 5.2.1: If we had instead retained the wt term at this step (so that ¯ the gravity-wave dispersion curves Eq. 5.1c is replaced by wt + pz /ρ¯ = −ρg/ρ), do not allow gravity waves for σ > Nb . Since in this book we are only interested in solutions for which σ  | f |  Nb (or, for switched-on winds, at times much larger than Nb−1 ), the resulting error in (7.5) is negligible.    For Rossby waves, σ 2 /c2 < σ 2R /c2 = 1 R 2 / Rˆ e2 f 2 /c2  f 2 /c2 . Neglecting 4

σ 2 /c2 in (7.2a), gives σ=−

k2

kβ , + + f 2 /c2 2

(7.6)

the well-known dispersion relation for Rossby waves. For a given σ and , there are Rossby waves for two values of k. This property is illustrated in the bottom-right panel of Fig. 7.2, where the horizontal dashed line intersects the dispersion curve at two locations, k1 and k2 . The two waves are referred to as long- and short-wavelength Rossby waves for k1 and k2 , respectively. The value of k that divides the two types occurs where r = 0, that is, at the critical wavenumber k R = −β/ (2σ R ) ≈ R −1 , that is, the reciprocal of the Rossby radius. Note that (7.6) implies that on the f -plane (β = 0) there are no Rossby waves, a property also apparent in both (7.2a) and (7.2b).

7.1.1.3

Evanescent Waves

In addition to trigonometric, gravity and Rossby waves, dispersion relation (7.2b) also allows evanescent waves in the frequency band, σ R < σ < σG , where r 2 < 0 and either k, , or both are complex. Such “evanescent Rossby/gravity waves” exist only in solutions that are bounded in some way, since otherwise they increase indefinitely in the direction opposite to the decay. Examples considered in this chapter (and elsewhere) occur when the wind forcing is bounded (Videos 7.3a and 7.3b) and along a meridional boundary (Sect. 7.2.2). They also exist along the edge of a region of trigonometric Rossby/gravity waves (Sect. 8.5).

7.1 Gravity and Rossby Waves

201

7.1.2 Phase and Group Velocities The arrows in the top panels of Fig. 7.2 point in the directions of the phase and group velocities for trigonometric, gravity and Rossby waves associated with several pairs of k and . According to (6.6), the direction of c p is the same as that of wavevector κ = (k, ), which is indicated by the arrows that extend from the origin to points on the circles. It follows from (6.16) that the direction of cg is the same as that of ∇ κ σ, which is directed perpendicular to the circles and points toward increasing σ; thus, arrows for the group velocity (thick) point away from the center of the circle for gravity waves and toward the center for Rossby waves. Because the center of the circle is so close to the origin for gravity waves, the directions of their phase and group velocities are almost always indistinguishable. (The exception occurs only for gravity waves very near the bottom of the bowl with σ ≈ f and k > kc , as illustrated in top-left panel of Fig. 7.2 for which σ/ f = 1.01.) In contrast, they can be quite different for Rossby waves; in particular, the zonal components of cg and c p , cgx and c px , have the same (opposite) directions for long-wavelength (short-wavelength) Rossby waves, a property that is particularly apparent for small  (Fig. 7.2, top-right panel). Finally, note that c px for Rossby waves is always directed westward (k < 0), whereas that for gravity waves can be either westward or eastward. The slopes of the lines in the bottom panels of Fig. 7.2 are the values of the zonal components of the phase and group velocities when  = 0. The phase speed of a wave with wavenumber k and frequency σ is c p = σ/k, the slope of the line that extends from (0, 0) to (σ, k). Its group speed, cg = σk , is the slope of the line tangent to the dispersion curve at the point (σ, k). For gravity waves, c px is infinite when k = 0 and decreases to ±c as k → ±∞. For Rossby waves, c px attains a finite maximum as k → 0 and decreases to 0 as k → −∞. For both wave types, cgx vanishes at the apex of the curves where k = kc . For gravity waves, cgx is directed eastward (westward) for k > kc (k < kc ), and vice versa for Rossby waves. Gravity and Rossby waves are dispersive since c p and cg are never exactly equal. On the other hand, gravity waves are nearly non-dispersive in the limit κ | f | /c, in which case (7.5) becomes σ = κc and c p = cg = (σ/κ) κ. Likewise, longwavelength Rossby waves are nearly non-dispersive in the limit |k| , ||  | f | /c, when (7.6) reduces to βc2 σ = − 2 k, (7.7) f κ = ki, and c p = cg = −βc2 / f 2 i. A visual indication of these properties in Fig. 7.2 (bottom panels) is that the slopes of the phase- and group-velocity lines of gravity and Rossby waves become nearly parallel in the above limits, the latter tending toward that of the slanted, dashed line (bottom-right panel). As discussed in later chapters, nearly non-dispersive Rossby waves play a prominent role in the largescale ocean circulation, among other things allowing the ocean to adjust to Sverdrup balance (Chaps. 12, 13, and 14). Note that their phase and group velocities increase equatorward (because of the f −2 in Eq. 7.7); this increase is obvious in observed waves and in many of the videos discussed in the book.

202

7 Midlatitude Waves

7.1.3 Extension to Variable f In the real ocean where f varies, solutions to (7.1c) are not plane waves. Since dispersion relation (7.2a) was derived for plane waves, its validity and all the wave properties derived from it are in question. On the other hand, f changes slowly on a spatial scale of the order of the earth’s radius Re , and ocean waves typically have a much smaller meridional wavelength λ (λ  Re ). Under these conditions, we can expect (7.2a) still to be valid to a good approximation, with wave properties changing smoothly in response to changes in f (Sect. 6.4). In support of this idea, properties of observed, off-equatorial, waves tend to fit those determined from (7.2a) very well regardless of latitude. To show that (7.2a) is valid even when f varies, we obtain an approximate solution to (7.1c) using the WKB approach (Bender et al. 1999; Ghatak et al. 1991). We look for the solution in the general form v(x, y, t) = V( y) exp[iθ(y)] exp(ikx − iσt),

( y) = θ y ,

(7.8)

where V and  are the wave’s amplitude and meridional wavenumber that can vary in y (Sect. 6.1). In (7.8), we introduced factor (defined below) to indicate that V and  are assumed to vary slowly with y. Inserting (7.8) into (7.1c) and reordering terms gives  V − 2i V y − i  y V − V yy 2

2

  f2 σ2 kβ 2 V ≡ F (y) V. (7.9) = −k − 2 + 2 − c c σ

Because the meridional scale of variations in  and V is Re (since it is determined by the variation in f ), the magnitudes of V y ,  y , and V yy are O(V/Re ), O(/Re ), and O(V/Re2 ), respectively. Then, setting  = 2π/λ it follows that, relative to the first term in (7.9), the terms proportional to V y and  y are O( ), where = λ/ (2π Re ), and the V yy term is O( 2 ). We look for a solution to (7.9) as a perturbation expansion. Neglecting the very small O( 2 ) term and separately balancing the O(1) and O( ) terms gives  2 = F

⇒

θ=±

y



F(y  ) dy  + θo , 

and 2iV y + i y V = 0

⇒

V = Vo

o , 

(7.10a)

(7.10b)

√ where θo and Vo o are constants of integration. Inserting these expressions for θ and V into (7.8) gives

7.1 Gravity and Rossby Waves

 v(x, y, t) = Vo

203

  o exp ±i 

y

    F(y ) dy exp(ikx − iσt + iθo ),

(7.11)

the approximate solution to (7.1c) accurate to O( ). Since our focus here is on trigonometric waves, we have assumed that F > 0 so that  is real in (7.11). For some values of k, σ, and f , though, F < 0 and then (7.11) describes a wave that grows or decays in y, rather than oscillates (see Sect. 8.5). For our purposes, the important property of (7.11) is expressed by the first equation of (7.10a): It is dispersion relation (7.2a) except that now  varies in y owing to the y-dependence of f and β in F. For waves that satisfy  1 (λ  2π Re ), then, all the constant- f wave properties derived above still apply locally even when f varies. 1 Note that wave amplitude V is not constant but varies with latitude like − 2 , so that (7.11) breaks down (goes to ∞) when  → 0. This breakdown is expected for the WKB solution, since the expansion parameter = λ/ (2π Re ) = 1/ (Re ) is not small in that limit. This problem is not serious, however, because it can be avoided using more sophisticated approximation methods and it does not occur in exact solutions (Ghatak et al. 1991; Sect. 8.5).

7.1.4 Critical Latitudes Consider a wave at a fixed frequency σ and a wavelength that satisfies λ  2π Re , so that (7.2b) is valid for variable f . Then, as noted above, trigonometric waves exist only if r 2 > 0, which is equivalent to the inequality f2
f , that is, only equatorward of a “critical latitude” where f = 2 sin θG = σ, θG = sin−1

σ . 2

(7.13a)

Similarly, for Rossby waves the last term in (7.12) is negligible and Rossby waves exist only equatorward of the critical latitude where f = 2 sin θ Rn = cn 2 cos θ Rn /(2Re σ), cn θ Rn = tan−1 . (7.13b) 2Re σ As for σ Rn , θ Rn varies with n through its dependence on cn , but we usually drop the subscript (θ Rn → θ R ) unless it is needed for clarity. Note that the concepts of critical

204

7 Midlatitude Waves

latitudes and frequencies are really different interpretations of the same inequality: the latter solving (7.12) for σ assuming f is fixed, and the former solving it for f for fixed σ. Solutions and videos in this book utilize two approximations for f : the midlatitude and equatorial, β-plane approximations (Eq. C.2 and neighboring text). In the former case, f = f o + βo (y − yo ) where f o and βo are f and β at an arbitrary latitude yo , and the critical latitudes are θG =

| f o | − βo |yo | σ − , Re βo Re βo

θ Rn =

| f o | − βo |yo | cn − . 2Re σ Re βo

(7.14)

In the latter, f = β y with β = 2/Re , they are θG =

σ , Re β

θ Rn =

cn . 2Re σ

(7.15)

Finally, θG and θ R expressed as distance from the equator rather than as angles are yG = R e θ G ,

y Rn = R e θ R ,

(7.16)

and usually y Rn → y R . Throughout the text, we refer to any of the above expressions (yG and y Rn as well as θG and θ R ) as “critical latitudes.” Figure 7.3 plots θG (magenta curve) and θ Rn as a function of P = 2π/σ, the latter shown for the n = 1–3 baroclinic modes (blue, red, and green curves) both when f is exact (solid) and when it is given by the equatorial β-plane approximation (dashed). Trigonometric waves of both types exist only equatorward of the curves (i.e., for |θ| < θG and θ Rn ). To illustrate the meaning of θG , suppose that the wave period is P = 1 or 2 days (10 or 20 days on the axis). Then, (7.13a) implies that gravity waves exist only at latitudes equatorward of θG = 30◦ or 14.5◦ , since only in those regions is σ > f . At larger periods, gravity waves are increasingly confined to the equator. (See Chap. 8 for a proper discussion of these equatorially-trapped, gravity waves). The dependence of θ Rn on P significantly impacts current variability in the northern NIO where θ  25◦N (horizontal dashed line), roughly the location of the NIO northern boundary. It indicates that n = 1 Rossby waves exist everywhere in the NIO at semiannual and annual periods (middle and right, vertical, dashed lines), but only equatorward of 9.2◦ at intraseasonal time scales (P < 60 days; left of the left-most, vertical, dashed line). For higher n values, Rossby waves become increasingly equatorial confined. The closeness of the θ Rn curves for both versions of f equatorward of 25◦N indicates the accuracy of the equatorial β-plane approximation in the NIO.

7.1 Gravity and Rossby Waves

205

Fig. 7.3 Critical latitudes for gravity (θG ; magenta curve) and Rossby (θ Rn ; blue, red, and green curves) waves as a function of wave period P = 2π/σ. For the Rossby waves, θ Rn curves are shown for the values of c1 (θ R1 ; blue), c2 (θ R2 ; red), and c3 (θ R3 ; green) listed in the second block of Table 5.1, and when f is exact (solid) and given by the equatorial β-plane approximation (dashed). Values of P for θG are 10 times smaller than indicated on the axis, that is, the axis extends from 0 to 40 days. The vertical dashed lines indicate where P = 60, 180, and 365 days. The horizontal dashed line is roughly the latitude of the northern boundaries of the Arabian Sea and Bay of Bengal

7.1.5 Videos Videos 7.1–7.2b show numerical solutions to Eqs. (C.1) when c = c1 and f is either constant (Videos 7.1 and 7.2a) or is specified by the midlatitude β-plane approximation (Eq. C.2; Video 7.2b). The solutions are all forced by initially imposed, circular pressure disturbances p located in the middle of the basin, which thereafter develop freely. Essentially, p is a packet of free waves at all possible frequencies and wavenumbers, and the videos follow the subsequent spreading of those waves throughout the domain. In Video 7.1, the initial pressure field is very narrow with a radius of r = 1◦ . Gravity waves radiate away from the initial disturbance, and eventually reflect from basin boundaries to propagate back into the interior of the domain. Consistent with the group-speed properties discussed above, gravity waves with shorter wavelengths spread more rapidly: The leading wave front is composed of the shortest wavelengths (of the order of r ) and it advances at a speed close to c1 . Subsequently, oscillations with increasingly longer wavelengths radiate from the basin center, as indicated by the expanding circles with different green shadings; these oscillations are weak because

206

7 Midlatitude Waves

disturbances with long wavelengths are not a significant part of the narrow, initial pressure field. After their radiation, an anticyclonic geostrophic current is present, circulating about the center of the initial disturbance, but that property is blurred by the gravity-wave reflections from the domain boundaries. The process in which steady, geostrophic flows are generated by the radiation of gravity waves is known as “geostrophic adjustment.” In Video 7.2a, the radius of the initial pressure field is increased to r = 10◦ . In this case the gravity waves, so apparent in Video 7.1, are only weakly present: They are visible only initially, as oscillating shades of green that radiate away from the initial disturbance, with shorter-wavelength waves preceding longer-wavelength ones. This change happens because short-wavelength, gravity waves do not “fit” well with the large-scale, initial disturbance, and hence cannot be strongly excited (see Sects. 12.3.2.2 and 15.1.4). Owing to the weakness of the gravity waves and their reflections, the anticyclonic, geostrophic flow is much more visible. In Video 7.2b with β = 0, the initial response is similar to Video 7.2a in that gravity waves radiate away from the initial region, leaving behind a geostrophicallybalanced, anticyclonic circulation. Subsequently, however, the geostrophic circulation propagates westward as a Rossby-wave packet. A striking aspect of the packet is its increasing northwestward tilt in time. Because the initial disturbance is large scale, most of the Rossby waves contained in the packet are nondispersive waves with the dispersion relation (7.7); their phase and group speeds are therefore close to cr = σ/k = −βc12 / f 2 , which increases equatorward like f −2 (e.g., with cr = 2.6 cm/s at 30◦N and 10.8 cm/s at 15◦N), thereby accounting for the tilt. Note that oscillations appear on the eastern edge of the Rossby packet as time progresses; they indicate the presence of dispersive Rossby waves in the packet, which appear east of the main packet because their group speed is slower than cr (see the discussion of Eq. 13.20 below). Interestingly, the gravity-wave response in Video 7.2b is very different than it is in Video 7.2a. Rather than spreading circularly from the initial disturbance, a packet of gravity waves propagates southward from the forcing region (more visible when the video is played slowly). It propagates to the southern boundary, where it reflects to return northward; subsequently, individual gravity waves reflect at their critical latitudes (defined above) to propagate southward again, and so on. This process is a striking example of the β-dispersion of inertial waves, a topic we consider in Sect. 10.1.2. To illustrate the impact of the Rossby-wave critical latitude θ R1 , it is necessary to excite waves at a single period P. To do that, we force the ocean with a wave generator, consisting of a longitudinally-narrow (0.4◦ wide), y-independent band of zonal wind τ x centered at 60◦E that switches on at t = 0 and thereafter oscillates at a period P. Videos 7.3a and 7.3b show the resulting solutions when P = 90 and 180 days and without horizontal mixing (νh = 0). The forcing drives an oscillating meridional jet along 60◦E, with coastal currents along the northern and southern boundaries; the latter are generated by Kelvin-wave propagation from the forcing band (Chap. 13), and provide a source and sink for the jet throughout the videos. In both videos, the Coriolis parameter f is given by the equatorial β-plane approximation, for which

7.2 Kelvin Waves

207

the values of θ R1 at the two periods are 14.7◦ and 29.4◦ , respectively. Consistent with these analytical predictions, trigonometric Rossby waves radiate from the generator only south of θ R1 in the equilibrium responses; in contrast, only evanescent waves are excited north of θ R1 , and the responses remain confined near the forcing region there. In Video 7.3b, the response south of θ R1 develops a remarkable structure, eventually including a quasi-checkerboard pattern. Given that νh = 0, this pattern could be numerical noise. On the other hand, because its wavelength is about 2◦ and so is well resolved by the 0.1◦ grid, it must be a real (dynamical) signal. Its cause is the excitation of near-resonant Rossby waves (Sect. 15.4.2). The group speed of such waves is very slow, accounting for the long spin-up time of the pattern. Video 7.3c is similar to Video 7.3b except with νh = 5×105 cm2 /s. Even with this small value, the horizontal mixing is strong enough to eliminate the small-scale features, suggesting that they do not appear in typical numerical models or the real ocean. On the other hand, the large-scale pattern west of the forcing band remains. That pattern is interesting in itself, as the Rossby-wave packet does not extend due west but rather also extends equatorward. We discuss the dynamics of this interesting feature in Sect. 7.3.3.

7.2 Kelvin Waves Coastal Kelvin waves are usually derived assuming that f is constant, in which case their structures are the same along zonal and meridional boundaries. In the NIO, however, Kelvin waves propagate considerable distances along boundaries that extend meridionally, so that f varies a lot. Here, we derive Kelvin-wave solutions for coasts that are oriented zonally (i.e., a northern or southern boundary of a rectangular basin) and meridionally (an eastern or western boundary). A key result is that periodic Kelvin waves along a meridional boundary exist only at latitudes |θ| poleward of the critical latitude for Rossby waves θ R . For generality, here we write solutions in such a way that they are applicable to all the cases: in either hemisphere ( f can have either sign); and for north, south, east, and west boundaries. In later chapters, solutions are written to apply in only one region, but they can be easily modified to apply to the others.

7.2.1 Zonal Coasts Along a zonally-oriented coast, we look for solutions in which v ≡ 0 everywhere and make no approximation for f (zonal Kelvin waves). To obtain the wave, we cannot use Eqs. (7.1), since they are degenerate when v = 0. Instead, we set v = 0 in Eqs. (6.1) to get

208

7 Midlatitude Waves

− iσu + ikp = 0,

f u + p y = 0,

−iσ

p + iku = 0. c2

(7.17)

Because f varies, the solution does not have the form of plane wave (6.2). Instead, we assume variables have the form q(x, y, t) = Q(y) exp(ikx − iσt), where the structure of Q(y) is to be determined. The first and third equations of (7.17) imply that σ = ±kc,

(7.18)

the dispersion relation for zonal Kelvin waves. With the aid of (7.18), and the first and second of Eqs. (7.17), it follows that Py = − f U = − f

f k P = ∓ P. σ c

(7.19)

Assuming that the coast is located at y = 0, solutions to (7.19) are   P(y) = Po exp ∓

y 0

 f (y  )  dy , c

(7.20)

where Po is an arbitrary amplitude. The complete solution for the zonal Kelvin wave is then      σ y f (y  ) dy  exp i (±x − ct) , (7.21a) p = Po exp ∓ c c 0 p u=± , c

v = 0,

(7.21b)

where we replace f with | f | and introduce the factor = f / | f | = ±1 in order to keep track of whether the Kelvin wave is in the northern ( = 1) or southern ( = −1) hemisphere. To be physically realistic, the Kelvin wave must decay away from the coast. Therefore, the upper (lower) signs apply when the coast corresponds to an equatorward (poleward) edge of the basin. When f is constant, the integral in the exponential in (7.21a) is just (| f | /c) y, so that    y |f| P(y) = Po exp ∓ y = Po exp ∓ (7.22) c R and the Kelvin wave decays offshore with an e-folding scale of the Rossby radius, Rn = | f | /cn . When f varies, we replace it with f → f + β y, where the values of f and β are those at the latitude of the boundary (an accurate approximation since the Kelvin wave decays rapidly offshore). In that case, the integral in (7.21a) is (y/R) 1 + 21 y/ Rˆ e , Rˆ e = | f | /β, so that

7.2 Kelvin Waves

209

   y y 2 . exp ∓ P(y) = Po exp ∓ R 2R Rˆ e

(7.23)

Because the first exponential requires that the Kelvin wave extends  only  offshore ˆ a distance of O(R), the argument of the second exponential is O R/ Re , which at midlatitudes is very small; therefore, the value of the second exponential is very nearly one, and the zonal Kelvin wave is essentially unchanged from its f -plane counterpart (i.e., the limit of Eq. 7.23 when β → 0 and Rˆ e → ∞). A key property of Kelvin waves is that their propagation direction is linked to the direction of their offshore decay: According to (7.21a), Kelvin waves in the northern (southern) hemisphere propagate in a direction such that the coast is on their right (left). Finally, the concepts of phase and group velocities also apply to Kelvin waves, in this case only in the alongshore direction (their propagation direction). It follows from (7.18) that c p = cg = ±c and hence they are nondispersive. Videos: Video group 7.4 shows numerical solutions to Eqs. (C.1) when c = c1 and f is constant. In Video 7.4a, the response is generated by initially specified, pressure P and zonal-velocity U fields, given by (7.22) and U = P/c1 along the southern boundary. As such, the initial disturbance contains only coastal Kelvin waves and, because Kelvin waves are nondispersive, the wave packet propagates along the coast with little distortion. As time progresses, however, a weak “tail” of short-scale Kelvin waves develops behind the primary one. It arises from numerical error owing to the jump in slope at the edges of the main packet (e.g., see the discussion of Fig. 3.4 in Mesinger and Arakawa 1976). There is a similar tail just behind the leading edge of the Kelvin wave, not clearly visible because it lies inside the main packet. In Video 7.4b, the initial pressure field is specified as in Video 7.4a but U = 0. Consequently, the initial packet is not a pure Kelvin wave, but rather contains gravity waves and a geostrophic circulation as well. The gravity and Kelvin waves quickly radiate away from the forcing region, leaving behind the geostrophic flow that circulates around a patch of high pressure. Note that, similar to Video 7.1, the wavelength of the gravity waves increases in time, and the Kelvin wave and geostrophic circulation are distorted by multiple reflections of gravity waves from the basin boundaries. Video 7.4c is similar to Video 7.4a except that the initial packet is much narrower. In this solution, the tail is more evident because the jump in slope of the primary packet is sharper. Further, the tail behind the leading edge of the primary packet is now strong enough to distort its structure visually. Finally, the solution in Video 7.4d is like that in Video 7.4c except the basin is bounded. Since f is constant, the structure of Kelvin waves along a meridional coast is simply a rotated version of that for a zonal Kelvin wave. A reasonable expectation, then, is that when a Kelvin wave encounters a boundary (corner), it reflects smoothly from (propagates around) the corner with little distortion, but that is not the case. Rather, a gravity-wave packet is generated, and consequently the

210

7 Midlatitude Waves

reflected Kelvin wave develops a tail (Buchwald 1968; Packham and Williams 1968). Multiple reflections intensify the tail, and add additional trailing oscillations.

7.2.2 Meridional Coasts Along a meridional boundary, we look for coastally-trapped solutions to Eqs. (6.1) when u = 0 only at x = 0. Since a general solution is too difficult to obtain, we restrict our analysis to waves with a frequency much less than | f | (σ  | f |) and a meridional length scale L y much greater than the Rossby radius of deformation R = c/ | f | (L y R). Under these restrictions, u t is negligible, and Eqs. (6.1) simplify to v=

1 px , f

u=−

1 1 pxt + p y , 2 f f

px xt −

f2 pt + β p x = 0 c2

(7.24)

(see Appendix B and Sect. 13.1). In the second equation, v in (6.1b) is eliminated using the first of (7.24). In the third equation, u and v in (6.1c) are eliminated using the first two of Eqs. (7.24).

7.2.2.1

Dispersion Relation

We again look for solutions of the form, q(x, y, t) = Q(y) exp(ikx − iσt), this time allowing for k to be complex so that the solution decays offshore. The last of Eqs. (7.24) gives   f2 (7.25) σ k 2 + 2 + kβ = 0, c the dispersion relation for the waves in the system. Note that (7.25) is almost the same as (7.6), the dispersion relation for midlatitude Rossby waves, differing in that  = 0 and f and β can vary. It follows that k depends on y since f varies. Critical Frequency and Latitude: Equation (7.25) is a quadratic equation for k, with the roots ⎞ ⎛  β ⎝ σ2 f 2 ⎠ k(y) = − (7.26) 1∓ 1−4 2 2 . 2σ β c For a given f and β, there is a critical frequency σ R below which k is real so that evanescent (coastally trapped) waves do not exist and the response consists of Rossby waves. That frequency occurs when the argument of the radical is set to zero, and the resulting expression gives the same σ R as in (7.4). Alternately, for fixed σ setting the argument to zero gives the same critical latitude θ R as in (7.13b), (7.14), and (7.15). Thus, for a given σ coastally trapped waves exist only for |θ| > θ R , and we refer to these waves as “meridional” or “β-plane” Kelvin waves. (There is no critical

7.2 Kelvin Waves

211

frequency and latitude corresponding to σG and θG in Eqs. 7.4 and 7.13a since gravity waves are filtered out of Eqs. 7.24.) The θ Rn curves in Fig. 7.3 also illustrate whether β-plane Kelvin waves exist in the NIO: They exist at a particular frequency provided that θ Rn is less than the latitude of NIO northern boundary (i.e., θ Rn  25◦N). At P = 60 days (left, vertical dashed line), Kelvin waves can exist north of 10◦N in the NIO for all n values. At longer periods, their presence becomes increasingly limited. At P = 180 days (middle, vertical dashed line), there is no n = 1 Kelvin wave, and the n = 2 and 3 modes waves exist only at latitudes θ > 18.0◦ and 11.5◦ . At P = 365 days (right, vertical dashed line), the n = 1 and 2 Kelvin waves no longer exist, and the n = 3 wave is present (just barely) only for θ > 22.5◦ .

7.2.2.2

Solution

To satisfy the boundary condition that u = 0 at x = 0, the second of Eqs. (7.24) with p = P(y) exp (ikx − iσt) requires that Py +

  σ kP = e− e P y = 0, f 

where (y) =

y yo

σk  dy f

(7.27a)

(7.27b)

is an integrating factor and yo an arbitrary reference latitude with the same sign as y. Equation (7.27a) has the solution P(y) = Po e−(y) ,

(7.28)

where Po is the coastal pressure at y = yo . The complete solution for the coastal wave is then   y  ky σk  ik dy + ikx − iσt , p, u = −i x p, v= p = Po exp − f f yo f (7.29) where the velocities are determined from the first two of Eqs. (7.24) with u also using p in (7.29). Note that u = 0 at x = 0 as required by the coastal boundary condition, but it does not vanish everywhere. In this form, (7.29) has little resemblance to familiar, f -plane Kelvin waves. To see the similarity, we rewrite p using (7.26) to eliminate k when |θ| > θ R . In that case, the argument of the radical in (7.26) is negative and k is complex. We can then write it as |f| β k(y) = − +i , (7.30a) 2σ σ

212

7 Midlatitude Waves

where , the local meridional wavenumber of the wave, is given by ⎞ ⎛ σ β ⎝ σ2 f 2 σ (y) = ± 4 2 2 − 1⎠ = ± ϑ(y), | f | 2σ β c c With the aid of Eqs. (7.30) and the relation that

 1 tan2 θ R 2 ϑ(y) = 1 − . tan2 θ

(7.30b) 1 2 , it follows β/ f dy = ln f / f (2 ) ( ) o yo

y

    21  y β σ f  − + i  dy = − ln ϑdy  , ± i (y) = 2f fo yo yo yo c (7.31) where is defined after (7.21b). Inserting (7.30) and (7.31) into (7.29) gives 

 p = Po

y

f fo

σk  dy = f

 21



y

     y |f| β σ ϑx exp −i x ∓ i ϑdy  − iσt , exp ∓ c 2σ yo c

(7.32)

and u and v are known in terms of p. The signs are chosen to ensure that p decays away from an eastern (lower signs) and western (upper signs) boundary. As a check, note that when β = 0, so that k = ±i | f | /c, k y = 0, ϑ = 1, f o / f = 1, solution (7.32) simplifies to have the form of the f -plane Kelvin waves found in the previous subsection, except located along eastern and western boundaries. The β-plane Kelvin waves differ from their f -plane counterparts in several ways: They exist only at latitudes poleward of θ R ; the amplitude of p increases with lat1 itude like | f | 2 ; and the magnitudes of their offshore decay scale (c/ | f |) ϑ−1 and alongshore wavelength λ = 2π (c/σ) ϑ−1 are larger by the factor ϑ−1 . Rewriting the first of Eqs. (7.30b), their dispersion relation is σ=±

c (y) , ϑ(y)

(7.33)

so that the phase and group speeds of β-plane Kelvin waves, c p = ±σ/ = ±cϑ−1 and cg = ±σ = ±cϑ−1 , are also increased by the factor ϑ−1 . Note, however, that since c p = cg the waves are still non-dispersive. A final difference is that β-plane Kelvin waves propagate westward with the wavenumber β/ (2σ) as well as decay offshore. How visible is this propagation? The zonal structure of p in (7.32) is exp (αx − iκx), where α = ∓ (| f | /c) ϑ and κ = β/ (2σ). So, whether the decay or offshore propagation is apparent depends on the ratio of κ to α, ε = κ/ |α| = Pβc/ (4π | f | ϑ) , (7.34) with offshore propagation (decay) being dominant for larger (smaller) values of ε. To illustrate, with c = 264 cm/s, P = 2π/σ = 180 days, and using the exact expression for f , it follows that ε = 0.98 at 40◦N; with P = 60 days, ε = 0.23 and 1.06 at 40◦N and 15◦N, respectively. Based on these values, we expect western propagation to be

7.2 Kelvin Waves

213

apparent at all latitudes when P = 180 days, but only at low latitudes when P = 60 days. These expectations are confirmed in the videos discussed next.

7.2.2.3

Videos

Video group 7.5 illustrates eastern-coastal responses forced by a wave source along the southern boundary that oscillates at periods P = 60, 180, and 365 days, when f is specified by the midlatitude β-plane approximation. Video group 7.6 is similar, except showing the western-coastal responses forced by a wave source along the northern boundary. In both groups, the responses differ markedly with P through its impact on the Rossby-wave critical latitude θ R given in (7.14). Here, we comment on the Kelvin-wave part of the solutions north of θ R , delaying a discussion of their Rossby-wave properties south of θ R until the next section. East coast: In Video 7.5a (P = 60 days), the critical latitude is θ R = 6.7◦ , so that no trigonometric Rossby waves exist in the basin since ys = 10◦ . In the equilibrium solution, then, the response consists entirely of Kelvin waves that radiate eastward from the source along the southern boundary, northward along the eastern boundary, and westward along the northern boundary. Consistent with (7.32), the width of the eastern-boundary, β-plane Kelvin waves decreases to the north as f increases; further, they propagate, as well as decay, offshore, although that feature is clearest near the southern boundary where the factors of f −1 and ϑ−1 in (7.34) increase ε. Weak, transient, long- and short-wavelength Rossby waves are also present in the video. The long-wavelength waves are generated along the eastern boundary by the passage of the initial Kelvin-wave front. Subsequently, they propagate across the interior ocean, where they are visible as alternating bands of green shading. They propagate faster closer to the equator, since their phase and group speeds tend to be inversely proportional to f 2 (when k 2 + 2  f 2 /c2 in Eq. 7.6); as a result, the slope of bands increasingly flattens farther to the west. Short-wavelength Rossby waves radiate eastward from the wind-forced region near the southern boundary. Both wave types interfere with the southern-boundary coastal jet, causing it to pulse with a zonal scale of 3–4◦ . Note that the long-wavelength, transient Rossby waves propagate across the interior ocean at all latitudes, whereas the equilibrium (periodic) solution cannot because θ R < ys (Sect. 7.3). The transient waves can do so because the transient packet contains waves at periods other than P. In response to the passage of the eastern-boundary Kelvin wave, pressure essentially jumps to a non-zero value everywhere along the coast. Thus, the initial coastal response is close to that of a step function and, according to Fourier theory, it can be viewed as being composed of periodic Rossby waves at all periods, including the limit that P → 0. It is the waves in the transient packet with periods greater than PR = 2π/σ R that are visible in Video 7.5a. In Video 7.5b (P = 180 days), β-plane Kelvin waves exist in the equilibrium solution only north of θ R = 26.3◦ , with Rossby waves south of that latitude. The offshore propagation of the β-plane Kelvin wave north of θ R is more clear than in Video 7.5a, a consequence of the larger ε value due to the increase in P. As in Video

214

7 Midlatitude Waves

7.5a, during the spin-up of the equilibrium solution transient Rossby waves radiate from the coast at all latitudes. In Video 7.5d (P = 365 days), the critical latitude is θ R = 55.7◦ and, consistent with the analytic solution, no β-plane Kelvin waves exist along the coast. West coast: As for Video 7.5a, in Video 7.6a (P = 60 days, θ R = 6.7◦ ) β-plane Kelvin waves exist everywhere in the basin. In the equilibrium response, Kelvin waves radiate westward from the source along the northern boundary, southward along the western boundary, and eastward along the southern boundary. Consistent with (7.32), the width of the western-boundary, β-plane Kelvin waves increases to the south as f decreases and they are associated with onshore (westward) phase propagation, the latter property being clearest only near the southern boundary where ε in (7.34) is increased by ϑ. A transient packet of long-wavelength (westwardgroup-velocity) Rossby waves is generated by the northern-boundary forcing, and by January, 2005, it has propagated to the western boundary. In addition, a transient packet of short-wavelength (eastward-group-velocity) Rossby waves initially extends offshore, particularly visible in the southern half of the domain; as time passes, it appears as weak signals with westward phase propagation along the outer edge of the southern-boundary Kelvin wave, which gradually propagate eastward across the basin. In Video 7.6b (P = 180 days, νh = 0), β-plane Kelvin waves are present only north of θ R = 26.3◦ , and their onshore propagation is much more apparent than in Video 7.6a, owing to the increase of ε with P. In Video 7.6d (P = 365 days, νh = 0), consistent with theory no β-plane Kelvin waves are present along the coast.

7.3 Boundary-Generated Rossby Waves Remarkable structures develop in the videos south of θ R . Although solution (7.29) represents their properties near the coast, it breaks down offshore (Sect. 7.3.1). It is possible, however, to use wave-group theory to explain their prominent offshore features (Sect. 7.3.3).

7.3.1 Solution When |θ| < θ R , k is real in solution (7.29), and consequently the integral in (7.29) is real so that the coastal response has no alongshore propagation ( = 0). These coastal signals are offshore-propagating Rossby waves, a property that is clearly seen in the for simplicity that f = β y. Then, with the aid of the relation limit √ |θ|  θ R . Assume 1 − ξ = 1 − 21 ξ, ξ  1, (7.26) simplifies to

7.3 Boundary-Generated Rossby Waves

k( 1 ) = − 2

215

σ f2 β , − β c2 σ

(7.35)

when |θ|  θ R , the wavenumbers for low-frequency, long- and short-wavelength Rossby For the eastern-boundary wave, it follows that  =  waves,   respectively.  − 21 σ 2 /c2 y 2 − yo2 so that   σ2  2 2 y − yo exp (ik1 x − iσt) , p = Po exp 2c2 

u = 2i

σx p, c2

σy p, c2 (7.36a)

v = −i

whereas for the western-boundary wave,  = − ln ( f / f o ) and hence p = Po

f exp (ik2 x − iσt) , fo

u = 0,

v=i

1 p. yσ

(7.36b)

Solutions (7.36a) and (7.36b) differ in their offshore structures with k1 (longwavelength) and k2 (short-wavelength) Rossby waves emanating from the eastern and western boundaries, respectively, and also in their alongshore structures through differences in the factors exp (). As demonstrated in the videos discussed next, solution (7.29) captures basic features of the Rossby-wave response to the exact equation set, (6.1) or (7.1). On the other hand, it violates the criterion that L y R, which is necessary for its governing Eqs. (7.24) to be accurate, in two regions. First, because the solution decays (propagates) offshore poleward (equatorward) of θ R , variables necessarily jump across θ R . As a result, the inequality is violated near θ = θ R so that the solution is inaccurate there (except right at x = 0 where there is no jump). Second, phase lines for the Rossby-wave part of the solution increasingly tilt as the wave propagates offshore, thereby developing an ever smaller, meridional scale that eventually violates L y R. The first error impacts the solution only locally and not seriously as the jump can be avoided in a couple of ways. For example, McCreary (1980) included damping γ in (7.17), thereby smoothing the jump away. Alternately, Grimshaw and Allen (1988) sought to address the issue by adding a boundary layer across θ = θ R to join the Kelvin-wave and Rossby-wave solutions smoothly. The second error impacts the large-scale response sufficiently far offshore because the propagation direction of the Rossby-wave packet is misrepresented in approximate solution (7.29). Specifically, the group velocity for Rossby waves is entirely westward because dispersion relation (7.25) is independent of  (i.e., cgy = σ = 0), whereas in exact solutions, like those in the videos discussed next (and elsewhere in the book), it also has a meridional component (Sect. 7.3.3).

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7.3.2 Videos As might be expected, the offshore responses in eastern- and western-coastal videos differ markedly, given that they are composed of long- and short-wavelength Rossby waves. At the same time, there are surprising similarities in the overall patterns (envelopes) of corresponding inviscid solutions. East coast: In Video 7.5b (P = 180 days), the near-equilibrium Rossby-wave response south of θ R has a remarkable offshore structure. Specifically, energy associated with larger-scale waves develops several oscillations, with “quasi-focal points” on the southern boundary near 103◦E, 65◦E, and 33◦E, a pattern traceable to the Rossby waves propagating meridionally as well as westward (Sect. 7.3.3 and 15.2.3). There is also small-scale variability, which develops more gradually as Rossby waves with smaller wavenumbers and slower group velocities propagate into the interior ocean. Its presence is sensitive to the strength of horizontal mixing in the model. Video 7.5c is comparable to Video 7.5b except with νh increased from 0 to 5×105 cm2 /s, and the mixing eliminates the smaller-scale signals. Videos 7.5d and 7.5e contrast solutions when P = 365 days with νh = 0 and 5×105 cm2 /s, respectively. At this period, the near-equilibrium responses have a large-scale structure with a single focal point near 65–70◦E. Video 7.5d also has narrowly-separated bands, which in Video 7.5e are damped away by viscosity. West coast: In Video 7.6b (P = 180 days), short-wavelength Rossby waves spread away from the western boundary, much more slowly than in Video 7.5b because their group speed is considerably less than that of long-wavelength waves (see the discussions of Eqs. 13.20 and 13.31 below). In the equilibrium response, however, the wave-energy pattern has a large-scale structure (envelope) similar to the one in Video 7.5b: Both have a periodicity with quasi-foci located at similar distances from their respective boundaries. This similarity is not coincidental: It happens because ray paths of the boundary Rossby waves are mirror images of each other when the boundary is oriented meridionally (Sect. 7.3.3). In typical OGCMs and the real world, however, an elegant pattern like that in Video 7.6b never arises, as with P = 180 days short-wavelength Rossby waves are too short to survive mixing processes. Video 7.6c, comparable to Video 7.6b except with νh = 5×105 cm2 /s, illustrates its demise, with the mixing damping the waves before they can propagate very far offshore. Since there are no large-scale Rossby waves in the offshore flow field of Video 7.6b, the response in Video 7.6c remains trapped near the western boundary. Video 7.6d (P = 365 days) shows the inviscid response when short-wavelength Rossby waves radiate from the western boundary at all latitudes. The time development of the offshore structure is slow, so much so that even at the end of the video (80 years) it is not yet near equilibrium. At that time, the Rossby-wave packet has a distinct eastern edge that intersects the southern boundary about 50–55◦E, and farther east there is an indication that another energy oscillation is beginning to develop. This point is located at the same distance from the coast as the focal point in Video 7.5d, a consequence of ray-path symmetry (Sect. 7.3.3). In Video 7.6e, a small amount of

7.3 Boundary-Generated Rossby Waves

217

mixing eliminates the offshore pattern, replacing it with a structure similar to that of a quasi-steady Munk layer (Sect. 11.2.1.3).

7.3.3 Meridional Energy Propagation The existence of meridional propagation of Rossby-wave energy in the exact video solutions contrasts with approximate solution (7.36), in which it radiates only westward. Here, we use the wave-group theory of Sects. 6.3 and 6.4 to explain this property and the patterns that result from it. Our discussion follows that of Schopf et al. (1981), who used the same approach to discuss the eastern-boundary Rossby-wave packet in their equatorially-forced problem (Sect. 15.2.1).

7.3.3.1

General Solution

The approach assumes that Rossby waves in the neighborhood of any point (x, y) within the packet can be represented locally as a plane wave, exp (ikx + iy − iσt) .

(7.37)

Subsequently, this “local Rossby wave” propagates in the direction of its group velocity (6.16) defined by Rossby-wave dispersion relation (7.6), rewritten here as k 2 + 2 +

f2 kβ =− . c2 σ

(7.38)

From (7.38), the zonal and meridional components of the group velocity are cgx

σ2 = σk = kβ

  β 2k + , σ

cgy = σ =

σ2 2. kβ

(7.39)

The Rossby wave propagates through a medium that changes only in y through the dependence of (7.38) on f and β. Equations (6.17a) therefore state that, as the Rossby wave propagates,  changes in y but k remains constant, the latter an essential simplification for finding the solution below. The slope of the path taken by the local Rossby wave, y (x), is given by     cgy dy f2 2 kβ 2 2 = =± −k − 2 . − = dx cgx 2k + β/σ 2k + β/σ σ c

(7.40)

In the last expression, we eliminated  using (7.38), and added the ± sign to indicate that , and hence dy/d x, can have both positive and negative values. To allow for

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7 Midlatitude Waves

an analytic solution to (7.40), we assume that f is given by the midlatitude β-plane approximation expressed as f = f o + βo (y − yo ) = βo y − (βo yo − f o ) = βo (y − y) ≡ βo η,

(7.41)

where η = y − y and y = yo − f o /βo , in which case (7.40) can be rewritten     kβo β 2 η2 2 dη 2 − =± − k − o2 . dx 2k + βo /σ σ c

(7.42)

Despite its nonlinearity, (7.42) has a simple structure, owing to k and βo being constant so that all dependence on η is contained in the term βo2 η 2 /c2 . Given its structure, we expect the solution to (7.42) to have the form η = A cos (bx + θo ), where A, b, and θo are constants to be determined. After inserting this expression into (7.42), it is straightforward to show that c η (x) = y − y = βo

 −

  2 βo kβo − k 2 cos x + θo . σ c 2k + βo /σ

(7.43)

(There is no longer a need for the ± sign since the cosine takes on both signs.) According to (7.43), energy associated with a Rossby wave that has a zonal wavenumber k propagates along a sinusoidal ray path. In all the videos in this book, Rossby-wave packets follow trajectories of this sort, so that their energy propagates meridionally as well as zonally.

7.3.3.2

Ray Paths from Boundaries

For Rossby waves generated along eastern and western boundaries, it is possible to simplify the solution further. Consider a ray path that emerges from either an eastern or western boundary at an arbitrary point P  = 0, η  equatorward of θcr . (Fig. 7.4, below, plots eastern-boundary ray paths from a number of such points.) Because the solution doesn’t oscillate along x = 0 equatorward of θcr (Sect. 7.3.1), its meridional wavenumber  must vanish there.Setting  = 0 in (7.38) and using (7.41), it follows that the value of k at P  , k  0, η  , is given by β 2 η 2 + o2 = 0 k +k σ c 2

 βo

⇒

 βo βo η 2 ± k =− 1− 2 , 2σ 2σ ηR 

(7.44)

where η R = c/ (2σ). Since k  remains constant everywhere along this ray path, we can set k → k  in solution (7.43), and use (7.44) to eliminate k  for η  . Applying the condition that η = η  at x = 0 then gives θo = 0, so that

7.3 Boundary-Generated Rossby Waves

219

  η(x) = η  cos κ x ,

κ =

2 σ

, c 1 − η 2 /η 2

(7.45)

R

describes the initial energy pathway followed by Rossby waves that leave the boundary at point P  . As suggested in the videos (and evident in Fig. 7.4), boundary-generated Rossby waves undergo multiple reflections from the closed southern boundary at y = ys = 10◦N. The reflected wave packets must ensure that v = 0 at ys in the total response (i.e., sum of the incoming and reflected waves). Therefore, their σ and k must be the same as for the incoming packet, but the sign of  is reversed so that cgy has the opposite sign and its energy propagates away from the boundary (Eq. 7.39). It follows that the reflected ray paths have the same form as (7.45), differing only in that they are shifted in x. Let ηs = ys − y be the value of η at ys , and consider an eastern boundary for which the ocean lies in the region x < 0. Then, the location offshore, x = −x1 , where ray path (7.45) first intersects ys is   ηs = cos κ x1  η

⇒

x1 = −

ηs 1 cos−1  , κ η

(7.46)

and the first reflected ray path is given by (7.45) with x → x + 2x1 : It rises sinusoidally from ηs at x = −x1 to a maximum of η  at x = −2x1 and returns to ηs at x = −3x1 . Farther west, a second reflection occurs in the region −5x1 < x < −3x1 , which is given by (7.45) with x → x + 4x1 . Extending to the nth reflection gives    ηn x, k  = η  cos κ (x + 2nx1 ) ,

− (2n + 1) x1 < x < − (2n − 1) x1 , (7.47a) where n = 0, 1, 2, · · · . The complete pathway with all reflections is 

η(x, k ) = y − y =

∞ !

ηn (x, k  ),

(7.47b)

n=0

but the upper limit can be truncated to a low integer since the ray paths are plotted only for a finite offshore extent (like that in Fig. 7.4). Two noteworthy properties of the ray paths are evident in solution (7.47). First, they are physically sensible only for η  < η R when the radical of κ is real; from the second of Eqs. (7.14) it follows that this inequality is equivalent to |θ| < θ R so that Rossby-wave ray paths in (7.47) exist only equatorward of θ R , as they should. Second, since the cosine is symmetric in x, ray paths for eastern and western boundaries are mirror images of each other, that is, they are unchanged when x and x1 are reversed in sign. This symmetry is surprising given that the paths are determined from very different Rossby waves: long-wavelength waves in the former, and short-wavelength ones in the latter. It is in fact a special case, which is lost when the boundary is slanted (Sect. 15.2.4).

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Fig. 7.4 Ray paths extending from the eastern boundary at various latitudes y  , when there is a southern boundary along 10◦N and P = 180 (top panel) and 360 (bottom panel) days. They are determined from (7.47) with η  = y  − y. The black-dashed line indicates the critical latitude for 180-day Rossby waves. The offshore responses in Videos 7.5b–e, 7.6b–e result from rays that leave the boundary at all latitudes (blue, red, and dashed-gray curves), whereas those in Videos 13.4b– e, and 13.5b–e are determined only by rays north of 20◦ , the southern edge of the wind forcing (blue and dashed-gray curves). Curves for a western boundary are the same as shown with the axis reversed (x → −x)

Figure 7.4 plots pathways from an eastern boundary determined from (7.47b) for several values of y  = η  − y, when there is a southern boundary along 10◦N and P = 180 (top panel) and 360 (bottom panel) days. In both cases, because κ increases with η  and y  , reflections occur more often for rays that originate farther from the southern boundary. For P = 180 days, rays leave the boundary only south of the critical latitude for 180-day Rossby waves (dashed line), whereas for P = 360 days they leave at all latitudes since y R = 55.7◦ lies north of the northern boundary of the video domain. Consistent with the second property in the previous paragraph, plots

7.4 Waves Along a Slanted Coast

221

of ray paths from a western boundary (not shown) are the same as in Fig. 7.4, except with the sign of x reversed. Collectively, the pathways exhibit structures that reproduce prominent features in the videos. For example, in the bottom panel of Fig. 7.4 ray paths from the eastern boundary combine to form a distinct western front (outer edge, envelope), and that front is very clear in Video 7.5d. Ray paths that have reflected from the southern boundary also combine to have an outer edge in the northeast corner of the bottom panel of Fig. 7.4. A similar front is developing in Video 7.5d, but very slowly because the front is composed of slowly-propagating Rossby waves with a small wavelength. For eastern-boundary Rossby waves, k  increases with y  . Consistent with this property, in Videos 7.5b,c and 7.5d,e features with smaller (larger) zonal scales appear farther from (closer to) the southern boundary. Moreover, note that the ray paths in Fig. 7.4 with larger zonal scales (red curves) tend to intersect the southern boundary near the same location; these pathways clearly outline the large-scale structures in Videos 7.5b,c and 7.5d,e, explaining why their equilibrium responses have quasifocal points. In Video 7.5b, the dashed-gray pathway emerges from the coast near y R , so that the Rossby wave associated with it is nearly resonant (Sect. 15.4.2); these waves have small zonal wavelengths, and account for the small-scale signals visible near y R . For western-boundary Rossby waves, although k  does not decrease with y  , the red curves (reversed in x) delineate the large-scale structures (envelopes) present in Videos 7.6b and 7.6d. In particular, the quasi-focal points of ray paths in Fig. 7.4 are at the same locations as those in the videos. Further, in Video 7.6b the northern edges (envelopes) of the blue curves in Fig. 7.4 correspond to bands of variability that extend westward and poleward from the second and third quasi-focal points.

7.4 Waves Along a Slanted Coast Real coasts are not oriented meridionally. How are properties of boundary-generated waves altered from those discussed above when the coast is slanted from a meridian? To address this question, we follow two approaches: for the Kelvin-wave regime by finding a solution in a coordinate system rotated so that its alongshore axis is parallel to the coast (Clarke 1991); and for the Rossby-wave regime by considering how ray paths are modified (Schopf et al. 1981). For simplicity, we consider a coast defined by y = −x cot α,

(7.48)

a straight-line barrier slanted at an angle α from north-south (Fig. 7.5, thick black line); see the above references for extensions of the theories that allow for coasts with weak curvature. As we shall see, properties of solutions are similar to those for a meridional coast, the key difference being that the critical latitude y R occurs at a lower latitude.

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7 Midlatitude Waves

Fig. 7.5 Schematic diagram of a coast slanted at angle α (thick black line), with an unrotated coordinate system (x, y) and a rotated one (x, ˜ y˜ ) with the y˜ parallel to the coast. Both axes intersect the origin at (0,0), the blue arrow indicating the y˜ axis is shifted slightly so it is distinct from the coastline. P  is an arbitrary point along the boundary; its coordinates are (x  , y  ) and (0, y˜ ) in the unrotated and rotated coordinate systems, respectively

7.4.1 Kelvin Waves In the rotated system, let: x˜ (across-shore) and y˜ (alongshore) be its coordinates; p˜ its pressure; and u˜ and v˜ velocity components, and L x˜ and L y˜ spatial scales, in the x˜ and y˜ directions, respectively. Because the x˜ and y˜ axes are linear and perpendicular to each other, the equations of motion have the same form as (6.1) and, under the restrictions that ( f T )−1  1, L x˜ is of order R = c/ f , and L y˜  L x˜ , they simplify to v˜ +

1 px˜ = 0, f˜

u˜ = −

1 1 p˜ xt˜ − p y˜ = 0, f˜2 f˜

p˜ t + u˜ x˜ + v˜ y˜ = 0 c2

(7.49)

(see Appendix B). This equation set is the same as (7.24) for the unrotated system, except that the Coriolis parameter f (y) = f˜(x, ˜ y˜ ) varies with x˜ as well as y˜ . Inserting the first two expressions in (7.49) into the third gives p˜ x˜ xt˜ −

f˜2 p˜ t − 2 c2

f˜x˜ p˜ xt˜ − f˜x˜ p˜ y˜ + f˜y˜ p˜ x˜ = 0. f˜

(7.50)

The magnitudes of f˜x˜ and f˜y are both [ f /Re ], where Re is the earth’s radius. Then, with the above restrictions on L x˜ and L y˜ , the f˜x terms are negligible  with ˜ ˜ respect to other terms in (7.50): Since Re L x = R the magnitude of 2 f x / f p˜ xt ,    [P/ (Re RT )], is much less than that of f˜2 /c2 p˜ t , P/(R 2 T ) ; similarly, because

7.4 Waves Along a Slanted Coast

223

   L y˜ R the scale of f˜x˜ p˜ y˜ , P/ Re L y˜ , is much less than that of f˜y˜ p˜ x˜ , [P/ (Re R)]. Neglecting the small f˜x˜ terms, (7.50) reduces to p˜ x˜ xt˜ −

f˜2 p˜ t + β cos α p˜ x˜ = 0. c2

(7.51)

In (7.51), we used the relation f˜y˜ = f y cos α = β cos α, which is apparent from the geometry of the two coordinate systems (Fig. 7.5). For a meridional coast, we obtained the solution to (7.24) by representing it in the plane-wave form, exp (ikx − iσt). In the rotated system, this approach generally fails because, owing of f˜ on x, ˜ the solution to (7.51) can’t be  to the dependency  represented as exp i k˜ x˜ − iσt . In the Kelvin-wave regime, however, the response is ˜ y˜ ) is effectively constant across the solution, allowing so narrow (L x˜ = R) that f˜ (x, it to be replaced with its boundary value f˜ (0, y˜ ). With this simplification to f˜, Eqs. (7.49) and (7.51) are identical to Eqs. (7.24), except that β → β cos α. It follows that the solution is the same as (7.29), with k in (7.26) modified to ⎛ ˜ y˜ ) = − β cos α ⎝1 ∓ k( 2σ



⎞ 2 ˜ f ⎠ 1−4 2 , β cos2 α c2 σ2

(7.52)

and the replacements x → x, ˜ y → y˜ . According to (7.52), Kelvin waves only exist when the argument of the radical is negative, so that k has an imaginary part and the solution decays offshore. Using the identity f˜(x, ˜ y˜ ) = f (y) to evaluate the radical in unrotated coordinates, it follows that Kelvin waves exist only poleward of a “coastal” critical latitude y Rc defined by f (y Rc ) =

βc cos α. 2σ

(7.53)

According to (7.53), y Rc is smaller than the critical latitude in the open ocean or for a meridional coast by the factor cos α. Clarke (1991) argued that this result, as well as the Kelvin-wave solution, remain valid for realistic coasts with curvature, provided the curvature is sufficiently small.

7.4.2 Rossby-Wave Pathways Once they leave the coast, boundary-generated Rossby waves behave like open-ocean waves: Their only knowledge of the coast is the value of the wavevector that is set there. Thus, it is not useful (is physically awkward) to consider their response in a rotated coordinate system. Following Schopf et al. (1981), then, we determine how their ray paths are modified in the unrotated system.

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Recall from Sect. 7.3.3 that the Rossby-wave response is specified by the set of ray paths (7.43) that originate at all points P  along the boundary where their zonal wavenumber is k(x  , y  ) ≡ k  . For a meridional boundary, we found k  by setting the meridional wavenumber of the Rossby wave  ≡ (x  , y  ) = 0, which represents the observed property that along-boundary phase differences are small (due to rapid adjustments by coastal Kelvin waves). For the slanted coast, we again ˜ the require that the along-boundary phase change is small, but in this case set , alongshore wavenumber in the rotated system, to 0. A unit vector parallel to the coast is m = − (sin α) i + (cos α) j . To ensure that phase is constant along the boundary (˜ = 0), the coastal wavevector, k = k  i +  j , must be directed offshore, so that m · k = −k  sin α +  cos α = 0

⇒

 = k  tan α.

(7.54)

Setting  = k  tan α in (7.38) gives k 2 sec2 α +

f 2 kβ = 0, + c2 σ

(7.55)

where f  = f (y  ), in which case ⎛ k  (y  ) = −

β cos α ⎝ 1∓ 2σ 2

 1−4

f 2

⎞ σ2

c2 β 2 cos2 α

⎠.

(7.56)

Note that (7.56) differs from (7.52) by an additional factor of cos α outside its radical; this inconsequential difference happens because k˜ is directed perpendicular to the coast whereas k  is directed east-west. The existence of Rossby waves requires that k  is real. According to (7.56), that occurs only for ray paths that originate along the boundary equatorward of the coastal critical latitude y Rc defined in (7.53). It is noteworthy that the same critical latitude is obtained for both the Kelvin- and Rossby-wave regimes (as should be the case), despite the methods that lead to that result being very different. The agreement is traceable to both methods requiring that along-boundary phase changes are small: in the Kelvin-wave regime by assuming L x R, and the Rossby-wave regime by setting ˜ = 0. Although not necessary for our purposes here, by restricting f to be linear it is possible to find a ray-path solution like (7.47) when the boundary is slanted. See Sect. 15.2.4 and Fig. 15.2 for a derivation and illustration of such paths when f is given by equatorial β-plane approximation.

7.5 Observations

225

7.5 Observations Midlatitude Rossby and coastal (Kelvin) waves are present throughout the NIO. Prominent examples are Rossby waves that propagate off the east coasts of the Bay of Bengal and Arabian Sea, and shelf waves (Kelvin-like waves) that propagate around their perimeters (Video 1.1; Sects. 4.7 and 4.9).

7.5.1 Critical Latitude Key properties of theoretical Rossby waves are that they exist only at frequencies σ ≤ σ Rn and latitudes |θ| ≤ θ Rn , where σ Rn and θ Rn are critical values given in (7.4) and (7.13). In the past few decades, satellite altimetry has made it possible to check if low-frequency, ocean variability satisfies these inequalities. In his analysis of satellite data, for example, Fu (2004) showed that much of the variability satisfied σ ≤ σ R1 , the critical frequency of the first baroclinic mode. Interestingly, he also found westward-propagating signals at latitudes higher than θ R1 , arguing that many of them can still be interpreted as Rossby waves with higher-than-expected speeds due to the impact of vertical shear in z background mean flow U (e.g., Killworth et al. 1997). To check if low-frequency signals in the NIO satisfy θ ≤ θ Rn in the interior ocean, we analyzed a 27-year record of daily, meridional, geostrophic, surface velocity v computed from ADT distributed by AVISO. Figure 7.6 plots spectra of v, Sv , as a function of latitude θ. Superimposed on the plot are are critical-latitude curves, θ Rn (θ), for the first (solid) and second (dashed) baroclinic modes. Consistent with theory, large Sv amplitudes tend to be confined equatorward of θ R1 , but there is also variability that lies outside θ R1 (σ). As noted in the previous paragraph, a possible reason for the latter is that they are Rossby waves with faster propagation speeds due to U . Another possibility is that they are locally-forced responses to the wind, rather than free waves. A third is that they are associated with mesoscale eddies: Chelton et al. (2007) concluded that more than 50% of the variability over much of the global ocean is accounted for by such eddies. There is also general support for the existence of a coastal critical latitude, in that coastally-trapped signals only exist poleward of boundary-generated Rossby waves (Video 1.1). On the other hand, given the complexity of realistic boundary shapes, it will be difficult to obtain observations that confirm its value is given by y Rc in (7.53).

7.5.2 Meridional Energy Propagation A distinctive property of theoretical, boundary-generated Rossby waves is that their energy propagates meridionally as well as westward (Videos 7.5b–7.5e and 7.6b– 7.6e). Despite their prominence in solutions, it is difficult to find observational support

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7 Midlatitude Waves

Fig. 7.6 Plot of the spectral density of surface, meridional, geostrophic velocity v determined from daily, AVISO sea-level data during 1993–2019 (Sv , shading, cm/s) with a spatial resolution of 0.25◦ , showing Sv as a function of latitude θ and frequency σ. Spectra of v are calculated by averaging Fast Fourier Transforms (FFTs) of v along 8 latitude lines extending from 2–20◦ north and south of the equator along 64◦E, 69◦E, 84.5◦E, and 89.5◦E. Along each section, the FFTs are obtained for three, 9-year subsets of the record: 1993–2001, 2002–2010, and 2011–2019, so that the number of FFTs averaged to obtain Sv is 24. The range of σ covers variability with periods of about 12 days to 9 years. Tick marks and letters along the top of the figure indicate noteworthy periods: annual (A), semiannual (S), and 90, 60, 30, and 15 days. The arrow indicates the period range of intraseasonal oscillations. Curves for θ R (σ) are included for the first (solid) and second (dashed) baroclinic modes

for this process in the NIO. One reason is the narrow widths of the Bay of Bengal and Arabian Sea, which prevent the outer edges of lower-frequency, Rossby-wave packets from being clearly visible. Another is the strong forcing in both basins, which blurs the boundary signals. A third possibility is nonlinear instabilities, which weaken or destroy the smaller-scale parts of the packets. For example, Cheng et al. (2017) noted that much of the small-scale variability in the northern bay is eddy-like (Sect. 4.7.2). Qiu et al. (2013) and Xia et al. (2020) explored the stability of boundary packets using a nonlinear, 1 21 -layer model, finding that their small-scale components were unstable to triad interactions that lead to eddy formation. In the solutions shown in Videos 7.5b, 7.5e, 7.6b, and 7.6e, the impact of instabilities on small-scale Rossby waves is parameterized as damping by linear horizontal viscosity.

Video Captions

227

7.5.3 Zero-Group-Velocity Resonance A noticeable feature in Fig. 7.6 is that higher-amplitude signals appear to align with θ R1 (green areas slightly above the solid curve) at periods of 60–100 days. In their analysis of altimeter data, Lin et al. (2008; 2014) noted a similar property throughout the world ocean, namely, that spectra of variability at periods from about 20–160 days tend to peak near θ R1 . They argued that this intensification happened because the group velocity of n = 1 Rossby waves vanishes at θ R1 : The frequency of Rossby waves at θ R1 is σ = σ R1 and the dispersion curve has zero slope at this value (as illustrated by the horizontal blue line in the bottom-right panel of Fig. 7.2). Consequently, wave energy near θ R1 doesn’t radiate efficiently from a wind-forced region, allowing the amplitude of the response to build up over time. (Video 7.3b provides an idealized example of a near-resonant Rossby wave.) Lin et al. (2008; 2014) suggested further that much of the variability, which Chelton et al. (2007; 2011) had concluded were eddies generated by unstable background currents, were near-resonant Rossby waves of this sort. This idea was questioned by Hughes and Williams (2010), who found that observed spectral peaks often occurred at frequencies higher than allowed by linear theory (as they do in Fig. 7.6), and attributed the shift to nonlinear instability of background currents. We return to this issue again in Sects. 8.6 and 15.4.2.

Video Captions Gravity/Rossby Waves Video 7.1 Gravity waves generated by an initially specified, small-scale (δ-functionlike) pressure field, p(x, y, 0) = p0 (1 + cos πr/R) /2, where p0 /g = 10 cm, r 2 = (x − xm )2 + (y − ym )2 , x m = (xm , ym ) = (90◦E, 10◦N) is the midpoint of the basin, and R = 1◦ . The Coriolis parameter is constant and evaluated at 10◦N. Closed conditions are applied at basin boundaries. Video 7.2a As in Video 7.1, except for an initially-specified, large-scale, pressure field with x m = (60◦E, 30◦N) and R = 10◦ . The Coriolis parameter is constant and evaluated at 30◦N. Open eastern and western boundary conditions are imposed as described in Appendix C. Video 7.2b As in Video 7.2a, except that the Coriolis parameter is specified by the mid-latitude β-plane approximation (C.2). Video 7.3a Response generated by a meridionally-narrow, y-independent band of zonal wind τ x , oscillating at a period of P = 90 days. Profile X(x) is given by (C.7b) with τo = 10 dyn/cm1 , x = 0.4◦ , and Y(y) = 1. The Coriolis parameter f is given by the equatorial β-plane approximation, and open boundary conditions are imposed as described in Appendix C.

228

7 Midlatitude Waves

Video 7.3b As in Video 7.3a, except with P = 180 days. Video 7.3c As in Video 7.3b, except with νh = 5×105 cm2 /s.

Kelvin Waves Video 7.4a Coastal Kelvin wave generated by initially specified pressure and zonalvelocity fields, p(x, y, 0) = po X(x)Y(y) and u(x, y, 0) = p(x, y, 0)/c1 , where: p0 /g = 10 cm; X(x) = cos(πx  /x) θ(x 2 /4 − x 2 ) with x  = x − xm , xm = 80◦ , and x = 10◦ ; and Y(y) = exp[(y − 100◦ )/R] with R = c1 / f = 105 km, the acrossshore structure of a theoretical Kelvin wave. The Coriolis parameter is constant and evaluated at 10◦N. Cyclic conditions are imposed. Video 7.4b As in Video 7.4a, except showing the response when u(x, y, 0) = 0. Video 7.4c As in Video 7.4a, except showing the response for a narrower initial state with x = 2◦ . Video 7.4d As in Video 7.4a, except in a bounded basin.

Eastern-boundary Waves Video 7.5a Eastern-boundary response generated by a zonal wind τ x along the southern boundary, oscillating at a period of P = 60 days. Profile X(x) is given by (C.7b) with τox = 1.5 dyn/cm2 , xm = 82.5◦E, and x = 5◦ ; profile Y(y) = exp(10◦ − y)/R with R = c1 / f (10◦N ) = 105 km; and T(t) = sin(σt)θ(t) with σ = 2π/P. The Coriolis parameter is specified by the midlatitude β-plane approximation (C.2). The eastern boundary is closed, and open conditions are imposed at the western boundary as described in Appendix C. Video 7.5b As in Video 7.5a, except with P = 180 days, τo = 2 dyn/cm2 , xm = 30◦E, and x = 20◦ . Video 7.5c As in Video 7.5b, except with νh = 5×105 cm2 /s. Video 7.5d As in Video 7.5b, except with P = 365 days and νh = 0. Video 7.5e As in Video 7.5d, except with νh = 5×105 cm2 /s.

Western-boundary Waves Video 7.6a Western-boundary response generated by a zonal wind τ x along the northern boundary, oscillating at a period of P = 60 days. Profiles X(x) and Y(y) are

Video Captions

229

given by (C.7b) with τox = 1.5 dyn/cm2 , x m = (10◦E, 50◦N), x = 10◦ , and y = 5◦ ; and T(t) = sin(σt)θ(t) with σ = 2π/P. The Coriolis parameter is specified by the midlatitude β-plane approximation (C.2). The western boundary is closed, and open conditions are imposed at the eastern boundary as described in Appendix C. Video 7.6b As in Video 7.6a, except with P = 180 days. Video 7.6c As in Video 7.6b, except with νh = 5×105 cm2 /s. Video 7.6d As in Video 7.6a, except with P = 365 days. Video 7.6e As in Video 7.6d, except with νh = 5×105 cm2 /s.

Chapter 8

Equatorial Waves

Abstract Near the equator, free-wave solutions are found under the assumption that the Coriolis parameter is given by f = βy (the equatorial β-plane approximation), which allows them to be represented as expansions in Hermite functions φ j , j = 0, 1, 2, . . .. Consequently, equatorial Rossby and gravity waves form a discrete set, with each wave corresponding to a specific j value. The j = 0 wave is a new type of wave, the mixed Rossby/gravity (Yanai) wave, which, depending on its zonal wavenumber, has properties similar to a Rossby or gravity wave. An equatorial Kelvin wave also exists. Solutions for the structures of these waves and their dispersion relations are obtained. Similarities between midlatitude and equatorial Rossby/gravity waves are noted: The two sets describe the same waves, differing only because of the approximation of f used to obtain them. Keywords Hermite functions · Rossby/gravity waves · Yanai and Kelvin waves · Phase and group velocities · Critical frequency and latitude · Similarity to midlatitude waves Near the equator f is no longer a slowly varying variable, since variations in f are as large, or larger than, f itself. As a result, equatorial waves cannot be represented in the simple form (7.8), that is, as a plane wave with a slowly-varying, meridional wavenumber  and amplitude V. On the other hand, it is still possible to find wave solutions analytically when f and β are set to their low-latitude limits, f = βy,

β = 2/Re

(8.1)

(the equatorial β-plane approximation). A limitation is that solutions are accurate only where (8.1) is a reasonable approximation; however, this restriction is not severe in the NIO since both f and β in (8.1) differ from their exact values by less than 10% for θ < 26◦ .

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_8. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_8

231

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8 Equatorial Waves

When f = βy, it is convenient to replace y in (7.1c) with a non-dimensional variable  β η= y, (8.2) cn in which case it can be written −

  1 vttt + vx xt + α02 ∂ηη − η2 vt + βvx = 0, c2

(8.3)

√ where α0n = β/cn . Because η is not constant, solutions to (8.3) are not plane waves. Instead, given the form of the operator in parentheses in (8.3), their meridional structures must be expressed in terms of Hermite functions, rather than have the simple form exp (iy). √ −1 = R0n = cn /β for mode Note that α0n defines a characteristic length scale, α0n n. As we shall see, the meridional scale of equatorial waves is determined by R0n . As such, it is analogous to the midlatitude Rossby radius (Rn = cn / f ), and is referred to as the “equatorial Rossby radius of deformation” √ for the nth vertical mode. With the estimate that cn ≈ 250/n cm/s, R0n ≈ 330/ n km, considerably larger than midlatitude values of Rn . We begin with a review of properties of Hermite functions that are used throughout the book (Sect. 8.1). Then, we derive expressions for equatorial, gravity and Rossby waves (Sect. 8.2), mixed Rossby-gravity wave (Sect. 8.3), and Kelvin wave (Sect. 8.4), and comment on their similarity to midlatitude waves (Sect. 8.5). We conclude with a brief review of observations of equatorial waves in the NIO and elsewhere (Sect. 8.6).

8.1 Hermite Functions Consider the solutions (eigenfunctions) φ j (η) of   ∂ηη − η2 φ j = λ j φ j ,

(8.4a)

subject to the far-field condition lim φ j (η) = 0.

η→±∞

(8.4b)

Solutions exist only when the constants λ j (eigenvalues) have the specific values λ j = −(2 j + 1),

j = 0, 1, 2, . . . ,

(8.4c)

8.1 Hermite Functions

233

as they blow up as η → ±∞ for any other choices of λ j and, hence, are not physically realistic. Their amplitudes are not specified by the above equations. They are set by the additional (arbitrary) constraint 

∞ −∞

φ 2j dη = 1,

(8.4d)

the normalization condition for the functions. The solutions defined by Eqs. (8.4) are known as Hermite functions. The Hermite functions have a number of useful properties. Following the same steps as for the vertical modes ψn (see the discussion of Eqs. 5.11 and 5.12), Eqs. (8.4a), (8.4b), and (8.4d) imply that they are orthogonal (orthonormal), that is,  ∞ φ j φ j  dη = δ j j  , (8.5) −∞

where δ j j  is the Kronecker-delta symbol (δ j j  = 1 when j = j  and is zero otherwise). Further, they are determined by the generating function j (−1) j 2 η2 /2 d e e−η . φ j (η) =   21 √ j dη 2 j j! π

(8.6)

Finally, they have the recursion relations, 

 j +1 j φ j+1 + φ j−1 , ηφ j = 2 2   j +1 j φ j+1 + φ j−1 , φ jη = − 2 2

(8.7a)

(8.7b)

essential relations for manipulating expressions involving Hermite functions. Since Eqs. (8.4) comprise a Sturm-Liouville problem, we know that the Hermite functions form a complete set, which means that any function q(η) can be represented by ∞  q(η) = q j φ j (η). (8.8a) j=0

With the aid of (8.5), the expansion coefficients q j are given by  qj =

∞ −∞

q(η)φ j (η)dη,

(8.8b)

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8 Equatorial Waves

and then from (8.7) that  (ηq) j =   qη j =



j +1 q j+1 + 2 j +1 q j+1 − 2

 

j q j−1 , 2

(8.9a)

j q j−1 , 2

(8.9b)

where the derivation of (8.9b) requires an integration by parts. Figure 8.1 plots Hermite functions φ j for j = 0–5 (top panels) and for j = 2 and 40 (bottom panels). Note that index j is the number of zero crossings for each function, and that they alternate in j between being antisymmetric (top-left panel) and symmetric (top-right panel) about the equator. The symmetry   property follows 1 from the lowest-order Hermite function, φ0 = π − 4 exp −η2 /2 , being symmetric and the higher-order Hermites all being related by  φ j+1 =

  1 ηφ j − φ jη , 2 ( j + 1)

(8.10)

which results from differencing Eqs. (8.7): Since φ0 is symmetric, (8.10) ensures that all the higher-order φ j ’s are symmetric or antisymmetric and that φ j+1 and φ j have opposite symmetry. The Hermite functions are less equatorially trapped asj increases. This property can be understood by rewriting (8.4a) in the form φ jηη = η2 − (2 j + 1) φ j , which shows that φ j oscillates when |η| is less than ηj =



2j + 1

(8.11)

and decays exponentially for |η| > η j . For this reason, η j is referred to as the “turning latitude” of φ j . It measures the meridional width of φ j , which slowly increases with j. Figure 8.1 (bottom panel) illustrates the structure of a typical, higher-order Hermite function, the j = 40 mode for which the turning latitude is η40 = 9. With R01 = 330 km, η = 10 corresponds to a latitude of 30◦ (3300 km) so that for the n = 1 mode φ40 extends well off the equator. Note that between its turning latitudes φ40 is sinusoidal with an amplitude and wavelength that increase slowly with |η|. Thus, φ40 (and other higher-order Hermite functions) has a structure like that in (7.8), except that it is cut off at the turning latitudes η j . It is, in fact, possible to extend WKB theory to obtain useful, approximate expressions for high-order, and even low-order, Hermites (Ghatak et al. 1991; see Example 5.1 in their Sect. 5.3).

8.2 Equatorial Gravity and Rossby Waves

235

Fig. 8.1 Plots of Hermite functions φ j for j = 0–5 (top panels) and j = 2 and 40 (bottom panel) as a function of η. Index j is the number of zero crossings for each function, the functions alternate in j from being between antisymmetric (top-left panel) to symmetric (top-right panel) about the equator. For the j = 40 function, the solution varies approximately sinusoidally between the turning latitudes (η40 = 9), with an amplitude and wavelength that increase slowly with |η|. Adapted from Fedorov and Brown (2009; top panels) and Ascani (2000, priv. comm.; bottom panel), respectively

8.2 Equatorial Gravity and Rossby Waves We look for solutions to (8.3) of the form v j (x, y, t) = V j φ j (η) exp(ikx − iσ t).

(8.12a)

According to (8.12a), v j has the form of plane wave (6.2) except with exp (iy) replaced by the Hermite function φ j (η), and V j is an arbitrary constant amplitude. The corresponding u j and p j fields are then found from (7.1a) and (7.1b). Inserting (8.12a) into these equations, and using recursion relations (8.7) to eliminate the ηφ j and φ jη terms, gives uj =

σ0 V j −j (η, k, σ ) exp(ikx − iσ t), iσ

pj σ0 = V j +j (η, k, σ ) exp(ikx − iσ t), c iσ

(8.12b)

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8 Equatorial Waves

where σ0n =

√

βcn and  ±j (η, k, σ )

=

j + 1 φ j+1 (η) ± 2 ck/σ − 1



j φ j−1 (η) . 2 ck/σ + 1

(8.12c)

As we shall see, σ0n provides a characteristic frequency for equatorial inertial oscillations (Sect. 10.2.1.1), and for this reason it is referred to as the “equatorial inertial frequency” for vertical mode n. Note that, owing to the factors of i −1 in (8.12b), u j and p j are in quadrature with v j with their phase lagging by π/2 radians. In addition, the meridional structures of u j and p j , ±j (η, k, σ ), are not proportional to φ j (η) but rather involve combinations of φ j−1 (η) and φ j+1 (η), and hence they have a different symmetry than v j . Equatorial waves with symmetric u j and p j are referred to as being “symmetric.” With this definition, equatorial waves for odd (even) values of j are symmetric (antisymmetric). Figure 8.2 plots the meridional structures φ j (black curves), −j (red curves), and +j (blue curves) for several equatorial waves when c = 265 cm/s (the value used in most of the videos), showing: j = 1 Rossby waves with periods P = 180 (top-left panel) and 32.9 (top-right panel); and a j = 1 gravity wave with P = 5.65 days (bottom-left panel). Wavenumbers associated with each period are not arbitrary, but rather are obtained from dispersion relation (8.13a) discussed next, and frequency/wavenumber pairs for each of the waves are indicated by the bullets that lie on red curves in Fig. 8.3. Figure 8.2 can also be interpreted as showing the meridional structures of v1 , u 1 , and p1 /c by multiplying the ± 1 structures by the factor σ01 /σ : It increases them by factors of σ01 /σ = 18.7 and 3.41 for the 180-day and 32.9-day Rossby waves, respectively, and decreases them by 0.59 for the gravity wave.

8.2.1 Dispersion Relation The dispersion relations for equatorial, Rossby and gravity waves are obtained by inserting (8.12a) into (8.3), and using (8.4a) to eliminate the η-dependent terms, φηη and η2 φ. The result is  σ2 2 2 (8.13a) σ k + α j − 2 + kβ = 0, c where α 2j = α02 (2 j + 1), so that there is a curve for each j value. 8.2.1.1

Non-dimensional Dispersion Curves

The solid curves in Fig. 8.3 plot the dispersion relations for the first 3 Rossby and 11 gravity waves (colored and gray curves). The curves are solutions to (8.13a)

8.2 Equatorial Gravity and Rossby Waves

237

Fig. 8.2 Meridional structures of several equatorially-trapped waves, plotting φ j (black curves), −j (red curves), and +j (blue curves). The panels show curves for: j = 1 Rossby waves with P = 180 (top-left panel) and 32.9 (top-right panel) days; the j = 1 gravity wave with P = 5.65 days (bottom-left panel); and the j = 0 Yanai wave for all periods (bottom-right). The characteristic speed is c = 264.67 cm/s, and values of k needed to define ±j for the Rossby/gravity waves are obtained from dispersion relation (8.13a) with σ = 2π/P. For the Yanai wave, only − 0 is plotted + =  , and the structures are valid for all P since they don’t depend on σ or k. The plots since − 0 0 can be reinterpreted as showing the meridional structures of v j , u j , and p j /c fields by multiplying the ± 1 structures by σ0 /σ

with real k (i.e., for trigonometric waves). (The red-dashed curves plot the real and imaginary parts of the j = 1, evanescent wave, discussed at the end of this subsection.) As is commonly done, the curves are plotted in the non-dimensional coordinates, σ  = σ/σ0 and k  = k/α0 , in which case (8.13a) simplifies to   σ  k 2 + 2 j + 1 − σ 2 + k  = 0.

(8.13b)

The advantage is that (8.13b) contains no factors of cn , so it is possible to plot curves for all vertical modes in a single diagram. To illustrate how (8.13b) applies to all vertical modes, consider the equatorial waves that are possible at P = 15 days. The horizontal dashed lines in Fig. 8.3 plot √ σ  = σ/σ0 = σ/ βcn , σ = 2π/P, for the n = 1, 5, 9, 14, 20, 25, and 50 modes

238

8 Equatorial Waves

Fig. 8.3 Dispersion curves for: the j = 1 (red), 2 (blue), and 3 (green), equatorially-trapped, Rossby/gravity waves, with additional gravity-wave curves for j = 4–11 (gray); the Yanai wave, which intersects the point, k  = 0, σ  = σ0 (Sect. 8.3); and equatorial Kelvin wave with the dispersion relation σ  = k  (Sect. 8.4). The curves are plotted in the non-dimensional coordinates, σ  = σ/σ0 and k  = k/α0 , which allows the dispersion curves for all vertical modes to be plotted in a single diagram. The horizontal dashed lines indicate the values of σ  for the n = 1, 5, 9, 14, 20, 25, and50 vertical modes when P = 15 days. The black and red, dashed curve is the hyperbola  σ  = −1/ 2k  , which intersects the gravity- and Rossby-wave dispersion curves at their extreme (critical) values. The red-dashed curves are the real and imaginary parts of the j = 1 evanescent wave. Bullets indicate the four waves displayed in Fig. 8.2 plus the Kelvin wave at P = 15 days

using cn values determined from the realistic Nb (z) profile in Fig. 5.1. (Given that the cn values in Table 5.1 are much the same for both profiles, the horizontal lines inversely are similarly located for the constant-Nb profile.) Because cn is roughly √ proportional to n, the σ  lines shift upwards approximately like n (Sect. 5.2.4). Possible equatorial waves occur wherever a σ  line intersects a dispersion curve. For the n = 1 and 5 modes, σ  doesn’t intersect any Rossby- or gravity-wave curves, but only those for the Kelvin and Yanai waves (discussed below). For the n = 9 mode, two gravity waves are possible for the j = 1 curve. For the n = 14 mode, two waves are possible for both the j = 1 and 2 curves (just barely for n = 2). For the n = 20 and 25 modes, two waves exist for the j = 1, 2, and 3 curves (and two more for the j = 4 curve are barely missed for n = 25). At periods longer than 15 days, the σ  lines in Fig. 8.3 shift downwards (see Fig. 16.1). For example, when P = 30 days the σ  lines in Fig. 8.3 are closer to the k  -axis by a factor of 2 (Fig. 16.1, top panel), still no Rossby waves are possible and gravity waves exist only for n ≥ 33. When P = 60 days, the σ  lines are closer by a factor of 4 (Fig. 16.1, middle panel), the n = 1–4 lines now intersect Rossby-wave

8.2 Equatorial Gravity and Rossby Waves

239

dispersion curves, and gravity waves exist only for very high n values (roughly n ≥ 4×33). At P = 180 days (Fig. 16.1, bottom panel), many Rossby waves are possible as σ  lines intersect their dispersion curves for n ≤ 34. Note that the midlatitude dispersion curves (7.2a) are identical to those of the jth equatorial wave in (8.13b) with the replacements  = 0, f → f j = σ0 η j where η j is given in (8.11), and β = 2/Re . Thus, the set of Rossby/gravity curves in Fig. 8.3 corresponds to a set of midlatitude curves in Fig. 7.2 plotted with f = f j and β = 2/Re . Note that the separation of the gravity and Rossby bands in Fig. 8.3 is much less than at midlatitudes (Fig. 7.2, bottom panels); the smaller separation happens because the values of f j in Fig. 8.3 are much smaller than the value of f at 15◦ used in Fig. 7.2.

8.2.1.2

Critical Frequencies

As for midlatitude Rossby/gravity waves (Sect. 7.1), it is useful to rewrite (8.13a) in the form  β 2 = r 2, k+ 2σ

r 2 = α02

 βc σ2 . − j + 1) + (2 4σ 2 βc

(8.14)

According to (8.14), trigonometric waves exist (r 2 is real) only when σ is small enough (Rossby waves) or large enough (gravity waves) for the sum of the first and second terms in brackets to be larger than the third. For a fixed j there are two critical frequencies, σ R j and σG j , such that Rossby (gravity) waves are trigonometric when σ < σ R j (σ > σG j ). They are determined by setting r 2 = 0 in (8.14), and the resulting roots are 

 σ0 

σG j =√ j +1± j , (8.15) σR j 2 where σ0 =

8.2.1.3

√ βc is the equatorial inertial frequency (also see Sect. 10.2.1).

Critical Latitude

Again setting r 2 = 0 in (8.14), but this time for a specified σ and solving for j = jcr , gives √ σ 2 1 a − a −1 , a= 2 , (8.16) jcr = 4 σ0 which generally is not an integer. From its definition, r 2 is positive and Rossby/gravity waves exist only if the value of j for a particular wave satisfies j ≤ jcr . It is possible to interpret jcr in terms of a critical latitude, ycr . Recall from (8.11) that the maximum latitudinal extent of a Hermite function φ j is measured by its turning latitude η j . Since all Rossby/gravity waves have j ≤ jcr , it follows that η j

240

8 Equatorial Waves

evaluated with j = jcr provides a measure of the critical latitude. Thus, for a fixed σ trigonometric Rossby/gravity waves exist only at latitudes y such that y < 2

2 ycr

ηj 2 jcr + 1 1 ≡ 2 = = 2 2 α0 α0 α0

  1 a 2 + a −2 −1 2 +1 = . a−a 2 2α02

(8.17)

This inequality is the equatorial analog of (7.12) for midlatitude waves. Note that with f = βy in (7.12), the two inequalities are the same. Because midlatitude gravity and Rossby waves are highly separated in frequency (σ R  σG ), in Sect. 7.1.4 we defined separate critical latitudes for Rossby and gravity waves, y R and yG (or θ R and θG ). For equatorial waves, the corresponding inequality (σ R j  σG j ) does not hold for small j values, and so we don’t make that separation for either jcr or ycr in this book. Nevertheless, whether ycr ( jcr ) refers to the critical latitude ( j value) of an equatorial Rossby or gravity wave is obvious from the context of the discussion.

8.2.1.4

Evanescent Waves

Finally, as at midlatitudes, evanescent equatorial waves are possible when r 2 < 0 in (8.14), that is, for frequencies that satisfy σ R j < σ < σG j . In Fig. 8.3, these waves lie between Rossby- and gravity-wave curves with the same j value. As for trigonometric waves, their dispersion relation is given by (8.12a) and (8.12b), the only difference being that k is complex. Expressed in non-dimensional variables, the real          2 1 − 4σ 2 j + 1 − σ 2 , and and imaginary parts of k are − 1/σ and ± 1/σ the red-dashed curves in Fig. 8.3 plot both parts for j = 1. For a fixed σ , evanescent waves exist at latitudes poleward of ycr . So, in forced problems evanescent waves are generated whenever the wind extends poleward of that latitude (Chaps. 13 and 15). They are also excited along eastern and western ocean boundaries, where at latitudes higher than the critical latitude they superpose to form β-plane Kelvin waves (Moore 1968; Sects. 7.2.2 and 15.2).

8.2.2 Phase and Group Velocities The phase and group velocities of equatorial, gravity and Rossby waves are also similar to their midlatitude counterparts. In particular, the phase velocities of Rossby waves are always directed westward, whereas their group velocities are either westward or eastward for long- and short-wavelength Rossby waves, respectively. In Fig.  8.3,  the two wave types are located to the right and left of the curve k  = −1/ 2σ  (dashed curve). As for midlatitude waves, the Rossby-wave dispersion relation reduces to a nondispersive form

8.2 Equatorial Gravity and Rossby Waves

σ =−

241

c k ≡ cr j k 2j + 1

(8.18)

in the limit of small |k|. Eliminating σ from Eqs. (8.12) using (8.18), and for convenience replacing wave amplitude for v j with one for p j by the replace  the arbitrary ment V j → ikcr j /cσ0 P j , the solution becomes vj = −



Pj ikeik (x−cr j t ) φ j , σ0 (2 j + 1)

where

1 ˆ ±j (η) = − √  2 2



P j ik (x−cr j t ) ∓ uj ˆ j (η) e =  p j /c c

φ j+1 φ j−1 ∓ √ √ j +1 j

(8.19a)

.

(8.19b)

ˆ ±j , a key simplification for the Note that, in comparison to (8.12c), there are no k’s in  derivation in the next paragraph. Because equatorial winds have a large zonal scale, nearly-nondispersive, equatorial Rossby waves like (8.19a) are prominent in the ocean (Sect. 8.6). Further, they are the only waves present in the “long-wavelength” approximation of the equations of motion (Chap. 14). For future reference, we note that packets of wave solutions (8.12) that satisfy dispersion relation (8.18) can be combined into a simple form. To see this property, we follow a derivation similar to the one that results in (6.11). Suppose that the initial (t = 0) zonal structure of the packet is P j (x), which has the Fourier transform P˜ j (k) (Sect. 9.14). With the replacement P j → P˜ j (k) / (2π ) in (8.19a), the wave represents the wave in the packet with wavenumber k. The complete packet is then the sum over waves for all possible k’s,

vj = − 

1 2π



∞

−∞

ik P˜ (k) e−ikcr j t eikx dk



φj , σ0 (2 j + 1)



 ˆ±  ∞ j 1 uj =− P˜ (k) e−ikcr j t eikx dk . p j /c 2π −∞ c

(8.20a)

(8.20b)

  The terms in brackets are just the inverse Fourier transforms of P˜ (k) exp −ikcr j t   and ik P˜ (k) exp −ikcr j t , which can be easily evaluated using the transform pairs −iak ↔ q(x − a), q(k)e ˜

−iak ik q(k)e ˜ ↔ qx (x − a),

(8.21)

with q˜ = P˜ j and q = P j (see the derivation of Eq. 9.38). Applying (8.21) to (8.20) gives 1 Px (x − cr j t)φ j , vj = − σ0 (2 j + 1)



1 uj ˆ ±j . = P(x − cr j t) p j /c c

(8.22)

242

8 Equatorial Waves

According to (8.22), the signal propagates westward at speed cr j without any distortion, owing to the nondispersive nature of its component waves. Note that, as required, the zonal structure of p j at t = 0 is P(x).

8.2.3 Videos Videos 8.1a–8.1d show numerical solutions of j = 1 Rossby waves for a spread of σ and k values that illustrate their basic properties: for long- and short-wavelength Rossby waves; nearly non-dispersive waves; and when σ = σ R . The solutions are all generated by specifying an initial Rossby-wave packet, consisting of the structure of a j = 1 Rossby wave given by (8.12) with a specified period P = 2π/σ , wavenumber k determined from (8.13a), and modulated by a zonally-broad envelope X(x). This initial state is then allowed to develop freely in time. As demonstrated in the videos, the resulting solution is dominated by a primary wave with period P and wavenumber k; however, because the forcing is switched on and X(x) is bounded, it also contains secondary waves with other periods and wavenumbers. Video 8.1a shows the solution in response to an initial packet composed primarily of long-wavelength Rossby waves with a period P = 45 days. Consistent with theory, its group speed is westward and its meridional structure is close to that in Fig. 8.2. Because the wind is switched on abruptly, inertial oscillations are excited (Chap. 10) and they are visible throughout the video as higher-frequency oscillations. Further, because the sinusoidal wave is modulated by X(x), the packet also contains j = 1 Rossby waves with wavenumbers different from k. As time passes, these secondary waves disperse from the main packet, the waves with longer wavelengths and faster group speeds leading the packet and vice versa. As a result, the original packet gradually broadens. Video 8.1b shows the solution when the packet is composed of short-wavelength waves at P = 45 days. In this case, although phase within the packet still propagates westward, the packet itself propagates eastward, consistent with the eastward group velocity of short-wavelength Rossby waves. Because the packet is better formed (contains more primary wavelengths within the envelope), secondary waves are weaker and the spreading is much less than in Video 8.1a. Video 8.1c shows a packet of long-wavelength Rossby waves in the low-frequency, large-zonal-wavenumber (nearly non-dispersive) limit. In this case, the packet propagates westward with little distortion, as in (8.22) since it is composed almost entirely of non-dispersive Rossby waves (8.18). Nevertheless, the packet also contains some dispersive Rossby waves owing to the finite extent of X(x). As a result, the main packet develops a “tail” of dispersive waves with a slower group speed. Finally, Video 8.1d shows a packet of j = 1 Rossby waves when σ = σ R (P = 31.94 days). In Fig. 8.3, this wave is indicated by the left-most bullet, located at the top of the red curve. It is unique in that it has zero group velocity (the curve has zero slope). Consistent with this property, the main packet remains in the central ocean. As time passes, however, other (secondary) Rossby waves in the packet, due

8.3 Mixed Rossby-Gravity Wave

243

to finite extent of X(x), gradually broaden the initial disturbance, with longer waves extending westward and vice versa. We consider zero-group-velocity waves again in Sect. 15.4.2. A similar set of solutions exists for j > 1 Rossby waves. To illustrate, Video 8.1e shows a j = 2 Rossby waves in the low-frequency, long-wavelength limit. Consistent with theory, the packet propagates westward with little distortion and at a speed slower by the factor cr 2 /cr 1 = 3/5 than the j = 1 Rossby-wave in Video 8.1c. Note that, since j is even, the wave is antisymmetric, that is, u and p (v) are antisymmetric (symmetric) about the equator.

8.3 Mixed Rossby-Gravity Wave The j = 0 solution is a new type of wave not present at midlatitudes. Its uniqueness is apparent in (8.15), in that σG = σ R when j = 0, a statement that the gravity- and Rossby-wave bands are no longer separated. With j = 0, (8.13a) factors into   β σ σ k+ − = 0. k+ c σ c

(8.23)

The first root, k = −σ/c, is not a physically realistic solution since |u 0 | and | p0 | in (8.12b) are not well defined (the denominator of the φ j+1 terms is zero).1 The other root, σ β (8.24) k=− + , σ c does not have this problem and so is acceptable. Figure 8.3 also plots dispersion curve (8.24). For small (large) values of σ  , values of k  are close to those for short-wavelength Rossby (gravity) waves. For this reason, the j = 0 wave is referred to as a “mixed Rossby-gravity” wave; it is also often called the Yanai wave, after the scientist who first discovered the wave in the atmosphere. Note that the slope of the dispersion curve is always positive, so that the group velocity of Yanai waves is always directed eastward. In contrast, their phase velocity can be directed either eastward or westward, depending on whether σ  is less or greater than 1 (σ ≶ σo ), respectively. Yanai waves are prominent in the equatorial Indian Ocean at intraseasonal time scales, and they exhibit vertical, as well as horizontal, propagation (Sect. 16.1). The structure of a Yanai wave is given by (8.12a) and (8.12b) with j = 0 and k given by (8.24), that is,

1

When the basin has northern and southern boundaries, however, this root is realistic. It corresponds to an antisymmetric pair of the zonal Kelvin waves discussed in Sect. 7.2.1, one located on each boundary (Philander 1977; Cane and Sarachik 1979).

244

v0 = V0 φ0 exp (ikx − iσ t) ,

8 Equatorial Waves

1 σ u0 = i √ V0 φ1 exp (ikx − iσ t) , 2 σ0

p0 = cu 0 .

(8.25) Figure 8.2 (bottom-right panel) plots the meridional structures, φ0 and ± = φ 1 , for 0 the Yanai-wave fields. Because φ0 and φ1 do not depend on σ or k, the structures are and the same for all Yanai waves. On the other hand, the relative amplitudes of u 0 √ p0 /c with respect to v0 do change, being multiplied by the factor σ  = (σ/σ0 ) / 2, which for P = 15 days is 0.64. Videos: Videos 8.2a–8.2c show Yanai waves with the dominant periods of P = 15, 6, and 9.34 days, respectively. Similar to Videos 8.1a–8.1d, the solutions are generated by specifying an initial Yanai-wave packet that is subsequently allowed to develop freely in time. The packet consists of a Yanai wave (8.25) with a specified frequency σ , wavenumber k(σ ) determined from (8.24), and modulated by a zonallybroad envelope X(x). Consistent with the group speed of all Yanai waves being eastward, the envelopes of the wave packets propagate eastward in all the videos. In contrast, the direction of phase propagation for waves within the envelopes depends on P. In Video 8.2a, σ = 1 2π/(15 days) = 4.9×10−6 s−1 is less than σ0 = (βc) 2 = 7.8×10−6 s−1 ; therefore, k < 0 and the theoretical phase speed of the Yanai waves is westward, a property apparent in the numerical response. In contrast, in Video 8.2b with σ = 2π/(6 days) = 12.2×10−6 s−1 , σ/σ0 > 1 and the phase speed of Yanai wave is eastward. Finally, in Video 8.2c, σ = 2π/(9.34 days) = σ0 , and the theoretical Yanai wave has k = 0 and infinite phase speed. Consistent with this property, the Yanai waves in the primary packet simply flip their sign across the equator with a period close to 9.34 days. As time passes, the primary waves in each video are modified by secondary waves, owing to Yanai waves being dispersive. They tend to broaden the main packet, as secondary waves with larger (smaller) group velocity propagate ahead (behind) the primary one. In Video 8.2a, secondary waves with wavelengths longer (shorter) than 2π/k lead (lag behind) the main packet because k < 0. Conversely, in Video 8.2b for which k > 0, the opposite tendency happens. In Video 8.2c with k = 0, secondary waves with eastward (westward) phase propagation lead (lag behind) the main packet; as a result, after some time the main packet appears to split into parts during each cycle.

8.4 Equatorial Kelvin Wave A final, equatorially trapped wave is the equatorial Kelvin wave. As for the zonal Kelvin wave at midlatitudes, it has v ≡ 0 and so was missed in the preceding derivation. To find it, we set v = 0 in (6.1), and look for a free-wave solution in which u and p have the form q(x, y, t) = Q(y) exp(ikx − iσ t). In this case, the equations of motion simplify to

8.4 Equatorial Kelvin Wave

245

− iσ U + ikP = 0,

f P + P y = 0,

−iσ

P + ikU = 0. c2

(8.26)

The first and third equations imply σ = ±kc,

(8.27)

the dispersion relations for two possible waves. With the aid of (8.27), the first and second of Eqs. (8.26) give Py = − f U = − f

f k P = ∓ P = ∓α02 yP, σ c

(8.28)

which has the solution p=

Po

1 2 2 exp ∓ α0 y exp (ikx − iσ t) . 2 

(8.29)

The solution for σ = −kc grows exponentially in y, and must be discarded.2 The solution for σ = kc, however, the equatorial Kelvin wave, is acceptable. Its complete solution is  σ  p K = Po φ0 (η) exp i (x − ct) , c

uK =

k pK pK = , σ c

v K = 0,

(8.30)

where the y dependency in (8.29) is replaced by π 4 φ0 (η), Po = π 4 Po absorbs the 1 factor π 4 , and subscripts “K ” are included to differentiate the Kelvin wave from other equatorially trapped waves. The meridional structure of u K and p K , φ0 (η), is shown in Fig. 8.1 (top-right panel); it is the same structure as that of v0 for the Yanai wave in Fig. 8.2 (bottom-right panel). Individual Kelvin waves can be combined to form a spatially confined, wave packet pK , v K = 0, uK = (8.31) p K = P(x − ct)φ0 (η), c 1

1

which propagates eastward without any distortion. The derivation of (8.31) is essentially the same as that for a packet of non-dispersive Rossby waves (Sect. 8.2.2). Videos: Video 8.3a shows the solution in a cyclic ocean, generated by an initially specified, Kelvin-wave packet given by (8.31). Consistent with theory, a nondispersive Kelvin-wave packet is produced. As for f -plane coastal Kelvin waves (Videos 7.4), its zonal structure gradually develops a tail of short-wavelength oscillations due to numerical error. 2

With northern and southern basin boundaries, however, the discarded root (the “anti-Kelvin wave”) is realistic, corresponding to a symmetric pair of the zonal Kelvin waves (Sect. 7.2.1) on each boundary (Cane and Sarachik 1979). Note that a combination of both discarded waves with the same amplitude gives a coastal Kelvin wave along only one of the boundaries.

246

8 Equatorial Waves

Video 8.3b is the same as Video 8.3a, except showing the response when only the initial pressure is initially specified and u K = 0. As a result, in addition to the Kelvin wave the initial state also contains j = 1 Rossby and gravity waves, which also involve φ0 (η). The Kelvin-wave packet (red patch) quickly propagates eastward from its initial position, passing through the eastern edge of the video toward the end of January. Similarly, the westward-propagating Rossby-wave packet passes through the western boundary near the end of February. The short-period oscillations are gravity waves, with both eastward and westward phase and group velocities. Eventually, it becomes difficult to identify individual waves, because they interfere so strongly with each other.

8.5 Relationship to Midlatitude Waves Throughout this chapter, we have emphasized similarities between equatorial and midlatitude waves. Indeed, the gravity and Rossby waves in two regions are dynamically the same, differing only in the approximations of f and the meridional extent of the domain used to obtain them. Likewise, the equatorial Kelvin wave is dynamically similar to a coastal Kelvin wave along a zonal boundary in that both have v ≡ 0. The only equatorial wave for which there is no obvious midlatitude counterpart is the Yanai wave, as it mixes properties of both Rossby and gravity waves. One obvious difference between Rossby/ gravity waves at the equator and midlatitudes is that the former are a discrete set indexed by j whereas the midlatitude ones vary continuously in the meridional wavenumber . This difference, though, is not fundamental: It happens because equatorial waves satisfy boundary conditions (8.4b), whereas no boundary conditions were applied to obtain the midlatitude waves. To illustrate this point, suppose that we had found the constant- f waves of Section 7.1 in a channel with walls at y = ys and yn where v = 0, rather than in an unbounded ocean; then, to satisfy those boundary conditions, the continuously-varying factor exp (iy) must be replaced by a discrete set of (eigen)functions φ j (y) = sin  j (y − ys ), where  j = jπ/ (yn − ys ) and j = 1, 2, · · · . The WKB solution of Sect. 7.1.3 can also be extended to show how discreteness arises. There, we considered solution (7.11) in an oceanic region where F > 0 and  is real. At higher latitudes, however, f can become large enough to change the sign of F across the latitude y  where  = 0 (the “turning” latitude). For solution (7.11) to be applicable in the entire ocean, then, its trigonometric (y < y  ) and evanescent (y > y  ) parts must be “matched” across y  in such a way that the unphysical, exponentiallygrowing, evanescent wave is eliminated. This matching is equivalent to applying boundary conditions (8.4b) for the Hermite functions, and it requires  to be a discrete, rather than continuous, variable. (See Sect. 5 of Ghatak et al. 1991, for a detailed discussion of the matching and discretization.) To summarize, midlatitude and equatorial, Rossby and gravity waves are the same waves, differing only in the approximations used to obtain them and in the regions where they are usually applied. Given their close relationship, however, midlatitude-

8.6 Observations

247

wave properties are often assumed to hold at low latitudes (5–10◦N), and in principle equatorial-wave solutions are applicable even well off the equator. The choice of which formalism to use depends on the specific problem under consideration.

8.6 Observations Remarkably, all the equatorially trapped waves discussed above were first predicted theoretically and only later observed. Before satellite observations, their detection was difficult and sporadic, owing to sparse data coverage; for example, see Wunsch and Gill (1976), Luther (1980), Weisberg et al. (1979), and Knox and Halpern (1982) for early identifications of gravity, Yanai, and Kelvin waves. In the Indian Ocean, prominent examples of equatorial Kelvin and Rossby waves are associated with the Yoshida Jets and their reflection from the eastern boundary (Sect. 4.4.1), and examples of Yanai waves are associated with intraseasonal variability (Sect. 4.4.4). Figure 8.4 provides a recent, comprehensive analysis of satellite altimeter data for the Pacific Ocean, which is designed to see how closely variability there fits theoretical dispersion curves for equatorial waves (Farrar 2020, priv. comm.; Farrar and Durland 2012). The figure plots sea-level spectral density as a function of zonal wavenumber (deg−1 ) and frequency (day−1 ), and the data are filtered to show signals that are symmetric (left panel) or antisymmetric (right panel) about the equator. Also plotted are dispersion curves for the n = 1 (solid) and 2 (dashed) vertical modes, with red curves indicating waves that have the appropriate symmetry for each panel. Consistent with theory, regions of higher spectral density tend to concentrate along dispersion curves, indicating that a part of the variability results from equatorial waves. For gravity waves, symmetric j = 1 waves (left panel, dashed lines) and antisymmetric j = 2 waves (right panel, solid lines) are present for both n = 1 and 2 modes. At lower frequencies, there are indications of a tongue of higher spectral density along the n = 1 Kelvin-wave curve (left panel) and along the n = 1 and 2 Yanai-wave curves (right panel). Higher levels also cover the Rossby-wave curves. At periods P greater than about 50 days and wavelengths λ larger than about 20◦ (σ  0.02 day−1 and 0 > k  0.05 deg−1 in both panels), the high levels likely result from Rossby waves. Outside this range (P  50 days and λ  20◦ ), however, most of the variability is associated with Tropical Instability Waves (TIWs; Legeckis 1986) rather than Rossby waves. TIWs are nonlinear signals with properties similar to Rossby waves, which are generated by instability of equatorial currents (Philander 1976; 1978a; Yu 1992; Yu et al. 1995). They don’t occur in the Indian Ocean, due to the absence of quasi-steady, equatorial currents there. As for midlatitude waves, we might expect that equatorial waves at the critical frequencies, σG j and σ R j , can be strongly excited by the wind: At these frequencies, the dispersion curves have zero slope (are located at the bottom/top of the gravity/Rossby-wave curves in Fig. 8.4), so that the possible waves have zero group

248

8 Equatorial Waves

Fig. 8.4 Plots of the log of sea-level spectral density as a function of zonal wavenumber (deg−1 ) and frequency (days−1 ) determined from altimeter data over the Pacific region. The two panels show spectra for signals that are symmetric (left) and antisymmetric (right) about the equator. Also plotted are dispersion curves for cn = 280 (solid) and 170 (dashed) cm/s, which are representative of the characteristic speeds for the n = 1 and 2 vertical modes. In each panel, red curves indicate the waves with the appropriate symmetry for the panel, and black curves can be ignored. The white areas are at wavenumbers and frequencies that are too high to permit spectral estimates from the data. Courtesy of Farrar (2020, priv. comm.)

velocity. Although there are signals at these frequencies in Fig. 8.4, their amplitude is similar to, or smaller than, signals at other locations. See Sect. 15.4.2 for a discussion of why these resonant waves are not prominent.

Video Captions Rossby Waves Video 8.1a A j = 1, long-wavelength Rossby wave when P = 45 days. It is generated by specifying an initial Rossby-wave packet, consisting of the real parts of

Video Captions

249

v j , u j , and p j in solutions (8.12), with: x → x  = x − xm , xm = 90◦ ; t = 0; j = 1; c = 264 cm/s; σ = 2π/P; k = k (1) from (15.3b); and the amplitudes for each field determined by po /g = (σ0 /σ ) cV1 /g = 10 cm. Further, the three fields are modulated by a broad envelope X(x) = cos(π x  /x)θ (x 2 /4 − x 2 ) with x = 100◦ . The Coriolis parameter is specified by the equatorial β-plane approximation, and cyclic boundary conditions are imposed. Video 8.1b Similar to Video 8.1a, except for a packet of short-wavelength, j = 1 Rossby waves at P = 45 days. The packet is generated as in Video 8.1a, except that k = k (2) from (15.3b). Video 8.1c Similar to Video 8.1a, except for a packet of nearly non-dispersive, longwavelength, j = 1 Rossby waves. The packet is generated as in Video 8.1a, except that P is set to a value large enough for the Rossby-wave dispersion curve to be nearly non-dispersive, and x = 80◦ . Video 8.1d Similar to Video 8.1a, except for a Rossby wave with P = 31.94 days, the critical period corresponding to the critical frequency σ R1 (P = 2π/σ R1 ) for the value of c1 used in the video (c1 = 264.67 cm/s). Video 8.1e As in Video 8.1c, except for a packet of nearly non-dispersive, longwavelength, j = 2 Rossby waves.

Yanai Waves Video 8.2a Yanai wave when P = 15 days. It is generated by specifying an initial Yanai-wave packet, consisting of the real parts of v0 , u 0 , and p0 in (8.25), with: x → x  = x − xm , xm = 30◦ ; t = 0; c = 264 cm/s; σ = 2π/P; k is given √ by(8.24); 2g = 10 and the amplitudes of each field determined by po /g = (σ0 /σ ) cV0 / cm. Further, the three fields are modulated by a broad envelope X(x) = cos(π x  /x) θ (x 2 /4 − x 2 ), with x = 60◦ . The Coriolis parameter is specified by the equatorial β-plane approximation, and cyclic boundary conditions are imposed. Video 8.2b As in Video 8.2a, except with P = 6 days. Video 8.2c As in Video 8.2a, except with P = 9.37 days, xm = 50◦E, and x = 100◦ .

Kelvin Waves Video 8.3a Equatorial Kelvin wave generated by initially specified pressure and zonal-velocity fields, p(x, y, 0) = po X(x)Y(y) and u(x, y, 0) = p/c1 , where:

250

8 Equatorial Waves

  po /g = 10.2 cm; X(x) = cos(π x  /x)θ x 2 /4 − x 2 , with x  = x − xm , xm =   1 60◦ , and x = 20◦ ; and Y(y) = exp − 21 η2 , with η = y/R0 and R0 = (c1 /β) 2 = 351 km. The Coriolis parameter is specified by the equatorial β-plane approximation, and cyclic conditions are applied. Video 8.3b As in Video 8.3a, except showing the response when u (x, y, 0) = 0 initially.

Part IV

Forced Solutions

Chapter 9

Overview

Abstract This chapter provides an overview of the wind-forced solutions obtained in the rest of the book. It first notes common aspects among all the solutions: the equations solved, boundary and initial conditions imposed, and forcing structures used. Then, it reviews mathematical concepts involved in finding many of the solutions: the Dirac δ-function and its properties, and Fourier and Laplace transforms. To conclude, solution methods used in later chapters are illustrated by solving a simplified version of the LCS equations, one with f = 0 and no meridional wind. Solutions are obtained for both switched-on and periodic winds, using both direct and transform methods. Keywords Equations of motion · Boundary conditions · Initial and radiation conditions · Wind forcing · Dirac δ-function · Solution methods · Laplace and Fourier transforms In Part IV, we discuss basic processes that underlie most of the oceanic phenomena discussed in Chap. 4. Our general approach is to obtain a suite of wind-driven, analytic solutions to the LCS modal equations (5.16) under idealized settings, in which each solution isolates a particular process. Observations, as well as more realistic solutions, can then be interpreted as a combination of these simpler ones. The chapters are organized so that the discussion proceeds from simpler-to-more-complex concepts and mathematics: Ekman drift and the excitation of inertial oscillations, both at midlatitudes and near the equator (Chap. 10); steady-state Sverdrup circulation and its closure by boundary currents (Chap. 11); the ocean’s response to both switched-on and oscillatory winds, at midlatitudes, along coasts, and near the equator (Chaps. 12–15); vertically-propagating waves and undercurrents (Chap. 16); and the Indian Ocean’s shallow overturning cells (Chap. 17). In this introductory chapter, we first discuss common aspects among all the analytic solutions (Sect. 9.1). The next three sections introduce mathematical concepts

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_9. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_9

253

254

9 Overview

and methods used in later chapters, providing overviews of the Dirac δ-function (Sect. 9.2), transform methods (Sect. 9.3), and examples of solutions to a simplified equation set (Sect. 9.4).

9.1 Common Aspects Equations of motion for most of the analytic solutions are Eqs. (5.16) for a single baroclinic mode without horizontal mixing, u t  − f v + px = F,

(9.1a)

vt  + f u + p y = G,

(9.1b)

pt  + u x + v y = 0, c2

(9.1c)

where for notational convenience subscripts n are neglected, ∂t  = ∂t + γ , γ = γn = ¯ and G = τ y / (ρH) ¯ are forcings by zonal and meridγn = A/cn2 , and F = τ x / (ρH) ional wind stress. Horizontal mixing acts to broaden small-scale features of solutions; so, for almost all the large-scale solutions of interest here, its impact is small enough to be ignored, the exceptions being for those that describe the structure of narrow western-boundary currents (Sects. 11.2.1 and 13.2.3.2). In contrast, we often retain vertical mixing because it impacts several solutions significantly; since it has the form γ q in each of Eqs. (9.1), it can be absorbed into the operator ∂t  . Following the steps used to obtain (7.1), Eqs. (9.1) can be rewritten as u t  t  − c2 u x x = f vt  + c2 v yx + Ft  ,

(9.2a)

pt  t  − c2 px x = −c2 f vx − c2 v yt  − c2 Fx ,

(9.2b)

 f2 f 1  1 vt  t  t  + vx xt  + v yyt  − 2 vt  + βvx = 2 Ft  − Fx y − 2 ∂t  t  − c2 ∂x x G. 2 c c c c (9.2c) The two equation sets are equivalent. We obtain solutions using one set or the other, in some cases mixing equations between them. −

9.1.1 Approximations Despite the simplifications built into Eqs. (9.1), solutions are still difficult to obtain analytically. Therefore, most solutions are found to versions of Eqs. (9.1) that are simplified in one way or another. For some solutions, f is set to a constant (Sect. 10.1)

9.1 Common Aspects

255

or to βy (Sects. 10.2, and Chaps. 14 and 15). In others, the equations are modified from (9.1) by dropping one or the other of the acceleration/damping terms, u nt  and vnt  (Chaps. 13 and 14) or both of them (Chap. 12). As we shall see, the advantage of their neglect is that gravity waves are filtered out of the system, thereby simplifying the mathematics needed to obtain solutions. The limitation is that time-dependent solutions are valid only at low frequencies or (for switched-on forcing) at long times, and when their zonal and/or meridional scale is much larger than the Rossby radius of deformation Rn (Appendix B). For our purposes, these restrictions are not a problem since our focus is on low-frequency, large-scale, baroclinic (upper-ocean) phenomena. We note, however, that the large-scale requirement does not hold for the barotropic response for which R0 ∼ 3000 km; thus, solutions to the approximate equation sets in Chaps. 12, 13, and 14 are not applicable to time-dependent problems for which the barotropic mode is the major part of the response (e.g., tsunamis and tides).

9.1.2 Basin and Boundary Conditions Most analytic solutions are obtained either in an unbounded domain or with a single boundary along x = 0 or y = 0, the exception being for the solutions in Chap. 11 that have both. Where there is a boundary, conditions applied there are either the first or both of n · u = 0, n· (k × u) = 0, (9.3) where n and k are unit vectors in the x-y plane perpendicular to the boundary and in the direction of z, respectively. The first condition states that there can be no flow across the boundary (no-normal-flow condition). The second requires that there is no flow tangential to the boundary (no-slip condition), and is used only in ocean models that include horizontal mixing. Both conditions are applied in the video solutions and in most OGCM solutions. Because most analytic solutions obtained in this book neglect horizontal mixing, they impose only no-normal flow, the exceptions being for the boundary layers discussed in Sects. 11.2.1 and 13.2.3.2.

9.1.3 Radiation and Initial Conditions The construction of most solutions also requires application of the radiation condition there are no sources of energy other than the wind.

(9.4)

This constraint is needed in solutions that involve integrals in either x (Sect. 9.4, Chaps. 12, 14, and 15) or y (Chap. 13): It determines their lower limits, ensuring that

256

9 Overview

energy radiates out of the forcing region not into it. In coastal solutions (Chap. 13), the condition also requires that coastally-reflected waves either decay away from the coast or radiate energy offshore. When the forcing is switched on, additional initial conditions require that the ocean is in a state of rest before the wind switches on, that is, u(x, y, t) = v(x, y, t) = p(x, y, t) = 0,

t ≤ 0.

(9.5)

Solutions to the exact equation set, (9.1), increase smoothly from this initial state as time progresses. In solutions to the approximate equation sets that drop u t  or vt  , conditions (9.5) are still imposed; however, in these cases one or more of u, v, or p jumps to a non-zero value just after the wind switches on (i.e., at time t = 0+ ≡ lim→0 ).

9.1.4 Wind-stress Forcing To allow for useful analytic solutions, the wind-stress forcing has the variableseparable form (9.6) τ α = τoα X(x)Y(y)T(t), where α = x or y and τoα is the forcing amplitude (Appendix C). The structure functions, X(x) and Y(y), are usually bounded, so that the forcing is a wind patch confined within the interior ocean; exceptions are solutions that isolate Ekman drift in which X(x) = Y(y) = 1 so that the wind is spatially uniform (Chap. 10) and for a meridional coast when X(x) = 1 and the forcing is a wind band (Chap. 13). For all but one solution, the time dependence, T(t), is either steady (equal to 1), switchedon (a step function θ equal to 1 if t ≥ 0 and 0 for t < 0), or periodic (e−iσ t ), the exception being a solution forced by a switched-on periodic wind (θ e−iσ t ). These temporal variations are idealizations that allow for interpretable analytic solutions. In solutions forced by switched-on winds, for example, adjustment processes (transients) occur consecutively (separately) and hence are readily identifiable. In solutions forced by periodic winds, the same adjustments happen continuously and so are less apparent; however, in this case wave energy radiates from the forcing region along distinct ray paths (Sects. 6.3, 6.4, 7.3.3, 15.2.3, and 16.1). Of course, observed wind variations are never completely steady, instantly switched-on, or perfectly periodic. Nevertheless, the idealizations are sufficiently realistic to describe many (most) wind variations that occur in the NIO reasonably well, with prominent examples being the Southeast Trades (quasi-steady), episodic events like ENSO and the IOD (switched-on), and the monsoon annual cycle (periodic).

9.2 Dirac δ-function

257

9.2 Dirac δ-function The Dirac δ-function has proven to be extremely useful in mathematics and physics, and it appears in the derivations of many of the solutions obtained in this  book. It is defined by the properties that: δ(ξ ) = 0 if ξ = 0, is infinite at ξ = 0, and − δdξ = 1 even in the limit that  → 0. Given its unusual properties (infinitesimally narrow and infinite in amplitude), it is mathematically better to refer to δ(ξ ) as a “generalized function” or “distribution,” but it is still commonly referred to as a “function.” A proper mathematical treatment of the subject is beyond the scope of this book. In this brief introduction, we only review its properties that are useful for our purposes. One obvious representation of a δ-function is δ(ξ ) = θξ (ξ ),

(9.7)

the derivative of a step function. To visualize (9.7) better, it is helpful to represent θ in terms of traditional (smooth) functions in the limit that their width  tends to zero. One smooth choice (there are many others) is θ (ξ ) = 0 for ξ ≤ 0, is (1 − cos πξ/) /2 for 0 < ξ ≤ , and is 1 for ξ > . Its derivative is  π sin(π ξ/), 0 0+ , ensuring that the latter reduces to the former in the limit that t → 0+ . Small-time Response To obtain the small-time solution, we integrate (9.24) from t = 0 to 0+ to get u t (x, 0+ ) − u t (x, 0) − c2



0+

u x x dt = Fo X(x).

(9.25a)

0

The first of initial conditions (9.5) ensures that u t (x, 0) = 0 and, assuming that u does not involve an initial δ-function (verifiable at the end of the derivation), the timeintegral term is vanishingly small compared to the other terms. Therefore, (9.25a) simplifies to u t = Fo X and the small-time solution is u = Fo X(x)t

0 < t ≤ 0+ .

(9.25b)

Note that the constraint, u t = Fo X , also follows directly from the first of Eqs. (9.22) since p = 0 at t = 0; indeed, it is equivalent to (a replacement for) the initial condition on p. Long-term Solution For times t ≥ 0+ , δ = 0 and (9.23) no longer has any forcing. Thus, the large-time solution is a combination of the two free waves allowed by the homogeneous form of (9.23), namely, westward- and eastward-propagating gravity waves with the general form u = A(x + ct) + B(x − ct). (9.26) Imposing the initial condition that u(x, 0) = 0 gives A(x) + B(x) = 0

⇒

A x (x) + Bx (x) = 0.

(9.27a)

9.4 Examples

263

To ensure that the limit of u in (9.26) as t → 0+ is (9.25b), we impose the equivalent condition that u t = Fo X at t = 0+ . Using the property that qt (x ± ct) = ±cqx (x ± ct) to evaluate the time derivative of (9.26), it follows that c A x (x) − cBx (x) = Fo X(x)

(9.27b)

at t = 0+ . Combining Eqs. (9.27) yields A x (x) = −Bx (x) = Fo X(x)/ (2c), and integrating in x then gives A(x) =

Fo 2c



x

X(x  )d x  ,

B(x) = −

L

Fo 2c



x L

X(x  )d x  ,

(9.28)

where L and L  are, as yet unspecified, constants of integration. With A and B known at t = 0, their values at all other times are given by the replacements x → x ± ct so that A(x + ct) and B(x − ct) are known. The solution for t ≥ 0+ is then u(x, t) =

Fo 2c



x+ct

X(x  )d x  −

L

Fo 2c



x−ct L

X(x  )d x  .

(9.29a)

The choices for A and B in (9.28) ensure that u x (x, 0) = 0, not that u(x, 0) = 0: To satisfy the latter condition requires that L  = L. To determine L, we impose radiation condition (9.4), which requires that u vanishes in the far field. Consistent with this condition, the choices, L = L  = −∞, ensure that u vanishes both as x → −∞ and ∞. With these choices for the lower integration limits, we can define a new integration variable, ξ = x  ∓ ct, and rewrite the solution as u(x, t) =

Fo 2c



x

−∞

X (ξ + ct)dξ θ (t) −

Fo 2c



x −∞

X (ξ − ct)dξ θ (t).

(9.29b)

In this form, the solution is more easily seen to represent packets of free waves propagating away from the wind patch. By adding the step function θ (t) to (9.29b), the solution is valid for all times (not just for t ≥ 0+ ), a property that is easily verified by inserting (9.29b) into (9.24). Figure 9.1 (top panel) schematically illustrates (9.29b) at a time t after the integrands no longer overlap, showing: the x-structure of the forcing (green curve); and the two parts of the solution (dashed, red and blue curves) and their sum (solid black curve). Both parts have constant, non-zero values as x → ∞; however, because they have the same amplitude but opposite sign, their sum cancels for sufficiently large x. Thus, the total solution spreads from the forcing region realistically, with wavefronts advancing at the speeds ±c and u = Fo /2c between the fronts.

264

9 Overview

Fig. 9.1 Schematic illustrations of solutions (9.29b) and (9.32) in the top and bottom panels, respectively. They show the x-structure of the forcing (green curve), the propagating parts of the solutions (red- and blue-dashed curves), the steady-state part of p (C, magenta curve), and their sum (solid black curve) at a time after the integrands of the parts no longer overlap. The dashed curves are shifted slightly to ensure they don’t interfere with each other, their sum, or the x-axis

p Field With u known, it is straightforward to find p from the second or first of Eqs. (9.22). Using the second gives pt = −c2 u x = −

cFo cFo X(x + ct)θ (t) + X(x − ct)θ (t). 2 2

(9.30)

Note that pt vanishes as t = 0, and hence pt initially grows linearly from zero. To integrate (9.30) in time, we split pt into two parts p1t = −

cFo X(x + ct)θ (t), 2

p2t =

cFo X(x − ct)θ (t) 2

(9.31)

( pt = p1t + p2t ). Then, with the aid of the identity qt (x ± ct) = ±cqx (x ± ct), it follows that   Fo x Fo x p=− X (ξ + ct)dξ θ (t) − X (ξ − ct)dξ θ (t) + C(x), (9.32a) 2 −∞ 2 −∞ where C(x) is a time-independent function determined next. Finally, to ensure that p vanishes at t = 0, we must set  C(x) = Fo

x

−∞

X (ξ )dξ θ (t).

(9.32b)

9.4 Examples

265

(As in the derivation of u, the lower limits of the above integrals are set to ensure that all the boundary conditions are satisfied.) Figure 9.1 (bottom panel) illustrates solution (9.32). The figure plots the three pieces of the solution: the two gravity waves (dashed, red and blue curves), the response to local forcing (term C, dashed magenta curve), and the total response (solid black curve). As for u, although the individual pieces of p extend to ∞, their sum does not. At the time of the plot, p has already developed a tilt to balance Fo X , and the gravity-wave fronts continue to propagate away from the forcing region (solid black curve). Discussion The existence of an x-independent current u between the wave fronts is noteworthy. Why is it required? Integrating the second of Eqs. (9.22) over the domain gives the constraint  ∞  ∞  2 2 ∞ u x d x = −c u −∞ = 0, P= pd x. (9.33) Pt = −c −∞

−∞

According to (9.33), the total amount of p in the basin, P, is conserved. As the wave fronts propagate, they continually decrease (increase) p in the western (eastern) ocean. Given (9.33), the only way these changes can occur is for p to be transferred continually from the western to eastern ocean, and that transfer is accomplished by u. A dynamically similar jet occurs in the equatorial ocean, the “bounded Yoshida Jet” (Chap. 14); as in solution (9.29b), it develops via the radiation of waves from the forcing region, differing in that the waves are equatorial Kelvin and Rossby waves rather than gravity waves.

9.4.1.2

Transform Method

From the first of Eqs. (9.19), it follows that L [δ(t)] = 1. Then, taking the Fourier and Laplace transforms of (9.24) gives   2  u = Fo   u = (s − ikc) (s + ikc)  X. s + k 2 c2 

(9.34)

Solving for   u and expanding the denominator into partial fractions gives   u=

X X Fo  Fo   = 2 2 2 2 2ikc c k + s /c



 1 1 − . s − ikc s + ikc

(9.35)

Essentially, the solution is complete at this point: It only remains to find the inverse transforms of   u . Note that all the issues using the direct approach (i.e., applying initial conditions, δ-function forcing, the need to find small-time and large-time solutions) are avoided by taking the Laplace transform, a major advantage of this solution method.

266

9 Overview

To invert   u , first find its inverse Laplace transform (the order of the inversion does not matter). Note that (9.35) consists of two terms, each having an s-dependence of form (s − a)−1 . Applying the inverse transform, L−1 [1/ (s − a)] = eat θ (t), to each term in (9.35) gives  Fo  X  ikct  u= e − e−ikct θ (t). (9.36) 2ikc In (9.36), the response is clearly seen to be composed of the two gravity waves, and that property is traceable to the two terms in (9.35) being inversely proportional to (s ± ikx). In complex analysis, factors like (s ± ikx)−1 are referred to as “poles” of order 1. In later chapters, transforms appear as the sum of similar poles, thereby identifying the waves present in the response and making inversions easy to obtain. To invert  u , first eliminate the factor of (ik)−1 using (9.18a), in which case ux =

 X  ikct Fo  e − e−ikct θ (t) 2c

(9.37)

is the Fourier transform of u x . Then, using convolution f˜(k) =  ∞ with  −ika  theorem (9.16) −ika −1 ik(x−a) ˜ e = 1/ (2π ) −∞ e and g(k) ˜ = X (k), and the relation F dk = e δ (x − a) that follows from (9.9), gives the transform pair e

−ika

X˜

 ←→

∞

−∞

δ(x − a − x  )X(x  )d x  = X(x − a).

(9.38)

Applying (9.38), the inversion of (9.37) is Fo [−X(x − ct) + X(x + ct)] θ (t), 2c

(9.39)



 x  x Fo − X(x  − ct)d x  + X(x  + ct)d x  θ (t). 2c L L

(9.40)

ux = which integrates to u=

To satisfy the initial and radiation conditions, L and L  are specified as described after (9.29a), and with these choices solution (9.40) is the same as (9.29b) obtained by the direct approach.

9.4 Examples

267

9.4.2 Periodic Forcing When F = Fo X(x)e−iσ t , Eqs. (9.22) become − iσ  u + px = Fo X(x),

−i

σp + u x = 0, c2

(9.41)

where σ  = σ + iγ and, for convenience, factors of e−iσ t are dropped. Combining Eqs. (9.41) yields σ 2 iσ  Fo X(x). (9.42) uxx + 2 u = c c2 Since the impact of damping appears only in the parameter σ  , there is no mathematical or conceptual advantage in restricting the solution to be inviscid: The inviscid response is simply the viscid solution in the limit that σ  → σ . Conversely, there is in fact a need to retain damping, as it bypasses difficulties with using the transform method (see the final paragraph in the chapter).

9.4.2.1

Direct Approach

Guess a solution to (9.42) of the form u = Ae−i (σ /c)x 



x

ei (σ /c)x X(x  )d x  + Bei (σ /c)x

L









x L

e−i (σ /c)x X(x  )d x  , 



(9.43) where the lower limits remain to be determined. Note that each of the terms on the right-hand side of (9.43) are solutions to the related differential equation, u x ± i(σ  /c)u = X , so the terms are reasonable guesses for the two parts of (9.43). Differentiating (9.43) twice gives uxx = −

σ 2 σ u − i (A − B) X + (A + B) X x . c2 c

(9.44)

A comparison of (9.44) to (9.42) shows that the differential equation is satisfied provided that σ iσ  Fo , (9.45) A + B = 0, −i (A − B) = c c2 the first expression eliminating the X x term in (9.44) and the second setting the amplitude of A − B. Combining Eqs. (9.45) leads to A = −B = −Fo / (2c), and hence

268

9 Overview

u=−

Fo −i (σ  /c)x e 2c



x

ei (σ /c)x X(x  )d x  + 



L

Fo i (σ  /c)x e 2c



x L

e−i (σ /c)x X(x  )d x  . 



(9.46) The first term on the right-hand side of (9.46) describes the excitation and radiation of waves with westward group velocity and decay. Radiation condition (9.4) requires that these waves don’t appear anywhere east of the wind forcing, which is satisfied by setting L = ∞. Conversely, the second term involves waves with eastward group velocity and decay, and condition (9.4) leads to L  = −∞. With these choices, and after replacing the neglected factor of e−iσ t , (9.46) is u=−

Fo −iσ t −i (σ  /c)x e e 2c



x ∞

ei (σ

 /c

)x  X(x  )d x  + Fo e−iσ t ei (σ  /c)x 2c



x −∞

e−i (σ

 /c

)x  X(x  )d x  .

(9.47a) Note that, although no initial conditions are needed for periodic forcing, radiation condition (9.4) is still required to determine L and L  . The p field associated with (9.47a) is p=

Fo −iσ t −i (σ  /c)x e e 2



x ∞

e i (σ

 /c

)x  X(x  )d x  + Fo e−iσ t ei (σ  /c)x 2



x −∞

e−i (σ

 /c

)x  X(x  )d x  ,

(9.47b) which is readily obtained from the second of Eqs. (9.22). In contrast to the response to switched-on forcing in which waves are transients, solutions (9.47) describe the continual radiation of gravity waves from an oscillating forcing. The solutions found in later chapters are dynamically similar, except they involve radiation of Rossby and Kelvin waves rather than gravity waves.

9.4.2.2

Transform Method

The Fourier transform of (9.42) is k 2 u˜ −

   σ iσ  Fo σ 2 σ k + u˜ = − 2 X˜ . u ˜ = k − 2 c c c c

(9.48)

Algebraically solving for u˜ gives u˜ = −

  i Fo ˜ 1 X˜ iσ  Fo 1 = − − , (9.49) X c2 (k − σ  /c) (k + σ  /c) 2c k − σ  /c k + σ  /c

providing the solution without any “guesswork.” Each term in (9.49) has an order-1 pole of the form (k − a)−1 , and so we can anticipate that its inversion will describe the wind-forced excitation of one of the gravity waves. To invert (9.49), we begin with the transform pair F −1



1 k−a

 ≡

1 2π



+∞ −∞

eikx dk = ±ieiax θ (±x), k−a

Im a ≷ 0,

(9.50)

9.4 Examples

269

which follows from Cauchy’s residue theorem of complex analysis. Then, with the aid of (9.50), an application of convolution theorem (9.16) with f˜(k) = (k − a)−1 and g(k) ˜ = q(k), ˜ gives the transform pair q(k) ˜ k−a

 ↔

x

ieiax ∓∞



e−iax q(x  )d x  ,

Im a ≷ 0.

(9.51)

Applying (9.51) with a = ±σ  /c and q(k) ˜ = X˜ (k) to (9.49) gives Fo u= 2c



x

e −∞

i (σ  /c)(x−x  )





X(x )d x −



x

e ∞

−i (σ  /c)(x−x  )



X(x )d x



 ,

(9.52)

which, after replacing the factor of e−iσ t , is the same result found by the direct method. Interestingly, there is no need to apply radiation condition (9.4) to determine L and L  , as their values appear “automatically” through the factor of θ (±x) in (9.50). To conclude, we note that damping is an important part of the preceding derivation. For the transform method to work, u˜ = F(u) must exist: Without damping, solution (9.52) doesn’t weaken as x → ±∞ and the Fourier integral F(u) is undefined; with damping, however, the solution decays exponentially in the far field so that F(u) exists. Although some damping is needed for the derivation to hold, its magnitude is unimportant. Therefore, there is no conceptual problem with defining the limit of (9.52) as γ → 0 to be the inviscid response. This approach is analogous to defining the integral in (9.9) to be the limit as  → 0 of the one in (9.10).

Chapter 10

Ekman Drift and Inertial Oscillations

Abstract Solutions that illustrate Ekman drift and inertial oscillations are obtained under a variety of settings: for a single mode of the LCS model; as a function of depth z both with and without a surface mixed layer; and for constant and variable f . At midlatitudes, the steady Ekman drift associated with a single mode of the LCS model is oriented to the right (left) of the wind in the northern (southern) hemisphere and, when the modes are summed, the solution converges to the classic Ekman spiral; however, if the Ekman drift is confined to a surface mixed layer as is commonly observed, the spiral structure is lost for sufficiently (realistically) strong vertical mixing. Ekman drift also exists near the equator, remaining finite there because pressure is involved in the dynamical balance. In response to zonal winds, Ekman drift diverges from the equator (equatorial Ekman pumping), generating a zonal jet that continuously accelerates (the Yoshida Jet). Inertial oscillations are generated whenever winds are switched on. When f varies, their energy propagates efficiently away from the latitude where they were generated along ray paths predicted by their dispersion relation, a process known as β-dispersion. Keywords Midlatitude and equatorial Ekman drift · Inertial oscillations · Yoshida Jet · Single-mode and z-dependent responses Ekman drift and inertial oscillations (gravity waves with frequencies near f ) are prominent everywhere in the world ocean. They are fundamental to ocean dynamics, the first response to forcing by surface wind stress. In this chapter, we find idealized solutions to (9.1) that illustrate their basic properties, both at midlatitudes (Sect. 10.1) and near the equator (Sect. 10.2). The solutions are dynamically similar in both regions, and we highlight their similarities (and differences) throughout Sect. 10.2 and at the end of the chapter (Sect. 10.3). The most striking difference is that the equatorial response forced by τ x has an accelerating zonal jet (the Yoshida Jet), a swift current that is observed in all equatorial oceans. Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_10.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_10

271

272

10 Ekman Drift and Inertial Oscillations

To isolate the phenomena, we assume: the ocean is unbounded; and the wind forcing is spatially uniform and switched on, that is, X(x) = Y(y) = 1 and T(t) = θ(t). (The impacts of ocean boundaries and spatially-confined winds are significant, and they are considered in later chapters.) We obtain solutions in a variety of situations: for constant and variable f ; for a single baroclinic mode; and in models with explicit z-dependence.

10.1 Midlatitude Ocean 10.1.1 Constant f At midlatitudes, the simplest solutions that illustrate both phenomena assume f is constant (β = 0). We obtain constant- f solutions for both a single vertical mode (Sect. 10.1.1.1) and for the vertical structure of the complete response (Sect. 10.1.1.2). We write down the responses forced by both τ x and τ y . Because f is constant, however, no terms in Eqs. (9.1) depend on direction, and the two solutions are the same after a rotation by ±90◦ : Specifically, the τ y solution is the τ x solution with u → v, v → −u, and τ x → τ y ; and the τ x solution is the τ y solution with u → −v, v → u, and τ y → τ x .

10.1.1.1

Single-mode Response

Since the forcing is spatially uniform and f is constant, the ocean’s response must also be spatially uniform. So, we look for a solution that neglects all spatial derivatives, in which case (9.2c) simplifies to vt  t  + f 2 v = (∂t + γ)2 v + f 2 v = − f Fo θ(t) + γG o θ(t) + G o δ(t),

(10.1)

y

¯ and G o = τo / (ρH). ¯ We obtain the solution to (10.1) using the where Fo = τox / (ρH) direct approach illustrated in Sect. 9.4.1.1. (The solution using transform methods, which follows steps similar to those in Sect. 9.4.1.2, provides an instructive exercise for interested readers.) Solution To deal with the impulse forcing, we first find the solution in the time interval from t = 0 to 0+ , that is, until just after the wind switches on (Sect. 9.4.1.1). A time integral of (10.1) from 0 to t gives   vt = −2γv − γ 2 + f 2



t 0

vdt  − ( f o Fo − γG o ) t + G o ,

(10.2)

10.1 Midlatitude Ocean

273

where initial conditions (9.5) ensure that the lower limits of the vtt and 2γvt integrations vanish. At small times, and assuming that v is finite (i.e., does not involve a δ-function), all the terms on the right-hand side of (10.2) are O(t) or less except the last one. In the limit that t → 0+ , then, (10.2) reduces to vt = G o so that the small-time solution is 0 < t ≤ 0+ . (10.3) v = G o t, Note that initial condition vt = G o , and hence solution (10.3), also follows directly from (9.1b) after imposing (9.5). For times t > 0+ , that is, after the wind has switched on, we can set θ(t) = 1 and drop the forcing term proportional to δ(t) in (10.1). Then, its general solution is v=−

f Fo − γG o + Ae−γt e−i f t + Be−γt ei f t , f 2 + γ2

(10.4)

where the first term on the right-hand side is a particular solution to (10.1) and the latter two terms are solutions to its homogeneous version with constants, A and B, to be determined. To ensure that (10.4) reduces to (10.3) at small times requires that v = 0 and vt = G o initially. Applying these two conditions to (10.4) gives the relations A+B =

f Fo − γG o , f 2 + γ2

− (γ + i f ) A − (γ − i f ) B = G o ,

(10.5)

which yield A=

1 + iγ/ f f Fo + i f G o , 2 f 2 + γ2

B=

1 − iγ/ f f Fo − i f G o . 2 f 2 + γ2

(10.6)

Using (10.6), (10.4) becomes v(t) = −

f Fo − γG o 1 + iγ/ f f Fo − i f G o −γt −i f t 1 − iγ/ f f Fo − i f G o −γt i f t + e e + e e . f 2 + γ2 2 f 2 + γ2 2 f 2 + γ2

(10.7) Solution (10.7)  can be simplified by expressing the complex numbers, 1 ± iγ/ f , in polar form e±iφ 1 + γ 2 / f 2 , where φ = tan−1 (γ/ f ), giving v(t) = −

f Fo − γG o 1 θ(t) +  e−γt [Fo cos ( f t − φ) + G o sin ( f t − φ)] θ(t). f 2 + γ2 f 1 + γ2/ f 2

(10.8a) Factors of θ(t) are included in (10.8a) to indicate that it is in fact valid for all times t ≥ 0: With the aid of relations q(t)δ(t) = q(0)δ(t) and q(t)δ  (t) = q(0)δ  (t), which are valid since δ = δ  = 0 unless t = 0, it is straightforward to demonstrate that solution satisfies (10.1).

274

10 Ekman Drift and Inertial Oscillations

Neglecting spatial derivatives, either (9.1b) or (9.2c) can be rewritten as u = − (vt + γv) / f + G o , which together with (10.8a) gives u(t) =

f G o + γ Fo 1 θ(t) +  e−γt [Fo sin ( f t − φ) − G o cos ( f t − φ)] θ(t). f 2 + γ2 f 1 + γ2/ f 2

(10.8b)

Finally, (9.2b) simplifies to p(t) = 0,

(10.8c)

so that pressure is unaffected by the spatially independent forcing. (It is affected, however, when f varies; see Sect. 10.1.2.) Note that solutions (10.8a) and (10.8b) do in fact satisfy the rotational symmetry mentioned above. Equations (10.8) describe the spin-up of a steady current, u¯ =

f G o + γ Fo , f 2 + γ2

v¯ = −

Fo f − γG o , f 2 + γ2

(10.9a)

the Ekman drift associated with a vertical mode when there is damping. When γ = 0, (10.9a simplifies to Go Fo u¯ = , v¯ = − , (10.9b) f f and is directed at right angles to the wind. As γ increases the angle decreases, becoming 45◦ to the right of the wind when γ = f and tending to zero as γ → ∞. Recall that γ = A/cn2 increases like n 2 (since cn ∝ n −1 for large n). For realistic choices of A, γ is near zero for the low-order modes, becoming dominant only for very high-order ones. To illustrate, suppose that A = 2.6×10−4 cm2 /s3 (the value used for the solution in Video 16.8b), f = 4×10−5 s−1 (its value near 15◦N), and cn = 250/n cm/s. Then, γ = f for n = 98 and is less than 0.1 f for n ≤ 31. A prominent feature of the spin-up are gravity waves oscillating at frequency f (inertial oscillations), with the velocity vector rotating clockwise (counterclockwise) in the northern (southern) hemisphere. Initially, they have an amplitude as large as the steady-state Ekman drift, but with damping they gradually decay in time. Because damping is weak for low-order baroclinic modes, however, inertial oscillations are prominent in ocean current records. Videos Videos 10.1a and 10.1b provide numerical versions of solution (10.8) in zonally cyclic basins with northern and southern boundaries (Appendix C). At those boundaries, gravity waves are excited to ensure there is no flow across them (v = 0 there). It is in fact possible to obtain analytic solutions for these waves (e.g., Anderson and Gill 1979; Gill 1982), and their properties are consistent with all the ones in the videos. Because the waves are weak (much weaker than in the variable- f discussed below) and difficult to analyze, we don’t review those solutions here.

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Video 10.1a shows the spin-up of inertial oscillations and eastward Ekman drift in response to a switched-on, uniform τ y when f has its value at θ = 30◦N. Consistent with solution (10.8), velocity vectors, v = ui + v j , rotate clockwise about the mean, eastward, Ekman drift, u¯ = (G o / f ) i, with a period of P = 2π/ f = 1 day. Initially, the response is spatially uniform everywhere except near the northern and southern boundaries. Subsequently, gravity waves spread away from the boundaries with shorter-wavelength waves preceding longer-wavelength ones (indicated by the motion of zero contours located between green bands), causing latitudinal differences to appear throughout the interior ocean. Almost everywhere the amplitude of the vectors, |v|, decreases nearly to zero each cycle, a result of destructive interference between the eastward Ekman drift, u¯ = G o / f , and an equal westward flow associated with the negative (u < 0) phase of the inertial oscillations. The exception is near the boundaries, where the amplitude of the inertial oscillations is weakened by the coastal boundary condition. Note that narrow bands of high and low sea level (pressure) quickly develop along the northern and southern boundaries, respectively, and remain throughout the video. They are generated during the first half of the first inertial cycle, when v moves water northward across the entire basin, causing a surplus (deficit) along the northern (southern) coast. This initial motion is reversed during the second half of the cycle, bringing the coastal responses back near to their initial value (zero). As time progresses, this cycle repeats. As a result, the amplitudes of the coastal sealevel bands oscillate throughout the video, but their sign never reverses. As we shall see, similar shifts of water happen across the equator in response to switched-on τ y forcing (Sects. 10.2.1.2 and 14.2). Video 10.1b is the same as Video 10.1a, except forced by a switched-on, uniform τ x . As expected, the interior-ocean response is much the same as in Video 10.1a, except that the Ekman drift, v¯ = −Fo / f , is rotated clockwise by 90◦ . A striking difference, however, is that strong coastal currents are generated (note the increase in color scale from that in Video 10.1a). In this case, the mean Ekman drift causes water to diverge from (converge onto) the northern (southern) boundary continuously. As a consequence, sea level continues to rise (drop) along the northern (southern) boundary, generating accelerating, geostrophic, eastward coastal currents; at the end of the video, the maximum velocity along both boundaries reaches 199 cm/s. (See Sect. 13.2.1 for a detailed discussion of this coastal process.).

10.1.1.2

Solutions with z-dependence

Solution (10.8) provides the response of a single vertical mode n, and so does not describe the structure of surface-confined Ekman drift. In principle, the structure can be obtained by adding up the responses to many vertical modes, specifically by: replacing all the neglected subscripts n in (10.8a) and (10.8b); multiplying vn and u n by ψn (z); and then summing N terms numerically as in (5.15a). The problem with this “brute-force” approach is that, because the Ekman layer is thin, N must be very large, requiring hundreds (or even thousands) of vertical modes for convergence. (An

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exception happens when Nb and γn are constants, in which case the vertical modes are cosines. Then, the summations can be evaluated analytically to obtain the Ekman spiral discussed below.) Here, we consider three cases that allow for analytic solutions: (i) when ν = γn = 0 and the wind enters the ocean as body force; (ii) when ν has a constant, non-zero value and the wind enters the ocean as a surface stress condition; and (iii) a version of (ii) in which ν = 0 only in a surface mixed layer. Body-force Case In case (i), solution (10.8) is independent of cn since γn = 0, so the n-dependence of vn and u n is determined only by the factor of Hn−1 in Fo and G o . Since, according to (5.17), Hn−1 = Z n is the expansion coefficient of Z (z), it follows that, after adding up all the modes, the vertical structure of the complete response is simply Z (z). Specifically, the solutions for v and u are ∞ 

 τx 1  x τ cos ( f t) + τ y sin ( f t) Z (z), Z (z) + ρ¯ f ρ¯ f n=0 (10.10a) ∞   1  x τy Z (z) + τ sin ( f t) − τ y cos ( f t) Z (z). (10.10b) u n ψn (z) = u(t) = ρ ¯ f ρ ¯ f n=0 v(t) =

vn ψn (z) = −

According to (10.10), both the Ekman drift and inertial oscillations remain trapped to the body-force layer, and the steady current is directed at right angles to the wind. Constant ν and No Ocean Bottom In case (ii), a more efficient solution method bypasses the modal expansion altogether, solving Eqs. (5.6) directly. Even then, the response is difficult to obtain analytically if ν depends on z. (For variable ν, a WKB approach like that used in Sect. 7.1.3 can be used to obtain an approximate solution, a complexity not required here.) Assuming that ν is constant and neglecting terms with horizontal derivatives, Eqs. (5.6a) and (5.6b) simplify to u t − f v = νu zz and vt + f u = νvzz , which can be summarized as (10.11a) qt + i f q = νqzz , where q = u + iv. We look for the solution to (10.11a) when the ocean has no bottom, subject to the boundary and initial conditions, νqz (0, t) =

τ θ(t), ρ¯

q (−∞, t) = 0,

q (z, 0) = 0,

(10.11b)

where τ = τ x + iτ y is the wind-stress forcing. When f = 0, (10.11a) has the form of a standard diffusion problem, in which the surface response continually mixes into the deep ocean (see Sect. 10.2.2). When f = 0, the Coriolis force alters the diffusion pattern significantly, allowing the

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response to adjust to a steady state. It is possible to obtain a time-dependent solution to Eqs. (10.11) that shows the adjustment: It consists of inertial oscillations that quickly diffuse away (as in the modal solution 10.8). To ignore the oscillations we set qt = 0 in (10.11a), and solve only for the steady-state response. A general solution is (10.12) q(z) = Aemz + Be−mz , √ where m = (1 ± i) /r , r = 2ν/ | f |, and the upper (lower) sign applies in the northern (southern) hemisphere. To ensure that the solution vanishes as z → −∞, we must set B = 0. Combining (10.12) at z = 0 with the first of conditions (10.11b) gives A = τ / (mν), and it follows that q=

τ mz τ e = √ e z/r e±i z/r ∓iπ/4 , ρmν ¯ ρ¯ ν | f |

(10.13)

the famous Ekman solution (Ekman 1905). According to (10.13), the Coriolis force has two remarkable impacts. First, it limits r to small values, so that the response is surface trapped. For example, with ν = 10 and 100 cm2 /s (typical values of mixing coefficients in the surface mixed layer) and with f = 3×10−5 s−1 , r = 8.2 and 25.8 m, respectively. Second, in the northern (southern) hemisphere the surface current is directed 45◦ to the right (left) of the wind and its direction spirals clockwise (counterclockwise) with increasing depth. Finally, the vertical integral of the Ekman drift is 

0 −∞

q dz = ∓

τ τ i = − i, ρ¯ | f | f

(10.14)

and is directed at right angles to the wind, just as in solution (10.10). Figure 10.1 (left panel) plots the velocity components of the Ekman spiral forced by τ y in the northern hemisphere, for which u = Re (q) = −U e z/r cos (z/r + π/4) ,

v = Im (q) = V e z/r sin (z/r + π/4) , (10.15)  √  where U = V = τ y / ρ¯ ν f . The figure is plotted in the non-dimensional coordinates, z  = z/r , u  = u/U , and v  = v/V , and as such is generally applicable. Note that u = v at z = 0, so that the surface drift is directed 45◦ to the right of the wind. Consistent with a clockwise rotation of the flow vector with depth: v goes to zero first at a shallower depth (z/r = −π/4) and u is zero at a deeper one (z/r = −3π/4). To apply the plot to dimensional cases, suppose τ y /ρ¯ = 1 cm2 /s2 and f = 3×10−5 −1 s . Then, with ν = 10 cm2 /s, z  = −6 (the bottom of the plot) corresponds to a depth 6r = 49 m and u  = v  = 0.8 (the right-hand edge of the plot) corresponds to a speed 0.8U = 46 cm/s. When ν is increased to 100 cm2 /s, the values are 6r = 155 m and 0.8U = 15 cm/s. With ν = 1000 cm/s, they are 6r = 490 m and 0.8U = 4.6 cm/s.

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Fig. 10.1 Ekman flows for a bottomless ocean in the non-dimensional coordinates defined in the text (left panel), and for a surface layer of thickness H = 50 m with τ y /¯ρ = 1 cm2 /s2 and ν = 10, 100, and 1000 cm2 /s (right panels). In each panel, u and v are the red and blue curves, respectively

Constant ν in a Surface Mixed Layer In the real ocean, strong mixing ν is confined primarily to a surface mixed layer of thickness H with weak mixing at greater depths. To represent this mixing structure, in case iii) we assume that ν = 0 for z < −H and replace the middle expression in (10.11b) with the zero-stress condition νqz (−H ) = 0.

(10.16)

To obtain the solution, we keep B = 0 in (10.12), impose the top and bottom stress conditions in (10.11b) and (10.16), and solve the resulting two equations for A and B to get 1 τ , B = e−2m H A. (10.17) A= ρmν ¯ 1 − e−2m H Setting A and B in (10.12) to these values gives q(z) =

τ em(z+H ) + e−m(z+H ) , ρmν ¯ em H − e−m H

(10.18)

the Ekman solution in a layer with a slippery bottom. In this form, it is straightforward to verify that qz satisfies the surface and layerbottom stress conditions. In addition, from either a depth integration of (10.18) or, equivalently, the governing equation i f q = νqzz , the depth-integrated Ekman flow is  0 τ , (10.19) qdz = −i ρ¯ f −H

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the same as in (10.14) for the bottomless solution, the agreement a consequence of imposing no-stress condition (10.16) at the layer bottom. It is instructive to evaluate solution (10.18) in the limits of small and large ν. When ν → 0, the factor exp (−m H ) → 0 because m has a positive real part. In this limit, the solution simplifies to (10.13 ), the Ekman spiral without a bottom: Essentially the vertical extent of the spiral is so small (r H ) that the solution never “feels” the bottom. Conversely, when ν → ∞ (strong mixing), m → 0 and all the exponentials in (10.18) can be approximated using the Taylor expansion, exp(1 + x) = 1 + x + 1 2 x + 16 x 3 + · · · , with x = ±2m H . Keeping terms to second (third) order in the 2 numerator (denominator) of (10.18), it follows that lim q =

ν→∞

1 + 21 m 2 (z + H )2 τ . ρm ¯ 2ν H 1 + 16 m 2 H 2

(10.20a)

Using the approximation that lim x→0 (1 + x)−1 = 1 − x and neglecting terms higher than order m 2 in the numerator then gives lim q = i

ν→∞

τ τ 1 + (z + H )2 − H 2 . ρ¯ f H 2ρν ¯ H 3

(10.20b)

As checks, note that (10.20b) is a solution to the problem posed (i.e., it satisfies Eq. 10.11a with qt = 0 as well as the boundary conditions imposed at z = 0, −H ) and that its depth integral is still given by (10.19). According to (10.20b), the flow consists of a depth-independent current at right angles to the wind (first term on the right hand side) plus a parabolic shear flow in the wind direction (second term). It is striking how much the confinement of mixing to a surface mixed layer alters Ekman flow. Without a mixed layer, its structure is the classic Ekman spiral (10.13), 1 and both its velocity components weaken with mixing like ν − 2 . In contrast, with a mixed layer and when ν is large enough for the spiral to feel the bottom (r > H ), the flow is given by (10.20b): It has no spiral, its velocity component parallel to τ weakens like ν −1 , and its perpendicular component has the magnitude τ / (ρ¯ f H ), which is independent of ν. In the limit of very strong mixing (ν → ∞), the Ekman structure is entirely at right angles to τ , essentially the same response as (10.10) when forcing in introduced as a body force. Figure 10.1 (right three panels) illustrates how the vertical structure of (10.18) changes as ν increases from small to large values, plotting u = Re (q) and v = Im (q) for τ y forcing (τ = iτ y in Eq. 10.20b). In all three panels, τ y /ρ¯ = 0.5 cm2 /s2 , f = 3×10−5 s−1 , and H = 50 m. From left to right, they show the responses when ν = 10, 100, and 1000 cm2 /s for which r/H = 0.16, 0.52 and 1.6, respectively. For the smallest mixing, the solution does not reach the layer bottom, and the response is essentially the same as in the bottomless case (left panel). For the middle value, the solution begins to feel the bottom, but is not yet close to (10.20b). For the largest value, the response is virtually indistinguishable from (10.20b), differing almost everywhere by less than 1%.

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10 Ekman Drift and Inertial Oscillations

10.1.2 Variable f In the real ocean, f varies with y. How does solution (10.8) change when β = 0? The primary impact is that, because f decreases with latitude, velocity vectors rotate more slowly closer to the equator. Consequently, convergences and divergences develop very quickly across the interior ocean that alternately deepen and shallow the thermocline. This process, known as β-dispersion, excites gravity waves much more efficiently than when f is constant: They are generated across the entire basin, rather than just at boundaries. As a result, β-dispersion allows the adjustment to steady Ekman flow to happen much more rapidly than on the f -plane. For variable f there is no simple, analytic solution like (10.8) that describes the complete adjustment. There are, however, partial solutions that illustrate the basic properties of the initial response and of meridional energy propagation.

10.1.2.1

Videos and Dynamics

Video 10.2a is comparable to Video 10.1a, except that f is specified by the midlatitude β-plane approximation. The initial response is remarkably different from the f -plane case, having prominent southward-propagating oscillations with a wavelength and phase speed that decrease in time. Energy associated with the waves also propagates southward, and as a consequence about halfway through the video oscillations are almost absent north of 30◦N. Eventually, though, the waves reflect from the southern boundary to propagate northward with northward phase velocity, and in the latter half of the video oscillations begin to reappear at higher latitudes. Video 10.2b is similar to Video 10.2a, except forced by a switched-on, spatially uniform τ x . The impact of β-dispersion is the same as in Video 10.2a, generating gravity waves that propagate to, and reflect from, the southern boundary. As in Video 10.1b, because the Ekman flow is directed southward, water is continuously removed from (piles up on) the northern (southern) boundary, causing meridional pressure gradients that drive accelerating, eastward, geostrophic currents; in this case, at the end of the video their maxima are 170 and 240 cm/s along the northern and southern boundaries, respectively, the different values due to f being different along each boundary. Further,  Ekman flow is divergent across the interior ocean, since  the v y = − (τ x / f ) y = β/ f 2 τ x > 0. As a result, water is continuously lost from the interior ocean so that pressure gradually lowers there, more so in the south where f is smaller. This process is an example of open-ocean Ekman pumping, a key aspect of wind-forced, ocean dynamics (Sect. 12.2 and elsewhere). To illustrate the impact of the gravity waves more clearly, Video 10.2c shows the solution forced by a switched-on, band of τ y confined from 40–45◦N, so that any motion outside the band is caused by gravity-wave radiation. As time passes, the gravity waves form a distinct packet that propagates away from the forcing region. The packet propagates to the southern boundary, where it reflects as another packet composed of waves with northward group and phase velocity. Later, the northward-

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propagating packet returns to the latitude band of the forcing region, where it reflects to form another southward-propagating packet. The packets broaden in time, because the gravity waves within them are dispersive. Note that sea-level bands occur along the edges of the forcing region. They exist for the same dynamical reason as the southern- and northern-boundary bands in Video 10.1a: They are generated during the first half of the first inertial cycle, and never completely cancelled.

10.1.2.2

Initial Response

The equation that describes v in Videos 10.2a and 10.2b is (9.2c) with ∂x = γ = 0, vtt − c2 v yy + f 2 v = −Fo θ (t) + G o δ(t).

(10.21)

Equation (10.21) differs fundamentally from (10.1) in that it retains the c2 v yy term: Despite F and G being spatially uniform, v varies with y because f does and, hence, the term cannot be neglected. Nevertheless, the constant- f solution is still a valid solution to (10.21 ) initially, as long as c2 v yy remains small with respect to the one of the other terms on its left-hand side, say, f 2 v. Specifically, it is valid provided meridional wavenumbers  in the response satisfy the inequality 2 f 2 /c2 or, equivalently, that their wavelengths satisfy λ2 4π 2 c2 / f 2 . Assuming that c2 v yy is small, the solution for both u and v is the constant- f solution (10.8) with γ = 0, which can be summarized as q=

 Go  1 − e−iφ θ(t), f

(10.22)

where Go = G o − i Fo , u = Re (q), v = Im (q), and φ (y, t) = f t. The phase of the wave part of (10.22) is φ, and then (6.4) and (6.6) give its local frequency, wavenumber, and phase speed, σ = −φt = f,

 = φ y = −βt,

c py =

σ f =− .  βt

(10.23)

Equations (10.23) accurately describe the general properties of the initial gravitywave responses in Videos 10.2a and 10.2b: Their local frequency is f ; and both its wavelength λ = 2π/ = −2π/ (βt) and phase speed c py decrease in time. Further, theoretical and video values also agree quantitatively: In both videos, f is given by the midlatitude β-plane approximation with β = 2×10−13 cm−1 s−1 , its value at 30◦N; at noon on January 5, 2000, for example, the wavelength of the oscillation is λ = 7.7◦ close to the theoretical prediction of 8◦ from (10.23). At later times, λ increases until it is close to 2πc/ f (2◦ at 30◦N), and solution (10.22) is no longer valid.

282

10.1.2.3

10 Ekman Drift and Inertial Oscillations

Meridional Energy Propagation

The wave-group theory in Sects. 6.3 and 6.4 provides a means for understanding the trajectory followed by the wave packet in Video 10.2c. The approach assumes that in the neighborhood of any point y within the packet the solution can be represented as a plane wave, exp (iy − iσt) , (10.24) and subsequently, this “local gravity wave” propagates meridionally in the direction of its group velocity. The derivation closely follows the one in Anderson and Gill (1979). According to (10.21) the dispersion relation for the waves is,

and then from (6.16) cgy

− σ 2 − c2 2 + f 2 = 0,

(10.25)

 c σ2 − f 2 c2 = = σ = σ σ

(10.26)

is the wave’s group velocity. Consider the waves generated at a latitude y  within the wind band, where f  = f (y  ) and the local inertial frequency is σ = f  . In Video 10.2c, f is given by the midlatitude β-plane approximation, which can be written in the form (7.41). Then, the packet will follow a trajectory y(t) that satisfies yt = ηt = cgy =

c



 f 2 − f 2 c η 2 − η 2 = , f η

(10.27)

where η = y − y, η  = y  − y, and y = yo − f o /βo . From the η-structure of the right-hand side of (10.27), we expect the solution has the form η = A cos (bt + θo ) and, after inserting this expression into (10.27), it is straightforward to show that η = y − y = η  cos



c , t + θ o η

(10.28)

is a general solution. For the initial packet, the arbitrary phase θo is determined by imposing the initial condition that the packet is located at y = y  (η = η  ) at t = 0, so that θo = 0. Thereafter, the packet follows trajectory (10.28) to the southern boundary, arriving there at time t = ts . For Video 10.2c parameters are c = 265 cm/s, yo = 30◦N, f o and βo are evaluated at yo , and y = −342.4 km. Then, with y  = 42.5◦N (the middle of the boundary wind band), η  = 5060 km, and η at the  southern    is ηs = 1452 km. Solution (10.28) with θo = 0 then gives ts = η  /c cos−1 ηs /η  = 33.3 days (February 3). In the video, we estimate ts by the time when the middle of the packet reaches the southern boundary. A measure of that time is when there is a standing oscillation

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283

near the southern boundary, that is, when half of the incoming and reflected packets overlap. That criterion gives 32–33 days, consistent with the theoretical value. The pathway followed by the reflected packet is given by (10.28) with θo = −2cts /η  and t > ts . Accordingly, the middle of the packet follows the reverse path back to y  arriving there at time 2ts ; then, it returns southward, again reaching the southern boundary at 3ts = 100 days (April 10). Consistent with the latter time, standing oscillations appear near the southern boundary about April 8–10 at the end of the video.

10.1.3 Observations There have been many studies of surface Ekman layers and inertial oscillations in the real ocean. Their properties tend to support those of the idealized solutions discussed in this chapter. Ekman Drift: To illustrate one of the studies, Price and Sundermeyer (1999) investigated the vertical structure of near-surface currents at three locations: 34◦N, 70◦W in the Atlantic, 36◦N in the California Current, and 10◦N in the Pacific. They found that currents decayed with e-folding depths of 10–15 m in the two subtropical areas and ∼30 m at the tropical site. Further, their vertical profiles had a spiral shape in which velocity rotated clockwise and its magnitude decayed with depth. The authors fit the idealized Ekman solution (10.13) to the data, finding best fits when the mixing coefficient was ν = 100±20, 175 ± 25, and 500 ± 150 cm2 /s at the three sites, respectively, all reasonable values for a surface mixed layer. Finally, the depth-integrated transport at the two subtropical sites was consistent with (10.14) but, for reasons that were not clear, at the tropical site was about 30% lower. Despite qualitative agreement with (10.15), there are always differences between observed current structures and Ekman theory, and resolving those differences remains an active field of research. Price and Sundermeyer (1999), for example, noted that the along-wind velocity was 2–4 times weaker than the across-wind velocity, so that the spiral appeared “stretched” in the along-wind direction. They suggested that, in the subtropics and under fair-weather conditions, this difference happens because the mixed-layer thickness H varies in response to diurnal warming and cooling. Subsequent efforts to reconcile theory/data differences proposed extensions to the Ekman solution (10.15) in which ν was allowed to vary in depth and time (e.g., Elipot and Gille 2009; Wenegrat and McPhaden 2016). Inertial Oscillations: Spectra of observed, near-surface, velocity data peak at frequencies equal to, or slightly higher than, local f (usually between f and 1.2 f ), with values falling off sharply to either side (e.g., Alford et al. 2016; their Figure 1). Consistent with the solutions discussed in this chapter, these near-inertial waves are generated by any wind stress that varies abruptly, that is, with a time scale of the order of or less than an inertial period, such as gusts, the passage of fronts, and during storms (D’Asaro et al. 1995). Furthermore, near-inertial waves migrate equatorward through β-dispersion, as illustrated in Video 10.2c. At lower latitudes,

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they appear as waves at frequencies higher than the local f , and a significant part of the internal-wave spectral energy can be attributed to this process (Garrett and Munk 1979; Staquet and Sommeria 2002). In the Bay of Bengal, Mukherjee et al. (2013) traced the source of internal-wave energy in their data to stormy weather in the north (see their Figure 2).

10.2 Equatorial Ocean Similar processes (i.e., the establishment of Ekman flow and excitation of inertial oscillations) also occur near the equator. The major difference is that the response forced by an x-independent τ x includes an accelerating jet centered on the equator. Yoshida (1959) wrote down the first equatorial solution to this forcing, and in his honor the accelerating jet is now called the Yoshida Jet. The solution derived in Sect. 10.2.1 is the more general solution obtained by Moore (Moore and Philander 1978).

10.2.1 Single-mode Response To allow for the vanishing of f , we adopt the equatorial β-plane approximation, setting f = β y = σ0 η. It is then possible to represent the forced response obtained here (and the more general, equatorial solutions obtained in later chapters) as expansions of Hermite functions, as in (10.30) below. As for the vertical-mode expansions (Eqs. 5.15), the Hermite sums should extend to ∞ but in practice can be truncated at a finite value J for physically realistic problems. For simplicity, we neglect damping (γ = 0) since the key physical ideas are contained in the inviscid response; impacts of damping are considered in Sect. 10.2.2. Finally, we make no restriction on the meridional structure of the wind Y(y) during the derivation, only setting it to 1 when discussing solution properties. Although straightforward, the derivation is lengthy. It is useful to obtain the solution in two steps: first for v, and then to use v to obtain u and p.

10.2.1.1

v Field

With the above restrictions, (9.2c) simplifies to −

  σ0 Go 1 vtt + α02 vηη − η 2 v = Fo 2 ηY(η)θ(t) − 2 Y(η)δ(t). 2 c c c

(10.29)

10.2 Equatorial Ocean

285

We look for the solution to (10.29) as an expansion of Hermite functions v(x, y, t) =

J 

v j (t)φ j (η).

(10.30)

j=0

First, insert (10.30) into (10.29), multiply by φ j  , and integrate from −∞ to ∞. Then, with the aid of (8.4a), the equation for the coefficients v j is −

σ0 Go 1 v jtt − α2j v j = Fo 2 (ηY ) j θ(t) − 2 Y j δ(t), c2 c c

(10.31)

where α2j = α02 (2 j + 1), (ηY ) j and Y j are the Hermite expansion coefficients of ηY and Y (Eqs. 8.9), respectively, and we have replaced the dummy index j  with j. As for the midlatitude solution, we find the solution using the direct approach (Sect. 9.4.1.1). A time integration of (10.31) from t = 0 to 0+ gives v jt = G o Y j , which with condition (9.5) integrates to v j = G o Y j tθ(t),

0 < t ≤ 0+ ,

(10.32)

the small-time solution to (10.31). The condition, v jt = G o Y j , also follows directly from (9.1b) after setting u(y, 0) = p(y, 0) = 0 and expanding the resulting equation into Hermite functions. For times t > 0+ when θ = 1 and δ = 0, (10.31) has the general solution v j (t) = −F j + Ae−iσ j t + Beiσ j t ,

F j = Fo

σ0 (ηY ) j σ 2j

,

(10.33)

where σ 2j = c2 α2j = σ02 (2 j + 1). For solution (10.33) to reduce to (10.32) at small times requires that A=

Gj 1 Fj − , 2 2i

B=

Gj 1 Fj + , 2 2i

G j = Go

Yj , σj

(10.34)

in which case   v j (t) = −F j 1 − cos σ j t θ(t) + G j sin σ j t θ(t),

(10.35)

to complete the solution for v. As for solution (10.8a), factors of θ(t) are included in solution (10.35) to indicate that it satisfies (10.31) for all times t ≥ 0. Note that (10.35) has the same form as (10.8b) with γ = 0, a consequence of their arising from the same physical process.

286

10 Ekman Drift and Inertial Oscillations

Oscillating Part According to (10.30) and (10.35), v contains an oscillatory part, v  ≡ F j cos σ j t + G j sin σ j t. Frequency σ j is referred to as the “inertial frequency” associated with each Hermite function. In Fig. 8.3, the inertial frequencies are given by the intersections of the gravity-wave and Yanai-wave dispersion curves with the σ-axis (where k = 0). Note that, despite the vanishing of f at the equator, the smallest value of σ j is cm/s, the not zero, but rather σ0 , the equatorial inertial frequency. With c√n ≈ 250/n √ 9.6 √n days, “equatorial inertial period” for mode n is P0n = 2π/σ0n = 2π/ βcn = √ and the periods of equatorial inertial waves are then P jn = 2π/σ jn = 9.6 n 2 j + 1 days, of the order of 10 days and longer. The similarity of solution (10.35) to (10.8a) with γ = φ = 0 suggests there is a close connection between σ j and the midlatitude inertial frequency f . To see this connection, √ recall that the amplitude of φ j is largest near its turning latitude, y j = η j /α0 = √2 j + 1/α0 , and the value of f at this latitude is σ j , that is, f (y j ) = β y j = σ0 2 j + 1, which is equal to σ j . In a wave packet like (10.30), then, the inertial oscillations visible at a given latitude will have frequencies close to local f (as in Videos 10.3a below). Steady Part

The v field also contains a steady part, v¯ ≡ − ∞ j=0 F j φ(η). Away from the equator and with Y(y) = 1, v¯ simplifies to midlatitude Ekman drift, −F/ f . To demonstrate that it does, we show equivalently v¯ +F → 0 at high latitudes.  Expressed

that I = f ∞ −F I = σ ηφ + F Y φ as a Hermite expansion, I = ∞ j 0 j o j j . Therefore, j=0 j j=0 because only high-order Hermites contribute at high latitudes, we need to demonstrate that I j → 0 for large j. This property is shown below in the discussion of the convergence of u a /t beginning with (10.40). In contrast, near the equator v¯ differs considerably from −F/ f , and in particular doesn’t blow up as y → 0. As such, it provides a definition of the meridional component of Ekman drift v¯ that extends to the equator.

10.2.1.2

u and p Fields

With v known, it is straightforward to find u and p. Neglecting zonal derivatives in (9.1a) and (9.1c) gives 

t

u= 0



(σ0 ηv + F) dt ,

p =− c



t

σ0 vη dt  ,

(10.36)

0

where, according to (9.5), we applied the initial conditions that u(y, 0) = p(y, 0) = 0. Replacing v with (10.30) and (10.35) and setting Y = ∞ j=0 Y j φ j gives

10.2 Equatorial Ocean

u=

∞   j=0

287

∞ ∞     σ0 σ0  −σ0 F j ηφ j + F j φ j t + F j sin σ j t − G j cos σ j t ηφ j G j ηφ j + σj σj j=0

j=0

(10.37a)

∞ ∞ ∞     p σ0 σ0  F j sin σ j t − G j cos σ j t φ jη (10.37b) F j σ0 φ jη t − G j φ jη − = c σj σj j=0

j=0

j=0

where we have separated variables q (u and p) into their accelerating qa (proportional to t), steady q, ¯ and oscillatory q  parts. It is noteworthy that p = 0, a property that arises from the variability of f . Note that the accelerating parts are driven only by τ x . Oscillating Part Setting k = 0 in (8.12c) and with the aid of (8.9a), it follows that  ±j (η, 0, σ)

=−

j +1 φ j+1 (η) ± 2



  j φ j−1 (η) = φ jη , −ηφ j . 2

(10.38)

Then, the oscillatory parts of (10.37), u  and p  , can be rewritten 

u p  /c



∞    σ0  π π  ∓ = + G j sin σ j t +  j (η, 0, σ). F j cos σ j t + σ 2 2 j=0 j

(10.39) In this form, it is clear that u  and p  are sums of the u j and p j fields of the equatorial gravity waves associated with v j = F j cos σ j t + G j sin σ j t, that is, the oscillatory part of the complete response is composed of equatorial inertial oscillations. Steady Part



 Solutions (10.37) also contain steady parts, u¯ = ∞ σ0 /σ j G j ηφ j and p¯ = j=0  

− ∞ j=0 c σ0 /σ j G j φ jη , both driven by meridional winds. Away from the equator and with Y(y) = 1, u¯ converges to G/ f . This convergence is ensured by the property that f u¯ − G → 0 at high latitudes, which can be shown by a procedure similar to the one used below to demonstrate the convergence of u a /t. At the equator, u¯ remains finite, which is possible because it is linked to p¯ near the equator (as discussed next). Pressure p¯ arises, not from the time integral of v¯ in (10.36) as might be expected, but rather from the lower limit of the integral of the oscillatory term, G j sin σ j t. As such, the cause of p¯ is dynamically similar to that for the pressure bands that exist at basin boundaries in comparable midlatitude videos (Videos 10.1a and 10.2a): There is a northward shift of water across the equator during the first half of the first inertial period that is never reversed. At the equator, the need for p¯ is apparent in that, according to (9.1b), the steady-state balance p¯ y = G must hold. Even off the equator, although f u¯ = 0 in (9.1b) it does not completely balance G, so p¯ y and hence p¯ are not zero there as well. Indeed, with the aid of (8.4a) and the definitions

288

10 Ekman Drift and Inertial Oscillations

of G j and σ j , it is straightforward to confirm that u¯ and p¯ satisfy balance (9.1b), f u¯ + p¯ y = G, with the contribution of p¯ y decreasing at higher latitudes. Together, then, u¯ and p¯ define the zonal component of Ekman drift near the equator, with p¯ necessary to ensure that u¯ remains finite. Accelerating Part The parts of u and p that grow linearly in time, u a and pa , are forced only by zonal winds. They exist because there is a divergence of Ekman transport v¯ at all latitudes owing to the variation of f (see the discussion of Video 10.2b in Sect. 10.1.2). Although not obvious in Eqs. (10.37), the accelerations intensify for lower j values; in this regard, the j = 0 contribution is an order of magnitude larger than any of the other terms, so that u a and pa are highly equatorially trapped (discussed next). This strong equatorial response illustrates the process of “equatorial Ekman pumping,” in which there is a convergence (divergence) of Ekman drift onto the equator in response to eastward (westward) winds. (See Sect. 14.2.1 for additional discussion of equatorial Ekman pumping, and Sects. 12.2 and 13.2.1 for discussions of Ekman pumping in the interior ocean and along coasts.) Following Moore and Philander (1978), we illustrate the equatorial trapping for the special case Y(y) = 1, which allows u a and pa to be rewritten in a way that makes the trapping clear. With the aid of (8.7a) and (8.9a), u a /t becomes

∞ ∞     (ηY ) j ua − = ηφ j + Fo Y j φ j −F j σ0 ηφ j + Fo Y j φ j = t 2j + 1 j=0 j=0       ∞  (ηY ) j j +1 j φ j+1 + φ j−1 + Y j φ j . = Fo − (10.40a) 2j + 1 2 2 j=0 √ √ Since Y = 1, it follows √ from (8.9b) that √ j + 1Y j+1 = jY j−1 , and then from (8.9a) that (ηY ) j = 2 ( j + 1)Y j+1 or 2 jY j−1 . Then,

∞  ua ( j + 1) Y j+1 φ j+1 + jY j−1 φ j−1 − = Fo + Yjφj t 2j + 1 j=0 ⎡ ⎤  ∞ ∞   ∞        j + 1 Y φ  j Yj φj j j = Fo ⎣− − + Yjφj⎦ , −1  + 3 2 j 2 j j=−1 j=0 j  =1

(10.40b)

where j  = j + 1 and j  = j − 1. Note that we can extend the first sum to j  = 0 owing to the j  factor in the numerator, and start the second sum at j  = 0 since j  + 1 is zero for j  = −1. Then, replacing both dummy indices with j gives

∞ ∞   j 2 ua ( j + 1) − = Fo − + 1 Y j φ j = −Fo Yjφj. t 2j − 1 2j + 3 (2 j + 3) (2 j − 1) j=0 j=0 (10.40c)

10.2 Equatorial Ocean

289

Because Y(y) is symmetric about the equator, only even Hermite functions contribute to u. ¯ So, (10.40c) can be rewritten ∞  ua 2 = −Fo Y2m φ2m , t (4m + 3) (4m − 1) m=0

(10.40d)

√ √ where j = 2m. A final step is to evaluate Y2m . From j + 1Y j+1 = jY j−1 and the √ 1 ∞  ∞ −1  2  property that Y0 = −∞ φ0 dη = −∞ π 4 exp −η /2 dη = 2π 4 , it follows that Y2m =

√ √ √ √ 2m − 1 2m − 3 1 (2m)! √ 1 · · · √ Y0 = m 2π 4 , √ √ 2 m! 2m − 2 2m 2

(10.41)

which shows that Y2m < Y2m−2 < Y0 . From the forms of the fraction in (10.40d) and of Y2m , it is clear that the Hermite series in (10.40d) converges rapidly. To illustrate,the ratio  of the amplitudes for √ 2/21 / (2/3) = 0.1 and that the m = 0 and m = 1 contributions is |u¯ 1 | / |u¯ 0 | = √  3/2/77 / (2/3) = 0.016. for the m = 0 and m = 2 contributions is |u¯ 2 | / |u¯ 0 | = Thus, the structure of u¯ is dominated by the φ0 (η) contribution, and hence is highly equatorially trapped. Note also that the sign of the m = 0 term is positive, so that u¯ 0 is in the same direction as the wind; in contrast, the signs for all the other terms are negative. A similar derivation gives ∞  4m + 2 pa /c = −Fo Y2m φ2m , t (4m + 3) (4m − 1) m=0

(10.42)

for the accelerating part of the pressure field. Like u, ¯ it is also dominated by the φ0 contribution and hence is equatorially trapped. This property must be the case since u a and pa are geostrophically linked; indeed, with the aid of (8.4a), it is easy to demonstrate directly that f u a = − pay from their definitions in (10.37).

10.2.1.3

Videos

Video 10.3a shows the solution forced by a switched-on, spatially uniform (X = Y = 1), meridional wind τ y . Away from equator, there are (midlatitude) inertial oscillations with frequencies near local f , which rotate clockwise (anticlockwise) in the northern (southern) hemisphere. Because f varies, β-dispersion is active, and by January 18, 1991, the flows poleward of ±15◦ are almost entirely composed of mean Ekman drift; however, inertial oscillations subsequently reappear apparently due to gravity waves arriving from the opposite hemisphere (see below). Near the

290

10 Ekman Drift and Inertial Oscillations

equator, a discrete set of equatorial inertial waves is excited. The dominant member of the set is the j = 0 wave, for √ which pressure is proportional to φ1 and the inertial period is P1 = 2π/σ1 = 2π/ βc is about 10 days. This wave is visible in the video as the sea-level response with a maximum and minimum near ±3◦ . Note that the oscillation is always positive (red) and negative (blue) in the northern and southern hemisphere, respectively, a consequence of the response being a superposition of the j = 0 inertial wave and the mean p¯ y that exists to balance G. Video 10.3b highlights the gravity-wave response, showing the solution when τ y is confined to a band from 20–25◦N. As in Video 10.2c, the waves form a distinct packet. It propagates southward, crosses the equator, and eventually reaches 20–25◦S near the beginning of February. Note that the rotational sense of the waves changes along this pathway, varying from clockwise to north-south to anticlockwise as the packet crosses the equator. At later times, the packet returns northward to 20–25◦N and then propagates southward again. Because the equatorial solutions also satisfy (10.21), we expect prominent features in the equatorial videos to be dynamically similar to those in comparable midlatitude videos. In Videos 10.3a and 10.3b, for example, (10.23) still describes the decrease in time of the meridional wavelength and phase speed of their initial responses. Further, in Video 10.3b (10.28) still describes the trajectory of the wave packet. Because f is given by the equatorial β-plane approximation ( f = β y), yo = f o = 0 so that η = y and η  = y  . Then, applying the initial condition that y = y  at t = 0, (10.28) simplifies to  c y = y  cos t . (10.43) y According the wave packet oscillates between y  and −y  with a period of   to (10.43), P = 2π y /c . With c = 265 cm/s and y  = 22.5◦N (the middle of the wind band), the time it takes the packet to reach the southern boundary is ts = P/2 = 69 days, in good agreement with solution in Video 10.3b. Video 10.4 shows the comparable solution forced by τ x . Although the initial response is similar to that in Video 10.3a, the solution is quickly dominated by the accelerating Yoshida Jet. (To ensure that small velocities remain visible in the video, the magnitudes of the plotted vectors are the square root of current values. Thus, arrows with a length of 5 correspond to a current speed of 25 cm/s, and so on.) Toward the end of January, 1991, the jet is well formed, with edges (where u switches sign) near ±9◦ . Note that, similar to Video 10.2b, Ekman drift causes water to diverge continuously from the northern and southern boundaries, thereby creating meridional pressure gradients associated with accelerating, eastward, geostrophic coastal currents; in addition, pressure gradually lowers at all latitudes outside the Yoshida Jet, in response to the Ekman divergence (Ekman pumping) there.

10.2 Equatorial Ocean

291

10.2.2 Solutions with z-dependence When solutions (10.30) and (10.37) are summed over many vertical modes, the vertical structure of the steady response away from the equator is essentially the same as at midlatitudes (Sect. 10.1.1.2), either converging to an Ekman spiral (when γ = 0 and the wind enters the ocean as a stress condition) or contained entirely within the surface mixed layer (when γ = 0 and the wind enters as a body force). What, then, is the vertical structure of the Yoshida Jet? It is not possible to obtain a simple solution that answers this question for all latitudes, but it is on the equator (η = 0). In the following, we obtain solutions for the three cases considered in Sect. 10.1.1.2.

10.2.2.1

Body-force Case

Because σ0 and σ j in (10.37a) both depend on n through cn , the overall n-dependence of u n is complicated. When η = 0 and the wind enters the ocean as a body force Z (z), however, (10.40a) simplifies to un =

∞ 

Fn j φ j t =

j=0

τx t, ρH ¯ n

(10.44)

so that the only n-dependence is contained in Hn−1 = Z n . Following the argument at the end of Sect. 10.1.1.2, the z-dependent u field is then u=

∞ 

u n ψn (z) =

n=0

τx Z (z)t, ρ¯

(10.45)

an accelerating equatorial jet confined to the body-force layer with the same vertical structure as that of the body force.

10.2.2.2

Constant ν and No Ocean Bottom

When γ = 0 and τ x enters the ocean as a stress condition, we look for a direct solution for u(x, 0, t) without expanding into vertical modes. At the equator ( f = 0), Eq. (5.6a) with νh = 0, constant ν, and no x-derivative terms simplifies to u t = νu zz ,

(10.46a)

and appropriate boundary and initial conditions are νu z (0, t) =

τx θ(t), ρ¯

u(−∞, t) = 0,

u(z, 0) = 0.

(10.46b)

292

10 Ekman Drift and Inertial Oscillations

Note that Eqs. (10.46) comprise a traditional diffusion problem, in which u mixes downward from the surface. The method of Laplace transforms (Sect. 9.3.2) provides an easy way to find the solution to (10.46). Taking the Laplace transform of (10.46b) with the initial condition u(z, 0) = 0 gives s uˆ = ν uˆ zz , which leads to the general solution uˆ = Ae

√

√

+ Be−

s/νz

s/νz

.

(10.47)

Since s has a positive real part, we set B = 0 to ensure that the solution vanishes ¯ /s, which at depth. The transform of the surface stress condition is ν uˆ z = (τ x /ρ) combined with (10.47) gives uˆ z (0, s) =

 s  21 ν

A=

so that uˆ =

τx ρνs ¯

τx 3 2

ρs ¯ ν

1 2

e

⇒

√

s/νz

A=

τx 3

1

ρs ¯ 2ν2

.

(10.48)

(10.49)

We invert uˆ using the transform pair, s

√ − 23 −k s

e



↔

 t −k 2 /t k e 2 − k erfc √ , π 2 t

where 2 erfc (ξ) = 1 − √ π



ξ

k ≥ 0,

  exp −ξ 2 dξ  ,

(10.50a)

(10.50b)

0

is the complementary error function, which can be found in most tables of inverse 1 Laplace transforms. Using (10.50) with k = −z/ν 2 , uˆ inverts to u=

   z2 z t −z exp − + 1 erfc √ . 2 π 4νt 2 νt ν2

 

τx 1

ρν ¯ 2

(10.51)

Note that u satisfies the bottom and initial conditions, that is, as u = 0 when z → −∞ and t → 0, the latter since erfc(∞) = 0. Further, from either a depth integration of 0 (10.51) or, more easily, from a depth integral of u t = νu zz , it follows that −∞ udz = τ x t; thus, the depth integral of the jet accelerates linearly in time, just as it does for an individual mode when γ = 0. From the arguments of √ the exponential and error functions, a measure of the vertical scale of u is r = 2 νt. A general form of (10.51), expressed in terms of non-dimensional coordinates, is

10.2 Equatorial Ocean

293

Fig. 10.2 Yoshida Jet solutions for a bottomless ocean in the non-dimensional coordinates defined in the text (left panel), and for a surface layer of thickness H = 100 m with τ x /¯ρ = 1 cm2 /s2 and ν = 100 cm2 /s (right panel). In the right panel, each curve corresponds to a different time nt: n = 1–5 and t = 0.1 day (red curves); n = 1–4 and t = 1 day (blue curves); and n = 2–7 and t = 2.5 days (black curves)



    1 exp −z 2 + erfc −z  (10.52) π √ where z  = z/r and u  = u/U , U = 2 (τ x /ρ) ¯ t/ν. Figure 10.2 (left panel) plots (10.52), showing that it decreases monotonically with depth. To illustrate how this general structure changes in response to t and ν, suppose that τ x /ρ¯ = 1 cm2 /s2 (τ x = 1 dyn/cm2 ). Then, with t = 1 day and ν = 1 cm2 /s, z  = −2.5 (the bottom of the plot) corresponds to a depth of 2.5r = 14.7 m and u  = 0.6 (the right-hand edge of the plot) to a speed of 0.6U = 353 cm/s. With t increased √ to 30 days and ν unchanged, the response deepens and accelerates by the factor t, in which case the edge depth and speed in the plot increase to 2.5r = 80.5 2 m and 0.6U = 1930 cm/s. When ν is increased √ to 100 cm /s and t remains at 1 day, r decreases and U increases by the factor ν, and 2.5r = 147 m and 0.6U = 35.3 cm/s. 

u =

10.2.2.3

Constant ν in Mixed Layer

When viscosity is confined to a surface mixed layer of thickness H , coefficient B must be retained in (10.47) and the zero-stress condition, ν uˆ z (−H ) = 0,

(10.53)

replaces the Laplace transform of the second of conditions (10.46b). Combining the transform of the first of conditions (10.46b) and (10.53) with (10.47) gives

294

10 Ekman Drift and Inertial Oscillations

A=

τx

1

ρs ¯ ν 1− 3 2

1 2

√ e−2 s/ν H

leading to uˆ =

τx 3

e 1

ρs ¯ 2ν2

√

B = e−2s/ν H A,

,

s/ν(z+H ) √ e s/ν H

(10.54)

√

+ e− s/ν(z+H ) √ . − e− s/ν H

(10.55)

As a check, it is straightforward to show that uˆ z satisfies the imposed surface and bottom, stress conditions. In addition, an integration of uˆ over the layer gives 0 √ √ τ x e s/ν(z+H ) − e− s/ν(z+H )  √ √ uˆ dz = 2   ρs ¯ e s/ν H − e− s/ν H −H



0

−H

=

τx ρs ¯ 2

 ⇒

0

−H

u dz =

τx t, ρ¯ (10.56)

consistent with an integration of (10.46a) and the previous two solutions. Transform uˆ in (10.55) is too complex to invert analytically. So, we invert it only in the limits of small and large s, which provide  √ thelarge- and √small-t limits of u, respectively.1 For s large enough that exp − s/ν H exp s/ν H , the second terms in the numerator and denominator of (10.55) are negligible with respect to the first ones. In that case, uˆ simplifies to (10.49) so that its inversion is (10.51), the response without a bottom: In this small-t limit, the response has not had enough time to diffuse throughout the layer, and so does not yet “feel” the bottom. For small s, the arguments of all the exponentials are small. Expanding them in a Taylor series about s = 0, and following the approach used to obtain (10.20b), leads to

τx 1 τx 1 2 2 lim uˆ = 2 + (z + H ) − H , s→0 ρs ¯ H ρsν ¯ H2 3

(10.57)

which has the inversion

τx 1 τx 1 2 2 t+ lim u = (z + H ) − H . t→∞ ρH ¯ ρν ¯ H2 3

(10.58)

According to (10.58), the depth-independent current grows linearly in time, whereas its vertical variation is parabolic and constant in time. To check the reasonableness of the solution, it is straightforward to show that (10.58) satisfies (10.46a) and the boundary conditions at z = 0, −H , and that it has the depth integral (10.55). Figure 10.2 (right panel) shows the numerical solution to (10.46) with νu z (−H ) = 0 and H = 100 m, plotting curves from that solution at small, moderate, and large times. Initially (0 < t < 0.5 days, red curves), the response does not feel the bottom, and has a structure like that in a bottomless ocean (left panel). At moderate times These limit relationships between q and its Laplace transform qˆ have been shown to be useful in many situations (including the one under discussion here), but they do not have general applicability and can lead to unrealistic approximations. See, for example, van der Pol and Bremmer (1964), Sneddon (1972), and Pipkin (1991) for discussions of this issue.

1

10.2 Equatorial Ocean

295

(10 < t < 40 days, blue curves), although the response feels the bottom, it has not yet adjusted to the long-term solution (10.58). At longer times (t ≥ 50 days, black curves), it is essentially indistinguishable from (10.58).

10.2.3 Observations The most prominent feature of equatorial Ekman drift and pumping is the accelerating Yoshida Jet. Yet, it is difficult to isolate this idealized response (i.e., describable by Eq. 10.58 and illustrated in the right panel of Fig. 10.2) in the real ocean, because real winds are not x-independent; as a result, the acceleration of the jet is quickly eliminated by the excitation of equatorial waves (see the discussion of the bounded Yoshida Jet in Sect. 14.2). Notwithstanding, using data from an equatorial current meter at 165◦E, McPhaden et al. (1988) investigated whether the non-accelerating (“shear-flow”) part of (10.58) was consistent with observed shear in the near-surface (z > −125 m) region with weak stratification. Defining a measure of the shear to be the difference in u between 10 and 100 m, u, which eliminates the accelerating part, (10.58) implies that φ H u = 0.4 τ x ≡ (10.59) ρν ¯ ν   when H = 125 m. Figure 10.3 plots u t  from the observations versus φ(t) with t  = t + 3 days, the lag between u and τ x allowing for the spin-up time of the shear flow in response to the wind (seen in Fig. 10.2). Consistent with theory, the scatter

Fig. 10.3 Regression of  daily estimates of u t  and φ(t), t  = t + 3 days during May, 1986. Data are from a current meter located on the equator at 165◦E. Variable D on the y-axis label corresponds to H in the text, and its value is D = 125 m. Vertical viscosity A is the slope of the regression line (dashed), and corresponds to ν in the text. From McPhaden et al. (1988)

296

10 Ekman Drift and Inertial Oscillations

of data points demonstrates that u and τ x are correlated. Further, the slope of the regression line is ν = 98 cm2 /s, a reasonable value for a well-mixed layer. It is also difficult to identify equatorial inertial oscillations. One reason is that many such waves, corresponding to vertical and horizontal modes with different n and j values, are possible. Another is that the equator is a collection point for off-equatorial inertial waves that propagate to the equator as internal gravity waves (illustrated in Video 10.3b). Eriksen (1980, 1985) notes these issues in his analyses of equatorial gravity waves based on current-meter data from the Indian and Pacific Oceans. On the other hand, identifying equatorial inertial oscillations is perhaps simpler in sea-level data, because they are dominated by responses from low-order vertical modes (Wunsch and Gill 1976). Three examples are visible in Fig. 8.4 as the spectral peaks with frequencies near 0.15, 0.19, and 0.25 cycles per day.

10.3 Review In this chapter, we obtained and contrasted solutions that isolate Ekman drift and inertial oscillations (i.e., the responses forced by switched-on winds with X = Y = 1) at midlatitudes and the equator. The equatorial solution reduces to the midlatitude one away from the equator, where both consist of oscillations generated at frequencies near local f and steady flow at right angles to the wind. Near the equator, however, these features are modified: The inertial oscillations become a discrete set, the smallest of which has a finite period; the steady-state Ekman drift involves pressure, allowing to remain finite at the equator; and there is a continuously accelerating, equatorial jet driven by τ x (the Yoshida Jet). Given these properties, it is useful to view the steady parts of solutions (10.35) and (10.37) as defining “equatorial Ekman flow” and the accelerating parts as resulting from “equatorial Ekman pumping.” They provide expanded definitions of Ekman flow and pumping that are valid even at the equator. The reason for the existence of the Yoshida Jet is apparent in the zonal-momentum equation. Neglecting px in (9.1a) gives u t − f v = F − γu.

(10.60)

Without damping (γ = 0) and sufficiently far from the equator, the ocean can adjust to the steady-state Ekman balance, − f v¯ = F. In contrast, at the equator where f = 0, u satisfies, u t = F and must accelerate indefinitely in the direction of the wind. With damping, the equatorial jet can adjust to the steady state u¯ = F/γ, but for realistic values of γ the resulting speed is much too large. Other factors (x dependence of the wind and wave radiation) must be taken into account to ensure that the equatorial jet is limited to realistic speeds (McCreary 1981a; Chap. 14).

Video Captions

297

Similarly, the need for a meridional pressure gradient p y follows from the meridional-momentum equation. Setting γ = 0 in (9.1b) gives vt + f u + p y = G.

(10.61)

Sufficiently far from the equator, p y is small and (10.61) quickly adjusts to the steadystate, Ekman balance f u¯ = G. At the equator, where that balance is not possible, there is an initial acceleration of v (vt = G) that establishes the steady-state balance p¯ y = G. What is the dividing latitude between the midlatitude and equatorial regimes of Ekman flow? Based on the equatorial solution obtained here, it is measured by the width of the Yoshida Jet,√that is, by the equatorial Rossby radius of deformation 1 R0n = (β/cn )− 2 ≈ 330/ n km. We note that McPhaden (1981) explored the width in a steady-state model with vertical viscosity and Rayleigh damping. In that model, the width scale of the model’s equatorial zonal jet depends on mixing parameters in addition to the background stratification.

Video Captions Midlatitude Ocean Video 10.1a Spin-up of inertial oscillations and eastward Ekman drift in response y to a switched-on, uniform meridional wind, τ y = τo θ(t). The Coriolis parameter f is constant and evaluated at 30◦N, and cyclic conditions are applied. Video 10.1b As in Video 10.1a, except for forcing by a switched-on, uniform zonal wind, τ x = τox θ(t). Video 10.2a As in Video 10.1a, except with f specified by the midlatitude β-plane approximation. Video 10.2b As in Video 10.2a, except for forcing by a switched-on, uniform zonal wind, τ x = τox θ(t). Video 10.2c As in Video 10.2a, except for a switched-on, narrow band of meridional y wind stress τ y = τo Y(y)θ(t), where Y(y) = θ[(y − 40◦ )(45◦ − y)] is a “top-hat” function confined from 40–45◦N.

Equatorial Ocean Video 10.3a Spin-up of equatorial, inertial oscillations and Ekman drift in response y to a switched-on, uniform, meridional wind, τ y = τo θ(t). The Coriolis parameter is specified by the equatorial β-plane approximation, and cyclic conditions are applied.

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10 Ekman Drift and Inertial Oscillations

Video 10.3b As in Video 10.3a, except for a switched-on, narrow band of meridional y wind stress τ y = τo Y(y)θ(t), where Y(y) = θ[(y − 20◦ )(25◦ − y)] is a “top-hat” function confined from 20–25◦N. Video 10.4 As in Video 10.3a, except for a switched-on, uniform, zonal wind, τ x = τox θ(t). To reduce the range in the length of the vectors, their amplitude is reduced by the replacement (C.9).

Chapter 11

Sverdrup Flow and Boundary Currents

Abstract In his original paper, Sverdrup obtained a steady-state solution to the depth-integrated fluid equations without momentum advection and mixing. That fundamental response is now called a “Sverdrup flow” and the balance of terms that generates it a “Sverdrup balance.” A mode of the LCS model can also be in a state of Sverdrup balance, and it is useful to refer to that response as also being a Sverdrup flow. In response to forcing by zonal winds, Sverdrup flow extends west of the forcing region, owing to Rossby-wave radiation. In contrast, when forced by meridional winds Sverdrup flow is confined to the forcing region, because the total Ekman pumping cancels out across the region; however, when there is vertical diffusion, the cancellation isn’t complete, allowing the response to extend west of the region. Sverdrup flows that extend to the western boundary of the basin, are closed by western-boundary currents confined to narrow boundary layers. Solutions are obtained for the well-known, frictional, western-boundary layers obtained by Stommel and Munk. Dynamically similar boundary layers exist on eastern, northern, and southern basin boundaries, and in the interior ocean along edges of forcing regions. Keywords Depth-integrated and single-mode responses · τ x verses τ y forcing · Western-boundary layer · Northern- and Southern-boundary layers · Stommel and Munk layers Another fundamental concept of large-scale, ocean circulation is Sverdrup flow, which describes the ocean’s response to mean winds remarkably well. As defined in Sverdrup’s (1947) paper, it is the depth-integrated solution to the steady-state version of Eqs. (5.6a), (5.6b), and (5.6d), without horizontal viscosity (νh = 0), namely, ¯ − f V + Px = τ x /ρ,

f U + Py = τ y /ρ, ¯

Ux + Vy = 0,

(11.1)

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_11. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_11

299

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11 Sverdrup Flow and Boundary Currents

0 0 0 where U = −D udz, V = −D vdz, and P = −D pdz. Balance (11.1) also exists for u n , vn , and pn in the steady-state and inviscid (γn = γn = νh = κh = 0) version of modal Eqs. (5.16). Under these conditions, it is possible and useful to refer to the response associated with any vertical mode as being in a state of “Sverdrup balance,” and we do that often in subsequent chapters. Here, we first find the Sverdrup-balanced response of a mode in an ocean with a single eastern boundary (Sect. 11.1). Then, we extend that solution to include other boundaries (Sect. 11.2), obtaining solutions for boundary currents along a western boundary (Sect. 11.2.1) and commenting on their structures along eastern, northern, and southern boundaries (Sect. 11.2.2).

11.1 Interior Circulation 11.1.1 Solution Since the wind is assumed steady, so is the ocean’s response. Dropping the pt  term, (9.1c) becomes u x + v y = 0, allowing the introduction of a streamfunction ψ to represent the velocity field (u = −ψ y , v = ψx ). Without the time derivatives and mixing terms, (9.2c) then reduces to   τ 1 curl , βψx = G x − Fy = H ρ¯

(11.2)

y

where curl τ ≡ τx − τ yx . The solution to (11.2), labelled ψ S , is  ψ S (x, y) =

x xe

τ 1 curl d x  , βH ρ¯

(11.3a)

where we set ψ S (xe , y) = 0 to satisfy the boundary condition that u S (xe , y) = −ψ Sy (xe , y) = 0 along the eastern boundary at x = xe . With ψ S known, u S and v S are then given by u S = −ψ Sy and v S = ψ Sx . The p S field follows from the inviscid and steady-state version of (9.1a), p Sx = F + f v S . With the aid of (11.3a), it follows that p S (x, y) = pe (y) +

f2 βH





x

curl xe

τ ρ¯ f



d x ,

(11.3b)

where pe (y) = p S (xe , y) is the value of p S along the boundary. Along the eastern ¯ which integrates to boundary where u = 0, (9.1b) reduces to p y = τ y / (ρH),  pe (y) = peo +

y yo

τy  dy , ρH ¯

(11.3c)

11.1 Interior Circulation

301

where we take yo to be any latitude south of the southern edge of the wind. The constant peo = pe (yo ) is an undetermined pressure at the point (xe , yo ); it shifts the value of pressure uniformly throughout the domain, but the shift isn’t dynamically important since only derivatives of p appear in Eqs. (9.1). Solution (11.3) defines the Sverdrup flow for each mode of the LCS model. Note that ψ S is driven by the wind-stress curl, curl τ , whereas p S is determined by the curl of the Ekman transport, curl (τ /ρ¯ f ) ≡ wek , the “Ekman pumping velocity” (Sect. 12.2). Further, the solution is independent of cn , and depends on n only through Hn . As a result, the horizontal structures of all the modes, including the barotropic mode, are the same. Moreover, after summing the modes according to Eqs. (5.15), the vertical structure of the complete response is Z (z), the same as that of the windforced layer (see the discussion of solution 10.10). This surface-trapped, Sverdrup response, however, almost never exists because the contributions of higher-order modes are reduced by damping and by long Rossby-wave adjustment times (Sect. 12.3.1); a possible exception is for cross-equatorial flow driven by a τ x ∝ y (Sect. 17.3.4).

11.1.2 Videos Videos 11.1a–11.2b show the responses to switched-on patches of westerly τ x and southerly τ y winds in a domain with an open western boundary along x = 60◦E. (Video 11.1b is discussed in Sect. 11.1.4.) For this discussion, we are interested in the steady-state responses visible in the latter half of the videos, which are adjusted close to Sverdrup-balanced states. See Sect. 12.3.1 for a detailed discussion of their transient responses. τ x Forcing: Toward the end of Video 11.1a, the solution consists of a double-gyre circulation, with eastward flow (in the direction as the wind) between the two gyres, flanked by westward currents (“countercurrents”) on either side. This structure agrees x , so with that of solution (11.3), for which the zonal current is u S = −ψ Sy ∝ −τ yy that eastward (westward) currents occur where the wind-stress curvature is negative (positive). The countercurrents are supplied by wind-forced meridional currents, v S ∝ τ yx , that diverge from the central eastward flow. The sea-level field, d = p/g (color shading) is thinner (thicker) in the latitude band where curl (τ / f ) = − (τ x / f ) is positive (negative). In addition to spatial structure, the amplitude of the analytic solution also agrees well with the numerical response. For example, it is straightforward to show that the maximum and minimum of p S in (11.3b) occur at and west of the western edge of x = 0, at 25◦N and 35◦N, respectively. For the wind patch at the latitudes where τ yy the model and forcing parameters used in the video, extrema of model sea-level, d S = p S /g, are 23.8 and –39.5 cm, close to their values of 23.4 cm and −38.8 cm in Video 11.1a. τ y Forcing: Video 11.2a is similar to Video 11.1a, except showing the response to a switched-on patch of southerly wind stress τ y . The steady-state, Sverdrup response has a very different structure from that in Video 11.1a, consisting of a single gyre con-

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11 Sverdrup Flow and Boundary Currents

fined to the interior ocean, even though transient Rossby waves still radiate from the forcing region: In steady state, the gyre has a sea-level maximum at 60◦E, 30.6◦N, with values of 20.4 cm and 20.0 cm in the analytic and numerical solutions, respectively. The difference between Videos 11.2a and 11.1a happens because the integrands of both (11.3a) and (11.3b) are proportional to X x for τ y forcing, and the integrals over the positive and negative parts of X x cancel west of the wind patch. To illustrate the importance of the cancellation, Video 11.2b shows the response to τ y when A = 5×10−4 cm2 /s3 , a value corresponding to a damping time scale of c12 /A = 4.4 years; in this case, the damping prevents the cancellation, and a westward extension of the main gyre remains due to Rossby-wave radiation (Sect. 12.3.1). For τ y forcing, ψ S ∝ τ y and p S ∝ f τ y . The resulting flow field is therefore y y u S = −ψ Sy ∝ −τ y and v S = ψ Sx ∝ τx , which defines a clockwise (anticlockwise) circulation about a wind patch of positive (negative) τ y . The clockwise gyre in Video 11.2a agrees with the predicted rotation. The maximum value of p S occurs at the longitude where τ y attains its maximum and the latitude where f τ y is maximum, which, since f increases northward, lies a bit north of the wind maximum. For the model and forcing parameters used in Video 11.2a, max(d S ) = 20.3 cm occurs near 30.6◦N, in good agreement with the video values of 20.5 cm and 30.6◦N.

11.1.3 Pycnocline Response To visualize the pycnocline response that corresponds to the sea-level patterns in the videos, it is useful to interpret solution (11.3) in terms of a 1 21 -layer model, since it’s layer thickness h corresponds to the depth of the top of the pycnocline (Sect. 5.3). To do so, we apply (5.27) to the video variables with H = 200 m, in which case velocity vectors are increased by the factor H1 /H = 1.39 and h = φd1 , where φ = gH1 /c12 = 391. Using the latter relation, steady-state h in the τ x forced solution of Video 11.1a thickens in regions where d1 > 0 to a maximum of h = H + φ max(d1 ) = 291 m, and thins in regions where d1 < 0 to a minimum of h = H + φ min(d1 ) = 48.7 m. Similarly, steady-state h in the τ y -forced solution of Video 11.2a deepens to a maximum of H + φ max(d1 ) = 278 m. For northerly y winds (τo < 0), the signs of all the fields in Video 11.2a are reversed, and h thins to a minimum of H − φ max (d1 ) = 122 m. For the above parameter choices, h never thins to zero thickness but for other choices it can (e.g., with H = 100 m), in which case solution (11.3) breaks down and subsurface water reaches the surface. Further, in the real ocean there is typically a mixed layer with a thickness h m < H of the order of 50 m, say, so h only needs to thin to h m before subsurface water reaches (is mixed to) the surface. This process, by which subsurface water is lifted into the surface layer, is known as “open-ocean upwelling.” It is an important process in the real ocean that, among other things cools SST and brings nutrients into the euphotic zone. (See Chaps. 12–14 for detailed discussions of open-ocean upwelling and its analogs along coasts and the equator.)

11.2 Boundary Currents

303

11.1.4 Observations Do baroclinic Sverdrup circulations like those in equilibrium states of Videos 11.1a and 11.2a exist in the NIO? As discussed later (Sects. 12.3.1, 13.2.3, and 14.2), it takes time for flows to adjust to Sverdrup balance, namely, the time for Rossby waves to propagate across the zonal width of the forcing region. For the barotropic response, the adjustment time is very short because the propagation speed of barotropic Rossby waves is so fast. In contrast, the adjustment by slower-propagating baroclinic Rossby waves takes much longer. Consequently, in the NIO where winds are highly variable, most observed (baroclinic) circulations are not in Sverdrup balance. An exception is the circulation driven by the Southeast Trades, which have a strong steady (annual-mean) component. Video 11.1b is similar to Video 11.1a, except the forcing region is shifted to the southern hemisphere and the wind direction reversed (τox < 0), a simple representation of the Southeast Trades. (See Video 17.3b for a similarly-forced solution to a 2 21 -layer model.) The observed SETR corresponds to the shallow region of the steady-state response, and the westward and eastward branches to either side correspond to the SEC and SECC (Sects. 4.3.1, 4.3.2, and 4.3.3).

11.2 Boundary Currents 11.2.1 Western Boundary A striking aspect of the general ocean circulation is the existence of strong westernboundary currents, like the Gulf Stream and Kuroshio in the Atlantic and Pacific Oceans. Western-boundary currents in the NIO are the EACC, EICC, EArCC, and the SC. A problem with solutions to (11.2) is that they cannot represent such currents: Since (11.2) is a first-order equation in x, only one boundary where u = 0 can be included, which in solution (11.2) is the eastern boundary. Stommel (1948) and Munk (1950) extended Sverdrup dynamics to allow for other ocean boundaries by including viscosity, either damping terms γu and γv (Stommel) or horizontal mixing terms νh ∇ 2 u and νh ∇ 2 v (Munk). Both Stommel (1948) and Munk (1950) obtained exact solutions to their respective equations sets. Here, we obtain simpler, but still highly accurate, versions of their solutions using a boundary-layer approach. (See Sect. 13.2.3.2 for further discussion of western-boundary currents, including their spin up to steady state.)

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11.2.1.1

11 Sverdrup Flow and Boundary Currents

Boundary-Layer Approximation

Boundary layers are common in fluid mechanics, occurring whenever processes involving higher-order derivatives (either mixing or nonlinear terms) are small in the interior of the fluid but significant near domain boundaries. The complete boundarylayer method expands both interior and boundary-layer solutions in perturbation series, matching them term-by-term in the region where they overlap (Bender et al. 1999). The approximate solutions found here essentially retain only the lowest-order terms in the perturbation expansions. This approach is nevertheless highly accurate because the expansion parameter (the ratio of the width of the western-boundary current to that of the basin) is so small. Including the above forms of viscosity in Eqs. (9.1) and neglecting diffusion (γ  = 0), the steady-state ψ equation becomes γ∇ 2 ψ − νh ∇ 4 ψ + βψx =

curl τ . ρH ¯

(11.4)

Following the boundary-layer approach, we divide the total solution into two parts, ψ(x, y) = ψ  (x, y) + ψ  (x, y),

(11.5)

where ψ  and ψ  are the interior and western-boundary responses, respectively. For ψ  , we assume the wind is of sufficiently large scale for the mixing terms in (11.4) to be negligible; in this case, the equation for ψ  is unchanged from (11.2) and the interior solution is still ψ S in (11.3a). For ψ  , we assume the western-boundary current is very narrow in x, in which case the impact of wind forcing is negligible; in addition, since ψ  is still broad in y we can ignore y-derivative terms in (11.4). With these approximations in (11.4), the resulting equation for ψ  is − βψ  = −γψx + νh ψxx x ,

(11.6)

after dropping an x-derivative from each term. To complete the solution, we match ψ  to ψ  = ψ S at x = xw to satisfy appropriate boundary conditions. One of those conditions is that there is no flow through any of the basin boundaries, which is ensured by setting ψ = 0 along all of them. The western-boundary condition for ψ  is therefore ψ  (xw , y) + ψ  (xw , y) = 0.

(11.7)

If νh = 0, a second boundary condition is needed, typically a no-slip condition (see the discussion of the Munk layer below). Note that by including ψ  , ψ = ψ  + ψ  no longer exactly satisfies the easternboundary condition ψ(xe , y) = 0. On the other hand, given the boundary-layer (physically realistic) restriction that ψ  is very narrow, ψ(xe , y) essentially does satisfy the condition, since ψ  has decayed enough to be negligible at the eastern boundary.

11.2 Boundary Currents

305

Since ψ  xboundaries, it follows that the net meridional transport across the  x= 0 at both basin, xwe v d x = xwe ψx d x = 0. Therefore, the western-boundary-current transport “closes” the interior Sverdrup circulation, that is, at any latitude its transport is the negative of the interior meridional transport across that latitude.

11.2.1.2

Stommel Layer

For Stommel’s case, (11.6) with νh = 0 is βψ  = γψ  ,

(11.8)

which has the general solution   x − xw , ψ  = A(y) exp − rs

rs =

γ . β

(11.9)

Applying boundary condition (11.7) gives A(y) = −ψ S (xw , y), and the solution is then   x − xw   . (11.10) ψ = ψ + ψ = ψ S (x, y) − ψ S (xw , y) exp − rs According to (11.10), the interior circulation ψ  is closed by a western-boundary current of width rs to form a basin-scale gyre. With ψ known, the velocity fields are given by u = −ψ y and v = ψx . To find p = p S + p  , we obtain p  by integrating the steady-state and unforced version of (9.1a), px = f v  − γu  = f ψx + γψ y , from x to xe . Since the contribution of     γψ y relative to f ψx is O γ/L y / ( f /rs ) = O rs /L y (γ/ f )  1, it can be neglected and then p  = f ψ  . In the LCS model, realistic parameter choices are, cn = 250/n cm/s, A = 5×10−4 cm2 /s3 or less and hence γn = A/cn2 = 8×10−9 n 2 , which for n = 1 corresponds to a decay time of γ1−1 = 4 years or longer. With these choices and −1 −1 the width of the western-boundary current, β = 2.28×10−13  s , however,  cm 2 rsn = γn /β = A/cn /β = 351n 2 m is unrealistically narrow for low-order modes. The resolution of this problem is that γn in the LCS model represents viscosity due to vertical mixing, whereas in western-boundary currents viscosity arises from a much stronger process, namely, horizontal mixing by eddies. To apply solution (11.10) to the real world, then, one represents the latter process by setting rsn to a realistic width scale, say, rs = 50 km, so that in the western boundary γn = rs β = 1.14×10−6 s−1 , a much larger value corresponding to a decay time scale of only γn−1 = 10.2 days.

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11.2.1.3

11 Sverdrup Flow and Boundary Currents

Munk Layer

For Munk’s problem, (11.6) reduces to − νh ψxx x + βψ  = 0,

(11.11)

which has the general solution ψ  = A (y) exp [α1 (x − xw )] + B(y) exp [α2 (x − xw )] ,

where α1,2 = −γh

(11.12)

√  1 3 ∓i = −γh exp (±iπ/3) 2 2

(11.13) 1

are the roots of −νh α3 + β = 0 that decay eastward and γh = (β/νh ) 3 . In this case, two boundary conditions are required to determine A and B. One is again that ψ(xw , y) = 0 so that there is no across-boundary flow. For the other condition, we choose the no-slip condition ψx (xw , y) = 0, ensuring that v  vanishes at x = xw . (A more complete condition is to set v = v  + v  = 0 at x = xw . Because v  in the narrow western-boundary than v  in the broad interior  current is so much stronger   flow, that is, v  v , the weaker condition that v = ψx = 0 there is adequate.) Imposing ψx (xw , y) = 0 in (11.12) requires that α1 A + α2 B = 0. Without loss of generality, we define A = Aα2 in which case B = − (α1 /α2 ) A = −α1 A so that √ 

 3 π ψ  = A(y) α2 eα1 (x−xw ) − α1 eα2 (x−xw ) = A(y)e−γh (x−xw )/2 sin γh (x − xw ) + . 2 3

(11.14) Imposing ψ(xw , y) = 0 requires that A(y) = −ψ S (xw , y), and the complete solution is then √  3 π   −γh (x−xw )/2 γh (x − xw ) + ψ = ψ + ψ = ψ S (x, y) − ψ S (xw , y)e sin . 2 3 (11.15) As for the Stommel solution, (11.15) also satisfies the condition ψ(xe , y) = 0 very well, since, for realistic values of νh , the width of the Munk layer rm = 1 γh−1 = (νh /β) 3  xe and ψ  (xe , y) is negligible. Likewise, p  satisfies px = f v  − νh u x x = f ψx + νh ψ yx x ≈ f ψx , since the magnitude of νh ψ yx x relative to f ψx is     O νh γh2 /L y / ( f γh ) = O rm /L y (rm /Re )  1, and it follows that p  = f ψ  .

11.2 Boundary Currents

11.2.1.4

307

Videos

Videos 11.3 are similar to Video 11.1a, except showing the responses in a smaller basin with a western boundary along x = 20◦E and an eastern damping region. Videos 11.3a, 11.3b, and 11.3c contrast the responses when the horizontal mixing coefficients are νh = 5×105 , 5×106 , and 5×107 cm2 /s, respectively. In the steady-state responses of each video, the interior ocean remains close to Sverdrup balance, and there are narrow boundary layers (Munk layers) along the western boundary. Figure 11.1 shows the structure of the Munk layers in the three solutions. As predicted, the boundary layer weakens and oscillates offshore. A convenient measure of the current width is the distance offshore x where v first reverses (goes to zero). From (11.14), that distance is

Fig. 11.1 Plots of v that illustrate the structures of the western-boundary Munk layers in Videos 11.3a (top), 11.3b (middle), and 11.3c (bottom), for which νh = 5×105 , 5×106 , and 5×107 cm2 /s, respectively. The plots are all taken at times after the solutions have adjusted to steady state. The red lines in each panel indicate the theoretical distance offshore determined from (11.16)

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11 Sverdrup Flow and Boundary Currents

√ x = 2πrm / 3;

(11.16)

it is indicated by the red vertical lines in each panel of Fig. 11.1, and agrees very well with x in the solution.

11.2.2 Other Boundaries Typically, viscous boundary layers are also present along the eastern, northern, and southern basin boundaries in solutions. Additionally, they can occur in the interior ocean across the boundaries (edges) of a Sverdrup circulation, where the flow field jumps from non-zero values to zero.

11.2.2.1

Eastern Boundary

Along an eastern boundary, solution (11.3a) sets u = 0 at x = xe , ensuring that there is no flow through the boundary, but it does not ensure that the no-slip condition, v = 0, is satisfied there. In a model with Laplacian mixing, then, there is an eastern1 boundary solution, ψ  = C(y) exp [−α3 (x − xe )], where α3 = (β/νh ) 3 is the third 3  root of −νh α + β = 0, and C is set to ensure that v (xe , y) + ψ Sx (xe , y) = 0 yielding C(y) = −ψ Sx (xe , y)/α3 . Because the wind forcing has a large zonal scale, curl τ is weak in solution (11.3a) so that ψ Sx (xe , y) is small; as a result, this easternboundary current is typically insignificant, much weaker than its western-boundary counterpart. On the other hand, prominent eastern-boundary currents are present in the real ocean (Chap. 4) and ocean models. They require an additional process not included in system (11.4) for their existence, namely, coastal upwelling (Chap. 13).

11.2.2.2

Northern and Southern Boundaries

When the wind extends to the northern or southern boundaries of the basin, boundary layers are needed to ensure that there is no across-boundary flow (v = 0) and, in the case of Laplacian mixing, no alongshore flow (u = 0). As such, they are dynamically similar to the western-boundary layers discussed above. The difference is that along these east-west coasts, y-derivatives dominate in the Laplacian terms of (11.4), so that the boundary-layer response satisfies γψ yy − νh ψ yyyy + βψx = 0,

(11.17)

defining “zonal” Stommel (νh = 0) and Munk (γ = 0) layers. Because (11.17) involves both x and y derivatives, it is difficult find analytic solutions to (11.17) in simple forms: They can be expressed in terms of error and hypergeometric func-

11.2 Boundary Currents

309

tions, respectively. Nevertheless, a key property of their structures can be inferred directly from the equation itself. Let y measure the width of the boundary layer; then, (11.17) implies that 1 1 (11.18) y ∼ (γ/β) 2 (xe − x) 2 for a zonal Stommel layer, and 1

1

y ∼ (νh /β) 4 (xe − x) 4 ,

(11.19)

for a zonal Munk layer, so that in both cases the width of the boundary layer broadens to the west.

11.2.2.3

Interior-Ocean Boundaries

Boundary layers in the interior ocean smooth out jumps in velocity and velocity shear across the edges of a Sverdrup circulation. They differ from boundary layers along basin boundaries in that they have two distinct parts on either side of the edge (a “double” layer), each part decaying away from the edge. To illustrate, consider a zonal Stommel layer (γ = 0, νh = 0) along the northern edge of a Sverdrup flow ψ  (x, y) from (11.3a). (Similar boundary layers exist along southern, eastern, and western edges of interior circulations.) The two parts of the boundary layer are the two independent solutions to (11.17) with γ = 0, νh = 0, which we label ψ − (x, y) for the southward-decaying solution and ψ + (x, y) for the northward-decaying one. Then, assuming the edge is located at y = 0, the complete solution has the form ψ = ψ  + Aψ − θ(−y) + Bψ + θ(y),

(11.20)

where the two boundary solutions are confined south and north of the edge, respectively, and A and B are constant amplitudes. Two independent equations are needed to determine A and B, and they are obtained by matching ψ and ψ y across y = 0, ψ  (x, 0) + Aψ − (x, 0) = Bψ + (x, 0),

+ ψ y (x, 0) + Aψ − y (x, 0) = Bψ y (x, 0). (11.21) A similar procedure can be followed to determine a zonal Munk layer (νh = 0, γ = 0) along the boundary. In that case, there are four independent solutions to (11.17), two decaying to the south and two to the north, and their amplitudes are found by matching ψ, ψ y , ψ yy , ψ yyy across y = 0. Boundary layers of this sort are present in video solutions with νh = 0 (see below).

310

11.2.2.4

11 Sverdrup Flow and Boundary Currents

Videos

Videos 11.4a, 11.4b, and 11.4c illustrate the structures of zonal Munk layers along northern and southern, basin boundaries for νh = 5×105 , 5×106 , and 5×107 cm2 /s, respectively. To isolate the Munk layers from the forced response, solutions are forced by a τ x of the form (C.7a) but with Y(y) = f / f o . With this choice, (τ x / f ) y = 0 and so the forcing has no Ekman pumping (Sect. 12.2.1). Consequently, the steady-state y responses consist almost entirely of constant Ekman drift, v¯ = τo X(x)/ f o , across the interior ocean, with Munk layers along the boundaries. In response to the switched-on wind, the Ekman drift is quickly established across the basin. If the video is slowed down, the excitation of inertial waves and β-dispersion is evident (Sect. 10.1.2). Thereafter, coastal Kelvin waves radiate westward (eastward) along the northern (southern) boundary, generating coastal jets that provide a source (sink) for the Ekman drift (Sect. 13.2.2). The northern Kelvin wave propagates down the western boundary and eastward along the southern boundary, eventually eliminating the southern-boundary jet east of the wind band. As time progresses, horizontal mixing gradually broadens the coastal jets until they have the structures of a zonal Munk layer along the northern and southern boundaries and a meridional one along the western boundary. Figure 11.2 provides snapshots of the northern-boundary layers in the three solutions at the end of each video, and the westward broadening is clear. The red curves in each panel plot (11.16) with xe = 80◦E. Isolines of each solution tend to follow this curve, confirming the structure of the coastal current is that of a zonal Munk layer. In Video 11.4c, the presence of zonal Munk layers along the northern and southern edges of the steady-state response is indicated by currents extending beyond the limits of the forcing region (north of 40◦N and south of 20◦N). Similar extensions occur in Videos 11.4b and 11.4a, but they are less visible because νh is smaller. The existence of interior zonal Munk layers in other solutions is also noted in Sects. 14.3.2 and 17.3.2.2.

11.2.3 Observations In the real ocean, snapshots of boundary currents commonly do not have simple structures like those in Figs. 11.1 and 11.2, that is, extend for long distances along a coast and weaken rapidly offshore. Instead, they often appear as a collection of eddies, which are generated by nonlinear instability of the otherwise narrow coastal currents (see the discussion of an inviscid, western-boundary layer in Sect. 13.2.3.2, which continuously thins). On the other hand, when averaged over times that are long with respect to the time scale of the eddies, boundary-layer structures with properties similar to those in Figs. 11.1 and 11.2 appear. Given this consistency,

11.2 Boundary Currents

311

Fig. 11.2 Plots of sea-level anomalies, d = p/g  , that illustrate the structures of the northernboundary Munk layers in Videos 11.4a (top), 11.4b (middle), and 11.4c (bottom), for which νh = 5×105 , 5×106 , and 5×107 cm2 /s, respectively. The plots are all taken at times after the solutions have adjusted to steady state. The red curves in each panel measure the theoretical width of the boundary layer determined from (11.19)

parameterizations of mixing by horizontal viscosities like the above, and extensions of them that allow for variable viscosity coefficients (e.g., Smagorinsky 1963, 1993), are commonly used in numerical models. It is important to keep in mind, though, that the resulting boundary-layer structures only approximate the ones that develop in the real ocean.

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Video Captions Sverdrup Flow Video 11.1a Solution driven by a switched-on, zonal wind stress τ x given by (C.7) with x = y = 20◦ , x m = (60◦E, 30◦N), and T(t) = θ(t). The Coriolis parameter is specified by the mid-latitude β-plane approximation, and open western boundary conditions are imposed as described in Appendix C. Video 11.1b As in Video 11.1a, except with x m = (60◦E, −12.5◦N) and τox = −1.5 dyn/cm2 . Video 11.2a As in Video 11.1a, except driven by a meridional wind stress τ y . Video 11.2b As in Video 11.2a, except that A = 5×10−4 cm2 /s3 .

Western Boundary Current Video 11.3a Solution driven by a switched-on, zonal wind stress τ x given by (C.7) with x = y = 20◦ , x m = (40◦E, 30◦N), and T(t) = θ(t). Horizontal mixing is included with νh = 5×105 cm2 /s. The Coriolis parameter is specified by the midlatitude β-plane approximation. The western boundary is closed, and open conditions are imposed at the eastern boundary as described in Appendix C. Video 11.3b As in Video 11.3a, except with νh = 5×106 cm2 /s. Video 11.3c As in Video 11.3a, except with νh = 5×107 cm2 /s.

Northern and Southern Boundary Currents Video 11.4a Solution forced by a switched-on, band of zonal wind τ x given by (C.7) with xm = 80◦ , x = 20◦ , and Y(y) = f / f o . Horizontal mixing is included with νh = 5×105 cm2 /s. The Coriolis parameter is specified by the mid-latitude β-plane approximation. The western boundary is closed, and open conditions are imposed at the eastern boundary as described in Appendix C. Video 11.4b As in Video 11.3a, except with νh = 5×106 cm2 /s. Video 11.4c As in Video 11.3a, except with νh = 5×107 cm2 /s.

Chapter 12

Interior Ocean

Abstract Wind-forced solutions are found to a simplified version of the LCS equations that neglects the acceleration and damping terms in the zonal and meridional momentum equations. When the wind is switched on, the Coriolis parameter f is constant, and there is no vertical diffusion, Ekman flow continuously drains (piles up) water to the left (right) of the wind axis in the northern hemisphere, and vice versa in the southern hemisphere, a process known as open-ocean Ekman pumping. When the wind is switched on and f varies, Ekman pumping is stopped by the radiation of Rossby waves, and without mixing the response adjusts to a steady-state Sverdrup flow. When the wind is periodic, these processes vary continuously. Keywords Simplified equation set · Constant- and variable- f solutions · Switched-on and periodic forcing · Open-ocean Ekman pumping · Adjustment to Sverdrup balance In this chapter, we obtain idealized solutions to a simplified version of the LCS equations (Sect. 12.1) that illustrate the ocean’s response to forcing by time-dependent winds confined to the interior of the basin (i.e., away from coasts and the equator). We then obtain solutions to the simplified set for both constant (Sect. 12.2) and variable (Sect. 12.3) f , the former isolating Ekman pumping driven by spatially-varying winds and the latter illustrating adjustment to Sverdrup balance by the radiation of Rossby waves. Solutions are obtained when the wind is switched-on, oscillatory, and in one case for a switched-on oscillatory wind (end of Sect. 12.2.2).

12.1 Simplified Model Equations A limitation of solutions under the midlatitude approximation (Chap. 7 and Sect. 10.1) is that f and β are constants, whereas in many regions of interest (e.g., the NIO) the domain is large enough for their values to vary substantially. An approximate Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_12. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_12

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system that allows for variable f and β sets u t  = vt  = 0 in (9.1a) and (9.1b), in which case the equations of motion for a baroclinic mode simplify to − f v + px = F,

(12.1a)

f u + p y = G,

(12.1b)

(∂t + γ)

p + u x + v y = 0, c2

(12.1c)

and the equation in v alone reduces to −

f2 f vt  + βvx = 2 Ft  − Fyx + G x x . c2 c

(12.2)

Because of its simplicity (and reasonable accuracy), this system continues to be used to understand ocean variability today. The advantage of Eqs. (12.1) is that the vt  t  t  , vx xt  , and v yyt  terms, present in the exact v-equation (9.1b), no longer appear. The absence of v yyt  simplifies (12.2) to a differential equation in only x and t, so that y appears as a parameter in solutions via f and β. The lack of vt  t  t  filters gravity waves out of the system, so that solutions only involve the low-frequency adjustment processes of primary interest here. Further, the lack of all three terms ensures that the only free waves in the system are nondispersive Rossby waves. Because (12.2) lacks the v yyt  term, free-wave solutions have the form v = V(y) exp (ikx − iσt) . (12.3) Inserting (12.3) into (12.2), and neglecting forcing and damping terms then gives σ = −kβ

c2 = −kβ R 2 , f2

(12.4)

dispersion relation (7.7) for a non-dispersive Rossby wave. The slanted dashed line in Fig. 7.2 (bottom-right panel) plots (12.4) when f and β have their values at ±15◦ . Under what conditions are Eqs. (12.1) a valid approximation to the complete Eqs. (9.1)? An informal way to address this question is to compare the equations in v alone, (9.2c) and (12.2), that result from each equation set. (A formal way to address it is provided in Appendix B.) One difference is that the three terms, vt  t  t  , vx xt  , and v yyt  , present in (9.2c) are absent in (12.2). Therefore, solutions to equation set (12.1) can be accurate only when those terms are negligible with respect to one of the remaining terms on the left-hand side of (12.2), say, ( f /c)2 vt  . Let T be the time scale of the solution’s temporal variability and damping γ −1 , and L x and L y its zonal and meridional scales. Then, the vt  t  t  term is small provided that f2 1 tt |  |v |vt  | t c2 c2

⇒

|v| f2 |v|  T3 T

⇒

T 2  f −2 ,

(12.5)

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315

and vx xt  and v yyt  are small when |vααt  | 

f2 |vt  | c2

⇒

|v| f2 |v|  L 2α T c2 T

⇒

L 2α 

c2 = R2, f2

(12.6)

where α = x or y. Thus, the approximation is accurate only when the wind forcing, and hence the ocean’s response, is large scale and slowly varying, the situation of interest to us. Figure 7.2 (lower-right panel) graphically illustrates these constraints: Dispersion curve (12.4) is accurate only where the solid and slanted-dashed curves overlap, that is, for σ  f and k R  1. Regarding damping, constraint (12.5) can be rewritten as T −2 = γn2 = A2 /cn4  f 2 ; for realistic choices of parameter A, the inequality is satisfied for low-order baroclinic waves, which typically dominate the response. Forcing term −G t  t  /c2 in (12.1) is also absent in (12.2). It is important during the spin-up of Ekman drift and inertial oscillations, and in the contributions of veryhigh-order baroclinic modes (n  1) that are needed to allow solutions to develop an Ekman spiral (Sect. 10.1). Consequently, its absence has essentially no impact on the slowly-varying, large-scale solutions of interest here. Another issue inherent in Eqs. (12.1) is that, because u t and vt are absent, we can anticipate that only one of the model variables increases smoothly from zero in response to switched-on forcing. Since the system retains pt , a logical choice is p. As we shall see, with this choice both u and v jump to non-zero values just after the wind switches on; specifically, when γ = 0 and for variable f , they jump to the steady parts of (10.8b) and (10.8a). In essence, approximate system (12.1) collapses all the inertial-wave adjustments discussed in Chap. 10 to initial jumps in u and v.

12.2 Constant- f Solutions In this section, we report solutions to (12.2) when f is constant, and the forcing is by a spatially bounded wind patch that is either switched on or periodic. These solutions highlight the Ekman pumping that is caused by spatial variations in the wind (wind curl) rather than by variations in f (as in Sects. 10.1.2 and 10.2). Looking ahead to Sect. 12.3, however, we note that when β = 0 the f -plane solutions found here are modified significantly by the radiation of Rossby waves. On the other hand, because Rossby waves propagate slowly at midlatitudes, responses similar to the f -plane solutions are often observed in the real ocean.

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12.2.1 Switched-On Forcing With f constant and β = 0, (12.2) can be rewritten vt  = vt + γv = −

 Fo c2  Fo X Y δ − γ X Y θ − 2 G x − Fy x . f f f

(12.7)

We obtain the solution to (12.7) using the direct approach (see Sect. 9.4.1.1). (By following similar steps to those in Sect. 9.4.1.2, it is equally easy to find the solution using the method of Laplace transforms.)

12.2.1.1

Solution

To find the small-time response to the impulse forcing, we integrate (12.7) in time and apply (9.5) to evaluate the lower limit of the integral of vt , to get 

t

v = −γ 0

vdt  −

 Fo c2  Fo X Y θ − γ X Y t − 2 G x − Fy x t. f f f

(12.8)

Assuming that v is finite (does not involve a δ-function), then all the terms on the right-hand side of (12.8) are O(t) or higher except the second. Therefore, in the limit that t ≤ 0+ (12.8) simplifies to v=−

F , f

0 < t ≤ 0+ ,

(12.9)

so that v immediately jumps to the Ekman velocity. Solution (12.9) also follows directly from (12.1a) after setting p(x, y, t) = 0 for t ≤ 0+ , a statement that pressure doesn’t jump but rather increases smoothly from zero. For t > 0+ (when δ = 0 and θ = 1), the general solution to (12.7) is v=−

 c2  F − G + Vo e−γt , − F x y f γ f2

(12.10)

where the first two terms on the right-hand side of (12.10) are steady-state, forced responses (the particular solution), and the last term is a solution to the equation, vt + γv = 0, with an amplitude Vo to be determined (homogeneous solution). We reduces to (12.9) at small times, which requires that choose Vo to ensure   that (12.10)  Vo = c2 /γ f 2 G x − Fy . The solution is then   c2 τ 1 − e−γt F curl , v=− − f fH ρ¯ f x γ

(12.11a)

12.2 Constant- f Solutions

317

where curl (τ / f ) / (ρH) ¯ = (G/ f )x − (F/ f ) y . (Constant factors of ρ¯−1 and f −1 are brought inside the curl operator to make the term agree with the Ekman-pumping velocity defined below in Eq. 12.12b.) Since initial condition (12.9) is satisfied, solution (12.11a) is valid for all times t ≥ 0. Equations (12.1a) and (12.1b), rewritten as px = f v + F and u = − p y / f + G/ f , then give   τ 1 − e−γt c2 curl , p=− H ρ¯ f γ

  G c2 τ 1 − e−γt u= + curl . f fH ρ¯ f y γ (12.11b) Note that in the limit γ → 0 the time dependence of each field simplifies to t and the response grows continuously in time. The flow field in Solution (12.11) consists of Ekman drift (the first terms on the right-hand sides of u and v) plus a geostrophic flow (the remaining terms). The Ekman drift switches on immediately (at time t = 0+ ), a consequence of u t and vt being dropped from Eqs. (12.1) so that the system lacks inertial waves; moreover, it lacks the modifications due to γ present in (10.9a), and so occurs at right angles to the wind. Neither of these distortions is a problem, since we are primarily interested in the response at times much greater f −1 and in the contributions of lower-order baroclinic modes for which γn2  f 2 . 12.2.1.2

Videos

Video 12.1a plots sea level d = p/g (contours) and velocity v (arrows) from a numerical solution when γ = 0, forced by a patch of eastward wind stress τ x (see Appendix C and the video caption at the end of this chapter for details). Although Ekman drift is too weak to be evident in the video, its impact is clear: It continuously drains water from the northern half of the wind patch and piles it up in the southern half, thereby generating regions of low and high pressure, respectively, that are associated with counter-rotating gyres that increase linearly in time. At the end of the video (4 years), the extreme values of d in the gyre centers are dmax = −dmin = ±91.5 cm, close to their values of ±94.1 cm predicted by the first of Eqs. (12.11b) using the video’s forcing and model parameter values ( f = 7.27×10−5 s−1 , c = 265 cm/s, and H1 = 279 m). Video 12.1b shows the response when τ x is replaced by a similar patch of northward wind stress τ y . As for the midlatitude constant- f solution in Sect. 10.1, when β = 0 there are no terms in Eqs. (12.1) that depend on direction, so the τ y -forced solution is the same as the τ x -forced one except rotated counterclockwise by 90◦ .

12.2.1.3

Dynamics

To discuss the dynamics of solutions (12.11) further, it is useful to interpret them in terms of a 1 21 -layer model, for which p is determined by changes in layer thickness

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Fig. 12.1 Response of solution (12.11) forced by the τ x field in Video 12.1a when the layer lacks (left panel) and has (right panel) a mixed layer of thickness h m . The panels show the y-structure of the wind (green curves) and values of h along x = 60◦E, after 0.5 (blue), 1.0 (red), and 1.63 (magenta) years. Without a mixed layer, the solution breaks down when h < 0 for t > 1.63 years. With a mixed layer, h < h m for t > 1.09 years and thereafter subsurface water upwells into the mixed layer to supply the Ekman drift in the mixed layer

h (a proxy for pycnocline depth). The transformation is accomplished by applying (5.26) to solutions (12.11), and by setting p = g  h (from Eq. 5.24) and c2 = g  H . Figure 12.1 (left panel) illustrates the 1 21 -layer version of the modified solution when γ = 0, H = 150 m, and the forcing is the same as in Video 12.1a, showing a meridional section along 60◦E. As time passes, the Ekman flow (v = −F/ f ) continuously transfers upper-layer water from the northern to southern half of the patch, and mass conservation requires that h thins (thickens) in the northern (southern) part of the layer. The resulting pressure-gradient field (g  h y ) drives zonal currents that are part of the accelerating, counter-rotating geostrophic gyres. Ekman Pumping This sequence of processes, in which divergence of surface Ekman drift due to wind curl impacts the pycnocline, defines “open-ocean Ekman pumping.” The pumping process itself does not involve damping, and is succinctly summarized by the h equation, obtained either directly from Eqs. (12.1) or the first of Eqs. (12.11) with f constant and γ = 0, (12.12a) h t = −wek , where wek = curl

τ =H ρ1 f



G f



 −H x

F f

 = ∇ · U ek ,

(12.12b)

y

and U ek = (Uek , Vek ) is the vertically-integrated Ekman velocity. According to (12.12a), h thins (the bottom of the layer rises) when wek > 0 in response to the divergence of U ek , and vice versa. Note that wek has the unit of velocity, and for this reason it is referred to as the “Ekman-pumping velocity.” Although wek is derived here assuming constant f , it has the same form even when f varies (Sect. 12.3.1). Similar adjustments occur in continuously stratified models, with wek impacting p by raising (or lowering) subsurface isopycnals.

12.2 Constant- f Solutions

319

Vanishing of h A problem with the inviscid, Ekman-pumping response is that eventually h < 0 and the solution breaks down. For example, d in Video 12.1a can be rescaled in terms of h using the relation h = φd1 with φ = 391 (Sect. 11.1.3). Then, after 4 years and with H = 150 m, the minimum value of h = H − φdmin = −208 m < 0. Thus, long before 4 years have passed, the solution is no longer valid (sensible). (A similar breakdown occurs in the LCS model. Let ρb be the density difference of ρb between the top and bottom of the ocean. Then, inequality |ρ|  ρb in the LCS model corresponds to inequality h < 0 in the layer model; in that case, unrealistic density inversions occur in the total density field, ρ + ρb , and the LCS solution breaks down.) How long does it take h → 0? From (12.12a) we have h = H − wek t, and setting h = 0 gives a surfacing time of ts = H/ max (w  ek ). For the sinusoidal wind structure (C.7b) used in the video, max (wek ) = τox / f (π/y) and, using the video values for τox , f , and y, ts = 1.6 years (magenta curve in Fig. 12.1, left panel). In the NIO, wind forcings can be stronger and narrower, and ts smaller. For example, ts reduces y to 1 month for the Findlater Jet for which τo ∼ 7 dyn/cm2 and y ∼ 5◦ . From this analysis, then, we expect solution (12.11) to be valid for times of the order of months to a year or so, depending on wind parameters. Damping and Entrainment To extend the solution to longer times requires an additional process that can counteract the thinning of h. One way to do that is to allow γ = 0 in (12.1c), which corresponds to including the damping term, γ(h − H ), in the h-equation. This term entrains water into (detrains water from) the layer in regions where h < H (h > H ); eventually, after h deviates sufficiently far from H , the entrainment is strong enough to provide a source/sink for all the Ekman drift, allowing the solution to reach steady state. In that state, p in (12.11b) adjusts to a minimum value, which expressed in terms of layer thickness is h = H − max(wek )/γ. It follows that the damping time scale of the diffusion (γ −1 ) needed to prevent h from thinning to zero is γ −1 < H/ max (wek ) = ts days. Damping coefficient γ corresponds to the vertical diffusion coefficient for the first (or other low-order) baroclinic mode of the LCS model, γ1 = A/c12 . With A = 1.3×10−4 cm2 /s3 (the value used for the EUC videos in Sect. 16.2), γ1−1 = 17 years, much too long to prevent h from vanishing. An alternate, and more realistic, way to represent mixing in a 1 21 -layer model is to include a surface mixed layer of thickness h m in which mixing is strong enough to keep h near h m < H despite the upwelling due to wek . To represent this process, we rewrite (12.1c) in terms of h and replace −γ(h − H ) with entrainment (5.20) to get h1 − hm θ(h m − h), (12.13) h t + H (u x + v y ) = we = − δt where the stepfunction ensures that entrainment is active only when h < h m . In the limit that δt → 0, the entrainment is strong enough to keep h very close to h m so that h t = 0 and h stops rising. Thereafter, we = H (u x + v y ) = wek , a statement that

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enough water upwells into the mixed layer to supply all the water for the divergent Ekman drift. Figure 12.1 (right panel) illustrates the process. This form of openocean upwelling is often invoked to explain the appearance of subsurface water at the ocean surface. Variable f A third mechanism for limiting the thinning of h occurs when f varies (β = 0), which allows the existence of Rossby waves and the adjustment to Sverdrup balance. We discuss these processes in Sect. 12.3.

12.2.2 Periodic Forcing 12.2.2.1

Solution

For periodic forcing with T(t) = e−iσt , the amplitude of the Ekman-pumping response is limited simply because the wind reverses periodically. Since the governing equations are linear, the model variables are also proportional to e−iσt . To emphasize this time dependence, we redefine F, G and τ by the replacements F(x, y, t) → F(x, y)e−iσt , G(x, y, t) → G(x, y)e−iσt , and τ (x, y, t) → τ (x, y)e−iσt , so that they represent only spatially-varying, forcing amplitudes. Since the solution is periodic, the operator ∂t  in (12.2) can be replaced by the complex number −iσ  = −iσ + γ = −i (σ + iγ), and the solution to (12.2) with β = 0 is then simply F −iσt c2 G x x − Fyx −iσt F −iσt c2 curl (τ / f )x −iσt−iφ+iπ/2  e e + 2 e = − − e , f f iσ  f f ρH ¯ σ2 + γ 2 (12.14a)   iφ where, in the last expression, σ is written in its polar form, σ + iγ = e σ2 + γ 2 −1 −iσt −iσt with φ = tan (γ/σ). From px = f v + Fe and u = − p y / f + Ge / f , it follows that v=−

G c2 curl (τ / f ) y −iσt−iφ+iπ/2 c2 curl (τ / f ) −iσt−iφ+iπ/2   e , u = e−iσt + e . ρH ¯ f f ρH ¯ σ2 + γ 2 σ2 + γ 2 (12.14b) According to (12.14), the circulation consists of Ekman drift that oscillates in phase with the wind, and an Ekman-pumping response that lags the wind by π/2 − φ. The amplitude of the Ekman-pumping response weakens with σ (is inversely proportional  to σ 2 + γ 2 ), since σ limits the time available (P/2 = π/σ) to spin up the response. In the low-frequency (σ → 0) limit, it is possible and instructive to compare the real part of solution (12.14) to (12.11): In this limit, the time dependence of the real part of the forcing is T(t) = limσ→0 cos (σt) = 1, corresponding to a wind that has switched on. In this limit and with γ = 0, it follows that φ → π/2 and e−iσt = e−iσt−iφ+iπ/2 → 1, in which case the real part of solution (12.14) reduces to p=−

12.2 Constant- f Solutions

321

the steady-state (t → ∞) limit of (12.11), as it should. In the low-frequency limit with γ = 0, however, φ → 0, e−iσt → 1, and e−iσt−iφ+iπ/2 → i + σt to first order in t; then, the real part of solution (12.14), obtained by ignoring i in i + σt, simplifies to the γ → 0 limit of (12.11).

12.2.2.2

Videos

Video 12.2a shows the response to a switched-on, periodic, zonal wind patch τ x , when T(t) = sin(σt)θ(t), (12.15) σ = 2π/P, P = 180 days, and A = γ = 0. The response to purely oscillatory forcing, T(t) = sin (σt), is the negative of the imaginary part of (12.14). So, one might expect the response forced by (12.15) to oscillate from positive to negative values, but surprisingly it does not. Consider the response in Video 12.2a in the northern half of the wind patch. There, the response decreases to a minimum (blue shading), returns only to near-zero values, and never increases to positive values. In contrast, in Video 12.2b, the same as Video 12.2a except with damping (A = 5×10−4 cm2 /s3 ), the response quickly begins to swing to positive values. The explanation for the difference is that the solution with T(t) given by (12.15) includes a transient part Q(x, y)e−γt (a solution to the homogeneous version of Eqs. 12.1), in addition to the oscillations in (12.14), with Q being set so that the total solution satisfies the initial condition p = 0 at t = 0. The resulting pressure field forced by (12.15) is p=

c2 curl (τ / f )   cos φ e−γt − cos (σt + φ) , 2 2 ρH ¯ σ +γ

(12.16)

and there are analogous expressions for u and v. With γ = 0, the sign of p never reverses. With γ = 0, however, its oscillatory part described by (12.14) becomes more apparent as the transient part of (12.16) decays, and positive (negative) values of p appear ever more strongly in the northern (southern) half of the forcing region.

12.2.3 Observations The impacts of open-ocean Ekman pumping wek are evident in regions where the wind-curl amplitude is large (Fig. 3.1, top-right panel; Video 3.1), with the thermocline rising and sea level decreasing where wek > 0 and vice versa where wek < 0 (Videos 1.1, 1.2, and 1.3). In the northern hemisphere, prominent locations of strong Ekman pumping occur during the summer, with: wek > 0 around the tip of India and off southeastern Sri Lanka (Sects. 4.8.1 and 4.8.3) and north/northwest of the

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axis of the Findlater Jet over the Arabian Sea (Sects. 4.9.5 and 4.9.6); and wek < 0 south/southwest of the Findlater-Jet axis (Sect. 4.9.4). In the southern hemisphere north of 10◦S, the most prominent feature is positive wek (negative curl) along the northern flank of the Southeast Trades, most obviously so during the boreal winter (Sects. 4.3.1 and (4.3.2). When f varies (discussed next), the idealized (constant f ) Ekman-pumping response discussed above is altered by Rossby-wave radiation. Nevertheless, the impacts of the initial Ekman pumping remain clear.

12.3 Variable- f Solutions The f -plane responses discussed in Sect. 12.2 are fundamentally changed by the radiation of Rossby waves when β = 0. Although it is possible to solve (12.2) with mixing, the basic spin-up processes are easier to interpret in the inviscid response; for simplicity, then, we set γ = 0, leaving the derivation of the γ = 0 solution to interested readers.

12.3.1 Switched-On Forcing With γ = 0 and T(t) = θ(t), (12.2) can be written vt + cr vx = −

 c2  Fo X Y δ − 2 G x − Fy x , f f

(12.17)

where, according to (12.4), cr = σ/k = −βc2 / f 2 , is the propagation speed of a nondispersive Rossby wave. As for the f -plane solution (Sect. 12.2.1), we use the direct approach (Sect. 9.4.1) to obtain the solution to (12.17).

12.3.1.1

Solution

Following the steps in the derivation of (12.9) leads to the small-time solution v=−

F , f

0 < t ≤ 0+ .

(12.18)

According to (12.18), v jumps to the Ekman velocity even for variable f . For times t ≥ 0+ (when θ = 1 and δ = 0), the general solution to (12.17) is v=

G x − Fy + V(x − cr t, y), β

(12.19)

12.3 Variable- f Solutions

323

a steady-state response (particular solution) plus a free Rossby wave (homogeneous solution). Ensuring that (12.19) reduces to (12.18) at small times (t = 0+ ) requires that   G x − Fy f Y Gx F = , (12.20) − V(x, y) = − − f β β f y β which determines V at t = 0. With the replacement that x → x − cr t in the arguments of all variables in (12.20), V is known at all times. Solution (12.19) is then  v(x, y, t) = Fo X(x − cr t)

Yy Y − β f

 θ(t) − Fo X(x)

−G o [X x (x − cr t) − X x (x)]

Y θ(t). β

Yy θ(t) β (12.21a)

Including factors of θ in each of the terms indicates that (12.21a) is valid for all times t ≥ 0. From (12.1a), px = f v + F, it follows that  f2 p = Fo χ+ (x − cr t) − χ+ (x) β

  Y fY θ(t), θ(t) − G o [X(x − cr t) − X(x)] f y β

and then (12.1b), u = − p y / f + G/ f , gives

(12.21b)

    x  Fy Yy f Y θ(t) + Fo cr y t X(x − cr t) θ(t) + dx β y β f y β y ∞ 

    G Y Y Y θ(t) − G o cr y t X x (x − cr t) θ(t) − + , (12.21c) + G o X (x − cr t) β y f β β y

u = −Fo χ+ (x − cr t)



x where χ+ (x) = ∞ X(x  )d x  . The lower limit of χ+ (x) does not emerge directly from the solution, but rather is required by radiation condition (9.4): Because the group velocity of non-dispersive Rossby waves is westward, (9.4) requires that no Rossby waves appear east of the forcing; thus, the lower limit must be set to any longitude east of the wind patch, a constraint satisfied by setting it to ∞. Finally, note that the terms proportional to cr y in (12.21c) are proportional to t and hence grow continuously. This property follows from the relation q y (x − cr t) = −cr y tqx (x − cr t),

(12.22)

with q being X or χ+ . The existence of q y results from the increasing tilt of phase lines to the west, owing to the equatorward increase of cr (see the end of Sect. 7.1); as a result, the meridional scale of the solution decreases as the Rossby wave propagates, and geostrophic u strengthens.

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12 Interior Ocean

12.3.1.2

Dynamics

Solution (12.21) describes the complete spin-up of the model ocean to steady state. The response at small times (t  0) can be found with the aid of the Taylor-series expansion q(x − cr t) = q(x) − cr tqx (x) + · · · = q(x) +

c2 β tqx (x) + · · · . f2

(12.23)

With this replacement in (12.21) and keeping terms only to first order in t, it is straightforward to show that the solution reduces to (12.11) with γ = 0. Subsequently, the solution is altered by the radiation of Rossby waves, that is, by the terms proportional to q(x − cr t). The final response (t → ∞) is solution (12.21) after the Rossby waves have propagated out of the domain, in which case the only remaining terms are those for a steady-state Sverdrup flow (11.3). Note that, because of the Rossby-wave adjustment, γ is no longer needed for the solution to reach a steady state. Figure 12.2 (left panels) schematically illustrates the x-structures of the response for τ x forcing (G = G o = 0 in Eqs. 12.21) when τox > 0. The structures are plotted for a zonal section located in the northern half of the wind patch where τ yx < 0, and at a time after the Rossby waves have propagated entirely west of the wind patch. Each curve corresponds to one of the terms in solutions (12.21). For v (top panel), the response consists of two parts (black curves): a Rossby packet proportional to X(x − cr t) and a steady response proportional to X(x). Further, the Rossby packet

Fig. 12.2 Schematic diagrams illustrating the zonal structures of the interior-ocean response to τ x (left panels) and τ y (right panels) forcing, showing v (top panels), p (middle panels), and u (bottom panels). As described in the text, individual curves correspond to various terms in solution (12.21). The dashed curves show the structures for the Rossby wave and steady responses, and the black curve is their sum. The red-dashed curves in the middle- and bottom-left panels are shifted slightly to ensure they don’t interfere with the black curves

12.3 Variable- f Solutions

325

is the sum of two pieces: one proportional to Y y /β (blue-dashed curve), which is the negative of the steady part; and another proportional to Y/ f (red-dashed curve). When t → 0, the two parts overlap, the steady and blue-dashed curves cancel, leaving only Ekman drift v = −F/ f (red-dashed curve). For p (middle panel), the Rossby (blue-dashed curve) and steady (red-dashed curve) responses are proportional to the zonal integrals χ+ (x − cr t) and χ+ (x), respectively, and hence separately both extend (unrealistically) to −∞. Because the two curves have the same amplitude and opposite sign, however, their sum (black curve) cancels west of the western edge of the Rossby packet, creating a wave front that propagates westward at the group speed of non-dispersive Rossby waves cr . At t = 0, the sum of the red- and blue-dashed curves cancel, consistently with the initial condition that p = 0 everywhere. The structure of u (bottom panel) is essentially the same as that of p, except for the additional contribution to the Rossby packet from the term proportional to cr y (magenta-dashed curve); because this contribution is proportional to t X(x − cr t), u still has the properties that it vanishes west of the Rossby-wave front at all times and everywhere at t = 0. Figure 12.2 (right panels) shows the adjustment for τ y forcing (F = Fo = 0 in y Eqs. 12.21) when τo > 0. Although the adjustment processes for τ y forcing are the x same as for τ , the zonal structures of the two responses differ because χ+ → X and X → X x in the τ y solution; as a result, the steady-state response does not extend westward, but rather is confined to the wind-forced region. For v and p, the Rossby and steady parts have the opposite sign and so cancel at t = 0. For u, the blue-dashed curve is the negative of the steady part, the magenta curve vanishes at t = 0, and hence initially the sum of the two parts is Ekman drift, u = G/ f (red-dashed curve). In summary, when β = 0 the ocean adjusts from the f -plane response (12.11) to a state of Sverdrup balance via the radiation of a Rossby wave. This sequence of processes is summarized by the h equation obtained from Eqs. (12.1) with β = 0 and γ = 0, (12.24) h t − cr h x = −wek . The initial Ekman-pumping state is characterized by the balance h t = −wek , the Rossby-wave adjustment by h t = cr h x , and the final Sverdrup state by cr h x = wek . Note that wek is the driving force for both the initial and final states, the difference being that wek is balanced by different terms.

12.3.1.3

Videos

Videos 12.3a, 12.3b, and 12.3c are identical to Videos 11.1a, 11.2a, and 11.2b, respectively. The first two are also comparable to Videos 12.1a and 12.1b, differing in that f is specified by the midlatitude β-plane approximation. Consistent with the analytic solution, the initial responses are like those on the f -plane. They are quickly altered, however, by westward-propagating, Rossby-wave packets and, after their passage, the responses are adjusted to Sverdrup balance (Sect. 11.1). In Video 12.3a forced by τ x , the p and u fields extend west of the forcing region behind the

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12 Interior Ocean

Rossby-wave front, which eventually propagates beyond the left edge of the video. In Video 12.3b forced by τ y , the Rossby-wave packet separates completely from the forcing region, and the inviscid, steady-state (Sverdrup) response to a τ y wind patch is a gyre confined entirely to the forcing region. In both packets, the predicted meridional tilt of the Rossby-wave front, due to equatorward increase of cr (for c = c1 = 265 cm/s cr = 2.6 cm/s at 30◦N and 10.8 cm/s at 15◦N), is apparent. Recall that the separation of the Rossby-wave packet for τ y forcing happens because the responses to forcing by positive and negative wind curl cancel west of the wind patch. To illustrate, Video 12.3c includes damping ( A = 5×10−4 cm2 /s3 ) that prevents the cancellation, and the steady gyre develops a westward-extending “tail.” Alternately, Video 12.3d shows the response when the zonal structure of τ y doesn’t weaken west of its maximum value, that is, when X(x) = 1 for x ≤ xm in (C.7b). This change eliminates the positive wind-curl forcing entirely so that there is no cancellation at all, and, as in Video 12.3a, the solution extends westward from the forcing region.

12.3.2 Periodic Forcing As for the f -plane solution in Sect. 12.2.2, we replace ∂t  in (12.2) with the complex number −iσ  , σ  = σ + iγ, in which case it can be rewritten   G x − Fy x F , vx − ikr v = ikr + f β

(12.25a)

  where kr = −σ  f 2 / βc2 = σ  /cr . Note that the impact of damping is contained entirely in kr through σ  .

12.3.2.1

Solution

It is useful to rewrite (12.25a) in the form, e

ikr x

        −ikr x  G f F ikr x −ikr x F e e v x = −e + − . f x β f xx f yx

(12.25b)

Then, dividing (12.25b) by exp (ikr x), integrating in x, and redefining F and G to be forcing amplitudes, gives the solution

 v=

x ∞

e−ikr x



f β



G f



 x

−

F f

  y

 d x  eikr x−iσt − x

F −iσt e . f

(12.26a)

12.3 Variable- f Solutions

327

Consistent with radiation condition (9.4), the lower limit of the integral is set to ∞ to ensure that no Rossby waves appear east of the wind patch. Variable p can be found from px = f v + Fe−iσt and (12.26a), yielding

 p=

x

e

−ikr x 

∞

f2 β



G f



 x

−

F f

 

 dx



eikr x−iσt .

(12.26b)

y

To obtain (12.26b) from (12.26a), we used the identity, 

x

eikr x



e−ikr x qx  d x  = q(x) + ikr eikr x

L



x

   qe−ikr x d x  = eikr x

L

x L



e−ikr x q d x 

 , x

(12.26c) where L = ±∞; it results from an integration of the first expression by parts under the restriction that q(±∞) = 0. Finally, from u = − p y / f + Ge−iσt and (12.26b), ⎧ ⎨

x





G f





⎫ ⎬

G eikr x−iσt + e−iσt ⎭ f ∞ y y

 

     x    f G F −ikr x  x −x + ikr y e − (12.26d) d x  eikr x−iσt . β f f y ∞ x

u=−

⎩

e−ikr x



f β

x

−

F f

 

dx

Note that the kr y term in (12.26d) has a part that is proportional to x (the x part of the x  − x factor), and so increases indefinitely west of the forcing region; it is analogous to the term proportional to t in (12.21c), and results from the tilting of phase lines west of the forcing region (Sect. 7.1.2).

12.3.2.2

Dynamics

Solutions (12.26a), (12.26b), and (12.26d), describe oscillating Ekman drift and Ekman pumping, and the continual adjustment toward Sverdrup balance by Rossbywave radiation. These interpretations are clear in the limits of high and low σ, when one or the other of the processes is dominant.   High- and Low-σ   Limits     Consider the high-σ   limit, defined by the inequality |kr |L x = σ  /cr  L x  1, where L x is the zonal width of the wind patch. In this limit, the effects of radiation are negligible: Either Rossby waves do not have enough time to propagate across the wind region before the wind reverses sign (σL x /cr  1), they are damped before they can do so (γ L x /cr  1), or both. After two integrations by parts, the integrals in (12.26) can be rewritten

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12 Interior Ocean

 e

ikr x

x

∞

e

−ikr x 

     e−ikr x q −ikr x  q x  −ikr x  q x  x  qd x = e + e −e dx −ikr kr2 x  kr2 ∞ x   q(x) 1 q , =− +O (12.27) ikr |kr |L x kr 



ikr x



x

and, given the above inequality, the term iq/kr dominates the others. Using relation (12.27) without the small contribution from the last term, it is straightforward to show that solution (12.26) simplifies to (12.14), that is, to the f -plane response for an oscillatory wind. (Note that when Eq. 12.27 is applied to the kr y term  in Eq. 12.26d, leading term vanishes because x  − x → 0.) For even larger σ   (the  the  limit that σ   → ∞), even Ekman pumping has no time to become established, and the response is just Ekman   drift.   Conversely, when σ   is low enough for |kr |L x = σ  /cr  L x  1, the exponentials in (12.26) can be replaced by 1, and the term proportional to kr y is negligible. In this limit, Rossby waves propagate across the wind region before the slowly-varying wind changes to any degree (σL x /cr  1) and damping is weak (γ L x /cr  1). Setting the exponentials to 1, the solution simplifies to v=  p=

x

∞

f2 β



G f



 x

−

F f

 

G x − Fy −iσt e , β

d x  e−iσt ,

(12.28a) 

u=−

y

x



∞

G x − Fy β



d x  e−iσt , y

(12.28b) a quasi-steady Sverdrup flow. Radiation Amplitude As suggested by (12.27) and  solution (12.28), the amplitude of the radiation varies markedly with frequency σ   and, equivalently, |kr |: Depending on whether |kr |L x x is greater or less than 1, it is measured by q/ |kr | and ∞ qx ∼ q L x , respectively, and so is much weaker for large |kr |. Expressed in another way, the amplitude of the radiation depends on how well the wavelength (λr = 2πcr /σ) and decay scale (δr = cr /γ) of the Rossby wave “fits” the zonal structure of the wind: If both λr and δr are greater than L x , the Rossby wave will be strongly excited; conversely, if either of the two scales is smaller than L x the radiation will be weak. An equivalent way to discuss the impact of kr on radiation strength is in terms of X˜ (k), the Fourier transform of X(x): Radiation with wavenumber kr will be promi      nent only if  X˜ (kr ) is sufficiently large. To illustrate, suppose X(x) = θ L 2x /4 − x 2 , a top-hat function. Then, X˜ (k  ) =



L x /2 −L x /2



X(x)e−ik x d x = L x

sin k  , k

k =

kLx , 2

(12.29)

12.3 Variable- f Solutions

329

˜  a sinc    function. According to (12.29), X (k ) decreases montonically from L x to 0 as k  goes from 0 to π, and for larger values has weak side lobes. Therefore, for Rossby-wave radiation to be appreciable kr = kr L x /2 < π, which is equivalent to λr > L x . At midlatitudes, the Rossby-wave speed for the baroclinic modes is slow enough  that most observed circulations fall into the high-σ   (high-|kr |) regime with weak radiation. This statement even applies to the annual cycle. For example, in the Bay of Bengal or Arabian Sea at 15◦N, the basin width is roughly L x = 1500 km. With c1 = 250 cm/s and cr = −10.8 cm/s, it follows that at the annual cycle, |kr | L x = |σ/cr | L x = 2.8  1. Since currents measured in the NIO are typically highly baroclinic (Chap. 4), their annual circulation is never in a state of quasi-steady Sverdrup balance, a common misconception. (On the other hand, wave speeds for the barotropic mode are very high. So, if the annual barotropic, i.e. depth-integrated, response were measured, it would be close to Sverdrup balance.) Breakdown Because solutions to (12.2) are approximate, they break down or are not applicable in some regions, failing to satisfy one or the other of the inequalities, L x , L y  R, that are required for Eqs. (12.1) to be valid. This failure can occur in two ways: (i) Since the Rossby-wave dispersion relation is (12.4), there is no critical latitude θ R and so solutions exist at all latitudes; and ii) sufficiently far to the west, the response develops small meridional scales caused by the ever-increasing tilt of phase lines due to Rossby waves propagating faster closer to the equator. The error from i), however, is not serious because for realistic forcing λr  L x at latitudes near and greater than θ R so that Rossby-wave radiation is weak. Regarding (ii), the error happens because Eqs. (12.1) don’t allow energy to propagate equatorward (Sect. 7.3.3), a limitation that is apparent in the following videos.

12.3.2.3

Videos

Video 12.4a has the same forcing as in Video 12.2a (i.e., is forced by patch of zonal winds with T given by Eq. 12.15 and P = 180 days), differing only in that f is specified by the midlatitude β-plane approximation. In this case, the non-oscillating part of the f -plane solution (12.16) does not remain in the forcing region, but rather propagates westward as a transient Rossby wave, leaving behind a local response that now oscillates from negative to positive values. (In Video 12.2a, the only process that can eliminate the transient part is weak damping.) Because the wind oscillates, it should continuously radiate Rossby waves to the west. Such Rossby waves, however, are not prominent in the response because part of the forcing region lies poleward of the Rossby-wave critical latitude (for θ > θ R = 26◦N) and because the wavelength of the Rossby waves are too short to be strongly excited by the large-scale wind (for θ < θ R ). Towards the end of the video, very weak Rossby waves are indicated by green bands that radiate from the forcing region south of 30◦N.

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12 Interior Ocean

To illustrate the dependence of the radiation amplitude on the forcing period P and L x , Videos 12.4b–12.4d show the response for other parameter choices. The transient responses are similar in all the videos, but their equilibrium states differ considerably. Video 12.4b shows the response when P is increased to 365 days. In this solution, θ R = 57◦N, so that Rossby waves can radiate from all latitudes. Nevertheless, in the equilibrium response Rossby-wave radiation is weak because λr  2πL x , being visible mostly in the southern part of the forcing region where λr is longest. In Video 12.4c, P = 365 days and L x is reduced by a factor of four to 5◦ . In equilibrium, prominent Rossby waves radiate from the forcing patch at all latitudes, because the decrease in L x ensures that the wavelengths λ of all the Rossby waves “fit” well with the wind, that is, they satisfy λr  2πL x . Interestingly, the Rossbywave packets in both Videos 12.4b and 12.4c bend equatorward because their group velocity has a southward component (Sect. 7.3.3). In Video 12.4d, P is increased to 1460 days and L x is the same as in Video 12.4a. Again, Rossby waves radiate from the forcing region at all latitudes, in this case because the resulting increase in λr ensures that Rossby waves to satisfy λr > L x at all latitudes. The equatorward bending of the Rossby-wave packet is much less than in Video 12.4c, because the southward component of its group velocity decreases with P (Sect. 7.3.3).

12.3.3 Observations The above sequence of processes (i.e., Ekman pumping followed by the adjustment toward Sverdrup balance by the radiation of Rossby waves) occurs throughout the NIO. Because the propagation speed of baroclinic Rossby waves is slow, however, the adjustment is almost never completed. (The exception is the Sverdrup circulation driven by the annual-mean component of the Southeast Trades; see Sect. 11.1.4.) On the other hand, Sverdrup flows can almost set up because the wind fields in the Arabian Sea and Bay of Bengal are zonally narrow (narrower than the basin widths); consequently, Rossby waves don’t have far to propagate and the adjustment time is not long. In the Arabian Sea, the most obvious examples are the responses to wind curl associated with the offshore weakening of the monsoon winds along the Somali and Arabian coasts (Sect. 4.9.4; Videos 3.1 and 4.4). During the summer monsoon, the region of high sea level (deep thermocline) in the western Arabian Sea is the ocean’s response to the offshore weakening of the Findlater Jet and it is adjusted close to Sverdrup balance (Sect. 17.3.3.1). During the winter monsoon, the wind-forced response is dynamically similar, but is weaker and of opposite sign. In the Bay of Bengal, there are similar responses to the across-basin weakening of the monsoon winds away from the Indian west coast, which help to drive the seasonal, basin-wide gyres (Sect. 4.7.1; Videos 3.1 and 4.2). Additionally, the Sri Lanka dome, and other western-boundary gyres, are generated by offshore wind curl and arguably adjusted close to Sverdrup balance (Sect. 4.7.4; Video 4.3).

Video Captions

331

Video Captions Constant- f Solutions Video 12.1a Ekman pumping in response to a switched-on, zonal wind stress τ x given by (C.7) with x m = (50◦E, 30◦N), x = y = 20◦ , and T(t) = θ(t). The Coriolis parameter is constant and evaluated at 30◦N, and open eastern and western boundary conditions are imposed as described in Appendix C. Video 12.1b As in Video 12.1a, except forced by a meridional wind τ y . Video 12.2a As in Video 12.1a, except that T(t) = sin (σt) θ(t), σ = 2π/P, and P = 180 days. Video 12.2b As in Video 12.2a, except with A = 5×10−4 cm2 /s3 .

Variable- f Solutions Video 12.3a As in Video 12.1a, except f is specified by the midlatitude β-plane approximation. Video 12.3b As in Video 12.3a, except forced by a meridional wind τ y . Video 12.3c As in Video 12.3b, except that A = 5×10−4 cm2 /s3 . Video 12.3d As in Video 12.3b, except that the zonal structure of τ y doesn’t weaken west of its maximum value, that is, X(x) = 1 for x ≤ xm in (C.7b). Video 12.4a As in Video 12.2a with P = 180 days, except that f is specified by the midlatitude β-plane approximation. Video 12.4b As in Video 12.4a, except with P = 365 days. Video 12.4c As in Video 12.4a, except with P = 365 days and the width of the forcing region is reduced to x = 5◦ . Video 12.4d As in Video 12.4a, except with P = 1460 days.

Chapter 13

Coastal Ocean

Abstract Wind-forced solutions along eastern and western coasts are found to a simplified version of the LCS equations that neglects the acceleration and damping terms in the zonal momentum equation. All are forced by a zonally-independent band of meridional wind stress τ y that is either switched-on or periodic. Most are discussed in terms of a one-layer, reduced-gravity model (a 1 21 -layer model) with layer-thickness h. For switched-on winds, solutions are obtained: i) in two dimensions (x, h) when the Coriolis parameter f is constant; and in three dimensions (x, y, h) when (ii) f is constant and (iii) f varies. In case (i) and without vertical mixing, h continuously thins at the coast, a process known as “coastal Ekman pumping.” In case (ii), the thinning is weakened or eliminated by Kelvin-wave radiation, which establishes an alongshore pressure gradient that balances τ y . In case (iii), the coastal response is further modified by Rossby-wave propagation, which: from an eastern coast carries the coastal currents completely offshore; and from a western coast continuously narrows the currents (without viscosity) or adjusts them to a Stommel or Munk layer (with viscosity). These solutions are modified to provide a simple representation of the coastal response forced by river outflow. Keywords Simplified equation set · 2-d and 3-d solutions · Constant- and variable- f solutions · Switched-on and periodic forcing · Coastal Ekman pumping · Adjustment to Sverdrup balance · Western verses eastern coasts · Slanted coasts In the NIO, coastal circulations are locally driven by the monsoon winds and, to a much lesser extent, by river outflow. Further, the response to these forcings involves the excitation of Kelvin and Rossby waves that significantly impact the ocean elsewhere. To illustrate these processes, we find solutions when the coast is a vertical wall along x = 0, with ocean present either in the region x < 0 (an eastern-ocean boundary) or for x > 0 (a western-ocean boundary). We also comment on modifications to periodically-forced solutions when the coast is slanted as in Fig. 7.5 (Sect. 13.3.4). Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_13. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_13

333

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13 Coastal Ocean

Real coasts, of course, are more complex, having continental shelves and coastlines with curvature, and including either of these complications typically makes finding solutions significantly more difficult. On the other hand, basic processes present in our simpler configurations also occur under these more complex settings. So, understanding the simpler solutions discussed here provides a useful (essential) dynamical foundation for further study. It is well known that at low frequencies (σ < f ) the coastal ocean responds strongly to the alongshore component of the wind but weakly to its across-shore component. This asymmetry happens because alongshore winds drive an acrossshore Ekman flow that generates a source or sink of water at the coast, and the coastal ocean must adjust to this mass change. In contrast, across-shore winds drive an alongshore Ekman flow, which does not impact the coastal mass balance. Here, we consider only the coastal response to alongshore winds. (See Crepon and Richez 1982, and McCreary et al. 1989, for discussions of the response forced by acrossshore winds.) Specifically, forcing is by an alongshore wind stress of the form τ y (y, t) = τoy Y (y) T(t),

(13.1)

where X (x) = 1 and Y(y) has equatorward (y = y1 ) and poleward (y = y2 ) edges, so that the forcing is a wind band. The advantage of this structure is that, because the y wind has no curl (τx = 0), the coastal response is driven entirely by onshore/offshore Ekman drift. It is possible to obtain more general solutions with X (x) = 1 (a coastal wind patch). Such solutions can be viewed as a combination of the solutions found here with the interior-ocean ones obtained in the previous chapter (see the discussion of Video 13.2b below). As for the interior ocean (Chap. 12), solutions are found to a simpler equation set that filters inertial oscillations from the response (Sect. 13.1). This simplification allows them to be obtained under a variety of conditions: for switched-on (Sect. 13.2) and periodic (Sect. 13.3) winds, in both 2-d and 3-d settings, and for both constant and variable f . In Sect. 13.4, we note that the processes active in these solutions are clearly identifiable in NIO observations. To conclude, we briefly consider river-driven coastal flows, noting their dynamical similarities to the wind-driven circulations just obtained (Sect. 13.5).

13.1 Simplified Model Equations A limitation of interior equations (12.1) is that they cannot simulate circulations with small zonal scales, a deficiency that prevents them from being able to represent flows along a north-south coast. A simplification of (9.1) that overcomes this limitation only sets u t = 0, so that without forcing by zonal winds (F = 0) the model equation set is

13.2 Switched-On Forcing

335

− f v + px = 0,

(13.2a)

vt  + f u + p y = G,

(13.2b)

pt  + u x + v y = 0, c2

(13.2c)

y

¯ Almost all solutions are obtained where G = G o Y (y) T (t) and G o = τo / (ρH). using Eqs. (13.2), the exception being the two solutions found in Sect. 13.2.3. Note that in this approximation the alongshore current v is in geostrophic balance, a property consistent with observed coastal flows. Further, Eqs. (13.2) are the same as (7.24) for free waves along a north-south coast, except they retain forcing and damping terms. Solving (13.2) for an equation in v alone gives vx xt  −

f2 vt  + βvx = G x x = 0, c2

(13.3)

which is unforced since X(x) = 1. It differs from (9.2c) in that it lacks the vt  t  t  /c2 and v yyt  terms. As a consequence, gravity waves are eliminated from the system and f only appears as a parameter in solutions, thereby allowing for simple analytic solutions even when f varies with y. On the  otherhand, solutions are accurate only when both terms are small with respect to f 2 /c2 vt  , which requires that the time T and meridional space L y scales of the response satisfy the inequalities T 2  f −2 and L 2y  R 2 (also see Appendix B); these inequalities are usually, but not always (Sect. 13.3), valid for the low-frequency, large-scale forcings of interest here. On the positive side, because (13.3) retains the vx xt  term, there is no restriction on the zonal scale of solutions L x . Finally, the lack of the G t  t  forcing term in (13.3) is not a problem for the same reasons noted in Sect. 12.1. Because u t is dropped from (13.2a), we expect that not all the model variables increase smoothly from zero in response to switched-on winds. As we shall see, only v and p do so, with u jumping to steady-state Ekman velocity due to the absence of inertial waves.

13.2 Switched-On Forcing A striking aspect of the coastal response to switched-on alongshore winds is a shift in the depth of the thermocline, a predominantly first-baroclinic-mode (n = 1) response. To focus attention on this important feature, we write down and discuss solutions in terms of a 1 21 -layer model, for which h corresponds to the depth of the top of the thermocline (Sect. 5.3.3). Further, for clarity we obtain most solutions without damping (γ = 0), only commenting on its impacts at the end of each subsection.

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13 Coastal Ocean

The exception is for the western-boundary solutions in Sect. 13.2.3.2, which include mixing to allow for finite-width, western-boundary currents. Solutions differ fundamentally depending on whether the model is: (1) 2-d, that is, ignores y variations; or 3-d and obtained on the (2) f -plane or (3) with variable f . We consider these cases sequentially, as each succeeding solution adds an important process not present in its predecessor.

13.2.1 Two-Dimensional Response Consider the coastal response when alongshore variations are ignored and f is constant. Setting β = 0, v = px / f , and p = g  (h − H ), (13.3) becomes h  − R 2 h x x = 0, where h  = h − H and R =

13.2.1.1

(13.4)

√  g H / | f | is the Rossby radius of deformation.

Solution

Equation (13.4) has two linearly independent solutions proportional to exp (±x/R). Radiation condition (9.4) requires that only one of them applies at a boundary, namely, the solution that decays away from the coast. The general solution of (13.4) is then h = H + A (t) e±x/R ,

(13.5)

where A can be a function of t but not x, and the upper (lower) sign here and elsewhere applies at an eastern (western) boundary. To evaluate A, we impose the boundary condition that u = 0 at x = 0. From  (13.2b) with γ = p y = 0 and v = g  / f h x , it follows that u=−

g G = 0 @ x = 0, h xt + f2 f

(13.6)

y

where G = G o θ (t) and G o = τo / (ρ1 H ). Inserting (13.5) into (13.6) gives At = ±R

fG g

y

⇒

A = ±R

f Go τo t = ± tθ(t) , g ρ1 c

(13.7)

√ where c = g  H and the factor  = f / | f | = ±1 keeps track of whether the wind is located in the northern or southern hemisphere. The solution for h is then

13.2 Switched-On Forcing

337 y

τo ±x/R te h = H ± θ(t) , ρ1 c

(13.8a)

which, with the aid of (13.2a) and (13.2b), gives the currents y  vt τo  G − = 1 − e±x/R θ(t) . f f f ρ1 H (13.8b) According to (13.8b), the response is coastally trapped, weakening exponentially offshore with an e-folding width scale R.

g τo hx = te±x/R θ(t) , f ρ1 H y

v=

13.2.1.2

u=

Dynamics

This famous solution, first obtained analytically by Charney (1955) and numerically by O’Brien and Hurlburt (1972), illustrates the essential dynamics of how alongshore winds drive coastal upwelling. Consider the eastern-coastal response (upper y sign) in the northern hemisphere ( = 1) for northerly winds (τo < 0). Figure 13.1 y (left panel) illustrates the thinning of h when τo = −1 dyn/cm2 , ρ1 = 1 gm/cm3 , f = 10−4 s−1 , c = 250 cm/s, and R = c/ f = 25 km. Immediately after the winds switch on, offshore Ekman drift u is established, which drains water from the coast. Subsequently, h thins at the coast in order to provide a source of water for this loss, and a geostrophically-balanced alongshore current v accelerates in the direction of the wind. As time passes, h gradually thins (blue and red curves) and eventually surfaces (h → 0, magenta curve), so that at later times the solution breaks down. Setting y h = x = 0 in (13.8a) and H = 100 m, the surfacing time is ts = −ρ1 cH/τo = 29 d. As for the interior solutions (see the end of Sect. 12.2.1), an additional process that can counteract the thinning of h is needed to extend the solution to later times.

Fig. 13.1 Schematic diagram illustrating the 2-d (y-independent) response of h forced by upwelling-favorable winds that drive offshore Ekman drift (arrows). The vertical axis is z, so that the h curves are plotted increasing downward. In the left panel, h thins from an initial value of H (blue and red curves), and eventually h → 0 (the thermocline surfaces), at which time the solution breaks down (magenta curve). In the right panel, there is a mixed layer of h m < H and h stops rising when it thins to h m ; thereafter, enough water upwells into the mixed layer to supply all the water for the offshore Ekman drift, and the system adjusts to a steady state with upwelling

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13 Coastal Ocean

One possibility is to set γ = 0, in which case solution (13.8) is altered only by  the replacement t → 1 − exp (−γt) /γ, allowing the layer to adjust to a steadyy state thickness h = H − τo / (γρ1 c) e±x/R . For the parameter values in the previous paragraph, however, γ must be large enough for γ −1 < ts = 29 days to keep h > 0, an unrealistically large damping since γ −1 = cn2 /A for low-order baroclinic modes is of the order of a decade or longer. A better way to include damping is to allow for strong mixing in a surface mixed layer of thickness h m , as in (5.20) and (12.13) with small δt. As a result, h stops rising at x = 0 when it thins to h m at x = 0, and thereafter enough water is entrained into the mixed layer at the coast to supply all the water for the offshore Ekman drift. Figure 13.1 (right panel) illustrates the resulting steady-state solution when δt → 0 so that in effect h never becomes less than h m . To summarize, the basic process that drives a coastal response in solution (13.8) is the divergence of water from the coast driven by Ekman drift. As such, it is dynamically similar to the open-ocean form of Ekman pumping discussed in Sect. 10.2, and is commonly referred to as “coastal Ekman pumping.”

13.2.2 Three-Dimensional Response, β = 0 The preceding 2-d solution is altered dramatically when τ y has a meridional structure Y (y). Specifically, coastal Kelvin waves are then possible, and they allow h to adjust to a steady state that (in many situations) doesn’t require γ = 0 or mixed-layer entrainment.

13.2.2.1

Solution

Even allowing for y-variations, the h-equation is still (13.4), and its general solution is (13.5) with A(t) replaced by A(y, t). Keeping p y in (13.2b) and setting G o → G (y), boundary condition (13.6) is altered to u=−

g g G = 0 @ x = 0. h xt − h y + 2 f f f

(13.9)

Combining (13.5) and (13.9) gives a differential equation for A, y

At τo + Ay = Y (y) θ(t) ,  c ρ1 c2

c = ±c,

(13.10)

where, as mentioned above, the factors ± and  = f / | f | in c ensure that the solution holds at eastern (upper sign) and western (lower sign) boundaries and in either hemisphere. The general solution to (13.10) is

13.2 Switched-On Forcing

339

  τo A= Y (y) θ(t) +  y − c t , ρ1 c2



y

Y (y) =

y yˆ

Y dy  ,

(13.11)

the sum of particular (steady-state) and homogeneous (Kelvin-wave) solutions. The lower limit yˆ is determined by radiation condition (9.4), which in the present situation is a statement that: Along an eastern-ocean (western-ocean) coast, there can be no response equatorward (poleward) of the forcing band because the Kelvin-wave group velocity is poleward (equatorward). This constraint is satisfied by setting yˆ = y1 (y2 ), that is, to the equatorward (poleward) edge of the wind band. To determine , we impose the initial condition that h = H and hence A = 0 at t = 0+ , justafter the wind has switched on. Evaluating (13.11) at that time gives y (y) = −τo / ρ1 c2 Y(y) and then, with the replacement y → y − c t,  is known for all times. It follows that h=H+

y   τo  Y (y) − Y y − c t e±x/R θ(t) , 2 ρ1 c

(13.12a)

and, with the aid of (13.2a) and (13.2b), that v=

y   g τo  h x = ± Y (y) − Y y − c t e±x/R θ(t) , f ρ1 cH

(13.12b)

  g τo G vt − hy + = Y (y) 1 − e±x/R θ(t) f f f ρ1 f H

(13.12c)

y

u=−

Note that u has no propagating part, consistent with the wave packet being composed of f -plane Kelvin waves for which u ≡ 0; moreover, except for the factor of Y (y), u is unchanged from its 2-d counterpart.

13.2.2.2

Dynamics

Solution (13.12) describes the response at all  times. The response just after the wind switches on can be found by expanding Y y − c t in a Taylor series  about t = 0  and keeping terms only to O(t), in which case Y (y) − Y y − c t → c Y (y) t = c Y (y) t. With this replacement, solution (13.12) is just the 2-d, coastal Ekman response (13.8), except with an additional factor of Y (y). Subsequently, the solution is altered by the radiation of the Kelvin-wave packet along the coast. The final steady-state response (t → ∞) is solution (13.12) without the Y (y − ct) terms, that is, after the Kelvin-wave packet has propagated out of the domain. Figure 13.2 (left panel) illustrates solution (13.12) at x = 0, the view looking westward from an eastern coast in the northern hemisphere. The solution is forced by y a northerly (τo < 0, upwelling-favorable) wind located from y1 = 0 to y2 = L with a meridional structure Y (y) indicated by the green curve. Let z (y) = − [H + aY (y)], y a = τo / (ρ1 cH ). Then, the left panel plots z (y) (red-dashed curve) and z (y − ct)

340

13 Coastal Ocean

Fig. 13.2 Plot of solution (13.12a) at x = 0 (left panel), and when it is modified by the presence of a mixed layer of thickness h m (right panel). The vertical axis is z, so layer-thickness curves are all plotted increasing downward. Both images look offshore from a northern-hemisphere eastern boundary, so that y increases to the right and horizontal velocity vectors are directed southward. Forcing is by a band of upwelling-favorable winds with the meridional structure Y(y) defined by (C.7b) (green curve). Parameter values used to prepare the plot are c = 250 cm/s, H = 100 m, y τo = −1 dyn/cm2 , L = 1000 km, and the response is shown at t = 7.72 days so that d = 1667 km. In both panels, the curves are the negatives of the Kelvin-wave (blue-dashed curves) and steady (red-dashed curves) parts of (13.12a), as well as their sum (black curves). The dashed-colored curves are shifted upwards slightly so that they don’t interfere with other plot elements. In the right panel, h only thins to the bottom of the mixed layer at z = −h m (black-dashed curve)

(blue-dashed curve), the negatives of the two parts of (13.12a), as well as their sum (black curve), and shows the solution at a time t = 7.72 days after all of the Kelvinwave part z (y − ct) has propagated to a location d = ct north of the wind band (d > L). Note that z (y) and z (y − ct) are the same except shifted in y and with opposite signs. Therefore, their sum cancels everywhere north of y = d + L, it has a wave front in the region d ≤ y < d + L, and is given by z (y) for y < d. After the passage of the wave front, h is adjusted to a steady state where the coastal pressure field balances the wind stress, that is, p y = g  h y = G o Y (y) =

y

τo Y (y) @ x = 0. ρ1 H

(13.13)

For the parameters listed in the figure caption, h ≥ 36 m, so that h never thins to zero thickness and all coastal upwelling is eliminated by the Kelvin-wave adjustment. Further, the time it takes the Kelvin wave to cross the wind band and for (13.13) to be locally established is t = L/c = 4.6 days, so that the transient (upwelling) phase of the coastal adjustment, during which h thins (h t = 0), shuts off very quickly. Thereafter, a southward coastal jet v accounts for all the water needed to balance the offshore Ekman drift u across the wind band, and since h t = 0 there is no upwelling. To confirm this result, an integral of (13.13) from y1 to y2 gives y y  δh = τo / ρ1 g  H y12 Y dy; then, with the aid of this expression and the property that v is geostrophically balanced, the jet transport north of the wind band (y ≥ y2 )

13.2 Switched-On Forcing

341

0 y   y is V = H −∞ v d x = g  H/ f δh = τo / (ρ1 f ) y12 Y dy, a statement that V equals the net Ekman transport across the wind band. For other reasonable parameter choices (e.g., H = 75 m and c = 100 cm/s), h can thin to zero thickness at some latitude within the wind band, and the solution breaks down. As for the 2-d solution, it is possible to prevent h from thinning to zero by including damping, but the required value of γ is unrealistically large. Alternately, we can include a mixed layer with entrainment we as in (12.13), and Fig. 13.2 (right panel) schematically illustrates the resulting response. The h field thins to h m at latitude y  and is equal to h m for y ≥ y  . In this case, the coastal jet attains a maximum  y y transport of V = τo / (ρ1 f ) y1 Y dy for y ≥ y  , less than the total Ekman transport since y  < y2 . As a result, V balances the net offshore transport only south of y  , whereas in the rest of the wind band where we = 0 (y  ≤ y < y2 ) upwelling supplies the additional water needed to balance the offshore Ekman drift.

13.2.2.3

Video

Video 13.1 shows a numerical solution of sea-level anomaly d = p/g = h/φ, φ = gH1 /cn2 (Eq. 5.27b). It is comparable to the solution in the left panel of Fig. 13.2, except forced by a band of southerly winds so that the response is flipped in sign and Y (y) = θ [(y − 15◦N) (30◦N − y)], a top-hat structure. Initially, onshore Ekman drift increases d (thickens h = H + h) near the coast as in 2-d solution (13.8). As time progresses, two Kelvin-wave fronts propagate northward along the coast from the edges of the wind band, as in solution (13.12). About 0500 on January 8, the southern wave reaches the northern edge of the wind band, and thereafter d and h tilt to balance the wind as in (13.13). On January 11, the northern front reaches the northeast corner of the basin, and subsequently propagates westward along the northern boundary, generating a coastal jet there; its offshore structure is the same as the eastern one, but it appears much narrower in the video owing to the different extents of the x- and y-axes. On January 17, the trailing edge of the southern front reaches the northern boundary, and thereafter the eastern-coastal flow is in steady state. Throughout the video inertial oscillations, filtered out of solution (13.8), are also visible; although they are circular as in Video 10.1a, they appear almost linear owing to the different axis ranges (see the discussion of Eq. C.8). The “bump” in sea level along y = 30◦N happens for the same reason as for the sea-level bands along the northern boundary in Video 10.1a and northern edge of the wind stress in Video 10.2c.

13.2.3 Three-Dimensional Response, β  = 0 When β = 0, the large-time limit of solution (13.12) is no longer the steady-state response, owing to the offshore radiation of Rossby waves. Unfortunately, there are no simple solutions to Eqs. (13.2) that illustrate the complete evolution of the solution

342

13 Coastal Ocean

from t = 0 to a state that involves Rossby waves. Here, then, we obtain solutions to simpler equation sets that show the response after the coastal adjustments of (13.12) have taken place. These solutions capture basic properties of the Rossby-wave response but, owing to distortions of the Rossby-wave coastal dispersion relation in the simpler systems, they also have unrealistic features. Because the properties of eastern- and western-coastal Rossby waves are so different, we consider each case separately.

13.2.3.1

East-Coast Solution

At an eastern coast, we obtain the solution to the inviscid (γ = 0) version of Eqs. (13.2) without the vt  term, that is, to the interior-ocean equation set (12.1) forced by wind stress (13.1). The corresponding v-equation is (13.3) without the vx xt  term. Setting vx xt  = 0, v = px / f , p = g (h − H ) and γ = 0 in (13.3), the equation for h  = h − H is (13.14) h t + cr h x = 0,   where cr = −β/ f 2 /c2 is the phase speed of non-dispersive, long-wavelength Rossby waves. Solution The general solution to (13.14) is h = H + A(x − cr t, y) .

(13.15)

At the boundary, we impose the condition that u = 0 at x = 0, which, after setting vt  = 0 in (13.2b), is g G = 0 @ x = 0. u = − hy + f f

(13.16)

Inserting (13.15) into (13.16) gives an equation for A y at x = 0, which integrates to the solution Go Go (13.17) A(−cr t, y) =  Y(y) θ(t) =  Y(y) θ(−cr t) , g g where Y (y) is defined in (13.11) and the replacement θ(t) → θ(−cr t) is valid since cr < 0. With A known at x = 0, it is known for all x with the replacement −cr t → x − cr t. The h field of the solution is then h=H+

Go Y (y) θ(x − cr t) . g

(13.18a)

13.2 Switched-On Forcing

343

Its velocity components, v= u=

g Go hx = Y (y) δ (x − cr t) , f f

G G Go Go − h y = [1 − θ(x − cr t)] + Y (y) cr y tδ (x − cr t) , f f f f

(13.18b)

(13.18c)

are readily obtained from h. Dynamics: According to solution (13.18), immediately after the wind turns on (t = 0+ ): h adjusts to the steady-state balance (13.13) right at the coast (x = 0) and is equal to H just offshore (x = 0− ); v consists of a δ-function coastal jet; and u is Ekman drift everywhere offshore (x < 0− ), dropping to zero only right at the coast (x = 0). This initial state is essentially the steady-state response of solution (13.12), except that the offshore decay scale R of the coastal circulation is zero; in effect, the neglect of vt in (13.2b) modifies the Kelvin-wave speed to infinity, thereby allowing the coastal response to develop instantly. Subsequently, Rossby waves carry the coastal response offshore, and east of the wave front the ocean is adjusted everywhere to a state of rest in which (13.13) holds. This steady-state solution is the Sverdrup-balanced (steady, inviscid) response to wind stress (13.1). As for the 2-d and β = 0 responses, a wind that drives offshore Ekman drift can be strong enough to cause h to thin to zero thickness at a latitude y  within the forcing region, and this situation can be avoided by including (unrealistically large) damping or a mixed layer of thickness h m . In the latter case, the steady-state response for h is the same as that for coastal h illustrated in the right panel of Fig. 13.2 (red-dashed curve) except that, owing to the offshore radiation of Rossby waves, the coastal h structure extends across the entire ocean: Specifically, for all x (13.13) holds for y < y  and h = h m for y ≥ y  . As a result, the interior ocean is at rest at all longitudes for y < y  (the offshore Ekman drift is cancelled by an onshore geostrophic current) but Ekman drift remains for y  ≤ y < y2 ; in the latter latitude band, the source water for the Ekman drift is provided by coastal upwelling (entrainment) as in the 2-d solution (13.8). Given offshore Rossby-wave propagation, a fundamental question is: Why do eastern-boundary currents exist at all? Several mechanisms have been proposed to account for them, all of which limit offshore Rossby-wave propagation in some way. One approach is to damp Rossby waves before they can propagate very far offshore (McCreary 1981b; Philander and Yoon 1982), a process we considered for the 1 21 layer model in the previous paragraph and discuss for the LCS model in Sect. 16.2.3.1 below. Two other trapping processes are: nonlinearities that reverse the propagation direction of long-wavelength Rossby waves (McCreary et al. 1992); and the presence of a continental shelf (e.g., Csanady 1978; Weaver and Middleton 1989, 1990; Furue et al., 2013). A discussion of these trapping processes, however, is beyond the scope of this book.

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13 Coastal Ocean

Videos Video 13.2a is comparable to Video 13.1, except that f is specified by the midlatitude β-plane approximation and Y (y) is given by (C.7b). In this video, all three of the adjustments discussed above (2-d, Kelvin-wave, and Rossby-wave adjustments) occur. Initially, the response is similar to that in Video 13.1, differing in that the meridional spreading of inertial waves is more apparent due to β-dispersion. Consistent with solution (13.18), the coastal response then propagates offshore as a packet of long-wavelength Rossby waves. A striking difference is that the Rossby-wave front in the video is not a step function; rather, it consists of a series of undulations that become increasingly narrow to the east, a consequence of the Rossby-wave packet containing dispersive Rossby waves (discussed below). The undulations extend south of the wind band, reflect from the southern boundary, and create a checkerboard-like pattern there. (In a version of Video 13.2a with νh = 5×105 cm2 /s, not shown, these smaller-scale features are damped out.) Eventually, the coastal h structure (i.e., h tilts to balance the wind) is present throughout the interior ocean, the Sverdrup-balanced state for this wind forcing. Video 13.2b is comparable to Video 13.2a, except that the wind forcing decays offshore, a more realistic offshore structure than a wind band. Specifically, τ y has the zonal structure X(x) given by (C.7b) with xm = 100◦E and x = 20◦. As a result, the solution is driven by offshore wind curl as well as alongshore coastal winds. To understand the contribution of each forcing type, note that the total forcing can be viewed as the sum of two parts: the wind band used to force Video 13.2a, which has no offshore wind curl; and the negative of the wind used for Video 12.3d shifted eastward until its eastern edge lies along the eastern boundary, which has no alongshore winds. Thus, although both forcings individually extend zonally to −∞, their sum cancels out everywhere west of 90◦ . In Video 13.2b, despite the offshore weakening of τ y , it is striking just how much forcing by coastal alongshore winds impacts pressure throughout the basin. More Dynamics: To understand the cause of the Rossby-wave undulations, it is useful to view the analytic and numerical packets as Fourier compositions of individual Rossby waves with wavenumber k and frequency σ (k). Then, the packet in solution (13.18) is composed entirely of non-dispersive Rossby waves with dispersion relation (7.25), whereas the video packet video contains dispersive Rossby waves with dispersion relation (7.6). Figure 7.2 (bottom-right panel) plots dispersion curves for both types when  = 0: non-dispersive (dashed straight line) and dispersive, longwavelength (solid curve for k < k R ≈ f /c) Rossby waves (see the discussion after Eq. 7.6).   The group velocity, cg = cgx , cg y , for non-dispersive Rossby waves determined from (12.4) is c2 cgx = σk = −β 2 , cg y = σ = 0. (13.19) f Because cg is independent of k and  the initial step-function structure of coastal h propagates offshore without modification. In contrast, dispersion relation (7.6) for dispersive Rossby waves gives the group speed

13.2 Switched-On Forcing

345

 k 2 − αˆ 2  2 2 cgx = β  2 θ αˆ − k , 2 2 k + αˆ

  2 2k 2 cg y = β  2 θ αˆ − k , 2 2 k + αˆ

(13.20)

where αˆ 2 = 2 + f 2 /c2 . Step functions are added to each equation, since here we ˆ are considering long-wavelength Rossby waves for which cgx > 0 and k < α. According to (13.20), cg reduces to (13.19) in the limit that k and  tend to 0, as it should. Note also that the magnitude of cgx decreases montonically to zero as k increases to α. ˆ In the video packet, then, the initial step-function structure is modified as it propagates offshore, because dispersive Rossby waves with a smaller   cgx  increasingly lag behind. Since waves with smaller cgx also have smaller λ, the local wavelength of the undulations decreases away from the front. Finally, cg y = 0 because  = 0; consequently, Rossby-wave energy propagates southward, as well as westward, from the eastern boundary (Sect. 7.3.3). The southward propagation is stronger for the shorter-wavelength waves (with larger ); eventually, these waves reflect from the southern boundary to create the checkerboard-like interference patterns visible in the southern parts of Videos 13.2a and 13.2b.

13.2.3.2

West-Coast Solutions

At a western-ocean coast, we obtain solutions to Eqs. (13.2) without the pt  term and with horizontal viscosity in the from νh vx x , an approximation of −νh ∇ 2 v appropriate for a narrow western-boundary current (Sect. 11.2.1). Setting p = g (h − H ) in (13.2a) and (13.2b), it is straightforward to show that h xt + βh  = −γh x + νh h x x x ,

(13.21)

where h  = h − H . The advantage of (13.21) is that it allows only short-wavelength Rossby waves with eastward group velocity (a consequence of the neglect of pt  ), which are the waves generated at a western boundary. We obtain two solutions to (13.21): (i) one that is time-dependent and inviscid (∂t = 0, γ = νh = 0), and (ii) another that is steady-state with mixing (∂t = 0, γ = 0 or νh = 0). To obtain simple solutions, we also assume that f = β y with β constant (the equatorial β-plane approximation). Inviscid Solution The method of Laplace transforms (Sect. 9.3.2) provides a straightforward way to solve (13.21) for the inviscid, time-dependent solution. With γ = νh = 0 and the initial condition that h (x, y, 0) = 0, it follows from (9.21) that a general solution for the Laplace transform of (13.21) is s hˆ x + β hˆ  = 0

⇒

ˆ y) exp − xβ , hˆ (y, s) = A(s, s

(13.22)

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13 Coastal Ocean

and the transform of boundary condition (13.9) is uˆ = −s

g ˆ  g ˆ  GoY = 0 @ x = 0. − h h + x f2 f y sf

(13.23)

ˆ Inserting (13.22) into (13.23) leads to a differential equation and solution for A, Aˆ y β − 2 Aˆ = f f

 GoY Aˆ =  f g sf

⇒

y

Go y Aˆ =  gs



y y2

  Y y dy  , y

(13.24)

so that the Laplace transform of hˆ  is Go y hˆ (x, y, s) =  g



y y2

  Y y exp (−xβ/s) . dy  y s

(13.25)

To complete the solution, we invert hˆ  using the transform pair,

1 k exp − s s

↔

√  J0 2 kt ,

(13.26)

which can be found in most tables of Laplace transforms. For our case, k = xβ, and so   

  Go y y Y y  2 dy J βxt , (13.27) h (x, y, t) = H +  0 g y y2 where J0 is a Bessel function of order zero. Description: Figure 13.3 (top panel) plots J0 (ξ) (black curve). Its structures for large and small values of ξ can be shown to be  J0 (ξ) ∼



1 2 cos ξ − π , πξ 4

1 J0 (ξ) = 1 − ξ 2 , 4

ξ  1,

ξ  1.

(13.28a)

(13.28b)

The blue- and red-dashed curves plot the two approximations, and it is apparent that they are quite good for ξ  2 and ξ  2, respectively. According to (13.28a), J0 (ξ) 1 is close to a sinusoidal oscillation for ξ  2, except that it weakens like ξ − 2 .

13.2 Switched-On Forcing

347

Fig. 13.3 (upper panel) Plots of exact J0 (ξ) (black curve) and its approximations (Eq. 13.28a; bluedashed curve) and (Eq. 13.28b; red-dashed curve), which are valid for large and small ξ. To be able to distinguish the curves, the approximate √ curve for large (small) ξ is shifted upwards (downwards) by 0.05. (bottom panel) Plots of J0 (2 βxt) at t = 3 (red curve) and 6 (blue curve) months. The structures of the curves are the same when the blue curve is stretched in x by a factor of 2

According to (13.28a), ξ is the phase of the offshore oscillation. With the aid of √ (6.4) and with ξ = 2 βxt, the local wavenumber, wavelength, and frequency of the oscillation are  √ 2π √ x, (13.29a) k = ξx = βt/ x ⇒ λ = √ βt σ = ξt =

 √ βx/ t.

(13.29b)

 Figure  √13.3 (bottom panel) plots the offshore structure of solution (13.27), J = J0 2 βxt , as a function of x at y = y1 (the southern edge of the wind band) when t = 3 months (red curve) and 6 months (blue curve). Consistent with (13.29a), J  oscillates and decays offshore, with its local wavelength λ increasing offshore. √ At later (earlier) times, J  thins (broadens) in x, a consequence of the factor t in ξ. More generally, consider its structure at two times t1 and t2 : J  at time t2 has the same value at the location (t1 /t2 ) x as J  at t1 does at location x. In agreement with this relation, note that the red and blue curves are the same, except the x-scale of the former is shrunk by a factor of 2 over that of the latter. Alternately, consider the response at longitude x = x1 ; as time passes and J  thins, the solution oscillates 1 at a frequency (13.29b) that decreases in time like t − 2 .

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13 Coastal Ocean

Dynamics: As for the east-coast solution, a useful way to understand the properties of solution (13.27) is to view the packet as being a Fourier composition of individual Rossby waves with wavenumber k and frequency σ (k). Equation (13.21) only allows for waves with the dispersion relation σ = −β/k, and hence the group velocity cgx = σk =

σ2 βλ2 β . = 2 = β k 4π 2

(13.30)

According to (13.30), waves with larger values of λ and σ have larger cgx , and so are present farther from the western boundary than waves with smaller values. Specifically, at time t we expect that waves with wavenumber k and σ = −β/k   frequency  will appear in the wave packet at the location x = cgx t = σ 2 /β t = βtλ2 / 4π 2 . Indeed, solving these expressions for σ and λ gives precisely their local values in Eqs. (13.29). √  An unrealistic property of solution (13.27) is that, because J0 2βxt = 1 at t = 0, h extends to x = ∞ immediately after the wind switches on. This feature happens because the dispersion relation obtained from (13.21), σ = −β/k, allows waves with k = 0 (λ = ∞), which according to (13.30) have infinite zonal group velocity. The zonal group velocity from dispersion relation (7.6), however, is  k 2 − αˆ 2  2 2 cgx = σk = β  2 θ k − αˆ , k 2 + αˆ 2

(13.31)

the same as in (13.20) except the argument of the step function is reversed to indicate that here cgx applies for short-wavelength Rossby waves. According to (13.31), cgx = 0 both when k = αˆ and as k → ∞, and therefore attains a maximum value for some intermediate value. To determine√its maximum, we set ∂cgx /∂k = 0 to   find that it occurs at the wavenumber km = 3α, ˆ and hence max cgx = 18 β/αˆ 2 =   1 βc2 / f 2 + 2 c2 , less than one eighth of the group speed of long-wavelength 8 Rossby waves. Therefore, western-boundary Rossby waves spread offshore at a finite speed, and much more slowly than eastern-boundary Rossby waves do. Another unrealistic property of inviscid solution (13.27) is that, because the offshore structure thins forever, λ → 0 as t → ∞. (Mathematically, this unrealistic property happens because in steady state and without damping Eq. 13.21 is first order in x, thereby eliminating the possibility of a steady-state, western-boundary layertolerate such small spatial scales: The resulting intense and narrow currents are nonlinearly unstable, leading to the generation of eddies that, on average, broaden the currents. In models, it is common to represent this process in a linear way by retaining horizontal viscosity, a process we first considered in Sect. 11.2.1 and discuss further after the following video discussion. Video: Video 13.3a shows the western-coastal response without viscosity (νh = 0), and its properties agree well with the preceding analytic results. As in solution (13.27), a packet of short-wavelength Rossby waves spreads offshore from the coast.   In theory, its front advances at the speed of the maximum group velocity, max cgx ,

13.2 Switched-On Forcing

349

which is inversely proportional to f 2 (assuming 2 c2 is small with respect to f 2 ); consequently, the front should spread faster at lower  latitudes, and that property is evident in the video. For example, at 10◦N max cgx = 2.1 cm/s so that after one year, say, the front should be located 5.9◦ offshore, and its location in the video is close to this value. Near the front, the theoretical local wavelength is λm = 2π/km , which at 10◦N is 3.0◦ both theoretically and in the video. Other key properties shared by (13.27) and the video solution are that: at a fixed time local wavelengths in the packet decrease toward the coast; and at a fixed location they decrease in time. As time progresses, however, wavelengths near the coast become so short that they can no longer be represented on the model grid (i.e., λ  2x = 0.2◦ ); consequently, a near-shore region develops without any Rossby waves, and its width broadens in time. Throughout the adjustment, there is a westward current along the southern boundary (purple and blue shading). It is generated by the offshore Ekman drift from the western boundary, which drives a northward, western boundary current that extends first to the equator and then eastward along the southern boundary via the propagation of coastal Kelvin waves. In this solution with νh = 0, however, the western-boundary current is not realistically simulated owing to the model’s resolution, among other things being too narrow to be visible (see Videos 13.3b and 13.3c below). Viscous Solution As noted above, solution (13.27) never reaches a steady state, but rather continues to thin forever. We can avoid this problem by keeping the mixing terms in (13.21). The resulting steady-state equation for h  is βh  = −γh x + νh h x x x , that is, (13.21) with h xt = 0. We consider the solutions with damping and viscosity separately. With only damping (γ = 0, νh = 0), the solution follows the same steps as in Eqs. (13.22)–(13.25), except with the replacements s → γ, hˆ  → h  , and G o /s → G o , yielding    Go y y Y y h (x, y) = H +  dy  e−xβ/γ . (13.32a)  g y y2 According to (13.32a), the steady-state, western-boundary layer has the zonal structure of a Stommel layer, decaying offshore with the e-folding scale r S = γ/β, (Sect. 11.2.1). The layer extends equatorward of the wind band, thereby providing a source (sink) for the offshore (onshore) Ekman transport within the band. In the videos discussed below, mixing is provided by Laplacian viscosity (νh = 0) without damping (γ = 0). A more relevant analytic solution than (13.32a), then, is the solution to βh  = νh h x x x = 0, and it is straightforward to show that it is h (x, y) = H +

Go y g 1



y y2

  sin Y y dy  e−γh x/2  y

 √   3/2 γh x + π/3 sin (π/3)

,

(13.32b)

where γh = (β/νh ) 3 . Solution (13.32b) is the same as (13.32a), except that its zonal structure is replaced with that of a Munk layer.

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13 Coastal Ocean

Dynamics: It is noteworthy that the structures of h along eastern and western coasts (at x = 0) differ: For an eastern coast, coastal h in (13.18a) is proy   portional to Y(y) = y1 Y y  dy  , a statement that balance (13.13) holds so that coastal h  tilts  y to  balance the wind; for a western coast, Y(y) is replaced by Yw (y) = y y2 Y y  /y  dy  in solutions (13.32), and coastal balance (13.13) is modified. This difference is traceable to the implementation of the coastal boundary condition, u = 0, in the two cases. The condition requires that (13.2b) holds with u = 0, that is, the balance of terms at x = 0 is p y + vt + γv − νh vx x = G,

(13.33)

where for the present discussion we include Laplacian viscosity νh vx x . For inviscid, eastern-coast solution (13.18), the offshore propagation of long-wavelength Rossby waves ensures that v = vt = vx x = 0 at x = 0, and the remaining two terms in (13.33) then require that balance (13.13) holds. In contrast, for a western coast the offshore propagation of short-wavelength Rossby waves does not eliminate the coastal current; consequently, for each of solutions (13.27), (13.32a), and (13.32b), one of the terms vt , γv, or νh vx x remains in (13.33), and balance (13.13) is necessarily altered. We can go one step farther. Expressed in terms of h  , condition (13.33) is h y +

γh  νh h x x G h x + − = . f f f g

(13.34)

For the three western-boundary solutions, (13.34) separates into the three separate conditions, h y +

G h xt = , f g

h y +

G γh x = , f g

h y −

G νh h x x x = , f g

(13.35)

respectively. Likewise, (13.21) splits into the three equations, h xt β = − h, f f

γh x β = − h, f f

−

νh h x x x β = − h, f f

(13.36)

Combining (13.35) and (13.36) shows that the coastal structures of h  in all the solutions satisfy the same coastal balance h y −

β  h = f f



h f

= y

G , g

(13.37)

and therefore they have the same coastal structure Yw (y). Videos: Videos 13.3b and 13.3c show solutions when νh = 5×105 and 5×106 cm2 /s, respectively. Together with Video 13.3a, the sequence shows a gradual change from an inviscid response to one dominated by viscosity. In Video 13.3c, for example,

13.3 Periodic Forcing

351

there is little indication of the offshore propagation of short-wavelength Rossby waves. Instead, the initial Kelvin-like response is followed by a rapid adjustment to a Munk-layer structure that changes sign offshore.

13.3 Periodic Forcing The three types of solutions obtained in Sects. 13.2.1–13.2.2 can also be obtained for periodic forcing when T (t) = e−iσt . Here, though, we obtain a general periodic solution to Eqs. (13.2) without any additional approximations, noting that three solutions analogous to the previous ones follow as special cases. In order to compare the solution more easily to the free-wave solution obtained in Sect. 7.3.1, we write it in terms of pn for the LCS model, rather than h.

13.3.1 Solution For periodic forcing, the solution is also periodic and the operator ∂t  = ∂t + γ is just the complex number −iσ  , σ  = σ + iγ. Viscid and inviscid solutions therefore have the same form, differing only in whether parameter σ  is σ + iγ or σ. With the aid of (13.2a), (13.3) becomes − iσ  px x + iσ 

f2 p + β px = 0. c2

(13.38)

Solutions to (13.38) are free waves of the form exp (ikx + iy − iσt) with the wavenumbers ⎛ ⎞  2 f 2 1 σ β k ( 2 ) (y) = −  ⎝1 ∓ 1 − 4 2 2 ⎠ , (13.39) 2σ β c the same wavenumbers as in (7.26) except with damping (σ → σ  ). Radiation condition (9.4) requires that only one of the waves applies to each boundary, namely, the wave that decays or has a group velocity away from the boundary. With this restriction, the general solution to (13.38) at each boundary is p = P (y) eik(y)x−iσt ,

(13.40)

where k = k (1)(k (2) ) is used for an eastern (western) boundary. Boundary condition (13.9) with ∂x → ik and ∂t → −iσ  holds, leading to the differential equation

σ k ∂y + P = G. f

(13.41)

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13 Coastal Ocean

To solve (13.41), it is useful to define the quantity (integrating factor) 

yo

σ k  dy , f

y



y

(y) =

(13.42)

in which case the solution to (13.41) is e

−





e P

 y

=G

⇒

τo −(y) e P(y) = ρH ¯

y yˆ

   e( y ) Y y  dy  ,

(13.43)

where yˆ is defined after (13.11). Note that the lower limit of the integral in (13.42) is arbitrary because terms yo cancel in P(y) owing to the multiplication of    involving exp [−(y)] and exp  y  . The complete solution is then p = Po(y) e

−(y) ikx−iσt

e

v=

ik p, f

y

,

τo Po(y) = ρH ¯ u=



y yˆ

   e( y ) Y y  dy  ,

 ky G 1 − eikx − i x p, f f

(13.44a)

(13.44b)

the solutions for v and u following from (13.2a) and (13.2b). With p written as (13.44a), it is easy to compare solution (13.44) with its free-wave counterpart (7.29). Because values of k are the same in (7.26) and (13.39), the two solutions differ only in their amplitudes, with the constant Po in (7.29) replaced by variable Po(y) in (13.44a). Here, Po(y) is adjusted to cancel Ekman drift at the coast, whereas for the free wave setting the amplitude to a constant Po ensures that u = 0 everywhere at the coast.

13.3.2 Dynamics Solution (13.44) describes the continual adjustment of the coastal ocean to oscillatory forcing. For weak damping, properties of the radiation patterns are essentially the same as those discussed in Sect. 7.2.2: Specifically, the radiation consists of β-plane Kelvin waves and Rossby waves poleward and equatorward of the critical latitude θ R , respectively (defined in Eqs. 7.13, 7.14, and 7.15). In the next two paragraphs, we discuss the responses in both regions when γ = 0, in which case k is real (complex) for |θ| less than (greater than) θ R . The discussion complements the similar one in Sect. 7.2.2.2.

13.3 Periodic Forcing

13.3.2.1

353

Kelvin-Wave Limit

When |θ| > θ R , the derivation leading to (7.32) applies, yielding

21



 y β σ f ϑdy  − iσt , exp ± ϑx exp −i x ± i c 2σ yo c (13.45a)    1 y  y y   fo 2 σ τo Po(y) = ϑdy  Y y  dy  , exp ∓i (13.45b) ρH ¯ f (y  ) yˆ yo c

p = Po(y)

f fo

the same as (7.32) except with Po replaced by Po(y). The corresponding f -plane and 2-d solutions are special cases of (13.45). On the f -plane, β = 0, f o = f , k = ∓i f /c, k y = 0, ϑ = 1, and (13.45) simplifies to 

σ 

σ    f    exp ∓i y Y y dy exp ± x exp ±i y − iσt , c c c yˆ (13.46) which is solution (13.12) except forced by an oscillatory wind. To recover the 2-d limit from (13.46), we assume that σ is large enough for the distance the Kelvin wave propagates during a wave period, measured by δ y = |c/σ|, is much smaller than the width scale of Y (y), y (δ y/y  1). After two integrations by parts (see Eq. 12.27), it follows that 

y

τo p= ρH ¯



y





σ  y

σ    δy c    . (13.47) exp ±i y exp ∓i y Y y dy = ∓ Y (y) 1 + O c c iσ y yˆ Then, after dropping the small last term in (13.47), (13.46) simplifies to

y τo c f p=∓ Y (y) exp ± x exp (−iσt) , ρH ¯ σ c

(13.48)

which is solution (13.8) except for an oscillatory wind. (Alternately, this solution can be found by rederiving the solution after dropping the P y term from Eq. 13.41.)

13.3.2.2

Rossby-Wave Limit

When |θ| ≤ θ R , k is real, there is no meridional propagation, and the response is Rossby-like. This is evident in the low-frequency limit (σ → 0), when   property k1 → − (σ/β) f 2 /c2 , k2 → −β/σ, and δ y/y  1. For an eastern boundary with k = k1 , the preceding restrictions imply that (y) → 0, and hence (13.44a) reduces to y  y   τo Y y  dy  exp (ik1 x − iσt) , (13.49a) p= ρH ¯ yˆ

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13 Coastal Ocean

describing the continuous offshore propagation of long-wavelength Rossby waves. a western boundary with k = k2 , the restrictions imply that (y) → For y f ) dy  = ln ( f o / f ) so that e = f o / f , and (13.44a) becomes (β/ yo y

τo f (y) p= ρH ¯



y yˆ

  Y y dy  exp (ik2 x − iσt) , f (y  )

(13.49b)

describing the continuous offshore propagation of short-wavelength Rossby waves. Note the similarity of these expressions to the solutions obtained in Sect. 13.2.3, which were obtained by imposing the low-frequency limit from the outset. In particular, at the coast the amplitude of (13.49a) is the same as for solution (13.18a), and that of (13.49b) is the same as those in solutions (13.32). Properties of the video (exact) solutions discussed below are consistent with solutions (13.48) and (13.49) near the coast. As for solutions (7.36), however, they break down sufficiently far offshore where the slopes of Rossby-wave phase lines invalidate the criterion L 2y  R 2 . A result of the slope is that Rossby-wave energy propagates meridionally as well as zonally (Sect. 7.3.3), a process not present in solutions (13.49).

13.3.2.3

Radiation Amplitude

As for interior solution (12.26), the amplitude of solution (13.44) involves an integration across the forcing region, and the form of the factor e in the integrand determines the radiation strength. In the Rossby-wave limit,  is real and small, and consequently has a minor impact on the amplitude. the Kelvin-wave limit (13.45b), however,    In y  the factor includes the term exp ∓i yo (σ/c) ϑdy , which oscillates meridionally with the wavenumber  = ∓ (σ/c) ϑ. Following the discussion in Sect. 12.3.2.2 (with x → y and kr → ), the radiation amplitude is largest when || L y  1, that is, when the wavelength of the Kelvin wave, λ y = 2π/ || = c P/ϑ, P = 2π/σ, is of the order of, or greater than, the width of the wind band L y . For low-order baroclinic modes and forcing periods of interest here, the inequality holds and Kelvin-wave radiation is prominent. (For example, with c = c1 ≈ 250 cm/s, P = 60 days, L y = 20◦ , and for simplicity setting ϑ = 1, then λ y = 117◦  L y .)

13.3.3 Videos Video groups 13.4 and 13.5 illustrate the eastern- and western-boundary responses forced by switched-on bands of oscillating meridional wind, τ y y = τo Y(y) sin(σt)θ(t), where Y(y) is a top-hat function from 20–40◦N. Within each group, the forcing periods are P = 2π/σ = 60, 180, and 365 days. The videos are

13.3 Periodic Forcing

355

similar to those in video groups 7.5 and 7.6, differing only in the nature of their forcing, and it is useful to compare solutions in the two sets. In all the videos, transient Rossby-wave packets propagate away from the coast at all latitudes north of the southern edge of the wind. In contrast, their equilibrium responses change markedly with P: The critical latitudes associated with the above periods are θ R = 6.7◦ , 26.3◦ , and 55.7◦ , respectively, so that the responses are everywhere coastally trapped (P = 60 days), consist of β-plane Kelvin waves and Rossby waves north and south of θ R (P = 180 days), and are composed of Rossby waves at all latitudes (P = 365 days). Offshore, Rossby-wave packets propagate energy along ray paths that undergo one or more reflections from the southern boundary (Sect. 7.3.3), resulting in complex radiation patterns in the interior ocean. East-coast reflections: In Video 13.4b (P = 180 days), the offshore response develops a pattern (envelope) of 3–4 oscillations of reflected wave energy, although it is blurred by small-scale signals; a clearer pattern emerges in Video 13.4c, in which smaller-scale signals are weakened by viscosity (νh = 5×105 cm2 /s). In both videos, the patterns have weaker amplitudes and different structures than in Videos 7.5b,c. They are weaker because the Rossby waves are forced by winds in a narrow latitude band from 20◦N to the critical latitude θ R = 26.3◦N, whereas in the earlier videos they are forced in a broader band extending from the southern boundary (10◦N) to θ R . Their structures differ because the ray paths that form them arise from different coastal locations: for Videos 13.4b,c from the red pathways in Fig. 7.4, and in Videos 7.5b,c from both the red and yellow pathways. In contrast, the patterns in Videos 13.4d,e (P = 365 days) are more similar to those in Videos 7.5d,e, because θ R = 55.7◦N so that the location and latitudinal extent of the ray paths that create them are more comparable. West-coast reflections: In contrast to the east-coast solutions, the solutions in Videos 13.5a–13.5e are almost the same as their counterparts in Videos 7.6a–7.6e, owing to the similar latitude range of their forcings: Because coastal information propagates southward along a western-ocean coast, offshore radiation occurs everywhere south of θ R in both solution sets. In Video 13.5a (P = 60 days), the equilibrium response consists entirely of β-plane Kelvin waves that continually propagate equatorward along the western boundary; in addition, a transient packet of short-wavelength Rossby waves is initially generated, with a structure similar to that in Video 13.3a. In Video 13.5b (P = 180 days), the coastal response consists of short-wavelength Rossby waves south of θ R = 26.3◦ , which propagate offshore to generate a pattern like that in Video 7.6b; because the pattern is composed of short-wavelength waves, most of it is eliminated with a small amount of horizontal viscosity (νh = 5×105 cm2 /s; Video 13.5c). At P = 365 days, the response consists of Rossby waves at all latitudes, which without viscosity develop an offshore pattern like that in Video 7.6d (Video 13.5d) and with viscosity (νh = 5×105 cm2 /s) forms a western-boundary region with a structure similar to, but narrower than, that in Video 13.5d (Video 13.5e).

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13 Coastal Ocean

13.3.4 Solutions for a Slanted Boundary When the coast is slanted as in Fig. 7.5, the coastal response in the Kelvin-wave regime is similar to (13.44). The interior-ocean response is still a periodic Ekman drift, u ek = G/ f , in the direction of x, and at the coast (x˜ = 0) the coastal response must cancel the component of u ek normal to the boundary, G cos α/ f . Steps in the derivation of the boundary solution are an extension of those in Sect. 7.4.1, modified to allow for forcing. They are also the same as (13.38)–(13.43), except with the y y replacements: q → q˜ for variables and coordinates; G → G cos α; τo → τo cos α; β → β cos α; and f˜(x, ˜ y˜ ) → f˜(0, y˜ ), which is valid in the Kelvin-wave regime (Sect. 7.4.1). The resulting boundary solution is (13.44) without the interior Ekmandrift term, ˜˜ ˜ , p˜ = P˜ o( y˜ ) e−( y˜ ) ei k x−iσt

v˜ =

i k˜ p, ˜ f˜

y

τo cos α P˜ o( y˜ ) = ρH ¯



y˜ yˆ

u˜ = −G cos α + i

  ˜  e( y ) Y˜ y  dy  ,

k˜ y˜ x˜ p, ˜ f˜

(13.50a)

(13.50b)

where k˜ is given by (7.52), that is, (13.39) with β → β cos α and f → f˜(0, y˜ ). The i, where i,  i, and j are complete solution is p, ˜ v˜j , and the vector sum of u ek i + u˜ unit vectors in the directions of x, x, ˜ and y˜ . As discussed in Sect. 7.4.1, the radical of k˜ implies that the critical latitude y R is modified to y Rc in (7.53), so that solution (13.50) still has Kelvin-wave ( y˜ > y Rc ) ˜ y˜ ) ≈ f˜(0, y˜ ), and Rossby-wave regimes ( y˜ < y Rc ). Given the constraint that f˜(x, solution (13.50) is valid in the Kelvin regime because it decays rapidly offshore, but is not valid in Rossby regime since it extends offshore. On the other hand, the Rossby-wave response is well defined by the set of ray paths that depart the boundary equatorward of y R (Sect. 7.4.2).

13.4 Observations The idealized solutions in this chapter identify a sequence of coastal processes forced by alongshore winds τ y : initial coastal Ekman pumping (Sect. 13.2.1 and Fig. 13.1); followed by the radiation of coastal waves (Sect. 13.2.2 and Fig. 13.2) and Rossby waves (Sect. 13.2.3). For upwelling-favorable τ y , the first process thins h (corresponding to raising the thermocline in the real ocean), whereas the second limits the change in h. Whether h thins to the mixed-layer thickness h m (so that subsurface water can upwell into the mixed layer) also depends on the background depth of the thermocline H , with larger H inhibiting upwelling and vice versa. In the NIO,

13.5 River-Driven Circulations

357

prominent regions forced by alongshore winds are the coasts of Somali, Arabia, and India during the monsoons. The responses in each region differ considerably, y depending on the wind strength τo and H . Along the east coast of India (Sect. 4.7.3.4), upwelling-favorable winds during summer thin h along the southern part of the coast (Videos 3.1 and 1.1), but the thinning does not usually lead to the uplift of cool subsurface water to the ocean surface and, hence, SST remains warm (Fig. 2.4 and Video 2.1). One possible reay son is that τo is weak enough for coastal waves to prevent h from thinning to the mixed-layer thickness h m (as in Fig. 13.2, left panel). Another is the propagation of a downwelling-favorable coastal wave around the perimeter of the Bay of Bengal: After its arrival, it thickens H everywhere along the coast, thereby counteracting the wind-driven thinning of h. Along the west coast of India (Sect. 4.9.2.4), the summertime winds also have an upwelling-favorable component (Video 3.1), which starts to thin h as early as May. As the monsoon advances, h thins along the coast and SST cools, albeit weakly (Videos 1.1 and 2.1). Although part of this coastal response is locally forced, a significant part results from upwelling-favorable, remote signals that are generated around Sri Lanka and along the Indian east coast: They decrease H along the Indian west coast, allowing the local winds to thin h to h m more easily (as in Fig. 13.2, right panel). During the summer monsoon, the alongshore winds off Somalia and Arabia are intense and upwelling-favorable, particularly off Somalia (Fig. 3.1, top panels, and Video 3.1; Sects. 4.9.5 and 4.9.6.3). Given the wind strength, we expect h to thin to h m over much of the coast (as in Fig. 13.2, right panel). This rise does occur, as evidenced by sea level decreasing and SST cooling there (Figs. 4.7 and 2.4; Videos 1.1 and 2.1). Off Somalia, however, the response is so strong (nonlinear) that the coastal circulation does not remain coastally confined, but rather forms gyres and cold water upwells in wedges between them (Figure 4.33 and Video 2.1). Another remarkable consequence is that the net upwelling of subsurface water throughout the season is much weaker than suggested by the solutions discussed in this chapter. The reason is the offshore decay of the wind forcing (Findlater Jet): The resulting offshore wind curl generates a downwelling-favorable Rossby wave that, when it arrives at the coast, thickens H enough to prevent h from thinning to h m and thereby eliminates upwelling (Chatterjee et al. 2019; see Sect. 17.3.3.1 for a detailed discussion of this process).

13.5 River-Driven Circulations Consider the response of a 1 21 -layer model to outflow from a coast along x = 0. Suppose the outflow has the form R(y) = Ro Y (y), where Y (y) describes its acrossstream structure, Ro is its maximum transport/width (cm2 /s), and the integral of R across the outflow is its total transport R. For the moment, assume that the density of river water is the same as that the model’s active layer, ρ1 . In that case, the winddriven solutions obtained in this chapter are easily adapted to simulate river-driven ones.

358

13 Coastal Ocean

Recall that the wind-driven solutions are composed of two parts: an interior Ekman y drift u e = G/ f = G o Y (y) / f , G o = τo / (ρ1 H ); and a coastal response in which the zonal current at the coast is u c = −u e so that the two sum to zero at the coast. Suppose now that u e > 0 and is dropped from the response. Then, we can view the modified solution as being driven by a flow u c from the land to the ocean, that is, by outflow at the velocity u r = u c . Specifically, the outflow-driven solutions are (13.8), (13.12), (13.18), and (13.44) without the number 1 in the expressions for u (to eliminate the interior Ekman drift) and with G/ f → R/H . In this simple model, then, the dynamics of wind- and outflow-driven flows are very similar, differing only in the existence of interior Ekman drift in the former. To illustrate, imagine adding −u e to the flow field in Video 13.2a: The addition eliminates the eastward-flowing Ekman drift west of the Rossby-wave front, replacing it with westward flow that extends eastward from the front to the coast. Of course, river water is fresher than ocean water (ρr < ρ1 ), and the river-driven solution in the previous paragraph lacks this important aspect. A discussion of this topic is beyond the scope of the book. We note, however, that one way to proceed is to use a variable-density, 1 21 -layer model that allows the layer-1 density ρ1 to vary in response to ρr . This approach was followed by McCreary et al. (1997), who found that river plumes can go “upstream,” that is, against the propagation direction of coastal Kelvin waves. Matano and Palma (2010, 2013) reported similar upstream propagation in their studies of coastal plumes using an OGCM.

Video Captions Switched-On Forcing Video 13.1 Eastern-boundary response forced by a band of meridional wind stress y of the form τ y = τo Y(y)θ(t), where Y (y) is given by (C.7b) with y = 20◦ and ym = ◦ 30 . The Coriolis parameter is constant and evaluated at 30◦N. The eastern boundary is closed, and open boundary conditions are imposed at the western boundary as described in Appendix C. Video 13.2a As in Video 13.1, except that f is specified by the midlatitude β-plane approximation. Video 13.2b As in Video 13.2a, except that X(x) is given by (C.7b) with xm = 100◦ E, so that τ y decays away from the eastern boundary. Video 13.3a As in Video 13.2a, except showing the western-boundary response when νh = 0. The western boundary is closed, and open conditions are imposed at the eastern boundary as described in Appendix C. Video 13.3b As in Video 13.3a, except that νh = 105 cm/s. Video 13.3c As in Video 13.3a, except that νh = 106 cm/s.s

Video Captions

359

Periodic Forcing Video 13.4a Eastern-boundary response forced by a band of meridional wind stress y oscillating with a period P = 60 days. The wind has the form τ y = τo Y(y)T(t), where: T (t) = sin (σt) θ(t) with σ = 2π/P; and Y (y) is given by (C.7b) with y = 20◦ and ym = 30◦ . The Coriolis parameter f is specified by the midlatitude β-plane approximation. The eastern boundary is closed, and open boundary conditions are imposed at the western boundary as described in Appendix C. Video 13.4b As in Video 13.4a, except with P = 180 days. Video 13.4c As in Video 13.4b, except with νh = 5×105 cm2 /s. Video 13.4d As in Video 13.4a, except with P = 365 days. Video 13.4e As in Video 13.4d, except with νh = 5×105 cm2 /s. Video 13.5a As in Video 13.4a, except showing the western-boundary response when νh = 0. The western boundary is closed, and open boundary conditions imposed at the eastern boundary as described in Appendix C. Video 13.5b As in Video 13.5a, except with P = 180 days. Video 13.5c As in Video 13.5b, except with νh = 5×105 cm2 /s. Video 13.5d As in Video 13.5a, except with P = 365 days. Video 13.5e As in Video 13.5d, except with νh = 5×105 cm2 /s.

Chapter 14

Equatorial Ocean: Switched-On Forcing

Abstract Near-equatorial solutions forced by switched-on winds are found to a simplified version of the LCS equations that neglects the acceleration and damping terms in the meridional momentum equation (the “long-wavelength” approximation). They are found under the assumption that the Coriolis parameter is given by f = βy, allowing them to be represented as expansions in Hermite functions. In an unbounded ocean, Ekman drift and Ekman pumping quickly establish an accelerating jet, the Yoshida Jet. Subsequently, the radiation of equatorial Rossby and Kelvin waves adjusts the response to a Sverdrup-balanced state plus a zonally-independent, equatorial jet that does not accelerate, the “bounded” Yoshida Jet. With an eastern boundary, the equatorial Kelvin wave reflects as a packet of Rossby waves, which has a characteristic wedge-shaped pattern as it radiates offshore. With both boundaries, signals are present with a period P close to the time it takes an equatorial Kelvin wave to cross the basin and the lowest order ( j = 1) Rossby wave to return; period P is a natural “ringing” time of the basin, and is the basis for equatorial basin resonance discussed in the next chapter. Keywords Long-wavelength approximation · Switched-on forcing · Equatorial Ekman pumping · Adjustment to Sverdrup balance · Bounded Yoshida jet · Boundary reflections When the model domain includes the equator, analytic solutions forced by timedependent winds can be obtained provided that f is approximated by the equatorial βplane approximation f = βy, which allows solutions to be represented as expansions in Hermite functions. In this chapter, we discuss the response when the forcing is switched-on, finding solutions to a simplified equation set that has been used in a number of equatorial problems (Sect. 14.1). We first find the response in an unbounded ocean (Sect. 14.2). Next, because equatorial waves propagate rapidly and quickly reach basin boundaries, we discuss their reflections from a single eastern or

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_14. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_14

361

362

14 Equatorial Ocean: Switched-On Forcing

western boundary (Sect. 14.3) and from both of them (Sect. 14.4). We conclude by noting aspects of the solutions that are evident in observations (Sect. 14.5). As shown in Sect. 14.2, the solution simplifies to a succinct form that reflects the simplicity of its underlying dynamics. Those dynamics are essentially the same as those for the interior-ocean (Chap. 12) and coastal (Chap. 13) solutions, namely, Ekman drift and Ekman pumping followed by an adjustment to steady state via wave radiation. The major difference is that the steady-state response in an unbounded ocean is a Sverdrup flow plus an x-independent equatorial jet, the “bounded” Yoshida Jet, which arises because the adjustment involves an equatorial Kelvin wave as well as equatorial Rossby waves. A state of Sverdrup balance is achieved only when the basin includes an eastern or western boundary, in which case the jet is eliminated by boundary-reflected waves. Although the final solutions in this chapter (and also Chap. 15) have dynamically simple forms, their derivations are complicated and lengthy, owing to the manipulations of Hermite functions required. Given this complexity, we expect readers will find it helpful to review derivations obtained elsewhere (e.g., Cane & Sarachik 1976, 1977, 1981; McCreary 1980; Boyd 2018).

14.1 Simplified Model Equations The simplified equation set neglects vt  , u t  − f v + px = F,

(14.1a)

f u + p y = G,

(14.1b)

pt  + u x + v y = 0, c2

(14.1c)

in which case (9.2c) simplifies to v yyt  −

 f2 f 1 1  vt  + βvx = 2 Ft  − 2 Fx y − 2 ∂t  t  − c2 ∂x x G. 2 c c c c

(14.2)

As for the equatorial Ekman response (Sect. 10.2.1), we look for solutions to (14.2) in the form J  v(x, η, t) = v j (x, t)φ j (η), (14.3) j=0

where J is large enough to ensure the solution is well converged. Following the steps that led to (10.31), the equation for v j is

14.2 Interior Solution

363

 σ0 α0   1  (14.4) (ηF) jt  − 2 Fη j x − 2 ∂t  t  − c2 ∂x x G j , 2 c c c   where α 2j = α02 (2 j + 1), and (ηF) j , Fη j , and G j are the Hermite expansion coefficients of ηF, Fη , and G, respectively. The advantages of (14.4) are apparent by comparing it to (9.2c), the v j equation that results from the complete equation set (9.1), which shows that (14.4) lacks v jt  t  t  , v j x xt  , and v j yyt  terms. The lack of v jt  t  t  filters gravity waves out of the system, so that solutions only involve the low-frequency adjustment processes of primary interest here. The lack of all three terms ensures that the only free waves in the system are longwavelength, non-dispersive Rossby waves with the dispersion relation (8.18), and for this reason equation set (14.1) is known as the “long wavelength” approximation to system (9.1). It is the non-dispersive nature of the Rossby waves in the system that allows for simple, analytic solutions to switched-on winds. The lack of the v jt  t  t  and v j x xt  terms also means that solutions are valid only when both terms are small compared to the α 2j v jt  = α02 v jt  = (β/c) v jt  term. Let T be the time scales of the temporal variability and the damping, and L x be the zonal scale of the solution. Then, the neglected terms are small provided that − α 2j v jt  + βv j x =

1  σ02 , T2

1  α02 L 2x

(14.5)

(also see Appendix B). The first inequality requires that the wind forcing is low frequency damping is weak (σ 2  σ02 and γn2 ), and the second that it is large scale   2 and−2 in x L x  α0 . For realistic parameter values, the two inequalities are satisfied for lower-order baroclinic modes. For sufficiently high-order modes, however, the sense √ of the first inequality can reverse since σ0n = βcn and γn = A/cn2 increase with n, but this reversal is not a problem for any of the solutions shown here, which are all dominated by low-order baroclinic modes. Because (14.2) retains the v yyt  term there is no limit on the meridional length scale L y , Eqs. (14.1) are able to describe typically narrow equatorial jets. Because only vt is dropped from Eqs. (14.1), one might expect that, in response to switched-on forcing, solutions can be found in which u and p increase smoothly from zero. For τ x forcing that expectation is correct, but for τ y forcing none of the variables increases smoothly. This necessity is suggested by (14.1b): Because vt  is absent, when G switches on so must either u, p, or both.

14.2 Interior Solution Here, we consider the equatorial ocean’s response to spatially-bounded patches of zonal and meridional wind in an unbounded domain. Given the complexity in finding the solution, some readers may find it useful first to skip to the discussion of the solution’s dynamics (Sect. 14.2.2.4) and to view the videos (Sect. 14.2.2.5) before delving into mathematical details of its derivation.

364

14 Equatorial Ocean: Switched-On Forcing

Although it is possible to obtain solutions with mixing, we set γ = 0 because the basic spin-up processes are easier to interpret in the inviscid response. With γ = 0 and T (t) = θ(t), (14.4) simplifies to Gj σ0 α0   1 1 Fη j x θ − 2 δ  + G j x x θ, v jt + v j x = 2 (ηF) j δ − cr j βc β βc β

(14.6a)

where cr j = −β/α 2j = −c/ (2 j + 1), and equations for u and p in terms of v derived from (14.1a) and (14.1c) are u tt − c2 u x x = f vt + c2 v yx + Fδ,

(14.6b)

px = f v − u t + Fθ,

(14.6c)

Note in (14.6a) that the forcing includes terms not only proportional to δ(t) = θ(t) but also to δ (t) = θtt , the latter a derivative of a δ-function: δ (t) is a double-peaked function, one representation of which is the derivative of (9.8). For notational convenience, in Eqs. (14.6) and all subsequent equations in this chapter, we redefine forcing terms Q(F and G) without time dependence, that is, Q(x, y, t) → Q(x, y) θ(t). This modification allows many forcing terms to be represented more concisely as Qθ(t), and we use the expanded form Q o X(x)Y (y)θ(t) only when necessary. We use the direct approach (Sect. 9.4.1.1) to solve Eqs. (14.6a), as the small-time response itself is dynamically interesting.

14.2.1 Small-Time Response To find the complete solution, we only need to evaluate the small-time response at t = 0+ . It is useful, however, to extend it a bit farther in time to include terms of O(t), so that it can be compared to the x-independent solution discussed in Sect. 10.2. In that solution, the terms proportional to t illustrate the process of equatorial Ekman pumping, and analogous terms are present in the small-time solution found here.

14.2.1.1

v Field

To obtain the small-time solution for v accurate   to O(t), we integrate (14.6a) three times in time, while dropping terms of O t 2 and higher. The integral of (14.6a) is 1 vj = − cr j



t 0

v j dt



 + x

  Gj σ0 1 G Fη j x t, θ − δ + − α (ηF) j x x 0 j βc2 βc2 β (14.7a)

14.2 Interior Solution

365

and the subsequent two integrations of (14.7a) are 1 cr j 1 cr j



t



v j dt = −

  t

0

0

 t 0

t





t

 

v j dt dt



+

0

v j dt dt = −

x

  t 0

0

t 0



t

Gj σ0 (ηF) j t − 2 θ, βc2 βc  



v j dt dt dt

0



− x

Gj t. βc2

(14.7b)

(14.7c)

Since the lowest-order forcing term on the right-hand side of (14.7a) is O(δ), the equation implies that the lowest-order termin v j is also O(δ). Therefore, in (14.7c) the triple integral contains terms only of O t 2 and  higher and so is negligible, and hence the double integral equals −cr j G j t/ βc2 to O(t). With the double integral known, (14.7b) provides the single integral. Then, eliminating the single integral from (14.7a) gives v=

∞  G jx 1 (ηF) j 1 1 Gj φj − θ+ θ + δ(t) σ0 2 j + 1 σ02 2 j + 1 (2 j + 1)2 β j=0

  ∞  4 j ( j + 1) 1 (ηF) j x   − F φj, + cr j t G + j x x η jx β (2 j + 1)2 σ0 2 j + 1 j=0

(14.8)

where cr j = −c/ (2 j + 1).

14.2.1.2

u and p Fields

To find u at small times, we integrate (14.6b) twice in time to get u = c2

 t 0

t 0

u x x dt  dt  + f



t 0

vdt  + c2

 t 0

t

v yx dt  dt  + Ft,

(14.9)

0

and insert v from (14.8). Because the lowest-order terms in the integrals of f v and term of u is O(1). It c2 v yx in (14.9) are O(1) and O(t), respectively, the lowest-order   follows that the double integral of c2 u x x is necessarily O t 2 , and hence is negligible. Consequently,

u=



∞ ∞ ∞    (ηF) j ηφ j φ jη G jx 1 Gj − t ηφ j θ + t + ηφ + + F φ j j j σ 2j + 1 α 2j + 1 2j + 1 (2 j + 1)2 j=0 0 j=0 0 j=0

(14.10a)

366

14 Equatorial Ocean: Switched-On Forcing

to O (t). The solution for p is obtained from (14.6c) using a similar procedure, giving

∞ ∞ ∞   (ηF) j ηφ j φ jη G jx 1  Gj p t + =− φ jη θ − + φ jη t 2 c σ0 j=0 2 j + 1 α0 2 j + 1 (2 j + 1) 2j + 1 j=0 j=0 (14.10b) to O (t).

14.2.1.3

Dynamics

Although solutions (14.8) and (14.10) are algebraically complicated, each of their terms corresponds to one of the processes discussed earlier in Chaps. 10 and 12. First, consider the terms in solutions (14.8) and (14.10) that are not proportional to x-derivatives of the forcing terms or to δ(t). They are just the steady and accelerating terms in the x-independent, equatorial solutions, (10.35) and (10.37), except that F and G include factors of X(x) since the forcing is bounded. As discussed in Sect. 10.2, they describe the equatorial extensions of Ekman drift (steady terms) and equatorial Ekman pumping (accelerating terms). The term proportional to δ(t) in (14.8) describes an instantaneous shift of water in the direction of τ y . It is needed to account for the jumps in p and u in the first terms on the right-hand sides of solutions (14.10). Interestingly, even though similar jumps occur in solutions (10.37), there is no term proportional to δ(t) in (10.35); nevertheless, a similar shift still occurs in solution (10.35), but it is associated with a time integral of the inertial oscillations rather than a δ-function (Sect. 10.2.1.2). The terms proportional to t in solutions (14.8) and (14.10), including those involving x-derivatives of F and G, describe the response of the equatorial ocean to Ekman pumping. Note that they are the same as the Ekman-pumping (accelerating) terms in solutions (10.37) forced by x-independent forcing, except with additional terms proportional to G x . They also correspond to the Ekman-pumping terms in the midlatitude solution (12.11) in the limit that γ → 0, their more complicated forms in (14.8) and (14.10) ensuring that Ekman pumping remains well defined even at the equator where f = 0. In summary, solutions (14.8) and (14.10) describe the response shortly after the wind switches on. As such, they provide definitions of equatorial, Ekman drift (steady terms) and Ekman pumping (accelerating terms), neither of which blows up at the equator because p = 0. A comparison of the solutions to their x-independent counterparts, (10.35) and (10.37), shows that they correctly represent the spin-up of the non-oscillatory parts of the response: Essentially, the spin-up with inertial waves in the latter is collapsed to an initial pulse in v and jumps in u and p in the former due to the neglect of vt in (14.1b).

14.2 Interior Solution

367

14.2.2 Long-Time Solution To find the long-time (t ≥ 0+ ) solution, we set θ = 1 and δ = δ  = 0 in Eqs. (14.6), obtain general solutions for v, u, and p, and then ensure that each matches solutions (14.8) and (14.10) in the limit that t → 0+ .

14.2.2.1

v Field

The general solution to (14.6a) when t ≥ 0+ is vj =

  G jx α0   − Fη j + R j x − cr j t β β

t ≥ 0+ ,

(14.11)

the sum of steady (particular) and free Rossby-wave (homogeneous) responses. To determine R j , we require that (14.11) reduces to the small-time solution evaluated at time t = 0+ , that is, to the first two terms on the right-hand side of (14.8). It follows that G jx G jx σ0 (ηF) j α0   Fη j + R j (x) = − − , 2 β β βc 2 j + 1 β (2 j + 1)

t = 0+ ,

(14.12)

which can be solved for R j (x) at t = 0+ . Making the replacement that x → x − cr j t in R j (x) then gives its value at all times,

(ηY ) j         α0 G o j ( j + 1) Fo − Yη j X x − cr j t − j0 4 R j x − cr j t = − j0 Y j X x x − cr j t . β 2j + 1 β (2 j + 1)2

(14.13)   Note that R0 = 0 since (ηY )0 − Yη 0 in the first term and j = 0 in the numerator of the fraction in the second; to “remember” that property we introduce the factor j0 ( j0 = 0 if j = 0 and is 1 otherwise), deleting it at the end of a derivation when it is no longer  needed (see the discussion of solution u h j below). With R j x − cr j t given by (14.13), the complete solution for v at all times is: (14.11) with factors of θ(t) included; plus the δ-function part of (14.8), which exists only for t < 0+ and so is not part of (14.11). Summing the two parts gives v=

∞  G jx

  α0   α0 G j Fη j θ + R j x − cr j t θ + δ φj β β βσ0 2 j + 1 j=0

∞    G x − Fy α0 G j R j x − cr j t θ + θ+ δ φj. = (14.14) β βσ0 2 j + 1 j=0 θ−

According to (14.14), the solution consists of an initial pulse proportional to δ(t),  a Rossby-wave adjustment R j (x − cr t), and a steady-state response G x − Fy /β,

368

14 Equatorial Ocean: Switched-On Forcing

the latter  being the  meridional velocity of a Sverdrup flow. As a check, by expanding R j x − cr j t into Taylor series (12.23) and keeping terms only to O(t), it is straightforward to verify that (14.14) reduces to (14.8).

14.2.2.2

u Field

To obtain u for time t > 0+ , we insert v from (14.14) into (14.6b), set θ = 1, and drop all terms on the right-hand side proportional to δ and δ  , to get

∞      G jx α0   − Fη j φ j ∂tt − c2 ∂x x u = σ0 η∂t + α0 c2 ∂ηx R j + β β j=0   ∞  ηφ j G x x − Fyx + φ jη R j x + c2 , (14.15) = σ0 c 2j + 1 β j=0 where, since R j is a function of x − cr j t, we used the property that R jt = −cr j R j x . It is convenient to solve for u as the sum of separate pieces, u = us +

∞ 

ur j + u h ,

(14.16)

j=0

where: u s and u r j are particular solutions to the steady and Rossby-wave forcings proportional to G x x − Fyx and to R j x , respectively; and u h is a homogeneous solution to (14.15) included to ensure that u satisfies radiation and initial conditions, (9.4) and (9.5). u s Solution The steady-state part of (14.15) satisfies   G x x − Fyx , ∂tt − c2 ∂x x u s = −c2 ∂x x u s = c2 β

(14.17)

which has the particular solution us = −

+ G y − Fyy

β

=

∞  j=0

us j ,

us j = −

 α0  G j − Fη+ φ jη , β

(14.18)

x where F + = ∞ Fd x  . Setting the lower limit of the integral of F to ∞ ensures that no information is present east of the forcing region, and with this choice solution (14.18) is just the zonal velocity of a Sverdrup flow (Sect. 11.1). We also write u s as a Hermite expansion, because u s j is useful in the derivation of u h below.

14.2 Interior Solution

369

u r j Solutions The Rossby-wave parts of (14.15) satisfy   ∂tt − c2 ∂x x u r j = σ0 c



 ηφ j + φ jη R j x . 2j + 1

(14.19)

Since R j x is a function of x − cr j t, u r j is as well. Therefore, u r jt = −cr j u r j x and     j ( j + 1) ∂tt − c2 ∂x x u r j = cr2j − c2 u r j x x = −4c2 ur j x x , (2 j + 1)2

(14.20)

where cr j = −c/ (2 j + 1). With the aid of (14.20), two integrations of (14.19) in x gives the particular solution ur j = −

 σ0 (2 j + 1)  ηφ j + (2 j + 1) φ jη R+j , 4c j ( j + 1)

(14.21)

x where R+j = ∞ R j d x  . The choice of lower limit ensures that no Rossby waves are present east the forcing region, which must be the case since the group speed of long-wavelength Rossby waves is westward. Using Eqs. (8.7) to eliminate ηφ j and φ jη , u r j can be rewritten ur j

  φ j+1 φ j−1 σ0 2 j + 1 σ0 ˆ −j R+j , (14.22) −√ + √ R+j = − (2 j + 1) =− √ c 2 2 c j +1 j

ˆ −j is (8.19b), the meridional structure function for long-wavelength Rossby where waves. Comparing (14.22) to (8.22), u r j can be seen to be the zonal velocity of the long-wavelength Rossby wave associated √ with vr j = R j (x − cr t). Finally, note that terms inversely proportional to j and j in (14.21) and (14.22) are not a problem when j = 0 since R+ 0 = 0. u h Solution The general solution to the homogeneous version of (14.15) is u h = A(x − ct, η) + B(x + ct, η) ,

(14.23)

which consists of waves propagating eastward and westward at speed c and an xindependent response. Equation (14.1) allow no waves with phase speed −c, and only one wave with speed c: the eastward-propagating, equatorial Kelvin wave (8.31). For u h to be physically valid, then, we must set B = 0 and find that the only propagating signal is a Kelvin wave. To determine u h , we impose the initial condition that u in (14.16) evaluated at t = 0+ must reduce to the small-time solution (14.10a) evaluated at that time, that

370

14 Equatorial Ocean: Switched-On Forcing

is, to (14.10a) without the terms proportional to t. Expressed mathematically, and with u s written in the expanded form of (14.18), that constraint is u=

∞ ∞     1 Gj ηφ j , ur j + u s j + u h = σ 2j + 1 j=0 j=0 0

t = 0+ .

(14.24)

   + In the following paragraphs, we evaluate ∞ j=0 u r j + u s j at t = 0 , and then u h at + t = 0 is known from (14.24). Given its complexity, it is useful to split u r j + u s j into y parts forced separately by meridional (u r s j ) and zonal (u rxs j ) winds (i.e., u r j + u s j = y u r s j + u rxs j ). y Solution u r s j is the sum of: (14.21), when R+j is determined from R j in (14.13)   with Fo = t = 0; and (14.18) when Fη+ j = 0. Then, y ur s j

 G j j ( j + 1) α0 σ0 2 j + 1  − G j φ jη ηφ j + (2 j + 1) φ jη −4 j0 =− 4c j ( j + 1) β (2 j + 1)2 β σ0 G j σ0 α0 ηφ j + j0 G j φ jη − G j φ jη , = j0 βc 2 j + 1 βc β j0 G j 1 1 Gj ηφ j − G 0 φ0η = ηφ j , = (14.25) σ0 2 j + 1 σ0 σ0 2 j + 1

where we used: the identities βc = σ02 and α0 /β = σ0−1 ; and in the last line the relation, φ0η = −ηφ0 , which follows from Eqs. (8.7) with j = 0. Note that, because factors of j in the numerator and denominator of the first line cancel out, j0 is needed in the second and third lines to know that those terms do not contribute when j = 0. Summing (14.25) over j gives u rys =

∞  j=0

y

ur s j =

∞  α0 j=0

Gj ηφ j . β 2j + 1

(14.26)

y

According to (14.26), u r s is equal to the right-hand side of constraint (14.24), a property that makes determining u h easy (see below). Solution u rxs j is the sum of (14.22) with G 0 = t = 0 in R+j and (14.18) with G j = 0, yielding u rxs j = −

σ0 2 j + 1 √ c 2 2





    (ηY ) j φ j+1 φ j−1 α0 X +(x) − Yη j − √ − j0 Fo √ β 2j + 1 j +1 j   α2 (14.27) + 0 Fo X +(x) Yη j φ jη . β

  After expanding (ηY ) j , Yη j , and φ jη using Eqs. (8.9) and (8.7b), (14.27) simplifies (seemingly miraculously) to

14.2 Interior Solution

371

u rxs j =

 Fo +  X (x) −Y j+1 φ j+1 + j0 Y j−1 φ j−1 . 2c

(14.28)

Summing this expression over all j gives u rxs =

∞  j=0

∞

u rxs j =

 Fo + Fo +   X (x) χ (x) Y0 φ0 , −Y j+1 φ j+1 + j0 Y j−1 φ j−1 = 2c 2c j=0

(14.29) that is, all but one of the terms cancels out in (14.29). Again, factor j0 is needed to keep track of terms that are zero when j = 0. y y Since u r j + u s j = u r s j + u rxs j and u r s equals the right-hand side of (14.24), constraint (14.29) reduces to u h (x) = −u rxs = −

Fo + X (x) Y0 φ0 , 2c

t = 0+ ,

(14.30)

which provides solution u h at t = 0+ . According to (14.23), u h at other times has the form A(x − ct, η) since B(x + ct, η) is unphysical. Therefore, u h at all times is (14.30) with x → x − ct, u h (x − ct) = −

 Fo  Fo Y0 X +(x − ct) φ0 = Y0 χ − X −(x − ct) φ0 . 2c 2c

(14.31)

Since an equatorial Kelvin wave has eastward group velocity, it cannot appear west X + as the difference terms, X − =  ∞  of  x of the wind patch. So, we rewrite  two ∞   −∞ X(x − ct) d x and χ = −∞ X(x − ct) d x = −∞ X x d x , x = x − ct, − where X is the Kelvin-wave response and χ is the x-independent term discussed next. Total u From (14.18), (14.22), and (14.31), and after replacing θ(t) functions since the solution is valid for all times, the complete u field, u = u s + ∞ j=0 u r j + u h , is ∞  −  σ0  ˆj θ− u=− (2 j + 1) R+j x − cr j t θ β c j=1    +∞    Fo Fo − Y0 φ0 − Y0 φ0 X (x − ct) θ + X x d x θ, 2c 2c −∞ + G y − Fyy

(14.32)

where the summation starts at j = 1 since R+ 0 = 0. According to (14.32), u consists of a Sverdrup flow (first term after the equal sign), a set of long-wavelength, equatorial Rossby waves proportional to R+j , and an equatorial Kelvin wave proportional to X −(x − ct). A final part is a steady, zonally uniform jet proportional to φ0 (term in braces), the “bounded Yoshida Jet.” Interestingly, the jet is generated only by zonal

372

14 Equatorial Ocean: Switched-On Forcing

winds, an indication that it is driven by equatorial Ekman pumping, as is the case for the continuously-accelerating Yoshida Jet in the x-independent solution (10.40). (A dynamically similar jet exists in the simple solution found in Sect. 9.4.1.1.)

14.2.2.3

p Field

It is possible to obtain p by solving (9.2b) with γ = 0, following the same steps used to obtain u from (14.6b). Since both u and v are known, however, a shorter method is to solve (14.6c), which can be integrated in x to get p. Inserting (14.14) and (14.32) into (14.1a) gives px = f v − u t + Fθ = f

∞    G x − Fy α0 G j θ+ δ σ0 ηφ j R j x − cr j t θ + β βσ0 2 j + 1 j=0

−σ0

∞  j=1

    F ˆ − − o Y0 φ0 X(x − ct) − u 0+ δ + Fθ, (14.33) R j x − cr j t θ j 2

where f v is on the first line and −u t + Fθ is on the second. To evaluate u t , we used the relation that R+jt x − cr j t = −cr j R+j x = cR j / (2 j + 1) and χt−(x − ct) = −cX(x − ct). Collecting like terms together gives ⎡ ⎤  ∞   + G G x − Fy α j 0 + F θ +⎣ ηφ j − u 0 ⎦ δ px = f β β j=0 2 j + 1     ∞    j +1 j Fo − ˆj + +σ0 φ j+1 + φ j−1 θ − Y0 φ0 X (x − ct) θ, R j x − cr j t 2 2 2 j=0 

(14.34)   where we used the identity (uθ )t = u t θ + u(t) δ(t) = u t θ + u 0+ δ(t) and expanded ηφ j on the second line using (8.7a). The terms proportional to R j in (14.34) can be rewritten using the relation,  ˆ −j

+

j +1 φ j+1 + 2



j ˆ +j , φ j−1 = − (2 j + 1) 2

(14.35)

  ˆ ±j in (8.19b). In addition, u 0+ is equal to which follows from the definitions of (14.10a) evaluated at t = 0+ , so the two terms proportional to δ cancel. With these replacements, (14.34) simplifies to px =

f2 β



Gx − f



F f

  θ + F − σ0 y

∞  j=0

 +  F ˆ θ + o Y0 φ0 X(x − ct) θ. (2 j + 1) R j x − cr j t j 2

(14.36)

14.2 Interior Solution

373

Finally, integrating (14.36) from +∞ to x gives 



 

∞   + σ0  ˆjθ (2 j + 1) R+j x − cr j t c j=1 y    +∞    Fo Fo − (14.37) Y0 φ0 − Y0 φ0 χ (x − ct) θ + X x d x θ, 2c 2c −∞

f2 p = c cβ

G − f

F+ f

θ−

where the integral of X in the last term of (14.36) is modified by the replacement X +(x − ct) = X −(x − ct) − χ . Note that solution (14.37) has the same form as u, consisting of pressure terms for: a Sverdrup flow;   a set of long-wavelength, equatorial Rossby waves proportional to R+j x − cr j t ; an equatorial Kelvin wave; and the bounded Yoshida Jet.

14.2.2.4

Dynamics

It is instructive to compare solutions (14.14), (14.32), and (14.37) to their midlatitude counterparts in Eqs. (12.21). Both solutions describe the same dynamical adjustments: Ekman drift being quickly established; followed by a response proportional to t (Ekman pumping) before waves propagate across the forcing region (0 < t < L x /c); and a final adjustment toward Sverdrup balance as the waves radiate from the region. A key difference in the two solutions is that the steady-state response of solutions (14.32) and (14.37) contains the x-independent, bounded Yoshida Jet in addition to a Sverdrup flow, owing to the eastward group velocity of equatorial Kelvin waves. Another difference is that the additional term in (14.14) proportional to δ is not present in (12.21a); it isn’t needed because (12.21b) has no jump in p since there is no boundary in the interior ocean analogous to the equator. Given the dynamical similarities between the equatorial and interior solutions, the schematic diagrams in Fig. 12.2 still apply to the equatorial ones, except they illustrate the response for index j rather than at a particular latitude. For τ x forcing Fig. 12.2 (left panel) illustrates the westward extension of the local response behind the wave fronts of the jth Rossby wave; further, with the x-axis reversed, the figure also represents the eastward expansion of the equatorial jet by the equatorial Kelvin wave. For τ y forcing, Fig. 12.2 (right panel) shows how the Rossby-wave packet separates from forcing region, and there is no Kelvin-wave response. A final difference is that the Ekman-pumping (accelerating) response is commonly observed at midlatitudes, because Rossby waves propagate slowly enough there that the adjustment to Sverdrup balance takes considerable time. In contrast, Ekman pumping is not a distinct process near the equator where equatorial Rossby and Kelvin waves propagate rapidly; for example, with L x = 2000 km and cn = 250/n, the acceleration phase at the equator lasts only for δt  L x /c = 9.3n days.

374

14.2.2.5

14 Equatorial Ocean: Switched-On Forcing

Videos

Videos 14.1a, 14.1b, and 14.1c illustrate the responses forced by τ x and τ y winds centered on the equator. To eliminate boundary reflections, the model basin includes eastern and western damping regions (Appendix C). The latter two videos are comparable to Videos 11.1a and 11.2a, which show the responses to similar forcings confined off the equator. To isolate the development of the bounded Yoshida Jet, the solution in Video 14.1a is forced by a y-independent band of zonal wind. As soon as the wind switches on, the equatorial jet accelerates in response to equatorial Ekman pumping. Within a month or so, however, wind-generated Kelvin and Rossby waves have propagated across the wind patch, establishing a zonal pressure gradient that balances the wind and stops the acceleration. Subsequently, the radiation of Kelvin and Rossby waves extends the jet farther from the forcing region, more rapidly in the eastern ocean since the Kelvin wave speed is three times larger than the fastest ( = 1) Rossby wave. Toward the end of the video, after most transients have radiated from the domain (the exceptions being in the western corners), the solution is adjusted to a steady state consisting of two parts: a y-independent pressure field with a zonal gradient that balances the wind ( p y = F) and no flow (u = v = 0), the Sverdrup state (11.3) for this wind field; and an x-independent, bounded Yoshida Jet. The solution in Video 14.1b is forced by a more realistic τ x that is confined near the equator. Nevertheless, similar adjustments happen as in Video 14.1a, resulting in a steady state consisting of a Sverdrup-balanced double gyre plus the bounded Yoshida Jet. Note that p does not change sign between the northern and southern gyres owing to the factor of f in (11.3b); at the same time, the currents have the same structure that they do at midlatitudes since (11.3a) depends only on β and not f . In addition, the equatorial current is no longer x-independent, since west of the forcing region it includes eastward flow from the Sverdrup circulation as well as the Yoshida Jet. In Video 14.1c there are no strong, equatorial flows because τ y winds do not generate an equatorial Kelvin wave or Yoshida Jet. Instead, after transient longwavelength Rossby waves and Yanai waves radiate from the patch, the response adjusts to a Sverdrup-balanced state consisting of a single clockwise gyre. Note that p changes sign across the equator owing to the factor of f in (11.3b), whereas the current structure remains unchanged from midlatitudes since (11.3a) depends only on β not f . Transient Yanai waves are not in the analytic solution because since they are filtered out of Eqs. (14.1), but they exist in Video 14.1c because it shows a solution to the complete equation set (9.1). To understand their interesting properties, it is useful to view the transient packet as being a superposition of sinusoidal Yanai waves with different wavelengths k and frequencies σ . Because the Yanai waves that can be strongly excited by the large-scale wind have wavelengths of the order of the zonal width of the wind patch (Sect. 15.1.4), k must be close to zero in which case σ is near σ0 (σ0 in Fig. 8.3); consequently, the transient packet consists predominantly of waves with periods near P = 2π/σ0 = 9.4 days. One result of this constraint

14.3 Reflections from a Single Boundary

375

is that the phase speed of the Yanai waves reverses in time. The reversal happens because positive and negative wavenumbers are possible for the waves with σ  σ0 and σ  σ0 , respectively. Because the group velocities of Yanai waves with positive k (eastward phase velocity) are greater than those with negative k (westward phase velocity), the phase velocity of waves in the packet are initially eastward and later turn westward. Solutions to (9.1) also contain transient, short-wavelength, Rossby waves with eastward group velocity. These waves, however, are not seen in the videos because their wavelengths are so short that they cannot be significantly excited by winds with a large zonal scale (Sect. 15.1.4).

14.3 Reflections from a Single Boundary Because equatorial waves propagate so rapidly, radiation from the unbounded-ocean response (Eqs. 14.14, 14.32, and 14.37) quickly reaches, and reflects from, ocean boundaries. Here, we consider the processes involved in reflections from both eastern (Sect. 14.3.1) and western (Sect. 14.3.2) boundaries. These processes are similar to those for the β = 0 solutions along midlatitude coasts considered in Sect. 13.2.3.

14.3.1 East-Coast Reflections 14.3.1.1

Reflection of an Equatorial Kelvin Wave

Consider the reflection generated at an eastern boundary when the ocean is forced by a switched-on, zonal-wind patch that does not extend to the boundary. In the solution obtained above, the boundary will subsequently be affected by the arrival and reflection of an equatorial Kelvin wave. The u field of the Kelvin wave in solution (14.32) is u K (x − ct) =



 +∞   Fo Y0 + Fo Y0 −χ −(x − ct) + χ (x − ct) φ0 . X x  d x  φ0 = − 2c 2c −∞

(14.38) For an eastern boundary at x = 0, then, the u field due to this incoming signal is u K (−ct) = −

Fo Y0 + χ (−ct) φ0 ≡ U K (t) φ0 , 2c

(14.39)

and it must be cancelled by a packet of equatorial Rossby waves with westward group velocity.

376

14 Equatorial Ocean: Switched-On Forcing

In system (14.1), all the Rossby waves have dispersion relation (8.18). Inserting this expression into Eqs. (8.12b) and (8.12c), the solution for the u field of each Rossby wave in the boundary packet can be written u r j = Ur j



   φ j−1 φ j+1 . x − cr j t √ −√ j j +1

(14.40a)

Then, since u r 1 contains a term proportional to φ0 , we √ can use it to cancel u K by φ / setting Ur 1(−cr 1 t) = −U K (t). In so doing, a term U K 2 √ √ 2, is generated, which can 3/ 2 U K , thereby generating be cancelled using u r 3 by setting Ur 3(−cr 3 t) = − √ √  √ 3/ 2 U K θ4 / 4 that can be cancelled by u r 5 . Continuing this another term procedure, an infinite chain of symmetric Rossby waves, for which the amplitude of the jth wave is √ √ √ √ 3 5   j Ur j −cr j t = − 1 √ √ · · · √ U K (t) , j − 1 2 4

j = 1, 3, 5, . . . ,

(14.40b)

is generated that cancels u everywhere along the coast. With the aid of (8.22), the complete set of reflected waves is then vj =

 cr j   Ur j x − cr j t , σ0



     φ j+1 φ j−1 uj 1 = Ur j x − cr j t ± √ , √ −√ p j /c j +1 j 2 2

(14.41)   where Ur j is the x-derivative of Ur j x − cr j t . This method, first carried out by Moore (1968), is now referred to as the Moore’s “chain rule.” Note that, because information in the chain passes from lower-to-higher j values, it extends to increasingly higher latitudes. Thus, the chain rule is consistent with the physical property that coastal waves propagate poleward along an easternocean boundary (Chap. 13). Note, however, that there is no indication of poleward propagation in solution (14.41), that is, it has no terms that depend on y − ct; their absence results from the neglect of vt in (14.1b), which essentially adjusts the speed of coastal Kelvin waves to infinity. (The lack of poleward propagation in solution (13.18) happens for the same reason.) Although in principle the chain is infinite ( j → ∞), useful solutions can be obtained even when j is truncated at a sufficiently large, finite value J . Figure 14.1 schematically illustrates the reflection when T (t) = θ (t). Each box (blue lines) indicates the area covered by the √ jth Rossby wave at time t, extending poleward to the turning latitudes ±η j = ± 2 j + 1 defined in (8.11) and offshore to x = cr j t = −c/ (2 j + 1). Because higher-order (larger j) Rossby waves extend to increasingly higher latitudes and propagate more slowly offshore, the front of the Rossby-wave packet has a characteristic wedge-shaped pattern, indicated by the dashed curve x = −ct/η2 . As a result of the reflection, the zonal current of the incoming Kelvin wave splits to flow along the edges of the packet.

14.3 Reflections from a Single Boundary

377

Fig. 14.1 Schematic diagram illustrating the reflection of a Kelvin-wave that arrives at the eastern boundary at t = 0. The red lines along η = ±1 indicate the width of the Kelvin wave. The blue lines indicate the areas covered by the first 6 symmetric Rossby waves, for which j = 1, 3, . . . , 11

Videos: Video 14.2a illustrates the eastern-boundary reflection of a Kelvin-wave packet generated by a switched-on zonal wind located in the western ocean. The wind forcing is the same as in Video 14.1b, except shifted westward in order to highlight the eastern-boundary reflections. The Kelvin-wave front arrives at the eastern boundary at the beginning of February. Consistent with the analytic solution and Fig. 14.1, a sequence of reflected Rossby waves then emerge from the eastern boundary. The first is the j = 1 Rossby wave, visible as two westward-propagating (darker red) patches on either side of the equator, which arrives at the western edge of the forcing region (40◦E) about May. The rest of the reflected packet extends farther from the equator and has increasingly more variations in latitude closer to the eastern boundary, indicating the presence of higher-order, symmetric (odd j values) Rossby waves. Fronts of the contributions of individual waves to the overall packet are best indicated by relative maxima (darker red areas) along its edges. Video 14.2b is the same as Video 14.2a except with the wind in the central ocean to highlight the impact of the reflection on the Yoshida Jet and Sverdrup responses. After the passage of the eastern-boundary Rossby waves, the Yoshida Jet is eliminated and the final steady state is just the double-gyre Sverdrup circulation. In that steady state, sea level has risen (is more red) throughout the basin, a result of flow into the basin along the northern and southern edges of the Rossby-wave packet.

378

14.3.1.2

14 Equatorial Ocean: Switched-On Forcing

Reflection of a Yanai Wave

In solutions to the exact Eqs. (9.1), Yanai waves can also be significantly generated by interior winds, and they can reflect from an eastern boundary. In this case, u at the boundary has the form u(0, η, t) = UY (t) φ1(η)

(14.42)

at the eastern boundary, its structure proportional to φ1(η), rather than φ0(η). This u field must be cancelled with an infinite chain of antisymmetric boundary waves that begins with j = 2. Unfortunately, because Yanai waves are dispersive, the reflected boundary waves don’t have a simple form like (14.40a) and a simple analytic solution for the reflection like (14.41) is not possible. (A simple analytic solution is possible for the reflections of periodic Yanai waves, a topic we consider in Sect. 15.2.) Video: Video 14.2c is the same as Video 14.1c, except with an eastern boundary. The transient Yanai waves begin to reflect from the eastern boundary by the end of January. They generate a sequence of antisymmetric, equatorial waves (even values of j ≥ 2) along the boundary. In contrast to Video 14.2b, the reflected waves are evanescent, and they superpose to generate coastally trapped, β-plane Kelvin waves (Sect. 7.2.2). This difference happens because the dominant waves in the Yanai-wave packet have σ ≈ σ0 , and the critical latitudes for those waves are very close to the equator (Sect. 15.2.1). Eventually, all the transient waves leave the basin as Kelvin waves along the northern and southern coasts.

14.3.1.3

Boundary Forcing

If the wind forcing extends to the eastern boundary, boundary-reflected Rossby waves are generated, not only by Kelvin and Yanai waves arriving from the interior ocean, but also by local forcing. In this situation, the boundary-wave packet must cancel the terms on both the top (local forcing) and bottom (remote forcing) lines of (14.32). At the boundary, the u field driven by local forcing can be written as the Hermite expansion ∞  U j (t) φ j (η) . (14.43) u(0, η, t) = j=0

The U0 terms are cancelled as described above for U K . To cancel the remaining U j ( j ≥ 1) terms, a separate chain of long-wavelength Rossby waves (14.40a) must be applied to each φ j term. Consider cancelling the φ j  term in the series. Since j  > 0, there are two directions the chain rule could be applied, either toward increasing or decreasing j. To be consistent with the poleward propagation of coastal waves along an eastern boundary (Chap. 13) the correct choice is the former, since Hermite

14.3 Reflections from a Single Boundary

379

functions for larger j extend to higher latitudes (Fig. 8.1). Furthermore, a decreasing chain is impossible to implement: It ends in uncanceled terms proportional either to φ0 or φ1 (depending on whether j  is even or odd) and there are no waves with westward group velocity that can cancel them. Videos: Videos 14.2d and 14.2e are similar to Videos 14.1b and 14.1c, except that there is an eastern boundary and the forcings are shifted eastward to be centered there. In both solutions, similar boundary adjustments occur, involving multiple chains of reflected equatorial waves as described above. In Video 14.2c forced by τ x , the response looks much the same as in Video 14.2b, except with the final Sverdrupbalanced state shifted to the eastern boundary. By contrast, in Video 14.2d forced by τ y an antisymmetric pressure field fills the basin, a consequence of the boundary now being forced by alongshore winds. As a result, the circulation there responds much as the midlatitude coastal ocean does by creating a coastal pressure field which: first locally balances the wind; then extends to higher latitudes through coastal Kelvinwave propagation; and finally propagates offshore as a Rossby-wave packet (see the discussion of Video 13.2b in Sect. 13.2.3). In this case, the reflected wave packet contains antisymmetric Rossby waves (even values of j > 0), and individual waves are visible by the regions where |d| has relative maxima along its edges.

14.3.2 West-coast Reflections For a western boundary located at x = 0, (14.43) still applies, and each of its terms must be cancelled by a chain of Rossby/gravity waves, like those in the easternboundary set (14.40a) except with eastward group velocity or decay. In the absence of damping (γ = 0), however, system (14.1) has no western-boundary waves, since it does not allow for short-wavelength Rossby waves. To include a western boundary, then, we need to use a boundary-layer approach as we did in Sects. 11.2.1 and 13.2.3.2, that is, add on a western-boundary solution to a more complete system that includes these waves. The structure of each boundary wave is more complicated than it is in (14.40a), because the waves are dispersive (e.g., see the discussion of Eq. 15.30 below). Although not necessary for our purposes, we note that it is possible to write down a set of boundary waves analogous to (14.41) under the approximations considered in Sect. 13.2.3.2. Even though a simple solution is not possible, Moore’s chain rule still applies. Consider cancelling the φ j  term in (14.43). Since coastal signals propagate equatorward along a western boundary (Chap. 13), the required chain must be in the direction of decreasing j. Therefore, the chain contains Rossby/gravity waves in which j decreases until the only uncanceled terms are proportional to φ0 or φ1 , depending on whether j  is even or odd. These last contributions must be cancelled by an equatorial Kelvin wave or Yanai waves for even or odd j  , respectively. We expect each Rossby wave in the chain to have properties similar to the westernboundary solutions discussed in Sect. 13.2.3.2: for weak mixing, a boundary layer that first propagates offshore, then thins in time, and finally adjusts to a narrow, vis-

380

14 Equatorial Ocean: Switched-On Forcing

cous boundary layer; and for strong mixing an immediate adjustment to Munk (or Stommel) layer. Therefore, the overall western-boundary response should have these properties as well. Videos: Videos 14.3a–14.3d illustrate the impact of western-boundary reflected waves generated by a variety of wind forcings. Each solution includes horizontal viscosity with νh = 5×105 cm2 /s. As at midlatitudes (Sect. 13.2.3.2), viscosity allows the structure of western-boundary currents to adjust to that of a Munk layer, and weakens the transient, short-wavelength, equatorial Rossby waves as they propagate offshore. (Solutions with νh = 5×106 cm2 /s are similar, except that the transient waves are weaker and their shortest components are eliminated.) Video 14.3a is similar to Video 14.1b except the solution has a western boundary. During February, wind-driven Rossby waves begin to arrive at the western boundary, first a j = 1 wave followed several months later by a j = 3 wave. Both waves reflect as packets of short-wavelength Rossby waves that carry information eastward. Because the waves are symmetric about the equator, the final term of each boundary packet is proportional to φ0 , thereby generating an equatorial Kelvin wave. Note that the propagation of the boundary-generated Kelvin wave across the basin eliminates the Yoshida jet. At later times, a weak packet of transient, short-wavelength Rossby waves, visible as short-scale variations in shading that propagate from the western boundary and eventually cross the basin. Video 14.3b is comparable to Video 14.1c except with a western boundary. In this case, wind-driven Rossby waves (primarily the j = 2 and 4 waves) reflect from the western boundary, generating chains of short-wavelength Rossby waves ending in the excitation of Yanai waves. The Yanai waves are weak, however, and not visually apparent in the basin due to the prominent inertial oscillations. As time progresses, the short-wavelength Rossby waves radiate farther offshore, where they are visible as the small-scale features centered near ±5◦ . Near the end of the video, the response is dominated by the Sverdrup gyre. In addition, there are narrow bands just outside the northern and southern edges of the gyre (green and yellow shading); they indicate the presence of zonal Munk layers due to horizontal viscosity, which are generated to smooth the jump in u across the boundaries (Sect. 11.2.2). There are similar bands in Video 14.3a, but they are less visible because the zonal Munk layers are weaker (they only need to smooth u y ) and blurred by western-boundary signals. Videos 14.3c and 14.3d are comparable to Videos 14.3a and 14.3b, respectively, except the wind forcings are centered on the western boundary. In both videos, the boundaries are impacted immediately after the winds switch on, so that boundarygenerated waves are more intense. As a result, the transient Kelvin wave in Video 14.3c “wiggles” due to the presence of equatorial inertial oscillations, and the offshore propagation of short-wavelength Rossby waves is much more evident. In Video 14.3d, short-scale Yanai waves are also present in the response, much more so than in Video 14.3b. The reason is that in Video 14.3b the zonal scale of the Yanai waves is determined by that of the wind, whereas in Video 14.3d it has no such restriction. Essentially, the presence of the boundary cuts off the forcing sharply in a step function θ(x) at the coast, and a Fourier decomposition of θ(x) contains all wavenumbers k.

14.5 Observations

381

14.4 Reflections from Both Boundaries When the basin contains both boundaries, the adjustment to steady state is more complex as it involves multiple boundary reflections. One result is that it also exhibits periodic variability. A prominent period is Tn =

4L x , cn

(14.44)

where L x is the zonal width of the basin. Time Tn is the sum of the time it takes a Kelvin wave for mode n to cross the basin (L x /cn ) and a j = 1 long-wavelength, Rossby wave to return (3L x /cn ). It is a natural ringing time for the equatorial ocean, and for the parameters used in the video it is Tn = 156 days. Video: Video 14.4 is similar to Video 14.1b, except the basin has both eastern and western boundaries. Basic features of the response are a combination of Videos 14.2a and 14.3a. In addition, periodic pulses of enhanced sea level, the result of multiple reflections of equatorial Kelvin and Rossby waves, are visible radiating across the basin along the equator. Note that these pulses are not present in Video 14.2a, which lacks a western boundary, confirming that they result from multiple reflections in a closed basin. Based on the arrival times of positive (or negative) pulses at the eastern boundary, the oscillation period is about 165 days, roughly consistent with (14.44). The pulses weaken after several reflections owing to energy loss by poleward radiation along the eastern boundary. Finally, note that the steady-state background sea level (i.e., external to the area covered by the Sverdrup circulation) is much less affected in Video 14.4 than it is in Videos 14.2a and 14.2b (colored yellow-orange rather than red). The reason is that mass is conserved in Video 14.4 because the basin is closed, whereas in the other two videos it continues to enter the basin through the open western boundary throughout the spin-up.

14.5 Observations Although the monsoon winds that force the NIO are largely periodic, solutions forced by switched-on winds nevertheless illustrate essential properties of observed equatorial flows. In particular, the theoretical response to forcing by equatorial τ x is very clear in the real ocean. Without boundaries, the theoretical response is equatorial Ekman pumping (Sect. 14.2.1) followed by wave radiation that rapidly adjusts the ocean to a steady-state, Sverdrup-balanced circulation plus an x-independent, bounded Yoshida Jet (Sect. 14.2.2; Video 1.1). Further, the jet is very strong even for weak winds, a consequence of f vanishing at the equator. In contrast, the response to equatorial τ y is much weaker because it lacks a Yoshida Jet.

382

14 Equatorial Ocean: Switched-On Forcing

The development of the Wyrtki Jets (Sect. 4.4.1) is a response of this sort. They are forced by the semiannual westerlies that occur on the equator in the central ocean during the intermonsoons (Video 3.1): The eastward-propagating, downwellingfavorable signals during April and October in Video 1.1 are equatorial Kelvin waves driven by those winds, and they establish the jets in the eastern ocean. There is also an annual component to equatorial τ x associated with the seasonally-reversing, cross-equatorial monsoon winds. During the winter, they have an easterly component that generates the upwelling-favorable, equatorial Kelvin wave that crosses the basin during January (Videos 3.1 and 1.1). During the summer, the winds also have an easterly component, albeit much weaker, resulting in a weak upwelling-favorable, equatorial Kelvin wave during August/September (Sect. 4.4.2). The bounded Yoshida Jet response is disrupted by the reflection of Rossby waves from the eastern boundary and, because the propagation speed of Rossby waves slows away from the equator, the reflection forms a wedge-like pattern (Fig. 14.1; Videos 14.2a and 14.2b). The development of such patterns in the NIO is a clear response to all the Kelvin-wave signals noted in the previous paragraph (Video 1.1). In addition, the reflection also contains coastal waves that carry the response around the perimeter of the Bay of Bengal and down the east Indian coast. This coastal signal is most evident during the reflection of the fall Wyrtki Jet, when it also appears to continue around Sri Lanka, the perimeter of the Arabian Sea, and even into the Gulf of Oman (Sects. 4.9.2, 4.9.3, and 4.10.2). Coastal signals associated with other eastern-boundary reflections are not as clear in Video 1.1, as they are blurred along their pathway by local forcing. The impact of western-boundary reflections of interior flows is not visually apparent in Video 1.1. One reason is that mixing damps short-wavelength Rossby waves before they propagate very far from the boundary. Another is that, although the reflection typically generates an equatorial Kelvin wave that can propagate across the basin, that signal is difficult to pick out from all the others present in the interior ocean. Finally, the ringing in the solution with both boundaries (Video 14.4) is not present in the observations, because observed winds do not switch on abruptly; on the other hand, ringing appears in another form because climatological winds are periodic, namely, as an equatorial basin resonance (see Sect. 15.4.1).

Video Captions Unbounded Basin Video 14.1a Equatorial response forced by a switched-on, zonal wind stress τ x , with X (x) given by (C.7), x = 20◦ , xm = 60◦E, Y (y) = 1, and T (t) = θ(t). The Coriolis parameter is specified by the equatorial β-plane approximation, and open eastern and western boundary conditions are imposed as described in Appendix C.

Video Captions

383

Video 14.1b As in Video 14.1a, except that Y (y) is given by (C.7b) with y = 20◦ , ym = 0◦ . Video 14.1c As in Video 14.1b, except forced by a meridional wind stress τ y .

Eastern Boundary Video 14.2a Equatorial response forced by a switched-on, zonal wind stress τ x . Its spatial structure is (C.7) with x = 20◦ , xm = 30◦E, y = 20◦ , ym = 0◦ , and its time dependences is T (t) = θ(t). The Coriolis parameter is specified by the equatorial β-plane approximation. The eastern boundary is closed, and open conditions are imposed at the western boundary as described in Appendix C. Video 14.2b As in Video 14.2a, except that the center of τ x is shifted eastward to xm = 60◦ , the middle of the basin. Video 14.2c As in Video 14.2b, except forced by a meridional wind τ y . Video 14.2d As in Video 14.2b, except with x m = (100◦ , 0◦ ) so that the center of τ x is located at the eastern boundary. Video 14.2e As in Video 14.2c, except with x m = (100◦ , 0◦ ) so that the center of τ y is located at the eastern boundary.

Western Boundary Video 14.3a Equatorial response forced by a switched-on, zonal wind stress τ x . The spatial structure of the wind is (C.7) with x = 20◦ , xm = 60◦E, y = 20◦ , and ym = 0◦ , and its time dependence is T (t) = θ(t). The Coriolis parameter is specified by the equatorial β-plane approximation, and νh = 5×105 cm2 /s. The western boundary is closed, and open conditions are imposed at the eastern boundary as described in Appendix C. Video 14.3b As in Video 14.3a, except forced by a meridional wind τ y . Video 14.3c As in Video 14.3a, except that the center of the forcing region is shifted westward to x m = (20◦ , 0◦ ) so that it intersects the western boundary. Video 14.3d As in Video 14.3c, except forced by a meridional wind τ y .

384

14 Equatorial Ocean: Switched-On Forcing

Both Boundaries Video 14.4 As in Video 14.1a, except in a basin with closed western and eastern boundaries.

Chapter 15

Equatorial Ocean: Periodic Forcing

Abstract A near-equatorial solution forced by periodic winds is obtained to the complete LCS equations. It is found assuming that the Coriolis parameter is f = β y, allowing it to be represented as an expansion in Hermite functions. The same processes identified in the previous chapter for switched-on winds occur in the periodic one, except that they happen continuously rather then sequentially. When there is an eastern boundary, the wind-forced equatorial Kelvin wave reflects as a packet of Rossby and evanescent waves, with Rossby waves existing only equatorward of the critical latitude θcr , and evanescent ones superposing to form a β-plane Kelvin wave north of θcr . With both boundaries, two types of resonant responses are possible: “equatorial basin resonance,” which is linked to the natural ringing time of the basin; and “zero-group-velocity” resonance that occurs when the ocean is forced at the critical frequency of an Rossby or gravity wave. Keywords Periodic forcing · Boundary reflections · Meridional and slanted boundaries · Meridional energy propagation · Equatorial resonances In this chapter, we discuss solutions that illustrate the response of the equatorial ocean to forcing by periodic winds. We begin by obtaining an analytic solution in an unbounded ocean (Sect. 15.1). We find the solution to the exact equation set (9.1), because a prominent part of forcing in the equatorial NIO occurs at higher frequencies (e.g., intraseasonal winds) for which Eqs. (14.1) are not valid. Then, we discuss reflections of equatorial waves from single eastern and western boundaries that are oriented meridionally (Sect. 15.2), commenting on the impact of boundaries slanted as in Fig. 7.5 in Sect. 15.2.4. As we shall see, similar to off-equatorial boundaryreflected waves (Sect. 7.3.3), packets of boundary-reflected equatorial waves propagate meridionally as well as westward, carrying their energy to, and across, the equator. Next, we review observational evidence in the NIO that supports solution properties (Sect. 15.3). To conclude, we discuss two types of equatorial resonances, which potentially can cause an enhanced equatorial response (Sect. 15.4): one when Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_15. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_15

385

386

15 Equatorial Ocean: Periodic Forcing

the basin contains both eastern and western boundaries; and another when the ocean is forced at the critical frequencies, σG and σ R j (Eq. 8.15).

15.1 Interior Solution As for the solution in Sect. 14.2, the derivation for periodic forcing is lengthy and complex, owing to the required manipulations of Hermite functions. Some readers may therefore find it helpful first to read the discussions of the solution’s dynamics (Sect. 15.1.4) and videos (Sect. 15.1.5) before studying the derivation in detail. Since both the forcing and variables are proportional to e−iσt , the ∂t  operators in (9.1) can be replaced by the complex number −iσ  , σ  = σ + iγ. With this modification, Eqs. (9.2a) and (9.2b) become − uxx −

σ 2 σ σ F u = −i f v + v − i , yx c2 c2 c2

(15.1a)

σ 2 p = − f vx + iσ  v y − Fx , c2

(15.1b)

− px x −

and, after following the steps that led to (10.31) and (14.4), the Hermite expansion of (9.2c) is − v jxx

    α0 Fη j x β σ 2 σ0 i c2 2  − i  v j x + α j v j − 2 v j = − 2 (η F) j + i − + ∂ σ G j. x x σ c c σ c2 σ

(15.1c) In the following derivations, we often replace the forcing terms with their expanded forms, F = Fo X(x)Y (y) e−iσt and G = G o X(x)Y (y) e−iσt . When we do that, for notational convenience we drop the factor e−iσt , replacing it only at the end of the derivation. We first solve (15.1c) for v, which then becomes a known forcing term when solving (15.1a) and (15.1b) for u and p. We obtain the solution using the method of Fourier transforms (Sect. 9.4.2.2), as it is somewhat easier (shorter) than the direct approach (Sect. 9.4.1.1). Before proceeding, we therefore recommend that readers review Sect. 9.4.2.2, where a much simpler solution is derived following the same steps used here.

15.1.1 v Field Let q˜ designate the Fourier transform of q. Then, with the replacement ∂x → ik, the Fourier transform of (15.1c) multiplied by k is      kβ σ 2 (2) k − k v˜ j = F˜ j (k) , k k 2 +  + α2j − 2 v˜ j = k k − k (1) j j σ c

(15.2a)

15.1 Interior Solution

387

where the forcing is    G o Y j 2 σ 2

α0 ck   ˜ ˜ F j (k) = −k Fo k − 2 X x (k) (ηY ) j +  Yη j X(k) + c σ σ c ˜ ≡ A j (k) X(k) + B j (k)

X x (k) . (15.2b) Including the additional factor of k in (15.2a) enables the replacement ik X˜ →

Xx in the meridional forcing terms of F˜ j (k), a form that simplifies the solution for u and p (see the comment following Eq. 15.17 below). (There is an arbitrariness to replacements of the sort, ik q(k) ˜ → qx (k). They are equivalent to an integration by parts of the inverted solution q(x), and don’t fundamentally alter the solution.) (2) Wavenumbers k (1) j and k j are the roots of      kβ σ 2 (2) 2 k − k = 0, k +  + α j − 2 = k − k (1) j j σ c 2

(15.3a)

namely,

    σ 2 β σ 2 (1) (2) kj ,kj = −  1 ± 1 − 4 2 2j + 1 − 2 , 2σ σ0 σ0

(15.3b)

(2) where α2j = α02 (2 j + 1), and k (1) j and k j correspond to equatorial waves with westward and eastward, group velocity or decay, respectively. Equation (15.3a) is dispersion relation (8.13a) for equatorial waves except with damping (σ → σ  ). For future use, note that identities (2)  k (1) j + k j = −β/σ ,

k (1) k (2) = α2j − σ 2 /c2

(15.4)

follow directly from (15.3a), and it is then straightforward to show that     (1) (2) 2 k−1 ± k j = 2α0 k−1 ± k j

j , j +1

(15.5)

where k−1 = σ  /c. The solution proceeds by solving Eqs. (15.2) for v˜ j , v˜ j =

A j(k) B j (k) ˜   X(k)  

+  X x (k) , (2) (1) k − kj k − k (2) k k− k k − kj j 

k (1) j

(15.6)

and then expanding the coefficients of each term (15.6) into partial fractions (Sect. 9.4.2.2). Note that A j (k) and B j (k) are polynomials of order N = 2 in k and that the number of factors of the form (k − b) in the denominator is M = 3. In that case,

388

15 Equatorial Ocean: Periodic Forcing

we can apply the general expansion theorem, N

n n=0 an (k) M m=1 (k − bm )

=

N

M  m=1

(k −

n n=0 an (b )  M−1 bm ) =1 (b (=m)

− bm )

,

N < M,

(15.7)

to make the expansions. Although (15.7) appears complicated, it states the following simple rule: To find the coefficient of the term 1/ (k − bm ) in the expansion, replace k with bm everywhere else it appears. (2) Applying (15.7) with bm = k (1) j , k j , and 0 to each term in (15.6) gives (1)

(2)

˜ ˜ A(1) A(2) X x (k) B j (0)

j X(k) + B j X x (k) j X(k) + B j X x (k)   +   + v˜ j = , (1) (2) (1) (1) (1) (2) (2) (2) kk j k j k j k j k − k j k j k j k − k j

(15.8)

(2) (1) (2) m m m m where k (1) j = −k j = k j − k j , A j ≡ A j (k j ), B j ≡ B j (k j ), and the last term follows from the property that A j (0) = 0. From the definitions of A j and B j , and with some algebra, (15.8) can be rewritten,

  S˜ mj k mj , k σ

X x (k)   − G o Y j 2 (1) (2) , v˜ j = m m m c kj kj k m=1 k j k j k − k j 2 

(15.9a)

where 

m S˜ m j kj ,k



α0 = −k m j Fo c

(ηY ) j +

ck m j  σ





Yη j

GoY j ˜ X(k) + σ





2 σ 2 km − 2 j c



x (k) . X

(15.9b) Note that the k-dependence of each of the terms in (15.8) and (15.9b) is an order-1 −1  −1  (2) , k − k , or k −1 ; they invert to produce expressions pole, either k − k (1) j j for wind-driven, long- and short-wavelength Rossby waves and a non-propagating response, respectively. Using transform pair (9.51) in the form, q˜ (k) k−a

 ↔

ie

x

iax

   e−iax q x  d x  ,

(15.10)

L

to invert each term of (15.8) and replacing the neglected factor of e−iσt , (15.9a) inverts to 2  σ G j m vj = V mj eik j x−iσt − i 2 (1) (2) , (15.11a) c kj kj m=1

15.1 Interior Solution

389

where

 V mj =

x

i

  S mj k mj , x  k mj k mj

Lm

m 

e−ik j x d x 

(15.11b)

is an x-dependent wave amplitude and 

m Sm j kj ,x

that is,



S mj

α0 = −k m j Fo c

(ηY ) j +

ck m j  σ



Yη j



GoY j X (x) + σ





2 σ 2 km − 2 j c

 X x (x) ,

(15.11c)     m m m ˜ ˜

k j , x is S j k j , k with the replacements X(k) → X(x) and X x (k) →

X x (x). One neglected factor of e−iσt is replaced in the first term on the right-hand side of (15.11a) and another is contained within G j . The integration limits L m are L 1 = ∞ (L 2 = −∞), ensuring that waves with westward (eastward) group velocity or decay are present only west (east) of the forcing region. The summation, v=

J 

v j φ(η) ,

(15.12)

j=0

then provides the solution for v, where J is sufficiently large to ensure convergence.

15.1.2 u Field Although the derivation of u j follows the same steps used to obtain v j , it is algebraically more complex (an exercise in bookkeeping) because individual equations contain so many terms. Taking the Fourier transform of (15.1a) gives ∞ ∞     2    Fo 2 ˜ iα0 −k−1 ηφ j + kφη v˜ j − ik−1 Y j φ j X(k) . k − k−1 u˜ = c j=0 j=0 (15.13) For convenience, we define u˜ j by u˜ ≡

∞ 

u˜ j ,

(15.14)

j=0

which allows summation symbols to be dropped from subsequent equations. (Note that defined in this way u˜ j is not the Hermite expansion of u, ˜ but rather just a bookkeeping device.) Inserting (15.14) into (15.13), dropping summation symbols, and using Eqs. (8.7) to eliminate ηφ j and φ jη , leads to

390

15 Equatorial Ocean: Periodic Forcing

 u˜ j = −iα0

j + 1 φ j+1 v˜ j + iα0 2 k − k−1



j φ j−1 ik−1 Fo ˜ Y j φ j X(k) v˜ j − 2 . 2 2 k + k−1 c k − k−1 (15.15)

Then, inserting v˜ j from (15.6) gives  u˜ j = −iα0  + iα0

(1)

˜ A(1) j +1 j X(k) + B j X x (k)    φ j+1 2 k (k − k ) k − k (1) k − k (2) −1

˜ A(2) j X(k)

j

j

B (2) j X x (k)

˜ + Fo Y j φ j −ik−1 X(k) j  +  φ j−1 . (1) (2) 2 c (k − k−1 ) (k + k−1 ) k − kj k (k + k−1 ) k − k j (15.16)

The next step is to expand the three terms in (15.16) into partial fractions. ˜ The coefficients of each X(k) and

X x (k) term satisfies N < M, the requirement for the validity of (15.7). Using that theorem, it is straightforward, albeit tedious, to show that (15.16) expands into   ⎧ ⎫ 2 2 ˜m ⎨  ⎬  α0 j+1 α0 2j φ j−1 S˜ mj 2 φ j+1 S j    +i    u˜ j = −i ⎩ m m m m m m k − k mj k − k mj ⎭ m=1 k j k j k j − k−1 m=1 k j k j k j + k−1   ⎛ ⎞ ˜0 α0 j+1 α0 2j φ j−1 S˜ 0j 2 φ j+1 S j ⎝ ⎠ + i +i (2) (2) k k−1 k (1) k k−1 k (1) j kj j kj  + −i

−



α0



j+1 ˜+ 2 φ j+1 S j (k−1 , k)

k−1 k−1 − k (1) j





k−1 − k (2) (k − k−1 ) j

−i



j ˜− 2 φ j−1 S j (−k−1 , k)    k−1 + k (2) k−1 k−1 + k (1) (k + k−1 ) j j

α0

!   " # Fj Fj i i ˜ ˜ ≡ u˜ r j + u˜ n j + u˜ k j . Y j φ j X(k) Y j φ j X(k) + k − k−1 2c k + k−1 2c

(15.17)

  X x , and S˜ ±j = In (15.17), S˜ mj is defined in (15.9b), S˜ 0j = S˜ mj (0, k) = −G o Y j σ  /c2

$ %   S˜ mj (±k−1 , k) = ∓Fo k−1 (α0 /c) (ηY ) j ± Yη j . The absence of any terms proportional to G o in S˜ ±j results from the particular form of F˜ j in (15.2b); specifically, the

terms proportional to k 2 − σ 2 /c2 are zero when k = ±k−1 . Equation (15.17) has three blocks of terms: The block in braces (u˜ r j ) has the poles −1 −1   and k − k (2) , the one in parentheses (u˜ n j ) the pole k −1 , and the one k − k (1) j j

in brackets (u˜ k j ) the poles (k − k−1 )−1 and (k + k−1 )−1 . The inversion of these blocks leads, respectively, to long- and short-wavelength Rossby waves, a non-propagating (local) response, and waves with the wavenumbers ±k−1 . Before proceeding with the inversion, it is useful to simplify the terms.

15.1 Interior Solution

391

u˜ r j and u˜ n j Parts

15.1.2.1

The u˜ r j part of (15.17) can be rewritten as, u˜ r j

2 S˜ mj σ0    = −i  σ m=1 k m k m k − k m j j j



φ j+1 j +1 − 2 ck mj /σ  − 1



φ j−1 j 2 ck mj /σ  + 1

2 S˜ mj   σ0    −j η, k mj , σ , =  iσ m=1 k m k m k − k m j j j



(15.18)

where −j is equal to (8.12c), and u˜ n j simplifies to u˜ n j =

iα0 S˜ 0j k

(2) k−1 k (1) j kj

&

j +1 φ j+1 + 2



j φ j−1 2

' = −i

Xx α0 G o Y j

ηφ j , (15.19) (1) c k k j k (2) j

the last expression following from (8.7a).

15.1.2.2

u˜ k j Waves

Since the only physically realistic wave with wavenumbers ±k−1 is the equatorial Kelvin wave with k = k−1 , u˜ k j must simplify considerably. With the aid of Eqs. (8.7), it follows that √ $ √   % α0 j√ + 1Y j+1 ˜S ± = ∓k−1 α0 Fo (ηY ) j ± Yη = ∓ 2k−1 Fo , (15.20) j

jY j−1 c c j0 where the factor j0 ( j0 = 0 if j = 0 and is 1 otherwise) is included to “remember” that the last term vanishes when j = 0. Then, using (15.5) and (15.20), u˜ k j simplifies to u˜ k j = i

Fo Y j+1 φ j+1 − Y j φ j ˜ Fo j0 φ j−1 Y j−1 − Y j φ j ˜ X(k) − i X(k) . 2c k − k−1 2c k + k−1

(15.21)

When summed over j from 0 to ∞, all terms cancel except the φ0 term in the first expression on the right-hand side of (15.21), yielding u˜ k =

∞  j=0

u˜ k j = −i

˜ Fo X(k) Y0 φ0 , 2c k − k−1

(15.22)

which corresponds to the equatorial Kelvin wave. Note that without j0 in Eq. 15.21, it would be easy to miss the fact that its term is zero when j = 0.

392

15.1.2.3

15 Equatorial Ocean: Periodic Forcing

Total u

Combining u˜ r j , u˜ n j , and u˜ k j gives   2   S˜ mj k mj , k ˜

x Fo Y0 φ0 X(k) σ0  α0 G o Y j ηφ j X   −j η, k mj , σ − i u˜ j =  . − iδ j0 (1) (2) k m m m iσ c 2c k − k−1 kj kj m=1 k j k j k − k j

(15.23) With the aid of (15.10), and after replacing neglected factors of e−iσt , (15.23) inverts to 2  m σ0  m −  α0 G j V j  j η, k mj , σ eik j x−iσt + ηφ j + δ j0 Uo(x) φ0 eik−1 x−iσt , (2)  iσ m=1 c k (1) k j j (15.24a) where  Fo Y0 x    −ik−1 x   Uo(x) = X x e dx (15.24b) 2c −∞

uj =

is an x-dependent wave amplitude. The solution for u is then u=

∞ 

u j;

(15.25)

j=0

no function φ j (η) is needed in (15.25) since, as for u˜ j , u j isn’t the Hermite coefficient of u.

15.1.3

p Field

To find p, we solve (15.1b) using the same procedure used to obtain u. Fortunately, the steps are so similar that there is no need to  repeat the derivation in detail. Taking ˜ j , and using (8.7) to eliminate the Fourier transform of (15.1b), defining p˜ = ∞ j=0 p ηφ j and φη , leads to 

j + 1 φ j+1 v˜ j − iα0 2 k − k−1



j φ j−1 Fo ik ˜ Y j φ j X(k) v˜ j − 2 . 2 2 k + k−1 k − k−1 c (15.26) A comparison of (15.26) to (15.15) shows that the two expressions are the same with the replacements: u˜ j → p˜ j /c, the sign of the term proportional to φ j−1 reversed, and k−1 → k in the numerator of the fraction of the last term. With these changes, Eqs. (15.16)–(15.23) are changed only in that −j → +j , ηφ j → φ jη in the nonpropagating term, and the sign of Y j φ j in the last term of (15.21) is positive. The inversion of p˜ j /c is then p˜ j = −iα0 c

15.1 Interior Solution

393

2  m pj α0 G j σ0  m +  V j  j η, k mj , σ eik j x−iσt − φ + δ j0 Uo(x) φ0 eik−1 x−iσt , =  (2) jη c iσ m=1 c k (1) k j j (15.27) the same as u j except with the replacements −j → +j and ηφ j → −φ jη . As for u, the summation ∞  p= pj (15.28) j=0

gives the solution for p.

15.1.4 Dynamics A comparison of the propagating terms in solutions (15.11a, b), (15.24a, b), and (15.27) with free-wave solutions (8.12) and (8.30) identifies them to be Rossby/gravity and Kelvin waves with x-dependent amplitudes that involve the forcing functions. Thus, we interpret them as forced responses, which describe both the continual generation and radiation of these waves. There are two Rossby/gravity waves for each j, corresponding to waves with westward (m = 1) and eastward (m = 2) group velocity. (This property contrasts with solutions to the interior and long-wavelength equation sets in Chaps. 12 and 14, which only allow for m = 1, nondispersive, Rossby waves.) Recall that when j = 0 only the m = 2 Rossby/gravity wave (the Yanai wave) exists, since the j = 0, m = 1 wave is undefined (Sect.  8.3); consistent with this property, S01 = 0 since k0(1) = −σ  /c and (ηY )0 = − Yη 0 , so that the unrealistic wave is not present in the solution. Note that the radiation terms have the same integral form that they do for interior solutions (12.26). As such, one process that determines their amplitudes is how well the wavelength of a particular wave “fits” the forcing, being large (small) when |k| L x 1 (|k| L x  1), where k = k mj or k−1 . (Another process is zero-groupvelocity resonance discussed in Sect. 15.4.2.) At midlatitudes, Rossby waves propagate slowly enough so that often |kr | L x  1 and radiation is weak (see the discussion after Eqs. 12.28). Near the equator, however, both Rossby and Kelvin waves propagate fast enough for |k| L x 1 to hold at prominent (e.g., annual and semiannual) forcing frequencies, and so radiation is strong. The remaining terms in the solutions are non-propagating (local) parts proportional to e−iσt . They correspond to the Ekman-drift (steady-state) terms in solutions (10.35) and (10.37) driven by τ y winds. As such, they define the equatorial Ekman drift that results from periodic τ y . It is possible to show that at very low frequencies and with weak damping (σ  → 0) the periodic solutions simplify to the large-time (t → ∞) response of solutions (14.14), (14.32), and (14.37), that is, with R j (x − cr t) = 0, δ(t) = 0, and θ(t) = 1. Thus, in this limit the periodic response simplifies to a quasi-steady, Sverdrup

394

15 Equatorial Ocean: Periodic Forcing

circulation plus a bounded Yoshida Jet, as should be the case. The derivation of this property, however, is tedious so we leave it as an exercise for dedicated readers.

15.1.5 Videos Videos 15.1a–15.1d and 15.1e–15.1h illustrate the responses to forcing by zonal and meridional winds oscillating at a range of periods P observed in the equatorial Indian Ocean: the quasi-biweekly (10 days), intraseasonal (45 days), seasonal (90 days), and semi-annual (180 days) time scales. These periods cover the range of high-σ to low-σ solutions discussed above. Solutions are obtained in an unbounded basin, that is, one with both eastern and western damping regions. τ x forcing: In Video 15.1a, P = 10 days and the critical index from (8.16) is jcr = 0.081 < 1 so that no periodic Rossby waves exist. Only the equatorial Kelvin wave is possible, visible propagating to the east. Its wavelength is λ−1 = 20.5◦ , close to the width of the wind stress L x = 20◦ ; therefore, |k−1 | L x ≈ 2π, the response doesn’t fit either of the limits just discussed, and the amplitude of the Kelvin wave is of moderate strength. For Video 15.1b (P = 45 days), jcr = 2.41 and so j = 1 and 2 Rossby waves are possible; however, since the forcing is symmetric about the equator, only the j = 1 wave can be generated. Throughout the video, equatorial Kelvin and j = 1, long-wavelength, Rossby waves are prominent, radiating east and west of the forcing region. The wavelength of the Kelvin wave is λ−1 = 92.3◦ , much longer than L x , and hence its amplitude is greater than the wave in Video 15.1a; in contrast, the ◦ wavelength of the long-wavelength Rossby wave is λ(1) 1 = 26.5 , only somewhat greater than L x , and the wave has moderate amplitude. Short-wavelength, j = 1, ◦ Rossby waves can also be excited by the wind. Because their wavelength (λ(2) 1 = 4.7 ) is considerably less than L x , their amplitude is small. As a result, they are visible only as ripples on the Kelvin wave during the latter half of the video, their slow development a consequence of their small group velocity (cgx = 9.5 cm/s = 26.8 deg/year). In Videos 15.1c (P = 90 days) and 15.1d (P = 180 days), jcr = 11.04 and 45.7 and Rossby waves with odd j values less than these limits are possible. Nevertheless, only a few low-order (primarily j = 1 and 3) long-wavelength Rossby waves radiate from the forcing region, because the wind structure Y (y) is meridionally narrow and composed mostly of low-order Hermite functions. There is no indication of shortwavelength ( ( Rossby waves in the eastern ocean because their wavelengths are so short ( ( that (k (2) j ( L x  1 holds, and hence their radiation from the forcing region is very weak (Sect. 15.1.4). Note in Video 15.1d that the flow is unidirectional along the equator twice a year, an indication that at this long period a bounded Yoshida Jet almost sets up. τ y forcing: Properties of Videos 15.1e–15.1h are similar to those in Videos 15.1a– 15.1d, except that Yanai waves replace Kelvin waves in the eastern ocean. In Video

15.2 Reflections from a Single Boundary

395

15.1e (P = 10 days), the wavelength of the Yanai wave is 148◦ , very long since P is close to 9.4 days for which λ0 = ∞, and because σ/σ0 = 0.94 < 1 it has westward phase velocity. In the other videos, antisymmetric Rossby waves are possible, and in the western ocean long-wavelength Rossby waves associated with low-order j values (primarily j = 2 and 4 waves because Y (y) is narrow) are prominent. By contrast, in the eastern ocean periodic Yanai and short-wavelength Rossby waves are weak or absent, because their wavelengths are short with respect to L x . In Video 15.1f (P = 45 days), a transient packet of large-scale Yanai waves first radiates from the forcing region. Thereafter, weak, periodic Yanai waves (λ0 = 4.2◦ ) and short◦ wavelength, j = 2 Rossby waves (λ(2) 1 = 5.8 ) radiate eastward, combining with each other to produce a distinctive interference pattern. The pattern develops slowly because their group speeds are small (cgx = 23.9 and 17.8 deg/year, respectively). The responses in Videos 15.1g (P = 90 days) and 15.1h (P = 180 days) are similar, except that in the eastern ocean their near-equilibrium responses are essentially zero because λ0 and λ(2) 1 are so small.

15.2 Reflections from a Single Boundary Periodic equatorial waves that radiate from the interior ocean reflect from eastern and western boundaries in the same way they do for the switched-on waves (Sect. 14.3), by generating chains of reflected waves. Let the location of either boundary be at x = 0. Then, similar to (14.43) for the transient solution, the u field of solution (15.24a, b) at the boundary can be summarized as the Hermite expansion u(0, η, t) =

∞ 

U j (σ) φ j (η) e−iσt .

(15.29)

j=0

This u field must be cancelled by a chain of boundary waves of the form &  u rmj = Urmj −

j + 1 φ j+1 + 2 ck mj /σ − 1 2



' φ j−1 j m eik j x−iσt 2 ck mj /σ + 1

+ δm2 UY φ1 eik0 x−iσt + δm2 U K φ0 eik−1 x−iσt ,

(15.30)

where: j ≥ 1; k−1 = σ/c; Urmj (σ), UY (σ) and U K (σ) are amplitudes for Rossby/gravity, Yanai, and Kelvin waves, respectively; and m = 1 (m = 2) for an eastern (western) boundary to ensure that the group velocities of the reflected waves are directed offshore.

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15.2.1 Eastern Boundary To cancel u everywhere along an eastern boundary, a separate chain of m = 1 waves is required to cancel each φ j in (15.29), with the direction of each chain extending toward larger j values. Consider the cancellation of φ j  . The required chain of waves begins with j = j  + 1 in (15.30). The value of j  + 1 can be less or greater than the critical value jcr (defined in Eq. 8.16). In the former case ( j  + 1 < jcr ), some boundary waves are trigonometric and radiate offshore as Rossby/gravity waves; however, eventually j > jcr in the chain so that these boundary waves are evanescent and hence coastally trapped. In the latter case ( j  + 1 > jcr ), all the boundary waves are coastally trapped. Moore (1968) proved that the evanescent waves for j > jcr sum to generate a β-plane Kelvin wave like the ones discussed in Sects. 7.2.2 and 13.3. Thus, the eastern-boundary response consists of Rossby/gravity waves at latitudes south of the critical latitude ycr (defined in terms of jcr in Eq. 8.17), and a β-plane Kelvin wave north of ycr . Videos for τ x forcing: Videos 15.2a–15.2d illustrate the impact of an eastern boundary on the unbounded solutions forced by τ x . In Video 15.2a (P = 10 days) and all the other videos in group 15.2, transient Rossby waves initially radiate off the eastern boundary. In the periodic response, however, all the boundary-reflected waves are evanescent and superpose to form β-plane coastal Kelvin waves. In the other τ x -forced videos, reflected Rossby waves are possible. Because the reflected waves are generated by an incoming, symmetric Kelvin wave, they must also be symmetric, that is, have odd j values ( j = 1, 3, 5, . . .). In Video 15.2b (P = 45 days), the critical index is jcr = 2.41 and only the j = 1 boundary-reflected wave is a Rossby wave. In the interior ocean, it combines with the equatorial Kelvin wave to produce a distinctive pattern. In Videos 15.2c (P = 90 days) and 15.2d (P = 180 days), jcr = 11.04 and 45.7 so that symmetric, boundary-reflected waves with odd j values less than these values are possible. From (8.17), their critical latitudes are located at ycr = 14.7◦ and 29.4◦ , well off the equator. Equatorward of ycr , then, packets composed of a number of Rossby waves radiate into the interior ocean. As at midlatitudes (Sect. 7.3.3), individual waves interfere with each other to develop complex patterns, a consequence of Rossby-wave energy propagating meridionally as well as westward (Sect. 15.2.3 below). Videos for τ y forcing: In Video 15.2e (P = 10 days), the reflection of the windgenerated Yanai wave leads to the formation of β-plane Kelvin waves that have the opposite sign in each hemisphere. At the other periods, Yanai-wave radiation east of the wind is weak or absent. To generate an eastern-boundary response, then, in Videos 15.2f–15.2h the wind is shifted to be centered on the eastern boundary, so that the coastal response is generated by the local alongshore wind rather than incoming radiation. The resulting solutions develop patterns that are similar to their counterparts in Videos 15.2b–15.2d, except they are antisymmetric about the equator: In Video 15.2f (P = 45 days), the interior response is a single j = 2, long-wavelength Rossby wave, whereas in Videos 15.2g (P = 90 days) and 15.2h (P = 180 days) complex patterns arise from packets of several Rossby waves (Sect. 15.2.3).

15.2 Reflections from a Single Boundary

397

15.2.2 Western Boundary At a western boundary, the cancellation of each φ j in (15.29) requires a chain of m = 2 Rossby/gravity waves (15.30) that begins with j = j  − 1 and extends to either j = 0 or 1, depending on whether j  is even or odd; then, the last element in the chain is cancelled by either an equatorial Kelvin (for j = 0) or Yanai ( j = 1) wave. As for the eastern-boundary reflection, western-boundary chains can, in principle, contain contributions with j values both greater and less than jcr , in which case the western-boundary response consists of m = 2 Rossby/gravity waves for |y| < ycr and a coastally trapped, β-plane Kelvin wave for |y| > ycr . (If the forcing is confined to the interior ocean, the only waves that can propagate to the western boundary are trigonometric Rossby waves for which j < jcr . The boundary response is then present only for |y| < ycr and has no β-plane Kelvin waves.) At sufficiently low frequencies, the offshore wavelength of the inviscid boundary layer, λ = 4πσ/β becomes small enough to be overwhelmed when there is damping or horizontal mixing, in which case the boundary-layer structure approaches that of a quasi-steady, Stommel or Munk layer (Sects. 11.2.1 and 13.2.3.2). Videos for τ x forcing: Videos 15.3a–15.3d show solutions forced by τ x when the basin has a western boundary. When P = 10 days, interior winds cannot generate a western-boundary response because no Rossby waves exist that can carry signals there. So, in Video 15.3a the forcing is shifted to the western boundary. As in Video 15.2a, the primary impact of the boundary is to cut off the western half of the forcing; as a result, the Kelvin-wave response is similar to that in Video 15.1a, except with about half the amplitude. A secondary feature is a transient packet of short-wavelength, j = 1 Rossby waves (Sect. 13.2.3.2). As time passes, the packet extends farther across the basin, with longer waves leading shorter ones; eventually, it vanishes near the western boundary, where wavelengths are so short that they cannot be resolved by the model grid. In Video 15.3b (P = 45 days), j = 1 Rossby waves are possible. Long-wavelength Rossby waves radiate from the forcing region to the western boundary, and then ◦ reflect as a chain consisting of a j = 1, short-wavelength (λ(2) 1 = 4.7 ) Rossby wave and an equatorial Kelvin wave. As time progresses, the short-wavelength Rossby waves slowly advance across the basin at their group speed (cgx = 26.8 deg/year). In Videos 15.3c and 15.3d (P = 90 and 180 days), the solutions are similar to the one in Video 15.3b, except that horizontal viscosity (νh = 5×106 cm2 /s) is included to ◦ ◦ damp reflected, short-wavelength Rossby waves (λ(2) 1 = 2.1 and 1.0 ) before they propagate very far offshore. In all the videos, the boundary-reflected Kelvin wave is not visible, apparently because its amplitude is sufficiently weak to be overwhelmed by the other signals. Videos for τ y forcing: As in Video 15.3a, the forcing in Video 15.3e (P = 10 days) is centered on the western boundary. Because half of the forcing is removed by the boundary, the excited Yanai wave has about half the amplitude of the one in Video 15.1e. There are also hints of a weak transient packet of j = 2, shortwavelength Rossby waves propagating from the boundary.

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15 Equatorial Ocean: Periodic Forcing

In Video 15.3f (P = 45 days), several Rossby waves are possible but j = 2 Rossby waves dominate owing to the wind being confined near the equator. Longwavelength j = 2 Rossby waves radiate from the forcing region to the western boundary, where they reflect as a chain composed of only a j = 2, short◦ wavelength (λ(2) 2 = 5.8 ) Rossby wave and a Yanai wave. As time progresses, the short-wavelength Rossby wave gradually advances across the basin at its group speed (cgx = 17.8 deg/year). Similar to the previous videos, the boundary-reflected Yanai wave is weak enough not to be visible in the overall response. At the longer periods, the equilibrium responses with νh = 5×106 cm/s are like those in Videos 15.3c and 15.3d, with boundary-reflected waves that quickly decay offshore. To highlight the properties of inviscid boundary reflections, Videos 15.3g (P = 90 days) and 15.3h (P = 180 days) show solutions forced by a τ y centered on the western boundary; additionally, the wind is y-independent (Y = 1) in order to allow for off-equatorial reflections. Without mixing, the boundary-reflected waves propagate across the basin; further, they develop large-scale patterns (envelopes) that are remarkably similar to the ones in Videos 15.2g and 15.2h except reversed in x, a consequence of the meridional propagation of Rossby-wave energy discussed next.

15.2.3 Meridional Energy Propagation The wave-group theory discussed in Sect. 7.3.3 also provides an explanation for the offshore patterns of boundary-reflected, equatorial Rossby waves. The steps that lead to (7.45) also apply when f is given by the equatorial β-plane approximation f = β y, which is recovered from the midlatitude β-plane approximation by setting yo = f o = 0 and βo to its equatorial value. With these replacements, solution (7.45) simplifies to   2 σ κ =  , (15.31) y (x) = y  cos κ x , c 1 − y 2 /y 2 R

where y R = c/ (2σ) and y  is the latitude of a particular ray path at the eastern boundary. Since the equator is not a solid boundary, wave energy does not reflect from it; consequently, (15.31) describes the complete pathways followed by boundarygenerated Rossby waves. Figure 15.1 plots ray paths determined from (15.31) for several values of y  when P = 90 (top) and 180 (bottom) days. The pathways shown extend westward from an eastern boundary equatorward of the latitudes in both hemispheres (dashed   critical lines). Given the symmetry of cos κ x , pathways from a western boundary are the mirror images of those in Fig. 15.1, that is, with x → −x in each panel. As for Fig. 7.4, the pathways collectively exhibit prominent patterns similar to features in the videos. In particular, ray paths that leave either boundary closer to

15.2 Reflections from a Single Boundary

399

Fig. 15.1 Ray paths extending from the eastern boundary at various latitudes y  when P = 90 (top panel) and 180 (bottom panel) days. They are determined from (15.31). The black-dashed lines in each hemisphere indicate Rossby-wave critical latitudes. Rays from the southern (northern) hemisphere are red (blue). Rays closer to the equator tend to focus onto points given by (15.33), whereas those farther away (dashed curves) do not. Curves for a western boundary are the same as shown with the axis reversed (x → −x)

the equator (solid curves) tend to focus at specific distances from the boundary. This property is apparent from solution (15.31) in its low-latitude limit  σ  y = y  cos 2 x , c

y 2 y R2 .

(15.32)

According to (15.32), low-latitude ray paths focus onto the equator at the same points   σ 1 π 2 xn = n + c 2

⇒

    1 cP 1 πc xn = n + = n+ . 2 2σ 2 4

(15.33)

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15 Equatorial Ocean: Periodic Forcing

Consistent with (15.33), quasi-focal points are apparent in Videos 15.2c, 15.2d, 15.2g, 15.2h, and 15.3g, 15.3h, at locations somewhat closer to the boundary than predicted, owing to the contribution of rays from higher latitudes (dashed curves).1 Focal points of this sort have not been observed in the real ocean. For westernboundary reflections, the reason is because packets of short-wavelength Rossby waves are readily eliminated by mixing and instability processes. For easternboundary reflections, one reason is that the complete response is a superposition of the responses from several vertical modes: As discussed in Chap. 16, multi-mode periodic solutions exhibit vertical propagation, blurring the presence of focal points associated with individual modes. Another possibility is that, because higher-latitude eastern-boundary waves have small zonal wavenumbers k, they can become nonlinearly unstable (Qiu et al. 2013; Xia et al. 2020).

15.2.4 Reflections from a Slanted Boundary Moore’s chain rule fails along a slanted coast because the amplitudes of the set of reflected equatorial waves can only be determined along a single latitude. On the other hand, for periods P long enough for the critical latitude y R to be located well off the equator, the theory of Sect. 7.4 can still be used to obtain the ray paths for reflected waves. With f = β y, the general ray-path solution (7.43) simplifies to c y (x) = β



  2 kβ β 2 − k cos x + θo . − σ c 2k + β/σ

(15.34)

Now consider a ray path that leaves a slanted coast, like the one in Fig. 7.5, from an    The value of arbitrary boundary  P located at (x , y ) in unrotated coordinates.   point    k at P , k = k x , y , is given by (7.55) or (7.56) and k = k  everywhere on the ray path (Sect. 7.3.3). With the aid of (7.55) and (7.56) and setting f  = β y  , (15.34) can be rewritten as ⎤ ⎡

⎥ ⎢ c2 k 2 2 (σ/c)  x + θo ⎥  tan2 α + y 2 cos ⎢ y=  ⎦ ⎣ 2 β 2 cos2 α tan2 α ± 1 − y 2 /y Rc (15.35a)

1

Schopf et al. (1981) applied the same wave-group theory to the long-wavelength equations (14.1), which only contain nondispersive Rossby waves. In that case, k 2 is absent in (7.38), which gives k  β/σ = −β 2 y 2 /c2 , and (7.39)–(7.43) no longer contain the 2k and k 2 terms. Thus, in the longwavelength approximation (15.32) is valid without any restriction on y  , so that all eastern-boundary energy focuses onto the single point xn . This property is present in the Cane and Sarachik (1981) and Cane and Moore (1981) solutions, which also assume the long-wavelength approximation.

15.2 Reflections from a Single Boundary

where y Rc =

401

c cos α 2σ

(15.35b)

and the upper (lower) sign is chosen for long-wavelength (short-wavelength) Rossby waves from an eastern (western) boundary. Note that (15.35a) is real only when y  ≤ y Rc , so that y Rc defines a “coastal” critical latitude that is smaller by the factor cos α than for Rossby waves generated in the interior ocean. As a check, also note that when the boundary isn’t slanted (α → 0) θo → 0 and solution (15.35a) simplifies to (15.31), as it should. It remains to choose θo . One constraint is that (15.35a) must intersect P  . To ensure that it does, in (15.35a) we set y = y  and (from the coastline definition in Eq. 7.48) x = y  tan α, to get ⎛

⎞  y 2 (σ/c) y  tan α ⎠− ,  θo = cos−1 ⎝    2 2 c2 k 2 /β 2 tan α + y 2 cos2 α tan2 α ± 1 − y 2 /y Rc (15.35c) which almost determines θo as a function of y  . Its specification, however is not yet complete because the inverse cosine is multi-valued. Let δθ be either of the two values of the inverse cosine in (15.35c) with the smallest magnitudes: They are equal in magnitude but have opposite signs, and all other values are equivalent to this pair as they differ from one or the other by ±2nπ. The proper choice of δθ ensures that the slope of the ray path at P  , dy/d x| P  , has the correct sign. Inserting k = k  and  =  from (7.56) and (7.54) into (7.39), the components of the group velocity for the Rossby wave leaving point P  are ( σ2 cgx ( P  =  kβ

& '   σ β y 2  2 2 2k + =  cos α tan α ± 1 − 2 , σ k y Rc

( σ2 tan α. cgy ( P  = 2 β

( ( Then, since dy/d x| P  = cg y ( P  / cgx ( P  , we have  sgn

(15.36)

& ( ' (  (  (  (     cgy ( P  dy (( ( = sgn = sgn cgy ( P  sgn cgx ( P  = sgn (α) sgn cgx ( P  . ( ( d x P cgx P 

(15.37) Another independent expression for dy/d x| P  can be found by taking the x-derivative of (15.35a), and evaluating it at point P  (x = y  tan α). With the aid of the first of Eqs. (15.36), and using (15.35c) with the inverse cosine replaced by δθ to eliminate θo , that expression simplifies to ( dy (( σσ = −2  ( d x P ck

  c2 k 2 /β 2 tan2 α + y 2 ( sin δθ. cgx ( P 

(15.38)

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15 Equatorial Ocean: Periodic Forcing

Neglecting positive-definite terms, the sign of this expression is then (     (   (  dy (( = −sgn (δθ) sgn k  sgn cgx ( P  = sgn (δθ) sgn cgx ( P  , sgn ( d x P (15.39) the last term resulting from the property that sgn(k  ) is negative for all Rossby waves. Combining (15.37) and (15.39) gives 

sgn (δθ) = sgn (α) .

(15.40)

According to (15.40), the correct value for the sign of δθ is determined by the sign of α, which completes the specification of θo .

Fig. 15.2 As in Fig. 15.1, except when the eastern boundary is slanted at the angle α = 45◦ . The ray paths are plots of solution (15.35a) with k  given by (7.56) and using the upper signs. The horizontal dashed lines in each hemisphere indicate the coastal critical latitudes, ±y Rc . The dashed curves are ray paths that extend offshore to a latitude greater than y Rc (blue-dashed curve) or intersect the coast (red-dashed curve)

15.3 Observations

403

Figure 15.2 plots the ray paths determined from (15.35) for an eastern boundary when α = 45◦ and P = 90 and 180 days. The collection of paths is similar to those in Fig. 15.1, for example, with the quasi-focal points occurring near the same locations (Schopf et al. 1981). A striking difference is that the coastal√ critical latitudes ±y Rc (horizontal dashed lines) are reduced by a factor of cos α = 1/ 2 from their positions in Fig. 15.1 when α = 0. Another is that the slopes of all pathways have a negative slope at P  , consistent with (15.37) since α > 0 and cgx < 0. As a consequence, pathways that leave the coast near y = y Rc extend poleward of y Rc (blue-dashed curves), but only to latitudes that are less than the open-ocean critical latitude y R = c/(2σ). Conversely, pathways that leave the coast near y = −y Rc eventually intersect the coast (red-dashed curves). Another impact of the boundary slope is that ray paths from a western boundary are no longer mirror images of those in Fig. 15.2 under the transformation x → −x. An indication of this lack is that when α = 0 the amplitude and wavenumber of solution (15.35a) depend on wavenumber k  (through the ± sign). Since reflected waves with different k  are used at eastern and western boundaries (long- and shortwavelength Rossby waves, respectively), ray paths from each boundary necessarily have different structures. A thorough exploration of western-boundary ray paths when α = 0 is extensive and, since a modest amount of horizontal viscosity replaces the inviscid response with a Munk layer, we don’t pursue this solution further.

15.3 Observations Processes that determine the periodic response of the equatorial ocean are much the same as those for the switched-on response, except that they happen continuously. Therefore, the observations noted in Sect. 14.5 in support of the switched-on solution also support the periodic one. The periodic response, however, allows for additional aspects: the significant excitation of Yanai waves; eastern-boundary reflections in which critical latitudes, meridional propagation, and focal points are dynamically important; and annual and semiannual tilts to coastal sea level forced by alongboundary winds. Yanai Waves: As for other waves, Yanai waves are efficiently excited by periodic winds in the interior ocean only when their wavelength λ is of the order of, or larger than, the zonal width of the forcing region L x (λ  L x ), so that the wave “fits well” with the zonal structure X(x) of the wind (Sects. 12.3.2.2 and 15.1.4). For Yanai waves, this fit occurs for waves near the frequencies σ ≈ σ0n for which the zonal wavenumber k ≈ 0. For modes n = 1–4, these frequencies correspond to the quasi-biweekly periods Pn = 9.38, 11.79, 14.91, and 17.64 days. In agreement with this result, Sengupta et al. (2001, 2004) demonstrated that quasi-biweekly Yanai waves exist in the central Indian Ocean near these periods, but not at longer periods; moreover, the authors showed that the Yanai waves were forced by the local winds. Yanai waves at longer periods and smaller λ, however, can be excited by westernboundary forcing. The reason is that the boundary acts as a point-source forcing in

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15 Equatorial Ocean: Periodic Forcing

which X(x) is delta-function-like and L x ≈ 0, so that the inequality in the previous paragraph is always satisfied. A prominent example of such forcing is the SC as it crosses the equator: The current develops an instability with a period of the order of P = 26 days, and this instability is an efficient radiator of Yanai waves (Kindle and Thompson 1989; Chatterjee et al. 2013). Similarly, longer-period Yanai waves can be forced at a western boundary by alongshore winds τ y (Sect. 15.2.2; Videos 15.3e– 15.3h), and this process accounts for a significant part of the observed Yanai waves in the western Indian Ocean (Chatterjee et al. 2013). Eastern-boundary Reflections: The reflection of equatorial Kelvin waves at the eastern boundary of the Indian Ocean is a major forcing for variability in the Bay of Bengal (and Arabian Sea), and its nature depends on the critical latitude for Rossby waves y Rn (Sect. 7.1.4) or its modification for radiation from a slanted boundary y Rc (Sect. 7.4). At the annual and semiannual periods, values of y Rn for the n = 1 and 2 modes (that dominate the sea-level response), lie poleward of the NIO northern boundary except for the mode-2, semiannual wave for which y R2 = 18.0◦N. Consistent with theory, Rossby waves at these periods are prominent in sea-level variability at all latitudes in the NIO (Video 1.1; Fig. 7.6). At shorter periods, the critical latitudes lie well within the NIO (Figs. 7.3 and 7.6): for the n = 1(2) modes and with P = 30, 60, and 90 days, they are y Rn = 4.9◦ (3.1◦ ), 9.7◦ (6.2◦ ), and 14.4◦ (9.2◦ ), respectively. According to theory, reflected Rossby waves should be confined to latitudes lower than y Rn . This prediction is supported by the regression plot of intraseasonal (30–120 day) variability in Fig. 4.21: Coastal waves are visible circulating around most of the perimeter of the Bay of Bengal, and a wedge-shaped pattern of reflected Rossby waves is present at low latitudes. Similar features are visible in Videos 15.3a and 15.3b. When looking at intraseasonal variability in finer detail, the simple response suggested by Fig. 4.21 is no longer apparent. For example, Cheng et al. (2013) noted that variability within the 90–120 day and 30–60 day period bands was concentrated in distinct bands, with weak energy levels equatorward of them (Bands B and D in Fig. 4.22). This bandedness differs from the video responses, and its cause remains unclear. One possibility is the complexity of the boundary shape. Another is nonlinear effects: The variability in the 90–120 day band is better described as being composed of nonlinear eddies rather than Rossby waves (Sect. 4.7.2), and the 30–60 day period band shows clear evidence of nonlinear interactions between 30–60 day and 90-day waves (Cheng et al. 2017). Further research is needed to identify the underlying processes. Other prominent features in idealized solutions are quasi-focal points on the equator (Videos 15.2c and 15.2d), but such focal points are not apparent in the real Indian Ocean. One reason they aren’t is the narrow width of the Bay of Bengal and Arabian Sea, which cuts off the outer edge of the reflected packets. Another reason is that, when the contributions of a number of vertical modes are summed, energy associated with the reflected waves propagates downward, eliminating the focal points associated with individual modes (Sect. 16.1). Boundary Forcing: The solutions in Videos 15.2f–15.2h and 15.3g–15.3h illustrate the response to forcing by alongshore winds τ y at the eastern- and western-ocean

15.4 Equatorial Resonances

405

boundaries. The eastern-boundary response in Video 15.2h is a simple representation of the semiannual response to alongshore winds along the coasts of Myanmar and Indonesia (Sect. 4.5.2). Similarly, the western-boundary solution in Video 15.3h illustrates the response to monsoon winds along Africa and Somalia (although the coastal response is blurred because the short-wavelength Rossby waves are undamped). In both solutions, coastal sea level tilts to balance or nearly balance τ y (Eqs. 13.13 and 13.37), and so has opposite signs on either side of the equator. This antisymmetry occurs in the real ocean (Video 1.1) and realistic solutions (Videos 1.2 and 1.3). We consider the western-boundary response in greater detail in Sect. 17.3.3.

15.4 Equatorial Resonances Equatorial-ocean resonances have been invoked to account for peaks in ocean spectra that are stronger than might be expected from corresponding wind spectra. The two resonance types discussed here happen because radiation of wave energy from the forcing region is inhibited, thereby allowing energy to accumulate: either owing to the reflection of equatorial waves from boundaries in a closed basin, or because the forcing excites waves with zero group velocity.

15.4.1 Equatorial Basin Resonance When the basin contains both eastern and western coasts, the adjustment to steady state is more complex than with a single boundary (Sect. 15.2), as it involves multiple wave reflections from both of them. One consequence of multiple reflections is the generation of equatorial-basin (EB) resonances. In many physical systems that exhibit resonance, the resonant response is sharply peaked at a particular frequency and, in the absence of damping, becomes infinitely large. In contrast, EB resonance is broad and remains finite even without damping, a consequence of energy loss along the eastern boundary. EB resonances occur when the equatorial ocean is forced by a zonal wind stress τ x with a period P close to an integral multiple of the ringing period Tn in (14.44), that is, when 4L x m P = Tn = (15.41) cn and m is a positive integer. Anti-resonance can also occur when m is replaced by m  = m − 21 . Cane and Moore (1981) and Gent (1981) derived condition (15.41) in an idealized solution to the long-wavelength equation set (14.1) in a basin with eastern and western boundaries but unbounded in y. More recently, Han et al. (2011)

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15 Equatorial Ocean: Periodic Forcing

explored EB resonances in solutions to numerical models with realistic Indian-Ocean boundaries, finding that (15.41) still holds well when L x is the width of the basin at the equator. To visualize the physics behind condition (15.41) in a simple way, suppose that τ x consists of a periodic set of impulses, τix = τox X(x) Y (y) δ(ti ), where ti = i P and i is an integer, so that each impulse is separated from its neighbors by time P. At each time ti , the forcing generates packets of equatorial waves. Consider the response to the impulse at time t0 = 0: One part is a packet of equatorial Kelvin waves (K 0 ) and another is a packet of long-wavelength, j = 1 Rossby waves (R0 ). Packet R0 propagates to the western boundary where it reflects as a Kelvin-wave packet (K 0 ), and K 0 propagates across the basin to reflect from the eastern boundary as another Rossby-wave packet (R0 ). Provided that (15.41) holds, R0 returns to the forcing region in phase with the locally-forced, Rossby-wave packet (Rm ) generated by the wind impulse at a time tm = m P, thereby intensifying the overall response. Conversely, when m → m  , R0 is out-of-phase with Rm and the response is diminished. (Similarly, the propagation and reflection of K 0 leads to a western-boundary-reflected Kelvin wave that can be in-phase or out-of-phase with K m .) The preceding explanation for (15.41) only considers multiple reflections of the equatorial Kelvin wave and j = 1 Rossby wave. In the complete response, other Rossby waves are generated, as well as coastal Kelvin waves along the eastern boundary. Because of the latter, not all the reflected energy is returned to the forcing region, some of it being lost via Kelvin-wave radiation along the eastern boundary due to Moore’s chain rule. This continual energy loss accounts for the relative weakness of EB resonance. Videos: To illustrate EB resonance, Videos 15.4a–15.4c contrast the responses when the ocean is forced: at the resonant period P = T1 (m = 1) in a bounded basin (Video 15.4a, resonant response) and with an open western boundary (Video 15.4b, no resonance); and at the anti-resonant period P = 2T1 (m = 21 ) in a bounded basin (Video 15.4c). For the parameters used in the video (L x = 80◦ , c = 264.7 cm/s) the resonant period is P = 155.6 days, close to the ringing period identified in Video 14.4. The difference in magnitude of the solutions in Videos 15.4a and 15.4b is evident, the amplitude of the eastern-ocean response without basin resonance being 0.35 times smaller then with resonance. Video 15.4c shows the response at the anti-resonant period P = 311 days, and the eastern-ocean response is again 0.35 times weaker than the resonant one. Observations: In the equatorial Indian Ocean, EB resonances of the second (n = 2) baroclinic mode with m = 1 and 2 have been invoked to explain the strengths of the 180-day (Wyrtki Jets) and 90 day responses, respectively (Jensen 1993; Han et al. 1999, 2011; Han 2005). In both cases, the wind appears to be too weak to account for the response: The 180-day equatorial current is considerably stronger than the annual one, despite the semiannual and annual components of τ x having comparable amplitudes; likewise, the 90-day and 30–60 day signals have similar strengths, even though the wind amplitude is much stronger in the latter frequency band. Recently, Huang et al. (2018) investigated the Indian Ocean’s Equatorial Intermediate Current

15.4 Equatorial Resonances

407

(the zonal flow from just below the thermocline to a depth of about 1200 m), concluding that resonance of the n = 2 mode was the dominant contributor to its semiannual variability.

15.4.2 Zero-Group-Velocity Resonance Recall that Rossby and gravity waves have critical frequencies, σcr = σ R j or σG j , (2) at which k (1) j = k j = −β/ (2σcr ) ≡ kcr and the slopes of their dispersion curves are zero (Fig. 8.3 and Eq. 8.15). The latter property ensures that these waves have zero group velocity (ZGV). When the ocean is forced at (near) these frequencies, we can expect the response to be large (resonant) since energy radiation from the spatially-limited forcing region is inhibited. Note that the Rossby/gravity-wave parts of solutions (15.11a, b), (15.24a, b), and (15.27) have amplitudes V mj that are inversely (1) (2) (2) proportional to k (1) j = k j − k j = −k j . This property is the mathematical representation of ZGV resonance, as each of the terms tends to ∞ without damping (σ  = σ) when σ → σcr and k j → 0. There are, however, conceptual problems with the solutions when they are evaluated at σ = σcr . For one thing, they are physically unrealistic in that their amplitude is infinite. For another, the resonant waves radiate far from the forcing region (they extend to ±∞): Since they have zero group velocity, how is such radiation possible? One way to interpret the resonant solutions is to view them only in the limit that σ (2) is close, but not equal, to σcr . Then, since k (1) j and k j have slightly different values, the response is finite and the two waves have non-zero group velocities that radiate from the forcing region. Near-resonant waves of this sort are visible in the videos discussed next. They are also present in videos discussed elsewhere, particularly in several of the videos reported in the next chapter (Videos 16.2a and 16.2e; 16.2b and 16.2f; and 16.3a). Videos: Another way to investigate ZGV-resonant solutionsis to spin them up from a state of rest. Video groups 15.5 and 15.6 illustrate the spin-up of near-resonant n = 1, j = 1 gravity and Rossby waves, respectively. In all the videos, eastern- and western-boundary dampers are included to simulate a zonally-unbounded ocean. Solutions are forced by switched-on zonal winds that oscillate at periods P near the theoretical resonant periods Pcr = 5.48 and 31.94 days for gravity and Rossby waves (Eq. 8.15). Apparently due to finite-difference errors, the numerical resonant periods Pcr are shifted somewhat from Pcr : In a suite of test solutions, we estimated Pcr to be close to 5.55 and 32.45 days. For the gravity-wave solutions, the zonal width of the forcing region is L x = 20◦ whereas for the Rossby-wave solutions it is decreased to L x = 10◦ (see below). Video 15.5a shows the development of the near-resonant, gravity-wave response when P = 5.55 days. The solution grows nearly linearly throughout the video, and in the latter half its wavelength λ is everywhere close to the theoretical resonant

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15 Equatorial Ocean: Periodic Forcing

value, λcr = 65.61◦. To illustrate the strength of the resonance, Videos 15.5b and 15.5c show the responses at nearby, off-resonant periods, P = 5.4 and 5.7 days. In contrast to Video 15.5a, both solutions quickly adjust to a quasi-steady state that does not grow in time. At P = 5.4 days, trigonometric gravity waves are possible. In the western and eastern oceans, their wavelengths are roughly −30◦ and 110◦ , the minus sign indicating westward phase propagation; these values are close to the theoretical (1) ◦ ◦ wavelengths at P = 5.3 days: λ(2) 1 = −26.2 and λ1 = 114.3 . At P = 5.7 days, only evanescent gravity waves are possible, and they quickly damp away from either side of the forcing region. Properties of the Rossby-wave solutions are similar to their gravity-wave counterparts. The near-resonant, Rossby-wave response when P = 32.45 days (Video 15.6a) increases nearly linearly throughout the video and its wavelength is close to the theoretical value, λcr = 11.26◦ , both east and west of the forcing region. At the nearby, off-resonant period P = 34 days (Video 15.6b), Rossby waves exist. In the equilibrium response, they extend east and west of the forcing region, and consistent with theory have a longer wavelength in the west. An equatorial Kelvin wave is also present. (A similar Kelvin wave is present in Video 15.6a, but it is too weak to be visible.) At P = 31 days (Video 15.6c), the two Rossby waves in Video 15.6b are replaced by evanescent waves that decay away from the forcing region. As for the gravity-wave solutions, the weaker amplitude of the off-resonant solutions is striking. Observations: Surprisingly, given their prominence in the analytic solutions and videos, ZGV resonant waves are not prominent in the real ocean. In Fig. 8.4, for example, there is little power near the gravity-wave resonance points (bottoms of the gravity-wave curves) and, although there is power near Rossby-wave resonance points, it is believed to be generated by current instabilities (associated with TIWs) rather than by resonant waves (Sect. 8.6). Recall that Lin et al. (2008, 2014) suggested that off-equatorial spectral peaks may be Rossby waves intensified by ZGV resonance (Sect. 7.5). Those off-equatorial resonances are in fact dynamically the same as the equatorial ones discussed here. They differ onlyin that  their frequency is √ 2 −1 2σ0 = cn / (2σ), the high enough that a 1 in (8.17): In this limit, ycr → a / same expression as y Rn in (7.16) for midlatitude waves. As noted in Sect. 7.5, however, there is evidence that off-equatorial spectral peaks also result from instabilities, rather than ZGV resonance. If ZGV resonances exist in the real ocean, they are surprisingly weak. Why? One possible reason is mixing. In the LCS model, the inclusion of vertical mixing (γ = 0) (2) ensures that k (1) j and k j have imaginary parts with opposite signs, so that k j  = 0 at resonance and even solutions with σ = σcr have finite amplitudes. On the other hand, for the low-order-mode (n = 1 or 2) resonances of interest here, realistic damping strengths (e.g., with A = A = 2.6×10−4 cm2 /s3 , the value used in Sect. 16.2.3.2 for the EUC) still allow the amplitude to be unrealistically large. Similarly, in test solutions with νh = 5×106 cm2 /s, resonant solutions reach an equilibrium amplitude that is too large. Thus, mixing is not likely the reason for weak ZGV resonances. Another possibility is that the forcing used to make the videos is too idealized. First, in the real ocean the amplitude of the forcing at resonant periods Pcr is weaker

Video Captions

409

than the value used in the videos (τox = 1.5 dyn/cm2 ), perhaps by an order of magnitude or more. Second, the forcing does not continue indefinitely with a single amplitude or without any phase change. Third, the zonal width of the idealized forcing L x is likely unrealistic. Recall that for radiation to be significantly generated by the wind, its wavelength λ must satisfy the inequality λ  L x (Sect. 12.3.2). For the gravity-wave resonance (Video 15.5a), this inequality is satisfied since L x = 20◦ and λ ∼ 65◦. For the Rossby-wave resonance (Video 15.6a), however, it is barely satisfied with L x = 10◦ and λ ∼ 11◦ but not with L x = 20◦, which is why we reduced L x to 10◦ for Video 15.6a. In the real world, L x is typically larger than 20◦, thereby limiting the strength of ZGV resonance for Rossby waves. Finally, in regions where the background currents are significant, the Rossby-wave dispersion relation is modified. Impacts are that the propagation speed of Rossby waves can be altered (Killworth et al. 1997) and their structure distorted (Proehl 1990). It is possible that the dispersion relation is so severely distorted that it no longer has a region where its slope vanishes so that its group velocity never vanishes, thereby eliminating ZGV resonance completely.

Video Captions Unbounded Basin Video 15.1a Equatorial response forced by a zonal wind stress τ x oscillating with a period of P = 10 days. The spatial structure of the wind is (C.7b) with x = 20◦ , y = 30◦ , and x m = (40◦ E, 0◦ ). Its time dependence is T(t) = sin(σt)θ(t) where σ = 2π/P. The Coriolis force is given by the equatorial β-plane approximation, and open eastern and western boundary conditions are imposed as described in Appendix C. Video 15.1b As in Video 15.1a, except with P = 45 days and x m = (60◦E, 0◦). Video 15.1c As in Video 15.1a, except with P = 90 days. Video 15.1d Video As in Video 15.1a, except with P = 180 days. Video 15.1e As in Video 15.1a, except forced by a meridional wind τ y with P = 10 days. Video 15.1f As in Video 15.1e, except that P = 45 days. Video 15.1g As in Video 15.1e, except that P = 90 days. Video 15.1h As in Video 15.1e, except with P = 180 days.

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15 Equatorial Ocean: Periodic Forcing

Eastern Boundary Video 15.2a Eastern-boundary response forced by a zonal wind stress τ x oscillating with a period of P = 10 days. The spatial structure of the wind is (C.7b) with x = 20◦ , y = 30◦ , and x m = (30◦ , 0◦ ). Its time dependence is T(t) = sin(σt)θ(t) with σ = 2π/P. The Coriolis force is given by the equatorial β-plane approximation. The eastern boundary is closed, and an open conditions are imposed at the western boundary as described in Appendix C. Video 15.2b As in Video 15.2a, except with P = 45 days. Video 15.2c As in Video 15.2a, except with P = 90 days. Video 15.2d As in Video 15.2a, except with P = 180 days. Video 15.2e As in Video 15.2a, except forced by a meridional wind τ y with P = 10 days. Video 15.2f As in Video 15.2e, except that P = 45 days and τ y is shifted eastward to be centered at x m = (100◦ , 0◦ ) so that it intersects the eastern boundary. Video 15.2g As in Video 15.2f, except that P = 90 days. Video 15.2h As in Video 15.2f, except with P = 180 days.

Western Boundary Video 15.3a Western-boundary response forced by a zonal wind stress τ x oscillating with a period of P = 10 days. The spatial structure of the wind is (C.7b) with x = 20◦ , y = 30◦ , and x m = (20◦ , 0◦ ) so that the wind intersects the western boundary. Its time dependence is T (t) = sin(σt)θ(t) with σ = 2π/P. The Coriolis parameter is given by the equatorial β-plane approximation, and an eastern damping region is included. Video 15.3b As in Video 15.3a, except with P = 45 days and x m = (80◦ , 0◦ ). Video 15.3c As in Video 15.3a, except with P = 90 days, x m = (80◦ , 0◦ ), and νh = 5×106 cm2 /s. Video 15.3d As in Video 15.3a, except with P = 180 days, x m = (80◦ , 0◦ ), and νh = 5×106 cm2 /s. Video 15.3e As in Video 15.3a, except forced by a meridional wind τ y with P = 10 days. Video 15.3f As in Video 15.3e, except with P = 45 days and x m = (80◦ , 0◦ ). Video 15.3g As in Video 15.3e, except with P = 90 days and Y (y) = 1. Video 15.3h As in Video 15.3e, except with P = 180 days and Y (y) = 1.

Video Captions

411

Equatorial Resonances Equatorial Basin Resonance Video 15.4a Equatorial response forced by a zonal wind stress τ x oscillating with a period of P = 155.6 days. The spatial structure of the wind is (C.7b) with τox = 1.5 dyn/cm2 , x = 20◦ , y = 20◦ , and x m = (60◦ E, 0◦ ). Its time dependence is T(t) = sin(σt)θ(t) where σ = 2π/P. The Coriolis force is given by the equatorial β-plane approximation. The basin is closed, and horizontal mixing (νh = 5×106 cm2 /s) is included to prevent the spreading of reflected, short-wavelength Rossby waves from the western boundary. Open conditions are imposed at the western boundary as described in Appendix C. Video 15.4b As in Video 15.4a, except that open conditions are imposed at the western boundary as described in Appendix C. Video 15.4c As in Video 15.4a, except with P = 311.2 days. Zero-Group-Velocity Resonance Video 15.5a As in Video 15.4a, except that P = 5.545 days. Video 15.5b As in Video 15.5a, except with P = 5.7 days. Video 15.5c As in Video 15.5a, except with P = 5.4 days. Video 15.6a As in Video 15.4a, except that P = 32.45 days, and x = 10◦ . Video 15.6b As in Video 15.6a, except that P = 34 days. Video 15.6c As in Video 15.6a, except that P = 31 days.

Chapter 16

Beams and Undercurrents

Abstract The complete solutions to the LCS model are superpositions of the responses to many modes. In these solutions and for periodic forcing, wave energy propagates both vertically and horizontally along ray paths determined by the wave’s dispersion relation. Provided the ray-path slopes are independent of wavenumbers, the responses have a beam-like structure. Solutions are found that illustrate beams associated with coastal and equatorial Kelvin waves, Yanai waves, and equatorial Rossby waves. For constant background stratification Nb , the beams are very clear, but when Nb varies with depth they are blurred by reflections of wave energy in regions where Nbz = 0. For switched-on forcing and when vertical mixing (damping) is sufficiently large, solutions adjust to steady-state, coastal and equatorial circulations that have realistic undercurrent structures. Keywords Vertically propagating waves · Ray paths · Energy and phase propagation · Primary and secondary beams · Steady-state undercurrents In previous chapters, we have discussed solutions for a single baroclinic mode. The complete response to the LCS model, however, is a superposition of many modes (Sect. 5.2.5). Here, we discuss multi-mode solutions for periodic (Sect. 16.1) and steady (Sect. 16.2) forcings along coasts and the equator. For periodic forcing, a remarkable property is that wave energy can form beams that propagate vertically as well horizontally. For steady forcing, solutions with damping have subsurface structures similar to those of observed coastal and equatorial currents.

16.1 Beams Because each baroclinic mode exists throughout the water column, it is not obvious that waves associated with several modes can interfere to produce a beam-like response. In this section, we first demonstrate they can do so using wave-group theSupplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_16. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_16

413

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16 Beams and Undercurrents

ory, and also derive other basic beam properties (Sect. 16.1.1). We then report videos that illustrate beams along coasts and the equator (Sect. 16.1.2). McCreary (1984), Rothstein et al. (1985), Moore and McCreary (1990), and Miyama et al. (2006) provide additional discussions of equatorial beams, the latter two studies considering Indian-Ocean phenomena. Further, with some restrictions (constant Nb , and simplified wind structures), Romea and Allen (1983) and Burkhardt (2019, priv. comm.) derive closed analytic expressions for beams that don’t involve mode summations.

16.1.1 Properties To demonstrate beam-like behavior, a first step is to replace the discrete set of vertical modes ψn (z) of the LCS model with an alternate version ψ(m, z) that depends on a continuously varying, local wavenumber m(z). It is then possible to write wave solutions in a form that involves continuously varying, zonal and vertical wavenumbers, k and m, in which case the wave-group theory in Sects. 6.3 and 6.4 can be applied with the replacement  → m. We ignore damping throughout the discussion, recognizing that its impact is primarily to decrease the amplitude of beams in the direction of their energy propagation (Sect. 6.5).

16.1.1.1

Continuous Vertical Modes

Function ψ(m, z) is a solution to (5.10a), 

1 ψz Nb2

 =− z

1 ψ c2

(16.1)

without imposing boundary conditions (5.10b), so that c is a continuous variable rather than the discrete set cn . Let L z = Nb /Nbz measure the vertical scale of Nb (z). Then, following the WKB approach used to obtain (7.11), an approximate solution to (16.1) that is valid when |m| L z  1 is   z    ψ (m, z) = A(z) exp i m z  dz  , with A(z) =



Nb (z),

c=

Nb (z) . |m(z)|

(16.2a)

(16.2b)

Inequality |m| L z  1 is a statement that the vertical scale of the oscillation, |m|−1 , is much less than that of the background variation L z . The same inequality is required for the validity of wave-group theory (Sects. 6.3 and 6.4), ensuring that the wavegroup analysis below is also valid. (In the paragraph after Eq. 16.10, we briefly discuss the impact on solutions when the inequality does not hold.)

16.1 Beams

16.1.1.2

415

Approximate Wave Solutions

To be consistent with the equatorial solutions discussed next, we consider the solution for a coastal Kelvin wave along a southern boundary at y = 0 in the northern hemisphere, noting the impact of this restriction in the discussion at the end of this subsection. Its horizontal structure is given by (7.21) with  = 1 and using the upper signs. The Kelvin-wave solution, the product of its horizontal structure and vertical structure (16.2), is therefore

   y    z    Nb f  |m| dy eikx−iσt , exp i m z dz exp − q(x, t; k, σ) = Qo No N b 0 (16.3) where x = (x, y, z), k = (k, m), q = p = uc, No is a measure of the magnitude of Nb (z) included to make the argument of the radical nondimensional, and Qo(m) is the wave’s amplitude. (For simplicity, a factor of ρ¯ is absorbed into p, as well as into p j below.) Its dispersion relation is σ = ck =

k Nb , |m|

(16.4)

the last term following from the second of Eqs. (16.2b). Similarly, equatorial Rossby/gravity and Kelvin waves have the horizontal structures, (8.12a) and (8.30), respectively, and the complete waves can be summarized as

   z    Nb exp i m z dz φ j  (η) eikx−iσt , (16.5) q j (x, t; k, σ) = Qoj No √ √ where x = (x, y, z), η = β/c y = β |m| /Nb y and Qoj (m) is the wave amplitude. For equatorial Kelvin waves, j is the label K , q K = u K = p K /c, QoK = UoK , and j  = 0. For Rossby/gravity and Yanai waves, j = 0, 1, . . . , and either: 1) q j = v j , Qoj = Voj , and j  = j; or 2) q j = u j = p j /c, Qoj = Uoj , and φ j  (η) → ∓j  (η, k, σ) defined in (8.12c) where the minus (plus) sign is for u j ( p j ). Written in terms of k and m (c = Nb / |m|), dispersion relations for each wave type are k Nb (16.6a) σ= |m| for equatorial Kelvin waves, k=

σ |m| β − Nb σ

(16.6b)

for Yanai waves, and σ=−

k Nb , |m| 2 j + 1

j = 1, 2, 3, . . .

(16.6c)

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16 Beams and Undercurrents

for non-dispersive Rossby waves. We restrict the analysis to non-dispersive Rossby waves, because only those Rossby waves combine to form an ideal beam.

16.1.1.3

Ray Paths

Forcing by a spatially confined wind field generates packets of waves (16.3) and (16.5), that is, superpositions of the waves integrated over the continuous variables, k and m. For these packets, the wave-group theory of Sects. 6.3 and 6.4 applies with the replacement  → m, so that packets propagate at the group velocity cg = σk i + σm k.

(16.7)

Using dispersion relations (16.4) and (16.6a, b, c), the group velocities for coastal and equatorial Kelvin waves, Yanai waves, and non-dispersive Rossby waves are       σ σ σ k , φY i ∓ k , −φ R i± (2 j + 1) k , cg = φ K i ∓ Nb Nb Nb

m ≷ 0,

(16.8)  −1 where φ K = Nb / |m|, φY = |m| /Nb + β/σ 2 , and φ R = (Nb / |m|) (2 j + 1)−1 . According to (16.8), energy associated with the three wave types propagates along ray paths with slopes s = dz/d x = σm /σk given by s=∓

σ σ σ , ∓ , ± (2 j + 1) , Nb Nb Nb

m ≷ 0.

(16.9)

A key property of (16.9) is that the slopes do not depend on k and m; consequently, when individual waves are combined, energy in the resulting packet also propagates with the slope s, a necessary property for the packet to have a beam-like structure. According to (16.9), if phase propagates upwards (m > 0 and upper signs), energy associated with coastal and equatorial Kelvin waves and with Yanai waves slopes downwards to the east and, surprisingly given their very different dispersion relations, at the same slope. In contrast, energy associated with non-dispersive Rossby waves slopes downward to the west at angles that are steeper by the factor 2 j + 1. Conversely, if phase propagates downwards, energy associated with Kelvin and Yanai (Rossby) waves slopes upward to the east (west).

16.1.1.4

Phase Propagation

From (6.6), the phase velocity of the waves is cp =

k2

σ (ki + mk) . + m2

(16.10)

16.1 Beams

417

For coastal and equatorial Kelvin waves and for non-dispersive Rossby waves, it follows from (16.10), (16.7), and dispersion relations (16.4) and (16.6), that c p · cg = 0.

(16.11)

According to (16.11), c p is perpendicular to cg so that lines of constant phase are parallel to the propagation direction of the packet. Note that (16.11) holds regardless of the values of k and m, and hence it holds for packets of these waves. Consider, for example, a packet in which energy propagates downward at slope s; then, lines of constant phase have the same slope and phase propagates upwards across the packet as it descends. This property is very clear in the Kelvin- and Rossby-beam videos discussed below. For Yanai waves, c p · cg = 0 owing to the additional term −β/σ in (16.6b), and there isn’t a simple relationship between c p and cg .

16.1.1.5

Velocity Vectors

In Kelvin and Yanai beams, current vectors, u = ui + wk, are also parallel to cg . This property follows from the relation u = p/c = p |m| /Nb , which holds for both wave types (Eqs. 7.21b, 8.25, and 8.30). The w field associated with the waves is given by (5.8d) with A = κh = 0, that is, w=

iσ pz , Nb2

(16.12)

where p is given by (16.3) or (16.5) with QoK → PoK . The z-derivative of p is determined by that of ψm ,  ψmz =

 Az + im ψm ≈ imψm , A

(16.13)

since, under the WKB restriction that |m| L z  1, (A (z) /A) ψm is negligible with respect to imψ. With the aid of (16.13) and the above relation between u and p, w in (16.12) becomes w = −σ

m σ m σ p=− u = ∓ u, 2 Nb |m| Nb Nb

m ≷ 0,

(16.14)

Thus, velocity vectors have the form u = ui + wk = u [i ∓ (σ/Nb ) k] .

(16.15)

Note that the slopes of the vectors, w/u = ∓σ/Nb , are independent of k and m. Therefore, when individual waves are combined into a beam, the slopes of velocity vectors in the beam itself are the same. A comparison of (16.15) with the first of (16.8) shows that u cg .

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16 Beams and Undercurrents

In contrast, current vectors in Rossby beams are not parallel to beam slopes. This difference happens because u = p/c, a consequence of the meridional structures of u and p for Rossby waves not being the same (due to the ± signs in Eqs. 8.12c). On the other hand, for symmetric (odd j values), non-dispersive Rossby waves, u vectors still have a characteristic slope that depends only σ/Nb on the equator (η = 0), where they are evaluated in most of the videos. Recall that for non-dispersive Rossby waves ∓j  (η, k, σ) simplifies to the form (8.19b), 1 ˆ ±j (η) = − √  2 2



φ j+1 φ j−1 ± √ √ j +1 j

 ,

(16.16a)

η = 0.

(16.16b)

and from (8.9a) it follows that on the equator φ j+1 = − j/ ( j + 1)φ j−1 ,

Combining these expressions with u j and p j from (16.5) gives

    z    uj Nb ˆ ∓j eikx−iσt = Uoj exp i m z dz  p j /c No

     z    2j + 1 Nb j ± ( j + 1) ikx−iσt e , = Uoj exp i m z dz φ j−1 √ √ = Uj No −1 2 2 j ( j + 1) (16.17) 

where U j is shorthand for all the terms except for the numerator of the fraction involving j. Note that u j and p j are related by u j = − (2 j + 1) p j /c. The arguments that lead to (16.14) are modified only slightly for w j , yielding wj =

iσ p j z σ uj m σ m uj =± , = −σ 2 p j = 2 Nb |m| 2 j + 1 Nb 2 j + 1 Nb Nb

m ≷ 0. (16.18)

The velocity vector on the equator is then   σ u j = u j i + w j k = U j (2 j + 1) i± k , Nb

m ≷ 0.

(16.19)

According to (16.19), the slope of u j is sj =

wj s σ/Nb = =± , uj 2j + 1 (2 j + 1)2

(16.20)

where s is the group-velocity slope for non-dispersive Rossby waves from (16.9). Thus, s j is smaller in magnitude than s by the factor (2 j + 1)−2 . Relationship (16.20) appears to hold well in the videos that have well-formed Rossby beams.

16.1 Beams

16.1.1.6

419

Discussion

The above theory considers beams in an ocean without a surface and bottom. When these surfaces are included, beams reflect from them. Consider, for example, a downward-propagating beam generated at the surface by wind forcing. It eventually reflects from the bottom as an upward-propagating beam, which in turn reflects from the ocean surface as a second downward-propagating beam, and so on. Provided |s| is sufficiently small, the set of reflected beams don’t overlap, and individual downward- and upward-propagating beams are separate from each other throughout most of the water column. On the other hand, if |s| is large enough they overlap significantly, and beam structure is blurred; for example, in the limit that |s| → π/2 (σ → ∞), beam properties are lost and the response appears as a standing wave. In deriving the coastal Kelvin-wave beam, we considered a southern boundary in the northern hemisphere. Since beam properties depend only on the dispersion relation having a simple form like (16.4), coastal beams also exist along a northern boundary except that they extend westward rather than eastward. Along a meridional coast, however, dispersion relation (7.30b) is more complex, only simplifying to the required form (σ = ±c,) for latitudes θ located poleward of the critical latitude θ R (Sect. 7.1.4). Nevertheless, well-formed coastal beams exist along meridional boundaries because enough high-order (n > 1) vertical modes ψn (z) contribute to them: Recall that θ R decreases with n because cn does; consequently, as n increases the critical latitude shifts ever closer to the equator until θ > θ R , and the coastal response for these modes is Kelvin-like. Nethery and Shankar (2007) showed that coastal beams exist along a meridional coast everywhere north of 6◦N (an idealized version of the Indian west coast), which was possible because the contributions of almost all the modes were Kelvin-like. Finally, recall that a condition for the validity of the above theory is that the ocean’s stratification changes slowly (i.e., |m| L z  1). In many locations, however, the stratification has a sharp near-surface pycnocline, where this condition doesn’t hold. What happens to a beam that encounters such a pycnocline? In this situation, although wave energy still propagates through the pycnocline, some of it also reflects from it (Philander 1978b; Rothstein et al. 1985). Similar reflections occur in the video solutions discussed next that are obtained using modes derived from the realistic background stratification Nb (z) in Fig. 5.1.

16.1.2 Videos The following videos illustrate beams along an eastern/northern coast and the equator, which are generated by winds oscillating at periods P = 15, 30, 60, and 180 days. (We also obtained an analogous suite of solutions along a western/southern coast, but don’t include them because their beam properties are so similar to those along the eastern/northern coast. In contrast, for quasi-steady undercurrents the responses differ considerably between the two coasts; see Sect. 16.2.3.1.) The solutions are

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sums of the responses of either N = 25 or 99 baroclinic modes, and are found using both the constant and realistic, stratification profiles Nb (z) in Fig. 5.1. To highlight the inviscid properties of beams, solutions are obtained without mixing, the exceptions being for solutions found in basins with a closed western boundary (Sect. 16.1.2.3). In all the solutions, the Coriolis parameter is given by the equatorial β-plane approximation ( f = β y). In most solutions N = 25, a value sufficient to resolve the beams present in them. Cutting off the mode summation at this (or any other) value causes truncation error, that is, the existence of unrealistic oscillations with N zero crossings across the water column. In solutions using realistic Nb , this error is usually barely visible in the videos, because the wind-coupling coefficients Hn−1 decrease so rapidly with n that modes with n ≈ N are weakly excited (Table 5.1). In solutions with constant Nb , however, truncation error is apparent because Hn−1 decreases slowly. In most of these solutions, the error remains confined to the wind-forcing region, and doesn’t interfere with the beams; in the few solutions where it does interfere, we increase N to 99 and the error is much reduced. To minimize truncation error, it might seem better simply to set N = 99 in all the solutions. For equatorial solutions, there are two problems with this approach. First, transient Kelvin, Yanai, and Rossby waves associated with high-order modes propagate so slowly that very long integrations are required to reach a quasi-equilibrium state. Second, high-order-mode gravity and evanescent waves are then possible, and they can introduce noise large enough to overwhelm (blur) beams. In OGCMs and the real ocean, contributions from high-order vertical modes are damped out by vertical mixing. Since here we wish to highlight inviscid beam dynamics, we don’t adopt this approach and just accept the presence of some truncation noise.

16.1.2.1

Coastal Beams

Coastal solutions are forced by an x-independent band of τ y confined from 15– 25◦N or, when P = 15, 30, 60, and 180 days, from 15–35◦N. For the above periods, critical latitudes for the first baroclinic mode are θ R1 = 2.4◦ , 4.9◦ , 9.7◦ , and 27.2◦ , respectively, and those for the other modes are less than or equal to these values. Thus, in the equilibrium responses contributions from almost all the modes are Kelvin-like north of 25◦ , the only exception being for the n = 1 mode when P = 180 days. Because coastal Kelvin waves continue to propagate around the perimeter of the basin, the videos show the responses first along the eastern boundary (left half) and then along the northern boundary with the x-axis reversed (right half). Along the eastern boundary, they plot v (shading) and v = (v, w) velocity vectors (arrows), and along the northern boundary they are −u and v = (−u, w). Constant-Nb Solutions: Videos 16.1a–16.1d show solutions for constant stratification. In each case, transient Kelvin waves radiate from the forcing region, and only after their passage solutions are adjusted to an equilibrium state. Because the speeds cn of the higher-order Kelvin waves are slow, the propagation of the transients can take some time. In Videos 16.1a–16.1c, for example, transient waves for

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the higher-order vertical modes don’t cross the basin until the beginning of year 3. In Video 16.1d with N = 99, the propagation of the transient waves is much slower, and they are still visible as narrow bands above the beam throughout the video. Consistent with wave-group theory, the equilibrium states consist of packets (beams) of Kelvin waves that propagate vertically as well as horizontally. Beam slopes s are close to their theoretical values in (16.9), ±σ/Nb , so that |s| is smaller for longer periods (smaller σ); as a result, in all but one of the videos (Video 16.1d) beam slopes are large enough to reflect from the bottom and top of the ocean. Also consistent with theory, phase lines and velocity vectors are parallel to cg , with phase propagating upward (downward) for downward (upward) extending beams. Finally, note that there is a phase shift of 2π across each beam; this shift is a general property of all Kelvin beams regardless of the specific meridional structure of the wind Y (y), provided that Y rises and falls montonically (Burkhardt 2019; priv. comm.). All four videos exhibit truncation error. In the equilibrium responses of Videos 16.1a–16.1c, the error is confined to the forcing region. In a version of Video 16.1d with N = 25 (not shown), truncation error spreads across the basin; it does so because at P = 180 days the wavelength of the mode-25 K wave, λ25 = Pc25 = 16◦ , “fits” the width of the forcing region L x = 20◦ (i.e., λ25 ≈ L x ) so that the wave is strongly excited (Sect. 13.3.2). In Video 16.1d with N = 99, λ99 = Pc99 = 4◦ L x and truncation error, although still visible, is much reduced. Variable-Nb Solutions: Videos 16.1e–16.1g are similar to Videos 16.1a–16.1d, except solutions use modes determined using the realistic stratification. The solutions also contain beams, but their structure is more complex. Similar to the constantNb solutions, the responses contain “primary” beams that extend directly from the wind-forced region. Since beam slopes, s = ±σ/Nb (z), now depend on z, the primary beams follow curved pathways that are steeper at depth where Nb is smaller. Nevertheless, because the value of Nb for the constant stratification is equal to the depth average of Nb (z), Nb , the bottom and top reflections occur at the same locations x in constant-Nb and variable-Nb videos that have the same P. This property follows from an integration of s −1 across the water column: 0 0   x = −D s −1 dz = (1/σ) −D Nb dz = D Nb /σ . In contrast to the constant-Nb solutions, the variable-Nb ones also contain “secondary” beams. They are generated by multiple reflections of wave energy from the sharp near-surface pycnocline, as indicated by the surface-trapped bands of high velocity, and some of this surface-trapped signal leaks into the subsurface ocean to form secondary beams. One impact of the secondary beams is to blur (broaden) the primary ones, making them harder to identify. Video 16.1h is the variable-Nb version of Video 16.1d with P = 180 days. Beam slopes are so small in this solution that the primary beam is confined to the upper ocean. Note that at any location along the coast, currents rise as time passes. Such surfacing of subsurface currents is observed to occur along many coasts, and is a certain indicator that the signal is a remotely-forced coastal wave that carries surfacegenerated energy downward. Video 16.1i shows an across-shore (x, z) section from the solution at 25◦N, plotting v (shading) and u = (u, w) velocity vectors (arrows).

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The v field for the n > 1 modes is coastally trapped (within the Rossby radius Rn for each mode), and the packet exhibits upward phase propagation. For the n = 1 mode, however, Rossby waves continually radiate offshore in the equilibrium response since 25◦N is equatorward of the critical latitude for the n = 1 mode. Near the surface (z  −100 m), the u field is dominated by Ekman drift, with onshore (offshore) flow occurring during times when τ y is positive (negative).

16.1.2.2

Equatorial Beams

Video groups 16.2, 16.3, and 16.4 illustrate beams of equatorial Kelvin, Yanai, and Rossby waves, respectively, which are forced by periodic, zonal or meridional (for Yanai waves) winds centered on the equator. The responses are more complex than along the coast, because other types of equatorial waves are possible and they can interfere with the beams. Overview The solid curves in Figs. 8.3 and 16.1 show non-dimensional, dispersion curves for equatorial waves with real wavenumbers when P = 15 days (Fig. 8.3) and P = 30, 60, and 180 days (Fig. 16.1, top, middle, and bottom panels). The horizontal dashed lines plot nondimensional frequencies, σn = σ/σ0n = (2π/P) / (βcn )1/2 , for a range of n values. In each panel, the intersection of the nth line with a dispersion curve indicates a possible equatorial wave at period P associated with mode n (see the discussion of Fig. 8.3 for more details). A beam-like response is possible whenever a number of waves of a particular type exist (i.e., when a number of horizontal lines intersect a particular dispersion curve), in which case they form a packet that is describable by wave-group theory. In addition to the waves of interest here (Kelvin, Yanai, and Rossby waves), gravity waves and evanescent waves can also be generated by wind forcing. When P = 15, the existence of gravity waves is apparent in that horizontal lines in Fig. 8.3 with n > 8 intersect gravity-wave curves; at the three longer periods (Fig. 16.1), gravity-wave intersections occur only for n > 32 when P = 30 days and for n > 100 when P = 60 and 180 days. Evanescent waves exist for each j at σ  values that lie between the jth Rossby- and gravity-wave dispersion curves, that is, within the bands, σ R j < σ  < σG j , where the limits are the critical frequencies defined in (8.15). To illustrate, the red-dashed curves in Figs. 8.3 and 16.1 plot the real and imaginary parts of the j = 1 evanescent waves when P = 15 and 30 days, respectively. In the P = 30 panel, for example, all the lines for 1 ≤ n ≤ 32 lie within the j = 1 band and so intersect two evanescent waves for each j, one decaying to the east and the other to the west. For our purposes, gravity and evanescent waves are noise that blur the beam responses. It is always possible to eliminate gravity waves by setting N to a sufficiently small value; for example, no gravity waves exist at P = 30 (60 and 180) days when N = 25 (25 or 99), the N values used in the videos. These values are large enough for solutions to contain enough modes that wave-group theory applies and,

16.1 Beams

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Fig. 16.1 Similar to Fig. 8.3, except highlighting dispersion curves for: the j = 1 (red), 2 (blue), 3 (green), and 4 (magenta) equatorially trapped, Rossby waves. Dispersion curves for the j = 1, gravity (red, top panel) and evanescent (red-dashed, top panel) waves, and for Yanai, and Kelvin waves (solid black curves) are also plotted. The horizontal dashed lines indicate values of σ  for a range of vertical modes n when P = 30 (top), 60 (middle), and 180 (bottom) days

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hence, for packets of Kelvin, Yanai, and Rossby waves to form distinct beams. At P = 15 days, however, gravity waves exist for n > 8 (Fig. 8.3), and it is not possible to choose N such that the solution both lacks gravity waves (N < 8) and has well-formed beams. Regarding evanescent waves, because they exist for moderate values of n, they generally cannot be eliminated by reducing N without significantly impacting the beams. Kelvin-Wave Beams Videos 16.2a–16.2h highlight the generation of equatorial Kelvin beams, plotting u (shading) and u = (u, w) velocity vectors (arrows) along the equator east of the forcing region. Since at a given P equatorial Kelvin waves exist for all vertical modes (all horizontal dashed lines in Figs. 8.3 and 16.1 intersect the Kelvin-wave dispersion curve), they combine to form well-defined beams in the quasi-equilibrium responses. Specifically, transient Kelvin waves associated with different baroclinic modes radiate across the basin, and the Kelvin beam forms after their passage. Given their dynamical similarities, solutions for equatorial Kelvin beams share many properties with those for coastal ones: They follow the same paths, have the same phase structure, and their velocity vectors u are parallel to cg (compare the equatorial videos to their counterparts in Videos 16.1a–16.1h). In the variable-Nb solutions, they also develop secondary beams, due to pycnocline reflections of wave energy. Finally, transient waves exist in all the videos, most obvious in Videos 16.2b– 16.2d where they appear as wave packets with a distinct trailing edge. The equatorial solutions differ from the coastal ones in that waves other than Kelvin waves are also present. At P = 15 days (Videos 16.2a and 16.2e), equatorial gravity waves with odd j values from 9 ≤ j ≤ 25 also exist (Fig. 8.3). (Waves with even j values don’t have the proper symmetry to be excited by the symmetric forcing.) For the constant-Nb solution (Video 16.2a), these waves are so prominent that they eventually completely destroy the beam pattern. For the realistic-Nb solution (Video 16.2e), the beam remains visible owing to the weaker excitation of higher-order modes: Only the near-resonant, j = 1, n = 8 gravity wave has significant amplitude (Fig. 8.3; Sect. 15.4.2). At P = 30, 60, and 180 days, the beams are clear because no gravity waves are possible for the N values used. At P = 30, 60, and 180 days, horizontal lines intersect (or almost intersect) Rossby-wave dispersion curves. At P = 30 days, the σ1 line almost touches the j = 1 Rossby wave curve. As a result, the wind generates two evanescent waves with properties similar to those of the nearby, resonant, n = 1, j = 1 Rossby wave, except that they decay weakly to the east or west (Fig. 16.1; Sect. 15.4.2). The eastwarddecaying wave is visible in the quasi-equilibrium responses of Videos 16.2b and 16.2f as a mode-1 signal with westward phase propagation. (Also visible in Video 16.2b are small-vertical-scale signals, which extend across the basin and exist even near the end of the video. They appear to be noise associated higher-order evanescent waves, generated by truncation error within the forcing region.) At P = 60 days, Rossby waves exist for n ≤ 4, and n = 3 and 4 waves are visible in Videos 16.2c and 16.2g, primarily in the latter.

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Video 16.2i provides an across-equatorial (y-z) section of the beam in Video 16.2h at 90◦E. There is no response until February, 2000, after the arrival of equatorial Kelvin waves from the wind-forced region. Note that there is no meridional velocity at any time (v = 0 in the velocity vectors), consistent with the response being a superposition of Kelvin waves. In the equilibrium response, the upward propagation of the currents is striking. Upward propagation is a commonly observed feature in eastern, equatorial oceans, supporting the existence of a Kelvin beam generated remotely farther west (Sect. 16.1.3). Yanai-Wave Beams Videos 16.3a–16.3f show the along-equatorial responses forced by a meridional wind oscillating at periods P = 15, 30, and 60 days, respectively. Given the numerical model’s grid, solutions plot meridional velocity v (shading) along y = 0 but velocity vectors u = ui + wk (arrows) along y = −y/2, the y value closest to the equator for u and w. (Since Yanai waves are antisymmetric, u = w = 0 at y = 0 but not at y = −y/2, allowing u vectors to be visible in the videos.) As for Kelvin waves, Yanai waves exist for all n values regardless of P (Figs. 8.3 and 16.1), and therefore superpose to form well-defined beams. Further, they follow the same pathways as Kelvin beams (Eq. 16.8) and velocity vectors u are parallel to cg (Eq. 16.15). Because c p · cg = 0, however, their phase lines are not parallel to the group-velocity vector but rather tend to be almost flat, a consequence of the requirement that k ≈ 0 in Yanai-wave beams (see the discussion of Eq. 16.21 below). At P = 15 days and with constant-Nb (Video 16.3a), a Yanai beam initially forms but is later overwhelmed by slower-propagating, near-resonant, j = 2, gravity waves with n ≈ 14 (Fig. 8.3; Sect. 15.4.2). With variable-Nb (Video 16.3d), the beam is visible throughout the video because the gravity-wave response is weak owing to the decrease of Hn−1 with n. At P = 30 days (Videos 16.3b and 16.3e) and 60 days (Videos 16.3c and 16.3f), no gravity waves are present, and Yanai beams are well-formed for both stratifications. At P = 60 days and with N = 99, the vertical wavelength λz of waves in the beam is very small. (In a version of this solution with N = 25, not shown, no beam develops because there aren’t enough vertical modes to resolve their small-scale structure.) Given this small-scale structure, a 60-day Yanai beam (or any Yanai beam with P much longer than 30 days) is not likely to exist in the real ocean, owing to damping by vertical mixing (Miyama et al. 2006). For realistic Nb , the solutions at P = 15 (Video 16.3d) and 30 (Video 16.3e) days show the impact of secondary beams: In the former, they broaden the beams; and in the latter they generate additional signals east of the primary beam due to pycnocline reflections. At P = 60 days (Video 16.3f), however, secondary signals are present west, as well as east, of the primary beam, and they cannot result from pycnocline reflections. Their likely cause is truncation error: N = 99 is still not large enough to resolve the very small-scale, near-surface oscillations within the beam. To illustrate the across-equatorial structure of Yanai beams, Video 16.3g plots u (shading) and v = (v, w) velocity vectors (arrows) for the solution in Video 16.3e (P = 30 days) along a meridional section at 45◦E. At this longitude, the primary beam is descending into the ocean and hence the signal exhibits upward phase propagation.

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Note that u and w are antisymmetric about the equator, consistent with the signal being composed of antisymmetric Yanai waves. The decrease in λz with P is striking. Each beam is composed of a spectrum of vertical modes ψn (z), and the decrease in λz means that the central modenumber n¯ in that spectrum shifts with P towards higher values (Miyama et al. 2006). Recall that one criterion for a wave to be strongly excited is that its zonal wavelength λ is greater than the zonal width of the wind L x or, equivalently, that its wavenumber k is sufficiently small (Sects. 12.3.2 and 15.1.4). For Yanai waves, k = 0 when √ σ0 = σ/ βcn = 1 (Figs. 8.3 and 16.1), and so we expect n¯ to be close to the n value determined by this condition. To illustrate, suppose Nb (z) has a constant value Nb . Then, the characteristic speed of the nth mode is cn = Nb D/ (nπ) and, using √ condition σ = βcn to eliminate cn , gives n¯ =

Nb D Nb D β . = cn π π σ2

(16.21)

At P = 15 days and using the value of Nb in Fig. 5.1 (blue line in the left panel), n¯ = 2.8; consistent with this property, the dominant mode in the beam in Video 16.3a appears to be n = 3, as indicated by the number of zero crossings in the beam from the top to the bottom of the ocean. At P = 30 and 60 days, n¯ = 11.0 and 44.2, and the numbers of zero crossings in Videos 16.3b and 16.3c are 11 and 42, respectively. At P = 180 days, n¯ = 398 so that the vertical scale of the beam is much too small to survive in the real ocean (and for us to attempt to generate a numerical solution). Rossby-Wave Beams Videos 16.4a–16.4f highlight the generation of Rossby beams, plotting u (shading) and u = ui + wk velocity vectors (arrows) along the equator west of the forcing region. Since no Rossby waves are possible when P = 15 (Fig. 8.3) and only one (almost) exists when P = 30 days (Fig. 16.1), the videos show the responses for P = 60, 180 and 365 days. Note that the P = 60- and 180-day videos are westward continuations of corresponding videos for equatorial Kelvin beams. As discussed above, packets of non-dispersive, equatorial Rossby waves superpose to form beams. Their general properties are: lines of constant phase are parallel to the group velocity (Eq. 16.11) but velocity vectors are not (Eq. 16.20); and, provided that X(x) rises and falls monotonically, the phase shift across them is 2π (Burkhardt 2019; priv. comm.). Even though Rossby waves in the exact video solutions are not restricted to be nondispersive, packets of them contain enough nearly-nondispersive Rossby waves to have a beam-like character. At P = 60 days (Videos 16.4a and 16.4d), j = 1 Rossby waves exist for the n = 1–3 vertical modes and nearly exist for the n = 4 mode. With this few number of available waves, a distinct beam structure is not generated. Nevertheless, constructive and destructive interference among them is still sufficient to generate a beam-like pattern with several bottom and surface reflections; it is most visible near the end of the video, after many of the short-vertical-scale transient waves have crossed the

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basin. Although j = 3 Rossby waves for n = 1 and 2 also have the correct symmetry to be excited by the wind, they are not visible in either of the videos. At P = 180 days (Videos 16.4b and 16.4e), j = 1 Rossby waves exist for the n = 1–34 vertical modes, and so they interfere to create a well-defined beam. Its structure is similar to that of a Kelvin beam at this period, except that the beam extends west of the forcing region and along ray paths at steeper angles, θ = ±(σ/Nb )/3. Also visible is another, weaker, Rossby beam composed of j = 3 Rossby waves, which propagates along ray paths with the angles, θ = ±(σ/Nb )/7. With constant Nb (Video 16.4b), beams consist of a primary one plus a series of secondary ones at slightly steeper angles, owing to the contributions of dispersive Rossby waves: In the bottom panel of Fig. 16.1, most horizontal lines intersect the dispersion curves where they have significant curvature. In addition, transient Rossby waves are apparent throughout the video. There are two distinct packets consisting of j = 1 and j = 3 Rossby waves: The trailing edge of the former passes through the western edge of video during March, 2007, and that of the latter is located near 45◦E at the end of the video. At P = 365 days (Videos 16.4c and 16.4f), the beams are similar to those in the 180-day solutions, except the j = 1 and 3 beams extend along pathways at half the slope. In addition, there are hints of weak, j = 5 and 7 Rossby beams and their bottom reflections, extending along ray paths with angles, θ = ±(σ/Nb )/11 and ±(σ/Nb )/15. In the constant Nb solution (Video 16.4c), the primary beam is clear with no secondary ones, because it is composed almost entirely of non-dispersive Rossby waves: At P = 365 days, horizontal lines in the bottom panel of Fig. 16.1 are shifted downward by a factor of two, and then the n ≤ 25 lines intersect the dispersion curves where they have little curvature. Video 16.4g provides an across-equatorial section at 40◦E, showing u (shading) and v = (v, w) velocity vectors (arrows) for the solution in Video 16.4e (P = 180 days). Since the forcing is by a symmetric zonal wind, u and w are symmetric about the equator and v is antisymmetric. At 40◦E, the response has three parts: a j = 1 Rossby beam, descending into the ocean with upward phase propagation; a j = 3 beam, which has arrived back near the surface after reflecting from the bottom; and a surface-trapped signal due to pycnocline reflections. For the j = 1 beam, u has two zero crossings at each depth, because u n for each of the Rossby waves in the beam involves a second-order Hermite function φ2 (Figs. 8.1 and 8.2). Similarly, the j = 3 beam, visible in the upper 1000 m later in the video, has four zero crossings since u n involves φ4 (Fig. 8.1).

16.1.2.3

Bounded Basins

To illustrate boundary effects, Videos 16.5a–16.5d show solutions forced by a zonal wind in mid-basin oscillating at P = 180 days, when the basin is unbounded and has an eastern boundary, western boundary, and both boundaries, respectively. When the basin includes a closed western boundary, the simplified form of horizontal viscosity in (C.4) is included near the western boundary so that the western-boundary current

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has a Munk-like structure that doesn’t radiate offshore to interfere with the interior flow field (Sects. 11.2.1 and 13.2.3.2). As a reference solution, Video 16.5a (a combination of Videos 16.2h and 16.4e) shows the along-equatorial response without any boundaries, and it has no deep flow in the eastern half of the basin. When an eastern boundary is included (Video 16.5b), the wind-generated Kelvin beam in Video 16.5a reflects from the eastern boundary, filling the eastern basin with waves. The reflection of each wave within the Kelvin beam generates a chain of j Rossby waves for each n (Sect. 14.3.1). Rossby waves associated with a specific j value then superpose to form beams with the same properties noted above for Rossby beams generated directly by wind forcing. In the quasi-equilibrium response, the boundary-reflected, j = 1 beam is clear, reflecting from the bottom in mid-basin. The j = 3 beam is also visible, reflecting from the bottom near 85◦E and the top near 65◦E. There are also indications of Rossby beams for j ≥ 5, which reach the bottom at longitudes east of 85◦E. In the western ocean, these beams interfere with the Rossby beams directly generated by the wind (Video 16.5a). With just a western boundary (Video 16.5c), the Rossby beams for each j in Video 16.5a reflect from the western boundary. Each mode n in the jth Rossby beam generates a chain of boundary waves that ends in a mode-n, reflected Kelvin wave, and the n Kelvin waves superpose to form a Kelvin beam that extends eastward across the basin. The Kelvin beam generated by the j = 1 Rossby beam is most visible in the video. It is vertically thick because the j = 1 Rossby beam that generates it is also thick, and it reflects from the bottom in the eastern ocean. With both eastern and western boundaries, Videos 16.5d, 16.5e, and 16.5f show solutions along and across the equator, the latter at x = 60◦E and 90◦E. Given the relative weakness of the western-boundary-reflected Kelvin beams in Video 16.5c, the along-equator flow field (Video 16.5d) is similar to that in Video 16.5b. In both meridional sections, the equatorial flow field spins up in the same manner, with upwardpropagating signals associated with the arrival of eastern-boundary-reflected Rossby waves. At later times, the deep j = 1 Rossby beam is prominent at 60◦E (Video 16.5e), but there are so many interfering beams at 90◦E (Video 16.5f) that individual ones are difficult (impossible) to detect. At both longitudes, the off-equatorial signals are predominantly associated with the n = 1 mode. Interestingly, they propagate in different directions: equatorward at 90◦E and poleward at 60◦E. The cause of this interesting difference is evident in Video 15.2d, which plots the n = 1 response: It occurs because the two sections are in very different positions relative to the focal point (Sect. 15.2.3).

16.1.3 Observations Given data scarcity, it is difficult to observe beams in their entirety. In the equatorial Pacific Ocean, Kessler and McCreary (1993) were able to do that using all hydrographic data from 10◦ S–10◦ N, from which they constructed a climatological

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annual cycle of temperature anomalies T . Remarkably the annual,  = 1 Rossby beam was visible in the climatology and, based on solutions to their LCS model, the observed beam followed predicted pathways. In the Indian Ocean, however, efforts to identify beams are based on observations from sparsely-distributed current meters, and the presence of beams can only be inferred indirectly. Coastal beams: Along both the east and west Indian coasts, vertically-propagating signals are present in data from individual current meters data (e.g., Amol et al. 2014; Mukherjee et al. 2014). Such signals are clear indicators of remotely-forced signals and, hence, support the existence of beams. Based on this data, Amol and Mukherjee (2015, priv. comm.) attempted to trace beams along both coasts with mixed results. Although they were unable to find any beams along the east coast, they could find evidence for west-coast beams at the annual cycle. Video 4.5 shows alongshore currents at the four west-coast moorings in Fig. 4.26, the data filtered to show the annual cycle. Note that the currents at three of the sites (Kanyakumari, 7◦ N; Goa, 9◦N; Mumbai, 20◦N) exhibit vertical propagation throughout the year, and that signals at more southern locations are shallower than those to the north. This variation of current amplitude and phase with depth and latitude roughly matches the predicted Kelvin ray path. The researchers concluded that detection of coastal beams is possible only if the remote forcing that generated the beam is significantly stronger than local forcing along the ray path. Regarding beams at higher (intraseasonal) frequencies, they have larger beam angles and so can only be detected by moorings that are not as far apart as in Fig. 4.26 and that cover the entire water column. Interestingly, coastal signals are present with both downward, as well as upward, phase propagation. They are particularly interesting since downward phase propagation indicates that their energy is propagating upwards from a subsurface source. In a recent modelling study, Dhage (2020) demonstrated that such signals might arise from the reflection of a horizontally-propagating Kelvin wave (or coastally-trapped wave) from a subsurface ridge in the form of an upward-propagating beam. Equatorial beams: Vertically-propagating signals have also been observed along the Indian-Ocean equator. In the western Indian Ocean, Luyten and Roemmich’s (1982) observations of upward-propagating, semiannual signals are likely due to a semiannual,  = 1 Rossby beam (Sect. 4.4.3). In the eastern ocean, Masumoto et al. (2005) reported upward phase propagation in the equatorial u field that was confined to the upper ocean, which were likely caused by a Kelvin beam arriving from the central basin (Video 16.5a). Recently, Huang et al. (2018b) analyzed currentmeter data from moorings at 83◦E and 90◦E, as well as an ocean-reanalysis product (ORAS4), concluding that much of the mid-depth signals at the current meters was generated by the arrival an  = 1 Rossby beam, which had been generated by the reflection of a Kelvin beam at the eastern boundary. Amol et al. (2021) analyzed bottom currents in current-meter observations at 77◦E, 83◦E, and 93◦E, finding that their dominant frequency increased from west to east. They interpreted this shift as indicating the presence of several eastern-boundary-generated Rossby beams: Beams with higher frequencies propagate away from the eastern boundary at steeper angles, and so arrive at the bottom closer to the boundary (as seen in Videos 16.5b and 16.5d).

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16.2 Undercurrents Undercurrents are prominent features of both equatorial and coastal circulations. They flow in a direction opposite to that of the surface current, and typically lie in a depth range from 100–300 m. In regions of the world ocean where the forcing wind has a strong mean component, undercurrents exist throughout the year. In the NIO, where the winds are dominated by annual and semiannual cycles, they are not permanent features. Here, we discuss properties of steady-state undercurrents present in the LCS model, including the dynamical adjustments that lead to them. We expect that similar dynamics apply to low-frequency (nearly quasi-steady) undercurrents in the NIO.

16.2.1 Conceptual Explanations Simple explanations, based on physical principles rather than ocean models, have been proposed for the existence of steady undercurrents. Despite their neglect of process details, they are essentially correct and so provide a useful, lowest-order description of undercurrent dynamics. Cromwell (1953) proposed such an explanation for the Pacific EUC. He argued that the easterly trades drive westward equatorial flow within the surface, wind-driven layer. As a result, surface water piles up in the western ocean, generating an eastward pressure force along the equator that balances the wind (as illustrated by the videos discussed in Sect. 14.2.2.5). Owing to mixing, some of that force still exists below the surface layer, providing a source of momentum to drive the eastward-flowing EUC. Cromwell (1953) further suggested that midlatitude Ekman theory could be applied near the equator (see Sect. 10.2), in which case easterly winds on either side of the equator drive a divergence of surface water there, causing equatorial upwelling. That upwelling raises subsurface water to the surface layer, thereby providing a sink for the eastward-flowing EUC and a source for the westward-flowing surface current. Similar ideas provide an explanation for coastal undercurrents (CUCs) along zonal and eastern boundaries. For example, consider an eastern coast in the northern hemisphere forced by northerly winds. The winds drive offshore Ekman drift, leading to an equatorward coastal current in the surface wind-driven layer, and a poleward, pressure-gradient force along the coast that balances the wind (see the discussion of Eq. 13.13); with mixing, part of that force extends below the surface layer to drive a poleward-flowing CUC. Further, the coastal-wind divergence due to the offshore Ekman drift can be strong enough to cause upwelling (discussion of Fig. 13.2), allowing some CUC water to upwell into the surface layer to join the equatorward-flowing surface flow or the offshore Ekman drift. One might expect the same explanation to result in an undercurrent along a western-ocean coast but it doesn’t (see below). Another useful interpretation of undercurrents is that they are simply beams when σ → 0. In this limit, beam angles tend to zero (Eq. 16.9) and so beams cause surface-

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trapped flows. Such “steady” beams, however, are infinitesimally thin (a δ-function in z) since their bottom and top edges overlap. Some other process, such as vertical mixing, is required to broaden them into the subsurface ocean.

16.2.2 Dynamics Ocean models are able to represent and extend the above conceptual ideas within a mathematical framework. In OGCM solutions, undercurrent dynamics are typically analyzed by considering balances of terms in the equations of motion. The LCS model provides an alternate way of thinking about undercurrent dynamics, by considering how the balance of terms varies with individual vertical modes rather than by location. For an OGCM, consider such an analysis for either a steady EUC that is symmetric about the equator or a steady CUC along a zonally-oriented coast. In either case, along the boundary (equator or coast), v = u t = 0 and, neglecting horizontal mixing (νh = 0), the zonal-momentum equation (5.1a) simplifies to uu x + wu z +

1 px = (νu z )z . ρ¯

(16.22)

(Horizontal mixing is typically small in solutions along all boundaries except western coasts.) Generally, all the terms in (16.22) are important somewhere, with the overall balance among them changing with longitude (e.g., Wacongne 1989). Under the same restrictions as above, the along-boundary, zonal-momentum equation for the LCS model (5.16a) simplifies to pnx =

τx − γn u n . ρH ¯ n

(16.23a)

Realistic undercurrent structures only occur when the equations for each mode include vertical mixing (γn = A/cn2 and γn = A /cn2 ), which strengthens with n as cn decreases (McCreary 1981a, b). Further, they exist only when the mixing coefficients, A and A , are set so that damping is negligible for the low-order modes. In that case, the mode balances differ markedly depending on whether the value of n is high, low, or intermediate. For high-order modes (n  8 or so in the following videos), damping is large ¯ n ). and their responses are essentially in the local, 2-d balance (γn u n = τ x /ρH Contributions from these modes superpose to generate the wind-driven upwelling/downwelling (v,w) circulation in the overall response, and their alongboundary flow u n is weak because γn is large. Conversely, for the lowest-order modes (n ≈ 1–3) damping is weak enough for their responses to adjust close to the ¯ n ); for these modes, pn has a zonal tilt that inviscid Sverdrup balance ( pnx = τ x /ρH balances the forcing and u n ≈ 0. This low-mode adjustment is crucial, as it limits the amplitude of the boundary currents: In a solution to an x-independent model

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in which all the modes are in the above 2-d balance noted above, boundary currents are unrealistically large by orders of magnitude, because γn is so small for the low-order modes (McCreary 1981a, b). Since u n is weak in both extreme balances, along-boundary currents are generated mostly by contributions from intermediate modes (n ≈ 3–8); the vertical scales for these modes is 200–300 m, thereby setting the depth scale of the undercurrent. Along meridional coasts, the along-boundary momentum equation for the LCS model is τy − γn vn . (16.23b) pny = ρH ¯ n ¯ n ), as in For high-order modes, the balance of terms in (16.23b) is 2-d (γn vn = τ y /ρH the previous paragraph. For low-order modes, however, the balance differs between eastern and western coasts. Along an eastern coast, the offshore propagation of loworder, long-wavelength Rossby waves weakens the alongshore current vn , allowing ¯ n ) for these modes. In contrast, the Sverdrup balance to be established ( pny = τ y /ρH along a western boundary the propagation of low-order, short-wavelength Rossby waves does not weaken vn , and the coastal response adjusts to a balance in which all ¯ n ) (see the derivation of Eq. 13.32a); three terms are important, ( pn /y) y = τ y / ( f ρH further, the coastal circulation lacks a CUC (discussed further below). The LCS model highlights the importance of two linear processes in undercurrent dynamics: vertical mixing and adjustment to Sverdrup balance. An issue with this dynamics, however, is that mixing coefficients must be set to values that are arguably too large for modeled undercurrents to compare favorably with observations (Benthuysen et al. 2014). In effect, the mixing compensates for neglected processes, such as the nonlinear terms in (16.22). Nonlinearites not only impact the strength and structure of boundary currents, they can destabilize them to generate smaller-scale features: squirts, jets, and eddies near coasts (e.g., Kosro 1987; Brink 1987; Narimousa and Maxworthy 1989; McCreary et al. 1991; Fukamachi et al. 1995), and Tropical Instability Waves along the equator (e.g., Legeckis 1986; Philander 1976, 1978a; Yu 1992; Yu et al. 1995). Further, they can reverse the propagation direction of Rossby waves, thereby preventing eastern-boundary currents from propagating offshore (McCreary et al. 1992). Along coasts, boundary currents are also significantly impacted by continental shelves. A discussion of these processes is beyond the scope of this book.

16.2.3 Videos The following videos illustrate the spin-up of CUCs and the EUC forced by switchedon winds. The solutions are sums of N = 25 baroclinic modes obtained using the realistic Nb (z) plotted in Fig. 5.1, which ensures their convergence. In all the solutions, the Coriolis parameter is f = β y. To isolate the impacts of damping, solutions are found both without and with vertical mixing.

16.2 Undercurrents

16.2.3.1

433

Coastal Undercurrents

Video groups 16.6 and 16.7 show coastal solutions generated along eastern and western boundaries. All solutions are forced by switched-on, x-independent bands of winds: northerly and confined from 15–35◦N for the eastern coast; southerly and confined from 25–45◦N for the western coast. As for the coastal-beam videos (Sect. 16.1.2.1), videos of the alongshore flow field continue along the adjacent zonal boundary, that is, eastern (western) boundary frames continue along the northern (southern) boundary. East coast: Videos 16.6a and 16.6b illustrate the inviscid response (νh = A = A = 0) near an eastern/northern coast. During the first half year, a southward, accelerating alongshore current develops in the direction of the wind (Video 16.6a). As Kelvin waves propagate poleward along the coast, an alongshore pressure gradient develops to balance τ y , the surface current stops accelerating, and a CUC appears at depth. As it strengthens, the CUC rises in the water column, a consequence of coastal adjustments associated with slower-propagating, higher-order-baroclinic Kelvin waves. In the across-shore section of the solution at 25◦N (Video 16.6b), the offshore radiation of the n = 1 Rossby wave (no zero crossing in the upper 1000 m) from the coast is clear. After this initial growth period, the coastal circulation weakens near the eastern boundary throughout the rest of the videos, owing to the continual offshore propagation of higher-order Rossby waves: The n = 1 Rossby wave is followed by the n = 2 wave (one zero crossing), the n = 3 wave (two zero crossings), and higher-order Rossby waves are visible trailing behind (Video 16.6b). Near the end of the Video 16.6b (3 years) a coastal circulation still remains, consisting of several weak bands extending offshore; however, as time passes they weaken further as higher-mode Rossby waves propagate offshore. Along the northern boundary the coastal currents remain large (Video 16.6a). Eventually, they will also be eliminated by Rossby-wave propagation from the eastern boundary; however, because the Rossby-wave speed is so slow there (50◦N), even at the end of the video they are weakened only very near the northeast corner. Videos 16.6c and 16.6d are comparable to Videos 16.6a and 16.6b, except that mixing is included with A = A = 1.3×10−3 cm2 /s3 , the value chosen to ensure the CUC develops a realistic vertical structure. The spin-up of the coastal currents is initially the same as for the inviscid solution; however, subsequently higher-order Rossby waves are damped before they can propagate offshore. As a result, the coastal circulation does not continuously weaken, but rather adjusts to a steady state with a surface jet in the direction of the wind, an opposing southward CUC, and a weak northward flow at greater depths. In the latitude band of the wind, there is a subsurface, onshore, geostrophic flow generated by the damped Rossby waves (Video 16.6d); at the coast, part of this inflow upwells to feed the offshore Ekman drift, another part bends northward to supply water for the CUC, and the rest supplies water for weak coastal downwelling below about 150 m. The coastal currents extend well beyond the northern edge of the forcing band (35◦N), owing to Kelvin-wave propagation

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(Video 16.6c). There is still a jump in the response at the corner, owing to the slow Rossby-wave adjustment along the northern boundary. As noted in Sect. 13.2.3.1, a fundamental issue about eastern-boundary currents is that, given the offshore propagation of Rossby waves, why they exist at all. The answer explored here for the CUC is Rossby-wave damping by vertical diffusion (McCreary 1981b; Philander and Yoon 1982), which damps higher-order Rossby waves before they can do so. On the other hand, the strength of the diffusion imposed here is arguably too strong (Benthuysen et al. 2014). If so, other processes, such as nonlinearities (McCreary et al. 1992) and a continental shelf (Csanady 1978; Weaver and Middleton 1989, 1990; Furue et al. 2013), must also contribute to easternboundary trapping. West coast: Videos 16.7a and 16.7b are the western-boundary counterparts of Videos 16.6c and 16.6d. Because coastal Kelvin waves propagate anticlockwise around the basin perimeter, they show the responses first along the western boundary with the y-axis reversed (left half) and then along the southern boundary (right half). Given the y-axis reversal, the plotted variables along the western boundary are −v (shading) and v = (−v, w) velocity vectors; along the southern boundary, they are u (shading) and u = (u, w) vectors. Video 16.6d shows the across-shore response at y = 35◦N. The solutions all include damping with A = A = 1.3×10−3 cm2 /s3 and horizontal viscosity with νh = 5×106 cm2 /s, the latter ensuring the westernboundary current adjusts to have a Munk-layer structure. Corresponding videos with A = A = 0 (not shown) are almost unchanged from Videos 16.7a and 16.7b, owing to the dominance of the horizontal viscosity. In contrast to the eastern-coast solution, the western-coast one has no CUC. Further, the vertical structure of its coastal jet is close to that of the wind forcing Z(z), and is exactly Z(z) when A = A = 0. These surprising properties happen at a western boundary because low-order modes also contribute to the coastal currents, since Rossby waves do not carry them offshore. McCreary and Kundu (1985) suggested that western-boundary undercurrents, such as those observed along the Somali, Omani, and east-Indian coasts, must be caused by the arrival of offshore-generated Rossby waves, rather than forcing by alongshore winds. McCreary et al. (1993) traced the origin of observed Somali undercurrents to Rossby waves that radiate from the west coast of India or are generated by Ekman pumping in the interior of the Arabian Sea. The onshore-propagation of Rossby waves has an even more surprising effect: They can weaken or even eliminate western-coastal upwelling (Chatterjee et al. 2019). See Sect. 17.3.3 for a discussion of the dynamics of this phenomenon.

16.2.3.2

Equatorial Undercurrent

Video group 16.8 shows solutions forced by switched-on, equatorial easterlies τ x in the central ocean (40◦E < y < 80◦E). To highlight the near-equatorial response driven by τ x rather than wind curl −τ yx , Y (y) of the wind is given by (16.24), which is constant within 10◦ of the equator.

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Video 16.8a shows the solution in an unbounded basin and without mixing (νh = A = A = 0), plotting an equatorial (x, z) section of u (shading) and u = (u, w) velocity vectors (arrows). Initially, a Yoshida Jet accelerates at the ocean surface (10.2.1), there is surface upwelling driven by divergent Ekman drift in the mixed layer, and there are prominent inertial oscillations that extend throughout the water column. As Kelvin and Rossby waves propagate across the forcing region, a nearsurface pressure gradient is established that stops the Yoshida Jet from accelerating to form a bounded Yoshida Jet (14.2.2.2), the subsurface part of which is the EUC. As time passes, the radiation of slower-propagating, higher-order waves continually thins and weakens the EUC. At the end of the video, the response is nearly xindependent, with a strong surface jet, a weak EUC, and weaker small-vertical-scale currents at greater depth, the latter due to truncation error. Recall that, in the steadystate analytic version of this numerical solution, all the flow is confined to the winddriven, surface layer (Sect. 10.2.2), an ideal state the numerical solution cannot attain owing to the finite integration time and truncation error. Video 16.8b shows a solution like Video 16.8a, except that it includes vertical mixing with A = A = 2.6×10−4 cm2 /s3 . The initial response is the same as in Video 16.8a. Subsequently, though, it adjusts to a steady state with a well-defined EUC that has a thickness of 100–150 m, a consequence of the damping of the radiation associated with higher-order modes. Note the dynamical similarity to the easternboundary CUC: In both cases, a steady-state undercurrent exists only due to the damping of radiation. Video 16.8c shows a solution like Video 16.8b, except with a closed eastern boundary. After the arrival of wind-driven Kelvin waves, Rossby waves associated with different vertical and meridional modes reflect from the eastern boundary, generating a sequence of westward-propagating pulses in the interior ocean. Collectively, the reflected waves weaken the surface jet and produce a westward current beneath the EUC, the model’s Equatorial Intermediate Current (EIC). Video 16.8d is similar to Video 16.8c, except the solution is found in a basin with both an eastern and western boundary, and mixing (C.4) is included so that the quasi-steady, western-boundary current adjusts to have a Munk-like layer. A primary impact of the western boundary is to reflect incoming Rossby waves as an equatorial Kelvin wave, which further weakens the EUC and surface jet. A secondary one is that some transient, short-wavelength Rossby waves escape from the westernboundary damping region and, in addition to inertial oscillations, they contribute to time-dependent “noise” in the interior flow. Video 16.8e provides a meridional (y, z) section of this solution, showing u (shading) and u = (v, w) vectors at x = 60◦E. After the inertial oscillations settle down, a near-surface overturning circulation remains, the Tropical Cell (TC), consisting of: equatorward, geostrophic flow into the core of the EUC; equatorial upwelling; poleward drift at the ocean surface; and off-equatorial downwelling confined to the tropics. The development of the EIC is also visible in Video 16.8e: It begins to develop in May, 2000, after the arrival of the fastest-propagating, eastern-boundary-reflected Rossby wave (n = 1, j = 1), and thereafter its structure changes with the arrival of the slower-propagating ones.

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In the real ocean, although some of the upwelled water participates in a TC, most of it arises from subduction in the subtropics, and is associated with a much broader, overturning circulation, the STC (Chap. 17). The dominance of the TC in Video 16.8e points toward a deficiency in the representation of vertical diffusion of the form, (κρz )z , in the LCS model (Eq. 5.6e): Because it depends on the density anomaly ρ rather than the total density, surface water cools as it sinks, thereby allowing tropical downwelling that is unrealistically strong (McCreary 1981a).

16.2.4 Observations Do undercurrents exist in the NIO? The answer depends on just how they are defined. The undercurrents illustrated above are steady-state, near-surface flows, which require mixing to develop a realistic vertical structure. Under this strict dynamical definition, and given the prominent wind variability, undercurrents are rare or non-existent in the NIO. On the other hand, an alternate and commonly-used definition is that undercurrents are any near-surface current (vertical extent of several hundred meters) that flows in a direction opposite to an overlying surface current (e.g., Sect. 4.4.3). Under this relaxed definition, NIO undercurrents are common. Indeed, undercurrents of the latter sort develop in response to most of the timedependent forcings considered in this book: For switched-on winds, they develop in the time it takes Kelvin waves to cross the forcing region of the wind (e.g., Videos 16.6a and 16.8a); for periodic winds, there is always a reverse flow just underneath and nearby the directly wind-driven surface one (e.g., Videos 16.1e–16.1h, and Videos 16.2e–16.2h). Most of the observed NIO undercurrents can be explained in this way, the exception being those along western boundaries where CUCs cannot be generated by low-frequency local forcing (Video 16.7a).

Video Captions Beams Coastal Beams Video 16.1a Multi-mode (N = 25) solution near an eastern/northern coast, using vertical modes for the constant stratification shown in Fig. 5.1 (Nb = 2.2237×10−3 s−1 ). The solution is forced by a switched-on, oscillating meridional wind of the y y form τ y = τo Y (y) sin(σt)θ(t), where: σ = 2π/P, P = 15 days; τo = 1.5 dyn/cm2 ; ◦ ◦ and Y (y) is given by (C.7b) with ym = 20 N and y = 10 . Along the eastern boundary, the video plots v (shading) and v = (v, w) velocity vectors (arrows); since the x-axis is reversed along the northern boundary, the plotted variables there

Video Captions

437

are −u and v = (−u, w). There is no horizontal viscosity or damping (νh = A = 0). The Coriolis parameter is specified by the mid-latitude β-plane approximation. The eastern boundary is closed, and open conditions are imposed at the western boundary as described in Appendix C. Video 16.1b As in Video 16.1a, except with P = 30 days. Video 16.1c As in Video 16.1a, except with P = 60 days. Video 16.1d As in Video 16.1a, except with P = 180 days, N = 100, ym = 25◦N, and y = 20◦ . With y = 10◦ the beam is very thin, and the increase in y allows the beam’s structure to be seen more easily. Video 16.1e As in Video 16.1a with P = 15 days, except using vertical modes determined from the realistic stratification Nb (z) shown in Fig. 5.1. Video 16.1f As in Video 16.1e, except with P = 30 days. Video 16.1g As in Video 16.1e, except with P = 60 days. Video 16.1h As in Video 16.1e, except with P = 180 days, ym = 25◦N, and y = 20◦ . Video 16.1i As in Video 16.1h, except showing an across-shore (x, z) section at 25◦N.

Equatorial Kelvin-Wave Beams Video 16.2a Multi-mode (N = 25) response along the equator, using vertical modes for the realistic stratification shown in Fig. 5.1. The solution is forced by a switchedon, oscillating zonal wind τ x = τox X (x)Y (y) sin(σt)θ (t), where: σ = 2π/P, P = 15 days; τox = 1.5 dyn/cm2 ; and X(x) and Y (y) are given by (C.7b) with x = 10◦ , y = 30◦ , and x m = (30◦E, 0◦ ). The video plots an equatorial (x, z) section of u (shading) and (u, w) vectors. There is no horizontal mixing or damping (νh = A = 0). The Coriolis parameter is specified by the equatorial β-plane approximation. Open conditions are imposed at the eastern and western boundaries as described in Appendix C. Video 16.2b As in Video 16.2a, except with P = 30 days. Video 16.2c As in Video 16.2a, except with P = 60 days. Video 16.2d As in Video 16.2a, except with P = 180 days, N = 99, and x = 20◦ . With x = 10◦ the beam is very thin, and the increase in x allows the beam’s structure to be seen more easily. Video 16.2e As in Video 16.2a with P = 15 days, except using vertical modes determined from the realistic stratification Nb (z) shown in Fig. 5.1.

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Video 16.2f As in Video 16.2e, except with P = 30 days. Video 16.2g As in Video 16.2e, except with P = 60 days. Video 16.2h As in Video 16.2e, except with P = 180 days, and the forcing width is increased to x = 20◦ . Video 16.2i As in Video 16.2h, except showing an across-equatorial section at 90◦E.

Yanai-Wave Beams Video 16.3a As in Video 16.2a, except showing the multi-mode (N = 25) response forced by a switched-on, oscillating meridional wind, τy = y y 2 τo X(x)Y(y) sin(σt)θ(t), with σ = 2π/P, P = 15 days, τo = 1.5 dyn/cm . The video shows an equatorial section of v and u = (u, w) vectors. Given the model’s grid, v is plotted along y = 0 and u along y = −y/2. Video 16.3b As in Video 16.3a, except with P = 30 days. Video 16.3c As in Video 16.3a, except that P = 60 days, N = 99, and the forcing width is increased to x = 20◦ . Video 16.3d As in Video 16.3a with P = 15 days, except using vertical modes determined from the realistic stratification Nb (z) shown in Fig. 5.1. Video 16.3e As in Video 16.3d, except with P = 30 days. Video 16.3f As in Video 16.3d, except with P = 60 days and x = 20◦ . Video 16.3g As in Video 16.3e, except showing an across-equator section at 45◦E.

Equatorial Rossby-Wave Beams Video 16.4a As in Video 16.2a, except with P = 30 days and xm = 90◦E. Video 16.4b As in Video 16.4a, except with P = 60 days. Video 16.4c As in Video 16.4a, except with P = 180 days and x = 20◦ . Video 16.4d As in Video 16.4a with P = 30 days, except using vertical modes determined from the realistic stratification Nb (z) shown in Fig. 5.1. Video 16.4e As in Video 16.3c, except with P = 60 days. Video 16.4f As in Video 16.3c, except with P = 180 days. Video 16.4g As in Video 16.4f, except showing an across-equatorial section at 40◦E.

Video Captions

439

Bounded Basins Video 16.5a As in Video 16.2a, except with P = 180 days, x m = (60◦E, 0◦ ), and x = 20◦ . Video 16.5b As in Video 16.5a, except with a closed eastern boundary. Video 16.5c As in Video 16.5a, except with a closed western boundary. Video 16.5d As in Video 16.5a, except with closed eastern and western boundaries. Video 16.5e As in Video 16.5d, except showing an across-equator section at 90◦E. Video 16.5f As in Video 16.5d, except showing an across-equator section at 90◦E.

Undercurrents Coastal Undercurrent Video 16.6a Multi-mode (N = 25) response near an eastern/northern coast for the realistic stratification shown in Fig. 5.1. The solution is forced by a switched-on y y meridional wind of the form τ y = τo Y(y)θ(t), where: τo = −1.5 dyn/cm2 ; and Y(y) is given by (C.7b) with ym = 25◦N and y = 20◦ . Along the eastern boundary, the video plots v (shading) and v = (v, w) velocity vectors (arrows); since the x-axis is reversed along the northern boundary, the plotted variables there are −u and v = (−u, w). There is no horizontal viscosity or damping (νh = A = 0). The Coriolis parameter is specified by the mid-latitude β-plane approximation. The eastern boundary is closed, and open conditions are imposed at the western boundary as described in Appendix C. Video 16.6b As in Video 16.6a, except showing an across-shore (x, z) section of v (shading) and (u, w) vectors at 25◦N. Video 16.6c As in Video 16.6a, except that A = 1.3×10−3 cm2 /s3 . Video 16.6d As in Video 16.6c, except showing an across-shore (x, z) section at of v (shading) and (u, w) vectors at 25◦N. Video 16.7a As in Video 16.6a, except showing the response near a western coast when ym = 35◦N and with horizontal mixing (νh = 5×106 cm2 /s). Along the southern boundary, the video plots u (shading) and u = (u, w) velocity vectors (arrows); since the y-axis is reversed along the western boundary, the plotted variables there are −v and v = (−v, w). There is damping with A = 1.3×10−3 cm2 /s3 . The Coriolis parameter is specified by the mid-latitude β-plane approximation. The western boundary is closed, and open conditions are imposed as described in Appendix C. Video 16.7b As in Video 16.7a, except showing an across-shore (x, z) section of v (shading) and (u, w) vectors at 35◦N. Note that the sign of v is not reversed, as it is in Video 16.7a.

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Equatorial Undercurrent Video 16.8a Multi-mode (N = 25) response along the equator, using vertical modes for the realistic stratification shown in Fig. 5.1. The solution is forced by a switchedon, zonal wind τ x = τox X(x)Y(y)θ (t), where: τox = −1.5 dyn/cm2 ; X(x) is given by (C.7b) with x = 40◦ and xm = 60◦ ; and ⎧   1 y − ym ⎪ ⎪ 1 + cos 2π , ⎨ y Y (y) = 2 |y| < ym 1, ⎪ ⎪ ⎩ 0, otherwise,

ym < |y| < ym +

y 2

(16.24)

where ym = 10◦ and y = 20◦ . According to (16.24), Y (y) is constant in a band from the equator to ±ym , and it decays sinusoidally to zero in a distance y/2 outside that band. The video shows an equatorial (x, z) section of u (shading) and currents (u, w) vectors. There is no mixing (νh = A = A = 0). The Coriolis parameter is given by the equatorial β-plane approximation. Open conditions are imposed at the eastern and western boundaries as described in Appendix C. Video 16.8b As in Video 16.8a, except with vertical mixing (A = A = 2.6×10−4 cm3 /s2 ). Video 16.8c As in Video 16.8b, except with a closed eastern boundary. Video 16.8d As in Video 16.8b, except with closed eastern and western boundaries. Video 16.8e As in Video 16.8d, except showing a meridional (y, z) section of zonal velocity u (shading) and currents v and w (arrows) at x = 60◦E.

Chapter 17

Cross-Equatorial and Subtropical Cells

Abstract The Cross Equatorial (CEC) and Subtropical (STC) Cells are the major shallow-overturning cells of the Indian Ocean, corresponding to the North and South STCs in the Pacific and Atlantic Oceans. Their meridional (overturning) structure is illustrated by two-dimensional (y, z) streamfunction plots from two solutions to ocean general circulation models (OGCMs). The plots show that the cells’ sinking branches are located in the southern hemisphere, whereas their upwelling branches are located in the southern hemisphere along the South Equatorial Thermocline Ridge for the STC, and in the northern hemisphere for the CEC. The cells’ three-dimensional (x, y, z) structures are determined by following pathways of model drifters in solutions to two layer models and an OGCM. Transports of the cell branches in each solution are compared to each other and to available observations. A suite of solutions to an idealized two-layer, reduced-gravity model (a 2 21 -layer model) is used to isolate the basic processes (wind forcing, upwelling, and detrainment) that generate the cells. In addition to the CEC and STC, the solutions have an equatorial “roll,” a small-scale overturning feature confined to the upper 50–100 m within a few degrees of the equator; its dynamics are discussed, and observational support for its existence noted. Keywords Shallow overturning cells · Cross equatorial cell · Subtropical cell · Upwelling and subduction · Two-dimensional streamfunctions · Three-dimensional pathways · β-plume · Equatorial roll In this final chapter, we discuss the structure and dynamics of the Indian Ocean’s shallow overturning circulations: the CEC and STC. It is a fitting way to end the book, as the cells encompass the entire Indian Ocean and their dynamics involve (and extend) many of the processes considered in earlier chapters. We begin with an overview of the cells, comparing them to their counterparts in the Pacific and Atlantic Oceans and noting their impacts (Sect. 17.1). Given the historical scarcity of observations, analyses of solutions to ocean models have played Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/978-981-19-5864-9_17. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9_17

441

442

17 Cross-Equatorial and Subtropical Cells

a prominent role in developing a complete picture of the CEC and STC circulations. We therefore describe their structure in several solutions (Sect. 17.2), first from a simplified 2-d perspective (Sect. 17.2.1) and then for their complete 3-d pathways (Sect. 17.2.2). In the rest of the chapter, we report idealized solutions that illustrate the basic processes that generate the cells and determine their properties (Sect. 17.3). Observational support for model results are provided at several locations in the text.

17.1 Overview As noted in Sect. 4.1.1, the defining characteristics of all overturning circulations are: sinking in one region due to surface cooling and evaporation, rising in another by vertical diffusion or wind-driven upwelling, and the surface and subsurface, horizontal flows that connect the two regions; further, the cells are of two types differing in the processes that drive their sinking and horizontal flows, namely, deep convection or subduction. In the Indian Ocean, the cells generated by deep convection are associated with RSW and PGW, whereas those generated by subduction are the CEC, STC, and the cell associated with NASHSW. See Chap. 4 for descriptions of the overturning circulations associated with RSW, PGW, and NASHSW. The CEC and STC are analogous to the North and South STCs in the Pacific and Atlantic Oceans, in which water downwells (subducts) in the northern- and southernhemisphere subtropics, flows to the equator within the thermocline, and upwells in the eastern, equatorial ocean (Schott et al., 2001, 2004). In contrast, in the Indian Ocean the CEC and STC upwelling branches are located in off-equatorial regions (in the northern hemisphere for the CEC and along the SETR for the STC), a consequence of the Indian Ocean lacking sustained equatorial easterlies. Further, because the Indian Ocean does not extend into the northern subtropics, most subduction for the cells happens in the southern hemisphere. As a result, since the upward and downward branches of the CEC lie in opposite hemispheres, its surface and subsurface branches necessarily cross the equator, accounting for its name. In the other oceans, the STCs are of well-known importance: Their equatorial upwelling branches impact climate by affecting SST and biology through the upward transport of nutrients into the euphotic zone; in addition, the cells are an essential part of the heat budget in each basin, with the atmospheric heat flux into the equatorial upwelling regions balanced by poleward (equatorward) advection of warm (cool) water in their surface (subsurface) branches. The Indian Ocean overturning cells have similar impacts. In particular, summertime upwelling in the northern hemisphere lowers SST, causing an annual-mean heat input into the the ocean from the atmosphere (e.g., McCreary et al. 1993; Schott and McCreary 2001): The CEC balances that input by carrying warm surface water out of the NIO and replacing it with cool subsurface water from the southern hemisphere.

17.2 Structure

443

17.2 Structure 17.2.1 Two-Dimensional Overturning In ocean models, a useful way to illustrate the overturning part of a cell is to plot its zonally-integrated meridional and vertical currents, thereby providing a 2-d (sideways) view without any zonal flow. The meridional streamfunction provides a convenient method for doing that.

17.2.1.1

Streamfunction Definition

Integrating the continuity equation (u x + v y + wz = 0) across the basin, and applying Leibniz’s rule to bring the derivatives outside the x-integration, gives Vy + Wz + N (xe , y, z) − N (xw , y, z) = 0,

(17.1a)

 xe

 xe

where: V = xw vd x and W = xw wd x are the zonally-integrated currents; xw (y, z) and xe (y, z) are the zonal locations of the western and eastern boundaries, which are functions of y and z in basins with realistic coastlines and shelves; and N (ξ, y, z) = u(ξ, y, z) − v(ξ, y, z) ξ y − w(ξ, y, z) ξz ,

ξ = xe or xw (17.1b)

is proportional to the amplitude of the normal velocity at each boundary. Since closed boundary conditions require that N (xe , y, z) = N (xw , y, z) = 0, (17.1a) simplifies to (17.1c) Vy + Wz = 0. Equation (17.1c) implies that a streamfunction ψ(y, z) exists with the properties V = ψz ,

W = −ψ y .

(17.2a)

Integrating the first of (17.2a) with respect to z, and applying the bottom boundary condition that ψ = 0 at z = −D (which ensures that W = ψ y = 0 there) gives  ψ(y, z) =

z

−D









V y, z dz =



z

−D



xe

  v x, y, z  d xdz  .

(17.2b)

xw

(In principle, it is also possible to obtain ψ by integrating the second of Eqs. 17.2a in y, but typically there isn’t a convenient boundary condition where v is known.) Eqs. (17.2a) can be rewritten as V = k × ∇ψ, where is the vector V = (V, W ), from which it follows that V is parallel to isolines of ψ and directed to the right of ∇ψ.

444

17.2.1.2

17 Cross-Equatorial and Subtropical Cells

Annual-Mean ψ

Streamfunction plots for the Indian Ocean have been made in a number of studies (see Schott and McCreary 2001, for a review). Figure 17.1 shows the annual-mean ψ for OGCM solutions obtained at the Japan Marine Science and Technology Center (JAMSTEC; Ishida et al. 1998; top panel) and discussed by Garternicht and Schott (Garternicht and Schott 1997; GS97; bottom panel). A single streamfunction that spans the latitude range of the entire Indian Ocean is not possible, owing to the gap in the Indonesian passages near 10◦S where N (xe , y, z) = 0. For this reason, the JAMSTEC streamfunction extends only to 8◦S. In contrast, the GS97 streamfunction does span the entire basin; the extension is obtained by obtaining exact streamfunctions both north and south of the gap, and then modifying the southern part by subtracting out a streamfunction associated with the ITF transport so that the two parts join smoothly. As such, although ψ is approximated south of 10◦S, it still illustrates the structure of the overturning there. The JAMSTEC plot shows water rising to the surface from 3–20◦N and from 8◦S to the equator with maximum transports of 6 and 11 Sv, indicating the upward branches of the CEC and STC, respectively. In contrast, the two cells in the GS97 plot have maxima that are about 1 and 12 Sv. It is noteworthy that its CEC upwelling branch is very weak, never reaching the levels in other models or inferred from observations (e.g., Table 17.1). Reasons for this marked difference are not clear; a possible cause is that the summertime, upwelling-favorable winds in the western Arabian Sea (the Findlater Jet) are too weak in the GS97 solution. Another is that its Somali upwelling is weakened too much by the onshore propagation of Rossby waves (Sect. 17.3.3.1). The GS97 plot shows the presence of subtropical subduction south of 17◦S, with 11 Sv providing the source waters for the STC and another 5 Sv (unshaded region near the southern boundary) departing the basin at depth. The plot also has a deep overturning cell, in which bottom water enters the Indian Ocean, is made less dense by vertical mixing with overlying water, and departs the basin as deep water (Sect. 4.1.3). Another overturning cell (unshaded region in both panels) is located at middepth, and the part of this cell above 500 m appears to be present in the JAMSTEC plot as well. The cause of this cell is not clear: It may in fact not be real, but rather an artifact of expressing ψ in terms of z instead of density coordinates, analogous to the Deacon Cell in the ACC (Döös et al. 2008). A final overturning feature in Fig. 17.1 is an “equatorial roll,” in which the offequatorial, southward, surface flow dips down to about 50 m as it crosses the equator, and the near-equatorial, surface flow is northward. It is referred to as a “roll” rather than an “overturning cell,” because it is largely confined to the surface mixed layer and so involves diapycnal processes only weakly or not at all. The theoretical possibility of the roll was known for some time before observational evidence confirmed its existence (see Sect. 17.3).

17.2 Structure

445

Fig. 17.1 Transport streamfunction plots, showing 2-d overturning circulations in the Indian Ocean from the JAMSTEC (upper panel) and GS97 (lower panel) solutions. In the lower panel, the dashed lines are the 35 and 35.2 psu contours from the zonally-averaged, annual-mean salinity field. Adapted from Miyama et al. (2003) and Garternicht and Schott (1997)

446

17.2.1.3

17 Cross-Equatorial and Subtropical Cells

Seasonally-Varying ψ

Because the continuity equation lacks a time-derivative term, (17.2b) is valid at all times. If the streamfunctions in Fig. 17.1 are plotted throughout the year, they have a large seasonal cycle, in which the sense of rotation of the CEC reverses. It is tempting to conclude that this cycle represents a seasonal change in the strength of the overturning cells, but it does not. Rather, it is caused by a back-and-forth, isopycnal “sloshing” of water, with isopycnals rising (deepening) in upwelling (downwelling ) regions, and involves little or none of the diapycnal changes that occur in an overturning cell (Miyama et al. 2003; Schott and McCreary 2001; Han 2021). In contrast, the overturning present in the annual-mean ψ cannot occur without diapycnal changes, and hence it is a good measure of cell strength.

17.2.1.4

Videos

Video 17.1a provides a quasi-2d (averaged from 50–90◦E) view of the near-surface, cross-equatorial flow in solutions to the JAMSTEC (left panel) and LCS (right panel) models forced by realistic (Hellerman and Rosenstein 1983) wind stresses, τ x and τ y , and obtained in realistic basins (Fig. 17.2, upper panels; Fig. 17.3). The two solutions illustrate the annual cycle of the CEC surface branch in Fig. 17.1 near the equator. During the summer, the cross-equatorial flow is strong and southward; it dominates the currents throughout the rest of the year, thereby accounting for the annual-mean flow being southward in Fig. 17.1. In contrast, during the winter the current is relatively weak and northward; at that time, it likely accounts for some SECC water bending northward to the equator (Sect. 4.3.3). During both seasons, the currents are very shallow, only dipping down a bit near the equator in response to the equatorial roll. The two solutions are strikingly similar, indicating that basic processes causing the cross-equatorial flow and equatorial roll are the same in both models. Video 17.1b provides clues into their dynamics, showing LCS solutions forced by both τ x and τ y (left), τ x alone (middle), and τ y alone (right): The latter two solutions suggest that the cross-equatorial flow is driven by τ x and the roll by τ y . We consider these processes in detail in Sects. 17.3.4 and 17.3.5. A noteworthy difference between the solutions is that the flow field in the JAMSTEC solution extends to somewhat deeper levels. The causes of this difference are not clear. One possibility is that mixing strengths in the two models are not the same; however, Miyama et al. (2003) were not able to eliminate the difference by adjusting mixing in the LCS model, and therefore ruled out this cause. Another possibility is that the background density field ρ in the JAMSTEC solution is not independent of x, y, and t, as is assumed for ρb (z) in the LCS model. A third is advection of ρ in the JAMSTEC solution, in which downwelling extends the response to deeper levels.

17.2 Structure

447

Fig. 17.2 Pathways of model drifters for surface (right) and subsurface (left) CEC branches from the MKM (top two rows) and TOMS models (bottom row). In both the MKM and TOMS solutions, the CEC and STC surface (subsurface) branches are contained in layer 1 (layer 2). In the left (right) panel, drifters were placed in layer 1 (layer 2) of the prominent, NIO upwelling regions and traced backwards (forwards) in time. Adapted from Miyama et al. (2003)

17.2.2 Three-dimensional Pathways The 3-d flow field of the cells is more complex than the simple circulation in Fig. 17.1, because their upwelling and downwelling branches are in different locations and hence require subsurface and surface currents to join them. To illustrate these pathways, Figs. 17.2 and 17.3 plot typical tracks of model drifters from the annual-mean solutions to the three models discussed in Miyama et al. (2003): 2 21 -layer (MKM; McCreary et al. 1993) and 4 21 -layer (TOMS; Jensen 1993) regional models confined to the Indian Ocean in Fig. 17.2; and the JAMSTEC OGCM (Ishida et al. 1998) of global extent in Fig. 17.3. To obtain the tracks, model drifters were placed in the northern-hemisphere upwelling regions (Somalia, Arabia, and near India) and then tracked either back-

Fig. 17.3 Pathways of backward-tracked drifters (left panels) and forward-tracked drifters (right panels) from the JAMSTEC solution. In both panels, model drifters were tracked from an original location in the prominent, NIO upwelling regions at a depth of 50 m (open circles). Adapted from Miyama et al. (2003)

448 17 Cross-Equatorial and Subtropical Cells

17.2 Structure

449

ward in time by using the negative of the annual-mean currents (left panels) or forward in time (right panels). In this way, following the forward tracks illustrates where surface water goes after it upwells (right panels), and following the backward tracks in reverse shows how subsurface water flows to the upwelling regions (left panels). The color shading indicates time in years since the drifter was introduced into the model, and colors that repeat along a track are separated by 10 years. The basic structure of the pathways is similar in all the solutions, the differences among them due primarily to their different parameterizations of vertical mixing and to the specification of open (southern and Indonesian Throughflow) boundary conditions. In addition, pathways differ between the layer and JAMSTEC models in that drifters in the former are confined to a specific layer, whereas those in the latter are free to move vertically in the water column.

17.2.2.1

Layer-model Pathways

Figure 17.2 illustrates sets of CEC pathways for which one end lies in the upwelling regions off Somalia (top), Arabia (middle), and India (bottom). Although there are no explicit STC pathways, they can be inferred from the CEC ones. Backward-Tracked Drifters Following the layer-2 trajectories backwards in time provides a picture of the CEC ’s subsurface pathways (Fig. 17.2, left panels). They all begin in the southeastern Indian Ocean, flow westward across the basin and around Madagascar within the deep part of the SEC, and then continue northward in the EACC. Note that the pathways are closely packed in the southern hemisphere, a feature that results from backwards tracking in the presence of subduction: The subduction causes the layer-2 flow field to be divergent, which, when tracking backwards, is a convergence. Further, all the tracks extend close to the southern boundary. This property does not mean that all water for the subsurface flow originates there (or outside the basin), but only some of it: Much of the layer-2 water in the cells arises from the subduction of layer-1 water into layer 2 farther north in the southeastern Indian Ocean (Sect. 17.2.3). Most subsurface pathways cross the equator near the western boundary. This property is expected since the strong horizontal mixing in the western boundary current allows water parcels to reverse the sign of their potential vorticity (Sect. 5.4). An exception is for some pathways in the MKM solution, which cross the equator as far east as 60◦E. This curious pathway happens because the MKM solution has a welldeveloped SECC in layer 2. EACC water bends offshore to flow eastward within the SECC, but instead of turning southward to upwell in the SETR, it bends northward. Water parcels along this trajectory gain the positive potential vorticity they need to cross the equator either because the impact of weak horizontal mixing is significant along the longer pathway or by rectification of the annual cycle. The TOMS solution

450

17 Cross-Equatorial and Subtropical Cells

doesn’t have this pathway, possibly because its layer-2 SECC is located somewhat farther northward, with its northern flank nearly reaching the equator. After crossing the equator, layer-2 water flows to the upwelling areas off Somalia, Arabia, and India/Sri Lanka. The pathways that join them are sequentially linked: Pathways that end off Arabia first pass Somalia, and those that end off India pass both Somalia and Arabia. In the TOMS solution, the pathways that join India/Sri Lanka to Arabia are complex with several current reversals, an indication of the weakness of the layer-2 currents; in the MKM solution (not shown), the corresponding pathways are similar but with only one reversal. Forward-Tracked Drifters Layer-1 drifters illustrate the CEC’s surface branch (Fig. 17.2, right panels). They all advect southward and eastward from the upwelling regions, crossing the equator increasingly farther to the east for the Somali, Arabian, and Indian drifters. Subsequently, the drifters turn westward near 10◦S, flowing across the basin either to the coast of Madagascar or to the southern boundary. In the Somali and Arabian trajectories, they diverge at Madagascar to flow to the African coast either north (top-right panel) or south (middle-right panel) of the island before exiting the basin in the Agulhas Current. For Indian trajectories, drifters in the TOMS solution flow to the southern boundary in the interior ocean, where they diverge to flow east and west (bottom-right panel); in contrast, similar drifters in the MKM solution follow a pathway like that in the middle-right panel, the difference likely due to the MKM model damping the u field near the southern boundary whereas TOMS does not. As for the layer-2 trajectories, the fact that layer-1 trajectories end at the southern boundary (or Madagascar) does not mean that all water parcels end up there, as some of them subduct into layer 2 elsewhere along the path (Sect. 17.2.3). STC Pathways Although Fig. 17.2 does not contain any explicit STC trajectories, it is easy to infer what they must be for the MKM solution. Layer-2 trajectories are similar to those in the top- and middle-left panels, except without their northern half. Their SEC branches lie somewhat farther north and their SECC branches somewhat farther south than the trajectories in the panels, with the latter bending southward toward the shallowest part of the SETR (∼7.5◦S) instead of northward to cross the equator. Layer-1 trajectories begin near 7.5◦S within the region surrounded by blue trajectories in the top-right panel. Some drifters reach the African coast far enough south that they bend southward as in the top-right panel. Most, however, reach the coast far enough north to bend northward at the coast, flow north to Somalia, and then follow a blue pathway to exit the basin in the Agulhas Current. Trajectories in the TOMS solution are harder to visualize because the flow field is noisier, but they must be similar to those for the MKM solution.

17.2 Structure

17.2.2.2

451

JAMSTEC-Model Pathways

Miyama et al. (2003) obtained a large number of drifter pathways for the JAMSTEC solution. Figure 17.3 plots 8 of them, the left and right parts of each panel providing their horizontal and vertical pathways. They are representative of typical pathways with: start points near Somalia, Arabia, and India; and end points in the Indonesian Seas, the Agulhas Current, and near Australia. Backward-Tracked Drifters For the backward-tracked drifters (left panels), subsurface trajectories originate either in the Indonesian Seas or in the subtropical South Indian Ocean. The two pathways from the Indonesian passages either: cross the Indian Ocean at depths shallower than 200 m in only 2–3 years (top-left panels); or first upwell off Indonesia, then subduct near 90◦E, and thereafter follow a longer pathway to the tip of India (upper-middle, left panel). The two pathways from the South Indian Ocean, both subduct there and take 20 or more years to reach Arabia or India (lower two panels of Fig. 17.3). Note that, in contrast to the layer models, these pathways lack a direct connection from Somalia to Arabia (bottom-middle, left panel) whereas there is one from Somali to India (bottom-left panel). Forward-Tracked Drifters The forward-tracked drifters (right panels) first advect to the equator where they descend to 50–100 m due to the equatorial roll (Fig. 17.1; Video 17.1a). Some drifters then upwell south of the equator and continue to move southward (upper three, right panel). Others are entrained into the roll, and undergo 1–2 loops as they are advected eastward (bottom-right panel). In the southern Indian Ocean, trajectories tend to fall into four distinct groups, in which drifters: (1) subduct in the western half of the basin and leave the basin via the Agulhas current (top-right panel); (2) remain in the southern hemisphere at depths shallower than 200 m (upper-middle, right panels); (3) move to the eastern boundary and exit the basin via the Leeuwin current (lower-middle, right panel); and (4) subduct in the eastern half of the basin, eventually returning to the northern hemisphere within the CEC (bottom-right panel). STC Pathways As for the layer models, none of the trajectories in Fig. 17.3 lead to the STC upwelling region. Given the structure of ψ for the JAMSTEC model (Fig. 17.1, top panel), however, we know that such pathways must exist. Consider trajectories that end at an arbitrary location P within the SETR (i.e., along ∼7.5◦S). Then, backward-tracked

452

17 Cross-Equatorial and Subtropical Cells

trajectories from P must extend to the African coast near 5◦S at depths of 300–400 m, and thereafter bend southward to follow pathways like those south of 5◦N in the two bottom-left panels of Fig. 17.3. Forward-tracked trajectories from P intersect Africa near 10◦S at shallow depths, and bend northward within the EACC to follow paths like those in the right, upper-middle and bottom panels of Fig. 17.3.

17.2.3 Transports Table 17.1 lists transports relevant to various STC and CEC branches determined from the above solutions. Values are divided into three blocks divided by vertical lines: for the subsurface branches of the STC (left block) and CEC (middle block); and for their surface branches (right block). For the layer models, the surface and subsurface branches of the cells are in layers 1 and 2, and their upwelling and downwelling branches are determined by area integrals of the entrainment/detrainment velocity w1 between the two layers. For the JAMSTEC model, the surface (layer 1) and subsurface (layer 2) branches are defined by the depth ranges 0–80 m and 80–500 m, and w1 is given by w at 80 m. These boundaries were chosen because they lie near the center and bottom of the 2-d overturning circulations in Fig. 17.1; further, the 80-m depth is deep enough to ensure that the equatorial roll is contained entirely within layer 1.

Table 17.1 Annual-mean transports relevant to the STC and CEC Model 29S2

SUB

ITF2

MZC2 UTR

USJ

EQ2

USM

UAR

UIN

EQ1

MKM 3.1

–11.7

0

–5.2

8.0

–0.1

3.3(2.5)

2.7

1.9

1.8

5.4(–11.2) 0

TMS

6.5

–2.0

–2.7

–1.6

4.9

2.7

6.2(1.2)

1.5

1.3

1.2

3.5(–8.7)

–2.4

–12.7

JAM

9.7

–4.7

–5.1

–7.7

7.5

0.6

6.6(–0.5) 3.1

1.0

1.2

1.9(–7.1)

–5.7

–18.2

–9.5

–5

–7

9.2

7.3

1.0

1.3

3.5(–6.5)

–10

OBS

6.0

ITF1

17S1 –11.8

Annual-mean transports (Sv) for the STC (left block) and CEC (middle block) subsurface branches and their surface (right block) branches, where the blocks are divided by thicker vertical lines. Signs of the transports are positive (negative) for northward or upward (southward or downward) flows, and the ITF transports are negative. Labels for upwelling and subduction transports have no subscript. Labels for section transports have a subscript that indicates the layer in which the transport value is calculated: For the JAMSTEC model (JAM), the dividing depth between layers 1 and 2 is 80 m; for the observations (OBS), it is either 100 m for EQn or 500 m for ITFn . Transport values for solutions are obtained by integrating annual-mean currents within the areas and across the sections shown in Fig. 17.4, and observed transports are reported for the same regions as described in the text. Transport 17S1 is the sum of the annual-mean transport across 17◦S east and west of Madagascar. Transports EQn are within (no parentheses) or east of (parentheses) the western-boundary region. Transports ITFn are evaluated through the Indonesian passages for the JAMSTEC model, specified in an eastern-boundary condition south of Indonesian for TOMS, and zero for the MKM model.

17.2 Structure

453

Fig. 17.4 Areas and sections where the transport values in Table 17.1 are obtained. Their labels are the same as in the table, and subscript n is a layer index (n = 1, 2). Curve xb(y) is a parabolic representation of a layer-2 streamline in Fig. 17.2

Figure 17.4 indicates sections and areas where the transports in Table 17.1 are calculated. The western edge of Region B, xb (y), is defined by the annual-mean, layer-2 trajectory that extends from the point on Madagascar where the East Madagascar Current diverges (near 17◦S), the “bifurcation trajectory.” For the layer models, xb is similar to the trajectories in the left panels of Fig. 17.2, except shifted to originate near 17◦S; for the JAMSTEC model, it is defined by the trajectory that leaves the Madagascar divergence point at a depth of 350 m. With these definitions, layer-2 water northeast of xb participates in the cells, whereas water southwest of it exits the Indian Ocean in the Agulhas Current (Fig. 17.2).

17.2.3.1

STC Layer-2 Transports

Layer-2 water arises from two primary sources: flow across the southern boundary (29S2 ) and subduction of layer-1 water into layer 2 (SUB). The split between these two sources varies markedly among the models, a consequence of model-dependent factors such as the southern-boundary conditions applied (for MKM and TOMS but not JAMSTEC) and the mixing parameterizations that determine upwelling and subduction. These source waters flow northward across 17◦S and then westward to

454

17 Cross-Equatorial and Subtropical Cells

the African coast, along the way joined by water from the deep part of the ITF (ITF2 ) and losing water to southward flow in the Mozambique Channel (MZC2 ). At the African coast, the source waters join the northward-flowing EACC, some of which first bends offshore to feed the SECC and then upwells in the SETR (UTR) and Sumatra/Java (USJ). In the MKM and JAMSTEC solutions, the dominant upwelling transport is UTR; however, USJ is much larger in TOMS, possibly because the TOMS solution is not in equilibrium. Interestingly, the total upwelling in the JAMSTEC solution (UTR + USJ = 8.1 Sv) is significantly less than its value in the 2d streamfunction (11 Sv, Fig. 17.1), the difference happening because the transports in Table 17.1 are evaluated across 80 m and so don’t include overturning confined above that depth.

17.2.3.2

CEC Layer-2 Transports

Source waters for the CEC occur mostly in the layer-2 western boundary current (EQ2 , no parentheses; EQW2 ). The MKM solution also has considerable crossequatorial flow east of the western boundary current (EQ2 , parentheses; EQE2 ), owing to part of the SECC bending northward to cross the equator in the interior ocean (Fig. 17.2, top-left panels). The offshore component in the TOMS solution exists because meandering of the SECC extends to the equator (Fig. 17.2, bottomleft panel): The transport associated with the first southward meander is absorbed into EQW2 , leaving a net negative positive contribution in EQE2 . Most of this inflow upwells in the three upwelling regions off Somalia (USM), Arabia (UAR), and the southern tip of India (UIN). The upwelling transports in the three subregions vary considerably among the solutions, indicating that regional upwelling is sensitive to model-dependent parameterizations. Note that total upwelling for the JAMSTEC solution (USM + UAR + UIN = 5.3 Sv) is somewhat less than the maximum overturning of the CEC in the streamfunction plot (6 Sv; Fig. 17.1), again because it doesn’t include any overturning above 80 m.

17.2.3.3

Layer-1 Transports

Water that upwells in the northern hemisphere flows southward across the equator in the interior ocean (EQ1 , parentheses; EQW1 ). Because there is a recirculation that crosses the equator within layer 1 (the Tropical Gyre), the net southward transport associated with the CEC surface branch is the sum of the two EQ1 values. That sum is close to the total of the northern-hemisphere upwelling transports, as expected. Farther south, the surface branches of the overturning cells are joined by the surface branch of the ITF (ITF1 ). Together, they flow southward across 17◦S into the subtropics (17S1 ), with much of the water in the eastern part subducting into layer 2 (SUB) and the rest leaving the basin in the Agulhas Current.

17.3 Dynamics

455

17.2.4 Observations Overall, there is good agreement between the flow paths of the CEC and STC circulations inferred from observations with those depicted in Figs. 17.2 and 17.3 (Schott et al. 2002; Nagura et al. 2018). Table 17.1 (bottom row) lists transports for the cell branches estimated from observations: Values are all taken from Schott et al. (2002) except for ITF1 and ITF2 , which are the annual-mean transports, ITFW and IIW, reported in Chap. 4. Given the large variation in the modeled transports and errors in the observed estimates, the values agree as well as might be expected. There are no data that directly measure the listed vertical (upwelling and subduction) transports. Schott et al. (2002) estimated upwelling transports using Ekman transports determined from the NCEP (1990–98) and European Remote-sensing Satellite-1/2 (ERS-1/2) scatterometer (1992–98) data sets: The Ekman divergence from a coast or area was estimated during the upwelling season (June–September) for both data sets; and the two values were averaged and divided by 3, to obtain an effective, annual upwelling rate. Transports UTR and UIN are the annual-mean Ekman divergences out of boxes UTR and UIN in Fig. 17.4, and transports USM and UAR are the Ekman divergences from the Somali and Arabian coasts, that is, along the western edges of boxes USM and UAR in Fig. 17.4. Schott et al. (2002) estimated subduction using the method of Marshall et al. (1993). Transport SUB is the resulting, annual-mean subduction rate in the southeastern Indian Ocean for surface densities σθ ≤ 25.7 kg/m3 ; water in this density range occurs within the SC at the equator at depths shallower than 150 m, and typically upwells off Somalia. The resulting upwelling and subduction rates are generally consistent with those from the solutions, except that observed USM is significantly larger than the modeled values. The reason for this difference is that not all the offshore Ekman transport generates upwelling (see Sect. 17.3.3).

17.3 Dynamics To isolate key parts of the wind forcing that drive the cells, we obtained a set of idealized solutions to a linear, 2 21 -layer model, the simplest system that can represent a complete overturning circulation in the upper ocean. We begin with a model overview (Sect. 17.3.1), which describes the experimental design for each solution (17.3.1.1) and notes conditions that must hold for solutions to develop overturning (Sect. 17.3.1.2). Then, we report idealized solutions that illustrate the processes that determine the STC (Sect. 17.3.2) and the CEC (Sect. 17.3.3), and that allow crossequatorial flow in the interior ocean (Sect. 17.3.4). We conclude with a discussion of the dynamics of the equatorial roll (Sect. 17.3.5).

456

17 Cross-Equatorial and Subtropical Cells

17.3.1 Model Overview 17.3.1.1

Experimental Design

Governing equations for the 2 21 -layer model are a linear version of Eqs. (5.19) extended to have two active layers, u nt − f vn +

τx pnx = δ1n + νh ∇ 2 u n , ρn ρ1 Hn

(17.3a)

vnt + f u n +

pny τy = δ1n + νh ∇ 2 vn , ρn ρ1 Hn

(17.3b)

  h nt + Hn u nx + vny = (−1)n−1 (we + wd ),

(17.3c)

where subscript n is a layer index (n = 1, 2), Hn is the background thickness of layer n in a quiescent ocean, and δ1n is the Kronecker-delta symbol (δn1 = 1 when n = 1 and is zero otherwise).1 Horizontal mixing with the coefficient νh = 5×107 cm2 /s is included to allow solutions to have a reasonably broad, western-boundary current. Layer Pressures: Expressions for pn and sea level d are p1   = g21 h 1 + g32 h, ρ1

p2  = g32 h, d = 21 h 1 + 32 h, ρ2

(17.3d)

  = gρnm /ρ3 , nm = gnm /g, h 1 = h 1 − H1 , and h = h − H with h = where gnm h 1 + h 2 and H = H1 + H2 . (A factor of ρ1 /ρ2 ≈ 1 is dropped from the expression for p2 .) They follow from a derivation like that for the 1 21 -layer model (Eqs. 5.21– 5.24), except beginning with pressures defined on level surfaces within layers 1, 2, and the deep ocean (layer 3) and assuming that ∇ p3 = 0. In all the solutions,   = 5 cm/s2 , and g32 = 1.25 cm/s2 . With these choices, H1 = 100 m, H2 = 400 m, g21 the characteristic speeds of the n = 1 and 2 vertical modes of the system are c1 = 286 cm/s and c2 = 175 cm/s, similar to those for the LCS model (Table 5.1). Entrainment and Detrainment: For an overturning cell to exist, solutions must be able to transfer layer-2 water into layer 1 (upwelling) and layer-1 water into layer 2 (subduction). A simple way to parameterize these processes is to include entrainment we and detrainment wd velocities with the h 1 -dependent forms,

Many studies have investigated the dynamics of shallow overturning cells using 2 21 -layer models (e.g., Rhines and Young 1982; Luyten et al. 1983; McCreary and  Lu 1994).  Typically, their equation sets differ from (17.3) in that Hn → h n in (17.3a) and Hn u nx + vny → (h n u n )x + (h n vn ) y in (17.3c), so that the equations are nonlinear. The most prominent nonlinear impact on solutions is that subsurface (layer-2) cell branches bend equatorward, rather than extending due westward. Otherwise, the basic processes that determine solutions in the two models are the same, and equations (17.3a) are a useful way to identify them.

1

17.3 Dynamics

h 1 − Hd θ(h 1 − He ) θ(ysb − y) . td (17.3e) Velocity we > 0 represents an upwelling rate into a surface mixed layer of thickness He whenever wind forcing thins layer 1 until h 1 < He (see the discussion of Eq. 5.20). Conversely, velocity wd < 0 specifies that subduction occurs whenever h 1 gets thicker than Hd > He and south of ysb = 12.5◦S. Parameter values are He = 50 m, Hd = 200 m, te = td = 0.1 day, the small values of te and td ensuring that h 1 never becomes much less than He or much greater than Hd . Recall that subtropical subduction occurs in the poleward part of subtropical gyres, when the thick wintertime mixed layer thins during the spring (Section 4.1.1), and velocity wd parameterizes those processes in a season-independent way. The factor of θ (ysb − y) limits subduction to the southern-hemisphere subtropics, thereby preventing the northern-hemisphere subduction that generates PGW, RSW, and NASHSW, or that allows upwelled water to subduct within the NIO. Regions of non-zero we and wd are the locations of the upwelling and downwelling branches of an overturning cell. Forcing: Forcing is by winds and, for one solution (Video 17.2), by a prescribed entrainment field we . The wind forcing has three parts: idealized versions of the background Southeast Trades (SET), the Findlater Jet (FJ and FJ ), and zonal winds of the Southwest Monsoon (ZW). Each forcing has the separable form (C.7a). Their spatial structures, X and Y, are indicated by the red arrows in the videos, and are precisely described in the video captions. Wind SET is switched on, whereas FJ, FJ , and ZW are either switched on or their time dependence has the annually-periodic form tm − t 1 θ[(tm − t1 ) (t2 − tm )] , (17.4) T (t) = sin π t2 − t1 we = −

h 1 − He θ(He − h 1 ) , te

457

wd = −

where tm = mod(t, 365) and t are measured in days, t1 = 365/4 days, and t2 = 365×(3/4) days. Equation (17.4) is a simple representation of the life cycle of the FJ and ZW winds during the Southwest Monsoon (May through October; contrast summer and winter wind fields in Video 3.1). Wind strengths for SET, FJ, and ZW are y τox = −1.0 dyn/cm2 , τo = 10.0 dyn/cm2 , and τox = 1.5 dyn/cm2 , respectively. The we forcing has the form we = weo X(x) Y (y) θ(t), where weo = 5×10−4 cm/s and X and Y are given by (C.7b) with xm = (86◦ E, −12.5◦ S), x = 20◦ , and y = 5◦ . Basin and Boundary Conditions: For simplicity, solutions are found in a rectangular basin either with closed, no-slip conditions imposed on all boundaries or an open eastern boundary. As we shall see, solutions in the closed basin develop CEC and STC pathways similar to those in Figs. 17.2 and 17.3. On the other hand, it cannot represent the pathways that extend outside the Indian Ocean into the Indonesian Seas or across the southern boundary. Solutions: Solutions are integrated for 5–20 years, depending on how long it takes them to approach equilibrium. The videos show h 1 and h in the left and right panels, respectively. Variable h 1 is useful because it allows upwelling (h 1 < He − H1 = −50 m) and downwelling (h 1 > Hd − H1 = 100 m) regions to be

458

17 Cross-Equatorial and Subtropical Cells

identified easily. Variable h is useful because it is proportional to p2 , so that its isolines are parallel to streamlines of layer-2 geostrophic flow. Reported transports are annual-mean values evaluated during the last year of the integrations.

17.3.1.2

Requirements for Overturning

Even with parameterizations (17.3e), solutions need not develop an overturning circulation. To understand when they can, consider the following sequence of thought problems in a closed basin (Solutions 1–4). Without entrainment and detrainment in (17.3c), wind-forced solutions will adjust to a steady state in which h 1 has minimum h 1 min and maximum h 1 max values (Solution 1). Even when entrainment and detrainment are included, Solution 1 will be unchanged if h 1 min > He and h 1 max < Hd , since in that case we and wd remain zero (Solution 2). Suppose now that initially h 1 min < He so that we = 0 (similar arguments hold if we begin with h 1 max > Hd ). Then, water entrains into layer 1, thickening it throughout the basin by an average amount h 1 ; consequently, the minimum value of h 1 increases to h 1 min + h 1 , thereby decreasing h 1 − He and we . There are two possible final states, depending on the magnitude of h 1 max − h 1 min ≡ δh 1 relative to Hd − He . If δh 1 < Hd − He , h 1 will continue to increase until h 1 min + h 1 > He and we = 0, h 1 max + h 1 will still be less than Hd , and wd will remain zero (Solution 3). On the other hand, if (17.5) δh 1 > Hd − He , the value of h 1 that sets we = 0 ensures that h 1 max + h 1 > Hd so that wd = 0. In this case, an equilibrium solution is achieved for a value of h 1 such that both we and wd are non-zero and the annual-mean, entrainment and detrainment transports balance, that is,     (17.6) We h 1 + Wd h 1 = 0, where We and Wd are the annually-averaged, basin-wide, area integrals of we and wd , respectively (Solution 4). Note that adjustments throughout the basin must occur in order to establish (17.6). In effect, the average thermocline depth (z = −h 1 ) is altered by h 1 . Similar basin-wide adjustments occur in the establishment of all overturning circulations. Whether (17.5) is satisfied is sensitive to model and forcing parameters: the back ; background layer thicknesses, H1 and H2 ; entrainment ground stratification, gmn and detrainment parameterizations; and wind strength. Given this sensitivity, it is not surprising that the STC and CEC strengths differ significantly among model solutions (Table 17.1).

17.3 Dynamics

459

17.3.2 STC Dynamics Before considering the impact of wind forcing on the STC, it is useful to consider first the simpler response forced by a prescribed entrainment we (or detrainment when we < 0). The ocean’s response to we develops a characteristic pattern, a β-plume (e.g., Talley 1979; Davey and Killworth 1989; Spall 2000; Belmadani et al. 2013; Schloesser 2015), which is also visible in the more-complex, wind-forced solution.

17.3.2.1

Response to we Forcing

Basic properties of a β-plume are illustrated by the solution to − f Hn vn +

Hn pnx = 0, ρn

f Hn u n +

Hn pny = 0, ρn

  Hn u nx + vny = (−1)n−1 we ,

(17.7) the steady-state, inviscid version of Eqs. (17.3) forced only by we . For simplicity, we assume that we is spatially limited and the ocean is unbounded. It is useful first to solve for the depth-integrated flow. Let Qn = H1 q1 + H2 q2 , where qn = u n , vn , or pn /ρn . Summing the layer-1 and layer-2 equations in (17.7) gives f U + P y = 0, Ux + V y = 0. (17.8) − f V + Px = 0, Since Eqs. (17.8) are unforced, radiation condition (9.4) requires that the solution is U = V = P = 0. With the aid of (17.3d), P = 0 implies that g32 h = − and then

H1 g21 h 1 , H

H2 p1 g21 h 1 , = ρ1 H

p2 H1 = − g21 h 1 ρ2 H

(17.9)

(17.10)

give the pressures in terms of h 1 . With p1 and p2 known, it is straightforward to obtain the solution in each layer. Solving the layer-1 equations for a single expression in p1 gives f 2 we p1x   = g21 h 1 + g32 h = − . (17.11) ρ1 β H1 Using (17.9) to eliminate h from p1 , (17.11) can be solved for h 1 to give 

L2

h 1 = H1 + x

f 2 we d x , β g21 H¯

(17.12)

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17 Cross-Equatorial and Subtropical Cells

where H¯ = H1 H2 /H and L 2 is any longitude east of the forcing region. Because the Rossby waves that establish (17.12) have westward group velocity, this choice of L 2 ensures that the solution does not extend east of the forcing region. With h 1 known, h 2 and pn follow from (17.9) and (17.10), and then u n and vn are known from geostrophy to complete the solution. According to (17.12), when we > 0 (entrainment) h 1 is thicker than H1 in, and west of, the forcing region. In the southern hemisphere, the β-plume is therefore associated with an anticlockwise, geostrophic gyre. To provide a sink for the layer2 water that upwells into layer 1 in the forcing region, the transport of the gyre’s westward branch must be stronger than the eastward one. Let L 1 be any longitude west of the western edge of we . With the aid of (17.10), (17.12), and an integration by parts, the zonal transport in layer 1 at x = L 1 is 

 ∞ p1y  h 1y  dy = − H¯ g21 dy ρ f −∞ −∞ 1 −∞ f    2   ∞  L2  1 f2 1 f β f2   we d x dy = − we + 2 we d x  dy  f β f β f β −∞ L 1 y y

∞  ∞  L2 

1 f 2 L2 = − we d x 

− we d x  dy  = −We . (17.13) f β L1 −∞ L 1 −∞

U1 (L 1 ) =  =−



∞

−∞

L2 L1

∞

H1 u 1 dy  = −H1



∞

Equation (17.13) is a statement that the total transport of both branches is the negative of the upwelling transport, as it must be to ensure mass conservation. A similar calculation shows that the total layer-2 transport is U2 (L 1 ) = We . Video: Video 17.2 illustrates the spin-up of a β-plume in a numerical solution to Eqs. (17.3) forced by we defined after (17.4) when the eastern boundary is open. The video shows h 1 = h 1 − H1 (color shading) and u1 = (u 1 , v1 ) (vectors) in the left panel and h = h − H and u2 = (u 2 , v2 ) in the right panel. Initially, we thickens h 1 in the forcing region, generating an anticlockwise circulation around the patch, and as time passes Rossby waves extend the region of high h 1 across the basin. Near the end of the video when the response is close to steady state, the circulation extends across the basin. The entrainment transport from the solution is We = 2.51 Sv, close to its theoretical value weo (2/π )2 L x L y = 2.50 Sv. The upwelled water is carried across the basin by U1 , then to the equator in a western-boundary current (Chap. 11), and finally eastward in an equatorial current generated by equatorial Kelvin-wave radiation. The transport of the equatorial current is Ueq1 = 2.52 Sv, essentially the same as We . (Arrows indicating the swift western-boundary current are not plotted in this, and subsequent, videos.) Consistent with theory, the layer-2 circulation (right panel) mirrors that in layer 1.

17.3.2.2

Response to SET Forcing

In response to forcing by SET winds, Ekman pumping thins (thickens) h 1 in the northern (southern) half of the forcing region (Sect. 12.3.1). Provided the wind is

17.3 Dynamics

461

sufficiently strong, h 1 thins to He and thickens to Hd north and south of the wind axis, respectively, initiating entrainment we and detrainment wd . The steady-state, layer-1 response can be viewed as containing both a wind-driven part and two β-plumes of opposite signs driven by we and wd , whereas the layer-2 response contains only the β-plumes. As a consequence, the layer-1 and layer-2 circulations no longer mirror each other. Videos: Video 17.3a shows the numerical solution driven by SET when the eastern boundary is open. White curves, which indicate the isolines where h 1 = He and Hd , appear by March, 2002, and thereafter we and wd occur in the regions enclosed by them. In the near-equilibrium solution, much of the water recirculates horizontally in subtropical (anti-clockwise) and tropical (clockwise) gyres. Their strength is measured by the depth-integrated (layer 1 plus layer 2) transports U of the zonal currents in the western ocean. The three zonal currents lie between the latitudes y1 = 30◦S, ◦ y3 = 7.5◦S, y4 = 0◦ , so that the transport between any latitude pair y2 = 21.25  yS, n is Umn = ym U(x, y) dy. Then, the strengths of the subtropical and tropical gyres at 45◦E are measured by U12 = 19.5 Sv and U34 = 32.0 Sv, and the strength of their common branch (the model South Equatorial Current) is U23 = − (U12 + U34 ) = −51.4 Sv. (Neglecting horizontal mixing, the depth-integrated, interior circulation satisfies Eqs. 17.8 with τ x /ρ1 added to the right-hand sides of the first equation. These equax   yn tions describe a Sverdrup flow, for which Umn (x) = − xe τ yx /β d x ; see Sect. ym

11.1. For the SET wind, this expression gives 18.7, 32.7 Sv, and –51.3 for the current transports at 45◦E, in good agreement with their values in the video solution.) The upwelling We = 5.41 Sv and subduction Wd = −10.30 Sv transports force βplume circulations, which superpose on the horizontal gyres. Their imbalance, We + Wd = −4.89 Sv, leads to equatorial currents in layers 1 and 2 with the transports, Ueq1 = −4.86 Sv and Ueq2 = 4.89 Sv, which provide a source and sink for the net subduction. Note that the northern gyres extend across the equator in both layers. In particular, the across-equatorial, layer-2 transport across the basin in the interior ocean (east of 45◦E) is V2e = 0.56 Sv, significantly greater than zero. In both layers, crossequatorial flow is possible because it is part of a zonal Munk layer generated along the northern edge of the forcing region (Sect. 11.2.2): The horizontal mixing in the Munk layer breaks the potential-vorticity constraint, which would otherwise prevent cross-equatorial flow (Sect. 5.4). Video 17.3b is similar to Video 17.3a, except with a closed eastern boundary. In this case, the equatorial currents in Video 17.3a reflect from the eastern boundary as Rossby-wave packets that lead to h 1 thinning throughout the basin. As time passes, the thinning is visible by the increasing blue shift in the shading of the background h 1 field (outside the forcing region). It strengthens we , weakens wd , and eventually allows the system to adjust to a near-steady state (year 19). In that state, background h 1 has thinned by h 1 = −21.9 m, and We = 9.21 Sv almost balances Wd = −9.27 Sv. Since they are depth-integrated transports, the strengths of the horizontal gyres are unchanged.

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17 Cross-Equatorial and Subtropical Cells

The steady-state circulation in Video 17.3b is the model STC. It exists only within and west of the forcing region. Water subducted into layer 2 circulates around the subtropical gyre to flow to the western boundary in its northern branch (the model SEC). There, some of it doesn’t recirculate but rather flows northward in the westernboundary current to join the northern branch of the tropical gyre (the model SECC). As it flows eastward, some of it bends southward to upwell in the region of thin h 1 (the model SETR). Finally, the upwelled water returns to the subduction region, either flowing there directly or first recirculating in the layer-1 subtropical gyre before doing so. Similar STC pathways are present in Figs. 17.2 and 17.3.

17.3.3 CEC Dynamics Offshore flow driven by the summertime Findlater Jet is a major cause of upwelling in the Indian-Ocean northern hemisphere. Given the wind strength, though, the amount of upwelling is surprisingly weak (Schott et al. 2002). Chatterjee et al. 2019 concluded that the weakness resulted from the onshore propagation of a downwelling-favorable Rossby wave. To investigate this process in detail, we first discuss a suite of numerical solutions forced only by idealized versions of the Findlater Jet. We then report a solution forced by FJ + SET, which develops an overturning circulation that includes a CEC.

17.3.3.1

Response to FJ Forcing

Upwelling off Somalia and Arabia depends critically on the offshore structure and time dependence of the Findlater Jet. To illustrate, we obtained numerical solutions in which FJ is either x-independent or weakens offshore and T (t) is either switched-on or given by (17.4). All solutions are found with an open eastern boundary. Videos: Video 17.4a shows the solution forced by FJ when T (t) = θ(t) and X(x) = 1. Initially, h 1 thins at the coast due to offshore Ekman drift. Quickly, h 1 becomes less than He along part of the coast, entrainment begins (we = 0), and shortly thereafter the solution adjusts to a steady state. The white line along the western boundary indicates where h 1 = He and coastal upwelling occurs. The initial adjustment of coastal h 1 is determined by coastal-Kelvin-wave propagation, as discussed in Sect. 13.2 for an eastern coast and illustrated in Fig. 13.2: Along a western coast, differences are that the Kelvin-wave response propagates in the opposite direction (toward lower latitudes), and separates from the forcing region as a distinct wave packet (rather than as a wave front). Figure 17.5 illustrates y the steady-state, western-coast responses for weak (τo = 1 dyn/cm2 ; left panel) and y strong (τo = 10 dyn/cm2 ; right panel) winds. For weak winds, h 1 never thins to He , there is no layer-2 flow, and a northward-flowing, layer-1 current with a structure like that in (13.32b) is the source for all the offshore Ekman drift. For strong winds, h 1 thins to He in the latitude band, y1 ≤ y ≤ y2 , the upwelling that occurs there provides

17.3 Dynamics

463

Fig. 17.5 Plot of solutions forced by FJ winds, showing the response along the western boundary y when τo = 3 dyn/cm2 (left panel) and 10 dyn/cm2 (right panel). The solutions are to a simpler, steady-state version of Eqs. (17.3) in which the coastal approximation, vn = pnx / f , is imposed; they are obtained for the parameter values listed in the text. The images in both images look onto the boundary from offshore, so that y increases to the right and horizontal velocity vectors are directed northward. The panels show Y(y) for the wind (green curve), and the negatives of h 1 (red curve) and h (blue curve) at the coast (x = 0); to conserve space, −h and −H are shifted upwards by δy = 250 m, that is, the plotted curves are h + δh and H + δh. Arrow lengths indicate approximate current strengths

some of the water for the offshore Ekman drift, and it is supplied by a layer-2 current.  y  = y 2 τ y / f dy  , is The net entrainment transport (upwelling from layer 2), We = Uek 1 y therefore less than the total offshore Ekman drift, Uek = y12 τ y / f dy  , a consequence of h 1 first having to thin to He before entrainment can occur. The steady-state response in Video 17.4a is close to that in the right panel of Fig. 17.5. The total offshore Ekman transport is Uek = 41.1 Sv, whereas the net upwelling, confined between y1 = 5.9◦N and y2 = 18.9◦N, is We = 19.96 Sv, less than half of Uek . The coastal currents extend southward to the equator and then eastward along the equator across the basin. The transport of the layer-2 current at 90◦E is Ueq2 = −19.91 Sv, consistent with its being the source of the upwelled water. Because the layer-1, equatorial current overlaps the Ekman drift, we define its  25◦ N transport by the difference Ueq1 = H2 30◦ S u 2(90◦ E, y) dy − Uek = −21.2 Sv; this value is consistent the layer-1 current providing the part of offshore Ekman drift not due to upwelling, Uek − We = 21.1 Sv. In striking contrast, Video 17.4b shows the solution when X(x) weakens offshore. In this case, the initial upwelling is completely eliminated by the onshore propagation of downwelling-favorable Rossby waves generated by offshore Ekman pumping (Sect. 12.3.1). Specifically, after only a few month’s the layer-1 flow field adjusts y to a Sverdrup flow driven by offshore τx , which is closed by a northward western boundary current. Because h 1 thickens to a maximum greater than 400 m just offshore, coastal h 1 never thins to h m , thereby eliminating all entrainment of layer-2 water into layer 1.

464

17 Cross-Equatorial and Subtropical Cells

Given the previous solution, how is any upwelling along the western coast possible? The answer lies in the lag between the onset of coastal upwelling and its elimination by the Rossby waves. To illustrate, Video 17.4c shows the solution when X(x) weakens offshore and the forcing has the time dependence T (t) in (17.4). The time variability ensures that some upwelling occurs each summer before the arrival of the Rossby waves. When averaged over the last year of the integration, a net upwelling remains with an annual-mean transport of We = 2.09 Sv. In a comparable solution with T(t) given by (17.4) and X(x) = 1 (not shown), which lacks the Rossby waves that weaken upwelling, the annual-mean upwelling transport is We = 5.25 Sv, and the difference We − We = 3.15 Sv measures the Rossby-wave impact. Similarly, the annual-mean Ekman transport is Uek = 13.1 Sv, and Uek − We = 7.8 Sv measures the impact of h 1 having to thin to He before upwelling can begin (as in the right panel of Fig. 17.5). In the near-equilibrium state of Video 17.4c, the equatorial currents reverse annually. In layer 1, the equatorial current is directed westward during the summer, in order to supply water for the increase in h 1 offshore; it is eastward during the rest of the year, providing a sink for the region of high h 1 that is carried to the coast by Rossby waves. In layer 2, the summertime equatorial current is westward to supply water for coastal upwelling, and is eastward and variable at other times in response to onshore Rossby-wave propagation. Despite their large seasonal variation, the transports of the annual-mean equatorial currents are Ueq2 = −Ueq1 = −2.09 Sv, and so balance We .

17.3.3.2

Response to FJ + SET Forcing

In a closed basin, the adjustments to steady state of the solution forced by FJ + SET are essentially the same as those for the SET solution. Specifically, the reflection of equatorial jets from the eastern boundary as Rossby-wave packets leads to a change in the background h 1 field by h 1 . These adjustments continue until the value of h 1 is such that the mass balance,       Wes h 1 + Wer h 1 + Wd h 1 = 0,

(17.14)

    holds, where Wes h 1 and Wer h 1 are the annually-averaged upwelling transports off Somali/Arabia and in the model SETR. Videos: Video 17.5a is forced by FJ + SET in a bounded basin, using T (t) in (17.4) for FJ. The adjustments discussed above for the individual SET and FJ solutions are evident in the video. In the near-equilibrium state, balance (17.14) holds with Wes = 2.64 Sv, Wer = 7.15 Sv, Wd = −9.77 Sv, and h 1 = −10.8 m. Subducted water flows into the northern hemisphere in layer 2 of the western boundary current, and upwells off Somalia/Arabia. The upwelled water recirculates offshore in the flow y driven by τx , returns to the western boundary at the southern edge of the forcing region, and flows into the southern hemisphere in layer 1 of the western-boundary

17.3 Dynamics

465

current. Finally, it flows eastward in the model SECC and then southward to the subduction region forced by SET, to close the model CEC. The CEC return pathway in Video 17.5a differs from the pathways in Figs. 17.2 and 17.3, which cross into the southern hemisphere in the interior ocean. To allow for some interior, cross-equatorial flow, Video 17.5b shows the solution when τ y extends south of the equator (wind FJ ). In this case, all the upwelled water crosses the y equator offshore in the southward, τx -driven, flow. Let the dividing longitude between the western-boundary current and the interior ocean be 45◦E. Then, annual-mean, cross-equatorial transports within (x ≤ 45◦E) and east of (x > 45◦E) the westernboundary current are V1w = 3.45 Sv and V1e = −5.79 Sv, respectively. Transport V1w is positive because it is part of the northward boundary current that closes the Sverdup gyre driven by FJ . The sum V1w + V1e = −2.34 Sv is close to the upwelling transport Wes = 2.35 Sv, as required by mass conservation in layer 1. Pathways of this sort are visible in the top-right panel of Fig. 17.2.

17.3.4 Cross-Equatorial Flow A distinctive feature of the CEC is that much of its southward, surface branch crosses the equator in the interior ocean (Figs. 17.2 and 17.3; Video 17.1a). Video 17.5b suggests that cross-equatorial flow in the western ocean is driven by wind curl associated with the offshore weakening of the Findlater Jet. What forces cross-equatorial flow farther east?

17.3.4.1

Conceptual Ideas

Several researchers have noted that τ x tends to be antisymmetric about the equator during both the summer and winter monsoons, so that the midlatitude expression for Ekman flow (−τ x / f ) has the same sign in each hemisphere (Schopf 1980; Levitus 1987; Jayne and Marotzke 2001; Godfrey et al. 2001; Miyama et al. 2003). Further, they proposed that Ekman flow might in fact extend across the equator, allowing surface water to cross the equator in the interior ocean. Of course, a conceptual difficulty with this idea is that one expects midlatitude Ekman flow to break down near the equator where f → 0, so that the precise nature of the cross-equatorial flow is not clear (see Sect. 10.2). Godfrey et al. (2001) and Miyama et al. (2003) noted several properties of solutions forced by this special wind, which suggest that the idea might have merit. Near the equator, antisymmetric τ x is roughly proportional to the distance from the equator (i.e., τ x = ay). One consequence is that, since f = βy near the equator, τ x / f = a/β remains finite even when y → 0, so arguably the midlatitude definition of Ekman flow is sensible even at the equator. Second, the Ekman-pumping velocity is identically zero, that is, wek = − (τ x / f ) y = − (a/β) y = 0. Consequently, isopycnals are never shifted vertically at all by this forcing (Sect. 12.2.1), so no geostrophic

466

17 Cross-Equatorial and Subtropical Cells

currents are generated and v is entirely Ekman drift; it follows that the adjustment to equilibrium doesn’t involve Rossby-wave radiation, but rather happens very rapidly in a time scale of the order of the equatorial inertial period. Third, for this wind −τ x / f = −τ yx /β, the value for a Sverdrup-balanced v field (Sect. 11.1); thus, the v field for this special wind field can be described as an “Ekman/Sverdrup” flow. Finally, since isopycnals are unaffected, the Ekman/Sverdrup flow must be very shallow, with most or all of its transport contained in the surface mixed layer. All of these properties indicate that antisymmetric τ x is very efficient for driving cross-equatorial flow, consistent with the solutions in Video 17.1b.

17.3.4.2

Theory

The midlatitude solutions of Chap. 12 provide theoretical support for these conceptual ideas. When τ x = ayθ(t) and τ y = 0 (so that G o = G = (Y/ f ) y = Y y / f − Y/ f =   Y yy = Fy /β y = 0), solution (12.21) simplifies to u n = pn = 0, vn = −

Fny Fn θ(t) = − θ(t) . β f

(17.15)

According to (17.15), the response instantly switches on to a steady-state Ekman/ Sverdrup flow, which is possible since inertial oscillations are filtered out of its governing equations (12.1). Likewise, for a periodic wind, τ x = aye−iσ t , solution (12.26) reduces to (17.15) with θ(t) → e−iσ t , an oscillating Ekman/Sverdrup flow. The equatorial solutions in Chaps. 14 and 15 confirm that the conceptual ideas also apply at the equator. When Y (y) = ay = α0 aη, it follows that (ηY ) j =

a α0



+∞ −∞

η2 φ j dη = 

Yη

a α0 

 j

=



+∞ 

−∞ +∞ 

−∞

a φ jηη + (2 j + 1) φ j dη = (2 j + 1) α0

a η α0



a φ j dη = α 0 η



+∞ −∞

φ j dη.



+∞

−∞

φ j dη,

(17.16a)

(17.16b)

x Then,  for the  switched-on wind, τ = ayθ(t), applying (17.16) to (14.13) gives R j x − cr j t = 0. After dropping this term, applying the simplifications noted just before (17.15), and setting Y0 = 0 because τ x is antisymmetric, solutions (14.14), (14.32), and (14.37) directly reduce to (17.15). Similarly for periodic forcing, τ x = aye−iσ t , solutions (15.11a), (15.24a), and (15.27) simplify to (17.15) with T (t) = relations to satisfy e−iσ t , provided σ  is small enough for the Rossby-wave  dispersion  the long-wavelength approximation so that k 1j = − σ  /c / (2 j + 1). With the aid of identities (17.16), it then follows that S (1) j = 0 in (15.11c) so that the integrals for m = 1 in the solutions vanish. Similarly, when σ  is small, k 2j = −β/σ  is large; consequently, the m = 2 integrals can be simplified using (12.27) with kr → k (2) j , leading to (17.15) with θ (t) → exp (−iσ t).

17.3 Dynamics

467

What is the vertical structure of the currents driven by this special τ x ? As noted in the previous paragraph, to a good approximation Eqs. (17.15) hold even at the equator. ¯ /Hn . Note that vn in (17.15) depends on n only through the factor Hn−1 in Fn = (τ x /ρ) Since Hn−1 = Z n (Eq. 5.17), it follows that after summing over all the modes the vertical structure of v is Z(z), that is, v is contained entirely within the wind-forced layer. This shallow structure is evident in Fig. 17.1 and Videos 17.1a and 17.1b.

17.3.4.3

Observations

How similar are off-equatorial Ekman transports to the equatorial Sverdrup transport inferred from observed winds? To explore this question, Schott et al. (2002) reported values for the NCEP and ERS-1/2 winds noted above. For NCEP winds, the annual-mean Ekman transports across 3◦N and 3◦S are –6.9±3.6 Sv and –6.0±3.9 Sv, respectively, and the cross-equatorial Sverdrup transport is –6.5±0.9 Sv. The simiy larity of the values suggests that: the contribution of τx in the western ocean to the cross-equatorial Sverdrup transport is small; and that τ x is nearly proportional to y from 3◦S to 3◦N. The same averages for the ERS-1/2 winds yield Ekman transports of –7.4±4.0 Sv and –9.3±6.4 Sv and an across-equatorial Sverdrup transport of –6.4±2.6 Sv; the larger differences among the values possibly result either from y the contribution of τx being larger or from τ x for the ERS-1/2 winds having more curvature near the equator.

17.3.4.4

Videos Forced by ZW

Video 17.6a shows the solution forced by ZW when T (t) = θ(t) in a closed basin. Southward flow develops almost instantly throughout the basin, readily crosses the equator, and is contained entirely within layer 1. These properties indicate the flow field is largely an Ekman/Sverdrup response to the antisymmetric part of τ x . Because Y (y) has curvature away from the equator, however, ZW also has Ekman pumping; it thins (thickens) h 1 in the northern (southern) hemisphere, generating eastward (westward) geostrophic currents. As a result, the pathway of the southward flow is modified into an arc. In response to alongshore winds along the northern boundary, h 1 rapidly becomes less than He there (white line), and coastal upwelling begins (Sect. 13.2.2 and the right panel of Figure 13.2 with y → −x). Later, Ekman pumping thins and thickens h 1 to He and Hd in the interior ocean (closed white curves), ensuring that upwelling and subduction occur there as well. Throughout the integration, upwelling dominates subduction, owing to the strong northern-boundary upwelling; consequently, h 1 continuously thickens, as indicated by the shading in the eastern ocean changing from green to orange as time passes. Averaged over the last year of the integration, the background h 1 field is thicker by h 1 = 22.1 m. At this time, the system is close to, but not yet at, equilibrium, as the annual-mean upwelling and subduction transports, We = 9.24 Sv and Wd = −8.98 Sv, don’t quite balance.

468

17 Cross-Equatorial and Subtropical Cells

As for the SET-forced solution, the near-equilibrium response in Video 17.6a can be viewed as consisting of two parts, which in this case cover the entire basin: a wind-forced Sverdrup circulation, and an overturning cell driven by we ≈ wd . The cross-equatorial flow has contributions from both parts. In layer 2, it occurs almost entirely in the western-boundary current, with transports V2w = 9.16 Sv and V2e = −0.06 Sv west and east of xwb = 45◦E. Consider a box in layer 2 that covers the northern hemisphere. Mass conservation in the box requires that V2 = V2w = We ,

(17.17)

a statement V2w supplies all the water for the northern-hemisphere upwelling. In agreement with (17.17), the values of V2w and We in the solution are close. In layer 1, the cross-equatorial transports within and east of the western-boundary region are V1w = 10.58 Sv and V1e = −19.68 Sv. Consistent with theory, V1e is close  xe tox the ¯ zonally-integrated, cross-equatorial Sverdrup transport VS = −H1 / (ρβ) xwb τ y d x = −19.61 Sv. Mass conservation in a northern-hemisphere, layer-1 box then leads to the balance (17.18) V1 = V1e + V1w = VS + V1w = −We . According to (17.18) V1w = |Vs | − We , a statement that V1w provides the water for the horizontal gyre that is not supplied by upwelling; its value in the solution is close to this theoretical prediction (10.37 Sv). Video 17.6b is similar to Video 17.6a, except with T (t) given by (17.4) so that ZW has an annual cycle. During each summer, the response develops a southward Ekman/Sverdrup response, and h 1 thins and thickens in opposite hemispheres in response to Ekman pumping. During the rest of the year, the regions of thin and thick h 1 propagate westward as packets of free Rossby waves. Because Rossby waves propagate slowly at the higher latitudes, however, the packets cannot propagate across the basin before the following summer; consequently, thinning and thickening of h 1 intensifies for several years before reaching an equilibrium state. Eventually, h 1 thickens to Hd in the southern hemisphere and subduction begins there. As for Video 17.6a, the system is still not at equilibrium after 20 years of integration, with We = 1.87 Sv, Wd = −0.87 Sv, and h 1 = 18.7 m. During the last year of the integration, the annual-mean, cross-equatorial transports are V2w = 1.42 Sv, V2e = 0.009 S, V1w = 4.86 Sv, and V1e = −6.28 Sv. Consistent with (17.17), V2e is close to We , the difference between them due to the solution not yet being in equilibrium. Sverdrup Consistent with (17.18), V1e is close to the annual-mean,  xe across-basin x ¯ τ d x = −6.24 Sv, and transport for the annual-mean winds, V¯ S = −H1 / (ρβ) y xwb V1e + V1w = −1.42 Sv is close to −We .

17.3 Dynamics

17.3.4.5

469

Video Forced by SET + FJ + ZW

Video 17.7 shows the solution forced by SET + FJ + ZW in a closed basin, when T (t) is given by (17.4) for FJ and ZW. In the steady-state response, the annual-mean mass balance         Wes h 1 + Wer h 1 + Wen h 1 + Wd h 1 = 0,

(17.19)

holds.  Inthe solution, the four terms during the last year  of the integration  are Wes h 1 = 3.62 Sv, Wer h 1 = 7.33 Sv, Wen h 1 = 2.81 Sv, Wd h 1 = −13.74 Sv, with h 1 = −12.1 m; their sum is very small (0.02 Sv), indicating that the response is near equilibrium. The solution contains an STC and CEC. The STC is not much changed from the one forced by SET alone (Video 17.3b), except that its strength Wer is somewhat weaker. The CEC differs in both the strength and structure of its cross-equatorial flows from that forced by FJ alone (Video 17.5b). In agreement with (17.17), V2w = 6.180 Sv is close to the northern-hemisphere entrainment Wes + Wen = 6.43 Sv. Transport V2e = 0.23 Sv, however, is significantly different from zero, its expected value, largely due to the zonal Munk layer on the northern edge of the SET-driven is close circulation (Sect. 17.3.2.2). As in (17.18), V1e = −11.85 Sv  to the annual xe y x mean, Sverdrup transport east of xeb , VS = H1 / (ρβ) d x = −12.10 ¯ xwb τx − τ y Sv. Transport VS has two parts, one driven by τx (FJ ) and the other by τ yx (ZW). The across-equatorial transport due to FJ is VF J  = H1 / (ρβ) ¯ τ y (xwb , 0) = −5.86 ◦  Sv, and it is confined west 60 E, the eastern edge of FJ . The transport forced by ZW, VZ W = −6.24 Sv, is spread across the basin. y

17.3.5 Equatorial Roll Wacongne and Pacanowski (1996) first reported an equatorial roll in a solution to their Indian-Ocean OGCM. They hypothesized that it was a response to forcing by cross-equatorial τ y during the monsoons (Figure 3.1, top-left panel; Video 3.1), an idea consistent with the idealized OGCM solution of Philander and Delecluse (1983). McCreary (1985) noted that equatorial rolls forced by τ y existed in solutions to simpler models (the LCS model and constant-density, surface-layer model of Stommel 1960), and compared them to roll in the Philander and Delecluse (1983) solution. Such rolls are also apparent in Videos 17.1a and 17.1b. Miyama et al. (2003) explored roll dynamics further in a suite of solutions to the LCS model forced by a τ y wind patch symmetric about the equator, with different thicknesses of the body-force structure Z(z) and mixing strength γ = γ  = A/cn2 . Figure 17.6 shows 6 members from the set. For each solution, Z(z) is the ramp function in (C.6) and z 1 = h m /2, the thickness of the constant layer above the ramp. The left panels of Fig. 17.6 illustrate the sensitivity of the roll to z 1 with A = 52×10−4

470

17 Cross-Equatorial and Subtropical Cells

Fig. 17.6 Meridional sections of v from LCS solutions forced by antisymmetric τ y , showing the responses for different values of mixing strength A and mixed-layer thickness z 1 = h m /2. All the solutions are obtained in a domain that extends from 20◦S to 20◦N, 40–100◦E, with realistic boundaries. The background Nb (z) used to determine the vertical modes is determined from the annualmean density field in the JAMSTEC solution averaged from 5◦S to 5◦N and 40–100◦E (similar to the observed Nb profile in Figure 5.1), and Z(z) is the ramp function in (C.6) with z 1 = h m /2. The wind has the form (C.7) except with X = sin [2π (x − xm ) /x] θ and Y → sin [2π (y − ym ) /y] θ; y parameter choices are τo = 1 dyn/cm2 , x = 40◦ , y = 20◦ , and x m = (70◦E, 0◦ ). In the left panels from top to bottom, A = 52×10−4 cm2 /s3 and z 1 = 10, 40, and 80 m. In the right panels from top to bottom, z 1 = 10 m and A = 0.8, 6.5, and 208×10−4 cm2 /s3 . The contour interval for v is 5 cm/s, and negative contours are shaded

cm2 /s3 , and the right panels show its dependence on A with z 1 = 10 m. As z 1 increases (left panels), the roll thickens and weakens. As A decreases (right panels and top-left panel), the roll attains a maximum strength when A = 52×10−4 cm2 /s3 , weakens monotonically for smaller values, and vanishes completely when A = 0 (not shown). Recall that without mixing ( A = 0) the steady-state response is a Sverdrup flow, which for forcing by southerly winds is a clockwise circulation around the wind region (Sect. 11.1 and Video 14.1c); in this solution, although there is cross-equatorial y flow where τx = 0, its vertical structure is Z(z) with no subsurface reversal. Observations: Given its sensitivity to mixing processes in models, does the equatorial roll exist in the real ocean? Schott et al. (2002; their Sect. 7) reported evidence

Video Captions

471

for the roll in the western basin, based on shipboard ADCP sections taken during the summer monsoon of 1995. In addition, based on a climatology of surface drifters, they noted that the longitudinally-averaged profile of v had the expected variation of a roll: During winter (December–February), v was northward on both sides of the equator and southward on the equator, whereas during summer (June–September) v was southward to either side of the equator and zero on the equator. The authors also reported observations where the roll was not evident. Horii et al. (2013), based on more than 8 years of equatorial current-meter observations at 80.5 and 90◦E, reported that during the summer there were subsurface southward currents at depths from 10–80 m (30–115 m) at 90◦E (80.5◦E); further, northward surface flow was clearly observed at 80.5◦E but not at 90◦E. In summary, there is evidence for the existence of equatorial rolls during some seasons and some locations, but not all. Given the sensitivity of the roll to mixing strength and mixed-layer thickness, this variation is expected.

Video Captions Two-Dimensional Overturning Video 17.1a Solutions to the JAMSTEC (left panel) and LCS (right panel) models, forced by climatological Hellerman and Rosenstein (1983) wind stress, obtained in a realistic basin (upper panels of Fig. 17.2), and averaged from 50–90◦E. The background stratification ρb (z) for the LCS model is the horizontal and temporal average of density from the JAMSTEC solution over the near-equatorial ocean (5◦S– 5◦N, 40–100◦E). The ocean depth is D = 4000 m; the vertical mixing strength is A = A = 52×10−4 cm2 /s3 ; and the number of vertical modes is N = 50, a value sufficient to ensure the solution is well converged. (Courtesy of T. Miyama, 2003.) Video 17.1b As in Video 17.1a, except showing solutions to the LCS model forced by both τ x and τ y (left panel; Solution C10), τ x alone (middle panel; Solution C11), and τ y alone (right panel; Solution C10–C11). The LCS solution forced by τ x and τ y is the same solution as in the right panel of Video 17.1a, except plotted over an expanded region (Courtesy of T. Miyama, 2003.)

SET Forcing Video 17.2 Solution to the 2 21 -layer model forced by a switched-on vertical velocity we that transfers water from layer 2 into layer 1, showing h 1 = h 1 − H1 (m) and layer-1 currents (cm/s) in the left panel and h = h − H (m) and layer-2 currents (cm/s) in the right one. The amplitude of we is weo = 5×10−4 cm/s and its spatial structure is given by (C.7b) with x m = (85◦E, 12.5◦S), x = 20◦ , and y = 5◦ . To ensure that the western-boundary current is easily visible, the horizontal viscosity coefficient is set to νh = 5×107 cm2 /s. The Coriolis force is given by the equatorial

472

17 Cross-Equatorial and Subtropical Cells

β-plane approximation, and open conditions are imposed at the eastern boundary as described in Appendix C. Video 17.3a [I3a]As in Video 17.2, except forced by a switched-on τ x . Its amplitude is τox = −1.0 dyn/cm2 . Its spatial structures are centered on x m = (xm , ym ) = (85◦E, 12.5◦S), with:   x − xm x 2 2 X (x) = cos 2π θ − (x − xm ) , x 4

(17.20)

and x = 60◦ (∞) if x ≥ xm (x < xm ); and Y (y) is a modified version of (C.7b) with y = 20◦ (35◦ ) if y ≥ ym (y < ym ). With these definitions, τ x is present everywhere in the basin south of 2.5◦S. The white curves indicate the h 1 = −50 m and 100 m contours within which entrainment and detrainment are active. Video 17.3b As in Video 17.3a, except that the eastern boundary is closed.

FJ Forcing Video 17.4a Solution to the 2 21 -layer model forced by a switched-on τ y , showing h 1 = h 1 − H1 (m) and layer-1 currents (cm/s) in the left panel and h = h − H y (m) and layer-2 currents (cm/s) in the right one. The wind amplitude is τo = 10 dyn/cm2 , X (x) = 1, and Y (y) = cos 2π

  y − ym y 2 θ − (y − ym )2 , y 4

(17.21)

where ym = 15◦N and y = 40◦ . With these choices, the wind forcing is a band that extends from 5◦N–25◦N. The white line along the western boundary indicates where h 1 = −50 m and, hence, the existence of entrainment (coastal upwelling). Video 17.4b As in Video 17.4a, except X(x) is given by (C.7b) with xm = 40◦E and x = 40◦ . Video 17.4c As in Video 17.4b, except with T (t) given by (17.4). Video 17.5a Solution to the 2 21 -layer model forced by a sum of the winds in Videos 17.3a and 17.4c. The eastern boundary is closed. Video 17.5b As in Video 17.5a, except the parameters of (17.21) are changed to ym = 10◦N and y = 60◦. With these choices, τ y extends from 5◦S–25◦N.

Video Captions

473

ZW Forcing Video 17.6a Solution to the 2 21 -layer model forced by a switched-on τ x , showing h 1 = h 1 − H1 (m) and layer-1 currents (cm/s) in the left panel and h = h − H (m) and layer-2 currents (cm/s) in the right one. The wind amplitude is τox = 1.5 dyn/cm2 . Structure X(x) is the same as in Video 17.3a. Structure Y (y) has three parts, ⎧ y ⎪ sin 2π , y≥0 ⎪ ⎪ ⎪ yn ⎪ ⎨ y ym < y < 0 Y(y) = γ sin 2π y , (17.22) s ⎪   ⎪ ⎪ γ y − y m ⎪ ⎪ 1 + cos 2π , y ≤ ym ⎩− 2 y where yn = 200◦ , ys = 60◦ , γ = ys /yn , and ym = −15◦ . The choice of γ ensures that Y y is continuous across y = 0. The white line along the northern boundary indicates where h 1 = −50 m and, hence, entrainment (coastal upwelling) is active, and the white curve in the southern hemisphere designates the h 1 = 100 m contour within which detrainment occurs. Video 17.6b As in Video 17.6a, except with T(t) given by (17.4). Video 17.7 Solution to the 2 21 -layer model forced by a sum of the winds in Videos 17.3a, 17.4c, and 17.6b. The eastern boundary is closed.

Appendix A

List of Acronyms

AABW AAIW ACC ADCP ADT AGCM ALACE AMOC APDRC Argo ASHSW AVISO BBD BBSW CEC CGCM CSIR CUC CTD CWEIO DJF DOC EACC EArCC ECMWF EEIO EEPO

Antarctic Bottom Water Antarctic Intermediate Water Antarctic Circumpolar Current Acoustic Doppler Current Profiler Absolute Dynamic Topography Atmospheric General Circulation Model Autonomous Lagrangian Circulation Explorers Atlantic Meridional Overturning Cell Asian Pacific Data Research Center Array for Real-time Geostrophic Oceanography Arabian Sea High Salinity Water Archiving, Validation, and Interpretation of Satellite Oceanographic Data Bay of Bengal Dome Bay of Bengal Surface Water Cross Equatorial Cell Coupled General Circulation Model Council of Scientific and Industrial Research Coastal Undercurrent Conductivity Temperature Depth Central/western Equatorial Indian Ocean December/January/February Deep Overturning Cell in the Indian Ocean East Africa Coastal Current East Arabia Coastal Current European Center for Medium-range Weather Forecast Eastern Equatorial Indian Ocean Eastern Equatorial Pacific Ocean

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9

475

476

EICC EIO ENSO ERS EUC FJ FJ GDP GRACE IATTC IR IDW IEW IIW IIOE IOD IPRC ITCZ ITF ITFW JAMSTEC JAMSTEC JJA JJAS KPP KT LCDW LCS LH LL LPS MDT MLT MKM MJO MOM MY NASHSW NADW NASA NCAR NCEP NECC NEMC

Appendix A: List of Acronyms

East Indian Coastal Current Equatorial Indian Ocean El Niño Southern Oscillation ESA European Remote-sensing Satellite Equatorial Undercurrent Findlater Jet for 2 21 -layer model extended Findlater Jet for 2 21 -layer model Global Drifter Program Gravity Recovery and Climate Experiment Inter-American Tropical Tuna Commission Infrared Indian Deep Water Indian Equatorial Water Indonesian Intermediate Water International Indian Ocean Expedition Indian Ocean Dipole International Pacific Research Center Intertropical Convergence Zone Indonesian Throughflow Indonesian Throughflow Water Japan Marine Science and Technology Center OGCM used in Miyama et al. (2003) June/July/August June/July/August/September K-profile mixed-layer parameterization Kraus-Turner mixed-layer model Lower Circumpolar Deep Water Linear, Continuously Stratified model Lakshadweep High Lakshadweep Low Low pressure system Mean Dynamical Topography Mixed Layer Thickness 2 21 -layer model used in Miyama et al. (2003) Madden-Julian Oscillations Modular Ocean Model Mellor/Yamada mixed-layer model North Arabian Sea High Salinity Water North Atlantic Deep Water US National Aeronautics and Space Administration US National Center for Atmospheric Research US National Climate for Environmental Prediction North Equatorial Counter Current Northeast Madagascar Current

Appendix A: List of Acronyms

NICW NIO NIO-Goa NIOSS NOAA NPISV NSF OFES OGCM OLR PBL PCC PGW PP PV QBM QuikSCAT PWP RAMA RSW SC SCOW SeaWiFS SEC SECC SEMC SET SETR SICW SJC SJUC SLA SLD SLP SMC SSH SSS SST STC TC TIW TMI TMPA TOMS

477

North Indian Central Water North Indian Ocean National Institute of Oceanography, Goa, India NIO-Goa summer school in 2010 US National Oceanic and Atmospheric Administration Northward-Propagating Intraseasonal Variability US National Science Foundation Ocean Model For the Earth Simulator Ocean General Circulation Model Outgoing Longwave Radiation Planetary Boundary Layer Pakistan Coastal Current Persian Gulf Water Pacanowski/Philander mixing parameterization Potential Vorticity Quasi-Biweekly Mode NASA quick scatterometer Price-Weller-Pinkel mixed-layer model Research Moored Array for Afr./Asian/Aust. Monsoon Analysis and Pred. Red Sea Water Somali Current Scatterometer Climatology of Ocean Winds NASA Sea-viewing Wide Field of view Sensor South Equatorial Current South Equatorial Counter Current Southeast Madagascar Current Southeast Trades for 2 21 -layer model South Equatorial Thermocline Ridge South Indian Central Water South Java Current South Java Undercurrent Sea-Level Anomaly Sri Lanka Dome Sea-Level Pressure Summer Monsoon Current Sea Surface Height Sea Surface Salinity Sea Surface Temperature Subtropical Cell Tropical Cell Tropical Instability Waves TRMM Microwave Imager TRMM Multi-satellite Precipitation Analysis 4 21 -layer model used in Miyama et al. (2003)

478

TRMM UGOS VGOS WEPO WICC WKB WJ WMC WOCE ZGV ZW

Appendix A: List of Acronyms

Tropical Rain Measuring Mission Zonal geostrophic velocity Meridional geostrophic velocity Western Equatorial Pacific Ocean West Indian Coastal Current Wentzel-Kramers-Brillouin approximation method Wyrtki Jet Winter Monsoon Current World Ocean Circulation Experiment Zero Group Velocity Zonal wind for 2 21 -layer model

Appendix B

Simplified LCS Equations

In Sect. 7.4 and Chaps. 12–14, we obtained solutions to simplified versions of the LCS modal Eqs. (9.1), which neglect either u t , vt , or both terms. Under what restrictions does their neglect still allow for accurate solutions? In the main text, we determined validity criteria informally by comparing v-equations derived from the exact and approximate equation sets. Here, we determine them formally, by carrying out a scale analysis of (9.1). The approach first writes all variables and coordinates of (9.1) in non-dimensional form by the replacement q → [Q] q, ˆ where [Q] is the scale (magnitude) of q and qˆ is the non-dimensionalized variable, which leads to a scaling [Sm ] for each term in a particular equation. Then, one of the scalings [S0 ] is divided into the others [Sm ], thereby defining a set of non-dimensional parameters πm = Sm /S0 . Finally, depending on restrictions appropriate for the physical situation under consideration, the magnitude of each πm can be seen to be significant (order 1) or negligible (order   1).

B.1

Non-dimensional Parameters

Scaled versions of Eqs. (9.1) are  

   U P ˆ uˆ tˆ − [ f V ] fˆvˆ + pˆ xˆ = Fo F, T Lx

(B.1a)

   V P ˆ pˆ yˆ = G o G, vˆtˆ + [ f U ] fˆuˆ + T Ly

(B.1b)



P T c2



    pˆ t  U V uˆ xˆ + vˆ yˆ = 0, + 2 cˆ Lx Ly

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9

(B.1c) 479

480

Appendix B: Simplified LCS Equations

where variable scalings Q are obviously defined and each term is multiplied by a factor [Sm ]. In each equation, we choose [S0 ] to be the scaling of a term  that is always important in the overall balance, namely, [ f V ], [ f U ], and V /L y . (Other choices are possible.) Dividing by each [S0 ] gives  

     1 U P Fo ˆ ˆ pˆ xˆ = uˆ tˆ − f vˆ + F, fT V f V Lx fV

(B.2a)

     P Go ˆ 1 V ˆ pˆ yˆ = vˆ ˆ + f uˆ + G, fT U t f U Ly fU

(B.2b)



P Ly T c2 V



  Ly U pˆ tˆ uˆ xˆ + vˆ yˆ = 0, + cˆ2 Lx V

(B.2c)

defining 8 non-dimensional πm parameters.

B.2

Parameter Choices

The next step is to set the parameter magnitudes that apply to each of the simplified equation sets. There are general scalings that apply to them all, as well as specific ones applicable for each set. For notational simplicity, we drop brackets from the parameters.

B.2.1

General Scalings

In all the equation sets, P P = = 1, f V Lx f U Ly

Go Fo = = 1, fV fU

Ly U = 1. Lx V

(B.3)

The first and second expressions state that geostrophic and Ekman balances are O(1) processes. (Another scaling for the Ekman terms assumes that Fo /( f V ) = G o /( f U ) = , resulting in the “quasi-geostrophic” equations.) The last scaling states that u x contributes to the velocity divergence as much as v y , a restriction needed to recover familiar wave solutions. Finally, in all the simplified equation sets 1 = , fT

(B.4)

a statement that the time scale T of the response is much longer than an inertial period.

Appendix B: Simplified LCS Equations

B.2.2

481

Interior-Ocean Scaling

In the interior ocean, there is no process that establishes a difference between meridional and zonal, spatial scales; further, both scales are large with respect to the Rossby radius of deformation R = c/ f . These restrictions lead to the scalings √ R R = = , Lx Ly

Ly = 1, Lx

(B.5)

√ the reason for the choice , rather than , is evident in (B.7). From the last relation in (B.3) and (B.5), it follows that U = V . Then, with the aid of (B.4), the magnitudes of the uˆ tˆ and vˆtˆ terms in (3.2) are U =1 V

1 U 1 V = = , fT V fT U

⇒

(B.6)

both negligibly small. Using the first of Eqs. (B.3), (B.4), and (B.5), the magnitude of the pˆ tˆ term is f V Lx Ly 1 Lx Ly P Ly = = = 1, (B.7) 2 2 Tc V Tc V f T R2 that is, O(1). So, under scalings (B.4) and (B.5) the interior-ocean equation set is (9.1) with u t  = vt  = 0, that is, set (12.1) solved in Chap. 12.

B.2.3

Coastal-Ocean Scaling

Consider an eastern or western coast, which can be oriented either meridionally or at an angle α (Fig. 7.5). In the latter case, we assume that the y-axis is parallel to the coast and x-axis is directed across-shore. Near coasts, the across-shore scale of the circulation is often coastally trapped with a scale of the order of the Rossby radius, whereas its alongshore scale, set by the wind forcing, is large. These restrictions lead to the scalings L x = R,

Lx = . Ly

(B.8)

Using the last relation in (B.3) and (B.8), the magnitudes of uˆ tˆ and vˆtˆ are then U = V

⇒

1 U = 2, fT V

1 V = 1, fT U

and with Eqs. (B.3), (B.4), and (B.8), the magnitude of pˆ tˆ is

(B.9)

482

Appendix B: Simplified LCS Equations

P Ly 1 Lx Ly = 1. = 2 Tc V f T R2

(B.10)

Under scalings (B.4) and (B.8), then, the simplified equation set is (9.1) with only u t = 0, namely, Eqs. (13.2) solved in Chap. 13.

B.2.4

Equatorial-Ocean Scaling

Near the √ equator, the√meridional scale L y of circulations is the equatorial Rossby radius c/β (L y = c/β ≡ R) and, provided R is defined by this value, general scalings (B.3) still√hold.√ Similarly, a measure of f near the equator is the equatorial inertial frequency cβ ( cβ ≡ f = 0), which, using c1√= 265 cm/s from Table 5.1, corresponds to an equatorial inertial period of P1 = 2π/ c1 /β = 9.4 days. In Chap. 14, we are interested in times much longer than P1 , and hence (B.4) applies. On the other hand, in Chap. 15 and Sect. 16.1 we consider forcing periods of this order, (B.4) doesn’t apply, and solutions must be found to the unapproximated equations. The zonal scale of the response L x set by the wind is typically much larger than L y and, with this restriction, appropriate scalings are L y = R,

Ly = . Lx

(B.11)

Using the last relation in (B.3) and assuming that (B.4) and (B.11) hold, the magnitudes of uˆ tˆ and vˆtˆ are then V = U

⇒

1 U = 1, fT V

1 V = 2, fT U

(B.12)

and from (B.3), (B.4), and (B.11) the magnitude of pˆ tˆ is P Ly 1 Lx Ly = = 1. T c2 V f T R2

(B.13)

Therefore, under scalings (B.4) and (B.11) the equations of motion simplify to (9.1) with vt = 0, that is, the long-wavelength Eqs. (14.1) solved in Chap. 14.

Appendix C

Video Overview

In this overview, we report general properties that hold for most videos. Exceptions needed for specific videos are provided in their captions at the end of each chapter. All the videos are downloadable from the websites noted in Sect. 1.3. Numerical Model: The videos are numerical solutions to u nt − f vn + pnx =

  τx − γn u n + νh ∇ 2 u n + −γd u n − γk u n , ρH ¯ n

(C.1a)

τy − γn vn + νh ∇ 2 vn + [νhd vnx x ] , ρH ¯ n

(C.1b)

vnt + f u n + pny =

  pnt pn pn pn + u nx + vny = −γn 2 + −γd 2 − γk 2 , cn2 cn cn cn

(C.1c)

a version of the LCS modal Eqs. (5.16) with γn = γn = A/cn2 , κh = 0, and constant νh . For simplicity, and to be consistent with their analytic counterparts, solutions to (C.1) are obtained in Cartesian coordinates. They are obtained on a C-grid, that is, with p, v, and u located at the center, northern edge, and eastern edge of square grid boxes of dimension x = y ≡ ξ . The model time step t is adjusted to be as large as possible but still to satisfy the CFL condition t < ξ/c1 . For most solutions, νh = γn = 0. Exceptions are for: some solutions with a closed western boundary, where νh = 0 to ensure the layer has a finite width; and multimode solutions of undercurrents, in which γn = 0 is needed so that they develop a realistic vertical structure. The additional mixing terms in brackets are dampers (described below), which are included in some solutions to eliminate (weaken) radiation from or along basin boundaries. The baroclinic modes used are determined from either the constant or realistic Nb (z) profiles in Figure 5.1. For each profile, values of cn , Hn , and other parameters are listed in Table 5.1 for a number of modes. Most solutions show the response of © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9

483

484

Appendix C: Video Overview

the n = 1 baroclinic mode, exceptions being for the multimode solutions in Chap. 16. Coriolis Force: Solutions are obtained using three different parameterizations of the Coriolis parameter f . One choice sets f to a constant value f = 2 sin θo . Another choice is the midlatitude β-plane approximation, f = f o + βo (y − yo ) ,

(C.2)

where f o = 2 sin θo , yo = Re θo , and βo = (2 /Re ) cos θo . In both versions, θo is set to the middle latitude of the solution domain. A final choice is the equatorial β-plane approximation, f = βy, where β = 2 /Re ; it is the special case of (C.2) with θo = 0. For all solutions, Re = 6370 km so that β = 2.28 × 10−13 cm−1 s−1 . Boundary Conditions: All solutions are obtained in rectangular basins. In most cases, closed, no-slip conditions are imposed on all sidewalls, the exception being for a few solutions in which cyclic conditions are applied that link the eastern and western boundaries. Generally, basins with a zonal width L x ≥ 80◦ have a grid step of ξ = 0.25◦ , where ξ is x or y, and for smaller basins ξ = 0.1◦ . Dampers: In many solutions, we would like the eastern boundary, western boundary, or both boundaries to be open, that is, to pass incoming radiation without any reflections. To simulate an open boundary, damping regions are included within the basin that absorb the incoming radiation with minimum reflections. Specifically, for an open eastern (western) boundary, the domain is expanded eastward (westward) by L d , and dampers are added to the u n and pn equations in these regions. Let either boundary be located along x = 0. Then, the dampers include the γd terms in (C.1a) and (C.1c), where γd (x) =

2  j=1

γ j

 δx j − |x|  θ δx j − |x| , δx j

(C.3)

a “double-ramp” damper in which the two dampers differ markedly in slope and strength. (Moore and McCreary, 1990, used this damper to minimize reflections from an eastern-ocean boundary, the gently-sloped damper absorbing short-wavelength Rossby waves with little reflection into long-wavelength ones, and the steep one eliminating Kelvin and Yanai waves before they reach the eastern boundary.) In most solutions, L d = 20◦ , the steeply-sloped damper has δx1 = 500 km and γ1 = cn /(86.2 km), and the gently-sloped one has δx2 = 1500 km and γ2 = cn /(750 km). In some equatorial solutions, this damper still allowed significant reflected some waves; to minimize these reflections, the damper was smoothed (broadened and weakened) by replacing the above parameters with L d = 40◦ , γ1 = cn /(500 km), δx2 = 4000 km, γ2 = cn /(2000 km). For some solutions with a closed western boundary, it is useful to weaken the offshore radiation of short-wavelength Rossby waves but otherwise not to impact the interior flow field. For this purpose, we include the term νhd vx x in the vn equation, where

Appendix C: Video Overview

485

vhd (x) = νho cos π



x − xw δx θ xw + −x , δx 2

(C.4)

νho = 5×106 cm2 /s, xw is the longitude of the western boundary, and δx = 10◦ . According to (C.4), νhd weakens offshore from a maximum of νho = 5×106 cm2 /s at the coast to zero δx degrees offshore. For some solutions with a closed, eastern boundary, it is useful to damp Kelvin waves along the northern and southern boundaries of the domain. Let either boundary be located at y = 0. Then, dampers proportional to γk are added to the u n and pn equations where δy − |y| γk (x) = γko θ(δy − |y|) , (C.5) δy −1 = 0.5 days, and δy = 0.5◦ . According to (C.5), the dampers weaken linearly to γko zero a distance δy offshore. Wind Forcing: Wind stress enters the ocean as a body force with the vertical structure ⎧ 1, z ≥ −h m /2 2 ⎨ −h m ≤ z < −h m /2 , 2 (z + h m ) / h m , Z(z) = (C.6) 3h m ⎩ 0, z < −h m

where h m = 100 m. According to (C.6), Z(z) is constant from the surface to a depth of 50 m, decreases linearly to zero at 100 m, and its integral over the water column is 1. The values of Hn listed in Table 5.1 are determined from this Z(z) using (5.17). Unless specified otherwise, the wind-stress forcing has the form τ α = τoα X(x) Y (y) T (t) ,

(C.7a)

where α = x or y, the default value of τoα = 1.5 dyn/cm2 , the spatial structure functions are 

 2 x − xm x 1 2 1 + cos 2π θ − (x − xm ) , X(x) = 2 x  4  (C.7b) 2 y − ym y 1 1 + cos 2π θ − (y − ym )2 , Y(y) = 2 y 4 and T (t) is either θ(t) or sin (σ t) θ(t) so that the wind switches on to a constant value or an oscillation. Other choices for X and Y are described in the captions. In particular, most videos in Chap. 13 set X(x) = 1, so that forcing is by an xindependent band of winds. Figure C.1 illustrates the above forcing structures for zonal (left panel) and meridional (right panel) winds when τoα = θ (t) = 1. According to (C.7b), the forcing is confined inside a rectangular region with sides of length x and y and its center at x m = (xm , ym ) (black dot). Structures X(x) and Y (y) are indicated by the blue and red curves on the eastern and northern edges of the right panel, respectively.

486

Appendix C: Video Overview

Fig. C.1 Schematic diagram illustrating the structures of most τ x (left panel) and τ y (right panel) fields. They are confined inside rectangular regions with their center at x m = (xm , ym ) (black dot) and sides of length x and y. Functions X(x) and Y(y) are indicated by the blue and red curves. Amplitudes of the wind-stress vectors (arrows) are X Y . The wind-curl structures (shading) are −y X Y y (left panel) and x X x Y (right panel), which increase from zero at the edges to a maximum amplitude of π . The shading interval is 0.5

Also plotted are the wind-curl structures (shading): −y X Y y for τ x (left panel) and x X x Y for τ y (right panel), where factors y and x are included so that the amplitude of the structures is π . The wind-stress vectors (arrows) are proportional to X Y . Vectors: In videos that include vectors (α,β), the plotted vectors are usually (α,β  ), where L x Py φ= , (C.8) β  = φβ, L y Px L x and L y are the lengths of horizontal and vertical axes in plot units, and Px and Py are the axis lengths as they appear in the video image. With this choice for φ, slopes of plotted vectors are appropriate for the image. The exception is for wind vectors in videos of solutions that are forced by a single wind component: In these videos, no correction on β = τ y is needed. In a few videos, the range in vector magnitudes is very large. To ensure that all vectors are visible, they are plotted using the square root of their amplitude, that is, vectors are replaced by     α, β  (C.9) α, β  →  1 . α 2 + β 2 4 1  According to (C.9), the amplitudes of the vectors are reduced to α 2 + β 2 4 but their direction is unchanged.

Appendix C: Video Overview

487

In videos with vectors, a calibration vector is located in the lower-left corner of   1  each video frame. It is labelled either “ α, β  ” or, if (C.9) is applicable, “ α, β  2 .” In solution videos, the label for wind calibration vectors is “(τ x , τ y )” since correction (C.8) is not made to τ y . Video Format: Most videos show sea  level and current vectors for the n = 1 mode, d1 = p1 /g (contours) and u1 = u 1 , v1 (arrows). Videos that show multimode solutions are summed over either N = 25 or 99 modes, depending primarily on how visible truncation noise is the solution and on the vertical scale of its prominent features. A time counter, located in the top-left corner of each video, indicates the passage of each video frame. The time interval between frames differs among the videos, depending on the time scale of the highlighted process. Video Labels: Within the text, video labels have one of the forms C.n or C.nα, where C is the chapter number and n is an integer. Several videos that highlight a specific process form groups, in which case n is the group label and α (a, b, c, etc.) indicates a specific video within the group. The video files themselves have labels C.n (text) or C.nα (text), where “text” provides information about their parameters.

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Index

A Ageostrophic currents, 94, 138, 139, see Ekman drift/flow Air-sea fluxes, 43–45 evaporation, 44 latent heat, 44 measurements of, 43 sensible heat, 44 wind stress, 43 Andaman Sea, 107–114 Antarctic Bottom Water, see Water masses Antarctic Intermediate Water, see Water masses Arabian Sea, 128–147 Arabian Sea High Salinity Water, see Water masses

B Baroclinic modes continuous, 414 discrete, 99, 100, 147, 166, 173, 176, 180, 199, 204, 225, 254, 255, 274, 303, 314, 315, 317, 319, 329, 330, 335, 338, 354, 363, 406, 413, 420, 424, 432, 433, 483, 484 Barotropic mode, 110, 166, 173, 176, 181, 255, 303, 329 Barrier layer, 67, 69 Bay of Bengal, 114–125 Bay of Bengal Surface Water, see Water masses Beams coasts, 133, 135, 136, 419, 420

equator, 103–105, 417–419, 422 boundary reflections, 427 Kelvin waves, 424 Rossby waves, 426 Yanai waves, 425 observations, 428 properties, 414 phase propagation, 416 ray paths, 416 velocity vectors, 417 β-dispersion, 206, 280, 283, 289, 310, 344 β-plume, 459–461 Book overview goals, 5 organization, 5 videos, 6 Boundary conditions cyclic, 484 far field, 232, 246 interior ocean, 309 open, 484 sidewalls, 166, 211, 246, 255, 265, 275, 300, 304–306, 308, 336, 338, 346, 350, 351, 443, 449, 453, 457, 484 surface and bottom, 165, 170, 171, 173, 175, 276, 279, 292, 294, 443 Boundary layers, 255, 303, 311 approximation method, 304 eastern boundary, 308 interior ocean, 309 northern and southern boundaries, 308, 310 observations, 310

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 J. P. McCreary and S. R. Shetye, Observations and Dynamics of Circulations in the North Indian Ocean, Atmosphere, Earth, Ocean & Space, https://doi.org/10.1007/978-981-19-5864-9

513

514 western boundary, 303, 307, 310, 348, 349, 379, 397, 483 Boundary waves Kelvin-wave limit, 211, 353, 354, 396 Rossby-wave limit, 214, 353, 354, 396 Bounded Yoshida Jet, see Yoshida Jet Boussinesq approximation, 165 β-plane Kelvin waves, 210–214, 240, 352, 355, 378, 396, 397 Breakdown of analytic solutions, 319, 329, 344, 348 C Characteristic speed, cn definition, 171 dependence on modenumber, 172 impact on damping, 175 values, 173, 180, 237, 248, 426, 456 Coastal processes 2-d adjustment, 337 Kelvin-wave radiation, 339 Rossby-wave radiation eastern boundary, 343, 350 western boundary, 350 Continental shelf/slope, 5, 108, 121, 140, 170, 334, 343, 432, 434, 443 Convection atmosphere, 2, 16, 18–20, 22, 24, 26–30, 33, 35, 37, 40, 91, 142 SST requirement for, 13, 16, 22, 24, 27, 29, 40, 49, 91 ocean, see Subduction Convergent series LCS model, 176, 275, 289, 389, 432 Coordinate systems Cartesian coordinates, 164, 483 non-dimensional, 237, 277, 292, 479 OGCMs, 166, 177 rotated, 222 Critical frequency equator, 239, 247, 386, 407, 422 midlatitudes, 199, 210, 211, 225 observations, 225 Critical latitude equator, 239, 240, 378, 396, 398 midlatitudes, 203, 204, 206, 207, 210, 211, 213, 214, 220, 221, 223, 329, 352, 355, 356, 396, 400, 403, 419, 420, 422 observations, 225, 404 slanted coast, 223–225, 401, 403 Cross Equatorial Cell, 51, 71, 73, 95, 441, 442, 444, 446, 449–452, 454, 455, 457, 458, 462, 465, 469

Index 2-d overturning, 443 3-d pathways, 446–452 dynamics, 462–465 observations, 455, 467 transports, 444, 452, 454 Cross-equatorial flow atmosphere, 1, 20, 35, 382 conceptual ideas, 465 dynamics, 465–469 importance of mixing and forcing, 182, 461, 465, 466 ocean, 143, 145, 446, 454, 455, 461, 465, 467–470 theory, 466 D Dirac δ-function, 257, 262, 265, 273, 316, 343, 364, 366, 367, 431 Detrainment, 55, 67, 178, 179, 452, 456, 458, 459, 461 Dispersion relations equator Kelvin wave, 244 modification by background current, 409 non-dispersive Rossby waves, 240, 363, 376, 466 Rossby/gravity waves, 236, 387 vertically-propagating waves, 415 Yanai wave, 243 general, 189, 190, 193 midlatitudes gravity waves, 199, 282 in variable medium, 203 Kelvin wave, zonal coast, 208 meridional coast, 210 non-dispersive Rossby waves, 201, 206, 314, 329, 344 Rossby/gravity waves, 196, 197 Rossby waves, 200, 344, 348 Rossby waves, short-wavelength approximation, 345, 348 vertically-propagating Kelvin wave, 415, 419 Diurnal variability, see Sub-annual variability Downwelling, 55, 65, 117, 159, 431, 433, 435, 436, 446, 457 overturning cell, 149, 442, 444, 447, 452, 457 Downwelling-favorable waves, 91, 92, 106, 109, 111, 114, 123, 125, 135, 147, 152, 157, 357, 382, 462, 463

Index Drifter currents, 84, 85, 92, 94, 97, 110, 114, 117, 118, 120, 124–127, 129, 132, 138, 139, 447–451, 471 E East Africa Coastal Current, see Surface currents East Arabia Coastal Current, see Surface currents East India Coastal Current, see Surface currents Eddies, 3, 5, 7, 55, 67, 93, 116, 118, 119, 138, 139, 151, 153, 157–159, 167, 176, 225, 227, 305, 310, 348, 404, 432 Ekman drift/flow divergence/convergence, 55, 91, 280, 288, 290, 318, 338, 430, 435, 455 equatorial solutions LCS model, 286, 287, 296, 366, 373, 393, 435 z-dependent, 290, 295 mixed-layer turbulence, 54 midlatitude solutions, constant- f LCS model, 274, 275, 310, 316–320 spiral, 276, 277, 279, 283, 291, 315 z-dependent, 275, 279 midlatitude solutions, variable- f β-dispersion, 280 initial response, 281 LCS model, 322, 325, 327, 328, 334, 335, 337–341, 343, 349, 352, 356, 358, 362, 422, 430, 433 meridional energy propagation, 282 observations, 21, 63, 67, 91, 110, 112, 115, 124, 126–129, 139, 141, 283, 295 transport, 139, 288, 301, 341, 349, 455, 463, 464, 467 Ekman pumping equatorial solutions, 288, 290, 296, 362, 364, 366, 372–374, 381, 393 midlatitude solutions, coastal, 337, 338, 356 midlatitude solutions, open ocean, 55, 280, 310, 313, 315, 318, 320, 322, 325, 327, 328, 330, 434, 460, 463, 467, 468 observations, 21, 50, 65, 91, 92, 114, 118, 123–125, 128, 138, 141, 144, 151, 152, 159, 280, 295, 321 velocity wek , 91, 301, 317, 318, 465 Ekman suction, see Ekman pumping Ekman/Sverdrup flow, see Sverdrup circulation

515 El Niño Southern Oscillation (ENSO), see Interannual variability Entrainment, 21, 22, 24, 55, 57, 59, 60, 62, 63, 65, 140, 153, 178, 179, 319, 338, 341, 343, 452, 456–463, 469 Equations of motion 1 21 -layer model, 178, 180, 181, 226, 302, 317–319, 335, 343, 357, 358, 456 linear, 180 nonlinear, 178 2 21 -layer model, 6, 164, 177, 178, 303, 455, 456 equations, 456 requirements for overturning, 458 coastal ocean, 334 equatorial ocean, 362 interior ocean, 313 LCS modal equations, 173 LCS model, 167 OGCM, 164 scale analysis of, 479–482 Equatorial basin resonance, 48, 99, 382, 405–407 observations, 406 Equatorial region, 95–105 Equatorial roll dynamics, 469 models, 444, 446, 451, 452, 469 observations, 470 Equatorial waves dispersion relations, 236, 243, 244 inertial frequency, 236 meridional structure, 235, 243, 245 observations, 247 reflections of, 375–381, 395–398 chain rule, 376, 379, 395–397 ringing period, 381, 405, 406 relationship to midlatitude waves, 246 Rossby radius, 232 Evanescent waves, 188, 189, 197, 200, 207, 210, 237, 238, 240, 246, 378, 396, 408, 420, 422–424

F Forcing, climatological fluxes, 48–53 evaporation minus precipitation, 50 heat flux, 51–53 wind stress, 48 wind-stress curl, 50 Fourier transform, 190, 241, 258–260, 266, 268, 328, 386, 389, 392 transform pairs, 241, 266, 268, 269, 388

516 Frequency definition, see General wave Freshwater fluxes, 21, 67, 119, 147 land, 46 precipitation, 45 rivers, 46

G General wave, 186 dispersion relation, 189, 190, 193 frequency, 187 group velocity, 192 uniform medium, 191 variable medium, 192 period, 188 phase velocity, 188 wavelength, 188 wavenumber, 187 Geostrophic currents, 92, 94, 110, 151, 157, 280, 466, 467 adjustment to, 206 alongshore balance, 335 from sea-level observations, 90 Gravity waves atmosphere, 23, 37 inertial oscillations, see Inertial oscillations ocean, see Rossby/gravity waves Great Whirl, see Sea-level features Group velocity definition, see General wave equator Kelvin waves, 245, 371, 373 Rossby/gravity waves, 240, 387, 389, 393, 395 Rossby waves, 227, 375 Yanai waves, 243, 375 meridional propagation, 217, 282, 401 midlatitudes Kelvin waves, 209, 212, 339 non-dispersive Rossby waves, 201, 344 Rossby/gravity waves, 201, 215, 460 Rossby waves, short-wavelength approximation, 345, 348 vertically propagating waves, 416 zero-group-velocity waves, 227, 242, 248, 407–409 Gulf of Aden, 157–160 Gulf of Oman, 151–153

Index H Hadley circulation, 12, 13, 15, 16, 30 Hermite functions, 189, 232, 234, 239, 246, 284–286, 289, 361–363, 368, 378, 386, 389, 392, 394, 395, 427 recursion relations, 233 structures of, 234 turning latitude, 234 Hyrdostatic approximation, 165, 167, 179, 200

I Indian Deep Water, see Water masses Indian Equatorial Water, see Water masses Indian Ocean Dipole, see Interannual variability Indonesian Intermediate Water, see Water masses Indonesian Throughflow Water, see Water masses Inertial oscillations equator, 73, 148, 236, 242, 284, 286, 287, 289, 296, 366, 380, 435, 466 midlatitudes, 54, 274–277, 289, 315, 334, 341 observations, 274, 283 Initial conditions, 256, 260–262, 265, 268, 273, 276, 286, 290–292, 317, 321, 325, 339, 345, 368, 369 Instabilities barocinic, 117 baroclinic, 168 barotropic, 117 convective, 13, 16, 24, 33, 54 Kelvin-Helmholtz, 54, 58 low-pressure systems, 23 MJOs, 34 Rayleigh-Bernard, 54 shear, 56, 58, 63, 141, 158, 227, 247, 310, 400 Somali Current, 105, 404 Interannual variability, 2–4, 25, 82, 91, 92, 96, 97, 102, 117, 118, 123, 133, 142, 144 ENSO dynamics, 27 history, 26 impacts, 28 IOD connection to ENSO, 31 properties, 28 Intertropical Convergence Zone

Index annual migration, 14 global properties, 12, 13 South Asian annual migration, 15, 16 Hadley circulation, 15 MJO impact, 35 SST impact, 15 Intraseasonal variability, 2, 3, 31–33, 35, 63, 92, 96, 97, 104, 105, 107, 110, 111, 114–116, 118, 123, 127, 132, 133, 135, 136, 149, 204, 226, 243, 247, 385, 394, 404, 429 Andaman Sea, 110 Bay of Bengal, 115 East India coast, 118, 123 equator, 104 Persian Gulf, 149 South Equatorial Thermocline Ridge, 92 Summer Monsoon Current, 127 West India Coastal Current, 132, 136 Inverse Fourier transform, 241, 258, 259 Inverse Laplace transform, 260 K Kelvin waves atmosphere, 29, 33 equator dispersion relation, 244 meridional structure, 245 midlatitudes meridional coast, 210, 211, see βplane Kelvin waves slanted coast, 222 zonal coast, 208 L Lakshadweep High and Low, see Sea-level features Laplace transform, 260 transform pairs, 260, 292, 346 Layer models, 177–181 1 21 -layer model, see Equations of motion 2 21 -layer model, see Equations of motion LCS model, 167–177 Longwave radiation, 29, 41, 42, 49, 51, 53 Lower Circumpolar Deep Water, see Water masses M Madden-Julian oscillations, see Sub-annual variability

517 Marginal seas, 72, 147 Meridional energy propagation from boundaries, 217, 218, 353, 398 interior ocean, 217, 280 observations, 225 Midlatitude waves dispersion relations, 196, 197, 199–201, 203, 208, 212 inertial oscillations, 274, 275 observations, 225 Rossby radius, 199, 200, 208, 210 Mixed layer Arabian Sea, 63–67 basin-wide properties, 58–63 salinity, 61 temperature, 59 thickness, 58 Bay of Bengal, 67–69 models, 55–58 Kraus-Turner (KT) model, 56 Price-Weller-Pinkel (PWP) model, 58 processes, 53–55 convective overturning, 54, 73 detrainment, 55 dynamical processes, 55 entrainment, 55 turbulence, 54 wind mixing, 54, 73 Mixed Rossby/gravity wave, see Yanai wave Mixing between water masses, 78, 79, 82, 153, 160 boundary layers/currents, 20, 255, 304, 307–311, 348, 349, 379, 397, 432, 435, 456 convective overturning, 72, 140 entrainment, 319, 338 horizontal, 165, 254, 255, 303–305, 307, 308, 310, 311, 336, 349, 397, 400, 408, 431–433, 435, 449, 456, 461 impact on group velocity, 193 Laplacian, 308, 349 LCS modal equations, 175 γ and γ  , 175, 176, 178 LCS model, 169, 170 ν and κ parameterizations, 169 mixed layer, see Mixed layer, processes OGCMs, 165 planetary boundary layer, 20 resonance, 408 undercurrents, 431

518 vertical, 165, 254, 277–279, 283, 297, 305, 319, 408, 425, 430–432, 444, 449, 469, 470 Monsoon Currents, 125–128 Multi-layer models, 181 Munk layer interior ocean, 309, 461 meridional, 217, 306–308, 310, 349, 380, 397 zonal, 308–310, 469

N Nonlinear effects, 65, 116, 141, 146, 176, 226, 227, 247, 310, 343, 348, 357, 400, 404, 434 terms, 5, 116, 176, 178, 180, 181, 218, 226, 304, 432 North Arabian Sea High Salinity Water, see Water masses North Atlantic Deep Water, see Water masses North Indian Central Water, see Water masses

O OGCMs, 4, 5, 35, 116, 125, 128, 146, 148, 149, 151, 156, 164–167, 169, 176, 177, 181, 216, 255, 358, 420, 431, 444, 447, 469

P Pakistan Coastal Current, see Surface currents Period definition, see General wave Persian Gulf, 82, 148–151 Persian Gulf Water, see Water masses Phase general wave, 186 plane wave, 186 Phase velocity definition, see General wave equator Kelvin waves, 245 Rossby/gravity waves, 240 Yanai waves, 243, 375 midlatitudes Kelvin waves, 209, 212 non-dispersive Rossby waves, 201 Rossby/gravity waves, 201 vertically propagating waves, 416

Index Plane wave, 186, 187, 189, 190, 192, 196, 202, 208, 217, 231, 232, 235, 282 Planetary boundary layer, 20, 39, 43 Potential vorticity, 20, 181, 182, 449 Precipitation active and break periods, 34 climatology, see Videos connection to ITCZ, 12 evaporation minus precipitation, 50, 54 impact of ENSO, 28 impact of IOD, 31 linkage to SST, 22 low-pressure systems, 22 measurements of, 45 orographic, 18 Q Quasi-biweekly mode (QBM), see Subannual variability R Radiation amplitude, 328, 330, 354, 393 Radiation condition, 255, 261, 263, 266, 268, 269, 323, 327, 336, 339, 351, 368, 459 Radiative fluxes back radiation, 42 downward radiation, 40 Ray paths from boundaries, 216, 218, 219, 221, 223, 224, 256, 355, 356, 398–400, 402, 403 general, 191, 192 meridional propagation, 217, 218, 398 vertically propagating waves, 136, 416, 427 Red Sea, 153–157 Red Sea Water, see Water masses Rigid-lid approximation, 166, 171–173 River-driven circulations relationship to wind-driven coastal solutions, 357 Rossby/gravity waves equator dispersion relation, 236 non-dispersive Rossby waves, 240 spatial structure, 235 midlatitudes dispersion relation, 196, 197, 203 non-dispersive Rossby waves, 201, 314, 353 Rossby radius

Index equator, 199, 232, 255, 297, 482 midlatitudes, 105, 199, 200, 208, 210, 232, 255, 422, 481 Rossby waves atmosphere, 19, 24, 29, 35 ocean, see Rossby/gravity waves Runoff, river and land, 2, 21, 39, 45–48, 61– 63, 67, 83, 109, 117, 119, 125, 147, 148 Russell, Bertrand, 7

S Sea-level features Andaman Sea, 109 Bay of Bengal Dome, 124 Great Whirl, 143 Lakshadweep High, 128 Lakshadweep Low, 129 overview, 86 Socotra Eddy, 144 South Equatorial Thermocline Ridge, 90 Southern gyre, 143 Sri Lanka Dome, 124 Shelf waves, 3, 5, 225 Shortwave radiation, 39, 41, 42, 49, 51 Slanted coast, 221, 356 Socotra Eddy, see Sea-level features Solution methods direct approach, 208, 211, 262, 267, 272, 285, 300, 316, 322, 336, 338, 342, 351, 364 Fourier transform, 259, 265, 268, 386 Laplace transform, 260, 265, 292, 293, 345 Somali Current, see Surface currents South Asian summer monsoon impact of SST and ocean processes, 21 non-orographic rainfall, 22 precipitation and orography, 18 winds, 19 South Equatorial Countercurrent, see Surface currents South Equatorial Current, see Surface currents South Equatorial Thermocline Ridge, see Sea-level features South Indian Central Water, see Water masses South Java Current, see Surface currents Southern Gyre, see Sea-level features Southern-hemisphere circulations, 90–95 Sri Lanka dome, see Sea-level features

519 Stommel layer interior ocean, 309 meridional, 305, 349, 380, 397 zonal, 308, 309 Stommel/Arons model, 78 Streamfunctions 2-d overturning, 443, 444, 454 horizontal flow, 300 seasonally varying, 446 Sub-annual variability Diurnal variability, 37 MJOs, 32 dynamics, 34 impacts, 34 QBM, 35 Sub-weekly oscillations, 36 Sub-weekly oscillations, see Sub-annual variability Subduction, 72, 73, 77, 79, 81, 95, 140, 436, 442, 444, 449, 452, 453, 455–457, 461, 462, 465, 467, 468 Subtropical Cell, 72, 73, 91, 436, 442, 444, 447, 449–452, 455, 457–459, 462, 469 2-d overturning, 443 3-d pathways, 446 dynamics, 458–462 observations, 455 transports, 444 Sumatra/Java coast, 105–107 Summer Monsoon Current, see Surface currents Surface currents East Africa Coastal Current, 93–95, 143– 145, 303, 449, 452, 454 East Arabia Coastal Current, 128, 140, 141, 157, 159, 303 East India Coastal Current, 62, 69, 114, 116–121, 123–125, 127, 129, 130, 135, 303 overview, 84 Pakistan Coastal Current, 128, 137 Somali Current, 2, 35, 55, 79, 82, 95, 105, 128, 142–147, 160, 303, 404, 455 South Equatorial Counter Current, 90, 94, 95, 97, 147, 303, 446, 449, 450, 454, 462, 465 South Equatorial Current, 90, 92–95, 303, 449, 450, 462 South Java Current, 106 Summer Monsoon Current, 62, 63, 124, 126–128, 147

520 West India Coastal Current, 62, 69, 127– 132, 135–138, 147 Winter Monsoon Current, 62, 126–128, 147 Wyrtki Jets, 21, 96–101, 106, 110, 124, 125, 127, 128 Sverdrup circulation 1 21 -layer model, 302 adjustment to, 201, 303, 320, 324, 325, 327, 330, 344, 373, 381, 431, 432, 463 body-force layer, 467, 470 definition depth-integrated, 177, 299 LCS model, 300 Ekman/Sverdrup flow, 466–468 observations, 88, 94, 146, 303, 330 solutions, 301, 305, 307–309, 324–326, 328, 329, 343, 362, 368, 371, 373, 374, 377, 379–381, 393, 432, 461, 466, 468 transport, 467–469 T Temperature inversions, 55, 60, 67, 69 Thermohaline structure bottom layer, 74 deep layer, 77 intermediate layer, 79 overview, 72 upper layer, 80 U Undercurrents conceptual explanations, 430 dynamics, 431 observations, 101, 106, 119, 123, 126, 127, 131, 132, 143–147, 430, 436 equatorial, 101–104 relationship to beams, 430 solutions, 177 coasts, 433 equator, 434 Upper Circumpolar Deep Water, see Water masses Upwelling coastal, 58, 308, 337, 340, 341, 343, 356, 430, 433, 434 East Arabia, 21, 24, 55, 60, 81, 82, 95, 141, 146, 177, 357, 447, 449, 450, 454 Indian east coast, 21, 118, 357 Indian west coast, 21, 135, 357 Pakistan, 137, 140

Index Somalia, 21, 24, 55, 60, 81, 82, 95, 142, 143, 146, 147, 177, 357, 444, 447, 449, 450, 454 deep ocean, 78 ENSO, 26, 92 equatorial, 110, 430, 435 Indian Ocean, 100, 101, 110 Pacific and Atlantic, 442 Pacific and Atlantic Oceans, 49 IOD, 30, 92 open ocean, 65, 302, 320 overturning cell, 91, 442, 444, 447, 452, 455, 457, 460 South Equatorial Thermocline Ridge, 81, 91, 92 Sri Lanka, 21, 81, 95, 127, 447, 449, 450, 454 Upwelling-favorable waves, 27, 91, 101, 109, 114, 124, 157, 159, 357, 382 Upwelling-favorable winds, 135, 137, 337, 339, 340, 356, 357, 444 V Vertical normal modes, 167, 170, 171 Vertically propagating waves dispersion relations, 415 group velocity, 416 phase velocity, 416 ray paths, 416 reflections of, 419 structure, 415 velocity vectors, 417 Videos download site, 6 observations sea level and surface currents, 90, 94, 109, 110, 112, 115, 117, 120, 123, 124, 126–129, 135, 137–139, 141–144, 151, 152, 157–160, 330 sea level anomaly, 3–5, 7, 21, 86, 97, 98, 106, 109, 144, 166, 167, 225, 321, 357, 381, 382, 404, 405 SST, 16, 21, 29, 38, 357 WICC currents, 133, 160 wind and rain, 18, 19, 21–23, 37, 50, 51 wind stress and curl, 48, 50, 69, 109, 125, 135, 137–139, 141, 157, 321, 330, 357, 382, 457, 469 solutions beams, coastal, 420, 421, 436, 437 beams, equatorial, 407, 422, 424– 429, 436–439

Index CEC and STC, 2-d overturning, 471 CEC and STC, FJ forcing, 472 CEC and STC, SET forcing, 303, 471, 472 CEC and STC, ZW forcing, 472, 473 coastal ocean, 334, 341, 344, 345, 348–350, 354, 355, 358, 359, 379 Ekman drift and inertial oscillations, 274, 275, 280–283, 286–290, 296–298, 341 equatorial ocean, periodic forcing, 117, 394–398, 400, 404–411 equatorial ocean, switched-on forcing, 374, 377–384 equatorial waves, 242–246, 248–250 interior ocean, 317–319, 321, 325, 326, 329–331, 344 midlatitude waves, 200, 205–207, 209, 213, 214, 216, 220, 221, 225–229, 245 Sverdrup flow and boundary currents, 301–303, 307, 310, 312 undercurrents, coastal, 433, 434, 436, 439 undercurrents, equatorial, 274, 435, 436, 440 W Walker circulation, 26, 27, 29 Water masses Antarctic Bottom Water, 74 Antarctic Intermediate Water, 79 Arabian Sea High Salinity Water, 83 Bay of Bengal Surface Water, 83 Indian Deep Water, 77 Indian Equatorial Water, 82 Indonesian Intermediate Water, 80 Indonesian Throughflow Water, 82 Lower Circumpolar Deep Water, 74 North Arabian Sea High Salinity Water, 83 North Atlantic Deep Water, 77 North Indian Central Water, 82 Persian Gulf Water, 82 Red Sea Water, 80 South Indian Central Water, 80 Upper Circumpolar Deep Water, 77 Wave groups impact of mixing, 193 non-uniform medium, 192

521 uniform medium, 189 Wavelength definition, see General wave Wavenumber definition, see General wave West India Coastal Current, see Surface currents Whitman, Walt, 6 Wind forcing, 256 Wind stress bulk formula for, 45 climatological, 48, see Videos friction velocity, 57 general structure in solutions, 485 Wind-stress curl, 50 climatological, see Videos Winds climatological, see Videos connection to ITCZ, 12 ENSO, 28 IOD, 31 orographic impacts, 1, 20 summer monsoon, 19 Winter Monsoon Current, see Surface currents WKB approximation, 195, 202, 246, 414 Wyrtki Jets, see Surface currents

Y Yanai waves dispersion relation, 238, 243 dynamics, 36 group velocity, 243, 375 meridional structure, 243 observations, 35, 36, 105, 247, 403 reflections of, 378, 379 Yoshida Jet, 271, 284, 290, 291, 296, 297, 381, 435 bounded Yoshida Jet, 265, 295, 371–374, 377, 382, 394, 435 equatorial trapping, 288 observations, 247, 295

Z Zero-group-velocity resonance, 407–409 observations, 227, 248, 408 Zero-group-velocity waves, 242, 407 observations, 248