Heaving, Stretching and Spicing Modes: Climate Variability in the Ocean [1st ed.] 9789811529405, 9789811529412

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Rui Xin Huang

Heaving, Stretching and Spicing Modes Climate Variability in the Ocean

Heaving, Stretching and Spicing Modes

Rui Xin Huang

Heaving, Stretching and Spicing Modes Climate Variability in the Ocean

123

Rui Xin Huang Woods Hole Oceanographic Institution Woods Hole, MA, USA

ISBN 978-981-15-2940-5 ISBN 978-981-15-2941-2 https://doi.org/10.1007/978-981-15-2941-2

(eBook)

Co Publisher ISBN: 978-7-04-054255-4 Jointly published with Higher Education Press The print edition is not for sale in Chinese mainland. Customers from Chinese mainland please order the print book from: Higher Education Press. © Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain 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

For his pioneering and profound contributions to modern physical oceanography His 1962 paper opened up the field of spicity and its application —Dedicated to the Centennial Anniversary of Henry M. Stommel

Preface

This book is a collection of topics related to some fundamental aspects of the wind-driven circulation and climate change. We discuss methods used in the water mass analysis and the climate study; in particular, we discuss the heaving modes induced by the adiabatic adjustment of the wind-driven circulation to wind stress perturbations. First, we can use simple reduced gravity models. Second, we can examine the climate variability in terms of vertical coordinates based on water properties, such as the potential density, the potential temperature and the salinity. These vertical coordinates based on the material properties of water parcels are Lagrangian coordinates. The combination of such material coordinates with the traditional fixed Eulerian coordinates in the horizontal directions gives rise to the Eulerian–Lagrangian hybrid coordinates, which can be used to identify heaving signals from climate data generated from observations or computer models. Some of the previous studies of climate based on isopycnal analysis were focused on climate variability on isopycnal surfaces. The promise of this approach is that climate signals can be separated into two components: vertical movement of isopycnal layers and water property changes on isopycnal surfaces. However, it is more accurate to study the climate variability associated with isopycnal layers; i.e., climate signals are examined in terms of changes of the layer depth, the thickness, the spicity and others. This method is called isopycnal layer analysis. Water mass analysis has been the backbone of physical oceanography, and the most commonly used tools are the Theta-S diagram and the isopycnal analysis. Recently, a potential spicity function whose contours are orthogonal to those of potential density has been defined. This opens up a new approach based on the sigma–pi diagram, which can be used as an additional tool in the water mass analysis. The sigma–pi diagram may provide new insights for the water mass analysis, the ocean circulation and the climate change. In particular, the introduction of an orthogonal coordinate system makes it possible to define the distance in the parameter space. With the exact measure of distance, many aspects of the water mass analysis and the climate change can be accurately quantified. These basic materials have been presented through numerous seminars in many institutions and discussed in some workshops, including “Climate Variability in the World Oceans: Heaving and Isopycnal Analysis” (Institute of Oceanography, CAS, Qingdao, China, March 2016) and “Workshop on Adiabatic Motions, Heaving Modes and Isopycnal Layer Analysis” (Xiamen vii

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Preface

University, Xiamen, China, April 2017). I am grateful for the encouragement from my colleagues and young friends. I have been benefited from discussions with my colleagues both in the USA and in China. Drs. Xiangsan Liang, Jim Price, Bill Dewar, Bo Qiu, Ray Schmitt, Dezhou Yang and Shengqi Zhou read parts of the draft and provided many critical suggestions. Dr. Xiaolin Zhang, Ms. Mary Zawoysky and Dr. Ruihe Huang read the early version of the whole manuscript and pinpointed numerous mistakes. Drs. Joe Pedlosky, Quanan Zheng and Zijun Gan have been a continuous source of scientific stimulation. Woods Hole, USA March 2020

Rui Xin Huang

Contents

1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Roles of Wind in Climate Variability . . . . . . . . . . . . . . 1.2 Main Thermocline in the World Oceans . . . . . . . . . . . . 1.3 Reduced Gravity Model, Advantage and Limitation . . . 1.3.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Reduced Gravity in the World Oceans . . . . 1.4 Layer Outcropping: The Physics and the Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

1 1 2 6 6 11

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12 16

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2 Climate Variability Diagnosed from the Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Climate Variability Diagnosed in the z-Coordinate . . . . . . . . 2.2 External/Internal Modes in Meridional/Zonal Directions . . . . 2.2.1 Heat Content Anomaly . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Salinity Anomaly. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Density Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Adiabatic Signals in the Upper Ocean . . . . . . . . . . . . . . . . . 2.3.1 Adiabatic Adjustment in the Upper Ocean . . . . . . . . 2.3.2 Adiabatic Wave Adjustment in the Meridional Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Regulation of MOC (MHF) by Wind Stress and Buoyancy Anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Surface Density Anomaly . . . . . . . . . . . . . . . . . . . . . 2.4.3 Correlation Between Surface Forces and MOC . . . . . 2.4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Adiabatic Heaving Signals in the Deep Ocean . . . . . . . . . . . 2.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Heaving, Stretching, Spicing and Isopycnal Analysis . . . . 3.1 Heaving, Stretching and Spicing Modes . . . . . . . . . . . . 3.1.1 Adiabatic and Isentropic Processes . . . . . . . . . . 3.1.2 Heaving, Stretching and Spicing Modes . . . . . . 3.1.3 External Heaving Modes Versus Internal Heaving Modes . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 28 28 33 35 39 41 43 46 46 46 48 52 53 57 59

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61 61 61 62

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Contents

3.1.4 Wave Processes Related to Adiabatic Internal Heaving Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Local Versus Global Heaving Modes . . . . . . . . . . . . 3.2 Potential Spicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Define Potential Spicity by Line Integration . . . . . . . 3.2.3 Define Potential Spicity in the Least Square Sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Solve the Linearized Least Square Problem . . . . . . . 3.2.5 Potential Spicity Functions Based on UNESCO EOS-80 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Potential Spicity Functions Based on UNESCO TEOS_10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 r–p Diagram and Its Application . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Meaning of Spicity . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Density Ratio Inferred from the Density–Spicity Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 The r–p Plane as a Metric Space . . . . . . . . . . . . . . . 3.4 Isopycnal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 The Lagrangian Coordinate . . . . . . . . . . . . . . . . . . . . 3.4.2 Isopycnal Analysis in the Eulerian Coordinate . . . . . 3.4.3 Isothermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Heaving Modes in the World Oceans . . . . . . . . . . . . . . . . . . . . . 4.1 Heaving Induced by Wind Stress Anomaly . . . . . . . . . . . . . 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 A Two-Hemisphere Model Ocean . . . . . . . . . . . . . . . 4.1.3 A Southern Hemisphere Model Ocean . . . . . . . . . . . 4.1.4 Adiabatic MOCs of the World Oceans with Rectangular Basins . . . . . . . . . . . . . . . . . . . . . . 4.1.5 MOC/MHF Simulated by a RGM in the World Oceans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Heaving Induced by Anomalous Freshwater Forcing . . . . . . 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Model Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Results from Numerical Experiments . . . . . . . . . . . . 4.2.4 Experiment for 40 Year Continuing Freshening of the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Heaving Induced by Anomalous Wind, Freshening and Warming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 A Simple Generalized Reduced Gravity Model . . . . . 4.3.3 Numerical Experiments Based on This Reduced Gravity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 70 73 73 74 76 78 79 83 84 84 94 110 137 138 147 152 159 161 161 161 165 175 184 190 195 195 198 198 208 209 209 209 210

Contents

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4.4 Heaving Induced by Convection Generated Reduced Gravity Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Model Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Results from Numerical Experiments . . . . . . . . . . . . 4.4.4 Numerical Experiments with Sinusoidal Reduced Gravity Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Heaving Induced by Deep Convection Generated Volume Loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Results of Numerical Experiments. . . . . . . . . . . . . . . 4.6 ENSO Events and Heaving Modes . . . . . . . . . . . . . . . . . . . . 4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Variability of Heat Content and Horizontal Heat Fluxes Due to ENSO Diagnosed from the GODAS Data . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Meridional Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . 4.6.4 Zonal Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5 Vertical Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.6 A Two-Hemisphere Model Ocean Simulating ENSO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Heaving Signals in the Isopycnal Coordinate . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Casting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 FDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 MDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Separating the Signals Into External and Internal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Statistics in the Density Space . . . . . . . . . . . . . . . . . . 5.2.5 External Signals in Terms of Layer Thickness . . . . . 5.3 Projecting Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Difference Between the Casting Method and the Projecting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Isopycnal Layer Analysis for the World Oceans . . . . . . . . . . 5.5.1 External Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Heaving Modes for r1 ¼ 30:9  0:05 kg/m3 . . . . . . . 5.5.3 Horizontal Distribution of Climate Variability for r1 ¼ 30:9  0:05 kg/m3 . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 The Heaving Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Regional Anomaly Patterns . . . . . . . . . . . . . . . . . . . . 5.5.6 A Meridional Section Through 60.5° W . . . . . . . . . . 5.5.7 A Zonal Section Along the Equator . . . . . . . . . . . . . 5.5.8 A Zonal Section Along 45.17° N . . . . . . . . . . . . . . . 5.6 Isopycnal Layer Analysis Based on r0 . . . . . . . . . . . . . . . . .

216 216 218 219 230 230 230 234 234 241 241

242 245 248 251 257 261 263 263 265 265 266 267 269 270 272 274 276 276 280 283 285 288 291 300 305 308

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Contents

5.7 Heaving Signals for the Shallow Water in the Pacific-Indian Basin . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Application of the Casting Method to the GODAS Data . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Isopycnal Layer Analysis of the Equatorial Dynamics Based on Projecting Methods . . . . . . 5.8 Heaving Signal Propagation Through the Equatorial Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Connection Between the MDC and the FDC . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Climate Signals in the Isohaline Coordinate . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Casting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 FSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 MSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Separating the Signals into External and Internal Modes . . . 7.3.1 FSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 MSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Analysis Based on the GODAS Data . . . . . . . . . . . . . . . . . . 7.5 Shallow Salty Water Sphere in the Atlantic Ocean . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

373 373 374 375 376 376 377 377 377 379 386

6 Heaving Signals in the Isothermal Coordinate . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Casting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 FTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Casting Method Applied to the GODAS Data . . . . . . . 6.3.1 The Choice of Temperature Scale . . . . . . . . . . . 6.3.2 Statistics in the Temperature Space . . . . . . . . . . 6.4 Projecting Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Isothermal Layer Analysis for the Layer of h ¼ 20  0:5  C . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Structure in the Pacific Basin . . . . . . . . . . . . . . 6.5 Signals of Layer Depth and Zonal Velocity in the Pacific Basin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Z-Theta Diagram and Its Application to Climate Variability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Connection Between the MTC and the FTC . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

1

Basic Concepts

1.1

Roles of Wind in Climate Variability

The ocean is subjected to the following forces: wind, heat/freshwater fluxes and tidal dissipation. In the old paradigm the thermohaline circulation is driven by surface thermohaline forcing. However, in the new paradigm the oceanic circulation is a dissipation system, maintenance of which requires external sources of the mechanical energy from the wind and the tidal dissipation. From another angle, the air-sea interaction can be classified in two categories: mechanical and thermohaline. Mechanical interaction includes the atmospheric pressure and the momentum flux at the sea level. Thermohaline interaction includes different kinds of heat flux and freshwater flux through the air-sea interface. In a broad framework, it should also include other tracer (oxygen and other nutrients) fluxes through the air-sea interface. Therefore, from both angles wind forcing plays key roles in regulating oceanic stratification and circulation, and this is the focus in our study. In a quasi-steady state, the wind stress plays two important roles. First, the wind stress drives currents in the upper ocean. The circulation in the upper ocean is mostly due to the dynamical effect of wind stress. In physical oceanography, these currents are collectively called the wind-driven circulation (Fig. 1.1a). Wind-driven circulation has been discussed in many classical papers and textbooks. Second, the wind stress provides the mechanical energy which is vitally important for

maintaining mixing in the ocean, including the mixed layer and the subsurface ocean. Mixing in the mixed layer affects the air-sea heat/freshwater fluxes; thus, it is also a key factor regulating the thermohaline circulation (Fig. 1.1b). For the time-dependent problems, the wind stress also plays a key role in regulating the general circulation. In the ocean interior, changes in the wind stress can induce the basin-scale adiabatic movement of the isopycnal layers. For example, if the wind stress is reduced, the main thermocline in the basin interior becomes shallower and moves upward, and the corresponding isopycnal layers above the main thermocline move upward; such motions of isopycnal layers are called heaving. In general, the heaving motions are indicated by the deformation of the isopycnal surfaces from the black curve to the red curve in Fig. 1.1c. Heave and heaving mode can induce large variability in terms of temperature and salinity at fixed points in the ocean interior. Therefore, they are the dominant sources of climate variability and are also the primary focus of this book. In addition, wind stress changes can lead to the shifting of the outcrop lines, depicted as the transition from the black curve to the blue curve in Fig. 1.1c. The shifting of the outcrop line and changes of the mixed layer properties lead to different partitions of the water mass formation/erosion in the density category, these processes are the most important factors inducing the thermohaline circulation variability in the world oceans.

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_1

1

2

1 Basic Concepts

Wind

Wind

Wind Surface mixing

Subsurface mixing (a) Driving the currents

(b) Maintain mixing

(c) Heaving/shifting of outcrop lines

Fig. 1.1 Roles of wind stress in the oceanic general circulation and climate change

In summary, the wind stress changes can induce many aspects of changes in the oceanic circulation; it is a great challenge to identify such changes from climate data. However, among many kinds of climate signals the heaving motions induced by the wind stress change alone are the dominating component; so that it is desirable to identify such signals. The most clear-cut manifestation of heaving signals is the shifting of the main thermocline in the ocean caused by wind stress changes. Such signals often propagate in the forms of Rossby waves and Kelvin waves. The common wisdom in the theory of the oceanic general circulation is that meridional overturning circulation (MOC) and the meridional heat flux (MHF) are directly linked to the diabatic heating/cooling across the air-sea interface and interior isopycnal/isothermal surfaces. Similarly, the thermohaline circulation in the ocean also induces the zonal overturning circulation (ZOC), the zonal heat flux (ZHF), and the vertical heat flux (VHF). However, adiabatic movements produced by wind stress anomalies on seasonal, annual, interannual and decadal time scales can generate anomalous MOC, MHF, ZOC, ZHF, and VHF. As will be shown in this book, about 85%– 90% of the climate variability identified from climate data sets obtained from observations or numerical simulations is heaving in nature. Thus, studying the role of heaving motions in the ocean and separating the contributions from heaving and non-heaving motions are vitally important for our understanding of the climate variability in the real world.

1.2

Main Thermocline in the World Oceans

The ocean is stratified; however, the stratification in the ocean is not uniform. In fact, there is a layer of strong vertical temperature gradient in the ocean, which is called the main thermocline. The suffix “cline” means a layer with a strong vertical gradient. For example, there are thermoclines, pycnoclines, haloclines and lysoclines. The pycnocline is a layer with a maximum density gradient, and the halocline is a layer associated with a maximum salinity gradient. The lysocline is a term used in geology, geochemistry and marine biology to denote the depth in the ocean below which the rate of dissolution of calcite increases dramatically. There are many types of thermoclines, including the diurnal thermocline, the seasonal thermocline and the permanent (main) thermocline. By definition, the main thermocline is a layer with a maximum temperature gradient. In order to identify the main thermocline, it is much more convenient to search the depth of the local maximum of the vertical temperature gradient by starting the search from the deep ocean and moving upward. Since density is the physical property directly linked to dynamics, it is more appropriate to use the term “pycnocline”. However, in most parts of the ocean, density in the upper ocean is primarily controlled by temperature, with the salinity playing a minor secondary role. Hence, the main thermocline and the main pycnocline are very

1.2 Main Thermocline in the World Oceans

close to each other in location; people often uses the term “main thermocline” because it is directly linked to heating/cooling and changes in the climate system. As shown in Fig. 1.2a, there is clearly a strong thermocline along the Equator, and it is called the equatorial thermocline. In the Pacific and Atlantic Oceans, it slopes down westward; but it slopes up westward in the Indian Ocean. Such patterns are primarily due to the fact that there are easterlies in both the Pacific and Atlantic basins; however, wind stress in the Indian Ocean has a very strong annual cycle and the annual mean wind is westerly. In the meridional sections, the main thermocline near the Equator appears in the form of a dumbbell, i.e., the thermocline is shallow near the equator, but is deep in the middle latitudes because of the bowl-shaped wind-driven subtropical gyres in both hemispheres (lower panels in Fig. 1.2).

3

The equatorial thermocline is asymmetric to the equator, and this phenomenon is associated with the strength of the zonal equatorial currents and the cross-equator flow in each basin. In the Atlantic basin, the strong meridional overturning circulation associated with the thermohaline circulation manifests in the form of a northward warm current moving across the equator in the upper ocean. In the Pacific basin, wind stress over the latitude band of the Australian continent creates the Indonesian Throughflow, which manifests in the form of a cross-equatorial current bringing the warm water from the midlatitudes in the Southern Hemisphere into the Northern Hemisphere. In fact, most of the water masses entering the Equatorial Undercurrent in the source region of the western Pacific basin come from the Southern Hemisphere. Zonal wind stress in the equatorial Indian Ocean also gives rise to cross-equator flow.

Fig. 1.2 Typical annual mean temperature profiles in the world oceans, based on WOA09 climatology

4

For examples, we show a case of using the vertical temperature gradient to identify the main thermocline from WOA09 climatology (Fig. 1.3). The upper panel shows the temperature contours. The corresponding vertical temperature gradient is shown in the lower panels and the main thermocline obtained after smoothing is marked by the red curved line. We also show a horizontal map of the main thermocline depth in the world oceans diagnosed from WOA09 climatology (Fig. 1.4). The search for the main thermocline requires a slightly complicated code. First, we start the search from the 1500 m level and move upward, setting the base of the main thermocline as the first grid with a vertical temperature gradient larger than dT/dz = 0.01 °C/m. When the base of the thermocline is located, we search for the exact depth above this grid where the vertical temperature

1 Basic Concepts

gradient is maximal. Using a single criterion of dT/dz for the entire world oceans is, of course, not accurate, and this is modified as follows. At the equatorial band, the temperature gradient is much larger, so that we modify the criterion as dT=dz ¼ 0:01½1 þ 4ð1  j/j=20Þ for j/j\20, where / is the latitude in degrees. For midlatitudes, this criterion is modified as dT=dz ¼ 0:01½1 þ 0:025ð1  j/j=20Þ for j/j [ 20. Since the main thermocline outcrops along strong zonal currents, such as the Gulf Stream, Kuroshio and ACC, the main thermocline is confined to lower and mid-latitudes only. It is clear that the main thermocline in the five subtropical basins appears as the local bowlshaped maximum. As will be discussed shortly, the zonal slope of the squared thermocline depth is linearly proportional to the Ekman pumping rate divided by the reduced gravity. This simple

Fig. 1.3 Identifying the main thermocline along 179.5° E from WOA09; the red curve indicates the position of the maximum vertical temperature gradient, i.e., the main thermocline

1.2 Main Thermocline in the World Oceans

relation gives rise to some interesting phenomena identifiable from Fig. 1.4. In fact, the main thermocline in the North Atlantic Ocean is much deeper than that in the North Pacific Ocean. This is primarily due to the high salinity in the North Atlantic Ocean; consequently, water in the upper ocean is dense. Consequently, the reduced gravity is relatively small; as a result, the thermocline in this basin is deep. The thermocline in the Indian Ocean is deepest in the world oceans, with a maximal depth of more than 900 m (Figs. 1.2b and 1.4). Wind stress perturbation induced adiabatic motions (The adiabatic motions induced by the wind stress perturbation) can penetrate to the base of the main thermocline; as a result, the wind stress perturbations in the world oceans can generate adiabatic signals below 1000 m. Furthermore, as will be shown in Chap. 2, adiabatic motions may penetrate to the whole depth of the water column in the world oceans. Such adiabatic motions in the deep ocean may be linked to the

5

variability of the Antarctic Circumpolar Currents, the strength of which is primarily regulated by the Southern Westerly. Figure 1.4 also shows that the thermocline depth along the western coast of Australia is much deeper than that along the other eastern boundaries of the world oceans. As an example, the temperature and potential density distributions along 29.5° S are shown in Fig. 1.5. It is readily seen that the main thermocline depth along the eastern boundary of the Indian basin is about 750 m, much deeper than the corresponding depth of 100 m in the South Pacific and 150 m in the South Atlantic. This may be linked to the unusually strong poleward eastern boundary current in the South Indian Ocean, the Leeuwin Current. When the strength of the Leeuwin Current changes, the thermocline depth along the eastern boundary of the South Indian Ocean may change in response. Even if the wind stress and other parameters for the wind-driven circulation in the South Indian Ocean remain unchanged, the depth

Fig. 1.4 Main thermocline depth in the world oceans, based on WOA09

6

1 Basic Concepts

Fig. 1.5 Zonal maps of temperature (a) and potential density (b) along 29.5° S, based on WOA09 climatology

of the main thermocline must change as well. Therefore, the wind-driven circulation in the South Indian Ocean is intimately linked to the strength of the Leeuwin current.

1.3

Reduced Gravity Model, Advantage and Limitation

For the wind-driven circulation in the ocean, the depth of the main thermocline can be described by a simple reduced gravity model as discussed below; while the detailed dynamical description of wind-driven circulation is referred to Huang (2010).

1.3.1 Model Formulation A reduced gravity model is formulated as follows. The oceanic stratification is simplified as a two-layer fluid environment. The upper and lower layers have the density q0 and q0 þ Dq; the upper layer thickness is denoted as h, and the lower layer is assumed to be infinitely thick, and thus motionless. In the ocean interior the frictional and inertial terms are negligible. For simplicity, we discuss a model formulated on the mid-latitude beta plane, and we will assume wind stress in a simple form: sx ¼ sx ðyÞ; sy ¼ 0. For a steady state, the momentum equations are reduced to

1.3 Reduced Gravity Model, Advantage and Limitation

fhv ¼ g0 hhx þ sx =q0

ð1:1aÞ

fhu ¼ g0 hhy

ð1:1bÞ

where g0 ¼ gDq=q0 is the reduced gravity. In addition, the model satisfies the continuity equation ðhuÞx þ ðhvÞy ¼ 0

ð1:1cÞ

Note that a major assumption made in the traditional level model based on the quasigeostrophic approximation is that the change in stratification in the horizontal direction remains a small fraction. As shown above, a major feature of the reduced gravity model is to include the finite amplitude variability of the layer thickness in both the continuity equation and the momentum equation. As such, the reduced gravity model can simulate the nonlinear mechanism associated with finite layer depth change, which is one of the most important dynamical features of basin-scale wind-driven circulation in the ocean. Cross-differentiating and subtracting (1.1a) and (1.1b) lead to the vorticity equation bhv ¼ sxy =q0

ð1:2Þ

This equation is called the Sverdrup relation. Substituting (1.2) into (1.1a) gives rise to a firstorder ordinary differential equation hhx ¼ 


x f2 s g0 q 0 b f y

ð1:3Þ

This equation is a key to understanding the adjustment of wind-driven circulation: in a steady state the slope of the squared thermocline depth is proportional to the Ekman pumping rate and inversely proportional to the reduced gravity. As climate conditions change, the wind stress induced Ekman pumping rate and the buoyancy of the upper ocean vary accordingly. As a result, thermocline slope changes, warm water in the upper ocean is redistributed, and the circulation

7

and sea water properties in the upper ocean go through a three-dimensional adjustment, which is the major focus of this book. Integrating Eq. (1.3) leads to the equation regulating the layer thickness, which is valid in the basin interior. In order to balance the vorticity over the whole basin, the integration should be started from the eastern boundary, as discussed in Huang (2010). The zonal integration leads to the squared layer thickness for the basin interior 2

h ¼

h2e


 2f 2 sx þ 0 ðxe  xÞ g q0 b f y

ð1:4Þ

This equation can also be written in terms of the Ekman pumping in the following form h2 ¼ h2e 


x 2f 2 we s x ð  x Þ; w ¼  e e g0 b f q0 y ð1:5Þ

For the given wind stress profile, one can use these two equations to calculate the layer thickness in the basin interior. However, there are two approaches.The first approach is to specify the upper layer depth along the eastern boundary and then integrate westward. This approach has been widely used in many textbooks. This approach is so popular that many young people entering our field would think that this is the only approach. The second approach is quite interesting and strongly relevant to climate study. This approach is to assume that the total amount of warm water above the main thermocline is constant, and we want to find the solution which satisfies the constraints in Eq. (1.4) (or Eq. 1.5) and Eq. (1.1c). In fact, Eq. (1.1c) is the steady version of the more general form of the continuity equation ht þ ðhuÞx þ ðhvÞy ¼ 0

ð1:1c0 Þ

Integration of Eq. (1.1c′) over the whole basin leads to the conservation law

8

1 Basic Concepts

ZZ

00

hðtÞ dxdy ¼ const:

ð1:1c Þ

Namely, the total amount of warm water in the upper layer remains constant in time. In this case, we can no longer specify the layer thickness along the eastern boundary apriori; instead, we need to find a solution that satisfies the integral constraint of a constant volume of warm water above the main thermocline. In fact, most numerical models are based on such a constraint: we start the model for an initial state of rest, with a given mean layer thickness over the whole basin. Since the wind-driven numerical models conserve the total volume of the warm water, when the model reaches a final state, we have a solution which satisfies the volume conservation constraint. For example, we run a simple model subjected to these two constraints. Assume the Ekman pumping rate over the subtropical basin (from /s ¼ 15 N to /n ¼ 40 N) is we ¼ Epf 

106 sin½pð/  /s Þ=ð/n  /s Þ ðm/sÞ, (Epf is the Ekman pumping factor, which can vary from 0 to 2.5), and g0 ¼ 0:01 ðm/s2 Þ. To get more accurate solutions the calculation is based on the corresponding equations in the spherical coordinate. In Fig. 1.6, layer depths under two different amplitudes of forcing for these two experiments are displayed side by side. Under the same amplitude of wind forcing, the maximum layer depth (near the western boundary) and the minimum layer depth (along the eastern boundary) for these two model constraints are different. If we fix the upper layer thickness along the eastern boundary at the value of he ¼ 300 m, the zonal slope of the thermocline and the layer thickness in the basin interior increase with the enhancement of Ekman pumping, as shown by the transition from the dashed blue curves to the solid blue curve in Fig. 1.7c. In particular, the layer thickness maximum along the western boundary increases rapidly, as shown by the

(a) h (m), Epf=0.5, he=300 m

(b) h (m), Epf=2.375, he=300 m

40N

40N

35N

450 35N

30N

30N

400

25N

800 700 600

25N

500

350 20N 15N

20N

0

10E

20E

30E

40E

50E

60E

(c) h (m), Epf=0.5, V=V0

40N

300 15N

400 0

10E

20E

30E

40E

50E

60E

(d) h (m), Epf=2.375, V=V0

500 40N

800 700

35N

35N 450

30N 25N

400

600

30N

500 400

25N

300

20N

20N

200 100

350 15N

15N 0

10E

20E

30E

40E

50E

60E

300

0

10E

20E

30E

40E

50E

60E

Fig. 1.6 Layer thickness (in m) in the case of constant layer thickness along the eastern boundary (upper panels) and the case of constant volume of warm water (lower panels)

1.3 Reduced Gravity Model, Advantage and Limitation

9

Fig. 1.7 a Total volume of the upper layer; b eastern boundary layer thickness as a function of the Ekman pumping factor; c layer thickness along the mid-latitude (30° N)

solid blue curve in Fig. 1.7b; while the layer thickness along the eastern boundary (dashed blue curve) remains unchanged, as required by the model constraint. As a result, the total amount of warm water in the ocean increases gradually, the blue curve in Fig. 1.7a. In this case, the change in the total volume of warm water in the upper layer implies a source of warm water; it could come from the lateral boundaries or the bottom boundary of the model. In any case, the model run discussed above cannot be considered as adiabatic or isolated from other parts of the ocean. If the total volume of warm water in the upper layer is fixed (V ¼ V0 , black curves in Fig. 1.7), the situation can be considered to be adiabatic. In this case, the zonal slope of the thermocline in the basin interior increases with the enhancement of Ekman pumping (the black curves in Fig. 1.7c). The layer thickness maximum along the western boundary also increases; however, the layer thickness along the eastern boundary declines quickly (the solid and dashed black curves in Fig. 1.7b). When wind forcing is very strong (Epf ! 2.5), the layer thickness along the eastern boundary vanishes. If wind forcing is stronger than this critical value, the upper layer thickness near the eastern boundary is zero. By definition, layer thickness cannot be negative, so that the upper layer should outcrop near the eastern

boundary. Outcropping is a very important physical phenomenon in the ocean, and the simple reduced gravity model described by Eqs. (1.1a, b, c′, c′′–1.5) is no longer valid. In fact, dealing with outcropping requires special methods in analytical and numerical models, as will be discussed shortly. In the case of a fixed amount of warm water, enhancement of Ekman pumping leads to the three-dimensional redistribution of warm water in the upper ocean, as shown in Figs. 1.8 and 1.9. Figure 1.8 shows that warm water is transported from both the low and high latitudes to the middle latitudes (Fig. 1.8a); warm water is moved from the eastern boundary to the western boundary (Fig. 1.8b); and warm water is removed from the upper ocean and pumped down to the deep part of the ocean, Fig. 1.8c. Such a three-dimensional shifting of water mass can be clearly demonstrated in terms of the shifting of the mass center (Fig. 1.9). When Ekman pumping is enhanced, the center of the warm water mass in the upper ocean moves westward (Fig. 1.9a), northward (Fig. 1.9b) and downward (Fig. 1.9c). From the volumetric distribution of warm water, one can infer the volumetric anomaly profiles in the meridional, zonal and vertical directions, defined as the deviation from the pivotal case of Epf = 1.0 (Fig. 1.10). For example, if the Ekman pumping factor is reduced to

10

1 Basic Concepts

Fig. 1.8 Warm water mass shifting due to the enhancement of Ekman pumping subjected to the fixed volume. a Meridional distribution of warm water volume (1015 m3/0.25°); b Zonal distribution of warm water volume (1015 m3/ 0.3°); c Vertical distribution of warm water volume (1015 m3/5 m)

(a) Xc (degreee)

(c) Zc (m)

(b) Yc (degree)

28.5N

24E

200

25E Depth (m)

28.0N

26E 27E

27.5N

28E 29E 30E

220

240

260 0

2 1 0.5 1.5 Ekman pumping factor

2.5

27.0N

0

0.5 1 1.5 2 Ekman pumping factor

2.5

0

0.5 1 1.5 2 Ekman pumping factor

2.5

Fig. 1.9 Migration of mass center due to the enhancement of Ekman pumping

Fig. 1.10 Modes of warm water volumetric anomaly, relative to the pivotal case of Epf = 1.0

Epf = 0.5, the volumetric anomaly in the meridional direction is shown as the blue curve in Fig. 1.10a. In such a case, warm water volume

at middle latitudes is reduced, while it is increased at high/low latitudes. On the other hand, if Ekman pumping is enhanced to Epf =

1.3 Reduced Gravity Model, Advantage and Limitation

1.5, 2.0 and 2.5, the warm water volume is increased at middle latitudes, but it is reduced at high/low latitudes. For the zonal profiles, when Ekman pumping is reduced to Epf = 0.5, the warm water volume anomaly is negative in the western basin, but it is positive in the eastern basin, as shown by the blue curve in Fig. 1.10b. On the other hand, when Ekman pumping is enhanced, warm water is pushed towards the western basin, leading to positive anomaly there and negative anomaly in the eastern basin. Among these features, the most important one is the downward push of the warm water associated with the enhancement of the Ekman pumping rate. Since the adjustment process is adiabatic, the total heat content (HC) is conserved during the adjustment; therefore, in the case of a simple reduced gravity model, the HC anomaly profile must be in the form of a first baroclinic mode (Fig. 1.10c). As shown by the blue curve in Fig. 1.10c, if Ekman pumping is reduced to Epf = 0.5, there is more warm water between 300 and 400 m, but there is less warm water between 400 and 600 m. The reduction of the warm water volume at a given depth implies that there is cooling at this level. Accordingly, the baroclinic modes of the warm water volume could be interpreted as the warming/cooling signals diagnosed at the corresponding geopotential levels. When the Ekman pumping rate is enhanced, the warm water is pushed downward, inducing warming at deeper levels and cooling at shallower level, as indicated by the black, red and green curves in Fig. 1.10c. Baroclinic modes of the HC anomaly induced by changes in the Ekman pumping rate (associated with wind stress perturbations) under the adiabatic conditions are the most important features often diagnosed from climate datasets. It is important to notice that for the second model, the assumption of constant volume is equivalent to assuming that the adjustment of the model is under the adiabatic constraint. In the ocean, for a relatively short time scale, the amount of warm water in the upper ocean can be treated as nearly constant. As such, wind stress

11

perturbations would induce changes in the slope of the main thermocline. When Ekman pumping is enhanced, the layer thickness near the eastern boundary declines, and the layer thickness along the western boundary increases, as shown by the black curves in Fig. 1.7b. Such an adjustment process is carried out by the anomalous currents and waves, and during the adjustment of the wind-driven gyre, the warm water in the upper ocean goes through a three-dimensional redistribution. Although such motions are adiabatic, they can carry substantial amounts of the HC anomaly with them; as a result, such adiabatic motions can cause temporal heat transport in three dimensional space. These motions are the prototype of heaving motions in the ocean and they are the primary subject of this book.

1.3.2 The Reduced Gravity in the World Oceans The common way of formulating a reduced gravity model is to assume that the reduced gravity is a constant in space and time. By definition, g0 ¼ gDq=q0 , where q0 and q1 ¼ q0 þ Dq are the mean density of the upper and lower layers. There are two important issues. First, in the calculation one should use the potential density, not the in situ density. In the following example, we use r1 , i.e., the potential density using 1000 db as the reference pressure. The reason for using r1 , instead of r0 , is as follows. In the Atlantic basin, r0 is not monotonic in the deep ocean; thus, the calculation of reduced gravity might be somewhat inaccurate. Second, by carrying out the calculation of reduced gravity for each water column, one realizes that the reduced gravity is a horizontally distributed function. As an example, we show the result of using 459 m as the interface to calculate the mean potential density for the upper and lower parts of the water column and the inferred reduced gravity for the world oceans. The calculation is based on climatology obtained from the GODAS data (Behringer and Xue 2004) as shown in Fig. 1.11. It is clear that in the Pacific basin, the reduced gravity varies within the range of 0.015–

12

1 Basic Concepts

Fig. 1.11 Reduced gravity (in cm/s2) for the world oceans, inferred from the GODAS data

0.03 m/s2; on the other hand, it is smaller in the Atlantic basin, within the range of 0.01– 0.02 m/s2. In the Indian basin, the reduced gravity is in the middle range. We can also estimate the reduced gravity model, using an isopycnal surface as the interface to calculate the mean potential density for the upper and lower parts of the water column and the inferred reduced gravity for the world oceans. For example, one can use r0 ¼ 26:8 ðkg/m3 Þ as the interface, and the result is quite similar to that shown in Fig. 1.11. In the commonly used reduced gravity model, the reduced gravity is assumed to be constant in time and space; the typical value used in these models varies over the range of 0.01–0.02 m/s2. In most cases, people specify this value as more or less arbitrary. However, as discussed above, the reduced gravity can be calculated from the climatological data in an accurate way. It is important to emphasize that the reduced gravity model is a highly truncated model in the density coordinate. As such, it cannot describe the circulation very accurately. However, the reduced gravity model excludes the complicated thermohaline processes and the fast signals associated

with the external gravity waves; hence, this type of model is much easier to study either analytically or numerically. Since the high speed barotropic waves, such as the barotropic Rossby waves, are excluded, the reduced gravity models are not suitable for the study of short time scales. It is possible to extend the commonly used reduced gravity model into a model including the thermodynamics and the horizontally nonconstant reduced gravity. The relevant information can be found in Huang (1991, 2010), McCreary and Yu (1992). The modified model is called the generalized reduced gravity model, in which the reduced gravity is a function of horizontal coordinates and time. In Chap. 4, such models will be used in the study of the circulation variability in the connection with density anomaly induced by heating/cooling or freshening.

1.4

Layer Outcropping: The Physics and the Numerical Method

As shown in Fig. 1.2, most isothermal layers outcrop at high latitudes; similarly, most isopycnal layers also outcrop at high latitudes. Layer

1.4 Layer Outcropping: The Physics and the Numerical Method

outcropping is a phenomenon that happens in the stratified ocean, and our models should simulate such phenomenon accurately. However, in the early stages of model development in history, layer models or isopycnal models did not take layer outcrop into consideration. The nonlinearity associated with layer outcropping is of critical importance. For a simple layer model, the continuity equation is ht þ ðhuÞx þ ðhvÞy ¼ 0

ð1:6Þ

Using the scale thickness H and velocity scale U, we define the following non-dimensional variables h ¼ Hð1 þ dh0 Þ; ðu; vÞ ¼ Uðu0 ; v0 Þ; t ¼ t0 H=U ð1:7Þ The non-dimensional continuity equation is as follows h0t0 þ ðu0x þ v0y Þ þ dðu0 h0x þ v0 h0y Þ ¼ 0

ð1:8Þ

where the non-dimensional layer thickness parameter is d ¼ DH=H  Oð1Þ

ð1:9Þ

Hence, the nonlinearity of layer thickness change is O (1). In comparison, the nonlinearity associated with advection terms in momentum equations is Ro  O (1). Therefore, for large-scale wind-driven circulation associated with large layer thickness change or outcropping the nonlinearity in the continuity equation is the most important dynamical issue and should not be ignored. For technical convenience, many modelers set a lower bound of layer thickness on the order of 10 m. Such models ignored the critically important roles of density fronts in the ocean circulation; thus, the model results were dramatically different from reality. The importance of simulating the density front accurately was recognized in the 1980s, and subsequent development in isopycnal models has made special effort in dealing the density front (or isopycnal outcropping). In terms of numerical

13

calculation, to simulate the outcrop line the socalled positive definite scheme should be used in the layer depth calculation. Most numerical models are based on the so-called central finite difference scheme in calculating the layer thickness. However, such a scheme may lead to a negative layer thickness near the outcrop line; hence, it is not suitable for a case with outcropping. To deal with the outcrop line, one has to use the so-called positive-definite finite difference schemes; the essential feature of such scheme is to guarantee that layer thickness is never negative. A simple choice is the well-known upwind scheme, which can guarantee the layer is never negative; many other higher-order accuracy schemes are also available. A layer model or isopycnal coordinate model including the outcropping line can provide much more accurate information related to density fronts, such as the Gulf Stream and Kuroshio Current; accordingly, they can simulate the oceanic circulation and climate changes more accurately. The first rigorous treatment of the outcrop line can be traced back to the pioneering study of Parsons (1969), who pointed out the critically important physical constraint related to the outcrop line: in a reduced gravity model, the outcrop line should be both the zero thickness line and a streamline. It is clear that the lowest order dynamics regulating the wind-driven circulation in the ocean interior cannot satisfy both constraints. In order to satisfy both dynamical constraints, there should be a thin boundary layer in which the higher-order dynamical terms playing the role of satisfying both constraints along the outcrop line. A condensed description of Parson’s solution can be found in Huang (2010). One example is shown in Fig. 1.12, where an outcrop window appears in the northwest corner of the basin; next to the outcrop line, there is an interior boundary layer. Under the assumption of a fixed amount of warm water in the upper layer, if the wind stress forcing is further increased, the outcrop window will extend further. The

14

1 Basic Concepts

Fig. 1.12 A solution of the subtropical gyre with an outcrop window, the horizontal/vertical axes and layer depth are in nondimensional unit

1

0.2 Outcrop window

0.6 0.8

0.8

0.2

0

0.7

0.4

0.6 8 . 0

0. 8

1

1.2

1

0.6

0.6

0

0.9

0.4

0.8

1.2

1.4

1.2

y

1 0.5

1.

4

0.4

1

1.4

0.3 0.2

1.2

0 0

0.8

0.1

1 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

expansion of the outcrop window implies that the warm water in the upper ocean is redistributed. The redistribution of the warm water in the ocean implies a three-dimensional redistribution of mass, heat and salt; thus, the wind stress anomaly can give rise to an anomalous transport of mass, heat and salt in the meridional, zonal, and vertical directions. In Chap. 4 we will discuss this issue in details, using several simple reduced gravity models for an idealized two-hemisphere model ocean, an idealized model for the Southern Hemisphere, including a periodical channel mimicking the Antarctic Circumpolar Current, and a reduced gravity model for the world oceans. In the traditional view, the meridional heat flux is directly tied to the thermohaline circulation and the heat exchange across the air-sea interface or isothermal surfaces. Heaving modes discussed in this book may be the other mechanism giving rise to the sizeable three dimensional

transport of mass, heat and freshwater on the interannual and decadal time scales. Parson’s work can be easily extended into the case of double gyres in a single hemisphere. When there is enough warm water, the upper layer covers the entire two-gyre basin. However, when the amount of warm water is not enough to cover the whole basin, the outcropping window first appears in the middle of the subpolar basin near the western boundary, as shown in the sketch in Fig. 1.13. As the wind stress is enhanced further, the outcrop window expands and eventually extends into the subtropical basin. In the case of strong wind forcing and a small amount of warm water, the upper layer is confined to a rectangular region in the southwest corner of the basin, very much like the warm pool in the Pacific basin. Parson’s model is much easier to be solved by numerical integration. As an example, we show one set of solutions obtained by a reduced gravity

1.4 Layer Outcropping: The Physics and the Numerical Method Fig. 1.13 The nondimensional position of the outcropping window for an idealized two-gyre basin, where k is a non-dimensional number indicating the nonlinearity of the model

15

1 0.9 0.8 = 0.2

= 0.4

= 2.0

0.7

Y

0.6 0.5 0.4 0.3 0.2 = 10

0.1 0

0

0.1

0.2

model based on spherical coordinates with a realistic coastal geometry of 1°  1° resolution and forced by the climatological annual mean wind stress obtained from the GODAS data. These cases were obtained by running the model for 200 years by starting from an initial state of rest. Two cases with the initial layer thickness of

0.3

0.4

0.5 X

= 8.0

0.6

0.7

0.8

0.9

1

250 and 75 m were run. In the first case, there is enough warm water, so that most of the subtropical basins are covered by the upper layer (Fig. 1.14). On the other hand, the upper layer outcrops in the most part of the ACC and a large part of the subpolar gyre in the Pacific and Atlantic basins.

Fig. 1.14 The upper layer thickness for the world oceans forced by the climatological annual mean wind stress of the GODAS data, with the initially uniform thickness of 250 m

16

1 Basic Concepts

Fig. 1.15 The upper layer thickness for the world oceans forced by the climatological annual mean wind stress of the GODAS data, with the initially uniform thickness of 75 m

However, as the amount of warm water is reduced, the upper layer can no longer cover the subtropical basins. As a result, there are outcrop windows in the subtropical basins which gradually expand (Fig. 1.15). Although these experiments were run under the assumption of different initial upper layer thickness and forced by the same wind stress, one can also run the model with the same amount of warm water in the upper layer, but gradually change the strength of wind forcing. As discussed above, under such an assumption, the outcrop window will gradually expand, very much like the situations shown in Figs. 1.14 and 1.15.

References Antonov JI, Seidov D, Boyer TP, Locarnini RA, Mishonov AV, Garcia HE, Baranova OK, Zweng M, Johnson DR (2010) World Ocean Atlas 2009, Volume

2: Salinity. In: Levitus S (ed) NOAA Atlas NESDIS 69. U.S. Government Printing Office, Washington, DC, p 184 Behringer DW, Xue Y (2004) Evaluation of the global ocean data assimilation system at NCEP: The Pacific Ocean. In: Eighth symposium on integrated observing and assimilation systems for atmosphere, oceans, and land surface, AMS 84th Annual Meeting, Washington State Convention and Trade Center, Seattle, Washington, pp 11–15 GODAS data, Provided by the NOAA-ESRL Physical Sciences Division, Boulder Colorado from their Web site at https://www.esrl.noaa.gov/psd/ Huang RX (1991) A note on combining wind and buoyancy forcing in a simple one-layer ocean model. Dyn Atmos Oceans 15:535–540 Huang RX (2010) Ocean circulation, wind-driven and thermohaline processes. Cambridge Press, Cambridge, 791 pp McCreary JP, Yu Z (1992) Equatorial dynamics in a 2½layer model. Prog Oceanogr 29:61–132 Parsons AT (1969) A two-layer model of Gulf Stream separation. J Fluid Mech 39:511–528

2

Climate Variability Diagnosed from the Spherical Coordinates

The spherical coordinates are commonly used in oceanography; in particular, most climate studies and datasets are based on the z-coordinate. Therefore, in the first part of this chapter we examine the climate variability using the z-coordinate; at the end of this chapter, we will also explore using longitudinal and latitudinal coordinates to diagnose the climate variability. In these coordinates climate signals can be separated into the external and internal modes. The external modes indicate the net change of heat content (or salt/density content) integrated over the world oceans; these modes are directly linked to anomalies in net external forcing, such as heat flux (or freshwater/density flux). On the other hand, the internal modes represent the internal exchanges of heat, salt and density between different layers (latitude/longitude bands). These processes may involve internal diabatic components, which give rise to changes of heat (salt/density) in individual layers; however, the global net contribution of these internal diabatic processes must be zero by definition. Therefore, separating the climate signals into external and internal modes in these coordinates may reveal interesting phenomena hidden in climate changes. The dynamical/thermodynamic processes leading to such internal modes of variability may take place primarily within isopycnal layers, which are slanted in the traditional spherical coordinates. Hence, to explore the cause of climate variability observed in the ocean, one should also look at climate variability from

different angles. Isopycnal/isothermal analysis is one of such tools, and it will be examined in details in this book. Although most studies of climate changes are focused on the upper ocean, with the limited information available about the deep ocean, one can also infer the climate variability in the deep ocean, from climate datasets generated from either observations or numerical simulations. Due to the server limits of data availability for the deep ocean, any result related to the deep ocean is at best speculation only. Nevertheless, such studies may reveal some important dynamics related to climate changes. Our discussion below is based on the GODAS data (Behringer and Xue 2004).

2.1

Climate Variability Diagnosed in the z-Coordinate

In the following discussion, we calculate the total heat content anomaly for the world oceans, with the mean annual cycle removed. Note that the GODAS data is a monthly mean climatology which began in 1980. The horizontal resolution is 1
 0:333
; there are 40 non-uniform layers in the vertical direction, and the center of the lowest layer is at the depth of 4478 m. Since the vertical grid in the GODAS data is non-uniform, we will use heat content per unit thickness, in units of J/m. In addition, figures are plotted in a

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_2

17

18

2 Climate Variability Diagnosed from the Spherical Coordinates

stretched vertical coordinate, with fine resolution for the upper one kilometer. The heat content anomaly is calculated as follows:
 dhc ¼ q0 Cp hði; j; k; mÞ   hði; j; k; maÞ dxðjÞdydzðkÞ

ð2:1Þ where q0 Cp ¼ 1035  4184 J/m3 =
C is the unit conversion constant, h is the potential temperature, m = 1, …, 420 is the time in month, ma is the corresponding month in the mean annual cycle. The heat content anomaly for each level can be calculated by horizontally integrating dhc defined in Eq. (2.1) over the world oceans ZZ Hc ðk; mÞ ¼

dhc dxdy

ð2:2Þ

A

This variable can be further separated into the barotropic (external) and baroclinic (internal) modes as follows
 Hc ðk; mÞ ¼ HcBT ðmÞ þ HcBC ðk; mÞ dzðkÞ ð2:3Þ The barotropic mode is defined as the vertical mean HcBT ðmÞ ¼

40 X

Hc ðk; mÞ=D

ð2:4Þ

k¼1

where D = 4736 m is the effective depth of the model ocean in the GODAS data. Therefore, the barotropic mode is the net change of the vertical mean heat content anomaly integrated over the world oceans. The baroclinic modes are defined as HcBC ðk; mÞ ¼ Hc ðk; mÞ=dzðkÞ  HcBT ðmÞ ð2:5Þ Both the barotropic and baroclinic modes are defined as the net heat content anomaly per meter. The global sum of the external mode in the z-coordinate should be the same as the global integration of the external mode in the density coordinate. The current modeling practice is to

use the thermal isolating boundary conditions around the lateral and bottom surfaces; so that heating/cooling of the model ocean must go through the air-sea interface. The time evolution of the corresponding summation reflects the general warming/cooling of the world oceans due to the anomalous air-sea thermal interaction. On the other hand, the internal modes in the density coordinate and the baroclinic modes in the z-coordinate diagnosed from data may have quite different patterns. These differences reflect the dynamic nature of the internal modes in these two different coordinates; such differences can provide useful information regarding the nature of climate variability in the world oceans, and this is the reason to use different coordinates for mapping out the climate variability. As shown in Fig. 2.1, patterns of the time evolution of total signals and the baroclinic mode signals are quite similar. Since the total signals are the sum of barotropic and baroclinic signals, the similarity in the patterns of total signals and baroclinic signals indicates that the contribution due to the barotropic signals is a small fraction only. There are clearly high frequency oscillatory components in the upper ocean. For the deep ocean, the signals are predominantly on decadal time scales. The Root-Mean Square (RMS) of the baroclinic mode is shown by the dashed red line in Fig. 2.1c, and its magnitude (0.1  1020 J/m) is much smaller than that of the baroclinic modes. The time evolution of the barotropic mode is shown in Fig. 2.1d. As noted above, the amplitude of this mode is much smaller than the baroclinic modes. This mode reflects the total heat content change in the world oceans, and such change is entirely due to the external heat flux variability associated with the air-sea interaction. In order to reveal the relation between these signals, we plot the time evolution of these signals in Fig. 2.2. It is clearly seen that the amplitude of the barotropic mode is much smaller than the baroclinic modes, especially near the surface. For example, the RMS amplitude of the barotropic mode is 0.115  1020 J/m,

2.1 Climate Variability Diagnosed in the z-Coordinate

19

Fig. 2.1 Heat content anomaly for the world oceans inferred from the GODAS data, in 1020 J/m. a The total heat content; b the baroclinic modes; c the RMS of the baroclinic modes (black curve) and barotropic mode (red dashed line); d the barotropic mode

while the RMS amplitude of the baroclinic mode signal in the top layer is 1.51  1020 J/m, about 13 times larger than that of the barotropic mode. In the surface layer (at 5 m depth), the heat content anomaly (the red curve in Fig. 2.2a) went through the period of increase before 1996; afterward, there was a period of oscillation without much increase. Similarly, the baroclinic mode at 105 m depth also shows a period of no systematic increase between 2005 and 2015. A period of no temperature increase is often called a hiatus, which has been widely discussed in recent literatures. Conceptually, the barotropic mode is controlled by the air-sea heat exchange anomaly; on the other hand, the baroclinic modes are controlled by internal processes. In the surface layer the baroclinic mode is ten times stronger than the barotropic mode; in fact, the heat content anomaly in the surface layer is primarily regulated by internal dynamical processes, and the anomalous air-sea heat flux and solar insolation play minor roles only. Therefore, although the heat content anomaly of the surface layer, such as that associated with the recent hiatus, can be

important to human society, it does not necessarily reflect the overall change of heat content in the world oceans; one should also examine the status of the barotropic mode. In the subsurface layers, the heat content anomaly trend is quite different. The heat content anomaly over the depth range of 949–1193 m increased over the whole period of the data record, as shown in Fig. 2.2b. In fact, the heat content increase rate was high before 1994. It slowed down during the period of 1995–2007; afterwards, the heat content of these subsurface layers increased with a speed doubled. It is particularly interesting to note that the heat content of the entire depth of the world oceans was actually negative for the period of 1997–2006, i.e., the heat content anomaly over these middle depth layers has signs opposite to that of the depth-integrated heat content of the world oceans. Such baroclinic modes of the heat content anomaly are due to the adiabatic adjustment of isothermal layers in the ocean, and this will be discussed in detail in the following chapters. On the other hand, in the deep ocean, especially near the sea floor, the amplitude of the

20

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.2 Time evolution of the heat content anomaly for the world oceans in units of 1020 J/m, including the barotropic mode (black curves) and baroclinic modes on different levels: a the upper ocean; b the middle layers; c the bottom layers, and d RMS of the baroclinic mode for each month

barotropic mode, the baroclinic mode, and the total heat content anomaly are of the same order of magnitude (Fig. 2.2c). The most important information revealed by this figure is that the baroclinic mode anomalies at the level of 3483 m and 3972 m are negative over the past ten years, while the total heat content anomaly is positive for the same time period. Accordingly, although the world oceans are warming up during this period, the deep ocean is actually cooling down. Since the baroclinic mode can be generated through internal adiabatic processes only, the cooling down of the deep ocean should be interpreted in terms of the adiabatic motions of the isothermal surfaces; in this book, we will call such motions as heaving motions. The apparent

cooling of the deep ocean will be discussed at the end of this chapter. In most previous studies, the dynamic effect of the wind-driven circulation is examined with a focus on the upper ocean only. For example, Huang (2015) discussed the heaving modes in the world oceans, and the analysis is mostly confined to the motions related to the main thermocline in the world oceans. Since the main thermocline in the world oceans is confined to the upper one kilometer, whether the wind stress anomaly can induce heaving motions in the deep ocean remains unclear from Huang’s (2015) study. However, our analysis here suggests that heaving motions are not limited to the upper ocean only. The heaving signals in the deep

2.1 Climate Variability Diagnosed in the z-Coordinate

21

ocean identified above suggest that wind stress anomalies can penetrate the whole depth of the world oceans and induce heaving motions in the bottom of the ocean. This seemingly counterintuitive phenomenon is of great importance for climate study. The apparent cooling of the deep ocean over the past decades has been discussed in previous studies, e.g., Liang et al. (2015). However, the dynamic reason for such deep cooling remains unexplained until now. Overall, baroclinic signals of the heat content anomaly are much stronger than the barotropic signals. To compare their strength, we calculated the RMS of baroclinic mode signals for each month and plotted them against the barotropic signals in Fig. 2.2d. It is clear that the baroclinic signals are much stronger than the barotropic signal. The RMS of the barotropic signals is 0.115  1020 J/m, and that of the RMS (over time) of the monthly RMS (over depth) baroclinic signals is 1.02  1020 J/m; thus, the baroclinic signals consist of 90% of the total signals. Such a large signal ratio implies that study of the baroclinic signals is of vital importance for our understanding of climate variability.

The structure of the baroclinic modes is further illustrated in Fig. 2.3, where the vertical structure of the mode for four selected months is included. It is readily seen that the amplitude of the barotropic mode is much smaller than that of the baroclinic modes, especially in the upper ocean. Each baroclinic mode shown in this figure has numerous zero crossings, and the number of zero crossings can be used to classify these baroclinic modes. For example, there are three zero crossings in December 2010; thus, the corresponding baroclinic mode can be called a third baroclinic mode. The structure of the baroclinic modes is a complicated functional of the winddriven and the thermohaline circulation in the world oceans, and it can vary greatly. The time evolution of the baroclinic modes over four consequent months is shown in Fig. 2.4. It is readily seen that there is a rapid warming at the depth of 50 m, but there is a strong cooling at 150 m; this is a strong indication of the baroclinic mode of heat exchange between different levels in the upper layer—note that the regular seasonal cycle has been removed. From the heat content anomaly, we can infer the vertical heat flux subjected to the zero heat

Fig. 2.3 The barotropic and baroclinic modes of the heat content anomaly for the world oceans at four time snapshots, in units of 1020 J/m

22

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.4 Time evolution of the barotropic and baroclinic modes of the heat content anomaly for the world oceans during four consequent months, in units of 1020 J/m

flux conditions on the sea floor, and the upward heat flux is defined positive. The ocean was warmed up during the first three time snaps, indicated by the negative barotropic heat flux in Fig. 2.5a–c. The ocean was cooled down in December 2010, indicated by the positive heat flux in Fig. 2.5d. On the other hand, the baroclinic mode signals can be much stronger than the barotropic mode signals; as a result, the total vertical heat flux can have signs opposite to the barotropic heat flux. December 2006 is a typical case (Fig. 2.5c). Although the barotropic heat flux is negative, suggesting warming of the whole water column on average, the total heat flux below 200 m is positive, suggesting an upward heat flux, namely cooling of the deep ocean. In general, the temperature stratification is stable, i.e., the bottom temperature is colder than that of the layer above. In addition, the model ocean is subjected to the thermal insulated condition on the sea floor. As a result, the heat flux in connection with vertical mixing can only warm up the bottom water. Consequently, the rapid cooling of the deep layer must be due to the advective heat flux associated with adiabatic migration of water masses in the

deep ocean. We will come back to this issue shortly. The adiabatic changes of deep water properties can be diagnosed from other water properties as well. For example, applying the same algorithm discussed above to the salt content in the GODAS data reveals similar phenomena (Fig. 2.6). In this analysis we also separate the signals into the barotropic and baroclinic modes. The time evolution of the total signals is shown in Fig. 2.6a, the baroclinic mode is shown in Fig. 2.6b, The RMS of the baroclinic mode is shown in Fig. 2.6c, and the RMS of the barotropic mode (0.18  1013 kg/m, red dashed line) is smaller than the baroclinic mode. The time evolution of the barotropic mode is shown in Fig. 2.6d. Accordingly, the total salt content in the world oceans declined continuously over the 35 years of model time. In reality, the total salt content in the world oceans should not change over such a relatively short time. The decline of the total salt content in this model ocean is owing to the Boussinesq Approximations; because of the volume conservation approximation used in formulating the model,

2.1 Climate Variability Diagnosed in the z-Coordinate (d) Dec. 2010

(c) Dec. 2006

(b) June 1996

(a) VHF June 1980

Depth (100m)

23

0 1 2

0 1 2

0 1 2

0 1 2

3

3

3

3

5

5

5

5

10

10

10

10

20

20

20

20

30

30

30

30

Total Barotropic

40 −2

−1 PW

40

40

40 0

−1.5

−1 −0.5 PW

0

−1

1 0 PW

2

0

2 PW

4

Fig. 2.5 Time evolution of the total and barotropic components of the vertical heat flux (positive means upward) in units of PW (1015 W)

Fig. 2.6 Time evolution of the salt content anomaly for the world oceans in units of 1013 kg/m. a The total salt content; b the baroclinic mode; c the RMS of the baroclinic mode; d the barotropic mode

freshwater input to the world oceans due to glacier melting is interpreted as the decline of salinity in the model. Accordingly, the decline of total salt content in the model is caused by the increase of fresh water in the world oceans.

With this caution in mind, we now examine the time evolution of different modes diagnosed from the GODAS data. As shown in Fig. 2.7, the barotropic mode continuously declined over the past 36 years, and it was negative in the second

24

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.7 Evolution of the salt content anomaly for the world oceans in units of 1012 kg/m, including the barotropic mode (black curves) and baroclinic modes: a the upper layers; b the middle layers; c the bottom layers, and d RMS of the monthly baroclinic modes

half of this time period. On the contrary, the baroclinic mode at the 3972 m level became positive at the end of this time period (Fig. 2.7c), i.e., the bottom water became slightly saltier. At the sea surface, the baroclinic mode oscillated with high frequency (Fig. 2.7a); around the 1000 m level, the salinity signals were positive for the first half of the time period, but became negative for the second half of the time period, and this is consistent with the trend of the barotropic mode. To compare the strength of barotropic and baroclinic salt content signals, we calculated the RMS of baroclinic mode signals for each month and plotted them against the barotropic signals in Fig. 2.7d. It is clearly seen that the amplitude of baroclinic signals are comparable with that of the

barotropic signals. The RMS of the barotropic signals is 1.82  1012 kg/m, and that of the RMS of the monthly RMS of the baroclinic signals is 2.38  1012 kg/m; thus, the baroclinic signals consist of 56% of the total signals. For the salt content anomaly, all these modes are of comparable amplitude (Fig. 2.8). The reason why the ratio of amplitude for the temperature and salt perturbations is so different may be explained as follows. Stratification in the upper ocean is primarily regulated by the temperature, so that the main thermocline is nearly the same as the main pycnocline. Wind stress perturbations can induce heaving modes, and hence a large amplitude of baroclinic modes in terms of temperature. On the other hand, the halocline is not directly linked to the pycnocline;

2.1 Climate Variability Diagnosed in the z-Coordinate

Depth (100m)

(a) S, June 1980

25

(b) June 1996

(c) Dec. 2006

(d) Dec. 2010

0

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

5

5

5

5

10

10

10

10

20

20

20

20

30

30

30

40

40

40

−0.5

0 0.5 1013kg/m

1

−0.2

0 0.2 1013kg/m

0.4

30

BT mode BC mode −0.4 −0.2 0 1013kg/m

40 0.2

−0.6 −0.4 −0.2 0 1013kg/m

0.2

Fig. 2.8 Evolution of the barotropic and baroclinic modes of the salt content anomaly for the world oceans in units of 1013 kg/m

as a result, baroclinic signals in terms of salinity are not as strong as those in terms of temperature. The adiabatic change of the deep water mass can be diagnosed from the density content anomaly as well. For example, applying the same

algorithm discussed above to the density (r0 ) content in the GODAS data reveals similar phenomena (Fig. 2.9). In this case, we also separate the signals into the barotropic and baroclinic modes. The time

Fig. 2.9 Density (r0 ) content anomaly for the world oceans in units of 1013 kg/m. a The total density content; b the baroclinic mode; c the RMS of the baroclinic mode; d the barotropic mode

26

2 Climate Variability Diagnosed from the Spherical Coordinates

evolution of the total signals is shown in Fig. 2.9a; the baroclinic modes are shown in Fig. 2.9b. The RMS of the baroclinic mode is shown in Fig. 2.9c; as shown in this panel, the RMS of the barotropic mode (0.17  1013 kg/m, red dashed line) is much smaller than the baroclinic mode. The time evolution of the barotropic mode is shown in Fig. 2.9d. Accordingly, the total density content in the world oceans declined continuously over 35 years. In reality, the total density content in the world oceans should not change over such a relatively short time. The decline of total density content in this model ocean is owing to the Boussinesq Approximations assumed in the model. Since the volume conservation approximation is used in formulating the model, both global warming and

freshwater input to the world oceans due to glacier melting are manifested as the decline of density in the model. In fact, the warming of the world oceans should not change the total mass; because of the melting of land-based glaciers, the total mass of water in the world oceans should be slightly increased. Thus, the meaning of the decline of the total density content in the model data should be interpreted as the increase of net heat content and fresh water in the world oceans, as discussed above in connection with Fig. 2.1 (the heat content anomaly) and Fig. 2.6 (the salt content anomaly). With this caution in mind, we now examine the time evolution of different modes diagnosed from the GODAS data. As shown in Fig. 2.10, the barotropic mode continuously declined over the

Fig. 2.10 Time evolution of the density content anomaly for the world oceans in units of 1012 kg/m, including the barotropic mode (black curve), the baroclinic modes (blue and red curves): a in the surface layer; b the middle layers; c the bottom layers and d the RMS of the monthly baroclinic mode

2.1 Climate Variability Diagnosed in the z-Coordinate

27

past 36 years, and it was negative in the second half of this time period. On the contrary, the baroclinic mode at 3972 m level became positive at the end of this time period, i.e., bottom water becomes slightly denser. This is consistent with the negative heat content anomaly associated with the baroclinic mode in the deep ocean shown in Fig. 2.1b, and it is also consistent with the salty signals for the bottom water shown in Fig. 2.6b. On the other hand, at the sea surface, the baroclinic mode oscillated with high frequency, quite similar to the heat content signals. To compare the strength of the barotropic and baroclinic density content signals, we calculated the RMS of monthly baroclinic mode signals and plotted the result against the barotropic signals in Fig. 2.10d. It is clear that the amplitude of baroclinic signals is much larger than that of the barotropic signals. The RMS of the barotropic signals is 1.76  1012 kg/m, and that of the RMS (in time) of the monthly RMS (in depth) baroclinic signals is 6.20  1012 kg/m; therefore, the baroclinic signals consist of 78% of the total signals. For the density anomaly, the amplitude of the baroclinic modes are much larger than that of the

barotropic mode; this can be seen clearly from the vertical profile for four months (Fig. 2.11). Our discussion above is based on the GODAS data. Similar phenomena can be identified in other climate datasets as well. For example, we analyzed the SODA data (Carton and Giese 2008) for the period of 1958–2008. As shown in Fig. 2.12, the barotropic mode of the heat content anomaly continuously declined over the first decade. Starting in 1970, it continuously increased and became positive after 1990. The structure of the baroclinic modes can be seen clearly in Fig. 2.13. At the end of the data record, there is a clear sign of a hiatus in terms of the surface temperature, in particular after year 2000 (Fig. 2.13a). In contrast, the whole water column was warmed up as depicted by the solid black curve, but over the past 15 years there was a clear sign of cooling down at the middle depth levels (Fig. 2.13b). Similarly, the deep levels have also cooled down (Fig. 2.13c). To compare the strength of the barotropic and baroclinic signals, we calculated the RMS of the baroclinic mode signals for each month and plotted the result against the barotropic signals in

(a)

Depth (100m)

0

0,

June 1980

(d) Dec. 2010

(c) Dec. 2006

(b) June 1996 0

0

0

1

1

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BT mode BC mode

40 −1

0 1013kg/m

40 1

−1

40

40 0 1013kg/m

1

−1

0 1013kg/m

1

−2

0 −1 1013kg/m

Fig. 2.11 The barotropic/baroclinic modes of the density content anomaly for four months for the world oceans

28

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.12 Heat content anomaly for the world oceans inferred from the SODA data, in units of 1020 J/m. a The total heat content; b the baroclinic mode; c the barotropic mode

Fig. 2.13d. It is clear seen that the amplitude of the baroclinic signals is much larger than the barotropic signals. The RMS of the barotropic signals is 0.138  1020 J/m, and that of the RMS of the RMS monthly baroclinic signals is 1.43  1020 J/m; consequently, the baroclinic signals consist of 91% of the total signals. This result is quite similar to the result diagnosed from the GODAS data, as discussed above in connection with Fig. 2.2. Figure 2.13d also reveals the following interesting phenomenon. During 1964–1976, the barotropic signals were negative and had a large amplitude, and the RMS of baroclinic signals was quite strong. Similarly, during 2003–2007, the barotropic signals were positive and the amplitude was large; at the same time, the baroclinic signals had a large RMS value. Therefore, strong barotropic warming or cooling on a decadal time scale is closely linked to the strong baroclinic variability of the heat content anomaly.

2.2

External/Internal Modes in Meridional/Zonal Directions

Similar to the projecting of climate signals onto the external/internal modes in the z-coordinate discussed above, climate signals can also be projected onto the external/internal modes in the meridional/zonal directions.

2.2.1 Heat Content Anomaly In the spherical coordinates, the heat content anomaly is defined as (Eq. 2.1)  dhc ¼ q0 Cp ½hði; j; k; mÞ hði; j; k; maÞ dxðjÞdydzðkÞ

From this anomaly, one can define the meridional mean of the zonally and vertically integrated heat content anomaly: ZZ Hc

meri

ð/; tÞ ¼

dhc dxdz

ð2:6Þ

2.2 External/Internal Modes in Meridional/Zonal Directions

29

Fig. 2.13 Time evolution of the heat content anomaly for the world oceans (based on the SODA data) in units of 1020 J/m, including the barotropic mode (black curves) and baroclinic modes on different levels: the surface levels (a); the middle levels (b); the bottom levels (c) and the RMS of the monthly baroclinic mode (d)

This time series can be further separated into the external and internal modes as follows meri Hcmeri External ðtÞ ¼ Hc

/

meri Hcmeri ð/; tÞ  Hcmeri Internal ðtÞ ¼ Hc External ðtÞ

ð2:7Þ This approach is applied to the GODAS data, and the result is shown in Fig. 2.14. Note that the external mode has the same shape as the external mode diagnosed for the z-coordinate discussed above, except in a different unit because the external mode discussed here is defined as the total heat content anomaly per degree of latitude, instead of the heat content anomaly per meter in the vertical direction. Similar to the cases discussed above, the external mode of the heat content anomaly is

entirely due to the net air-sea heat exchange, while the internal modes are owing to the meridional redistribution of heat in the ocean interior. The time evolution of the external and internal modes is shown in Fig. 2.15. It is clear that at low latitudes the internal meridional modes include high frequency interannual oscillations, which might be related to the long Rossby waves associated with the ENSO-like climate variability, Fig. 2.15a; on the other hand, internal modes at high latitudes are dominated by decadal trends. In particular, the long term trend of warming at 65° S is remarkable. However, at 41.5° S, there is a clear sign of decadal variability. In comparison, the climate variability at 58.5° N has a rather low amplitude. The fine structure of the external and internal meridional modes is shown in Fig. 2.16. It is clear that in general, the internal modes have an

30

2 Climate Variability Diagnosed from the Spherical Coordinates (a) ∫ Hc (1020J/°), Total

60N 40N 20N 0 20S 40S 60S

(b) ∫ Hc (1020J/°), Internal mode

80

80

60

60

40

40

20

20

0

0

−20 −40

−20 −40 −60

1980 1985 1990 1995 2000 2005 2010

1980 1985 1990 1995 2000 2005 2010 (c) ∫ Hc (1020J/°), External mode 10 0 −10 1980 1985 1990 1995 2000 2005 2010

Fig. 2.14 Meridional modes of heat content anomaly for the world oceans. a The total heat content; b the internal mode; c the external mode

Fig. 2.15 Time evolution of meridional modes of heat content anomaly for the entire water column, including the external (red) and internal modes at different latitudes: a Equator and low latitudes; b high latitudes

amplitude larger than the external mode, indicating that the heat exchange between different meridional bands plays an important role in generating climate signals at different latitudes. For example, in June 1980 the whole ocean was warmed up, as indicated by the red line in Fig. 2.16a. However, the internal meridional mode for the Southern Ocean (south of 40° S) was cooled down. Since the high latitude ocean is

colder than the low latitude ocean, such cooling signals cannot be generated by the lateral heat diffusion. The only likely mechanism responsible for this phenomenon is the lateral advection leading to adiabatic redistribution of water masses in the ocean. The details of such lateral movement of water masses is left for further study. This mode decomposition method can also be used for an individual layer defined for a certain

2.2 External/Internal Modes in Meridional/Zonal Directions (b) ∫ Hc, June 1996

(a) ∫ Hc, June 1980 60N

31 (c) ∫ Hc, Dec. 2006

(d) ∫ Hc, Dec. 2010

Internal Extrenal

40N 20N 0 20S 40S 60S −60 −40 −20 0 1020J/°

20 40 −60 −40 −20 0 1020J/°

20 40 −40 −20

0 20 1020J/°

40 −40 −20

0 20 1020J/°

40

Fig. 2.16 External and internal meridional modes at four time snaps

(a) ∫ Hc (1020J/°), 0~282m, Total

(b) ∫ Hc (1020J/°), 0~282m, Internal mode 40

40

30

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2010

(c) ∫ Hc (1020J/°), 0~282m, External mode

1985

1990

1995

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2010

Fig. 2.17 Meridional modes of the heat content anomaly for a layer between 0 and 282 m. a The total heat content; b the internal mode; c the external mode

depth (Figs. 2.17 and 2.18). Note that the external modes for such cases are defined for the individual layers; as such, they have different shapes and magnitudes. In a relatively shallow layer, the signals are mostly confined to the equatorial band and characterized by the interannual variability,

reflecting the dominating contributions associated with the ENSO cycles. The signals at high latitudes have a relatively low amplitude and are characterized by the decadal signals (Fig. 2.18). In the relatively deeper layers, however, the signals spread more towards the high latitudes; in addition, there is clear decadal variability even

32

2 Climate Variability Diagnosed from the Spherical Coordinates (a) ∫ Hc (1020J/°), 665~1071m, Total

60N 40N 20N 0 20S 40S 60S 1980

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(b) ∫ Hc (1020J/°), 665~1071m, Internal mode 30

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(c) ∫ Hc (1020J/°), 665~1071m, External mode

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Fig. 2.18 Meridional modes of the heat content anomaly for the layer between 665 and 1071 m. a The total heat content; b the internal mode; c the external mode

term signals also contain relatively short bursts of strong signals which do not seem to move much. Such propagation is much slower than typical wave motions, such as Kelvin waves; the dynamics involved in this phenomenon remains unclear at this time. Furthermore, there are weak signs of cross-basin eastward propagations of

for the equatorial band; such variability may be associated with the long term trend of the ENSO cycles (Fig. 2.18). Similarly, we can define the zonal external/ internal modes (Fig. 2.19). It is interesting to note that there seem to be clear signs of slow eastward propagations in each basin; however, such long (a) ∫ Hc (1020J/°), Total

1980

1985

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1995

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(b) ∫ Hc (1020J/°), Internal mode

20

30E 0 30W 60W 90W 120W 150W 180 150E 120E 90E 60E 30E 2010

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(c) ∫ Hc (1020J/°), External mode

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Fig. 2.19 Zonal modes of the heat content anomaly for the entire water column. a The total heat content; b the internal mode; c the external mode

2.2 External/Internal Modes in Meridional/Zonal Directions

2.2.2 Salinity Anomaly

signals from the Indian Ocean to the Pacific Ocean on the decadal time scale; such signals may go through the narrow channel among the gaps in the Maritime Continent and the Antarctic Channel. On the other hand, there is no clear sign of signal propagation from the Pacific to the Atlantic Ocean, suggesting that the Antarctic Channel here is too narrow for transporting large amplitude signals. This method can be used for an individual layer defined for a certain depth (Figs. 2.20 and 2.21). Note that the external modes for such cases are defined for the individual layers; as such, they have different shapes and magnitudes. At the relatively shallow layer, the signal pattern (Fig. 2.20) is similar to that in the deeper layers shown in Fig. 2.21. Here again, there is no clear sign of signal propagations from Pacific Ocean to the Atlantic Ocean. For a deeper layer (665–1071 m) there are clear signs of eastward migrations of signals in the Indian-Pacific basin (Fig. 2.21); in fact, two bands of opposite signals can be easily identified from this figure. Most interestingly, however, there seems to be some signal propagations from the Pacific Ocean to the Atlantic Ocean at this depth range. The dynamics remains unclear at this time.

(a) ∫ Hc (1020J/°), 0~282m, Total

1980

1985

1990

1995

2000

2005

The approach applied to the heat content anomaly can also be applied to the salinity anomaly. In general, global salinity signals are dominated by the general trend of freshening (Fig. 2.22a, c). There is a remarkable difference between the two hemispheres—the freshening in the Southern Hemisphere is much more pronounced than that in the Northern Hemisphere. With the external signals separated, the internal signals become clear. In the first 10– 15 years the Southern Ocean was saltier, and in the last 10 years it was fresher. On the other hand, the trend in the Northern Hemisphere is of the opposite sign (Figs. 2.22b and 2.23). The time evolution of the external and internal meridional modes of the salt content anomaly is shown in Fig. 2.24. In general, the amplitude of the external mode is comparable with that of the internal modes. We can also diagnose the zonal modes of the salt content anomaly. The zonal mode of the total signals is dominated by the decadal change of global freshening, which is manifested as the positive signals in the last century and negative signals in this century (Fig. 2.25a). After

(b) ∫ Hc (1020J/°), 0~282m, Internal mode

15

30E 0 30W 60W 90W 120W 150W 180 150E 120E 90E 60E 30E 2010

33

15

10

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(c) ∫ Hc (1020J/°), 0~282m, External mode

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Fig. 2.20 Zonal modes of the heat content anomaly for a layer between 0 and 282 m. a The total heat content; b the internal mode; c the external mode

34

2 Climate Variability Diagnosed from the Spherical Coordinates (a) ∫ Hc (1020J/°), 665~1071m, Total

1980

(b) ∫ Hc (1020J/°), 665~1071m, Internal mode

8

30E 0 30W 60W 90W 120W 150W 180 150E 120E 90E 60E 30E

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(c) ∫ Hc (1020J/°), 665~1071m, External mode

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Fig. 2.21 Zonal modes of the heat content anomaly for a layer between 665 and 1071 m. a The total heat content; b the internal mode; c the external mode (a) ∫ S (1013kg/°), Total

(b) ∫ S (1013kg/°), Internal mode

25

60N

15

20 40N

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15 10

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5 0

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(c) ∫ S (1013kg/°), External mode 10 0 −10 1980

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Fig. 2.22 Meridional modes of the salinity anomaly for the entire water column. a The total salt content; b the internal mode; c the external mode

separating the signals into the external and internal modes the difference between the three basins becomes more clear (Fig. 2.25b).

This method can be used for individual layer defined for certain depths (Figs. 2.26 and 2.27). Note that the external modes for such cases are

2.2 External/Internal Modes in Meridional/Zonal Directions

35

Fig. 2.23 Meridional modes of the salt content anomaly for the selected latitudinal bands

Fig. 2.24 Time evolution of external and internal modes of the salt content anomaly for the world oceans in units of 1013 kg/°

defined for the individual layers; as such they have different shapes and magnitudes.

2.2.3 Density Anomaly Similarly, we can also diagnose the meridional mode of the density anomaly. The meridional mode of total signals is dominated by the decadal

changes associated with the global warming, which is manifested as mostly positive signals in last century and negative signals in this century (Fig. 2.28a). After separating the signals into the external and internal modes the patterns become more clear. As in the case of the vertical modes, the external modes are dominated by positive signals in the last century and negative signals in this

36

2 Climate Variability Diagnosed from the Spherical Coordinates (a) ∫ S (1013kg/°), Total

(b) ∫ S (1013kg/°), Internal mode

10

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(c) ∫ S (1013kg/°), External mode

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Fig. 2.25 Zonal modes of the salinity anomaly for the entire water column. a The total salt content; b the internal mode; c the external mode (a) ∫ S (1013kg/°), 0~282m, Total

(b) ∫ S (1013kg/°), 0~282m, Internal mode

2.0

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(c) ∫ S (1013kg/°), 0~282m, External mode 0.2 0 −0.2 −0.4 1980

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Fig. 2.26 Zonal modes of the salinity anomaly for a layer between 0 and 282 m. a The total salt content; b the internal mode; c the external mode

century, and this is owing to the general trend of global warming and freshening associated with the melting of the land-based glaciers. For the internal modes the Southern Ocean became lighter over the past 10 years; this is may be due to the freshening associated with land-based glaciers, as shown in Fig. 2.22b.

Changes in total density at different latitude bands are shown in Fig. 2.29. In general, over the past 10 years the northern hemisphere density anomaly is positive, but the southern hemisphere density anomaly is negative. The time evolution of the external and internal meridional modes of the density content anomaly

2.2 External/Internal Modes in Meridional/Zonal Directions

37

Fig. 2.27 Zonal modes of the salinity anomaly for a layer between 665 and 1071 m. a The total salt content; b the internal mode; c the external mode

(a) ∫

0

(1013kg/°), Total

(b) ∫ 30

60N

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(1013kg/°), Internal mode 20 15

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(c) ∫ 5 0 −5 −10 1980

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(1013kg/°), External mode

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Fig. 2.28 Meridional modes of the density anomaly for the entire water column. a The total density content; b the internal mode; c the external mode

is shown in Fig. 2.30. In general, the amplitude of the external mode is comparable with that of the internal modes. We can also diagnose the zonal modes of the density anomaly. Similar to the case shown in

Fig. 2.25, the zonal mode of total signals is also dominated by the decadal change of global warming. As shown in Fig. 2.31, there were positive signals in the last century and negative signals in this century. After separating the

38

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.29 Meridional modes of the density content anomaly for the entire water column

Fig. 2.30 Time evolution of the external and internal modes of the density content anomaly for the world oceans in units of 1013 kg/°

signals into external and internal modes the difference between the three basins becomes more clear (Fig. 2.31b). The difference between the three basins can be explored further by examining the density anomaly at different depth ranges, as shown in

Figs. 2.32 and 2.33. In particular, over the depth range of 665–1071 m (Fig. 2.33) the eastern Indian Ocean and Eastern Atlantic Ocean become heavier, but the eastern Pacific Ocean and Western Atlantic Ocean become lighter. Such climate trends are worth further exploration.

2.3 Adiabatic Signals in the Upper Ocean (a) ∫

0

39

(1013kg/°), Total

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Fig. 2.31 Zonal modes of the density anomaly for the entire water column. a The total density content; b the internal mode; c the external mode

Fig. 2.32 Zonal modes of the density anomaly for a layer between 0 and 282 m. a The total density content; b the internal mode; c the external mode

2.3

Adiabatic Signals in the Upper Ocean

Part of the climate changes on decadal time scale can be interpreted in terms of adiabatic motions associated with the adjustment of the wind-

driven circulation; such motions are called the heaving of isopycnal layers; thus, heaving can include contributions from adiabatic and diabatic processes. However, using a simple reduced gravity model to study climate variability means the focus is on the adiabatic motions.

40

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.33 Zonal modes of the density anomaly for a layer between 665 and 1071 m. a The total density content; b the internal mode; c the external mode

Heat content changes in the ocean, including a hiatus of the global surface temperature and other phenomena, can be interpreted in terms of heaving associated with the adjustment of the wind-driven circulation induced by the decadal variability of wind. A simple reduced gravity model is used to examine the consequence of the adiabatic adjustment of the wind-driven circulation. Decadal changes in wind stress can induce a three-dimensional redistribution of the warm water in the upper ocean. In particular, wind stress changes can generate baroclinic modes of the heat content anomaly in the vertical direction. In fact, changes in stratification observed in the ocean may be induced by wind stress change in the local region or in the remote parts of the world oceans. Many studies have been focused on the internal variability of the atmosphere-ocean coupled system, such as studies of the basin modes, the oceanic response to stochastic wind and the oceanic teleconnections. Johnson and Marshall (2004) explored the global teleconnections of the meridional overturning anomaly. They came to the conclusion that the MOC anomaly on decadal and shorter time scales is confined to the hemisphere basin where the

perturbations originated. However, as will be discussed shortly, the adjustment of the thermocline induced by wind stress changes in a basin is not necessarily limited to the local region; instead, our study will show that the dynamical consequence of the stress variability in any part of the ocean can propagate into other parts of the world oceans. Recently, a hiatus in global warming has become a hotly debated issue related to the global climate changes. Over the past several decades, the global sea surface temperature (SST) kept increasing; in addition, there is a decadal variability of the world ocean heat content. However, ocean warming is far from uniform. For example, in the North Atlantic, the tropics/subtropics have warmed, but the subpolar ocean has cooled. These changes can be linked to NAO, and they are also directly linked to the gyre scale circulation changes and the meridional overturning circulation (Lozier et al. 2008, 2010). In particular, over the past 10–15 years, changes in global SST have more or less leveled off, i.e., this is a hiatus in the global SST record; on the other hand, the subsurface heat content keeps increasing, at seemingly an even higher rate (Meehl et al. 2011; Balmaseda et al. 2013).

2.3 Adiabatic Signals in the Upper Ocean

As will be shown in this book, changes in the oceanic heat content can be studied in terms of a simple reduced gravity model, which is formulated with an idealized geometry and simple wind stress perturbations; the focus of such a model study is on the dynamical consequence of the large-scale adjustment of wind-driven gyres. One of the most important consequences of such adjustments is the basin scale transport of the warm water mass above the main thermocline. We assume that such movements take place within a relatively short time, on the order of inter-annual and decadal time scales. Neglecting the contributions due to the surface thermohaline forcing and internal diapycnal diffusion, the total amount of warm water above the main thermocline can be considered as constant. Since the free surface of the ocean is kept to the same value within the order of one meter, the horizontal transport of the warm water above the main thermocline is compensated for by the return flow of the cold water below the main thermocline. Therefore, the isopycnal movements induced by wind stress perturbations discussed above can be idealized as adiabatic, and they are commonly called heaving. Heaving can induce a redistribution of warm water in the three dimensional space, and hence the changes in the horizontal/vertical density/temperature stratification and the corresponding anomalous meridional, zonal and vertical overturning circulation and heat/freshwater transport.

2.3.1 Adiabatic Adjustment in the Upper Ocean A change in wind can induce shifting of the warm water in the upper ocean and thus a baroclinic mode of heat content anomaly, as illustrated in Fig. 2.34. Since the Coriolis force vanishes near the equator, the zonal momentum balance is primarily between the zonal wind stress and the zonal pressure force associated with the zonal sea level gradient. If the equatorial easterlies are enhanced, more warm water is pushed westward, leading to a higher sea level in the western basin.

41

Because the total amount of warm water is constant, the thermocline near the eastern boundary shoals in compensation. As a result, there is a three dimensional redistribution of the warm water in the basin (red curves and arrows in Fig. 2.34a), leading to a new balance between the sea level pressure gradient and the zonal wind stress. On the equatorial x–z plane, the anomalous circulation appears in the form of a zonal overturning cell rotating anticlockwise. As shown in Fig. 2.34a, warm water within a wedge near the eastern boundary (defined by the black curve and the red curve above) is moved to a similar wedge area near the western boundary. Consequently, the layer moves down near the western boundary, but it moves upward near the eastern boundary. Hence, there is a westward shifting of the heat content (depicted by the westward horizontal red arrow) and a downward shifting of the heat content (depicted in Fig. 2.34b). The westward transport of the warm water above the main thermocline and the eastward return flow of the cold water below the main thermocline give rise to the anomalous ZOC, depicted by the black arrow in Fig. 2.34a. As discussed above, the upper layer moves upward near the eastern boundary at the equator. In general, there is no strong eastern boundary current; thus, the upper layer thickness is nearly constant along the eastern boundary, i.e., the upper layer moves upward along the whole eastern boundary, as depicted by the red arrows in Fig. 2.34c. Since wind stress perturbations are confined to the equatorial band, the Ekman pumping rate at middle latitudes remains unchanged. The square layer depth obeys Eq. (1. 4); accordingly, with a shallower layer depth along the eastern boundary, the thermocline at middle latitudes moves upward, as depicted by the dashed red curve and arrows in Fig. 2.24c. The uniform upward movement of the main thermocline at middle latitudes pushes the warm water above the main thermocline towards the equator, as depicted by the dashed red arrow in the left part of Fig. 2.34c. Accordingly, in the western part of the basin the equatorial main thermocline is pushed downward further; i.e., the enhancement of the equatorial easterlies

42

2 Climate Variability Diagnosed from the Spherical Coordinates North Middle latitude Meridional heat flux

Equatorward mass/heat fluxes

Z Easterly anomaly West

East

Z

Zonal heat flux Meridional heat flux

Vertical heat flux Cold

Warm

Eastern boundary

South Equatorial section (a) Shifting of the thermocline and 3D transport (b) Downward shifting of of mass/heat induced by the enhanced easterly heat in 1st barolinic mode

(c) Shifting of warm water towards low latitudes

Fig. 2.34 Sketch of heaving motions induced by the enhanced easterlies, including the subtropical gyres: a the shifting of thermocline; b the shifting of heat content in the vertical direction; c the shifting of heat content in the meridional direction

induce the warm water to flow westward in the equatorial band and flow equatorward from mid-latitudes. The nearly uniform shoaling of the thermocline along the eastern boundary will be demonstrated through simple numerical models, as discussed in Chap. 4, in particular Fig. 4.7. The equatorward flow of the warm water above the main thermocline combines with the compensating return flow of the cold water below the main thermocline, giving rises to the anomalous meridional overturning circulation and the corresponding meridional transport of heat (and freshwater) as depicted by the solid red arrows and black curved arrows in Fig. 2.34a. Since the conceptual model discussed above is adiabatic, the heat content in the whole basin has zero net gain or loss; thus, in the vertical direction the heat content anomaly must appear in the form of baroclinic modes. These baroclinic modes of the heat content anomaly are entirely due to the adiabatic adjustment of the winddriven circulation in the ocean, i.e., heaving. A similar argument also applies to the thermocline associated with the wind-driven gyre at mid-latitudes. When the Ekman pumping rate is

enhanced, the E-W slope of the thermocline increases; hence, the warm water in the upper ocean is pushed westward, leading to deepening of the thermocline in the western basin. Since the total amount of warm water is assumed to be constant, the thermocline in the eastern basin moves upward in compensation. As a result, there is an anticlockwise rotating zonal overturning cell, which looks like the cell depicted by the black curved arrow in Fig. 2.34a. The baroclinic modes of the heat content anomaly discussed above may provide a simple explanation of the hiatus discussed in many recent studies. As shown in Fig. 2.1d, the world oceans have warmed up after 2007. At the sea surface, there has been a hiatus of the surface temperature, as shown in Fig. 2.2a; in contrast, at the 1000 m level, the ocean has warmed up at twice the speed, as shown in Fig. 2.2b. The changes in the heat content in the world oceans can be idealized as follows. We assume that the global warming of the oceanic component can be roughly represented by a barotropic mode of the heat content anomaly shown in the upper left parts of Fig. 2.35. Due to the intensification of the equatorial easterlies over the past

2.3 Adiabatic Signals in the Upper Ocean

43

Fig. 2.35 The combination of a barotropic and baroclinic modes can give rise to a surface warming hiatus or a surface speeding warming

Uniform warming in barotropic mode

Accelerating warming in surface layers

Warming Hiatus in surface layers Accelerating warming in subsurface layers

Warming Hiatus in subsurface layers

HC anomay in 1st baroclinic mode

(a) Strong easterly anomaly

15–20 years, there is a baroclinic mode of the heat content anomaly, with a cold anomaly in the upper part of the water column and a warm anomaly in the lower part of the water column. As shown in Fig. 2.35a, the combination of these two modes gives rise to a nearly zero heat content anomaly in the upper part of the water column, i.e. a surface warming hiatus; while the heat content anomaly in the lower part of the water column doubles, i.e. a subsurface accelerating warming. Therefore, the intensification of the equatorial easterlies can induce cooling in the upper layer and warming in the subsurface layer, and this may offer an explanation for the hiatus of global surface temperature and the accelerating subsurface warming over the past 10–15 years. Furthermore, the meridional transport of warm water in the upper ocean can lead to sizeable transient meridional overturning circulation, poleward heat flux and vertical heat flux. Thus, heaving plays a key role in the oceanic circulation and climate. It is readily seen that if the equatorial esterlies are weakened, opposite processes should take place, i.e., the surface layer will be warmed up with double speed. As shown in the right part of Fig. 2.35, the baroclinic heat content anomaly now consists of double warming anomaly in the surface layer and cooling anomaly in the subsurface layers. The combination of the quasi

(b) Strong westerly anomaly

barotropic mode and this baroclinic mode of heat content anomaly would give rise to accelerating warming in the surface layer and warming hiatus in the subsurface layers. More in-depth discussion about heaving modes in the upper ocean will be discussed in details in Chaps. 3 and 4.

2.3.2 Adiabatic Wave Adjustment in the Meridional Direction Wind stress is one of the most important forces driving the oceanic general circulation. In particular, wind stress perturbations play a vital role in regulating the climate variability on a broad time scale. In addition, the surface density anomaly is also linked to the meridional overturning circulation. In fact, according to the old paradigm the thermohaline circulation is driven by surface density difference. Although the new paradigm has substantially softened this statement, there is clearly a close connection between the surface density difference and the thermohaline circulation. Therefore, in this section we explore the correlation between the zonal wind stress, the surface density and the meridional overturning circulation (and meridional heat flux) using the GODAS data.

44

2 Climate Variability Diagnosed from the Spherical Coordinates

intensified, the equatorial main thermocline in the western basin moves down, and the signals propagate eastward in the form of the Equatorial Kelvin waves. At the eastern boundary these waves bifurcate into the poleward coastal Kelvin waves moving along the eastern boundary. Since the Coriolis parameter gradually increases poleward, the coastal Kelvin waves gradually shed their energy in the form of westward long Rossby waves. Due to the total volume conservation of the warm water, the main thermocline in the western subtropical basin moves upward in compensation, as depicted by the solid red curve in Fig. 2.36b. The redistribution of warm water implies the transport of heat and mass; therefore, there is anomalous MOC (MHF) as depicted by the dashed curved arrows in Fig. 2.36b. Of course, there is also an anomalous ZOC (ZHF) and a vertical heat content shifting; however, these components of climate variability cannot be included in such a simple two-dimensional sketch. The case shown in Fig. 2.36 is a simple example; in fact, for wind stress changes taking place at other latitudes, either in the subtropical basin or in the subpolar basin, similar adjustments can be induced. The major difference is

The meridional overturning circulation (MOC) and meridional heat flux (MHF) are regulated by the wind stress and buoyancy forcing through complicated dynamic processes. These processes can be identified by carefully analyzing results from numerical models. For example, Polo et al. (2014) discussed the influence matrix for the AMOC at 26° N. The basic idea is that the oceanic general circulation is regulated by the combined effects of the wind stress and buoyancy forcing. When the wind stress or buoyancy forcing in any parts of the world oceans is perturbed, the circulation in the upper ocean changes in response. Such changes propagate in the forms of Kelvin waves and long Rossby waves through the world oceans, and eventually bring about changes in the MOC, MHF, ZOC, ZHF, and the vertical redistribution of the heat content and other parts of the ocean circulation. Since both Kelvin and long Rossby waves travel with finite speed, the response of the circulation to the perturbed forcing is delayed, and the exact time of delay depends on the type of perturbations and the dynamical pathway of the signal propagations. Figure 2.36 is a sketch of the oceanic response to the wind stress anomaly in the equatorial band. If the Equatorial Easterlies are

Fig. 2.36 Wind stress anomaly can induce the adjustment of the circulation in a two-hemisphere basin: a the Equatorial Kelvin waves (KW), the coastal Kelvin waves and the long Rossby waves (RW); b the adjustment of the main thermocline

W W Intensified easterly RW KW

S

RW

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RW KW

KW

RW N

E (a) Anomalous wind induces Kelvin/Rossby waves 60S

45S

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45N

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N (b) Adjustment of the main thermocline in the western basin

2.3 Adiabatic Signals in the Upper Ocean

45

that away from the equator, signals induced by wind stress changes in the basin interior cannot propagate in the form of Kelvin waves; instead, they propagate in the form of westward long Rossby waves only. These long Rossby waves eventually reach the western boundary; part of their energy is reflected in the form of eastward short Rossby waves, but these waves are mostly trapped near the western boundary and dissipated locally. The rest of the energy is converted into the equatorward coastal Kelvin waves, which reach the equator and turn into the eastward Equatorial Kelvin waves and follow the pathway depicted by the red arrows in Fig. 2.36a. Therefore, the adjustment is quite similar to the case with the wind stress anomaly imposed along the equatorial band as discussed above. Similarly, buoyance anomaly can also induce changes in the upper ocean circulation. As an example, the oceanic response to the buoyance anomaly at high latitudes is illustrated in Fig. 2.37. Even if the wind stress is not changed, the variability in surface buoyance forcing can induce changes in the wind-driven circulation in the upper ocean. The perturbations propagate westward in the form of long Rossby waves, and Fig. 2.37 Buoyance anomaly can induce adjustment of the circulation in a two-hemisphere basin: a the Equatorial Kelvin waves, the coastal Kelvin waves and the westward long Rossby waves; b the adjustment of the main thermocline

the complete pathway of such waves is depicted by the red arrows in Fig. 2.37a, very much the same as described in Fig. 2.36a. Similar to the case associated with wind stress perturbations discussed in Fig. 2.36, buoyance anomaly can also induce the three dimensional redistribution of heat and mass. The corresponding anomalous MOC (MHF) is depicted in Fig. 2.37b. The dynamical analysis of the response to surface buoyance anomaly will be explained in details in Chap. 4. It is important to emphasize that dynamical processes, such as wave processes, in the ocean take time to complete, i.e., some of these processes are time delayed. The delay time depends on the specific processes involved. In general, the time delay associated with the buoyancy anomaly can be longer than that associated with a wind stress anomaly. This is due to the factor that changes in wind stress, in particular the zonal wind stress, often take place over the whole basin; as a result, perturbations can reach the western boundary quickly, without much time delay. On the other hand, buoyance anomaly can be confined to a regional ocean, so that the corresponding perturbations can take some time to

W KW RW

RW KW

S

RW

RW KW

KW

Buoyance anomaly RW N

E (a) Anomalous buoyance induces Kelvin/Rossby waves 60S

45S

30S

15S

0

15N

30N

45N

MOC(MHF)

Z

S

60N

N (b) Adjustment of the main thermocline in the western basin

46

2 Climate Variability Diagnosed from the Spherical Coordinates

reach the western boundary, and thus there is a delay in the circulation response in the whole basin.

2.4

The Regulation of MOC (MHF) by Wind Stress and Buoyancy Anomalies

2.4.1 Introduction The dynamic processes regulating the MOC (MHF) are quite complicated, and they are the focus of climate studies. Exploring these dynamical processes can be carried out using simple analytical models, simple numerical models and the state-of-art numerical models. The focus of this section is to diagnose the correlation between the zonal wind stress (surface density) and MOC (MHF). Since the circulation is a complicated three-dimensional phenomenon evolving with time, the correlations should be studied in a four dimensional space-time domain; in particular, the time delay correlations are the crucial parts of such studies.

2.4.2 Surface Density Anomaly The surface density qs ¼ qs ðSs ; Ts Þ anomaly at each station can be separated into two components, the corresponding density deviations can be defined as follows   qSsbar ¼ qs Sbar s ; Ts ;

dqSs

bar

bar

¼ qSs

 qSs bar

t

ð2:8Þ qTs

bar



 bar

¼ qs Ss ; Ts

;

dqTs

bar

¼ qTs

bar

 qTs bar

t

ð2:9Þ t

t

¼ Ss and Tsbar ¼ Ts depict the cliwhere Sbar s matological mean surface salinity and temperature at each station in the GODAS data. Therefore, the RMS of these three density variables can be calculated from the GODAS data. The horizontal distribution of these quantities is shown in Fig. 2.38. It is clearly seen that the

surface density anomaly is primarily confined to the subtropical gyres in the Northern Hemisphere, in particular the western and central parts of the subtropical gyres. The variability in the Southern Hemisphere is much weaker. In most parts of the world oceans, the variabilities are primarily controlled by the temperature anomaly. The role of the salinity anomaly is noticeable in two regions: the region of the Gulf Stream separation from the coast and the Bay of Bangor where the freshening induced by strong precipitation and river runoff may induce large anomalous signals in density. In order to show the relative contributions of the temperature and salinity anomalies, we calculated the ratio of RMS density anomaly defined as R¼

RMSðdqSs bar Þ RMSðdqSs bar Þ þ RMSðdqTs bar Þ

ð2:10Þ

As shown in Fig. 2.39, R is larger than 0.7 for most parts of the world oceans, indicating that the temperature anomaly is the primarily contributor to the surface density anomaly. There are, however, places where the salinity anomaly can play a major role in regulating the surface density anomaly, such as the warm pools in the Pacific-Indian Ocean and the Atlantic-Pacific Ocean, the Bay of Bangor and along the edge of Antarctica. The actual contributions due to temperature and salinity perturbations for three zonal sections are shown in Fig. 2.40. As shown in the top two panels (along 10° N and the Equator) the salinity anomaly plays the dominating role in generating the surface density anomaly. In particular, near the eastern boundary of each basin, the density anomaly is large, and the contributions owing to salinity perturbations also play a vital role; this suggests that simulating the salinity anomaly accurately is a key to simulating the circulation and the climate in these regions. Along the edge of the Antarctica, the salinity contributions to the density anomaly are the dominating term. This is due to the fact that at low temperature the thermal expansion

2.4 The Regulation of MOC (MHF) by Wind Stress …

47

Fig. 2.38 Root-mean square of the surface density anomaly (in units of kg/m3), based on the GODAS data

coefficient is quite small, on the order of 0.1 kg/m3/°C; therefore, salinity plays the dominating role of controlling the density, as shown in Fig. 2.40c. In particular, the salinity variabilities play the vitally important role in regulating water density near the site of bottom water formation. As an example, we show the temperature, salinity and density anomalies at the sea surface for a station in the Weddell Sea (73° S, 30.5° W). At this station, the mean temperature is very low, −1.29 °C. The variability of the surface density (black curve) basically follows that of the salinity variability (blue curve) (Fig. 2.41). At such low

temperature, density is primarily controlled by the salinity anomaly. Only when temperature perturbations appear in the form of sharp positive spikes, the density variability responds in the form of small negative spikes. This phenomenon can be seen in a clearer way. As shown in Fig. 2.42, on interannual and decadal time scales the density anomaly induced by the salinity anomaly alone (blue curve) can capture most parts of the density anomaly (black curve) recorded at this station. On the other hand, on a seasonal time scale strong spikes in the temperature anomaly also play an important role in regulating the density anomaly.

48

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.39 Ratio of the RMS surface density anomaly, based on the GODAS data

2.4.3 Correlation Between Surface Forces and MOC Our discussion in this section will be focused on the relation between the surface forcing and the MOC. The MOC is a complicated threedimensional phenomenon; at least it consists of several important components, including the Ekman transport in the upper ocean, which is directly linked to the wind stress, the geostrophic component below the surface layer and the western boundary currents. Each of these components involves complicated dynamical processes in the ocean. Different components of the circulation are linked to the surface wind stress and the surface density anomalies. Exploring such connections is a challenging job. In this section, we explore the statistical correlations between three meridional transport indexes and other field properties. As shown below, these three indexes reflect different aspects of the meridional circulation and their connection with the wind stress and the surface density anomalies is also different. The first one is the zonally integrated meridional transport over the top 1100 m, and this will be labeled as the MOC:

Z0 MOC ¼

Zke r cos /vdk

dz H

ð2:11Þ

kw

where H = 1071 m is the lower interface of the 31st level grid box used in the GODAS data, kw and ke are the western and eastern boundary of the corresponding basins, r is the radius of the Earth, / ¼ 26:5
N or / ¼ 26:5
S is the latitudes of the zonal sections. The second index is the MOC subtracting the Ekman transport defined by the corresponding monthly mean zonal wind stress GeoMOC ¼ MOC  MEkman

Zke

¼ MOC þ kw

r cos /sk dk ð2:12Þ 2q0 x sin /

The third index is the subsurface geostrophic transport, defined as the meridional transport from a subsurface level (90 m) to the 1071 m deep level, i.e., it is defined as follows Z90m SubSurMOC ¼

Zke r cos /vdk ð2:13Þ

dz 1071m

kw

2.4 The Regulation of MOC (MHF) by Wind Stress …

49

Fig. 2.40 Contribution to the density anomaly from temperature and salinity perturbations along three zonal sections, based on the GODAS data

We start with the correlation between the zonal wind stress and the AMOC (MOC in the Atlantic Ocean) at 26.5° N, and the correlation with no time lag is shown in Fig. 2.43. The strongest correlation is approximately −0.6 for the zonal wind stress in the latitudinal band of 20° N–35° N in the subtropical North Atlantic Ocean, indicated by the blue color in Fig. 2.43a. The zonal wind stress at high latitudes in the North Atlantic Ocean also appears as a region of maximum positive correlation. These correlation extrema are clearly

linked to the Ekman transport induced by the surface zonal wind stress; this point is confirmed by the fact that the correlation between taux and the GeoAMOC/SubSurAMOC over these regions is substantially reduced or even the sign is flipped, as shown in Fig. 2.43b, c. It is also interesting to see that the zonal wind stress north of the equatorial Pacific and within the ACC band in the South Indian Ocean are also positively correlated to the AMOC, as shown in Fig. 2.43a. This correlation may reflect the role

50

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.41 Surface density, salinity and temperature anomalies at (73° S, 30.5° W)

Fig. 2.42 Time evolution of the surface density and contributions from the salinity/temperature anomalies at (73° S, 30.5° W)

of the Indonesian Throughflow and the winddriven gyre in the South Indian Ocean; these two factors can affect the Agulhas current and its leakage to the Atlantic Ocean, as discussed by Beal et al. (2011). On the other hand, the pattern of correlation between the zonal wind stress and SubSurAMOC shown in Fig. 2.43c looks quite different than that shown in Fig. 2.43a, indicating that the

dynamic effect of the zonal wind stress north of the equatorial Pacific and within the ACC band in the South Indian Ocean is mostly limited to the upper branch of the AMOC (at 26.5° N section), and has no influence on the subsurface geostrophic component at the 26.5° N section. Next, we examine the correlation between surface density and the AMOC at 26.5° N, and the correlation with no time lag is shown in

2.4 The Regulation of MOC (MHF) by Wind Stress …

51

Fig. 2.43 Correlation between the zonal wind stress and AMOC/GeoAMOC/SubSurAMOC (at 26.5° N) in the world oceans

Fig. 2.44 Correlation between the surface density and AMOC/GeoAMOC/SubSufAMOC (at 26.5° N) in the world oceans

Fig. 2.44. The strongest positive correlation is approximately 0.5 for the density anomaly in the latitudinal band of the ACC, in particular near the Weddell Sea, indicated by the red color in Fig. 2.44a–c. Apparently, the surface density anomaly in the ACC is correlated to all three meridional transports in quite similar ways: the higher density anomaly in the Weddell Sea promote the stronger AMOC. The surface density anomaly in the Northern Hemisphere appears as a region of maximum negative correlation. This reflects the fact that the strong AMOC can transport more heat northward and thus makes the surface density lighter. The negative correlation between surface density

anomalies in the North Atlantic Ocean is much smaller for the subsurface geostrophic component of the AMOC because the subsurface currents have much less thermodynamic effects on the surface density in this region. Similarly, one can examine the correlation of the zonal wind stress vs. the PMOC (MOC in the Pacific Ocean) at 26.5° N, and the correlation of the surface density vs. the PMOC at 26.5° N. The corresponding figures are excluded. Our discussion above is focused on the correlation between these time series with no delay. In reality, dynamic signals take time to move through the world oceans, and the typical signals are in the form of long baroclinic long Rossby waves and

52

2 Climate Variability Diagnosed from the Spherical Coordinates

Kelvin waves. Hence, it is vitally important to examine the correlation between different field variables which are subjected to time delays. For climate study the meridional heat flux (MHF) is of great interest; accordingly, we explore the correlation between the MHF and the zonal wind stress and the surface density. Z0 MHF ¼

Zke rCp q0 h cos /vdk

dz H

ð2:14Þ

kw

where Cp is the specific heat capacity under constant pressure, q0 is the constant reference density, T is the temperature. We begin with the MHF in the Northern Hemisphere. As shown in Fig. 2.45, the correlation of the zonal wind stress vs. the AMHF (MHF in the Atlantic Ocean) and the PMHF (MHF in the Pacific Ocean) is quite similar to the correlation between the zonal wind stress and the AMOC shown in Fig. 2.43a. The similarity of patterns for the correlation between the wind stress (surface density) and the MOC and that between the wind stress (surface density) and the MHF is expected because in general the MHF is primarily controlled by the volumetric flux of the warm water in the upper ocean, i.e., MOC; hence, these two types of

correlation are closely linked to each other, with minor differences.

2.4.4 Conclusion In this section we explored the statistical correlation between the zonal wind stress (surface density) and the MOC (MHF). The correlation pattern can be used as a tool to diagnose the dynamical processes linking these variables. Since the zonal wind stress is directly linked to the meridional Ekman transport in the upper ocean, it is a simple index for the wind-driven circulation. Another important index is the Ekman pumping rate, which can be directly calculated from the wind stress; in addition, the zonally integrated Ekman pumping rate is more directly linked to the gyre scale circulation and thus to the MOC (MHF) and the density structure in the upper ocean. However, the zonal integration of Ekman pumping is subjected to time delay due to the finite speed of the first barotropic long Rossby waves; hence, the calculation of the corresponding zonally integrated Ekman pumping rate is slightly more complicated. Nevertheless, it is doable with care taken to such technical details. Recently, the MOC in the Atlantic Ocean (AMOC) has received much attention. Although

Fig. 2.45 Correlation between the surface zonal wind stress and the MHF (26.5° N) in the two basins of the Northern Hemisphere

2.4 The Regulation of MOC (MHF) by Wind Stress …

53

the impact of the PMOC (PMHF) and the IMOC (IMHF) is smaller than that of the AMOC, these volumetric fluxes and heat fluxes play critical roles in regulating the climate variability of the world, in particular in eastern Asian and China. Studies of the South China Sea (SCS) often used the terms of the atmospheric bridge and the oceanic bridge, i.e., climate variability in the SCS can be carried out through the dynamical processes in the atmosphere and ocean. This concept may be extended to both the North Pacific Ocean and South Indian Ocean. For the North Pacific Ocean, the climate variability in the ocean induced by processes away from this region through both the atmospheric bridge and the oceanic bridge is as follows. For example, ENSO events are predominately confined to the equatorial region. Through atmospheric processes, the regions outside of the equatorial band are affected by changes in the atmospheric circulation. Such changes can affect the wind stress, heat flux and freshwater flux through the air-sea interface at the middle and high latitude ocean. These changes in atmospheric conditions affect the oceanic circulation at middle and high latitudes; accordingly, they can be classified as contributing to the atmospheric bridge. Furthermore, ENSO events can also affect the oceanic meridional transport of both mass and heat content from low latitudes to middle and high latitudes: such transports can be classified as the oceanic bridge. Changes of the oceanic conditions at middle and high latitudes can thus be classified in terms of the atmospheric bridge and the oceanic bridges. Similarly, the change of the oceanic circulation in the Indian Ocean can be linked to climate changes in other parts of the world oceans. For example, during ENSO events, the Walker circulation changes, and hence affects the atmospheric circulation in the Indian Ocean sector; this can be called the atmospheric bridge. On the other hand, ENSO events can also affect the Indonesian Throughflow and South China Sea Throughflow in terms of the volumetric transport and heat/freshwater transport through the corresponding channels. ENSO

events can also affect the Antarctic Circumpolar Current (ACC)—through upstream/downstream exchange they can influence both the Atlantic and Pacific sectors. These processes can be classified as the oceanic bridges. Hence, the climate variability in the Indian Ocean may be separated into two large categories, the atmospheric bridge and the oceanic bridge. Such a separation may help us to understand the climate variability in a better way.

2.5

Adiabatic Heaving Signals in the Deep Ocean

Although the deep ocean is sheltered from the direct impact of the surface wind stress, there are still strong climate variability signals, as shown by the RMS signals at 3972 m depth, diagnosed from the GODAS data (Fig. 2.46). The density signal maximum appears in the equatorial Atlantic basin; the salinity signal maximum appears in the Atlantic sector and the equatorial band. On the other hand, the potential temperature signal maximum appears in the Indian Ocean. The difference in the bottom temperature anomaly between the equatorial Atlantic Ocean and the Pacific/Indian Ocean implies that the inter-basin mode discussed in Huang (2015) may play a key role in the establishment of the deep ocean cooling anomaly. The inter-basin mode will be further discussed in Chap. 4. Climate variability in the deep ocean is dynamically linked to the wind stress anomalies applied to the world oceans. As shown in Fig. 2.47, the westerlies over the Southern Ocean induce a strong eastward ACC. The ACC is a geostrophic current sustained by the balance between the northward Coriolis force and the southward pressure gradient force. On the surface, the strong westerlies (panel a) and the corresponding strong sea level gradient (panel b) indicate strong eastward currents. As shown in Fig. 2.47, there are strong fronts in the density, salinity and temperature. Over very broad scales in time and space, the strength of ACC and these fronts continuously adjust

54

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.46 Root-mean square of the water property at 3972 m level, diagnosed from 36 years (1980–2015) of the GODAS data

themselves in response to changes in the wind stress forcing, thermohaline forcing and other forcing. In general, a slower zonal velocity corresponds to weaker fronts and thus gentler slope of the isopycnal (isohaline or isothermal) surfaces. As a result, water mass properties, such as temperature, salinity and density, in the deep

ocean are redistributed in the vertical and horizontal directions. These motions are called heaving motions. Under the continuously varying forcing conditions in the ACC, the zonal currents, temperature, salinity and density in the deep ocean change in response (Fig. 2.48). In particular,

2.5 Adiabatic Heaving Signals in the Deep Ocean

55 (c) U (mm/s) & T (°C)

(a) x(N/m2)

14

0.2

12

3.2

0.1

10 3.4 Depth (km)

0 −0.1 (b) SSH(m)

0.5

0

3.8

−4 −6

60S 55S 50S 45S 40S 35S 30S 25S 20S

34.75 3.6

34.74 34.71

(kg/m3)

14

34.73

10 8

60S 55S 50S 45S 40S 35S 30S 25S 20S

27.85

27.88

6

6 4

27.885 27.86

0 −2

2 0

27.89

−4 34.72

12

10

2

34.7

4.2

0

−8

8 4

4.0

(e) U (mm/s) &

12

3.4

0 −2

0.8

0.4

14

34.755

3.2

Depth (km)

1 0.6

4.4 (d) U (mm/s) & S (psu)

4.4

2

0.2

−1.0

3.8

4

−0.2

4.2

60S 55S 50S 45S 40S 35S 30S 25S 20S

8 6

3.6

4.0 −0.4

0 −0.5

−1.5

1.2

−2

27.875 27.87

−4

−6

−6

−8

−8

60S 55S 50S 45S 40S 35S 30S 25S 20S

Fig. 2.47 Climatological zonal-mean fields for the deep ocean in the Southern Hemisphere: a the zonal wind stress; b the sea level; the zonal velocity (color map) overlaid by c temperature contours (white); d salinity contours (white); e density contours (white)

both the temperature and density vary greatly in the equatorial band, as shown in Figs. 2.48b, d. The salinity variability maximum is located at 40° S. The dynamic explanation of this figure remains unclear at this time. All these changes are closely linked to the zonal wind stress changes; of course, the ACC is associated with many eddies playing a vitally important role; as such, the strength of the ACC including the fronts, is not a simple linear function of the zonal wind stress. Nevertheless, there is a strong positive correlation between the zonal wind stress and the zonal velocity, as shown in Fig. 2.49. The time evolution of properties is shown in Fig. 2.50. The strength of the ACC in the deep ocean is represented by the red curve, and the strength of the southern westerlies is represented by the pink curve. There is a visible correlation between these two curves. For example, the

positive peak of wind stress in 1987–1988 corresponds well with the zonal velocity peak. Most importantly, the zonal wind stress went down slightly after 1998, and the corresponding zonal velocity also went down, in particular after 2006, as shown on the right part in Fig. 2.50. As shown in Fig. 2.50, the bottom temperature went down noticeably, but the bottom salinity and density went up noticeably for the same time period. Therefore, the baroclinic modes of anomalous potential temperature, salinity and potential density in the deep ocean, as shown in Figs. 2.1, 2.2, 2.3, 2.6, 2.7, 2.8, 2.9, 2.10 and 2.11, are closely linked to the anomalously strong zonal velocity in the deep ocean induced by the westerly anomalies in the core of the ACC. The connection between the zonal current and the bottom temperature variability is as follows.

56

2 Climate Variability Diagnosed from the Spherical Coordinates (a) Mean| U|(mm/s)

(b) Mean|

|(°C)

3.0

0.07

Depth (km)

3.0

0.06

2.5

3.2

0.05

3.4

2.0

0.04

3.6

0.03

1.5

3.8 4.0

0.02 1.0

4.2

0.01

4.4 60S 50S 40S 30S 20S 10S 0 10N 20N 30N 40N

0.5

(c) Mean| S|(0.001 psu)

60S 50S 40S 30S 20S 10S 0 10N 20N 30N 40N (d) Mean|

0|(0.001

kg/m3)

5.5 4.0

Depth (km)

5.0 3.0

4.5

3.2

4.0

3.5 3.0

3.5

3.4

2.5

3.0

3.6

2.0

2.5

3.8

2.0

4.0

1.5

1.5 1.0

1.0

4.2

0.5

0.5

4.4 60S 50S 40S 30S 20S 10S 0 10N 20N 30N 40N

60S 50S 40S 30S 20S 10S 0 10N 20N 30N 40N

Fig. 2.48 Zonal mean property perturbations (absolute value) in the deep ocean: a the zonal velocity (South of 10° S only); b the temperature; c the salinity; d the density

When the Southern Westerlies vary, the strength of the ACC fronts changes in response. Assuming that the wind stress declines, then the front is weakened; as such, the slope of the isopycnal surfaces in the ACC band becomes gentler. Consequently, at the southern edge of the ACC, the relatively warm water is pushed down, indicated by the red arrow in Fig. 2.51. As the total volume of water mass in each density category is constant, the cold bottom water near the sea floor must move upward north of ACC, mostly near the equator and in the Northern Hemisphere, depicted by the blue arrows. This mechanism can be illustrated using the GODAS data. As an example, we plot the temperature difference over the 8-year period,

December 2006 to December 2014. The zonally mean temperature difference is shown in Fig. 2.52a, and the corresponding heat content anomaly is shown in Fig. 2.52b. It is clear that the temperature became warmer south of the ACC, but it became cooler north of the ACC. The meridional profiles of the heat content anomaly at two depths are shown in Figs. 2.52c, d. These two panels show the thermal anomalies induced by the weakening of the ACC, and illustrate changes in the thermal structure in response to the variability of the velocity fronts associated with the ACC. Therefore, the apparent cooling of the deep ocean over the past ten years is directly linked to the slow-down of the ACC, and the decline in the southern westerlies.

2.6 Final Remarks

57

Fig. 2.49 Correlation between the zonally mean zonal wind stress (red triangles) and the zonal velocity in the Southern Ocean

2.6

Final Remarks

As discussed above, in the traditional spherical coordinates, the climate variability can be separated into the external (barotropic) and the internal (baroclinic) components. The external component represents the change of the total heat content, salt content and density content in the world oceans. Such signals must be directly linked to the external thermohaline forcing, including the heat flux, and the freshwater flux through the air-sea interface, sea-ice interface or the land-sea interface.

On the other hand, internal modes represent the heat (salt or density) content exchanges between different vertical levels (latitudinal or longitudinal bands). Although we can identify the internal (baroclinic) mode climate signals, this signal separation does not necessarily reveal the nature of such internal signals, and within such an approach itself it is difficult to further explore the cause of these baroclinic signals. In most cases, mixing plays a relatively minor role in generating climate signals discussed above, it is most likely that advection (both in the horizontal and vertical direction) may play the key role in generating such signals.

58

2 Climate Variability Diagnosed from the Spherical Coordinates

Fig. 2.50 Time evolution of the zonal velocity and bottom (3972 m) properties (salinity, temperature, and density), where dU ¼ dU ð51:50
S  44:83
S; 3972 mÞ, dsx ¼ dsx ð51:50
S  46:50
SÞ; all curves subjected to 13 month moving smoothing

North

ACC

Isopycnal

Fig. 2.51 Sketch illustrating the heat content changes in the deep ocean in response to the weakening of the ACC associated with decline of the Southern Westerlies

For example, over the past decade there is a clear sign of the so-called hiatus of the sea level temperature. Such a phenomenon is difficult to understand from an analysis based on the zcoordinate. However, it might be much easier to understand it, if we examine the climate variability from a different angle. In fact, the main goal of this book is to explore the possible approach based on isopycnal layer analysis in order to further separate the climate variability into the heaving, stretching and spicing components.

2.6 Final Remarks

59

Fig. 2.52 Changes of the temperature and heat content in the deep ocean over 8-year time (2006–2014) period

References Balmaseda MA, Trenberth KE, Källén E (2013) Distinctive climate signals in reanalysis of global ocean heat content. Geophys Res Lett 40:1754–1759. https://doi. org/10.1002/grl.50382 Beal LM, Ruijter WPM De, Biastoch A, Zahn R (2011) On the role of the Agulhas system in ocean circulation and climate. Nat 472, 429–436. https://doi.org/10. 1038/nature09983 Behringer DW, Xue Y (2004) Evaluation of the global ocean data assimilation system at NCEP: The Pacific Ocean. Eighth symposium on integrated observing and assimilation systems for atmosphere, oceans, and land surface, AMS 84th Annual Meeting, Washington State Convention and Trade Center, Seattle, Washington, pp 11–15 Carton JA, Giese BS (2008) A reanalysis of ocean climate using Simple Ocean Data Assimilation (SODA). Mon Weather Rev 136:2999–3017 Huang RX (2015) Heaving modes in the world oceans. Clim Dyn. https://doi.org/10.1007/s00382-015-2557-6

Johnson HL, Marshall DP (2004) Global teleconnections of meridional overturning circulation anomalies. J Phys Oceanogr 34:1702–1722 Liang X-F, Wunsch C, Heimbach P, Forget G (2015) Vertical redistribution of oceanic heat content. J Clim 28:3821–3833 Lozier MS, Leadbetter S, Williams RG, Roussenov V, Reed MSC, Moore NJ (2008) The spatial pattern and mechanisms of heat-content change in the North Atlantic. Science 319:800–803 Lozier MS, Roussenov V, Reed MSC, Williams RG (2010) Opposing decadal changes for the North Atlantic meridional overturning circulation. Nat Geosci 3:728–734 Meehl GA, Arblaster JM, Fasullo JT, Hu A, Trenberth KE (2011) Model-based evidence of deep-ocean heat uptake during surface-temperature hiatus periods. Nat Clim Change 1:360–364 Polo I, Robson J, Sutton R, Balmaseda (2014) The importance of wind and buoyancy forcing for variations and the geostrophic component of the AMOC at 26oN, J. Phys. Oceanogr 44:2387–2408. https://doi. org/10.1175/JPO-D-13-0264.1

3

Heaving, Stretching, Spicing and Isopycnal Analysis

3.1

Heaving, Stretching and Spicing Modes

3.1.1 Adiabatic and Isentropic Processes Climate variability in the ocean can be generated by two basic mechanisms: the internal nonlinear dynamics or the external forcing anomalies. The later can be further separated into three basic terms: anomalies in momentum, heat and freshwater fluxes through the air-sea interface. Tidal dissipation anomalies can also induce climate variability; however, such variability is characterized by much longer time scales, and thus will not be discussed here. Climate processes under the condition of no heat exchange are called isothermal, and they are often called adiabatic. Similarly, processes generating climate variability under the condition of no salt flux anomaly may be called isohaline. The other major cause of climate variability is the wind stress anomaly. Wind-driven circulation is mostly confined to the upper 1–1.5 km of the ocean, so that climate variability induced by wind stress anomalies can penetrate into the depth of 1.5 km in the ocean. There are also evidences indicating that the effects of wind stress perturbations can penetrate into the deep ocean, as discussed in Chap. 2. Motions in the ocean generated under the conditions of no heat/salt flux anomalies are sometime called isentropic. However, this may not

be an accurate term to describe such processes. In fact, the entropy of seawater is a complicated function of salinity, temperature and pressure. Although for a long time there was no standard subroutine available for calculating entropy of seawater, the situation is now quite different. In fact, one can use the standard subroutines in the TEOS_10 to calculate seawater thermodynamic variable, including entropy and others. Seawater entropy is a function of temperature, salinity and pressure; at the sea level, entropy changes are primarily owing to the variability of temperature. However, salinity can also affect entropy. In fact, entropy under isothermal condition can change slightly due to the increase of salinity. Therefore, using the term “adiabatic” to describe motions induced by wind stress perturbations alone is inaccurate. Furthermore, even the commonly used word of isentropic is not accurate. In fact, through a process of weak mixing of temperature and salinity, both temperature and salinity of a water parcel can be changed, but the entropy remains unchanged. Accordingly, a process induced by wind stress change, but under isothermal and isohaline conditions, cannot be simply called an “isentropic” process. In this book, we will occasionally use a new term “isoTS” process to indicate that the process is reversible and it is under both isothermal and isohaline conditions, so that it is also isentropic. In order to understand climate variability, it is desirable to separate the causes of such variability, i.e., whether it is induced by an anomaly

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_3

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in heat flux, freshwater flux or wind stress. It is to emphasize that motions in the ocean are the results of combined thermohaline forcing and wind stress. Since the relevant dynamics are strongly nonlinear, in principle it might be impossible to completely separate contributions from thermal forcing, haline forcing and wind stress. Nevertheless, it is still of great interest to separate these signals as cleanly as possible.

density compensated. For a long time, it has been realized that to map out the variability on an isopycnal it is desirable to use a second thermodynamic variable. Ideally, this thermodynamic variable should be “orthogonal” to the density, so that an orthogonal curvilinear coordinate system can be established and used for water mass analysis and climate study (Stommel 1962). This thermodynamic function is now called spiciness or spicity. Despite much effort into searching such thermodynamic function, no function satisfies the orthogonality constraint was found. However, resent study shows that such functions that satisfy the orthogonality constraint almost perfectly do exist, e.g., Huang et al. (2018). We will discuss this in detail in the next section. In the 1990s, heave and heaving modes were introduced into physical oceanography, e.g., Bindoff and McDougall (1994); since then these terms have been used in the study of climate change related to the circulation and (T, S) properties. When it was first introduced in physical oceanography, heaving was a term to describe the “adiabatic motion” of the isopycnal, without the effect of heating/cooling or salinification. For example, Bindoff and McDougall (1994) discussed the wind stress anomaly’s effect on the isopycnal outcrop line; although the wind-driven circulation adjustment through Rossby waves was mentioned, no dynamical details were presented. It is well known that wind stress changes can induce adiabatic motions in the ocean. As discussed above, however, changes in the wind stress can also affect diapycnal and isopycnal mixing in the ocean. Consequently, it is unclear how to identify the adiabatic motions from climate data obtained either from in situ observations or numerical simulations of the oceanic circulation and climate. Hence, defining the heaving signals as adiabatic motions in the ocean poses a technically challenging task as how to apply such a definition in climate data analysis. In this chapter the concept of heaving is reexamined in terms of isopycnal analysis. Isopycnal analysis is based on using potential density as the vertical coordinate. Accordingly, climate variability is cast onto isopycnal

3.1.2 Heaving, Stretching and Spicing Modes In Chap. 2, we have discussed climate variability using the spherical coordinates, and the vertical coordinate is the z-coordinate. Such approach is the Eulerian coordinate, with the analysis focused on the fixed grids in space. Although such approach can reveal a wealth of climate signals, it is difficult to separate the contribution due to the isoTS processes discussed above. There are other approaches for analyzing climate variability; in particular, one can use coordinates based on the physical properties of sea water. In fact, the observers can follow the movement of isopycnal surfaces or isopycnal layers. In previous publications, such movements are called heave or heaving motions in the oceans. As discussed by Huang (2015), isoTS motions in individual basins of the world oceans are all linked through complicated wave/current processes induced by wind stress perturbations. To better understand the isoTS motions induced by the wind stress anomaly, the term adiabatic heaving/stretching modes can be used. In accordance with the common usage, from now on we will use the adjective “adiabatic” to mean both isothermal and isohaline, or isoTS, unless explicitly stated otherwise. Since tracer properties, such as potential temperature, salinity and potential density, are nearly conserved during large-scale adiabatic motions, isopycnal analysis has been used as a convenient tool in water mass and climate variability analysis. By definition, potential density is constant on a given isopycnal surface; therefore, potential temperature and salinity on this surface must be

Heaving, Stretching, Spicing and Isopycnal Analysis

3.1 Heaving, Stretching and Spicing Modes

Heaving

Stretching

Fig. 3.1 Heaving and stretching modes of an isopycnal layer

coordinates in the form of three components: heaving, stretching and spicing. Heaving signals represent the vertical movement of individual isopycnal surfaces (dashed curves in Fig. 3.1). If heaving modes of all isopycnal surfaces over the entire depth of the ocean are described, the information is complete. However, if we are concentrated on an individual isopycnal surface, heaving motion of this surface itself is not adequate for the description of climate variability in the vicinity of this isopycnal surface; in addition, stretching of this isopycnal layer is the necessary supplementary information. Stretching signals represent the stretching/ compressing of individual isopycnal layers, as defined by the upper and lower interfaces in the isopycnal coordinates (vertical double-arrowed lines in Fig. 3.1). The third term is spicing, that represents the change in spicity. The time evolution of a tracer, such as spicity, obeys the following equation, where the left-hand side includes the time dependent term and the horizontal advection term:
   @ ðhpÞ þ rq  h! u p ¼ rq  kq rq ðhpÞ @t @ _ Þ þ Hp ðqhp  @q ð3:1Þ

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The first term on the right-hand side is the contribution in connection with the isopycnal mixing. The second term represents the contribution owing to the diapycnal mixing. The last term, Hp is the spicity source term, including the diapycnal mixing of spicity, such as the contribution due to the air-sea interaction associated with mode water or deep water formation and the air-sea interaction in the cold tongue in the equatorial Pacific Ocean; in addition, this term includes the spiciling effect associated with the horizontal/vertical mixing. Both the potential density and the potential spicity are conservative quantities, i.e., they will conserve their original value during ideal fluid motions, without mixing. However, because of diapycnal mixing or horizontal mixing, neither potential density nor potential spicity would be conserved due to the nonlinear nature of the equation of the state of seawater. Such a phenomenon associated with density mixing is called cabbeling; in parallel, mixing of potential spicity also lead to a change of spicity, which can be called spiciling. The meaning of spicity and spiciling will be discussed in detail in the next section. The continuity equation in the isopycnal coordinates can be formally written as follows
 @ @ _ Þ h þ r q  h! u ¼  ðqh @t @q

ð3:2Þ

where q_ denotes the diapycnal velocity. Accordingly, the amount of water within each isopycnal layer is regulated by the difference in diapycnal velocity crossing the upper and lower interfaces of each isopycnal layer. As discussed above, diapycnal velocity is closely linked to the vertical mixing of tracer (temperature, salinity, spicity), and the cabbeling effect due to the lateral mixing of tracers. However, the water mass formation and the erosion associated with surface thermohaline forcing are the most important regulators for water masses in the world oceans. The contribution of water mass formation/erosion is somewhat hidden in the continuity equation discussed above.

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3.1.3 External Heaving Modes Versus Internal Heaving Modes

and thermal/haline mixing may be nearly impossible. As a result, if we define heaving as the adiabatic motions of isopycnal layers, such a strict definition may be technically so challenging that it is not practical. To overcome this problem, we adapt the following slightly more flexible working definition. Heaving motions are defined as the movement of the isopycnal layers (in potential density coordinates), the isothermal layers (in potential temperature coordinates) or the isohaline layers (in salinity coordinates). Heaving signals defined in this way can be identified from data, using the methods discussed in this book. It is important to emphasize that heaving signals defined in this way may include contributions in connection with the wind stress perturbations and the thermohaline forcing anomaly; the exact separation of the combined nonlinear dynamic effects is left for further study. The oceanic circulation and the sea water property distribution can be described in terms of water mass formation and transportation. The mixed layer in the upper ocean is the major site of water mass transformation. In the subsurface ocean, water mass balance is established through three basic processes as shown in Fig. 3.2:

Wind stress anomalies can play complicated roles in generating climate anomalies in the ocean. First, wind stress induced Ekman pumping can change the wind-driven circulation in the upper ocean. Changes in Ekman pumping induce compression or stretching of the water column, leading to the so-called internal adiabatic heaving mode in the ocean interior. The difference in heaving motions between adjacent isopycnal surfaces gives rise to the stretching mode. In addition, changes in the wind-driven circulation leads to the shifting of the outcrop lines. As a result, subduction and obduction rate distribution in the density coordinates can be changed, and this leads to the perturbations in the water mass volumetric partition in density categories. It is important to emphasize that in this book, density is referred to as potential density, unless specifically stated otherwise. Therefore, changes of the in situ density associated with vertical adiabatic migration of water parcels are excluded in our study. Second, wind stress perturbations lead to changes in mechanical energy sustaining mixing in the ocean, in particular mixing in the mixed layer and the upper ocean. As a result, water mass transformation in the ocean is changed. Consequently, due to the nonlinear nature of the oceanic circulation a clear separation of the pure adiabatic processes in connection with wind stress perturbations and the diabatic processes associated with anomalous wind energy input

Heaving, Stretching, Spicing and Isopycnal Analysis

Water Mass Balance ¼ subduction  obduction þ transformation ðerosionÞ

In the ocean, water mass is formed through the following processes: bottom water formation, deep water formation and subduction (formation of light water masses). Subduction is the process

External heaving mode ρ2

Internal heaving mode

ρ1

ρ1

ρ2

Mixed layer

Obduction

ρ1 ρ2

Subduction Deep water formation Bottom water formation

Water mass transformation due to mixing

Fig. 3.2 Sketch of the external and internal heaving modes in the world oceans

3.1 Heaving, Stretching and Spicing Modes

describing water parcels leaving the base of the mixed layer and entering the permanent pycnocline irreversibly. Water masses are transformed (or eroded) through many processes, such as obduction or mixing in the subsurface ocean. In particular, obduction is defined as a process opposite to subduction, i.e. the water mass irreversibly moves from the permanent pycnocline into the mixed layer and thus loses its original identity. The basic concepts of subduction and obduction are referred in Qiu and Huang (1995) and Huang (2010). Diapycnal mixing in the subsurface ocean is another major process that transforms water masses. In addition, due to the nonlinearity of the equation of state, the density of sea water parcels after mixing becomes heavier than the mean of the parental water parcels before mixing. This is called the cabling effect, e.g., Huang (2010). Cabling associated with isopycnal mixing is also an important form of water mass transformation. In this book, we define heaving as the movement of isopycnal (or isothermal, isohaline) layers. For a station in the ocean, the vertical movement of isopycnal layers, or the heaving motions, can be separated into two components: the external heaving modes and the internal heaving modes (Fig. 3.2). The external heaving modes represent the uniform expansion (contraction) of the isopycnal layers, i.e. the thickness increase (decrease) of the corresponding isopycnal layer associated with the net volume change of the given water mass density category. Such changes are due to the variation in the volumetric distribution in density space, and are closely linked to the changes in the thermohaline circulation. Note that external heaving modes must be directly linked to diabatic (or salinification) processes; in fact, without diabatic (or salinification) processes, it is impossible to change the density of sea water. The internal heaving modes represent the horizontal inhomogeneity of the isopycnal layer thickness in the circulation system, i.e., the local expansion (contraction) of isopycnal layer thickness. By definition, the global integration of the internal heaving mode is zero; therefore, the global mean thickness of the isopycnal layers

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associated with the internal modes remains unchanged. Heaving mode generation is primarily through two processes: First, the surface forcing anomaly, including the wind stress anomaly and the surface thermohaline forcing anomaly, such as the surface heating/cooling anomaly and the freshwater flux anomaly, can alternate water properties in the mixed layer near the sites of subduction/ obduction. Furthermore, wind-driven gyres may be modified by changes in a basin-scale wind stress anomaly; such changes can affect the density stratification near the site of water mass formation/erosion, in particular the shape and location of the outcrop lines. These changes affect the water mass formation/erosion rate and its partition in the density category. The corresponding layer thickness anomaly can be separated into the external mode (red) and the internal mode (blue) of heaving signals, as shown in Fig. 3.3a. After the formation of the internal mode perturbation, it is carried by currents, waves and eddies along the isopycnal surfaces. Ideally, such motions are adiabatic and isohaline, or isoTS. The well-known cases for this type of heaving mode generation include the abnormal formation of mode water or deep water due to eventful cooling snaps. For example, the cooling signals associated with subduction can generate outstanding signals in the subsurface ocean, and propagate along isopycnal surfaces, e.g., Deser et al. (1996). Similarly, an excessive amount of freshwater input from the Arctic Ocean or from the atmosphere can give rise to a sudden expansion of water mass volume in the salinity coordinate. As for the climate variability associated with the salinity anomaly, the Great Salinity Anomaly event (1968–1982) that took place in the North Atlantic is a good example for the heaving modes generated in the salinity coordinate (Dickson et al. 1988). Second, internal heaving modes can be generated by the adiabatic motion induced by the wind stress anomaly in the basin interior in the forms of layer stretching/compression. The dynamics associated with the formation of

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Fig. 3.3 Two of the most important sources of internal heaving modes

heaving modes related to the wind-driven circulation and its variability has been discussed in many previous publications, e.g., Huang and Pedlosky (1999) and Huang (2010). In general, the barotropic long Rossby waves leave no change in the stratification; the first baroclinic mode of Rossby waves alternate the depth of the main thermocline, and the high mode baroclinic Rossby waves change the high mode structure of the stratification. Note that the climate variability induced by the wind stress perturbation in the background of a stratified ocean with current appears in the forms of the so-called dynamical thermocline modes, which are quite different from the commonly used normal modes; on the other hand, the so-called normal modes are defined for the stratified ocean with no motion. These internal modes can appear in forms of a layer thickness anomaly or a layer depth anomaly, as shown in Fig. 3.3b. After the generation of the internal heaving mode anomaly, the corresponding layer thickness/depth perturbations are carried by the currents, waves and eddies to move within isopycnal layers. The well-known

cases of this type of internal heaving mode signal formation include the water mass redistribution in the upper ocean induced by wind stress perturbations and other perturbations, as will be discussed in Chap. 4. In the classical paradigm of the oceanic general circulation, the wind-driven circulation and the thermohaline circulation are conceptually treated in separation. Accordingly, wind stress perturbations can induce only adiabatic motions in the ocean. However, our understanding of the oceanic general circulation has evolved rapidly over the past decades. According to the new paradigm of the oceanic circulation, the thermohaline circulation needs external sources of mechanical energy to balance the friction and the dissipation involved in the circulation system. In particular, the wind stress energy plays a vital role in maintaining diapycnal mixing in the ocean interior. Consequently, changes in wind stress must alternate with diapycnal mixing in the oceanic interior, and thus change the thermohaline circulation through the regulation of heat and freshwater diffusion.

3.1 Heaving, Stretching and Spicing Modes

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In contrast, changes in diapycnal (or isopycnal) mixing associated with wind stress perturbations may not be entirely negligible. Such changes should appear as part of the external mode owing to the global mean effect of anomalous mixing and internal heaving signals due to the localized anomalous mixing in the climate dataset obtained from the in situ observations or the numerical simulation of the oceanic circulation. However, the corresponding components may be of secondary importance in the climate data, and their separation and identification remain a great challenge. It is important to emphasize that isopycnal layers have a non-uniform thickness; accordingly, the control equation for tracers is in the form of Eq. (3.1). For the climate datasets based on observations or monthly mean numerical model output, the sub-grid scale flux, the source term and the vertical advection of flux terms are difficult to identify. Therefore, most studies of the climate variability are focused on the contribution associated with the horizontal advection terms, r  ðh~ upÞ. In particular, climate study is often focused on the contribution owing to the horizontal advection term. From Eq. (3.1) this terms can be further divided in the following ways:     rq  ðh~ upÞ ’ rq  h~ up þ rq  ðh~ uÞ 0 p   þ rq  h~ up0 ð3:3Þ The first term on the right-hand side represents the steady term, and the climate anomaly manifests in the form of time-mean tracer carried out by the layer-integrated flow perturbations, plus the timemean layer-integrated flow carrying the tracer perturbations. Since the layer thickness varies greatly in space and time, it is a critically important factor regulating climate variability of tracer. It is important to emphasize that the causes of external and internal heaving modes can be very complicated; due to the nonlinear nature of the oceanic circulation a clear explanation of these modes may be difficult. In fact, these are the most challenging tasks in climate study. Hence, our

goal in this book is focused on the identification of the external and internal heaving modes, and the exact mechanisms leading to these modes are left for further study. In the following section we illustrate the basic concept of the climate variability associated with the heat exchange and the heaving in the potential density coordinate or the potential temperature coordinate. For simplicity, we ignore the role of salinity, i.e., we will assume that salinity is constant, so that it does not affect the change in density. As shown in Fig. 3.4, the surface air-sea heat exchange anomaly can lead to expansion (compression) of the water mass volumetric distribution in the density/temperature coordinates. For example, the warming anomaly and the cooling anomaly combined can lead to expanding of the water mass volume between h1 and h2(h1 > h2) and shrinking of water mass volume between h2 and h3. Assuming that temperature is linearly distributed between h1 and h2 isothermal surfaces, for an observer sitting on the original h2 surface the local temperature is increased, and the h2 surface is moved to the new position (red curve) and this is a climate variability induced by the heat flux anomaly. As discussed above, there are other types of climate variability caused by the adiabatic motions of the isopycnal/isothermal layers. In fact, the wind stress anomaly can induce the stretching of the isothermal layers in one part of the ocean. However, for the adiabatic adjustment of the wind-driven circulation the total volume of water mass within each isopycnal layer remains unchanged, so that the isopycnal layer of our concern must be compressed in other parts of the ocean to compensate. Therefore, such changes must appear in the form of internal heaving modes, as shown in Fig. 3.3b. Note that wind stress perturbations can lead to heaving motions, heating/cooling can also lead to the migration of the isopycnal layers. In this sense, although we may be able to identify the heaving signals from climate data, in order to pin down the dynamical cause of heaving, whether it is due to the wind stress perturbation or heating/cooling, we need to go through a detailed dynamical analysis, including the traditional water mass

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Warm anomaly θ1 θ2

Cold anomaly

θ3

Expansion of isothermal layer

Shrinking of isothermal layer

Fig. 3.4 Sketch illustrating climate variability induced by the surface heat flux anomaly Table 3.1 Barotropic/baroclinic modes versus external/internal modes z-coordinate

Isopycnal coordinate

Barotropic mode Due to anomalous diabatic force, such as air-sea interfacial heat/freshwater fluxes

External heaving modes dh, dS, dh, dD as functions of density ! they are the results of anomalous diabatic forcing

Baroclinic modes Owing to shifting of water masses in space; they are primarily the results of adiabatic processes

Internal heaving modes dh, dS, dh, dD as functions of (x, y); however, the integration over the whole world oceans is zero They are primarily associated with adiabatic processes induced by wind stress anomaly

analysis in the second step; this is, however, beyond the scope of this book. Similarly to the case discussed above, for an observer originally sitting on the h2 isothermal surface, local temperature is changed, in a way seemingly the same as in the previous case. For the local observer, it may not be easy to tell the difference between the local temperature anomaly caused by the heat flux anomaly and by the adiabatic motions induced by the wind stress anomaly. As will be explained shortly, this is the limitation of using a coordinate fixed in space. In such a coordinate, one can easily define the climate anomaly, but there is no way to further separate such climate variability into the external and internal heaving modes. In this book our focus is on the heaving modes and the isopycnal layer analysis, although

we may include a brief description in terms of the z-coordinate. For these two coordinates we will use slightly different terminology as listed in Table 3.1.

3.1.4 Wave Processes Related to Adiabatic Internal Heaving Modes As discussed in the previous section, one of the most important sources of internal heaving modes is the adiabatic adjustment induced by the compression/stretching of isopycnal/isothermal layers; hence, these heaving modes can be called “adiabatic internal heaving modes”. The typical cases for the generation and propagation of such modes are the adiabatic Rossby and

3.1 Heaving, Stretching and Spicing Modes

Kelvin waves generated in response to changes in the wind stress applied to the upper ocean. The response to wind stress perturbations has been discussed in many classical papers, e.g., Anderson and Gill (1975). Most importantly, such adjustments in the basin interior are carried out by Rossby waves, including the barotropic and baroclinic Rossby waves. These solutions consist of three parts, a wave front starting from the eastern boundary, a second wave front reflecting from the western boundary, and the middle one representing the undisturbed field. These solutions show that the Sverdrup relation is built up after the arrival of the Rossby wave front originating from the eastern boundary, a brief description of such a wave solution is referred in Huang (2010). For the adjustment in response to the wind stress anomaly in a specific region in the world oceans, the dynamic processes involve the propagation of both the Rossby waves and Kelvin waves. A typical case is shown in Fig. 3.5. If the deep water formation rate in the North Atlantic Ocean is changed, the signals move away from the source region in the form of

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coastal Kelvin waves, which move equatorward along the western boundary of the North Atlantic Ocean (segment 1). When these waves reach the equator, they cannot cross the equator, and they move eastward as the equatorial Kelvin waves (segment 2). At the eastern boundary, they become the poleward Kelvin waves in both hemispheres, sending the climate variability signals westward in the form of Rossby waves in both the North and South Atlantic Oceans, depicted by the thin blue arrows A. The southern branch of Kelvin waves moves along the western coast of Africa (segment 3), turns around at the Cape of Agulhas, and continues to travel northward along the eastern coast of Africa (segment 4). When these waves reach the equator in the Indian Ocean, they become the eastward equatorial Kelvin waves (segment 5). These waves bifurcate into the poleward coasttrapped Kelvin waves when they hit the eastern boundary. These Kelvin waves send the westward long Rossby waves in the Indian basin (blue thin arrows with label B). There might be a small portion of energy leaking through the Maritime Continent, not shown in this sketch;

Fig. 3.5 Schematic pathways of the Kelvin and Rossby waves generated by a source of deep water formation in the North Atlantic Ocean

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however, most of the energy may be carried by the waves/currents around the Australia continent (segment 6 and 7). These waves eventually reach the equatorial Pacific and move eastward in the form of equatorial Kelvin waves (segment 8), and reach the western coast of the South American continent, where they bifurcate into the poleward Kelvin Waves. These poleward moving Kelvin waves send climate signals in the form of westward long Rossby waves in the Pacific basin (blue thin arrows with label C). The southern branch of these waves (segment 9) reaches the southern tip of South American and turn around, continuing their movement along the eastern coast of South American continent (segment 10). They eventually reach the Equator in the Atlantic Ocean and become the eastward equatorial Kelvin waves, and thus complete the whole loop. The entire route of these waves is depicted in Fig. 3.5. The most important point of this figure is that climate signals in the world oceans are carried out by both the Rossby waves and Kelvin waves. For a specific region in the ocean interior climate change signals are generated at the corresponding eastern boundary after the arriving of the Kelvin waves. Due to the dynamic and kinematic constraints of these waves it may take many years or decades for the climate signals to propagate into the specific regions in the world oceans. For example, along the eastern coast of South Africa, climate change signals are sent out in the form of Rossby waves, labeled by X. However, south of the Cape of Agulhas, signals should come from the southern tip of Tasmania, characterized by blue arrows labeled by Y. For a station at this latitude and off the coast of South America, it takes a long time for the long Rossby waves to travel through this distance because these waves have rather low group velocity at the high latitudes. The station near the southern tip of the South America and off the coast is an extreme case. As shown in Fig. 3.5, climate change signals at this latitude must be sent out at the tip of the South America (arrows labelled by Z). Consequently, it

takes many decades for climate signals to reach this part of the world oceans. In summary, inter-basin climate change signals propagate in the forms of Kelvin waves. On the global scale signals move eastward in the following order: Atlantic basin ! Indian basin ! Pacific basin ! Atlantic basin On the other hand, within each basin the signals can move eastward via equatorial Kelvin waves or westward via Rossby waves. This rule of climate signal propagation is quite useful, when we interpolate climate signals diagnosed from datasets, as will be explained in the remaining part of this book. Although in the numerical experiments based on a simple reduced gravity model with linear bottom friction, the adjustment of the circulation in response to the wind stress perturbation is on the order of less than 100 years, as will be shown in Chap. 4, numerical experiments based on more sophisticated models reveal that it may take much longer time, on the order of up to several hundred years for the world oceans to reach a quasi-steady state, e.g., Allison et al. (2011), Jones et al. (2011) and Samelson (2011). Hence, we conclude that the adiabatic adjustment induced by wind stress perturbations may take a long time to be completed. As a result, the adiabatic heaving signals induced by wind stress perturbations in some parts of the ocean may persist for a long time, on the order of decades or longer.

3.1.5 Local Versus Global Heaving Modes In the following sections we will introduce methods for identifying heaving signals in the world oceans from climate datasets. It is important to notice that heaving modes are not locally defined. Instead, heaving modes in general are defined for the world oceans. For any specific choice of vertical coordinate, such as potential

3.1 Heaving, Stretching and Spicing Modes

density, potential temperature or salinity, heaving modes are defined over specific coordinate surfaces, based on the assumption that water masses are interchangeable within each layer defined by these specific surfaces for the whole singleconnected basin, i.e., water masses with their conserved properties can be transported from one part of the basin to the other parts through wave and current activity. Therefore, identifying heaving modes must be based on data covering the entire single-connected oceanic domains. In the following discussion we will assume that water within each coordinate layer, either in a coordinate based on the potential density, the potential temperature or the salinity, is connected for the whole oceanic domain of concern. This is our working assumption, and this assumption makes the analysis much simpler. In reality, however, water masses on different parts of the ocean are not necessarily linked to each other during the adjustment of the wind-driven and the thermohaline circulation. In fact, warm water in each basin, such as the Pacific basin, the Atlantic basin, and the Indian basin, are not directly connected to each other because the corresponding isopycnal/isothermal layers outcrop within lower and middle latitudes. Thus, in principle heaving modes for such

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shallow layers should be diagnosed for individual single-connected domain. For example, the warm water of high temperature in the warm pool in the Pacific-Indian basin is separated by the continental boundary with similar warm water in the Atlantic basin, Fig. 3.6, based on the GODAS data (Behringer and Xue 2004; also through the website: https:// www.esrl.noaa.gov/psd/data/gridded/data.godas. html). The most important channels connecting the water masses include the ACC and the Arctic Ocean and the Indonesian Throughflow channel. In addition, the depth of the sills may be important in determining the exact type of water that can be connected through heaving motions. For the potential temperature coordinate, the water mass warmer than 20 °C should be divided into two major basins: The Pacific-Indian basin, and the Atlantic basin. The Indonesian throughflow channel is relatively shallow; as a result, it may not allow the communication for the cold part of the temperature range. In fact, the water colder than 20 °C is likely to be exchanged via the ACC channel, and hence the corresponding heaving modes should be calculated over the entire world oceans. In this sense, heaving modes can be classified as local heaving modes and global heaving modes.

Fig. 3.6 Annual mean surface temperature in the world oceans, based on the GODAS data

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Fig. 3.7 Annual mean surface density (r1, in units of kg/m3) in the world oceans, based on the GODAS data

Similarly, the connectivity of water mass in world oceans on isopycnal surfaces (r1) is also subjected to the bathymetry of the oceans (Fig. 3.7). Roughly speaking, the heaving modes for the water mass with a density lighter than 29 kg/m3 can be separated into the local modes for the Pacific-Indian basin and the local modes for the Atlantic basin. On the other hand, heaving modes for the water heavier than 29 kg/m3 should belong to the global modes.

From Figs. 3.6 and 3.7, it is clearly seen that warm and light water in the upper ocean is separated into two single-connected basins: The Pacific-Indian basin and the Atlantic basin. Last, we plot the annual mean surface salinity map, as shown in Fig. 3.8. The most outstanding feature is the high salinity in all subtropical basins. It is obvious that the high salinity surfaces in these regions are mutually disconnected; any attempt to define the global heaving modes for

Fig. 3.8 Annual mean surface salinity in the world oceans, based on the GODAS data

3.1 Heaving, Stretching and Spicing Modes

these high salinity surfaces has no physical basis. In fact, since they are more or less confined into individual basins, one can define the local heaving modes for these high salinity surfaces in each basin. The corresponding heaving modes for salinity surfaces below these high salinity surfaces may or may not be a global mode. In Chap. 7, we will discuss the global external and internal heaving modes for the salinity anomaly; however, due to the complicated nature of salinity distribution in the world oceans, a simple approach may not be accurate enough. Therefore, we made another attempt to diagnose the heaving modes for the shallow salty water sphere in the Atlantic Ocean in Chap. 7. Hence, although dealing with the anomalous salinity signal is complicated, it is not hopeless. In oceanographic application, people often have to deal with data collected for a regional ocean only. As a result, survey and water mass analysis are often focused on part of the world oceans. It is, thus, important to distinguish the difference between the global and local modes. As shown in the sketch of Fig. 3.9, the idealized world oceans consist of two basins, A and B. For a given month, the volume anomaly for a certain water mass, either in terms of density or temperature, is depicted by the blue curve. The global external heaving mode is depicted by the red line, and the internal heaving mode is depicted by the distance between the red line and the blue curve. If our focus is on basin A only, the external heaving mode obtained from data within basin A is depicted by the black dashed line, and the corresponding internal heaving mode is depicted by the distance between the black dashed line and the blue curve. It is clear that the external mode calculated in this way should be called the

Local external mode

Basin A

Global external mode

Basin B

Fig. 3.9 Global mode versus local mode

73

local external mode. Both the local external mode and the internal mode defined in this way are different from the globally defined modes, depicted by the distance between the red and blue curves.

3.2

Potential Spicity

3.2.1 Introduction The concept of a thermodynamic variable whose contours are “orthogonal” to potential density contours in the potential temperature-salinity (h– S) space has been discussed in many previous publications, e.g. Stommel (1962), Veronis (1972), Mamayev (1975) and Munk (1981). The orthogonality between this variable and potential density is very important because this variable can describe temperature and salinity information not included in potential density. In the early study by Veronis (1972) , the concept of constructing a curvilinear orthogonal coordinates system in the h–S plane based on potential density and potential spicity was first proposed. Mamayev (1975) went through a lengthy discussion about the advantage of introducing such orthogonal coordinates. Munk (1981) proposed to name such a thermodynamic variable ‘spiciness’, whose contours are orthogonal to potential density contours; he argued that such a variable should give a measure of the strength of the intrusions. Jackett and McDougall (1985) argued that in theory finding a differential function whose contours are exactly orthogonal to those of potential density is not trivial. Therefore, they abandoned the orthogonality constraint postulated by pioneers in the searching for such a function in previous studies. Their approach was followed up in the subsequent studies, such as Flament (2002), and McDougall and Krzysik (2015). Note that in most previous studies, such as Flament (2002), Huang (2011) and McDougall and Krzysik (2015), the term “spiciness” or “spicity” has been used in the paper titles and through most of the text. For water mass analysis, it is much more accurate to call this function

74

3

Heaving, Stretching, Spicing and Isopycnal Analysis

as the potential spicity (p). By definition, contours of potential spicity are orthogonal to those of the potential density referenced to the same pressure. Accordingly, a more accurate terminology, potential spicity, is adapted in this book. Similarly to Jackett and McDougall (1985), McDougall and Krzysik (2015) “have opted to have the isopycnal variations of spiciness be proportional to the isopycnal water-mass variations, expressed in terms of density. Because of the nonlinear nature of the equation of state of seawater, this water-mass variation constraint cannot be simultaneously satisfied with any definition of orthogonality. We have argued that there is no theoretical justification for any meaning of orthogonality”. Flament (2002) postulated a spiciness function whose contours have a slope of the same magnitude as those of the potential density, but the signs are opposite in the h–S diagram. Flament did not discuss the technical details associated with solving the nonlinear least square problem in his approach. The debate about how to define a useful thermodynamic variable in addition to the potential density has been focused on two issues. First, whether we can define a thermodynamic variable whose contours are orthogonal to potential density contours. Second, what is the physical meaning and advantage of such a function. The first question had no positive answer for a long time. Despite of several attempts, no potential spicity function satisfying the orthogonality constraint had been found. Regarding the second question, Veronis (1972) and Mamayev (1975) theorized that orthogonality makes it possible to construct a new orthogonal coordinates for water mass study. Since such a function was not available for a long time, no specific advantages of using such a potential spicity function have been discussed. Therefore, until now most studies have been based on the non-orthogonal functions, e.g. the spiciness function postulated by Flament (2002) and McDougall and Krzysik (2015). Huang (2011)

tried to define a potential spicity function having contours orthogonal to those of potential density; however, the defined function is not really orthogonal. Without orthogonality, the physical meaning of potential spicity would be less clear. Note that the orthogonality we defined here is based on a fixed aspect ratio of the axes; if the aspect ratio of the axes is changed, any orthogonal coordinates will become non-orthogonal; consequently, this definition of “orthogonal” depends on the relative scales chosen for the axes of the h–S diagram, similar to Veronis (1972). The advantages of the orthogonal coordinates are fairly obvious: many of the complications associated with non-orthogonal no longer exist and both coordinates can be regarded as independent. Our discussion in this section follows the approach by Huang et al. (2018).

3.2.2 Define Potential Spicity by Line Integration First of all, to rigorously define the orthogonality for two families of contours, we must use the same dimension for both axes of the h–S diagram, so that we can compare gradients at different directions in the h–S space. Furthermore, a square grid box in such a phase space should have the same length in the same unit. The orthogonality is meaningful only for a fixed aspect ratio between the horizontal and vertical axis lengths. Therefore, we will use the following pair of variables. Most importantly, they have the same dimensions as the potential density @ 1 @ ¼ @x q0 b0 @S @ 1 @ y ¼ q0 a0 h; ¼ @y q0 a0 @h x ¼ q0 b0 S;

ð3:4Þ

where a0 ¼ a and b0 ¼ b are the mean values of thermal expansion and saline contraction coefficients averaged on the domain of h = [−2, 40]

3.2 Potential Spicity

75

(oC) and S = [10, 40] (psu). In these coordinates the potential density gradient operator is equal to: @r ~@r ~ 1 @r ~ 1 @r þj ¼i þj rr ¼ ~i @x @y q0 b0 @S q0 a0 @h b a ¼ ~i ~j b0 a0 ð3:5Þ On the sea surface, the net pressure is zero, p = 0, thus the thermodynamic state of seawater can be expressed in terms of two variables (h, S). In the traditional way, potential density is used as a dynamical variable, the best choice of the other thermodynamic variable should be called potential spicity, which varies along the potential density isopleths and it is denoted as p. The basic requirement is that the gradients of potential density and potential spicity be perpendicular rr  rp ¼ 0

ð3:6Þ

A simple way to construct such a function is to define it as follows px ¼ ry ; py ¼ rx

ð3:7Þ

If we can find such a function, and if it is differentiable twice, then it is readily seen that the vector field associated with the gradient of potential spicity is non-divergent, i.e.: r  ðrpÞ ¼ 0

ð3:8Þ

From Eq. (3.8), one can derive the following constraint  ! ! r  ðrrÞ ¼ r  py i  px j ¼ rxx þ ryy ¼ 0 ð3:9Þ The left hand side of this equation can be called the “source” of the potential density. As discussed by Huang et al. (2018), the commonly used function of potential density does not satisfy such a constraint. The “source” is primarily due to the contribution of the second term associated with the increase of the thermal expansion coefficient with temperature. At low salinity and temperature, the

first term of the “source” is quite large compared with its mean value, and this is the major obstacle in constructing the desirable potential spicity function orthogonal to potential density. In general, a vector ! g ¼ ðgx ; gy Þ is called a conserved vector when its components satisfy the constraint: @gy =@x ¼ @gx =@y. For a conserved vector, there exists a scalar function Hðx; yÞ whose gradient equals this vector, i.e. rHðx; yÞ ¼ ~ g. The construction of such a scalar function can be carried out by the integration of Hðx; yÞ along chosen paths, and the results are independent of the choice of the integral path. However, if the constraint @gy =@x ¼ @gx =@y is not satisfied, a differentiable function that satisfies constraint rHðx; yÞ ¼ ~ g does not exist. In such a case, although one can construct a function by integrating the differential relation along certain selected paths, the results do not lead to a differentiable function. Most importantly, the results of such integration will depend on the selected paths. Therefore, potential spicity (spiciness) functions obtained by following different integral paths are different, and they should also have different physical meanings. Since in the coordinate discussed above the potential density satisfies the differential relation b a rr ¼ ~i ~j b0 a0

ð3:10Þ

the desirable potential spicity function should satisfy the following differential relation 

a~ b~ rp ¼ c iþ j a0 b0

ð3:11Þ

where c ¼ cðq0 a0 h; q0 b0 SÞ is an arbitrary function. In previous studies, a different differential expression was proposed for searching spicity function  rp ¼

a~ b~ iþ j a0 b0

ð3:110 Þ

Since a function that satisfies the constraint (3.11′) does not exist, many efforts have been

76

3

focused on finding the solution by projecting constraint (3.11′) onto certain curves, and calculating the potential spicity function by line integration. The potential spicity (spiciness) functions constructed through these different integral paths must be different. Most importantly, although the spicity function satisfies the constraints obtained by projecting the differential relation (3.11′) onto the integral path, such functions do not satisfy the original differential relations in directions other than the tangent of the integral path. As a   result, rp 6¼ a ~i þ b ~j in general; in particu-

pðx; yÞ that can satisfy constraint (3.6) in the least-square sense over the domain of h–S diagram, i.e.:

a0

b0

lar, contours of potential spicity function defined in this way are not orthogonal to the potential density contours. Consequently, the meaning of potential spicity functions defined in this way remains unclear. In fact, although potential spicity had been used in many studies in the past, the physical meanings of this variable remain poorly understood. A careful examination of pursuing constraint (3.11′) reveals that the coefficients for gradient and scaling are incorrect, and there is no reason a priori to set cðq0 a0 h; q0 b0 SÞ  1. As discussed above, setting such a constraint leads to difficulties in finding the desirable orthogonal function. Accordingly, a logical step to take is to release such a non-necessary constraint and search for the desirable function, including the function of cðq0 a0 h; q0 b0 SÞ 6¼ 1.

3.2.3 Define Potential Spicity in the Least Square Sense As discussed above, a simple constraint defined by Eq. (3.8) does not lead to a differential function; thus, such a constraint should be modified. In fact, we can slightly modify our goal of the exact orthogonality as the target function of the following least square problem. By definition, the angle between ∇r and ∇p is k ¼ arccos   rrrp jrrjjrpj , and the ‘non-orthogonality’ can be measured in terms of the deviation from the target value of 90°. We will search a function

Heaving, Stretching, Spicing and Isopycnal Analysis





 rr  rp ¼ Minimum Dk ¼ RMS arcsin jrrjjrpj ð3:12Þ In the common practice, seawater properties are defined in terms of high-order polynomials, the task defined by Eq. (3.12) is a least square problem in multi variables, with up to tens of variables. Such a nonlinear least-square problem is rather difficult to be solved. We will solve this least square problem in two steps. First, a target vector ðry ; rx Þ is created by rotating the gradient vector ∇r; our goal is to search a scalar function p, the components of whose gradient vector match the lengths of this target vector. As discussed above, it is impossible to find a perfect match. For this reason, we modify our goal as follows: searching for a scalar function p, whose gradient vector matches the length of the target vector in the least-square sense: DR ¼

X h


rx  py

2


2 i þ r y þ px

¼ Minimum

ð3:13Þ

We will denote this scalar target function for p as f, and will choose to fit this function in terms of a polynomial. To illustrate the basic idea, we use a 4th order polynomial in the following discussion; hence, our goal is to define the following fitting function, f, as f ¼ a1 x þ a2 y þ a3 x2 þ a4 xy þ a5 y2 þ a6 x3 þ a7 x2 y þ a8 xy2 þ a9 y3 þ a10 x4 þ a11 x3 y þ a12 x2 y2 þ a13 xy3 þ a14 y4 ð3:14Þ where ak (k = 1, 2, 3, …, 14) is the fitting coefficients to be found. This function can be used to calculate the potential spicity at each grid point (i, j) in the h–S diagram. We assume that  
 the vector ~ A ¼ Ax ; Ay ¼ ry ði; jÞ; rx ði; jÞ i;j

i;j

3.2 Potential Spicity

77

perpendicular to potential density is given for each grid point. We introduce the following notations: dx ¼ xi þ 1  xi1 ; dy ¼ yj þ 1  yj1

dE=dak ¼ 0; k ¼ 1; 2; 3; . . .; 14

For example, at grid (i, j) the first constraint dE=da1 ¼ 0 leads to 

Dx2i ¼ x2i þ 1  x2i1 ; Dy2j ¼ y2j þ 1  y2j1 Dx3i Dx4i

¼ ¼

x3i þ 1 x4i þ 1

 

x3i1 ; x4i1 ;

Dy3j Dy4j

¼ ¼

y3j þ 1 y4j þ 1

 

y3j1 y4j1

ð3:15Þ

þ a10 Dx4i þ a11 Dx3i yj þ a12 Dx2i y2j þ a13 dxy3j

ð3:16Þ Dfi;jy ¼ a2 dy þ a4 xi dy þ a5 Dy2j þ a7 x2i dy þ a8 xi Dy2j þ a9 Dy3j þ a11 x3i dy þ a12 x2i Dy2j þ a13 xi Dy3j þ a14 Dy4j

ð3:17Þ 

Our goal is to fit the given vector ~ A¼  y x Ai;j ; Ai;j . For the x-component of the vector,

Dfi;jx ¼ Xi;j ; where Xi;j ¼ dx  Axi;j

ð3:18Þ

Therefore, the corresponding error in fitting this component of the vector is ex ¼ Dfi;jx  Xi;j

ð3:19Þ

Similarly, the errors in fitting the other component of the vector is e ¼

Dfi;jy

 Yi;j ; where Yi;j ¼ dy 

Ayi;j

ð3:20Þ

Accordingly, the target of least square problem (3.13) is to minimize the following function E¼

X 

Dfi;jx

 Xi;j

ð3:23Þ

X i;j

Dfi;jx dx ¼

X

Xi;j dx

ð3:24Þ

i;j

where the left side is a linear function of ak (k = 1, 2, 3, …, 14). The second example is for the constraint dE=da7 ¼ 0. At a grid (i, j) this constraint leads to Dfi;jx Dx2i yj þ Dfi;jy x2i dy ¼ Xi;j Dx2i yj þ Yi;j x2i dy ð3:25Þ Summarizing over all grids leads to X

ideally we should have

y

 Dfi;jx  Xi;j dx ¼ 0

Summarize over all grids leads to the following equation

Using the central difference scheme, we have the following finite differences Dfi;jx ¼ a1 dx þ a3 Dx2i þ a4 yj dx þ a6 Dx3i þ a7 yj Dx2i þ a8 dxy2j

ð3:22Þ

2

þ



Dfi;jy

 Yi;j

2 

 Dfi;jx Dx2i yj þ Dfi;jy x2i dy i;j X
 ¼ Xi;j Dx2i yj þ Yi;j x2i dy

where the left side is a linear function of ak (k = 1, 2, 3, …, 14). The final set of linear equations for the coefficients ak (k = 1, 2, 3, …,14) is F ~ a ¼~ e

ð3:21Þ The constraints of minimizing this target function are

ð3:27Þ

where F is a 14  14 matrix, both ~ a and ~ e are 14-dimensional vectors; the construction of both F and ~ e are discussed above. From Eq. (3.27), we can calculate vector ~ a by simply inversing the matrix F ~ a ¼ F 1 ~ e

i;j

ð3:26Þ

i;j

ð3:28Þ

With the vector ~ a calculated, the potential spicity function in terms of a 4th -order polynomial is completely determined.

78

3

Although the result of this calculation gives us a fitting function f, whose gradient vector satisfies the least square constraint (3.13), the error of angle deviation from orthogonality is still rather large. It is obvious that such a scalar function is not the desirable solution for the least square problem (3.12). In order to find a solution with a smaller angle deviation from orthogonality, in the second step we will use an iteration process to solve the original least square problem (3.12), and the solution obtained from solving the least square problem (3.13) can be used as a good candidate for an initial solution at the beginning of the iteration process discussed below.

Following the notation used in the previous section, we have:

3.2.4 Solve the Linearized Least Square Problem For a small angle, the least square problem defined in Eq. (3.12) can be reduced to 

rr rp  Dk ¼ RMS ¼ Minimum ð3:29Þ jrrj jrpj Directly solving this nonlinear least square problem is difficult. However, this problem can be further reduced to the following linear least square problem: 

rr  rp Dk ¼ RMS jrrjjrp0 j

¼ Minimum ð3:30Þ

where p0 is the initial result of the approximate potential spicity function obtained in the previous iteration, and p is the potential spicity function calculated as the new corrected solution. In fact, the solution obtained from Eq. (3.27) can be used as a good first initial solution. The target function of the problem defined in Eq. (3.30) is 2 1
C¼ rx px þ ry py R    R ¼ r2x þ r2y p20;x þ p20;y

ð3:31Þ

Heaving, Stretching, Spicing and Isopycnal Analysis

px ¼ Dfi;jx =2Dx; py ¼ Dfi;jy =2Dy

ð3:32Þ

The constraints of minimizing this target function are dC=dak ¼ 0; k ¼ 1; 2; 3; . . .; 14:

ð3:33Þ

For example, the constraint upon the first parameter, a1, leads to an equation:  2
rx px þ ry py 0 ¼ dC=da1 ¼ R  rx ry x dDfi;j =da1 þ dDfi;jy =da1 2Dx 2Dy  rx
rx px þ ry py dx ¼ 0 or RDx

ð3:34Þ

This relation should apply to each grid point, and the summation over all grid points leads to the following equation X rx
 rx px þ ry py dx ¼ 0 RDx i;j

ð3:35Þ

Note that R in the denominator is specified from
 the previous iteration; px & py in rx px þ ry py are treated as unknowns; thus, they are the linear function of the unknowns ak (k = 1, 2, 3, …, 14). Similarly, we obtain the corresponding equations for parameter a2, a3, …, a14. In this way, we obtain a equations system of 14 variables ak (k = 1, 2, 3, …, 14) and 14 equations, similar to Eq. (3.27): 2

b1;1 6 b2;1 6 4 : b14;1

b1;2 b2;2 : b14;2

30 1 : b1;14 a1 B C : b2;14 7 7B a2 C ¼ 0 ð3:36Þ 5 @ : : : A : b14;14 a14


 where the matrix B ¼ bi;j is defined in a way similar to the procedure leading to Eq. (3.27) discussed above. However, the right side of this equation system is zero; therefore, the only

3.2 Potential Spicity

79

solution is trivial of zero. To find meaningful solutions we choose the following iterative approach, and solutions obtained from the linear least square problem defined in Eq. (3.27) can be used as the first initial solution. The original equation system (3.36) is separated into two parts. First, we rewrite the first equation in the following form
 a11 ¼  b1;2 a02 þ b1;3 a03 þ    þ b1;14 a014 =a11 ð3:37Þ
 Using a02 ; a03 ; . . .; a014 from the previous solution, we can calculate a11 , the first parameter of the new solution. When this new value a11 is obtained, we can calculate the new values of other parameters by using the following equation system, that is a subset of the original equation system (3.36) 2

30 1 1 b2;3 : b2;14 a2 B a1 C b3;3 : b3;14 7 7B 3 C : : : 5@ : A b014;3 : 1 b14;14 a114 b2;1 B b3;1 C C ¼ a11 B @ : A b14;1

b2;2 6 b23;2 6 4 : b14;2

ð3:38Þ

Solving this equation can lead to an improved solution with smaller RMS angle errors over the h–S diagram. This approach can be applied iteratively. Since the original least square problem is defined for a multi-dimensional parameter space, there occur many solutions corresponding to localized minima in the parameter space; among these solutions we choose the one with the smallest RMS error of angle deviation from orthogonality.

3.2.5 Potential Spicity Functions Based on UNESCO EOS-80 This method is applied to finding potential spicity functions based on the UNESCO EOS-80 equation of state (Unesco 1981, 1983). In order to cover a wide range of salinity, the calculation is based on a 10th order polynomial. In parallel to the available Matlab code for calculating

potential density, the Matlab program of the potential spicity function is in the form of sw_pspi(s, t, p, pr), where (s, t) is the in situ salinity (psu) and temperature (°C), p is the in situ pressure (db), and pr is the reference pressure. These potential spicity functions are defined over the domain of h = [−2, 40] (°C) and S = [10, 40] (psu); the domain of definition is broad enough, so that these functions are applicable for oceanographic application in the open ocean, including the Arctic. For the convenience of application, we define the potential spicity function at seven reference levels: pr = 0, 500, 1000, 2000, 3000, 4000, 5000 db, respectively. The two-step linearization method turns out to be an efficient way to find solutions. The rootmean square angle errors (deviation from orthogonality) of these solutions decline rapidly within just a few iterations. As an example, the decline of root-mean square angle errors in the case of using a reference pressure of pr = 0 db is shown in Fig. 3.10. Panel a shows the result obtained from the first step, i.e. the solution for Eq. (3.13). It is obvious that the root-mean square errors (27.68°) are too large. The largest angle errors appear in the domain of low temperature and low salinity, where the thermal expansion coefficient is nearly zero or even negative. Thus, such a solution cannot be used for sea water analysis. However, with the implementation of the second step, the angle errors decline rapidly. In fact, with just one iteration, the root-mean square angle error is reduced from 27.68° to 0.01071°; the corresponding error is further reduced to 0.0009066° and 0.0009034° at iteration 2 and 3. Consequently, with just a few iterations, the root-mean square angle errors become so small that the corresponding solutions are acceptable for practical applications. In addition to the orthogonality constraint, the final definition of the potential spicity function requires two additional constraints. First, we set the potential spicity to be zero at the mass center ð S; T Þ for each reference level, based on the WOA09 data (Antonov et al. 2010), as listed in Table 3.2. Second, the amplitude of potential spicity satisfying the orthogonality is arbitrary; hence,

80

3

Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.10 Angle errors, deviation from orthogonality. Iteration 0 is for the solution of Eq. (3.13) Table 3.2




 S; T based on the WOA09 data at different reference pressures pr

pr(db)

0

500

1000

2000

3000

4000

5000

S (psu)

34.5914

34.6567

34.5997

34.7311

34.7468

34.7336

34.7266

o

18.1534

7.3949

4.2768

2.3580

1.7434

1.4064

1.3184

T ( C)

we rescale the potential spicity function. The main domain of definition adapted in this study is h = [−2, 40] (°C) and S = [10, 40] (psu). However, away from the Arctic Ocean, most sea water is confined to a much smaller domain in the h–S space. For this reason, a core domain h = [0,30] (°C); S ¼ ½32; 38ðpsuÞ is defined, and this core domain covers the most part of the sea water in the open ocean application. We use the core domain to rescale the potential spicity function, so that within the core domain at each

reference level, the averaged amplitude of the gradient ratio is: cðh; SÞ

core domain

core domain

¼ jrpj=jrrj

¼1 ð3:39Þ

In addition, warm and salty water should correspond to high spicity. With these three constraints, the potential spicity function is completely determined.

3.2 Potential Spicity

81

Table 3.3 Aspect ratio of the ðr; pÞ contour map in the h–S plane Reference pressure (db)

0

500

1000

2000

3000

4000

5000

a=b

0.2936

0.3060

0.3184

0.3436

0.3690

0.3944

0.4196

Aspect ratio

Whole domain

h ¼ ½2; 40; S ¼ ½10; 40

0.4111

0.4284

0.4458

0.4810

0.5166

0.5522

0.5874

Core domain

h ¼ ½0; 30; S ¼ ½32:5; 37:5

1.7616

1.8360

1.9104

2.0616

2.2140

2.3664

2.5176

The orthogonality of the (r, p) contours in the h–S diagram is defined for a fixed aspect ratio as listed in Table 3.3. Using an aspect ratio (height/width) different from the nominal value can lead to an incorrect map of the (r, p) contours.

For example, the potential spicity function defined at sea surface pressure is shown in Fig. 3.11a. It is readily seen that the potential density and potential spicity contours are orthogonal. In particular, at low salinity and temperature the thermal expansion coefficient is

(a) σ 0 & π 0 contours 40 22

20

18

16

14

12

10

2 4

6

−3.45

15

20

25 S (psu)

−3.

3

26

24

−3.4

−3

30

−1

−3.445

10

30

21 0

−3.44

0

26

28

22

20

18

16

14

12

4

10

8

θ (°C)

8 6

.4

10 5

24

1 −1

−2

.43

3

−3

−3

−3

−3.

20 15

−2.5

25

0

30

18 14

8

4 2

35

− .5 2

−2

35

40

(b) Deviation from orthogonality, in 0.001° 40

10

35 5

30 RMS=0.00090°

θ (°C)

25

0

20 15

−5

10 −10

5 0 10

15

20

25 S (psu)

30

35

40

−15

Fig. 3.11 a Potential density (r0) and potential spicity (p0) contours in the h–S plane over the definition domain of h ¼ ½2; 40ð CÞ; S ¼ ½10; 40ðpsuÞ; b angle of deviation from orthogonality

82

3

negative; consequently, the potential density contours are convex towards low salinity. As a result, the potential spicity contours in this vicinity are convex upwards. Whether two families of contours are orthogonal or not may be hard to judge by the naked eye; thus, we calculate the angle of deviation from the 90°, with the results shown in Fig. 3.11b. In this case the RMS of the angle of deviation from orthogonality is quite small, with a value of 0.0009°. In fact, the largest errors appear along the edge and corners of the domain; however, within the interior of the domain, the angle deviation from orthogonality is on the order of only 0.0001°. The most important character of spicity is that warm and salty water corresponds to high spicity, or it is spicier; on the other hand, cold and fresh water corresponds to low or negative spicity. Such small deviations from orthogonality indicate that the errors in the distance defined in terms of the r–p coordinates are much smaller than the errors induced in the in situ salinity and temperature measurements; as a result, the newly defined potential spicity function can be accepted as a nearly perfectly orthogonal to the potential density.

As discussed above, the major difference in the spicity (spiciness) functions defined in previous studies and the newly defined spicity function is that the new function includes an amplitude function cðh; SÞ ¼ jrp0 j=jrr0 j which is not a constant value of one unit. As shown in Fig. 3.12, the amplitude of this function varies over more than two orders of magnitude. In fact, this ratio is on the order of 0.005 for the lower left region of low temperature and salinity. Such a low amplitude ratio can be inferred from the contour map in Fig. 3.11a. The contours of potential density are more or less uniformly distributed over the whole domain, indicating that the amplitude of the gradient is roughly on the same order. On the other hand, the distribution of potential spicity contours is highly non-uniform; in particular, at the lower left corner of Fig. 3.11a, the gradient of potential spicity is quite low, and this leads to the very small amplitude ratio shown in lower left corner of Fig. 3.12. Over the core domain, the amplitude function has a value near one unit. Specifically, along the heavy solid curve, the amplitude function equals one unit, i.e., the gradient of both functions has the same value.

0|

| 40

Heaving, Stretching, Spicing and Isopycnal Analysis

/|

0|

4

35 30

3

θ (°C)

2

1.5

1

0.8

0.6

0.4

0.3

0.2

0.1

0.05

0.025

0.01

25 20 15 10

2 1.5

1

0.8

0.6

0.4

0.3

0.2

0.1

0.05

5

0.02

0.01

5

0.00

5 0 10

15

20

25 S (psu)

30

35

40

Fig. 3.12 Amplitude function cðh; SÞ ¼ jrp0 j=jrr0 j in the h–S plane over the definition domain of h ¼ ½2; 40ð CÞ; S ¼ ½10; 40ðpsuÞ and the core domain (box)

3.2 Potential Spicity

83

3.2.6 Potential Spicity Functions Based on UNESCO TEOS_10 The most updated equation of state is defined by UNESCO TEOS_10 (McDougall and Barker 2011). Accordingly, the suitable thermodynamic variables are the conservative temperature, denoted as H (oC), and the absolute salinity, denoted as SA(g/kg). Similar to the case of UNESCO EOS-80, the definition of the potential spicity function needs two additional constraints. First, we set the potential spicity to be zero at the mass center
  SA for each reference level, which is calcuH; lated from the WOA09 data, as listed in Table 3.4. When plotting the (r, p) contour map in the H–SA plane, the aspect ratio of the contour map must be equal to the corresponding value listed in Table 3.5. In comparison with UNESCO EOS-80, the potential spicity defined in UNESCO TEOS_10 has an angle of deviation slightly larger. Nevertheless, these errors of the angle deviation from the orthogonality are quite small, and they are negligible compared to the relative errors associated with those found in the in situ observations. Following the currently used notation in the Matlab code based on TEOS_10, the potential spicity function is defined in the form of gsw_pspi (SA, H, pr), where (SA, H) is the

absolute salinity (g/kg) and the conservative temperature (°C), and pr is the reference pressure (db). These potential spicity functions are defined over the domains of H = [−2, 40] (°C) and SA = [10, 40] (g/kg). For the convenience of application, we also define potential spicity at 7 reference levels: pr = 0, 500, 1000, 2000, 3000, 4000, 5000 db, respectively. As an example, we show the contours of r0 and p0 in Fig. 3.13. It is readily seen that potential spicity contours are orthogonal to the potential density contours. The exact error of the angle of deviation from orthogonality is shown in Fig. 3.13b. In this case the RMS error is about 0.00216°; this is slightly larger than the corresponding value of 0.0009° defined in terms of UNESCO EOS-80. In conclusion, it is possible to define a potential spicity function, which contours are (in the least square sense) orthogonal to those of potential density. Introducing such functions opens up a new approach in oceanography. First of all, it can be combined with potential density to form an orthogonal coordinate system. The r–p diagram provides another diagnostic tool for water mass analysis in the world oceans; in particular, the orthogonality of the coordinates enables us to define the distance between water masses and thus the radius of signal and the radius of the state.

Table 3.4 ðSA ; HÞ based on the WOA09 data at different reference pressure pr pr (db)

0

500

1000

2000

3000

4000

5000

SA (g/kg)

34.7552

34.8231

34.7693

34.9045

34.9219

34.9090

34.9031

H (°C)

18.1532

7.3420

4.1973

2.2145

1.5201

1.0894

0.8894

Table 3.5 Aspect ratio the ðr; pÞ contour map in H–SA plane Reference pressure (db)

0

500

1000

2000

3000

4000

5000

a=b

0.2911

0.3046

0.3181

0.3448

0.3712

0.3971

0.4224

H ¼ ½2; 40; SA ¼ ½10; 40

0.4075

0.4264

0.4453

0.4827

0.5197

0.5559

0.5914

H ¼ ½0; 30; SA ¼ ½32:5; 37:5

1.7466

1.8276

1.9086

2.0688

2.2272

2.3826

2.5344

Aspect ratio

84

3

Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.13 a Potential density (r0) and potential spicity (p0) contours in the H–SA plane over the defintion domain of H = [−2, 40] (°C); SA ¼ ½10; 40ðg/kgÞ; b angle of deviation from orthogonality

3.3

r–p Diagram and Its Application

3.3.1 The Meaning of Spicity 3.3.1.1 The r–p Diagram The introduction of potential spicity provides a new tool in the study of oceanography. By definition, the contours of potential spicity are orthogonal to those of potential density by a very good approximation. As a result, using these two variables one can construct the desirable orthogonal coordinates, the r–p coordinates. Water mass analysis has been based on the classical h–S diagram, now we can also use the

new tool, the r–p diagram. As a first step we show the (h, S) contours on the r0–p0 diagram (referenced to 0 db pressure at sea surface). Although (r, p) contours are orthogonal on the h–S plane, the h and S contours are nonorthogonal on the corresponding r–p plane, as shown in Fig. 3.14. As an example, the volumetric distribution of water masses in the world oceans is shown in both the traditional h–S diagram and the r2–p2 diagram (2000 db is chosen as the reference pressure). Worthington (1981) pioneered water mass censure of the world oceans by plotting the volumetric distribution of water masses in the world oceans. With the availability of climatological data, such figure can be easily reproduced

3.3 r–p Diagram and Its Application

85

Fig. 3.14 h and S contours on the r0–p0 plane

as shown in the left side of Fig. 3.15. This figure also includes the integrated volumetric distribution for fixed temperature (panel a) and salinity (panel e). Water mass properties on the r2–p2 diagram appear in a slanted form. This figure also includes the integrated volumetric distribution for fixed potential density (panel c) and potential spicity (panel f). The detailed information shown in panel d provides another angle of water mass analysis in term of its density-spicity properties.

The Mediterranean Water appears as a separated small cluster of high spicity on the lower-right side in panel d. Note that due to the nonlinearity of the equation of state, stratification inferred from potential density based on a fixed reference pressure is inaccurate, if we use potential density to study water mass properties at a pressure level far away from the reference pressure. Since potential spicity is also defined for the corresponding reference pressure, far away from the

86

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Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.15 Volumetric distribution on the h–S diagram with resolution of 0:1  C  0:02 psu (left panels) and r2–p2 diagram with resolution of 0:05 ðkg/m3 Þ  0:05 ðkg/m3 Þ (right panels) for the world oceans Fig. 3.16 Boundaries between different oceans

reference pressure, the information inferred from the r–p diagram is inaccurate. For example, if we are interested in deep water properties, a deep reference pressure should be used in constructing the r–p diagram. As an example, we examine the water mass properties in the deep ocean, using a deep reference pressure of 5000 db. We separate the world oceans into the major basins, as shown in Fig. 3.16. The Arctic Ocean is north of 67.5° N; the boundaries between the Indian, Pacific and Atlantic Oceans are marked by solid red lines. Most of the land-locked seas, such as the Mediterranean Sea and Black Sea, are excluded. The structure of the r5–p5 diagram for water masses below 3500 db is shown in Fig. 3.17. The

water masses in the world oceans are shown in panel a; there are clearly two branches, the primary branch on the lower-left represents the densest water masse (Antarctic Bottom Water) and other water masses produced through mixing in the other parts of the abyssal ocean. This primary branch represents the major portion of the water masses in the world oceans, and this branch can be further separated into the corresponding components in three major basins. The Atlantic Ocean contributes to the right branch because the high spicity is related to the high salinity; the Pacific Ocean contributes to the left branch because low spicity is related to the low salinity; the Indian Ocean contributes to the middle branch.

3.3 r–p Diagram and Its Application

87

Fig. 3.17 Fine structure of r5–p5 diagram for the oceans, in units of log10V (m3)

In addition, there is a small secondary branch representing the slightly light and high spicity North Atlantic Deep Water, as shown in panels d and f.

3.3.1.2 The Connection with Compressibility In fluid dynamics the Bulk Modulus Elasticity is defined as the reciprocal of the compressibility under constant entropy, i.e.: El ¼ v

dp dp ¼q dv dq

ð3:40Þ

The commonly used unit is Pa (N/m2). However, in this book we will use the notation E for elasticity as defined below. In oceanography the compressibility under constant entropy is defined as Kg ¼ where c ¼ which

is

 1 @q 1 ¼ 2 ð1=PaÞ q @p g;S qc

ð3:41Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð@p=@qÞg;S is the sound speed, approximately

1560 m/s.

The

commonly used pressure unit is 1 db ¼ 104 N/m2 ¼ 104 Pa; and for sea water in the world oceans this variable is within the range of ð4  4:6Þ  106 =db. For the convenience of use, we will introduce a term “elasticity”, defined as E ¼ 1000 db  qKg ¼

1000 db ðkg/m3 Þ c2 ð3:42Þ

and it has the same unit as the density, i.e., its unit is [kg/m3]. Thus, elasticity is the density increase due to 1000 db (roughly 1000 m) change in pressure through an adiabatic and isohaline movement of water parcel. Note that elasticity is an important thermodynamic variable that affects the thermohaline circulation. In general, elasticity can be defined as a function of in situ temperature, salinity and pressure. However, one can also use the alternative definition of the so-called potential elasticity, in which elasticity is defined as a function of potential temperature referred to the specific reference pressure, salinity and in situ pressure.

88

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Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.18 a Potential density, potential spicity and potential elasticity as functions of potential temperature and salinity, using sea level pressure as the reference pressure; b relation between potential spicity and elasticity

By definition, elasticity isopleths are not perpendicular to the density isopleths. As one example, we plot these three thermodynamic variables as functions of potential temperature and salinity at sea level, Fig. 3.18. Although elasticity is not perpendicular to density isopleths, its isopleths can also provide information complementary to the density isopleths. The elasticity of sea water is closely linked to spicity—cold and fresh water is more compressible than the warm and salty water, i.e., spicy water is less compressible. Accordingly, spicity is also a good indicator for the compressibility of sea water.

3.3.1.3 Sources of Deep Water in the World Oceans Sea water is a complicated two-component thermodynamic system. In addition to the

temperature, there is a second variable, the salinity. With the addition of salinity, sea water thermodynamics are rich in nonlinearity. If there were no hydrological cycle, salinity in the ocean would be uniform, and density would be controlled by temperature alone. In such an ocean, there would only be the thermal circulation, but no thermohaline circulation. For the pure thermal circulation, the density controlled by temperature determines the vertical overturning circulation; in particular, densest water in the upper ocean would sink to the bottom of the ocean. However, with the addition of the nonuniformly distributed salinity, the densest water formed at the sea surface does not necessarily sink to the sea floor. The role played by the addition of salinity determines the spicity or the elasticity. In fact, water mass with low spicity or

3.3 r–p Diagram and Its Application

89

Fig. 3.19 Potential spicity on the sea level base on two reference pressures: a 0 db; b 5000 db

high elasticity is more compressible and thus it can sink to the bottom of the world oceans. The water at low latitudes in the Atlantic Ocean is warm and salty, giving rise to high spicity; at high latitudes it is fresh and cold, giving rise to negative spicity (Fig. 3.19). There is a delicate balance and competition between water masses formed in the North and South Atlantic Oceans as discussed below. At the sea level the patterns of p0 and p5 are slightly different due to the difference in defining the zero points of the spicity function and the nonlinearity of the equation of state of seawater. As shown in Fig. 3.20a, surface density (excluding the Mediterranean Sea) is maximal in the North Atlantic Ocean. However, due to the

nonlinearity of the equation of state of seawater, the surface water in the North Atlantic Ocean cannot sink to the bottom of the world oceans. On the other hand, the surface water formed at the edge of the Weddell Sea has a rather high density. Although at the sea level it is slightly less dense than that formed in the surface North Atlantic Ocean, it is cold and fresh; consequently, it is much more compressible than the surface water formed in the North Atlantic Ocean. As a result, it can sink to the bottom of the world oceans. As shown in Fig. 3.20b, using 5000 db as the reference pressure, the surface water formed in the Weddell Sea is the heaviest water mass, and hence it can sink to the bottom of the world oceans.

Fig. 3.20 Potential density at the sea surface, with reference pressure of 0 and 5000 db

90

3

Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.21 Water mass properties in the North Atlantic Ocean (upper panels) and South Atlantic Ocean (lower panels) Table 3.6 Contrast between the properties of surface density maximum in the Atlantic Ocean, in units of kg/m3; the corresponding stations with maximum density of the sea surface are marked by the white crosses Location of r0 Max

r0

p0

E0

r5

N. Atlantic

74.5° N, 4.5° E

27.8353

−2.6175

4.6894

49.9962

S. Atlantic

75.5° S, 61.5° W

27.7625

−3.3346

4.8194

50.5270

This phenomenon is clearly shown in Fig. 3.21 and Table 3.6. Since our focus is on the open ocean, the high density waters in the Mediterranean and Red Sea are excluded. In the North Atlantic, surface water reaches its maximal density r0 ¼ 27:8353 kg/m3 at station (74.5° N, 4.5° E); In comparison, surface density maximum in the Weddell Sea, r0 ¼ 27:7625 kg/m3 is smaller, and the corresponding spicity (elasticity) is relatively low (high). Because of the nonlinearity of the equation of state, the location of maximum density of r5 is located at a location slightly different from the corresponding location of density maximum at the sea surface, as shown in the lower panels. The maximal density of r5 is located at (75.5° S, 61.5° W)—this is at the western edge of the Weddell Sea, and due to the smoothing of the Matlab contour program, this maximum is not shown in Fig. 3.21g. Nevertheless, the density

maximum in the Weddell Sea is much larger than that in the North Atlantic Ocean (panel d). Accordingly, at the sea level water mass formed in the North Atlantic Ocean is heavier than that formed in the Weddell Sea; however, owing to the large elasticity in the Weddell Sea, the dense water formed there can sink to the bottom of the ocean and becomes the dominant source of bottom water in the world oceans.

3.3.1.4 Spicity as a Tracer for Climate Variability on Isopycnal Surfaces It is well known that large scale motions mostly take place along isopycnal surfaces; thus, isopycnal surface analysis has been widely used in oceanography. In many previous studies, temperature and salinity anomalies on isopycnal surfaces have been examined. However, by definition, on a given isopycnal surface temperature

3.3 r–p Diagram and Its Application

91

Fig. 3.22 Layer depth, potential spicity, potential temperature and salinity on an isopycnal surface r0 ¼ 26:5 kg/m3

and salinity must compensate each other, and the best thermodynamic variable for evaluating climate signals on isopycnal surfaces is the spicity. As an example, we examine the properties on the isopycnal surface r0 ¼ 26:5 kg/m3; the corresponding depth, temperature and salinity distributions are shown in Fig. 3.22a, c, d. As shown in Fig. 3.22c, d, the patterns of temperature and salinity are the same—this is due to the density compensation discussed above. Accordingly, the best choice of describing variability on an isopycnal surface is to use potential spicity. As shown in Fig. 3.22b, potential spicity has the same pattern as that of temperature and salinity, with some minor differences. By definition, on the isopycnal surfaces the variances of potential temperature and salinity should be density compensated; however, it does not mean they should vary in the same way. As shown in Eq. (3.11) there is a differential relation between variance of potential spicity, potential temperature and salinity: dp ¼ q0 cðh; SÞðbdh þ adSÞ. Therefore, these three variables should

vary in different ways. For example, the amplitude of the gradient of potential spicity, potential temperature and salinity on the isopycnal surface r0 ¼ 26:5 kg/m3 (within a selected region of the North Atlantic Ocean) is shown in Fig. 3.23. It is clear that the spatial distribution of the normalized gradients are different. Hence, for climate study, using spicity as the single thermodynamic variable to study climate variability on an isopycnal surface is the most efficient and accurate way.

3.3.1.5 Spicity Anomaly and Climate Variability The time evolution of spicity obeys the Eq. (3.1) discussed in Sect. 3.3.1.
   @ ðhpÞ þ rq  h! u p ¼ rq  kq rq ðhpÞ @t @ _ Þ þ Hp  ðqhp @q In this subsection we discuss some of the implication in this equation.

92

3

Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.23 Normalized gradient distribution of potential temperature, salinity and potential spicity on isopycnal surface r0 ¼ 26:5 kg/m3, within a selected region in the North Atlantic Ocean and normalized by the corresponding maximum in this region

Spicity Anomaly Due to Advection In the open ocean, the adiabatic advection of spicity may be a major source of spicity variability. In fact, the pure horizontal advection of a tracer can lead to a change in tracer concentration, as shown in Fig. 3.24. In the initial state, layer depth is constant and tracer concentration is a linear function in the x-coordinate, depicted by black lines. Assuming the layer is like a plastic

3.0

h0

2.5

h1

tube of toothpaste, the action of squeezing the tube leads to the movement of the tracer, as shown by the blue curves, and the net change of the tracer is depicted by the red curve. Spicity Anomaly Due to Mixing It is important to emphasize that although potential spicity is a conserved quantity, it is not conserved during mixing because the nonlinearity

(a) Layer depth

2.0 1.5 1.0 0.5 0.0 (b) Tracer concentation 2.0 C0 1.5

C1

1.0

C1 C0

0.5 0.0 −1.0

−0.8

−0.6

−0.4

−0.2

0.0 x

0.2

0.4

0.6

0.8

Fig. 3.24 Changes of tracer concentration by cause of horizontal convergence/divergence

1.0

3.3 r–p Diagram and Its Application

93 (b) Spiciling

(a) Cabbeling 30

20 0

24 8 2

25 1

2

4

20 E

0

8 22

26

C

θ (°C)

θ (°C)

−1

4

24

0

15

6

22 1

B

24

20

24 8

25

6

22

20 0

G 15

−1

8 22

26 1

1

10

10 A

0

−2

−2

−1

−1

−2.5

0 32

34

35 S (psu)

36

37

−2

26

−3

−2.5

33

28

30

−2

26

−3

−2.5

5 28

5

0

F

38

0 32

30

30

−2.5

33

34

35 S (psu)

36

37

38

Fig. 3.25 Cabbeling and spiciling due to mixing of two water masses with same density (a) or the same spicity (b)

of the equation of state of seawater. First of all, during water mass formation both density and spicity of masses are set up by air-sea interaction. In the ocean interior mixing of different water masses can also modify spicity. In addition, owing to the nonlinearity of the equation of state, mixing of spicity is not a linear process, as explained below. The cabbeling phenomenon is well known. If two water parcels (A & B) with the same density but different temperature and salinity mix, the final product (C) should have a density larger than the density of the original water parcels. As shown in Fig. 3.25a, if two water parcels labeled as A and B mix, the final product should have a density higher than that of the parent water parcels. A similar phenomenon also happens for spicity mixing. As shown in Fig. 3.25b, if we mix two water parcels (E & F) with the same spicity but different temperature and salinity, the final product (G) should have a spicity higher than that of the original water parcels, and this is

called spiciling. At very low temperature and salinity the mixing may produce spicity lower than that of the original water parcels. Spicity Anomaly Diagnosed from Model Simulation As another example, both the variability of thickness of the isopycnal layer r0 ¼ 26  0:05ðkg/m3 Þ and the corresponding spicity at the central surface of r0 ¼ 26 ðkg/m3 Þ along the equator (0.1661° N), diagnosed from the GODAS data, is shown in Fig. 3.26. It is clearly seen that both the layer thickness and the spicity anomaly appear in the forms of eastward propagating large packages, especially at west of the dateline. Such eastward signals are closely linked to the zonal advection associated with the strong interannual-decadal cycles of ENSO events; in addition, the mixing of spicity and the mixing induced spiciling can also contribute. It is speculated that the meridional convergence/divergence flow may also contribute

94

3

Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.26 Changes of layer thickness and potential spicity diagnosed from an array of equatorial station, based on the GODAS data

to the amplitude change of these moving packages. The question related to the exact percentage of adiabatic advection remains to be examined.

3.3.2 Density Ratio Inferred from the Density–Spicity Diagram One of the most important applications of the r–p diagram is the study of double-diffusion and water-mass properties. In previous studies, the double-diffusion related water mass properties were examined in terms of the double-diffusive phase diagram, as discussed by Ruddick (1983) and You (2002). The diagram was constructed by plotting the data points in terms of aDh=Dz and aDS=Dz. The construction of such a diagram requires the calculation for each pair of data point in the water column, and the results are often

presented in terms of either the Turner angle or the density ratio for each vertical grid point pair. In such a double-diffusion diagram the density ratio or the Turner angle obtained for each grid pair has no obvious connection to that of the next pair in the same vertical column. Because of errors in measurements and turbulence such diagrams are quite noisy. Therefore, the application of such a diagram is rather cumbersome. However, using the r–p diagram, the bulk density ratio can be easily displayed.

3.3.2.1 Density Ratio Inferred from r–p Diagram Introducing the r–p diagram opens up a new approach for mixing and water mass analysis. In fact, the distribution of the water mass of a vertical column from each station in the r–p diagram can be used to study the double-diffusion propensity as follows.

3.3 r–p Diagram and Its Application

95

First, the density gradient (Eq. 3.10) and the spicity gradient (Eq. 3.11) can be rewritten in the following form b a b a rr ¼ ~i ~j ¼ b0~i  a0~j; b0 ¼ ; a0 ¼ b0 a0 b0 a0 ð3:43Þ
0  ð3:44Þ rp ¼ c a~i þ b0~j where c ¼ cðh; SÞ is the gradient ratio of these two functions. The projection of these two gradients on a line segment rl ¼ e~i þ f~j is: Dr ¼ b0 e  a0 f; Dp ¼ cða0 e þ b0 fÞ

(1) Along the right part of the horizontal axis: Dr ¼ 0; k ¼ 0

ð3:50Þ

So that the density ratio is: Rq ¼ 1; Tu ¼ 90

ð3:51Þ

(2) Along the vertical axis: Dp ¼ 0; k ¼ 1; Rq ¼ 1; Tu ¼ 0 ð3:52Þ (3) Between these two lines and along the line: k ¼ a0 =cb0

ð3:45Þ

ð3:53Þ

DS ¼ 0 and Rq ¼ aDh=bDS ¼ 1 ð3:54Þ

The density ratio is defined as: aDh=Dz aDh a0 f ¼ ¼ Rq ¼ bDS=Dz bDS b0 e

ð3:46Þ

The slope of a curve in the r–p diagram can be estimated as: k ¼ Dr=Dp ¼

a 0 f  b0 e cða0 e þ b0 fÞ

Rq ¼

1 þ cka0 =b0 1  ckb0 =a0

Dr ¼ 0; k ¼ 0; Rq ¼ 1; Tu ¼ 90 ð3:55Þ

ð3:47Þ

Hence, these two parameters are related to each other Rq  1  k¼
0 c b Rq =a0 þ a0 =b0

(4) On the left-hand side of the diagram, there are two lines: Along the horizontal axis, we have the relationship:

ð3:48Þ ð3:49Þ

These two relationships include an important fact:the density ratio inferred from the r–p diagram is independent of the scaling factor introduced as the last step in defining the spicity. For example, if the spicity function is rescaled with a factor of 2, i.e., p ! 2p, then the slope k is reduced in half, but the fact ck remains unchanged.

Between the horizontal and vertical axes and along the line k ¼ b0 =ca0

ð3:56Þ

we have: Dh ¼ 0 and Rq ¼ aDh=bDS ¼ 0

ð3:57Þ

Therefore, the dynamical zoning of the r–p diagram is quite similar to the corresponding double-diffusive phase diagram discussed by Ruddick (1983) and You (2002). In fact, Fig. 3.27 can be obtained by rotating Fig. 1 of Ruddick (1983) 135° anti-clockwise, or by rotating Fig. 3 of You (2002) 45° clockwise. As shown in Fig. 3.27, the phase diagram is separated into four regions in the counterclockwise direction. First, from 0° to nearly 45°, it is

96

3

R =0, =0 k= '/c '

=0 R = 1, Tu=0° , S=0 R= k= '/c ' Doubly stable

Diffusive

Finger R =1, Tu=90°

R =1, Tu= 90°

=0

Fig. 3.27 Different modes of double-diffusion mixing identified from the r–p diagram

the finger region, where salt finger activity may take place. From nearly 135° to 180° is the diffusive region, where diffusive layering propensity is strong. Between these two regions of double diffusion, there is the region of doubly stable. Due to the existence of other factors in Eq. 3.49, the slope of the r–p curve does not exactly correspond to the density ratio; however, the slope of the r–p curve can be used as an approximate indicator of the density ratio. When the domain of potential thermohaline intrusion is identified from the r–p diagram, the exact density ratio or the Turner angle for the domain in concern can be calculated from Eq. 3.46. In the past, the information about doublediffusive stability of water masses was obtained by calculating the density ratio for each observational pair. In addition, the density ratio can be unbounded when the vertical salinity difference

Heaving, Stretching, Spicing and Isopycnal Analysis

is small, as discussed by You (2002). With the introduction of the r–p diagram, however, the bulk density ratio can be much more easily extracted by plotting the water masses properties of a water column in the r–p diagram. One of the most important applications of h–S diagram is the study of mixing in the world oceans. Many classical studies of mixing between two water masses are based on the assumption that mixing takes place along the straight line between these two water masses, as depicted by the dashed straight line in Fig. 3.28a, e.g., Mamayev (1975). However, Schmitt (1981) suggested that due to the turbulent processes involved in the two-component thermodynamic system of sea water the ocean mixing between water masses in a water column is more likely to be in the vicinity of the constant density ratio; and this appears in the form of an arc in the h–S diagram (right part of Fig. 3.28a). Another form of mixing is the isopycnal mixing taking place along an arc (left part of Fig. 3.28a). On the other hand, the dominating forms of mixing in the oceans appear in the form of straight lines in the r–p diagram, as shown in Fig. 3.28b. First, the isopycnal mixing takes place along lines parallel to the horizontal axis; second, the double-diffusive mixing takes place along straight lines of constant slope (density ratio) in the r–p diagram. These processes are plotted in terms of both the h–S diagram and the r–p diagram using the

Fig. 3.28 Two prototypes of mixing in the h–S diagram and r–p diagram

3.3 r–p Diagram and Its Application

97

Fig. 3.29 Hypothetical mixing between two water masses in the h–S diagram and the corresponding part in the r–p diagram

recently defined potential spicity function. According to the classical theory mixing between two water masses takes place along the straight line linking these water masses, as shown in Fig. 3.29a. The corresponding hypothetical mixing processes should take place along the curves shown in Fig. 3.29b. Apparently, there is no observational evidence that mixing would take place along such curves in the r–p diagram. Thus, mixing along straight lines in the h–S diagram postulated in many classical books about water mass analysis may not actually happen in the world oceans. On the contrary, mixing in the ocean may take place in the form of a constant density ratio, e.g. Schmitt (1981). As an example, hypothetical mixing along several straight lines in the r–p diagram is shown by the colored lines in Fig. 3.30b; the corresponding processes are shown by the color curves in Fig. 3.30a. It is readily seen that these processes taking place along arcs in the traditional h–S diagram, not

straight lines as postulated in classical text books about water mass analysis and the h–S diagram. For this reason, using the r–p diagram may help us to identify mixing processes in the world oceans. It is important to emphasize that although potential spicity is a conserved quantity, it is not conserved during mixing because of the nonlinearity of the equation of the state of seawater. Spiciling due to mixing can also change the spicity. Therefore, the straight lines in Fig. 3.30b represent the infinitesimal mixing processes associated with turbulence, they do not mean mixing of two water parcels with finite difference in potential density and potential spicity would take place along the finite length of such lines.

3.3.2.2 Application to the Central Water in the World Oceans We begin with the central water in the world oceans. In the classical work on mixing and

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Heaving, Stretching, Spicing and Isopycnal Analysis (b) − diagram

(a) −S diagram 30

18

20 0

24 8 2

25

19 20

6

22

21

1 4

22

(°C)

0

15

−1

228

26

(kg/m3)

20

24 25

1

26

10 0

−2

27

−1

−2.5

−2 −2.5

26

−3 33

34

35 S (psu)

36

37

30

28

28

5

0 32

23

0

24

29 38

30

−2

0 0

2 (kg/m3)

4

6

Fig. 3.30 Hypothetical mixing between two water masses in the r–p diagram (right panel) and the corresponding part in the h–S diagram (left panel)

water masses, the straight line between two points in the h–S diagram has been used to theorize on the mixing of two water masses. Such mixing is based on the assumption that the temperature and salt mixing are kept in constant ratio. The linearly and proportionally mixing of temperature and salt has been the cornerstone for inferring mixing from the h–S diagram. However, mixing in the ocean does not necessarily follow such an idealized rule. The application of the h–S diagram to the central water in the world oceans is a good example. Central water belongs to a special type of water mass that exist in the upper kilometer of the world oceans, and their properties are characterized by a very tied h–S relationship, as discussed by Schmitt (1981). With the help of the new r–p diagram, the special characters of central water can be seen much more clearly; the following figure is based on the WOA09 data. As shown in Fig. 3.31, the water mass properties of central water appear in the form of an arc in the

traditional h–S diagram; however, they appear mostly as constant sloped lines in the r–p diagram. For the water mass properties taken from roughly the same location as reported by Schmitt (1981), the data points fit the same density ratio line in the r–p diagram, i.e. 1.95 for the N. Atlantic, 1.89 for the S. Atlantic, and 2.58 for the S. Pacific. The only exception is for the N. Pacific, Schmitt (1981) reported a density ratio of 3.82 based on GEOSECS data; however, the best fit for the WOA09 data is 5.0. The reason for such a difference is unclear, and it may be due to the difference in the GEOSECS data (a snapshot) and the WOA09 data (a climatology). Nevertheless, this figure demonstrates that the r–p diagram can be used to clearly show the density ratio in the regions characterized by double diffusion. Another example is the application of the r–p diagram to the Atlantic Ocean along the 20.5° W section, based on the WOA09 data (Fig. 3.32). Panel a shows the traditional h–S diagram for this

3.3 r–p Diagram and Its Application

99 (b) − diagram

(a) −S diagram 21 25 22

(kg/m3)

23

15

24

0

(°C)

20

25 10

N. Atlantic S. Atlantic N. Pacific S. Pacific

5 34

35

36 S (psu)

37

R =5.0

R =2.56 R =1.89

26

R =1.95

27 28

−2

0

2 0

4

(kg/m3)

Fig. 3.31 Water mass properties at four stations in the North and South Atlantic and Pacific, based on the WOA09 data. N. Atlantic (25.5° N, 35.5° W), S. Atlantic (17.5° S, 31.5° W), N. Pacific (15.5° N, 161.5° E), South Pacific (23.5° S, 127.5° W)

Fig. 3.32 Water mass properties along 20.5° W: a h–S diagram; b, c r5–p5 diagram

section, in which data points for all of the stations along this section are included. As a comparison, the corresponding data at five stations (as listed inside panel a) are shown in the r–p diagrams in panel b and c. Our discussion is focused on the deep ocean; accordingly, the reference pressure is set at 5000 db, and the data within the upper 100 m is not

included. It is clear that water mass properties along this section are mostly confined within the wedge above Rq ¼ 2:45, i.e., well within the region of the salt finger. The fine structure shown in panel c also reveals that at high latitudes, water mass properties are close to the diagonal line with the slope equal to 1. The meaning of this phenomenon remains to be explored.

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3.3.2.3 Application to the Atlantic Ocean We begin with examining the vertical structure of potential spicity. All figures shown in the following three subsections are based on the WOA09 data. As shown in Fig. 3.33, the vertical map of potential spicity (p2) is somewhat similar to that of potential density (r2), except it is high spicity in the equatorial band near the surface, and this is due to the warm and salty water. At high latitudes, owing to the relatively low temperature and salinity, spicity is low or even negative. This low spicity water sinks to the deep

ocean and leads to the low spicity water in the abyss of the world oceans. It is emphasized that in the vertical direction spicity is not necessary monotonic, and this reflects the nature of lateral advection of spicity. Along the 30.5° S zonal section, the potential density and potential spicity maps look quite different from those taken along the meridional section (Fig. 3.33). As shown in Fig. 3.34, the vertical map of spicity is somewhat similar to that of potential density, and they all share the same kind of sloping contours. In the potential density map, there are low value contours near

Fig. 3.33 Meridional maps of potential density and potential spicity in the North Atlantic Ocean

Fig. 3.34 A zonal section map of potential spicity along 30.5° S

3.3 r–p Diagram and Its Application

101

200 (b) Salinity

(a) Pot. Temp.

Depth (m)

250 300 350 400 450 500

8

10

12

14

35

16

35.5 S (psu)

T (°C)

200

36

(d) Pot. Spicity

(c) Pot. Density

Depth (m)

250 300 350 400 450 500 26.6

26.8

27.0 0

27.2

(kg/m3 )

−2

−1

0 0

1

(kg/m3 )

Fig. 3.35 Staircases in the temperature (a), salinity (b), potential density (c) and potential spicity profiles based on in situ observations (Schmitt et al. 2005)

the western boundaries at the upper ocean and high value contours near the eastern boundaries in the deep ocean. In contrary, the distribution of potential spicity has opposite trend. This reflects the fact that low spicity water sinks to the deep ocean and leads to the low spicity water in the abyss of the world oceans. Another example of the application of this new spicity function is to analyze the data collected during a recent study of salt finger in the North Atlantic Ocean (Schmitt et al. 2005). In order to study the salt finger phenomenon in the subtropical gyre of the North Atlantic Ocean, high resolution data were collected and carefully analyzed. The potential temperature, salinity, potential density and potential spicity profiles are plotted in Fig. 3.35. The outstanding features of staircases of potential temperature, salinity, potential density and potential spicity are clearly visible in these panels. In particular, the slightly unstable staircase of potential density shown in Fig. 3.35c indicates that convective activity was happening during the time of the measurements. The corresponding h–S diagram and r–p diagram are shown in Fig. 3.36. The data points representing water mass properties appear in the

form of a slightly curved arc in the traditional h– S diagram (Fig. 3.36a). On the other hand, these data points fall onto a straight line with a bulk density ratio of Rq ¼ 1:47; hence, the density ratio inferred from this data set is in the vicinity of this value. As reported by Schmitt et al. (2005), ‘The density ratio … was 1.4 to 1.7 across most interfaces, which indicates a strong propensity for salt fingering.’

3.3.2.4 Application to the Southern Ocean The vertical structure of water mass properties in the Southern Ocean is quite interesting; as an example, we show one set of maps taken along 64.5° S (Fig. 3.37). In this zonal section, temperature distribution is in the form of sandwich: it is relatively cold in the surface layer and the deep ocean; however, it is relatively warm between 200 and 2000 m. The salinity is low at the surface and near the sea floor, but it is relatively high in the subsurface layer. The temperature and salinity stratification leads to a sandwich distribution of spicity, Fig. 3.37d. A close examination of the r–p curves at four selected stations is shown in Fig. 3.38. We are

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Fig. 3.36 h–S profile and r0–p0 profile for the water column between 200 and 500 m, based on in situ observations (Schmitt et al. 2005)

Fig. 3.37 Property maps along the 64.5° S section. a Potential temperature; b salinity; c potential density; d potential spicity

focused on the near surface part of the water column; accordingly, the pr = 0 is used as the reference pressure. It is clear that in the deep ocean the slope of the r0–p0 curves is smaller than k ¼  dr dp ¼ 1, and this indicates that salt fingering may be expected for this part of the water column. In the layer above, the slop is less

than k ¼  dr dp ¼ 1, and this suggests that diffusive layering may be expected in this depth range. Accordingly, double diffusive layering and thermohaline intrusion may appear in the subsurface layers in the Southern Ocean. Since this depth range is well below the top 100 m of the sea surface, thermohaline intrusions may be

3.3 r–p Diagram and Its Application

103

Fig. 3.38 r0–p0 curves along the 64.5° S section

easily observed without the distortion of powerful surface waves in the upper ocean. Water mass properties in this high latitude oceanic region is rich in thermohaline intrusions, which can be easily identified from in situ measurements. As an example, the data from a single Argo profile is shown in Fig. 3.39. From the upper panels of Fig. 3.39, at the depth range of 200–700 db a remarkable pattern of complicated water mass interleaving can be seen clearly. This phenomenon can be seen in the traditional h– S diagram in Fig. 3.39e. However, the thermohaline interleaving phenomenon can be much better seen in the r0–p0 diagram; in particular, many segments of r0–p0 curve are with slope or k ¼  dr less than k ¼  dr dp ¼ 1 dp ¼ 1 (Fig. 3.39f), suggesting the potential activity of salty finger or diffusive layering.

3.3.2.5 Application to the Arctic Ocean The Arctic Ocean is characterized by low temperature and relatively low salinity. This region of ocean poses a big challenge for spicity function. In fact, spiciness (spicity) functions were used in the study of water masse changes in the Arctic Ocean, e.g., Timmermans and Jayne (2016). However, the spiciness function defined in previous publications is not suitable for use as the second coordinate because in the Arctic environment, the slope of the spiciness contours is so close to that of the potential density contours, as shown in the contour maps of density and spicity in Flament (2002) and McDougall and Krzysik (2015). Using the WOA09 data and the newly defined spicity function, water mass properties in the Arctic Ocean (north of 70° N) in terms of the h–

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Fig. 3.39 Water mass properties observed by Argo Float 7900676, Cycle #103, taken from http:/www.ifreme.fr/.; the data were collected in Jan. 31, 2019 at station (106.277° W, 61.49° S): a h; b S; c r0 ; d p0 ; e h–S diagram overlaid with potential density and potential spicity contours; f r0–p0 diagram (the dashed lines indicate slope k = 1 and k = −1)

S diagram and r–p diagram are shown in Fig. 3.40. The two major branches in the r–p diagram represent low salinity water near the Siberia coast and the salty water originating from the North Atlantic Ocean. The core of the Arctic Ocean water mass will be discussed shortly. In the polar regions the vertical structure of spicity is quite interesting. We begin with the vertical maps of water mass properties in the Arctic Ocean, Fig. 3.41. In this cross-pole section, temperature distribution is in the form of sandwich, i.e. it is relatively cold in the surface layer and the deep ocean; however, it is relatively warm between 300 and 800 m. Note that our

focus is on the depth range around 300db; thus, the reference pressure is selected as 250 db for both potential density and potential spicity in the discussion below. The temperature stratification leads to a sandwich distribution of spicity, Fig. 3.41d. A close examination of the r0.25–p0.25 curves at four selected stations is shown in Fig. 3.42. It is clear that in the deep ocean the slope of the r0.25–p0.25 curves is roughly less than k ¼  dr dp ¼ 1. As discussed above, this indicates that salt fingering may be expected for this part of the water column. In the layer above, the slop is less than k ¼  dr dp ¼ 1, and this suggests that

3.3 r–p Diagram and Its Application

105

Fig. 3.40 h–S diagram (left) and r–p (right) diagram for the Arctic Ocean, based on the WOA09 data

Fig. 3.41 Section maps for the Arctic Ocean. a Potential temperature; b salinity; c potential density; d potential spicity

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Fig. 3.42 Selected r0.25–p0.25 curves for the Arctic Ocean

diffusive layering may be expected in this depth range. Consequently, thermohaline intrusion is expected for the subsurface layers in the Arctic Ocean, as will be discussed further in the subsequent sections. The Arctic Ocean is characterized by rich activities associated with eddies, internal waves and double diffusion. In particular, thermohaline intrusion and double diffusion interleaving can be found in high resolution CTD data collected by the Ice-Tethered Profiler (ITP). For example, the upper part of the first 350 temperature and salinity profiles observed by ITP38 (downloaded from www.whoi.edu/itp/data) is shown in the upper panels of Fig. 3.43. There are clearly bandlike structures in both temperature and salinity. Apparently, there are layers of different water mass properties in the form of interleaving cold and warm layers; however, the nature of such phenomena is not clearly shown in terms of the traditional h–S properties pattern. On the other

hand, the characteristics of water mass property are clearly shown in terms of the spicity distribution (lower panels of Fig. 3.43). In fact, relatively warm and salty water corresponds to high spicity water; there is an outstanding high spicity layer around 300 db. The spatial structure of thermohaline interleaving can be easily identified from the vertical gradient of the spicity, as shown in Fig. 3.43d. Hence, both the spicity and its vertical gradient maps can be used as a convenient tool in identifying the existence of thermohaline layer interleaving from in situ high resolution CTD observations. As a concrete example, we examine three profiles from ITP38: Profile 51 (89.0706° N,145.6487° W), Profile 61 (89.0888° N, 149.02618° W), and Profile 71 (89.1157° N,161.2505° W). Water properties from these stations are shown in the traditional h–S diagram (Fig. 3.44a) and the corresponding r–p diagram (Fig. 3.44b). Stratification at the low temperature

3.3 r–p Diagram and Its Application

107

Fig. 3.43 Water mass properties taken from ITP39: a temperature; b salinity; c spicity, based on the reference pressure of 250 db; d vertical gradient of spicity

range is primarily salinity controlled, and the temperature change from the surface to the deep ocean can be seen on the left panel. The spatial variability of the double diffusive staircase in the Arctic Ocean was discussed by Shibley et al. (2016). With spicity as a new tool, we examine water mass properties from a slightly different angle. Since water is mostly stably stratified, the zigzag change of spicity in each profile can be seen clearly. The upper part of the water column is characterized by low temperature and salinity, and the bulk density ratio is on the order of 0.36, indicated by the dashed cyan line, which is well within the range of the double diffusive region. At middle depth the water

column is characterized by staircases or zigzags in the r–p diagram, indicating double diffusive layering, where the bulk density ratio can change from 0.54 (the dashed green line) to 1.85 or 1.62 (the dashed black line or the dashed magenta line). Therefore, the r–p diagram is a useful and convenient tool in displaying the general characteristics of double diffusive mixing in the ocean. Although one may use spiciness or spice functions defined in previously published studies, those function are not orthogonal to the density function. As such, the density ratio inferred from the r–p diagram based on such functions can be quite different from the real values.

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Fig. 3.44 h–S diagram and r–p diagram for three selected profiles observed in ITP38

Thermohaline intrusions observed at these stations also provide an interesting application of the spicity function. Munk (1981) first introduced the term “spiciness”, and he argued that a thermodynamic function whose contours are orthogonal to those of density should be quite useful in describing the intrusion/interleaving associated with internal waves. However, no such function was constructed over the past 38 years, and Munk’s postulation has remained unchecked. (a)

0.25

0.25

With the newly defined spicity function, we can now reexamine Munk’s proposal. As shown in Fig. 3.45a, although the intrusion phenomenon is clearly visible, it is hard to identify accurately and quantitatively. The major reason is that thermal expansion and saline contraction coefficients vary greatly in the h–S space; as a result, it is difficult to quantify the intrusion. With the introduction of spicity function, it is straightforward to identify intrusion phenomena,

diagram

(b) (

2 0.25)

(c) VPE

29.03 29.04 29.05 3 0.25(kg/m )

29.06 29.07 29.08 29.09 29.10 29.11 29.12 29.13

−1.72

−1.70

−1.68 3 (kg/m ) 0.25

−1.66

−1.64

0 (

0.5 1 2 3 2 6 ) (10 kg /m ) 0.25

0

2 VPE (10 4J/m3)

4

Fig. 3.45 a The r–p diagram; the dashed curve indicates the background r–p curve; b the squared spicity perturbation; c the virtual potential energy

3.3 r–p Diagram and Its Application

109

as shown in Fig. 3.44b. To quantify intrusion/interleaving, we need to define the socalled mean state of the “background”. Since the circulation changes with time rapidly, for such small scale phenomenon the socalled climate-mean state is not suitable as a candidate for the mean state. Instead, a smoothed profile based on an instantaneous measurement might be a good choice. Of course, the exact definition of such a mean profile is debatable. The intrusion and interleaving identified from in situ CTD measurements can be quantified in terms of virtual potential energy as follows. For scale on the order of 100 km or smaller, the available potential energy associated with the density anomaly in the ocean can be defined as ZZZ APE ¼


2 q  qref g dv dqh =dz

ð3:58Þ

where qh is the background potential density. Accordingly, we can define the virtual potential energy (vpe) for intrusions in the r–p diagram as


2 ZZZ p  pref vpe ¼ g ; VPE ¼ vpe dv ð3:59Þ dqh =dz The corresponding unit for vpe is J/m3. This definition introduces the dynamic effect of the background stratification; a weak stratification can be more favorable to the development of double diffusive instability. In the ocean, perturbations are mostly adiabatic in nature, in particular the motions associated with meso-scale eddies and internal waves. As such, spicity anomaly on a specified isopycnal surface is the result of the horizontal advection of spicity, which is closely linked to the movement of isopycnal layers. Consequently, the spicity anomaly on the isopycnal is also closely linked to the movement of the isopycnal layers. Because of the vertical movement of isopycnal layers, density at the climatological mean location of the isopycnal surface changes accordingly. Hence,

 p  pref is dynamically linked to q  qref . Therefore, the VPE is also dynamically linked to

the APE, although the exact relationship depends on the dynamical detail of the circulation. For example, a density range r0:25 ¼ ½29:0328; 29:1302 ðkg/m3 Þ is selected for a close examination. The first step is to construct a reference state profile from the instantaneous profile. Due to the strong variability of the circulation on such short time scales and small spatial scales, the so-called climatological state is not suitable for the use as the reference state. As a working assumption, we postulate the following approach. A uniform grid of 101 points over this density range is set up. The spicity anomaly and pressure are interpolated onto this grid, and then smoothed repeatedly; the smoothed spicity profile obtained is shown as the dashed curve in Fig. 3.45a. Using the difference between the smoothed spicity and the non-smoothed spicity, the spicity anomaly related to this ‘reference state’ can be calculated; this also leads to the VPE as discussed above. The corresponding results are shown in Fig. 3.45b, c. The sum of the virtual potential energy integrated over this pressure range is 0.0112 J/m2. The squared spicity anomaly and VPE and its vertical sum can be used as an integral measure of the strength of intrusion at this station. In addition, we can also estimate the available potential energy from the density perturbations. The first step is to construct a reference state density profile. There may be different way to construct this profile, here we construct a profile defined by smoothing over a relatively large horizontal scale; in fact, this can be done by smoothing the time evolution of the density section maps. The density anomaly is defined as the difference between this large-scale mean along the section and the instantaneous profile, as shown in Fig. 3.46a. Using the difference between the mean density and the instantaneous density we can calculate the square density anomaly and APE as discussed above. The corresponding results are shown in Fig. 3.46b, c. The sum of the virtual potential energy integrated over this pressure range is 0.0018 J/m2. The squared density anomaly and APE and its vertical sum can be used as an integral measure of the

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Fig. 3.46 a The r0:25 profile; the dashed curve indicates the background profile; b the squared density perturbation; c the available potential energy

strength of intrusion in terms of density perturbations at this station.

3.3.3 The r–p Plane as a Metric Space The concept of distance is fundamental for many branches of physical sciences. We would like to quantify the difference between two water masses, water properties at two stations, water properties of two states of the circulation. In many cases, the difference between two points or two sets of data can be expressed in terms of distance. In order to do so, we need to establish a metric space and introduce the so called distance in such a metric space. As discussed by Huang et al. (2018), the newly defined potential spicity (p) function in combination of potential density (r) can be used to establish an orthogonal curvilinear coordinates. The corresponding r–p plane consists of a two-dimensional metric space, in which the distance between two points can be rigorously defined. With the introduction of distance in physical oceanography, many problems can be accurately quantified, and this opens up a door for better understanding of physical processes in the ocean.

3.3.3.1 Meaning of the Orthogonal Coordinates The superiority of the orthogonal coordinate system in comparing with the non-orthogonal ones can be clearly illustrated in the following example. From the basic formula in trigonometry, the lengths of the three sides of a triangle (Fig. 3.47) satisfy the following relationship c2 ¼ a2 þ b2  2ab cos c

ð3:60Þ

Accordingly, the distance between two points A and B in a plane satisfies the above equation. Hence, c2 a2 þ b2 , and the equal sign holds only when c ¼ 90 , i.e., when OB is orthogonal to OA. It is common knowledge that the distance

Fig. 3.47 An example for a triangle in a rectangular plane

3.3 r–p Diagram and Its Application

111

Fig. 3.48 A orthogonal ðr; pÞ and non-orthogonal ðr; gÞ system in the h–S plane

between two points satisfies the simple relationship c2 ¼ a2 þ b2 if and only if the coordinate system is an orthogonal system. In general, in orthogonal coordinates the signals can be separated into two components to the maximum degree. For example, orthogonal coordinates have no off-diagonal terms in their metric tensor, such as the 2ab cos c term in (3.60). In other words, the infinitesimal squared distance can always be written as the sum of the squared infinitesimal coordinate displacements. Assuming the origin of the coordinate system is at point O (Fig. 3.48); we have two coordinate systems: (r, p)—an orthogonal system based on the spicity defined above, and (r, η)—a nonorthogonal system based on the spiciness defined in previous studies. For a water parcel at point A, its position is h _ i determined by arc elements Dp ¼ OC and h _ i Drp ¼ OB in the (r, p) coordinates; conversely, in the ðr; gÞ coordinates, its position is h _ i determined by arc elements Dg ¼ OD and h _ i Drg ¼ OE . In this case, we have Drg [ Drp , but Dg\Dp. Since ðr; pÞ is an orthogonal system, signals are optimally projected onto the coordinates. Compared with projection in the ðr; gÞ coordinates with the projection in the ðr; pÞ coordinates, Dg\Dp indicates that climate

signals projected onto the r-axis may contain passive signals; on the other hand, Drg [ Drp , i.e., climate signal projected onto the η-axis may contain dynamic signals. Therefore, the ðr; gÞ coordinates cannot provide a clear separation between dynamic and passive climate signals; for this reason, this system is less desirable. Furthermore, the “distance” between points O and A can be concisely expressed in terms of " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# _

OE ¼

_ 2

_ 2

OC þ OB

because ðr; pÞ is an

orthogonal system. On the other hand, in the ðr; gÞ coordinates, the distance between these two points including a complicated metric term involving the local angle between two axes, so that it cannot be expressed in the same simple qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi _

form; instead, we have OA \

_ 2

_ 2

OD þ OE .

3.3.3.2 The Distance in Terms of the r–p Coordinates With the introduction of potential spicity we now have a dual pair coordinate system, i.e. h–S and r–p. Thus, in addition to the traditional h–S diagram used in oceanography, one can also use the r–p diagram for water mass analysis. A major advantage of using the r–p coordinates is that we can introduce the concept of distance. In the traditional h–S plane, it is rather difficult to quantify how much the signals spread

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3

because these two axes have different dimensions; furthermore, the thermal expansion and haline contraction coefficients vary in the parameter space. With the introduction of the new orthogonal curvilinear coordinates r–p, the situation is quite different because now we can define the distance between two water parcels, with properties ðh1 ; S1 Þðh2 ; S2 Þ; using the sea surface as the reference level, we have

 r0;1 ; p0;1 r0;2 ; p0;2 . The corresponding distance between these two water parcels are

at location B vary with a much smaller amplitude than A (upper panels of Fig. 3.49). However, it is rather difficult to quantify the signal spreading in the traditional h–S diagram. Using the r–p coordinates, we can calculate the distance between two water masses and the rootmean square distance from the center of mass. It is important to emphasize that in plotting r–p diagram, both axes should be on the same scale. As shown in the lower panels of Fig. 3.49, the radius of signal at location A is 0.8665 kg/m3, which is much larger than that at location B, 0.2194 kg/m3. This example demonstrates the usefulness of calculating the distance between water masses and the radius of signal.

D1;2 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2ffi r0;1  r0;2 þ p0;1  p0;2 ð3:61Þ

A simple and vitally important application of the concept of distance is the radius of signal. First, we calculate the mean potential density and the mean potential spicity ðr; pÞ corresponding to the mass center of signals. Then the radius of signal is defined as the root-mean square distance from each data point ðri ; pi Þ to the mass center ðr; pÞ Rs ¼ RMS

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 þ ðpi  p  Þ2 ðri  r

ð3:62Þ

The radius of signal can be used to evaluate the data scattering. Based on the monthly mean GODAS data from 1980 to 2015, we select two stations A (179.5° E) and B (100.5° W) along the equator (0.5° S); both locations are at the depth of 145 m. Since the location A is well above the equatorial main thermocline, its water mass properties (such as h and S) carry strong variability related to ENSO events. As a result, the h–S properties at station A scatter greatly in the h–S plane (Fig. 3.49a). On the other hand, location B is near the eastern boundary and below the main thermocline, the variability in water mass properties is much less affected by the ENSO processes; consequently, the water mass properties

Heaving, Stretching, Spicing and Isopycnal Analysis

3.3.3.3 The Distance Between r–p Curves Assume that we have (T, S) observations at two stations, A & B. The comparison of water properties can be described as follows. Based on the traditional h–S diagram, we can use the potential temperature as a coordinate. First, we select the common temperature domain defined as: hmin ¼ h0 and hmax ¼ h1 ; within this domain one can define a uniform grid h ¼ ½h0 : ðh1  h0 Þ=ðimt  1Þ : h1 . Second, salinity data at these two stations can be interpolated onto this grid; i.e., we have two salinity arrays:s1 ¼ s1 ð1 : imtÞ and s2 ¼ s2 ð1 : imtÞ. Using these two arrays, the distance between two h–S profiles can be defined as DR ¼ q0 b0 RMSðs1  s2 Þ

ð3:63Þ

There is conceptually a problem because the thermal expansion coefficient varies greatly; in addition, the saline contraction coefficient also varies. Consequently, the so-called distance in the traditional h–S space cannot be accurately defined. With the introduction of the r–p diagram, the distance between two r–p curves can be accurately defined. First, a uniform grid in the density range common for both stations is selected;

3.3 r–p Diagram and Its Application

C)

(a)

113

S diagram, location A

24

26

22

24

20

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18

20

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18

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16 35.1

35.2 (c)

35.4 35.3 S (psu)

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S diagram, location B

(b)

28

35.6

12

34.9

diagram, location A

35 (d)

35.1 S (psu)

35.2

35.3

diagram, location B

23.0 22.5 23.5 23.0

RS=0.8665 (kg/m3)

RS=0.2194 (kg/m3)

24.0 24.5 24.0 25.0

0

(kg/m3)

23.5

24.5 25.5 25.0 26.0 25.5 1

2 3 0 (kg/m )

3

26.5

−1

0 0

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(kg/m3)

Fig. 3.49 h–S (upper panels) and r–p (lower panels) diagrams for two grid points taken at a depth of 145 m along the equatorial ocean, with respective longitude of 179.5° W (left panels) and 100.5° W (right panels), based on the GODAS data. The red dots correspond to the mean values and the red circles refer to the radius of signal (Huang et al. 2018)

spicity at these two stations is interpolated onto this density grid; then the RMS of spicity difference between these two stations is defined as the distance between these two r–p curves DR ¼ RMSðp1  p2 Þ

ð3:64Þ

The distance between two profiles is in units of kg/m3. Note that the range of potential density can be manually selected or computed as

rm ¼ min½maxðprofile AÞ; maxðprofile BÞ ð3:65aÞ rn ¼ max½minðprofile AÞ; minðprofile BÞ ð3:65bÞ For examples, we applied this method to diagnose the similarity and difference between typical water mass properties in the South China Sea, the Pacific and the Atlantic Oceans, based

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S diagram

(b)

2

2

diagram

30 25

(kg/m3)

C

20 15

5 0

2

10 SCS NP SP SA NA

−5 33.5 34.0 34.5 35.0 35.5 36.0 36.5 37.0 37.5 S (psu)

SCS NP: 1.1414 SP: 1.8855 SA: 3.5521 NA: 4.0560

26 27 28 29 30 31 32 33 34 35 36 37 0

2

4 2

6 (kg/m3)

8

10

12

Fig. 3.50 Typical vertical profile of water properties in the Pacific Ocean and Atlantic Oceans and the South China Sea: a the traditional h–S curves; b the r–p curves, including the distance (in units of kg/m3) between the r–p curves in these four ocean areas and the South China Sea

on the WOA09 data. A station at (15.5° N, 115° E) is chosen as a representative for the South China Sea. The following stations are selected as representative for each basin: N. Atlantic: 25.5° N, 35.5° W; S. Atlantic: 17.5° S, 31.5° W; N. Pacific: 15.5° N, 161.5° E; S. Pacific: 23.5° S, 127.5° W; In this example, we calculate the distance between two r–p curves, one taken from SCS and another one taken from the open oceans, over the common density range only (Fig. 3.50). Of course, we can also calculate distance for some selective density ranges. For example, one can focus on the deep part or the shallow part of the profiles. It is clear seen that the water property profile in the South China Sea is closest to that of the North Pacific Ocean. The second closest is the water profile in the South Pacific Ocean. The South Atlantic Ocean is ranked at the third place. Finally, the North Atlantic Ocean is the least similar ocean. This ranking is quite consistent with the deep water circulation in the world oceans. Therefore, the distance based on the r–p

diagram gives us an accurate numerical statement of the water mass properties in the world oceans. Similarly, one can use the r–p diagram to estimate distance between water mass properties taken along a zonal section in the North Pacific Ocean (Fig. 3.51). From this figure, the water property at the representative station (115° E) in the South China Sea is quite similar to that at a station at 132° E. As we move eastward and away from the Luzon Strait, the water mass profiles become less similar. However, at stations in the East North Pacific Ocean, the corresponding profiles become more and more similar to that of the South China Sea. The similarity of water mass property profiles implies important information about mixing and circulation in the ocean. Water mass property distribution at a station is always an important issue in oceanography study. The basic question is how we can identify the source region of a water mass. A simple way is to trace water mass properties, such as potential temperature, salinity or potential vorticity. However, mixing, either diapycnal mixing or lateral mixing, can change water mass properties greatly along the path of the water masses; so that, identifying the source of water mass can be

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115

Fig. 3.51 Typical vertical profiles of water properties along the 15.5° N section in the North Pacific Ocean and the South China Sea. a The traditional h–S curves; b the r–p curves; c the distance (in units of kg/m3) between the r–p curves at these stations and that in the South China Sea

a rather complicated issue. A commonly used method is to compare water properties taken from two stations. In the traditional h–S diagram, there is probably no exact measure to quantify such problem. However, using the recently developed r–p diagram, one can quantify such problem rather conveniently and accurately. In particular, the distance between two r–p curves can be used as an index for tracing the source of water masses. In the traditional water mass analysis, the h–S relation is used. For example, people can use temperature signals on certain isopycnal surfaces to trace the source of water masses. With the newly available r–p diagram, one can trace the water mass as follows. Instead of tracing temperature or salinity anomaly, it is much better to trace the spicity anomaly, which combines the information in both temperature and salinity. In addition, using the least square distance defined as the squared spicity anomaly over an density range gives an integral measure of the distance between two r–p curves or two water masses over the corresponding density range. This discussion can be extended into the whole North Pacific Ocean. We select a station in the South China Sea, marked by the red stars in Fig. 3.52. At this station and at the depth of 1000 db r2 ¼ 36:4565 kg/m3, and this is chosen as the lower boundary of the upper ocean; at the depth of 2000 db,r2 ¼ 36:8118, and this is chosen as

the upper boundary of the deep ocean. We calculate the distance between r–p curve at this station and at all stations in the open North Pacific Ocean for two cases: the upper ocean (1000 db), defined by the isopycnal surfaces of r2 \36:4565 kg/m3 and the deep ocean (below 2000 db), defined by isopycnal surfaces of r2 [ 36:8118 kg/m3. As shown in Fig. 3.52a, the distance between the r2–p2 profiles in the upper ocean has a pattern closely resembling the circulation in the upper ocean, which is dominated by the winddriven circulation. Accordingly, the DR contour starting from the sample station in SCS moves along the western boundary of the North Pacific Ocean, and eventually separates from the western boundary and moves eastward in the middle of the basin. Therefore, water mass properties represented by the r2–p2 profile are nearly conserved along the western boundary current. In contrast, the distance between the r2–p2 profiles in the deep ocean (below 2000 db) has a quite different pattern (Fig. 3.52b). The contour map shown in Fig. 3.52b suggests that deep water in the SCS originates from the westward movement of water mass along a slightly slated band between 10° N and 25° N. As another example, the distance between r2–p2 profiles in the deep part of the South China Sea is examined in order to infer the deep circulation. The distance between a pivotal station

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Fig. 3.52 Distance (in units of kg/m3) between the r2–p2 profiles taken from a station in SCS (red dot) and stations in the North Pacific Ocean. a For the upper ocean, r2 \36:4565 kg/m3; b the deep ocean, r2 [ 36:8118 kg/m3

at the northern part of the deep basin and other stations suggest that the deep circulation starts from the northern basin, but it bifurcates into two branches in the middle of the deep basin (Fig. 3.53a, b). These figures show that the left branch of the circulation seems to penetrate to the southwestern corner, as shown by the distance contour patterns in panels a, b and c. Panel d suggests that the deep circulation moves northward from the southern edge of the deep basin. Wang et al. (2011) discussed the deep circulation inferred by the geostrophic calculation based on high-resolution hydrographic data. They postulated that the circulation in the deep basin of the South China Sea consists of two gyres: a cyclonic gyre in the northeast part of the deep basin

and an anticyclonic gyre in the south-west part of the deep basin. The boundary separating these two gyres is located around 115° E. Accordingly, the pattern shown in Fig. 3.53 is consistent with the deep circulation postulated by Wang et al. (2011). An interesting application of this method is to identify the source region of mesoscale eddies in the ocean. These mesoscale eddies in the ocean can carry water masses from their original place to travel over large distances on the order of 1000–2000 km. Simple methods of tracing water properties, such as temperature, salinity or potential vorticity, carried by these eddies may not provide clear information about their source region. However, the distance between two r–p curves is a much more conserved tracer, so that it

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117

Fig. 3.53 Distance (in units of kg/m3) between the r2–p2 profiles (below 2000 db) taken from a pivotal station (red dots) and other stations in the South China Sea

may provide much more clear pictures as shown above. Note that for identifying the source region of mesoscale eddies, the contour map may appear disrupted, such as the map shown in Fig. 3.53c. Nevertheless, such maps can provide useful information about the potential trajectories or the remote connection of water masses.

3.3.3.4 The Radius of Climate and the Radius of Seasonal Cycle Similar to the situation in the atmosphere, the oceanic environment also changes with time;

hence, we can introduce a new concept—radius of climate, which can be used to evaluate how much water mass properties at a given grid point vary over a long time period. By definition, climate means variability over many decades or even many centuries. Due to the limitation of data availability, here this concept is applied to some oceanic climate data sets currently available. As an example, our discussion in this section is confined to the application to the monthly mean GODAS data. Although climate might be defined in terms of the variability of both the temperature and salinity, it is technically difficult

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to quantify the climate variability accurately. However, with the introduction of the ðr; pÞ diagram we can define the climate variability in terms of a single index, the radius of climate. At a fixed grid point in the world oceans, the long-term means of both potential density and potential spicity can be calculated, and the corresponding root-mean square distance from the instantaneous potential density and potential spicity to the long-term means is defined as the radius of climate for this grid point:

three major components: heaving, stretching and spicing. As will be shown at the last section of this chapter, the contributions associated with heaving and stretching are dominating components. We can also define the climate variability associated with the climatological mean seasonal cycle

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 þ ðp  p Þ2 RC ¼ RMS ðr  r

Heaving, Stretching, Spicing and Isopycnal Analysis

RSC ¼ RMS

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrSC  rSC Þ2 þ ðpSC  pSC Þ2 ð3:66bÞ

ð3:66aÞ

For the discussion in this section, both the potential density and the potential spicity are referred to the sea surface pressure, pr = 0. It is clear that the radius of climate is defined in the same way as the radius of signal discussed in the previous section. The only difference is that the radius of signal is referred to the length of the data record in our concern, and the length of record may vary greatly for different cases. In contrary, the radius of climate is defined to the entire length of historical data record. It is to emphasize that climate signals observed at a fixed location are due to the complicated processes in the ocean. As discussed in the beginning of this chapter, climate signals observed at a fixed location can be separated into

where the subscript SC denotes the seasonal cycle. For the monthly mean climatological data, the seasonal cycle has only 12 monthly mean data points for the potential temperature, salinity, potential density and potential spicity. As an example, the horizontal distribution of the radius of climate at the depth of 135 m in the world oceans is shown in Fig. 3.54. As shown in Fig. 3.54b, at this depth the radius of climate reaches maxima in the equatorial band of the Pacific Ocean, two centers are near 150° W, and the third one is near the western boundary. In addition, there are maxima near the western boundaries in the Atlantic Ocean and Indian Ocean. The radius of climate maxima in these locations may be directly linked to the frequent ENSO like events.

Fig. 3.54 Radius of seasonal cycle (a) and radius of climate (b) at depth of 135 m in the world oceans

3.3 r–p Diagram and Its Application

119

Fig. 3.55 A zonal-depth section of the radius of seasonal cycle (a) and radius of climate (b) along 5.5° N, overlaid with potential density contours (white, in units of kg/m3)

In comparison, maxima of the radius of seasonal cycle are mostly confined to the off equatorial band of 0–10° N in the Northern Hemisphere (Fig. 3.54a). In the Pacific Ocean, the maximal radius of seasonal cycle appears in the central equatorial band only. In both the Atlantic and Indian Oceans they appear near the western boundaries. The zonal distribution of the radius of climate is shown for a section along 5.5° N (Fig. 3.55b). In the equatorial Pacific Ocean, there are two bands of maximum, one next to the western boundary and another one east of the dateline. A careful examination reveals that the radius of climate maximum gradually moves upward in the density coordinate, from r0 ¼ 24:5 kg/m3 near the western boundary to r0 ¼ 22 kg/m3 near the eastern boundary. The vertical migration (in density coordinate) of the radius of climate with longitude may reflect the vertical migration of the zonal long Rossby waves associated with the ENSO like events in a stratified environment. Similarly, radius of climate maxima can be identified in both the Atlantic and Indian Oceans. The distribution of the radius of seasonal cycle is shown in Fig. 3.55a. In contrast to the pattern shown in Fig. 3.55b, radius of seasonal cycle is much smaller than the radius of climate. This fact indicates that climate variability along this section is mostly owing to the interannual variability of

temperature and salinity, while the seasonal variability plays the secondary role. Most importantly, the large radius of climate maxima near the western boundaries in the Pacific and Atlantic Oceans completely disappear in Fig. 3.55a, indicating that climate variability in these regions is entirely due to the interannual variability, with very limited contribution from the seasonal cycle. The meridional distribution of the radius of climate along the 149.5° W section is shown in Fig. 3.56b. In the equatorial band, climate variability is mostly confined to the subsurface ocean, near the main thermocline, and there are three maxima located within the density range of r0 ¼ ð22:5  24:5Þ kg/m3. This feature again reflects the fact that climate variability in this part of the ocean is mostly linked to the vertical motions of isopycnal surfaces; since motions at this depth range is mostly adiabatic, the associated climate variability is mostly adiabatic in nature, i.e., they are closely linked to heaving modes discussed in this book. On the other hand, away from the equatorial band, radius of climate is maximal near the surface. It is clear that within these latitude bands the air-sea interaction taking place at the upper surface of the ocean must play an important role. As such, climate variability in these areas must include substantial contributions due to the diabatic processes taking place near the air-sea interface.

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Fig. 3.56 A meridional-depth section of radius of seasonal cycle (a) and radius of climate (b) along 149.5° W, overlaid with potential density contours (white, in units of kg/m3)

Fig. 3.57 The horizontal distribution of the maximum radius of seasonal cycle (a) and radius of climate (b) for the world oceans

The meridional distribution of the radius of seasonal cycle along the same section is shown in Fig. 3.56a. The pattern is quite similar to that shown in Fig. 3.56b; the corresponding amplitude is similar, but slightly smaller, indicating the important role of the seasonal cycle. We can also find the horizontal distribution of the maximal value of radius of climate for the entire water column and the corresponding depth at each station (Figs. 3.57 and 3.58). The radius of climate reaches maxima within the equatorial

band and the subtropical gyres in the Northern Hemisphere (Fig. 3.57b). In the equatorial Pacific Ocean there are three zonal bands of maximal radius of climate. The northern band is near 10° N, the southern band is near 5° S, and the middle bans is near 5° N. These bands of radius climate maximum are closely linked to the westward long Rossby waves associated with ENSO events. On the other hand, large radius of climate at mid-latitude bands in the Atlantic and Pacific

3.3 r–p Diagram and Its Application

121

Fig. 3.58 The depth of the maximum radius of seasonal cycle (a) and radius of climate (b) for the world oceans

Oceans may be the result of the strong air-sea interaction associated with the mode water formation and the adjustment of thermocline at this latitude band. The distribution of radius of seasonal cycle maxima is shown in panel a. Along the equatorial band, the amplitude is much smaller than that shown in panel b, implying that climate variability maximum in the equatorial band is mostly owing to the strong interannual variability associated with ENSO like events; while seasonal cycle plays a small and secondary role. The depth of the maximal radius of climate in the world oceans is shown in Fig. 3.58b. At lower latitudes it is in the range of 100–250 m, and it slopes down westward. In addition, it has a dumbbell like shape, suggesting that it is close to the local main thermocline depth. On the other hand, in the off equator regions, the corresponding depth is nearly zero, in fact, it is roughly 5 m, the depth of the top grid in the GODAS data. This is consistent with the pattern shown in Fig. 3.56. Therefore, at the mid-latitude bands the large radius of climate appears in the surface ocean, where the air-sea interaction is the major player in climate change. The pattern of the depth of the maximal radius of seasonal cycle (panel a) is quite similar to that shown in panel b; thus, the mechanism of selecting the depth of maximum is similar for both cases.

Furthermore, we can calculate the total climate variability for each station in the world oceans, and we will use the integration term R0 defined as H RC dz (Fig. 3.59b). The global maximum is located near the western boundary in the North Atlantic Ocean, and it is directly linked to the mode water formation in this area. Similarly, there are local maximum in the South Atlantic Ocean and North Pacific Ocean, both these local maxima are closely linked to the mode water formation in these locations. On the other hand, there are local maxima near the western boundaries of the equatorial bands in the Pacific, Indian and Atlantic Oceans, and these regions of high climate variability may be due to the strong ENSO like events in the equatorial band. The pattern of the total seasonal cycle variability in the world oceans is shown in panel a. There are two regions of great interest. First, in the northwest Atlantic Ocean, there is a global maximum of total climate variability as shown in panel b; however, the corresponding integral associated with the variability of the seasonal cycle is quite small. This fact indicates that climate variability in this region of the ocean is primarily induced by the strong interannual or decadal variabilities in the ocean; these variabilities are connected with mode water formation in this vicinity.

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Fig. 3.59 The vertically integrated seasonal cycle variability (a) and climate variability (b) for the world oceans

On the other hand, the integral of the seasonal cycle reaches the global maximum near the Somali coast. Comparing panels a and b for this region, it is clear that the climate variability in this region is primarily due to the seasonal cycle, while the contribution associated with interannual variability is of secondary importance. The relative contribution of seasonal cycle to the climate variability at each station is defined as R0 R0 the percentage ratio 100  H RSC dz= H RC dz (Fig. 3.60a). The maximum of this ratio is located in the Arabian Sea, in particular the eastern

and western boundaries. This indicates that seasonal cycle in this part of the world oceans is the dominating player of climate variability. In the equatorial Pacific Ocean and Atlantic Ocean this ratio is on the order of 50%–60%; hence seasonal cycle in these areas also plays important role in generating climate variability. However, for most part of the world oceans, the contribution owing to the seasonal cycle is quite small; this fact can be seen much better in Fig. 3.60b. In fact, near the edge of the Antarctic and region of mode water formation in the North

Fig. 3.60 The percentage of the vertically integrated seasonal cycle variability versus the climate variability for the world oceans

3.3 r–p Diagram and Its Application

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Fig. 3.61 The contributions due to temperature and salinity variances to the vertically integrated climate variability in the world oceans

Atlantic Ocean, the climate variability is primarily due to the strong interannual variability, as shown in both the left and right panels of this figure. We can further separate the contributions to the radius of seasonal cycle associated with the temperature variability (or salinity variability) alone. At each grid point we can calculate the following two indexes

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
 2 
 2
 RSC h; S ¼ RMS r hSC ; SSC  rSC þ p hSC ; SSC  pSC

ð3:67aÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
 2 
 2 RSC ðh; SÞ ¼ RMS r hSC ; SSC  rSC þ p hSC ; SSC  pSC

ð3:67bÞ To explore the meaning of these two terms we calculate the vertical integration of each terms and compare them with the vertical integration of the radius of seasonal cycle RSC ðh; SÞ (Fig. 3.61). It is clearly seen that for the most part of the world ocean, the seasonal cycle associated with temperature variability alone is dominating; while the salinity variability is important for the regions associated with bottom water formation near the edge of the Antarctic, as discussed in Sect. 2.4.2. In addition, the Berlin Sea, the site of mode water formation in the North Atlantic Ocean, is also characterized by substantial contribution due to salinity variability. Furthermore, there is a small

region around 10° N–20° N in the Atlantic Ocean, where the salinity variability makes a noticeable contribution to the seasonal cycle. The dynamical processes related to such phenomenon in these regions remains to be explored. In the rest part of this section we select three stations in the world oceans to examine the radius of climate and the radius of seasonal cycle in the traditional (h, S) diagram and the new ðr; pÞ diagram. Our analysis here is focused on the depth of 135 m. The first example is taken from station A (55.5° W, 15.1660° N). The data points spread over the traditional (h, S) plane, but the degree of data spreading is difficult to be quantified in this phase diagram (Fig. 3.62a, b). On the other hand, in the ðr; pÞ plane, the corresponding data spreading is quite easy to be quantified by the radius of climate (Fig. 3.62c) and the radius of seasonal cycle (Fig. 3.62d). For this case, the radius of seasonal cycle is about a half of the radius of climate, indicating that seasonal cycle is important, but it is not the dominating component of climate variability. The seasonal cycle can be further separated into the components associated with temperature and salinity variability alone. The corresponding radius of seasonal cycle is shown in the lower panels of Fig. 3.63. If we set the salinity as the

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Fig. 3.62 The (h, S) diagram (upper panels) and (r0–p0) diagram (lower panels) for the radius of climate (left panels) and the radius of seasonal cycle (right panels) at station A ð55:5 W; 15:1660 NÞ; the red stars indicate the mean value and the dashed red circles indicate the radius of signal

seasonal cycle mean, the corresponding radius of seasonal cycle is 0.1517 kg/m3; on the other hand, if we set the potential temperature as the seasonal cycle mean, the corresponding radius of climate is 0.0205 kg/m3. The large difference between these two values indicates that the seasonal cycle at this grid point is primarily due to temperature variability, with the salinity variability plays a very small role. The second example is taken from station B ð67:5 W; 41:8325 NÞ (Fig. 3.64). This is at the

location of mode water formation, and the signals are characterized by a strong interannual variability. As a result, the radius of climate is about 3 times of the radius of seasonal cycle. The contrast between variability at these two stations also demonstrates the usefulness of the ðr; pÞ diagram. In the traditional ðh; SÞ diagram, it is rather difficult to quantify/compare the climate and seasonal cycle at these two stations; however, with the help of the ðr; pÞ diagram, we can accurately quantify these differences.

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125

Fig. 3.63 The (h, S) diagram (upper panels) and (r0, p0) diagram (lower panels) for the radius of seasonal cycle associated with thermal variability alone (left panels) and the radius of seasonal cycle associated with haline variability alone (right panels) at station A ð55:5 W; 15:1660 NÞ; the dashed red circles indicate the radius of signal

The contribution to the radius of seasonal cycle due to the temperature and salinity can be clearly quantified in the ðr; pÞ diagram (Fig. 3.65). Once again, the temperature variability is the dominating player for the seasonal cycle, while the salinity variability is a small and secondary contributor. The third example is taken from station C ð149:5 W; 4:1661 NÞ. This is located in the

equatorial band of the Pacific Ocean, where the strong ENSO like events prevail. This site is characterized by the large radius of climate (1.3837 kg/m3) and radius of seasonal cycle (1.1108 kg/m3) (Fig. 3.66). The contribution to the radius of seasonal cycle owing to the temperature and salinity variability is shown in Fig. 3.67. Since the contribution owing to the salinity variability is so much

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Fig. 3.64 The same as Fig. 3.62, but for station B ð67:5 W; 41:8325 NÞ

smaller than that due to the temperature variability, the range of spicity and density used in panel d is much smaller than those used in panel c. Comparing the lower panels of Figs. 3.65 and 3.67, the radius of seasonal cycle by cause of temperature at station C (1.1113 kg/m3) is about 7 times of that at station B (0.1548 kg/m3). In comparison, both the seasonal and interannual variability of temperature at station C is very strong, and thus the corresponding radius of climate and radius of seasonal cycle associated with temperature are large.

3.3.3.5 The Radius of State for the World Oceans One of the most important applications of the r–p coordinates is to define the radius of state, which can be used as an index for the strength of the oceanic general circulation. This index includes contributions due to the spreading of water masses in both the potential density axis and the potential spicity axis; the corresponding unit is kg/m3. The strength of mixing/stirring associated with the oceanic general circulation can be evaluated in terms of the radius of state

3.3 r–p Diagram and Its Application

127

Fig. 3.65 The same as Fig. 3.63, but for station B ð67:5 W; 41:8325 NÞ

based on the root-mean square distance from the reference state of a homogeneous ocean to the physical state of the oceanic circulation. The radius of state can also be used as a measure of the distance between two states of the oceanic circulation. The world oceans are a turbulent system characterized by mixing/stirring. In a general sense, the transports associated with the largescale circulation can be called large-scale stirring; hence, turbulence and currents on broad

scales from basin scale to molecular scale can be collectively called mixing/stirring, and this encompasses all important aspects of the oceanic general circulation, including the wind-driven circulation and the thermohaline circulation. Surface buoyancy forcing, including heat flux and freshwater flux through the air-sea interface, is among the major factors regulating the oceanic general circulation. On the other hand, wind stress directly regulates the wind-driven circulation; in fact, the mechanical energy supplied by

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Fig. 3.66 The same as Fig. 3.62, but for station C ð149:5 W; 4:1661 NÞ

wind stress and tidal dissipation is the major source of energy supporting the dissipation associated with the oceanic general circulation. Obviously, mixing/stirring and the oceanic general circulation are complicated phenomena and they can be described in many different ways. Therefore, in the study of the oceanic circulation and climate it is desirable to have simple indexes to evaluate the strength of the oceanic general circulation and mixing/stirring. First, the horizontal transport rate of the winddriven circulation and the meridional overturning

circulation rate in the ocean can be used. There are many similar indexes describing the strength of the oceanic general circulation in the world oceans, including the transport of the Gulf Stream and Kuroshio Current and the global meridional overturning circulation, the GMOC, or the corresponding part in the Atlantic Ocean, or the AMOC. These indexes based on the circulation rate have been used widely; however, there are also some shortcomings as well. For example, the circulation is a complicated phenomenon in three dimensional space. As such, an

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Fig. 3.67 The same as Fig. 3.63, but for station C ð149:5 W; 4:1661 NÞ

index based on the maximum streamfunction in the x–y plane or the y–z plane does not contain complete information about the threedimensional circulation. Second, the strength of the buoyancy fluxes across the air-sea interface can be used as indexes for the description of the oceanic general circulation. However, buoyancy forcing is a forcing condition at the upper surface only; strong surface forcing does not necessarily mean that the oceanic general circulation is strong.

Thus, it is desirable to define an index that reflects the strength of mixing/stirring and circulation in the ocean. In this section, we postulate the radius of state based on the water mass properties distribution in the r–p space. States of the Oceanic General Circulation The radius of the state is a simple index to evaluate the state of ocean, and it is defined as the root-mean square distance of each water parcel to the mass center of water masses in the

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ocean. By definition, a state of a homogeneous ocean, in which both the potential temperature and the salinity are uniform, has a zero radius of state. Such an ocean is dead in the scientific sense; there is no motion, and no life in such an ocean. If we want to search life on some very remoted planets, light spectrum can be collected and analyzed. If the light spectrum representing the ocean is a monotonic single line in the spectrum, such an ocean cannot sustain life. On the other hand, if the corresponding spectrum is wide, chance of life existing is large. On the other hand, the larger the radius of state is, the more active is the ocean. Accordingly, the radius of state is a simple measure for the activity of the ocean state. For example, an ocean at rest without horizontal advection is vertically homogeneous, except for the skin layer in the upper ocean, where air-sea interaction may set up a horizontally differential temperature and salinity. Nevertheless, the radius of state for such an ocean is practically zero. To demonstrate the meaning of the radius of state, we discuss how does the radius of state change with the strength of mixing/stirring in the ocean. From the climatological mean state inferred from the WOA09 data, we can find the global extreme value of potential temperature
 bot and salinity on the sea floor hbot min ; Smax . Using the climatological mean surface potential tem  surf perature and salinity hsurf ; S at each grid real real

point, we can construct the temperature and salinity distribution at each station of an idealized ocean as follows   surf bot þ h  h h ¼ hbot min min expðz=DH Þ real   ð3:68Þ surf bot S ¼ Sbot max þ Sreal  Smax expðz=DH Þ where DH is the penetration depth, a free parameter used to indicate how deep the surface signals can penetrate. For a small DH, the idealized ocean is close to a homogeneous ocean, where the ocean interior is close to the extreme
 bot values hbot min ; Smax ; for a large DH, the oceanic state is close to reality as defined by the WOA09 data. For the different penetrating depths, the corresponding radius of state, and the contribution due to spreading of the potential density and the potential spicity is shown in Fig. 3.68. Note that the radius of state defined by density or spicity is the same as the standard deviation (STD) for these two variables. As DH is increased gradually, the variance of density and spicity increases, and the radius of the state of the idealized ocean also increases accordingly. Accordingly, the radius of state can be used as an index to evaluate the spreading of both temperature and salinity in the parameter space. An ocean with a large radius of state is an ocean that is dynamically more active.

Fig. 3.68 Radius of state of the idealized ocean, depending on the scale depth of penetration

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Fig. 3.69 Time evolution of the mean (left) and variance (right) of potential temperature (in units of °C) in the major basins and the world oceans

Time Evolution of the State of the World Oceans The radius of state is a simple index to evaluate the state of the ocean, and it can be used to evaluate the time evolution of the state of the world oceans. Our discussion here is based on the GODAS data; therefore, the Arctic Ocean is excluded in the discussion. In the following analysis we will separate the world oceans into three major basins; the Mediterranean Sea, the Black Sea and other small marginal seas are excluded in the calculations for the three basins. First, according to the GODAS data the global mean potential temperature went through a gradual decline from 1980 to 2005, but then went up quickly from 2006 to 2011; afterward it declined again and stayed relatively flat during the years of 2012–2016 (Fig. 3.69). The mean trend averaged over 37 years is a slight increase. The GODAS data is based on the traditional volume conservation approximation, plus data assimilation; as such, the model does not conserve heat content accurately; consequently, the

total heat content change in the model may be different from reality. As shown in Fig. 3.69, according to the GODAS data, the mean potential temperature increases for all basins from 2006 onward, although there are clear signs of hiatus after 2010. Over the past 37 years, the mean temperature in the Atlantic and Pacific Oceans went down slightly; however, it increased in the Indian Ocean and the world oceans. On the other hand, the temperature variance in all three major basins and the world oceans increases over the past 37 years (right panels of Fig. 3.69). Thus, although the mean temperature in a basin may decline, the corresponding variance can increase, i.e., the ocean is thermally more active. The mean salinity in all basins and the world oceans went down steadily; this is mostly due to melting of land-based glaciers (left panels in Fig. 3.70). Note that the downward trend is relatively flat in all basins after 2007; the reason for such a flat rate in the GODAS data remains unclear.

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Fig. 3.70 Time evolution of the mean and variance of salinity (in units of psu) in the major basins and the world oceans

On the other hand, the salinity variance behaves in the opposite sense. As shown in the right panels of Fig. 3.70, the salinity variance in all basins and the world oceans has increased over the past 37 years. Combined with the left panels of Fig. 3.70, this indicates that although mean salinity is reduced owing to melting of land-based glaciers, the mixing/stirring of salt is enhanced. As a result, the salinity variance in all basins is greatly increased, indicating more active hydrological cycle and saline mixing in the world oceans. We now look at the mean potential density. To evaluate the whole depth of the ocean the 2000 db was used as the reference pressure. As shown in the left panels of Fig. 3.71, the mean potential density in all basins went down over the past 37 years; such a continuous decline of the mean density is due to the combination of warming and melting of land-based glaciers. However, the trend of the potential density variance is the opposite. As shown in the right

panels of Fig. 3.71, the density variance in all major basins and the world oceans has increased over the past 37 years. Density variance is the most important index for the dynamic activity in the ocean; hence, according to the GODAS data, the world oceans are dynamically more and more active. The changes in the potential spicity are quite interesting. As shown in the left panels in Fig. 3.72, the mean spicity in the Atlantic and Pacific Oceans has declined over the past 37 years, and this is owing to the combination of freshening (Fig. 3.70a, c) and slightly cooling (Fig. 3.69a, c). In contrast, the spicity is increasing in the Indian Ocean. This is the result of warming combined with salinification in this basin. Overall, the mean spicity in the world oceans has declined, apparently due to the combination of global warming and freshening. On the other hand, the trend of the spicity variance is different, as shown in right panels of Fig. 3.72. In parallel to the common usage of

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Fig. 3.71 Time evolution of the mean and variance of potential density (in units of kg/m3) in the major basins and the world oceans

Fig. 3.72 Time evolution of the mean and variance of potential spicity (in units of kg/m3) in the major basins and the world oceans

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stratification, we can use a new terminology of spicification, which is a measure of the gradient of spicity in either the horizontal or vertical directions. The direct measure of spicification is the spicity variance, as shown in right panels of Fig. 3.72. In all three major basins and the world oceans, the spicity variance becomes larger and larger, although the mean spicity itself declines in the Atlantic Ocean, Pacific Ocean and the world oceans. The increase of spicity variance may be interpreted as the strengthening of the thermohaline circulation in the ocean. These data are also plotted in the forms of a h–S diagram and a r2–p2 diagram, as shown in Fig. 3.73. These figures clearly show the migration of the water mass centers in the oceans. Note that the linear trend can be different than what can be inferred from the slope of the line linking the blue dot and the red dot in each panel; thus, the linear trends by least-square fitting are inserted in each panel. Overall, the spicity variance for the world oceans increases with time, indicating that the hydrological cycle is enhanced with time, and the thermohaline circulation is more active in general. Finally, we evaluate the radius of state for the oceans. As shown in Fig. 3.74, the radius of state in all three major basins and the world oceans all increased over the past 37 years. This indicates that due to global warming and melting of landbased glaciers, the oceanic general circulation becomes dynamically more active in general. The oceanic general circulation is a very complicated system, including many elements in the three dimensional space; thus, it is rather difficult to compare the circulation in the world oceans. However, introducing the radius of state provides a simple index for such a complicated system. As an example, one may ask the following question: which component really regulates the variability of the oceanic general circulation? To answer this question, we draw the composed figures in Fig. 3.75. To begin with, the key parameters to regulate the variability of the circulation are the standard deviation, not the mean values. As shown in Fig. 3.75a, although

the temperature variability is the largest in the Indian Ocean, the largest variability in salinity is in the Atlantic Ocean. This gives rise to the largest variability in spicity in the Atlantic Ocean, as shown in Fig. 3.75b. The largest variability in spicity leads to the largest radius of state in the Atlantic Ocean. The circulations in the Atlantic and Pacific Oceans are different. The stratification in the Pacific Ocean is characterized by strong stratification, 20% larger than the Atlantic Ocean. On the other hand, the Atlantic Ocean is characterized by stronger spicification, 20% larger than the Pacific Ocean, as shown in Fig. 3.75b. There is a long time misunderstanding of the haline circulation induced by changes in freshwater flux through the air-sea interface. According to the common wisdom about the Goldsbrough-Stommel theory of the saline circulation, the rate of circulation induced by the surface freshwater flux is on the same order as the freshwater flux, which is rather small. However, as will be emphasized in Chap. 4, in the world oceans there is strong mixing driven by the external mechanical source of energy from wind and tidal mixing, the corresponding haline circulation appears in the form of very strong baroclinic circulation, whose strength is on the same order as those of the wind-driven circulation and the thermal circulation. Since the oceanic general circulation is a complicated system, it is unclear what really regulates the overall variability of the circulation: is it the temperature or the salinity. Most previous studies were focused on the role of thermal forcing anomaly in regulating the circulation; such results may lead to the somewhat incorrect conclusion that the variability in thermal forcing or temperature in the ocean is the dominating factor in regulating the circulation. However, Fig. 3.75 reveals the most important dynamical factor that instead of the temperature variability, it is the salinity variability that plays the dominating role in regulating the overall variability in the oceanic circulation.

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Fig. 3.73 h–S diagram (upper) and r2–p2 diagram (lower) for the oceans, blue dots mark year 1980, and red dots mark year 2016

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Fig. 3.74 Time evolution of the radius of state (in units of kg/m3) of the major basins and the world oceans

Fig. 3.75 Standard deviation of potential temperature and salinity (a) and radius of state (b) for the world oceans and three main basins. Numbers in panel b indicate the corresponding radius of state (in units of kg/m3) for the individual basins and the world oceans

3.4 Isopycnal Analysis

3.4

Isopycnal Analysis

In the common practice, climate data are presented in the spherical coordinates, in particular the z-coordinate is widely used as the vertical coordinate. However, a major portion of largescale motions in the ocean is directly linked to the movement of isopycnal surfaces and the stretching/compressing of isopycnal layers; such motions can be called heaving and stretching. Therefore, using isopycnal analysis can help us to extract the heaving/stretching components and the relatively small residuals (the spicing signals) from the climate signals. The isopycnal layer movements, including stretching/compressing, are regulated by the oceanic general circulation. Thermohaline forcing at the sea surface and diapycnal/alongisopycnal mixing can change the water mass distribution in the density space. In addition, the wind-driven circulation can change the shape of isopycnal outcropping and hence affect the water mass formation and erosion. Furthermore, the wind-driven circulation can lead to the adiabatic motion of isopycnal layers in the upper ocean, and thus lead to the change of the water mass distribution in space. Although it would be nice to separate the movements of isopycnal layers into the so-called adiabatic and non-adiabatic components; practically, this is a very challenging task. In this section we will be content with the goal of separating the climate signals into the heaving/stretching components and the residuals. Assume that we have long term observations at one station; so that, the climatological mean
 potential density profile r ¼ r S; h; p; pr at this station can be defined, where the salinity and potential temperature profiles are the climatological mean of observations at this station, shown as the black curves in Fig. 3.76. In the commonly used Matlab code, potential density is defined as a function of salinity, in situ temperature, pressure and reference pressure, i.e.,r ¼ rðS; T; p; pr Þ. In the following discussion we will use the potential density based on the sea surface pressure, i.e., r ¼ r0 ¼ r ðS; T; p; 0Þ.

137

In most cases, the original data points are scattered in the h–S plane, and the purpose of data analysis is to separate such signals into different components and thus reduce the scattering of data. In order to compare the degree of signal scattering in water property data it is desirable to have an exact measure of data scattering. The radius of signal introduced in Sect. 3.3 can be used as such a measure. A large radius of signal indicates that there is a lot of variability due to the processes not-yet identified. Each time we reduce the radius of signal, some dynamical processes are separated, hence leading to a better understanding of the climate variability. Since temperature and salinity are based on different units, it is more convenient to define the radius of signal in the r–p diagram Rs ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þ2 þ ½RMSðp  p Þ2 ½RMSðr  r ð3:69Þ

where r is the potential density and p is the potential spicity. As discussed above, spicity is defined as a thermodynamic variable which contours are “perpendicular” to the density contours. It is well known that in situ density is not a conserved thermodynamic variable because it increases greatly with the downward motion of water parcels. However, potential density is a conserved quantity, and potential spicity is also a conserved quantity; thus, both of them can be used as good tracers. By definition, potential density is constant on potential density surfaces; hence, potential spicity is an ideal variable to describe climate variability on potential density surfaces. The main focus of this section is to introduce isopycnal layer analysis, which can be used to separate climate signals into three components: the heaving/stretching components and the spicity component. Since the density is directly linked to the stratification and the pressure gradient force in the vertical and horizontal directions, the isopycnal component is directly related to large-scale

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Fig. 3.76 Isopycnal analysis at one station in the Lagrangian coordinate

dynamical processes in climate variability; therefore, density signals can be called dynamical climate signals. On the other hand, spicity is not directly linked to large-scale dynamics, so that it is a passive tracer and the spicity signals can be called the passive climate signals. However, spicity may be linked to other processes in the oceanic general circulation, as discussed in Sect. 3.3. Isopycnal analysis for climate variability can be carried out in two coordinates, i.e. the Lagrangian and Eulerian coordinates, as follows.

3.4.1 The Lagrangian Coordinate Imagine that an instrument equipped with a CTD moves along a selected isopycnal surface in the ocean. The only data recorded by the instrument is temperature, salinity and pressure. Since the instrument can exactly follow the isopycnal, the (h, S) collected by the instrument must be density compensated, i.e., they automatically satisfy the constraint of constant potential density; hence, the most valuable thermodynamic variables identifiable on the isopycnal surface are spicity and pressure. Such signals are incomplete for climate study; in order to find the complete information necessary for climate study one must

retrieve some relevant information from the background climate profile as follows. (a) For a given time, the vertical profiles of water property ðSð pÞ; T ð pÞ; rð pÞÞ are given, the blue curve in Fig. 3.76. For the specific isopycnal surface r we can find the corresponding pressure level, (potential) temperature and salinity ðSins ; Tins ; pins Þ by interpolation of the instantaneous profiles, and this point is depicted by the red triangle in Fig. 3.76. (b) The reference state is defined by tracing the climatological mean profile from the red triangle to the red circle at level pins, and it
 is denoted as Sref ; Tref ; pref ¼ pins ; rref . Based on the water property defined on the climatological mean profile, the climate
signals are Sins  Sref ; Tins  Tref ; pins ¼ pref ; rins  rref Þ. (c) Similar to the method applied to the central surface of the isopycnal layer discussed above, using the new locations of the upper/lower layer interfaces pupper and plower ins ins we can identify the climate signals for the layer upper and lower interfaces: upper lower lower rupper Lag;ins  rLag;ref ,rLag;ins  rLag;ref , where the subscript Lag indicates it is based on Lagrangian coordinate. Therefore, the distance between the upper and lower layer lower interfaces is dDrLag ¼ rupper Lag;ref  rLag;ref .

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(d) The climate variability associated with an isopycnal layer has three terms: First, the heaving term, defined as
 drHeaving ¼ rins  rref  rins  rref ð3:70Þ Second, the stretching term, defined as drStretching ¼ dDrLag  dDrLag

ð3:71Þ

Third, the spicing term, defined as dpSpicing ¼ pins  pins

ð3:72Þ

The complete information on climate variability in the isopycnal coordinates includes three terms: heaving, stretching and spicing. Heaving is associated with the vertical movement of the central isopycnal surface; stretching is associated with the compression/stretching of the individual isopycnal layer; finally, spicing is associated with the temperature and salinity variability on the central surface of the isopycnal layer. By definition, the temperature and salinity variability on an isopycnal surface must be compensated. Therefore, using temperature (salinity) variability as an index to discuss climate variability on an isopycnal surface is inaccurate because some climate variability is also linked to salinity (temperature) change; however, using both temperature and salinity variability is redundant. In contrast, using the spicity anomaly on an isopycnal surface can describe climate variability most concisely and accurately. This method is applied to the equatorial Pacific Ocean, and the GODAS data are used in the following calculation. As an example, we use the 37year record at an equatorial station (179.5° W, 0.167° S). The climatological mean potential temperature, salinity, potential density, potential spicity profiles at this station are shown in Fig. 3.77. The time evolution of climate signals in the upper ocean at this station is shown in Fig. 3.78. The strong variability in potential temperature, salinity, potential density and potential spicity is closely linked to the ENSO cycles.

The isopycnal analysis based on the Lagrangian coordinate is as follows. Our calculation here is focused on the isopycnal surface of
 r0 ¼ 25 kg/m3 . Over 37 years the corresponding reference potential density and pressure varied greatly (Fig. 3.79). There are clearly signs of decadal variability in both terms, reflecting the decadal variability in the ENSO cycles. In particular, the strong ENSO events in 1982–1983, 1986–1987, 1992–1993, and 1997–1998 are outstanding (Fig. 3.79). On the other hand, ENSO cycles behaved in quite different ways in the 21 century. By definition, in the Lagrangian coordinate the water parcel properties ðhins ; Sins Þ observed on a specific isopycnal surface must be potential density compensated (Fig. 3.80a). Of course, this is not the complete information for climate variability. We must use some additional information to make up the loss of climate signals due to the constraint that (h, S) data are collected on this fixed isopycnal surface. We propose to use the (h, S) properties inferred from the climatological mean profile and identified from the same pressure level as the reference values ðhref ; Sref Þ; the corresponding data are presented in Fig. 3.80b. The difference between the data on the left-hand side and the data on the right-hand side provides the missing information for data collected by the observer siting on the isopycnal surface. With the addition of the data inferred from the climatology, we arrive at the modified data ðhins  href ; Sins  Sref Þ, which are labeled as the total climate signals shown in the left panel of Fig. 3.81. The original data collected by the observer sitting on the isopycnal ðhins ; Sins Þ shown in the right panel of Fig. 3.81 is really the climate perturbations. As discussed above, (h, S) data collected on the isopycnal surface must be density compensated. Thus, the only signal left behind is the spicity, so that these data points must fall on an isopycnal contour. The few data points slightly off the main line in Fig. 3.81b are due to the numerical errors induced by linear interpolation in the calculation. The power of isopycnal analysis is much better illustrated in terms of the r0–p0 diagram,

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Fig. 3.77 The climatological mean profiles

Fig. 3.82. The original data collected at a fixed isopycnal level are modified by subtracting the corresponding climatological mean ðr0;ins  r0;ref ; p0;ins  p0;ref Þ, Fig. 3.82a. The radius of signal is 0.6979 (kg/m3). The original data collected by the observer sitting on the isopycnal ðr0;ins ; p0;ins Þ is really the climate perturbations. Since (h, S) data collected on isopycnal surface must be density compensated, the only signals left behind is the spicity. The corresponding signals in isopycnal coordinates are rins  rins ; pins  pins , as shown in Fig. 3.82b. The radius of signal after isopycnal analysis is 0.0663 kg/m3. Note that the scattering in the potential spicity axis is also substantially reduced after the isopycnal analysis; in fact, it is reduced from 3 to 0.35 kg/m3, as shown in Fig. 3.82. Hence, isopycnal analysis based on the Lagrangian coordinate is able to substantially compress the climate signals into a narrow band in the r–p diagram. The corresponding radius of signal is reduced from 0.6979 to 0.0663 (kg/m3). The time evolution of temperature and salinity
 identified from isopycnal r0 ¼ 25 kg/m3 is shown in Fig. 3.83.

Note that for an observer sitting at a fixed depth at a station, the climate signals consist of the instantaneous potential temperature, salinity, and potential density. In addition, the observer can also infer the isopycnal layer thickness. On the other hand, for an observer following the isopycnal movement, the value of potential density is always the same as the pre-specified value. The real climate signals the observer can report are the time evolution of the depth of the isopycnal surface and the mean spicity in this layer. In addition, the observer may measure the thickness of an isopycnal layer, as shown in Fig. 3.84. The time evolution of the isopycnal layer thickness (Fig. 3.84b) is an important part of climate signals and it has not received much attention until now. Physically, the vertical migration of isopycnal layers can be seen as the result of the vertical integration of the layer thickness anomaly in the density coordinate. As shown in Fig. 3.84b, there was a maximum peak associated with the 1982–1983 El Nino events; such a peak is an indication of the passage of the Equatorial Kelvin waves. However, other strong ENSO events do not seem to induce such a large peak in the thickness of this

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141

Fig. 3.78 Time evolution of potential temperature, salinity, potential density and potential spicity at an equatorial station for the upper ocean

isopycnal layer; hence, in the stratified ocean, the Kelvin waves associated with individual ENSO events may pass through the water column at different density range. In order to determine climate variability, one needs to use some climatological data, as discussed above. Accordingly, climate study based on the Lagrangian coordinate must be combined with data collected in the Eulerian coordinate. Our understanding of the interplay of these two different coordinates still remains preliminary.

A common problem in demonstrating climate variability, such as Figs. 3.83 and 3.84, is that the variability of each variable has different units. For example, temperature is in units of °C, salinity in the units of psu, pressure in the units of db and potential spicity in the units of kg/m3. Results of climate study would be much easier to understand, if we convert all variabilities in the same density unit. There are two important issues in presenting the complete picture of climate variability. First,

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 Fig. 3.79 Time evolution of the reference potential density and pressure for isopycnal r0 ¼ 25 kg/m3

Fig. 3.80 (h, S) properties: a ðhins ; Sins Þ identified from the isopycnal surface; b the corresponding value ðhref ; Sref Þ from the climatological profile

all relevant information should be converted into variables based on the same unit; second, all three terms, the heaving, stretching and spicing should be included. As discussed above, the heaving term is defined as drHeaving ¼
 rins  rref  rins  rref , the stretching term is

defined as drStretching ¼ dDrLag  dDrLag , and the spicing term is defined as dp ¼ pins  pins . As shown in Fig. 3.85, both heaving and stretching terms dominate the climate variability, while the spicing term consists of a very small part of climate variability.

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143


 Fig. 3.81 (h, S) diagram for a Lagrangian observer on isopycnal surface of r0 ¼ 25 kg/m3 : a climate signals ðhins  href ; Sins  Sref Þ; b signals left behind after isopycnal analysis ðhins  hins ; Sins  Sins Þ


 Fig. 3.82 r–p diagram for climate signals collected by a Lagrangian observer on isopycnal surface r0 ¼ 25 kg/m3 ;
 the red circles indicate the radius of signal. a Climate signals r0;ins  r0;ref  r0;ins  r0;ref ; p0;ins  p0;ref 
 p0;ins  p0;ref ; b spicity signals left behind after isopycnal analysis ðrins  rins ; pins  pins Þ

It is readily seen that the variability associated with heaving and stretching are the dominating components of climate variability; on the other hand, the spicing terms is only a rather small fraction of the total climate variability. This

figure shows that heaving and stretching are closely linked to major ENSO events, such as 1982–1983, 1987–1988, 1992–1993, 1997– 1998; afterward there are no major peaks, and this may be due to the decadal changes of the

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 Fig. 3.83 Time evolution of temperature and salinity identified from isopycnal r0 ¼ 25 kg/m3

equatorial dynamics, such as the new types of ENSO events. On the other hand, the spicing term does not seem to be linked to the strong ENSO events, and this suggests that spicity anomaly may be regulated by mechanisms different from that producing strong ENSO events. Although the stretching signal is an important part of the signals, it is not well defined because it is linearly proportional to the initial thickness

of the layer. Accordingly, to characterize the importance of climate variability one may focus on the heaving contribution. It is also important to include the contribution due to spicity as well; thus, instead of using the potential density signals alone, the spicity signals should be included in the evaluation of the state. Therefore, the radius of signal before and after isopycnal analysis can be used as a good index for climate variability as follows.

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145


 Fig. 3.84 Climate signals for a Lagrangian observer on a fixed isopycnal surface r0 ¼ 25 kg/m3 : a the depth of the
 isopycnal surface; b the thickness of an isopycnal layer with r0 ¼ 25  1 kg/m3 ; c mean potential spicity of this layer

The radius of signal before and after the isopycnal analysis is R0s and R1s , defined as 
 ; R0s ¼ Rs r0; ins  r
0;ref  r0;ins   r0;ref p0;ins  p0;ref  p0;ins  p0;ref


R1s ¼ Rs p0;ins  p0;ins



RSðr0;ins  r0;ref Þ
 ¼ RMS r0;ins  r0;ref  r0;ins  r0;ref

ð3:73Þ ð3:74Þ

From this pair of variable we define the heaving R0

ratio,HR ¼ R0 þs R1  100%. For the present case, s

The RS (radius of signal) for potential density variability:

s

the mean heaving ratio is 91.3% (Fig. 3.86); accordingly, the dominating part of the climate signal is heaving in nature. The other way is to compare the contributions from the heaving and non-heaving terms, omitting the contribution from the stretching term as follows.

ð3:75aÞ The RS for potential spicity: RSðp0;ins  p0;ref Þ
 ¼ RMS p0;ins  p0;ref  p0;ins  p0;ref ð3:75bÞ The RS for the combination of potential density and potential spicity:

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Fig. 3.85 Climate signals include three terms, all in the same density unit: the heaving term ðdrins Þ, the stretching term ðdDrÞ, and the spicing term ðdpÞ

Fig. 3.86 Heaving ratio diagnosed from the upper water column of an equatorial station, based on the GODAS data, subjected to 7-month smoothing

3.4 Isopycnal Analysis

h RSðr0 ; p0 Þ ¼ RSðr0;ins  r0;ref Þ2 i1=2 þ RSðp0;ins  p0;ref Þ2

147

ð3:75cÞ

The RS for the in situ potential spicity (signals left behind after isopycnal analysis):
 RSðp0;ins Þ ¼ RMS p0;ins  p0;ins

ð3:75dÞ

This set of formula will be used for analyzing compositions of signals identified from isopycnal layer analysis in Chap. 5. For a given isopycnal surface, these terms can provide useful information regarding the horizontal distribution of heaving signals and the non-heaving signals left behind after isopycnal layer analysis based on Lagrangian coordinate.

3.4.2 Isopycnal Analysis in the Eulerian Coordinate The climate variability analysis based on the Eulerian coordinate is the method used by an observer siting at a fixed depth of a station. The climatological mean potential density profile at this station is depicted by the black curve in

Fig. 3.87 Sketch of isopycnal analysis at one station in the Eulerian coordinate

Fig. 3.87. The procedure of isopycnal analysis is as follows. (a) At a given time, the instantaneous profiles of temperature, salinity and potential density are depicted by the solid blue curve in Fig. 3.87. At the selected fixed pressure (depth) level, p = pins, using the sea level as the reference level pr ¼ 0, the corresponding potential density is rins ¼ rðSins ; Tins ; pins ; 0Þ, and depicted by the red triangle. (b) From this value of potential density rins we can search for radi ¼ rins along the vertical red line and find the corresponding pressure, salinity and (potential) temperature ðSadi ; Tadi ; padi Þ on the climatological mean profiles, depicted by the black triangle. (c) Thus, the water parcel with property ðSadi ; Tadi ; padi Þ may adiabatically move from pressure padi to pressure pins. During the movement, water properties might be changed slightly due to either mixing or climate change (including lateral movement of water parcels). (d) Using this isopycnal surface rins ¼ radi , the original climate signals ðSins ; Tins ; pins ; pins Þ are decomposed into two components: the isopycnal signals ðSadi ; Tadi ; padi Þ and the residual signals of ðSins  Sadi ; Tins  Tadi ; pins  padi Þ.

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(e) Similarly, one can diagnose the stretching term by diagnosing the corresponding density at the upper/lower interface of a column of fixed pressure range, denoted as rupper Eul;ins

heaving and stretching terms; however, the addition of this contribution makes the meaning of stretching somewhat vague. As an example, our calculation here is focused on the 135 m level, corresponding to pressure of 135.8 db. The potential density signal, r0;ins , at this level is taken from the original data. In order to find the climate signals we need to find the corresponding water mass from the climatological profile. Based on the climatological profile (the solid vertical red line in Fig. 3.87), the potential density radi ¼ rins is at pressure level padi . Over 37 years the corresponding adiabatic potential density and pressure varied greatly (Fig. 3.88). The corresponding temperature and salinity signals at the corresponding pressure level of padi are shown in the left panels of Fig. 3.89. Using the isopycnal analysis, the original signals are decomposed into two components. After subtracting the signals associated with isopycnal movement (labeled with subscript “adi”), the residual signals are shown in the right panels in Fig. 3.89. These strong climate signals in both T and S are clearly linked to the strong ENSO like events, such as the 1982–1983, 1987–1988, 1992–1993,

and rlower Eul;ins . Therefore, the density difference between the upper and lower layer interfaces lower is dDrEul ¼ rupper Eul;ins  rEul;ins (f) The climate variability associated with the heaving, stretching and spicing terms can be estimated as follows: First, the heaving term is defined as drHeaving ¼ rins  rins

ð3:76Þ

Second, the climate signal associated with isopycnal stretching is defined as drStretching ¼ dDrEular  dDrEuler

ð3:77Þ

Third, the climate signal due to residual signals after mapping is defined as dpSpicing ¼ pins  padi  pins  padi

ð3:78Þ

Note that we may try to include the contribution associated with changes of spicity in the

Heaving, Stretching, Spicing and Isopycnal Analysis

Fig. 3.88 Time evolution of the adiabatic potential density and pressure at pressure 135.8 db

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Fig. 3.89 Time evolution of temperature and salinity identified at pressure of 135.8 db: a potential temperature hadi , b potential temperature difference hins  hadi , c salinity Sadi , d salinity difference Sins  Sadi

and 1997–1998 events. At the beginning of this century the climate variability is noticeably reduced, and this phenomenon may reflect the changes of the air-sea interaction associated with the equatorial dynamics. The efficiency of the signal decomposition can be clearly seen if we plot the water mass properties in the (h, S) diagram (Fig. 3.90). On the left panel, the original signals ðhins ; Sins Þ are scattered in the h–S diagram; after removing the

isopycnal component, the residual signals ðhins  hadi ; Sins  Sadi Þ are mostly compressed onto a single line in the h–S diagram. Actually, it is much better to use the potential density and potential spicity (r0–p0) diagram. The in situ data ðr0;ins ; Dp0 ¼ p0;ins  p0;ins Þ are scattered around in the r0–p0 diagram, with a radius of signal of 0.7912 kg/m3 (Fig. 3.91). However, after using isopycnal analysis to separate the signals, the residual signal left behind is only the

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Fig. 3.90 h–S diagram for data collected by an observer sitting on a fixed pressure level of 135.8 db; a original signals; b signals left behind after isopycnal analysis

Fig. 3.91 r–p diagram for data collected by an observer sitting on a fixed pressure level of 135.8 db; the red circles indicate the radius of signal. a Total signals; b residual signals left behind after isopycnal analysis

spicity signals at a fixed potential density (which appears as a horizontal line in the r0–p0 plane),, with a much smaller radius of signal of 0.0824 kg/m3 (Fig. 3.91b). By definition, the meaning of the radius of signal is much more clear in the r0–p0 diagram. Note that the scattering in the potential spicity axis is also substantially

reduced after the isopycnal analysis; in fact, it is reduced from 2.5 to 0.45 kg/m3 (Fig. 3.91b). The stretching term cannot be exactly defined because it depends on the mean thickness of the layer of our concern. For an infinitely thin layer, this term is linearly proportional to the initial thickness of the layer; however, as the initial

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Fig. 3.92 Contribution of the heaving, stretching and spicing terms, subjected to 7-month moving smoothing

layer thickness increases gradually, there is a limit for this term. For this reason, we try to search for the layer with maximum RMS layer density changes. For the present case, we put the upper interface at 85 m and the lower interface at 185 m. The contribution due to the heaving, stretching and spicing terms are shown in Fig. 3.92, and the relative contribution of these terms are 49.5%, 43.3% and 7.2%, respectively. It is clear seen that both the heaving term and the stretching term dominate the climate variability, while the spicing term consists of less than 10% of the climate variability in this case. Although the relative contribution to climate variability may change from location to location, it is speculated that for most parts of the upper ocean in the world, climate variability is primarily associated with heaving and stretching, while spicing is a less important term in general. It is quite useful to introduce a signal ratio based on the radius of signal. As discussed above, climate signals can be separated into the heaving, stretching and spicing terms, and for most cases heaving/stretching signals are the dominating component of the climate signals.

The stretching term is not well defined because it is linearly proportional to the initial thickness of the layer. Thus, to characterize the importance of climate variability one may focus on the heaving contribution. It is also important to include the contribution due to spicity as well; therefore, instead of using the potential density signals alone, the spicity signals should be included in the evaluation of the state, i.e., it is more complete to use the radius of signal before and after isothermal analysis as follows. The radii of signal before and after the isopycnal analysis are R0s and R1s , defined as   R0s ¼ Rs r0;ins  r0;ins ; p0;ins  p0;ins

ð3:79Þ

R1s ¼ Rs ðpins  padi  pins  padi Þ

ð3:80Þ

From this pair of variable we define the R0

heaving ratio, HR ¼ R0 þs R1  100%. For the pres

s

sent case, the mean heaving ratio is 90.6% (Fig. 3.93); accordingly, the dominating part of the climate signal is heaving in nature. An advantage of using the Eulerian coordinate is that we can compare different climate variability indexes at the same location. For example,

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Fig. 3.93 Heaving ratio diagnosed from the upper water column of an equatorial station, based on the GODAS data, subjected to 7-month smoothing

we can compare available potential energy and virtual potential energy associated with climate variability, and they are defined as follows. ape ¼ g

ðrins  rins Þ2 ðpins  pins Þ2 ; vpe ¼ g q0 dqh =dz q0 dqh =dz ð3:81Þ

Applying these formulas, the corresponding terms for this station are shown in Fig. 3.94. These two terms are of the same magnitude for the present case. In general, they might be different. The connection between these two indexes remains to be explored. In this section, we examine how to apply isopycnal analysis for climate data in the ocean. In terms of isopycnal analysis, both the Eulerian and Lagrangian coordinates can be used, and original data can be separated into the isopycnal component and the spicity component. Our results indicate that a large part of the climate signal in the ocean is in forms of heaving signals, and the spicity signals left behind consist of a relatively small portion of the original signals.

Therefore, isopycnal analysis can substantially reduce the scattering of water mass properties in either the h–S diagram or the r–p diagram.

3.4.3 Isothermal Analysis Based on the long-term observations at one station, the climatological mean profile h ¼ hð pÞ; S ¼ Sð pÞ at this station can be defined, black curve in Fig. 3.95. In addition, the climatological profiles of potential density and potential spicity are defined accordingly. Imagine that a specially designed instrument is deployed in the ocean, and it moves along a fixed isothermal surface. Hence, the only data recorded by the instrument are salinity and depth (pressure) because (potential) temperature is kept fixed by the design of the instrument. Such signals are incomplete for climate study; in order to find the complete information necessary for the study of climate variability one must try to find some reference from the background climate profiles. Similar to the case discussed above there

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153

Fig. 3.94 Ape and vpe for this grid point, subjected to 7-month moving smoothing

Fig. 3.95 Sketch of isothermal analysis at one station in the Lagrangian coordinate

are at least two approaches: We can use the Eulerian coordinate or the Lagrangian coordinates. Our discussion here is focused on the Lagrangian coordinate, and the basic variables are functions of (x, y,h). Accordingly, we can analyze the climate signals as follows. (a) For a given time, the vertical profile of water property ðSðpÞ; hðpÞ; rðpÞÞ can be calculated, as shown by the blue curve in Fig. 3.95. For

the observer sitting on a specific isothermal surface hi the following data can be collected: the corresponding pressure, potential temperature and salinity ðSins ; Tins ; pins ; rins ; pins Þ. (b) From this value of potential temperature had ¼ hins ¼ hi , we search along the dashed vertical red line and find the corresponding pressure, salinity and potential temperature ðSad ; had ; pad Þ on the climatological mean

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profiles, depicted by the black triangle. This gives rise to the vertical pressure difference pins  pad . Since a Lagrangian observer is sitting on the same isothermal layer, the climate variability collected is ðSins  Sad ; hins ¼ had ; pins  pad Þ. Note that in this way, no climate signals related to the temperature variability are included. Missing such an important part of climate variability is undesirable. (c) One way to recover the relevant information is to use the pressure anomaly pins  pad . This means we are tracing the climatological mean potential temperature profile over the pressure difference of pins  pad , i.e., we will trace the climatological mean profiles from the black triangle to the red circle at pins level. Accordingly, we define a reference
 state Sref ; href ; pref ¼ pins ; rref ; pref . With the water property defined on the climatological mean profiles, the climate signals are
Sins  Sref ; hins  href ; pins  pad ; rins  rref ; pins  pref Þ, where rins  rref is the density anomaly, which can be used to represent the heaving term. (d) The stretching term is defined as follows. As long as the layer is not too thick, this term is linearly proportional to the initial layer thickness. In the following analysis we will set the layer thickness as 5 °C. The stretching term is defined as the density difference between the upper layer and lower layer interfaces. (e) The corresponding radius of signal must be defined in the r–p plane. Before the isothermal analysis, radius of signals in the original data is defined as

By definition, in the Lagrangian coordinate the water parcel properties observed on a specific isothermal surface are ðSins ; pins; rins ; pins Þ. Of course, this is not complete information for climate variability. We must use some additional information to make up the loss of information related to climate variability due to the constraint that data are collected on this fixed isothermal surface. As discussed above, we can use the (T, S) properties inferred from the climatological mean profile and identified from the same pressure level as the reference values ðhref ; Sref Þ. This method is applied to the 37-year GODAS data at an equatorial station (179.5° W, 0.167° S). With the addition of the data inferred from climate data, we arrive at the modified data ðhins  href ; Sins  Sref Þ, which are labeled as the climate signals shown in the left panel of Fig. 3.96. The original data collected by the observer sitting on the isothermal ðhins ; Sins Þ are really the climate perturbations, and we plot ðhins  hins ; Sins  Sins Þ in Fig. 3.96b. As discussed above, (T, S) data collected on the isothermal surface must have a constant temperature; thus, the only signals left behind are the salinity signals, Fig. 3.96b. The power of isothermal analysis can also be illustrated in terms of the (r0–p0) diagram, Fig. 3.97. As shown in this figure, the original data collected at a fixed isothermal level should be modified by subtracting the corresponding data on the climatological mean profile ref ðr0;ins  rref 0 ; p0;ins  p0 Þ, which are plotted in Fig. 3.97a (subtracting the mean value). The original data collected by the observer sitting on the isothermal ðr0;ins ; p0;ins Þ is really the climate perturbations. Note that the scattering in the potential spicity axis is also substantially reduced after the isothermal analysis; in fact, it is reduced from 2.5 kg/m3 in panel a to 0.2 kg/m3 in panel b. The radius of signal is substantially reduced from 0.6029 to 0.0406 kg/m3, as shown in Fig. 3.97. The time evolution of temperature and salinity identified from isothermal h ¼ 18  C is shown in Fig. 3.98.

R0s ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2ffi RMSðdr  drÞ þ RMSðdp  dpÞ

ð3:82Þ where dr ¼ rins  rref dp ¼ pins  pref . The radius of signal after isothermal analysis is defined as R1s ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fRMS½rins  rins g2 þ fRMS½pins  pins g2

ð3:83Þ

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155

Fig. 3.96 h–S diagram for climate signals collected by a Lagrangian observer for isothermal surface of h ¼ 18  C; a Climate signals ðhins  href ; Sins  Sref Þ; b signals left behind after isothermal analysis ðhins  hins ; Sins  Sins Þ

Fig. 3.97 r–p diagram
for climate signals for a Lagrangian isothermal surface of h ¼ 18  C: a climate 
observer tracing  signals r0;ins  r0;ref  r0;ins  r0;ref , p0;ins  p0;ref  p0;ins  p0;ref ; b spicity signals left behind after isothermal analysis (rins  rins ; pins  pins )

Note that for an observer sitting at a fixed depth at a station, the climate signals consist of the instantaneous potential temperature, salinity, and potential density. In addition, the observer can also infer the isothermal layer thickness.

On the other hand, for an observer following the isothermal movement, the value of potential temperature is always the same as the prespecified value. The real climate signals the observer can report are the time evolution of the depth of the isothermal surface and the mean

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Fig. 3.98 Time evolution of potential temperature and salinity identified from isothermal h ¼ 18  C

salinity in this layer. In addition, the observer may measure thickness of an isothermal layer, as shown in Fig. 3.99b. The problems associated with the information presented in Fig. 3.99 are that these curves are in different units, so that it is hard to compare their relative contribution. The heaving term is defined as drHeaing ¼ rins  rref , the stretching term is defined as DrStretch Isothermal ¼ rref ;upper  rref ;lower , and the spicing term is defined as dp ¼ pins  pins . The results

are shown in Fig. 3.100. This shows clearly that both heaving and stretching terms dominate the climate variability, while the spicing term consists of a very small part of the climate variability. As discussed above, climate signals can be separated into the heaving, stretching and spicing terms, and in most cases the heaving/stretching signals are the dominating component of the climate signals. The stretching term is not well defined because it is linearly proportional to the initial thickness of the layer. Therefore, to

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157

Fig. 3.99 Climate signals for an Lagrangian observer on a fixed isothermal surface h ¼ 18  C: a the depth of the isothermal surface; b the thickness of an isothermal layer with h ¼ 18  0:5  C; c salinity; d potential density; e potential spicity of this layer

Fig. 3.100 Heaving, stretching, and spicing signals diagnosed from the upper water column of an equatorial station, based on the GODAS data, subjected to 7-month smoothing

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Fig. 3.101 Heaving ratio diagnosed from the upper water column of an equatorial station, based on the GODAS data, subjected to 7-month moving smoothing

characterize the importance of climate variability one may focus on the heaving contribution. It is also important to include the contribution due to spicity; thus, instead of using the potential density signals alone, the spicity signals should be included in the evaluation of the state, i.e., it is more complete to use the radius of signal before and after isothermal analysis as follows. The radii of signal before and after the isothermal analysis are R0s and R1s , defined as h
 R0s ¼ Rs r0;ins  r0;ref  r0;ins  r0;ref ;
i p0;ins  p0;ref  p0;ins  p0;ref ð3:84Þ R1s

¼ Rs ðrins  rins ; pins  pins Þ

ð3:85Þ

From this pair of variables we define the heaving ratio, HR ¼

R0s R0s þ R1s

 100%. For the present case,

the mean heaving ratio is 94.2%, Fig. 3.101; accordingly, the dominant part of the climate signal is heaving in nature. The other way is to compare the contributions from the heaving and non-heaving terms,

omitting the contribution from the stretching term as follows. The RS (Radius of signal) for potential density variability: RSðr0;ins  r0;ref Þ
 ¼ RMS r0;ins  r0;ref  r0;ins  r0;ref ð3:86aÞ The RS for potential spicity: RSðp0;ins  p0;ref Þ ¼
 RMS p0;ins  p0;ref  p0;ins  p0;ref

ð3:86bÞ

The RS for the combination of potential density and potential spicity: RSðr0 ; p0 Þ h i1=2 ¼ RSðr0;ins  r0;ref Þ2 þ RSðp0;ins  p0;ref Þ2 ð3:86cÞ The RS for the in situ potential density and spicity (signals left behind after isothermal analysis):

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RSðr0;ins ; p0;ins Þ h
2
2 i1=2 ¼ RMS r0;ins  r0;ins þ RMS p0;ins  p0;ins

ð3:86dÞ This set of formula will be used for analyzing compositions of signals identified from isothermal layer analysis in Chap. 6. For a given isothermal surface, these terms can provide useful information regarding the horizontal distribution of heaving signals and the non-heaving signals left behind after isothermal layer analysis based on Lagrangian coordinate.

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Huang RX (2011) Defining the potential spicity. J Mar Res 69:545–559. https://doi.org/10.1029/2018JC014 306 Huang RX (2015) Heaving modes in the world oceans. Clim Dyn. https://doi.org/10.1007/s00382-015-2557-6 Huang RX, Pedlosky J (1999) Climate variability inferred from a layered model of the ventilated thermocline. J. Phys. Oceanogr 29:779–790 Huang RX, Yu LS, Zhou SQ (2018) New definition of potential spicity by the least square method. J Geophys Res Ocean 123:7351–7365. https://doi.org/10.1029/ 2018JC014306 Jackett DR, McDougall TJ (1985) An oceanographic variable for the characterization of intrusions and water masses. Deep-Sea Res 32:1195–1207 Jones DC, Ito T, Lovenduski NS (2011) The transient response of the Southern Ocean pycnocline to changing atmospheric winds. Geophys Res Lett 38:L15604. https://doi.org/10.1029/2011GL048145 Mamayev OI (1975) Temperature-salinity analysis of world ocean waters. Elsevier Scientific Publishing Company, Amsterdam, p 384 McDougall TJ, Barker PM (2011) Getting started with TEOS-10 and the Gibbs Seawater (GSW) Oceanographic Toolbox, 28 pp, SCOR/IAPSO WG127, ISBN 978-0-646-55621-5 McDougall TJ, Krzysik OA (2015) Spiciness. J Mar Res 73(5):141–152 Munk W (1981) Internal waves and small-scale processes. In: Evolution of physical oceanography. MIT Press, Cambridge, pp. 264–291 Qiu B, Huang RX (1995) Ventilation of the North Atlantic and North Pacific: Subduction Versus Obduction. J Phys Oceanogr 25:2374–2390 Ruddick B (1983) A practical indicator of the stability of the water column to double-diffusive activity. DeepSea Res 30:1105–1107 Samelson RM (2011) Time-dependent adjustment in a simple model of the mid-depth meridional overturning cell. J Phys Oceanogr 41:1009–1025 Schmitt RW (1981) Form of the temperature-salinity relationship in the central water: evidence for doublediffusive mixing. J Phys Oceanogr 11:1015–1026 Schmitt RW, Ledwell JR, Montgomery ET, Polzin KL, Tool JM (2005) Enhanced diapycnal mixing by salt fingers in the thermocline of the tropical Atlantic. Science 308:685–688 Shibley N, Timmermans ML, Carpenter JR, Toole J (2016) Spatial variability of the Arctic Ocean’s double-diffusive staircase. J Geophys Res 122:980– 994. https://doi.org/10.1002/2016JC012419 Stommel H (1962) On the cause of the temperaturesalinity curve in the ocean. Proc Natl Acad Sci 48:764–766 Timmermans ML, Jayne SR (2016) The Arctic Ocean Spicies Up. J Phys Ocean 46:1277–1284. https://doi. org/10.1175/JPO-D-16-0027.1

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Unesco (1981) The Practical Salinity Scale 1978 and the International Equation of State of Seawater 1980. Unesco technical papers in marine science 36, 25 pp Unesco (1983) Algorithms for computation of fundamental properties of seawater. Unesco technical papers in marine science 44, 53 pp Veronis G (1972) On properties of seawater defined by temperature, salinity and pressure. J Mar Res 30:227– 255 Wang GH, Xie SP, Qu TD, Huang RX (2011) Deep South China Sea circulation. Geophys Res Lett 38:L05601. https://doi.org/10.1029/2010GL046626

Worthington LV (1981) The water masses of the world ocean: some results of a fine-scale census. In Warren BA, Wunsch C (eds) Evolution of Physical Oceanography, MIT Press, Cambridge, MA, pp 42–69 You Y-Z (2002) A global ocean climatological atlas of the Turner angle: implications for double-diffusion and water-mass structure. Deep-Sea Res(I) 49:2075– 2093

4

Heaving Modes in the World Oceans

4.1

Heaving Induced by Wind Stress Anomaly

4.1.1 Introduction Circulation in the world oceans is a complicated system, and it involves the transport of water masses, heat and freshwater in the three dimensional space. To describe such complicated phenomena, many two-dimensional maps, onedimensional profiles or zero-dimensional indexes have been used. For example, to describe the transport of water mass and heat, the meridional overturning circulation (MOC), the meridional heat flux (MHF), the zonal overturning circulation (ZOC), the zonal heat flux (ZHF) and the vertical heat flux (VHF) have been widely used in previous studies. The thermohaline circulation plays a crucial role in regulating the MOC and MHF; in addition, the quasi-horizontal circulation associated with wind-driven gyres in a stratified ocean also play a role in transporting heat poleward, as discussed in many previous studies, e.g., Schmitz (1996a, b), Ganachaud and Wunsch (2000), Talley et al. (2011), Talley (2013) and Huang (2010). Our discussion in this chapter is focused on the change and transport of heat content in the ocean. The same method can be used to analyze the change and transport of salt (or freshwater) content in the ocean, and the transport of salt is closely related to the hydrological cycle in the ocean.

In a steady state, the MHF is intimately linked to heat transport through the air-sea interface or between water parcels. As such, MHF is directly linked to the diabatic processes in the ocean. The basic pattern of air-sea heat flux and the implied horizontal heat fluxes in the world oceans is shown in Fig. 4.1, based on the annual mean climatology of the GODAS data (Behringer et al. 1998). As shown in Fig. 4.1, the net air-sea heat flux in the world oceans is non-uniformly distributed. The most outstanding features are the strong heat flux into the cold tongues in the equatorial Pacific and Atlantic Oceans, plus the strong heat release to the atmosphere in the Kuroshio, the Gulf Stream and the Labrador Sea. This map implies that ocean currents transport strong heat fluxes in both the meridional and zonal directions. The corresponding strong MHF and ZHF are shown in Fig. 4.1b, c. The MHF in the Northern Hemisphere reaches the maximum value of 1.5 PW (1 PW = 1015 W). Because the world oceans are separated into three major basins, except the periodic channel between 40° S and 70° S, we calculate the ZHF for each basin. The ZHF for each basin is defined as zero at the east most oceanic grid of each basin, and the eastward heat flux is defined as positive. As shown in Fig. 4.1c, in the steady state, there is a huge ZHF in each basin. In both the Pacific and Atlantic Oceans, the ZHF reaches the maximum amplitude of −1.0 and −0.40 PW respectively; whereas the ZHF is positive in the

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_4

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Fig. 4.1 Annual mean net air-sea heat flux climatology based on the GODAS data (a), the heavy dashed lines indicate the boundaries separating three basins; the MHF (b) and the ZHF (c)

middle of the Indian Ocean, with a maximum value of 0.22 PW. The east-west asymmetric feature of the net air-sea heat flux and the implied ZHF is a critically important component in the climate system. In the steady state, both the MHF and ZHF are closely linked to thermal diffusion across the airsea interface or between water parcels; in other words, these fluxes are closely linked to diabatic processes in the ocean. However, in a transient state, both the MHF and the ZHF may vary due to other dynamical or thermodynamic processes. As discussed above, perturbations induced by wind stress anomalies are mostly in the form of adiabatic motions, without the exchange of heat/salt across the interfaces; these motions are called adiabatic heaving. Wind stress changes take place in many different parts of the world oceans, and the

corresponding adjustments of the wind driven circulation evolve with time in rather complicated ways. As a result, it is a great challenge to identify the individual adjustment processes and their sources from climate datasets for the world oceans. The goal of this chapter is to use simple reduced gravity models to catch the major consequences induced by the anomalies in wind stress, heat flux and freshwater flux. A reduced gravity model has one active layer only, and it can be used as a tool in exploring the vertical movement of the main thermocline; thus, the model can reveal the heaving modes in the ocean. However, such a one-active layer model cannot reveal the compression or stretching of the layers in the subsurface ocean; as a result, the model is incapable of exploring the stretching modes in the ocean interior. In order to explore the stretching modes in the ocean interior, a

4.1 Heaving Induced by Wind Stress Anomaly

model with at least two active layers is required. Such a model is beyond the scope of this book, and is left to the reader to explore. Recently, a hiatus in global warming has become a hotly debated issue related to global climate changes. Over the past several decades, the global sea surface temperature (SST) kept increasing; in addition, there is a decadal variability of global ocean heat content, e.g., Levitus et al. (2005, 2009, 2012), Watanabe et al. (2013). However, ocean warming is far from uniform. For example, in the North Atlantic, the tropics/subtropics have been warmed, but the subpolar ocean has been cooled. These changes can be linked to NAO, and they are also directly linked to the gyre scale circulation changes and the MOC (Lozier et al. 2008, 2010). In particular, over the past 10–15 years, the change in global SST has more or less leveled off, i.e., the socalled hiatus in the global SST record; on the other hand, the subsurface heat content keeps increasing, at a seemingly even higher rate, e.g., Meehl et al. (2011), Balmaseda et al. (2013), Kosaka and Xie (2013) and Chen and Tung (2014). As discussed in previous studies, the hiatus of global SST may be due to many mechanisms. However, our focus is to explore the possible mechanism directly linked to the oceanic circulation, in particular the linkage to variability in the wind-driven circulation. In fact, early studies suggested that this may be an important connection, e.g., McGregor et al. (2012), Jones et al. (2011), England et al. (2014). As will be explained in this chapter, a hiatus of the global SST record in combination with accelerating subsurface heat content increase may be explained in terms of the general trend of the warming of the whole water column and the stratification changes induced by the adjustment of the wind-driven circulation in response to global decadal wind stress perturbations. We will use idealized geometry and simple wind stress perturbations and focus on the dynamical consequence of the large-scale adjustment of wind-driven gyres (Huang 2015). One of the most important consequences of such adjustment is the basin scale quasi-horizontal

163

transport of water mass. We assume that such movements take place within a relatively short time, on the order of inter-annual and decadal time scales. Neglecting the contributions due to the surface thermohaline forcing and internal diapycnal diffusion, such movements can be idealized as adiabatic, and they are commonly called heaving. Heaving can induce changes in the basin mean vertical stratification and the corresponding mean vertical heat content profile. The basic idea illustrated in Fig. 2.34 can be extended to a model ocean with multiple gyres. For an idealized two-hemisphere basin subjected to zonal wind stress s, there are several winddriven gyres, including the subpolar gyres, the subtropical gyres, and the equatorial gyres; the fundamental structure of the thermocline in a quasi-steady state is sketched by black curves in Fig. 4.2. Assuming the amount of warm water in the upper ocean remains unchanged, during the adjustment of the wind-driven circulation the warm water from one gyre is redistributed to other gyres in the model ocean. For example, if the equatorial easterlies relax, the equatorial thermocline moves up in response. The upward movement of the equatorial thermocline leads to the poleward transportation of warm water in the upper ocean (red arrows in Fig. 4.2a). A fundamental assumption made in reduced gravity models is that the lower layer is very thick; consequently, the pressure gradient is very small and the velocity is small and negligible. This assumption leads to the simple formulation of pressure gradient in the upper layer. However, a small velocity multiplied by a very large layer thickness can lead to a finite amount of transport in the lower layer. In the situation discussed above, since the total water column height at each station is nearly constant, movement of the upper layer in the model implies that cold water in the lower layer should move toward the Equator (blue arrows) to fill up the space left behind by the poleward transport of warm water in the upper layer. Thus, the movement of water in the upper and lower layers implies three important physical processes, depicted in Fig. 4.2. First, the poleward flow in the upper layer and the equatorward

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Heaving Modes in the World Oceans

Fig. 4.2 Symmetric/asymmetric heaving modes induced by wind stress changes in a two-hemisphere basin. Black curves depict the undisturbed thermocline ðs ¼ s0 Þ and black arrows indicate vertical movements of thermocline; red curves indicate the disturbed thermocline ðs ¼ s0 þ DsÞ and red arrows indicate the movement of warm water above the thermocline; blue arrows indicate the movement of cold water below the thermocline (Huang 2015)

flow in the lower layer give rise to an anomalous MOC and MHF. Second, there is an anomalous ZOC/ZHF, that is not included in the twodimensional sketch in Fig. 4.2; but it is shown in the three dimensional sketch in Fig. 2.34. Third, stratification in the ocean is changed due to such exchanges. Since the wind-driven circulation is considered as adiabatic, the total heat content for the whole ocean must be constant. Consequently, the basin mean vertical heat content anomaly must appear in the form of baroclinic modes. In the simple case illustrated in Fig. 2.34, heaving motions can induce a first baroclinic mode; for general cases involving multiple wind-driven gyres, the situation becomes more complex, and second or even higher baroclinic modes can be generated as well. The second example is for a case with wind stress forcing change in the subtropical basin of the Northern Hemisphere (Fig. 4.2b). If Ekman pumping in the subtropical basin is reduced, the thermocline in the subtropical basin shoals; this

leads to the warm water transport toward both high and low latitudes, depicted by the red arrows. In compensation, cold water in the lower layer flows toward the subtropical basin of the Northern Hemisphere, depicted by the blue arrows. Similarly, if wind forcing in the subpolar basin in the Northern Hemisphere is enhanced, the cyclonic gyration is intensified, and the thermocline moves upward. The cyclonic gyre can no longer hold up the same amount of warm water; hence, warm water is transported from the northern subpolar basin southward to the rest of the basin. As a result, the thermocline in other parts of the basin deepens in response (Fig. 4.2c). In addition, wind stress in both hemispheres can change simultaneously. Such changes can be idealized in terms of the symmetric and asymmetric modes induced by symmetric and asymmetric wind stress perturbations, sketched in Fig. 4.2d, e.

4.1 Heaving Induced by Wind Stress Anomaly

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4.1.2 A Two-Hemisphere Model Ocean

ðhvÞt þ fhu ¼ g0 hhy þ sy =q0  jv þ Am r2 ðhvÞ ð4:5Þ

4.1.2.1 Model Set Up With the focus on the wind-driven circulation mostly confined above the thermocline, the density structure in the ocean can be idealized as a step function in the density coordinate. This two-hemisphere (labeled as 2H hereafter) model ocean consists of two layers: the upper (lower) layer has a constant density of q0 ðq0 þ DqÞ; the upper layer thickness is denoted as h, and the lower layer is infinitely thick and motionless. The model is based on the rigid-lid approximation. As such, the effect of free surface f 6¼ 0 is replaced by a non-constant hydrostatic pressure p ¼ pa at the flat surface z ¼ 0. The pressure in layers beneath can be calculated using the hydrostatic relation. The pressure gradient for a reduced gravity model is rp ¼ g0 rh

ð4:1Þ

where g0 ¼ gDq=q0 is the reduced gravity. In approximation, the sea surface level is linked to the upper layer thickness rf ¼ g0 rh=g

ð4:2Þ

Hence, the sea level can be inferred from the thermocline depth, with an unknown constant determined by requiring the basin-integrated sea surface level to be zero, i.e.: ZZ fdxdy ¼ 0

ð4:3Þ

A

Since the model has one active layer only, the time dependent momentum and continuity equations for the active upper layer are ðhuÞt  fhv ¼ g0 hhx þ sx =q0  ju þ Am r2 ðhuÞ ð4:4Þ

ht þ ðhuÞx þ ðhvÞy ¼ 0

ð4:6Þ

where ðsx ; sy Þ are the zonal and meridional wind stress, j ¼ 0:0015 m/s and Am ¼ 1:5  104 m2 =s are the vertical/horizontal momentum dissipation coefficients. An interfacial friction linearly proportional to the velocity shear is used, and this form of friction can be interpreted as a crude parameterization of baroclinic instability. The model ocean is formed on an equatorial beta-plane, with the Coriolis parameter defined as the linear function of latitude f ¼ by

ð4:7Þ

The 2H model is 150° wide in the zonal direction, and extends from 70° S to 70° N. Following the common practice of non-eddy resolving modeling, the model is based on the B-grid, with 152  142 grids and 1°  1° resolution, with grid size is set to 110 km. In this model, we chose b ¼ 2:2367  1011 =m/s, so that for a grid point at 35° N the Coriolis parameter f35 N ¼ 8:36552  105 =s is the same as the Coriolis parameter in the spherical coordinates. In this section, we will be focused on the role of zonal wind stress only. The undisturbed zonal wind stress profile (in N/m2) applied for the 2H model is sk ¼ 0:02  0:08 sinð6j/jÞ  0:05n ½1  tanhðh10j/jÞ io p  0:05 1  tanh 10  j/j ð4:8Þ 2 where / is the latitude. The numerical model is based on the traditional leap-frog scheme with a time step of 3153.6 s for both the momentum equations and the continuity equation. The detail of this numerical model can be found in Huang

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(1987). To prevent the upper layer outcropping in the subpolar basin, a relatively large layer depth of 350 m is specified in the initial state of rest. According to the WOA09 data (Antonov et al. 2010), the annual mean potential density referred to the sea level pressure, averaged over a depth of 0–300 m for the global oceans, is estimated at r0 ¼ 25:96 kg=m3 , but the corresponding value over depth of 400–5500 m is estimated at r0 ¼ 27:66 kg/m3 ; thus, g0  0:017 m/s2 . Hence a round off value of g0 ¼ 0:015 m/s2 is used in this model. The annual mean potential temperature averaged over depth of 0–300 m is 13.36 ° C, and the corresponding value over depth of 400–5500 m is 2.555 °C; as a result, the temperature difference between the upper and lower layers is 10.8 °C. Hence, in this model the temperature difference is set at 10 °C. The corresponding temperature in the upper and lower layers is set to Tupper ¼ 15  C and Tlower ¼ 5  C respectively, and these values will be used in calculating the MHF and ZHF. The vertical heat content profile is calculated during the numerical integration of the model as follows. The model is separated into many horizontal sub-layers, each of 10 m thick; the corresponding interfaces are dðkÞ ¼ 10; 20; 30; 40; . . .ðmÞ

ð4:9Þ

Heaving Modes in the World Oceans

The corresponding heat content of each sub-layer is calculated as hcðkÞ ¼ Jc

X

Thi;j dA

ð4:10Þ

i;j

 Thi;j ¼

Tupper d ðkÞ; if hði; jÞ  d ðkÞ Tupper d ðkÞ þ Tlower ½hði; jÞ  d ðkÞ; if hði; jÞ\d ðkÞ

ð4:11Þ where dA = (110000)2 m2 is the horizontal area of each grid box (on a beta-plane), Jc is the constant factor for converting into heat content in units of Joules. The model was run for 300 years to reach a quasi-equilibrium reference state. This reference state is symmetric to the Equator, and it has three gyres in each hemisphere. The wind stress, the thermocline depth and the streamfunction of the reference state are shown in Fig. 4.3. To explain the vertical profile of the heat content anomaly, we also plot a meridional profile of the thermocline depth at 7 grid points (770 km) east of the western boundary; this profile can represent the layer depth maximum or minimum as a function of latitude, Fig. 4.3b. The relatively wide western boundary layer is due to the low horizontal resolution and large frictional parameters used in the model. In the subtropical basin, the maximum depth of the thermocline of the anticyclonic gyre is approximately 603 m, mimicking the situation in

Fig. 4.3 The reference state: wind stress (a), thermocline depth near the western boundary (b), maps of thermocline depth (c) and streamfunction (d) (Huang 2015)

4.1 Heaving Induced by Wind Stress Anomaly

the North Pacific Ocean. The strength of the subtropical gyre is about 24 Sv, somewhat weaker than in the North Pacific Ocean. This relatively weak gyre is owing to the idealized wind stress profile used in the model. In the subpolar basin, there is a weak cyclonic gyre, giving rise to a dome-shaped thermocline. Our main focus is to explore the fundamental structure induced by decadal variation of small amplitude wind stress perturbations; accordingly, the choice of the model parameters and wind stress profile should not qualitatively affect the main results from this model.

4.1.2.2 Zonal Wind Stress Variability in the World Oceans Wind stress in the world oceans varies over a broad spectrum in space and time. As an example, zonal wind stress in the central Pacific Ocean taken from the GODAS data is shown in Fig. 4.4. On decadal time scales the zonal wind stress in the central Pacific varies with amplitude on the order of 0.015–0.03 N/m2(Fig. 4.4b, c). Thus, it is reasonable to use decadal wind stress perturbations on the order to 0.015–0.02 N/m2 in numerical experiments exploring the dynamical consequences of heaving. 4.1.2.3 Numerical Experiments From the reference state, a series of numerical experiments was carried out. In each experiment,

167

the model was restarted from the reference state and forced by wind stress with small perturbations in the form of a Gaussian profile: 0

2

sx ¼ Dse½ðyy0 Þ=Dy

ð4:12Þ

where Ds ¼ 0:015 N/m2 is the amplitude, Dy ¼ 1100 km, y0 is the central latitude of the perturbations (Fig. 4.5). Note that because the amplitude of the wind stress anomaly is relatively small, perturbations to the wind-driven circulation are almost linearly proportional to the amplitude (including the sign) of wind stress anomaly; accordingly, the corresponding results for wind stress perturbations with opposite signs can be inferred from results presented in this section. In each experiment, wind stress perturbations were linearly increased from 0 at t = 0 to the specified strength at the end of 20 years; afterwards, the wind stress perturbations were kept constant and the model run for an additional 20 years. Such numerical experiments may represent typical cases for climate variability induced by decadal wind stress perturbations.

4.1.2.4 The Pivotal Case, Exp. 2H-A In this experiment, the equatorial easterlies are enhanced. The adjustment of the wind-driven circulation induces a three dimensional redistribution of warm water in the model ocean. The

Fig. 4.4 Zonal wind stress in the central Pacific Ocean (averaged over 140° E–140° W and after 61 month moving smoothing) (a) and its deviation from the 34-year mean (b); zonal wind stress difference (2011–1996) averaged over 180°–140° W (c); all based on the GODAS data (Huang 2015)

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Heaving Modes in the World Oceans

Fig. 4.5 Wind stress perturbations used in numerical experiments 2H-A, 2H-B, 2H-C, 2H-D, 2H-E, in units of 0.01 N/m2

Fig. 4.6 Exp. 2H-A: Time evolution of the meridional distribution of the volume anomaly (a), the MOC (c) and vertical distribution of the heat content anomaly (e); the corresponding profiles at the end of 40 year run are shown in (b, f), and the 40-year mean MOC (MHF) and VHF profiles are shown in (d, g) (Huang 2015)

time evolution of the volume anomaly and heat content anomaly is shown in Fig. 4.6. Because of the intensification of the equatorial easterlies, the slope of the equatorial thermocline is enhanced; warm water above the thermocline is pushed toward the Equator from both hemispheres. Consequently, the amount of warm water in the equatorial band increases with time, but it declines at middle/high latitudes in both hemispheres (Fig. 4.6a). At the end of the 40year experiment, the meridional distribution of the volumetric anomaly is shown in Fig. 4.6b. In the reduced gravity model, the lower layer is assumed to be infinitely thick; hence, the pressure gradient and velocity in this layer is negligible. In the ocean the poleward mass flux in the upper layer must be compensated by the equatorward return flow in the lower layer; thus,

the adjustment of wind-driven gyre in the upper layer should induce an anomalous MOC. The equivalent MOC can be diagnosed as follows Zxe Mmoc ðy; tÞ ¼

hðx; y; tÞvðx; y; tÞdx

ð4:13Þ

xw

where xw and xe are the western and eastern boundaries, v is the meridional velocity. By convention, the MOC induced by a northward flow of warm water in the upper layer is defined as positive. As shown in Fig. 4.6c, the wind-driven circulation adjustment induces a pair of anomalous MOC asymmetric to the Equator, with a maximum rate of more than 0.3 Sv at year 20, when the wind stress perturbations reach the peak amplitude. Afterward, the anomalous MOC

4.1 Heaving Induced by Wind Stress Anomaly

169

declines gradually. If the model were run for much longer, the model ocean would gradually reach a new quasi-equilibrium state, in which the anomalous MOC vanishes. Since the MOC varies so much during the adjustment, the MOC rate averaged over the entire 40 years is used, and the corresponding meridional profile, with maximum amplitude of 0.2 Sv, is shown in Fig. 4.6d. The MOC in the ocean is primarily associated with the surface thermohaline forcing and the thermohaline circulation; in a steady state the wind-driven circulation in combination of surface heating/cooling can also contribute to the MOC. Recent studies revealed a close link between the MOC and the surface thermohaline and wind forcing. For example, the NAO cycle plays a vital role in generating the variability of the MOC in the Atlantic Ocean, e.g., Lozier et al. (2010), Zhai et al. (2014). However, a major point here is that during the adiabatic adjustment of the wind-driven circulation the anomalous MOC appears, and this is not directly linked to the surface thermohaline forcing anomaly. Our numerical experiments indicated that, even taking the value averaged over the entire 40 years, the MOC associated with adiabatic adjustment of the wind-driven circulation may consist of a substantial portion of the variable MOC in the world oceans. The anomalous MOC inferred from the model also gives rise to the anomalous MHF defined as Zxe P ðy; tÞ ¼

qo Cp DT hðx; y; tÞvðx; y; tÞdx xw

ð4:14Þ where qo ¼ 1035 kg/m3 is the mean reference density, Cp ¼ 4186 J/kg= C is the mean heat capacity under constant pressure, DT ¼ 10  C is the temperature difference between the upper and lower layers in the model ocean. Due to the anomalous MOCs, there is a negative MHF in the Northern Hemisphere and a positive MHF in the Southern Hemisphere. In our simple model

the pattern of MOCs and MHFs is the same; the mean value averaged over the 40 years of the whole experiment is nearly 8.7 TW (1 TW = 1012 W) (Fig. 4.6d). The warm water is redistributed in the vertical direction as well, and the heat content anomaly (in units of J/m) in the model is defined as ZZ

  Tðx; y; z; tÞ  Tref ðx; y; zÞ dxdy

DHC ðz; tÞ ¼ Cp q0 A

ð4:15Þ where Tðx; y; z; tÞ is the instantaneous temperature and Tref ðx; y; zÞ is the temperature in the reference state. Since the model has one moving layer only, temperature T is set to the upper/lower layer temperature, i.e.  Tupper ; if z [ hðx; y; tÞ T¼ . The time Tlower ; if z\hðx; y; tÞ evolution of the heat content anomaly is shown in Fig. 4.6e, and the heat content anomaly at year 40 is shown in Fig. 4.6f. In the reference state, the thermocline near the western boundary at low latitudes is the deepest, approximately 603 m, Fig. 4.3b. Hence, positive thermocline perturbations at low latitudes induce a positive anomaly of the basin-mean heat content at this depth range. On the other hand, the thermocline along the eastern boundary and at high latitudes shoals, leading to a negative heat content anomaly at shallow depths, shown in the upper part of Fig. 4.6f. Since these motions are adiabatic, heat content anomalies must appear in the form of baroclinic modes. Taking the time derivative of the heat content anomaly gives rise to the mean heat content anomaly rate dDHC ¼ ½DHC ðz; T Þ  DHC ðz; 0Þ=T dt

ð4:16Þ

With T = 40 year, this gives rise to the mean rate (per meter) averaged over the 40-year experiment. The vertical integration of this time rate gives rise to the VHF

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4

Z0 V ðzÞ ¼

dDHC dz dt

ð4:17Þ

z

As shown in Fig. 4.6g, for the present case the VHF is on the order of −8 TW (the negative sign indicates a downward shifting of heat content). Dividing by the total area of the model ocean leads to the VHF per unit area, on the order of 0.03 W/m2. It is important to emphasize that such VHF is due to the adiabatic adjustment of the water mass in the ocean; if we look at the heat content profile through the one-dimensional potential temperature coordinate, there is no change at all. The rate of warming in the upper ocean has been discussed in many recent publications, e.g., Lyman and Johnson (2008), Lyman et al. (2010), Abraham et al. (2013), Chen and Tung (2014). Because of the relatively spare data coverage, the rate of warming remains uncertain and for the following discussion we will use the mean rate reported in the comprehensive review by Abraham et al. (2013). According to their analysis, over the period of 1970–2012, the planetary heat storage in the upper 700 m is 0:27  0:04 W/m2 , equivalent to an increase of heat content in the

Heaving Modes in the World Oceans

global upper ocean of 1:9  1023 J (or 27  1019 J/ m) or a temperature change of 0.2 ° C. This is a mean barotropic mode of warming, and this can be used as a benchmark. It is readily seen that the magnitude of the baroclinic modes of the heat content anomaly obtained from this set of experiments is about 1/3 of the mean global warming rate inferred from observations. At the end of the experiment, the horizontal structure of the perturbations to the wind-driven circulation is shown in Fig. 4.7. The equatorial thermocline deepens, in particular 10° off the Equator and near the western boundary, with maximum amplitude of 15 m (Fig. 4.7a); on the other hand, the thermocline shoals at high latitudes, with a maximum value of −9.8 m. Since dissipation along the eastern boundary is relatively low, the thermocline thickness perturbation is nearly constant along the entire eastern boundary. In this experiment, the layer thickness along the eastern boundary varies within the range of −4.65 to −5.15 m (right edge of Fig. 4.7a). This confirms the argument in Fig. 2. 34, i.e., in response to changes in the equatorial wind stress, the thermocline along the entire basin moves upward nearly uniformly. Since the wind stress and Ekman pumping at off equatorial

Fig. 4.7 Exp. 2H-A: Perturbations at the end of a 40-year experiment: a the thermocline depth, b the streamfunction anomaly and c sea level anomaly (Huang 2015)

4.1 Heaving Induced by Wind Stress Anomaly

latitudes remains unchanged, the thermocline at middle and higher latitudes shoals, pushing warm water in the upper ocean equatorward. The streamfunction anomaly appears as a pair of gyres asymmetric to the Equator, with maximum amplitude of 2 Sv. The streamfunction of the anomalous circulation in the Northern Hemisphere is positive, i.e., it is an anomalous anticyclonic circulation. As a result, the original anticyclonic circulation, including its western boundary current, is intensified, Fig. 4.7b. It is well known that the adjustment of the wind-driven circulation in a closed basin is carried out through wave motions. In particular, Kelvin waves and long baroclinic Rossby waves play vital roles in establishing the circulation, e.g., Anderson and Gill (1975), Wajsowicz and Gill (1986); Hsieh and Bryan (1996), Marshall and Johnson (2013). Furthermore, for simplified geometry, the adjustment of the global ocean has been studied by Huang et al. (2000), Primeau (2002), and Cessi et al. (2004). In the present case, the wind forcing anomaly along the Equator and low latitude band induces an enhancement of the east-west slope of the thermocline at low latitudes. The changes in thermocline depth move westward in the form of long Rossby waves; these signals reach the western boundary and form coast-trapped Kelvin waves, and these waves move toward the Equator and then propagate eastward along the Equator. After reaching the eastern boundary, these waves reflect and bifurcate into the poleward propagating waves. Although the waves move along the eastern boundary with a speed close to Kelvin waves, recent studies suggested that such waves should be interpreted in terms of long cyclonic Rossby waves, e.g., Marshall and Johnson (2013). On their poleward propagation path along the eastern boundary, these waves gradually shed their energy and mass, forming the westward baroclinic Rossby waves carrying the signals through the ocean interior. Due to the negative wind stress anomaly applied to the Equator and low latitude band, the thermocline slope anomaly in this region is positive. The sharp reduction of layer thickness

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perturbations near the western boundary indicates that the northward transport of the western boundary current at this latitude is enhanced. On the other hand, the thermocline anomaly is negative at high latitudes, Fig. 4.7a. Within the framework of the reduced gravity model, the corresponding sea level anomaly in the final state can be inferred from the upper layer depth anomaly Dh, Eq. (4.2). The sea level anomaly at the end of the experiment has the same pattern as the thermocline thickness anomaly. For the present case, a negative equatorial wind anomaly induces a positive sea level anomaly in the western part of the basin at low latitudes, and a negative sea level anomaly in the eastern part of the basin. Observations, e.g., Merrifield (2011), Qiu and Chen (2012), indicate that the sea level anomaly pattern in the Pacific Ocean over the past 10–20 years has a basin scale structure similar to that shown in Fig. 4.7c. Figure 4.8 shows the sea level anomaly over two time periods for the equatorial-subtropical Pacific Ocean with the corresponding zonal wind stress anomaly, based on the GODAS data. The pattern of sea level anomaly averaged over the period of 2003–2010 is basically the same as that shown in Fig. 4.7c. Since the adjustment in response to wind stress change takes time to complete, the corresponding zonal wind stress anomaly (lower panels) is calculated for the same time period, but with a two-year shifting. The patterns shown in Figs. 4.7c and 4.8a are quite similar to the linear tread in sea level from satellite observations over the period 1993–2010 (Qiu and Chen 2012). For the period of 2008–2015 zonal wind stress anomaly in the eastern (western) equatorial Pacific is positive (negative); the strong positive sea level anomaly was greatly reduced and became negative for part of the basin (Fig. 4.8b). Hence, the strong positive (negative) sea level anomaly in the western (eastern) North Pacific may be linked to the stronger than normal easterlies in the Equatorial Pacific. However, when the strong anomalous easterlies are relaxed, the sea level anomaly pattern in the Pacific Ocean may swing in the opposite direction.

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Heaving Modes in the World Oceans

Fig. 4.8 Sea level anomaly averaged over two time periods (upper panels) and the corresponding zonal wind stress over the equatorial band (5.16° S–5.16° N) averaged over the same time period plus a two-year time shifting, based on the GODAS data

4.1.2.5 Other Cases with Wind Stress Perturbations at One Latitudinal Band In Exp. 2H-B, a positive wind stress anomaly is applied to the Equator. Since wind stress perturbations applied to these experiments have a

small amplitude, results from Exp. 2H-B are very close to those in Exp. 2H-A, but with opposite signs. For example, the volume anomaly is now negative for the equatorial band, and it is positive at middle and high latitudes (solid black curve in Fig. 4.9a).

Fig. 4.9 Exp. 2H-B and 2H-C: Perturbations at the end of 40 year experiments: the meridional distribution of the volume anomaly (a) and the vertical profiles of the heat content anomaly (c); the mean profile averaged over the 40 year experiments: the MOC (MHF) (b) the VHF (d)

4.1 Heaving Induced by Wind Stress Anomaly

In Exp. 2H-C, a positive wind stress anomaly is applied to the 40° N band, and it creates a negative volume anomaly at a latitudinal band from 35° N to 60° N (dashed black curve in Fig. 4.9a). Note that the volumetric anomaly created in this case is much larger than in the previous two cases; in addition, there are other stronger anomalous features. In compensation, at other latitudes the volumetric anomaly is positive. The corresponding southward transport of warm water in the upper layer creates a southward MOC in Exp. 2H-C, with the maximum amplitude near the latitude of wind stress anomaly maximum (Fig. 4.9b). The MOC associated with the thermohaline circulation in such a two-hemisphere model basin is likely to be on the order of 10 Sv; thus, the amplitude of perturbations in Exp. 2H-C is about a few percentages of the climatological mean MOC. The anomalous MOC gives rise to a sizeable MHF in the basin (Fig. 4.9b). In particular, in Exp. 2H-C, the maximum southward heat flux is about 27.5 TW, and such a flux is a sizeable fraction of the heat flux variability for such a model basin. The heat content anomaly in the vertical direction in these cases appears in the form of first baroclinic modes, as shown in Fig. 4.9c. In Exp. 2H-B, the heat content anomaly is positive above the depth of 420 m, but it is negative below this depth. In Exp. 2H-C, the zerocrossing of the first baroclinic mode is moved upward to the depth of 390 m. In Exp. 2H-C the warm water at latitudes higher than 35° N is pushed toward low latitudes (dashed black curve in Fig. 4.9a). As a result, the thermocline at low latitudes deepens, so that heat content anomaly is positive below 390 m, but it is negative in the upper ocean (dashed black curve in Fig. 4.9c). Note that in terms of the vertical heat content anomaly, over most of the depth the perturbations created in Exp. 2H-C are of opposite signs, compared with 2H-B. It is clear that the shape of the heat content anomaly created by wind stress perturbations is the result of the delicate competition between the zonal mean thermocline depth and the wind stress perturbations applied to the model ocean.

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The strong heat content anomaly in the vertical direction corresponds to an equivalent VHF. As shown in Fig. 4.9d, the VHF averaged over 40 years in Exp. 2H-B is positive, indicating an upward shifting of heat content; the VHF is on the order of 8 TW. On the other hand, the VHF in Exp. 2H-C is −11 TW (−0.04 W/m2), and it is about 1/7 of the mean warming rate of 0:27  0:04 W/m2 estimated by Abraham et al. (2013). As schematically shown in Fig. 2.34, changes in the zonal wind stress can directly induce the zonal shifting of warm water in the upper layer and thus give rise to anomalous ZOC (Tai et al. 2015). Similar to the MOC discussed above, the zonal transport of warm water in the upper layer of a reduced gravity model leads to an equivalent anomalous ZOC defined as Zyn MZOC ðx; tÞ ¼

uðx; y; tÞhðx; y; tÞdy

ð4:18Þ

ys

where ys and yn are the southern and northern boundaries, and u is the zonal velocity. The ZOC associated with an eastward flow in the upper layer is defined as positive. Since the ZOC is associated with the zonal transport of warm water in the upper layer and the compensating return flow of cold water in the lower layer, there is a critically important quantity associated with the ZOC: the ZHF, that is defined as Zyn HZOC ðx; tÞ ¼

q0 Cp DTuðx; y; tÞhðx; y; tÞdy ys

ð4:19Þ Because all these three parameters are constant, the ZHF profile is the same as that of the ZOC, except for a different unit. The corresponding time evolution of the zonal volumetric anomaly and the corresponding ZOC in Exp. 2H-A is shown in Fig. 4.10. For the first 20 years when the amplitude of wind stress anomaly is increased, warm water is pushed

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Fig. 4.10 Time evolution of the zonal distribution of the volumetric anomaly (a) and corresponding profile at the end of the simulation (b) and the ZOC (c) and the mean ZOC(ZHF) averaged over 40 years (d) in Exp. 2H-A

toward the western boundary (Fig. 4.10a); the associated ZOC is negative, with a peak value of −0.6 Sv. As soon as the wind stress reaches its target value and increases no more, the ZOC becomes positive, indicating that the warm water partially returns to the eastern basin. Nevertheless, the mean maximum value of the ZOC averaged for 40 years is approximately −0.28 Sv, and the mean ZHF is about −8 TW.

4.1.2.6 Cases with Asymmetric/Symmetric Wind Stress Perturbations In this set of experiments, wind stress perturbations symmetric/asymmetric to the Equator are applied to two latitudinal bands, labeled as Exp. 2H-D and 2H-E (Fig. 4.5). If wind stress perturbations are symmetric to the Equator, the thermocline depth perturbations in the North and South Hemispheres are symmetric. When wind stress perturbations are applied to 40° N/40° S, the warm water volume is reduced at the latitude bands around 45° N/45° S, and warm water is pushed to other latitudes (solid black curve in Fig. 4.11a).

In Exp. 2H-E, asymmetric wind stress perturbations are applied to 40° N/40° S, the warm water volumetric anomaly is negative (positive) around the 45° N(45° S) latitude band (dashed black curve in Fig. 4.11a). The meridional transport of warm water induces the transient MOC and MHF, in Exp. 2H-D (Exp. 2H-E) it is asymmetric (symmetric) to the Equator, Fig. 4.11b. In Exp. 2H-D, the maximum anomalous MOC is 0.5 Sv, and MHF is 20 TW; in Exp. 2H-E, the maximum anomalous MOC is 0.8 S) and MHF is 32 TW. The combination of symmetric wind stress perturbations at two latitude bands induces a baroclinic mode of the heat content anomaly in the vertical direction, and the amplitude of this baroclinic mode is larger than the cases with a single wind perturbation band discussed above (Fig. 4.11c). These heat content anomalies correspond to a relatively large VHF (Fig. 4.11d). When asymmetric wind stress perturbations are applied (Exp. 2H-E), the zonal distribution of the volumetric anomaly and the ZOC are one order of magnitude smaller than those in Exp. 2H-D, as shown by the dashed black curves in Fig. 4.12.

4.1 Heaving Induced by Wind Stress Anomaly

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Fig. 4.11 Exp. 2H-D and 2H-E: Perturbations at the end of 40 year experiments: the meridional distribution of the volumetric anomaly (a) and vertical profiles of the heat content anomaly (c); the mean profile averaged over the 40 year experiments: the MOC (MHF) (b) the VHF (d)

Fig. 4.12 Zonal volume anomaly at the end of the simulation (a) and the mean rate ZOC averaged over 40 years (b) in Exp. 2H-D and 2H-E

4.1.3 A Southern Hemisphere Model Ocean Wind-driven circulation in the Southern Hemisphere has very special features because all subbasins in the Southern Hemisphere are linked through the Antarctic Circumpolar Currents (ACC). The existence of this periodic channel gives rise to some unique features of the heaving modes. As sketched in Fig. 4.13, in addition to the inter-gyre modes for a single basin discussed above, there are two new types of basic modes

unique to the Southern Ocean, including the annular modes and the inter-basin modes. There are two types of wind stress anomaly: anomaly in the Southern Ocean, that can generate annular modes; anomaly in individual basins at low latitudes, that can generate inter-basin modes. First, we assume that zonal wind stress in the Southern Ocean changes within certain latitudinal bands. For example, if zonal wind stress over the ACC is intensified, the slope of the front in the ACC increases and the thermocline shoals.

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4 Equator

Ind

Pac

At l

Heaving Modes in the World Oceans Ind

Pac

At l

Annular modes Wind stress anomaly in the Southern Ocean Equator Ind

Pac

At l

ACC

ACC

(b) Homogeneous mode

(c) Inhomogeneous mode

Equator ACC

Ind

Pac

At l

Ind

Pac

At l

(a) Basic state Inter-basin modes Wind stress anomaly in individual basins

ACC (d) Pacifc mode

ACC (e) Atlantic mode

Fig. 4.13 Sketch of the basic heaving modes in the Southern Hemisphere. The thin solid frames depict three subbasins and the annular channel; heavy solid horizontal lines indicate the basin-mean depth of the thermocline; red dashed lines indicate the position of the mean thermocline after adjustment; arrows depict the movement of the basinmean thermocline; dashed lines indicate the adjusted levels induced by wind stress perturbations

Note that further increase in wind stress may lead to the eddy saturation and stoppage of thermocline shoaling, e.g., Hallberg and Gnanadesikan (2006); however, such phenomena is beyond the scope of the low resolution reduced gravity model discussed here. As a result, the mean depth of the thermocline in the ACC declines, depicted as the change from the solid line to the dashed red line (Fig. 4.13b). Warm water in the ACC band is pushed toward the low latitudes, leading to slightly deeper thermocline in three sub-basins, depicted by the dashed red lines. A concrete example is as follows. We assume that the zonal wind near the southern boundary of the model ocean is intensified (red curve in Fig. 4.14). In response to wind stress change, the front in the ACC moves northward and becomes steeper. The change of the thermocline shape in the ACC pushes warm water in the upper ocean from the ACC toward the low latitudes. As a result, warm water volume in the ACC declines, but it increases in the middle and low latitudes (red curve in Fig. 4.14b). As discussed above, the meridional movement of warm water in the upper layer implies that

there must be a compensating return flow below the thermocline, and thus an equivalent MOC in the model ocean. Note that the MOC is zero at the beginning and end of the transition state associated with wind stress perturbations, and it is non-zero only during the transition state of the model. Therefore, we will use the rate averaged over the whole transition period to evaluate the equivalent MOC during the adjustment, the red curve in Fig. 4.14c. As discussed above, the wind stress change induced MOC also carries an equivalent MHF, that contributes to global climate change. Another important consequence of wind stress induced adjustment is the change of the vertical stratification or the heat content profile. Due to the intensification of the westerlies over the ACC, warm water is transported from the ACC to low latitudes. Since the thermocline is deep at low latitudes and shallow at high latitudes, such movement of warm water in the upper ocean induces a negative volume anomaly at shallow depths and a positive volume anomaly at the deep level (the red curve in Fig. 4.14d). The shifting of the heat content in the vertical

4.1 Heaving Induced by Wind Stress Anomaly

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Fig. 4.14 Examples illustrating the consequence of heaving in the Southern Ocean: a wind stress perturbations; b the meridional distribution of the volume anomaly; c the 120 year mean MOC (MHF); d the vertical distribution of the heat content anomaly; e the 120 year mean VHF (Huang 2015)

direction implies a VHF, and the corresponding profile averaged over the adjustment is shown in Fig. 4.14e. If the zonal wind stress is weakened, the opposite processes should take place, as depicted by the blue curves in Fig. 4.14. Since the amplitude of wind stress perturbations is relatively small, the anomalies produced have nearly the same patterns, but with opposite signs. This simple model provides insight for the cooling in the deep ocean discussed in Chap. 2. For a reduced gravity model, when the westerlies over the ACC are reduced (the case depicted by blue curves in Fig. 4.14), warm water above the main thermocline moves southward toward ACC. As a result, warm water is pushed downward in the ACC band. In compensation, cold water below the thermocline is pushed away from the ACC band. This implies that north of ACC cold water below the thermocline moves upward. As such, the deep ocean is cooled down. If we choose a very deep interface as the initial condition, or a multiple layer reduced gravity model, weakening of the Southern westerlies should give rise to cooling of the ocean near the sea floor away from the ACC band. Hence, such models can provide simple and intuitive insight for the cooling of deep layers diagnosed from the GODAS data, as discussed in Sect. 2.5. In the case discussed above, wind stress anomaly has a constant value over the whole

latitudinal circle, so that thermocline adjustment is nearly uniform in the Southern Ocean, and the corresponding adjustment in individual basins has similar amplitude; accordingly, we call this a homogeneous annular mode. However, wind stress anomaly may not be uniform over the latitudinal circle. For example, if a negative zonal wind stress anomaly is confined to the Pacific sector of the ACC, then the adjustment of the main thermocline in the Southern Ocean is depicted as the red dashed line in Fig. 4.13c. Main thermocline in each sub-basin moves up in response; however, such response is not the same in all basins. As shown in Fig. 4.13c, the perturbation in the Pacific basin is larger than that in the Indian and Atlantic basins. Accordingly, the adjustment of thermocline in this case belongs to the inhomogeneous annular mode. Second, we assume that zonal wind stress in individual sub-basins changes along certain latitudinal bands. For example, if zonal wind stress in the Pacific basin is weakened, the thermocline shoals, moving from the solid line to the dashed red line (Fig. 4.13d). Assuming the total volume of warm water in the upper layer of the world oceans remains unchanged, the thermocline in both the Indian and Atlantic basins should move downward in compensation, as shown by the dashed red lines in this panel. The thermocline in the ACC band may move slightly, but such change is excluded in this sketch. Similarly, if

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zonal wind stress in the Atlantic basin is weakened, the thermocline in the Atlantic basin moves upward, but thermocline in the Indian and Pacific basins moves downward in compensation. These cases are called the inter-basin modes.

4.1.3.1 Model Set Up Using the same basic equations in Sect. 4.1.2, a second model is formulated for an idealized Southern Hemisphere ocean (SH model hereafter). The model basin includes a 360° wide channel model subjected to the periodic conditions, and it extends from 60° S to the Equator. The northern part of the model ocean is divided into three sub-basins, separated by three continents. The Indian and Atlantic basins are 60° wide, and the Pacific basin is 150° wide; each continent is 30° wide in longitude (Fig. 4.15). The southern part of the model ocean is occupied by a 15° wide periodic channel, corresponding to the ACC. The model is also an equatorial beta-plane model, with b ¼ 2:1  1011 =m/s the corresponding Coriolis parameter matches that in the spherical coordinates at 45° S, the northern edge of the periodic channel. The other parameters of the model are: Am ¼ 1:5  104 m2 =s, j ¼ 0:005 m/s. For numerical stability, a rather high interfacial friction parameter is used for the SH model. To capture the deep thermocline in the Southern Ocean a relatively large amount of

4

Heaving Modes in the World Oceans

warm water is specified in the initial state, and the corresponding mean layer depth is 750 m. The reduced gravity model used here makes use of observations. According to the WOA09 data, the annual mean potential density, averaged over depth of 0–700 m for the global oceans, is estimated at r0 ¼ 26:61 kg/m3 , but the corresponding value over depth of 800–5500 m is estimated at r0 ¼ 27:72 kg/m3 ; so, g0  0:0111 m/s2 . Hence a round off value of g0 ¼ 0:01 m/s2 will be used in this model. The annual mean potential temperature averaged over depth 0–700 m is 9.55 °C, and the corresponding value over depth 700–5500 m is 1.87 °C; hence the temperature difference between the upper and lower layers is 7.68 °C. Hence, in this model the temperature difference is set at 7.5 °C. The zonal wind stress profile (Fig. 4.15a) is taken from the mean zonal wind stress averaged over the 51 years of SODA 2.1.6 data (Carton and Giese 2008). As will be shown shortly, due to the selection of parameters, the model ocean can reach a quasi-equilibrium state within 100 years. Actually, the model was run for 300 years to reach a quasi-steady state, and this will be used as a reference state. The thermocline depth and the streamfunction of the reference state are shown in Fig. 4.15d. This reference state has three subtropical gyres north of the periodic channel. The thermocline thickness is nearly constant along all the all eastern boundaries of the model ocean. Along the southern edge of the model the

Fig. 4.15 The steady reference state of a reduced gravity model for the Southern Ocean: a zonal wind stress; b layer depth along the western boundaries; c layer depth distribution; d streamfunction (Huang 2015)

4.1 Heaving Induced by Wind Stress Anomaly

ocean thermocline shoals to a depth of less than 200 m. The thermocline is the deepest in the Pacific basin, reaching 956 m (red curve in Fig. 4.15b). The subtropical gyres in the three sub-basins are rather weak, with a maximum transport of 11 Sv (Indian basin), 16 Sv (Pacific basin) and 11 Sv (Atlantic basin); the transport of the modelled ACC is 26 Sv (Fig. 4.15d). It is clear that these values are much smaller than the corresponding values in the world oceans. For example, diagnosis based on climatological hydrographic data indicates that the thermocline depth is the deepest in the South Indian Ocean; on the other hand, the transport of ACC is on the order of 100 Sv. Such large differences between our model ocean and the world oceans are due to the fact that the model is formulated for a rather idealized geometry with uniform reduced gravity, and forced by a simple zonal wind stress that is zonally constant over the whole Southern Ocean. Since the goal of this section is to explore the fundamental structure of the heaving modes, the difference in thermocline depth (or volumetric transport) between the model and observations should not qualitatively affect the basic physical processes simulated in the model. From this reference state, we carried out a series of numerical experiments. In the first set of experiments, the zonal wind stress was perturbed along certain latitudinal bands; such wind stress perturbations were chosen to explore the annular modes sketched in Fig. 4.13b. In the second set of experiments, the zonal wind stress perturbations were confined to individual basins; this choice of wind stress perturbations was aimed to explore the inter-basin modes depicted in Fig. 4.13d, e.

4.1.3.2 Exp. SH-A, SH-B and SH-C In this set of experiments, the model was restarted from the reference state and forced by wind stress including additional small positive perturbations in zonal wind stress, as defined in Eq. (4.12). The wind stress perturbations were linearly increased from 0 at t = 0 to the specified strength at the end of 20 years. Afterward, the

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wind stress perturbations were kept constant and the model run for an additional 100 years. The reason for running the experiments for 120 years is as follows. Adjustment of the winddriven circulation is carried out by wave like perturbations, mostly the baroclinic long Rossby waves; these waves move quite slowly at high latitudes, typically on the order of a few centimeters per second. Due to the large interfacial friction imposed in the model, the adjustment time scale of the wind-driven circulation in our model ocean is primarily determined by the high latitude basin crossing time of first baroclinic long Rossby waves. The theoretical speed for the baroclinic long Rossby waves is c ¼ bg0 H=f 2

ð4:20Þ

The southern tips of the continents in the model ocean is 45° S, where the Coriolis parameter and beta in our beta-plane model is f ¼ 1:0395  104 =s, b ¼ 2:1  1011 =m/s. Since g0 ¼ 0:01 m/s2 , assuming H ¼ 800 m gives a typical wave speed of 0.0168 m/s at high latitudes. For a beta-plane model ocean of 360 grids with a zonal grid size of 110 km, the corresponding time for the first baroclinic long Rossby waves to travel through the southern tip of the South Pacific basin in the model is about 31 years, and the corresponding time for the waves to travel around the whole Southern Ocean in the model is around 75 years. Thus, running the model for additional 100 years after the wind stress perturbation reaches its final amplitude is long enough for the circulation to reach a quasi-equilibrium state. Since the solutions vary greatly with time, it is also meaningful to examine the final states of these three experiments at the end of the 120 year experiments. The meridional distribution of volume anomaly is shown in Fig. 4.16a. In Exp. SH-A (weakened equatorial easterlies), the volume anomaly is negative north of 30° S, but it is positive south of 30° S. The meridional shifting of warm water implies a MOC. Although the MOC averaged over the 120 year experiments is much smaller than its peak value at year

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Fig. 4.16 Experiment SH-A, SH-B, SH-C: a Meridional distribution of the volume anomaly at the end of a 120 year run; b the 120 year mean MOC (MHF); c the vertical distribution of the heat content anomaly; d the 120 year mean VHF (Huang 2015)

20, it is still quite sizeable. For Exp. SH-A, the MOC minimum (averaged over 120 year run) is −0.059 Sv, Fig. 4.16b. Due to the anomalous MOC, there is a MHF, with a mean value of nearly −1.8 TW averaged over the 120 years of the experiment. In comparison, the volume anomaly in Exp. SH-B (wind stress perturbations placed at 30° S band) is much larger and it is of an opposite sign (blue curve in Fig. 4.16a). The MOC averaged over 120 years reaches its peak of 0.19 Sv around 30° S. This MOC also carries a sizeable MHF with a peak value of 5.7 TW (blue curve in Fig. 4.16b). In Exp. SH-C (wind stress perturbations placed at 60° S band), there is a negative volume anomaly in the periodic channel, giving rise to a MOC (maximum value of 0.23 Sv) and MHF (maximum value of 7.4 TW) (red curve in Fig. 4.16b). The anomalous MOCs and MHFs diagnosed from these experiments are much smaller than the corresponding values inferred from mean values of the long term observations. For example, Talley (2013) put the estimate of the global overturning cell associated with bottom water formation in the world oceans at 29 Sv, and the associated MHF on the order of 0.1–0.2 PW. Nevertheless, these transient MOC and MHF suggest that adiabatic adjustment of the winddriven circulation may contribute to a substantial portion of variability in the MOC and MHF

diagnosed from observations or numerical simulations of the world oceans. The adjustment of the wind-driven circulation also induces the vertical redistribution of heat content. At the end of a 120-year experiment, all the heat content anomaly appears in the form of first baroclinic modes (Fig. 4.16c). As defined in Eq. (4.17), the basin-scale redistribution of heat content in the vertical direction implies a VHF (Fig. 4.16d). The baroclinic mode structure shown in Fig. 4.16c is for the profile averaged over the whole model ocean; but the heat content profile in each sub-basin can be quite different, depending on the spatial distribution of wind perturbations. Due to the assumption of adiabatic motions, the entire model basin mean heat content anomaly must appear in the form of baroclinic modes. For each sub-basin, however, the net heat content anomaly is non-zero owing to the inter-basin shifting of warm water. For this reason, the heat content anomaly in a sub-basin can contain a barotropic component. In Exp. SH-A, the heat content anomaly in both the Atlantic and Indian basins (blue and green curves in Fig. 4.17a) has a single positive lobe below 750 m; the heat content anomaly in the Pacific basin (black curve in Fig. 4.17a) is positive between 750 and 830 m, but it is negative below 840 m. The warm water from the deep part of the Pacific basin is pushed toward the ACC and piled up at shallower levels

4.1 Heaving Induced by Wind Stress Anomaly

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Fig. 4.17 Experiment SH-A (a), SH-B (b), SH-C (c): Global mean heat content anomaly in the vertical direction (at the end of a 120 year run) (Huang 2015)

(red curve in Fig. 4.17a). The heat content anomaly in the ACC is zero above 160 m, and it is positive below. Heat content anomaly above 750 m in all sub-basins is zero; hence the global heat content anomaly profile (magenta curve) overlaps with the heat content anomaly profile in the ACC. Below this depth, the global heat content profile is dominated by the contribution from the Pacific basin (black curve). The heat content profiles in each sub-basin have quite different baroclinic structures in the present case. These results demonstrate that heat content anomaly profiles induced by adiabatic motions of the wind-driven circulation in the Southern Ocean can have complex baroclinic structures. The heat content anomaly in Exp. SH-B is quite different from Exp. SH-A. The maximum depth (>950 m) of the thermocline in the reference state is located near 30° S, where wind stress perturbations can induce a large positive heat content anomaly in all three sub-basins around the depth of 900 m. On the other hand, the heat content anomaly is mostly negative above 760 m (Fig. 4.17b). Thus, the baroclinic structure of global heat content is in the form of a first baroclinic mode. If wind stress perturbations are applied to the latitude band around 20° S, the global heat content anomaly appears in the form of a third baroclinic mode.

In Exp. SH-C wind stress perturbations apply to the 60° S latitude band where the thermocline is much shallower. The transport of warm water from high latitudes to low latitudes creates a positive heat content anomaly below 800 m. Although the basin mean heat content anomaly is in the form of a first baroclinic mode, the heat content anomaly profiles in all three sub-basins have no zero-crossing, and they seem to be a combination of a barotropic mode and a second baroclinic mode, but the heat content anomaly in the ACC appears in a form close to a first baroclinic mode (Fig. 4.17c). The transported warm water is mostly originated from the ACC at much shallower levels. Hence, the heat content anomaly above 630 m in the ACC is negative, with a peak near the 210 m level, indicating that the southern edge of the ACC loses a lot of warm water at the shallow level (red curve, that is overlaid by the magenta curve in Fig. 4.17c). The combination of heat content anomaly from these four sub-basins creates a first baroclinic mode with shape peaks at the 200 and 890 m levels (magenta curve in Fig. 4.17c). The corresponding time evolution of the volume anomaly in each sub-basin is shown in Fig. 4.18. It is well known that the adjustment of the wind-driven circulation in a closed basin is carried out through wave motions, in particular

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Fig. 4.18 Experiment SH-A (left panels), SH-B (middle panels), SH-C (right panels): Upper panels: time evolution of the volume anomaly (in units of 1013 m3) in each sub-basin; lower panels: the volume anomaly ratio (instant/final); the red dashed lines indicate the 95% or 105% level of the final value (Huang 2015)

the first baroclinic long Rossby waves. As discussed above, the time scale of adjustment for the model ocean is estimated at 75 years, and the volume anomaly ratio (instantaneous volume anomaly /final volume anomaly) gradually reaches the final value of 1 unit after a 100-year integration (low panels of Fig. 4.18). Although the total amount of warm water at low/middle latitudes declines in general, the situation in each sub-basin is different. In Exp. SHA, the Pacific basin loses warm water quickly, but the Atlantic basin actually gains warm water slowly. The Indian basin also loses warm water during the first 20 years, but it starts to gain warm water afterward and ends up with more warm water at the end of the 120 year run (Fig. 4.18a). In this case, the wind forcing anomaly along the Equator and low latitudes leads to a decline of the east-west slope of the thermocline at low latitudes. These signals reach the western boundary and form the coast-trapped Kelvin waves moving toward the Equator. They propagate eastward along the Equator. After reaching

the eastern boundary, these waves reflect and bifurcate into the poleward propagating waves. On their poleward propagation paths along the eastern boundaries, these waves gradually shed their energy and mass, forming the westward long Rossby waves carrying the signal through the ocean interior, e.g., Huang et al. (2000). In this experiment, the Pacific Basin sets the pace of the adjustment because of its large size. Only after the adjustment of the Pacific basin is nearly complete, the final signals propagate downstream, i.e., eastward, leading to the completion of the adjustment in the Atlantic basin, and then finally the Indian basin. In fact, the warm water volume anomaly in the Indian basin reaches 95% of its final value (at year 120) only after 74.7 years. Note that the warm water volume anomaly in the Atlantic basin actually overshoots the final value reached at year 120 (lower part of column in Fig. 4.18a). The oscillations shown in model solutions discussed here are similar to the oscillatory solutions discussed in previous studies, e.g., Cessi and Paparella (2001), Cessi et al. (2004). Due to the selection

4.1 Heaving Induced by Wind Stress Anomaly

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of parameters, however, these oscillations are strongly damped. In Exp. SH-B, all sub-basins gain warm water in the final state, except the ACC that loses warm water. The corresponding time evolution of the vertical volume anomaly is shown in Fig. 4.18b. The total volume of warm water in the Pacific basin actually overshoots and then turns back to the final value around year 80 (lower part of Fig. 4.18b). In Exp. SH-C, warm water in the upper ocean in the ACC band is pushed northward; thus, the volume anomaly in the Indian, Pacific and Atlantic basins increases, but it is greatly reduced in the ACC (Fig. 4.18c). Since wind stress perturbations are applied to rather high latitudes, where there is no blockage of continents and the thermocline adjustment in all basins are mostly synchronized and nearly completed within 60 years and without much delay in individual basins, as shown in the previous cases.

4.1.3.3 Exp. SH-D, SH-E In these experiments, the model was restarted from the reference state and forced by wind stress with small perturbations 0

2

2

sx ¼ Dseððxx0 Þ=DxÞ ððyy0 Þ=DyÞ

ð4:21Þ

where Ds ¼ 0:02 N/m2 is the amplitude, Dx ¼ 3300 km, Dy ¼ 1100 km, ðx0 ; y0 Þ are the longitude and latitude of the center of wind stress

perturbations. The wind stress perturbations were linearly increased from 0 at t = 0 to the full scale of the specified strength at the end of 20 years. Afterward, the wind stress perturbations were kept constant and the model run for an additional 100 years. The aim of carrying out these experiments is to explore the inter-basin modes associated with warm water adjustment associated with wind stress perturbations are applied to individual basins. In Exp. SH-D, positive zonal wind stress perturbations are applied to the Indian Ocean sector of the equatorial band,   ðx0 ; y0 Þ ¼ ð60 E; 0 Þ; in Exp. SH-E wind stress perturbations are applied to the Pacific sector of the equatorial band, ðx0 ; y0 Þ ¼ ð180 ; 0 Þ. These wind stress perturbations induce warm water volume decline at low latitudes (with the peak near 10° S) and warm water increase at high latitudes (Fig. 4.19a). The pattern of changes in the circulation is similar in both cases; however, when wind stress perturbations are applied to the Pacific basin (Exp. SH-E), the changes in the circulation are much stronger, almost double the amplitude in Exp. SH-D. Such a difference is due to the much larger zonal dimension of the Pacific basin. The meridional transport of warm water implies an anomalous MOC and MHF. When wind stress perturbations are applied to the Pacific basin, the amplitude of the MOC (−0.017 Sv) and MHF (−0.52 TW) are much larger than

Fig. 4.19 Experiment SH-D and SH-E: a Meridional distribution of the volume anomaly at the end of a 120 year run; b the 120 year mean MOC (MHF); c vertical distribution of the heat content anomaly at the end of the 120 year run; d mean VHF at the end of the 120 year run (Huang 2015)

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in the case when wind stress perturbations are applied to the Indian basin with the corresponding value of −0.0073 Sv and −0.23 TW (Fig. 4.19b). In addition, the heat content anomaly in the vertical direction also changes in response. When wind perturbations are applied to the Indian basin, the heat content anomaly appears in the form of a second baroclinic mode (black curve in Fig. 4.19c). However, when wind stress perturbations are applied to the Pacific basin, the heat content anomaly appears in the form of a first baroclinic mode, but with a much large amplitude; such a large amplitude is clearly due to a fact that the Pacific basin is much larger than the Indian basin. As a result, the corresponding VHF in Exp. SH-E is also much larger than that in Exp. SH-D (Fig. 4.19d). Changes in the vertical stratification in each sub-basin are different for these two cases. When wind stress perturbations are applied to a subbasin, the thermocline shoals, as indicted by the large negative heat content anomaly in the corresponding sub-basin. In Exp. SH-D, the heat content anomaly in the Indian basin is positive above 800 m; it is negative for depths of 800– 940 m (Fig. 4.20a). In contrast, in the Pacific and Atlantic basins it is positive over the depths of 750–950 m; in the ACC it is mostly positive over the depths of 170–900 m, with an extremely small negative lobe over the depths of 910– 960 m (not visible in Fig. 4.20a). The contributions from all sub-basins give rise to a global heat

Fig. 4.20 Experiment SH-D (a) and SH-E (b): The global mean heat content anomaly in the vertical direction (at the end of the 120 year run) (Huang 2015)

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Heaving Modes in the World Oceans

content anomaly in the form of a second baroclinic mode, indicated by the magenta curve in Fig. 4.20a. On the other hand, when wind stress perturbations are applied to the Pacific basin (Exp. SHE), the warm water volume anomaly is much stronger (Fig. 4.20b), basically doubling the amplitude in the previous case. Since wind stress perturbations are applied to the Pacific basin, they induce a large negative volume anomaly in the Pacific basin over the depths of 850–960 m, but it has a small positive value over the depths of 750–840 m. In both the Indian and Atlantic basins, the heat content anomaly is non-negative, and it has a positive value over the depths of 750–950 m.

4.1.4 Adiabatic MOCs of the World Oceans with Rectangular Basins In this section, a simple reduced gravity model is set up to explore how the zonal wind stress anomaly in the equatorial Pacific can affect the MOCs in the world oceans. The model is a simplified version of the world oceans with three basins represented by rectangular basins, where the Pacific basin is 150° wide, both the Atlantic and Indian basins are 60° wide; the Indian basin is closed around 14° N (Fig. 4.21). An idealized zonal wind stress profile

4.1 Heaving Induced by Wind Stress Anomaly

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Fig. 4.21 A simplified model ocean for the world ocean circulation: a the zonal mean wind stress (black curve), and the perturbed wind stress profile (red curve); b the layer depth for the reference state; c the streamfunction for the reference state

is used for the model (black curve in Fig. 4.21a). The model was initialized with a uniform depth of 500 m and run for 100 years to reach a reference state of quasi-equilibrium; the parameters of the model are the same as used for other studies discussed above. In the reference state there are five subtropical gyres, the ACC in the Southern Hemisphere and two subpolar gyres in the Northern Hemisphere (Fig. 4.21b, c).

4.1.4.1 Numerical Experiments Restarting from the reference state, we carried out several numerical experiments. The perturbed zonal wind stress is shown by the red curve in Fig. 4.21a, that includes zonal wind stress perturbations in the meridional direction, multiplied by a function in time and longitude   dsx ¼ ds exp ½ð/  /0 Þ=D/2 f ð xÞT ðtÞ ð4:22Þ where ds ¼ 0:02 N/m2 is the amplitude of the wind stress perturbations; D/ ¼ 10 is the scale width of the wind perturbations, f ð xÞ is a function of longitude and T ðtÞ is a function of time. In Exp. EQW-1, the model was restarted from the reference state and forced by the perturbed zonal wind stress described in Table 4.1. Since the applied zonal wind stress perturbations were with

enhanced easterlies over the equatorial Pacific, warm water in the upper layer is pushed towards the western part of the equatorial Pacific (Fig. 4.22a), the maximum layer depth increase is about 14 m; in the other parts of the Pacific Ocean and other basins, layer depth declines as required by the total volume conservation of warm water. The corresponding horizontal streamfunction perturbations are shown in Fig. 4.22b. The three dimensional redistribution of the warm water in the upper layer can be diagnosed in terms of the volume transport (or the overturning circulation) in the meridional/zonal/vertical directions. In this section, our analysis is focused on the anomalous MOC (MHF) in all three subbasins (Fig. 4.23). As shown in Fig. 4.22a, the layer thickness anomaly is positive in the warm pool, but it is negative for most parts of the world oceans. Accordingly, the MOC is negative north of the Equator and positive south of the Equator in the Pacific basin. Warm water everywhere is pushed towards the warm pool in the Pacific sector, and this induces negative MOCs in both the south Atlantic and Indian basins (Fig. 4.23b, c). To examine the meridional profile of the MOC (MHF), we plot the time snap shots at year 20 (Fig. 4.24). It is clear that the easterly anomaly can induce sizable perturbations in both the MOC and MHF. In particular, the response in

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Table 4.1 Definition of the functions f ð xÞ

T ðtÞ

Exp. EQW-1

¼ 1 in Pacific basin ¼ 0 in Indian/Atlantic basins

Linearly increases from 0 to 1 from t = 1–20 yr; ¼1 from year 20 to year 40

Exp. EQW-2

¼ 1 in Pacific basin ¼ −1 in Indian basin ¼ 0 in Atlantic basin

Exp. EQW-3

¼ 1 in Pacific basin ¼ −1 in Indian basin ¼ 0 in Atlantic basin

T ¼ sinð2pt=DT Þ for 0 t 60 yr, DT ¼ 30 yr

Fig. 4.22 Results from Exp. EQW-1. a the upper layer depth perturbations; b the streamfunction perturbations

the Atlantic basin is much stronger than that in the Indian basin, because the Indian basin is far away from the Pacific basin in the downstream direction. For the large scale circulation in the ocean separated by large continents, climate change signals propagate from one basin to the others through Kelvin waves around the southern tips of these continents (Fig. 3.5). Therefore, climate signals in the Pacific basin move to the Atlantic basin first via the southern tip of the South America continent; afterward, climate signals in the Atlantic basin move to the Indian basin via the southern tip of Africa. As such, climate signals in the Indian basin are much weaker than those in the Atlantic basin (Fig. 4.24). The MHF in both the Atlantic and Indian basin are southward, but in the South Pacific it is northward (Fig. 4.24). Consequently, the heat

content in the world oceans is pushed towards the equatorial Pacific Ocean – this leads to the warming up of the Pacific and cooling down of both the Atlantic and Indian Oceans. This interbasin heat transport induced by the equatorial easterlies is an interesting component of climate variability in the world oceans. In comparison, we also run the Exp. EQW-2 (Table 4.1), in which the zonal wind stress perturbations include anomalous westerlies in the Indian basin. The combination of anomalous easterlies in the Pacific basin and anomalous westerlies in the Indian basin mimic the effect of the anomalously strong Walker circulation centralized in the Marine Continent. The additional westerly anomaly in the Indian basin pushes warm upper layer water eastward in the Indian basin, leading to a negative layer depth anomaly in most parts of the Indian basin,

4.1 Heaving Induced by Wind Stress Anomaly

187

Fig. 4.23 Time evolution of MOCs in Exp. EQW-1 for the Pacific basin (a); the Atlantic basin (b) and the Indian basin (c)

especially the western part (Fig. 4.25a). Such anomalous flow manifested in the form of anomalous horizontal gyres in the Indian basin (Fig. 4.25b). As shown in Figs. 4.25 and 4.26, the additional anomalous westerlies in the Indian basin induced a stronger anomalous MOC and MHF in the Indian basin; at the same time, patterns of MOCs and MHFs in both the Pacific and Atlantic basins remain unchanged, with a slightly larger amplitude.

To examine the meridional profile of MOCs (MHFs), we plot time snap shots at year 20 (Fig. 4.27). In this case the response in the Indian basin is slightly stronger than that in the Atlantic basin because the Indian basin is directly forced. In addition, we also ran a third experiment, Exp. EQW-3, in which the model was restarted from the reference state and subjected to an easterly anomaly in the Pacific basin and westerly anomaly in the Indian basin, but the zonal wind stress perturbations contain a 30 year

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Fig. 4.24 The MOC (MHF) obtained from Exp.EQW-1 at year 20

Fig. 4.25 Results from Exp. EQW-2. a The upper layer depth perturbations; b the streamfunction perturbations

sinusoidal cycle. Such a long term wind stress anomaly mimic the climate variability related to the long term quasi-period cycle of PDO like events. As such, the MOC in the Pacific, Atlantic and Indian basins went through periodic cycles

(Figs. 4.28, 4.29, and 4.30). Under the periodic wind stress anomalies, anomalous MOC and MHF have larger amplitude, on the order of 0.4 Sv and 15 TW in both the Atlantic and Indian basins.

4.1 Heaving Induced by Wind Stress Anomaly

189

Fig. 4.26 Time evolution of MOCs in Exp. EQW-2 for the Pacific basin (a); the Atlantic basin (b) and the Indian basin (c)

In conclusion, the zonal wind tress anomalies in the Pacific and Indian basins can induce a sizable anomalous MOC and MHF in the Atlantic basin, and such a physical mechanism is worth further explorations in climate study. Since our model is a simple reduced gravity model, it is incapable of exploring the potential impact on deep and bottom water formation in the Atlantic Ocean. However, our results suggest that zonal wind stress anomalies in the Pacific and Indian basins might affect the AMOC, including the

formation of Antarctic Bottom Water (AABW) and North Atlantic Deep Water (NADW). To examine the meridional profile of MOCs (MHFs), we plot the time snap shots at years 37 and 52 (Figs. 4.29 and 4.30). In this case the amplitude of the response in the Indian basin is similar to that in the Atlantic basin because the system enters a periodic oscillation; thus, perturbations in the Atlantic basin become stronger, although this basin is not subjected to direct wind stress perturbations.

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Fig. 4.27 MOCs (MHFs) obtained from Exp. EQW-2 at year 20

4.1.5 MOC/MHF Simulated by a RGM in the World Oceans

in the equatorial Pacific can affect the MOCs in the world oceans.

4.1.5.1 Introduction As discussed above, for a steady-state ocean the meridional overturning circulation (MOC) is intimately related to diabatic processes taking place in the upper and subsurface ocean. For the ocean in transit, an anomalous MOC can also be induced by wind stress anomaly. Previous studies, e.g., Zhao and Johns (2014) and Yang (2015), have highlighted the importance of wind stress in the MOC anomaly on the seasonal time scale. On longer time scale, Huang (2015) has pointed out the importance of heaving modes on the three dimensional redistribution of warm water mass, and the related MOC (MHF) and the zonal overturning circulation (ZOC) and the related zonal heat flux (ZOC). In this section we will use a simple reduced gravity model, with realistic coastline and wind stress to explore how zonal wind stress anomaly

4.1.5.2 Model Formulation This is a simple reduced gravity model based on the spherical coordinates with the realistic coastline; a major difference between this model and that discussed in the previous section is that in this model the Indian Ocean is linked to the Pacific Ocean at the equatorial band by the Indonesian Throughflow. For this low-resolution model, the Indonesian Throughflow channel is somewhat larger than that in other similar models. For the large-scale circulation of low resolution on the order of one degree, the inertial terms are small and thus negligible. The basic equations are ðhuÞt  2x sin /hv ¼ g0 hhk =r cos / þ sk =q0  ju þ Am r2 ðhuÞ ð4:23Þ

4.1 Heaving Induced by Wind Stress Anomaly

191

Fig. 4.28 Time evolution of MOCs in Exp. EQW-3 for the Pacific basin (a); the Atlantic basin (b) and the Indian basin (c)

ðhvÞt þ 2x sin /hu ¼ g0 hh/ =r þ s / =q0  jv þ Am r2 ðhvÞ ð4:24Þ ht þ

1 1 ðhuÞk þ ðhvÞh ¼ 0 r cos h r

ð4:25Þ

where r ¼ 6370 km is the Earth’s mean radius, x ¼ 7:2921  105 =s is the angular velocity of Earth’s rotation, g0 is the reduced gravity, that is set to a constant value of 0.01 m/s2 in this

  section, sk ; s/ are the zonal and meridional wind stress, ðj; Am Þ are the parameters for the vertical and horizontal momentum dissipation. A bottom friction linearly proportional to the horizontal velocity is used as in previous models. The model has a one by one-degree resolution and it is subjected to the periodic boundary conditions on the E–W direction. Since the model runs include cases with the upper layer outcrop in the ocean, in particular the ACC, a positive definite numerical scheme is

192

Fig. 4.29 MOCs (MHFs) obtained from Exp. EQW-3 at year 37

Fig. 4.30 MOCs (MHFs) obtained from Exp. EQW-3 at year 52

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Heaving Modes in the World Oceans

4.1 Heaving Induced by Wind Stress Anomaly

193

Fig. 4.31 Layer depth (a) and streamfunction (b) for the reference state obtained from the RGM. White lines indicate the 26.5° N and 26.5° S sections for meridional transport plots shown in following figures

implemented in the continuity equation calculation. Our model is based on the spherical coordinates, the algorithm used here is a straightforward extension of the original FluxCorrected Transport (FCT) algorithms (Zalesak 1979; Huang 1987) based on the rectangular Cartesian coordinates to the spherical coordinates. Because a volume conservation numerical scheme is used for the layer thickness calculation, the total volume of the upper layer remains constant all the time, even when the upper layer outcrops in part of the model ocean. To limit the size of the outcropping window in the subpolar basins, a relatively large amount of warm water is specified in the initial state of rest; the corresponding mean layer depth is 1500 m. First, we run the model forced by the climatological of the GODAS data converted to 1  1 degree resolution for 100 years to guarantee that the model reaches a quasi-equilibrium reference state. The reference state has the fundamental structure of the main thermocline in the world oceans, including the bow-shaped main thermocline in the five subtropical basins (Fig. 4.31). In the subtropical basins, the maximum depth of the upper layer is about 1600 m or more, roughly corresponding to the depth of the winddrive gyres in the world oceans. In this setting, the layer thickness associated with the ACC is large enough to simulate the communication of

climate signals between individual sub-basins in the Southern Hemisphere. The strength of the subtropical gyre is about 15–20 Sv, somewhat weaker than the situation in the world oceans. These relatively weak gyres are due to the rather idealized setting of the model.

4.1.5.3 Results from the Numerical Experiments The model was restarted from the quasi-steady state discussed above and run for 108 years forced by the 36-year annual mean wind stress of the GODAS data (1980–2015); this 36-year wind stress was repeated three times, and the analysis below is based on the last 36 years of numerical experiment. Because the reduced gravity model excludes the contribution due to adiabatic processes associated with the thermohaline circulation, in a steady state the meridional transport across a fixed latitude is zero. There is an exception to this constraint: owing to the existence of the Indonesian Throughflow, in the steady state, the northward volume flux in the South Pacific Ocean is compensated by the southward volume flux in the South Indian Ocean. Accordingly, we will also show the meridional volume fluxes in the South Pacific Ocean and the South Indian Ocean, plus the sum of these two meridional fluxes.

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Heaving Modes in the World Oceans

Fig. 4.32 Time evolution of the meridional transports along 26.5° N and 26.5° S diagnosed from the reduced gravity model

MOCs diagnosed from 26.5° N and 26.5° S for the world oceans diagnosed from the reduced gravity model are shown in Fig. 4.32. It is clear that MOCs diagnosed from this model show strong interannual and decadal signals. As discussed above, MOCs for a steady state of the reduced gravity model should be zero; hence, the signals shown in this figure depict the contribution of adiabatic motions induced by wind stress perturbations. For the South Pacific and South Indian Oceans, the MOCs are non-zero (Fig. 4.32b). Since there is a non-zero contribution owing to the Indonesian Throughflow, MOCs in both the South Pacific and South Indian Oceans are correlated to each other. In order to show the contribution due to the time-dependent components of the flow, we include the meridional volume flux integrated over both the South Pacific and South Indian Oceans. As shown by the red curve in Fig. 4.32b, there is also strong inter-annual and decadal variability for the South PacificIndian Ocean. To explore the issue of how much the RGM can capture the MOCs in the world oceans, we put the MOCs diagnosed from the RGM and the GODAS data in the same Fig. 4.33. Note that the

GODAS data is a data-assimilation result, but the wind-stress data available from the GODAS data is for 36 years only. Using such relatively short data to run the RGM, it is difficult to simulate variability on a decadal time scale. Since the model cannot provide accurate information for phenomena over such long time period, the long term trend in the model results is an artifact of using the wind stress data that covers a relatively short time spans. Therefore, we plot the results with the mean value of the MOC removed. As shown in Fig. 4.33, MOCs diagnosed from the RGM can capture many features of the variation diagnosed from the GODAS data. For several events, the MOCs diagnosed from the RGM can approximately match those diagnosed from the GODAS data. MOCs diagnosed from the RGM are comparable with those diagnosed from the GODAS data. It is clearly seen that the two curves shown in this figure share some similarity; in particular, their anomalies have quite similar frequencies. The maximum around year 1987, 1992 and 2006–2007 are the similar patterns shown in both MOC profiles. The RGM is incapable of simulating the large amplitude of the AMOC obtained from the GODAS data from 1994 to 2016; such a large

4.1 Heaving Induced by Wind Stress Anomaly

195

Fig. 4.33 Time evolution of the meridional transports along 26.5° N in the North Atlantic Ocean and North Pacific Ocean diagnosed from the RGM and the GODAS data

amplitude of the AMOC may be due to the strong anomalous heat and freshwater fluxes in the world oceans; simulating these anomalous forcing is beyond the goal of the RGM. Of course, because of the contribution of thermohaline processes, the MOC diagnosed from the GODAS data also reflects an oscillation pattern quite different from that obtained from the reduced gravity model. In addition, we have calculated the equivalent MHF from the RGM and compared it with that diagnosed from the GODAS data. Since the MHF in the Atlantic Ocean contains a major component linked to the thermohaline circulation, it is expected that the MHF diagnosed from the GODAS data should be much larger than the MHF diagnosed from the adiabatic RGM. Nevertheless, the variability of the MHF diagnosed from the RGM still share many similar features with the MHF diagnosed from the GODAS data with the mean subtracted (Fig. 4.34). The comparison to the North Pacific Ocean is also quite similar.

4.2

Heaving Induced by Anomalous Freshwater Forcing

4.2.1 Introduction As discussed above, the wind-driven circulation adjustment induced by wind stress perturbations can induce adiabatic heaving in the ocean. Buoyancy forcing anomaly is another major player in generating climate variability. The diabatic processes can be separated into two categories, variability induced by heat flux anomaly, and by freshwater flux anomaly. Climate variability induced by haline forcing anomaly is quite interesting and it is the focus of this section. Dynamic processes in this category involve no anomalous heating/cooling across the air-sea interface or layer interface; accordingly, the corresponding perturbations can be called adiabatic as well. However, as discussed in

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Fig. 4.34 Time evolution of the meridional transports along 26.5° N diagnosed from the GODAS data and the reduced gravity model (RGM)

Chap. 3, “adiabatic” is not a very accurate term for oceanography. In the present case, since salinity is changed, it is not an isoTS process; nevertheless, we will use the term “adiabatic” in our discussion. Durack and Wijffels (2010) have analyzed climate variability related to the freshwater flux anomaly over the past half century. The linear trend of salinity change over the past 50 years is on the order of 0:2 psu. In the Atlantic basin, the linear salinity trend is 0.2/50 yr; in the western and northern North Pacific, the linear trend is −0.2/50 yr. Chen and Tung (2014) showed that the freshwater flux anomaly on decadal time scale may be linked to the heat content anomaly in the ocean. For example, North Atlantic subpolar (45° N–65° N) salinity varied greatly over the past 60 years. In particular, the salinity anomaly is positive and reaches the magnitude of 0.06 psu in this region. Apparently, the heat content anomaly in this region is in phase and closely linked to the salinity anomaly. However, they offered no clear dynamic explanation for this connection.

Contributions to the circulation due to surface freshwater flux have been discussed in previous studies. For example, Goldsbrough (1933) assumed an idealized distribution in which precipitation and evaporation is balanced for each latitude band, and he postulated a simple theory for the evaporation/precipitation-driven circulation in the ocean. By adding on the western boundary currents, Stommel (1957, 1984) was able to generalize this solution to a more realistic distribution of air-sea freshwater flux, with evaporation at low latitudes and precipitation at high latitudes. The corresponding solution is called the Goldsbrough-Stommel solution. However, for a long time no one paid much attention to the Goldsbrough-Stommel solution, because for a realistic parameter range such a circulation is one order of magnitude weaker than that associated with the wind-driven circulation and the thermally driven circulation. There is a misconception involved in the role of surface freshwater flux and the salinity anomaly. If there were no salt in the ocean, or if there were no strong mixing driven by external

4.2 Heaving Induced by Anomalous …

West

East

197

South

North

Subtropical gyre

Salinification

(a) Shifting of the main thermocline in the subtropical basin induced by salinity change

(b) Shifting of the main thermocline (from black to red) along the eastern and western boundaries, and the induced MOCs

(c) The induced vertical heat content shifting

Fig. 4.35 Sketch illustrating change of stratification induced by freshening of the upper layer in the subtropical gyre

mechanical energy in the ocean, then the relatively weak barotropic circulation predicted by the Goldsbrough-Stommel theory would be the only circulation driven by air-sea freshwater flux. However, for an ocean with salt and strong mixing driven by external sources of mechanical energy, freshwater anomalies can induce strong baroclinic circulations with strength comparable to those due to wind stress and thermal forcing, as discussed by Huang (2010). To explore the baroclinic circulation anomalies induced by freshwater anomalies one can use a simple reduced gravity model. In parallel to the study of the thermocline adjustment induced by wind stress perturbations, the freshwater induced density changes in the upper ocean can also induce adiabatic adjustment of the wind-driven circulation in the upper ocean. As a result, the main thermocline in the upper ocean adjusts, leading to three dimensional shifting of warm water in the upper ocean and many associated changes in the ocean. As discussed in Chap. 1 (Eq. 1.3), the slope of the squared thermocline depth is inversely proportional to the reduced gravity. Thus, if evaporation in the mid-latitude of a subtropical basin is enhanced, salinity in the upper layer increases; as a result, the corresponding reduced gravity declines, leading to the enhancement of the main thermocline slope and the downward

shifting of the thermocline, depicted by the red arrows and curve in Fig. 4.35. Since the thermocline depth along the eastern boundary is inversely proportional to the square root of the reduced gravity, a decline of the reduced gravity in the subtropical gyre leads to the thermocline deepening along the eastern boundary in the subtropical basin, i.e., the thermocline is moved from the black horizontal line to the red curve in the upper part of Fig. 4.35b. The combination of the thermocline deepening along the eastern boundary and the enhanced east-west slope of the thermocline give rise to a much deeper thermocline in the subtropical basin interior and along the western boundary at the mid-latitude (the bow-shaped red curve in Fig. 4.35b). Assuming the total volume of warm water above the thermocline in the whole basin does not change during such a relatively short time, the thermocline away from the mid-latitude must move upward. In compensation, cold water below the main thermocline should move horizontally in the opposite direction. As a result, there are anomalous MOCs induced by salinification at the midlatitudes of the subtropical gyre (dashed red curves with arrows in Fig. 4.35b). These anomalous MOCs consist of the movements of warm (cold) water above (below) the thermocline; consequently, there is an anomalous

198

meridional heat flux. In addition, there is the associated movements of warm water in the vertical direction, including cooling at the shallower levels and warming at the deeper levels (blue and red curves in Fig. 4.35c). Such vertical movement of warm upper layer water implies vertical shift of heat content and hence equivalent vertical heat flux (red arrow in Fig. 4.35c). In the case of anomalously excessive precipitation in the surface layer, the adjustment of the wind-driven gyre is opposite to the case discussed above. As a result, the main thermocline slope decreases, and the warm water in the upper ocean is redistributed. This redistribution also leads to a vertical heat content anomaly in the form of a baroclinic mode. The salinity anomaly in the subpolar basin can induce adjustment of the thermocline in a similar way. In general, surface freshening can lead to a change of the thermocline and warm water redistribution in the whole basin, in a way similar to that associated with wind stress perturbations.

4.2.2 Model Set Up In the commonly used reduced gravity models the layer above the main thermocline has a constant density. However, due to horizontal differential heating and freshwater forcing, density above the main thermocline is not constant within a closed basin. The equivalent reduced gravity distribution diagnosed for the world oceans was shown as Fig. 1.11. In addition, the traditional reduced gravity model treats the dynamics in isolation from the thermodynamics. In the ocean, dynamics and thermodynamics are closely coupled. It is desirable to build up a simple layer model that can overcome such shortcomings in the traditional reduced gravity model. This is a generalized reduced gravity model, so that there is only one active layer. The time dependent momentum and continuity equations are

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Heaving Modes in the World Oceans

  ðhuÞt  fhv ¼  0:5g0 h2 x þ sx =q0  ku þ Am r2 ðhuÞ   ðhvÞt þ fhu ¼  0:5g0 h2 y þ sy =q0  kv þ Am r2 ðhvÞ ht þ ðhuÞx þ ðhvÞy ¼ 0

ð4:26Þ ð4:27Þ ð4:28Þ

where g0 ¼ g q2qq1 ¼ g0 ðx; y; tÞ is the generalized 0

reduced gravity, that can vary with time within the model basin. In theory, the horizontal density gradient in the upper layer implies vertical shear of horizontal velocity; accordingly, the horizontal velocity calculated from such a model should be interpreted as the layer mean velocity. The relevant theoretical issues of such models have been discussed by Huang (1991), Ripa (1995), and Scott and Willmott (2002). The model ocean is the same as described in Sect. 4.1, including the zonal wind stress profile. To prevent the upper layer outcropping in the subpolar basin, the mean layer depth is set as 350 m. In the steady state, the layer thickness satisfies the following equation h2 ¼ h2e þ

2f 2 bq0

Zxe x

x

1 s dx g0 ðx; yÞ f y

ð4:29Þ

4.2.3 Results from Numerical Experiments Following the common practice in reduced gravity models, the dynamics are separated from the thermodynamics. In fact, we will specify the reduced gravity as a time varying function g0 ðx; y; tÞ, without explicitly calculating the time evolution of the buoyancy field from the thermodynamic equation for the upper layer. First, we run the model with constant g0 ¼ 0:01 m=s2 for 300 years to guarantee that the model reaches a quasi-equilibrium state. This basic reference state has a three-gyre structure in each hemisphere. The wind stress, the main

4.2 Heaving Induced by Anomalous …

199

Fig. 4.36 Wind stress, thermocline depth and streamfunction for the reference state

Fig. 4.37 Changes of the reduced gravity profile used in numerical experiments based on the generalized reduced gravity model

(b) Two Hemispheres

(a) Northern Hemisphere Undisturbed Exp. 1 Exp. 2 Exp. 3

60N 40N

Undisturbed Exp. 4 Exp. 5 Exp. 6

60N 40N

20N

20N

0

0

20S

20S

40S

40S 60S

60S 0

0.2

0.4

0.6

0.8

gpri (0.01m/s2)

thermocline depth and the streamfunction of the reference state are shown in Fig. 4.36 Second, we restart the model from the existing equilibrium state and subjected to a time evolving g0 ¼ g0 ðx; y; tÞ, with no change in wind stress plus a thermally insulation condition. The salinity induced change in the reduced gravity is gradually imposed as a linear function of time over the first 20 years of numerical experiment. At year 20, gravity perturbation reaches its target value, and the model is run for an additional 20 years, without further change in the reduced gravity. The 40 year of duration in these runs is long enough for the establishment of the quasisteady state, as confirmed by our numerical experiments.

1.0

1.2

0

0.2

0.4

0.6

0.8

1.0

1.2

gpri (0.01m/s2)

Six numerical experiments were carried out, and the corresponding final profile of the perturbed reduced gravity are shown in Fig. 4.37. As discussed by Durack and Wijffels (2010), over the past 60 years salinity changes in the world oceans are on the order of 0.2 psu. Assuming the saline contraction coefficient is approximately 8  10−4/psu, the corresponding density change is on the order of 0.15 kg/m3. Hence, we set the amplitude of reduced gravity variability to be 0.001 m/s2; this is in the acceptable range, and it corresponds to 10% of the undisturbed reduced gravity used in the reference run. The reduced gravity perturbation profiles used in numerical experiments are in the form of

200

sinusoidal function of latitude (Fig. 4.37). In Exp. 1, the salinity perturbation is positive, and the corresponding negative perturbation in reduced gravity is confined to the subtropical basin (10° N–40° N), depicted by the dashed red curve in Fig. 4.37a. In Exp. 2, the salinity anomaly is positive and the corresponding reduced gravity perturbation is in the form of a sinusoidal function confined to the subtropicalsubpolar basin (10° N–70° N), depicted by the blue curve in Fig. 4.37a. In Exp. 3, the salinity induced reduced gravity perturbation covers the latitudinal band of 10° N–70° N, depicted by the red curve, plus the dashed red curve, in Fig. 4.37a. The change in reduced gravity in the subtropical basin for this case is the same as in Exp. 1, i.e., the salinity anomaly in the subtropical basin is positive; on the other hand, in the subpolar basin the positive anomaly in reduced gravity indicates there is a freshwater anomaly. The freshening of the subpolar basin and salinification of the subtropical basin is a potential scenario linked to global warming, e.g., Curry and Mauritzen (2005), Curry et al. (2003). In addition, three more experiments were carried out to examine the dynamical consequence of the salinity anomaly over the entire two-hemisphere basin (Fig. 4.37b). In Exp. 4, salinity (reduced gravity) anomaly is positive (negative) for the whole basin, mimicking the salinification of the whole Atlantic. In Exp. 5, salinity (reduced gravity) anomaly is positive (negative) for the Southern Hemisphere, but the sign is opposite for the Northern Hemisphere. In Exp. 6, salinity anomaly has a constant negative value for the two-hemisphere model ocean, mimicking the freshening of the world oceans due to melting of sea ice and land-based glaciers; the corresponding reduced gravity anomaly has a positive value of 0.001 m/s2. Although in the commonly used reduced gravity model, layer thickness along the eastern boundary is nearly constant, it is no longer the case for the generalized reduced gravity model. As shown in Eq. (4.27), in a steady state and neglecting friction and meridional wind stress forcing, the no-normal flow condition along the

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Heaving Modes in the World Oceans

eastern boundary is reduced to the following constraint ðg0 h2 Þe ¼ const:

ð4:30Þ

Hence, for latitudes of constant reduced gravity, layer thickness along the eastern boundary remains constant; however, for the latitudes of varying reduced gravity, layer thickness along the eastern boundary is inversely proportional to the square root of the reduced gravity. In addition, the zonal gradient of squared layer thickness is inversely proportional to the reduced gravity used in the model, as shown in Eq. (1.3). Thus, the decline in reduced gravity in the subtropical basin should enlarge the layer depth along the eastern boundary and enhance the slope of the main thermocline in the subtropical basin, depicted by the red arrows and curve in Fig. 4.35a.

4.2.3.1 Experiments with Reduced Gravity Changes Confined to the Northern Hemisphere In Exp. 1, the reduced gravity is lessened in the subtropical basin. As a direct consequence, in the final state after adjustment the layer depth along the eastern boundary increases about 15 m at 20° N; however, it has a nearly constant value of −4 m at other latitudes. Within the same latitude band, the east-west slope of the thermocline is steepened, and the thermocline along the western boundary moves down, as shown by the black curve in Fig. 4.38a. Since the total volume of warm water remains unchanged, away from the subtropical latitude band in the Northern Hemisphere the thermocline depth along the western boundary on other latitude bands moves upward, as shown by the black curve in Fig. 4.38a. In Exp. 2, the layer depth perturbation along the eastern boundary reaches the maximum value around 40° N, where the reduced gravity is the lowest; the thermocline along the western boundary in the same latitude band moves down. On the other hand, the thermocline at other latitude bands moves upward, as shown by the black curve in Fig. 4.38b. In Exp. 3, layer depth

4.2 Heaving Induced by Anomalous …

201

Fig. 4.38 Changes of the thermocline depth along the eastern (red) and western (black) boundaries induced by freshening/salinification in Exp. 1, 2, and 3. Negative values indicate deepening of the layer

Fig. 4.39 Time evolution of volume anomaly and HC anomaly in Exp. 1

perturbation along the eastern boundary reaches the maximum value around 20° N, where the reduced gravity is minimum; correspondingly, the thermocline along the western boundary moves downward in the same latitude band. However, at 50° N the thermocline along the eastern boundary moves upward because the reduced gravity perturbation is maximal; the corresponding thermocline depth along the western boundary is reduced (black curve in Fig. 4.38c). The movement of thermocline along the eastern and western boundaries shown in Fig. 4.38a suggested that the warm water volume

at the corresponding mid-latitude band is increased (red color in Fig. 4.39a); in compensation, the warm water volume anomaly at other latitudes is negative, depicted by the blue color. At the end of the numerical experiment, the meridional distribution of the volume anomaly is shown in Fig. 4.39b. Due to the conservation of warm water volume, the HC anomaly signals in the vertical direction must appear in the form of baroclinic modes. Since the main thermocline is the deepest at this latitude band, the deepening of the main thermocline leads to a positive HC anomaly at this depth range, depicted by the red color in

202

Fig. 4.39c. In contrast, the warm water volume is reduced at shallower levels, indicated by the blue/green color in Fig. 4.39c. At the end of the numerical experiment, the HC anomaly profile appears in the form of a first baroclinic mode (Fig. 4.39d). The vertical shifting of warm water implies that there is a vertical heat flux from the shallower levels to the deeper levels, and such vertical heat flux is entirely owing to the adiabatic adjustment induced by salinification in the subtropical basin of the Northern Hemisphere. The other important implication of warm water transport in the upper layer is the MOC and MHF. The basic assumption of the reduced gravity model is that the lower layer is infinitely thick and motionless. However, the sea level variability is confined to the range within the order of one meter; for this reason, transport of the upper layer water implies that the cold water in the lower layer should move in the opposite direction in compensation. Therefore, both the MOC and MHF can be inferred from the time evolution of the upper layer water. The corresponding MHF is calculated by assuming the upper layer temperature is 10 °C and the lower layer is 5 °C. As shown in Fig. 4.40, there are large MOCs during the adjustment. Since warm water in the upper layer flows from other latitudes to the

4

Heaving Modes in the World Oceans

latitude band of 10° N–35° N, there is a negative MOC north of 30° N and a positive MOC south of 30° N. At the end of year 20, their amplitude reach more than −0.15  106 m3/s and 0.5  106 m3/s respectively. Even after averaging over the total length of 40 yr, the MOCs still have quite a large amplitude. These large MOCs also imply strong MHF in the model ocean, as shown in Fig. 4.40c, d. This example suggests that an adjustment of the thermocline in the upper ocean induced by salinification can induce a noticeable contribution to both the anomalous MOC and MHF. The perturbations of thermocline depth and the streamfunction at the end of the 40-year experiment are shown in Fig. 4.41. The thermocline deepens in the latitude band of 10° N–35° N; in the rest of the model basin, the thermocline shoals. The streamfunction perturbations appear in the form of many alternative anomalous cells, shown in Fig. 4.41b. In the first three experiments, Exp. 1, Exp. 2, and Exp. 3, warm water volumetric anomaly induced by salinity changes are confined to the northern hemisphere in the model basin, and results from these experiments share similar features (Figs. 4.42 and 4.43). The meridional distribution of the warm water volume anomaly is quite similar, except that the

Fig. 4.40 Time evolution of the MOC and MHF for Exp. 1

4.2 Heaving Induced by Anomalous …

203

Fig. 4.41 Perturbations of the thermocline depth and streamfunction at the end of Exp. 1

Fig. 4.42 Meridional distribution of the volume anomaly, MOC and MHF averaged over 40 yr, obtained from Exp. 1, 2, and 3

Fig. 4.43 Vertical HC anomaly in Exp. 1, 2, and 3

204

anomaly is slightly reduced and moved poleward in Exp. 2; these differences are due to the fact that the center of the reduced gravity anomaly is moved poleward and the meridional gradient of the density anomaly is reduced. In Exp. 3, the reduced gravity anomaly in the subpolar gyre is positive, implying a negative density anomaly in the upper layer, mimicking freshening at high latitudes as observations indicate, e.g., Curry and Mauritzen (2005), Curry et al. (2003). As a result of the decline in reduced gravity, thermocline doming in the subpolar basin is enhanced; thus, warm water volume in the subpolar basin is reduced more than in Exp. 1 and 2, as shown by the red curve in Fig. 4.42a. If the numerical experiment runs for a much longer time, the transient motions should die out, and leave no MOC and MHF in the new quasisteady state of the model ocean. However, for experiments run for a relatively short time, the transient transportation of warm water in the upper layer implies that both the MOC and MHF are non-zero. In all these three experiments, there are sizeable MOCs and MHFs even after averaging over the 40 years of experiments (Fig. 4.42b, c). The quasi-horizontal transport of warm water in the upper layer also leads to a HC anomaly in the vertical direction. In all of these three cases, the HC anomaly in the Northern Hemisphere

4

Heaving Modes in the World Oceans

appears in the form of first baroclinic modes, cooling in the depth range of 0−460 m and warming in the depth range of 460−650 m. On the other hand, the HC anomaly in the Southern Hemisphere is negative over the entire depth range; this is consistent with the layer depth change shown in Fig. 4.41. Because of the difference in salinification/ freshening pattern, the vertical profile of the HC anomaly varies slightly (Fig. 4.43). Since reduced gravity changes are confined to the Northern Hemisphere, the corresponding vertical HC anomaly in the Northern Hemisphere is relatively larger than that in the Southern Hemisphere. In addition to the meridional overturning discussed above, changes in the reduced gravity can also induce an anomalous zonal overturning circulation. The time evolution of the zonal volumetric anomaly, the ZOC and the ZHF for Exp. 1 is shown in Fig. 4.44. At the very beginning, there is a weak positive ZOC, indicating an eastward transport of warm water in the upper layer; this is quickly replaced by a negative ZOC transporting warm water westward (Fig. 4.44b). Although the ZOC quickly flips its sign right after the reduced gravity reaches its target value and increases no more, there is a positive volume anomaly near the western boundary at the end of the experiment (Fig. 4.44a). There is a negative ZHF, and its peak value is about −3 TW, but it gradually

Fig. 4.44 Time evolution of zonal volume anomaly and anomalous ZOC and ZHF in the model basin, Exp. 1

4.2 Heaving Induced by Anomalous …

205

diminished after the reduced gravity stopped increasing (Fig. 4.44c). For these experiments, the zonal volumetric anomaly at the end of numerical experiments is shown in Fig. 4.45. It is clearly seen that zonal volumetric anomaly in Exp. 3 is much larger than in Exp. 1 and 2.

(a) Exp.1

0

(b) Exp.2

(c) Exp.3

The corresponding ZOC averaged over the 40 year run is shown in Fig. 4.46. There is a negative ZOC with a value of −5  104 m3/s, averaged over the entire Exp. 1 (Fig. 4.46a). Once again, the maximal amplitude of the ZOC in Exp. 3 is 8  104 m3/s, much larger than that in Exp. 1 and 2.

(d) Exp.4

(e) Exp.5

(f) Exp.6

10E 20E 30E 40E 50E 60E 70E 80E 90E

−2

0 2 4 1012m3/1°

−2

0 2 1012m3/1°

4 −6 −4 −2 0 2 4 1012m3/1°

−5

−2

5 0 1012m3/1°

2 0 1012m3/1°

−5 0 5 10 1012m3/1°

Fig. 4.45 The final state of zonal volume anomaly in the model basin, Exp. 1, 2, 3, 4, 5 and 6

0

(a) Exp.1

(b) Exp.2

(c) Exp.3

(d) Exp.4

(e) Exp.5

(f) Exp.6

10E 20E 30E 40E 50E 60E 70E 80E 90E

0 −1

−5 104m3/s

0 104m3/s

1 −8 −6 −4 −2 0 104m3/s

−10 −5 104m3/s

0

0 0.5 1 1.5 104m3/s

Fig. 4.46 The anomalous ZOC averaged over 40 years of model run in Exp. 1, 2, 3, 4, 5 and 6

−10 −5 104m3/s

0

206

0

4

(a) Exp.1

(b) Exp.2

(c) Exp.3

Heaving Modes in the World Oceans

(d) Exp.4

(e) Exp.5

(f) Exp.6

10E 20E 30E 40E 50E 60E 70E 80E 90E

−1 −0.5 TW

0 −0.2

0 TW

−1 TW

0.2 −2

0

−3 −2 −1 TW

0

0

0.2 TW

0.4 −3 −2 −1 TW

0

Fig. 4.47 The anomalous ZHF averaged over 40 years of model run, Exp. 1–6

The corresponding ZHF averaged over the 40 year run is shown in Fig. 4.47. Once again, because of the larger ZOC in Exp. 3, the corresponding ZHF is much larger than that in Exp. 1 and 2.

4.2.3.2 Experiments with Reduced Gravity Changes in Both Hemispheres In Exp. 4 and 5, the salinity anomaly is distributed in both hemispheres. As discussed above, the meridional variability of reduced gravity leads to the non-uniform thermocline depth along the eastern boundary; the layer depth anomaly along the eastern boundary and the

(a) Δh, Exp.4

Δh (m)

Fig. 4.48 Changes of the thermocline depth along the eastern (red) and western (black) boundaries induced by freshening/salinification in Exp. 4 and 5. Negative values indicate layer deepening

western boundary obtained from these experiment is shown in Fig. 4.48. Since in Exp. 4 the reduced gravity anomaly is minimal around the Equator, the thermocline depth along the eastern boundary is deepest there; in addition, the thermocline depth along the western boundary is also maximal in the same latitudinal band (Fig. 4.48a). The corresponding layer depth anomalies for Exp. 5 can be explained in a similar way (Fig. 4.48b). In Exp. 4, the salinity anomaly is positive over the entire two-hemisphere basin, giving rise to a negative reduced gravity anomaly. Reduction in reduced gravity at low latitudes where the thermocline is the deepest induces deepening of the

(b) Δh, Exp.5

−20

−20

−15

−15

−10

−10

−5

−5

0

0

5

5

10

10

15 20 70S

15

Δhe Δhw

50S

30S

10S

20 10N 30N 50N 70N 70S

50S

30S

10S

10N 30N 50N 70N

4.2 Heaving Induced by Anomalous …

207

Fig. 4.49 Meridional distribution of volume anomaly, MOCs and MHFs averaged over 40 years, obtained from Exp. 4, 5, and 6

thermocline at low latitudes. In compensation, the warm water volume anomaly is negative for latitude bans poleward of 36° S or 36° N; on the other hand, the volumetric anomaly is positive in the low latitude band (black curve in Fig. 4.49a). In Exp. 5, the salinity anomaly is positive (negative) in the Southern (Northern) Hemisphere, giving rise to a negative (positive) reduced gravity anomaly there. Due to the reduction of reduced gravity in the subtropical gyre of the Southern Hemisphere, the slope of the thermocline is enhanced, and water mass redistribution pushes more warm water into the latitude band near 30° S. On the other hand, the positive salinity anomaly in the Northern Hemisphere produces a negative warm water volume anomaly near 30° N (blue curve in Fig. 4.49a). In Exp. 6, the salinity anomaly has a constant positive value over the entire two-hemisphere basin, giving rise to uniformly negative reduced gravity anomaly. Since reduced gravity is constant, the thermocline depth along the eastern boundary is nearly constant, and it is different from the red curves shown in Fig. 4.48. On the other hand, the perturbation solution is quite similar to that in Exp. 4, but with opposite signs (red curve in Fig. 4.49a). There is some minor difference near the poleward boundaries, as shown in this figure. During the thermocline adjustment the warm water transport in the meridional direction gives rise to a strong MOC and MHF. In Exp. 4, warm

water in the Northern Hemisphere is pushed towards low latitudes, giving rise to a negative MOC, with a mean value of 3  105 m3/s averaged over the 40 years. The MOC in the Southern Hemisphere is the exact opposite of that in the Northern Hemisphere (Fig. 4.49b). In Exp. 5, evaporation (precipitation) induces negative (positive) reduced gravity anomaly in the Southern (Northern) Hemisphere. During the thermocline adjustment, warm water in the upper layer is pushed towards the Equator and low latitudes. As a result, there is a large negative MOC, with a mean value of −6  105 m3/s. These large MOCs correspond to large MHFs. In Exp. 4, the amplitude of the mean MHF is 6:4 TW; in Exp. 5, the amplitude of the mean MHF is −12.7 TW (Fig. 4.49c). These MHFs represent a non-negligible portion of the total MHF carried by the world oceans in the climate system; consequently, the contribution associated with the adjustment of the thermocline due to salinification in the ocean is not negligible for climate study. The quasi-horizontal transport of warm water in the upper layer implies a heat content change in the vertical direction. Since in Exp. 4 reduced gravity declines everywhere in the basin, in particular the latitude bands of maximum thermocline depth in the undisturbed state, the thermocline slope increases. As a result, thermocline deepens where the undisturbed thermocline is the deepest. The HC anomaly is positive for the

208

4

0

(a) Δ HC, Exp.4 S.H. (or N.H.) Net

Depth (100m)

1

(b) ΔHC, Exp.5

0

S.H. N.H. Net

1

Heaving Modes in the World Oceans

0 1

2

2

2

3

3

3

4

4

4

5

5

5

6

6

6

7 −0.5

0

0.5

1

7 −1

−0.5

Δ HC (1021J/m)

0

0.5

Δ HC (1021J/m)

7 1 −1

(c) Δ HC, Exp.6 S.H. (or N.H.) Net

−0.5

0

0.5

Δ HC (1021J/m)

Fig. 4.50 Vertical HC anomaly in Exp. 4, 5 and 6

lower part of the wind-driven gyre, but it is negative at shallower levels (Fig. 4.50a). The profile of the reduced gravity anomaly in Exp. 5 is asymmetric with respect to the Equator. In each hemisphere, the HC anomaly appears in the form of a baroclinic mode, with a large anomaly below 250 m, and small anomaly of opposite sign above 250 m. The HC anomaly is somewhat asymmetric with respect to the Equator (Fig. 4.50b). However, due to the slight nonlinearity associated with the inversion of the reduced gravity, the HC anomaly is not exactly asymmetric. In fact, the lowest part of the water column gains more water in the Southern Hemisphere than that lost in the Northern Hemisphere, resulting in a small net gain of warm water. Above 550 m, the net HC anomaly is negative. The vertical HC anomaly profile in

Exp. 6 is quite similar to that in Exp. 4, with opposite signs (Fig. 4.50c).

4.2.4 Experiment for 40 Year Continuing Freshening of the Ocean Our last experiment mimics the global ocean freshening over the past half century. This is labeled as Exp. 7, in which the model ocean is restarted from the quasi-steady state and subjected to continued freshening over 40 years and the reduced gravity is linearly increased with time and at the end of the experiment it is increased to 0.011 m/s2. The final state at the end of the experiment is shown in Fig. 4.51. It is clear that the volumetric

Fig. 4.51 Time evolution of the meridional volumetric anomaly, the MOC anomaly and MHF anomaly for Exp. 7

4.2 Heaving Induced by Anomalous …

209

Fig. 4.52 Time evolution of the zonal volumetric anomaly, ZOC and ZHF anomaly for Exp. 7

anomaly, the MOC and MHF continue to increase until the end of the experiment, as shown in Fig. 4.51. The anomalous MOC reaches the amplitude of 3.3  105 m3/s, and the MHF reaches the amplitude of 6.8 TW. There is also a noticeable zonal distribution of the volumetric anomaly and a ZOC, as shown in Fig. 4.52a. The corresponding ZOC and ZHF for this case are also larger than in the previous experiments. Therefore, the freshening of the world oceans may induce sizable anomalous MOC, MHF, ZOC and ZHF.

4.3

Heaving Induced by Anomalous Wind, Freshening and Warming

4.3.1 Introduction Climate variability in the ocean can be induced by changes in wind stress and buoyancy forcing. In order to understand the climate variability they generated it is desirable to study the effect of each forcing in separation. Although such a linear decomposition cannot give an accurate description of the nonlinear dynamics involved in climate variability, this approach can provide useful insights about the physical processes involved. In particular, our interest is focused on the relative roles of wind stress, freshwater and heating/cooling.

Wind stress change is probably a factor dominating over a broad spectrum in time frequency domain. As discussed in Huang (2015) and Tai et al. (2015), wind stress perturbations on interannual to decadal time scales can induce large three dimensional change of stratification in the ocean, including an anomalous MOC (MHF) and ZOC (ZHF). In particular, the transport of warm water in the wind-driven gyre may induce baroclinic modes of the vertical heat content (HC) anomaly. Such baroclinic modes may help us to understand the vertical shifting of the HC anomaly identified from the climate record, either from observations or computer generated datasets. The buoyancy forcing anomaly is another major player in climate variability. Climate variability can be induced by a haline forcing anomaly alone, as discussed in the previous section. Thus, in this section we will compare climate variability induced by wind stress perturbations, freshwater flux anomaly and the warming of the ocean.

4.3.2 A Simple Generalized Reduced Gravity Model The model used in the previous section is used here. The wind stress, the main thermocline depth and the streamfunction of the reference state are shown in Fig. 4.36.

210

4

In the commonly used reduced gravity model the reduced gravity is assumed to be constant. The generalized reduced gravity model relaxes such a constraint, and it is formulated as follows. A. The reduced gravity can change with time: g0 ¼ gDq=q0 ; Dq ¼ q0 ðaDT þ bDSÞ; DT ¼ DT0 þ dT ðtÞ; DS ¼ DS0 þ dSðtÞ ð4:31Þ So that, the reduced gravity of the model is g0 ¼ g00 þ g½adT ðtÞ þ bdSðtÞ

ð4:32Þ

B. In the more generalized case the reduced gravity can be a function of time and space 0

0

g ¼ g ðx; y; tÞ

ð4:33Þ

For example, a case associated with salinity change in a single basin is discussed in the previous section. In this section we will focus on a case where the reduced gravity is a function of time only. There is another possible extension of the reduced gravity model, in which the total volume of the upper layer can change with time. However, in such a model, the upper and lower layers must communicate through the interfacial velocity. In such a model, the lower layer is in motion. Such a model with two moving layers has been explored in previous studies, e.g., Luyten and Stommel (1986), Pedlosky (1986), and Huang (1993); however, such models are beyond the scope of this book.

4.3.3 Numerical Experiments Based on This Reduced Gravity Model We restart the model from the existing equilibrium state and subjected to a time evolving reduced gravity and wind stress perturbations as described below. The evolution of the parameter setting is a linear function of time, i.e.:

Heaving Modes in the World Oceans

f ðtÞ ¼ t=T; T ¼ 40 year

ð4:34Þ

Exp. A. This is the experiment restarting with the wind stress perturbation: Dsk ¼ Ds0 exp½ð/  /0 Þ=D/f ðtÞ

ð4:35Þ

where Ds0 ¼ 0:01 N/m2 , / is the latitude, /0 ¼ 0, D/ ¼ 25 are the parameters chosen for this experiment. Exp. B. In this experiment, the reduced gravity of the model is increased with time as follows Dg0 ¼ Dg00 f ðtÞ

ð4:36Þ

where Dg00 ¼ 0:001 m/s2 is the parameter chosen for this experiment. Exp. C. In this experiment the reduced gravity and the upper layer temperature are increased as Dg0 ¼ Dg00 f ðtÞ; Tupper ¼ Tupper;0 þ DTupper f ðtÞ ð4:37Þ where Dg00 ¼ 0:001 m/s2 , Tupper;0 ¼ 10  C and DTupper ¼ 0:1  C represents the global warming in the upper layer. Exp. D. In this experiment the wind stress perturbation in Eq. 4.35 and the reduced gravity and upper layer temperature change defined in Eq. 4.37 are all imposed on the model, when it is restarted from the reference state.

4.3.3.1 Results from These Four Numerical Experiments We begin with Exp. A, that is characterized by zonal wind stress perturbations at the low latitude band, Eq. (4.35). The enhancement of the equatorial easterlies pushes warm water in the upper ocean westward. As a result, there is warm water piled up at the low latitude band and drainage of warm water from the high latitude band (red and blue color bands in Fig. 4.53a). In the zonal plane, this leads to warm water moving from the eastern basin and piling up in the western basin (red and blue color bands in Fig. 4.54a).

4.3 Heaving Induced by Anomalous …

211

Fig. 4.53 Time evolution of the volumetric anomaly (in units of 1013 m3/1°) distribution in the meridional direction for these four experiments

Fig. 4.54 Time evolution of the volumetric anomaly (in units of 1013 m3/1°) distribution in the zonal direction in Exp. A, B, C and D

Since the thermocline is deep at low latitudes (Fig. 4.36b), warm water piling up at the low latitude band implies that there is more warm water at deep levels (below 550 m), and this is accomplished by a decline of warm water at the shallower levels. The vertical shifting of warm water during the adjustment is clearly shown by the red and blue color bands in Fig. 4.55a. Because the total amount of warm water in the upper layer is assumed to be conserved, the positive and negative volumetric change of warm water is compensated for on the meridional plane, the zonal plane and the vertical plane. In

particular, the heat content anomaly in the vertical direction must appear in a form of the baroclinic mode (Fig. 4.55a). In Exp. B, the model run is subjected to a linear increase of reduced gravity only, but there is no change in the wind stress forcing and no change in the upper layer temperature. Such a change in the reduced gravity alone can be due to the freshening of the ocean because of melting of sea ice and land-based glaciers, as discussed in the previous section. Physically, the increase in the reduced gravity makes the thermocline slope smaller, and such an effect is somewhat

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Heaving Modes in the World Oceans

Fig. 4.55 Time evolution of the heat content anomaly (in units of 1020 J/m) distribution in the vertical direction in Exp. A, B, C and D

equivalent to weakening of the equatorial easterlies. As a result, there is less warm water in the low latitude band, but there is more warm water at a higher latitude band, as depicted by the blue and red color bands in Fig. 4.53b. As shown in the zonal plane, warm water is moved from the western basin and piled up in the eastern basin (blue and red color bands in Fig. 4.54b). Since the thermocline is deep at low latitudes, warm water drainage from the low latitude band implies that there is less warm water at deep levels (below 400 m) and more warm water at shallower levels. The vertical shifting of warm water during the adjustment is clearly shown by the blue and red color bands in Fig. 4.55b. Because the total amount of warm water is conserved, the heat content anomaly in the vertical direction appears in the form of the second baroclinic mode (Fig. 4.55b). In Exp. C, the model run is subjected to a linear increase of reduced gravity; in addition, the upper layer is subjected to a gradual warming, as specified in Eq. (4.37). The change in temperature can indirectly affect the dynamics of the system through the change in the reduced gravity. Hence, the volumetric anomaly in the meridional and zonal planes is exactly the same as in Exp. B (Figs. 4.53 and 4.54). On the other hand, the heat content anomaly in the vertical direction is different as shown in Fig. 4.55; in fact, temperature difference between

the upper and lower layers’ changes with time in Exp. C. In Exp. D, the model is subjected to additional wind stress perturbations. Although the meridional distribution of the volumetric anomaly is somewhat similar to Exp. C, the negative volumetric anomaly at the low latitude band is smaller than in Exp. C, implying that the zonal wind stress perturbations applied in Exp. D work against the reduced gravity anomaly. In fact, intensification of the low latitude easterlies tends to push warm water towards the low latitude band (Fig. 4.53a). Nevertheless, the dynamical effect of increasing the reduced gravity overpowers that due to wind stress perturbations and leads to a meridional volumetric anomaly pattern similar to that in Exp. C, with a slightly decline of the negative volumetric anomaly at the low latitude band as shown in Fig. 4.53d. The zonal volumetric anomaly in Exp. D is also the result of competition between the wind stress perturbations and the reduced gravity increase. In this case, the wind stress perturbations overpower the reduced gravity change. As a result, the zonal volumetric anomaly pattern is similar to Exp. A, with warm water piling up in the western basin and the negative warm water volumetric anomaly in the eastern basin. In terms of the vertical heat content anomaly, the effect of the reduced gravity increase overpowers the wind stress perturbations. As a result,

4.3 Heaving Induced by Anomalous …

213

Fig. 4.56 Time evolution of the MOC in Exp. A, B, C and D

in the vertical direction, the heat content anomaly is positive and it is negative in the deep level, as shown in Fig. 4.55d. The meridional shifting of warm water implies an anomalous MOC, as shown in Fig. 4.56. For Exp. A, there is a pair of MOC, negative in the Northern Hemisphere and positive in the Southern Hemisphere, reaching the amplitude of 0.1 Sv at the end of the numerical experiment (Fig. 4.56a). For Exp. B, C and D, the sign of MOCs flips, and there are positive MOCs in the Northern Hemisphere and negative MOCs in the Southern Hemisphere. Note that MOCs in Exp. B and C are exactly the same, with the amplitude of 0.25 Sv, because they are subjected to the same reduced gravity change; dynamically, what induced the reduced gravity change does not really matter for the MOC. In Exp. D, the effect of reduced gravity overpowers that of wind stress change, and leaves a pair of MOC, with the same pattern as in Exp. B and C, but with smaller amplitude, Fig. 4.56d. These anomalous MOCs imply poleward heat flux (MHF) with patterns similar to the corresponding MOC (Fig. 4.57). There is a minor difference in MHF between Exp. B and C. This minor difference is due to the fact that for the same meridional volumetric transport of warm water the temperature difference between the upper and lower layers is slightly larger in Exp. C because the upper layer temperature is gradually increased. However, the relative change in the magnitude of temperature difference is quite

small; it is on the order of less than 2%. As a result, such a difference in MHF is barely noticeable in Fig. 4.57b, c. Similar to the case of the MOC, the dynamic effect of wind stress perturbations is overpowered by the reduced gravity increase; consequently, the MHF pattern in Exp. D is similar to those in Exp. B and C, with a smaller amplitude (Fig. 4.57d). The zonal shifting of warm water presented in Fig. 4.54 leads to the anomalous ZOC (Fig. 4.58). In Exp. A, there is a negative ZOC transporting warm water from the eastern basin to the western basin, with the amplitude of 0.22 Sv (Fig. 4.58a). In Exp. B and C, over the major part of the basin there is a positive ZOC, transporting warm water eastward, with an amplitude of 0.12 Sv. In addition, there is a small negative ZOC near the eastern boundary, transporting warm water away from the eastern basin (Fig. 4.58b, c). Once again, the warming of the upper layer in Exp. C does not affect the ZOC, compared with that in Exp. B. In Exp. D, the competition between wind stress perturbation and reduced gravity increase results in the wind stress perturbation playing the dominant role. As a result, the ZOC pattern is similar to Exp. A (Fig. 4.58d). These results are summarized in the following three figures. Figure 4.59 includes the mean MOC averaged over the 40 year experimental runs and the corresponding MHF. As commented above, the MOCs for Exp. B and C are identical; as a result, the blue

214

4

Heaving Modes in the World Oceans

Fig. 4.57 Time evolution of the poleward heat flux in Exp. A, B, C and D

Fig. 4.58 Time evolution of the zonal overturning circulation in Exp. A, B, C and D

curve in Fig. 4.59a is completely overlain by the red curve. The MHF in Exp. C is only very slightly larger than that in Exp. B, so that the difference is invisible. Overall, the wind stress perturbation induced MOC and MHF (black curves) are smaller than those of the reduced gravity increase (red curves). The MOCs (MHFs) induced by these two factors have opposite signs; hence, in Exp. D, they work against each other, leading to a pair of MOC (MHF) depicted by green curves in Fig. 4.59. For the ZOC and ZHF, in Exp. A the wind stress perturbations in both hemispheres work in the same direction, creating a relatively large negative ZOC (ZHF). On the other hand, reduced gravity increase in the model ocean (Exp. B and C) induces two counter-rotating ZOCs and thus

leads to a relatively small positive mean ZOC (ZHF), as shown by the red curves in Fig. 4.60. In Exp. D, the wind stress perturbation effect overpowers the effect of the reduced gravity increase, leading to a relatively small negative ZOC (ZHF) (green curves in Fig. 4.60). These experiments lead to some interesting final states of the volumetric anomaly in the zonal direction (Fig. 4.61a). In Exp. A, the wind stress perturbations lead to a relatively large amplitude positive volumetric anomaly near the western boundary, but a negative volumetric anomaly near the eastern boundary. The volumetric anomaly induced by the positive reduced gravity anomaly (Exp. B and C) has signs opposite to that induced by wind stress perturbations, and the

4.3 Heaving Induced by Anomalous …

215

Fig. 4.59 Mean MOC and MHF in Exp. A, B, C and D 70N 60N 50N 40N 30N 20N 10N 0 10S 20S 30S 40S 50S 60S 70S

Fig. 4.60 Mean ZOC and ZHF in Exp. A, B, C and D

(b) Mean MHF

(a) Mean MOC A B C D

−0.2

0

−0.1

0.1

0.2

−4

2

0

−2

106m3/s

TW

(a) Mean ZOC

(b) Mean ZHF

4

0 A B C D

10E 20E 30E 40E 50E 60E 70E 80E 90E

−0.2

−0.1

0

106m3/s

Fig. 4.61 Final volumetric anomaly in the zonal direction (a) and heat content anomaly in the vertical direction (b); in Exp. A, B, C and D

0.1

−4

−2

0

TW

2

216

4

corresponding amplitude is relatively smaller. In Exp. D, the effect of wind stress perturbations overpowers that due to the positive reduced gravity anomaly, leading to a volumetric anomaly pattern similar to that induced by wind stress perturbations, but with a slightly smaller amplitude (the green curve in Fig. 4.61a). The most interesting phenomenon is the vertical heat content anomaly left behind after the 40-year numerical integration (Fig. 4.61b). In Exp. A, the intensified easterly leads to a baroclinic mode of the HC anomaly, with a cold anomaly above 450 m, and a positive HC anomaly below, depicted by the black curve. In contrast, the positive reduced gravity anomaly (Exp. B) leads to a HC pattern almost opposite to that due to wind stress perturbation, depicted by the blue curve. With the additional change in temperature (Exp. C), the HC anomaly is shifted toward the positive value for almost the entire water column, as depicted by the red curve. Note that the difference in the HC anomaly profile between Exp. C and Exp. B is represented by the dashed red line, i.e., uniform warming in the upper layer creates a uniform increase of HC in the model. Finally, in Exp. D, when the model ocean is subjected to all these forcings, the vertical HC profile in the final state is represented by the green curve. Since the model runs are within the range of small perturbations, it is easy to show that the green curve is the superposition of the black curve and red curve.

4.4

Heaving Induced by Convection Generated Reduced Gravity Anomaly

4.4.1 Introduction Buoyancy forcing anomaly is another major player in generating climate variability. As discussed above, the wind-driven circulation adjustment induced by wind stress perturbations can induce adiabatic (isothermal and isohaline) heaving in the ocean. On the other hand, an anomalous heat flux or haline flux can induce large changes in the upper ocean circulation.

Heaving Modes in the World Oceans

Although such motions are diabatic in nature, we can study the change of the oceanic circulation induced by the diabatic perturbations specified as initial perturbations to an otherwise steady state as follows. In the Labrador Sea, the strong anomaly in air-sea buoyancy flux exists, most likely linked to anomalous atmosphere-ocean interaction. The consequences of such an anomalous flux can be studied, for example, the strong air-sea heat flux from the ocean to the atmosphere (cooling) can be studied in terms of an idealized two-layer model, as sketched in the left panel of Fig. 4.62. Although such an anomalous air-sea interaction is definitely diabatic in nature, we can study the corresponding dynamical consequences by two idealized adiabatic models. First, we can idealize the cooling as a sudden decline of the reduced gravity of the upper layer (Fig. 4.62a). Before cooling, the upper layer and lower layer are of constant density, and the reduced gravity is constant. Due to cooling, density in the upper layer increases in proportion to the amount of heat loss (or salinification); and the change in density varies with geographic location. As a result, the reduced gravity in the model ocean is no longer constant; instead, it is declined from the original constant value (blue dashed curve in Fig. 4.62a). The wind-driven circulation readjusts in response to the changes in reduced gravity, and the dynamical details of such adiabatic adjustment are the focus of this section. Second, we can idealize the cooling as a sudden loss of warm water volume in the upper layer, shown as the difference between the black and blue dashed curves in Fig. 4.62b; however, we assume that the reduced gravity of the upper layer remains unchanged. The dynamical consequence of this initial perturbation will be discussed in Sect. 4.5. In the first case sketched in Fig. 4.62a, the dynamical adjustment after the initial perturbations in reduced gravity is as follows. The initial perturbations in reduced gravity are specified within the green ellipse, and the center of the peak value is marked by the blue color (Fig. 4.63a). In a steady state, the W–E slope of

4.4 Heaving Induced by Convection Generated Reduced …

217

Fig. 4.62 Sketch illustrating two possible ways to simulate the dynamic consequence of cooling induced change in the upper ocean: a change in the reduced gravity of the upper layer; b change in volume of the upper layer

0 (b) y−z view

(a) Horizontal view 70N

H 70N

Z

0

0

E

70S

70S

(c) x−z view

Z

(d) Vertical HCa Z

W

E

_

0

+

H

Fig. 4.63 a Sketch of the adjustment induced by a sudden change in reduced gravity in a simple generalized reduced gravity model; b anomalous meridional overturning cells; c anomalous zonal overturning cell; d vertical heat content anomaly

the thermocline in the upper layer is inversely proportional to the square root of the reduced gravity. Due to the sudden decline of reduced gravity within the green circle, the thermocline in the upper part of the subpolar gyre cannot hold

up the same amount of water set by the constant reduced gravity before perturbations. As a result, the excess amount of warm water in the upper layer is transported toward the western boundary in the form of the first baroclinic long Rossby waves, depicted by the black arrows in the western part of the subpolar basin. These long Rossby waves can be called cooling Rossby waves. Wave processes related to the adjustment in the ocean were discussed in details in Sect. 3.1.4. When these Rossby waves reach the western boundary, a small portion of their energy is reflected in the form of short eastward Rossby waves; these waves are mostly dissipated near the western boundary. The major part of the signals and energy is converted into the equatorward Kelvin waves turning into the eastward equatorial Kelvin waves after reaching the Equator. When these equatorial Kelvin waves reach the eastern boundary, they bifurcate and become the poleward coastal Kelvin waves. Due to the increase of the Coriolis parameter, these coastal Kelvin waves gradually shed their energy in the form of the westward baroclinic Rossby waves. The combination of these waves (black arrows in Fig. 4.63a) help the system to readjust towards a new state. In particular, in the Northern Hemisphere the removal of warm water from high latitudes to middle latitudes manifests as a

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4

negative MOC cell at the high latitude band; similarly, the poleward shifting of warm water at low latitudes manifests as a positive MOC at middle latitudes (Fig. 4.63b). Note that the adjustment involves a threedimensional redistribution of warm water in the upper ocean. In particular, there are anomalous ZOCs and heat content anomalies in the vertical direction (Fig. 4.63c, d).

4.4.2 Model Set Up The model is a two-hemisphere model ocean (Fig. 4.64). To limit the size of the outcropping window in the subpolar basin, a relatively large amount of warm water is specified in the initial state of rest; the corresponding mean layer depth is 600 m. The reduced gravity is specified as a time varying function g0 ðx; y; tÞ, without explicitly

(a)

x

70N

(b) h (100m)

60N

Heaving Modes in the World Oceans

calculating the time evolution of the buoyancy field from the thermodynamic balance equation for the upper layer. First, we run the model with constant g0 ¼ 0:01 m=s2 and an initial uniform layer depth of 600 m for 200 years to guarantee that the model reaches a quasi-equilibrium reference state. The reference state has a three-gyre structure in each hemisphere. The wind stress, the main thermocline depth and the streamfunction of the reference state are shown in Fig. 4.64. In the subtropical basin, the maximum depth of the main thermocline is about 780 m, mimicking the situation in the North Atlantic Ocean; in the western part of the subpolar basin the upper layer depth is about 200 m, mimicking the domeshape main thermocline of the subpolar gyre. The strength of the subtropical gyre is about 15 Sv, somewhat weaker than the situation in the North Atlantic Ocean.

7.5

(c)

(Sv)

15

7.0

50N

10

6.5

40N 30N

6.0

20N

5.5

5

10N

5.0

0

0 4.5

10S 20S

4.0

30S

3.5

40S

−5

−10

3.0

50S 2.5

60S 70S −1 0 1 0.1N/m2

−15 10E 20E 30E 40E 50E 60E

10E 20E 30E 40E 50E 60E

Fig. 4.64 Reference state of the 2-hemisphere model: a Zonal wind stress; b thermocline depth; c stream function

4.4 Heaving Induced by Convection Generated Reduced …

4.4.3 Results from Numerical Experiments We restart the model from the existing equilibrium state and subjected to a time evolving g0 ¼ g0 ðx; y; tÞ, with no change in wind stress plus a thermally insulation condition. The perturbed reduced gravity is in the following form 



y  ys x  xw dg0 ¼ Dg0 exp sin p sin p f ðt Þ yn  ys xe  xw

ð4:38Þ where Dg0 ¼ 0:005 m/s2 is the amplitude of reduced gravity perturbations, yn ¼ 69:5 N, ys ¼ 40:5 N are the northern and southern edges of the reduced gravity perturbations, xw and xe are the western and eastern edges of the reduced gravity perturbations; f ðtÞ is a function of time used to describe the time evolution of reduced gravity perturbations applied in each numerical experiment, as will be described in the following numerical runs. As an example, the spatial pattern of the reduced gravity anomaly for the case with xw ¼ 0:5 E and xe ¼ 29:5 E is shown in Fig. 4.65.

Fig. 4.65 Reduced gravity perturbation used in the model

219

4.4.3.1 Numerical Experiments with Different Duration of the Imposed Reduced Gravity Perturbations To explore the responding time scales for the MOC at subtropical latitudes to changes in reduced gravity specified in the western part of the subpolar basin, we carried out two sets of 20 year numerical experiments. In the first set of experiments, the longitudinal extend of the reduced gravity is specified as xw ¼ 0:5 E, xe ¼ 29:5 E, the reduced gravity perturbation was imposed from the beginning of the numerical experiments and lasts for a specified time listed in Table 4.2. As an example, the time evolution of the solution in Exp. D is shown in Fig. 4.66. The time evolution of the imposed reduced gravity anomaly is shown in Fig. 4.66d. The system responds immediately after the onset of reduced gravity perturbations, Fig. 4.66a. As shown in Eq. 4.29, due to the decline of reduced gravity at the middle of the subpolar gyre, the slope of the main thermocline there is enhanced. Consequently, the dome-shaped thermocline there cannot hold the same amount of warm water in

Reduced gravity perturbation (cm/s2) −0.05 60N −0.10

50N 40N

−0.15

30N

−0.20

20N 10N

−0.25

0 −0.30

10S 20S

−0.35

30S −0.40

40S 50S

−0.45

60S 10E

20E

30E

40E

50E

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4

Heaving Modes in the World Oceans

Table 4.2 Time duration of reduced gravity perturbations imposed in experiments Experiment

A

B

C

D

E

Time duration of the imposed reduced gravity anomaly (yr)

0.25

0.5

1

5

10

Fig. 4.66 Results from Exp. D. a Time evolution of the meridional distribution of the volumetric anomaly; b, c meridional section views at year 2.5 and 5; d time evolution of the specified reduced gravity anomaly

the upper layer. The adjustment of the cyclonic gyre leads to the warm water being expelled from the subpolar gyre into the subtropical gyre (blue and red color patches in Fig. 4.66a). Two meridional sections at year 2.5 and 5 are shown on the right-hand side panel (Fig. 4.66b, c). In the zonal direction, the adjustment of the wind-driven gyre leads to warm water piling up in a narrow band, the location of which changes with time (Fig. 4.67). As examples, two zonal sections at year 2.5 and 6.1 are shown in panel b and c of Fig. 4.67. The adjustment of the wind-driven gyre also leads to the shifting of the warm water in the vertical direction. The subpolar gyre loses warm water; because the mean depth of the thermocline

there is shallow, the volumetric loss appears at a relatively shallow depth (light blue color band over the depth range of 200–600 m in Fig. 4.68 a). Below this band, there is another band of the negative volume anomaly around 650 m and a positive volume anomaly band near 750 m, indicating deepening of the subtropical gyre at middle latitudes. Two section views at year 5 and 10 are plotted in Fig. 4.68b, c; it is clear that after the imposed perturbations were cut off, the system returns to the original structure with a strong positive volumetric anomaly appearing at the depth range of 650 m. The shifting of warm upper layer water implies an anomalous MOC and ZOC. As shown in Fig. 4.69, there is a strong MOC immediately

4.4 Heaving Induced by Convection Generated Reduced …

221

(a) ΔV (1013m3/1°)

(c) yr 6.1

(b) yr 2.5 3.0 2.5

10E

2.0 1.5

20E

1.0 30E

0.5 0

40E

−0.5 −1.0

50E

−1.5

cm/s2

60E

0

5

10 (d) dgpri

15

0

5

10

15

−2

2 0 1013m3/1°

−2

2 0 1013m3/1°

0

−0.5 20

yr Fig. 4.67 Results from Exp. D. a Time evolution of the zonal distribution of the volumetric anomaly; b, c zonal section views at year 2.5 and 6.1; d time evolution of the specified reduced gravity anomaly

Fig. 4.68 Time evolution of the vertical HC anomaly in Exp. D. a Time evolution of the vertical distribution of the volumetric anomaly; b, c vertical section view at year 5 and 10; d time evolution of the specified reduced gravity anomaly

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4

Heaving Modes in the World Oceans

Fig. 4.69 Time evolution of the MOC in Exp. D. a Time evolution of the meridional distribution of MOC anomaly; b, c meridional section view at year 2.5 and 7.6; d time evolution of the specified reduced gravity anomaly

after the onset of reduced gravity perturbations. For this experiment, the MOC reaches its maximum around year 2.5. After year 5, there is no more reduced gravity perturbation and the system starts to recover. The most important aspect of the recovery is the flipping sign of the MOC (blue color in Fig. 4.69a). Two time sections of the MOC at year 2.5 and 7.6 are shown in Fig. 4.69b, c. In the zonal direction, there are strong anomalous ZOCs immediately after the onset of reduced gravity perturbations. For this experiment, the ZOC reaches its minimum around year 3 (Fig. 4.70a). After year 5, there is no more reduced gravity perturbation and the system starts to recover, and there is a positive ZOC, implying warm water being transported eastward. Two time sections of the ZOC at year 3.5 and 8 are shown in Fig. 4.70b, c. The response of the system also depends on the time duration of the imposed reduced gravity. The sensitivity of the model is shown in Fig. 4.71. When the duration is too short, the system responds with a quick, but incomplete

adjustment, as shown in the top two panels in Fig. 4.71. When the duration of reduced gravity perturbation is set to 5–10 yr, the response of the system is complete. In fact, the MOC reaches its maximum at year 2.5, well before the reduced gravity perturbation is cut off; the reverse of the MOC takes place after the reduced gravity perturbation is cut off. We now focus on the MOC at subtropical latitudes, say around 29.5° N. For this set of experiments, the time evolution of the MOC is shown in Fig. 4.72. It is clearly seen that the initial stage of spin up of the MOC near the western boundary is almost the same because the reduced gravity perturbations were specified over the same location, with the only difference in the time duration. On the other hand, the establishment of the MOC is gradually postponed, as the corresponding amplitude of the MOC peak is increased with the longer duration of imposed reduced gravity perturbations; these features are closely linked to the specified time duration of the reduced gravity perturbation anomaly. The recovery of the system also depends on the

4.4 Heaving Induced by Convection Generated Reduced …

223

Fig. 4.70 Time evolutions of the ZOC in Exp. D. a Time evolution of the zonal distribution of the ZOC anomaly; b, c zonal section view at year 3.5 and 8; d time evolution of the specified reduced gravity anomaly

duration of the imposed reduced gravity perturbations, as shown in the central and right part of Fig. 4.72. It is also interesting to note that the amplitude of the negative MOC is nearly twice as large as the amplitude of the positive MOC peak during the first phase of spin-up.

4.4.3.2 Results from Numerical Experiments with Different Locations of Imposed Reduced Gravity Perturbations In the second set of experiments, the reduced gravity anomaly was imposed in time as a step function  1; if t 10 yr f ðt Þ ¼ ð4:39Þ 0; if t [ 10 yr This set of experiment is to test the sensitivity of the MOC response to the location of the reduced gravity perturbations. The longitudinal extend of the reduced gravity is specified as in Table 4.3.

The MOC in the model responds to such perturbations in reduced gravity in the first five experiments is shown in Fig. 4.73. For the first three runs, with the reduced gravity perturbations confined to the western 1/3 of the model basin, the response is very quick, but the amplitude is small, as shown in the top four panels. When the reduced gravity perturbations are extended eastward, the response takes longer time and the amplitude becomes larger, as shown in the lower two panels in Fig. 4.73. In all three of these experiments, the system starts to recover right after the reduced gravity perturbations drop off, as shown on the right part of Fig. 4.73. This figure shows clearly the MOC response, and it is sensitively dependent on the location of the imposed reduced gravity perturbations. In Exp. F, the reduced gravity perturbations (panel a) were specified next to the western boundary; consequently, the response is immediate, but the system also quickly turns to a new quasi-steady state through the development of a negative MOC around year 2 (panel b). At

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4

Heaving Modes in the World Oceans

Fig. 4.71 Time evolution of MOCs in Exp. A, C and E. Panels (a, c, e), reduced gravity perturbations; panels (b, d, f), the MOCs

Fig. 4.72 Time evolution of the MOC at 29.5° N in Exp. A, B, C, D, E

4.4 Heaving Induced by Convection Generated Reduced …

225

Table 4.3 Longitudinal locations of the reduced gravity perturbations in the second set of numerical experiments Experiment

F

G

H

I

J

K

L

M

xw

0.5° E

0.5° E

0.5° E

0.5° E

0.5° E

10.5° E

10.5° E

10.5° E

xe

4.5° E

9.5° E

19.5° E

29.5° E

39.5° E

39.5° E

49.5° E

59.5° E

Center

2.5° E

5.0° E

10.0° E

15.0° E

20.0° E

25° E

30° E

35° E

Response to convection loss 0 −0.2 −0.4

(a) 0

10E

20E

30E

40E

50E

60N 40N 20N 0 20S 40S 60S

60E 0.5 0 −0.5 −1.0

(b) 20

15

10

5

1

yr 0 −0.2 −0.4

(c) 0

10E

20E

30E

40E

50E

60N 40N 20N 0 20S 40S 60S

60E

1 0 −1

(d) 1

5

10

15

20

yr 0 −0.2 −0.4

(e) 20E

10E

0

30E

60E

50E

40E

1

60N 40N 20N 0 20S 40S 60S

0 −1 −2 (f) 1

5

10

15

−3 20

yr

Fig. 4.73 Time evolution of MOC in the second set of experiments (F, H, J). Panels (a, c, e): the zonal profile of reduced gravity along 55° N; panels (b, d, f): the time evolution of the MOC

the end of year 10, the reduced gravity perturbation was cut off, and the system came back to the original steady state through the adjustment almost opposite to what happened in the beginning of the experiment. As the width of the reduced gravity perturbation domain is enlarged, the MOC response pattern gradually evolved, as shown in the rest of Fig. 4.73. The response of the MOC at 29.5° N in experiments (F, H, J) is shown in Fig. 4.74. In these experiments the initial stage of spin-up

depends on the distance of the perturbation center, although the reduced gravity perturbations are imposed for the same length of time (10 yr). For example, in Exp. F the MOC is quickly established within less than 0.5 yr; afterward the system is moving towards a new equilibrium state corresponding to the modified reduced gravity distribution. For Exp. F the adjustment towards the new equilibrium state is nearly complete at the end of year 10. Thus, when the reduced gravity perturbations were cut

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4

Heaving Modes in the World Oceans

Fig. 4.74 Time evolution of the response of the MOC at 29.5° N in Exp. F, H, J

off, the readjustment process in the system is nearly opposite of that taking place in the first 10 yr, as shown by the thin black curve in Fig. 4.74. As the center of reduced gravity perturbations is gradually moved eastward, the spin-up process gradually slowed down, but the peak value of the MOC increases. In addition, the time when the MOC flips its sign is also postponed. Consequently, in Exp. (H, J) the system does not have enough time to reach a new quasi equilibrium state corresponding to the perturbed reduced gravity distribution, i.e., the MOC is still in the negative phase when the reduced gravity perturbations were cut off. As a result, the system went through the adjustment to the removal of the reduced gravity perturbations with a much large negative MOC peak, as shown by the dashed and thick black curves in Fig. 4.74. The delay of the MOC response along 29.5° N is a function of the distance between the center of reduced gravity perturbations and the western boundary, as shown in Fig. 4.75. In this figure, the horizontal axis is the distance between the western edge of the model basin and the center of convection in units of degree, and the delay time for the MOC minimum is defined as the time duration subtracting the same 10-year duration of the reduced gravity perturbations. This figure suggests that the delay of the MOC maximum is almost linearly proportional to the distance

between the center of the reduced gravity perturbation and the western boundary. What process controls the response of the MOC? Intuitive reasoning may suggest that the westward first baroclinic long Rossby waves emitted from the center of the reduced gravity perturbations may be responsible for this time delay. However, a simple scaling indicates that the corresponding signal speed is about ten times faster than the corresponding long Rossby wave traveling at latitude band of 55° N. The mean signal speed inferred from the MOC maximum is about 2.1 cm/s, as depicted by the dashed line in Fig. 4.75. The mean signal speed of 2.1 cm/s seems too fast for the First baroclinic long Rossby waves in the central latitude of the subpolar basin of this reduced gravity model. To examine this issue clearly, we plot Fig. 4.76, where the solid curve represents the mean speed of the first baroclinic long Rossby waves inferred for the model basin calculated as  x  C ð yÞ ¼ bg0 h0 ð yÞ=f 2 ð yÞ

ð4:40Þ

Rx  x where h0 ð yÞ ¼ x20:5 E1x0:5 E x0:520:5 E E h0 ðx; yÞdx is the mean layer thickness of the unperturbed solution averaged over the western part of the basin. Since our model is a beta-plane model, beta and g′ are both constant—we also omit the potential contribution due to variance in g′, as

4.4 Heaving Induced by Convection Generated Reduced … Fig. 4.75 Delay of the MOC in response to convection in Exp. F, G, H, I, J, K, L, M

227

Delay of MOC response to convection 6

(M )

Max(MOC) Min(MOC)

(L)

5 (K) 4

yr

(J) 3 (I) 2 (H) 1 (G) (F) 0

25

20

15

10

5

30

35

D (W.E. − convection center), in°

Fig. 4.76 Mean long Rossby wave speed inferred from Exp. I (Solid curve) and the mean signal speed inferred from these experiments (dashed line)

Mean Rossby wave speed 3.5 3.0 2.5

c (cm/s)

c=2.1 (cm/s), Inferred from model runs 2.0 1.5 1.0 0.5 0 30N

35N

40N

will be shown below, the control latitude is away from the latitude band of perturbed reduced gravity. We use the unperturbed layer depth averaged over the domain between the western boundary and 20.5° E. The corresponding long Rossby wave speed inferred from the model parameter is shown as the solid curve in Fig. 4.76. The signal speed inferred from these experiments (dashed line in Fig. 4.75) is represented by the dashed line in Fig. 4.76, i.e., it is approximately equal to 2.1 cm/s, and this is about 10 times faster than the corresponding long Rossby waves’ speed at the central latitude of

45N

50N

55N

60N

65N

70N

60° N. Therefore, we postulate that the adjustment process involved in this model is primarily controlled by long Rossby waves emitted from the central longitude and south of the inter-gyre boundary of the model. As shown in Fig. 4.75, if the cooling anomaly is located in the western part of the subpolar basin, it takes about 3–4 years for the anomalous MOC to reach the maximum. On the other hand, if the signals are originating from the middle basin or the eastern part of the subpolar basin, it may take 5–6 years for the establishment of the anomalous MOC.

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4

Heaving Modes in the World Oceans

Fig. 4.77 Time evolution of the AMOC anomaly (regular seasonal cycle removed) based on the GODAS data; the AMOC here is defined by the northward meridional transport over the depth of 100–1000 m

The comparison with reality in the North Atlantic Ocean is shown in Fig. 4.77. The time evolution of the AMOC (regular seasonal cycle removed), defined as the meridional transport over the depth range of 100–1000 m, is shown in the upper panel. It is clear that the AMOC showed positive signals from 1994 to 2004; afterward, the system was dominated by negative signals. The positive AMOC signal maxima first appears around the latitude of 45° N, and it takes about 3 years to propagate to the 26.5° N section of the RAPID line. In addition, it may take about 0.5 year for the AMOC at 45° N to respond to the buoyancy anomaly. Thus, the AMOC at 26.5° N section is about 3.5–4 years delayed relative to the buoyancy anomaly in the subpolar gyre. To compare with the current monitoring program of RAPID LINE, the corresponding

transport across 26.5° N is plotted in the lower panel of Fig. 4.77. Our model suggested that the anomalous buoyancy signals in the subpolar gyre could induce an anomalous AMOC at the subtropical latitudes. To illustrate this idea, we compose Fig. 4.78. First, we calculate the zonal mean density ðr0 Þ averaged over (52.17° N–58.17° N, 0–200 m); from this time series, the seasonal cycle is removed and the absolute value as a function of time and longitude is calculated. At each longitude, the maximum value of the time series is selected and plotted in the upper panel of Fig. 4.78. It is clearly seen that there are two zones of density anomaly extrema: the western basin and the central/eastern basin. To investigate the relation between the density anomaly in the subpolar basin and the AMOC anomaly in the subtropical latitudes, we

4.4 Heaving Induced by Convection Generated Reduced …

229

Fig. 4.78 a Maximum of the absolute value of density perturbation averaged over the top 200 m in the subpolar basin (in units of kg/m3); b phase relation between the density anomaly in the western basin and anomalous AMOC (dAMOC, in units of Sv) at 26.5° N; c phase relation between the density anomaly in the central/eastern basin and anomalous AMOC at 26.5° N. Curves in panel b and c are all normalized

calculated the basin mean density anomaly as follows. The density anomaly in the western subpolar basin is defined as the zonal mean over the longitude range of (52.17° N–58.16° N, 59.5° W–39.5° W, 0–100 m); the density anomaly in the central/eastern subpolar basin is defined as the zonal mean over the longitude range of (52.17° N–58.16° N, 29.5° W–19.5° W, 0– 100 m). Curves in Fig. 4.78b, c are all normalized. In addition, the anomalous AMOC curve is shifted 2.5 (5) year ahead in Fig. 4.78b, c.

With the phase shifting of 2.5 years of dAMOC shown in Fig. 4.78b, it is clear that there is a close relation between the positive density anomaly (during 1991–1996) in the western subpolar basin and the positive anomalous dAMOC (at 26.5° N) with phase shifting of 2.5 years. With the phase shifting of 5 years of dAMOC shown in Fig. 4.78c, it is clear that there is a close relation between the negative density anomaly (during 2001–2008) in the

230

4

central/eastern subpolar basin and the negative anomalous dAMOC (at 26.5° N) with a phase shifting of 5 years. In addition, there is a clear relation between the positive density anomaly (during 1987–1995) in the central/eastern subpolar basin and the positive anomalous dAMOC (at 26.5° N) with phase shifting of 5 years. Therefore, the 2.5-year time delay for density perturbations in the western basin and the 5-year time delay for density perturbations in the central/eastern basin shown in Fig. 4.75 is roughly consistent with the time delay inferred from the GODAS data.

4.4.4 Numerical Experiments with Sinusoidal Reduced Gravity Perturbations Climate variability in the North Atlantic Ocean is also characterized by quasi-period oscillations, such as the AMO. The AMO is a complicated phenomenon deeply rooted in the air-sea coupled climate system. There have been many studies devoted to the mechanisms responsible for the AMO; our focus in this section is to examine the adjustment of wind-driven circulation from the angle of adiabatic motions. We postulate the following scientific problem: how does the winddriven circulation responds to the periodic change of reduced gravity in the upper ocean? This experiment is called Exp. N, that is restarted from the existing quasi-steady state as described above, and run for 60 year subjected to the following reduced gravity perturbations 



y  ys x  xw dg0 ¼ Dg0 exp sin p sin p sinð2pt=DT Þ yn  ys xe  xw

ð4:41Þ ys ¼ 40:5 N, where Dg0 ¼ 0:005 m/s2 ,   yn ¼ 69:5 N, xw ¼ 0:5 E, xe ¼ 29:5 E; DT ¼ 30 yr is the period of the specified oscillations in reduced gravity perturbations. In response to the periodic change in the reduced gravity of the upper layer, the winddriven circulation goes through similar periodic

Heaving Modes in the World Oceans

adjustments, involving the three-dimensional shifting of warm water in the upper layer, as shown in Figs. 4.79, 4.80, 4.81, and 4.82. In the meridional direction, there are warm water volume anomaly maximum and minimum appearing at the mid-latitude band, as shown in Fig. 4.79. The meridional shifting of warm water in the upper layer implies a strong anomalous MOC and MHF, as shown in Fig. 4.80a. For the parameters used in our model, the peak values of the MOC and MHF are on the order of 3 Sv and 140 TW, and these are sizable contributions to the corresponding climate mean values. In the vertical direction, the warm water shifting leads to the strong HC anomaly (Fig. 4.81), that is in the form of baroclinic modes due to the basic assumption of adiabatic adjustment. In the zonal direction, the warm water shifting leads to banded patterns of positive and negative volumetric anomalies mostly confined to the western part of the model basin (Fig. 4.82). Such zonal shifting of warm water implies a strong ZOC and ZHF (Fig. 4.83); the amplitude of ZOC and ZHF is on the order of 2 Sv and 80 TW; they are clearly important contributions to climate variability.

4.5

Heaving Induced by Deep Convection Generated Volume Loss

4.5.1 Introduction As discussed in the previous sections, a strong buoyancy anomaly can induce a large density anomaly in the subpolar basin interior, and the dynamical consequence can be simulated by a generalized reduced gravity model in two ways. The first way is to treat the problem as a decline of reduced gravity in the upper layer, the focus in the previous section. The second way is to treat the problem as a sudden loss of warm water volume in the upper layer. As shown in Fig. 4.84a, the initial perturbation in upper layer

4.5 Heaving Induced by Deep Convection …

231

Fig. 4.79 Time evolution of the volume anomaly in Exp. N. a Time evolution of the meridional distribution of the volumetric anomaly; b, c meridional section view at year 28 and 39; d time evolution of the specified reduced gravity anomaly

Fig. 4.80 Time evolution of the MOC in Exp. N. a Time evolution of the meridional distribution of the MOC anomaly; b, c meridional section view at year 18 and 34; d time evolution of the specified reduced gravity anomaly

232

4

Heaving Modes in the World Oceans

Fig. 4.81 Time evolution of the vertical distribution of the HC anomaly in Exp. N. a Time evolution of the vertical distribution of the HC anomaly; b, c vertical section view at year 21 and 38; d time evolution of the specified reduced gravity anomaly

Fig. 4.82 Time evolution of the zonal volumetric anomaly in Exp. N. a Time evolution of the zonal distribution of volumetric anomaly; b, c zonal section view at year 22 and 41; d time evolution of the specified reduced gravity anomaly

4.5 Heaving Induced by Deep Convection …

233

Fig. 4.83 Time evolution of the ZOC in Exp. N. a Time evolution of the meridional distribution of the ZOC anomaly; b, c zonal section view at year 17 and 36; d time evolution of the specified reduced gravity anomaly

(a) Horizontal view

0

70N

(b) y−z view

H

Z

0

0

E

70S

70S

Z

(c) x−z view

70N

(d) Vertical HCa Z

_ +

W

E

0

H

Fig. 4.84 a Sketch of the adjustment induced by the sudden decline of the upper layer thickness in a simple reduced gravity model; b anomalous meridional overturning cell; c anomalous zonal overturning cell; d vertical heat content anomaly

thickness is specified within the green ellipse, and the center of peak value is marked by the blue color. The dynamical adjustment after the initial loss of warm water volume is as follows. In a steady state the W–E slope of the thermocline in the upper layer is linearly proportional to the Ekman pumping and inversely proportional to the square root of the reduced gravity. Since wind stress and reduced gravity in the upper ocean are kept unchanged during the experiments, the thermocline in the upper ocean of the subpolar gyre tries to recover the same shape defined in the unperturbed state. As a result, the warm water in low latitude is transported northward to refill the volume deficit left by the strong cooling anomaly. The signals propagate in the form of the first baroclinic long Rossby waves moving towards the western boundary in the subpolar basin, depicted by the black arrows in the western part of the subpolar basin.

234

When these long Rossby waves reach the western boundary, a small portion of their energy is reflected in the form of eastward short Rossby waves, mostly dissipated near the western boundary. The major part of the signals and energy is converted into the equatorward Kelvin waves; they turn into the eastward equatorial Kelvin waves after reaching the Equator. When these equatorial Kelvin waves reach the eastern boundary, they bifurcate and become the poleward coastal Kelvin waves. Because of the increase of the Coriolis parameter, these coastal Kelvin waves gradually shed their energy in the form of the westward first baroclinic long Rossby waves. The combination of these waves indicated by the black arrows in Fig. 4.84a helps the system to readjust towards a new state. In particular, in the Northern Hemisphere the movement of warm water from low latitudes to high latitudes manifests as a positive MOC cell in the whole basin (Fig. 4.84b). Note that the adjustment involves a threedimensional redistribution of warm water in the upper ocean. In particular, there are anomalous ZOCs and a heat content anomaly in the vertical direction (Fig. 4.84c, d). The details of threedimensional shifting of warm water will be discussed in details in this section.

4.5.2 Model Formulation The model is the same as the one used in the previous section. First, we run the model with constant g0 ¼ 0:01 m/s2 and an initial uniform layer depth of 600 m for 200 years to guarantee that the model reaches a quasi-equilibrium reference state.

4.5.3 Results of Numerical Experiments We restart the model from the existing equilibrium state and subjected to a volumetric loss dhðx; yÞ at the restarting time t = 0, with no

4

Heaving Modes in the World Oceans

change in wind stress, reduced gravity, and including a thermally insulation condition. 





y  ys x  xw p sin p dh ¼ Dh exp sin yn  ys xe  xw ð4:42Þ where Dh ¼ 250 m is the amplitude of volumetric loss, yn ¼ 69:5 N, ys ¼ 40:5 N are the northern and southern edges of the volumetric loss perturbations, xw and xe are the western and eastern edges of the volumetric loss perturbations. As an example, the spatial pattern of the volumetric loss because of cooling for the case with xw ¼ 0:5 E and xe ¼ 29:5 E is shown in Fig. 4.85. Five numerical experiments were carried out with the initial layer thickness loss specified as in Table 4.4. Results from experiments (A, C, E) are shown in Fig. 4.86, including the zonal profile of the initial layer thickness loss and the time evolution of the anomalous MOC. When the layer thickness loss, specified as the initial perturbation, is imposed near the western boundary, the anomalous MOC is established quickly, and corresponding amplitude is large, as shown in Exp. A and C. On the other hand, if the initial layer perturbation is specified in the central basin, the anomalous MOC is established much slowly and the corresponding amplitude is much smaller, as shown in Exp. E. The time evolution of the MOC can be seen clearly for the 29.5° N section, Fig. 4.87. The time delay of the anomalous MOC maximum is roughly proportional to the distance between the western boundary and the center of the layer thickness perturbations (Fig. 4.88). For Exp. A, B and C, the center of layer thickness perturbation is mostly confined to the western basin; the corresponding anomalous MOC maximum is within the range of 1–5 years. On the other hand, if the center of layer thickness loss is located in the central or eastern basin, the responding time is on the order of 10 years.

4.5 Heaving Induced by Deep Convection … Fig. 4.85 Upper layer thickness reduction due to strong convection

235 Thickness of layer lost to convetion (m)

70N −60

60N 50N

−80

40N

−100

30N 20N

−120

10N

−140

0 10S

−160

20S

−180

30S −200

40S

−220

50S 60S

−240

70S

10E

20E

30E

40E

50E

Table 4.4 Domain of the initial layer thickness perturbations Experiment

A

B

xw

0.5° E

0.5° E

xe

8.5° E

23.5° E

However, for the North Atlantic Ocean, volumetric loss due to deep convection takes place primarily within the western basin, the Labrador Sea; thus, our discussion below will be focused on the cases with convection induced volumetric loss confined to the western basin, i.e. the situation similar to Exp. A, B and C. The delay of 3–5 years shown in Exp. B and C is roughly the same as the time delay diagnosed from the GODAS data, as discussed in the previous section. It is speculated that the first baroclinic long Rossby wave may play a vital role in the adjustment process. To explore this issue, we plot the equivalent wave speed diagnosed from our experiments (dashed lines in Fig. 4.89). Although these two dashed lines indicate a slightly different phase speed for the long Rossby wave inferred from the model runs, it is clear that they are much faster than the equivalent first baroclinic long Rossby waves inferred from the layer thickness of the undisturbed solution. As

C

D

E

0.5° E

11.5° E

22.5° E

35.5° E

46.5° E

57.5° E

discussed above, the center of volumetric loss is mostly confined to the western basin; thus, the wave speed represented by Exp. B is more relevant. As such, the intergyre boundary is the critical latitude where the first baroclinic long Rossby waves play the role of setting the time scale of the adjustment induced by the initial volumetric loss. The volumetric loss specified at the beginning of these numerical experiments induces a rapid adjustment of the circulation. In a steady state the W–E slope of the thermocline is set by the wind stress pumping and the reduced gravity. When the model is restarted from the existing quasisteady state and subjected to a sudden loss of warm water from the upper layer, the circulation system quickly responds and tries to fill the volumetric deficit in the western subpolar basin. In such basin scale adjustment processes, the first baroclinic long Rossby waves and the boundary Kelvin waves play the vital roles of redistributing warm water in the upper layer; as a result, the

236

4

Heaving Modes in the World Oceans

Response to convection loss 0 −1 −2

(a) 0

10E

60N 40N 20N 0 20S 40S 60S

20E

30E

40E

50E

60E 2 1

(b)

0 20

15

10

5

1

yr 0 −1 −2

(c) 0

10E

60N 40N 20N 0 20S 40S 60S

20E

30E

40E

50E

60E 1.5 1.0 0.5

(d)

0

1

5

10

15

20

yr 0 −1 −2

(e) 20E

10E

0 60N 40N 20N 0 20S 40S 60S

30E

50E

40E

60E 0.8 0.6 0.4 0.2 0 −0.2

(f) 1

5

10

15

20

yr

Fig. 4.86 Exp. (A, C, E): Change of the MOC in response to convection volumetric loss of the upper layer: panels a, c, and e indicate upper layer thickness change (along latitude circle of 55° N) due to convection; panels b, d, and f show the time evolution of the MOC, in units of Sv

Fig. 4.87 Response of the MOC (at 29.5° N) to the convection volumetric loss for Exp. A, B, C, D, and E

4.5 Heaving Induced by Deep Convection … Fig. 4.88 Delay of the maximum MOC response as a function of the distance between the western boundary and the longitudinal center of the convection volumetric loss

237 Delay of MOC response to convection

15

E 10

yr

D

C

5 4 3 2 1

B A 0

5

10

35

30

25

20

15

40

45

D (W.E. − convection center), in°

Mean Rossby wave speed 3.5 3.0 2.5

c (cm/s)

Fig. 4.89 Mean long Rossby wave speed inferred from Exp. B and E; the solid curve corresponding to the speed of the first baroclinic long Rossby wave inferred from the reference state of the model ocean

2.0 c=1.50 (cm/s), inferred from Exp. B

1.5

c=1.17 (cm/s), inferred from Exp. E

1.0 0.5 0 30N

35N

40N

volumetric deficit in the subpolar basin is gradually refilled with warm water transported from other parts of the model ocean. As an example, the time evolution of zonally integrated meridional distribution of the volumetric anomaly is shown in Fig. 4.90, including the volumetric loss specified at the beginning of the numerical experiments, shown as the curve in Fig. 4.90b. Note that the volumetric anomaly shown in this figure and the following figures is referred to in the initial state of the numerical experiments. As time progresses, the volumetric deficit in the subpolar basin is gradually filled up. At year 20, the accumulation of the warm water in the subpolar basin is nearly filled up to compensate the initial volume deficit due to the sudden loss associated with strong cooling; as a

45N

50N

55N

60N

65N

70N

compensation, there is now a volumetric deficit away from the subpolar basin in the Northern Hemisphere (Fig. 4.90c). In the zonal direction, warm water in the upper layer is transported from east to west, so that there is a gradual pileup of water in the western basin (red color patch in Fig. 4.91). As time progresses, the volumetric deficit in the western basin, as shown in panel b, is gradually filled up. At year 20, the accumulation of the warm water in the eastern basin is nearly filled up to compensate for the initial volume deficit due to the sudden loss associated with strong cooling (Fig. 4.91c); in compensation, there is now a volumetric deficit in the central and eastern basins. In the vertical direction, there was a large volume deficit in the western subpolar basin

238

4

Heaving Modes in the World Oceans

Fig. 4.90 a Time evolution of meridional volumetric anomaly for Exp. B; b meridional distribution of the convection volumetric loss; c volumetric changes at the end of the 20-year experiment

Fig. 4.91 a Time evolution of the zonally volumetric anomaly for Exp. B; b zonal distribution of the convection volumetric loss; c volumetric changes at the end of the 20-year experiment

4.5 Heaving Induced by Deep Convection …

239

Fig. 4.92 a Time evolution of the vertical HC anomaly for Exp. B; b vertical distribution of the convection HC loss; c volumetric changes at the end of the 20-year experiment

specified at the beginning of the numerical experiment. Because the layer is relatively shallow in the western subpolar basin, this initial layer depth change is projected as a volumetric deficit at the shallow depth range, with a maximum at the 200 m level (Fig. 4.92b). As time progresses, the volumetric deficit at this shallow depth level in the subpolar basin is gradually filled up. At year 20, the accumulation of the warm water at the shallow level in the subpolar basin is partially filled up to compensate the initial volume deficit due to the sudden loss associated with strong cooling; in compensation, there is now a volumetric deficit at the 600 m level, corresponding to the base of the thermocline at the subtropical basin in both hemispheres. Note that we count the volumetric anomaly relative to the initial state, including the volume loss specified at the initial time. As the model is subjected to the adiabatic assumption, there is no net heat content anomaly. Consequently, the heat content anomaly should be in

the form of a baroclinic mode, i.e., the vertical integration of the heat content anomaly should be zero (Fig. 4.92c). The three dimensional transport of warm water can be projected into many different indexes of the circulation, such as the MOC and MHF. In fact, there is a strong anomalous MOC during the adjustment process, with a peak value of 1.7 Sv at year 2.7, and this also corresponds to a peak value of 0.07 PW (Fig. 4.93b). Even after averaging over the 20 years of experimentation, the mean MOC (MHF) is about 1 Sv (0.04 PW) (Fig. 4.93c). Since the climatological mean MHF at this latitude is on the order of 1 PW, the MHF perturbation induced by deep convection represents a non-negligible component of climate variability. The three dimensional transport of warm water also has a sizeable projection on the zonal direction. As shown in Fig. 4.94, there is a strong anomalous ZOC during the adjustment process, with a peak value of 1.4 Sv at year 2.7, and this

240

4

Heaving Modes in the World Oceans

Fig. 4.93 a Time evolution of the MOC for Exp. B; b meridional profile of the MOC (MHF) averaged over the 20year experiment

(a) ZOC (106m3/s)

(b) ZOC(ZHF), yr 2.7 (c) Mean ZOC(ZHF)

0

1.5

10E

1.0

20E

0.5

30E

0

40E

−0.5

50E

−1.0

60E

0

5

10 yr

15

20

−1.5 −1 −0.5

0

−1

106m3/s −60 −40 −20 TW

−0.5

0

106m3/s 0

−40

−20 TW

0

Fig. 4.94 a Time evolution of the ZOC for Experiment B; b meridional profile of the ZOC (ZHF) averaged over the 20-year experiment

4.5 Heaving Induced by Deep Convection …

also corresponds to a peak value of 0.05 PW (Fig. 4.94b). Even after averaging over the 20 years of the experiment, the mean ZOC (ZHF) is about 0.7 Sv (0.03 PW) (Fig. 4.94c); thus, these are non-negligible components of climate variability.

4.6

ENSO Events and Heaving Modes

Based on the GODAS data, the anomalous MHF (ZHF and VHF) induced by adiabatic motions consists of more than 80% of the climate variability signals on the inter-annual to decadal time scales. Hence, adiabatic heaving induced signals may consist of a major part of climate signals on decadal time scale and thus play an important role in the oceanic circulation and climate change. A simple reduced gravity model can be used to explore the heaving mode associated with ENSO events. However, a simple reduced gravity model represents the continuous stratification in density coordinates with a single active grid only; for this reason, the stretching mode and spicing mode cannot be accurately explored by such a simple model. Using a simple reduced gravity model, the anomalous MHF, ZHF and VHF induced by adiabatic heaving modes associated with idealized ENSO cycles in the world oceans are examined. For example, in a Pacific-like model basin the 4-year cycle of the equatorial easterlies can lead to anomalous MHF (ZHF and VHF) with the magnitude of 0.05 PW (0.3 and 0.6 PW).

4.6.1 Introduction Our interest here is focused on the Pacific Ocean and in particular the equatorial band. First, using the GODAS data we recalculate the meridional heat flux for the Pacific basin, as defined by the dashed lines in Fig. 4.95. In addition, the zonal heat flux in the Pacific basin is defined within the

241

equatorial band of 15.17° S to 15.17° N. As shown in Fig. 4.95, the net air-sea heat flux in the ocean is spatially non-uniformly distributed. The most outstanding features in this map are the strong heat flux into the cold tongues in the equatorial Pacific and the strong heat release to the atmosphere in the Kuroshio. This map implies that there should be a mechanism in the ocean that transports heat in both the meridional and zonal directions. In fact, there are strong MHF and ZHF in the Pacific Ocean (Fig. 4.95b, c). The MHF is calculated by integrating the surface heat flux, starting from the edge of Antarctica and moving northward. The MHF is poleward in both hemispheres. In the Northern Hemisphere, it reaches the maximum value of 0.84 PW. The ZHF in the Pacific basin discussed here is defined for the equatorial band, and it is set to zero at the east most grid point in each latitude; the eastward heat flux is defined as positive. As shown in Fig. 4.95c, in the steady state, there is a huge westward ZHF in the equatorial Pacific basin, and it reaches the maximum amplitude of −1.2 PW. The east-west asymmetric feature of the net air-sea heat flux and the implied ZHF is a critically important component in the climate system, that has not received adequate attention. However, we postulate that in the study of climate change, it would be of great interest to examine the variability of the air-sea heat flux and the associated ZOC and ZHF. As discussed above, in the steady state both the MHF and ZHF are closely linked to thermal diffusion across the interface of the oceanatmosphere or between water parcels; i.e., these are closely linked to turbulent diabatic motions in the oceans. In addition, there is also the VHF that is related to the turbulent diffusion in the vertical direction. ENSO cycles are one of the most important dynamic components of climate variability in the Earth system. Over the past couple of decades, many studies have been devoted to the exploration of ENSO dynamics, its generation and effects on global climate. There are many theories about ENSO dynamics. In particular, Jin

242

4

Heaving Modes in the World Oceans

Fig. 4.95 a Net air-sea heat flux into the ocean; b meridional heat flux inferred from the surface heat flux for the Pacific basin (defined by the heavy dashed lines in a); c zonal heat flux inferred from the surface heat flux for the equatorial Pacific basin

(1997) postulated the recharge paradigm of ENSO, in which the meridional heat advection plays a key role; this is further explained by Jin and An (1999) in their study emphasizing the important role of heat transport associated with the meridional and zonal advection. A similar idea, in particular in connection with the volumetric anomaly of the warm pool, has been further explored by Meinen and McPhaden (2000) and Clarke et al. (2007). In this section, we will explore the three dimensional heat fluxes associated with the quasi four-year cycle of the ENSO events from both the realistic view diagnosed from the GODAS data and the heaving mode associated with a simple reduced gravity model.

4.6.2 Variability of Heat Content and Horizontal Heat Fluxes Due to ENSO Diagnosed from the GODAS Data Since temperature observations in the upper kilometer are more reliable, our discussion about the HC anomaly will be focused on the upper kilometer only. As an example, we analyzed the HC anomaly for the Pacific Basin over the area defined by the dashed lines in Fig. 4.95a. Zke dH ðz; tÞ ¼ C

Z/n dk

kw

  q0 r 2 cos /Cp h   h d/

/s

ð4:43Þ

4.6 ENSO Events and Heaving Modes

243

Fig. 4.96 Time evolution of the baroclinic mode of the HC anomaly for the upper kilometer of the Pacific basin (a), and the corresponding equatorial easterlies (subjected to 7 month moving smoothing) in the central Pacific basin (b)

where h is monthly mean potential temperature. From this data array we further separated the barotropic mode of the HC anomaly for each month and the resulted monthly mean baroclinic mode of the HC anomaly field is shown in Fig. 4.96a. The corresponding time series of the equatorial easterlies, defined as the zonal wind stress averaged over the domain of (0.33° S– 0.33° N, 180°–140° W) subjected to 7-month smoothing is shown in Fig. 4.96b. This figure shows that the equatorial easterlies are closely related to the baroclinic mode of the HC anomaly. There is high frequency activity of the baroclinic mode of the HC anomaly, in particular in the upper 250 m. There are also strong baroclinic HC anomalous signals even at the 600 m level. To demonstrate such a

phenomenon, we plot the time evolution of the barotropic mode of the HC anomaly, maximum/minimum of the baroclinic mode of the HC anomaly, plus the baroclinic mode of HC anomaly at the 590 m level (Fig. 4.97a). First of all, the black curve in this figure shows the barotropic mode of the HC anomaly for the Pacific basin; it is clear that the whole Pacific basin had cooled down in 1980–1986; afterward, the whole basin was gradually warmed up. There are strong cold baroclinic signals for the whole time period. In particular, there were baroclinic signals at the 590 m level during 2004–2007 and 2012–2015, when the equatorial easterlies were strong. Due to the complicated oceanic dynamics leading to the baroclinic mode, the timing of the baroclinic mode maximum/minimum is not

244

4

Heaving Modes in the World Oceans

Fig. 4.97 Time evolution of the barotropic mode of the HC anomaly, baroclinic mode of the HC anomaly maximum/minimum, the baroclinic mode of the HC anomaly at 590 m, and the equatorial easterlies, all subjected to 7month moving smoothing

Table 4.5 Correlation between the baroclinic mode of the HC anomaly maximum/minimum and the zonal wind stress in the equatorial central Pacific C dHbc;max

C dHbc;min

C dHbc;590 m

Delay (month)

0

9

−3

Correlation

0.4577

0.3894

−0.1844

exactly synchronized with the equatorial easterlies. To reveal their statistical relation, we calculated their correlation as shown in Table 4.5. It is readily seen that the correlation between the baroclinic mode of the HC anomaly maximum/minimum is very high, with the value

C of 0.4577 (no time lag) for dHbc;max and 0.3894 C (with 9 month lag) for dHbc;min . The correlation C between dHbc;590 and the equatorial easterlies is m −0.1844; however, this maximum correlation is C realized for dHbc;590 m 3 months ahead of the equatorial easterly signals. Although the correlation of −0.1844 is relatively low, it implies the physical connection between these time series. As discussed above, in the steady state both the MHF and ZHF are closely linked to thermal diffusion across the interface of the oceanatmosphere or between water parcels; in another word, these are closely linked to diabatic

4.6 ENSO Events and Heaving Modes

motions in the oceans. However, heaving in the ocean induced by wind stress perturbations on inter-annual and decadal time scales may also generate horizontal heat fluxes in both the meridional and zonal directions, as discussed below.

245

dominated by strong annual cycle with the amplitude of 8 PW. By subtracting the climatological mean annual cycle, we obtain the interannual variability of MHF, denoted as HF dMSF ð/; tÞ: HF HF HF ð/; tÞ dMSF ð/; tÞ ¼ MSF ð/; tÞ  MSF

ð4:45Þ

4.6.3 Meridional Heat Flux Our second step is to calculate the MHF associated with the surface heat flux, and this is defined by the surface integration of the air-sea heat flux hsurf as follows Zke HF MSF ð/; tÞ

¼

Z/ dk

kw

h

surf 2

r cos /d/ ð4:44Þ

/s

where kw and ke are the western and eastern boundaries of the Pacific basin, as depicted by the heavy dashed lines in Fig. 4.95a; /s is the southern boundary of the basin, i.e. the edge of the Antarctica. Samples of the time series of MHF are shown in Fig. 4.98a, and this is

annual cycle

As shown in Fig. 4.98b, the typical amplitude of this term is 1.5 PW. In parallel to the MHF induced by surface heat flux, we can also calculate the MHF associated with meridional advection as follows Zke HF MVT ð/; tÞ

¼

Z0 q0 r cos /Cp vðh  h0 Þdz

dk kw

D

ð4:46Þ where Cp is the specific heat under constant pressure h is the potential temperature of sea water h0 is the reference potential temperature, and this is set to 0 °C in common practice. It is important to emphasize that if there is a net

Fig. 4.98 Time evolution of meridional heat fluxes, vertical axis is time in years: a Inferred by zonally integrated surface heat flux ZSHF; b inter-annual variability, by subtracting the climatological mean annual cycle from panel a; c Inter-annual variability of meridional advective heat flux; all fluxes in units of PW

246

4

meridional volume flux through a section, the choice of this reference may affect the advective heat flux through this section; thus, the selection of h0 should be a careful one. For the Pacific Ocean, there is the Indonesian through flow near the Equator; as a result, the calculation of meridional heat flux may depend on the choice of h0 . However, our focus in this chapter is the inter-annual variability of the meridional heat flux in the Pacific Ocean, and the separation of the contribution associated with the Indonesian Throughflow is left for further study. Therefore, in the discussion hereafter, we set h0 ¼ 0  C, following common practice. In a steady ocean state, the meridional heat advection term is entirely due to the surface airsea heat flux. In a non-steady state, however, this horizontal advection term is associated with both the diabatic and adiabatic motions in the ocean. Similar to the case of the MHF induced by the surface heat flux, we also have subtracted the climatological annual cycle of the MHF associated with meridional advection HF HF HF ð/; tÞ dMVT ð/; tÞ ¼ MVT ð/; tÞ  MVT

annual cycle

ð4:47Þ HF ð/; tÞ and samples of the time evolution of dMVT are shown in Fig. 4.98c. Because of the nonlinear dynamical interaction, however, it is impossible to separate contributions from the adiabatic and diabatic processes. Therefore, we will use the following terminology: the external mode and the internal mode of heat flux. The MHF inferred from the surface air-sea heat flux alone is called the external mode; on the other hand, MHF associated with the meridional advection term is called the internal mode because it is not directly linked to the surface air-sea heat flux. From Fig. 4.98 it is readily seen that the amplitude of the internal mode variability of the MHF associated with the meridional advection is about 3–4 time larger than that associated with the air-sea heat flux, i.e., the amplitude of the

Heaving Modes in the World Oceans

internal mode is about 3–4 times larger than that of the external mode. To demonstrate the relation between these two MHFs, we define the following terms: HF dMVT;max ðt Þ ¼ HF dMVT;min ðt Þ ¼ HF dMSHF;max ðt Þ ¼ HF dMSHF;min ðt Þ ¼



 HF dMVT ð/; tÞ





max

/s / /n

min

/s / /n

max



/s / /n

min

/s / /n

HF dMVT ð/; tÞ

HF dMSHF ð/; tÞ



ð4:48Þ ð4:49Þ ð4:50Þ

 HF  dMSHF ð/; tÞ ð4:51Þ

Therefore, these represent the maximum (minimum) value of the MHF, associated with the meridional advection flux (Eqs. 4.48 and 4.49) and the surface air-sea heat flux (Eqs. 4.50 and 4.51), over the whole meridional range of the model ocean. The time evolution of these terms (subjected to 7 month smoothing) is shown in Fig. 4.99. To clearly show their relative magnitude, their root-mean-square values (before 7 month smoothing) are also included as numbers and dashed lines in the same color in this figure. To show the connection with the ENSO dynamics, we also include the Oceanic Nino Index (ONI) taken from the Climate Prediction Center of the National Weather Service website: http://www.cpc.ncep.noaa.gov/products/ analysis_monitoring/ensostuff/ensoyears.shtml. This figure shows that the MHF maximum/ minimum are closely linked to the equatorial easterlies and the ENSO dynamics (as represented by ONI). For example, the strongest El Nino events on record (1982–1983)/(1997–1998) and the strongest La Nina event on record (1998– 2001) are closely tracked by the peaks of HF HF dMVT;max and dMVT;min . Dynamically, the peaks HF HF of dMVT;max and dMVT;min represent the contributions to meridional heat flux due to meridional transport of warm water associated with the ENSO cycle. Although most studies about ENSO dynamics have been focused on the zonal transportation of mass and heat, our analysis

4.6 ENSO Events and Heaving Modes

247

Fig. 4.99 Upper panel: Time series of the inter-annual variability of: the maximum and minimum of the meridional heat flux (MHF), the maximum and minimum of the meridional heat flux inferred from the meridional integration of surface heat flux (SHF); ONI is the Oceanic Nino Index. Lower Panel: equatorial zonal wind stress. All curves, except Nino Index, are subjected to 7 month smoothing

demonstrates that the meridional adjustment of warm water mass and the associated meridional heat flux associated with the ENSO cycle are also important components of the climate system. In addition, this figure also shows that the character of the ENSO events apparently changed around the beginning of this century, and this is also HF HF reflected in the ONI, the dMVT;max and dMVT;min . The close relationship between the equatorial easterlies and the MHF anomaly maximum and minimum is also reflected in terms of the correlation between them. As shown in the first two

columns of Table 4.6, the correlation is 0.3025 and −0.1739 (all passed the 95% confidence test), respectively. In Fig. 4.99 we also include the information related to the zonal maximum and minimum of the MHF inferred from the zonal integration of the surface air-sea heat flux, depicted by the black and green curves in the upper panel of Fig. 4.99. These curves are different from that HF HF representing dMVT;max and dMVT;min . Most importantly, the major EL Nino events on record (1982–1983)/(1997–1998) and La Nina event on

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Heaving Modes in the World Oceans

Table 4.6 Correlation between MHF anomaly maximum/minimum associated with meridional advection and the zonal wind stress in the equatorial central Pacific HF dMVT;max

HF dMVT;min

HF dMSHF;max

HF dMSHF;min

Delay (month)

4

−5

19

14

Correlation

0.3025

−0.1739

0.1786

0.1823

Table 4.7 Fluxes in units of PW, flux divergence in units of PW/1° Mode

MHF

ZHF

HF dMVT

HF dZUT

Max. Internal External Ratio (%)

1.510 0.675 69.0

Min.

Max.

1.408 0.685 67.3

record (1998–2001) do not appear as any major HF HF peaks in dMSHF;max and dMSHF;min . This fact that air-sea heat flux is not particularly strong during the strong ENSO events in history suggests that the air-sea flux associated with the ENSO cycle is not the dominating player in regulating the strength of the ENSO cycle. On the other hand, these two time series are also statistically related to the equatorial easterlies, although the correlation is small, 0.1786 HF dMSHF;max (with 19 month delay) and 0.1283 for HF (with 14 month delay). The much dMSHF;min longer delay reflects the fact that change in airsea heat flux is mostly a dynamical consequence of the meridional heat advection. HF As discussed above, dMVT ð/; tÞ and HF dMSF ð/; tÞ represent the internal mode and external mode of the MHF. To compare the relative magnitude of these variables, we calculated the root-mean square values of the corresponding maximum and minimum (before taking the 7month smoothing); the corresponding value of these root-mean square values are plotted as the dashed lines in Fig. 4.99. As listed in Table 4.7, the root-mean square values of the MHF maximum and minimum associated with the

Min.

2.476

2.586

0.389

0.405

86.4

86.5

meridional advection are much larger than that associated with the surface air-sea heat flux. Therefore, the MHF anomaly associated with the ENSO cycle is mostly (more than 67%) adiabatic motion of warm water in the upper ocean.

4.6.4 Zonal Heat Flux Our third step is to calculate the ZHF associated with the surface heat flux, that is defined by the surface integration of the air-sea heat flux hsurf as follows Zke HF ZSF ðk; tÞ

¼

Z/n dk

k

hsurf r 2 cos /d/

ð4:52Þ

/s

where /s is the southern boundary of the basin, i.e. the edge of Antarctica, /n ¼ 64:75 N is the northern boundary of the data domain, ke is the eastern boundary of the Pacific basin, as depicted by the heavy dashed line in Fig. 4.95a. Samples of the time series of ZHF are shown in Fig. 4.100a; these time series are dominated by a strong annual cycle with the amplitude of 3 PW. By subtracting the climatological mean annual

4.6 ENSO Events and Heaving Modes

249

Fig. 4.100 Time evolution of zonal heat fluxes in the equatorial Pacific (15.17° S–15.17° N), vertical axis is time in year: a Inferred by zonally integrated surface heat flux ZSHF; b inter-annual variability, by subtracting the climatological mean annual cycle from panel a; c Inter-annual variability of zonal advective heat flux; all fluxes in units of PW

cycle, we obtain the inter-annual variability of HF ZHF, denoted as dZSF ðk; tÞ: HF HF HF ðk; tÞ ðk; tÞ ¼ ZSF ðk; tÞ  ZSF dZSF

annual cycle

ð4:53Þ As shown in Fig. 4.100b, the typical amplitude of this term is 1.5 PW. In parallel to the ZHF induced by surface heat flux, we can also calculate the ZHF associated with zonal advection as follows Z/n HF ZUT ðk; tÞ

¼

Z0 q0 r cos /Cp u hdz

d/ /s

ð4:54Þ

D

In a steady ocean state, the zonal heat advection term is entirely due to the surface airsea heat flux. In a non-steady state, however, this horizontal advection term is also linked to both the diabatic and adiabatic motion in the ocean.

Similar to the case of the ZHF induced by surface heat flux, we have also subtracted the climatological annual cycle of the ZHF associated with zonal advection HF HF HF ðk; tÞ dZUT ðk; tÞ ¼ ZUT ðk; tÞ  ZUT

annual cycle

ð4:55Þ

HF Samples of the time evolution of dZUT ðk; tÞ are shown in Fig. 4.100c. Similar to the case of meridional heat flux, it is impossible to separate contributions from these two types of motion: the adiabatic and diabatic motions. Therefore, we will separate the zonal heat flux into external and internal modes. The ZHF inferred from the surface air-se heat flux alone is called the external mode; on the other hand, the ZHF associated with the zonal advection term is called the internal mode because it is not directly linked to the surface air-sea heat flux. From Fig. 4.100, it is readily seen that the amplitude of the internal mode variability of the ZHF associated with the zonal advection is about

250

4

4–6 times larger than that associated with the airsea heat flux, i.e., the amplitude of the internal mode is about 4–6 times larger than that of the external mode. To demonstrate the relation between these two ZHFs, we define the following terms: HF dZUT;max ¼ HF dZUT;min ¼

max



 HF dZUT ðk; tÞ





kw k ke

min

kw k ke

HF dZUT ðk; tÞ

ð4:56Þ ð4:57Þ

Heaving Modes in the World Oceans

HF ¼ dZSHF;max HF dZSHF;min ¼

max



kw k ke

min

kw k ke



HF dZSHF ðk; tÞ

HF dZSHF ðk; tÞ





ð4:58Þ ð4:59Þ

Therefore, these represent the maximum (minimum) value of the ZHF, associated with zonal advection (Eqs. 4.56 and 4.57) and the surface air-sea heat flux (Eqs. 4.58 and 4.59) over the whole meridional range of the model ocean. The time evolution of these terms (subjected to 7 month smoothing) is shown in Fig. 4.101. To

Fig. 4.101 Upper panel: Time series of the inter-annual variability of: the maximum and minimum of the zonal advective heat flux (ZHF) (subjected to 7 month moving smoothing), the maximum and minimum of the meridional heat flux inferred from the zonal integration of surface heat flux (SHF). Lower Panel: equatorial zonal wind stress (subjected to 7 month moving smoothing)

4.6 ENSO Events and Heaving Modes

251

clearly show their relative magnitude, their rootmean-square values (before 7 month smoothing) are also included as numbers and dashed lines in the same color in this figure. To show the connection with the ENSO dynamics, we also include the Oceanic Nino Index (ONI). This figure shows that the maximum/ minimum of the zonal advective heat flux (Eqs. 4.56 and 4.57) are closely linked to the equatorial easterlies and the ENSO dynamics (as represented by ONI). For example, during the strongest El Nino events on record (1982–1983)/ (1997–1998) and the strongest La Nino events on record (1998–2001) the advective ZHF were extremely strong. The character of the ENSO events apparently changed around the beginning of this century, and this is also reflected in the HF ONI and the dZUT;min . The close relationship between the equatorial easterlies and the advective ZHF anomaly maximum and minimum is also reflected in terms of the correlation between them. As shown in the first and second columns of Table 4.8, the correlation between them are 0.3077 and −0.4822 (all passed the 95% confidence test), respectively. On the other hand, the correlation between the ZHF associated with the surface heat flux and the equatorial wind is much smaller, with the values of 0.2025 and −0.1393 only. In addition, the delay time for the maximum surface heat flux is 9 months, i.e., 3 months more than the maximum of the advective flux maximum; this also suggests that there is further delay between the zonal advection of heat and the air-sea heat flux induced by changes in the surface sea temperature and the air-sea interaction afterward. Similar to the case of the MHF, the root-mean square values of the ZHF maximum and minimum associated with the zonal advection are much larger than that of the surface air-sea heat

flux. Therefore, the ZHF anomaly of the ENSO cycle is mostly (>86%) adiabatic motions of warm water in the upper ocean, as shown in the third and fourth columns in Table 4.7.

4.6.5 Vertical Heat Flux We begin with the vertical mass transport in the world oceans because it is closely linked to the ENSO; since the GODAS data is obtained from a model based on the Boussinesq approximations, we present the vertical volume flux (VVF) instead. ZZ VVF ¼

basin wr

2

dkd/

ð4:60Þ

For the major basins in the world oceans there are both upwelling and downwelling and the corresponding basin integrated VVF is shown in Fig. 4.102. Accordingly, the sum of upwelling and downwelling in each basin is huge, or the order of 108 m3/s. There is net upwelling on the order of 106 m3/s in most basins, except at the shallow depth of 100–1400 m in the Atlantic basin (Fig. 4.102e). Physically, in a volume conserving model there should be no net vertical volume flux. However, in most numerical models based on the hydrostatic approximation the vertical velocity is calculated from the divergence of the horizontal velocity. As such, vertical velocity in these models is quite noisy, and the vertical velocity provided in the monthly mean data sets, such as the GODAS data, may look smooth after some selected smoothing methods; however, such a vertical velocity field will not exactly satisfy such a physical constraint. Although one can certainly rescale the vertical velocity applying the zero net

Table 4.8 Correlation between the ZHF anomaly maximum/minimum associated with zonal advection and surface heat flux and the zonal wind stress in the equatorial central Pacific HF dZUT;max

HF dZUT;min

HF dZSHF;max

HF dZSHF;min

Delay (month)

6

6

9

7

Correlation

−0.3077

−0.4822

0.2025

−0.1393

252

4

Depth (km)

(a) Indian (108m3/s)

Heaving Modes in the World Oceans

(b) Pacific (108m3/s)

(c) Atlantic (108m3/s)

0

0

0

1.0

1.0

1.0

2.0

2.0

2.0

3.0

3.0

3.0

4.0

4.0

4.0

−2

−1

0

1

2

−4

−2 8

0

Depth (100m)

0 0.5 1.0

2.0

2.0

3.0

3.0

4.0

4.0 2

4

4

−2

−1

0

1

2

3

(e) Net (10 m /s)

(d) World Oceans (10 m /s) 4.0 0 1.0

−6 −4 −2

2

6

3

Indian Pacific Atlantic Global 0

5

10

Fig. 4.102 Integrated vertical volume flux for each ocean; red: net positive; blue: net negative; black: the algebraic sum

constraint, we do not apply such an additional procedure because we believe that such a correction may consist of less than one percent of the final vertical heat flux result; so that, it is not necessary. Our main interest is the inter-annual variability associated with the ENSO cycle; thus, we first define the mean seasonal cycle of the VVF (Fig. 4.103). There are clearly biannual cycles in all three basins; the reason of such biannual cycle is not clear at this time, and it may be due to the biannual crossing of the maximum solar radiation. In the Pacific basin the seasonal cycle is trapped within the top 2 km, in particular the top 300 m. In the Atlantic basin, it is also trapped within the top 2 km; but the seasonal cycle is quite strong even in the depth of 3 km. On the other hand, the seasonal cycle in the Indian basin is trapped within the top 3 km, with the maximal vertical heat flux appears at the depth of 1 km. The inter-annual variability of VVF for the Pacific, Atlantic and Indian Ocean is calculated by subtracting the mean seasonal cycle (Fig. 4.104). For comparison, we also include the zonal wind stress perturbation defined above

(Fig. 4.104b) and the net surface air-sea heat flux (Fig. 4.104a). The 1983–1984 ENSO event is the first major ENSO event in this time series; the VVF in the Pacific Ocean responded with a large positive signal; however, for subsequent ENSO events the Pacific VVF response gradually shifted in time and its phase became leading the ENSO events. In particular, the 1997–1998 ENSO event appeared in the form of a positivenegative pair, and was followed by a huge positive lobe around year 2001. Judging from the wind stress perturbations time series shown in Fig. 4.104b, the character of ENSO events apparently went through great change; in particular, there were no major strong western wind anomalies (that is defined for the equatorial band from 180° to 140° E) after 2001. In fact, many recent studies have been focused on the different dynamical characters of ENSO events, mostly in the 21 century, characterized by the so-called central Pacific mode of ENSO, e.g., Kao and Yu (2009), Yu and Kim (2011). In fact, there were no major positive VVF signals in the Pacific basin after 2006; instead, there were major negative VVF signals during 2009–2011,

4.6 ENSO Events and Heaving Modes

253

Fig. 4.103 Mean seasonal cycle of the vertical volume flux (a, c, e, g) and the corresponding annual mean (b, d, f, h); STD means the stand deviation, and it represents the strength of the seasonal cycle within a basin

those signals seem to be linked with the strong easterly anomaly for this period. VVF signals in the Atlantic basin seemed to be in opposite phase to that in the Pacific basin during the typical strong ENSO events, in 1982–1983, 1987–1988, 1997–1998. In fact, the influence of the 1997– 1998 event seems to persist till 2005. On the other hand, the VVF signals in the Indian basin seem relatively independent of those in the Pacific and Atlantic basins. Similar to the heat fluxes associated with the meridional and zonal advections, the VHF associated with the vertical advection is defined as: Z/n HF VWT ðz; tÞ

¼

Zke q0 r cos /Cp whdk ð4:61Þ

d/ /s

kw

In addition, we define the climatological mean annual cycle of the VHF associated with

advection and the VHF deviation from the climatological mean annual cycle HF HF HF ðz; tÞ ðz; tÞ ¼ VWT ðz; tÞ  VWT dVWT

annual cycle

ð4:62Þ In a steady state of the ocean and below the top 100 m, the thermal energy balance in the vertical direction is between the vertical heat flux associated with turbulent eddy diffusion and advection because the solar radiation cannot penetrate below the top 100 m. A simple one-dimensional heat balance argument may suggest that the downward diffusive heat flux is balanced by an upward advective heat flux. A careful examination reveals that in a three-dimensional ocean the balance is quite different. As Liang et al. (2015) pointed out, in the high-latitude North Atlantic and the Southern Ocean the diffusive heat flux is upward due to the vertical projection of isopycnal

254

4

Heaving Modes in the World Oceans

Fig. 4.104 Time evolution of: a the surface heat flux integrated over the Pacific Ocean; b equatorial wind stress anomaly; vertical volume flux anomaly (mean seasonal cycle removed) for c the Pacific; d the Atlantic; e the Indian Oceans

diffusion. As a result, the corresponding advective heat flux is downward required by the heat balance in a quasi-steady state. Using the GODAS data, the VHF associated HF with vertical advection, VWT , can be calculated. As shown in Fig. 4.105b, the annual mean VHF

in the Pacific basin is mostly negative (downward), with a maximum value of −1.36 PW at a shallow depth of 50 m. The deviation from the annual mean can be either positive or negative. In order to compare the contribution from vertical advection and the net surface heat flux,

4.6 ENSO Events and Heaving Modes

255

Fig. 4.105 Mean seasonal cycle of vertical heat flux in a global oceans; c Pacific; e Indian; g Atlantic, and the corresponding annual mean profiles (b, d, f, h)

we introduce the total surface heat flux integrated over the whole Pacific Ocean, the area is defined by the dashed lines in Fig. 4.95. Zke HSF ðtÞ ¼

Z/n dk

kw

hsurf r 2 cos /d/

ð4:63Þ

/s

Similarly, we define the total surface heat flux deviation from the climatological mean annual cycle dHSF ðtÞ ¼ HSF ðtÞ  HSF ðtÞ

annual cycle

ð4:64Þ

It is readily seen that the inter-annual variHF ability of dVWT ðz; tÞ has quite a large amplitude, on the order of 2 PW, and the maximum and minimum appear at the depth of 50–150 m, that is the core depth of the equatorial thermocline in the Pacific Ocean (Fig. 4.106c). In comparison, the maximum and minimum of dHSF ðtÞ is about

0.5–1 PW (Fig. 4.106a), much smaller than that associated with the VHF due to advection. FurHF ðz; tÞ thermore, the correlation between the dVWT maximum/minimum and the zonal wind stress in the equatorial central Pacific is 0.5630 and 0.5605, with a 10 months delay (Table 4.9). In contrast, the correlation between dHSF ðtÞ and equatorial zonal wind stress is 0.3114, much smaller and with a 20 month delay. The difference in correlation and delay time between the VHF due to vertical advection and the heat content anomaly indicate that the VHF associated with vertical advection is a fast and direct response to the wind stress anomaly; while the surface heat flux anomaly is an indirect response to the wind stress anomaly. This is quite consistent with the commonly accepted theory of ENSO dynamics. Accordingly, the zonal wind stress anomaly generates equatorial Kelvin waves, those transport warm water above the main thermocline in the zonal direction. As a

256

4

Heaving Modes in the World Oceans

Fig. 4.106 Time evolution of: a the surface heat flux ðdHSF ðtÞÞ integrated over the Pacific ocean; b equatorial wind stress anomaly; vertical heat flux anomaly (mean seasonal cycle removed) for c the Pacific; d Atlantic; e Indian Oceans

Table 4.9 Correlation between VHF anomaly maximum/minimum and the integrated SHF and the zonal wind stress in the equatorial central Pacific HF dVmax

HF dVmin

dH SHF

Delay (month)

10

10

20

Correlation

0.5630

0.5605

0.3114

result, the sea surface temperature changes, and this induces the air-sea interaction of cooling or heating of the upper ocean. The air-sea interaction anomaly induces changes in the atmospheric circulation, in particular the wind stress on the equatorial band, and thus starting the first phase of the ENSO cycle.

4.6 ENSO Events and Heaving Modes

The HC anomaly in the ocean is induced by diabatic and adiabatic processes. Our focus here is the inter-annual variability of the HC in the world oceans. Since temperature observations in the deep ocean are rare, model output, such as the GODAS data, is not very reliable for the deep ocean. Hence, our discussion is limited to the upper kilometer only. As shown in Fig. 4.107c, the HC anomaly appear in the form of baroclinic modes; in particular, during each ENSO event, the baroclinic mode structure of the HC anomaly in the upper 300 m is remarkably clear in the Pacific Ocean; it is also quite visible in the Atlantic Ocean. The baroclinic mode of the HC anomaly seems to have a biannual frequency component in the Indian Ocean, that may reflect the biannual oscillations in the Indian Ocean. The HC anomaly integrated over the global ocean also appears in the form of the baroclinic modes, but it is less frequently compared with that in the Pacific Ocean. On the other hand, the HC anomaly integrated over the global ocean often appears in the form of barotropic modes, at least for the upper kilometer, as shown in Fig. 4.107f. Such barotropic features are closely linked to the net heating or cooling of the world oceans, as indicated by the positive or negative net air-sea heat flux shown in Fig. 4.107a. One of the most outstanding features in this figure appeared after the year 2000. During the first half of this period, the upper ocean (shallower than 800 m) was cooled down. However, water below this layer was warmed up. In fact, there was a positive subsurface HC anomaly around 800 m appearing in 2011. These features have been widely discussed and debated in terms of the so-called hiatus, e.g., Kosaka and Xie (2013), England et al. (2014). The details of such phenomena is beyond the scope of this book and it is left for further study. As discussed by Huang (2015), there are global modes of heaving; these modes are induced by wind stress changes over the global oceans. Due to the changes of wind stress in different parts of the global ocean, warm water above the main thermocline is redistributed

257

among different parts of the world oceans. The dynamical processes involved are quite complicated, and they may take a long time to be completely understood. As a result, the time evolution of HC modes in the world oceans cannot be interpreted in terms of equatorial wind stress anomaly alone.

4.6.6 A Two-Hemisphere Model Ocean Simulating ENSO 4.6.6.1 Model Set Up As shown above, ENSO like events can induce large amplitude anomalous heat fluxes in the ocean; thus, the same simple reduced gravity used in Sect. 4.1.2.1 is used in this section to illustrate the fundamental physics. 4.6.6.2 Numerical Experiments In the first experiment to be discussed the model was restarted from the above mentioned quasiequilibrium state and forced by additional wind stress perturbations in the following form 2

s0x ¼ Dseðy=DyÞ sinð2pt=T Þ

ð4:65Þ

where Ds = −0.015 N/m2 and Dy = 1100 km, T = 4 year is the nominal period of ENSO. The numerical experiment was run for 32 years under the forcing of the steady wind specified in (Eq. 4.8), plus the additional zonal wind stress specified in Eq. (4.65). Since we are interested in the periodic nature of the solution, we omit the first five periods and select two periods, from year 20 to 28; however, for simplicity, this time period will be labelled as year 0–8 in the following figures. First, the ENSO cycle is associated with the strong MOC and MHF, as shown in Fig. 4.108. The idealized ENSO cycle begins with strong easterly anomaly, i.e. a negative zonal wind stress anomaly, as shown in Fig. 4.96c. Such wind stress perturbations induce a negative (positive) MOC in the Northern (Southern) Hemisphere, with roughly eight month delay.

258

4

Heaving Modes in the World Oceans

Fig. 4.107 Time evolution of: a the surface heat flux integrated over the Pacific Ocean; b equatorial wind stress anomaly; heat content anomaly in the upper kilometer (mean seasonal cycle removed) for c the Pacific; d Atlantic; e Indian; f the world oceans

4.6 ENSO Events and Heaving Modes

259

Fig. 4.108 The meridional overturning circulation (a); the instantaneous overturning rate at year 3.7 and the corresponding meridional heat flux (b); the zonal wind perturbations (c)

After the first year, the equatorial easterly anomaly declines and the corresponding MOCs flip signs; the corresponding MHF is now poleward in each hemisphere. As an example, the MOC and MHF profiles at year 3.7 are shown in Fig. 4.108b, with peak values of 1.57 Sv in the MOC and 0.068 PW in MHF. Around year 3, the easterly starts to intensify again, the corresponding MOC (MHF) flip signs again, and the system enters the cycle of the opposite phase. Second, the ENSO cycle is accomplished by the strong ZOC and ZHF cycles, as shown in Fig. 4.109. During the first half of the ENSO cycle, strong easterly anomalies drive a negative ZOC (ZHF) due to the westward transport of the warm water in the upper layer. The amplitude of the ZOC is 6.5 Sv, and that of the ZHF is 0.28 PW. As shown in Fig. 4.109c, the climate mean

ZHF in the Pacific sector is about 1.2 PW; therefore, the anomalous ZHF induced by the idealized ENSO cycle is quite a sizeable contribution to the climate variability. For the next half of the ENSO cycle, the ZOC and ZHF flip signs (Fig. 4.109). Third, the ENSO cycle also induces strong VHF and vertical HC (heat content) anomaly, similar to the cases for decadal wind stress perturbation induced adiabatic heaving modes in the world oceans, discussed in Sect. 4.1. The HC anomaly appears in the form of baroclinic modes (Fig. 4.110). As an example, the vertical profile of the HC anomaly, its vertical accumulation and the equivalent VHF at year 3.8 are shown in Fig. 4.110b–d. The peak value of the HC anomaly is approximately 13  1019 J/m, and the peak value of vertical heat flux is 0.48 PW.

260

4

Heaving Modes in the World Oceans

Fig. 4.109 The zonal overturning circulation (a); the instantaneous overturning rate at year 0.8 and the corresponding poleward heat flux (b); and the zonal wind perturbations (c)

Fig. 4.110 The vertical heat content anomaly (a); the vertical heat content anomaly at year 3.8 (b); the accumulated vertical heat content anomaly (c); the mean vertical heat flux (defined as the vertical heat flux inferred from the change of the vertical heat content over the past one year) (d); and the zonal wind perturbations (e)

4.6 ENSO Events and Heaving Modes

The amplitude of the HC anomaly induced by the idealized ENSO cycle is comparable with the mean global warming rate for the world oceans, and we conclude that the idealized ENSO cycle examined above can induce strong heat flux anomalies in the meridional/zonal and vertical directions; such anomalous heat fluxes consist of a noticeable portion of the climate variability, and hence they should be studied carefully.

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262 Levitus S, Antonov JI, Boyer TP, Locarnini RA, Garcia HE, Mishonov AV (2009) Global ocean heat content 1955–2008 in light of recently revealed instrumentation problems. Geophys Res Lett 36: L07608. https://doi.org/10.1029/2008GL037155 Levitus S, Antonov JI, Boyer TP, Baranova OK, Garcia HE, Locarnini RA, Mishonov AV, Reagan JR, Seidov D, Yarosh ES, Aweng MM (2012) World ocean heat content and thermosteric sea level change (0–2000 m), 1955–2010. Geophys Res Lett 39: L10603. https://doi.org/10.1029/2012GL051106 Lozier MS, Leadbetter S, Williams RG, Roussenov V, Reed MSC, Moore NJ (2008) The spatial pattern and mechanisms of heat-content change in the North Atlantic. Science 319:800–803 Lozier MS, Roussenov V, Reed MSC, Williams RG (2010) Opposing decadal changes for the North Atlantic meridional overturning circulation. Nat Geosci 3:728–734 Luyten JR, Stommel H (1986) Gyres driven by combined wind and buoyancy flux. J Phys Oceanogr 16:1551–1560 Lyman JM, Good SA, Gouretski VV, Ishii M, Johnson GC, Palmer MD, Smith DM, Willis JK (2010) Robust warming of the global upper ocean. Nature 465:334–337. https://doi.org/10.1038/nature09043 Lyman JM, Johnson GC (2008) Estimating annual global upper-ocean heat content anomalies despite irregular in situ sampling. J Clim 21:5629–5641. https://doi. org/10.1175/2008JCLI2259.1 McGregor S, Sen Gupta A, England MH (2012) Constraining wind stress products with sea surface height observations and implications for Pacific Ocean sealevel trend attribution. J Clim 25:8164–8176 Marshall DP, Johnson HL (2013) Propagation of meridional circulation anomalies along western and eastern boundaries. J Phys Oceanogr 43:2699–2717 Meehl GA, Arblaster JM, Fasullo JT, Hu A, Trenberth KE (2011) Model-based evidence of deep-ocean heat uptake during surface-temperature hiatus periods. Nat Clim Change 1:360–364 Meinen CS, McPhaden MJ (2000) Observations of warm water volume changes in the equatorial Pacific and their relationship to El Niño and La Niña. J Clim 13:3551–3559 Merrifield MA (2011) A shift in Western Tropical Pacific sea level trends during the1990s. J Clim 24:4126–4138 Primeau F (2002) Long Rossby wave basin-crossing time and the resonance of low-frequency basin modes. J Phys Oceanogr 32:2652–2665 Pedlosky J (1986) The buoyancy and wind-driven ventilated thermocline. J Phys Oceanogr 16:1077–1087 Qiu B, Chen SM (2012) Multidecadal sea level and gyre circulation variability in the Northwestern Tropical Pacific Ocean. J Phys Oceanogr 42:193–206. doi: http://dx.doi.org/10.1175/JPO-D-11-061.1 Ripa P (1995) On improving a one-layer ocean model with thermodynamics. J Fluid Mech 303:169–201

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Heaving Modes in the World Oceans

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5

Heaving Signals in the Isopycnal Coordinate

Dynamical processes in the ocean are directly linked to the density field. Potential temperature (temperature hereafter, unless stated explicitly) and salinity are dynamical tracers; although they can affect the dynamical processes, such effects must go through the link of density. In fact, the concept of heaving can be more rigorously defined in the density coordinate, or the isopycnal coordinate. Most importantly, for the largescale oceanic circulation and climate problems potential density is monotonic in the vertical direction; thus, it is an ideal alternative vertical coordinate. In this chapter we explore using potential density as the Lagrangian coordinate to identify heaving signals in the world oceans.

5.1

Introduction

Density has been used as a vertical coordinate in many previous studies in physical oceanography. There are many potential candidates, such as the in situ density, potential density, and the socalled neutral density. Depending on the specific focus, some of the desirable properties of a density used as the vertical coordinate include: (1) It is monotonic, so that a one-to-one transformation between the geopotential height coordinate and the isopycnal coordinate can be easily carried out. (2) It conserves the local vertical and horizontal stratification, so that dynamical processes can be described accurately in this coordinate.

(3) It is a conserved property, so that water parcels stay on the same coordinate surface during three-dimensional adiabatic movements in the ocean. Although it seems easy to satisfy these requirements, there is no isopycnal coordinate that can satisfy all these constraints. Searching a good isopycnal coordinate as a vertical coordinate has been a long and winding road in the history of oceanography. At first, the in situ density q ¼ qðS; T; pÞ seems a convenient choice; however, it is not a conserved quantity. In fact, the in situ density increases rapidly with pressure; a large part of this change is due to the increase of in situ pressure with depth. Increase of in situ density with pressure is mostly inert in terms of dynamics; as a result, in situ density cannot be used as a vertical coordinate. Montgomery’s (1938) pioneering work on water mass analysis was based on the rt surface. However, by definition, rt ¼ qðS; T; p ¼ 0Þ (where q is density, S is salinity, T is the in situ temperature, and p is pressure) is not a conserved quantity. Although Montgomery called his method isentropic analysis, what he used is not an isentropic surface. In fact, this density variable is no longer used in oceanography. In order to overcome problems associated with in situ density, potential density was introduced in dynamical oceanography, e.g., HellandHansen (1912). The potential density surface

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_5

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rp ¼ qðS; h; pr Þ (where h is potential temperature, pr is reference pressure) was introduced and widely used in hydrographic data analysis. Potential density is a conserved property, so that it satisfies the third constraint. However, a careful examination reveals that away from the reference level, water parcels having the same potential density may sit on different pressure levels of the local water column. Hence, using potential density to describe adiabatic motions in the ocean may introduce certain errors. This problem associated with using potential density is discussed in details by Huang (2014a). In fact, potential density can represent the vertical and horizontal stratification at the reference pressure only; away from the reference pressure level it does not represent the local stratification accurately. In some cases, potential density is not even monotonic in the vertical direction, as will be shown shortly. Therefore, potential density does not satisfy the first and second constraints. Due to the nonlinearity of the equation of state, using a non-local reference pressure to calculate the static stability can lead to values quite different from the truth. To overcome this problem, the patched potential density method was introduced, e.g., Reid (1994) or de Szoeke and Springer (2009).

5

Heaving Signals in the Isopycnal Coordinate

As an improvement, the neutral density surface was introduced. The basic concept of neutral density surface was to build up a surface whose tangent at each point matches the local potential density surface tangent, as first postulated by Forster and Carmack (1976); their idea has been generalized into a theory of neutral density and neutral density surface, e.g., McDougall (1987a), Jackett and McDougall (1997). Neutral density has been used in many recent studies; however, neutral density is not really neutral; as shown by Huang (2014b), water parcels moving along a neutral density surface are subjected to buoyancy force. It can be demonstrated that mixing of water parcels on the same neutral density surface can change the gravitational potential energy of the system. The most problematic issue related to the commonly used neutral density is that it is defined as a function of longitude and latitude. As such, neutral density is not a conserved quantity. The truly neutral surface is called the adiabatic density surface, as discussed by Huang (2014a). These surfaces were first discussed by McDougall (1987b) in connection with the propagation of the submesoscale coherent vortices. The basic properties of this new density surface are discussed by Huang (2014a). The adiabatic density surface is the truly conserved quantity in the ocean.

Fig. 5.1 Potential density sections in the Atlantic Ocean along 30.5° W, based on the WOA09 data. a r0 (pr ¼ 0); b r1 (pr ¼ 1000 db)

5.1 Introduction

However, the adiabatic density surface can have multiple sheets in the oceans; consequently, it is not monotonic in the vertical direction. As a compromise, we will use potential density as the Lagrangian coordinate in this book. It is well known that potential density based on a shallow reference pressure may not be monotonic in the deep ocean. As an example, we show the 30.5° W meridional section of potential density r0 , using the sea surface as the reference level. It is clear that potential density r0 below 2.5 km in the Atlantic Ocean is not monotonic (Fig. 5.1 based on the WOA09 data, Antonov et al. 2010). However, if we use r1 (using 1000 db as the reference pressure), there is no such a problem. Our main interest is to identify the heaving modes, which are mostly concentrated in the upper 1–2 km; accordingly, using r1 seems to be a good choice: it can accurately describe processes near the 1000 db level, including heaving processes in the deeper levels. Of course, other choices may be used as well. In fact, if we are only interested in the heaving modes for the upper 1 km of the ocean, using r0 is a good choice as well.

5.2

Casting Method

In the study of climate variability, the commonly used vertical coordinate is the geopotential coordinate, or simply the z-coordinate. As an alternative, the isopycnal coordinate can be used. The traditional approach is to use the climatological mean potential density as the vertical coordinate; by definition, this is a Eulerian coordinate, that is fixed in space. The other approach is to use the instantaneous potential density as the vertical coordinate, and this is a Lagrangian coordinate; combining the traditional Euler coordinate in the horizontal directions with the density coordinate gives rise to a EulerianLagrangian hybrid coordinates. Unfortunately, most oceanic general circulation models currently used in the community are based on the Boussinesq Approximations; one of

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the fundamental assumptions made in the Boussinesq models is to replace mass conservation with the volume conservation approximation. As such, the models can generate artificial sources/sinks of mass associated with surface heating/cooling or evaporation/precipitation. For example, if the ocean is warmed up, the total mass in the ocean or in a mass-conserving model ocean should remain unchanged. On the other hand, heating leads to a decline in water density; as a result, total mass in a volume conserving model declines, and this is an artifact of such models. In the case of precipitation, rains into the ocean add to the total mass in the ocean, but the total amount of salt in the ocean should remain unchanged. However, in a volume conserving model freshwater in forms of precipitation leads to a decline of salinity in the ocean, and the density in the ocean declines as well. Due to the volume conserving approximation assumed in these numerical models, the total mass and total amount of salt in the model ocean decline. Of course, such decline in total mass and salt in the models are artifacts. In conclusion, we should be cautious in interpreting the meaning of the mass anomaly signals diagnosed from models based on the volume conservation approximation. We will use a four-dimensional coordinate system ðx; y; r1 ; tÞ in the following analysis, where r1 is the potential density, and t is time. In fact, we will introduce two isopycnal coordinates: The Fixed Density Coordinates (FDC hereafter) and the Moving Density Coordinates (MDC hereafter), and we will use two methods to convert climate data from the traditional (x, y, z, t) coordinates into these two new coordinates.

5.2.1 FDC The FDC uses the climatological mean potential density at each spatial grid (i, j, k) in the spherical coordinates as the vertical coordinate. In addition, one may also separate the contribution due

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to the regular seasonal cycle by using the climatological mean potential density in each month as the vertical coordinate. At each grid point (i, j, k) in the spherical coordinates, the mass content anomaly in the mth month in the FDC is denoted as DW FDC ði; j; k; mÞ ¼ ðr1 ði; j; k; mÞ  r1 ði; j; kÞÞDxð jÞDyDzðkÞ

ð5:1Þ

where r1 ði; j; k; mÞ and r1 ði; j; kÞ are the potential density and the corresponding climatologic mean of the corresponding month in a year. Note that density is in the commonly used r unit, i.e., it is the potential density subtracting the constant value of 1000 kg/m3. Data is put into discrete density bins in the FDC according to the following rule: DW FDC ði; j; k; mÞ added to wFDC ði; j; n; mÞ a if r1;n  0:1\r1 ði; j; kÞ  r1;n þ 0:1; r1;n; ¼ 25:1; 25:3; . . .; 33:9ðkg/m3 Þ ð5:2Þ Thus, the climate variability data series is in the form
wFDC ¼ wFDC ði; j; n; mÞ; n ¼ 1; 2; . . .; 45; a a m ¼ 1; . . .; m month; in units of kg ð5:3Þ where m month is the total number of months in the data record. The physical meaning of this variable is as follows. The mass content anomaly at a spherical coordinate grid point (i, j, k) is calculated, and a positive (negative) value indicates that water in this grid point is heavier (lighter) than the climatological mean (seasonal cycle subtracted) density. Note that for climate signals at each spatial grid (i, j, k) there is only one density bin assigned with a non-zero value. Since for the large-scale circulation the stratification is always stable, this is a monotonic coordinate transformation. This time series can also be summed up into a time series of the global density content anomaly in the FDC as:

Heaving Signals in the Isopycnal Coordinate

WaFDC ðn; mÞ ¼

jmt imt X X

wFDC ði; j; n; mÞ a

ð5:4Þ

i¼1 j¼1

5.2.2 MDC The MDC uses the instantaneous potential density at each grid (i, j, k) in the spherical coordinates as the vertical coordinate. A simple way of this coordinate transform is based on the casting method described as follows. At each grid point in the spherical coordinates, the mass content is denoted as, wði; j; k; mÞ ¼ r1 ði; j; k; mÞDxð jÞDyDzðkÞ ð5:5Þ where r1 ði; j; k; mÞ is the potential density at the mth month. Note that density is in the commonly used r unit, i.e., it is the potential density subtracting the constant value of 1000 kg/m3. The resulting data is “cast” into the discrete bins in the isopycnal coordinates by storing the data in the discrete density bin n in the MDC, and denoted as wMDC ði; j; n; mÞ, using the instantaneous potential density to determine the right bin number n r1;n  0:1\r1 ði; j; k; mÞ  r1;n þ 0:1; r1;n ¼ 25:1; 25:3; . . .; 33:9 ðkg/m3 Þ

ð5:6Þ

As a result, the time series of the total mass content is in the form
W MDC ¼ wMDC ði; j; n; mÞ; n ¼ 1; . . .; 45; m ¼ 1; . . .; m month; in units of kg ð5:7Þ where m month is the total number of months in the data record. Due to the climate variability, density anomaly signals in a fixed spatial grid (i, j, k) at different times may be put in different density bins in the MDC. The advantage of this method is that it works regardless of whether the stratification profile is monotonic or non-monotonic. It is important to emphasize that the quality of the resulting dataset

5.2 Casting Method

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in the new isopycnal coordinates is directly linked to the original resolution in the (x, y, z) coordinates. Since the vertical resolution in the deep ocean is rather low in the commonly used OGCM, the resulting dataset in isopycnal coordinates may not have adequate resolution; as such, the maps produced from such a transformation may not accurately reflect the physical processes, especially for the high density range. To overcome such hidden problems, one can use data interpolation to increase the vertical resolution of the original data before the coordinate transformation. From the multi-year data, one can define the climatological mean annual cycle of the total mass content for 12 months of an annual cycle h Annual cycle Annual cycle W MDC ¼ wMDC ði; j; n; mmÞ; n ¼ 1; . . .; 45; mm ¼ 1; 2; . . .; 12 ð5:8Þ The corresponding anomaly is defined as wMDC ði; j; n; mÞ ¼ W MDC  W MDC a

Annual cycle

ð5:9Þ The physical meaning of this variable is slightly different from that defined in Eq. (5.3) for the FDC. In the FDC the mass content anomaly indicates that the mass content changes due to the density deviation from the climatological mean at the corresponding spatial grids. Because of strong climate variability, density deviation at a fixed spatial grid can be larger than 0.1 kg/m3, the half value of density intervals set for the FDC in this chapter. Nevertheless, all signals generated at a fixed spatial grid are put into the same density bin in the FDC. In the MDC density for each bin is fixed for a specified range (in the case discussed above, it is set to the 0.2 kg/m3 intervals); thus, the mass content anomaly indicates the mass content change in connection with volume increase (decrease) of water mass within this density category. Furthermore, due to climate variability the density anomaly signals generated in the same spatial grid at a different time may be put into

different density bins in the MDC. On the other hand, climate variability recorded in a specific density bin in the MDC may come from different spherical coordinate grids in different months. For each month, the total mass content anomalies integrated over the world oceans in the MDC are defined as WaMDC ðn; mÞ ¼

jmt imt X X

wMDC ði; j; n; mÞ ð5:10Þ a

i¼1 j¼1

5.2.3 Separating the Signals Into External and Internal Modes With the mass anomaly signals diagnosed from the world oceans, our next step is to separate these signals into the external and internal modes. The external mode signals are directly linked to diabatic processes, mostly likely the airsea heat flux and the freshwater flux anomaly. To a minor degree, wind stress anomaly can also contribute. The roles of wind stress anomaly are much more complicated, including horizontal shifting of the outcrop lines and alternating the mixing rates in the ocean. On the other hand, the internal modes can be induced by wind stress anomaly alone; such heaving signals can be classified as adiabatic heaving. In fact, our study in Sect. 4.1 based on simple reduced gravity models is mostly confined to cases of adiabatic heaving. Of course, the thermohaline forcing anomaly, especially at the site of subduction and deep (bottom) water formation, can also contribute to the formation of an internal mode. In addition, changes in diapycnal mixing may also contribute to a minor degree. Results from such analysis may provide useful clues for identifying the mechanisms generating such anomalies.

5.2.3.1 Use of the FDC As shown in Eq. (5.4), density anomaly signals in the FDC can be summarized as a twodimensional data array. By definition, the FDC is defined by the climatological mean density

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(with seasonal cycle subtracted); hence, it is not a Lagrangian coordinate. For example, if WaFDC ðn; mÞ [ 0 for a given index pair of (n,m), we can only say that in the mth month, density at grids with climatological mean density satisfying r1;n  0:1  r1 ði; j; kÞ  r1;n þ 0:1 increases. Such an increase in the instantaneous density might be due to thermal/saline exchanges with the neighboring grids or through air-sea interaction. In addition, such changes can also be the result of adiabatic motions induced by wind stress perturbations. Namely, the heaving motions in the world oceans can push different types of water mass from the neighboring grids to the grids of our concern. The clean signals identifiable from the data in the FDC are the global sum Gs WaFDC ðmÞ ¼

45 X

WaFDC ðn; mÞ

ð5:11Þ

n¼1

This quantity represents the change of the total mass content for the world oceans, and it is solely the result of the air-sea heat/freshwater changes, i.e., it is the net contribution due to the diabatic processes. Note that in the world oceans the total mass of water should be conserved, except for the mass sources owing to freshwater exchange through the upper surface and the rivers and landbased glaciers. However, in a Boussinesq model, such as the GFDL MOM2 model used in generating the GODAS data (Behringer and Xue 2004), heating/cooling or evaporation/precipitation can induce artificial sources/sinks of mass.

5.2.3.2 Use of the MDC The discussion for the MDC is much more elaborate; thus, some of the details are included in the Appendix. As stated in Eq. (5.30) in the Appendix, at each vertical grid point k at station (i, j) the net sum of anomalous signals in the MDC equals the signal in the FDC, i.e.:

nmt X

Heaving Signals in the Isopycnal Coordinate

MDC FDC Wa;k ði; j; n; mÞ ¼ Wa;k ði; j; n0 ; mÞ

n¼1

ð5:12Þ There is only one grid n0 in the FDC that has a non-zero value. In the MDC, signals in each density bin can be further separated into the external and internal components WaMDC ¼ Ex WaMDC ðn; mÞ þ In WaMDC ði; j; n; mÞ ð5:13Þ where Ex WaMDC ðn; mÞ ¼ WaMDC ði; j; n; mÞ

i;j

ð5:14Þ

is the external mode, defined as the volumeweighted mean of the mass content anomaly in each density bin. As discussed above, this is the total mass content anomaly for each density interval, that is directly linked to the thermal/haline exchange across all boundaries of this isopycnal layer, in particular including the air-sea interface. By definition, the internal modes integrated over the entire volume defined by the isopycnal layer of the r1;n bin must be zero, i.e., i;j

In WaMDC ði; j; n; mÞ ¼ 0

ð5:15Þ

Since its integration over an isopycnal layer is zero, this variable represents the local heaving signals of the mass anomaly in each isopycnal layer. The method discussed above is now applied to the GODAS data. The value of r1 covers quite a wide range, from 20 to 34 kg/m3; for climate study, most water masses are confined to the range of 25–34 kg/m3. Furthermore, a close examination reveals that warm and light water in the upper ocean may be separated by the continents, and heaving signal analysis for the warm

5.2 Casting Method

water masses should take this fact into consideration. Accordingly, heaving signals for warm and light water in the Pacific-Indian basin should be studied separately from the warm and light water in the Atlantic basin.

5.2.4 Statistics in the Density Space We begin with the external mode anomaly in both the MDC and the FDC. As shown in Fig. 5.2, most signals appear in the high density range of 29–34 kg/m3. More importantly, color bar on the left panels representing the strength of signals is about 100 times larger than that on the right panels, indicating that signals in the MDC are much stronger than in the FDC. The reason

269

why the signal ratio of the MDC/FDC is so large is as follows. In the FDC the magnitude of signals is estimated at DFDC ¼ Drmax DV; on the other hand, in the MDC the magnitude of signals is the difference between the maximum signal Dmax MDC ¼ rmax DV (DV is the typical size of grid box in the deep ocean) and the climatological mean signal, mean mean i.e., DMDC ¼ Dmax MDC  DMDC , where DMDC ¼  max  0:5  DMDC þ Dmin MDC is the climatological mean signal. In the extreme case, there might be no data or very little data falling into the specific density bin in the MDC in some specific months; as a result, the lowest limit of Dmin MDC is close to max zero, and DMDC  Dmean  0:5D MDC MDC . Assuming the corresponding DV in the FDC and the MDC

Fig. 5.2 The total volume integration of the external mode of weight content anomaly (seasonal cycle subtracted) in the MDC (left panels) and the FDC (right panels), using a lower-resolution grid Dr1 ¼ 0:2 kg/m3 in isopycnal coordinates. The lower panels show fine structure patterns for the high density range

270

is close to each other, the corresponding signal ratio is DMDC =DFDC  0:5rmax =Drmax . Assuming rmax  34 kg/m3 and Drmax  0:15 kg/m3 , this gives rise to a rough estimate of the maximum signal ratio:DMDC =DFDC  0:5  34=0:15  113,i.e., amplitude of signals in the MDC is about 100 times larger than that in the FDC. This dramatic difference in signal strength in the MDC and the FDC is also an important difference in the Eulerian and Lagrangian coordinates. Furthermore, the pattern of signals in the MDC is quite different from that in the FDC. As discussed above, positive (negative) signals in a MDC bin indicate that water mass volume in this density category increases (decreases) because of diabatic processes, such as heating/cooling or salinification. On the other hand, positive signals in a FDC bin indicate that water masses with the corresponding climatological mean density become heavier due to cooling or salinification. Negative signals can be interpreted in a similar way, but in terms of processes with opposite signs. Beginning with the year 1980, there was a strong positive signal band in the MDC over the density range of 32.1–32.7 kg/m3, with the central density of 32.5 kg/m3; thus, there was more water mass in this density range. There is a band of negative signals centered in 31.7 kg/m3, indicating that the water mass volume in this density category was smaller than the climatological mean. The amplitude of these centers of extrema is on the order of 20  1016 kg/m3 for the isopycnal layer of r1 ¼ 31:7  0:1 kg/m3 . On the other hand, the extrema in the FDC have a much lower amplitude, on the order of only 0:3  1016 kg/m3 . Starting from the year 1980, there was a band of positive signals centered at 32.1–32.3 kg/m3. This implies that water in this density range was heavier (most likely caused by cooling) and hence moved towards higher density range; the corresponding signals in the MDC were in the forms of positive and negative clusters. There is a minor negative signal band around 32.5 kg/m3 in the FDC, indicating lightening (due to warming) of water in this density range; however, the corresponding

5

Heaving Signals in the Isopycnal Coordinate

signals in the MDC appear to be overpowered by the overall enhancement of the water mass volume in the related density range. Starting from the year 2000, there is a clear trend of opposite signs. In the MDC there is a band of negative signals centered around 32.5 kg/m3, and there are two bands of positive signals centered at 31.9 and 32.7 kg/m3. These positive bands indicate the water mass volume enhancement associated with the strong warming (and lightening of water density) in the recent decade. On the other hand, in the FDC and starting from the year 2000, there are predominating negative signals within density range of 31.6–32.4 kg/m3, indicating the lightening of the water mass induced by the general trend of warming.

5.2.5 External Signals in Terms of Layer Thickness Our discussion above is based on signals in terms of mass anomaly; it is more interesting to see the perturbations in terms of the equivalent isopycnal layer thickness perturbations. For the external modes, their equivalent thickness perturbations can be calculated as follows dhMDC Ex ðn; mÞ ¼

jmt imt X X i¼1

WaMDC ði; j; n; mÞ=denðnÞ=areaðnÞ

j

ð5:16Þ where area(n) is the total horizontal area of the corresponding isopycnal layer, which can be inferred from the climatological mean density field of the GODAS data. The corresponding distribution in density space is shown in Fig. 5.3. The time evolution of isopycnal layer thickness perturbations is shown in Fig. 5.4. This figure is focused on the high density range. Note that due to the change of horizontal area with density, this figure provides a quite different view of climate changes in the isopycnal coordinates. As shown in Fig. 5.2, in terms of the total mass anomaly, the signal maximum for year 1980– 2000 is centralized at r1 ¼ 32:5 kg/m3 , and

5.2 Casting Method Fig. 5.3 Horizontal area occupied by each isopycnal layer of Dr1 ¼ 0:2 kg/m3 , based the annual mean climatology of the GODAS data

271

Horizontal area (log10 (area), m2)

15.0 14.5 14.0 13.5 13.0 12.5 12.0 11.5 11.0 10.5 10.0

25

26

27

28

29

30

31

32

33

34

3 1 (kg/m )

perturbations for density higher than this value seem rather small. The horizontal area occupied by the isopycnal layers of r1 [ 32:5 kg/m3 declines quickly, as shown in Fig. 5.3. As a result, the equivalent layer thickness perturbations for the high density range are actually much

larger than that for the relatively lower density range (Fig. 5.4a). We will focus on the time evolution of the external modes of layer thickness perturbations for 5 isopycnal layers. This figure shows that the external modes of layer thickness perturbations

Fig. 5.4 Isopycnal layer thickness perturbations of the external modes for the world oceans (a); thickness perturbation of 5 selected isopycnal layers, subjected to 13 month smoothing (b); based on the GODAS data

272

5

have quite a large amplitude, on the order of 100 m. In fact, the largest layer of thickness perturbations appears in these isopycnal layers: (1) r1 ¼ 32:5  0:1 kg/m3 (heavy black curve in Fig. 5.4b), with amplitude larger than 150 m at the end of the record; (2) r1 ¼ 33:1  0:1 kg/m3 (thin black curve in Fig. 5.4b); this represents the densest water mass in this model, and the layer thickness perturbation was more than 100 m in the beginning of the record. As discussed above, the variability in volumetric distribution of water masses reflects changes in global water mass formation and erosion, and they are the results of competition between wind stress, heat and freshwater exchange across the air-sea interface and isopycnal surfaces in the subsurface ocean. Thus, the time evolution of the external modes provides useful information for the global water mass balance and the relevant physical processes.

5.3

Projecting Method

Instead of the casting method discussed in the previous section, the projecting method discussed below is probably more convenient to use because this method can be used to extract multiple variables from the climate dataset. For the large-scale oceanic circulation, the stratification is always stable; accordingly, the one-to-one conversion from the z-coordinate to the isopycnal coordinate can always be carried out. Note that due to some complicated technical problems, potential density inferred from climate datasets may not be always stably stratified. In particular, for dataset generated from data assimilation, the situation of unstable potential density pairs can happen quite often. Therefore, when applying the projecting method, all unstable potential density pairs in the water column must be modified in order to get rid of these unstable pairs. Assuming that density stratification at a station is stable, one can apply the projecting method as follows. For a given time, the corresponding instantaneous potential density (black)

Heaving Signals in the Isopycnal Coordinate ρ

T

z4

z3 z2

h2

z1

T T1 T2 T3 ρ 4

T4 ρ 3

ρ2

ρ1

Fig. 5.5 Sketch of the projecting method applied to potential density (black) and potential temperature (red) profiles

and potential temperature (red) profiles at a station are schematically shown in Fig. 5.5. The instantaneous density is used as the material coordinate; hence, this method is based on the Lagrangian coordinate. The corresponding density bins are q1  Dq; q2  Dq; q3  Dq; q4 Dq; by linear interpolation (or other forms of interpolation), the corresponding vertical position of the centers of these density bins can be calculated and they are labeled as z1 ; z2 ; z3 ; z4 . From the locations of these centers, the corresponding temperature are T1 ; T2 ; T3 ; T4 . The corresponding thickness of density layer q2  0:5Dq is h2 . The volumetric contribution to the isopycnal layer q2 is Dv ¼ h2 dxð jÞdy. By interpolating water properties in each water column, one can find the corresponding properties for each density layer at each station: Mean layer thickness h Mean layer depth D Mean layer temperature T Mean layer salinity S Mean velocity U and V or the approximate Montgomery streamfunction Wm . By subtracting the climatological mean seasonal cycle, we obtain the perturbations: dh; dD; dT; dS; dU; dV ðor dWm Þ. In a similar

5.3 Projecting Method

273

way, one may obtain other field properties, such as the mean temperature stratification, mean salinity stratification, and mean value and their mean vertical gradients of oxygen and other nutrients in each isopycnal layer. Note that on an isopycnal surface, temperature and salinity perturbations are compensated by definition. As a result, there is only one independent thermodynamic variable, spicity p, as discussed in Chap. 3 and by Huang et al. (2018) . Spicity is a thermodynamic variable of sea water. Since density is constant on an isopycnal surface, spicity as a convenient variable independent of density is introduced, its contours are perpendicular to the density contours. Because we are dealing with potential density and potential temperature, the natural choice is potential spicity. In addition, for the large-scale circulation problems away from the equatorial regime, potential vorticity may be a better conservative quantity than the layer thickness perturbations. For the large-scale circulation, the relative vorticity is quite small and negligible. Since the Eulerian-Lagrangian coordinate system is used in the analysis, the horizontal coordinate is fixed in space. As a result, the Coriolis parameter is not changed with time, and potential vorticity and its perturbations are reduced to the following forms f f q ¼ ; dq ¼  2 dh h h

ð5:17Þ

streamfunction. Although geostrophic streamfunction can be defined on either the pressure surfaces or the specific volume anomaly surfaces, there is no exact form of geostrophic streamfunction on other surfaces, such as potential density surfaces, potential temperature surfaces, the salinity surfaces, etc. However, approximate forms of geostrophic streamfunction can be defined on potential density surface (Zhang and Hogg 1992); McDougall and Klocker (2009) discussed different forms of approximate geostrophic streamfunction on potential density surfaces and neutral density surfaces in detail. It is well known that on the specific volume anomaly surface, the Montgomery streamfunction is: wm ¼ pd

35;0

Zp 

Note that the first, second and fourth variables on this list are not completely independent of each other. In fact, for the geostrophic flow in the ocean, the geostrophic streamfunction is linked to the vertical stratification. Isopycnal analysis started from the pioneering work of Montgomery (1937, 1938). In 1937, Montgomery postulated a streamfunction which can be used to identify the geostrophic flow on specific volume anomaly surfaces; such a streamfunction is now called the Montgomery

ð5:18Þ

0 1 where d35;0 ¼ q1  qð35;0;p Þ is defined as the

specific volume anomaly relative to a pivotal state of S = 35, T = 0 °C and p = 0. For other commonly used coordinate surfaces, there is no exact form of geostrophic streamfunction, although some approximate forms of geostrophic streamfunction have been discussed in previous studies. For example, Zhang and Hogg (1992) postulated an approximate geostrophic streamfunction for potential density surfaces:

Therefore, in an isopycnal layer one can identify at least four climate variables: dhðdqÞ; dD; ðdT; dS; or dpÞ; ðdU; dV or dWm Þ

d35;0 dp0

wZH ¼ ðp   pÞd

35;0

Zp 

d35;0 dp0

ð5:19Þ

0

 can be chosen to where the constant pressure p reduce the errors in geostrophic streamfunction. Many other potential choices of approximate geostrophic streamfunction have been discussed, including the forms which can be used for the neutral density surface, e.g. McDougall and Klocker (2009). Our discussion above shows that there are close links between the geostrophic streamfunction and the vertical density profile. Therefore, the anomalous geostrophic velocity at each

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Heaving Signals in the Isopycnal Coordinate

take into account of the important facts that isopycnal depth and thickness varying with space and time; thus, the anomalous horizontal advection of tracers should be calculated with the following formula. For example, the anomalous spicity advection consists of two parts:  dFp ¼ r 

  !0 ! 0 hu  p þ r  hu  p ð5:20Þ

where the layer thickness term is a critically important factor in the calculation. On the other hand, the advection of dh (dq) and dD is governed by the eddy and mean flow interaction. Fig. 5.6 Three types of perturbations in an isopycnal layer and their movements: dp—spicity anomaly; dh (dq) —layer thickness (potential vorticity) anomaly; dD— layer depth anomaly

isopycnal surface is closely linked to the stratification anomaly, and hence the isopycnal layer thickness and depth anomaly. Unfortunately, such a dynamical link cannot be written out in a simple analytical form. On the other hand, when dealing with climate variability identified from climatological datasets generated from numerical models, velocity and its anomaly are available; so that, there is no need to infer the velocity field from the water mass properties or the Montgomery streamfunction. Nevertheless, it is important to remember that these three anomaly fields are linked through the dynamics. In summary, climate variability for the isopycnal layers contains at least the following signals: perturbations in the layer depth, layer thickness and spicity. Most importantly, these variables are functions of space and time, and the corresponding maxima/minimum of these variables sit on different parts of the isopycnal surface, and they move in different direction, as shown in Fig. 5.6. In addition, the velocity perturbations are also internally linked to the layer depth and thickness perturbations. In departure from the simple version of isopycnal analysis, it is important to emphasize that in this new isopycnal layer analysis we must



  !0 ! 0 dFpv ¼ r  hu  q þ r  hu  q ð5:21Þ Due to the complicated nature of the background flow field and the nonlinear interaction, the movements of the dh (dq) and the dD field are rather complicated.

5.4

Difference Between the Casting Method and the Projecting Method

The casting method can be directly applied to any dataset, even if the density stratification in the dataset is not always stable. This method can be applied to the original data without interpolation or other modifications. However, if this method is applied to the climate dataset generated from observations or computer models, the vertical resolution used in the deep ocean may not be adequate, when the data is casted onto the isopycnal coordinates. As a result, the climate variability for the heavy density range may not have adequate resolution, and some of the climate variability patterns generated in the MDC may appear in the forms of banded structure. As an example, we applied the casting method using Dr1 ¼ 0:1 kg/m3 (half of the grid size used to generate Fig. 5.2); the results are shown in Fig. 5.7. The banded structure in the MDC is quite noticeable. On the other hand, the pattern in

5.4 Difference Between the Casting Method …

275

(a) ΣWa in MDC (1016 kg/0.1 1 ) 30.8

(b) ΣWa in FDC (1016 kg/0.1 1 ) 60

31.0

30.8 31.0

40 31.2

0.10 31.2

20

31.4

0.05

31.6

0

31.8

31.8

−0.05

32.0

−20 32.0

32.2

32.2

31.6 0

1

(kg/m3)

31.4

0.15

32.4 32.6

−40

−60

1980 1985 1990 1995 2000 2005 2010

−0.10 −0.15

32.4 32.6

−0.20

1980 1985 1990 1995 2000 2005 2010

Fig. 5.7 The total volume integration of the external mode of the weight content anomaly (seasonal cycle subtracted) in the MDC (a) and the FDC (b), using a high-resolution grid Dr1 ¼ 0:1 kg/m3 in isopycnal coordinates, based on the same 40 non-uniform z-grids in the GODAS data

the FDC remains smooth without such a banded structure. The limitation of crude vertical resolution used in a climate dataset also exists for other applications of the casting method based on directly using the vertical resolution in the original dataset. For example, if we apply the casting method to analyze climate variability on certain horizontal density surfaces, the resulting maps may contain some band-like structures. However, if we interpolate the GODAS data onto a fine uniform grid of 475 layers (10 m for each layer and linearly interpolating the original data on the non-uniform 40 grid), the banded structure (for the range of r1 ¼ 31:2 32:2 kg/m3 ) in the MDC with lower vertical resolution mapping will be replaced by somewhat smoother patterns, as shown in Fig. 5.8. In comparison, patterns in Fig. 5.7b and Fig. 5.8b are basically the same, i.e., the pattern in the FDC is insensitive to the vertical resolution used in calculation and mapping. However, generating

another dataset with higher vertical resolution means using some kind of interpolation method to create the new dataset, which can also change the details of the climate variability, and this must be taken into consideration in climate study. A reminder to the reader is that although the fine resolution results generated in the method discussed above gives rise to a rather smooth pattern, this does not necessarily mean climate variability in the real world is smooth. In fact, if we have climate data with much high resolution generated either from observations or numerical models, fractal patterns somewhat similar to that shown in Fig. 5.7 may appear. However, this is speculation only, and fine resolution climate datasets generated in the near future may deliver a clean-cut answer to this problem. On the other hand, the projecting method is based on interpolation of the original dataset. Using interpolation means this method can avoid the artificial banded structure shown in Fig. 5.7. Furthermore, this method assumes that the

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5

3 1 (kg/m )

(a) ΣWa in MDC (1016kg/0.1 1)

Heaving Signals in the Isopycnal Coordinate (b) ΣWa in FDC (1016kg/0.1 1)

30.8

40

30.8

0.15

31.0

30

31.0

0.10

31.2

20

31.2

31.4

10

31.4

31.6

0

31.6

31.8

−10 31.8

−0.05

32.0

−20 32.0

−0.10

32.2 32.4 32.6

−30 −40 −50

1980 1985 1990 1995 2000 2005 2010

0.05 0

32.2 −0.15 32.4 32.6

−0.20

1980 1985 1990 1995 2000 2005 2010

Fig. 5.8 The total volume integration of the external mode of the weight content anomaly (seasonal cycle subtracted) in the MDC (a) and the FDC (b), using a high-resolution grid Dr1 ¼ 0:1 kg/m3 in isopycnal coordinates and 475 uniform z-grid interpolation of the GODAS data

stratification in the original dataset is stable. Any unstable stratified part of the dataset must be corrected before this method can be used. This raises a question whether the result obtained from this method depends on exactly how the unstable stratified part of the dataset is corrected. In addition, different methods of interpolation can also lead to different results of climate variability. As shown above, although stratification for the large-scale circulation should be stably stratified, unstable stratification often appears in climate datasets based on observations or numerical simulations. Therefore, the problems listed above do exist, and the consequence of these problems is left for further study.

5.5

Isopycnal Layer Analysis for the World Oceans

In this section we apply the projecting method to the GODAS data for the world oceans. Using the projecting method, one can simultaneously

extract climate signals of multiple variables. We begin with the external mode variability of basic variables on isopycnal coordinate r1 (using 1000 db as the reference pressure).

5.5.1 External Modes In the following analysis, we will confine our interest to the density range of r1 ¼ ½25:0 33:7 kg/m3 , with a uniform layer thickness of Dr1 ¼ 0:1 kg/m3 . The variability of five variables, h (layer thickness), D (layer depth), T (layer mean temperature), S (layer mean salinity) and p1 (layer mean spicity) will be discussed here. The variability over the 35 years of data record is further separated into the multiyear mean value, the 35-year mean annual cycle and the interannual variability (with the annual cycle removed). Since signals within shallow density layers have frequency characters different from those in deep layers, these figures are further separated into two subpanels.

5.5 Isopycnal Layer Analysis for the World Oceans

277

Fig. 5.9 Mean thickness for each layer of Dr1 ¼ 0:1 kg/m3 : a the time mean thickness; b the mean annual cycle; c the interannual variability

As discussed above, variability in the external mode is directly linked to changes created by exchange of heat and freshwater across the layer interfaces, in particular the air-sea interface. Even the external modes of high density layers receive climate variability in the air-sea fluxes because these layers outcrop at high latitudes. Layer thickness and its variability are shown in Fig. 5.9. Positive (negative) values of layer thickness anomaly indicate the increase (decline) of water mass volume in the corresponding density category. The mean layer thickness increases gradually from a very small value in the upper ocean to the order of 600 m for the deep layers (panel a). The mean layer thickness has a noticeable annual cycle. In the light density range, there is a bi-annual cycle with a small amplitude of 0.6 m, and this may reflect the change of solar insolation at lower latitudes, especially within the latitudinal band of the Tropic of Cancer. On the other hand, in the heavy part of the density range, the annual cycle is dominated by a simple annual cycle, with a rather large amplitude, on the order of 10 m; this

may reflect the simple annual cycle of solar insolation at high latitudes. Interannual variability of layer thickness for different parts of the density range is also characterized by different frequencies. The light part of the density range is characterized by strong interannual variability with small amplitude of 4 m; on the other hand, the dense part of the water column is characterized by decadal variability with large amplitude on the order of 200 m, as shown in Fig. 5.9c. The layer thickness anomaly shown in the lower part of Fig. 5.9c indicates that during the 1980s the density layer of r1 ¼ 32:6  0:05 kg/m3 was 200 m thicker than the multiple year mean, and it was more than 200 m thinner during the 2000s. Layer depth signals exhibit frequency characters similar to those of layer thickness (Fig. 5.10). However, for the annual cycle the corresponding amplitude is 10 times larger. Layer depth variability tends to have the same sign as the layer thickness, in particular for the layers in the high density range. In fact, they either increase or decrease at the same time, i.e.,

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Heaving Signals in the Isopycnal Coordinate

Fig. 5.10 Mean depth for each layer of Dr1 ¼ 0:1 kg/m3 : a the time mean depth; b the mean annual cycle; c the interannual variability

layers move down (up) and become thicker (thinner) (Figs. 5.9 and 5.10). On the other hand, the interannual variability of layer depth is different from that of layer thickness. Although both of these variables show interannual variability for the light density water masses, for the density larger than r1 ¼ 27 kg/m3 the interannual variability of layer thickness and depth shows a strong sign of decadal signals. The corresponding amplitude of signals is also different. In particular, the layer depth signals for the dense water masses in 1980s are negative, but it becomes positive in the 2000s. In combination with Fig. 5.9, this indicates that these heavy density layers were thicker and shallower in 1980; but their variability flipped signs in 2000s. Changes in isopycnal layer thickness and depth are accomplished by changes in both layer mean potential temperature and salinity (Figs. 5.11 and 5.12). For most of the density range the annual cycle is in the form of a simple annual cycle in both temperature and salinity, without noticeable bi-annual components as in

the cases of layer thickness and depth. It is interesting to notice that for heavy density range and over most part of the time in this century both the mean temperature and salinity is reduced. The same sign for the interannual variability in both the temperature and salinity on the same isopycnal surface is required for density compensation. Since density in each layer is constant, variability in layer mean temperature and salinity must be compensated; thus, the most concise and accurate layer mean thermodynamic variable is potential spicity, and the corresponding signals are shown in Fig. 5.13. Apparently, the pattern of variability in spicity shows a finer structure, which may be useful in analyzing climate variability. The decline trend of spicity for dense water is consistent with the general trend of spicity changes in the world oceans, as shown in Fig. 3.72d. In order to show the climate variability in a clear way we include the time evolution of layer properties for three isopycnal layers (Fig. 5.14).

5.5 Isopycnal Layer Analysis for the World Oceans

279

Fig. 5.11 Mean temperature for each layer of Dr1 ¼ 0:1 kg/m3 : a the time mean temperature; b the mean annual cycle; c the interannual variability

Fig. 5.12 Mean salinity for each layer of Dr1 ¼ 0:1 kg/m3 : a the time mean salinity; b the mean annual cycle; c the interannual variability

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5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.13 Mean spicity for each layer of Dr1 ¼ 0:1 kg/m3 : a the time mean; b the mean annual cycle; c the interannual variability

Climate variability in the lighter isopycnal layers seems rather small (red curves in Fig. 5.14); however, climate variability for the deep isopycnal layers is much stronger (black curves in Fig. 5.14). Both the layer thickness and the depth anomaly increased over the past 30 years in the isopycnal layer of r1 ¼ 32  0:05 kg/m3 . In contrast, there are strong decadal variations of layer thickness and depth for the isopycnal layer of r1 ¼ 33  0:05 kg/m3 . The difference of climate variability in these two isopycnal layers reflects the complex nature of climate variability in the stratified ocean. The climate variability of temperature and salinity on the decadal time scale has patterns different from those of layer thickness and depth (Fig. 5.14c, d). The variability for the isopycnal layers of r1 ¼ 31  0:05 kg/m3 and r1 ¼ 32  0:05 kg/m3 is rather small. For the isopycnal layer of r1 ¼ 33  0:05 kg/m3 , the climate variability started from negative values in 1980,

increased until 1990; then oscillated and slightly went down afterward. The corresponding signals in potential spicity are shown in Fig. 5.14e. As commented above, the patterns of temperature, salinity and spicity are the same because of the density compensation constraint. These panels show that oceanic dynamic processes lead to different patterns for layer depth and thickness, in comparison with those of temperature, salinity and spicity.

5.5.2 Heaving Modes for r1 ¼ 30:9  0:05 kg/m3 We now examine climate signals for a specific isopycnal layer r1 ¼ 30:9  0:05 kg/m3 . The external modes of mean layer thickness, layer depth, temperature, salinity and spicity are shown in Fig. 5.15; these correspond to the sections through r1 ¼ 30:9 kg/m3 in Figs. 5.9,

5.5 Isopycnal Layer Analysis for the World Oceans

281

Fig. 5.14 Interannual variability of properties for three isopycnal layers

5.10, 5.11, 5.12 and 5.13. There are noticeable annual cycles of the external modes. In particular, the layer thickness has a bi-annual cycle. However, for the layer depth the bi-annual cycle is weak; instead, it is more like a simple annual cycle. For temperature, salinity and spicity there is also a simple seasonal cycle only. The interannual variability is shown on the right panels. As discussed above, climate

variabilities for such a dense isopycnal layer are dominated by decadal signals. From 1980 to 2000, the layer thickness did not change much; on the other hand, the layer depth was large (layer was deeper) in 1980, and then this layer became shallower from 1980 to 1990. However, the layer thickness oscillated with a rather large amplitude from 2002 to 2014; the layer depth also oscillated in phase with the

282

5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.15 Time evolution of the external modes of dh, dD, dT, dS and dp1 in the layer of r1 ¼ 30:9  0:05 kg/m3

layer thickness. Layer mean temperature, salinity and spicity oscillated in a way different from the

layer thickness and depth. The reason of such differences remain to be explored.

5.5 Isopycnal Layer Analysis for the World Oceans

5.5.3 Horizontal Distribution of Climate Variability for r1 ¼ 30:9  0:05 kg/m3

and the corresponding mean absolute value is calculated as jdDj ¼ jD0 j

We also plot the mean absolute value of climate variability defined as follows. For example, the layer depth can be separated into the mean annual cycle and the interannual variability (with annual cycle removed): D ¼ Dannual cycle þ D0

283

ð5:22Þ

The interannual variability is defined for each month over the total length of the data record,

420 month

ð5:23Þ

The horizontal distribution of mean properties for the isopycnal layer r1 ¼ 30:9  0:05 kg/m3 is shown in Fig. 5.16, including the layer depth, layer thickness, temperature and salinity. The layer depth map indicates the existence of winddriven gyres in the ocean, in particular the five bow-shaped subtropical gyres. The layer thickness maxima indicate the places with thick layer or low potential vorticity mode water. The top two places with maximum layer thickness are in

Fig. 5.16 Horizontal distribution of layer-mean fields for r1 ¼ 30:9  0:05 kg/m3 : a layer depth; b layer thickness; c salinity and d temperature

284

the North Atlantic Ocean and South Indian Ocean (Fig. 5.16b). In the eastern basin of the Pacific Ocean there is a layer thickness maximum, that may represent the so-called Eastern Subtropical Mode Water. The mean salinity and temperature in this isopycnal layer reflect the fundamental feature of the thermohaline circulation in the world oceans, with the warm and salty water in the Atlantic Ocean and the relatively cold and fresh water in the Pacific Ocean. The properties in the Indian Ocean are in the middle range (Fig. 5.16c, d). The horizontal distribution of mean absolute value of layer property perturbations for the isopycnal layer of r1 ¼ 30:9  0:05 kg/m3 is

5

Heaving Signals in the Isopycnal Coordinate

shown in Fig. 5.17, including the layer depth, layer thickness, temperature and salinity. In the top panels, the areas with red color indicate the regions characterized by a strong layer depth/thickness anomaly, suggesting strong heaving of this isopycnal layer. The most outstanding areas are in the central South Indian Ocean and the North Western Atlantic Ocean. In addition, there are also large signals in the North Pacific Ocean and South Atlantic Ocean. In terms of salinity and temperature, the maximum is located in the northwestern part of the Atlantic Ocean, as shown in the lower panels of Fig. 5.17. In addition, there are narrow strips of area southwest of Australia and southwest of

Fig. 5.17 Horizontal distribution of layer-mean absolute values of interannual climate variability for r1 ¼ 30:9  0:05 kg/m3 : a layer depth; b layer thickness; c salinity and d temperature

5.5 Isopycnal Layer Analysis for the World Oceans

285

Fig. 5.18 Horizontal distribution of a the mean potential spicity p1 ; b the layer-mean absolute values of interannual climate variability for r1 ¼ 30:9  0:05 kg/m3

the Atlantic Ocean where salinity and temperature anomaly signals are relatively strong. As discussed above, for data collected on isopycnal surfaces/layers temperature and salinity signals must be compensated, so that the corresponding water masses stay in the same potential density layers. There are many important dynamical climate signals, and some of them are hidden in the original signals collected on these isopycnal layers, including the depth and thickness of the isopycnal layers; these signals can be identified by referring to the climatology. On the other hand, in each isopycnal layer the only signals left behind are the passive climate signals in forms of spicity variability within each isopycnal layer. Therefore, a better way to present climate variability in isopycnal coordinates is to present spicity variability for individual isopycnal layers. In the present case, we show the mean potential spicity in this isopycnal layer in Fig. 5.18a. This figure shows the typical pattern of spicity distribution in the world oceans. Most importantly, the subtropical North Atlantic Ocean is characterized by high spicity associated with warm and salty water masses; similarly, spicity is high in the South Atlantic Ocean, the south-western part of the South Pacific Ocean and the Indian Ocean, plus the Arabian Sea. On the other hand, spicity is low in the North Pacific

Ocean and the Antarctic Circumpolar Current where water masses are relatively cold and fresh. Spicity anomalies are high in the North Atlantic Ocean near the site of the Gulf Stream recirculation (Fig. 5.18b), indicating the strong climate variability associated with mode water formation. In addition, there are two high spicity variability centers in the Southern Hemisphere: one in the Brazil–Malvinas Confluence Zone in the Atlantic Ocean and another one southwest of Australia, indicating strong air-sea interaction in these two regions.

5.5.4 The Heaving Ratio As discussed in Chap. 3, the climate signals over the past several decades are primarily associated with heaving or movements of the isopycnal layers. The state of circulation at each station in an isopycnal layer can be analyzed using the Lagrangian coordinate and described in terms of the radius of signal introduced in Sect. 3.4.1. Since the contribution associated with the stretching term is linearly proportional to the mean thickness of the isopycnal layer, we will focus on the relative contribution of heaving and spicing terms here. Using the notation defined in Sect. 3.4.1 (formula 3.74a, b, c, and d), the results are shown in Fig. 5.19, where the radius

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Fig. 5.19 Horizontal distribution of the radius of signal on r1 ¼ 30:9  0:05 kg/m3 associated with a the potential density; b the potential spicity; c the total radius of signal; d the radius of signal after isopycnal analysis

of signal (RS) is defined by the Root-Mean Square (RMS) of signals as follows. Panel a: RS for potential density variability:   RSðr1;ins  r1;ref Þ ¼ RMS r1;ins  r1;ref  r1;ins  r1;ref

ð5:24Þ Panel b: RS for potential spicity:   RSðp1;ins  p1;ref Þ ¼ RMS p1;ins  p1;ref  p1;ins  p1;ref

ð5:25Þ Panel c: RS for the combination of potential density and potential spicity: h RSðr1 ; p1 Þ ¼ RSðr1;ins  r1;ref Þ2 i1=2 þ RSðp1;ins  p1;ref Þ2

ð5:26Þ

Panel d: RS for the in situ potential spicity (signals left behind after isopycnal analysis):   RSðp1;ins Þ ¼ RMS p1;ins  p1;ins

ð5:27Þ

The RS of potential density and potential spicity is high near the site of the Gulf Stream and the Kuroshio; in addition, it is high in the equatorial band and the Brazil–Malvinas Confluence Zone. The RS for the total signal is high in these regions (Fig. 5.19a–c). Thus, climate variability in these regions is strong. Note that the amplitude of the RS associated with potential spicity is much larger than that associated with potential density; as a result, potential spicity signals dominate the climate signals. However, after isopycnal analysis, the only signals left behind are the in situ spicity signals

5.5 Isopycnal Layer Analysis for the World Oceans

(Fig. 5.19d). The strong signals appear to be confined mostly to the north of the Gulf Stream and the site of mode water formation in the North Atlantic Ocean. In addition, the RS is relatively high in the Brazil–Malvinas Confluence Zone in the Atlantic Ocean and southwest of Australia in the South Indian Ocean. At each station in this isopycnal layer, we can calculate the heaving ratio defined as follows. Since we use a relatively thin isopycnal layer, the contribution associated with the stretching term is much smaller than the corresponding term. Thus, in the following discussion we will focus on the heaving term and spicing term and omit the potential contribution associated with the stretching term. We will adapt the following definition of the heaving ratio HR ¼

RSðr1 ; p1 Þ    100% ð5:28Þ RSðr1 ; p1 Þ þ RS p1;ins

The map of the heaving ratio based on the 35 year mean climate data for r1 ¼ 30:9  0:05 kg/m3 is shown in Fig. 5.20. It is clear that for most parts of the world oceans climate signals

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in this isopycnal layer are predominately heaving in nature, as depicted by the red color. There are two places characterized by a low heaving ratio. First, the northwestern corner of the North Atlantic Ocean where temperature and salinity anomaly are strong (Fig. 5.17). Due to strong airsea interaction, temperature, salinity and density vary greatly, so is the spicity (Fig. 5.18b). Second, the ACC band is associated with a low heaving ratio; this is also associated with a strong temperature and salinity anomaly (Fig. 5.17). Third, there are four strips of oceanic area in the South Indian Ocean, where the heaving ratio is relatively low in two of the strips, but it is high in the other two strips. These strips correspond to strong air-sea interaction and heaving motions in the South Indian Ocean. In addition, there is clearly a band of a low heaving ratio located within the boundary between the subtropical and subpolar gyres in the North Pacific Ocean, extending from the western coast of Canada. This implies strong air-sea interaction; however, the exact meaning of this band remains unclear at this time.

Fig. 5.20 Heaving ratio for climate variability in isopycnal layer r1 ¼ 30:9  0:05 kg/m3 (b), the zonal mean (a) and meridional mean (c)

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5.5.5 Regional Anomaly Patterns 5.5.5.1 The Northwest Atlantic Sector The anomaly associated with the mean annual cycle in the Northwest Atlantic Ocean is shown in Fig. 5.21. The layer depth anomaly maximum (40 m) appears near the northern edge of the strong Gulf Stream system; the layer thickness anomaly maximum appears 10° eastward. These anomalies are far away from the continents; nevertheless, they may be linked to the cold break from the North American continent; in addition, they are probably associated with the strong seasonal cycle of the Gulf Stream system, induced mostly by wind stress and the

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thermohaline forcing anomaly. The temperature maximum has a rather low amplitude, and it appears much closer to the coast line, suggesting the potential role of air-sea heat flux, in particular, the latent heat flux associated with evaporation. Since the temperature and salinity anomaly observed on a fixed isopycnal should be density compensated, we also include the spicity anomaly in this figure. As expected, the spicity anomaly maximum appears in the same location as the temperature anomaly maximum. The corresponding anomaly associated with interannual variability is shown in Fig. 5.22, and its amplitude is about three times larger than that of the annual cycle. Note that the temperature

Fig. 5.21 Layer-mean absolute values of the annual cycle in the North Atlantic Ocean for r1 ¼ 30:9  0:05 kg/m3 : a layer depth; b layer thickness; c temperature and d spicity

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Fig. 5.22 Layer-mean absolute values of interannual variability in the North r1 ¼ 30:9  0:05 kg/m3 : a layer depth; b layer thickness; c temperature; d potential spicity

anomaly maximum now appears more northward, around 44° N–46° N, south of the Georges Bank. On the other hand, the spicity anomaly maximum is located slightly southward, around 40° N. The different locations of the temperature and spicity anomalies on the decadal time scale imply different degrees of temperature and salinity compensation on the decadal time scale. This phenomenon should be examined in details.

5.5.5.2 The South Indian Sector For comparison we present the anomaly associated with the mean annual cycle in the South

Atlantic

Ocean

for

Indian Ocean (Fig. 5.23). The layer depth anomaly maximum (40 m) appears near the ACC; the layer thickness anomaly maximum appears a few degrees northward. These are probably associated with the strong seasonal cycle of the ACC, induced by local wind stress and thermohaline forcing anomaly. The temperature/spicity maxima have a rather low amplitude, and they appear slightly southward; they may be more closely linked to the local thermohaline forcing anomaly. The corresponding anomaly associated with interannual variability has an amplitude two

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Fig. 5.23 Layer-mean absolute values of the annual cycle in the South Indian Ocean for r1 ¼ 30:9  0:05 kg/m3 : a layer depth; b layer thickness; c temperature; d potential spicity

times larger than that of the annual cycle (Fig. 5.24). Note that the layer depth and thickness anomaly now appear more northward than those associated with the annual cycle. Their locations seem to be in the center of the subtropical gyre, suggesting the role of a winddriven gyre adjustment. The corresponding temperature/spicity anomaly maxima appear more eastward, around 90° E–100° E, close to the western coast of Australia. These anomalies suggest the potential roles of strong air-sea interaction and horizontal advection on the interannual time scale in this vicinity.

5.5.5.3 Time Evolution of the Anomalies in the North Atlantic and South Indian Sectors In order to understand the character of interannual variability, we plot the time evolution of the interannual variability of the external mode (EX) and internal mode (IN) at three stations for the layer thickness, layer depth, temperature, salinity and spicity. As shown in Fig. 5.25, the amplitude of the external mode (red curves) is about several times smaller than the internal modes; in addition, the EX mode includes annual

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Fig. 5.24 Layer-mean absolute values of Interannual variability in the South r1 ¼ 30:9  0:05 kg/m3 : a layer depth; b layer thickness; c temperature; d potential spicity

frequency oscillations. The amplitude of the external mode of the layer thickness is much smaller than that of the layer depth, i.e., the layer may move up and down, but the total volume of the layer remains nearly unchanged. The internal mode of variability in the South Indian Ocean (blue and green curves) is characterized by strong signals on the decadal time scale. On the other hand, variability in the North Atlantic Ocean includes relatively weaker signals on both the decadal and interannual time scales.

Indian

Ocean

for

5.5.6 A Meridional Section Through 60.5° W We now examine the structure of heaving signals along individual sections, beginning with a section through 60.5° W in the North Atlantic Ocean. This section cuts through the mode water formation region and the core of the Gulf Stream recirculation; consequently, it can reveal information relevant to climate changes in terms of heaving. As shown in Fig. 5.26a, isopycnal layer

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Fig. 5.25 Interannual variability of the external mode (EX, multiplied by 10) and internal mode (IN), at three stations, of a layer thickness, b layer depth, c temperature, d salinity and e spicity

thickness has a local maximum near r1 ¼ 30:8  0:05 kg/m3 , indicating the existence of mode water between latitude 25° N and 35° N. This section is characterized by warm and salty water in the Atlantic Ocean (Fig. 5.26c, d); this combination of temperature and salinity gives rise to spicy water in the middle latitude

band, as shown in Fig. 5.26f. As for the mean potential density, at any given potential density surface, the instantaneous value should be the same as the nominal value; hence, we plot the mean of the reference potential density instead. It is interesting to notice that the mean potential density corresponding to each isopycnal layer is

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Fig. 5.26 Vertical distribution of the layer-mean fields along 60.5° W: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

different from the nominal climatological mean value; instead, it is somewhat biased to the target value (panel e), but the exact meaning of this bias remains unclear. Interannual variability of the layer properties in this section is shown in Fig. 5.27. Layer thickness perturbation maximum appears near density band r1 ¼ 30:8  0:05 kg/m3 , as shown in Fig. 5.27a. This is also consistent with layer depth perturbation maximum shown in Fig. 5.27b. The region of layer thickness and depth perturbation maximum indicates a zone of strong heaving activity in this section. On the other hand, both temperature and salinity perturbations have a maximum at high latitudes and within the less dense water layers, suggesting that climate signals in this region may have strong diabatic components (Fig. 5.27c, d). There is strong vertical movement of isopycnal layers in both the southern and northern ends of this section, as shown in Fig. 5.27e. On the other

hand, the strong spicity anomaly appears near 40° N. In this narrow region, although the temperature and salinity anomalies are compensated, lateral advection can leave behind a trace of climate variability in forms of the spicity anomaly. The relation between heaving and nonheaving signals can be further illustrated in terms of the radius of signal introduced above. In order to compare the contribution due to these components we show the composed Fig. 5.28. The radius of signal associated with potential density and potential spicity is shown in Fig. 5.28a, b. The signal maximum associated with potential density is located in two centers at 10° N and 40° N. However, the amplitude of radius of signal associated with potential density perturbations is rather low, as shown in Fig. 5.28a; consequently, these two centers do not seem to contribute to the maximum of the total radius of signal shown in Fig. 5.28c.

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Fig. 5.27 Vertical distribution of mean absolute values of interannual climate variability along 60.5° W: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

Fig. 5.28 Vertical distribution of radius of signal along 60.5° W: a associated with potential density; b potential spicity; c the total radius of signal; d radius of signal after isopycnal analysis

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Fig. 5.29 Distribution of the heaving ratio along 60.5° W (b), the meridional mean (a) and the vertical mean (c)

On the other hand, the potential spicity anomaly gives rise to a much larger radius of signal, and makes a vital contribution to the total radius of signal, as shown in Fig. 5.28b, c. The radius of signal for the spicity anomaly left behind after isopycnal analysis is shown in Fig. 5.28d, and its pattern is quite similar to that of the total radius of signal before isopycnal analysis. For each isopycnal layer at a station, we can calculate the heaving ratio using the definition introduced above. The distribution of the heaving ratio along this section is shown in Fig. 5.29. It is clear that for most part of this section, climate variability is dominated by the heaving signals. It is only in a small region north of 45° N that the non-heaving signals become dominating; accordingly, diabatic processes associated with air-sea interaction are the major contributor to climate variability in this region. However, overall most climate variability in this section is primarily heaving in nature.

As an example, we plot the climate variability for the r1 ¼ 30:7  0:05 kg/m3 isopycnal layer over the period of 1980–1987 (Fig. 5.30). As discussed above, the contribution due to the external mode is relatively small, and the pattern of the anomaly does not really change whether or not the external signals are separated. Therefore, we do not separate the external and internal modes of heaving signals. In this figure and similar figures shown later, the blank areas indicate that there is no data, implying that the corresponding isopycnal layers outcrop; as a result, there is no corresponding data projection. There are strong layer thickness perturbations, mostly appearing within latitude bands of 33° N and 38° N, indicating the potential connection with mode water formation. On the other hand, the layer depth anomaly appears at slightly higher latitudes, from 33° N to 42° N. There is extremely strong interannual variability, on the order of −300 to 250 m (Fig. 5.30b). The

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Fig. 5.30 Climate variability along this section and for the isopycnal layer of r1 ¼ 30:7  0:05 kg/m3 from 1980 to 1987: a layer thickness; b layer depth; c temperature; d spicity

temperature and spicity perturbations appear at higher latitude bands, from 35° N to 44° N. In addition, we plot the climate variability for the r1 ¼ 30:7  0:05 kg/m3 isopycnal layer over the period of 2005 to 2014 (Fig. 5.31). Climate variability patterns shown for this time period look quite different from those for the period of 1980–1987. Thus, it is clear that the climate variability along this section has very strong decadal signals.

5.5.6.1 A Meridional Section Through 64.5° E For an additional comparison, we examine the structure of heaving signals along 64.5° E in the Indian Ocean. This section cuts through the mode water formation and the core of the ACC, and it can reveal the relevant information about climate changes in terms of heaving. As shown in Fig. 5.32a, isopycnal layer thickness has a maximum near r1 ¼ 31 kg/m3 , indicating the

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Fig. 5.31 Climate variability along this section and for the isopycnal layer r1 ¼ 30:7  0:05 kg/m3 from 2005 to 2014: a layer thickness; b layer depth; c temperature; d spicity

existence of mode water between latitude 30° S and 40° S. Around 40° S, there are strong temperature, salinity, potential density and potential spicity fronts, which are associated with the strong currents in the ACC. The interannual variability of the layer properties in this section is shown in Fig. 5.33. The layer thickness perturbation maximum appears near the density band r1 ¼ 31:1 kg/m3 (Fig. 5.33a). This is also consistent with climate mean layer depth maximum shown in Fig. 5.32a. The regions of layer thickness and depth perturbation maxima in the subsurface layers indicate

the zone of strong heaving activity in this section. On the other hand, the strongest potential density signals appear in the subsurface ocean south of the equator (Fig. 5.33e); this region is characterized by very weak temperature and salinity perturbations (Fig. 5.33c, d). Therefore, this region is dominated by strong heaving signals. On the other hand, both temperature and salinity perturbations have their maxima at higher latitudes in the Southern Hemisphere and within the slightly dense water layers, suggesting that climate signals in this region may have

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Fig. 5.32 Vertical distribution of layer-mean fields along 64.5° E: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

strong diabatic components (Fig. 5.33c, d). The peaks of spicity perturbation signals appear near the surface layer, including a peak south of the Equator and a secondary peak around 40° S (Fig. 5.33f). The relation between heaving and nonheaving signals in this section is further illustrated in terms of the radius of signal. Figure 5.34 shows the comparison of contributions due to these components. In this figure, the radius of signal for the potential density component and the potential spicity component are shown in panels a and b, and the total signal is shown in panel c. Accordingly, the total signals are primarily controlled by the potential density component, with the contribution of potential spicity as the secondary source. On the other hand, the net signal after isopycnal analysis is shown in panel d; and the spatial pattern is quite different from those shown in panels a, b, and c.

The map of the heaving ratio in this section is shown in Fig. 5.35. Over the major part of this section climate signals are dominated by the heaving component. There is only one narrow tongue of the low heaving ratio, that stems from the sea surface and penetrates to deep isopycnal layers—this indicates strong air-sea interaction and mode water formation in this region. We now examine climate variability along a single isopycnal layer of r1 ¼ 31  0:05 kg/m3 isopycnal layer over the period of 1980–1987 (Fig. 5.36). In this figure the external and internal modes of heaving signals are not separated. As discussed above, the contribution due to the external mode is relatively small; accordingly, the pattern of the anomaly does not really change whether or not the external signals are separated. There are strong layer thickness perturbations, mostly appearing within the latitude bands of 35° S and 38° S; in fact, for this whole time period the layer thickness perturbation is negative,

5.5 Isopycnal Layer Analysis for the World Oceans

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Fig. 5.33 Vertical distribution of mean absolute values of interannual climate variability along 64.5° E: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

Fig. 5.34 Vertical distribution of the radius of signal along 64.5° E, associated with: a potential density; b potential spicity; c the total radius of signal; d radius of signal after isopycnal analysis

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Fig. 5.35 Distribution of heaving ratio along 64.5° E (b), and the meridional mean (a) and the vertical mean (c)

indicating the deficit of mode water formation over this 9-year period. On the other hand, there was a large amplitude layer depth anomaly appearing at a slightly lower latitude, around 30° S. There was extremely strong interannual variability of layer depth perturbation, on the order of −150 to 150 m (Fig. 5.36b). The temperature and salinity perturbations appear at a higher latitude band, on the edge of the outcrop line, south of 40° S. In comparison, climate variability for the r1 ¼ 31:0  0:05 kg/m3 isopycnal layer over the period of 2005–2014 is shown in Fig. 5.37. Climate variability patterns shown for this time period look quite different from those for the period of 1980–1987. For example, layer thickness perturbations in the latitude band of 35° S were mostly positive for this time period. In contrast, layer depth perturbations along the latitude band of 30° S–35° S were mostly negative for this time period. In addition, temperature and spicity anomaly along the edge of the outcrop line were larger

than those for the time period of 1980–1987. For this reason, it is clear that climate variability along this section has very strong decadal signals.

5.5.7 A Zonal Section Along the Equator As a comparison, we examine the structure of heaving signals along the Equator. We begin with the mean layer properties (Fig. 5.38). To show the pattern of layer thickness and depth clearly, the logarithm scale is used in panels a and b. The most noticeable feature is the relatively thick layers in the upper part of the equatorial Indian, Pacific and Atlantic Ocean clearly (Fig. 5.38a). Below the thick layers, the main pycnocline appears in the form of low thickness layers in these three basins, plus the cold tongues in both the Pacific and Atlantic basins. Interannual variability of the layer properties in this section is shown in Fig. 5.39. Our

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Fig. 5.36 Climate variability along this section and for the isopycnal layer r1 ¼ 31  0:05 kg/m3 from 1980 to 1987: a layer thickness; b layer depth; c layer temperature; d layer spicity

discussion is focused on the Pacific sector. There are clearly three patches of large layer thickness anomaly (Fig. 5.39a). On the top, the strong layer thickness perturbations are closely linked to the ENSO events within the upper layers. Below these layers of large thickness anomaly, there is a region of very weak layer thickness perturbations due to the strong stratification in the main pycnocline. There is a patch of strong layer thickness anomaly near the eastern boundary and in the density range of r1 ¼ 30:5  30:8 kg/m3 (mean depth of 150–250 m); this local maximum seems to be linked to the core of the cold tongue. On the

bottom, there is a layer perturbation maximum for the dense water masses in this section. The regions of layer depth perturbation maximum appear as a slanted patch extending from the central equatorial upper ocean toward the eastern boundary, where it reaches the maximum; there is a low layer depth anomaly in the cold tongue and the warm pool (Fig. 5.39b). Note that the location of the layer thickness perturbation maximum and the layer depth perturbation maximum are located in different places, although they also collapse in the central part of the upper ocean. These patches of strong

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Fig. 5.37 Climate variability along this section and for the isopycnal layer r1 ¼ 31:0  0:05 kg/m3 from 2005 to 2014: a layer thickness; b layer depth; c layer temperature; d layer spicity

anomaly in layer thickness and layer depth perturbations indicate the zone of strong heaving activity in this section. On the other hand, both temperature and salinity perturbations have their maxima mostly in the upper part of the cold tongue, suggesting that climate signals in this region may have strong diabatic components (Fig. 5.39c, d). In comparison, the temperature and salinity anomalies have rather low amplitudes in the warm pool and subsurface, indicating that the diabatic signals are quite weak in this part of the Pacific Ocean. In addition, there is the second maximum of temperature and salinity

anomaly in the subsurface layers near the western boundary of the Pacific basin. Finally, we look at the signals associated with the potential density anomaly, which is quite strong in the subsurface layers, as shown in Fig. 5.39e. Consequently, heaving activity is quite strong for this part of the water column. In comparison, signals associated with potential spicity have two peaks: one near the eastern boundary, that is clearly linked to the cold tongue; the second one is located in the subsurface layers and close to the western boundary of this section. This one might be linked to the source

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Fig. 5.38 Vertical distribution of layer-mean fields along the Equator: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

water from the South Pacific in connection with the Indonesian Throughflow. The relation between heaving and nonheaving signals in this section is further illustrated in terms of the radius of signal (Fig. 5.40). The radius of signal associated with potential density and potential spicity all have a peak in the subsurface layers in the eastern Equatorial Pacific. For the potential density, there is a secondary maximum west of the dateline. The contribution from these two components give rise to the pattern shown in Fig. 5.40c. On the other hand, the net signal after isopycnal analysis is shown in panel d; and the spatial pattern is quite different from those shown in panels a, b, and c. There are two peaks, one near the cold tongue and another one near the western

boundary, corresponding to the strong spicity anomaly signals displaced in Fig. 5.39f. As shown in Fig. 5.41, the climate signals over the major part of the equatorial section are dominated by heaving signals, all way from the surface layers to the bottom layers. There are two regions with a low heaving ratio: the cold tongue and a small patch near the western boundary (Fig. 5.41b). Therefore, climate signals along the equatorial band are primarily in the form of heaving motions. It is only within the two narrow regions where the diabatic processes associated with airsea interaction lead to strong non-heaving signals. Hence, we conclude that although climate variability in the equatorial region, such as the ENSO events, are closely linked to air-sea interactions, most of the processes are closely

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Fig. 5.39 Vertical distribution of mean absolute values of interannual climate variability along the Equator: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

Fig. 5.40 Vertical distribution of the radius of signal along the Equator: a associated with potential density; b potential spicity; c the total radius of signal; d the radius of signal after isopycnal analysis

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Fig. 5.41 Distribution of the heaving ratio along the Equator (b), and the zonal mean (a) and the vertical mean (c)

connected to the heaving motions, i.e., the movement of isopycnal surfaces.

5.5.8 A Zonal Section Along 45.17° N In addition, we show the structure of heaving signals along 45.17° N in the Atlantic Ocean. This section cuts through the core of the North Atlantic Current and the associated water masses, revealing the relevant information about climate changes in terms of heaving. As shown in Fig. 5.42a, the isopycnal layer thickness has a maximum near r1 ¼ 31:6 kg/m3 , indicating the existence of mode water. Mean temperature and salinity distribution along this section reveals the noticeable contrast between the northward warm and salty water masses and the southward cold and fresh water masses, indicated by the sharp front between 50° W and 40° W (Fig. 5.42c, d). The contrast between different water masses is clearly shown

by the low spicity for the southward water masses in the western part and the high spicity for the northward water masses (Fig. 5.42f). Interannual variability of the layer properties in this section is shown in Fig. 5.43. The layer thickness perturbation maximum appears near the longitudinal band of 50° W–40° W and over the density range of r1 ¼ 31:4  32:2 kg/m3 (Fig. 5.43a). This is also consistent with the layer depth perturbation maximum shown in Fig. 5.43b. The regions of layer thickness and depth perturbation maxima indicate the zone of strong heaving activity in this section. On the other hand, both temperature and salinity perturbations have a maximum near the western boundary of this section and within the slightly less dense water layers of r1 ¼ 30  31:5 kg/m3 , suggesting that heaving signals in this region may have strong diabatic components (Fig. 5.43c, d). Since these density layers are underneath the air-sea interface, the strong diabatic signals in this section are unlikely

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Fig. 5.42 Vertical distribution of layer-mean fields along 45.17° N: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

due to the local air-sea interaction; instead, they may be advected from the upstream regions by the currents. In comparison, there is no temperature and salinity anomaly maximum in the eastern part of this section, indicating that the diabatic heaving signals are quite weak in this part of the Atlantic sector. The patterns of the reference potential density anomaly and the potential spicity anomaly are different, although they both reach the maximum in the western part of this section. As commented above, these two panels show the regions of strong heaving motions and strong diabatic activity through this section (Fig. 5.43e, f). The relation between heaving and nonheaving signals in this section is further illustrated in terms of the radius of signal (Fig. 5.44). The radius of signal associated with potential

density peaks in the subsurface layers in the region west of 40° W and in the subsurface layers (Fig. 5.44a); however, the corresponding amplitude is small. On the other hand, the RS of the spicity peaks in slightly different locations, as shown in Fig. 5.44b, d. The corresponding amplitude is much larger. Accordingly, the total signals are primarily controlled by the potential spicity component, and with the contribution of potential density as the secondary source. The climate signals over the major part of this section are dominated by heaving signals, in particular in the eastern part (Fig. 5.45). The region near the western boundary is clearly dominated by the non-heaving component, which is closely linked to air-sea interaction in the processes of mode water formation.

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Fig. 5.43 Vertical distribution of mean absolute values of interannual climate variability along 45.17° N: a layer thickness; b layer depth; c temperature; d salinity; e reference potential density; and f potential spicity

Fig. 5.44 Vertical distribution of radius of signal along the 45.17° N section: a associated with potential density; b potential spicity; c the total radius of signal; d radius of signal after isopycnal analysis

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Fig. 5.45 Distribution of the heaving ratio along 45.17° N (b), and the zonal mean (a) and the vertical mean (c)

5.6

Isopycnal Layer Analysis Based on r0

In comparison with isopycnal layer analysis based on r1 , in this section we analyze the heaving signals in the world oceans using r0 . As an example, we examine climate variability for the isopycnal layer of r0 ¼ 26:8  0:05 kg/m3 . Climate variability is separated into the external mode and internal mode. Time evolution of the external mode is shown in Fig. 5.46. Both the layer thickness and temperature show a clear trend over the past 30 years, as shown in panels a and b. There are clearly relatively largeamplitude oscillations over the period of 2007– 2015. The evolution of layer depth variability seems different. Mean layer properties and their anomalies are shown in Figs. 5.47, 5.48 and 5.49. Note that due to the nonlinearity of the equation of state, isopycnal layer depths based on r0 and r1 cannot

be the same. As a result, the mean properties in these isopycnal layers are different. As shown in Fig. 5.47a, the mean depth of this isopycnal layer is somewhat similar to that for the isopycnal layer of r1 ¼ 30:9 0:05 kg/m3 , shown in Fig. 5.16a. On the other hand, mean layer thickness (Fig. 5.48a) in this isopycnal layer looks quite different from that shown in Fig. 5.16b. In terms of perturbations, both the layer depth and thickness perturbations have the global maximum in the Indian and Pacific sectors of ACC (Figs. 5.47b and 5.48b). In particular, the global maximum of layer thickness and its perturbation near the southwestern corner of Australia indicate the mode water formation and strong variability, which should be investigated more carefully. In addition, there are the secondary maxima at the latitude of the Gulf Stream and the Kuroshio. In these regions, the temperature anomaly has a rather low amplitude, implying the adiabatic nature of heaving signals in these areas

5.6 Isopycnal Layer Analysis Based …

309

Fig. 5.46 Global mean layer mean properties for the isopycnal layer of r0 ¼ 26:8  0:05 kg/m3

(Fig. 5.49). There is one exception: in the North Atlantic Ocean there is a global maximum of temperature anomaly at the latitude band of the Gulf Stream; this maximum is close to the region of layer depth and thickness perturbation maximum (Fig. 5.48b). This implies strong diabatic processes related to air-sea interaction and mode water formation. Since the temperature and salinity anomalies on an isopycnal surface must be density compensated, climate variability is manifested in the form of the spicity anomaly. Figure 5.50 shows the climatology of mean spicity and its anomaly in r0 ¼ 26:8  0:05ðkg/m3 Þ. The regions of high spicity anomaly are located in the places with a high temperature anomaly (Fig. 5.49b). The radius of signal associated with different terms is shown in Fig. 5.51. In the present case, the pattern of the radius of signal is primarily

controlled by the contribution associated with the spicity term. The patterns of these two terms are quite similar (Fig. 5.51a, b); however, the amplitude of the radius of signal associated with the spicity term is twice of that associated with potential density terms. As a result, the total radius of signal has a pattern similar to that of spicity (panel c). On the other hand, the radius of signal after isopycnal analysis is also quite large and its maximum is located in the northwest corner of the Atlantic basin (panel d). The heaving ratio is shown in Fig. 5.52. Similar to the previous maps, for most part of the world oceans, heaving signals are dominating. The most noticeable exceptions are the northwestern part and a narrow region north of the equator of the North Atlantic Ocean, where the low heaving ratio is closely linked to the strong air-sea interaction and other non-heaving

310

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Heaving Signals in the Isopycnal Coordinate

Fig. 5.47 Mean layer depth and absolute value of perturbation for the isopycnal layer of r0 ¼ 26:8  0:05 kg/m3

processes. In addition, there are several regions in the Southern Hemisphere where the heaving ratio is relatively low, indicating relatively strong non-heaving motions in these arears. Since the perturbation anomalies of layer depth, layer thickness and temperature are located next to each other, we show the refined figures for these variables in the northwestern North Atlantic Ocean (Fig. 5.53). It is clear that the depth perturbation and temperature perturbation maxima are located in slightly different places. To illustrate the detail of layer property perturbations, the time evolution of these properties at two selected stations is shown in Fig. 5.54. The first station is located at (41.14° N, 46.5° W),

corresponding to the layer depth maximum in the middle of Fig. 5.53b; the second station is located at (43.83° N, 50.5° W), corresponding to the temperature anomaly maximum. As shown by the blue curves in Fig. 5.54, at the first station both the layer depth and thickness perturbations have large amplitudes, but the temperature perturbations have a rather low amplitude. On the other hand, at the second station (red curves), the situation is the opposite. The strong decadal variability of heaving signals at these two stations is remarkable. For example, layer thickness perturbations at the first station can reach the magnitude of more than 300 m, and the layer core can move upward or

5.6 Isopycnal Layer Analysis Based …

311

Fig. 5.48 Mean layer thickness and absolute value of perturbation for the isopycnal layer r0 ¼ 26:8  0:05 kg/m3

downward for more than 200 m; on the other hand, temperature perturbations at the second station have quite a large amplitude on the order of 5 °C. Similarly, the layer thickness, layer depth and temperature perturbations for the South Indian Ocean are shown in Fig. 5.55. It is readily seen that in this isopycnal layer, the thickness and depth perturbations are correlated, but not at exactly the same location. In addition, the temperature anomaly maxima appear at different locations.

The time evolutions of layer properties for two stations in the Southern Hemisphere are shown in Fig. 5.55. The first station (38.5° S, 90.5° E) is located southwest of Australia; as depicted by the blue curves, both layer thickness and layer depth perturbations have rather large amplitudes, but the temperature perturbations have a rather low amplitude. The second station (43.17° S, 59.5° W) is located east of the South American continent, where the layer thickness and layer depth perturbations have amplitudes much smaller than

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Heaving Signals in the Isopycnal Coordinate

Fig. 5.49 Mean temperature and absolute value of perturbation for the isopycnal layer r0 ¼ 26:8  0:05 kg/m3

the corresponding parts at the first station (in the Indian Ocean). On the other hand, the temperature anomaly has a much larger amplitude. Therefore, the nature of heaving signals at these two stations seems quite different.

5.7

Heaving Signals for the Shallow Water in the Pacific-Indian Basin

Our discussion in the previous sections was based on r1 or r0 in the deep part of the world oceans. However, a closer examination reveals that the warm water in the Pacific-Indian basin is

disconnected with that of the Atlantic basin. Hence, the warm water in the Pacific-Indian basin should be studied in separation from that of the Atlantic basin. Because our study in this section is focused on the shallow water masses only, the natural choice is r0 . Water masses distribution in the r0 coordinate covers a rather wide range, as shown in Fig. 5.56. Most of the water masses appear in the density range of higher than 26 kg/m3. Those water masses appear at high latitudes and the deep ocean, and are thus not directly linked to the equatorial dynamics. For the study of equatorial dynamics, our focus is on water masses with density in the range of r0 = 21–26 kg/m3, as marked by the dashed lines in Fig. 5.57.

5.7 Heaving Signals for the Shallow Water …

313

Fig. 5.50 Horizontal distribution of mean potential spicity p1 and its root-mean square variability in the isopycnal layer r0 ¼ 26:8  0:05 kg/m3

As shown in Fig. 5.58, based on climatological annual density data, water masses with a density no larger than 25–26 kg/m3 are separated into two basins by the South American continent and the South African continent: the PacificIndian basin and the Atlantic basin. Accordingly, heaving modes for water satisfying r0  26:0 kg/m3 can be analyzed for the PacificIndian basin and the Atlantic basin separately. Actually, water masses with a density in the range of 25–26 kg/m3 may be exchangeable around the Cape of Agulhas. In this section we will ignore the effect on the heaving modes due to such possible connection for water masses heavier than 25 kg/m3. As shown in Fig. 5.59, water masses with density no larger than 26 kg/m3 in the PacificIndian basin are mostly confined to the upper 300 m. In fact, the corresponding shallow water masses in the Pacific sector are linked to that in the Indian sector through the narrow channel in the middle of the maritime continent.

5.7.1 Application of the Casting Method to the GODAS Data As explained above, this algorithm can be applied to an individual basin and to a selected range of isopycnal layers. Since our main interest is the shallow water in the Pacific-Indian basin, our analysis can be focused on the density range of r0 = 22–26 kg/m3. Using such a relatively narrow range of density, we can increase the resolution in density space to Dr0 ¼ 0:1 kg/m3 . This relatively high resolution can help us to resolve the dynamical process with a finer resolution. The Lagrangian coordinate based on r0 is called a MDC, and the anomaly diagnosed from the r0 coordinate is denoted as Da. The external mode of Da in the MDC is shown in Fig. 5.60a, and the corresponding sum of absolute values of heaving modes is shown in Fig. 5.60b. These two figures are without smoothing.

314

5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.51 Horizontal distribution of the radius of signal in the layer of r0 ¼ 26:8  0:05 kg/m3 associated with: a potential density; b potential spicity; c the total radius of signal; d the radius of signal after isopycnal analysis

The total integration of signals over the density space as a function of time is shown in the lower panels. This represents the total source of mass in the Pacific-Indian basin over the density range of r0 = 22–26 kg/m3. The maps shown in Fig. 5.60a, b seem quite noisy. To show the structure hidden behind, we applied the 13-month smoothing, and the results are shown in Fig. 5.61. In these new figures, the quasi four year oscillations in the system can be seen, in particular in the left panel. In the high density range, r0 = 25–26 kg/m3, the mass gain during the early 1980s and the mass loss after year 2002 are clearly visible. Of course, such mass sources and sinks in the model are due to

the volume conservation approximation made in the mode formulation. Physically, the source (sink) of mass should be interpreted as cooling (warming). In this case, the red color at the lower left corner in panel a should be interpreted as the extra amount of water in this density range in the early 1980s, and the blue color in the lower right corner of panel a indicates that there is less water in this density range. Since the patterns for the lower density range and higher density range are quite different, we replot Fig. 5.61a and separate it into two density segments (Fig. 5.62). Over the lower density range the external mode signals possess some quasi four year

5.7 Heaving Signals for the Shallow Water …

315

Fig. 5.52 Heaving ratio for the isopycnal layer r0 ¼ 26:8  0:05 kg/m3

Fig. 5.53 Root-Mean Square value of the anomaly of layer thickness (a); depth (b); temperature (c); salinity (d); spicity (e); heaving ratio (f) in the North Atlantic Ocean for the isopycnal layer of r0 ¼ 26:8  0:05 kg/m3

oscillations closely linked to the ENSO cycles (Fig. 5.62a). Before 1998, strong negative heaving signals appear in the density range of r0 = 21.2–23.4 kg/m3, and strong positive heaving signals appear in the density range of

r0 = 22.2–22.4 kg/m3. These strong heaving signals may be closely related to the strong airsea heat flux anomaly associated with ENSO cycles; in particular, during the La Niña phase of the ENSO cycle, the strong air-sea heat flux

316

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Heaving Signals in the Isopycnal Coordinate

Fig. 5.54 Time evolution of layer r0 ¼ 26:8  0:05 kg/m3 properties at two stations in the North Atlantic Ocean

(a) RMS ( h) (m) 30S

100

(c) RMS ( T) (

(e) RMS RMS (

)

80

35S

60

0.6

40S

40

0.4

45S

20

0.2

100

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80

20 70E 80E 90E 100E 110E

0.10 0.05

95 90

40

45S

0.15

0.15

60

40S

0.20

70E 80E 90E 100E 110E (f) Heaving ratio (%)

70E 80E 90E 100E 110E (d) RMS ( S) (spu)

30S

(kg/m3) 0.25

0.8

70E 80E 90E 100E 110E (b) RMS ( D) (m)

0)

70E 80E 90E 100E 110E

0.10

85

0.05

80 70E 80E 90E 100E 110E

75

Fig. 5.55 Root-Mean Square value of anomaly of layer thickness (a); depth (b); temperature (c); salinity (d); spicity (e); heaving ratio (f) in the South Indian Ocean for the isopycnal layer of r0 ¼ 26:8  0:05 kg/m3

5.7 Heaving Signals for the Shallow Water …

317

Fig. 5.56 Time evolution of layer properties (r0 ¼ 26:8  0:05 kg/m3 ) at two stations in the South Indian and South Atlantic Oceans

Total volume (m3)

18 17

log10V/0.1

16 15 14 13 12 11

21

22

23 0

24 25 (kg/m3)

26

27

28

Fig. 5.57 Annual mean volumetric distribution in the r0 coordinate, based on the GODAS climatology

anomaly due to the expansion of the cold tongues may be the main course of such strong external mode signals.

In contrast, for the high density range the quasi four-year oscillation pattern cannot be well defined (Fig. 5.62b). Instead, there is a clear sign of decadal variability. In the early 1980s, there is a clear cluster of positive signals, but after year 2000 there is clear sign of negative signals. As shown in Fig. 5.62a, b, for different parts of density range the external heaving signals are regulated by different physics. These signals can also be viewed in terms of the equivalent layer thickness perturbations, similar to the discussion in Sect. 5.3. For the external modes, their equivalent thickness perturbations can be calculated as follows dhMDC Ex ðn; mÞ ¼

jmt imt X X i¼1

RMDC ði; j; n; mÞ=denðnÞ=areaðnÞ a

j

ð5:29Þ

318

5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.58 Climatological annual mean potential density (r0 ) distribution on the sea surface, based on the GODAS data

Fig. 5.59 Climatological annual mean potential density, r0  26:0 kg/m3 , distribution along two meridional sections: a Pacific Ocean, 179.5° E; b Indian Ocean, 59.5° E, based on the GODAS data

where area(n) is the total horizontal area of the corresponding isopycnal layer within the PacificIndian basin; this area can be inferred from the climatological mean density field of the GODAS data (Fig. 5.63). For the lower density range, the equivalent layer thickness perturbations are large, in particular for the isopycnal layer of r0 ¼ 21:0  0:05 kg/m3 (Fig. 5.64).

As we move to the higher density range, the characteristics of the external modes change gradually. For density heavier than 25.0 kg/m3, the decadal variability becomes clearly defined (Fig. 5.65). In fact, for the isopycnal layer of r0 ¼ 26  0:05 kg/m3 (black curve in Fig. 5.65 b) the decadal variability is dominating, at the same time there is no or very little signals associated with the quasi four-year ENSO cycle.

5.7 Heaving Signals for the Shallow Water …

319

Fig. 5.60 The total volume integration of the external mode of density anomaly (seasonal cycle subtracted) in the MDC (a) and the total integration of signals over the density space (c); the integration of the absolute value of heaving signals in each density bin (b) and the total integration of signals over the density space (d)

Fig. 5.61 The signals shown in Fig. 5.60a, b after a 13 month smoothing

320

5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.62 A close view of patterns shown in Fig. 5.60 by separating it into lower density range and high density range

Fig. 5.63 Horizontal area occupied by isopycnal layers in the Pacific-Indian basin

5.7.2 Isopycnal Layer Analysis of the Equatorial Dynamics Based on Projecting Methods 5.7.2.1 Multiple Layer Isopycnal Analysis Our focus in this section is on the heaving signals for the model domain of the equatorial PacificIndian Ocean from 5° S to 5° N, and our approach is based on projecting the variability onto isopycnal coordinates, r0 , using 51 isopycnal layers with equal layer thickness of

Dr0 ¼ 0:1 kg/m3 . This is a horizontal region with open lateral boundaries, so the heaving signals discussed in this section are a combination of the external and internal modes. The 36year climatological mean layer properties averaged over this domain are shown in Fig. 5.66. We begin with the region-mean water properties averaged over the model domain for each isopycnal layer (Fig. 5.66). There are clear signs of variability on interannual and decadal time scales. To reveal such multiple year oscillations we plot the signals in two isopycnal layers, r0 ¼ 21  0:05 kg/m3 and r0 ¼ 26  0:05 kg/m3 (Fig. 5.67). It is clear that there is something like quasi 4 year oscillations superimposed on the background of decadal variability for variables in the surface layer (red curves). There is also strong variability in layer thickness, layer depth and zonal velocity for the lowest layer; however, there are very little signals for the temperature in the lowest layer. This difference suggests that the heaving signals in the lower layer are primarily adiabatic, while the heaving signals in surface layers contain both adiabatic and diabatic components. We now examine climate variability along the equatorial section, and our discussion in this

5.7 Heaving Signals for the Shallow Water …

321

Fig. 5.64 External mode signals in terms of the equivalent layer thickness perturbations (in units of m) for the low density range; the left panel for the contour map and the right panel for five selected isopycnal layers with a thickness of 0:1r unit

Fig. 5.65 External mode signals in terms of the equivalent layer thickness perturbation (in units of m) for the high density range; the left panel for the contour map and the right panel for five selected isopycnal layers with thickness of 0:1r unit

322

5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.66 Time evolution of layer mean values for the Pacific-Indian Ocean between 5° S and 5° N, with the annual cycle removed and subjected to a 13 month moving smooth: a layer thickness; b layer depth; c temperature; d zonal velocity

section is primarily focused on the Pacific Ocean. First, Fig. 5.68 shows the mean layer properties along the equatorial section. As shown in the middle part of panel a, the thickness of the upper layers is quite large, indicating the low stratification for the water in the warm pool and central equatorial Pacific. The low thickness zones for the middle density range indicate the main pycnocline in the equatorial band; in particular, the cold tongue is marked by the stratification minimum in this section. The corresponding mean layer depth is shown in panel b. It is clear that these layers cover the upper 200 m of the equatorial pycnocline in the

Pacific-Indian basin. The shallow layer depth on the right-hand side of panel b corresponds to the cold tongue in the East Equatorial Pacific, where the main pycnocline is quite shallow and some isopycnal layers actually outcrop. The mean temperature section is shown in panel c, where the basic feature of both the warm pool and cold tongue can be readily identified. The mean potential spicity is shown in panel d, where the core of high spicity is located in the central Pacific, indicating the warm and salty water masses. At the eastern edge of the Pacific section, there is a rather thin tongue of low spicity water.

5.7 Heaving Signals for the Shallow Water …

323

Fig. 5.67 Time evolution of layer mean values for the Pacific-Indian Ocean between 5° S–5° N, with the annual cycle removed and subjected to a 13 month moving smooth for two isopycnal layers: a layer thickness; b layer depth; c temperature; d zonal velocity

The circulation variability along the equatorial section is shown in Fig. 5.69. Here the variability is defined as the monthly mean deviation from the climatological mean. By definition, these variables include the contributions from both the external and internal modes. As shown in Fig. 5.69a, the layer thickness perturbations are mostly confined to the upper part of the water column and in the middle of the equatorial basin, away from the western and eastern boundaries. The maximum amplitude is on the order of 12 m. For the major part of the isopycnal layers below the surface, the amplitude of perturbations is relatively small. In particular, layer thickness perturbations within the main thermocline are

quite small, and it reflects the physical constraint that strong stratification can suppress climate variability. The amplitude of the layer depth perturbation has a pattern quite different from that of the layer thickness. As shown in Fig. 5.69b, the layer depth perturbation maximum appears in the form of a band slated from the upper ocean in the middle of the basin to the eastern boundary for the deep isopycnal layers. It is well known that the eastward equatorial Kelvin waves can be classified into two basic categories: the downwelling and upwelling Kelvin waves. Thus, this band indicates the strong isopycnal layer depth perturbations associated with the equatorial

324

5

Heaving Signals in the Isopycnal Coordinate

(a) Mean h (m)

21

20

0

(kg/m3)

22

15

23

10

24

5

25 26

(b) Mean D (m)

21

200

0

(kg/m3)

22

150

23

100

24

50

25 26

0

(c) Mean T ( )

21

0

(kg/m3)

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25

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20

24 25

15

26

(d) Mean

0

(kg/m3)

21

0

3

(kg/m ) 3

22

2

23 24

1

25

0

26

60E

90E

120E

150E

180

150W

120W

90W

−1

Fig. 5.68 Circulation properties along an equatorial section for the isopycnal layer of r0 ¼ 21  26 kg/m3 , with isopycnal layer thickness of Dr0  0:1 kg/m3 : a Climatological mean layer thickness h, b layer depth D, c layer temperature T and d potential spicity

Kelvin waves moving along the equatorial wave guide. The slated nature of the layer depth perturbations indicates that in the isopycnal coordinate the Kelvin waves move from shallow and light layers into the deep and dense layers in a stratified ocean. The dynamical nature of this downward shifting of Kelvin Waves remains unclear at this time. The patterns of the temperature anomaly are quite different from those associated with layer thickness and layer depth. Note that on density

surfaces temperature and salinity perturbations must be compensated. Therefore, dT and dS signals on the potential density surface are not independent, and one needs to plot either the temperature perturbations or the salinity perturbations only. The temperature anomaly maximum appears within the center of the cold tongue and near the western boundary at the deep layers (the meaning of this patch of temperature anomaly maximum remains unclear; it may be related to the response

5.7 Heaving Signals for the Shallow Water …

325

Fig. 5.69 Mean property perturbations along an equatorial zonal section for the isopycnal layer of r0 = 21–26 kg/m3, with isopycnal layer thickness of Dr0 ¼ 0:1 kg/m3 : a layer thickness |dh|, b layer depth |dD|, c layer temperature |dT|, d spicity |dp0|

of the cross equatorial flow from the eastern coast of Australia in response to ENSO events.) On the other hand, the temperature anomaly is quite weak in most part of the equatorial section (Fig. 5.69c). The most outstanding features in Fig. 5.69 are the separation of perturbations associated with layer thickness, layer depth, temperature and spicity. The climate variability in the Indian sector shares some similarity with that in the Pacific sector. The layer thickness perturbation is maximal in the center of the Indian sector, Fig. 5.69a.

However, the layer depth perturbations in the Indian sector do not have a slate band of maximum as in the Pacific sector. This may be due to the complicated nature of Kelvin waves in the Indian sector, where the biannual cycle and 2year cycle prevails and making the adjustment of the main thermocline quite different from that in the Pacific sector. In addition, there is no cold tongue in the Indian sector; as such, there temperature anomaly pattern in the Indian sector looks quite different from that in the Pacific sector. In the Indian sector, temperature perturbations maximum

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Heaving Signals in the Isopycnal Coordinate

Heaving ratio (%) along the Equator

21

90 80

22

0

(kg/m3)

70 23

60 50

24

40 30

25

20 26

60E

90E

120E

150E

180

150W

120W

90W

Fig. 5.70 Heaving ratio along Equator

appears in the center of the basin, and there is no temperature anomaly maximum near the eastern or the western boundaries (Fig. 5.69c). The mean of spicity perturbations is shown in Fig. 5.69d; the pattern is similar to that of temperature perturbations and salinity perturbations (not shown in the figure). Similar to the discussion above, the heaving ratio for the equatorial section is shown in Fig. 5.70. In this section, heaving signals dominate the climate signals, except for the western/eastern edges of the Pacific basin. The cold tongue near the eastern boundary is the primary site of air-sea coupling. There is a vertical column near the western boundary where the diabatic effect seems important. Figure 5.70 provides an important insight to the dynamic nature of heaving signals appearing in the equatorial section.

5.7.2.2 Single Layer Isopycnal Analysis We now turn to the time evolution of perturbations in a single isopycnal layer, and we choose the r0 ¼ 24  0:05 kg/m3 isopycnal layer and within the horizontal boundaries discussed above in the Pacific Ocean. For the mean state the layer thickness maximum is in the central equatorial Pacific around 140° W, corresponding to the low stratification

water above the equatorial main thermocline (Fig. 5.71a). Since the ENSO events take place above the main thermocline, we expect to see strong climate variability in this vicinity. The climate anomalous fields in this isopycnal layer are shown in Fig. 5.72. It is clear that anomaly maxima are located in different places, suggesting that they are somewhat independent of each other, although some of them are related. First, layer thickness variability is high southeast of the equatorial band (panel a). On the other hand, the layer depth variability maximum is located near 140° W, approximately the same location as the mean layer thickness maximum shown in Fig. 5.72a. The combination of layer mean thickness maximum and layer depth variability maximum indicates the location of strong isopycnal movements related to the ENSO events. As for the water mass properties, such as temperature, salinity and spicity, their variability maxima are located near the eastern boundary (the cold tongue) and the western boundary (the warm pool) (panels c, d, and e). The corresponding radius of signal is shown in Fig. 5.73. Apparently, the radius of signal of the potential density (panel a) is much larger than that of spicity (panel b), and the corresponding maximum is located in different places in this

5.7 Heaving Signals for the Shallow Water …

327 (a) Mean h (m)

4N

5 4

2N 0

3

2S

2

4S

1 (b) Mean D (m)

4N

140 120 100 80 60 40

2N 0 2S 4S (c) Mean T ( ) 4N

24

2N

23

0

22

2S

21

4S (d) Mean S (psu) 4N

35.5

2N

35.0

0 2S

34.5

4S (e) Mean

0

(kg/m3)

4N 2N 0 2S 4S 120E 130E 140E 150E 160E 170E 180 170W 160W 150W 140W 130W 120W 110W 100W 90W 80W

2.0 1.5 1.0 0.5 0

Fig. 5.71 Mean properties for the isopycnal layer of r0 ¼ 24  0:05 kg/m3

isopycnal layer. The radius of signal for the total signal before isopycnal analysis is primarily controlled by the contribution from the potential density in the cold tongue, but it is controlled by the contribution from spicity in the middle of the equatorial ocean, as shown in panel c. On the other hand, radius of signal after isopycnal analysis is characterized by a maximum in the

warm pool near the western boundary of the Pacific basin and the cold tongue (panel d). The heaving ratio for this isopycnal layer can be calculated (Fig. 5.74). For this isopycnal layer, most of signals is heaving in nature (dark red and red color). There are two small regions where the signals contain relatively strong nonheaving components. The first region is right on

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5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.72 Root-Mean Square of layer property perturbations for the isopycnal layer of r0 ¼ 24  0:05 kg/m3

the Equator and below the center of the surface warm pool. The second one is along the eastern boundary and underneath the surface cold tongue. Since this isopycnal layer does not outcrop, the non-heaving signals on this surface may be

due to the horizontal advection of spicity or downward propagation of thermohaline signals through anomalous diapycnal mixing; interpretation of these two areas remains to be further explored.

5.7 Heaving Signals for the Shallow Water …

329

Fig. 5.73 Radius of signal for the isopycnal layer of r0 ¼ 24  0:05 kg/m3 . a associated with potential density; b potential spicity; c the total radius of signal; d radius of signal after isopycnal analysis

Fig. 5.74 Heaving ratio distribution for the isopycnal layer r0 ¼ 24  0:05 kg/m3

330

5.8

5

Heaving Signal Propagation Through the Equatorial Section

Most of our discussion has been focused on climate mean properties; however, the most promising way of analysis is to apply this method for the instantaneous dynamic field. As an example, we analyze the time evolution of the depth perturbations along the Equatorial Pacific for two strong ENSO events (Fig. 5.75). The depth perturbation associated with the 1996–1997 event was initiated from the western boundary in the summer of 1996. The perturbations grow and moved eastward and downward to the higher density range (Fig. 5.75). It is thus interesting to know that Kelvin waves in the stratified ocean associated with ENSO events move across the density interfaces.

Heaving Signals in the Isopycnal Coordinate

Note that the total amount of water integrated over in each density category changes with time, this means there should be mass exchange in the meridional direction, not just a simple east-west transport of water masses. The situation for the 2009–2010 ENSO event is quite different. As seen in Fig. 5.76, the volume of thermocline water in the central and eastern parts of the equatorial Pacific increases, and our analysis indicates that such additional warm water must come from the off-equatorial band. This means that this ENSO event is initiated by the converging of warm water from the extra-tropical parts of the ocean, so it is quite different from the traditional ENSO events, when warm thermocline water is transported along the equatorial band, with much less communication with the extratropical ocean.

Fig. 5.75 Depth perturbations for the equatorial Pacific Ocean during July 1996 to June 1997, in units of m

Appendix: Connection Between the MDC …

331

Fig. 5.76 Depth perturbations for the equatorial Pacific Ocean during July 2009 to June 2010, in units of m

Appendix: Connection Between the MDC and the FDC These two coordinates are closely linked to each other, and we explore their connection in this section. In order to compare these two approaches, in this appendix we introduce slightly different variables as follows. The variables defined in the main text are divided by the factor of DV ¼ DxDyDz, i.e., we will use the density and its deviation from the seasonal cycle mean in the mth month, i.e. rði; j; k; mÞ and seasonal

rði; j; k; mÞ  rði; j; k; mÞ , as the new variables in the MDC and the FDC, and these variables will be denoted as wMDC ði; j; kÞ and a wFDC ði; j; kÞ. a An idealized example for illustration is shown in Fig. 5.77. In this case, at a station (i, j) and at the kth vertical level, there are density

observations in the month of April over the 10year period, the observed density (in sigma unit, and in the order of increasing density and denoted by black numbers) is: ½26:11; 26:23; 26:31; 26:37; 26:41; 26:44; 26:48; 26:51; 26:61; 26:68 These density observations fall into 7 density bins in the MDC with resolution of 0.1 kg/m3, i.e.: ½26:1  0:05; 26:2  0:05; 26:3  0:05; 26:4  0:05; 26:5  0:05; 26:6  0:05; 26:7  0:05 (denoted by the black numbers below the axis); Thus, the corresponding 10-year climatological m mean profile in the MDC, denoted as rMDC , is depicted by the blue lines and numbers. As an example, the observed density for a specific year is 26.23 kg/m3. This observation falls into the MDC bin of 26:2  0:05 kg/m3 , and the corresponding observation value for this year

332

5

Heaving Signals in the Isopycnal Coordinate

Fig. 5.77 Sketch of density anomaly in the FDC and the MDC

in other MDC bins is zero. Therefore, this year’s observation in the MDC is represented by the green number (26.23) and green segmented line in Fig. 5.77, and we denote this observation in the MDC as rMDC . The corresponding anomaly is defined by subtracting the climatological profile (blue segmented line) from this observation profile (green segmented line), i.e., rMDC ¼ a MDC MDC r r and it is depicted by the red segmented line in Fig. 5.77. Note that the observed signal appears in only one bin in the MDC, so that in the MDC rMDC has one non-zero point only; however, the anomalous signals rMDC have a 7 non-zero points in the MDC. The corresponding signals are positive for the 26:2  0:05 kg/m3 bin only, but are negative for all other bins. The net signal for this year in the MDC is the P algebraic sum of the red lines, 7i¼1 rMDC ði Þ ¼ a 3 0:185 kg/m . On the other hand, in the FDC (the upper panel in Fig. 5.77), the climatological mean density is r ¼ 26:415 kg/m3 , there is one non-zero data point r ¼ 26:23 kg/m3 kg/m3 only. As a result, r  r ¼ 28:185 kg/m3 . Hence, in

each year the net signal in the MDC, summed over the whole density range, is the same as the anomaly signal in the FDC, as depicted by the margent bar. For this specific year discussed above, the anomaly signal in the FDC is represented by the second margent bar from the left in the upper part of Fig. 5.77. Therefore, for each month we have the following constraint between variables in these two coordinates wMDC ði; j; kÞ ¼ a

lmt X

rMDC ði; j; k; lÞ ¼ wFDC ði; j; kÞ a a

l¼1

¼ ðr  rÞFDC ð5:30Þ

References Antonov JI, Seidov D, Boyer TP, Locarnini RA, Mishonov AV, Garcia HE, Baranova OK, Zweng MM, Johnson DR (2010) World Ocean Atlas 2009, Volume 2: Salinity. In: Levitus S (ed) NOAA Atlas NESDIS 69. U.S. Government Printing Office, Washington, DC, 184 pp

References Behringer, DW, Xue Y (2004). Evaluation of the global ocean data assimilation system at NCEP: The Pacific Ocean. In: Eighth symposium on integrated observing and assimilation systems for atmosphere, oceans, and land surface, AMS 84th Annual Meeting, Washington State Convention and Trade Center, Seattle, Washington, pp 11–15 de Szoeke RA, Springer SR (2009) The materiality and neutrality of neutral density and orthobaric density. J Phys Oceanogr 39:1779–1799 Forster TD, Carmack EC (1976) Frontal zone mixing and Antarctic bottom water formation in the southern Weddell Sea. Deep Sea Res 23:301–317 Helland-Hansen B (1912) The ocean waters, an introduction to physical oceanography. Intern Rev d Hydrobiol (Suppl) Bd. III, Ser 1, H. 2:1–84 Huang RX (2014a) Adiabatic density surface, neutral density surface, potential density surface, and mixing path. J Trop Oceanogr 33(4):1–19. https://doi.org/10. 3969/j.issn.1009-5470.2014.04.001 Huang RX (2014b) Energetics of lateral eddy diffusion/advection, Part I. Thermodynamics and energetics of vertical eddy diffusion. Acta Oceanologia Sinica 33:1–18 Huang RX, Yu LS, Zhou SQ (2018) New definition of potential spicity by the least square method. J Geophys

333 Res Ocean 123:7351–7365. https://doi.org/10.1029/ 2018JC014306 Jackett D, McDougall TJ (1997) A neutral density variable for the world’s oceans. J Phys Oceanogr 27:237–263 McDougall TJ (1987a) Neutral surfaces. J Phys Oceanogr 17:1950–1964 McDougall TJ (1987b) The vertical motion of Submesoscale coherent vorticies across neutral surfaces. J Phys Oceanogr 17:2334–2342 McDougall TJ, Klocker A (2009) An approximate geostrophic streamfunction for use in density surfaces, Ocean Modelling 32(3–4):105–117. https://doi.org/10. 1016/j.ocemod.2009.10.006 Montgomery RB (1937) A suggested method for representing gradient flow in isentropic surfaces. Bull Amer Meteor Soc 18:210–212 Montgomery RB (1938) Circulation in upper layers of southern North Atlantic deduced with use of isentropic analysis. Pap Phys Oceanogr Meteorol 6(2):1–55 Reid JL (1994) On the total geostrophic transport of the North Atlantic Ocean: flow patterns, tracers, and transports. Prog Oceanogr 33:1–92 Zhang HM, Hogg NG (1992) Circulation and water mass balance in the Brazil Basin J Marine Res 50:385–420

6

Heaving Signals in the Isothermal Coordinate

6.1

Introduction

In the isothermal coordinate the movements of isothermal layers are defined as heaving motions. Heaving modes can be separated into the external and internal modes. The external heaving modes describe the changes in the global volumetric distribution of water masses in the potential temperature coordinate, isothermal coordinate hereafter; thus, external heaving modes are due to diabatic processes, including the heat flux anomaly across the air-sea surface or interfaces. On the other hand, internal heaving modes characterize the local variability of the water mass volumetric anomaly, and such a local anomaly has zero contribution to the global water mass volumetric distribution in the isothermal coordinate. Although the adjustment of the winddriven circulation (including Rossby waves, Kelvin waves and currents) induced by wind stress perturbations is the primary cause of the internal heaving modes, the horizontal difference in anomalous thermal exchange across isothermal surface can also give rise to internal heaving modes. The further separation of the causes for the internal heaving modes from adiabatic processes induced by wind stress anomaly and that due to internal diabatic processes is difficult, and this is beyond the scope of this book. This chapter begins with the presentation of a rigorous method to separate thermal anomaly signals into the external heaving modes and the internal heaving modes. The goal of this chapter

is to build up a theoretical framework for the analysis of heaving modes. The core of the methodology is to use the instantaneous temperature as a Lagrangian coordinate.

6.2

Casting Method

In the study of climate variability, choosing a suitable vertical coordinate is a crucial step. The commonly used vertical coordinate is the geopotential coordinate, or simply the z-coordinate. Climate variability, such as the temperature anomaly, is identified as the deviation from the climatological mean at each spatial grid (x, y, z). The spatial distribution and temporal evolution of climate variability are then analyzed in the four dimensional space-time coordinates (x, y, z, t). The other approach is to use the temperature as the vertical coordinate. There are two potential approaches: using the temperature as the Eulerian coordinate or use it as the Lagrangian coordinate. Since the in situ temperature is not a conserved quantity, the potential temperature is used as the coordinate instead. Unfortunately, most oceanic general circulation models currently used by the community are based on the Boussinesq Approximations, and one of the fundamental assumptions made in the Boussinesq models is to replace the mass conservation with the volume conservation approximation. In such models, the tracer conservation equations use the divergence of tracer flux in

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_6

335

336

6

approximation. For example, the prognostic equation for the temperature uses the temperature flux term r  ð! u TÞ to replace the enthalpy flux divergence term r  ðq! u HÞ. Therefore, the quantity T calculated in this way is not exactly the potential temperature. However, this is the common practice in model simulations, and there is currently no other dataset available that is produced under the more physically sound enthalpy constraints discussed above. Hence, we will use the currently available datasets, and use the term “potential temperature” or simply “temperature” in the discussion below. We will use the four-dimensional coordinates ðx; y; h; tÞ in the following analysis, where h is (potential) temperature and t is time. Furthermore, we will introduce two isothermal coordinates: the Fixed Temperature Coordinate (FTC) and the Moving Temperature Coordinate (MTC). Climate variability in these two coordinates is defined as follows.

if hn  0:5\hði; j; kÞ  hn þ 0:5;

6.2.1 FTC The FTC uses the climatological mean temperature at each spatial grid (i, j, k) as the vertical coordinate. At each grid point (i, j, k) in the spherical coordinates, the heat content anomaly in the mth month in the FTC is denoted as:
 DH FTC ði; j; k; mÞ ¼ hði; j; k; mÞ  hði; j; kÞ Cp q0 DxðjÞDyDzðkÞ

Heaving Signals in the Isothermal Coordinate

hn ¼ 2; 1; 0; . . .; 36 ð CÞ

ð6:2Þ

One may also separate the contributions due to the regular seasonal cycle by using the multiple year mean temperature in each month hði; j; k; maÞ as the vertical coordinate, where ma is the corresponding month in the climatological mean annual cycle. Accordingly, the climate variability data series is in the form
HaFTC ¼ HaFTC ði; j; n; mÞ; n ¼ 1; 2; . . .; 39; m ¼ 1; 2; . . .; m month; in units of J ð6:3Þ where m month is the total number of months in the data record. The physical meaning of this variable is as follows. The heat content anomaly at a spatial grid point (i, j, k) is calculated, and a positive (negative) value indicates that water in this grid point is warmed up (cooled down) relative to the climatological mean (seasonal cycle subtracted) temperature. Note that for each spatial grid (i, j, k) there is only one temperature bin being assigned with a non-zero value; as a result, the coordinate transformation is a unique one-toone transformation. This time series can also be summed up into a time series of the global heat content anomaly in the FTC: HaFTC ðn; mÞ ¼

jmt imt X X

HaFTC ði; j; n; mÞ

ð6:4Þ

i¼1 j¼1

ð6:1Þ where hði; j; k; mÞ and hði; j; kÞ are the temperature and the corresponding climatological mean, Cp ¼ 4148 J/kg/K is the sea water heat capacity under constant pressure, and q0 ¼ 1035 kg/m3 is the mean density of sea water. Data is put into discrete temperature bins in the FTC according to the following rule: DH

FTC

ði; j; k; mÞ added to HaFTC ði; j; n; mÞ

6.2.1.1 MTC The MTC uses the instantaneous temperature at each spatial grid (i, j, k) as the vertical coordinate. At each grid point in the spherical coordinates, the heat content is calculated as, hði; j; k; mÞ ¼ hði; j; k; mÞCp q0 DxðjÞDyDzðkÞ ð6:5Þ where hði; j; k; mÞ is the (potential) temperature at the mth month. The resulting data is stored in

6.2 Casting Method

337

discrete temperature bin n in the MTC, and denoted as hMTC ði; j; n; mÞ, using the instantaneous temperature to determine the right bin number n hn  0:5\hði; j; k; mÞ  hn þ 0:5; hn ¼ 2; 1; 0; . . .; 36 ð CÞ

ð6:6Þ

Consequently, the total heat content time series is in the form
H MTC ¼ hMTC ði; j; n; mÞ; n ¼ 1; . . .; 39; m ¼ 1; . . .; m month; in units of J ð6:7Þ where m month is the total number of months in the data record. Due to the climate variability, the thermal anomaly signals at different times for a fixed spatial grid (i, j, k) may be put in different temperature bins in the MTC.From the multiyear data, one can define the climatological mean annual cycle of the total heat content for 12 months of the annual cycle h H MTC ¼ hMTC ði; j; n; mmÞ; n ¼ 1; . . .; 39; mm ¼ 1; 2; . . .; 12 ð6:8Þ After the annual cycle is subtracted, the corresponding heat content anomaly is defined as hMTC ði; j; n; mÞ ¼ H MTC  H MTC a

ð6:9Þ

The physical meaning of this variable is slightly different from that defined in Eq. (6.3) for the FTC. In the FTC, the heat content anomaly indicates the heat content change in connection with the temperature deviation from the climatological mean at each spatial grid. Due to strong climate variability, the temperature deviation at a fixed spatial grid can be larger than half degree (half of the temperature intervals set for the FTC in this chapter). Nevertheless, all signals generated at a fixed spatial grid are put into the same temperature bin in the FTC. In the MTC, the temperature range for each bin is fixed (in the current study, it is set to the

one-degree intervals); hence, the heat content anomaly indicates the heat content change in connection with volume increase (decrease) of water mass within this temperature bin. Furthermore, due to strong climate variability thermal anomaly signals generated in the same spatial grid (i, j, k) may be put into different temperature bins in the MTC. For each month, the total heat content anomalies integrated over the world oceans in the MTC are defined as HaMTC ðn; mÞ ¼

jmt imt X X

hMTC ði; j; n; mÞ ð6:10Þ a

i¼1 j¼1

6.2.1.2 Separating the Signals into External and Internal Modes Our final goal in analyzing the thermal anomaly signals diagnosed for the world oceans is to identify the potential mechanisms generating such anomalies. In particular, our interest is to separate the potential contributions due to external heaving modes and internal heaving modes. 6.2.1.3 FTC As shown in Eq. (6.4), the thermal anomaly signals in the FTC can be summarized as a twodimensional data array. By definition, the FTC is defined by the climatological mean temperature (or the seasonal mean temperature); for this reason, it is not a truly Lagrangian coordinate. For example, if HaFTC ðn; mÞ [ 0 for a given index pair of (n, m), we can only say that in the mth month, temperature at grids with climatological mean (or seasonal mean) temperature satisfying hði; j; kÞ ¼ hn increases on average. Such increases in the instantaneous temperature might be the result of thermal exchanges with the neighboring grids or through the air-sea interaction. It can also be due to heaving motions of isothermal layers induced by wind stress perturbations in the world oceans; anomalous wind can push warm water from the neighboring grids to the grids of our concern.

338

6

The clean signals identifiable from the data in the FTC are the global sum Gs HaFTC ðmÞ ¼

39 X

HaFTC ðn; mÞ

ð6:11Þ

n¼1

This variable represents the time evolution of the total heat content for the world oceans, and it is solely owing to the air-sea heat changes, i.e., it is the net contribution due to the external diabatic processes.

6.2.1.4 MTC The discussion for the MTC is much more elaborate; some of the details are included in the Appendix. As stated in Eq. (6.20), at each vertical grid point k at station (i,j) the net sum of anomalous signals in the MTC equals to the signal in the FTC, i.e., nmt X

Heaving Signals in the Isothermal Coordinate

interface and the interfaces between different isothermal layers. By definition, the internal modes integrated over the entire volume defined by the isothermal surfaces of each hn bin are zero, i.e.: i;j

In HaMTC ði; j; n; mÞ ¼ 0

ð6:15Þ

Since its integration over the upper and lower interfaces of this isothermal interval is zero, this variable represents the internal heaving components of the thermal anomaly signals in each isothermal layer.

6.3

Casting Method Applied to the GODAS Data

ð6:13Þ

Our analysis in this section is based on the casting method applied to the GODAS data (Behringer and Xue 2004), a monthly mean climate dataset covering the time period of 35 years (January 1980 to December 2014). The horizontal resolution of the data available for public access is 1/3° in latitude and 1° in longitude. As will be explained shortly, temperature stratification in the ocean is not always stable; accordingly, there is no strictly one-to-one transformation between the z-coordinate and the isothermal coordinate. Consequently, the heaving signals diagnosed in the isothermal coordinate should be interpreted with caution.

ð6:14Þ

6.3.1 The Choice of Temperature Scale

MTC FTC Ha;k ði; j; n; mÞ ¼ Ha;k ði; j; n0 ; mÞ ð6:12Þ

n¼1

where the subscript k indicates the vertical level k of our concern. There is only one grid n0 in the FTC that has a non-zero value. In the MTC, signals in each temperature bin can be further separated into the external and internal components HaMTC ¼ Ex HaMTC ðn; mÞ þ In HaMTC ði; j; n; mÞ

where Ex HaMTC ðn; mÞ ¼ HaMTC ði; j; n; mÞ

i;j

is the external mode, defined as the volumeweighted mean of the heat content anomaly in each temperature bin. As discussed in the Appendix, this is the total heat content anomaly for each temperature bin, which is directly linked to the thermal exchange across the boundaries of this isothermal layer, including the air-sea

In using temperature as the Lagrangian coordinate to record the heat content anomaly, choosing a suitable temperature scale is an important technical issue. The essential physical issue is that the reference point of enthalpy or the socalled heat content, is not uniquely defined. In many applications, people use a simple formula

6.3 Casting Method Applied to the GODAS Data

to calculate the heat content of a water parcel, such as HC ¼ qCp hDV, where q, Cp , h, DV are the density, specific heat under constant pressure, temperature and volume. Since the centigrade is the internationally accepted unit for temperature, using such a formula implies the enthalpy is zero at h ¼ 0  C. However, one may also use the Kelvin scale of temperature, in which the corresponding zero point is moved to the so-called absolute zero temperature. Physically, there is no reason apriori for the specific choice of the zero point of enthalpy. Using the commonly used centigrade as the temperature scale for the Lagrangian coordinate may be inconvenient because the generation of water masses with temperatures close to 0 °C would correspond to a nearly zero increase of heat content in the corresponding temperature bin; in addition, the generation of water masses with temperatures lower than 0 °C would be recorded as a negative value of the heat content anomaly in the corresponding temperature bin. Furthermore, as will be discussed shortly, in order to infer the equivalent thickness perturbations from the heat content signals the heat content anomaly is divided by the mean temperature in each bin; if the mean temperature is zero, the corresponding calculation cannot be carried out.

339

Thus, in order to overcome such problems we will use a nominal temperature scale in this section hnom ¼ h þ 10. Using such a nominal temperature is equivalent to moving the reference point of enthalpy to −10 °C. As discussed above, the choice of the reference point for enthalpy is not unique. A different choice for the reference point can lead to a different amplitude of heat content and its anomaly; nevertheless, such a selection should not affect the explanation of the relevant physics. However, all the figures shown below are still based on the centigrade scale of temperature. In addition, it is important to emphasize that climate variability in the FTC has nothing to do with the reference point of enthalpy; hence, the centigrade scale is used in the FTC by default.

6.3.2 Statistics in the Temperature Space Using the formulae discussed above, the net contribution of the heat content (HC) anomaly integrated over each temperature bin, i.e., the volumetric integration of the external modes of the HC anomaly, in the MTC and the FTC are calculated (Fig. 6.1). The most important difference between external heaving signals in these

Fig. 6.1 The total volumetric integration of the external mode of the HC anomaly (seasonal cycle subtracted) in the MTC (a) and the FTC (b)

340

two coordinates is that signals in the MTC are 10 times stronger than that in the FTC. In addition, signals patterns in these two coordinates are different, as will be discussed in details below. The difference in the signal strength in the MTC and the FTC can be explained in a way similar to the difference of signals strength in the MDC and the FDC discussed in Chap. 5. In the FTC, the magnitude of signals is estimated at DFTC ¼ Dhmax DV; on the other hand, in the MTC, the magnitude of signals is the difference between the maximum signal Dmax MTC ¼ hmax DV (DV is the typical size of grid box in the deep ocean) and the climatological mean signal, mean mean i.e., DMTC ¼ Dmax MTC  DMTC , where DMTC ¼ min 0:5  ðDmax MTC þ DMTC Þ is the climatological mean signal. In the extreme case, there might be no data or very little data falling into the specific density bin in the MTC in some specific months; thus, the lowest limit of Dmin MTC is close to zero, max  0:5D and DMTC  Dmean MTC MTC . Assuming the corresponding DV in the FTC and the MTC is close to each other, the corresponding signal ratio is DMTC =DFTC  0:5hmax =Dhmax . Assuming hmax  10  C and Dhmax  0:5  C, this gives rise to a rough estimate of the maximum signal ratio: DMTC =DFTC  0:5  10=0:5  10, i.e., the amplitude of signals in the MTC is about 10 times larger than that in the FTC. This dramatic difference in signal strength in the MTC and the FTC is also an important difference between the Eulerian and Lagrangian coordinates. For each month the integration over the entire temperature range [−2 to 36 °C] of the heat conP P tent anomaly n HaMTC ðn; mÞ and n HaFTC ðn; mÞ should be the same as discussed above; this quantity is the total heat content anomaly for the world oceans, and the time evolution of this variable is shown in Fig. 6.2. From the physical point of view, this quantity should match the time integration of the surface air-sea heat flux available from the GODAS data. However, our analysis shows that there is a large mismatch. There are many potential reasons for this mismatch.

6

Heaving Signals in the Isothermal Coordinate

First, the meridional accumulation of the zonally integrated air-sea heat flux is not zero at the northern boundary of this dataset; this suggests that there is a heat flux across the northern boundary of the data domain. Second, as in many currently used models, the numerical model used to generate this dataset does not conserve thermal energy. Third, the GODAS data is monthly mean; as a result, different terms may not match exactly in the process of making the monthly mean. Therefore, we do not include the comparison of the time evolution of the total heat content anomaly with the time integrated global sum of the air-sea heat flux from the GODAS data. As shown in Fig. 6.2, the world oceans went through 10 years of cooling, from 1980 to 1990. Afterward, the global oceans gradually warmed up with strong interannual and decadal variability. Using the formulae discussed above, the signals of climate variability in the MTC is presented in terms of the heat content anomaly. Another way to present the signals is to convert the signals to the equivalent layer thickness perturbations as follows Ex hMTC ðn; mÞ ¼ Ex HaMTC ðn; mÞ=Cp =q0 =areaðnÞ a =ðtemðnÞ þ 10Þ ð6:16Þ

Fig. 6.2 Total heat content anomaly diagnosed from the GODAS data using either the MTC or the FTC

6.3 Casting Method Applied to the GODAS Data Horizontal area (1014m2) 3.0 2.5 2.0 1.5 1.0 0.5 0.0

0

5

10

15 θ( )

20

25

30

Fig. 6.3 Horizontal area for each isothermal layer, based on the annual mean climatology of the GODAS data

where area(n) is the corresponding horizontal area occupied by the isothermal layer tem(n). Note that the nominal mean temperature hnom ¼ temðnÞ þ 10 is used in this formula. Using the annual mean climatology of temperature distribution from the GODAS data, the distribution of horizontal area can be calculated, as shown in Fig. 6.3. We now compare the fine structure of the heat content anomaly shown in Fig. 6.1, and our discussion below is focused on the lower temperature range of 0–13 °C. The most important information revealed by Fig. 6.1 is that the external heaving mode signals, or the diabatic signals, are mostly confined to the low temperature range, say near 3 °C, with tails extending to the slightly higher temperature of 4–10 °C. This indicates that the diabatic signals are closely linked to the deep water and bottom water formation/erosion regions. On the other hand, the diabatic signals for warm water regions seem much weaker. Thus, general cooling or heating of the world oceans takes place primarily through the connection with deep/bottom water formation/erosion. We emphasize here again: changes in water mass volumetric distribution in the isothermal coordinate are the direct consequence of variability in water mass formation through subduction and deep/bottom water formation; the role of water mass erosion through obduction and water

341

mass transformation is critically important as well. For example, the North Atlantic Deep Water (NADW) formation in the Atlantic Ocean has gone through great changes over the past decades, and this is a very important factor in regulating global water mass volumetric distribution. On the other hand, in the world oceans, there are sinks of NADW counterbalancing the formation of NADW; the sinks of NADW include water mass transformation along the path of NADW in the ocean, and finally the NADW is eroded through obduction most likely within the band of the Antarctic Circumpolar Current, e.g., Liu and Huang (2012). Therefore, the variability in water mass volumetric distribution should be examined through both water mass formation and erosion. Although the total heat content anomaly in each month is the same for both the MTC and the FTC, the patterns in the time-temperature space are quite different, as shown in Fig. 6.1. Heat content anomaly pattern in the MTC shows more variability than that in the FTC. The detailed information shown in Fig. 6.1 reflects the physical processes taking place in the world oceans. As shown in Fig. 6.4, the world oceans were relatively warm and gradually cooled down over a 10-year period (1980–1990). In the FTC, this is reflected as positive heat content anomaly, mostly in the temperature bin of 2 °C; in addition, there are two weak cooling branches centralized at 1 and 5 °C (Fig. 6.1a). As the ocean was gradually cooled down, the corresponding warming anomaly diminished, and almost disappeared in late 1990s. Within the same temperature range, there is a cooling trend starting from 2000 and persisting until the end of the data record. The 35-year time evolution of the thermal anomaly in this temperature bin is shown by the red curve in Fig. 6.4b. On the other hand, there are two warming branches, one centralized at 1 °C and one centralized at 4 °C (it seems to extend up to 10 °C, as shown in Fig. 6.4a). These two branches indicate the temperature warming over the past 15 years, as simulated by the model. Apparently, 2 °C is the only temperature bin that shows cooling after 2005, but all other temperature bins indicate warming

342

6 (a) HCa (10 23J/°C), in FTC 12 10

θ(°C)

8

Heaving Signals in the Isothermal Coordinate (b) HCa (10 23J/°C), in FTC

1.0 0.8 0.6

1.0

0.4 0.2

0.5

θ=0°C θ=1°C θ=2°C θ=3°C θ=4°C

0 6 4

−0.2 −0.4 −0.6

2 0 1980 1985 1990 1995 2000 2005 2010

0

−0.5

−0.8 −1.0

−1.0 1980 1985 1990 1995 2000 2005 2010

Fig. 6.4 The total volumetric integration of the HC anomaly in the FTC (a) and the corresponding signals in several isothermal layers (b)

(Fig. 6.4b). The total contribution from these two warming branches overpowers that of the cooling branch and gives rise to the total heat content variability (Fig. 6.2). The heat content anomaly map in the MTC provides a different view of the same processes, as shown by the quite different patterns for the same periods. For the period of 1980–1996, there is a primary positive branch centralized at 3 °C; in addition, there are two weak negative branches centralized at 2 °C and 5–7 °C (Fig. 6.5a). The positive branch centralized at 3 °C indicates that there was more warm water in this temperature range during this period. Combining this information with the warming branch centralized at 2 °C in the FTC suggests that water with climatological mean temperature of 2 °C is warmed up and the corresponding temperature becomes 3 °C and it appears as a positive anomaly in the 3 °C bin in the MTC. On the other hand, after the year 2000 a major negative branch in the MTC appears in the temperature range of 3 °C and a positive branch appears for the temperature range of 4–7 °C. This indicates that during this time period the total volume of 3 °C water is reduced, and the

corresponding water mass is warmed up and contributes to a positive heat content anomaly with a high temperature (Fig. 6.5a). The fine structure of the time evolution in several isothermal layers is shown in Fig. 6.5b. As discussed above, the amplitude of the heat content anomaly may depend on the choice of the reference point of the enthalpy. If we use the zero point of the Kelvin temperature scale, the value of positive and negative branches in the MTC may increase 10 times. As a result, the heat content anomaly presented in this way may include a subjective choice of the reference point of enthalpy. It is desirable to find another way to present the signals. As described in Eq. (6.16), one can use the equivalent layer thickness perturbations as the alternative, and the corresponding results are presented in Fig. 6.6. Physically, this indicates the layer thickness variability due to external heating/cooling. Of course, to a minor degree, such external modes of heaving can also be induced by the redistribution of water masses in connection with water mass transformation in the subsurface ocean. Because isothermal layers of lower temperature occupy a relative small area in the world oceans, the corresponding layer thickness

6.3 Casting Method Applied to the GODAS Data (a) HCa (10 23J/°C), in MTC

343 (b) HCa (10 23J/°C), in MTC

10

12 10

θ(°C)

8 6

5

0

10 5

θ =0°C θ =1°C θ =2°C θ =3°C θ =4°C

0 −5 −5

4 −10 2 0 1980 1985 1990 1995 2000 2005 2010

−10 −15 1980 1985 1990 1995 2000 2005 2010

Fig. 6.5 The total volumetric integration of the HC anomaly in the MTC (a) and the corresponding signals in several isothermal layers (b)

Fig. 6.6 The equivalent layer thickness perturbations for the external heaving modes in the MTC (a) and the corresponding signals in several isothermal layers (b)

perturbations are larger. As shown in Fig. 6.6, the largest layer thickness perturbation is in the isothermal layer of 0 0:5  C (blue curve in Fig. 6.5b), although the corresponding heat content anomaly is relatively small (Fig. 6.5).

Our discussion above has been primarily focused on the diabatic signals or the external heaving signals identified from the GODAS data. Our next step is to identify the internal heaving signals; in fact, this is our primary interest in

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6

climate study. In this section, the internal mode is defined by Eq. (6.13) rewritten in the following form In HaMTC ði; j; n; mÞ ¼ HaMTC  Ex HaMTC ðn; mÞ ð6:130 Þ Namely, the internal mode heaving signals can be obtained by subtracting the external mode diabatic signals from the anomalous signals at each grid point. It is important to emphasize that such perturbation signals are defined in the four dimensional time-space. The basic strategy of our approach is to identify the external and internal mode heaving signals in this fourdimensional space. As long as we calculate the diabatic signals (the external modes) and the total signals, this four-dimensional array is completely determined, and it can be used to analyze the heaving signals in many different ways. Among many different possible ways of presenting the internal heaving signals, we choose to calculate the sum of the absolution value of the internal heaving signals in each temperature bin as  P a function of time, i.e., i;j;k In HaMTC ði; j; n; mÞ, or the sum of the positive internal mode heaving signals in each temperature bin as a function of  P
time i;j;k In HaMTC ði; j; n; mÞ [ 0 . As shown in

Heaving Signals in the Isothermal Coordinate

Fig. 6.7, the amplitudes of these two variables are much larger than those shown in Fig. 6.4; this means that the internal heaving signals are much stronger than the external mode (diabatic) signals in general. In addition, now the temperature range of high amplitude signals is wider than that shown in Fig. 6.1, and they occupy the range of 1–8 °C, with the peak of signals maximum located around 2–3 °C. Therefore, internal heaving signals are mostly confined to the site of deep water formation/erosion; in addition, they also appear in the upper part of water column associated with warmer temperature. To quantify the phenomenon discussed above, we calculate the absolute heat content anomaly signals integrated over the entire temperature range for the MTC and the FTC and the total absolute value of the internal heaving signals integrated over the entire temperature range for the MTC (Fig. 6.8). As shown in Fig. 6.8a, in each month the absolute value of heat content anomaly integrated over the temperature range in the MTC (red curve) is much larger than that in the FTC (black curve), and the ratio of the corresponding root-mean-square value of ratio is 21:2:1:73  3:1. The contrast between the heaving signals is even remarkable. For each month the absolute value of the internal mode heaving signals

Fig. 6.7 The integration of the internal heaving signals in each temperature bin: the absolute value (a) and the positive value only (b)

6.3 Casting Method Applied to the GODAS Data

345

Fig. 6.8 The total absolute heat content anomaly signals integrated over the entire temperature range for the MTC and the FTC (a); the total absolute value of the internal mode heaving signals integrated over the entire temperature range for the MTC (b). The dashed lines and numbers above them depict the RMS of the corresponding variables

integrated over the entire temperature domain is shown as the red curve in Fig. 6.8b, and the black curve corresponds to the integration of positive signals only. The RMS of these two variables is shown as the dashed lines in Fig. 6.8b. The corresponding ratios of these four variables are 1.73:5.27:65.7:131.4 or approximately 1:3:39:77. Therefore, the strength of internal heaving signals in the MTC is approximately 40 times that of the diabatic signals identified from the FTC. We also carried out comparisons in terms of root-mean square signals for each temperature bin over the 420 months of data record. For each temperature bin in both the FTC and MTC, the corresponding RMS over the time domain is shown by the curves in Fig. 6.9; the dashed lines and numbers above them depict the RMS over the temperature domain for the corresponding variables. The corresponding ratio is quite large: 0.13:1.5:5.5:11.1, or approximately 1:11.5:42:85. Accordingly, internal heaving signals are statistically dominating. It is clear that although external signals are mostly confined to the low temperature range (below 2 °C), internal mode heaving signals are much more spreading and covering the range up to 15 °C.

6.4

Projecting Method

As a basic requirement for using the projecting method, potential temperature should monotonically decline in the downward direction, so that there is a one-to-one correspondence between the z-coordinate and the T-coordinate. Although this constraint is generally satisfied, there are quite a few exceptions where it is violated. As one example, we show both potential temperature and potential density sections along 60.5° W for three months (Fig. 6.10). It is readily seen that for the latitude band higher than 36° N thermal stratification in the range of h\19  C is unstable for these months. On the other hand, density stratification is stable in general, except for a few inversions north of 40° N. Theoretically, density stratification for large scale motions in the ocean should be stable because any unstable density stratification would be wiped out by strong vertical convection induced by the unstable stratification. The unstable stratification displayed in this figure may be some artifact in the GODAS data, and we ignore such seemingly unphysical phenomena in our discussion here.

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6 (a) Signal RMS, in FTC

1.0

rms (Ha), in MTC rms (|InHa|), in MTC rms [InHa>0], in MTC

25

0.8 0.7

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0.6

1023J/°C

1023J/°C

(b) Signal RMS, in MTC

30

rms(Ha), in FTC

0.9

Heaving Signals in the Isothermal Coordinate

0.5 0.4

15

11.1 10

0.3 0.2 0.1

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0 −2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 θ(°C)

Fig. 6.9 The RMS of the integrated thermal anomaly for both the FTC (a) and the MTC (black curve in b) and the heaving signals in the MTC (blue and red curves in b)

Depth (100m)

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), Sep., 2011

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Fig. 6.10 Potential temperature and potential density sections along 60.5° W

35N

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6.4 Projecting Method

We conclude that potential temperature may not be a monotonic function of depth; for this reason, using potential temperature as the vertical coordinate must be carried out with caution. Nevertheless, temperature stratification for the range of h [ 19  C is stable in general, and we will use isothermal layer analysis for temperature higher than 19 °C.

6.4.1 Isothermal Layer Analysis for the Layer of h ¼ 20 0:5  C Isothermal layer analysis based on the projecting method is quite similar to isopycnal layer analysis; in both methods, we choose a set of coordinate layer with specified density/temperature interval for each layer and examine the climate variability in these layers. In this section, our

347

analysis will be focused on two examples. The first example is isothermal layer analysis for the potential temperature layer of h ¼ 20 0:5  C, and the second example is for a section along the equator. We begin with climate variability analysis for the isothermal layer of h ¼ 20 0:5  C. This layer is within the warm water sphere in the upper ocean. As shown in Fig. 6.11, water in this isothermal layer can communicate between the three basins; thus, the heaving mode diagnosed is really a global mode. Climate variability in this layer can be seen in Figs. 6.11, 6.12, 6.13 and 6.14. In particular, heaving signals can be seen from layer depth and layer thickness perturbations shown in Figs. 6.11b and 6.12b. It is clear that these variabilities are closely linked to the climate changes of the subtropical gyres. In addition, layer stretching and vertical movement of the

Fig. 6.11 Depth of the 20 ± 0.5 °C isothermal layer: a the climatological mean; b the monthly mean absolute value of perturbations

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Heaving Signals in the Isothermal Coordinate

Fig. 6.12 Thickness of the 20 ± 0.5 °C isothermal layer: a the climatological mean; b the monthly mean absolute value of perturbations

entire layer has different characters, although they may locate near each other. These figures show clearly the site of subtropical mode water formation near the western boundaries of the subtropical gyres in the North Atlantic Ocean, North and South Pacific Ocean. Most interestingly, there are clear signs of the so-called eastern subtropical mode water formation sites in the North Atlantic Ocean, North and South Pacific Ocean. The eastern subtropical mode water formation is closely related to the so-called eastern subtropical gyres, which has been discussed in previous studies, e.g., Hautala and Roemmich (1998), Hanawa and Talley (2001), Liu and Huang (2012). On the other hand, the diabatic signals can be seen from the layer potential density r0 and layer

salinity perturbations shown in Figs. 6.13 and 6.14. The global maximum of diabatic perturbations is located in the northern North Atlantic Ocean. This suggests that heaving signals in this small region are strongly diabatic, but in the other parts of the world oceans heaving signals are mostly adiabatic. We also include the mean zonal velocity in this isothermal layer and its perturbations (Fig. 6.15). The 20 0:5  C layer is clearly associated with the core of the equatorial undercurrent for both the Pacific and Atlantic basins. This layer also includes a section of strong zonal velocity perturbations near the eastern boundaries in all three basins. Another character of this isothermal layer is the density thickness, i.e., the density difference

6.4 Projecting Method

349

Fig. 6.13 Potential density r0 of the 20 ± 0.5 °C isothermal layer: a the climatological mean; b the monthly mean absolute value of perturbations

dr0 between the upper and lower interfaces of this isothermal layer. This variable indicates the strength of density stratification for this isothermal layer. As shown in Fig. 6.16a, the large density increment means strong density stratification of this layer and vice versa; the typical cases are the cold tongue in the equatorial Pacific and a band west of the Australia continent. The other climate signal associated with this density difference dr0 is its anomaly ddr0 , as shown in Fig. 6.16b. First of all, in the northwest of North Atlantic Ocean there is a global maximum, that is closely linked with the layer thickness anomaly maximum. This seems to be linked to the strong diabatic signals associated with the air-sea interaction in this region. There

are three other local maxima in the southeast part of the Pacific, Indian and Atlantic Oceans. The dynamic meaning of these local maxima remains unclear at this time. The radius of signal of climate variability in this isothermal layer can be analyzed, using the Lagrangian coordinate, as discussed in Sect. 3.4.3 (Formula 3.86a, b, c and d). the corresponding maps of radius of signal are shown in Fig. 6.17. The radius of signal associated with potential density is high within the equatorial band, in particular near the eastern boundary in the Pacific basin (panel a). In comparison, radius of signal associated with spicity is much smaller in this area. On the other hand, it is quite large in the northwestern corner of the Atlantic basin (panel b).

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Heaving Signals in the Isothermal Coordinate

Fig. 6.14 Salinity of the 20 ± 0.5 °C isothermal layer: a the climatological mean; b the monthly mean absolute value of perturbations

The combination of these two factors gives rise to a two maxima structure for the radius of total signal in these two areas (panel c). The radius of signal left after isothermal analysis is mostly confined to the northwestern corner of the Atlantic basin (panel d). Therefore, in this isothermal layer, strong signals appearing in the equatorial band are mostly heaving in nature. In addition, in the northwestern corner of the Atlantic basin there are strong non-heaving signals, which are closely linked to air-sea interaction and mode water formation. The heaving signals can be conceptually estimated by the heaving ratio. As discussed in 0

Chap. 3, heaving ratio is defined as HR ¼ R0 Rþs R1  s

s

100%, where R0s is radius of signals before

isothermal analysis, and R1s is the radius of signals left behand after isothermal analysis. For most parts of the world oceans the heaving ratio in this layer is on the order of 90%; accordingly, most of the climate signals identified is associated with the heaving motion of the isothermal layer (Fig. 6.18). It is surprising to see that the heaving ratio in the northwest North Atlantic Ocean is quite low, indicating that climate signals here have rather strong diabatic components associated with airsea interaction. To reveal the details of these processes, we include close-up maps for this region in Fig. 6.19. This figure clearly shows that in the northwestern corner of the Atlantic basin the spicity

6.4 Projecting Method

351

(a) Umean (m/s) 0.8

4N

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0.5 4N 0.4 2N 0.3 0 0.2 2S 0.1 4S 30E

60E

90E 120E 150E

180

150W 120W 90W

60W

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0

Fig. 6.15 Zonal velocity of the 20 ± 0.5 °C isothermal layer: a the climatological mean; b the monthly mean absolute value of perturbations

variance is the dominating contributor to the radius of signal before the isothermal analysis. Furthermore, in the northwestern corner of the Atlantic basin there are strong non-heaving signals, which are closely linked to air-sea interaction and mode water formation. The next step is to examine the external heaving modes of this isothermal layer. The time evolution of basic layer properties is shown in Fig. 6.20. As discussed above, the external heaving modes are generated by the diabatic forcing anomaly, i.e., by the heat and salt exchange anomaly across the interfaces. The global mean layer thickness, layer depth, layer density and layer salinity are shown in the left panels of Fig. 6.20, where the red curves indicate

results after 13-month smoothing. The interannual variability with the mean annual cycles removed is shown in the right panels of Fig. 6.20. It is readily seen that heaving signals for layer thickness, layer depth, and layer density (salinity) have different characteristics on interannual and decadal time scales. We now examine heaving signals along the equatorial section in this isothermal layer (Fig. 6.21). The first choice is based on the time evolution of the layer depth anomaly along the zonal sections within the equatorial band. The annual cycle consists of a noticeable part of the signals in the equatorial band. However, our focus is on the interannual variability; for this reason, before showing the time evolution of the

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Heaving Signals in the Isothermal Coordinate

Fig. 6.16 Density difference over the 20 ± 0.5 °C isothermal layer: a the climatological mean; b the monthly mean absolute value of perturbations

layer depth anomaly, we separate the signals into the mean annual cycle and the signals with the annual cycle removed. In the Pacific basin, the annual cycle in the equator does not have a clear pattern of wave propagation (Fig. 6.21b); however, for the 5° N and 5° S sections, there is clearly a simple seasonal cycle, with signals propagating westward. In the Indian basin, the annual cycle appears in the form of biannual cycle; this is linked to the strong monsoon cycle in this basin. At the equator, wave signals propagate eastward; along 5° N and 5° S sections, wave signals propagate westward. In the Atlantic basin, there is a simple annual cycle, with wave signals propagating eastward along the equator, and wave signals propagating westward along both the 5° N and 5° S sections.

6.4.2 Structure in the Pacific Basin 6.4.2.1 Heaving Signals Along the Equator in the Pacific Basin The time evolution of the layer depth anomaly (with the annual cycle removed) is shown in Fig. 6.22. First of all, there are clearly signs of eastward propagating Kelvin waves. The positive signals indicate the downwelling Kelvin waves and the negative signals indicate the upwelling Kelvin waves. Some of the strong ENSO events are clearly shown in this figure. Most positive signals move eastward with a relative fast phase velocity, and they move across the Pacific basin within a time of 3–4 months, indicating that they belong to the simple adiabatic Kelvin waves. On the other hand, some large amplitude negative

6.4 Projecting Method

353

Fig. 6.17 The radius of signal associated with potential density (a), potential spicity (b), the total signals (c), and signal left after isothermal analysis (d) in the 20 ± 0.5 °C isothermal layer

Fig. 6.18 Heaving ratio of climate signals in the 20 ± 0.5 °C isothermal layer

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Heaving Signals in the Isothermal Coordinate

Fig. 6.19 The radius of signal associated with potential density (a), potential spicity (b), the signal left after isothermal analysis (c), and the heaving ratio (d) in the 20 ± 0.5 °C isothermal layer for the Western North Atlantic Sector

signals seem to move eastward relatively slowly, indicating that these Kelvin wave processes may involve with air-sea interaction. Apparently, these are not really free Kelvin waves; by involving air-sea interaction, they also share some characteristics of forced waves, and their speed can be quite different from the free waves. For example, in the 1982–1983 event, there was a strong positive signal in the eastern part of the basin in the late season of 1982. A strong negative signal patch started from the dateline at the end of 1982. This patch moved eastward with a relatively slow phase speed, and it became a strong negative patch in the central Pacific basin in the late season of 1983. Similarly, the strong 1997–1998 event also manifests as strong signal patches in this figure. In particular, there are two positive patches of signals in the summer of 1997. In addition, there is a strong negative signal patch that started at the end of 1997 near the western edge of the Pacific basin; in fact, the negative patch moved eastward with a phase

speed much slower than the typical Kelvin waves, suggesting that the air-sea interaction associated with this eastward Kelvin wave package is very strong. There are also other strong events shown in this figure. In particular, there are the strong positive-negative signals associated with the 2006–2007 and 2009–2010 events. One of the most important features shown in this figure is the strong decadal variability of the layer depth anomaly superimposed on the background of quasi-four year interannual variability. For example, during the period of 1998–2002, there is a clear sign of gradually building up of the layer depth, as shown in the second panel of Fig. 6.22. Furthermore, there is a clear sign of layer depth building up from 2010 all the way to the end of 2015; and the layer depth anomaly moves gradually from the western part to the eastern part of the Pacific basin. This multi-year process of building up the basin scale layer depth anomaly is an important sign of the decadal

6.4 Projecting Method

355

Fig. 6.20 External modes of the 20 ± 0.5 °C isothermal layer: top row for the layer thickness; second row for the layer depth; third row for the layer potential density ðr0 Þ; the fourth row for the salinity; left column for the monthly mean perturbations, and the red curves are the 13-month smoothed curves; right column for the signals with the annual cycle removed

variability of the equatorial dynamics in the Pacific and it is worth of further study. For the zonal section along the equator, the time evolution pattern of the layer depth remains quite similar in the case of the annual cycle signals retained or removed; this indicates that the annual cycle is only a relatively small component of the total signals. In another words, the heaving signals along the equator are dominated by interannual variability, as shown in Fig. 6.22.

6.4.2.2 Heaving Signals Along 5° N in the Pacific Basin We now examine the layer depth heaving signals along 5° N in the Pacific basin. As shown in Fig. 6.23, there are strong annual signals moving westward, and they take about one year to cross the Pacific basin; these are the first baroclinic long Rossby waves. With the mean annual cycle removed, the wave patterns become less well organized (Fig. 6.23). In

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Heaving Signals in the Isothermal Coordinate

Fig. 6.21 Mean annual cycle of layer depth heaving signals in the 20 ± 0.5 °C isothermal layer along 5° N (a), Equator (b) and 5° S (c)

general, there is no clear sign of organized wave propagation of these perturbations, although during the 1982–1983 event a positive anomalous signal patch started from the eastern boundary and moved toward the western boundary. At an even earlier time, there was a negative patch of signal starting from the western boundary and moved toward the middle of the basin. A similar situation also happened during the 1997–1998 event. The explanation of the many dynamic details shown in this figure remains unclear.

6.4.2.3 Heaving Signals Along 5° S in the Pacific Basin The layer depth heaving signals along 5° S in the Pacific basin are shown in Fig. 6.24; there are also strong annual signals moving westward, and

they take about one year to cross the Pacific basin; these are the first baroclinic long Rossby waves. This figure is quite different from Fig. 6.23 because there are large amplitude perturbations in the layer depth detected along 5° S. The interannual variability of heaving signals seems much stronger. For example, there are large patches of negative signals from 1997 to 2003; in addition, during the time period of 2005 to 2014 there were strong negative signal patches in the eastern basin and strong positive signal patches in the western basin. The large difference in layer depth perturbation patterns existing between the 5° N and 5° S sections reflects some fundamental differences in the dynamic processes taking place within these two sections.

6.4 Projecting Method

357

Fig. 6.22 Time evolution of layer depth heaving signals in the 20 ± 0.5 °C isothermal layer along the Equator in the Pacific Ocean, with the mean annual cycle removed

Fig. 6.23 Time evolution of layer depth heaving signals in the 20 ± 0.5 °C isothermal layer along the 5° N in the Pacific Ocean, with the mean annual cycle removed

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Heaving Signals in the Isothermal Coordinate

Fig. 6.24 Time evolution of layer depth heaving signals in the 20 ± 0.5 °C isothermal layer along the 5° S in the Pacific Ocean, with the mean annual cycle removed

6.5

Signals of Layer Depth and Zonal Velocity in the Pacific Basin

Finally, we show the monthly evolution of the layer depth and zonal velocity anomalous patches within the equatorial band (15° S–15° N, Figs. 6.25, 6.26, 6.27 and 6.28). These figures show the migration of the layer property anomaly in this isothermal layer during four major ENSO events, 1982, 1997, 2004 and 2009. It is readily seen that the layer depth anomaly has quite different patterns and moves in quite different ways. Note that climate variability inferred from monthly mean data is quite limited in terms of revealing the movement of climate variability; nevertheless, these figures may serve the purpose of illustrating the potential of heaving modes analysis based on the isothermal coordinates. We

also hope that isothermal layer analysis based on these patterns may lead to better understanding of the ENSO dynamics. The mostly outstanding feature is the eastward migration of the layer depth anomaly maximum from August to January of next year during the 1982–1983 event (Fig. 6.25) and 1997–1998 event (Fig. 6.26). However, during the 2004– 2005 event (Fig. 6.27), the migration of the layer depth anomaly was noticeably different from previous events. Furthermore, during the 2009– 2010 event (Fig. 6.28), the evolution of events also appeared in a way different from other major events discussed above. The difference between these events reflects the essential differences in the physics related to these strong ENSO events. On the other hand, the zonal velocity anomaly has a pattern quite different from that of the layer depth anomaly, and its movement is also quite different (Figs. 6.25, 6.26, 6.27 and 6.28).

6.6 Z-Theta Diagram and Its Application …

359

Fig. 6.25 Time evolution of the 20 ± 0.5 °C isothermal layer depth (the top and third rows) and the corresponding zonal velocity (the second and fourth rows) for the equatorial Pacific Ocean, Aug. 1982–Jan. 1983

6.6

Z-Theta Diagram and Its Application to Climate Variability Analysis

As discussed in Chap. 2, climate variability analysis based on the traditional z-coordinate does not reveal much information about the nature of climate variability. Instead, we have discussed the concepts of using either potential density or potential temperature as the Lagrangian coordinate to describe climate variability. A further extension of this method is to compile

data in the z-sigma or z-theta coordinates. Following the procedures discussed in Chap. 2 and the beginning of this chapter, we can construct a time series of the two-dimensional z-theta array as follows. The heat content for each spherical grid point is hði; j; k; mÞ ¼ hði; j; k; mÞCp q0 DxðjÞDyDzðkÞ ð6:17Þ where hði; j; k; mÞ is the (potential) temperature at the mth month. The resulting data is stored in discrete depth bin k and temperature bin n in the

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Heaving Signals in the Isothermal Coordinate

Fig. 6.26 Time evolution of the 20 ± 0.5 °C isothermal layer depth (the top and third rows) and the corresponding zonal velocity (the second and fourth rows) for the equatorial Pacific Ocean, Aug. 1997–Jan. 1998

MTC (Moving Temperature Coordinate), using the instantaneous temperature to determine the right bin number n hn  0:5\hði; j; k; mÞ  hn þ 0:5; hn ¼ 2; 1; 0; . . .; 36 ð CÞ

ð6:18Þ

Thus, the total heat content time series is in the form H ¼ ½hh;z ðl; k; mÞ; l ¼ 1; . . .; 39; k ¼ 1; . . .; kmt; m ¼ 1; . . .; m month; inunitof J ð6:19Þ

After subtracting the climatological mean, this leads to the time series of the two-dimensional data array. As an example, we show the heat content anomaly diagnosed from the GODAS data for the month of Jan. 2000 (Fig. 6.29). This figure is focused on the upper part of the water column. Integrating over the temperature range for each level leads to the vertical distribution of heat content anomaly shown in panel b. Integrating over the depth range for each temperature bin leads to the distribution of the heat content anomaly in terms of the temperature shown in panel c.

6.6 Z-Theta Diagram and Its Application …

361

Fig. 6.27 Time evolution of the 20 ± 0.5 °C isothermal layer depth (the top and third rows) and the corresponding zonal velocity (the second and fourth rows) for the equatorial Pacific Ocean, Aug. 2004–Jan. 2005

This time series can be analyzed in different ways. For example, we can plot changes in the heat content anomaly over certain time periods. In Fig. 6.30, we show the heat content change over a 4-year (Jan. 2004–Jan. 2008) period. The heat content anomaly change in the vertical direction is shown in Fig. 6.30b: the upper layers were cooled down and the layers below 120 m were warmed up. The integration over each temperature bin is shown in Fig. 6.30c, where the blackPcurve indicates the net heat content change ð h dHCÞ within this time period. By definition, this indicates the net change of heat content for the temperature bins and such change

must come from external heating/cooling over this temperature range. The red curve indicates the total signal of heat P content change, defined as h jdHCj. The difference between the red curve and the black curve indicates the heat content compensation due to adiabatic movement of the water masses in the vertical direction. As shown in Fig. 6.30a, the heat content anomalies are often compensated for, i.e. positive and negative anomalies for the same potential density bin can appear, and compensating for each other. The section view for the time evolution of the heat content anomaly in the theta coordinate for

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Heaving Signals in the Isothermal Coordinate

Fig. 6.28 Time evolution of the 20 ± 0.5 °C isothermal layer depth (the top and third rows) and the corresponding zonal velocity (the second and fourth rows) for the equatorial Pacific Ocean, Aug. 2009–Jan. 2010

the selected depth range or the time evolution of the heat content anomaly in the depth coordinate for the selected temperature range are quite interesting. Our focus here is on the heat content for the equatorial band. If the equatorial easterlies decline, the slope of the isothermals is reduced, depicted by the blue and red arrows. The blue curves indicate the initial positions of the isothermals and the red curves for their positions after the adjustment (Fig. 6.31a). Due to the communication between the equatorial band and the off-equatorial regions the total water mass in each temperature bin is not necessarily conserved during the adjustment in response to wind stress perturbations.

When the isothermal surface A (blue) moves to the new position A′ (red), there is a heat content shift in the upper ocean, indicated by green curve in panel b. Since there might be a communication between the equatorial band and the off-equatorial bands, the corresponding heat content profile is not a pure baroclinic mod, i.e., there might be some heat gain for this case. Similarly, when the isothermal surface B (blue) moves to the new position B′ (red), there is also a heat content shift in the lower part of the water of water column, as shown by the blue curve in panel b. Combining these two profiles in panel b gives rise to the net heat content anomaly profile, as

6.6 Z-Theta Diagram and Its Application …

Fig. 6.29 Heat content anomaly for January 2000

Fig. 6.30 Heat content difference over 4 years for the world oceans

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(a)

Heaving Signals in the Isothermal Coordinate

(b)

Z

(c)

A A'

B' B E Fig. 6.31 The adjustment of isothermal surfaces in the equatorial band in response to wind stress changes; a blue curves indicate the positions of the isothermal surfaces before adjustment and red curves for their positions after adjustment. Panel b and c depict the vertical profiles of heat content anomaly induced by the isothermal adjustment

shown by the curve in panel c. Thus, in the present case, wind stress anomaly driven adjustment of the isothermal layers, i.e. the heaving motions, leads to a warming (cooling) in the upper (lower) part of the water column. There is certain degree of heat compensation within the water column, but the compensation is incomplete. Note that the degree of signal compensation can vary greatly depending on the selection of temperature or depth range. In the following example, we select both depth and temperature ranges centralized by the water mass with low potential vorticity, or water masses with a relatively thick layer. If we select a depth range near the air-sea interface, the diabatic process may play a capital role and leads to less signal compensation. In order to reveal adiabatic motions in the ocean we can use the temperature sections of the z-theta diagram. For example, we can plot the time evolution of the heat content anomaly for certain isothermal layers, and Fig. 6.32 presents such a figure for the heat content anomaly integrated over the temperature range of (14–16 °C) for the equatorial Pacific (6° S–6° N). During the typical ENSO events, such as 1982–1983, 1986–1987, 1992–1993, 1997–1998, the positive anomaly (red patches, indicating more water in this temperature range) moves upward towards

shallow depths, Fig. 6.32a. There are also negative (blue patches, indicating less water in this temperature range) patches which seem to compensate for the positive anomaly signals within same temperature range. In the past 10–15 years, there were more ENSO events in the form of the so-called CP Mode; the CP Mode is quite different from the traditional EP Mode. How much the positive and negative heat content anomalous signals are compensated for can be revealed by vertically integrating signals within the isothermal layer. As shown in Fig. 6.32b, the red curve indicates the sum of R positive signals h dHC  dz; for dHC [ 0, while the blue curve indicates the sum of negative R signals h dHC  dz; for dHC\0. The black curve in Fig. 6.32b indicates the net R sum of signals h dHC  dz, that is the algebraic sum of the red and blue curves. By definition, this term represents the net external signals, which must come from an external source of water within this temperature range due to diabatic heating/cooling or exchange with the environment of this isothermal layer. Whenever the black curve has a zero value, there is no net external source, indicating that volume gain and loss are exactly compensated for in this isothermal layer. The non-zero value of the black curve indicates the amount of net volumetric gain/loss in connection with diabatic heating/cooling or

6.6 Z-Theta Diagram and Its Application …

365

Fig. 6.32 Heat content anomaly for the isothermal layer (14–16 °C) in the Pacific Ocean: a Time evolution of distribution in depth; b the heat content integrated over the depthR range, red shows the total positive signals, R blue shows  the total negative signals, and black is the net; c total signals, h jdHCj  dz, and the external signals,  h dHC  dz

water mass exchange through the side boundaries of this isothermal layer. To evaluate how much of the signals are adiabatic in nature we plot the total signals R jdHCj  dz (red curve) and the external signals Rh   dHC  dz (black curve) in Fig. 6.32c. The h ratio of amplitude of these two curves indicates how much of the signals are adiabatic in nature. As shown in Fig. 6.32c, during typical ENSO

events, the total signals are strong and a large percentage of the signals is adiabatic in nature. To present a better view of the heat content anomaly during the ENSO event we selected three time slots and plotted the time evolution of the heat content anomaly in Fig. 6.33. These panels clearly show the heat content signals during these events. In particular, the positive signals move towards a shallower depth during the ENSO

366

6

Heaving Signals in the Isothermal Coordinate

Fig. 6.33 The heat anomaly during three ENSO events in the isothermal layer (14–16 °C) in the Pacific Ocean. Upper panels for the time evolution of heat content distribution on depth; the lower panels for the integrated signals, the red curves for the total positive signals, the blue curves for the total negative signals and the black curves for the net signals

events. On the other hand, the negative signals do not seem to move vertically; in fact, they seem to stay on the same depth range, although during the intervals between ENSO events they seem to move towards shallower depth range, as shown in Fig. 6.32a. It is interesting to notice that during ENSO events, negative signals appear above and below the positive signals, indicating that during ENSO events the isothermal layer slope declines and thus pushes water within the isothermal layer to flow towards the middle depth of this isothermal layer. As shown in Fig. 6.33c, the ENSO event in 2009–2010 seems different from other classical events, like the 1982–1983 and 1997–1998 events; exploring the meaning of such differences is, however, beyond the scope of our discussion here. As discussed above, the net heat content anomaly signals for each isothermal layer

indicates the contribution from either the external thermal forcing or the mass exchange across the site boundaries of the domain. In the case of the Pacific basin, the Indonesian Throughflow is an important contributor to the climate signals. To explore the potential contribution of this interbasin flow we repeat the same calculation above for the Indian-Pacific basin, with the results shown in Figs. 6.34 and 6.35. These figures show pattern similar to those in Figs. 6.32 and 6.33. Therefore, by including the Indian basin does not change the fundamental structure of the heat content anomaly signals. Instead, it can only slightly modify the patterns. As discussed above, net heat content anomaly signals for each temperature bin indicates the contribution from either the external thermal forcing or the mass exchange cross the site boundaries of the domain. In the case of the

6.6 Z-Theta Diagram and Its Application …

367

Fig. 6.34 Heat content anomaly for the isothermal layer (14–16 °C) for the Indian-Pacific Ocean: a time evolution of distribution in depth; b integrated over the depth range, red is the total positive, blue is the total negative, and black is the net; c total signal and the external signals

Pacific basin, the Indonesian Throughflow is an important contributor to the climate signals. As shown by the red curve in Fig. 6.36, the net signals in the Indian-Pacific often have an amplitude smaller than those in the Pacific basin alone. This suggests that the Indonesian Throughflow plays a key role in the adiabatic transport of climate signals during the ENSO like events.

The z-theta diagram can also be examined for the fixed layer depth range. For example, the heat content anomaly signal distribution for a fixed depth range in the Indian-Pacific basin is shown in Figs. 6.36 and 6.37. These figures show pattern similar to those in Figs. 6.34 and 6.35. There are clear signs of two phenomena. First, the positive anomalous signals are sandwiched between the negative anomalous signals during

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6

Heaving Signals in the Isothermal Coordinate

Fig. 6.35 Heat anomaly during three ENSO events in isothermal layer (14–16 °C) for the Indian-Pacific Ocean. Upper panels for the time evolution of heat content distribution on depth; the lower panels for the integrated signals, the red curves for the total positive signals, the blue curves for the total negative signals and the black curves for the net signals

Fig. 6.36 Net heat content signals integrated for the isothermal layer over the whole depth range for the Indian-Pacific basin (red) and the Pacific basin (black)

6.6 Z-Theta Diagram and Its Application …

369

Fig. 6.37 Heat content anomaly for the depth range of (140–200 m) in the Indian-Pacific Ocean: a time evolution of the heat content distribution on the isothermal coordinate; b integrated over the depth range, red is the total positive, blue is the total negative, and black is the net; c total signal and the external signals

major ENSO events; second, positive and negative signals move towards a lower temperature range, as shown in Fig. 6.37. The refined picture for three typical ENSO events is shown in Fig. 6.38. Signal compensation during these major ENSO events is clearly shown in this figure. As shown in the sketch Fig. 6.31, the warm and cold signals are nearly compensated for. In Fig. 6.38 the nearly compensation of warm and cold signals is shown by the black curves in the lower panels.

Appendix: Connection Between the MTC and the FTC These two coordinates are closely linked to each other. In order to compare these two approaches, in this appendix we introduce slightly different variables as follows. The variables defined in the main text are divided by the factor of Cp q0 DxDyDz, i.e., we will use the temperature and its deviation from the seasonal cycle mean in the mth month, i.e.

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6

Heaving Signals in the Isothermal Coordinate

Fig. 6.38 Heat anomaly during three ENSO events in the depth range of (140–200 m) for the Indian-Pacific Ocean. Upper panels for the time evolution of heat content signals in isothermal coordinate; the lower panels for the integrated signals, the red curves for the total positive signals, the blue curves for the total negative signals and the black curves for the net signals seasonal

hði; j; k; mÞ and hði; j; k; mÞ  hði; j; k; mÞ , as the new variables in the MTC and the FTC, and these variables will be denoted as HaMTC ði; j; kÞ and HaFTC ði; j; kÞ. An idealized example for illustration is shown in Fig. 6.39. In this case, at a station (i, j) and at the kth vertical level, there are temperature observations in the month of April over the 10year period, the observed temperature in °C (in the order of increasing temperature and denoted by black numbers) is:

number below the axis); Thus, the corresponding 10-year climatological mean profile in the MTC, m

denoted as hMTC , is depicted by the blue lines and numbers. As an example, the observed temperature for a specific year is 12.3 °C. This observation falls into the MTC bin of 12 0:5  C, and the corresponding observation value for this year in other MTC bins is zero. Therefore, this year’s observation in the MTC is represented by the green number (12.3) and green segmented line in Fig. 6.39, and we denote this observation in the ½11:1; 12:3; 13:1; 13:7; 14:1; 14:4; 14:8; 15:1; 16:1; 1:6:8 MTC as hMTC . The corresponding anomaly is defined by subtracting the climatological profile These temperature observations fall into 7 tem- (blue segmented line) from this observation ¼ perature bins in the MTC with resolution of 1 °C, profile (green segmented line), i.e., hMTC a i.e.: ½11 0:5; 12 0:5; 13 0:5; 14 0:5; 15 hMTC  hMTC m and it is depicted by the red 0:5; 16 0:5; 17 0:7 (denoted by the black segmented line in Fig. 6.39. Note that the

Appendix: Connection Between the MTC and the FTC

371

Fig. 6.39 Sketch of temperature anomaly in the FTC and the MTC

observed signal appears in only one bin in the MTC, so that in the MTC hMTC has one non-zero point only; however, the anomalous signals hMTC a have 7 non-zero points in the MTC. The corresponding signals are positive for the 12 0:5  C bin only, but are negative for all other bins. The net signal for this year in the MTC is the algebraic sum of the red lines, P7 MTC ðiÞ ¼ 1:85  C. On the other hand, in i¼1 ha the FTC (the upper panel in Fig. 6.39), the climatological mean temperature is h ¼ 14:15  C, there is one non-zero data point h ¼ 12:3  C only. As a result, h  h ¼ 1:85  C. Hence, in each year the net signal in the MTC, summed over the whole temperature range, is the same as the anomaly signal in the FTC, as depicted by the margent bar. For this specific year discussed above, the anomaly signal in the FTC is represented by the second margent bar from the left in the upper part of Fig. 6.39. Therefore, for each month we have the following constraint between variables in these two coordinates

HaMTC ði; j; kÞ ¼

lmt X

hMTC ði; j; k; lÞ ¼ HaFTC ði; j; kÞ a

l¼1

¼ ðh  hÞFTC ð6:20Þ

References Behringer DW, Xue Y (2004) Evaluation of the global ocean data assimilation system at NCEP: The Pacific Ocean. Eighth Symposium on Integrated Observing and Assimilation Systems for Atmosphere, Oceans, and Land Surface, AMS 84th Annual Meeting, Washington State Convention and Trade Center, Seattle, Washington, pp 11–15 Hanawa K, Talley LD (2001) Mode waters. In Siedler G, Church J, Gould J (eds) Ocean circulation and climate. International Geophysical Series, Academy Press, New York, pp 373–386 Hautala SL, Roemmich DH (1998) Subtropical mode water in the northeast Pacific basin. J Geophys Res 103:13055–13066 Liu L-L, Huang RX (2012) The global subduction/obduction rates, their interannual and decadal variability. J Clim 25:1096–1115

7

Climate Signals in the Isohaline Coordinate

Salinity can be used as a Lagrangian coordinate to identify heaving modes of the isohaline surface in the world oceans. Most numerical models currently in use are based on the volume conservation approximation. Although in the GODAS website, the unit of salinity is in psu (practical salinity unit), it is inconceivable how a volume-conserving model can produce salinity in units of psu. Nevertheless, we will use psu as the salinity unit in analyzing the GODAS data in this chapter. Since salinity distribution is not monotonic in the vertical direction, treating the heaving mode of the isohaline surface as a global mode is questionable. Thus, the discussion presented in the first half of this chapter can serve as an illustration of the methodology only. However, for some specific regions in the world oceans, salinity is monotonic in the vertical direction, and it can be used as a vertical coordinate. As an example, in the second part of this chapter we will use salinity as a Lagrangian coordinate to study the shallow salty water sphere in the Atlantic Ocean. The identification of heaving modes of isohaline surfaces in the world oceans remains a great challenge and we hope this method will be modified and applied to the local isohaline heaving modes in the world oceans. In this

chapter we present a concise description of the isohaline coordinate and its application to climate study.

7.1

Introduction

Climate variability in the ocean can be analyzed using computer generated climatological data. One of the main focuses is the salinity anomalies, which are closely linked to the hydrological cycle and the melting of sea ice and land based glaciers. A major difficulty associated with salinity analysis is the inadequate coverage of historical salinity observations. With the development of the ARGO program and the satellite salinity mission, more salinity observations are made and the situation can be much more improved. In a recent study, Durack and Wijffels (2010) made an attempt to identify the heaving modes in salinity data by using a complicated formula including a spatially and temporally parametric model to fit the data. However, their approach lacks solid physical reasoning, and the results obtained may be different from the heaving modes defined here. Using salinity as a vertical coordinate may result in dealing with the vertically non-

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2_7

373

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Climate Signals in the Isohaline Coordinate

Fig. 7.1 Salinity sections in the world oceans, based on the WOA09 data

monotonic salinity distribution in the world oceans. As an example, we show several meridional salinity sections in the world oceans (Fig. 7.1). It is readily seen that there are relatively freshwater tongues associated with the Antarctic Intermediate Water in the Indian, Pacific and Atlantic Oceans. In the Atlantic Ocean, the North Atlantic Deep Water appears in the form of deep salty water tongues. Therefore, there is no one-to-one transformation between the z-coordinate and the isohaline coordinate. There might be some technical steps needed to be designed in the analysis of the salinity anomaly

in terms of heaving or saline diffusion related to freshwater exchange across the sea surface and between water masses in the subsurface.

7.2

Casting Method

Salinity in the world oceans varies over a very broad range. Near the mouth of rivers, it can be nearly zero; in some special regions, it can be more than 40 psu. However, for the climate study, we will be focused on the middle range. As shown in Fig. 7.2, most sea water is within

7.2 Casting Method Volume distribution in S coordinate 17 16 log10(V) (m3/0.1psu)

Fig. 7.2 Climatological mean volume distribution in the isohaline coordinate (based on the GODAS data), the dashed lines indicate the salinity domain S = [32.85– 37.45] (psu) of our focus in this chapter

375

15 14 13 12 11 30

the range of S = 32.85–37.45 psu. Thus, using this salinity range and an interval of 0.1 psu, we can get a better resolution in the isohaline coordinate for most water masses in the world oceans. We will use a four-dimensional coordinates ðx; y; S; tÞ in the following analysis, where S is salinity (in kg/m3), and t is time. Furthermore, we will introduce two isohaline coordinates: the fixed salinity coordinates (FSC) and the moving salinity coordinates (MSC) . Climate variability in these two coordinates is defined as follows.

31

32

33

34

35 S (psu)

36

37

38

39

where Sði; j; k; mÞ and Sði; j; kÞ are the salinity and the corresponding climatologic mean. Data is put into discrete salinity bins in the FSC according to the following rule: DSFSC ði; j; k; mÞ added to SFSC a ði; j; n; mÞ if Sn  0:05\Sði; j; kÞ  Sn þ 0:05; Sn ¼ 32:9; . . .; 37:4 ðpsuÞ

ð7:2Þ

Therefore, the climate variability data series is in the form
SFSC ¼ SFSC a a ði; j; n; mÞ; n ¼ 1; 2; . . .; 45;

7.2.1 FSC

m ¼ 1; 2; . . .; m month; in units of kg

The FSC uses the climatological mean salinity at each spatial grid (i, j, k) as the vertical coordinate. In addition, one may also separate the contribution due to the regular seasonal cycle by using the multiple year mean salinity in each month as the vertical coordinate. At each grid point (i, j, k) in the spherical coordinates, the salt content anomaly in the mth month in the FSC is denoted as, DSFSC ði; j; k; mÞ ¼ ðSði; j; k; mÞ  Sði; j; kÞÞq0 DxðjÞDyDzðkÞ ð7:1Þ

ð7:3Þ where m month is the total number of month in the data record. The physical meaning of this variable is as follows. The salt content anomaly at a spatial grid point (i, j, k) is calculated, and a positive (negative) value indicates that water in this grid point is getting saltier (freshened) relative to the climatological mean (seasonal cycle subtracted) salinity. Note that for each spatial grid (i, j, k) there is only one salinity bin being assigned with a non-zero value. This time series can also be summed up into a time series of the global salt content anomaly in FSC as follows

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7

SFSC a ðn; mÞ

¼

jmt imt X X

Climate Signals in the Isohaline Coordinate

The corresponding anomaly is defined as SFSC a ði; j; n; mÞ

ð7:4Þ

ði; j; n; mÞ ¼ SMSC  SMSC sMSC a

i¼1 j¼1

7.2.2 MSC The MSC uses the instantaneous salinity at each spatial grid (i, j, k) as the vertical coordinate. At each grid point in the spherical coordinates, the salt content is denoted as: sði; j; k; mÞ ¼ Sði; j; k; mÞDxðjÞDyDzðkÞ

ð7:5Þ

where Sði; j; k; mÞ is the salinity at the mth month. The resulting data is stored in discrete salinity bin n in the MSC, and denoted as sMSC ði; j; n; mÞ, using the instantaneous salinity to determine the right bin number n Sn  0:05\Sði; j; k; mÞ  Sn þ 0:05; Sn ¼ 32:9; . . .; 37:4ðpsuÞ

ð7:6Þ

Accordingly, the total salt content time series is in the form
SMSC ¼ sMSC ði; j; n; mÞ; n ¼ 1; . . .; 45; m ¼ 1; . . .; m month; in units of kg

The physical meaning of this variable is slightly different from that defined in Eq. (7.3) for the FSC. In the FSC, the salt content anomaly indicates the salt content change due to the salinity deviation from the climatological mean at the corresponding spatial grids. Because of strong climate variability, the salinity deviation at a fixed spatial grid can be larger than 0.1 psu (the salinity intervals set for the FSC in this chapter). Nevertheless, all signals generated at a fixed spatial grid are put into the same salinity grid in the FSC. In the MSC, the salinity for each bin is fixed (in the current study, it is set to the 0.1 psu intervals); accordingly, the salt content anomaly indicates the salt content change because of the volume increase (decrease) of water mass within this salinity bin. Furthermore, due to climate variability salinity anomaly signals generated in the same spatial grid may be put into different salinity grids in the MSC. For each month, the total salt content anomalies integrated over the world oceans in the MSC are defined as:

ð7:7Þ where m month is the total number of month in the data record. Due to the climate variability, the salinity anomaly signals in a fixed spatial grid (i, j, k) may be put in different salinity bins in the MSC. From the multiyear data, one can define the climatological mean annual cycle of the total salt content for 12 months of an annual cycle h SMSC ¼ sMSC ði; j; n; mmÞ; n ¼ 1; . . .; 45; mm ¼ 1; 2; . . .; 12 ð7:8Þ

ð7:9Þ

SMSC ðn; mÞ ¼ a

jmt X imt X kmt X

sMSC ði; j; n; mÞ a

i¼1 j¼1 k¼1

ð7:10Þ

7.3

Separating the Signals into External and Internal Modes

With these salt anomaly signals diagnosed for the world oceans, our next goal is to identify the mechanisms generating such anomalies. In particular, our interest is to separate contributions

7.3 Separating the Signals into External and Internal Modes

owing to heaving (mostly adiabatic and isohaline motions) and salinification processes.

377

nmt X

SMSC ði; j; n; mÞ ¼ SFSC a a ði; j; n0 ; mÞ

ð7:12Þ

n¼1

7.3.1 FSC As shown in Eq. (7.4), salinity anomaly signals in the FSC can be summarized as a twodimensional data array. By definition, the FSC is defined by the climatological mean salinity (with seasonal cycle subtracted) ; as such, it is not a truly Lagrangian coordinate. For example, if SFSC a ðn; mÞ [ 0 for a give index pair of (n, m), we can only say that in the mth month, salinity at grids with climatological mean salinity satisfying Sði; j; kÞ ¼ Sn increases on average. Such an increase in the instantaneous salinity might be the result of saline exchanges with the neighboring grids or through the air-sea interaction. It can also be due to heaving modes in the world oceans—which pushes salty water from the neighboring grids to the grids of our concern. The clean signals identifiable from the data in the FSC in the global sum Gs SFSC a ðmÞ ¼

45 X

SFSC a ðn; mÞ

There is only one grid n0 in FSC that has a nonzero value. In MSC, signals in each salinity level can be further separated into the barotropic (external) and baroclinic (internal) components SMSC ¼ Ex SMSC ðn; mÞ þ In SMSC ði; j; n; mÞ a a a ð7:13Þ where: ðn; mÞ ¼ SMSC ði; j; n; mÞ Ex SMSC a a

In SMSC ði; j; n; mÞ a

n¼1

7.3.2 MSC The discussion in MSC is much more elaborate; our analysis in this section is parallel to those associated with density and temperature as described in Appendix in Chaps. 5 and 6. At each grid point (i, j, k), the net signal in the MSC is equal to the signal in the FSC, i.e.,

ð7:14Þ

is the external mode, defined as the volumeweighted mean of the salt content anomaly in each salinity bin. This is the total salt content anomaly for each salinity interval, which is directly linked to the saline exchange across the boundary of this isohaline interval. By definition, the baroclinic mode integrated over the entire volume defined by the isohaline surfaces of the Sn bin is zero, i.e.:

ð7:11Þ

This quantity represents the change of the total salt content for the world oceans, and it is solely due to the freshwater flux anomalies associated with air-sea interaction and melting of ice in the ocean or land-based glaciers. In a Boussinesq model, these source of fresh water appear in the form of equivalent salt fluxes.

i;j

i;j;k

¼0

ð7:15Þ

Since its integration over the upper and lower interfaces of this isohaline interval is zero, this variable represents the heaving components of the haline anomaly signals in each isohaline layer.

7.4

Analysis Based on the GODAS Data

Our analysis in this section is based on the GODAS data. First, for each month the integration over the entire salinity range [32.85–37.45 P psu] of the salt content anomaly n SMSC ðn; mÞ a P FSC S ðn; mÞ should be the same as disand n a cussed above; this quantity is the total salt content anomaly for the world oceans, and the time

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Climate Signals in the Isohaline Coordinate

Fig. 7.3 a Total salt anomaly integrated over the whole model domain in both the FSC and the MSC; b the mean salinity anomaly, inferred from the MSC

evolution of this variable is shown in Fig. 7.3. Theoretically, the integrated SC anomaly, in the MSC and the FSC should be the same; there is apparently a minor difference, which might be due to the errors accumulated in the calculations. Based on the variability of the total salt content in the model, one can infer the changes in the mean salinity. As shown in Fig. 7.3b, the mean salinity declined about 0.007 psu over the 35 years of the model run. Assuming the mean salinity in the ocean is about 35 psu, and the mean depth of the world oceans is 3500 m, then a 0.007 psu change in the mean salinity is equivalent to adding on 70 cm of freshwater. The best estimate from observation is approximately on the order of 10 cm. Thus, there is a large mismatch. The reason of this mismatch is not clear at this time. Although the total salt content anomaly in each month is the same for both the MSC and the FSC, the signals in the MSC and the FSC are quite different. First of all, signals in the MSC is 100 times larger than those in FSC; note that the unit in Fig. 7.4a is 1017 kg/0.1psu; while in Fig. 7.4b, it is 1015 kg/0.1psu. In addition, the patterns in the time-salinity space are quite different (Fig. 7.4). In the FSC,

the anomaly pattern often appears as a signal positive or negative band. On the other hand, in the MSC there are multiple bands at the same time; furthermore, the strength of signals in the MSC is 100 times stronger than in FSC, as described above. As we have emphasized above, the meaning of the signals peaks in these two coordinates are quite different. In the FSC, there is a single band of positive signals 1980–1995 centralized around salinity S = 34.6 psu. In particular, the salt anomaly signals are all positive in 1980. This indicates all water masses in the world oceans were saltier during this period of time. After year 1996–1997, the sign of the signals changed. For this time period, the negative signal band slightly moved towards a lower salinity. At the same time, there is a small group of positive signals centered at salinity S = 35.3 psu. In contrast, the salinity content anomaly signals in the MSC are more complicated, characterized by multiple bands of positive/negative alternative patterns (left panels of Fig. 7.4). For the period of 1980–1990, there is a positive signal band centered at S = 34.7 psu, and a secondary positive band centered at S = 34.9 psu. Therefore, there was more water mass with

7.4 Analysis Based on the GODAS Data

379

Fig. 7.4 The total volume integration of the external mode of SC anomaly (seasonal cycle subtracted) in the MSC (left panels) and FSC (right panels). The lower panels show the fine structure pattern for the central range of salinity

salinity in these ranges. In addition, there is a negative signal band centered at S = 34.5 psu. Combining the information from Fig. 7.4b, water mass in the range of S = 34.4–34.9 psu became saltier and moved to the high salinity range. This is manifested as a sink in salinity range of S = 34.4–34.5 psu and source of water mass with high salinity (left panels of Fig. 7.4). A process in the opposite direction took place after year 2000, and the description of the patterns in the left and right panels is quite similar to the discussion above, with the opposite signs, and hence we will not repeat the discussion here.

7.5

Shallow Salty Water Sphere in the Atlantic Ocean

As discussed above, a major technical issue of using salinity as the vertical coordinate is that it is not monotonic; thus, the application of using the salinity as the Lagrangian coordinate must overcome this difficulty. Practically, one must find a region in the ocean where salinity is monotonic in the vertical direction. The salty water sphere in the Atlantic Ocean is a good example. The upper ocean in the Atlantic

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Climate Signals in the Isohaline Coordinate

Fig. 7.5 Annual mean salinity distribution: a at the sea surface; b along 20° W; all based on the GODAS climatology

Fig. 7.6 Total area covered by the individual isohaline layers in the Atlantic Ocean, based on the GODAS data

Ocean is salty, and salinity declines downward from the surface maximum to the depth of S = 35 psu (Fig. 7.5b). In addition, the salty water of salinity higher than S = 35.5 psu is further separated into the Northern branch and the Southern branch, with a separating boundary roughly along 5.5° N (Fig. 7.5a). Accordingly, we separate the salty water sphere in the upper part of the Atlantic Ocean into three domains: (1) North Atlantic upper layer: S  35:5 psu and north of 5.5° N; (2) South

Atlantic upper layer: S  35:5 psu and south of 5.5° N; (3) Atlantic upper layer: 34:5  S  35:5 psu. This layer covers the whole upper ocean of the Atlantic Ocean. The corresponding area occupied by each salinity layer (with salinity interval of 0.1 psu) is shown in Fig. 7.6. Note that for the lower salinity range, the corresponding area declines slightly, and this is may be due to the fact that some of the isohaline surfaces ground, i.e., they intercept the topography, leading to smaller area.

7.5 Shallow Salty Water Sphere …

381

Fig. 7.7 The time evolution of the total salt anomaly integrated over the whole density range

The algorithm discussed above was applied to these three regions in the Atlantic Ocean. Accordingly, the total salinity anomaly signals integrated over the whole density range are shown in Fig. 7.7. As commented above, the total amount of salt in the world oceans is likely to be constant over such a short time period (35 years); however, in Boussinesq models, the mass conservation is replaced by a volume conservation approximation. As such, the surface freshwater anomaly is intepreted as a sink of salinity in the model. Therefore, the decline of total salinity in the Atlantic Ocean obtained from this model should be intepreted as an increase of total mass and volume in the ocean. As shown in Fig. 7.7, the “net salt” in the model ocean was continuously reduced from 1980 to 1997; afterward, the “net salt” in the model ocean went up rapidly from 1997 to 2002. After 2002, the “net salt” in the Atlantic Ocean of the model ocean declines rapidly, and this may indicate that there is a large amount of freshwater entering the model Atlantic Ocean, either from the Arctic Ocean or from precipitation minus evaporation. In the following section we present the salinity anomaly signals. First, we show the external modes of heaving signals in the following domains:

D. Atlantic upper layer: 34:5  S  35:5 psu. This layer covers the whole upper ocean of the Atlantic Ocean (Fig. 7.8a). E. North Atlantic upper layer: S  35:5 psu and north of 5.5° N (Fig. 7.8b). Domain D corresponds to slightly deep isohaline surfaces for the whole North Atlantic Ocean. The external modes of the salinity anomaly have quite large amplitude, on the order of 5  1017 to 3  1017 ðkg=0:1psuÞ. The peaks are centralized at S = 34.9 psu and S = 35.0 psu. This result is converted into the equivalent layer thickness perturbations as discussed above dhMSC Ex ðn; mÞ ¼

jmt imt X X i¼1

WaMSC ði; j; n; mÞ

j

=salðnÞ=areaðnÞ ð7:16Þ As shown in Fig. 7.9, the external modes of layer thickness perturbations are quite large, on the order of 200 m, especially for the lower salinity range in this figure. In fact, for the isohaline layer of S = 34.9 psu the layer thickness anomaly (red curve) was 100 m in 1980, and went down to −100 m in 1996; eventually it overshot to 190 m at the end of the record. The trend of layer

382

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Climate Signals in the Isohaline Coordinate

Fig. 7.8 The total volume integration of the external mode of the SC anomaly, in units of salt anomaly (seasonal cycle subtracted): a Domain D; b Domain E

Fig. 7.9 The equivalent layer thickness perturbations of the external mode of the SC anomaly in Domain D, in units of m (seasonal cycle subtracted): a distribution in S-time space; b the distribution along four isohaline surfaces

thickness anomaly on S = 35.0 psu (black curve) behaves in a completely opposite way (Fig. 7.9). Such large layer thickness perturbations indicate that water mass volume distribution (for this density range) in the Atlantic Ocean went through great changes over the past 35 years, and

such changes imply climate variability (freshwater anomaly) with period of 30 plus years. On the other hand, the external modes of heaving signals for the density range of S > 35.2 psu have a rather low amplitude. In addition, there seems to be no decadal modulation; instead, there may

7.5 Shallow Salty Water Sphere …

383

Fig. 7.10 The equivalent layer thickness perturbations of the external mode of the salt content anomaly in Domain E, in units of m for a layer with thickness of 0.1 psu (seasonal cycle subtracted) : a the distribution in S-time space; b the distribution along four isohaline surfaces

be some high frequency oscillations for this salinity range. On the other hand, the external modes of layer thickness anomaly for the upper part of the salty water sphere are relatively small, on the order of 20 m (Fig. 7.10). In contrast with the strong decadal signals found for the lower salinity range, or relatively deep isohaline surfaces, the external modes of salinity signals are characterized by strong quasi four year signals, as shown by the red curve in Fig. 7.10b. Such high frequency oscillations imply that the freshwater anomaly in the subtropical North Atlantic Ocean may be closely linked to the quasi four-year cycle of ENSO events. There are many publications devoted to this potential connection, but the details of such connection are beyond the scope of this book. Similar to the discussion in the previous chapter, we could present the internal heaving signals for the salinity anomaly in the zonal or meridional sections; we can also present these signals in individual isohaline layers. In the following section, we choose to present the

meridional migration of the internal heaving signals. In fact, the zonally integration of the internal heaving signals can shoot light on the meridional transport of salinity anomaly in the ocean, and these signals are defined as follows dhMSC Ex ðj; n; mÞ ¼

imt X

WaMSC ði; j; n; mÞ=salðnÞ

i¼1

ð7:17Þ The corresponding anomaly is in units of volume per each 0.333° in the meridional direction and for each 0.1 psu isohaline layer. There are clear signs of signals moving from 30° N toward 20° N (Figs. 7.11 and 7.12). In fact, the equivalent meridional speed of migration is reduced from the layer of S ¼ 37  0:05 psu to the layer of S ¼ 36:8  0:05 psu; this may reflect that mean currents in the layer of S ¼ 37  0:05 psu are faster than that in the layer of S ¼ 36:8  0:05 psu. As discussed above, the internal heaving signals on these shallow layers contain strong high frequency signals; the period of these signals are on the order of approximately four years.

384

7

Climate Signals in the Isohaline Coordinate

V (1012m3/0.333°), S=37.0 (psu) 1.0

30N

0.5

0 20N −0.5

−1.0

10N 1980

1985

1990

1995

2000

2005

2010

Fig. 7.11 Time evolution of the zonally integrated volumetric anomaly in the isohaline layer of S ¼ 37  0:05 psu

Fig. 7.12 Time evolution of the zonally integrated volumetric anomaly in the isohaline layer of S ¼ 36:8  0:05 psu

In deeper isohaline layers the meridional migration character of the internal heaving signals is quite different from that in the shallow layers. As shown in Figs. 7.13 and 7.14, there are strong positive signals initiated in 1980 and propagated southward into both the subpolar and subtropical gyre interior. As shown in Fig. 7.13, the southward migration of the internal heaving signals

dominates the landscape. The meridional migration speed is somewhat inversely proportional to the distance from 60° N. There are alternative positive and negative signal bands (Fig. 7.13). In the even deeper isohaline layers the meridional migration character of the internal heaving signals is completely dominated by the signals from 60° N (Fig. 7.14). These signals

7.5 Shallow Salty Water Sphere …

385

Fig. 7.13 Time evolution of the zonally integrated volumetric anomaly in the isohaline layer of S ¼ 35:2  0:05 psu

Fig. 7.14 Time evolution of the zonally integrated volumetric anomaly in the isohaline layer of S ¼ 35:1  0:05 psu

reached 40° N in 1995, implying a movement along the sinking branch of the meridional overturning circulation in the North Atlantic. The positive volumetric signals in 1980 may represent the strong freshwater event induced by

excessive freshwater input into the Labrador Sea around 1980, the so-called great salinity event; the relevant information has been discussed in may published papers, e.g., Dickson et al. (1988) and Hakkinen (2002).

386

References Dickson RR, Meincke J, Malmberg S-A, Lee AJ (1988) The “great salinity anomaly” in the Northern North Atlantic 1968–1982. Prog Oceanogr 20:103–151 Durack PJ, Wijffels SE (2010) Fifty-year trends in global ocean salinities and their relationship to broad-scale

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warming. J Clim 23:4342–4362. doi: http://dx.doi.org/ 10.1175/2010JCLI3377.1 Hakkinen S (2002) Freshening of the Labrador Sea surface waters in the 1990s: another great salinity anomaly? Geophs Res Lett 29(24):2232. https://doi. org/10.1029/2002GL015243

Index

A Absolute salinity, 83 Adiabatic motions, 5, 11, 20, 39, 62, 64–68, 162, 180, 181, 194, 230, 241, 246, 251, 264, 268, 364 Adiabatic signals, 5, 39 Adiabatic signals in the upper ocean, 39 Adjustment, 7, 11, 39–45, 62, 67–71, 121, 162, 163, 167–171, 176, 177, 179–183, 195, 197, 198, 200, 202, 207, 211, 212, 216–218, 220, 222, 225–227, 230, 233–235, 239, 247, 290, 325, 335, 362, 364 Adjustment of the circulation, 44, 45, 70, 235 Adjustment of the main thermocline, 44, 45, 177, 325 Adjustment of the thermocline, 40, 198, 202, 207 Adjustment of the wind-driven circulation, 40, 67, 163, 167, 169, 171, 180, 181, 197 Adjustment of the wind-driven gyre, 11, 198, 220 Agulhas current, 50 Air-sea buoyancy flux, 216 Air-sea freshwater flux, 196, 197 Air-sea heat flux, 19, 161, 162, 216, 241, 242, 245–252, 257, 288, 315, 340 Air-sea interaction, 1, 18, 63, 93, 119, 121, 130, 149, 216, 249, 254, 268, 285, 287, 290, 295, 298, 303, 306, 309, 337, 349–351, 354, 377 Air-sea interface, 1, 2, 14, 18, 53, 57, 61, 119, 127, 129, 134, 161, 195, 268, 272, 277, 305, 338, 364 AMOC, 44, 49–53, 128, 189, 194, 195, 228, 229 Antarctic, 5, 14, 33, 46, 53, 86, 122, 123, 175, 189, 241, 245, 248, 285, 341, 374 Antarctic Bottom Water (AABW), 86, 189 Antarctic Circumpolar Current (ACC), 4, 5, 14, 15, 49–51, 53–56, 71, 175–181, 183–185, 191, 193, 285, 287, 289, 296, 297, 308, 341 Antarctic Intermediate Water, 374 Antonov, J. I., 79, 166, 265 Arctic Ocean, 65, 71, 80, 86, 103–107, 131, 381 Atlantic basin, 3, 11, 12, 15, 53, 70–72, 177–179, 182–184, 186, 187, 189, 191, 196, 251–253, 289, 300, 309, 312, 313, 348–352 Atlantic Ocean, 3, 5, 33, 38, 49–53, 69, 70, 73, 86, 89–92, 98, 100, 101, 104, 113, 114, 119, 121–123, 128, 134, 161, 169, 189, 195, 218, 228, 230, 235, 257, 264, 265, 284, 285, 287–289, 291, 292, 300, 305, 309, 310, 315–317, 341, 348–350, 373, 374, 379–383

Atmospheric bridge, 53 Available potential energy, 109, 110, 152 B Balmaseda, M. A., 40, 163 Barker, P. M., 83 Baroclinic signals, 18, 21, 24, 25, 27, 28, 57, 243 Baroclinic Rossby waves, 66, 69, 171, 217 Barotropic signals, 18, 21, 24, 27, 28 Behringer, D. W., 11, 17, 71, 161, 268, 338 Bindoff, N. L., 62 Brazil–Malvinas Confluence Zone, 285–287 Bulk density ratio, 94, 96, 101, 107 Buoyancy flux, 129, 216 C Cabbeling, 63, 93 Cape of Agulhas, 69, 70, 313 Carton, J. A., 27, 178 Casting method, 265, 266, 272, 274, 275, 313, 335, 338, 374 Cessi, P., 171, 182 Chen, X. Y., 163, 170, 171, 196 Central water, 97, 98 Coastal Kelvin waves, 44, 45, 69, 217, 234 Cold tongue, 63, 161, 241, 300–303, 317, 322, 324–328, 349 Conservative temperature, 83 Cooling of the deep ocean, 20–22, 56 Cooling Rossby waves, 217 Coordinate Eulerian coordinate, 62, 138, 141, 147, 151, 153, 265, 335 Eulerian-Lagrangian coordinate, 273 fixed density coordinate (FDC), 265–270, 275, 276, 331, 332, 340 fixed salinity coordinate (FSC), 375–379 fixed temperature coordinate (FTC), 336–342, 344–346, 369–371 isohaline coordinate, 373–375 isopycnal coordinate, 13, 63, 68, 139, 140, 263, 265–267, 269, 270, 272, 274–276, 285, 320, 324

© Higher Education Press and Springer Nature Singapore Pte Ltd. 2020 R. X. Huang, Heaving, Stretching and Spicing Modes, https://doi.org/10.1007/978-981-15-2941-2

387

388 isothermal coordinate, 335, 336, 338, 341, 358, 369, 370 Lagrangian coordinate, 138–141, 147, 152–154, 159, 263, 265, 268, 270, 272, 273, 285, 313, 335, 337–340, 349, 359, 373, 377, 379 moving density coordinate (MDC), 265–270, 274–276, 313, 319, 331, 332, 340 moving salinity coordinate (MSC), 375–379 moving temperature coordinate (MTC), 336–346, 360, 369–371 orthogonal coordinate, 73, 74, 83, 84, 110, 111 spherical coordinates, 15, 17, 28, 57, 62, 137, 165, 178, 190, 193, 265–267, 336, 375, 376 z-coordinate, 17, 18, 28, 29, 58, 62, 68, 137, 335, 338, 345, 359, 374 CTD, 106, 109, 138 D Density ratio, 94–98, 101, 107 Density flux, 17 Diapycnal mixing, 63, 65, 66, 114, 267, 328 Dickson, R. B., 65, 385 Diffusive layering, 96, 102, 103, 106, 107 Distance, 70, 73, 82, 83, 110–118, 127, 129, 138, 225, 226, 234, 237, 384 Distance between two sigma-pi curves, 112 Distance between two water parcels, 112 Double diffusion, 94, 96, 98, 106 Double diffusion diagram, 94 Durack, P.J., 196, 199, 373 E England, M., 163, 257 ENSO, 31, 32, 53, 93, 112, 118–121, 125, 139, 140, 143, 144, 148, 241, 242, 246–248, 251–253, 255–257, 259, 261, 301, 303, 315, 318, 352, 358, 364–370, 383 Equatorial dynamics, 144, 149, 312, 355 Equatorial easterly, 41–44, 163, 167, 168, 179, 186, 210, 212, 241, 243, 246–248, 251, 259, 362 Equatorial Kelvin waves, 44, 45, 69, 70, 140, 217, 234, 255, 323 Equatorial Pacific, 49, 50, 63, 70, 119, 120, 122, 139, 161, 171, 184–186, 190, 241, 242, 249, 303, 320, 322, 326, 330, 331, 349, 359–362, 364 Equatorial thermocline, 3, 163, 168, 170, 255 Eastern subtropical gyre, 348 Eastern subtropical mode water, 284, 348 F Flament, P., 73, 74, 103 Freshwater flux, 1, 53, 57, 61, 62, 65, 68, 127, 134, 162, 195, 196, 209, 267, 374, 377

Index G Giese, B. S., 27, 178 GODAS data, 11, 12, 15–19, 22, 23, 25–29, 43, 46–49, 53, 54, 56, 71, 72, 93, 94, 112, 113, 117, 121, 131, 132, 139, 146, 152, 154, 157, 158, 161, 162, 167, 171, 172, 177, 193–196, 228, 230, 235, 241, 242, 251, 254, 257, 268, 270, 271, 275, 276, 313, 318, 338, 340, 341, 343, 345, 360, 373, 375, 377, 380 Goldsbrough, G., 134, 196 Gulf Stream, 4, 13, 46, 128, 161, 285–288, 291, 308, 309 H Hakkinen, S., 385 Hallberg, R., 176 Hanawa, K., 348 Heat flux, 1, 2, 14, 17, 18, 21–23, 62, 67, 68, 127, 161, 162, 173, 195, 198, 202, 214, 216, 241, 245–261, 288, 335, 340 Heave, 1, 62 Heaving induced by anomalous freshwater forcing, 195 Heaving induced by anomalous wind, freshening and warming, 209 Heaving induced by convection, 216 Heaving ratio, 145, 146, 151, 152, 158, 285, 287, 295, 298, 301, 303, 305, 308–310, 315, 316, 326, 327, 329, 350, 353, 354 Heaving signals, 2, 20, 53, 62–65, 67, 70, 147, 152, 159, 263, 267–269, 291, 293, 295–298, 300, 303, 305, 306, 308–310, 312, 315, 317, 319, 320, 326, 328, 338, 339, 343–348, 350–352, 355–358, 381–384 Helland-Hansen, B., 263 Hiatus in global SST record, 40, 163 Huang, R. X., 6, 7, 12, 13, 20, 53, 62, 65, 66, 69, 73–75, 110, 113, 161, 163, 164, 166–168, 170, 171, 177, 178, 180–184, 190, 193, 197, 198, 209, 210, 257, 264, 341, 348 I Ice-Tethered Profiler (ITP), 106 Indian basin, 5, 12, 69–71, 179, 180, 182, 184–189, 191, 252, 253, 352, 366 Indian Ocean, 3, 5, 33, 38, 49, 50, 53, 69, 86, 118, 119, 131, 132, 134, 162, 179, 183, 186, 190, 193, 194, 252, 254, 256, 257, 284, 285, 287, 289–291, 296, 311, 312, 316, 318 Indonesian Throughflow (ITF), 3, 50, 53, 71, 190, 193, 194, 246, 303 Interannual variability, 31, 119, 121–124, 126, 245, 276–281, 283, 289–293, 295, 297, 300, 305, 351, 354–356 Inter-decadal variability, 340 Internal waves, 106, 108, 109 Isopycnal analysis in the Eulerian coordinates, 147 Isopycnal analysis in the Lagrangian coordinates, 139

Index Isopycnal layer analysis, 58, 68, 137, 147, 274, 276, 308, 320, 347 Isopycnal mixing, 62, 63, 65, 67, 96, 137 Isothermal analysis, 17, 151–155, 158, 350, 351, 353, 354 Isohaline surface, 373, 377, 380–383 J Jackett, D. R., 73, 264 Jin, F. F., 242 Johnson, H. L., 40, 171 K Kosaka, Y., 163, 257 Kuroshio Current, 13, 128 Krzysik, O. A., 73, 74, 103 L Labrador Sea, 161, 216, 235, 385 Layer outcropping from a simple model, 13 Layer thickness, 6–9, 11, 13, 15, 16, 41, 65–67, 93, 94, 140, 151, 154, 155, 163, 165, 170, 171, 185, 193, 198, 200, 226, 233–236, 270–274, 276–278, 280, 281, 283, 284, 288–308, 310, 311, 315–318, 320–326, 381–383 Least square problem, 74, 76–79 Leeuwin Current, 5, 6 Levitus, S., 163 Liang, X-F., 21, 253 Liu, L-L., 341, 348 Lozier, M. S., 40, 163, 169 Luyten, J. R., 210 Lyman, J. M., 170 M Mamayev, O. I., 73, 74, 96 Marshall, D. P., 40, 171 McCreary, J. P., 12 McDougall, T. J., 62, 73, 74, 83, 103, 264, 273 Mechanical interaction between atmosphere and oceans, 1 Mediterranean Sea, 86, 89, 131 Meehl, G. A., 40, 163 Meridional Heat Flux (MHF), 2, 14, 43–46, 52, 161, 162, 164, 166, 168, 169, 172–177, 180, 198, 183, 185–190, 192, 195, 202–204, 207–209, 213–215, 230, 239–242, 244–248, 249–251, 257, 259 Meridional Overturning Circulation (MOC), 2, 3, 40, 42–46, 48–52, 128, 161, 163, 164, 168, 169, 172–177, 179, 180, 183–185, 187–192, 194, 195, 197, 202–204, 207–209, 213–215, 218–220, 222–227, 230, 231, 234, 236, 237, 239, 240, 257, 259, 385 Metric space, 110

389 Mode asymmetric heaving mode, 164 baroclinic mode, 11, 18–29, 41–43, 55, 57, 66, 68, 164, 169, 170, 173, 174, 180, 181, 184, 198, 201, 202, 204, 208, 209, 211, 212, 216, 230, 239, 243, 244, 257, 259, 377 barotropic mode, 18–29, 42, 43, 68, 170, 181, 243, 244, 257 ENSO events and heaving modes, 241 external heaving mode, 64, 65, 68, 73, 335, 337, 341, 343, 351 external mode, 17, 18, 29–37, 39, 40, 65, 67, 73, 246, 248–250, 267–272, 275–277, 280–282, 290–292, 295, 298, 308, 313, 314, 317–319, 321, 338, 339, 342, 344, 355, 377, 379, 381–383 first baroclinic mode, 11, 66, 164, 173, 180, 181, 184, 202, 204 global mode, 72, 73, 257, 347, 373 heaving mode, 1, 14, 20, 24, 43, 62–68, 70–73, 119, 162, 164, 175, 176, 179, 190, 241, 242, 259, 265, 280, 313, 335, 337, 341, 343, 347, 351, 358, 373, 377 inter-basin mode, 53, 175, 176, 178, 179, 183 internal heaving mode, 64–68, 73, 335, 337 internal mode, 17, 18, 28–40, 57, 65, 66, 68, 73, 246, 248–250, 267, 268, 290–292, 295, 298, 308, 320, 323, 335, 337, 338, 344, 345, 376 isohaline heaving mode, 373 local mode, 72, 73 meridional modes of density content anomaly, 36, 38 meridional modes of heat content anomaly, 30 meridional modes of salt content anomaly, 33, 35, 383 second baroclinic mode, 181, 184, 212 symmetric heaving mode, 164 third baroclinic mode, 21, 181 vertical mode, 35 zonal modes of heat content anomaly, 32, 34 zonal modes of salt content anomaly, 33 Model ocean reduced gravity model for the world oceans, 14 southern hemisphere model ocean, 175 two-hemisphere model ocean, 14, 165, 200, 218 two-hemisphere model ocean simulating ENSO, 257 world oceans with rectangular basins, 184 Mode water formation, 121–124, 287, 291, 295, 296, 298, 300, 306, 308, 309, 348, 350, 351 Montgomery, R. B., 263, 272–274 Munk, W., 73, 108 N North Atlantic Current, 305 North Atlantic Deep Water (NADW), 87, 189, 341, 374 North Atlantic Ocean, 5, 49, 51, 69, 89–92, 100, 101, 104, 114, 121, 123, 195, 218, 228, 230, 235, 284,

390 285, 287–289, 291, 309, 310, 315, 316, 348–350, 381, 383 North Atlantic Oscillation (NAO), 40, 163, 169 O Obduction, 64, 65, 341, 371 Oceanic bridge, 53 P Pacific basin, 3, 11, 14, 33, 70, 71, 177–187, 189, 191, 241–243, 245, 248, 252–254, 302, 326, 327, 349, 352, 354–356, 358, 366–368, 371 Pacific Ocean, 5, 33, 38, 46, 51–53, 59, 63, 86, 114–116, 118–122, 125, 131, 132, 134, 139, 167, 171, 185, 186, 190, 193, 195, 241, 246, 252, 254–258, 284, 285, 287, 302, 318, 322, 326, 330, 331, 348, 357–362, 365–371 Parsons, A. T., 13 Pedlosky, J., 66, 210 PMOC, 51, 53 Potential density, 5, 6, 11, 12, 55, 62, 63, 64, 67, 71–77, 79, 81–85, 88, 89, 97, 100–105, 109, 110, 112, 113, 118–120, 126, 130, 132, 133, 137–142, 144, 145, 147–152, 155, 157, 158, 166, 178, 263–266, 272, 273, 285, 286, 292–294, 297–299, 302–304, 306, 307, 309, 314, 318, 324, 326, 327, 329, 345, 346, 348, 349, 353, 354, 355, 359, 361 Potential spicity, 63, 73–85, 88, 89, 91, 92, 94, 97, 100–102, 104, 105, 110–112, 118, 126, 130, 132, 133, 137, 139–141, 145, 147, 149, 150, 152, 154, 157, 158, 273, 278, 280, 285, 286, 289–291, 293–295, 297, 298, 299, 302–304, 306, 307, 313, 314, 322, 324, 329, 353, 354 Potential spicity functions based on the UNESCO EOS-80, 79 Potential spicity functions based 0n the UNESCO TEOS_10, 83 Potential spicity in the least square sense, 76 Potential temperature, 18, 53, 55, 62, 64, 67, 71, 73, 87, 88, 91, 92, 101, 102, 105, 112, 114, 118, 124, 130, 131, 136–141, 147, 149, 152–156, 166, 170, 178, 243, 245, 263, 264, 272, 273, 279, 335, 336, 345–347, 359 Projecting method, 272, 274–276, 320, 345, 347 Pycnocline, 2, 24, 65, 300, 301, 322 Q Qiu, B., 65, 171 R Radius of climate, 117–121, 123, 124–126 Radius of seasonal cycle, 117–121, 123–126 Radius of signal, 83, 112, 113, 118, 124, 125, 137, 140, 143, 144, 145, 149–151, 154, 158, 285, 286,

Index 293–295, 298, 299, 303, 304, 306, 307, 309, 314, 326, 327, 329, 349, 350, 351, 353, 354 Radius of state, 126, 127, 129, 130, 131, 134, 136 Rapid line, 228 Reduced gravity model formation of a reduced gravity model, 6, 11, 13, 14, 162, 165, 173, 177, 178 generalized reduced gravity model, 12, 198–200, 209, 210, 217, 230 reduced gravity, 4–7, 9, 11–14, 39–41, 70, 162, 163, 165, 168, 171, 173, 176–179, 184, 189–191, 193–202, 204–214, 216–227, 230–235, 241, 242, 257, 267 reduced gravity in the world oceans, 11 Ripa, P., 198 Roles of wind stress, 2, 209, 267 Rossby waves in ocean interior, 69, 70, 226, 235 Rossby wave speed, 227, 237 Ruddick, B., 94, 95 S Salt finger, 96, 99, 101, 102, 104 Schmitt, R. W., 96–98, 101, 102 Schmitz Jr, W. J., 161 Sea level anomaly, 170–172 Seasonal cycle, 21, 118–126, 228, 252–256, 258, 266, 268, 269, 272, 275, 276, 281, 288, 289, 319, 331, 336, 339, 352, 369, 375, 377, 379, 382, 383 Sea Surface Temperature (SST), 40, 163, 256 Shibley, N., 107 Sigma-pi (r–p) curve, 96, 101, 108, 112–115 Sigma-pi (r–p) diagram, 83, 84, 86, 94–99, 101, 104, 106–109, 111, 112, 114, 115, 137, 140, 143, 150, 152, 155 South China Sea, 113–117 Southern Ocean, 30, 33, 36, 53, 57, 175, 177–179, 181, 253, 380 Spiciling, 63, 93, 97 Squared layer depth in the ocean interior, 7 Stommel, H., 62, 73, 196, 210 Subduction, 64, 65, 267, 341, 371 Subpolar basin, 14, 44, 164, 166, 167, 193, 198, 200, 204, 217–219, 226, 227–230, 233, 235, 237, 239 Subpolar gyre, 15, 163, 185, 204, 217–220, 228, 233, 287 Subtropical basin, 4, 8, 14–16, 44, 72, 164, 166, 193, 197, 200, 202, 218, 239 Subtropical gyre, 3, 14, 42, 46, 101, 120, 163, 167, 178, 179, 185, 193, 197, 207, 218, 220, 283, 290, 347, 348, 384 Subtropical mode water, 348, 383 Surface density anomaly, 43, 46–48, 51 Surface warming, 43 T Tai, W., 173, 209 Talley, L. D., 161, 180, 348

Index Tasmania, 70 TEOS_10, 61, 83 Thermocline depth of the main thermocline in the world oceans, 193 depth of the thermocline from a simple model, 2 equatorial thermocline, 3, 163, 168, 170, 255 main thermocline, 1–8, 11, 20, 24, 41, 42, 44, 45, 66, 112, 119, 121, 162, 177, 193, 197–201, 209, 218, 219, 255, 257, 323, 325, 326 searching for the main thermocline, 2 two approaches of calculating the layer depth, 7 Thermohaline interleaving, 103, 106 Thermohaline intrusion, 96, 102, 103, 106, 108 (H–SA) plane, 83, 84 (h–S) curve, 114, 115 (h–S) diagram, 74, 76, 79, 81, 84, 86, 96–99, 101, 103–106, 108, 111, 112, 115, 134, 135, 149, 150, 152, 155 (h–S) plane (space), 73, 74, 80–82, 84, 108, 111, 112, 137 Time evolution mean potential density, 132 mean potential spicity, 112 mean potential temperature, 131, 139, 154, 243, 278 mean salinity, 131, 279, 376, 377, 378, 380 radius of state, 136 Turner angle, 94, 96 U UNESCO, 79, 83 V Variance of potential density, 133 Variance of potential spicity, 91, 133 Variance of potential temperature, 91 Variance of salinity, 132 Veronis, G., 73, 74

391 Vertical Heat Flux (VHF), 2, 21–23, 42, 43, 161, 168–170, 172–175, 177, 180, 183, 184, 198, 202, 241, 251–256, 259, 260 Virtual Potential Energy (VPE), 108, 109, 152 W Wang, G. H., 116 Warm pool, 14, 71, 185, 242, 301, 322 Water mass analysis, 62, 73, 83–85, 94, 97, 111, 115, 263 Water mass erosion, 341 Water mass transformation, 64, 65, 341, 342 Western boundary current, 48, 115, 171, 196 Wijffels, S. E., 196, 199, 373 WOA09 data, 79, 80, 83, 98–100, 103, 105, 114, 130, 166, 178, 264, 265, 374 X Xie, S. P., 163, 257 Xue, Y., 11, 17, 71, 268, 338 Y Yang, J. Y., 190 You, Y-Z., 94–96 Yu, Z., 12, 252 Z Zalesak, S. T., 193 Zhao, J., 190 Zonal Heat Flux (ZHF), 2, 42, 44, 161, 162, 164, 166, 173, 174, 190, 204, 206, 209, 214, 215, 230, 240–242, 244, 248–251, 259 Zonal Overturning Circulation (ZOC), 2, 41, 44, 161, 164, 173–174, 190, 204–206, 209, 213–215, 218, 220, 222, 223, 230, 233, 234, 239–241, 259, 260 Zonal wind stress variability, 167