Freezing of Lakes and the Evolution of Their Ice Cover [2 ed.] 3031256042, 9783031256042

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Matti Leppäranta

FREEZING OF LAKES AND THE EVOLUTION OF THEIR ICE COVER SECOND EDITION

Springer Praxis Books

This book series presents the whole spectrum of Earth Sciences, Astronautics and Space Exploration. Practitioners will find exact science and complex engineering solutions explained scientifically correct but easy to understand.Various subseries help to differentiate between the scientific areas of Springer Praxis books and to make selected professional information accessible for you.

Matti Leppäranta

Freezing of Lakes and the Evolution of Their Ice Cover Second Edition

Matti Leppäranta Institute for Atmospheric and Earth System Research (INAR) University of Helsinki Helsinki, Finland

Springer Praxis Books ISBN 978-3-031-25604-2 ISBN 978-3-031-25605-9 (eBook) https://doi.org/10.1007/978-3-031-25605-9 1st edition: © Springer-Verlag Berlin Heidelberg 2015 2nd edition: © Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the Second Edition

This second edition has been prepared eight years after the first one. There has been much progress in lake ice research during this period, and, consequently, a revision is well motivated. This edition contains new findings especially for lowlatitude lakes in Central Asian arid climate, physical limnology below ice, remote sensing, and lake ice climatology including climate change impact on lake ice. In addition, based on comments from several readers and students of the first edition, modifications, restructuring, additions, and technical improvements have been made in most sections. The recent progress in the research of lake ice has been largely connected to ice– water interaction, lake ice climatology, and the role of ice cover in lake ecology. Remote sensing has also much progressed together with increased availability of high-resolution images. The cryosphere has become more important in the Earth system science, and lake ice forms one specific part of the cryosphere with major local environmental and practical impacts. The author is grateful to his collaborators in the research of lake ice science. The author is especially thankful to the participants of Winter Limnology field courses in Lammi Biological Station, Finland, and Dalian University of Technology, as well as lecture courses in Saint Petersburg State University and Institute of Northern Water Problems in Petrozavodsk. Fruitful contacts have continued with the majority of the colleagues listed in the preface of the first edition. Others to thank are professors Yila Bai, Lars Bengtsson, Irina Fedorova, Ivan Mammarella, Yubao Qiu, Xiaohong Shi, Jia Wang, Lijuan Wen, and Johny Wüest, and Drs. Xiao Deng, Wenfeng Huang, Johanna Korhonen, Peng Lu, and Elena Shornikova. The list of my students in lake ice has increased by Joonatan Ala-könni, Xiaowei Cao, Salla Kuittinen, Marjan Marbouti, Shuang Song, Dongshen Su, Fang Yang, and Yaodan Zhang. The author also wants to thank the Springer team for good collaboration. Support to continue lake ice research has been available from the Academy of Finland, Chinese Academy of Sciences, and European Union 7th Framework Programme. Lahti, Finland February 2022

Emeritus Professor Matti Leppäranta

v

Preface to the First Edition

Ice cover is an essential element in cold climate lakes. To a large degree, it isolates the water body from the atmosphere and sunlight and slows down physical and biological processes. In the boreal zone, tundra, and mountain regions, the ice cover is seasonal, while in very high altitudes and high polar latitudes, perennially ice-covered lakes are found. Ice formation results in different ice types and stratigraphy, which further influence on the ice properties. Ice sheets of freshwater lakes are poor in impurities. However, they may possess liquid layers or inclusions, which serve as habitats of biota. Freezing of lakes also brings both practical advantages and problems to human living conditions in cold regions. It is of great interest to know the physics and ecology of lake ice, to monitor and predict ice conditions in lakes, and to evaluate what can be the impact of climate changes to lake ice seasons. The present book, Freezing of Lakes and the Evolution of Their Ice Cover, provides the status of knowledge in the physics of lake ice and the interactions between the ice cover and the liquid water body underneath. Historical developments in lake ice research are also discussed. Chapter 1 gives a brief overview and presents the research fields. The second chapter contains the classification of ice-covered lakes, and in Chapter 3, the structure and properties of lake ice are presented. Ice growth and melting are treated in Chapter 4, while the following chapter focuses on ice mechanics. Chapter 6 goes into the more exotic environment of pro-glacial lakes. The last three chapters consider important lake topics related to the presence of ice in lakes. Chapter 7 contains the physics of the water body beneath lake ice. Chapter 8 discusses the winter ecology of freezing lakes and the lake ice interface toward the society including the impact of climate change on lake ice seasons. The book ends into a brief closing chapter and list of references. Examples of research problems for student learning are listed throughout the book. The underlying idea behind the book has been to include the whole story of lake ice into a single volume. There is a crying need for such synthesis, as winter limnology research and applications are increasing and, apart from review papers, no comprehensive monograph exists on this topic in English. The author has contributed to lake ice research since the 1980s. In particular, his topics of interest have been lake ice structure and thermodynamics, light transfer in ice and snow, ice mechanics vii

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Preface to the First Edition

in large lakes, and lake ice climatology. Mathematical modeling of ice growth, drift, and decay are covered in this research. This book has grown from the author’s research, collaborative visits to several universities and research institutions, and intensive international courses—winter schools—in lake ice and winter limnology. These visits concern in particular the Estonian Marine Institute of Tartu University (Tallinn, Estonia), Institute of Limnology of the Russian Academy of Sciences (Sankt Petersburg, Russia), Institute of Low Temperature Science of Hokkaido University (Sapporo, Japan), Leibniz-Institute of Freshwater Ecology and Inland Fisheries (Berlin, Germany), Marine Systems Institute of Tallinn University of Technology (Tallinn, Estonia), Northern Water Problems Institute of the Russian Academy of Sciences (Petrozavodsk, Russia), and the State Key Laboratory of Coastal and Offshore Engineering of Dalian University of Technology (Dalian, China). The winter schools have been arranged in Lammi Biological Station and Kilpisjärvi Biological Station of the University of Helsinki, in the Saroma-ko field site of the Hokkaido University, and in the Dalian University of Technology. Very recently, the author joined the Global Ice Modeling Project of the Global Lake Ecological Observatory Network (GLEON). In the progress of his research, the author has learned about lake ice from a large number of colleagues. Especially, he wants to thank Prof. Erkki Palosuo as well as Professors Lauri Arvola, Nikolai N. Filatov, Robert V. Goldstein, Hardy B. Granberg, Timo Huttula, Toshiyuki Kawamura, John E. Lewis, Zhijun Li, Anu Reinart, Kalevi Salonen, Matti Tikkanen, Kunio Shirasawa and Juhani Virta, and Drs. Helgi Arst, Cheng Bin, Christof Engelhardt, Ants Erm, Glen George, Sergey Golosov, Sergey Karetnikov, Georgiy Kirillin, Esko Kuusisto, Nikolai M. Osipenko, William Rizk, and Arkady Terzhevik. He is also deeply thankful to his present and former students, in particular Ms. Elina Jaatinen, Mr. Juho Jakkila, Dr. Onni Järvinen, Ms. Anni Jokiniemi, Mr. Tom Kokkonen, Dr. Ruibo Lei, Ms. Elisa Lindgren, Ms. Shi Liqiong, Ms. Ioanna Merkouriadi, Dr. Jari Uusikivi, Dr. Caixing Wang, Dr. Keguang Wang, and Dr. Yu Yang. The scientists and technicians, who have participated in lake ice research programs from this institute and from several other organizations, are gratefully acknowledged, as well as students of the summer and winter schools. Ms. Elisa Lindgren is thanked for a very careful review of the manuscript, and Dr. Salla Jokela is thanked for several graphic products for this book. The home institute of the author since 1992, the Department of Geophysics (fused to Department of Physics in 2001) of the University of Helsinki, is deeply thanked for good working conditions and support. This work is a contribution of the Nordic Center of Excellence CRAICC (Cryosphere–atmosphere interactions in a changing Arctic climate), and the results are based on several research projects, primarily Ficca and Finnarp programs of the Academy of Finland and Clime project of the EU V Framework. The preparation of this book was made possible by financial support for a sabbatical year from the Finnish Cultural Foundation, which is gratefully acknowledged. Helsinki, Finland August 2014

Matti Leppäranta

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Lake Ice Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Ice Season . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 6 9 11

2

Freezing of Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Lake Types and Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Lake Basins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Water Budget of Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Physical Properties of Lake Water . . . . . . . . . . . . . . . . . . . . 2.2 Ice-Covered Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Freezing of Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Vertical Mixing and Seasons . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Seasonal Lake Ice Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Lakes with Perennial Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Climate and Water Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 General Regional Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Radiation Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Air Pressure, Temperature, Humidity and Wind . . . . . . . . 2.4 Remote Sensing of Lake Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Optical Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Thermal Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Microwave Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 18 18 20 25 28 28 30 33 36 39 39 41 45 50 50 53 54 56 58

3

Structure and Properties of Lake Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ice Ih: The Solid Phase of Water on Earth . . . . . . . . . . . . . . . . . . . . 3.1.1 Ice Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Ice Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 64 64 66 ix

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3.1.3 Ice Formation in Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.1.4 Physical Properties of Lake Ice . . . . . . . . . . . . . . . . . . . . . . 71 3.1.5 Snow Cover on Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.2 Lake Ice Types and Stratigraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.1 Ice Structure Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.2 Lake Ice Stratigraphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.2.3 Ice Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.3 Impurities in Lake Ice Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 Light Transfer Through Ice Cover . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.4.1 Radiance and Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.4.2 Optical Properties of Lake Ice . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.3 Light Transmittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4

5

Thermodynamics of Seasonal Lake Ice . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Heat Budget of Freezing Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Heat Budget and Radiation Balance . . . . . . . . . . . . . . . . . . 4.1.2 Solar Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Terrestrial Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Turbulent Heat Exchange with the Atmosphere and Water Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Boundary Layer Above Lake Ice . . . . . . . . . . . . . . . . . . . . . 4.2.2 Boundary-Layer Below Lake Ice . . . . . . . . . . . . . . . . . . . . . 4.3 Ice Growth and Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Thermodynamic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Congelation Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Snow on Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Frazil Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Ice Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Thermodynamic Models of Lake Ice . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Analytic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Numerical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Quasi-Steady Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Time-Dependent Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.6 Two-Phase Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 110 110 112 116

Mechanics of Lake Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Strain and Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Rheological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Ice Cover as a Plate on Water Foundation . . . . . . . . . . . . . . . . . . . . 5.2.1 Elastic Lake Ice Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Viscous Behaviour of Lake Ice . . . . . . . . . . . . . . . . . . . . . . .

159 160 161 162 164 167 167 168

117 117 122 123 123 125 133 137 139 143 143 144 147 148 149 152 155

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5.2.3 Thermal Cracking and Expansion . . . . . . . . . . . . . . . . . . . . 5.2.4 Displacements in Ice Cover . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Bearing Capacity of Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Ice Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Ice Load Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Estimation of Ice Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Drift Ice in Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Drift Ice Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Equations of Drift Ice Mechanics . . . . . . . . . . . . . . . . . . . . . 5.5.3 Models of Drift Ice Dynamics . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170 172 175 181 181 183 184 184 189 194 200

6

Proglacial Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Ice Sheets and Glaciers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Supraglacial Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Structure of Supraglacial Lakes . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Thermodynamics of Supraglacial Lakes . . . . . . . . . . . . . . . 6.2.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Epiglacial Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Occurrence of Epiglacial Lakes . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Thermodynamics of Epiglacial Lakes . . . . . . . . . . . . . . . . . 6.4 Subglacial Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Formation and Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Lake Vostok . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 206 212 212 214 219 223 223 227 228 229 231 231

7

Lake Water Body in the Ice Season . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Ice Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Cooling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Analytic Slab Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Mixed-Layer Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Turbulence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Thermal Structure and Circulation Under Ice . . . . . . . . . . . . . . . . . 7.2.1 Water Body During the Ice Season . . . . . . . . . . . . . . . . . . . 7.2.2 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Dynamics of Water Body Beneath Ice . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Circulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Shallow Water Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Melting Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Light Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Optically Active Substances . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Light Below the Ice Cover . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Quantum Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 236 236 242 245 247 249 249 252 255 255 259 262 266 266 267 268 271

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8

Ice-Covered Lakes Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Frozen Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Water Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.5 Nutrients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Winter Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Physical Limitations of Ecosystems Under Ice . . . . . . . . . 8.2.2 Primary Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Ecosystems Under Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Lake Ice Ecosystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Shoreline Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Lake Ice and Society . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Lake Ice Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Winter Traffic in Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Sports and Recreation on Lake Ice . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 278 278 279 281 286 286 287 287 289 290 292 292 293 293 295 299 302

9

Lake Ice Climatology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Ice Phenology and Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Ice Occurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Ice Phenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Ice Thickness and Extent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Mathematical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Linear Surface Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Ice Occurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Freezing and Breakup Dates . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Ice Thickness and Extent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Climate Change Impact on Lake Ice Season . . . . . . . . . . . . . . . . . . 9.4.1 Climate Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Model Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 307 312 313 315 315 320 320 321 323 326 328 328 330 333 335

10 Future of Frozen Lakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Role of Ice Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Science and Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 What Will Be? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

339 340 342 345 347

Annex: Ice Properties and Useful Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 349 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Chapter 1

Introduction

At a hole in lake ice, a painting from the year 1900 by Finnish artist Pekka Halonen (1865–1933), who is well known especially of his winter landscapes. In cold regions the normal life has adapted to the presence of frozen lakes. Holes in ice cover have been used for household water, flushing clothes, and cold bathing. © Finnish National Gallery, printed by permission

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_1

1

2

1 Introduction

A lake is a liquid water body in a depression on the surface of the Earth. It is associated with a water balance, which largely determines the lake volume and the water quality. The size of lakes ranges from small ponds to large intra-continental basins with an area more than 104 km2 . The global water storage of lakes is 125,000 km3 , and about 80% of that is fresh water,1 accounting for the fraction of 0.3% of the Earth’s freshwater resources (Henderson & Henderson, 2009). In the boreal zone, tundra, Central Asian cold and arid climate zone, and high mountain regions, lakes freeze over in winter. The ice season may last more than half a year, and the thickness of the ice can reach two meters. In high polar zones and high mountains, there are perennially ice-covered lakes, with ice cover several meters thick. Sub-glacial lakes are located at the base of the 4 km thick Antarctic ice sheet and represent an extreme case of lakes under a very thick ice cap on Earth.

1.1 History Frozen lakes have belonged to people’s normal life in the cold regions. Solid ice cover has been an excellent base for traffic across lakes, to travel, and to transport cargo, provided the ice has been thick enough (Fig. 1.1). Indeed, an ice cover isolates island residents from the mainland for periods when the ice is too weak to walk but still limits boating. These periods bear in Finnish language an old word kelirikko (from ‘keli’ = traffic conditions, and ‘rikko’ = broken); in Swedish this word is menföre, or in Russian it is ‘pacpytica’ (rasputica). One of the first records of lake ice events is ‘The battle on the ice’, a winter clash between the forces of the Norwegian king Áli and the Swedish king Aðals on the ice of Lake Vänern in Sweden, at about year 530 AD (Osborn & Mitchell, 2007). The legend of the battle on ice of Lake Peipsi in February 1246 was filmed by Sergey Eisenstein in his movie Alexander Nevsky (1938). The chart Carta Marina of Olaus Magnus Gothus (1539) over Northern Europe and his book on northern peoples (1555) showed drawings and descriptions of crossing over ice-covered lakes by skis, skates and horses and reported of horse races on ice in the Middle Ages. During the Little Ice Age (1450–1850), skating and games on lake ice spread all over to Central Europe. Winter roads on lake ice and river ice together with snow-covered overland sections formed a very important traffic network in northern Eurasia in the prehistoric times. These winter networks provided easy ways of crossing between hydrological drainage basins that is a strong limitation in boat traffic. Winter roads on lake ice are still in use, kept by road maintenance authorities as well as by private people and industry. Also, lake ice serves as an excellent landing platform for aircraft. In wood industry, logs have been stored in winter on ice to be dragged or floated along

1

Fresh water refers to natural water with very low concentration of dissolved substances. There is no strict limit but the concentration 0.5‰ (or 0.5 g L−1 , L denotes litre) serves as a convenient reference. This limit is also often used for the upper limit of the salinity of drinking water.

1.2 Lake Ice Science

3

Fig. 1.1 Lake Äkäslompolo in southern Lapland in winter. Photograph by the author

streams to factories after ice melt. Until about mid-twentieth century, when refrigerators were not common, the cold content of lake ice was utilized. Ice blocks were cut in winter, stored under sawdust or hay to protect from melting, and in summer used as a cooler for foods, especially in dairy farms. Openings are cut by ice-saws into lake ice covers for household water and cold bathing, which is in modern times claimed to be a healthy habit although in the Middle Ages these baths were used as a punishment. For ice-covered lakes, special techniques have been developed for domestic and commercial winter fishing. Recently, recreation activities have become important utilizers of frozen lakes, particularly for nature tourism, sport fishing, longdistance skating, skiing, ice sailing and many kinds of games. There, safety questions are a major issue.

1.2 Lake Ice Science The Earth’s surface waters are classified into lakes, rivers, and seas. In cold regions, ice occurs in these basins as lake ice, river ice, sea ice, and blocks of land ice2 forming the category of floating ice. Rivers have a dynamic ice cover due to their permanent and turbulent flow, resulting in abundance of frazil ice, anchor ice and ice jams. Sea ice forms of saline water and contains brine, and large sea ice basins are governed by 2

Land ice is of glacial origin, brought to water bodies by calving of glaciers.

4

1 Introduction

the presence of drift ice. Massive icebergs belong to the polar ocean environment, but small blocks of land ice can be found in proglacial lakes. Lake ice is the simplest form of floating ice because of the weaker role of dynamics as compared with sea ice and river ice. The research history of lake ice shows gains from sea ice and river ice research, where practical problems have often led to earlier innovations. For the background references, the status of sea ice physics is presented in Leppäranta (2011), Thomas (2016), Wadhams (2000) and Weeks (2010), and for river ice a good coverage is contained in Ashton (1986), Ferrick and Prowse (2002), Hicks (2016) and Shen (2006). In saline lakes the small-scale structure of ice resembles sea ice, and in very large lakes drift ice occurs as in marine basins. In freezing lakes with a notable and permanent throughflow, ice features such as in freezing rivers can be found. Scientific research on lake ice was commenced in the 1800s along with the birth of physical limnology and lake hydrology. Scientists in the European Alpine countries played a strong role in early physical limnology, and therefore it is natural that lake ice was an important topic from the beginning (e.g., Götzinger, 1909; Verescagin, 1925). Lake ice monitoring was commenced in northern Europe in the 1800s along with the rise of geosciences (Nordqvist, 1887; Simojoki, 1978). The ice season modifies the hydrological year in cold regions by its influence on the annual distribution of runoff and by the problems caused by ice to the management of water resources. Ice engineering problems were included in scientific research from the late 1800s (Barnes, 1928), concerning shipping, ice forces on structures, and the bearing capacity of ice. Lake ice phenology3 has been a major research topic, since ice seasons show large variability, which is a critical issue to the local population (Simojoki, 1940). Collection of ice phenology time-series of inland waters was commenced in the 1800s (Hällström, 1839; Levänen, 1894), and now they are of great importance in the research of the climate history. For example, nearly 200 years long lake ice time series are available from 1833 for Lake Kallavesi, Finland (Simojoki, 1959), and from 1850 for Lake Mendota, Wisconsin (Wing, 1943). Even longer series exist: e.g., from 1443 for Lake Suwa, Japan (Arakawa, 1954), and from 1696 for River Tornionjoki, Finland (Kajander, 1993). Records of lake ice winters in Central Europe have been used to examine the occurrence of extreme cold winters (Maurer, 1924; Takács et al., 2018). Winter temperature conditions under ice cover were also investigated in the early lake research (Homén, 1903), and ecology of frozen lakes gained more attention only later (see, e.g., Greenbank, 1945; Hutchinson, 1957; Vanderploeg et al., 1992). Research of ice-covered lakes remained for long, until about 1970s, scattered and occasional. The annual cycle of lakes was taken into the topics of the International Hydrological Decade 1965–1974 with ice season then considered in cold regions (Falkenmark, 1973). Thereafter, more effort was made on polar lakes as the access to reach them was better and automated measurement techniques had become feasible (Vincent & Laybourn-Parry, 2008). After the year 2000, lake ice science has regained much attention, due to environmental problems and due to climate change 3

Phenology refers to studies of the dates of recurring phenomena; it originates from the Greek word ϕα´ινω (phain¯o), ‘to make appear’.

1.2 Lake Ice Science

5

impact questions (Bengtsson, 2011; Duan & Xiao, 2015; George, 2009; Karetnikov et al., 2017; Kirillin et al., 2012; Leppäranta, 2014; Magnuson et al., 2000). Satellite remote sensing of lake ice has progressed due to the availability of high-resolution imagery (Duguay et al., 2015; Qiu et al., 2019). Winter biology of lakes has been an increasing research field (Melnik et al., 2008; Salonen et al., 2009; Shuter et al., 2012; Song et al., 2019). Oxygen is a major question since its renewal is limited in the presence of ice cover (Golosov et al., 2007; Huang et al., 2021). Questions of life in extreme conditions have brought in more research of lakes with perennial ice and pro-glacial lakes (Keskitalo et al., 2013; Leppäranta et al., 2020; Priscu, 1998; Vincent & Laybourn-Parry, 2008), especially in the Antarctic continent. The ecosystem of subglacial lakes has been largely unknown, but the first samples ever taken from these lakes showed signs of life, mainly bacteria (Christner et al., 2006; Rogers et al., 2013). Presently the leading research topics are the light transfer through ice, interaction between the ice and the water body, remote sensing, winter ecology, and lake ice climatology. Characteristics of lake ice seasons are quite sensitive to climate variations. For human living conditions, shorter ice seasons, while extending the open water season, with thinner ice would severely or even drastically limit traditional on-ice activities. Climate change issues also necessitate a better understanding of the role played by lake ice cover in the emission of greenhouse gases, especially methane, into the atmosphere, and in the global carbon budget. Mathematical modelling applications in lake ice research have concerned ice growth and decay, radiation transfer, ice forces, and ice drift or ice displacements. The models are used for basic science, forecasting, ice engineering, environmental research and protection, and lake ice climate research. Two classical, analytical models, which are still applicable as the first approximations, are the ice growth model by Stefan (1891) and the bearing capacity model by Hertz (1884). The former model predicts ice thickness to be proportional to the square root of the freezingdegree-days, and the latter model predicts the bearing capacity for a point load to be proportional to the squared ice thickness (Fig. 1.2). Also, semi-empirical ice phenology models were used in the past. Prediction of the freezing date was based on the weighted integral of the air temperature history (Rodhe, 1952), while ice breakupdate was modelled based on the integral of positive degree-days (Williams, 1971). More advanced thermodynamic approach to ice season was given by Pivovarov (1973). Until 1990s, thermodynamic lake ice models were mostly semi-analytic, based on the freezing-degree-days for ice growth and positive degree-days for melting (e.g., Ashton, 1986; Leppäranta, 1993, 2009). Also, numerical models were employed for the evolution of ice thickness and temperature (e.g., Croley & Assel, 1994; Leppäranta, 1983; Stepanenko et al., 2019; Yang et al., 2012). Numerical sea ice models had been developed earlier (Maykut & Untersteiner, 1971; Saloranta, 2000; Semtner, 1976) and they were utilized also for lake ice. The snow layer, which plays an important, interactive role in the evolution of the thickness of ice, especially in the boreal zone, forms the most difficult part of lake ice thermodynamic models.

6

1 Introduction

Fig. 1.2 The bearing capacity of lake ice is a major safety question for on-ice traffic. The photograph shows a rescue operation of a car in Lake Päijänne, Finland in spring 2005. Published in newspaper Etelä-Suomen Sanomat on March 31, 2005. Photograph by Mr. Pertti Louhelainen. Printed by permission

Mechanical models have been developed for ice forces in the scales of man-made structures (Korzhavin, 1962; Michel, 1978; Palmer & Croasdale, 2012) and for drift ice for whole lake basins (Leppäranta & Wang, 2008; Ovsienko, 1976; Wake & Rumer, 1983; Wang et al., 2005). The drift of ice is forced by winds and currents, and the mobility of ice is determined by its strength and the size of the basin. Ice drifts in very large lakes, such as the Caspian Sea and Lake Superior, as in marine basins, and in this field as well sea ice models were the basis of lake ice models (Coon, 1980; Doronin, 1970; Hibler, 1979; Wang et al., 2003; Wang et al., 2010).

1.3 Ice Season By far, most freezing lakes have a seasonal ice cover. The ice is normally thin as compared with the lake depth. The physical properties of ice show large random variability due to the high homologous temperature,4 the large size of individual crystals, and the presence of impurities. Lake ice shows a simple stratification pattern consisting of congelation ice formed of the lake water and snow-ice formed of eventual slush on ice (e.g., Adams & Roulet, 1980; Leppäranta & Kosloff, 2000; Michel & 4

The temperature of a medium as a fraction of its melting point temperature (Kelvin scale).

1.3 Ice Season

7

Ramseier, 1971; Palosuo, 1965). Snow accumulation on lake ice is common apart from cold and arid climate regions (e.g., Huang et al., 2012; Obryk et al., 2014). Even a thin layer of snow significantly weakens the heat exchange between the lake water body and the atmosphere, and the transfer of sunlight into the water. Frazil ice formation may occur in the first stage of freezing lakes, but later on, apart from large lakes, it seems to be rare. The evolution of a lake ice season is primarily a thermodynamic process (see Kirillin et al., 2012; Leppäranta & Wang, 2008; Michel & Ramseier, 1971). After the freeze-up, when the ice is thick enough, it spans a stable lid over the lake. Ice breakage and mechanical displacements may take place all winter in large lakes, but in smaller lakes only if the ice is thin or weak. Decay of a lake ice cover is a thermomechanical process, starting from the shoreline (Fig. 1.3). Melting of near-shore ice releases the ice from solid boundaries, and the ice cover may then shift as forced by winds and currents. Ice loses its strength and bearing capacity due to internal deterioration. Movement of ice causes further breakage of ice that speeds the decay process. Land-ice interaction due to thermal expansion and onshore ride-up or piling-up is an important environmental and practical issue. It has attracted geographers for more than one hundred years (Alestalo & Häikiö, 1979; Buckley, 1900; Helaakoski, 1912). Due to thermal and mechanical stresses, lake ice deforms shore areas and loads man-made structures at the shoreline. In shallow areas, bottom scouring by ice and freezing through the whole water column give rise to erosion of the lake bottom. Land–ice interaction has therefore geological and biological consequences.

Fig. 1.3 Ice break-up takes place in spring in boreal lakes. During a few weeks prior to the disappearance of ice, the ice cover is weak and patchy and impossible to cross by foot or boat. Lake Päijänne, Finland, end of March 2017. Photograph by the author

8

1 Introduction

In a lake water body, physical phenomena and processes are very different under ice cover from the open water conditions. The ice cover cuts the transfer of momentum from the wind to the water body that damps turbulence and mixing. We may say that the time slows down. The surface water temperature is at the freezing point, there is very little vertical transfer of heat, apart from geothermal lakes, and only a weak thermohaline circulation is present. The temperature structure and circulation are quite stable, and salinity stratification may be important even in freshwater lakes. In large lakes, the ice cover may experience episodic movements and disturb the water body. When available, solar radiation provides strong heating, in fact the strongest forcing given into fully ice-covered lakes, resulting in convective deepening of the mixed layer (Kirillin et al., 2012). During ice melting, the meltwater with its impurities is released into the water body. The volumetric changes in the liquid water body, associated with the formation and growth of ice are of minor consequence in deep lakes, but in very shallow lakes, substantial fractional reductions in the volume of liquid water may result that also effects the geochemistry of the lake water (Yang et al., 2016). In shallow lakes the bottom sediments have an important role in the heat budget. They collect large heat storage in summer and release the heat back to the water body in winter. Gas release from sediments forces vertical circulation, and impurities may dissolve from the sediments to enrich the near-bottom waters and create a thin bottom boundary layer from the resulting salinity stratification (Golosov et al., 2007; Lu et al., 2020). This near-bottom layer also has impact on the near-bottom living conditions. Lake ice formation and growth have influence on the water quality and on the lake ecology (e.g., Kirillin et al., 2012; Salonen et al., 2009; Song et al., 2019). The concentration of suspended matter decreases, since due to weak inflow and turbulence, sedimentation is not compensated by resuspension. At the time of ice breakup, the load of impurities peaks from the meltwater of the drainage basin. A critical factor is that the ice cover reduces the level of dissolved oxygen in the water body by cutting the influx from the atmosphere (Bengtsson, 2011; Golosov et al., 2007; Greenbank, 1945; Hargrave, 1969; Huang et al., 2021). At the beginning of the ice season, the oxygen content depends on the strength of the autumn mixing. In windy autumn, the whole water body may cool down even to 1 °C, and the cold water will then contain a large amount of oxygen. In winter, oxygen is consumed mostly at the lake bottom by bacterioplankton through organic matter mineralization. This results in long winters in anoxic conditions in the deep water and, as a consequence, fish kills (Fig. 1.4). Sunlight is the main limitation for primary production under ice cover. Nutrient levels are usually sufficient after the autumn mixing. Photosynthesis is paused for the polar night, and in the case of a thick snow cover on ice, the dark season is longer and extends to sub-polar latitudes. In the Central Asian arid climate, freezing lakes are snow-free and photosynthesis continues over all winter under ice (Song et al., 2019). In general, vertical stratification with a thin upper layer can be favourable for primary production beneath a bare ice cover, and an under-ice bloom may form if the bare ice period lasts long enough.

1.4 This Book

9

Fig. 1.4 Fish kill in Lake Äimäjärvi, southern Finland, March 2003. Ice season 2002–2003 began exceptionally early (late October), and the oxygen storage was not sufficient for the whole ice season. Photograph by Mr. Jouni Tulonen, printed with permission

1.4 This Book The second edition of Freezing of lakes and the evolution of their ice cover, presents an update (year 2022) to the first edition (2014) of the status of knowledge of the physics of lake ice with applications (Fig. 1.5). The text has been prepared eight years after the first one providing new material especially for low latitude lakes in Central Asian arid climate, physical limnology below ice, remote sensing, and lake ice climatology including climate change impact on lake ice. In addition, modifications, restructuring, additions, and technical improvements have been made in most sections. Recent progress in the research of lake ice has been largely connected to ice–water interaction, lake ice climatology, and the role of ice cover in lake ecology. Remote sensing has also much progressed together with increased availability of high-resolution images. The book has focus on freshwater lakes, with river ice and sea ice results utilized where necessary. The role of dissolved substances of the parent water is explained here based on results from sea-ice research, since research ice in saline lakes is still quite limited. A historical view is embedded with material from the more than 100 years long period in lake ice research. Earlier literature of the physics of icecovered lakes is quite sparse. The key lake ice books include Ashton (1986), Barnes

10

1 Introduction

Fig. 1.5 Learning of frozen lakes in the field is an essential corner stone of knowledge. Students of winter limnology field course learning to use an underwater robot below ice cover in Lake Lovojärvi, Finland. Photograph by the author

(1928), Michel (1978), Pounder (1965) and Shumskii (1956). Pivovarov (1973) presented a scientific monograph on freezing lakes and rivers with weight on the liquid water body, while Ashton (1986) had a more engineering point of view on this topic [see also Ashton (1980) for a condensed review paper on the topic]. More recent material is given in the article collections by George (2009), Vincent and Laybourn-Parry (2008), and in the reviews of Kirillin et al. (2012) and Shuter et al. (2012). Recently a book about mountain lakes in the Sierra Nevada of California was prepared by Melack et al. (2021). The first chapter introduces the topic with a brief historical overview. Chapter 2 presents lakes, their classification, the zones of ice-covered lakes, and remote sensing methods. Ice formation and the structure and properties of lake ice are treated in Chap. 3, including ice impurities and light transfer. Chapter 4 contains lake ice thermodynamics from the freezing of lakes to ice melting, with thermodynamic models included. Lake ice mechanics is the topic of Chap. 5 with engineering questions such as ice forces and bearing capacity of ice. A section is included on drift ice in large lakes. Glaciers and pro-glacial lakes are treated in Chap. 6. Quite exotic lakes are introduced, particularly lakes at the top and bottom of glaciers and ice sheets. Chapter 7 focuses on the water body beneath lake ice cover with water balance, stratification, and circulation. Ecology and environmental questions of ice-covered

References

11

lakes, and lake ice and society including practical questions are treated Chap. 8. The next chapter focuses on lake ice climatology looking into time series and analytic modelling of climate sensitivity and climate impact. The final closing words are written in Chap. 10, and the list of references follow in Chapter 11. Study problems with solutions are given as examples in the text. Properties of ice and liquid water and useful constants and formulae have been collected in the annex. Lake ice belongs to the cryosphere part, which closely interacts with human living conditions and industry. The ice cover has caused problems, but people have learnt to live with them and to utilize the ice. Especially, in the present time this is true for on-ice traffic, recreation activities, and tourism. Ice fishing has become a widely enjoyed winter hobby. Winter sports such as skiing, skating, and ice sailing are now popular activities on frozen lakes. The interaction between lake ice and human life is to stay to the future, but modifications may be brought along the evolution of the climate. The lake ice response to climate warming would appear as the shortening of the ice season due to increasing air temperature, but also the quality of the ice and ice seasons would change via the climate influence on the thickness and structure of the ice.

References Adams, W. P., & Roulet, N. T. (1980). Illustration of the roles of snow in the evolution of the winter cover of a lake. Arctic, 33(1), 100–116. Alestalo, J., & Häikiö, J. (1979). Forms created by thermal movement of lake ice in Finland in winter 1972–73. Fennia, 157(2), 51–92. Arakawa, H. (1954). Five centuries of freezing dates of Lake Suwa in Central Japan. Archiv für Meteorologie, Geophysik und Bioklimatologie, B, 6, 152–166. Ashton, G. (1980). Freshwater ice growth, motion, and decay. In S. Colbeck (Ed.), Dynamics of snow and ice masses (pp. 261–304). Academic Press. Ashton, G. (Ed.). (1986). River and lake ice engineering (p. 485). Water Resources Publications. Barnes, H. T. (1928). Ice engineering (p. 364). Montreal Renouf Publishing Co. Bengtsson, L. (2011). Ice-covered lakes: Environment and climate—Required research. Hydrological Processes, 25, 2767–2769. https://doi.org/10.1002/hyp.8098 Buckley, E. R. (1900). Ice ramparts. Transactions of the Wisconsin Academy of Sciences, Arts and Letters, 13, 141–157. Christner, B. C., Royston-Bishop, G., Foreman, C. M., Arnold, B. R., Tranter, M., Welch, K. A., Lyons, W. B., Tsapin, A. I., Studinger, M., & Priscu, J. C. (2006). Limnological conditions in subglacial Lake Vostok, Antarctica. Limnology and Oceanography, 51(6), 2485–2501. Coon, M. D. (1980). A review of AIDJEX modeling. In: R. S. Pritchard (Ed.), Proceedings of ICSI/AIDJEX Symposium on Sea Ice Processes and Models (pp. 12–23), University of Washington, Seattle. Croley, T. E., II., & Assel, R. A. (1994). A one-dimensional ice thermodynamics model for the Laurentian Great Lakes. Water Resources Research, 30(3), 625–639. https://doi.org/10.1029/ 93WR03415 Doronin, Yu. P. (1970). On a method of calculating the compactness and drift of ice floes. Trudy Arkticheskii i Antarkticheskii Nauchno-issledovatel’skii Institut, 291, 5–17. [English Trans. 1970 in AIDJEX Bulletin, 3, 22–39].

12

1 Introduction

Duan, A., & Xiao, Z. (2015). Does the climate warming hiatus exist over the Tibetan plateau? Science and Reports, 5, 13711. Duguay, C. R., Bernier, M., Gauthier, Y., & Kouraev, A. (2015). Remote sensing of lake and river ice. In M. Tedesco (Ed.), Remote sensing of the cryosphere (pp. 273-306). Wiley. Falkenmark, M. (Ed.). (1973). Dynamic studies in Lake Velen. International hydrological decade— Sweden, Report 31. Swedish Natural Science Research Council, Stockholm, Sweden. Ferrick., M. G., & Prowse, T. D. (Eds.). (2002). Special issue: Hydrology of ice-covered rivers and lakes. Hydrological Processes, 16(4), 759–958. George, G. (Ed.). (2009). The impact of climate change on European lakes (p. 607). Springer. Golosov, S., Maher, O. A., Schipunova, E., Terzhevik, A., Zdorovennova, G., & Kirillin, G. (2007). Physical background of the development of oxygen depletion in ice-covered lakes. Oecologia (online). https://doi.org/10.1007/s00442-006-0543-8 Götzinger, G. (1909). Studien über das Eis des Lunzer Unter- und Obersees. Int Rev Ges Hydrobiol Hydrogr, 2, 3–386. Greenbank, J. (1945). Limnological conditions in ice-covered lakes, especially as related to winterkill of fish. Ecological Monographs, 15, 343–391. Hällström, G. G. (1839). Specimina mutanti currente seculo temporis, quo glacies fluminun annuae dissolutae sunt. Acta Acad Sci Fenn, 1, 129–149. Hargrave, B. T. (1969). Similarity of oxygen uptake by benthic communities. Limnology and Oceanography, 14, 801–805. Helaakoski, A. (1912). Havaintoja jäätymisilmiöiden geomorfologisista vaikutuksista [Beobachtungen über die geomorfologischen Einflüsse der Gefriererscheinungen]. Suomalaisen Maantieteellisen Yhdistyksen julkaisuja, 9, 109 pp. Henderson, P., & Henderson, G. M. (2009). The Cambridge handbook of earth science data (p. 276). Cambridge University Press. Hertz, H. (1884). Über das Gleichgewicht schimmender elastischen Platten. Wiedmanns Annalen der Physik und Chemie, 22, 449–455. Hibler, W. D., III. (1979). A dynamic-thermodynamic sea ice model. Journal of Physical Oceanography, 9, 815–846. Hicks, F. (2016). An introduction to river ice engineering: For civil engineers and geoscientists (p. 170). CreateSpace, Amazon. Homén, Th. (1903). Die Temperaturverhältnisse in den Seen Finnlands. Förh. vid Nat. fork. mötet i Helsingfors 1902. Huang W, Zhang, Z., Li, Z., Leppäranta, M., Arvola, L., Song, S., Huotari, J., & Lin, Z. (2021). Under-ice dissolved oxygen and metabolism dynamics in a shallow lake: The critical role of ice and snow. Water Resources Research, 57(5). Huang, W.-F., Li, Z., Han, H., Niu, F., Lin, Z., & Leppäranta, M. (2012). Structural analysis of thermokarst lake ice in Beiluhe Basin, Qinghai-Tibet Plateau. Cold Regions Science and Technology, 72, 33–42. Hutchinson, G. E. (1957). A treatise on limnology (Vol. 1, p. 1015). Wiley. Kajander, J. (1993). Methodological aspects on river cryophenology exemplified by a tricentennial break-up time series from Tornio. Geophysica, 29(1–2), 73–95. Karetnikov, S., Leppäranta, M., & Montonen, A. (2017). Time series over 100 years of the ice season in Lake Ladoga. Journal of Great Lakes Research, 43(6), 979–988. Keskitalo, J., Leppäranta, M., & Arvola, L. (2013). First records on primary producers of epiglacial and supraglacial lakes in the western Dronning Maud Land, Antarctica. Polar Biology, 36(10), 1441–1450. https://doi.org/10.1007/s00300-013-1362-0 Kirillin, G., Leppäranta, M., Terzhevik, A., Granin, N., Bernhardt, J., Engelhardt, C., Efremova, T., Golosov, S., Palshin, N., Sherstyankin, P., Zdorovennova, G., & Zdorovennov, R. (2012). Physics of seasonally ice-covered lakes: A review. Aquatic Sciences, 74, 659–682. https://doi. org/10.1007/s00027-012-0279-y Korzhavin, K. N. (1962). Vozdeystviye l’da na inzhenertrye sooruzheniya [The action of Ice on engineering structures]. Izdatel’stvo Sibirskogo Otdel. Akademiya Nauk SSSR, Novosibirsk.

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Leppäranta, M. (1983). A growth model for black ice, snow ice and snow thickness in subarctic basins. Nordic Hydrology, 14(2), 59–70. Leppäranta, M. (1993). A review of analytical sea ice growth models. Atmosphere-Ocean, 31(1), 123–138. Leppäranta, M. (2009). Modelling of growth and decay of lake ice. In G. George (Ed.), Climate change impact on European lakes (pp. 63-83). Springer. Leppäranta, M. (2011). The drift of sea ice (2nd ed., 347 pp.). Springer-Praxis. Leppäranta, M. (2014). Interpretation of statistics of lake ice time series for climate variability. Hydrology Research, 45(4–5), 673–684. Leppäranta, M., & Kosloff, P. (2000). The thickness and structure of Lake Pääjärvi ice. Geophysica, 36(1–2), 233–248. Leppäranta, M., Luttinen, A., & Arvola, L. (2020). Physics and geochemistry of lakes in Vestfjella, Dronning Maud Land. Antarctic Science, 32(1), 29–42. Leppäranta, M., & Wang, K. (2008). The ice cover on small and large lakes: Scaling analysis and mathematical modelling. Hydrobiologica, 599, 183–189. Levänen, S. (1894). Solfläckarnes inflytande på islossingstiderna i Finlands floder och på vattenståndet i finska viken [The influence of sunspots to ice breakup in Finnish rivers and to water level elevation in the Gulf of Finland]. Fennia, 9(4). Lu, P., Cao, X., Li, G., Huang, W., Leppäranta, M., Arvola, L., Huotari, J., & Li, Z. (2020). Mass and heat balance of a lake ice cover in the Central Asian arid climate zone. Water, 12, 2888. Magnuson, J. J., Robertson, D. M., Benson, B. J., Wynne, R. H., Livingstone, D. M., Arai, T., Assel, R. A., Barry, R. G., Card, V., Kuusisto, E., Granin, N. G., Prowse, T. D., Stewart, K. M., & Vuglinski, V. S. (2000). Historical trends in lake and river ice cover in the northern hemisphere. Science, 289, 1743–1746. Errata 2001 Science, 291, 254. Maurer, J. (1924). Severe winters in southern Germany and Switzerland since the year 1400 determined from severe lake freezes. Monthly Weather Review, 52, 222. [Trans. from German original in Meteorologische Zeitschrift, 41, 85–86]. Maykut, G. A., & Untersteiner, N. (1971). Some results from a time-dependent, thermodynamic model of sea ice. Journal of Geophysical Research, 76, 1550–1575. Melack, J. M., Sadro, S., Sickman, J. O., & Dozier, J. (2021). Lakes and watersheds in the Sierra Nevada of California: Responses to environmental change (p. 203). University of California Press. Melnik, N. G., Lazarev, M. I., Pomazkova, G. I., Bondarenko, N. A., Obolkina, L. A., Penzina, M. M., & Timoshkin, O. A. (2008). The cryophilic habitat of micrometazoans under the lake ice in Lake Baikal. Fundamental and Applied Limnology, 170, 315–323. Michel, B. (1978). Ice mechanics (499 pp.). Laval University Press. Michel, B., & Ramseier, R. O. (1971). Classification of river and lake ice. Canadian Geotechnical Journal, 8(36), 36–45. Nordqvist, O. (1887). Om insjöarnes temperatur. Öfversigt af Finska Vetenskaps-Societetens Förhandlingar, 29, 23–32. Obryk, M. K., Doran, P. T., & Priscu, J. C. (2014). The permanent ice cover of Lake Bonney, Antarctica: The influence of thickness and sediment distribution on photosynthetically available radiation and chlorophyll-a distribution in the underlying water column. Journal of Geophysical Research: Biogeosciences, 119(9), 1879–1891. Osborn, M., & Mitchell, S. A. (2007). The battle on ice. ANQ, 20(3), 70–74. Ovsienko, S. (1976). Numerical modeling of the drift of ice. Izvestiya, Atmospheric and Oceanic Physics, 12(11), 1201–1206. Palmer, A., & Croasdale, K. (2012). Arctic offshore engineering (327 pp.). World Scientific Publishing. Palosuo, E. (1965). Frozen slush on lake ice. Geophysica, 9(2), 131–147. Pivovarov, A. A. (1973). Thermal conditions in freezing lakes and rivers (136 pp.). Wiley. [Trans. from Russian]. Pounder, E. R. (1965). Physics of ice. Pergamon Press.

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Qiu, Y., Xie, P., Leppäranta, M., Wang, X., Lemmetyinen, J., Lin, H., & Shi, L. (2019). MODISbased daily lake ice extent and coverage dataset for Tibetan Plateau. Big Earth Data, 3(2), 170–185. Priscu, J. C. (Ed.). (1998). Ecosystem dynamics in a polar desert: The McMurdo dry valleys Antarctica. Research Series (Vol. 72). American Geophysical Union. Rodhe, B. (1952). On the relation between air temperature and ice formation in the Baltic. Geografiska Annaler, 1–2, 176–202. Rogers, S. O., Shtarkman, Yu. M., Koçer, Z. A., Edgar, R., Veerapaneni, R., & D’Elia, T. (2013). (2013) Ecology of subglacial Lake Vostok (Antarctica), based on metagenomic/metatranscriptomic analyses of accretion ice. Biology, 2, 629–650. https://doi.org/10.3390/ biology2020629 Salonen, K., Leppäranta, M., Viljanen, M., & Gulati, R. (2009). Perspectives in winter limnology: Closing the annual cycle of freezing lakes. Aquatic Ecology, 43(3), 609–616. Saloranta, T. (2000). Modeling the evolution of snow, snow ice and ice in the Baltic Sea. Tellus, 52A, 93–108. Semtner, A. J. (1976). A model for the thermodynamic growth of sea ice in numerical investigations of climate. Journal of Physical Oceanography, 6, 379–389. Shen, H. T. (2006). A trip through the life of river ice—Research progress and needs. In Proceedings of the 18th IAHR International Symposium on Ice (pp. 85–91), Hokkaido University, Sapporo, Japan. Shumskii, P. A. (1956). Principles of structural glaciology (p. 497). Dover Publications, Inc. [in Russian]. [English Trans. 1964] Shuter, B. J., Finstad, A. G., Helland, I. P., Zweimüller, I., & Hölker, F. (2012). The role of winter phenology in shaping the ecology of freshwater fish and their sensitivities to climate change. Aquatic Sciences, 74(4), 637–657. Simojoki, H. (1940). Über die Eisverhältnisse der Binnenseen Finnlands. Mitteilungen des Meteorologischen Instituts der Universität Helsinki, 43, 194 pp. + Annexes. [in German]. Simojoki, H. (1959). Kallaveden pitkä jäähavaintosarja [A long series of ice observations for Lake Kallavesi]. Terra, 71, 156–161. Simojoki, H. (1978). The history of geophysics in Finland 1828–1918. The history of learning and science in Finland 1828–1918, No. 5b. Societas Scientarum Fennicae, Helsinki. Song, S., Li, C., Shi, X., Zhao, S., Li, Z., Bai, Y., Cao, X., Wang, Q., Huotari, J., Tulonen, T., Uusheimo, S., Leppäranta, M., & Arvola, L. (2019). Under-ice metabolism in a shallow lake (Wuliangsuhai) in Inner Mongolia, in cold and arid climate zone. Freshwater Biology, 00, 1–11. Stefan, J. (1891). Über die Theorie der Eisbildung, insbesondere über Eisbildung im Polarmeere. Annalen der Physik, 3rd Ser, 42, 269–286. Stepanenko, V. M., Repina, I. A., Ganbat, G., & Davaa, G. (2019) Numerical simulation of ice cover in saline lakes. Izvestiya, Atmospheric and Oceanic Physics, 55(1), 129–139. Takács, K., Kern, Z., & Pásztor, L. (2018). Long-term ice phenology records from eastern–central Europe. Earth System Science Data, 10, 391–404. Thomas, D. N. (2016). Sea ice (3rd ed., 664 pp.). Wiley-Blackwell. Vanderploeg, H. A., Bolsenga, S. J., Fahnenstiel, G. L., Liebig, J. L., & Gardner, W. S. (1992). Plankton ecology in an ice-covered bay of Lake Michigan: Utilization of a winter phytoplankton boom by reproducing copepods. Hydrobiologia, 243(244), 175–183. Verescagin, G. (1925). A selection of works from Lake Baikal expedition. Doklady Academy of Sciences SSSR, 12, 161–164. [in Russian]. Vincent, W. F., & Laybourn-Parry, J. (Eds.). (2008). Polar lakes and rivers. Limnology of Arctic and Antarctic aquatic ecosystems (322 pp.). Oxford University Press. Wadhams, P. (2000). Ice in the ocean (p. 364). Gordon & Breach Science. Wake, A., & Rumer, R. R. (1983). Great lakes ice dynamics simulation. Journal of Waterway, Port, Coastal and Ocean Engineering, 109, 86–102. Wang, K., Leppäranta, M., & Reinart, A. (2005). Modeling ice dynamics in Lake Peipsi. Verh. Internat. Verein. Limnol., 29, 1443–1446.

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Wang, J., Hu, H., & Bai, X. (2010). Modeling Lake Erie ice dynamics: Process studies. In Proceedings of the 20th IAHR International Symposium on Ice, University of Helsinki, Finland. Wang, K., Leppäranta, M., & Kouts, T. (2003). A model for sea ice dynamics in the Gulf of Riga. Proceedings of Estonian Academy of Sciences. Engineering, 9(2), 107–125. Weeks, W. F. (2010). On sea ice (p. 664). University of Alaska Press. Williams, G. P. (1971). Predicting the date of lake ice breakup. Water Resources Research, 7(2), 323–333. Wing, L. (1943). Freezing and thawing dates of lakes and rivers as phenological indicators. Monthly Weather Review, 71, 149–155. Yang, F., Li, C., Leppäranta, M., Shi, X., Zhao, S., & Zhang, C. (2016). Notable increases in nutrient concentrations in a shallow lake during seasonal ice growth. Water Science and Technology, 74(12), 2773–2883. Yang, Y., Leppäranta, M., Li, Z., & Cheng, B. (2012). An ice model for Lake Vanajavesi, Finland. Tellus A, 64, 17202. https://doi.org/10.3402/tellusa.v64i0.17202

Chapter 2

Freezing of Lakes

Terra/Aqua—MODIS (Moderate Resolution Imaging Spectroradiometer) image of the eastern Great Lakes region on February 28, 2004. White ice floes seen on Lake Huron, top left, and Lake Erie, bottom centre. Deeper Lake Ontario, right, remains free of ice, though the land around it is in winter’s icy grip. The mean depth and area are 19 m and 26,000 km2 in Lake Erie and 86 m and 19,000 km2 in Lake Ontario. NASA. Production by The Visible Earth team (http://visibleearth.nasa. gov/)

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_2

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2 Freezing of Lakes

2.1 Lake Types and Characteristics 2.1.1 Lake Basins Lakes are individuals. Each lake has its own geometry, bathymetry, drainage basin, and outflow. The first-order geometrical properties of lakes are based on their lateral or horizontal and vertical extent (Table 2.1). The lateral dimensions are taken as the length of the major axis l, the width w, defined as the maximum length perpendicular to the major axis, √ and the area A. The characteristic linear dimension of the lake is taken as l or A. The largest lake is the saline Caspian Sea (370,000 km2 ), and the largest freshwater lake is Lake Baikal by volume (23,620 km3 ) and Lake Superior by area (82,100 km2 ). Lake Ladoga is the largest freshwater lake in Europe (17,700 km2 ). There is no physical lower size limit for a lake, but in limnology very small lakes, with area less than 50,000 m2 , are rather called ponds. The vertical dimension of a lake is taken as its mean depth H˜ or its maximum depth Hmax . The deepest lake is Lake Baikal, the mean depth is 760 m,  and the maximum } depth is 1620 m. The five geometric dimensions A, l, w, H˜ , Hmax define four dimensionless characteristics: elongation γ , shape factor κ, steepness ξ , and the aspect ratio δ: γ =

l

w

≥ 1, κ =

H˜ A Hmax ≥ 1, δ = ≤ 1, ξ = lw l H˜

(2.1)

Normally δ ∼ 10−3 but goes up to δ ∼ 10−2 for small lakes. The hypsographic curve (H ) is equal to the fractional area of the lake deeper than H: Hmax (H ) = π (z)dz

(2.2)

H

where π is the spatial density of lake depth. Example 2.1 Take a cone-shape lake with radius R and maximum depth Hmax . Then   2 ˜ H max γ = 1, κ = π4 , ξ = 3, δ = H6R = 2R , and (H ) = π R 2 1 − HHmax .

Table 2.1 Classification of lakes for their lateral and vertical size Lateral l

Very large 1000 km

Large 100 km

Medium 10 km

Small 1 km

Very small 100 m

Vertical H˜

Very deep 500 m

Deep 100 m

Medium 20 m

Shallow 5m

Very shallow 1m

2.1 Lake Types and Characteristics

19

A single lake is a connected unit so that liquid water particles can circulate throughout. There may be sub-basins with straits between, sometimes taken as separate lakes. The geometry of lakes has fractal characteristics (Korvin, 1992). The length of the shoreline L increases with the measurement step s as a power law: L ∝ ( s)1−D , where D > 1 is the fractal dimension or the Hausdorff dimension of the shoreline; thus, L → ∞ as s → 0. Therefore, the length of the shoreline is not a well-defined geometrical property of a lake. Similarly, the number of lakes in a region depends on the resolution of the map and makes sense only if a minimum size is defined. Instead, the surface area and the lengths of the major and minor axes of a lake are well defined. Apart from the descriptive geometry, however, the fractal concept has not brought much new applicable information to lake research. The shoreline geometry is indeed an important factor during the ice season, but the length scale of this factor is finite and process dependent and there is no direct link to the Hausdorff dimension. A natural geological classification of lakes is according to the origins of the depressions, where lakes have formed (Table 2.2). The time scale of the depressions is usually very long, apart from those created by landslides and environmental engineering. The formation of glacial lakes has taken place during retreating of glaciers and ice sheets. In the Northern Europe and North America there are large lake districts, which originate from ice sheet–land processes after the Last Glacial Maximum about 25,000 years ago. Proglacial lakes receive their inflow mainly from living glaciers and ice sheets, and they are classified into epiglacial, supraglacial and subglacial lakes (Hodgson, 2012; Leppäranta et al., 2020). Supraglacial lakes are a particular category in that their base is ice rather than rock or soil. In the Antarctic dry valleys there are hypersaline lakes, such as Lake Vanda, with perennial lake ice (Gibson, 1999). Epishelf lakes are freshwater lakes dammed behind ice shelves, overlaying seawater or being connected to the sea by a conduit (Davies et al., 2017). Example 2.2 The Baltic Sea has a dynamic history since the retreat of the Weichselian ice sheet, beginning 15,000 years ago, due to the positive freshwater balance and rebound of the ground from the pressure of the ice sheet (see Leppäranta & Myrberg, 2009). First, the basin was freshwater Baltic Ice Lake, then (11,600–10,800 ago) brackish Yoldia Sea, connected via Närkes Strait to the Atlantic, and again (10,800–9000 years ago) freshwater Ancylus Lake (Fig. 2.1). About 9000 years ago, the Atlantic outflow changed to a strait in Denmark, and Ancylus Lake turned into a brackish marine basin, which has further developed to the present Baltic Sea. The land uplift is still in progress in the northern Baltic Sea, and in 1000 years the northern basin, the Bay of Bothnia, will be isolated from the oceanic connection and become the largest freshwater lake in Europe. Other large Holocene lakes in the Baltic Sea drainage basin are Ladoga, Onega, Vänern, Peipsi, and Saimaa.

20 Table 2.2 The origins of lake basins with examples from the cold climate zone

2 Freezing of Lakes Class

Mechanism for depression

Example

Tectonic

Plate shifts

Lake Baikal (Russia)

Volcanic

Caldera

Lake Shikotsu (Hokkaido, Japan)

Meteorite

Impact crater

Lake Lappajärvi (Finland)

Landslide

Land mass accumulation

Attabad Lake (Pakistan)

Reservoir

Man-made

Rybinsk reservoir (Volga basin, Russia)

Glacial

Ice sheet dynamics Lake Saimaa (Finland)

Epiglacial

Ice sheet dynamics Lake Mandrone (Italy)

Supraglacial

Solar radiation

Lake Suvivesi (Dronning Maud Land)

Subglacial

Geothermal heat

Lake Vostok (East Antarctica)

2.1.2 Water Budget of Lakes Lakes receive water from precipitation directly (P) and via inflow from the drainage basin (I), and they lose water by evaporation/sublimation (E) and outflow (O). The lake water balance presents the evolution of the water storage S = A H˜ : dS = (P − E) A + I − O dt

(2.3)

where t is time, and A is the area of the lake. Inflow comes from rivers and brooks, over land surface and from ground, and water flows out along a river and seeps through lake bottom to ground. Condensation (deposition) of water vapour is included as negative evaporation (sublimation), but these cases are occasional, minor contributors. A lake without outflow is closed and loses water only by evaporation or sublimation. There are bifurcation lakes with two outflow streams. Many lakes are regulated for inflow or outflow due to needs of hydropower plants and prevention of floods or drought. On a long term, e.g., over an annual cycle, the water storage is normally in equilibrium and the left-hand side of Eq. (2.3) is zero. Then the flushing time or the water renewal time of a lake provides the basic timescale of the water storage: tR =

S S = I + PA O + EA

(2.4)

The range is from less than one month to decades for most lakes. The inflow along land surface and the groundwater flux are the most difficult parts in the water balance.

2.1 Lake Types and Characteristics

21

Fig. 2.1 Ancylus Lake. Also shown are the largest freshwater lake of Europe, Lake Ladoga, which seceded from the Yoldia Sea about 10,000 years ago, and the Bay of Bothnia, the future largest freshwater lake in Europe. The contour inside the Ancylus Lake shows the present Baltic Sea. Ancylus Lake has been the largest freshwater lake on Earth after the last Glacial Maximum. Modified from Leppäranta and Myrberg (2009)

Water storage can be monitored from the water level elevation. Inflow and precipitation are external factors, independent of the lake. In turn, evaporation/sublimation and outflow depend on the state of the lake, the former on the lake surface temperature and the latter on the water level elevation. Precipitation data are available from most weather stations. They are given as the accumulation rate of an equivalent liquid water layer, normally in millimetres per time. There are large difficulties to determine the solid precipitation, especially during snowstorms due to the turbulent transport of snowflakes. Therefore, recorded precipitation data from snow accumulation are often biased down. In cold climate regions precipitation gauges are equipped with specific structures to manage the snowfall aerodynamics properly (Fig. 2.2). In cold air the saturation water vapour pressure is small, and therefore precipitation is normally low. Annual precipitation is around 500 mm in the boreal zone and half of that in tundra. Antarctic dry valleys have an extremely dry climate with no precipitation. High precipitation is recorded at windward sides of mountains, where

22

2 Freezing of Lakes

Fig. 2.2 Tretyakov model precipitation gauge. The metal slices around the cylinder constitute the specific design for aerodynamics to obtain representative snowfall data. Photograph by the author

the orography forces air masses upward. For example, in Norway, Rocky Mountains of North America, and Japanese Alps the highest annual precipitation levels are 2000–3000 mm. Freezing and melting do not influence the overall water budget, since they just change the phase of the water molecules. When the ice is floating, liquid water level elevation is unchanged, but grounding of ice necessarily lowers the water level. Inflow is usually low during the ice season since precipitation over the drainage basin is mostly solid and forms a temporary storage on land. If the soil surface layer is frozen, the groundwater storage is blocked from top, and liquid precipitation may show up in a significant surface runoff. A low inflow lowers the water surface level, and then the natural outflow also decreases. In regulated lakes the water level is kept above a designed minimum. Closer to the climatological edge of freezing lakes, liquid precipitation events occur all winter, and the water level elevation varies more smoothly. When the melting of snow begins, the inflow starts to increase and reaches a major peak. The water level elevation has influence on the ice–land interaction, such as shore erosion and lake bottom scouring by moving ice. Example 2.3 Specific cases of water balance. (a) Over the annual cycle, the net change of the water storage is normally close to zero and then (P − E)A = I − O. In dry regions, P ≈ 0 and O = 0, and, thus, at equilibrium AE ≈ I .

2.1 Lake Types and Characteristics

23

(b) A change of the water storage, S = A H˜ , can be expressed as the sum of the change in mean depth and the areal change: δS = Aδ H˜ + H˜ δ A. For a coneshape lake topography, the former term takes 1 /3 and the latter term takes 2 /3 of the change of the water volume. Figure 2.3 shows an example of the annual course of the water budget in a boreal lake. In winter, the inflow was low, and the water level was slowly falling. The inflow peaked after snowmelt in April–May, that showed up delayed and smoothed in the outflow. The monthly average inflow and outflow were 610 mm and 671 mm, respectively, spread over the lake, and the flushing time was about 12 months. The excess of outflow over inflow was balanced by the positive net atmospheric flux, precipitation higher than evaporation, while the contribution of groundwater flux was at most 1% of the total. The atmospheric fluxes showed strong annual cycle, and the average precipitation and evaporation/sublimation were 56.9 mm month−1 and 43.3 mm month−1 , respectively. The level of the estimated evaporation/sublimation was about 5 mm month−1 in winter and more than 100 mm month−1 in July–August. The quality of lake water is determined by the water balance and biogeochemical processes in the lake. Impurities are received by atmospheric deposition, resuspension of sediments, and inflow from the drainage basin. Their concentrations are changed internally by biochemical processes, and they are lost by sedimentation and outflow. The water balance Eq. (2.3) can be modified for concentrations of different Fig. 2.3 A case of annual course of the water balance in Lake Vanajavesi, southern Finland: a inflow and outflow; b lake–atmosphere water exchange. The area of the lake is 103 km2 , the mean depth is 7.7 m, and the area of the drainage basin is 2774 km2 . The data are from Jokiniemi (2011) except for sublimation taken from Yang et al. (2012)

24

2 Freezing of Lakes

Table 2.3 Classification of lakes based on the salinity (S) of the water Category

Salinity (‰)

Physics

Freshwater lakes

S < 0.5

Salinity has only marginal influence

Saline lakes • Brackish lakes

0.5 ≤ S ≤ 35 0.5 ≤ S < 24.6

Salinity at most at the oceanic level Freezing point < temperature of max density in brackish water

Hypersaline lakes

S > 35

Salinity greater than the oceanic level

substances, noting that evaporation and sublimation remove only pure water. The renewal time of each substance can be defined similarly to the flushing time of the water mass. Freezing has no influence on the annual budgets of impurities, but rejection of impurities in ice growth enriches the water under ice while melting has an opposite effect. This can be a major issue in very shallow lakes (Yang et al., 2016). On the other hand, precipitation is accumulated on the ice cover during an ice season, and this storage is released to the liquid water body during the melting period, which is much shorter than the ice growth period. In limnology, the concentration of dissolved salts is expressed as the fractional mass (salinity S) or as the mass per volume (cd ). The salinity is given in parts per thousand or per mill (‰) and the mass in grams or milligrams per liter. Clearly cd = ρw S, where ρw is the density of the water sample. Lakes are classified by salinity to fresh, brackish, saline, and hypersaline types (Table 2.3). Salinity is a conservative thermodynamic state variable of lake water and plays an important role in the stratification, mixing, circulation, and freezing in saline lakes. Brackish water salinity has the temperature of maximum density above the freezing point, while in hypersaline lakes, salinities can be even above 300‰ and the freezing points is very low, even down to − 50 °C. The chemical composition of lake water plays an especially significant role in hypersaline lakes, for which lake specific calibration for physical properties of water would be desirable. Saline lakes form when evaporation/sublimation plays a dominant role in the water balance, such as Lake Aral in Central Asia and Great Salt Lake in Utah. A specific class of saline lakes are tidal lakes and lagoons, which exchange water with the ocean (Dailidiene et al., 2011; Shirasawa et al., 2005). In the northern part of the Baltic Sea, due to glacial land uplift lakes becomes slowly isolated from the sea and the salinity consequently decreases (e.g., Lindholm et al., 1989). These lakes are called fladas. Example 2.4 During the recent decades, the Tibetan plateau lake system has undergone major changes due to the increased air temperature and precipitation (see Qiu et al., 2019). Climate warming has brought shorter ice seasons. But increased precipitation has brought higher water volumes and therefore lower salinities in the several saline lakes in the region, and, even more, retreating glaciers are releasing more freshwater than before that also lowers lake salinities. This desalination has increased the temperatures of the freezing point and maximum density that has impact on the freezing date in addition to the impact of increasing air temperature. What further

2.1 Lake Types and Characteristics

25

Table 2.4 Physical properties of pure water at the freezing point temperature and gauge pressurea zero and the variability of these properties in cold freshwater lakes

Value (T = 0 ◦ C, p = 0) Variability T < 4 ◦ C, S < 0.50/00 Property Molecular weight

18.015 g mol−1

Density

999.8 kg m−3

< 0.1% < 0.1%

1.79 ×

10−3

75.6 N

m−1

0.51 ×

10−10

1402 m

s−1

Specific heat

4.22 kJ

kg−1

Thermal conductivity

0.561 W m−1 °C−1

Viscosity Surface tension Compressibility Speed of sound

Ns

m−1

≈ 10%; 1.57 × 10−3 N s m−1 at 4 °C < 1%; 75.1 N m−1 at 4 °C

Pa−1

Very small ≈ 1%; 1421 m s−1 at 4 °C

°C−1

kg−1

< 0.1% ≈ 1%

Latent heat of freezing

333.5 kJ

Latent heat of evaporation

2.49 MJ kg−1

– < 1%

Electric conductivityb

< 1 µS cm−1

High; ≈ 100 µS cm−1 for S = 0.1‰

Relative permittivity

87.9

Sensitive to salinity

a Gauge

pressure is the pressure with reference to the ambient atmospheric pressure b In pure water the electric conductivity is extremely small but then rises fast with salinity

complicates the case are possible modifications in stratification, circulation, and evaporation in the lake. The key lake water properties in ecology are pH and the concentrations of dissolved oxygen, carbon dioxide, nutrients, and harmful substances. Ecological classification of lake waters is based on the trophic status: the ultra-oligotrophic, oligotrophic, mesotrophic, eutrophic, and hyper-eutrophic lakes. The trophic status is related to the concentrations of the main nutrients, i.e., phosphorus and nitrogen. Consumption of oxygen is faster with higher trophic status, and therefore eutrophic lakes and more likely to reach anoxic conditions in winter.

2.1.3 Physical Properties of Lake Water The physical properties of pure water depend weakly on the temperature (T ) and pressure1 (p) (Table 2.4). Pressure needs to be considered only in deep lakes. At the surface of high mountain lakes, the air pressure is low that has influence primarily on the saturation levels of dissolved gases. In natural waters, salinity modifies the physical properties but apart from electromagnetic properties its influence is small for fresh water (S < 0.50/00). The density of lake water, ρw = ρw (T , S, p) must be considered in detail, since even very small differences influence the stratification and circulation. The present 1

Usually pressure is given in bars; the SI unit is Pascal, 1 bar = 100 kPa.

26

2 Freezing of Lakes

advanced method to calculate the density is derived from the seawater standard TEOS-10 (Thermodynamic Equation of Seawater) based on the absolute salinity (see Millero et al., 2008). TEOS-10 mathematical algorithm is highly complicated and cannot be presented as an explicit formula, but there is free software available in the web for users. Since the ionic composition of water varies between lakes, in principle, each lake needs an individual equation of state (Boehrer et al., 2010). The TEOS algorithm was modified for lake water applications by Pawlowicz and Feistel (2012), and a practical solution was presented by Moreira et al. (2016). For a general approximation, the UNESCO 1980 equation of state of sea water (UNESCO, 1981) has been employed for limnological applications. The equation of state is then expressed in explicit form as: ρw = ρw (T , S, p) =

ρw (T , 0, 0) + (T , S, 0) 1 − K (T p,S, p)

(2.5)

where ρw (T , 0, 0) is the density of pure water at gauge pressure zero, (T , S, 0) is the salinity correction, and the denominator gives the pressure correction with the bulk modulus K (T , S, p). Usually, salinity is ignored in freshwater bodies and pressure is ignored in shallow lakes. However, even in freshwater lakes, salinity may dominate the stratification below ice cover since in cold water the influence of temperature on density is minimal. For many applications, such as first-order analysis and mathematical modelling, Eq. (2.5) is approximated by a second-order polynomial. A convenient form for cold water (T ≤ 10 ◦ C) with low salinity (S ≤ 100/00) and gauge pressure zero reads: ρw (T , S, 0) = ρw (0, 0, Tm0 ) − a(T − Tm0 )2 + (b0 − b1 T )S

(2.6)

where ρw (0, 0, Tm0 ) = 999.84 kg m−3 is the reference density, Tm0 = 3.98 ◦ C is the temperature of maximum density of pure water at p = 0, S is in ‰, and {a, b0 , b1 } are the parameters of the formula. Taking these parameters so that the approximation is exact for pure water at the temperatures 0 °C and Tm0 , and the depression of the temperature of maximum density with salinity is properly followed, we have a = 8.0 × 10–3 kg m−3 °C−2 , b0 = 0.82 kg m−3 , b1 = 3.6 × 10−3 kg m−3 °C−1 . Equation (2.6) resembles the approach for individual lakes by Moreira et al. (2016), who used lake-specific salinity parameters {b0 , b1 } in the salinity correction. Example 2.5 The average density of a human body is 1062 kg m−3 (Krzywicki & Chinn, 1966). If the water salinity is more than 75‰, the water is denser and a human body floats on the surface. There are many hypersaline lakes, which satisfy this requirement, a famous one is the Dead Sea (S ≈ 315‰, ρw ≈ 1240 kg m−3 ). Freezing lakes vary widely in type (Fig. 2.4). Freshwater lakes are vertically mixed at the temperature of maximum density, and therefore the depth of a lake is one of the principal characteristics to influence the timing of the freezing. The lateral size of a lake influences the mixing conditions, since long wind fetches create

2.1 Lake Types and Characteristics

27

Fig. 2.4 Lakes in the cold climate zone: a Ephemeral, Lake Müggelsee, Berlin; b Boreal, Lake Pääjärvi, southern Finland; c Tundra, Lake Kilpisjärvi, Lapland; d Deep, non-freezing lake, Lake Shikotsu, Hokkaido. Photographs by the author

more turbulence and higher waves and, consequently, deeper forced mixing. Thus, the deeper or larger the lake, the later will the freezing be. Shallow bays freeze first, and deep, large basins remain open to the latest. In saline lakes, mixing by cooling may be limited by salinity stratification, and the temperature of maximum density and the freezing point temperature depend on the salinity. The vertical pressure distribution in a lake is obtained from the hydrostatic law dp = −ρw g dz

(2.7)

where z is the vertical co-ordinate positive upward, and g = 9.81 m s−2 is the acceleration due to gravity. If the density is constant, we have p(z) = p(ζ ) + ρw g(ζ − z), where ζ is the surface elevation; we can take p(ζ ) = pa = air pressure or p(ζ ) = 0 = gauge pressure. A 10.3-m freshwater column adds one standard atmosphere (1013.25 m bar) to the pressure. In the equation of state (Eq. 2.5), the bulk modulus is K ≈ 20 kbar, and thus at 100 m depth the density is greater than at the surface by about 0.5 kg m−3 or 0.05%. Transfer of solar radiation into a lake acts as a source of heat and provides photons for primary production. The transfer depends on the concentration of coloured dissolved organic matter (CDOM) and turbidity of the water (Arst, 2003; Arst et al., 2008; Kirk, 2011). CDOM is a chemically complex mixture including humus, and

28

2 Freezing of Lakes

it can be visually observed in the degree of brown colour in the water. CDOM concentration is expressed by its absorption coefficient at a fixed wavelength (often 440 nm) measured from filtered lake water. Turbidity is related to the concentration of suspended matter and chlorophyll a. The colour and turbidity together determine the depth of visibility or the Secchi depth,2 which is more than 5 m in clear water lakes and less than 1 m in strongly brown, turbid, or eutrophic waters. The light penetration depth of lake water, defined as the e-folding distance of light intensity, provides the scale for the mixing depth in calm waters from solar heating. Secchi depth is about twice the light penetration depth. The euphotic depth reference is normally taken as the depth where the level of solar radiation is 1% from the incident radiation. In the presence of an ice cover, a large fraction of incident radiation is lost by reflectance, and the resulting euphotic depth is much less than in summer.

2.2 Ice-Covered Lakes 2.2.1 Freezing of Lakes The temperatures of the freezing point and maximum density of pure water are 0 °C and 3.98 °C, respectively, at the standard atmospheric pressure. With increasing salinity or pressure, these temperatures become lower. The temperature of maximum density can be mathematically obtained from the equation of state, but the expression is cumbersome and lower order fits are therefore employed. For seawater, the temperature of maximum density (Caldwell, 1978) and the freezing point (UNESCO, 1981) are, respectively, Tm = 3.982 − 0.2229 · S − 0.02004 · p(1 + 0.00376 · S)(1 + 0.000402 · p) (2.8a) T f = −0.0575 · S + 1.710523 × 10−3 S 3/2 − 2.154996 × 10−4 S 2 − 7.53 × 10−3 p (2.8b) where the temperature is in centigrade degrees, salinity is in per mill, and the pressure is the gauge pressure is bars. These formulae are normally used for lake water and serve for a general approach in this book. They are valid for the salinity up to 40‰ and can be inaccurate and biased for hypersaline lakes. The freezing point depends on the chemical composition of the dissolved substances. At high salinities, the freezing point depression is strongly non-linear, since different salts crystallize at their individual eutectic temperatures (for seawater, see Assur, 1958). Therefore, in hypersaline lakes the freezing point formula is more clearly lake specific. The freezing point can be very low. The Great Salt Lake of Utah 2

Secchi depth is the maximum depth where a white disk, diameter 30 cm, is visible from the surface.

2.2 Ice-Covered Lakes

29

does not freeze, except at freshwater inflows, although the winter weather is cold and snowy. The temperature of maximum density decreases with increasing salinity faster than the freezing point. In the surface water of freshwater lakes (S < 0.50/00), 3.87 ◦ C < Tm ≤ 3.98 ◦ C and −0.07 ◦ C < T f ≤ 0 ◦ C. In saline lakes, these temperatures are equal at (T = −1.28 ◦ C, S = 24.60/00), which is here taken as the upper limit of the salinity of brackish water, i.e., the maximum density of brackish water is above the freezing point (Fig. 2.5). However, the stable stratification at T < Tm is weaker the higher the salinity. Also, pressure has a significant influence on the temperature of maximum density and freezing point. For pure water, at 100 m depth, Tm = 3.78 ◦ C and T f = −0.07 ◦ C, and at 1 km depth they are Tm = 1.94 ◦ C and T f = −0.74 ◦ C. The temperature of maximum density decreases faster than the freezing point, but they become equal only at very high pressure. Example 2.6 Lake Baikal is a deep, freezing freshwater lake in Siberia. At its maximum depth of 1.6 km, the temperature of maximum density is 0.64 °C, the freezing point is − 1.18 °C and the density of water is 1008 kg m−3 . The depression of the temperature of maximum density with pressure has a key role in vertical Fig. 2.5 Density of water as a function of temperature for different salinities at gauge pressure equal to zero: a Pure water density as a function of temperature, b Density difference between the maximum density and the density at 0 °C in fresh and brackish water (salinities 0, 1, 5 and 10‰). The plots are based on the seawater equation of state

30

2 Freezing of Lakes

mixing in Lake Baikal (Piccolroaz & Toffolon, 2013). The sub-glacial Lake Vostok is located at 4 km beneath the Antarctic ice sheet (e.g., Siegert et al., 2001). The pressure is about 350 bar at the lake surface, and, consequently, if the water were fresh, then Tm = −4.02 ◦ C and T f = −2.64 ◦ C. A first-order zonation of potentially freezing lakes can be taken as the land areas where the January (Northern Hemisphere) or July (Southern Hemisphere) mean air temperature is less than the freezing point of fresh water (Fig. 2.6). This region is called the cold climate zone in the present book. There, at least very shallow freshwater lakes freeze in normal years. The cold climate zone covers most of Eurasia and North America down to around 40° N latitude at sea level (see Bates & Bilello, 1966; Hutchinson & Löffler, 1956). In the Central Asian mountains, the boundary is more south, below 30° N (Qiu et al., 2019), while in the Western Europe it is more north due to the heat received from the North Atlantic current. In the Southern Hemisphere, the only land masses between 45° S and the Antarctic continent are some small islands and the southern tips of New Zealand and South America with minor freezing inland water basins, such as Lake Alta, a small and deep cirque lake at 45° 03' S 168° 49' E, 1800 m above sea level in the New Zealand Southern Alps. In South America, the cold zone even reaches the Equator in the high Andes, where glacial lakes exist in front of mountain glaciers, e.g., Lake Palcacocha in Peru at 9.5° S, 77.5° W, 4.5 km altitude. In the cold climate zone there are lakes, which do not freeze. In fall, the surface temperature of lakes follows behind the decreasing air temperature due to heat storage collected in summertime to the water body and the bottom sediments. If the thermal time scale of a lake is longer than the cold winter period, the lake remains open. Such lake is, for example, Lake Shikotsu (mean depth 266 m), a caldera lake in Hokkaido, Japan, where the mean January air temperature is − 5 °C (Fig. 2.4d). Another possibility is geothermal heating which may keep the surface temperature high due to convective mixing. An example is Lake Hévíz in Hungary, where the water temperature is above 20 °C in winter although other lakes in the region are normally frozen. Lake Hévíz is the largest geothermal lake in the world (47,500 m2 area) and has been used for bathing at least since the Roman times. Finally, the salinity depression of the freezing point temperature may prevent lakes from freezing in the cold climate zone. At extreme, the hypersaline Don Juan Pond in McMurdo Dry Valleys in Victoria Land, Antarctica with salinity around 400‰ is unfrozen even at temperatures below – 50 °C (Meyer et al., 1962).

2.2.2 Vertical Mixing and Seasons The stratification of lake waters shows a stable or neutral density structure. Unstable stratification does not persist in natural conditions, and an increase of surface water density leads to convective mixing to the depth where a corresponding density is

2.2 Ice-Covered Lakes

31

Fig. 2.6 Mean a January and b July air temperature (°C) on the Earth’s surface, 1959–1997 (Global Climate Animations, Department of Geography, University of Oregon based on NCEP/NCAR Reanalysis Project 1959–1997 Climatologies Data). © Department of Geography, University of Oregon. Printed by permission

32

2 Freezing of Lakes

Fig. 2.7 Temperature stratification in dimictic and cold monomictic lakes. The dotted rectangle denotes for autumn and spring overturn of the water mass

found. In freshwater lakes, a full turnover of the water mass takes place at the temperature of maximum density with renewal of the oxygen storage. The annual frequency of the turnover events is the basis of limnological classification of lakes (Fig. 2.7). Most seasonally freezing freshwater lakes have annually two turnover events— in spring and fall—and thus are dimictic. The summer stratification shows warm upper layer and cold lower layer3 with a thermocline layer between, and in winter the stratification is reversed. The time between spring and fall turnover becomes shorter with colder climate, and finally the turnover events merge together—the lake is then cold monomictic. Turnover periods are especially strong in the cooling season. Convective mixing can continue by wind forcing to temperatures below the maximum density temperature (Homén, 1903; Pulkkanen & Salonen, 2013). In icecovered lakes, spring convection may begin under ice that limits oxygen renewal. Summer stratification forms after ice breakup, fast if the water body had warmed beneath the ice. In saline lakes, a permanent salinity stratification may form depending on the heat and salt balances. The surface salinity, and consequently density, may increase due to evaporation or salt rejection in freezing. A layer with a sharp change in salinity is called a halocline, analogous to the thermocline. There may be a system with several thermoclines and haloclines, in contrast to freshwater lakes with just one permanent seasonal thermocline. If a halocline is very weak, local turnover may take place by thermal convection or mechanical mixing. If this is not the case, the lake is meromictic, referring to the existence of a permanent halocline and anoxic bottom layer. E.g., in Finland there are many small meromictic lakes developed during the Holocene. Example 2.7 Consider a two-layer lake with temperature Tb and salinity Sb in the lower layer and salinity S0 < Sb in the upper layer. Turnover is possible in cooling, if ρw (Tm , S0 ) > ρw (Tb , Sb ). This inequality can be exactly solved from the equation of state, but a first-order approximation can be taken when S0 , Sb < 100/00 from Eq. (2.6). The turnover condition is: 3

In limnology the upper layer is called epilimnion and the lower layer is called hypolimnion.

2.2 Ice-Covered Lakes

33

Table 2.5 Classification of lakes according to the quality of the ice season Ice cover

Explanation

Surface temperature T0

Limnological typea

Ice-free

Ice does not form

Winter T0 ≥ Tm

DM, WM, MM

Ephemeral

Ice may form

Winter T0 < Tm or T0 = T f ≥ Tm

DM, MM

Seasonal

Ice forms and melts

Winter T0 = T f

DM, CM, PM, MM

Perennial

Multi-year ice

Summer T0 ≈ T f

CM, AM, PM, MM

a DM

dimictic, CM cold monomictic, WM warm monomictic, PM polymictic, AM amictic, MM meromictic

Sb − S0
24.60/00, this pre-winter disappears since convection continues until freezing. Years in ice-free, ephemeral, seasonal and perennial ice lakes are qualitatively different. The ephemeral zone has both ice-free years and ice years, and it forms the boundary zone between ice-free and regular seasonally ice-covered lakes. Between the perennial and seasonal zone there is a quasi-perennial zone, where lakes sometimes have ice-free summer. The quasi-perennial zone is quite narrow, but the ephemeral zone is wide. Kirillin and Leppäranta (2022) classified the lake ice zones based on the freezing-degree-days. A summary of the lake classification systems is given in Table 2.6. There have been approaches to classify the size of lakes based on physical processes. Kirillin et al., (2012) focused on the role of Coriolis acceleration, which lead to the baroclinic Rossby radius of deformation as the size criterion. The transition between “small” and “large” is at the scale of 1 km. Ashton (1980) classified freezing lakes into small lakes (lakes with stable ice cover), reservoirs (stable ice cover and significant through flow), and large lakes (lakes with drift ice), where the distinction between “small” and “large” is at the order of 10 km.

2.2.3 Seasonal Lake Ice Zone The zone of seasonally freezing lakes covers large areas of the continents between 40° and 80° latitudes (Fig. 2.8). Bates and Bilello (1966) estimated the southern boundary of the seasonal ice cover in the Northern Hemisphere to follow approximately the

34

2 Freezing of Lakes

Table 2.6 Lake classification systems Background

Object

Basis

Geological

Lake basin

Origin of the depression where the lake has formed

Physical

Size

Horizontal extent and depth

Geochemical

Salinity

Influence on water density and other properties

Limnological

Mixing

Frequency of convective overturns

Optical

Colour and turbidity

Optically active substances

Biological

Trophic status

Level of chlorophyll and nutrients

Cryospheric

Ice

Quality of ice season

latitude 40–45° N, higher in the Western Europe and lower in Central Asia and North America. This is close to the diagram of Hutchinson and Löffler (1956). In the Tibetan plateau there are freezing lakes at the latitude of 30° N but their altitude is above 4 km (Qiu et al., 2019). Basically, where the mean air temperature in the coldest month is below the freezing point, shallow freshwater lakes have potential to freeze over (see Fig. 2.5).

Fig. 2.8 The zone of seasonally freezing lakes in the northern hemisphere (Bates & Bilello, 1966) and the January 0 °C climatological isotherm taken from Fig. 2.6. The contours 100 and 180 refer to the mean length of the ice season in days

2.2 Ice-Covered Lakes

35

The seasonal lake ice zone can be further sub-classified into lakes with a stable or unstable ice cover. The stable case is a landfast ice cover (Fig. 2.9), while an unstable ice cover may break and does not necessarily reach complete extent. This characteristic is lake-specific, since ice formation is strongly dependent on the local climate and the depth, size, and morphology of the lake basin. In a medium-size lake (10 km scale), the thickness of ice needs to be more than 10 cm to span a solid, stable ice cover across the lake. For example, in the present climate, the lakes in Finland usually have a stable ice cover, but two large lakes nearby, Ladoga (Prokacheva & Borodulin, 1985) and Peipsi/Chudskoe (Reinart & Pärn, 2006) in Russia and Estonia, have an unstable ice cover. Many large lakes have numerous islands and, consequently, limited fetches that protect the ice from breaking. The Central Asian dry climate zone possesses many freezing lakes. The low altitude brings strong solar radiation all year, and in the dry climate there is very little snow, so that the lakes are covered by a transparent ice lid. Solar radiation flux through ice is large during winter, providing heat to the water body and photons for the primary production (Song et al., 2019; Su et al., 2019). There the lake water environment is quite different from high latitude conditions. Apart from the ephemeral zone, the mean annual maximum ice thickness in seasonally freezing lakes ranges from 0.2 m to around 2 m. The overall maximum

Fig. 2.9 Wuliangsuhai Lake in Inner Mongolia. Its latitude is low (40.5° N), but the lake freezes over annually, and the maximum annual ice thickness is about half meter. Photograph Dr. Yang Fang, printed with permission

36

2 Freezing of Lakes

Fig. 2.10 Lake Constance (Bodensee) froze over in winter 1963. The ice was thick enough for cars to drive on, and pressure ridges formed since wind fetches are long enough to build up high pressure in the ice cover (Zintz et al., 2009). Photograph by Mr. Julius Pietruske, printed by permission

is within 2–3 m: the sea ice data from the Siberian shelf show that seasonal sea ice can grow to approximately 2 m in one winter, and, as a theoretical upper limit, 2.7 m has been suggested based on the freezing-degree-days in the coldest place in the northern hemisphere, Oymakon, East Siberia (Kirillin et al., 2012). The ephemeral zone extends through eastern Europe, but in the western Central Europe, apart from high mountains, lake ice is quite rare. In this zone the nature as well as the peoples need to adapt life for both ice and ice-free winters. Winter 1963 was the latest very cold winter in Central Europe, and at this extreme, even the deep Lake Constance bordering Austria, Germany and Switzerland froze over (Fig. 2.10).

2.2.4 Lakes with Perennial Ice There are two kind of lakes with perennial ice. In very cold climate at high latitudes or altitudes, at least some of the ice grown in winter may survive over summer and become multi-year ice. In northern Canada and Alaska, there are lakes where the winter ice cover contains first-year ice and multi-year ice (Veillette et al., 2010). Occurrence of multi-year lake ice was observed in Colour Lake, Axel Heiberg Island, Nunavut (Adams et al., 1989) but it is there a rare event. Another lake with occasional multi-year ice is Teshekpuk Lake in Alaska (Arp & Jones, 2011). The partial summer ice cover is something delicate between fully ice-covered state and seasonal ice cover (Fig. 2.11). The conditions of occurrence must be quite specific. In cold summers ice may survive, e.g., in bays sheltered from direct solar radiation. To the author’s knowledge, in the present climate, the coldest place in the Northern Hemisphere— Siberia—contains only seasonally ice-covered lakes. Perennially fully ice-covered lakes exist in Antarctic continent in very cold and arid climate conditions, e.g., in the McMurdo Dry Valleys (Priscu, 1998). The ice cover is snow-free, clear ice. The surface heat balance shows a predominant loss of heat, while in summer solar radiation causes internal melting of the ice and,

2.2 Ice-Covered Lakes

37

Fig. 2.11 In Schirmacher oasis, Dronning Maud Land, Antarctica there are lakes where lake ice may survive over summer. The picture shows such lake at the Russian Novolazarevskaya Station (70° 49.3' S 11° 38.7 E) in late summer, February 4, 2005. Photograph by the author

penetrating the ice cover, provides heat to the water body underneath. Due to the low thermal conductivity of ice, the absorbed solar heat does not leak easily from the water body and the liquid phase survives over the winter. The case is similar to the so-called cool skin phenomenon in lakes and seas during calm and clear summer days. Thus, this kind of perennially ice-covered lake is at equilibrium state when solar radiation transfer through the ice cover keeps a permanent liquid water body beneath and low light absorption by ice keeps the ice cover. The thickness of ice has been observed to be several meters in these lakes, in Lake Vida as much as 20 m (Bar-Cohen et al., 2004). There are three types of proglacial lakes at glaciers and ice sheets (e.g., Menzies, 1995). Epiglacial lakes are found on the bare ground close to the boundary of land ice masses (Kaup, 1994; Leppäranta et al., 2020; Shevnina et al., 2021), subglacial lakes are very old water bodies at the base of an ice sheet (e.g., Leitchenkov et al., 2016; Siegert et al., 2001), and supraglacial lakes exist in blue ice regions in the glacial surface layer (Bajracharya & Mool, 2009; Leppäranta et al., 2013; Winther et al., 1996). Epiglacial lakes are as normal cold region lakes on ground, fed mainly by glacial melt water inflow. They are an important habitat of life in the polar world. Normally these lakes have seasonal ice cover but blocks of glacial ice may be present.

38

2 Freezing of Lakes

Subglacial lakes are one of the most exciting geographical findings of the twentieth century. The first evidence of their existence was from radio-echo soundings in the 1960s at the Vostok Station (77° 28' S 106° 50' E) of Soviet Union (see Priscu & Foreman, 2009). Subglacial lakes are not yet well known but they form very specific physical and microbiological systems at the bottom of the Antarctic ice sheet. The water temperature is at the melting point corresponding to the ambient pressure, about − 3 °C. The water body must be in thermal balance between the geothermal heat flux from the ground and conduction of heat through the overlaying ice sheet. The largest subglacial lake is Lake Vostok, located close to the Vostok station. Supraglacial lakes form and grow from the penetration of solar radiation into the ice (Fig. 2.12). For sufficiently strong penetration, the surface needs to be bare ice, and therefore these lakes form in glacial blue ice regions. In summer, after the glacial surface layer has warmed up, supraglacial lakes form from ice melting by up to 2–3 cm day−1 water production per unit area or of the order of one metre in one summer. These lakes can be seasonal or perennial, their base is ice, their water temperature is near the freezing point, and they may have an ice cover. They are extremely low in biota (Keskitalo et al., 2013). Supraglacial lakes are found on the Antarctic and Greenland ice sheets and in many mountain glaciers. In Greenland perennial supraglacial lakes have a major impact on the ice mass balance (Hoffman et al., 2011; Liang et al., 2012).

Fig. 2.12 Household water is pumped from the ice-covered supraglacial Lake Suvivesi for the Finnish Station Aboa, western Dronning Maud Land, Antarctica (Leppäranta et al., 2013). Photograph by the author

2.3 Climate and Water Budget

39

2.3 Climate and Water Budget 2.3.1 General Regional Climate The local weather determines to a large degree whether a lake freezes over or not. In practice, the air temperature must be below the freezing point for a period depending on the lake depth, and the heat loss from the lake must overcome any geothermal heating. The world’s climate conditions are classified according to the Köppen– Geiger system into five main classes depending on the monthly mean temperature T and precipitation P (Fig. 2.13): A Equatorial climates

Tmin > 18 ◦ C

B Arid climates

Threshold depends on Tmean and P

C Warm temperate climate

−3 ◦ C < Tmin < 18 ◦ C

D Snow climates

Tmin < −3 ◦ C

E Polar climates

Tmax < 10 ◦ C

Lakes freeze regularly in the snow climate and polar climate. Also, in cold and arid climate regions and at the cold boundary of the warm temperate climate zone freezing lakes are common. Apart from very large lakes, the size of lakes is small compared to weather systems, and therefore the atmospheric conditions over a given lake are quite homogeneous. However, more variability is seen in the drainage basins, which are often much larger than the lake itself. Large-scale atmospheric circulation characteristics are described by indexes based on the atmospheric pressure distribution. In Europe, the NAO (North Atlantic Oscillation) index is widely used. NAO-index equals the normalized anomaly of the difference in the height of the 500-mbar atmospheric pressure level between Ponta Delgada, Azores and Stykkisholmur, Iceland, telling of the intensity of the westerlies. When the index is positive, the westerly winds are stronger than normal and winters in Europe are warm, and vice versa. The index value 1 (− 1) means that the westerlies are one standard deviation stronger (weaker) than the average. Much of the variability of lake ice seasons in Europe is correlated with NAO (e.g., Blenckner et al., 2004). This is trivial but it is noteworthy that the correlation is quite high, and therefore the regional climate has a strong role in lake ice climatology not hidden by specific local conditions or individual lake properties. In the land regions around the Pacific Ocean, El Niño–Southern Oscillation (ENSO) is connected to anomalously warm and cold winters (e.g., Mishra et al., 2011). NAO cannot be predicted for future, but ENSO has a significant intra-seasonal memory so that seasonal forecasting of ice winters for the Great Lakes of North America can be performed (Wang et al., 2012). The influence of weather and climate on frozen lakes is given by the solar radiation and the exchange of mass, heat and momentum between lakes and the atmosphere. The air–lake fluxes depend on the lake surface temperature and air temperature, humidity, wind speed, cloudiness, precipitation, and air pressure. Standard weather

Dfa

Cwb Cwc

10

−40

−50

50

−60

−70

−80

BWh

Csb

Dsc

−90

BSk

90

Dsa

Af

Am

Cfc

Cwa

Aw

Af

Cfb

Csb

BWk

ET

Csa

Csb

ET

Cfc

BSk

Cfa

Cfb

Aw

Am BSh

−60

Af

As

EF

−60

−40

−40

−20

Am

Csa

Csb

0

Am

BWh

Cfb

20

EF

40

40

Cfb

Cwb

Cwa BSh BWh

60

Cfa

BSh

Dfb

60

80

100

Af

BSk

Dfc

100 ET

BWh

Aw

Cwb

Dfa

80

120

120

BWh

BWk

Dfd

160

160

Cwa

ET

Aw

Csa

140

Am

BSh

140

180

Aw

Cwb

Dwc

Dwd

180

W: desert S: steppe f: fully humid s: summer dry w: winter dry m: monsoonal

A: equatorial B: arid C: warm temperate D: snow E: polar

Aw

Af

Csa

Aw

Csb

BWk

20

ET

EF 0

Cwa

Csc

−20

Dwa Dwb Dwc Dwd

Cfb

−160 −140 −120 −100 −80

BWh

Aw

Aw

Cfb

Dfc Dfb Dfa Cfa

BSh

Dsd

Cfa

−160 −140 −120 −100 −80

Dsc

BSh

Dsb

BSk

Precipitation

Main climates

Csb

Csa

Af

Am

−80

Cfa

Cwa

Dwa

Dwb

80

BSk

Cfa

−60

Cfb

Af

60

−50

50

−40

40

−30

−20

−10

0

10

20

30

Version of April 2006

Kottek, M., J. Grieser, C. Beck, B. Rudolf, and F. Rubel, 2006: World Map of KöppenGeiger Climate Classification updated. Meteorol. Z., 15, 259-263.

−70

BWh

BSh

Aw

Dfc

70

h: hot arid F: polar frost k: cold arid T: polar tundra a: hot summer b: warm summer c: cool summer d: extremely continental

Temperature

Fig. 2.13 Köppen-Geiger classification of climates (Kottek et al., 2006). Temperatures in Table are monthly means. Printed by permission

http://gpcc.dwd.de http://koeppen-geiger.vu-wien.ac.at

−30

−20

−10

0

30

20

40

60

80

Dfd

BWk BWh

Dfc

70

Aw

Dfb

Resolution: 0.5 deg lat/lon

As

Am

Af

updated with CRU TS 2.1 temperature and VASClimO v1.1 precipitation data 1951 to 2000

World Map of Köppen−Geiger Climate Classification

40 2 Freezing of Lakes

2.3 Climate and Water Budget

41

station data include temperature and humidity at 2 m altitude, wind speed and direction at 10 m altitude, precipitation, and air pressure reduced to sea level altitude. Some stations also provide cloudiness in units of 1 /8 or per cent. Direct radiation measurements are rarely available, and therefore the cloudiness becomes the key variable to estimate the radiation balance. Table 2.7 presents climatological data from two stations: Jokioinen (snow climate, boreal forest) and Utsjoki (polar climate, tundra). Routine weather stations are located on land, and using their data over a lake needs particular consideration. True weather data over a lake can be obtained from floating stations (Fig. 2.14). In general weather and climate are the major factors in the mass and heat budget of lakes. During the ice season precipitation is accumulated on lake ice to add on the mass budget (see Sect. 2.1.2). The surface heat balance of freezing lakes consists of solar radiation, terrestrial radiation, turbulent sensible and latent heat fluxes, and heat transfer via precipitation. The last term can be significant in transient processes with phase changes, i.e., snowfall on open water or rain on snow or ice cover, but it is unlikely to show up as an important component in lake climate statistics. Solar radiation penetrates into snow, ice and water, but the other heat fluxes are pure surface fluxes.

2.3.2 Radiation Balance The radiation balance consists of solar and terrestrial radiation, also called, respectively, shortwave and longwave radiation. The net radiation provides the governing forcing to the annual course of freezing lakes. The physical basis of energy transfer by radiation is the Planck’s law of black body radiation, which gives the spectral distribution of the emitted electromagnetic radiation of a black body as a function of temperature (in degrees K): E B (λ; T ) =

8π hc02    λ5 exp λkhcB0T − 1

(2.9)

where λ is the wavelength, h is the Planck’s constant, c0 is the speed of light in vacuum, and k B is the Boltzmann’s constant. This distribution has a single peak and vanishes at λ → 0 and λ → ∞. With increasing body temperature, the level of radiation increases and the peak shifts to shorter wavelengths. The radiation spectrum of a grey body is εE B (λ; T ), where ε is the emissivity, 0 < ε < 1. Planck’s law integrated over the wavelength gives the Stefan-Boltzmann law, which tells that the radiative heat flux of a black body is proportional to the fourth power of the absolute temperature: Q = σ T04 , where σ = 5.6704×10−8 W m−2 K−4 is the Stefan-Boltzmann constant. For a grey body, we have Q = εσ T04

(2.10)

42

2 Freezing of Lakes

Table 2.7 Climatological data from the normal period 1981–2010 for Jokioinen (60° 48' N 23° 30' E, 104 m) and Utsjoki Kevo (69° 45' N 27° 00' E, 107 m) Month

Temperature

Precipitation

Solar radiation

Relative Cloudiness humidity

Wind speed

Wind direction

°C

mm

W m−2

%

m s−1

°

1/8

Jokioinen January

− 5.6

46

11

89

6.4

3.6

165

February

− 6.3

32

38

87

6.0

3.5

171

March

− 2.4

32

87

82

5.5

3.5

168

April

3.5

30

152

72

5.6

3.5

153

May

9.8

41

208

65

5.1

3.5

163

June

14.0

63

225

68

4.9

3.3

235

July

16.7

75

218

71

5.3

3.1

201

August

15.0

80

162

77

5.6

3.1

150

9.9

58

97

83

5.9

3.3

194

September

4.9

66

41

88

6.2

3.5

194

November

− 0.2

57

13

91

6.7

3.7

164

December

− 3.9

47

6

91

6.6

3.7

173

October

Utsjoki Kevo January

− 14.0

27

1

85

5.5

2.8

165

February

− 12.8

24

15

83

5.4

2.9

171

March

− 8.2

21

66

81

4.7

3.0

168

April

− 2.5

25

146

76

5.1

2.9

153

May

3.7

27

181

72

5.1

3.0

163

June

9.6

50

195

68

5.4

3.2

235

July

13.1

72

170

74

5.1

2.9

201

August

10.7

57

116

80

5.8

2.6

150

5.7

38

61

85

5.9

2.6

194

September October

− 0.5

39

29

88

6.0

2.7

194

November

− 8.3

28

1

88

5.8

2.6

164

December

− 12.3

25

0

86

5.8

3.7

173

The data are from Pirinen et al. (2012) expect cloudiness which is from 1971–1980 (FMI 1982) for Jokioinen and 2010–2020 for Utsjoki Kevo

The Sun radiates almost as a black body at a temperature of 5900 K, and the radiation peaks at 480 nm. The total solar power on the Earth is given by the solar constant, which is defined as the annual average of solar radiation incident on a plane perpendicular to the solar rays on the top of the atmosphere. Its value is equal to Q sc = 1367 W m−2 (e.g., Iqbal, 1983). The solar constant has been quite stable during the measurement history from about year 1900. Solar radiation incident on the

2.3 Climate and Water Budget

43

Fig. 2.14 Ice station Lotus, Lake Pääjärvi in southern Finland (e.g., Jakkila et al., 2009; Leppäranta et al., 2017; Wang et al., 2005). Instruments are deployed on and below the float. Solar power panels and atmospheric surface layer mast are seen in the photograph. Also included (not seen) are instrumentations for ice (temperature and light) and water body (temperature, light, conductivity, and flow velocity). Photograph by the author

Earth’s surface depends on the Earth–Sun distance, solar elevation, and attenuation in the atmosphere. The actual extra-terrestrial solar radiation has an annual cycle due to the ellipticity of the Earth’s orbit, from 1294 W m−2 in the June summer solstice to 1483 W m–2 in the December winter solstice. The solar elevation depends on the time and the latitude, obtained from astronomical formulae (see Annex). Incident solar radiation consists of the direct beam from the Sun and diffuse radiation from the whole sky due to scattering in the atmosphere. The contributions of direct and diffuse radiation depend on the solar elevation and cloudiness. On the way through the atmosphere, solar radiation is reduced by 30–80% due to the absorption and scattering by gas molecules, aerosols, and clouds, and finally hits the Earth’s surface as the incident solar radiation4 Q s , with a strong annual cycle on freezing lakes (Table 2.8). The fraction α Q s , where α(0 ≤ α ≤ 1) is the albedo, is reflected from the lake surface and scattered back from the lake surface layer. Algebraic estimators of the incident solar radiation are based on astronomical formula and routine weather data (see Sect. 3.5). A simple approximation for the daily mean value is 4

Total solar radiation incident on a horizontal plane, including direct and diffuse radiation, is global radiation.

44

2 Freezing of Lakes

Table 2.8 The bands of the electromagnetic radiation in satellite remote sensing of lake ice Window

Wavelengths

Sub-band

Wavelength

Ultraviolet

100–380 nm

UV-C UV-B UV-A

100–280 nm 280–315 nm 315–380 nm

Violet Blue Green Yellow Orange Red

380–450 nm 450–495 nm 495–570 nm 570–590 nm 590–620 nm 620–750 nm

Optical or light 380–750 nm PAR 400–700 nm (photosynthetically active radiation)

Applicability

Visual pictures of surface, shows ice/snow/water areas, fractures, ice forms

Infrared

750 nm–1 mm Near IR (IR-A) Short-wave (IR-B) Thermal (IR-C) Far IR

750–1400 nm IR-A as optical; IR-C: surface 1.4–3 µm temperature 3–15 µm 30–1000 µm

Microwaves

1 mm–1 m

1–10 mm 1–30 cm

Millimeter waves Radar Radiometer

Ice/water areas; radar also ice forms; radiometer data resolution limited

The band limits are not sharp and show minor variability in literature

Q s = Qˆ s sin(θ0 )td

(2.11)

where Qˆ s ∼ 200 − 600 W m−2 is a local tuning factor, which depends primarily on the cloud climatology, θ0 is the solar elevation at noon, and td (0 ≤ td ≤ 1) is the fractional sunlight time of the day. The grey body model (2.10) applies for the radiation emitted by the Earth’s surface. At the prevailing temperatures the radiation spectrum peaks at close to 10 µm. The emissivity of a lake surface is ε0 = 0.96 − 0.98. For ε0 = 0.97, the emitted thermal radiation is 226–306 W m−2 for the surface temperature between −20 and 0 ◦ C. Thermal radiation from the atmosphere is a complicated issue since it comes from different altitudes with different temperatures and emissivities. For parameterization from standard weather data, a grey body analogous formula is used, with air temperature at 2 m height as the representative temperature and an ‘effective emissivity’ εa ≈ 0.7 − 0.9 which depends on the air humidity and cloudiness. The net terrestrial radiation is:

  Q n L = Q La − Q L0 = ε0 σ εa Ta4 − T04 ≈ −ε0 σ (1 − εa )Ta4 + 4(Ta − T0 )Ta3 (2.12) where Q La and Q L0 are the terrestrial radiation by the atmosphere and Earth’s surface, respectively. The last equation is based on the binomial formula. Because of the low atmospheric emissivity, the net terrestrial radiation is mostly negative and well below zero: Q n L ∼ −50 W m−2 , mostly between − 80 and − 20 W m−2 . The

2.3 Climate and Water Budget

45

emitted radiation from the Earth’s surface can be calculated with good accuracy if the surface temperature is known, but for the atmospheric radiation indirect estimators based on the routine weather data have a limited accuracy. Cloudiness is the key factor to determine the emissivity, but it is available only at few weather stations, and good data are provided only by trained personnel. Example 2.8 A simple planetary climate model (no atmosphere) is based on the balance between the net solar radiation and the emitted radiation from the Earth’s surface: ε0 σ T04 =

1 (1 − α)Q sc 4

where α is the planetary albedo of the Earth, and the factor ¼ comes from averaging the solar radiation over the planet. Taking α = 0.3 and ε0 = 0.97, we have T0 = −16 ◦ C. For other planets, the albedo can be different, and the local solar constant is inversely proportional to the square of the distance between the planet and the Sun. We can extend this model by adding atmosphere with emissivity εa . Taking for simplicity ε0 = 1 and assuming that atmospheric radiation is equally lost to space and returned to Earth’s surface, the solution is  T0 (εa ) =

1 (1 − α)Q sc

· 1 4σ 1 − 2 εa

 14

In this case T 0 (0) = − 18°, and T0 (0.8) = 16 ◦ C, which is not far from the present global average (15 °C).

2.3.3 Air Pressure, Temperature, Humidity and Wind At the sea surface level, the air pressure is between 900 and 1100 mbar and the reference, standard atmospheric pressure is 1013.25 mbar. Using the ideal gas law for air, the equation of state of dry air is p = ρ Rd T , where Rd =287 J K−1 kg−1 is the gas constant of dry air. The consequent adiabatic lapse rate is  = g/c p = 9.8 K km−1 , where c p = 1004 J K−1 kg−1 is the specific heat of dry air at constant pressure. The lapse rate of moist air is less because condensation releases heat and partly compensates for the influence of pressure release. In the troposphere, the normal adiabatic lapse rate is  = 6.5 K km−1 (e.g., Curry & Webster, 1999; Holton, 1979). The vertical distribution of air pressure is obtained by the hydrostatic equation (Eq. 2.7) and the ideal gas law. The result is:   g z Rd  p(z) = p(0) 1 − T0

(2.13)

46

2 Freezing of Lakes

The power in Eq. (2.13) is 5.26 for  = 6.5 K km−1 . The scale of the pressure reduction is 100 mbar km−1 (10 kPa km−1 ) in the troposphere. The lateral length scale of air pressure variations is much more than the size of lakes, except for very large lakes. However, small horizontal pressure variations may induce seiches in ice-covered lakes (Kirillin et al., 2012, 2018). In mountain lakes the air pressure at the lake surface is lower than at sea level that has an impact on air density and the solubility of gases. The saturation pressure of dissolved oxygen decreases by about 1% per 100 m. The freezing point and the temperature of maximum density of the surface water are affected by less than 0.01 °C, even in high mountains. The radiation balance is modified since the atmosphere above lakes becomes thinner with altitude. Example 2.9 At 5 km altitude the air pressure is about 500 mbar, and at the freezing point temperature the air density is 0.64 kg m−3 . The adiabatic lapse rate provides the actual temperature decrease with altitude in neutral conditions: T (z) = T0 − z. At 5 km altitude, the temperature is 32.5 K lower than at the sea

surface level. If T0 > T f , the freezing point is reached at the altitude of T0 − T f ·  −1 , i.e., possible even in the equatorial climate in high mountains. Air temperature is a widely used climate characteristic in geosciences and ecology.5 It is quite stable, easy to measure with high accuracy, and there are longterm records available for many observation stations around the world. The annual cycle of air temperature largely follows the solar radiation with a delay of around one month (Table 2.8). In general, the amplitudes of the annual and diurnal cycles of air temperature are larger in land areas than in lake districts. A parcel of air contains a small amount of water vapour. The absolute humidity is given either by the mass proportion of the water vapour, i.e., the specific humidity q, or by the partial pressure of the water vapour, e. They are related by q = 0.622

e p

(2.14)

The amount of water vapour is limited by the saturation level, es = es (T ), which depends strongly on the temperature (Fig. 2.15; for the exact formulae, see Annex). At temperatures below the freezing point, the saturation pressure is given both for condensation into liquid water and deposition into snow crystals. E.g., at the standard atmospheric pressure, es (10 ◦ C) = 12.27 mbar, es (0 ◦ C) = 6.11 mbar, and es (−10 °C) = 2.86 mbar (condensation) or 2.60 mbar (deposition). The pressure is lower at deposition than at condensation, i.e., snow crystals are more likely to form than liquid droplets in cold clouds. The relative humidity is the fraction of the absolute humidity from the saturation level: 5

Celsius (°C) is a convenient temperature unit in study of freezing lakes and mostly used in this book. The absolute temperature (K) is the natural scale for thermodynamics and used here where considered preferable. These units transform by 273.15 K = 0 °C. Fahrenheit degrees are used in the U.S., T [◦ F] = 95 T [◦ C] + 32.

2.3 Climate and Water Budget

47

Fig. 2.15 Saturation water vapour pressure as a function of temperature at water surface and at ice surface

R=

e q = qs es

(2.15)

Annual variations in the relative humidity are usually small, but the absolute humidity follows largely the air temperature. Adiabatic cooling of rising air leads to cloud formation and eventually to precipitation. Atmospheric circulation dynamics, orography, and surface heating may trigger the necessary upward motion of air. Since the saturation water vapour pressure increases with temperature, the amount of water vapour is small in cold atmosphere and, consequently, precipitation is usually low in cold climate. When the water vapour in rising air reaches the saturation level, it is condensed or deposited into, respectively, liquid droplets or snow crystals. The shape of the snow crystals depends on the temperature and the degree of oversaturation in clouds (Fig. 2.16). The liquid droplets may further crystalize. If the atmosphere is cold down to the Earth’s surface, snow crystals land in the solid phase, but hailstones fall fast and can reach the ground even in warm summer. If the phase of precipitation is unknown, a practical approach is to take it solid when the air temperature is below a specified level, e.g., − 1 °C. Wind velocity over a lake shows large spatial variations depending on the surface roughness, vegetation, and stratification of the atmospheric boundary layer. Land– lake differences in wind velocity can be large, and sometimes correction factors are employed to estimate the wind velocity on a lake from measurements in a nearby land station. As a rule of thumb, the shadowing effect of the upwind coast extends out from the shore around twenty times the height of the shore topography and vegetation. Wind, air temperature and humidity determine together the turbulent transfer of momentum, sensible heat, and latent heat. The bulk formulae are for them, respectively,

48

2 Freezing of Lakes

Fig. 2.16 Snow crystal structure as a function of temperature and oversaturation (Magano & Lee, 1966)

τ a = ρa C D |U a − U 0 |(U a − U 0 )

(2.16a)

Q H = ρa c p C H (θa − θ0 )|U a − U 0 |

(2.16b)

Q E = ρa L E C E (qa − q0 )|U a − U 0 |

(2.16c)

where ρa is the air density, C D , C H and C E are the bulk exchange coefficients, U is the wind velocity, c p is the specific heat of air at constant pressure, θ is the potential temperature,6 L E is the latent heat of evaporation or sublimation, and the subscripts a and 0 refer to air and surface, respectively. We can assume that the humidity is at the saturation level at the lake surface, q0 = qs (θ0 ), and normally take U a − U 0 ≈ U a , since Ua U0 . The surface temperature can be measured or estimated indirectly from auxiliary data, but it is not available in routine data. The magnitude of the bulk exchange coefficients is 10−3 , and in general they depend on the reference height of the atmospheric data, stratification of the boundary layer, and surface roughness (see Sect. 4.1 for more details). In neutral stratification with 10 m reference height, C D = C H = C E = 1.2 × 10−3 is a good reference over ice (Andreas, 1998) and representative also over open lakes. In highly stable stratification, these coefficients are reduced by down to 50%, and in highly unstable stratification, they are increased by up to 25%.

6

Potential temperature is the temperature of dry air brought adiabatically to the surface. In the 10-m surface layer, it nearly equals the actual temperature, Ta ≈ θa (the difference is less than 0.01 °C).

2.3 Climate and Water Budget

49

In weather stations temperature and humidity are measured at the altitude of 2 m above the surface, while wind is recorded for its speed and direction7 at 10 m. In neutral stratification and with a typical surface roughness of lake ice, 0.1 mm, the wind speed and temperature and humidity changes from the surface are reduced by the factor of 0.86 when coming down to 2 m from 10 m level; thus, the heat transfer coefficients would change from 1.2 × 10−3 at 10 m to about 1.6 × 10−3 for 2 m reference height. In autumn, lakes are usually warmer than surface air layer, and the stratification is unstable with strong turbulent fluxes. After the freeze-up, insulation by the ice cover lowers the air–lake surface temperature and humidity differences reducing the turbulent fluxes, and when the ice is thick, the stratification is no more unstable. During ice melting, air is usually warmer than lake surface, the boundary layer is stable, and winds and turbulent transfer are weak. Example 2.10 (a) Assume neutral stratification and take the standard weather data with U a (10 m) = 5 m s−1 , T a = − 5 °C, and R = 80%. The momentum transfer is 0.039 N m−2 . Then Ua (2m) =4.3 m s−1 , and with T0 = −6 ◦ C, the sensible heat gain is 9.1 W m−2 . We have ea =4.22 mbar, e0 = es (T0 ) =3.69 mbar, and deposition of water vapour takes place with latent heat gain by 2.0 W m−2 . In case T0 = −4 ◦ C, the sensible heat loss is 9.1 W m−2 , sublimation takes place with the latent heat loss of 7.2 W m−2 . (b) For the latent heat flux Q E = −30 W m−2 , sublimation is E =1.0 mm day−1 . Assume that the relative humidity is 50% and wind speed is 10 m s−1 . For T0 = 0 ◦ C, we have Q E = −100 W m−2 or E ≈ 3 mm day−1 , and for T0 = −20 ◦ C, we have Q E = −20 W m−2 or E ≈ 0.6 mm day−1 . Evaporation/sublimation is limited by the air temperature and humidity, which define the moisture deficit in air, and by the available heat flux. The latent heat is in kJ kg−1 equal to 2494−2.2T (T ≥ 0 ◦ C) for evaporation or 2828−0.39T (T ≤ 0 ◦ C) for sublimation. The level of mass flux is 0.1–10 mm water or ice per day: evaporation by 1 mm water or sublimation by 1 mm ice per day at the freezing point temperature needs a heat flux of 28.9 W m−2 or 30.0 W m−2 , respectively. In dry and cold climate regions, although saturation water vapour pressure is low, sublimation can be a large term in the mass balance (Huang et al., 2012; Leppäranta et al., 2016). Occasionally atmospheric water vapour deposits as surface hoar crystals on the surface, sometimes observed as beautiful ‘frost flowers’ (Fig. 2.17).

7

Wind direction (WD, degrees) tells from where the wind blows in the compass angle, zero toward north and turning clockwise. Mathematical right-hand coordinate system has zero direction toward east (x-axis) and angle turns counter-clockwise. Wind vector direction (to where the wind blows) is in the right-hand system 270°-WD.

50

2 Freezing of Lakes

Fig. 2.17 Frost flowers on lake ice in Lake Vähä-Tahko, southern Finland, on January 16, 2014. The physical size of individual flowers is 5–10 cm. Photograph by Mr. Jari Toivonen, printed by permission

2.4 Remote Sensing of Lake Ice 2.4.1 Techniques Remote sensing has become a highly versatile set of tools to map lake ice properties and processes. It has a long history. Traditional visual observations of freezing and breakup of lakes were the first remote sensing products. Ground-based photographs have for a long time since illustrated interesting lake-ice phenomena, and presently a large variety of excellent lake ice photographs can be seen in the web, kept by nature clubs, fishing societies, tour skating associations etc. Aerial reconnaissance flights were commenced in large lakes in the 1930s and performed until 1990s, when they were replaced by satellite methods. The bird’s eye view opened a lake-scale picture of the ice cover to an observer that meant great progress. Reconnaissance flights and aerial photography were employed over Lake Ladoga for monitoring of ice extent in the period 1943–1994 (e.g., Karetnikov &

2.4 Remote Sensing of Lake Ice

51

Naumenko, 2011). On average, 19 reconnaissance flights were performed in one winter. Satellite remote sensing came in use stepwise, and since 1995 the ice mapping system in Ladoga has been solely based on that. Recently, drone systems have provided a feasible and flexible method for local studies, especially to monitor rapid changes in early and late winter. Figure 2.18 shows an example from Norway Pond, New Hampshire, where spots with circular lines show up in a new ice cover. In newly form ice, intensive air–ice–water interaction produces exciting features, which are often difficult to understand. In addition to optical cameras also other instruments have been used by drones. Divers have mapped ice bottom features and ice–water interaction processes, and the modern approach has introduced underwater robot cameras. Attempts have also been made to use acoustic signals for ice observations, in particular propagation of sound waves in ice cover (Shaw & Decleren, 2015). In the present century the spatial resolution of free satellite data has become good enough for mapping lakes down the medium-size scale (Duguay et al., 2015; Qiu et al., 2019), over a wide spectral range (Table 2.8). The sensors provide good spatial

Fig. 2.18 Drone photo over Norway Pond, NH, taken on December 16, 2019. Strange surface features show up in a new ice cover. Pond size is 0.16 km2 . © Steve Pope, printed by permission

52

2 Freezing of Lakes

coverage, but the temporal resolutions depend on the satellite orbit systems and swath width. For high-resolution systems the orbit repeat cycle is from days to weeks. Satellite sensors use natural or their own electromagnetic signals for observations. Firstly, they can record the reflected and scattered solar radiation from the Earth’ surface for the optical–near infrared band signals. This method is dependent on the weather and illumination condition. Secondly, satellite sensors can record the emitted thermal radiation from the Earth’s surface in the thermal infrared and microwave bands. The former works in clear sky conditions, but there are all-weather transparent bands for Earth’s surface mapping in the microwave window. Thirdly, satellites can have a laser or radar to transmit own signal to the target and measure the returning backscatter. The contrast between ice and open water surfaces is good in cold weather sorting out them, but when the surface temperature is at the freezing point, phase changes take place in the surface layer and liquid water is present weakening the contrasts in the recorded imagery. The Sun radiates almost as a black body at the temperature of its outer surface, 5900 K (Fig. 2.19). The solar spectrum peaks at 480 nm, has nearly all its energy in the 300–3000 nm wavelengths covering the ultraviolet, optical (light), and nearinfrared bands (Iqbal, 1983). Narrow Frauenhofer lines are seen in the spectrum due to absorption by elements in the solar atmosphere. At the Earth’s surface, the visible or sunlight spectrum is quite even which means that the Sun is white, however, due to atmospheric effects the Sun appears yellow to our eye. Even though the level of solar radiation on the Earth’s surface varies largely, especially due to solar altitude and cloudiness variations, the shape of the sunlight spectrum is quite stable. UV-C and most of UV-B are absent at the Earth’s surface, absorbed by ozone in the upper atmosphere. Wavelengths longer than 1200 nm have insignificant penetration depth into water and ice. Atmospheric gases have absorption bands in the infrared and microwave windows. Ozone (O3 ) absorbs in the ultraviolet, oxygen (O2 ) absorption peak shows up at 760 nm, and water vapour has IR-A absorption peaks at 950 and 1100 nm. In the wavelengths longer than 1200 nm water vapour and carbon dioxide have several absorption bands. Examples of multispectral satellite sensors for lake ice mapping with freely available data are the Thematic Mapper on the Landsat series of satellites, the European Space Agency’s Sentinel-2 satellites, and the Moderate-Resolution Imaging Spectroradiometer (MODIS) on NASA’s Earth Observing System Terra and Aqua satellites. Landsat and Sentinel-2 sensors provide 10–30 m spatial resolution at repeat cycles of several days whereas MODIS provides daily images over the whole Earth at 500 m resolution. The satellite methods have reached a quality high enough for monitoring to fully replace airborne observations. Detailed ice features are still occasionally mapped by aerial photography for specific studies or experiments. In microwave radiometry, airborne sensors can operate at much higher resolution than satellites. Airborne long wavelength (0.3–3 m) radars, so-called ground penetrating radars or georadars, can penetrate through lake ice and consequently be useful for collecting information about the ice thickness and vertical structure (Annan & Davis, 1977; Annan et al., 2016). These radars are often mounted on a surface-vehicle such as a sledge.

2.4 Remote Sensing of Lake Ice

53

Fig. 2.19 Solar radiation spectrum at the top of the atmosphere and at the sea surface level. Atmospheric absorption bands due to ozone (O3 ), oxygen (O2 ), water vapour and carbon dioxide are illustrated in the solar radiation band

In satellite remote sensing of lake ice, the lateral properties are usually well obtained since the contrast between ice (or snow) and open water is strong in all satellite measurement channels. These properties are ice coverage, fractures and leads, size and shape of ice floes, and, from sequential imagery, ice movement. But information of the vertical extent of ice and snow cover is limited. Snow surface can be distinguished from ice surface at optical and infrared channels, and in general a qualitative classification of ice type is possible. The mapping of ice thickness is presently the main lake ice remote sensing problem. The thickness is of major significance in determining the bearing capacity of ice and ice loads on shores and structures. There is no single feasible sensor solution to this problem.

2.4.2 Optical Remote Sensing Optical and near-infrared satellite sensors measure the reflected and scattered solar radiance from the Earth’s surface toward the sensor. Clouds can fully contaminate the signal. The recorded radiance is determined by the physical properties of the surface layer as well as the solar altitude, atmospheric conditions, and the sensor technique. The surface reflectance depends on the surface roughness, while the volume backscatter depends on the size and shape of ice crystals, gas and liquid water inclusions, and the quality and quantity of other impurities in the ice cover (Duguay et al., 2015; Warren, 1982; Warren & Clarke, 1990). Snow and ice surfaces are often assumed to be Lambertian, i.e., to have a perfectly diffuse reflectance. However, this

54

2 Freezing of Lakes

assumption is not always justified, and instead bi-directional reflectance distribution function (BRDF) models have been employed (e.g., Voss et al., 2000). Example 2.11 The black body radiation (Eq. 2.9) is used as a reference to define the colour temperature Tc of a light source: it is the temperature of an ideal black body radiator, which has the spectral energy distribution closest to the object in concern. It can be determined by measurement of power and absorbance A at two wavelengths: 1 1 A(λ1 , T ) kB −1

log − = −1 T Tc A(λ hc0 λ1 − λ2 2, T ) In the derivation, it is assumed that λk B T hc0 , which simplifies the Planck’s law to derive the direct solution. Optical images provide data at several channels that can be combined for photographic information. Open water, bare ice and snow-covered ice can be distinguished, as well as fracture systems and openings. Thus, lake ice phenology, ice coverage, ice type, and ice floes are obtained. With a series of images, horizontal displacements of ice floes can also be determined (Wang et al., 2005). Good satellite data for lake studies have been available since 1990s, and the data are now feasible for time series research (Qiu et al., 2019; Wang et al., 2012). Figure 2.20 shows Lake Baikal ice ring in a Landsat image. These rings occur in spring in Baikal, and field investigations have shown that they are connected to warm, clockwise eddies under the ice cover (Kouraev et al., 2016). The penetration depth of sunlight is 10–20 cm in snow and 1–5 m in ice, depending on the gas inclusions. The volume backscatter can be used for ice type classification, although classes are case dependent rather than universal. Since gas pockets are good scatterers, they increase the level of scattering, and since they are large compared with optical and near-infrared wavelengths, the spectrum is even. This is particularly evident from the bright and white optical signature of snow. In the far red and nearinfrared bands, absorption by water molecules becomes more significant, and the reflectance of ice and snow decreases. The colour of meltponds on Arctic sea-ice has been shown to be weakly connected to the ice thickness based on a two-stream radiative transfer modelling (Lu et al., 2018).

2.4.3 Thermal Remote Sensing Thermal infrared (TIR) is a powerful remote-sensing technique for revealing surface objects with different temperature or emissivity (Del Grande et al., 1991). This technique has a potential for lake ice mapping (Rees, 2005) since usually ice surface is colder than open water surface. Ice surface temperature provides information of ice thickness, since in general the colder is the surface, the thicker the ice is. This method also allows monitoring of the location of the melting front in spring, but at the melting stage the surface is isothermal, and all contrasts disappear.

2.4 Remote Sensing of Lake Ice

55

Fig. 2.20 Lake Baikal ice ring shown in a Landsat image on April 1, 2016, in the central part of the lake. The thin ice of the ring appears darker and more transparent than the whiter, thicker ice around. Source NASA Landsat Image Gallery

TIR method uses the thermal radiation of the Earth’s surface as the measurement signal. It is independent of sunlight and follows the black body law (Eq. 2.9) peaking at about 10 µm wavelength, which is used in TIR mapping. For a satellite sensor to receive thermal radiation from the surface, cloud-free sky is necessary. TIR mapping provides the radiant temperature TR , which is transformed to the real surface temperature T0 by T0 =

1 TR ε

(2.17)

where ε = ε(λ) is the emissivity of the surface. The emissivity at 10 µm is 0.998 for snow, 0.997 for ice, and 0.995 for liquid water (Steffen, 1985), and the radiant temperature is therefore lower than the real temperature by 0.5–1 °C. When a lake is partly covered by ice, in cold weather the surface temperature is T0 < T f in ice areas and T0 ≥ T f in open water areas. Thus, ice and open water areas can be distinguished. The ice or snow surface feels the warmer water underneath due to thermal diffusion and therefore the surface temperature depends on ice thickness (Leppäranta & Lewis, 2007). The method is applicable when the surface temperature of ice is less than about − 5 °C and the thickness of ice is less than half meter. The static heat conduction equation is based on the equilibrium between heat conduction through ice and the surface heat balance Q 0 . Then we have a bare ice cover

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ki

T f − T0 + Q0 = 0 hi

(2.18)

where h i is ice thicknesses and ki is the thermal conductivity of ice, T f > T0 and Q 0 < 0. Leppäranta and Lewis (2007) employed a linear equation for the surface heat balance, Q 0 = K 0 + K 1 (Ta − T0 ) (see Sect. 4.3.2), where Ta is air temperature and the parameters K 0 and K 1 are estimated from weather data or taken as calibration factors if any independent thickness information is available. An algorithm for ice thickness resulted as

ki T f − T0 (2.19) hi = − K 0 + K 1 (Ta − T0 ) In the presence of snow on ice, the left-hand side is replaced by h i + kksi h s , where h s and ks are the thickness and thermal conductivity of snow. In general, TIR remote sensing gives the joint insulation effect of snow and ice, and complementary data is needed to tell their individual fractions. Because the surface temperature may change fast, care needs to be taken when thermal mapping is performed. The surface heat balance should be nearly steady with weak solar flux. For example, the radiation balance at the surface may result in the “cold skin phenomenon” (Leppäranta et al., 2020). A good time for thermal mapping is before eventual sunrise when the surface temperature is at minimum. The method can be improved by using a time-dependent equation (see Sect. 4.3) that is desirable with satellite data since the time of day of overflights are fixed. In spite of its potential in the lake ice thickness problem, TIR method has not been widely utilized.

2.4.4 Microwave Remote Sensing Microwave window provides all-weather possibilities for satellite remote sensing of the Earth’s surface, that is very useful for targets such as lakes, which are changing at daily timescale. Atmosphere is practically transparent for several microwave channels. Microwave radiometry, or the passive microwave method, records the thermal radiation of the Earth’s surface as in the TIR method but using microwave channels. Radar, or the active microwave method, transmits own signal and records its return from the target. In radiometry, the measured radiation of the given channel is expressed as the brightness temperature TB , which is equal to the temperature of a black body that would provide the same radiance at the channel wavelength: E B (λ, TB ) = E(λ, T )

(2.20)

2.4 Remote Sensing of Lake Ice

57

where E is the spectral power density of the radiation from the given body. Black body radiates with the maximum power, and therefore TB ≤ T . In the microwave window, TB = εT . The detected surface layer is less than 1 cm for liquid water but more for dry ice or snow. Liquid water surface has much lower emissivity than ice surface that enables the distinction between open water and ice cover. However, in the melting period, interpretation of the signals is complicated, because the surface-layer is isothermal, and the emissivity of ice is influence by meltwater. Another drawback of the method is the poor spatial resolution in satellite remote sensing (kilometres at best), and therefore it is feasible only for large lakes. Wang et al. (2021) analysed lakes in the Tibetan Plateau for the ice phenology using long-term passive microwave data from several satellites. Radar transmitter and receiver can together employ a chosen wavelength and a configuration for polarizations. The quality and strength of the return signal depend on the transmitted signal, surface roughness and properties of the target medium. Synthetic Aperture Radar (SAR) technique provides a method for mapping with very high resolution by creating a synthetic antenna using the satellite movement (e.g., Rees, 2005). With short wavelengths of centimetres, the penetration depth SARs may be several meters in dry ice but much less in wet or moist ice. Radar data can in principle provide ice and open water distinction and information of ice quality (Surdu et al., 2015; Tom et al., 2020). Figure 2.21 shows a satellite SAR image over Lake Peipsi/Chudskoe as an example. The lake was completely ice-covered, so that the backscatter shows different ice types and man-made ice roads. However, the backscatter contrast between ice and open water may have either sign, and, therefore, no universal ice/open water algorithm can be set based on the backscatter of single pixels. Over the ice cover, radar can recognize different kinds of ice but because of the large difference in the relative permittivity between the liquid and solid phases of water, radar is highly sensitive to the wetness of the ice surface. The signal is also sensitive to roughness changes in the scale of the radar wavelength, such as frost flowers. Thus, interpretation of radar imagery can be challenging. Kouraev et al. (2009) used also radar altimeter in examining the ice season and sea level in the Aral Sea. Outcome from mechanical deformation such as fractures and ridges show up well in the backscatter and can be further used to examine ice mechanics processes and phenomena (Goldstein et al., 2009). With sequential imagery, repeated features can be used to track ice kinematics, as has been widely utilized for sea ice. Through interferometric processing the phase information also enables ice identification and highly precise change detection in surface topography or lateral displacements (van der Sanden et al., 2018).

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Fig. 2.21 Sentinel-1 SAR image on February 19, 2021, over Lake Peipsi/Chudskoe, located between Estonia and Russia. The lake length is 140 km and maximum width 50 km, location at around 58.5° N 27.5° E. The longitude interval 0.9° is here 52 km. Source European Space Agency

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Su, D., Hu, X., Wen, L., Lyu, S., Gao, X., Zhao, L., Li, Z., Du, J., & Kirillin, G. (2019). Numerical study on the response of the largest lake in China. Hydrology and Earth System Sciences, 23, 2093–2109. Surdu, C. M., Duguay, C. R., Pour, H. K., & Brown, L. C. (2015). Ice freeze-up and break-up detection of shallow lakes in Northern Alaska with spaceborne SAR. Remote Sensing, 7(5). Tom, M., Aguilar, R., Imhof, P., Leinss, S., Baltsavias, E., & Schindler, K. (2020). Lake ice detection from Sentinel-1 SAR with deep learning. In ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences (V-3-2020, pp. 409–416). UNESCO. (1981). The practical salinity scale 1978 and the international equation of state of seawater 1980. Technical papers in marine science (Vol. 36). van der Sanden, J. J., Short, N. H., & Drouin, H. (2018). InSAR coherence for automated lake ice extent mapping: TanDEM-X bistatic and pursuit monostatic results. International Journal of Applied Earth Observation and Geoinformation, 73, 605–615. Veillette, J., Martineau, M.-J., Antoniades, D., Sarrazin, D., & Vincent, W. F. (2010). Effects of loss of perennial lake ice on mixing and phytoplankton dynamics: Insights from High Arctic Canada. Annals of Glaciology, 51(56). Voss, K. J., Chapin, A., Monti, M., & Zhang, H. (2000). Instrument to measure the bidirectional reflectance distribution function of surfaces. Applied Optics, 39(33), 6197–6206. Wang, J., Bai, X., Hu, H., Clites, A., Holton, M., & Lofgren, B. (2012). Temporal and spatial variability of Great Lakes ice cover, 1973–2010. Journal of Climate, 25, 1318–1329. Wang, K., Leppäranta, M., & Reinart, A. (2005). Modeling ice dynamics in Lake Peipsi. Verhandlungen des Internationalen Verein Limnologie, 29, 1443–1446. Wang, X., Qiu, Y., Zhang, Y., Lemmetyinen, J., Cheng, B., Liang, E., & Leppäranta, M. (2021). A lake ice phenology dataset for the Northern Hemisphere based on passive microwave remote sensing. Big Earth Data. https://doi.org/10.1080/20964471.2021.1992916 Warren, S. G. (1982). Optical properties of snow. Reviews in Geophysics and Space Physics, 20(1), 67–89. Warren, S. G., & Clarke, A. D. (1990). Soot in the atmosphere and snow surface of Antarctica. Journal of Geophysical Research, 95, 1811-1816. Winther, J.-G., Elvehøy, H., Bøggild, C. E., Sand, K., & Liston, G. (1996). Melting, runoff and the formation of frozen lakes in a mixed snow and blue-ice field in Dronning Maud Land, Antarctica. Journal of Glaciology, 42(141), 271–278. Yang, F., Li, C., Leppäranta, M., Shi, X., Zhao, S., & Zhang, C. (2016). Notable increases in nutrient concentrations in a shallow lake during seasonal ice growth. Water Science and Technology, 74(12), 2773–2883. Yang, Y., Leppäranta, M., Li, Z., & Cheng, B. (2012). An ice model for Lake Vanajavesi, Finland. Tellus A, 64, 17202. https://doi.org/10.3402/tellusa.v64i0.17202 Zintz, K., Löffler, H., & Schröder, H. G. (2009). Der Bodensee. Ein Naturraum im Wandel. Thorbecke, Ostfildern.

Chapter 3

Structure and Properties of Lake Ice

Ice sampling work. Investigations of the structure and geochemistry of lake ice is based on ice samples. The ice stratigraphy and crystal structure are studied in a cold room from the samples and ice geochemistry is studied from the meltwater

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_3

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3 Structure and Properties of Lake Ice

3.1 Ice Ih: The Solid Phase of Water on Earth 3.1.1 Ice Crystal Structure In the Earth’s nature, water is the only substance, which occurs in all three phases: gas, liquid, and solid. Phase transitions take place continuously. The water molecule consists of one oxygen atom and two hydrogen atoms connected to the oxygen atom by covalent bonds (Fig. 3.1a), which are strong and keep the water molecule stable. The hydrogen atoms are not symmetrically located but the bonds H–O–H form an angle of 104.5° that makes the water molecule an electric dipole. The centre of each hydrogen atom is 0.0957 nm from the centre of the oxygen atom. The oxygen pole is negatively charged, and the hydrogen pole is positively charged. The dipolar structure is the cause of exceptional properties of water, e.g., liquid water is an excellent solvent. Water molecules are connected by hydrogen bonds into linear chains in liquid water. The oxygen atom of a water molecule and a hydrogen atom of a neighbouring water molecule form the hydrogen bond, which is 0.117 nm long. These bonds are much weaker than the covalent bonds, and they form and break continuously. The hydrogen bonds are the cause of relatively high freezing and boiling points of water, as compared with similar substances, and liquid phase having higher density than solid phase. There are about ten possible crystal lattice structures for the solid phase of water (Petrenko & Whitworth, 1999). But in the natural pressure and temperature conditions within 10 km up or down from the Earth’s surface, all ice belongs to the form Ice Ih, which has hexagonal crystal structure (Fig. 3.1b). Cubic ice (Ice Ic) occasionally

Fig. 3.1 a Water molecules in liquid state; b Ice Ih crystal lattice. The plane of the hexagons is parallel to the basal plane, and c-axis is normal to the basal plane. The distance between neighbouring molecules is 0.275 nm (at 0 °C). Based on Pounder (1965)

3.1 Ice Ih: The Solid Phase of Water on Earth

65

occurs in the upper atmosphere at low temperatures and low pressures. In this book, only the hexagonal ice is considered. The crystal lattice of Ice Ih (from now on, just ‘ice’) is tetrahedral where each ice molecule is joined by hydrogen bonds to four other ice molecules. The electron pairs are evenly spaced in their orbits and the covalent bonds H–O–H form an angle of 109.5°, which equals the tetrahedral angle in a perfect tetrahedron. The distance between neighbouring molecules is 0.275 nm, and the distance between oxygen and hydrogen atoms in one molecule is 0.095 nm. The lattice follows the Bernal– Fowler rules (e.g., Petrenko & Whitworth, 1999): (1) There are two hydrogen bonds connected to each oxygen atom, and (2) There is only one hydrogen atom in a bond. Exceptions to these rules are defects in the crystal lattice. The concentration of foreign molecules in the ice crystal lattice is extremely small and can be ignored in lake ice investigations. Ice crystal tetrahedra possess hexagonal symmetry (Fig. 3.1b). This can be visible, e.g., in the morphology of snow crystals and frost flowers. The arrangement of molecules is anisotropic in the crystal lattice. They are concentrated in a series of parallel planes, which are called basal planes. Normal to the basal plane is the crystal c-axis. The basal plane is isotropic but there is anisotropy between the basal plane and the c-axis direction. There are more hydrogen bonds in the basal plane than in the planes perpendicular to the basal plane, and therefore many physical properties of ice possess a corresponding anisotropy. Ice is optically uniaxial with the c-axis as the optical axis. Light goes through directly along the optical axis, while in other directions birefringence (double refraction) takes place. Thermal conductivity is highest along the basal plane that has impact on the initialization of ice growth on a water surface. Shear deformation is easiest parallel to the basal plane since then the number of bonds to be broken for shear is at minimum. Therefore, this plane is also called slip plane or shear plane in ice mechanics. Ice crystal lattice is an open structure, where molecules are more sparsely packed than in the liquid phase of water, and therefore ice is much less dense than liquid water. At the freezing point, the densities of liquid and solid phases are 999.84 kg m−3 and 916.7 kg m−3 , respectively. Thus, the density decreases by 8.3% at freezing. This large decrease in density at the change from liquid to solid phase is exceptional,1 and indeed water shows the largest decrease among all substances in the Earth’s nature. The density of freshwater ice, as normal, increases with decreasing temperature. The thermal volume expansion coefficient of ice is about 1.5 × 10–4 °C−1 , and the density of pure ice is 920.8 kg m−3 at − 30 °C. The best method to determine the bulk density of ice is the direct way: to measure the mass/volume ratio of a geometrically regular sample. The presence of pores in the ice gives biased estimates when the method of submersion of a sample in a liquid is employed. Example 3.1 Ice density from the crystal lattice (Pounder, 1965). The density of ice can be evaluated as the mass/volume ratio of one mole of ice. The mass is obtained 1

Bismuth, gallium and germanium, for example, become less dense in the liquid–solid phase change. But the change is by far not so large than in water.

66

3 Structure and Properties of Lake Ice

from the atomic weights of two hydrogen atoms and one oxygen atom, 18.02 g, and the volume is obtained from the geometry of the crystal lattice, which has been determined by X-ray diffraction. The crystal lattice is based on prism units with double layers of oxygen atoms on top and bottom. The prism volume contains six oxygen atoms but each one is shared between three hexagons (see Fig. 3.1b). At 0 °C, the area of the base hexagon is A = 0.1772 nm2 and the height of the prism or vertical separation between the oxygen atom double layers is b = 0.3683 nm (exact numbers from Pounder, 1965). The unit prism contains the mass of 6/3 = 2 oxygen atoms or two water molecules, and the volume of one mole is consequently 21 N A Ab = 19.65 cm3 , where N A is Avogadro’s number. The ice density at 0 °C becomes then 917.0 kg m−3 (the direct measurement is 916.7 kg m−3 ). Example 3.2 Ice density is a fundamental property of floating ice. According to the Archimedes law, the freeboard h f is given by hf =

ρw − ρi hi ρw

where ρi and ρw are the densities of ice and water, respectively. Thus, the freeboard of solid ice is 8.3% of the ice thickness (at 0 °C). If the gas content of freshwater ice is 1%, the freeboard would be 9.2% of the ice thickness. The saying that 1 /9 (11.1%) of an iceberg is visible relates to marine environment, where seawater has higher density than freshwater, 1025 kg m−3 . Furthermore, an ice block of volume V can support the floating of a mass of (ρw − ρi )V , 83 kg for each cubic meter in fresh water.

3.1.2 Ice Nucleation When liquid water is cooled, the nucleation temperature is below the freezing point (Fig. 3.2). Certain amount of supercooling is needed to start nucleation, and thereafter the ‘extra cold’ is recovered and the temperature returns to the freezing point. Then, in the solid state, temperature decreases if heat removal continues. Homogeneous and heterogeneous nucleation modes exist in the primary nucleation (see Michel, 1978). In the former case, very strong supercooling (40 K below freezing point) would be required in pure fresh water, but this is not the case in natural lakes. Instead, heterogeneous nucleation takes place with suspended particles acting as the crystallization seeds, and supercooling remains small. Also snow or ice crystals falling on a supercooled lake water surface are excellent seeds for nucleation. In the secondary nucleation, crystals grow on existing ice surfaces. The latent heat of freezing is large (L f = 333.5 kJ kg−1 ) and severely limits the volume of ice that can be produced in natural conditions. For a heat loss Q, an ice is produced in time t; e.g., for a large loss from lake layer of thickness h = ρQt iLf

3.1 Ice Ih: The Solid Phase of Water on Earth

67

Fig. 3.2 A schematic cooling curve of a liquid with heat being removed at a constant rate; T is the temperature, T0∗ is the temperature of phase equilibrium, ΔT0 is the degree of supercooling, t is time. Modified from Michel (1978)

surface, Q = 100 W m−2 , and t = 1 d, an ice layer of h = 28 mm thickness is produced. To build a crystal in a homogeneous fluid, Gibbs free energy ΔG is required to form the surface and the bulk particle. This energy and the minimum radius of stable crystals are (Michel, 1978), respectively, ΔG =

( )2 σcl T0∗ 16π σcl 3 ρw L f ΔT0

(3.1a)

σcl T0∗ ρw L f ΔT0

(3.1b)

rc = 2

where σcl is the surface energy of the crystal in contact with the liquid, T0∗ is the solid– liquid equilibrium temperature (273.15 K at the standard atmospheric pressure), and ΔT0 is the amount of supercooling. For pure water, σcl = 33 mJ m−2 , the amount of supercooling is 40 K, and rc = 1.35 nm is the critical radius of a crystal, containing about 100 molecules. In natural surface waters, with heterogeneous nucleation taking place on suspended particles, the amount of necessary supercooling is reduced from the homogeneous case. Assume that an n-faced prismatic ice crystal grows on a surface of a substrate. The presence of the substrate brings the surface energies between the substrate and liquid (σsl ) and substrate and crystal (σsc ) into the problem (see Michel, 1978). The nucleation temperature depends on the character of the substrate, and the key dimensionless parameter, γ , is defined by: cosγ =

σsl − σsc σcl

(3.2)

68

3 Structure and Properties of Lake Ice

The parameter γ is usually taken as the angle of contact between ice and the nucleating material. In heterogeneous nucleation, there is a critical crystal size after which the crystal grows freely, reducing the free energy of the system. The functional form of the critical radius is as in homogeneous nucleation, and the corresponding Gibbs free energy is ΔG = 4π (1 − cosγ )

( )2 σcl T0∗ tan(π/n) σcl π/n ρw L f ΔT0

(3.3)

Thus, the ratio of the supercooling in heterogeneous nucleation to the supercooling in homogeneous case equals, using the approximation tan(π/n) ≈ π/n, ΔTn = ΔT0

/

1 − cosγ 2

(3.4)

Ice as a substrate has γ = 0 and no supercooling is needed. Considering suspended particles, the best inorganic nucleators in nature have γ = 10◦ , and consequently the supercooling would be 3.5 K. This is, however, more than the observed supercooling in lakes. Field studies as well as laboratory experiments have shown that in calm, freshwater crystallization starts at temperatures at 0.5–1.5 K below the freezing point. In turbulent flow conditions, the surface water layer is well mixed, and supercooling takes place across the whole layer (Fig. 3.3). It has been observed that then the amount of supercooling is not more than about 0.1 K, much less than in spontaneous heterogeneous nucleation. Several theories have been proposed for the nucleation at such high temperatures. Michel (1978) suggested that near the lateral boundaries (shore or ice) turbulence is weaker, and steeper surface temperature gradient can be set up. Ice germs form at the surface, flow away to the turbulent flow, and serve for forced nucleation in the turbulent surface layer. The supercooling is in this situation

ΔT =

2σcl T∗ ρw L f rl 0

(3.5)

where rl is the radius of the seed particle. For clay particles, rl ∼ 1 μm, and the supercooling in forced nucleation becomes 0.05 K. The process of multiplication of crystals in this way is very fast. The border ice mechanism has been questioned, however. Osterkamp (1977) suggested a mass exchange-mechanism to be responsible, with ice particles, cold soil particles, etc. falling on supercooled turbulent water initiating the nucleation. Splashing of water particles in cold air could freeze them and generate nucleus for secondary nucleation. With ice particles in the flow, secondary nucleation would cause them to multiply and produce high concentration of frazil2 ice. 2

Frazil comes the French word fraisil, which refers to coal cinders.

3.1 Ice Ih: The Solid Phase of Water on Earth

69

Fig. 3.3 Snowfall brings ice nuclei onto lake surface, and in windy conditions a soft slushy layer is formed in the surface. The picture shows slush deformation structures in Lake Savojärvi, southwest Finland. Photograph Eeva Räihä, printed by permission

3.1.3 Ice Formation in Lakes Ice crystal lattices start to grow on the lake surface into needle or platelet shapes with the c-axis perpendicular to the axis of the needle or to the plane of the platelet, as favoured by the anisotropic thermal conductivity of ice crystals. Overlain together the platelets form macrocrystals—multiple crystals, which are optically like single crystals. Here the term ice crystal or grain refers to a single crystal or a macrocrystal equivalently. Lake ice crystals are classified in terms of their size, shape, and c-axis orientation (Table 3.1). These characteristics change vertically in a continuous manner, but their measures are absorbed into a few ideal classes (Table 3.1). The size of lake ice crystals ranges from less than a millimetre to one metre, usually it is within 0.1–100 mm. In general, the slower the ice grows the larger will the crystals be. The main classes of crystal shape are granular and columnar, but in the initial ice formation there is a multitude of shapes with granular crystals, needles, and plates with regular and irregular edges. The orientation of crystals can be at extremes dominantly vertical or dominantly horizontal. The origin of the water molecules in the ice crystals is the

70 Table 3.1 Classification of ice crystals based on their size, shape, and orientation (Michel, 1978; Michel & Ramseier, 1971)

3 Structure and Properties of Lake Ice Crystal size

Crystal shape

Fine

< 1 mm

Medium

1–5 mm

Large

5–20 mm

Extra large

> 20 mm

Giant

Meters

Needles, plates Granular (linear dimensions of the same magnitude) Columnar: elongated vertically

Crystal c-axis orientation

Vertical Horizontal (random/aligned in 2D) Random in 3D

lake water body or precipitation on ice cover. Ice formed of snowfall or rain is called meteoric ice. The first ice layer on lake surface is called primary ice (Michel & Ramseier, 1971). The physics of the primary nucleation is partly unclear (see above). Ice formation may initiate in three different modes of heterogeneous nucleation: nucleation on suspended particles into needles and plates at the surface in calm or laminar flow conditions, nucleation into fine or medium-size grains on suspended particles as finegrained frazil ice in turbulent flow conditions, or nucleation on snow or ice nuclei precipitating onto the water surface (Fig. 3.3). Secondary ice forms beneath the primary ice layer as congelation ice, where crystallization takes place at the bottom of the ice, or as frazil ice advected from upstream (Michel & Ramseier, 1971). In the secondary nucleation, no supercooling is needed. Congelation ice crystals are large, granular macrocrystals with c-axes perpendicular to the surface, if the primary ice has formed in calm conditions. Otherwise, crystals are columnar, and their c-axes are randomly oriented in the surface layer and turn horizontal deeper in the ice (Gow, 1986; Li et al., 2011). Superimposed ice forms on top of the primary ice. This includes snow-ice formation from slush,3 freezing of surface ponds originating from meltwater or liquid precipitation (Adams & Roulet, 1980; Palosuo, 1965), and surface hoar deposition (see Fig. 2.17). Slush formation by flooding is the main source of superimposed ice. In saline lakes, ice crystals are in size and shape much as freshwater ice crystals (Fig. 3.4). No ice structure analyses are known to the present author for hypersaline lakes. Most of the investigations of saline ice concern sea-ice showing that there are two important differences caused by the dissolved substances in the parent water. In sea-ice the crystal boundaries are jagged, and between the single crystal platelets inside macrocrystals, there are brine inclusions (Weeks, 1998, 2010). Brackish ice (with salinity less than 6‰) has been examined from field data in the Baltic Sea 3

Slush is water-saturated snow.

3.1 Ice Ih: The Solid Phase of Water on Earth

71

Fig. 3.4 Ice crystal structure of Baltic Sea landfast ice (left) and lake ice (right). Each connected colour spot corresponds to one ice crystal; the scale on top is in millimetres. The top layer is finegrained snow-ice, and the lower layer is columnar-grained congelation ice. According to Kawamura et al. (2002)

(Kawamura et al., 2001; Palosuo, 1961; Weeks et al., 1990), and the results show that the transition between freshwater and sea ice crystal structure type takes place when the salinity of the parent water is 1–2‰. It can be anticipated that in saline lakes the situation is much as in seawater. Weeks and Lofgren (1967) have also shown that ice formed of sodium chloride solution is similar to sea ice.

3.1.4 Physical Properties of Lake Ice The basic properties of freshwater ice with the sensitivity to temperature are given in Table 3.2. More details are shown in Annex. For most properties the sensitivity to temperature is not very high (less than 5% per 10 °C), and in many applications fixed values can be used. Congelation ice possesses anisotropy due to the alignment of c-axes, columnar ice also due to the crystal geometry. The bottom temperature of

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3 Structure and Properties of Lake Ice

Table 3.2 Physical properties of freshwater lake ice and liquid water Property

Ice at 0 °C

Ice at to − 20 °C

Liquid water at 0 °C

Density (kg m−3 )

916.7

920.3

999.8

Latent heat of melting (kJ

kg−1 )

333.5

–

–

Latent heat of sublimation (MJ kg−1 )

2.82

2.84

–

Latent heat of evaporation (MJ kg−1 )

–

–

2.49

kg−1

°C−1 )

2.11

1.96

4.22

Thermal conductivity (W m−1 °C−1 )

2.14

1.88

0.561

Young’s modulus (GPa)

8.7

10.0

2.15

Poisson’s ratio

0.33

0.33

–

Speed of sound (km s−1 )

3.08

3.30

1.45

Relative permittivity

91.6

97.5

87.90

Specific heat (kJ

lake ice is at the freezing point while the surface temperature depends on the surface heat balance and the thicknesses of snow and ice. Example 3.3 The density of freshwater lake ice, ρi = ρi (νa , T ), where νa is the gas content can be expressed as ρi (νa , T ) = (1 − νa )ρi (0, T ) This equation can be used to estimate the gas content from the measured ice density. When νa = 1%, , then (at 0 °C) ρi = 907.5 kg m−3 . Considering νa ∼ 1% as the scale of gas volume in congelation lake ice, 910 kg m−3 serves as its convenient reference density. For the gas volume of snow-ice, the reference density would be 870 kg m−3 , but measurements show a large variability.

3.1.5 Snow Cover on Ice Lake ice is often covered with snow, which differs from ice layers in many respects (Fig. 3.5). Compared with ice, snow cover is soft and very good thermal insulator, and snow cover possesses high reflectance and small light penetration depth. In all lakes the ice surface is bare at least in the early and late ice season, and in cold and arid climate regions the snow cover is absent or very thin all winter, e.g., in the Tibetan Plateau and dry valleys of Antarctica. In very cold climate the adhesion between ice and snow is weak, and winds can clear the snow away if the snow accumulation keeps on a modest level. This is a normal situation, e.g., in Lake Baikal. Observations and classification of snow on lake ice can be made following the methods in the international snow classification manual (Fierz et al., 2009). Granberg (1998) has written a review article of snow on sea-ice, serving lake ice research as

3.1 Ice Ih: The Solid Phase of Water on Earth

73

a

b

Fig. 3.5 a Schematic example of snow stratigraphy on a lake ice cover; b Snow fieldwork on lake ice with inspecting snow thickness stratigraphy and sampling vertical profile. Photograph by the author

74

3 Structure and Properties of Lake Ice

Table 3.3 Physical properties of seasonal snow (Fierz et al., 2009; Granberg, 1998; Male, 1980; Yen, 1981) Property

Value

Comments m−3

Reference value 250 kg m−3

Density

100–400 kg

Grain size

1–10 mm

Changes due to metamorphic processes

Liquid water content

0–100%

Significant only at 0 °C; water holding capacity is 5%

Thermal conductivity

0.03–0.4 W m−1 °C−1

Proportional to density squared

Albedo

0.5–0.9

Depends on liquid water content, grain size

Light penetration depth

5–15 cm

Depends on liquid water content, grain size

well. The structure and properties of snow undergo remarkable evolution in metamorphic processes during the ice season. Snow on lake ice has two specific characteristics. First, since the ice is floating, the lower part of snow layer may submerge and form slush with the lake water. Second, the presence water at the freezing point temperature beneath the ice acts as a heat source to cold snow and has an influence on snow metamorphosis. Particularly in cold weather conditions depth hoar may form in the lower layer of snow cover. The fundamental properties of snow on lake ice are the thickness, density, grain size and shape, liquid water content, and impurities (Table 3.3). The density of seasonal snow ranges from less than 100 kg m−3 for new, loose snow to 400 kg m−3 for wind-packed or wet snow, while the grain size is within 1–10 mm. The density of snow is the key property, and there are simple density evolution models forced by the weather conditions (Bartelt & Lehning, 2002; Yen, 1981). The density changes by wind packing, constructive and destructive metamorphosis, and snow melt-freeze metamorphosis. Density is the main factor to estimate snow strength and thermal properties. The liquid water content (νw , volume fraction) has a very large variability. The water holding capacity of snow is about νw∗ = 5% (DeWalle & Rango, 2008), and due to the relatively large size of snow crystals, the hydraulic conductivity is good so that the water exceeding the holding capacity, νw > ν ∗w , is filtrated down through the snow layer fast. On lake ice, this means the development of a slush layer at the snow–ice interface. In cold snow νw ≈ 0, and in water-soaked snow or slush νw ∼ 50%. The fundamental properties of snow are used for the parameterization of other physical properties. The heat capacity of snow is simply (ρc)s = ρs ci , where the subscript s is for snow. The thermal conductivity of snow, κs , is a highly important property for ice growth since snow is an effective insulator due to its high porosity. The classical model takes the proportionality κs ∝ ρs2 (Abel’s, 1893), and the presently widely used formula is (Yen, 1981) κs = 2.22362 · ρs1.885

(3.6)

3.2 Lake Ice Types and Stratigraphy

75

where κs is in W m−1 °C−1 and ρs is in g cm−3 . According to Yen’s formula, the conductivity typical in boreal snow is one order of magnitude lower than the conductivity of ice: for ρs = 0.25 g cm−3 , we have κs = 0.16 W m−1 °C−1 while κi = 2.1 W m−1 °C−1 at 0 °C. At the limit ρs → ρi , κs /κi → 0.88, and the power of the formula suggests that the conductivity of snow-ice is 5–10% lower than the conductivity of congelation ice. Due to the high albedo and small light penetration depth of snow, sunlight is largely blocked out from the ice and the water body under a snow cover (Warren, 1982). The optical properties of snow are highly sensitive to the liquid water content and to a lesser degree to the grain size, as well as to eventual light-absorbing particles. Dry snow albedo is 0.9 while dry ice albedo is 0.5, and therefore absorption of solar radiation by ice is five times that by snow. The light penetration depth of snow is one order of magnitude smaller than this depth of ice. With the low thermal conductivity and high attenuation of solar radiation, the snow layer protects the ice from the weather variations and delays the melting of ice to begin in spring.

3.2 Lake Ice Types and Stratigraphy 3.2.1 Ice Structure Analysis Research of the crystal structure of ice is based on ice samples, which are usually taken by core drills in cylindrical samples with diameter 10–20 cm (see Eicken et al., 2009). Not much can be said about the ice structure without manual processing. Ice samples are examined at the site for their visual features and immediate measurements (Fig. 3.6). Thick sections (1 cm) are prepared for the stratification and gas bubbles (Fig. 3.7). They are further processed in cold laboratory at − 10 °C or lower temperature. Attached to glass plates, using a plane or a microtome, the thick sections are processed for thin sections (less then 1 mm). The crystal structure is revealed from the thin sections in polarized light between crossed polarizers (Langway, 1958). Due to birefringence, crystals show different interference colours; only if the c-axis is perpendicular to the thin section, no light goes through the crossed polarizers and the crystal appears black (see Fig. 3.4). The thin sections can be rotated in a universal stage to determine the polar and azimuth angles of the c-axes (Langway, 1958). The size and shape of ice crystals and their c-axis directions define the crystal structure for geophysical investigations.

3.2.2 Lake Ice Stratigraphy Information about the ice growth history is stored in the layers and crystals and can be recovered by ice sample analyses (Gow, 1986; Michel & Ramseier, 1971). Table

76

3 Structure and Properties of Lake Ice

Fig. 3.6 An ice sample has been taken. At the site, the visible structure is recorded, and the temperature profile is measured. Photograph by the author

Fig. 3.7 A thick section of lake ice in normal light (Lake Pääjärvi, Finland). The sample floats in an open water spot and the snow-ice (white) and congelation ice (black) layers show up clearly. The total thickness of this ice sample is 35 cm. Photograph by the author

3.4 shows the classification of lake ice for the crystal structure and stratigraphy. In general, there are three vertical layers in a static lake ice sheet: primary ice, secondary ice, and superimposed ice. The primary ice layer can be thin (P1, P2), difficult to identify in vertical ice samples, or of finite thickness (P3, P4). The secondary ice layer is normally congelation (S1, S2, S3) ice but may have a sub-layer structure with congelation ice and frazil ice (S4, S5) layers. A multi-layered secondary ice is common in river ice. Superimposed ice is mostly snow-ice (T1, T2) but can also account for frozen ponds on the surface.

3.2 Lake Ice Types and Stratigraphy

77

Table 3.4 Lake ice stratigraphy and forms of ice (Gow, 1986; Michel & Ramseier, 1971) Ice type

Code

Grain size

C-axes

Conditions

Primary ice

P1

≥ Large

Vertical

Calm, slow growth

P2

≥ Medium

Vertical/Random

Calm, rapid growth

P3

Fine

Random

Turbulent (frazil ice)

P4

≤ Medium

Random

Snowfall

S1

≥ Large granular

Vertical

Beneath primary ice P1

S2

≥ Large columnar

Horizontal

Beneath primary ice P2

S3

≥ Large

Horizontal, aligned

Perennial lake ice

S4

≤ Medium

Random

Frazil ice transport

S5

≤ Medium

Random

Drained frazil slush

Secondary ice Congelation ice

Frazil ice Superimposed ice

T1

≤ Medium

Random

Wet snow

T2

≥ Medium

Random

Wet snow

Frozen pond

T3

As P & S

As P & S

Surface pond

Agglomerate ice

R

All size

All variants

Agglomeration of ice

Snow-ice

Primary Ice The texture of primary ice depends on the prevailing weather conditions (Michel & Ramseier, 1971). The crystal structure of primary ice influences the crystal structure of the secondary ice underneath (Gow, 1986). In calm or laminar flow conditions, quiet freezing (Gow & Govoni, 1983) takes place (P1 and P2), typically in a cold clear night with thermal radiation losses from the water body. A thin, supercooled surface water layer is formed, and needle-shaped crystals appear first on the surface with their main axis perpendicular to the c-axis. Ice skim grows horizontally in the supercooled surface layer, and the size of crystals varies largely depending on the concentration of nucleation seeds and the rate of the heat loss from the water. Slow heat loss means slow crystallization, and the needles expand into plate shape aligned with the surface and with c-axis perpendicular to the surface. At fast heat loss, ice crystals grow fast and remain small. The plates do not any more float only horizontally, but the orientation of c-axes becomes random. In quiet freezing the primary ice layer is very thin (less than 1 mm), and it may be easily lost by melting or sublimation. The freeze-up starts from the shoreline, where the lake is shallow and cools fast, and a near-shore ice border ice zone is formed. The ice cover extends further offshore if the conditions remain calm. Turbulent flow can break thin border ice to restart then in the next calm phase.

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3 Structure and Properties of Lake Ice

Frazil ice (P3) forms in turbulent conditions. The crystals are circular disks, they agglomerate into porous flocs, and the c-axes are randomly oriented in three dimensions. Frazil ice formation can be fast since the surface temperature is at the freezing point and the heat loss from the water body is very large at a low air temperature. When the buoyancy of the frazil ice overcomes the turbulence, the crystals will float up and form shuga and grease ice and gradually freeze into a solid ice layer. The thickness of the primary frazil layer can be several centimetres depending on the level of turbulence. Frazil ice aggregates may collide with each other and become rounded plates. This is called pancake ice due to its visual appearance, often observed in large lakes (e.g., Nghiem & Leshkevich, 2007). The size of the pancakes is up to a few meters. Frazil ice formation is a common phenomenon in rivers (see Ashton, 1986). Frazil flocs are effective in capturing impurities from the water. When ice formation is initiated by ice or snow nuclei precipitating onto water (P4), the new lake ice crystals remain small and randomly oriented. The solid precipitation may also come from the water body itself by frozen splashes or by evaporation–deposition mechanism. During heavy snowfall, the primary ice layer forms of congealed snow slush and can be much thicker than what results from quiet freezing (see Fig. 3.3). Secondary Ice The secondary ice layer grows down from the ice bottom into the water body as congelation ice (S1 and S2) or as an accumulation of frazil ice (S4 and S5) (Gow & Govoni, 1983; Leppäranta & Kosloff, 2000; Li et al., 2011; Michel & Ramseier, 1971; Shumskii, 1956; Sokol’nikov, 1957). Getting thicker congelation ice grows slower. Thus, the crystal size increases with depth, and typically large or extra-large crystals are found beneath a 10–20 cm thick transition layer (Gow & Govoni, 1983). The resulting congelation ice structure depends on the external conditions. In a single lake, both S1 and S2 type congelation ice can be found in different winters according to the assumed role of the primary ice to set the initial conditions (Gow & Gowoni, 1983; Leppäranta & Kosloff, 2000). If the c-axes of the primary ice layer are vertical (P1 and P2), this alignment is inherited to the congelation ice below, and ice crystals are giant macro grains of ~ 10 cm size (Gow & Govoni, 1983). If the c-axes are randomly oriented in primary ice, the crystal alignment turns horizontal with depth (P2, P3, P4). This process is called geometric selection, illustrated in Fig. 3.8 (Kolmogorov, 1949; see also MüllerStoffels et al., 2009; Pounder, 1965). Since a crystal grows faster perpendicular to the c-axis, the crystals with vertical c-axes are closed out during the growth process, and beneath the transition layer the c-axes are horizontal. In this case, congelation ice layer has a columnar crystal structure, and the crystal columns are 0.5–5 cm in diameter and 5–50 cm in height, size increasing with depth. In freshwater ice the directions of c-axes are usually random in the horizontal plane. Alignment of c-axes has been found in perennial lake ice (S3, Chambers et al., 1986) and in seasonal lake ice (Li et al., 2011). In marine water, preferred alignment in the direction of currents beneath the ice has been observed (Weeks & Gow, 1978), and the phenomenon was shown by laboratory experiments (Langhorne, 1983).

3.2 Lake Ice Types and Stratigraphy

79

Fig. 3.8 Geometric selection in lake ice growth when the c-axes are randomly oriented in the primary ice layer. Redrawn from Pounder (1965)

Another possible secondary ice type is frazil ice (types S4 and S5). Frazil ice crystals are fine flowing free with the water in turbulent flow, and further downstream may attach to the bottom of existing ice. Frazil ice layers are typical in river ice (Ashton, 1986; Leppäranta & Kosloff, 1987; Michel & Ramseier, 1971) and they are also common in Antarctic seas, where open water is abundant in the ice pack (Gow et al., 1982). In large lakes, semi-persistent open water spots are observed, serving as potential sites for the generation of frazil ice. However, available publications on lake ice crystal structure are too limited to state, apart from the primary ice layer, how rare frazil ice layers are in lake ice. Superimposed Ice The main form of superimposed ice is usually snow-ice or frozen slush (Adams & Roulet, 1980; Palosuo, 1965), while frozen surface ponds and hoar deposition have a minor contribution. The source slush for snow-ice is a mixture of snow and liquid water available from flooding, liquid precipitation, or snow melt water. Thus, ice crystals of snow-ice are meteoric ice or frozen lake water. Snow-ice needs the snow and therefore it is missing in arid climate. As far as it is known only a thin layer of ice is formed of frozen ponds on lake ice. Snow-ice crystals are fine or medium size (see Fig. 3.4), depending on the size of the parent snow crystals and voids between, and the orientation of the c-axes is random. The thickness of snow-ice layer is limited by the available snow and liquid water. Snow-ice may form a thick layer, at extreme 90% of the whole ice layer in Finland (Fig. 3.9). The snow layer experiences densification when it is mixed with lake water. When the mixing takes place by flooding, approximately half of the slush is liquid water and half of it is snow (Leppäranta & Kosloff, 2000). Snow-ice grows from the top of the slush down. At the start, a pocket of slush is captured between the new snow-ice and the old ice, and if the snow-ice growth is slow or stops, a slush pocket may persist for several weeks (Leppäranta, 2009), see Fig. 3.9. Snow-ice inherits air inclusions from the snow layer and has lower density than congelation ice, 860–890 kg m−3 corresponding to 3–6% air content4 (Palosuo, 1965). Because of the large air content, the thermal conductivity and strength are lower in snow-ice than in congelation ice, and snow-ice scatters light strongly appearing opaque. In Finland, the snow-ice fraction is typically about 1 /3 of the total ice thickness. In winter 2013 snow accumulation was much more than 4

In glaciology, when snow transforms to ice by compression, the density at the transition is taken as 830 kg m−3 (e.g., Paterson, 1999), corresponding to the gas content of 9.5%.

80

3 Structure and Properties of Lake Ice

Fig. 3.9 Ice sample from Lake Valkea-kotinen, southern Finland, 19 March 2013. Scale below in centimetres, top on the right. Total thickness 45 cm with 5 cm of congelation ice shown by the clear layer. Within 30–35 cm from the bottom there was a 2-cm slush layer illustrated by the cut in the sample. Photograph by the author

average in southern Finland and the timing was favourable for snow-ice formation (Fig. 3.7). As a result, the ice thickness was around 50 cm with up to 90% snow-ice, corresponding to close to 2 /3 of the snow thickness on land. Floating of snow-covered lake ice is described by the Archimedes law. The ice freeboard is ) ( ρi ρs hi − hf = 1− hs (3.7) ρw ρw where h i and h s are the thicknesses of ice and snow, respectively. The ice surface sinks down as more snow accumulates, and the ice is forced beneath the water surface i h i . Since ρw − ρ i ≈ 90 kg m−3 , for ρs = 250 level, when h f < 0, i.e. h s > ρwρ−ρ s kg m−3 the snow thickness must be at least about one-third of the ice thickness for the flooding to occur. Then water flows through eventual cracks to form slush with the snow on the ice. Cold freshwater ice may remain solid under a snow load, and in such situation a new drillhole acts as an artesian well with water flooding up fast. Fishermen often report that water bursts up when a hole is drilled through. Pressure heads up to 12 cm have been measured in Finland (Leppäranta et al., 2003b). In saline lakes the ice is more porous than freshwater ice, and lake water may penetrate through under pressure. Snow-ice formation is limited by the snow accumulation. In the flooding process, snow must be added on for further growth. When only snow-ice is produced, its thickness, h si , is limited by

3.2 Lake Ice Types and Stratigraphy

81

(

ρw − ρ i h si ≤ 1 + ρs

)−1

Hs ≈

3 Hs 4

(3.8a)

where Hs is the total snow accumulation during the winter. In practice, snow is densified when mixed with water, and therefore Eq. (3.8a) is a biased up by 10–15%. When there is a congelation ice layer, snow accumulation needs first to overcome the buoyancy of the congelation ice, and ‘extra snow’ if available can be used to grow snow-ice. In spring, the day-and-night melt-freeze cycle produces snow-ice. Snow melts in daytime and the mixture of meltwater and snow freezes in night-time. The resulting upper limit of snow-ice growth is from the mass balance h si ≤

1 ρs hs ≈ hs ρi 4

(3.8b)

When liquid precipitation, PL , acts as the water source for the slush formation, it will be added to the snow and the upper limit of snow-ice thickness is ) ( 1 h si ≤ min 2PL , h ≤ P 2 s

(3.8c)

This assumes that snow density is 250 kg m−3 , and wetted snow is compressed, and snow-ice contains fifty-fifty liquid precipitation and snowfall. Extra water forms an ice lens, and extra snow is unchanged, but at most the total precipitation is the overall limit. High mountains with heavy snow accumulation, such as Sierra Nevada of California, introduce an extreme case. The ice cover consists of slush and frozen layers and the total thickness of this mass can be 3 m (Melack et al., 2021). The ice is mostly a mixture of snow and lake water. Agglomerate Ice and Anchor Ice Agglomerate ice (Table 3.4) consists of broken ice pieces. Ice cover can be broken by winds, waves and changes in water level elevation, and ice blocks may drift and accumulate into vertical stacks, ridges. Agglomerate ice is common in large lakes that are partially frozen for a longer period during the ice season (Fig. 3.10). In sea ice research this ice type is called (mechanically) deformed ice and divided into further sub-classes such as rafted ice and ridged ice (see WMO, 1970, and later web updates). Frazil may also crystallize onto lake bottom to form anchor ice that is common in river environment (Ashton, 1986; Shen, 2006). Anchor ice is expected to form in turbulent, supercooled, shallow water conditions. In lakes, anchor ice is rare but has been observed in shallow water when primary ice was forming in windy conditions. In southern Finland, anchor ice formation has been observed in Lake Päijänne as a transient phenomenon. In conditions of supercooling and turbulent, shallow water, anchor ice was observed by a diver on several occasions (Fig. 3.11), but when the

82

3 Structure and Properties of Lake Ice

Fig. 3.10 Agglomerate ice produce by ice pressure in the coastal zone in Lake Baikal. Photograph by Oleg Timoshkin, printed by permission

conditions returned to normal, the anchor ice disappeared in a few days. This ice class is not included in the original classification of Michel and Ramseier (1971). As a follow-up, anchor ice can rise to the surface when its buoyancy overcomes the binding forces at the bottom. Then it will be part of the ice cover and can be visually recognised from the captured bottom sediments. Example 3.4 Ice formation in a medium-size lake, Lake Pääjärvi in southern Finland. The surface area of this oligo-mesotrophic lake is 13.4 km2 , and the mean depth is 15.3 m (Ruuhijärvi, 1974). The structure of the ice has been monitored for 20 consecutive years in connection with field courses. The ice cover was usually static but in one winter when the ice was relatively thin (33 cm), a minor mechanical shift was observed with a formation of a small ridge. The annual maximum ice thickness ranged from 24 to 80 cm, and the relative proportion of snow-ice ranged from a minimum of 5% to a maximum of 90% of the total ice thickness. The standard deviation of ice thickness was 12.5 cm, while for snow thickness it was 8.0 cm. Usually the ice formed in a cold calm night in late autumn, and the secondary ice was congelation ice with vertical crystal orientation and extra-large granular crystals (S1). The freeboard was within ± 5 cm, and snow-ice formed mainly by flooding. In one winter out of four, a several centimeters thick, semi-persistent slush layer developed within the snow-ice layer, and the presence of algae could be visually

3.2 Lake Ice Types and Stratigraphy

83

Fig. 3.11 Anchor ice in Lake Päijänne, southern Finland. Photograph by Teemu Lakka, printed by permission

observed. Leppäranta and Kosloff (2000) reported that the density of snow on Lake Pääjärvi is 0.23–0.33 g cm−3 . The statistics are (in cm):

Ice thickness

Congelation ice

Snow-ice

Snow

Mean

44.3

31.0

15.3

7.5

St. dev.

12.5

12.5

9.1

8.0

Min

24

23

0

0

Max

80

70

40

30

Freeboard 2.5 0.9 − 5.0 5.0

84

3 Structure and Properties of Lake Ice

Fig. 3.12 Platelet ice at the bottom of lagoon ice cover, Northern Hokkaido. The scale numbers show centimetres. The case was documented in a field course organized by Kunio Shirasawa, Hajo Eicken, and the present author. Photograph by the author

A specific form of ice is platelet ice which has been reported from Antarctica (see Hoppmann et al., 2020). There, rising meltwater from the bottom of ice shelves experiences supercooling and meeting the bottom of sea ice crystallized into platelets. The critical condition for plate formation is the under-ice supercooling. Similar phenomenon has been observed in Saroma-ko lagoon in the Sea of Okhotsk where inflowing river water becomes supercooled between ice bottom and saline lagoon water (Fig. 3.12).

3.2.3 Ice Mass Balance A lake ice cover consists of ice and snow layers. The mass per unit area, expressed in thickness of an equivalent water layer, is hw =

ρs ρi hi + hs ρw ρw

(3.9)

This formulation allows for porosity changes for snow and ice both. The mass of an ice cover changes by precipitation (P), sublimation/evaporation (E), flooding (F), drainage (D), and freezing and melting. The mass balance is

3.2 Lake Ice Types and Stratigraphy

d ρi db hw = +P−E+F−D dt ρw dt

85

(3.10)

where db/dt is freezing/melting at the ice bottom. This term is at most about 50 mm per day, and evaporation/sublimation is at maximum about 10 mm per day, but the other terms have no clear limitations. In arid regions, the ice budget is simple since precipitation and flooding are absent, and then, consequently, drainage is only for meltwater (Fig. 3.13). At equilibrium, ice growth is balanced by sublimation. Sublimation can become important in the mass balance, e.g., Huang et al. (2019) observed 30 cm sublimation during a half-year ice season in the Tibetan plateau. An important dimensionless characteristic of freezing lakes is the ratio of ice thickness to the lake depth, ζ = h/H . Normally ζ 1, and the ice cover is a minor part of the mass balance of the lake. But in very shallow lakes, ζ ∼ 21 ad thus the ice significantly reduces the volume of liquid water in the lake (Yang et al., 2016). Ice thickness is a function of space and time, h = h(x, y; t). In a lake, ice thickness field can be described by the cumulative thickness distribution (Thorndike et al., 1975)

Fig. 3.13 The sources of impurities and their catchment processes in lake ice. Usually, atmospheric fallout and flooding are the main sources in lakes, but where frazil ice and anchor ice forms, the concentration of impurities is at largest

86

3 Structure and Properties of Lake Ice

∏(ξ ) =

A{h ≤ ξ } A{h ≤ ∞}

(3.11)

where A is the surface area. The area function and the distribution function are continuous from the right, so that ∏(0) = 1 − I, ∏(∞) = 1, where I is the ice coverage or the ratio of ice cover to the total lake area. The conservation law of the thickness distribution is written (adapted from Thorndike et al., 1975) dh d ∏=− · dt dt

(

) ∂ ∏ +ψ ∂h

(3.12)

The first term on the right-hand side provides changes due to ice growth and melting, while the second term is for mechanical changes. Example 3.5 Assume a lake ice cover with constant thickness h 1 . Then ∏(ξ) = H(ξ − h 1 ), where H is the Heaviside function, H(x) = 0(1) for x < 0(x ≥ 0). (i) Due to rafting (overriding of ice sheets), let the ice coverage open by ΔI. The resulting ice thickness distribution has three steps: at ξ = 0 from 0 to ΔI, at ξ = h 1 from ΔI to 1 − ΔI, and at ξ = 2h 1 from 1 − ΔI to 1. (ii) Let ˙ the ice thickness change by )] the rate h. Then the thickness distribution changes as ( [ ˙ . d∏/dt = H ξ − h 1 + ht

3.3 Impurities in Lake Ice Cover Lake ice cover contains impurities between ice crystals. They originate from the water body, bottom sediments, or atmospheric deposition (Fig. 3.14) and consist of gas bubbles, liquid inclusions, and particles, which are called ice sediments. The impurities influence on the ice properties, particularly the electromagnetic properties are sensitive to them. Liquid inclusions and sediments are examined from ice meltwater samples as usually water samples are analysed (Leppäranta et al., 2003b). Ice cores are sectioned according to the stratigraphy, for a good resolution the slices are 5–10 cm thick. Fig. 3.14 Ice-covered lake in the Beiluhe Basin, Qinghai–Tibet Plateau at 4640 m altitude. This is in the Central Asian arid climate zone with no significant winter precipitation. Photograph by Zhijun Li

3.3 Impurities in Lake Ice Cover

87

In freshwater lakes, concentrations of impurities in ice are low compared with the lake water. In river environment, frazil ice particles are abundant and effective in harvesting suspended particles from the river water. There are not much published studies of ice impurities in saline lakes, but it is anticipated that dissolved substances make a significant contribution as they do in sea ice (Weeks, 2010). The accumulation period of impurities can be several months long, but their release to lake water takes place mostly during the melting period of some weeks. Gas Content The gas volume is νa ∼ 1–2% in congelation ice and νa ∼ 3–6% in snow-ice (e.g., Li et al., 2011; Palosuo, 1965). The influence of gas volume on ice properties is on the order of per cents. Volume-related properties are reduced proportional to the gas volume, such as the density and specific heat, but the reduction of area-related properties, such as strength and thermal conductivity, depend on the geometry of the gas inclusions and are reduced by a factor of 1 − b(νa ), where b is an empirical function. Assuming gas inclusions as randomly spaced spherical bubbles, by dimen2/3 sional analysis it is seen that b ∝ νa . If the inclusions were long vertical cylinders, 1/2 we have b ∝ ν a (Assur, 1958). Gas bubbles have a major influence particularly on the scattering of electromagnetic waves that is a major issue in light transfer through lake ice and backscatter in microwave remote sensing of lake ice. Gases are released from the water body and bottom sediments due to warming, because the saturation level of dissolved gases decreases with temperature, and due to biogeochemical processes. Gas bubbles accumulate beneath ice and are locked inside during ice growth. The size of gas bubbles is 0.1–10 mm, but occasionally larger ones are observed, e.g., methane bubbles tend to be large (Wik et al., 2011). The geometry of gas bubbles can be examined from thick sections using a microscope. In congelation ice, gas bubbles appear sometimes as layers, and the bubbles may extend vertically into cylinders if there is continuous supply of gas during an ice growth episode. Ice density decreases in proportion to the gas content (see Example 3.3), and therefore density measurements can be used to estimate the gas volume in ice. Figure 3.15 shows an example from Hongipao Reservoir, a freshwater basin in northeast China (Li et al., 2011) based on field samples. Most gas bubbles were less than 1 mm in diameter, and their shape was spherical with higher content in the upper ice layer. In the top 5 cm layer the bubbles were small and spherical, and in 12–18 cm layer the bubbles were large. Down in the middle layers, the bubbles were arrow-shaped while in the bottom layer (70–80 cm) they were cylindrical with smaller volume. The mean gas content was about 2%, and the bubble size and volume varied with the ice growth stage. The bubble size and volume were observed to be similar in a lake in the Tibetan plateau (Huang et al., 2012). Salinity When lake water freezes, dissolved substances are separated out from the solid phase (Weeks, 2010; Weeks & Ackley, 1986). The salinity of the ice is lower than the salinity of water, so that at freezing the surface water layer is enriched

88

3 Structure and Properties of Lake Ice

Fig. 3.15 Ice sample analysis for gas bubbles in winter 2008–2009 in Hongipao Reservoir, Heilongjiang Province, China. a Gas bubbles in lake ice cover, February 9, scale shows cm from top; b Bubble size and gas volume as functions of depth, Dec 19 and Feb 9. From Li et al. (2011). Photograph by Zhijun Li

with dissolved substances, and ice melting has an opposite consequence (Huang et al., 2021; Kirillin et al., 2012). The segregation coefficient κ provides the fraction of dissolved substances, which freezing captures from the parent water (Weeks & Ackley, 1986): Si = κ Sw

(3.13)

where Si and Sw are the salinities of ice and water, respectively. The segregation coefficient depends on the salinity and increases with the growth rate of ice. In freshwater lakes, κ ≈ 0.1 (Leppäranta et al., 2003b) but in sea water, κ ≈ 0.25 − 0.5 (Weeks & Ackley, 1986).

3.3 Impurities in Lake Ice Cover

89

When ice grows in saline water, a cellular ice–water interface forms due to constitutional supercooling (Weeks & Ackley, 1986), which is caused by the difference in the molecular diffusion of salt and heat in liquid lake water (Fig. 3.16). This interface can capture a large fraction of dissolved substances between crystal platelets. In fresh water, ice–water interface is planar, and the separation of salts from the solid phase is strong. The actual water salinity, where the transition between freshwater ice type and saline ice type takes place, is Sw =1–2‰ according to observations in the Baltic Sea (Kawamura et al., 2001; Palosuo, 1961; Weeks et al., 1990). Salts are stored in the ice between ice crystal platelets in the form of liquid pockets or salt crystals. Individual salts have their own crystallization temperature or eutectic temperature, and therefore the phase composition of saline ice depends on the chemical composition and temperature (Assur, 1958). The volume of liquid saline pockets, or the brine volume, depends on the temperature and salinity of ice and can be determined from the phase diagram of the lake in concern. The individual phase diagram of any freezing lake is not known to the author. Since seawater is a chemically uniform solution, one phase diagram has been used for all sea-ice (Assur, 1958). The salinity of liquid brine, Sb , within the ice corresponds to the freezing point at the ambient temperature T , and provides the brine volume: T = T f = T f (Sb ) νb =

ρi S i ρb S b

(3.14a) (3.14b)

where ρb ≈ 1000+0.82Sb ‰ kg m-3 is the density of brine. Equation (3.14b) assumes that all salts are in the brine, but in very cold ice a part of salts is crystallized and must be taken out (Assur, 1958; Weeks & Ackley, 1986). The brine volume changes with temperature and is the primary factor to influence the properties of saline ice. It may reach 10% and more when the ice is warm, and at νb ∼ 50% the ice loses

a b

Fig. 3.16 a Constitutional supercooling in saline ice, slow diffusion of salts keeps nonlinear salinity and freezing point. b Ice–water interface for freshwater ice and saline ice. S refers to the salinity of the water. Based on Weeks and Ackley (1986)

90

3 Structure and Properties of Lake Ice

its integrity. Brine flows out from ice into the water, especially when the ice warms up in spring. Solid salt crystals scatter light and therefore influence on the optical properties of ice. In highly humic lakes, liquid humus layers have been found in the snow-ice layer and humus pockets also in the congelation ice layer (Salonen et al., 2009) as shown in Fig. 3.17. Humus absorbs light at short wavelengths and has notable influence on the transparency and colour of the ice. Brine pockets can serve as habitats of biota, and they have been examined widely for sea-ice ecology (Horner et al., 1992; Kuparinen et al., 2007). Since the pockets contain liquid water and nutrients, sufficient conditions for primary production exist as soon as photon flux is available. Algae are captured into the brine pockets in the freeze-up process, and their growth is primarily light-limited. When the cellular interface forms (Fig. 3.16), the most active layer is the bottom layer, or so-called skeleton layer, which contains large fraction of brine and may become coloured brown-green by algae. In freshwater lakes, congelation ice does not provide an appropriate environment for living organisms, but algae can grow in eventual quasi-permanent slush layers between snow-ice and congelation ice, as has been observed in many lakes in Finland. Ice Sediments Sediment particles are captured from the water body in congelation ice growth, flooding of ice brings these into the slush layer on top, and one more source is atmospheric deposition. In congelation ice the concentration of sediments is low

Fig. 3.17 Liquid humus inclusions in lake ice in Lake Valkea-kotinen, southern Finland. a Humus layers in snow-ice, scale in centimetres from the top surface; b Humus pocket, size about 1 cm across. Photographs by the author

3.3 Impurities in Lake Ice Cover

91

but in snow-ice and lake water the sediment concentrations can be comparable. The bulk concentration in lake ice varies but the magnitude is as in water (Leppäranta & Kosloff, 2000; Leppäranta et al., 2003b). In the presence of frazil ice formation, the ice cover can be enriched with sediments as compared with water as is observed often in river ice (Leppäranta et al., 2003b). In a two-layer congelation ice–snow–ice lake ice cover, most of the sediments are due to the flooding and atmospheric deposition. Sediments may influence the properties of ice and they may also have a significant role in transporting matter if the ice drifts. Very shallow waters may freeze throughout to the lake bottom that gives another way to catch sediments to the ice. This ice may be lifted up when the water level rises that takes place normally in the melting period. Then ice also is often drifting, and the bottom sediment becomes transported away from shallow areas to other parts of the lake. Also anchor ice formation can remove bottom material in corresponding manner. With the buoyancy of (ρw − ρi )V g, where V is the volume, anchor ice can overcome cohesion between the ice and bottom. Example 3.6 Meltwater samples have been examined for the presence of impurities in four different types of lakes in southern Finland (Leppäranta et al., 2003b). The bulk concentration of dissolved substances was around 1 /5 of the concentration in water. Detail profiles showed that the segregation coefficient for congelation ice in freshwater lakes was about 1 /10 (Leppäranta & Kosloff, 2000). The level of suspended matter was close to each other in ice and water. Congelation ice is typically much cleaner than the water from which it forms whilst snow-ice includes impurities from the parent slush. The averages of the years 1997–1999 and nutrient concentrations in 1999 were: Conductivity μS cm−1 (at 25 °C)

Dissolved matter mg L−1

Suspended matter mg L−1

Organic fraction (%)

pH

Congelation ice fraction (%)

91.8

Päijänne, oligotrophic lake Ice

7.7

11.3

2.1

23

6.6

Snow

19.0

20.0

17.0

50

x

Water

78.7

31.3

1.4

49

7.0

Pääjärvi, oligo-mesotrophic lake Ice

13.0

14.3

2.1

36

6.7

Snow

16.5

15.0

4.2

38

x

Water

108.0

64.0

3.7

40

6.6

69.0

Vesijärvi, eutrophic lake Ice

7.0

12.7

2.0

33

6.6

Snow

28.0

23.5

9.9

54

x

Water

128.3

52.3

1.1

50

7.0

89.7

Tuusulanjärvi, hypereutrophic lake (continued)

92

3 Structure and Properties of Lake Ice

(continued) Conductivity μS cm−1 (at 25 °C)

Dissolved matter mg L−1

Suspended matter mg L−1

Organic fraction (%)

pH

Congelation ice fraction (%)

Ice

15.0

17.3

12.6

Snow

9.5

17.0

11.6

24

6.6

77.8

58

x

Water

208.3

143.0

11.5

22

6.8

Phosphorus (μg L−1 )

Nitrogen (μg L−1 )

P–PO4

Total

N–NO3

Total

Q w /K 1 , i.e., the air temperature must be low enough to make the heat transfer possible. When Q w = 0, the time evolution of ice thickness is

4.3 Ice Growth and Melting

h(t) =

/

133

t [h(0) + h a ] + 2

a2 S

a (t)

− h a , Sa =

[

] T f − (Ta + TR ) dτ

(4.40)

0

Comparing this with the Stefan case (Eq. 4.30), the surface temperature is replaced by Ta + TR , and the term h a ∼ 10 cm is the atmospheric buffering effect (see also Eq. 4.38). The Stefan formula is recovered when K 1 → ∞, physically corresponding to intensive turbulence that can take transfer up all heat coming from the ice. In polar winter, TR < 0 and in winter in low-latitude mountains TR > 0 that makes a significant modification to traditional freezing-degree-day formulae (Leppäranta and Wen, 2022). The atmospheric buffering effect is strong over thin ice. For h(0) = 0 and Sa (t) = 20 ◦ Cd, we have h = 7.9 cm while Stefan’s formula would give 14.8 cm. In the limit for large freezing-degree-days, the Stefan formula overestimates Eq. (4.40) by about h a .

4.3.3 Snow on Ice Snow accumulation on lake ice is common. Only in dry climate lakes are dominantly covered by bare ice. In the ephemeral lake ice zone, ice seasons are short, and precipitation is often in liquid phase so that snow-free ice surface may persist over long periods. A snow cover on lake ice has three roles: Firstly, the low thermal conductivity of snow slows ice growth down; Second, high albedo of snow cover protects ice from melting and delays the ice breakup; and third, mixing snow with liquid water leads to formation of slush and snow-ice. The mass balance of snow on lake ice is based on precipitation, sublimation, snow drift (Y ) and runoff (R). The snow cover is described by its thickness and density (e.g., Paterson, 1999; Leppäranta et al., 2013), and snow thickness changes due to net mass balance and densification: (∼ ) ∼ ρ s dh s d ρs = P − E + Y − R − hs (4.41) ρw dt dt ρw ∼

where ρ s is the vertical mean density of the snow cover. The terms are given in water equivalent (SWE) per time. The last term on the right-hand side is the snow densification that has a major influence on thermal conduction, since ks ∝ ρs2 (see Sect. 3.1.5). The heat conduction equation and the surface boundary condition are as for ice (Eqs. 4.26 and 4.28a), with thermal properties of snow, and at the ice–snow interface (z = s) we have the continuity of heat flow: ( ) ∂ ∂T ∂ ks + κs Q sδ eκs z (ρs ci T ) = ∂t ∂z ∂z

(4.42a)

134

4 Thermodynamics of Seasonal Lake Ice

| ∂ T || z = 0 : ks = Q 0 + ρs L f M0 ∂z |0− | | ∂ T || ∂ T || z = s : ki =ks ∂z |s − ∂z |s +

(4.42b) (4.42c)

Snow as an Insulator Consider snow-covered lake ice in steady polar winter conditions. Solar radiation and heat flux from water are ignored. Heat conduction equations through ice and snow read: ρi L f

T f − Ts Ts − T0 dh ≥0 + Q w = ki = ks dt h hs

(4.43)

Eliminating the ice surface temperature from the latter equation, the ice growth law is T f − T0 Qw dh ki ki + = · ;β = , dt ρi L f ρi L f h + βh s ks

(4.44)

Snow appears as the term βh s in the denominator and tells that the insulation effect of snow is β ∼ 10 times better than the ice insulation. In general, h s = h s (t), and there is no general explicit solution. Example 4.6 Consider the special case h s = λh(t), as one would expect that there is correlation between ice thickness and snow accumulation in the snow climate zone. Then h + βh s = (1 + λβ)h, and the Stefan solution can be directly used: / h(t) =

a2 S, a = h 2 (0) + 1 + λβ

/

2k i ,S = ρi L f

t

(

) T f −T 0 dτ

0

1 . In practice, λ is limited by the Thus, ice thickness is proportional to √1+λβ

i ∼ 13 , the snow load pushes the ice beneath the Archimedes law. If λ > ρwρ−ρ s lake water surface level, and flooding may take place (see Sect. 3.2.2). At λ = 13 , √ 1 ≈ 21 , i.e., effective snow insulation can reduce the ice growth to half of that 1+λβ in the case of snow-free ice.

Snow-ice Snow mixed with water becomes slush, and this slush may freeze and form snow-ice, the main type of superimposed ice on congelation ice (see Sect. 3.2.2). The water

4.3 Ice Growth and Melting

135

Fig. 4.9 Schematic cross-section of snow on lake ice. The top layer is above the water surface level, and below that snow-ice and slush layers are found

may come from flooding, precipitation, or melting. Flooding is usually the main source, and often melt-freeze cycles produce a small amount of snow-ice in spring. When a slush layer has formed, snow-ice grows down from its upper surface (Fig. 4.9). The vertical heat transfer equation of snow-ice is as Eq. (4.26) for congelation ice, but the boundary conditions are replaced by | | ∂ T || ∂ T || z = η1 : k i = ks ∂ z |η1− ∂z |η1+  T = Tf | z = η2 : dη2 = ρi Lki f ν · ∂∂zT |η+ dt

(4.45a)

(4.45b)

2

where η1 and η2 are the tops of the snow-ice and slush layers, respectively (see Fig. 4.9). The latent heat released is reduced since only the liquid water fraction ν of the slush needs to be frozen. Also, there is no heat flux from the slush to the bottom of the snow-ice. Observations suggest that typically ν ~ ½ when slush forms by flooding (e.g., Leppäranta & Kosloff, 2000). The boundary conditions give the quasi steady-state model for the snow-ice thickness h si and the solution as: ρi L f ν

T f − Tη1 Tη − T0 dh si = ki = ks 1 ≥0 dt h si hs T f − T0 ki dh si = · dt ρi L f ν h si + βh s

(4.46a) (4.46b)

Thus, with the same temperature gradient, snow-ice growth rate is ν −1 ≈ 2 times the growth rate of congelation ice. A new flooding event produces slush on top of the existing snow-ice, and thus it is possible that alternating slush and snow-ice layers exist. One semi-persistent slush

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4 Thermodynamics of Seasonal Lake Ice

layer is often found in boreal lakes (Leppäranta & Kosloff, 2000). Specific conditions develop if there is a heavy snowfall and in early ice season. The insulation effect of the snow cover is strong from the beginning, and the resulting ice has a large snow-ice fraction with variable ice thickness. Such situation occurred in winter 2013–2014 in Lake Kilpisjärvi, northern Lapland. Even though the winter was cold, weak spots remained in the ice cover all winter. If the underlying ice is very cold, the slush may also freeze at its bottom using the cold content of the ice. This gives the growth Δh si at the ice–slush interface, obtained from ρi L f νΔh si = −ρi ci T˜ h, where T˜ is the mean ice temperature. E.g., if the mean temperature of half-meter thick ice is –5 °C, we have Δh si = 3 cm. When the volume of meltwater or liquid precipitation is small, hard crust layers or ice lenses may result from freeze-melt metamorphosis. Example 4.7 After a flooding event, if h s = constant, snow-ice thickness is / h si (t) =

( )2 a2 S + βh s − βh s , S = ν

t

(

) T f −T 0 dτ

0

Take βh s ∼ 100 cm. If the snow surface temperature is –10 °C, after 10 days snow-ice thickness is 10.5 cm, provided the slush layer is at least this thick. Under the same forcing, a bare ice layer thicker than 20 cm would form in an open water spot. Example 4.8 Consider a continuous snowfall over a mountain lake. By Archimedes s h s , which can be slush, snowlaw, the depth of the submerged portion is H = ρwρ−ρ i ice, or a multi-layered slush–snow-ice system. To have continuous snow-ice growth at the surface, the condition is: a 2 T f − T0 dh si ρs dh s = · ≥ · dt 2ν βh s ρw − ρi dt For a constant rate of snowfall and surface temperature, continuous growth of snowice is possible until ) ) a 2 ρw − ρi ( dh 2s cm 2 ( ≤ · T f − T0 T f − T0 ≈ 1 ◦ dt νβ ρs Cd If the snow accumulation is 1 cm per day, we have h si ≤ 15 cm for T f − T0 = 10 ◦ C, and even snow accumulation of 0.5 cm per day and T f −T0 = 20 ◦ C, h si ≤ 60 cm. With decreasing snow accumulation rate, it is possible to get thicker snow-ice. But in cold climate mountains with heavy snow accumulation, such as the Sierra Nevada of California (Melack et al., 2021), a multi-layer system normally develops. Snow-ice formation can also take place due to melt-freeze cycles and liquid precipitation. In both cases the question is that how much snow is available to form the slush. If a slush layer is formed in the lower part of the snow layer, it will freeze into

4.3 Ice Growth and Melting

137

snow-ice. Snow-ice formation is easier than by flooding since the insulating effect of snow becomes reduced when snow is consumed in snow-ice formation. Data from a coastal landfast ice site in Sakhalin Island have shown that snow-ice formation in spring may add up to 20 cm to the ice thickness due to melt–freeze cycles and possibly also due to liquid precipitation (Shirasawa et al., 2005). Example 4.9 Consider the case of man-made ice techniques to speed up natural ice growth. At time zero, we have snow-covered congelation ice with ice and snow thickness h i0 and h s0 , respectively. We may try to increase the growth rate by (i) removing snow away, (ii) pumping water on bare ice. The result is √ (i) h = (h i0 + h a )2 + a 2 Sa − h a (ii) h = h i0 + ρKi L1 f Sa Case (ii) is clearly much more effective. Then ice grows proportional to Sa , the proportionality coefficient is ≈ 0.4 cm (°C·d)–1 , while in (i) congelation ice is generated. If h 0 = 20 cm and Sa = 100 ◦ Cd, case (i) gives h = 35 cm, and case (ii) gives h = 60 cm. In case there is snow on ice, pumping leads to snow-ice formation. Then the snow insulation decreases with snow-ice growth, but taking h s0 /2 as an effective insulation thickness, we have. /( )2 2 βh os (iii) h = h o + + aν Sa − βh2os 2 If h os = 10 cm, the whole snow cover is transformed into snow-ice by Sa ≈ 50 ◦ C d with h = 30 cm. After this, with pumping on bare ice we get 20 cm more ice. Thus, pumping water on snow cover is also effective but limited by the available snow. Although the strength of snow-ice is less than the strength of congelation ice, pumping lake water into snow cover produces stronger ice cover than reducing the insulation (case i). This pumping method was used in Finland, Lake Katumajärvi, to strengthen the ice in mild winters when the lake ice cover served as the starting point of a ski marathon (Palosuo, 1982). The bearing capacity question was not critical for the skiers, but it was so for the trucks, which collected the skiers’ bags to transport them to the goal area.

4.3.4 Frazil Ice Frazil ice forms in turbulent, slightly supercooled open water (e.g., Martin, 1981; Michel, 1978). Ice crystals flow free in turbulent eddies and are transported by the mean current. They have buoyancy, which overcomes the turbulence when the frazil mass is large enough. Then floating surface frazil ice pancakes form and may freeze together into a solid layer. In lakes frazil ice forms the primary ice layer when the initial ice formation takes place in turbulent conditions. In large lakes, there may be semi-persistent open water areas to serve as frazil ice source areas in winter.

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4 Thermodynamics of Seasonal Lake Ice

In the cooling process of a turbulent mixed layer, heat is lost to the atmosphere more than received from the deep waters. Cooling continues until minor supercooling, then frazil ice formation is commenced, and the surface water temperature is kept at the freezing point. Heat losses from the water body and, consequently, the rate of frazil accumulation can be large in cold weather. The thermodynamics is straightforward. The surface heat loss increases the volume of frazil and compensates for the heat flux coming up from the deep water. The heat budget equation is: ρi L f

] ( [ dh F ∗) = − Q 0 + 1 − e−κw H Q sδ + Q w dt

(4.47)

where h F represents the volume of frazil per unit area or the equivalent thickness of the frazil ice mass, and H ∗ is the mixed layer depth. As long as frazil ice crystals are free, they drift in the turbulent flow. Equation (4.47) has an equilibrium solution ] [ ( ∗) 0 ≤ Q w = − Q 0 (T0 ) + 1 − e−κw H Q sδ = −[K 0 + K 1 (Ta − T0 )]

(4.48)

where the last equation is the linearized surface heat balance. If T0 > T f , we have open water situation, where the surface heat loss is compensated by heat flux from deep water, while T0 = T f is the frazil ice case. In boreal winter, K 0 ∼ −50 W m–2 and K 1 ≈ 15 W m–2 °C–1 . In the presence of strong geothermal heat flux, say, 200 W m–2 we have T0 − Ta > 10 ◦ C, i.e., a freshwater lake may stay open down to the air temperature of –10 °C. In the case of a hypersaline lake, say, T f = −20 ◦ C and Q w = 0, we have T0 − Ta = −3 ◦ C at equilibrium. Thus, the lake is open down to the temperature of –17 °C. The volume of frazil can be directly integrated from Eq. (4.47) with T0 = T f since the heat fluxes are independent of the frazil volume. Thus, there is no self-insulating feedback mechanism as in the case of congelation ice growth, and because the surface temperature is at the freezing point, the heat loss continues to be large. Example 4.10 A simple model for frazil ice accumulation is ( ) ] 1 [ ( ) dh F =− K 0 T f + K 1 Ta − T f + Q w dt ρi L f ( ) K 0 T f ≈ (1 − α)Q s − ε0 (1 − εa )σ T f4 In the boreal zone winter, K 0 ∼ −50 W m–2 which corresponds to the frazil ice production by 1.4 cm d–1 , and for K 1 ≈ 15 W m–2 °C–1 the temperature difference term is as much when Ta − T f = −3.3 ◦ C. In very cold weather, say, Ta − T f = −20 ◦ C, the K 1 -term gives frazil ice as much as 8.5 cm d–1 . Congelation ice growth would grow about that much in the first day, but at 20 cm thickness its growth rate would be down at 3.5 cm d–1 .

4.3 Ice Growth and Melting

139

Fig. 4.10 Frazil ice accumulation and congelation ice growth at air temperature held constant of –15 °C. Normal case refers to climatological wind, and windy case refers to doubled wind speed

Figure 4.10 shows a comparison between the growth of snow-free congelation ice and frazil ice. When the wind is strong, turbulent heat losses and frazil ice accumulation become much more than congelation ice growth. If the heat loss is 100 W m–2 , the resulting frazil ice production rate is 2.8 cm day–1 . Accumulation of frazil ice in a turbulent boundary-layer has been modelled by Omstedt and Svensson (1984) using a one-dimensional model. Solving the full frazil ice problem, we need to consider also advection, and, consequently, a two-dimensional modelling approach. Frazil ice models have been largely worked on in river ice research (Lal and Shen, 1993). Frazil ice is not largely discussed in this book because of its likely unimportance in freezing lakes. For more information, the reader is referred to books and review papers in river ice and sea ice (Ashton, 1986; Martin, 1981; Michel, 1978; Wadhams, 2000; Weeks, 2010).

4.3.5 Ice Melting Melting of ice takes place in similar manner for all ice forms—congelation ice, snowice and frazil ice—depending on the crystal structure and impurities. Melting begins when the ice has warmed up to the melting stage and the heat balance continues to be positive (Fig. 4.11). Transient melting periods may occur all winter, but the final melting period is commenced when the radiation balance has turned dominantly positive. The timing of this turning depends on the latitude and the quality of the snow cover. Melting takes place at the top and bottom surfaces and by solar radiation inside the ice. According to recent field data (Jakkila et al., 2009; Leppäranta et al., 2010,

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4 Thermodynamics of Seasonal Lake Ice

Fig. 4.11 Decay of lake ice cover is a chaotic, thermodynamic–dynamic process driven by solar radiation, air temperature, precipitation, and wind. Photograph by the author

2016, 2019), surface melting, bottom melting and internal melting are of the same magnitude and the exact levels depend on the weather conditions during the melting season. Figure 4.12 shows results from Lake Kilpisjärvi, Arctic tundra from two melting seasons. Internal melting was strong in 2013 and the ice lost its integrity when the porosity was 40–50% and the thickness was still 20 cm. Melting of an ice cover shows clear horizontal characteristics. This appears as a patchy surface with snow, bare ice, and wet spots, and open water first forms near the shoreline and shallow spots. The patchiness develops due to positive feedback of melting to albedo: once albedo starts to decrease, melting further lowers the albedo and speeds up the decay process. In very shallow areas, the water beneath ice warms faster and melts ice from the bottom, as is observed. Melting of saline ice is more complicated than melting of freshwater ice since the brine dynamics need to be added into the system. The porosity and temperature change together at all temperatures dictated by the phase diagram of the ice in question. For sea ice case the problem has been largely investigated (see, e.g., Assur, 1958; Schwerdtfeger, 1963; Cox & Weeks, 1983; Leppäranta & Manninen, 1988; Wadhams, 2000; Weeks, 2010).

4.3 Ice Growth and Melting

141

Fig. 4.12 Evolution of ice thickness and porosity in 2013 and 2014. Ice thickness is shown by the red lines based on surface and bottom heat fluxes, and crosses (x) with dashed lines are for direct data. Porosity is given by blue lines based on measured absorption of sunlight in the ice. The slush layer based on direct data is shown for 2014, and the yellow mark shows breakup in 2013 (Leppäranta et al., 2019)

Ice Thickness and Porosity During Melting Period The state of melting lake ice is described with its thickness, temperature, and porosity as {h = h(t), T = T (z, t), ν = ν(z, t)}. The rates of surface melting, bottom melting, and internal melting are then obtained from the boundary conditions (Eqs. 4.28a, b) and integrating the solar radiation absorbed by the ice from the porosity Eq. (4.35): | [ ( ) ] Q0 dh || 1 max =− , 0 + E = M0 dt |0 1 − ν(0, t) ρi L f | | ) ( ki dh || 1 ∂ T || Qw = Mb = · − dt |b 1 − ν(b, t) ρi L f ∂z |b+ ρi L f ( ) κi Q sδ κi z ki ∂ T ∂ν ∂ = e + ∂t ρi L f ∂z ρi L f ∂z

(4.49a) (4.49b) (4.49c)

Surface melting is due to surface heat balance and sublimation, bottom melting is due to heat flux from water, and internal melting is due to solar radiation. First, snow cover is melted, analogously to Eq. (4.49a), and ice melting is commenced when snow cover is thin enough to let sunlight pass through.

142

4 Thermodynamics of Seasonal Lake Ice

Internal melting gives rise to structural defects starting from the crystal boundaries. The crystal structure is then revealed to the human eye. Due to its appearance, melting columnar-grained ice is called candled ice in common language. After breakage, ice floes and blocks drift with winds and surface currents, the heat from the water body to ice increases strongly, and ice disappears rapidly. Breakage increases the surface-tovolume ratio of the ice mass that also increases the melt rate. In the 1990s the author observed a 45-cm thick ice cover to disappear in two days in Lake Tuusulanjärvi, Finland; the ice was already porous to start with, risky to walk on. The very fast end of breakup has been sometimes phrased as ‘ice sinks overnight’ (Humphreys, 1934). At extreme, surface melting may be absent and ice decays by internal and bottom melting. The condition of cool skin in decaying ice-cover, T0 < T f , is Q 0 < 0 < Q 0 + Q sδ (Leppäranta et al., 2016), valid usually due heat losses by terrestrial radiation and sublimation. Beneath the ice bottom, solar radiation warms the water by e−κ i h Q sδ , and a fraction of this heat returns to the ice bottom adding on the heat flux Q w . This fraction was estimated as one-third for an Arctic tundra lake in Leppäranta et al. (2019). Ice melting can be further examined with two characteristics: ice thickness and ∼ mean porosity ν. These are coupled, since more porous ice takes less energy for melting and thinner ice gains less sunlight. When the cold content of ice disappears, we have a pair of equations for ice thickness and mean porosity as: [ ] max(Q 0 , 0) + Q w dh 1 +E =− ∼ · dt ρi L f 1− ν

(4.50a)

∼

) Q sδ ( dν = 1 − e−κi h dt ρi L f h

(4.50b)

The first equation takes out the pores from the melting to reduce the thickness of ice, assuming that vertical variations in the porosity are small that the mean porosity works there, and the second equation distributes the solar heat across the thickness of ice. In the porosity equation, the right-hand side is ∝ h −1 for h ≫ κi−1 and ≈ κρiiQL sδf

for h « κi−1 . In the former case, all sunlight is absorbed by the ice. Internal melting, by adding on porosity, weakens the ice, and at ν = ν ∗ ∼ 0.5 the ice collapses under its own weight (see Fig. 4.12). Thus, for the annual maximum ice thickness h max at time t = 0, the final ice breakup comes from melting from boundaries or via breakage, i.e., at ⎧ ⎨

t1

tb = min t1 , t2 | ⎩

(M0 + Mb )dt = h max , ν(t2 ) = ν ∗

⎫ ⎬ ⎭

(4.51)

0

Example 4.11 Consider melting of clear congelation ice with constant solar and surface fluxes. Take Q 0 = 50 Wm−2 , Q w = 10 W m−2 , Q sδ = 50 Wm−2 , κi = 1 m−1 , E = 0, and initial thickness h 0 = 0.5 m of solid ice. Then (from Eqs. 4.50)

4.4 Thermodynamic Models of Lake Ice

143

∼

m dν cm 1 dh =− ≈ c; m = 1.7 , c = 0.014 ∼, dt d d 1− ν dt ∼

∼

The porosity increases by the fixed rate, ν= ct, and ν= 0.5 at t = 35.7 d. Ice thickness decreases as h ≈ h0 + m

log(1 − ct) c

and h = 0 at t = 43.7 d. Thus, after 35 days ice thickness is 15 cm and it breaks under its own weight due to the critically high porosity. A special case of ice decay is sublimation. Even though latent heat fluxes are usually small in cold environment, in arid cold climate the situation can be different (e.g., Huang et al. 2012; Leppäranta et al. 2016). Ice decay due to sublimation is Qe dh ρa = = Ce (qa − q0 )Ua dt ρ Le ρ

(4.52)

The saturation specific humidity at 0 °C is 3.8·10–3 . Thus, for C E ∼ 10−3 , Ua ∼ 10 m s–1 and relative humidity of 50%, the sublimation rate is 2 mm d–1 . For 50 days, the loss of ice by sublimation would be about 10 cm. Sublimation also allows an equilibrium thickness of bare congelation ice where the growth rate equals sublimation.

4.4 Thermodynamic Models of Lake Ice 4.4.1 Basics Thermodynamic modelling of lake ice is based on the heat conduction equation. In the lake ice application (Eq. 4.26), solar radiation acts as a source term. The upper and lower boundaries are moving with flux boundary conditions, and the temperature is at the freezing point at the lower boundary (Eqs. 4.28a, b). A lake ice cover has a multi-layer structure. Snow cover is commonly on top, and snow accumulation may lead to slush and snow-ice formation. The surface heat balance with ice–atmosphere interaction needs a proper treatment since the surface temperature is highly sensitive to atmospheric conditions (e.g., Cheng et al., 2003; Maylut & Untersteiner, 1971; Stepanenko et al., 2019; Yang et al., 2012). The length (L) and time (T ) scales of thermal diffusion in ice are tied by the diffusion coefficient: D = 0.095m2 s−1 , [D] = L2 T . When the horizontal length scale is more than 10 m, grid points are independent, and a vertical thermodynamic model for each grid point is sufficient. Therefore, almost all lake ice thermodynamic models are one-dimensional. In vertical models the length scale is 0.1–1 m, and then

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4 Thermodynamics of Seasonal Lake Ice

the time scale of heat diffusion is T ∼ 0.1–10 d. In long time scales, quasi-steady models can be employed. In these models the thermal inertia is ignored, i.e., the heat capacity ρi ci → 0, and k

∂T + Q sδ eκ z = constant ∂z

(4.53)

Often the solar radiation term is absorbed in the surface boundary condition that is equivalent to taking κ → ∞, and the temperature is linear in layers with k = constant. In fact, in high-latitude winter, solar radiation is weak or absent. For snowcovered ice, we have ki ∂∂Tz = ks ∂∂Tz = constant. Quasi-steady models are coarse but fast and stable, and they can resolve the daily cycle in thin ice and synoptic time scale in thick ice. Here we focus on freshwater lakes. Thus, the thermal properties of ice—density, specific heat, and thermal conductivity—depend basically on the gas content, and for congelation ice and snow-ice layers they can be taken as constants. But the optical properties undergo strong evolution in the melting season. In saline ice, the dynamics of brine volume has an important role, and the thermal properties of ice depend on temperature and salinity. There are a few papers on saline lake ice models, but they are applications based on the sea ice models (e.g., Bitz & Lipscomb, 1999; Maykut & Untersteiner, 1971).

4.4.2 Analytic Models Analytic thermodynamic models of lake ice are mostly quasi-steady. The growth and melting periods are treated separately. Ice growth is obtained from heat conduction equation, while melting is taken as an energy balance problem, where the net absorption of heat is used for ice melting. Analytic models provide the scales of ice phenology and ice thickness for different lakes and environmental conditions. The main source of uncertainties is in the treatment of the snow cover. In arid climate, snow accumulation is weak and analytic models are then quite good. The insulating capacity of snow depends on snow accumulation and metamorphic processes. Ice Growth The classical problem of modelling lake ice thermodynamics is the seasonal thickness cycle of congelation ice. Stefan (1891) derived the first ice growth model, which was based on quasi-steady conduction of the latent heat released in freezing through the ice to the atmosphere. He examined the ice in the Arctic Ocean, but in fact his model applies even better for freshwater lake ice, since the thermal properties of saline ice are more variable. Stefan’s model was further developed by Barnes (1928) for different kind of growth conditions in Canadian freshwater bodies. Since then, these models with semi-empirical modifications have been applied for the first-order

4.4 Thermodynamic Models of Lake Ice

145

approximations for ice growth (e.g., Anderson, 1961; Ashton, 1986; Leppäranta, 1993). The Stefan formula (Stefan, 1891) gives the widely referred representative √ maximum annual ice thickness as a function the freezing–degree-days: h = a S. The improved formulation including the atmospheric surface layer buffering effect is (Anderson, 1961; Barnes, 1928; Zubov, 1945) h(t) =

/ a 2 S a (t) + h a2 − h a

(4.54)

which is reduced to the Stefan formula when h a = 0. This model is biased up since it ignores snow and heat flux from water. Therefore, an empirical freezing-degree-day coefficient a∗ is estimated from field data and used instead of the exact physical value ∗ a = 3.48 cm(◦ C · d)−1/2 as the local model parameter, 21  aa  1 (Fig. 4.13). ∗ The ratio a /a ranged, e.g., within 0.5–0.8 in Ashton (1989) and within 0.6–0.85 in Nolan (2013). Example 4.12 a. Using the air temperature data for Oymakon, East Siberia (the coldest location in the Northern Hemisphere) in the winter 1981–1982 (the coldest one for this region in the period 1971–2000) (Kirillin et al., 2012). Barnes model (Eq. 4.54, h a = 10 cm) gives the ice thickness of 2.5 m, which represents an upper limit for the thickness of seasonal lake ice. b. For a 100-day cold season with the mean air temperature –10 °C below the freezing point, Barnes model gives the ice thickness of 95 cm, which can be

Fig. 4.13 Stefan and Barnes models of ice thickness as a function of freezing-degree-days for snow-free ice (original models) and for effective snow insulation (a ∗ = a/2)

146

4 Thermodynamics of Seasonal Lake Ice

reduced in empirical modification to 47.5 cm for a ∗ = a/2 representing the case when snow insulation is very strong. c. In a shallow freshwater lake, ice growth reaches the lake bottom when h ≥ H , where H is the lake depth. In a shallow saline lake, salt rejection from growing ice increases the salinity of the liquid water body that lowers the freezing point. Assuming that the ice salinity is zero, the water salinity develops as Sw (t)Hw (t) = Sw (0)Hw (0), where Hw is the depth of the liquid layer, that lowers the freezing point. Starting with Sw (0) = 10‰ and H = 1 m, at Hw (t) = 10 cm the water salinity is 100 ‰ and the freezing point about –6 °C. Ice Decay Modelling the decay of lake ice is in principle straightforward since ice loss is due to the net gain of energy from external fluxes and heat conduction is less significant (see Eqs. 4.49a–c). Analytic melting models are simplified energy balance models, where the heat flux from the lake water is ignored. Parameterization of albedo and light transmittance can be made for a better model performance. Since ice melts also in the interior, ‘ice thickness’ represents the volume of ice per unit area. In many applications, the rate of melting is taken proportional to the air temperature above the freezing point. This results in positive-degree-day models, which can be generalized with the linear surface heat flux formula to ) ( K1 K0 dh , TR = = −A max Ta + TR − T f , 0 ◦ C , A = dt ρi L f K1

(4.55)

where A is the degree-day coefficient. For K 1 ≈ 15 W m–2 we have A ≈ 0.5 cm(◦ Cd)−1 . In the melting period, the parameter K 0 increases sharply due to the solar radiation coming up, while the parameter K 1 is quite stable. Mostly K 0 ≥ 0 and Ta − T f ≥ 0 ◦ C, and the solution can be written as: t h(0) − h(t) =

(

)

t

A Ta − T f dτ + 0

0

K0 dτ ρi L f

(4.56)

where t = 0 is the time of the maximum ice thickness, and the first term on the right hand side is the sum of positive-degree-days. Thus, melting consists of the degree-day term, and the net radiation term. Normally they both increase with time and are correlated. Therefore, by tuning the coefficient A, we can account for both, as has been done in empirical studies of ice melting. In High-Arctic Canadian lakes, the degree-day coefficient is around 1.5 cm °C–1 (Mueller et al., 2009), suggesting that the net radiation was stronger than the real degree-day effect in the ice breakup. At extremes, melting in supraglacial lakes accounted for 0.5–1 m during Antarctic summer although the sum of positive-degree-days was not more than around 5 °C d (Leppäranta et al., 2016), in Tibetan lakes solar radiation may melt lake ice

4.4 Thermodynamic Models of Lake Ice

147

cover before air temperature has reached the feezing point, and river ice breakup is dominated by melting and erosion at the bottom of the ice (Shen & Yapa, 1985). Example 4.13 We can estimate that in the melting season, April in southern Finland we have K 0 ~ 25 W m–2 and K 1 ∼ 15 W m–2 °C–1 . With the mean April temperature of 3 ◦ C, melting is 1.3 cm d–1 from the degree-day term and 0.7 cm d–1 from the net radiation term and, together 2.0 cm d–1 . Thus, the mean 50 cm thick ice would melt in about 25 days that agrees well with observations. Since net radiation and air temperature steadily increase in spring, the melt rate is higher if melting starts later and vice versa.

4.4.3 Numerical Models Numerical lake ice models are of two basic types: (1) Quasi-steady models, where the ice cover is divided into layers with a quasi-steady heat flow through the system; and (2) Time-dependent models, where the vertical equation of heat conduction is directly integrated. In both cases the boundary conditions are properly treated. The ice growth modelling problem possesses negative feedback to errors. The background Stefan’s formula implies that the squared ice thickness scales with to the freezing-degree-days S, and any error becomes damped with time. If at the time t = t 1 we have the model output h (t1 ) = h(t1 ) + e, where √h is the real thickness and e is the error, the Stefan solution continues as h (t) = (h(t1 ) + e)2 + a 2 S(t; t1 ), where the freezing-degree-days start from t1 . The error propagation follows as Δ

Δ

Δ

h (t) − h(t) ≈

h(t1 ) e → 0, as t → ∞ h(t)

(4.57)

Thus, ice growth modelling is a smooth problem. In contrast, modelling of ice melting is a rough problem as it possesses a positive feedback mechanism to errors. In a simple heat budget model, melting progresses with the positive degree-days and the radiation balance (Eq. 4.56). Assuming melting is independent of ice thickness, any error in ice thickness remains as such. However, when melting progresses, the positive feedback causes diverging errors. E.g., if there appears an albedo bias δα at time t2 , error propagation gives for t > t 2 Δ

t

h (t) − h(t) ≈ δα t2

Qs dτ → ±∞, as t → ∞ ρi L f

(4.58)

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4 Thermodynamics of Seasonal Lake Ice

4.4.4 Quasi-Steady Models In the quasi-steady approach, the ice cover is vertically divided into layers and the heat conduction is solved through all layers simultaneously (Leppäranta, 1983; Liston & Hall, 1995). This approach was used for sea ice modelling by Semtner (1976). This is in principle as in analytic modelling, but the boundary conditions can be properly treated. The solution at each time step is iterated to satisfy the continuity of the heat flux. The ice is vertically divided into n layers, with grid points h 0 , h 1 , . . . , h n so that h 0 is at the surface and h n is at the bottom and equals ice thickness. For the quasi-steady heat flux, the growth and melting are treated separately as, respectively: T f − Tn−1 T1 − T0 dh n + Q w = ki T0 < T f : ρi L f = · · · = ki dt h n − h n−1 h1 − h0 [ ] = − Q sδ + Q 0 (T0 ) ≥ 0 T0 = T f : ρi L f

[ ] dh n = − Q sδ + Q 0 (T0 ) + Q w ≤ 0 dt

(4.59a) (4.59b)

The surface boundary condition is given in general form, and the exact formulation is up to the modeller (see Sect. 4.1). The extremes are the full surface flux model and the linearized form (Eq. 4.36) with own parameters fixed for each time step. In the case of melting, we can often take T0 = T f . Fixed grid points can be used with variable size for the last grid cell, or the grid can deform with the ice with uniform size of the grid cells. The thickness and density of snow need to be modelled, since the thickness gives the size of the insulating snow layer and density gives the thermal conductivity of snow (Leppäranta, 1983; Liston & Hall, 1995). Snow accumulation is given in terms of precipitation (water equivalent). The density of new snow is assumed or parameterized, the magnitude is ρs ∼ 100 kg m–3 , and the thickness new snow is s = ρw P. When the snow becomes older, its density increases due obtained from ρs dh dt to snow metamorphosis, and old seasonal snow may reach densities of 400 kg m–3 (Anderson 1976; Liston & Hall, 1995; Yen, 1981; also see Sect. 3.1.5). Thermal conductivity of snow is proportional to density squared, and therefore snow densification is of concern in lake ice modelling. Liston and Hall (1995) employed a compaction model ρs h se −(T f −Ts )/ΔT −ρs /ρ1 ∂ρs = e e ∂t P1

(4.60)

where h se is the water equivalent of the overlaying snow, Ts is snow temperature, and (P1 , T1 , ρ1 ) are the model parameters; P1 = 0.11 mm d, ΔT = 12.5 ◦ C and ρ1 = 48 kg–1 m3 in Kojima (1967). Snow density changes also during melting. The melted snow reduces the snow depth and is redistributed through the snowpack until

4.4 Thermodynamic Models of Lake Ice

149

Fig. 4.14 a The thickness of congelation ice (black ice) and snow-ice as a function of a fixed snow accumulation. b The thickness of congelation ice and snow-ice as a function of density of new snow and packing rate of snow in a normal winter (mean black ice = 58 cm, mean snow-ice = 13 cm). The air temperature and snow accumulation correspond to Oulu in the northern Finland (Leppäranta, 1983)

a maximum density is reached, taken as 550 kg m–3 in Liston and Hall (1995). Any additional meltwater accumulates at the base of the snowpack as a slush layer. Leppäranta (1983) employed a quasi-steady model to examine snow-ice formation and sensitivity of ice thickness and stratigraphy to thermal properties of snow. He used a three-layer (congelation ice, snow-ice, and snow) model to examine the formation of snow-ice. A fixed packing rate with the upper limit of 450 kg m–3 for the snow density was assumed. Figure 4.14 shows the model outcome for the stratigraphy of the ice sheet and sensitivity of ice thickness to snow accumulation. The representative heat flux from the water was estimated from ice and atmosphere climatology. Later the quasi-steady approach was taken, e.g., by Liston and Hall (1995) and Duguay et al. (2003). In quasi-steady modelling, the daily cycle in conduction can be resolved up to the thickness of 20 cm, while for the synoptic time scale (~10 days) this thickness is 70 cm. In many models of ice-covered lakes, the ice is treated as a quasisteady layer to provide simple boundary conditions for the water body (Mironov et al., 2012; Omstedt, 2011).

4.4.5 Time-Dependent Models The full heat conduction equation of the snow and ice layers is solved in timedependent lake ice models for the evolution of temperature and thickness in a discrete

150

4 Thermodynamics of Seasonal Lake Ice

grid or in a finite element system (Eqs. 4.26 and 4.28a, b). The advantage over quasisteady models is a high space–time resolution and more advanced treatment of the boundary conditions. The present congelation ice models for lakes are largely based on the sea ice models Maykut and Untersteiner (1971) designed for congelation ice, further developed by, e.g., Flato and Brown (1996) and Bitz and Lipscomb (1999). A snow-ice layer was added on top of congelation ice for subarctic seas by Saloranta (2000), Cheng (2002), Cheng et al. (2003) and Shirasawa et al. (2005). An approach for a two-phase model, which includes temperature and porosity of ice for each grid cell, has produced a realistic solution to ice deterioration in the melting season (Leppäranta, 2009). The surface boundary condition is a delicate element in time-dependent modelling since the surface temperature is highly sensitive to the atmospheric surface layer dynamics. Turbulent air–ice heat transfer based on the Monin–Obukhov similarity theory was considered in the sea ice modelling by Cheng (2002) and Cheng et al. (2003). Time-dependent models have been introduced in lake ice modelling after the year 2000 (Leppäranta & Uusikivi, 2002; Leppäranta, 2010; Stepanenko et al. 2019; Yang et al., 2012, 2013). The system of equations is obtained from Sect. 4.3:

hs > 0 :

) ( ∂T ∂ ∂ k + Qs (ρcT ) = ∂t ∂z ∂z ⎧ | ⎨ ks ∂∂Tz | − = Q 0 + ρ L f M0 0 s ⎩ dh = dt

ρw ∼ (P ρs

 hs = 0 :

ki dh dt

 z=b:

(4.61a)

∼

hs d ρs ∼ ρ s dt

(4.61b)

ρw ∼ ρs

(4.61c)

− E + Y − R − M0 ) + |

∂T | ∂ z 0−

=

= Q 0 + ρ L f M0 s = + M0 ); dh dt

− ρρw (E

T = Tf dh = ρikLi f · dt

|

∂T | ∂z b+

−

1 ρi L f

Q

P

(4.61d) w

The thermal properties of the system may vary in the different layers. Snow metamorphosis can be included with consequent impact on snow thickness, density, and thermal conductivity. The treatment of the snow is largely technical, especially how to treat meltwater and rain. In saline ice, care should be taken with parameterization of the thermal properties of ice when the ice approaches the melting stage (see Maykut & Untersteiner, 1971; Schwerdtfeger, 1963). The model is one-dimensional, and the numerical solution is not difficult. Timedependent models have high vertical resolution (1–10 cm). For a grid size Δz ∼ 10 cm, the time scale of diffusion is τ ∼ 0.1 day, the time-step needed for resolution of daily cycles in numerical modelling. The boundary conditions need special treatment. Especially for a strongly stable atmospheric surface layer, the heat transfer at the surface is tough to solve (e.g., Yang et al., 2013). At the bottom of the ice, the heat flux from lake water can be prescribed or taken from an underwater thermodynamic

4.4 Thermodynamic Models of Lake Ice

151

model (see Kirillin et al., 2012; Liston & Hall, 1995; Petrov et al., 2006; Shirasawa et al., 2006; Stepanenko et al., 2019). The ice cover has in general separate congelation ice, slush, snow-ice, and snow layers. To obtain the snow-ice growth, slush formation (snow saturated with liquid water) is accounted for when liquid water is available, and snow-ice starts to form from the top of the slush. The solution is straightforward, but the equations become quite technical. Formation of slush is a sink for the snow layer, and formation of snow-ice is a sink for the slush layer. The four layers interact in a dynamic way: snow accumulation creates slush and snow-ice depending on the total thickness of ice, while the growth and decay of congelation ice depend on the snow and slush conditions. Slush and snow-ice may form a multiple layer structure with alternating layers of snow-ice and slush. In the model of Saloranta (2000), all slush was collected in the same box and used as one unit for the source of snow-ice. Flooding is assumed to follow from Archimedes law, i.e., lake water is let to penetrate the ice as soon as the ice surface is beneath the water surface level. A threshold snow overload can be specified for a flooding event to begin. Liquid precipitation is obtained from the forcing assuming that precipitation is liquid when the air temperature is above a given limit, say, Ta > 1 ◦ C. Melt water of snow comes from the model directly. The proportion of snow crystals in water-saturated snow is assumed to rise to a given level, say, 50%. As an example, a numerical modelling study of the ice cover in Lake Vanajavesi, southern Finland, is presented (Yang et al., 2012) based on the thermodynamics model HIGHTSI (Cheng, 2002). The model was used for an individual ice season 2008–2009 and for the climatological period 1971–2000 with good calibration data. The novel model features were advanced treatment of superimposed ice and turbulent heat fluxes, coupling of snow and ice layers, and snow modelled from precipitation. The albedo was critical for snow and ice melting. A parameterization suitable for the Baltic Sea coastal snow and land-fast ice was selected (Pirazzini et al., 2006). Albedo was 0.15 for ice thinner than 10 cm, 0.75 for snow depth more than 10, and otherwise fitted in between. The temperature criterion of Ta < 0.5 ◦ C was set for the precipitation to be solid. The integral interpolation method was applied to build up the numerical scheme (Cheng, 2002). A high vertical resolution (10 layers in snow and 20 layers in ice) ensured that the high frequency response of the snow and ice temperature to forcing and the distribution of the absorption of solar radiation near the surface are correctly resolved. The first day when the ice starts to grow successively is defined as the freezing date. The results are shown in Figs. 4.15 and 4.16. Modelled ice climatology showed growth by 0.5 cm day–1 in December–March and 2 cm day–1 melting in April. The tuned heat flux from the lake water to ice was low, 0.5 W m–2 . The diurnal weather cycle gave significant impact on the ice thickness evolution in spring. Ice climatology was highly sensitive to snow conditions, and the surface temperature showed strong dependency on thickness when the thickness was less than 0.5 m. The ice season responded strongly to air temperature: an increase by 1 or 5 °C of the air temperature level decreased the mean length of the ice season by 13 or 78 days, respectively, (from

152

4 Thermodynamics of Seasonal Lake Ice

Fig. 4.15 Time series of observed and modelled snow and ice thickness. The dark grey line and the asterisks are the observed snow thickness on land and on the lake, respectively. The circles are the observed average ice thickness, and the spatial standard deviation is indicated by the vertical bar. The black dashed and solid lines are modelled snow and ice thickness, respectively. The heat flux from water was 0.5 W m–2 (Yang et al., 2012)

152 days), and the thickness of ice by 6 cm or 22 cm (from 50 cm), respectively. The agreement with observations was good. In fact, the fit of such model simulations largely depends on the quality of the forcing data and the sub-model for air–ice interaction. The annual cycle of the ice thickness was well represented by the model and demonstrated that the model can be used to simulate the evolution of the ice cover under a range of meteorological conditions. The weakest components of the model appear to be formulation of snow cover evolution and the onset of melting.

4.4.6 Two-Phase Models Internal melting is an important process in the melting season. The implications are that the physical properties of the ice change: the strength of the ice strongly decreases, optical properties are continuously modified, and liquid water pocket are formed serving as habitats for biota. To properly treat the internal melting necessitates the inclusion of the ice porosity as a dependent model variable. A two-phase lake ice model was introduced by Leppäranta (2009) based on Eqs. (4.49a–c), where liquid phase and solid phase fractions considered in addition to the thickness and temperature of ice (Fig. 4.17). When the temperature reaches the

4.4 Thermodynamic Models of Lake Ice

153

Fig. 4.16 Time series of observed and simulated ice thickness based on varying heat flux from water, Q w = 0–5 W m–2 . Measurements of ice thickness are shown by open circles. (Yang et al., 2012)

freezing point, additional heat adds on the porosity. At the freezing point and zero porosity, further heat loss takes the temperature below the freezing point. A realistic structural profile was obtained in Leppäranta (2009) with locations of liquid water containing layers. In the case of saline ice, the same approach works but then the porosity changes at any temperature due to equilibrium requirement for the salinity of brine (see Maykut & Untersteiner, 1971). In warm ice and snow liquid water inclusions co-exist with the solid-state ice crystals. The ice and snow cover become a two-phase system, where the proportions of the solid and liquid phases change according to heat fluxes. The properties of this system also change with the phase proportions. In the two-phase model the phase proportions are simulated for each grid cell in addition to the temperature. This occurs usually in the melting season, but in the ephemeral lake ice zone, ice grows and melts in turn through all ice season. Internal melting gives rise to structural defects and finally the ice breaks into small pieces into the water with a rapid final decay. The model is based on the heat conduction equation with solar radiation as the source term and ice growth and melting resulting from phase changes. Ice becomes isothermal in the melting stage, heat conduction ceases, and grid cells in an ice cover contain ice and liquid water. [It (is assumed ) that]in the differential heat content of a grid cell dz, dH = (1 − ν)ρi c T − T f − L f dz (see Eq. 4.34): (i) When the temperature is below the freezing point, the liquid water content is zero and heat gains and losses are reflected in the temperature; and (ii) When the temperature is at the freezing point, heat gains and losses are reflected in the porosity. ( ) Denoting by q = ∂∂z κ ∂∂Tz + Q s the heat fluxes between cells, we have

154

4 Thermodynamics of Seasonal Lake Ice

Fig. 4.17 A two-phase model grid for the freshwater lake ice model. In each grid cell the temperature (T ) and porosity (ν) are determined; if temperature is below the freezing point, the porosity is kept zero, and at the freezing point temperature is fixed but porosity changes

T < T f : ν = 0,  T = Tf :

∂T q = ∂t ρi ci

ν = 0 and q < 0 : ∂∂tT = ρiqci , ν = 0 ν > 0 or q ≥ 0 : T = T f and dν = ρi qL f dt

(4.62a)

(4.62b)

The boundary conditions are as in the classical models, and positive ice surface and bottom heatings decrease the thickness of ice. Absorption of sunlight decreases with depth, but surface heat balance may be negative due to terrestrial radiation losses. As a result, the porosity will have a maximum beneath the surface, typically at 10–20 cm depth (Fig. 4.18). In the two-phase model slush layers are well reproduced and the physical representation of melting ice is realistic. Also, a mechanical model can be added to examine the breakage of ice, which is a critically important factor to the quality of the melting period. Mechanical problems have not been much discussed in lake ice seasonal models, although mechanical events are ice thickness and quality dependent and therefore sensitive to climate variations. If the modelled ice structure evaluation is correct, better estimates for ice loads and bearing capacity are obtained, ice breakup comes correctly, and biological habitats exist in the ice cover.

References

155

Fig. 4.18 A two-phase model outcome. The lines show increase of ice porosity in the melting period, peaking at 10–15 cm depth (Leppäranta, 2009)

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Schumann, U. (1988). Minimum friction velocity and heat transfer in the rough surface layer of a convective boundary layer. Boundary-Layer Meteorology, 44, 311–332. Schwerdtfeger, P. (1963). The thermal properties of sea ice. Journal of Glaciology, 4(36), 789–807. Semtner, A. J. (1976). A model for the thermodynamic growth of sea ice in numerical investigations of climate. Journal of Physical Oceanography, 6, 379–389. Shen, H. T., & Yapa, P. D. (1985). A unified degree-day method for river ice cover thickness simulation. Canadian Journal of Civil Engineering, 12(1), 54–62. Shirasawa, K., Leppäranta, M., Saloranta, T., Polomoshnov, A., Surkov, G., & Kawamura, T. (2005). The thickness of landfast ice in the Sea of Okhotsk. Cold Regions Science and Technology, 42, 25–40. Shirasawa, K., Leppäranta, M., Kawamura, T., Ishikawa, M., & Takatsuka, T. (2006). Measurements and modelling of the water—ice heat flux in natural waters. In Proceedings of the 18th IAHR International Symposium on Ice (pp. 85–91). Hokkaido University, Sapporo, Japan. Stefan, J. (1891). Über die Theorie der Eisbildung, insbesondere über Eisbildung im Polarmeere. Annalen der Physik, 3rd Ser, 42, 269–286. Stepanenko, V. M., Repina, I. A., Ganbat, G., & Davaa, G. (2019). Numerical simulation of ice cover in saline lakes. Izvestiya, Atmospheric and Oceanic Physics, 55(1), 129–139. Su, D., Hu, X., Wen, L., Lyu, S., Gao, X., Zhao, L., Li, Z., Du, J., & Kirillin, G. (2019). Numerical study on the response of the largest lake in China. Hydrology and Earth System Sciences, 23, 2093–2109. Tennekes, H., & Lumley, J. E. (1972). A first course in Turbulence (p. 320). MIT Press. Wadhams, P. (2000). Ice in the ocean (p. 364). Gordon & Breach Science. Wang, C., Shirasawa, K., Leppäranta, M., Ishikawa, M., Huttunen, O., & Takatsuka, T. (2005). Solar radiation and ice heat budget during winter 2002–2003 in Lake Pääjärvi, Finland. Verhandlungen Der Internationalen Vereinigung Für Theoretische Und Angewandte Limnologie, 29, 414–417. Weeks, W. F. (2010). On sea ice (p. 664). University of Alaska Press. Yang, Y., Leppäranta, M., Li, Z., & Cheng, B. (2012). An ice model for Lake Vanajavesi, Finland. Tellus A, 64, 17202. https://doi.org/10.3402/tellusa.v64i0.17202 Yang, Y., Cheng, B., Kourzeneva, E., Semmler, T., Rontu, L., Leppäranta, M., Shirasawa, K., & Li, Z. (2013). Modelling experiments on air-snow-ice interactions over an Arctic lake. Boreal Environment Research, 18, 341–358. Yen, Y.-C. (1981). Review of thermal properties of snow, ice and sea ice (CRREL Report 81-10). U.S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH. Zillman, J. W. (1972). A study of some aspects of the radiation and heat budgets of the Southern Hemisphere oceans. In Meteorological Studies (Vol. 26, 562 pp.). Bureau of Meteorology, Department of Interior. Zubov, N. N. (1945). L’dy Arktiki [Arctic Ice]. Izdatel’stvo Glavsermorputi, Moscow. [English translation (p. 491) 1963 by U.S. Naval Oceanographic Office and American Meteorological Society, San Diego].

Chapter 5

Mechanics of Lake Ice

An ice road across Lake Pielinen in northeast Finland, between the villages Vuonislahti and Koli. This is one of the longest ice roads in Europe, about seven kilometres. It is opened when congelation ice thickness is 40 cm. Traffic rules give the maximum weight of three tons, the maximum speed of 50 km h–1 , and the minimum distance of 50 m between cars. Cars are not allowed to pass or stop. The lake is 90 km long, and the ice road makes a major shortcut to driving distances. Photograph by Ms. Maritta Räsänen, Pohjois-Savo Centre for Economic Development, Transport and the Environment, Kuopio. Printed with permission

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_5

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5.1 Rheology The science of rheology examines the relationships between the internal stress, mechanical deformation, and material properties of media. Material properties are such as density, temperature, and porosity, while deformation is specified by strain and its time history. The present chapter begins with a general introduction into rheology. There are two principal reasons for going to the basics. First, natural ice has quite complicated rheology, and, secondly, for students and scientists in lake physics or limnology, rheological questions are not so familiar, because lake water is a linear Newtonian fluid, and its mechanics obeys the well-established Navier–Stokes equation. Natural lake ice is a polycrystalline medium, and its mechanical behaviour depends on the time and space scales under consideration. In mechanics, lake ice is normally considered as a continuum. The basic rheology models—elastic, viscous, and plastic media—are all applicable in lake ice processes (Fig. 5.1). For a continuous plate of polycrystalline ice, the temperature, size, shape and orientation of ice crystals and the impurities between crystals influence on the mechanical properties. Broken ice, i.e., an ice field consisting of ice blocks or floes, is a granular medium where individual blocks or floes represent the individual ‘grains’ (see Leppäranta, 2011). When the number of grains is large, a continuum approximation can be taken for length scales much longer than the grain size. Important applications of granular models are mechanics of ridge formation from ice blocks, ice jams in rivers, and drifting of ice floes on a lake surface. Mechanics is treated first in the three-dimensional space. The Cartesian coordinates are (x, y, z) with x directed east, y directed north, and z directed up. The corresponding unit vectors are denoted by i, j and k. The gradient operator is in the Cartesian system ∇ = i∂/∂ x + j∂/∂ y + k∂/∂z. Furthermore, we deal with three different mathematical quantities: scalars are numbers, vectors have magnitude and direction, and (2nd order) tensors have vector components on all the three coordinate planes. If we take a ball in hand and press it, a tensor is needed to describe

Fig. 5.1 Left A solid ice cover (Lake Saanajärvi, Finland, tundra lake); Right Pancake ice in the Arctic. Photographs by the author (left) and Jarmo Vehkakoski (right), printed with permission

5.1 Rheology

161

how the size and shape of the ball changes. In the Cartesian system, tensors can be represented as matrixes.

5.1.1 Stress Consider a material sample of a continuum (Fig. 5.2). An internal stress field σ exists within the sample depending on the external loading and the mechanical properties of the medium. The stress, a 2nd order tensor with the dimension of force per area, gives the distribution of forcing on surfaces of infinitesimal volume elements in the sample. For an arbitrary unit surface with unit normal n, the stress acts as a force vector σ ·n, which consists of the shear stress—the projection parallel to the surface— and the normal stress—the projection to the surface normal. In three-dimensional space, there are three independent surface orientations and three independent vector directions. Therefore, the stress tensor is given by 3 × 3 = 9 components (Fig. 5.2). In Cartesian form: ⎡ ⎤ σ xx σ xy σ xz σ = ⎣σ yx σ yy σ yz ⎦ (5.1) σ zx σ zy σ zz The diagonal components are the normal (compressive or tensile) stresses and the off-diagonal components are the shear stresses. The component σx x gives the normal stress in x-direction, the component σ yx gives the shear stress across y-axis

Fig. 5.2 Stress σ on a material element in a Cartesian system. σii are normal stresses and σi j are shear stresses, σi j = σ ji

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5 Mechanics of Lake Ice

in the x-direction, etc. The stress tensor is necessarily symmetric, since otherwise there would be net internal torques, which would not disappear when the particle size approaches zero (e.g., Hunter, 1976). Thus, σx y = σ yx , σx z = σ zx and σ yz = σ zy that reduces the number of independent stress components to six. The stress tensor can be uniquely decomposed into spherical and deviatoric parts σ S I and σ ' , respectively, by '

σ = σS I + σ ; σS =

1 trσ 3

(5.2)

where I is the unit tensor,  I pq = 1(0) for p = q( p /= q), and tr is the trace operator. For a matrix A = A pq , tr A = A x x + A yy + A zz . In Eq. (5.2), the factor 1 /3 comes from the three-dimensionality; in the two-dimensional (one-dimensional) theory the factor is 1/2 (1). The eigenvalues of A are the solutions (λ) of the eigenvalue equation det( A − λI) = 0, and the corresponding eigenvectors  are the solutions of the equation ( A − λI) = 0. The eigenvalues of the stress tensor are the principal stresses, and the corresponding eigenvectors give the directions of these stresses. Example 5.1 Hydrostatic pressure is a stress σ = − pI, where p > 0 is the scalar pressure. It is spherical with σs = − p, compressive (σs < 0), and in any direction, σ · n = − pn. In lake water, p = pa + ρw gd, where pa is the surface atmospheric pressure and d is the depth. At d = 10 m, p ≈ 2 pa , i.e., each 10 m of water adds the pressure equivalent of about one atmosphere to the total.

5.1.2 Strain and Rotation The techniques to describe deformation are given in books on continuum mechanics (e.g., Hunter, 1976; Mase, 1970). A brief summary is presented here. Let x stand for the reference configuration of a material body, and consider a change of the body to X, represented by a mapping x → X. The displacement d = X − x consists of translation, rotation, and strain. Translation and rotation correspond to rigid body motion, while strain corresponds to physical deformation of the body. Strain, like stress, is a 2nd order tensor and can be decomposed into change of size (compression/dilatation) and shape (shear). Strain and rotation are included in the displacement gradient ∇ d. The linear theory is used for small deformations (|∇ d| ≪ 1), and then the strain ε and the rotation ω are given by the symmetric and anti-symmetric parts, respectively, of the displacement gradient: ε=

1 ∇ d + (∇ d)T 2

(5.3a)

ω=

1 ∇ d − (∇ d)T 2

(5.3b)

5.1 Rheology

163

where the superscript T stands for the transpose, A Tpq = Aq p . Rotation has three independent components, which give the rotations around the co-ordinate axes (Fig. 5.3). Strain is symmetric and possesses six independent components, three representing the normal strains and three representing the shear strains, analogous with the stress components. A rotation of 1% corresponds to turning of the body by 0.01 rad ≈ 0.573º. There are two modes of strain: normal strain and shear strain (Fig. 5.3). Normal strain is extensive (positive) or contractive (negative), while shear strain changes the shape. Inversely, any physical deformation of a body can be decomposed into normal and shear strains. An extensive/contractive strain of 1% means that the corresponding material dimension has lengthened/shortened by 1%, and a shear strain of 1% means that the right angle has changed by 0.01 rad under shear. Velocity equals the rate of displacement, and, consequently, the strain-rate and the rotation rate or the vorticity are obtained from the velocity gradient just as strain and

Fig. 5.3 Strain modes and rotation, and their measures: upper graphs, extension and contraction; lower graphs, shear and rotation. Note that shear and rotation are drawn for positive signs

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5 Mechanics of Lake Ice

Fig. 5.4 The basic rheology models (in one dimension)—elastic, plastic and viscous—for the stress σ as a function of the strain ε and the strain-rate ε˙ ; E is the elastic modulus or Young’s modulus, η is the viscosity, and σY is the yield strength

rotation were obtained from the displacement gradient: ε˙ =  1 ∇u − (∇u)T . 2

1 2

 ∇u + (∇u)T , ω˙ =

x /= 0, with other displacement derivaExample 5.2 Simple shear is defined by ∂d ∂y x tives equal to zero. Then the shear strain in the x y-plane is 21 ∂d , and the rotation ∂y x around z-axis is − 21 ∂d . In pure shear, we have ∂y

∂dx ∂y

∂d y /= 0, with other displace∂ x

 ∂d 1 ∂dx x is 2 ∂ y + ∂ xy = ∂d and the ∂y

=

ment derivatives equal to zero. Then the shear strain rotation is zero. In both cases the spherical strain is zero.

5.1.3 Rheological Models The basic models are linear elastic or Hooke’s medium, linear viscous or Newton’s medium, and ideal plastic or St. Venant’s medium (e.g., Hunter, 1976; Mase, 1970). One-dimensional cases are illustrated in Fig. 5.4. The linear elastic model assumes that stress is proportional to strain, while in the linear viscous model stress is proportional to the strain-rate; the proportionality coefficients are, respectively, the elastic modulus or Young’s modulus, and viscosity. Rubber and water1 are good material examples of linear elastic and viscous media. An ideal plastic medium collapses once the stress achieves yield strength; children’s modelling wax serves as an example of a plastic medium. Mechanical analogues for these models include spring balance for linear elasticity, dashpot2 for linear viscosity, and static friction for plasticity. The rheology of a one-dimensional linear elastic beam reads σ = Eε 1

(5.4)

Stress in laminar flow of water is linear viscous. In turbulent flow, turbulent or Reynolds stresses overcome viscous stress by several orders of magnitude, they are properties of the flow, not properties of the medium. 2 Such as a door stroke compressor, which dampens the movements of a door at closing.

5.1 Rheology

165

where E is Young’s modulus, which appears as the slope angle in Fig. 5.4. For an isotropic two- or three-dimensional case, one additional parameter is required for shear. Usually this is treated using the Poisson’s ratio μ. For a beam in uniaxial (x) tension, the tensile strain εx x is seen in the loading direction but contraction appears in the perpendicular directions (y, z) given by the Poisson’s ratio as: μ=−

ε yy εzz =− εx x εx x

(5.5)

Once the Young’s modulus and Poisson’s ratio are known, the bulk modulus K and the shear modulus G are obtained from them. Linear elastic rheology is written in three-dimensional form as

1 σ = K (trε)I + 2G ε − (trε) I (5.6a) 3 K =

E E ,G = 3(1 − 2μ) 2(1 + μ)

(5.6b)

In isotropic media, the principal axes of the stress and strain tensors are parallel. For the two-dimensional case, 1 /3 is replaced by 1/2 in these formulae. Lake ice behaves in elastic manner in short-term events, in such cases as driving a vehicle on ice or loading a dock by drifting ice. In a viscous medium, the stress depends on the strain-rate. The linear viscous rheology is analogous with the linear elastic case (Eqs. 5.6a, b) with ε replaced by the strain-rate ε˙ :

1 σ = ζ (tr ε˙ )I + 2η ε˙ − (tr ε˙ )I (5.7) 3 where η is the (shear) viscosity and ζ is the bulk or the second viscosity. For an incompressible flow, such as water flow in Earth’s nature, tr ε˙ = ∇ · u = 0, where u = ui + vj + wk is the velocity, and thus the bulk viscosity is not relevant. In simple shear flow u = u(y), and the linear viscous stress is σx y = 2ηε˙ x y = η

du dy

(5.8)

In ice applications, non-linear viscous rheologies are used for long-term loading cases. They are obtained from Eq. (5.7) by allowing the viscosities to depend on the principal strain-rates. An ideal plastic medium is rigid at low stress levels but fails at the yield stress σY . In one dimension, ε = 0, σ < σY ; ε /= 0, σ = σY

(5.9)

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5 Mechanics of Lake Ice

In two (three) dimensions, the yield point is replaced by a yield curve (yield surface), which describes the yield stress as a function of the principal strains. At yielding, the actual strain evolution depends on the inertia and external forces acting on the system. In a stable plastic medium, strain hardening takes place. The yield strength increases under strain, and more stress is needed for further deformation. For example, more force is needed to make a snowball in hand smaller, and thus at a constant load, a stationary steady state results with a finite snowball. In an unstable plastic medium, strain softening takes place, and even if the forcing is lowered the strain may continue. The basic rheological models can be expanded for more complex ones (e.g., Mellor, 1986). First, linear models can be changed to general non-linear models. Second, models can be combined. E.g., linear elastic and linear viscous models in series give the Maxwell medium, while combining them parallel gives the Kelvin– Voigt medium. For a constant load, Maxwell medium has an immediate elastic response followed by linear viscous flow, while a Kelvin–Voigt medium flows in a viscous manner toward an asymptotic determined by the elastic part. The strength of a medium is defined as the maximum stress a test specimen can support, and it depends on the mode of deformation and the mode of failure. In brittle failure the specimen breaks but in ductile failure the strain increases in the specimen with no increase in the stress. Usually the former deals with fast loading cases when the response of is elastic, while the latter concerns long-term viscous or plastic flow. In the next section, the mechanical properties of lake ice are briefly presented. The discussion is limited to the general level. More detailed presentations are easily available in ice engineering literature (e.g., Ashton, 1986; Michel, 1978a, 1978b; Palmer & Croasdale, 2012). The goal here is to provide the basic information of the mechanical properties of lake ice and their variability based on a literature overview of laboratory experiments with ice samples, in situ field tests, rheological models, and semi-empirical equations of ice properties. There has been more work done on the mechanical properties of sea ice that also can be utilized for lake ice applications (e.g., Mellor, 1986; Palmer & Croasdale, 2012; Sanderson, 1988; Timco & Weeks, 2010). Lake ice is considered as a polycrystalline continuum, and the relatively large variability of its mechanical properties comes from the large crystal size, high homologous temperature, and impurities. In addition, test arrangements and the size and shape of ice specimens have caused artificial variability in test results. Isotropy and homogeneity are normally assumed. Granular ice is isotropic but columnar ice, which is a common ice type in lakes, has anisotropy between vertical and horizontal in the crystal structure (see Sect. 3.2). Homogeneity is not exactly true, since the temperature is at the freezing point at the bottom of lake ice but the surface temperature can be much lower. In situ field tests have the great advantage that ice is loaded in natural condition.

5.2 Ice Cover as a Plate on Water Foundation

167

5.2 Ice Cover as a Plate on Water Foundation 5.2.1 Elastic Lake Ice Cover In short-term loading, ice shows elastic behaviour (Ashton, 1986; Michel, 1978a, 1978b). Short term refers here to the time scale of 1–2 min or less. In a very short time scale (seconds) ice is perfectly elastic–brittle material, where it cracks once the strength limit has been reached. Long-term response is elastic–plastic or viscousplastic, where the ice yields after having reached the strength level. The elastic parameters of lake ice depend on the temperature, gas content, size and shape of crystals, and impurities (Ashton, 1986). Based on experimental data, the standard references for isotropic lake ice are Young’s modulus E = 9.0 GPa and Poisson’s ratio μ = 0.33 (Table 5.1). The bulk modulus and shear modulus are then K = 8.8 GPa and G = 3.4 GPa (Eqs. 5.6a, b). The assumption of isotropy is widely used, and experimental data show that the variations in Young’s modulus and Poisson’s ratio are within 0.5 GPa and 0.015 or 5% from the standard values. There is not much data available on the strength of porous lake ice. The strength of snow-ice was lower by about 25% compared with congelation ice in the lake ice field data of Butkovich (1954). The strength of ice is the fundamental quantity to evaluate ice loads, the bearing capacity of ice, and the stability of ice cover. In the elastic regime, the brittle strength of ice is 1–10 MPa, depending on the mode of loading (Table 5.1). The brittle compressive strength of lake ice is about 10 MPa at the temperature of –10 °C (Ashton, 1986). The strength of ice increases with decreasing temperature, and the brittle compressive strength is 20 MPa at temperatures about –30 °C. In various tests, at a fixed temperature and strain-rate, the strength has varied by a factor of two. Borehole tests have been made for the compressive strength with a cylindrical pressure meter. The maximum strain at failure is strength/elastic modulus ~ 10–3 . Compared with ice, Young’s modulus and compressive strength are one order of magnitude larger for wood and two orders of magnitude larger for steel. Michel (1978b) presented a semi-empirical model for the uniaxial compressive brittle strength of lake ice: σc = 9.4 · 10

4

1 0.78 Pa (d in m and T in ◦ C) √ + 3|T | d

Table 5.1 Elastic modulus and strength of freshwater ice (Ashton, 1986; Mellor, 1986)

(5.10)

Elastic modulus (GPa)

Strength (MPa)

Strain at breakage

Compression

9.0

10

1.1 × 10–3

Tension

9.0

1.5

0.17 × 10–3

Shear

3.4

1.0

0.29 × 10–3

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5 Mechanics of Lake Ice

where d is grain size. According to this formula, the compressive strength ranges within about 1–10 MPa, at highest for cold, fine-grained ice. Natural variations are large, but the formula shows the scales of the grain size and temperature effects. Multi-axial tests have indicated strength values even above 10 MPa. Tensile strength depends primarily on the grain size and less on the temperature, and it is not sensitive to strain-rate (Ashton, 1986). The level of brittle tensile strength has been 1– 2 MPa in experimental data. Shear is a special case of multi-axial loading (two principal stresses have equal magnitude and opposite signs). The shear strength depends primarily on the loading rate and not much on the temperature, and the test results have been mostly in the range 0.5–1.5 MPa. Lake ice lies on water foundation, where the water pressure at the ice bottom is ρw gh d , where h d is the ice draft. This pressure is balanced by the weight of ice. For small vertical displacements (ice is neither submerged nor raised out of water), the response of the foundation is proportional to the displacement. This behavior is analogous to elasticity, and therefore the theory of plate on elastic foundation can be utilized for floating ice. Flexural strength is a convenient reference of the strength of floating ice, because in situ tests can be easily performed in the specific natural conditions with the steep temperature gradient (Weeks, 1998). In laboratory tests such conditions cannot be reproduced. The flexural strength of freshwater lake ice is 1–1.5 MPa (Gow, 1977; Mellor, 1986). For a homogeneous medium, flexural strength is equal to tensile strength, since at bending tensile failure takes place. We can extend our understanding to saline ice from the results of sea ice investigations. As shown by Weeks (1998), in situ flexural strength drops by one order of magnitude when the brine volume (νb ) increases from νb = 0 to νb = 0.15 (Fig. 5.5). The brine content is highly sensitive to temperature in warm ice.

5.2.2 Viscous Behaviour of Lake Ice When the time scale of loading is more than the order of minutes and the stresses are below the brittle strength of ice, viscous behaviour results. This can be observed in the flow of glaciers and ice sheets (e.g., Paterson, 1999). Ductile strength is less than the brittle strength and decreases with strain-rate, reaching 1 MPa at the strain-rate of 10–7 s–1 ≈ 10–3 (3 h)–1 . The viscous regime is much more complicated than the elastic regime. Experiments of viscous ice flow are usually made by monitoring the deformation of the test specimen at a constant load. The strain history undergoes three consecutive regimes (Table 5.2; Fig. 5.6). In primary creep (I) the ice is hardening, while in tertiary creep (III) the ice starts to fail into an accelerating strain. Between them there is a linear regime (II). The strain inflection point is at about 0.01-strain level in the secondary creep stage (Ashton, 1986; Mellor, 1986). The viscosities and the time scales of the regions depend strongly on the ice temperature.

5.2 Ice Cover as a Plate on Water Foundation

169

Fig. 5.5 Flexural strength of sea ice as a function of brine content (νb ), measured data and a regression fit. Reproduced from Weeks (1998)

Table 5.2 Creep regimes in viscous deformation of ice under constant load (Ashton, 1986; Mellor, 1986)

Regime

Name

Viscous lawa

I

Primary creep

Sub-linear

II

Secondary creep

Linear

III

Tertiary creep

Super-linear

y(x) = is sub-linear for a < 1, linear for a = 1, and super-linear for a > 1 a Law

cx a

Fig. 5.6 Creep of ice (in per cent) under a constant load. The primary (I), secondary (II) and tertiary regimes (III) are illustrated. Drawn based on Ashton (1986)

Modelling these creep regimes is discussed in Schulson and Duval (2009) for glacier ice. The primary creep is modelled as (Glen, 1955) √ ε p (t) = ε0 + β 3 t + ε˙ min t

(5.11)

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5 Mechanics of Lake Ice

√ where ε0 is the instantaneous strain, the transient strain εt = β 3 t is made of recoverable delayed elastic strain εd and irreversible viscous strain, and ε˙ min is the minimum strain-rate (see Schulson & Duval, 2009). Delayed elastic strain can be an order of magnitude larger than immediate elastic strain, while transient creep can be two orders of magnitude larger than the elastic strain. The secondary creep is well described by the Glen’s (1955) law, which is widely used for glacier flow (see Paterson, 1999) ε˙ = Aσ n

(5.12)

where A and n are the flow law parameters. Data for the stresses in the range 0.2– 2 MPa support the value n = 3, and for lower stresses the power is less than 3 (Schulson & Duval, 2009). The parameter A depends primarily on the temperature, A~10–22 Pa–3 s–1 (Paterson, 1999). In the tertiary creep the strain-rate increases from the minimum achieved in the secondary creep. In Maxwell medium, elastic and viscous models are connected in series, and the model serves as the basis for the ductile behaviour of ice. The viscous part is taken as a nonlinear law, which is far preferable to the linear form. For a fixed temperature the model can be written in the form (see Ashton, 1986; Glen, 1955; Ramseier, 1971) ε˙ =

σ n 1 σ + MD E E

(5.13)

  where M and n are functions of strain, strain-rate and stress, and D = D0 exp − QRTS D is the self-diffusion coefficient for the water molecules in ice, D0 ≈ 9.1 · 10−4 m2 s–1 , Q S D = 59.8 kJ mol–1 is the activation energy for self-diffusion, R = 8.31 J mol–1 K–1 is the universal gas constant, and T is the absolute temperature. The parameter M shows a very large variability in experimental data.

5.2.3 Thermal Cracking and Expansion The thermal expansion coefficient of ice depends weakly on temperature. The linear coefficient is 5.0 · 10–5 °C–1 , i.e., 5.0 cm across 1 km distance for each one-degree temperature change. It is clear that the strongest temperature variation take place near the ice surface. Thermal cracks are common features in bare ice surface, but snow cover hides much of them from visual observations (Fig. 5.7). When the temperature changes fast, the elastic response dominates as there is not time enough for viscous adjustment. In a cooling situation, contraction in the surface layer of ice cover takes place and tensile cracks are formed, while in warming the ice cover expands. However, contraction and expansion are not symmetric: at cracking water comes up to fill the cracks and freeze in, and then at warming, a net expansion results with the ice cover spreading out (Fig. 5.8). In the absence of snow, ice surface layer cools fast, and thermal cracking can be intensive. When the pressure is released

5.2 Ice Cover as a Plate on Water Foundation

171

Fig. 5.7 Dry thermal crack in a boreal lake. Photograph by the author

by cracking, wave motion takes place in the ice cover and the crack propagation can also be heard. A whining sound results since high frequencies travel faster than low frequencies in ice cover. Metge (1976) classified thermal cracks into dry, narrow and wide categories (see also Ashton, 1986). Dry cracks, which are the most common ones, do not penetrate the ice cover and act as bellows with temperature changes. Narrow wet cracks can freeze rapidly and consequently add material to the ice cover. Most of the contraction is concentrated at 1–2 fractures, which often form between two tensile stress risers, e.g., between two headlands. They can be up to 20 cm wide (Metge, 1976). In a long channel they form at regular intervals, while other cracks skirt the shore from headland to headland. Wide cracks do not freeze through in one night but thin ice bridges form, and if the daytime temperature is high enough the bridges will break. This breakage is usually along shear of inclined plane, and one side of the crack slips over the other that may initiate a pressure ridge. The rate of strain can be obtained from the temperature change as ε˙ = α T˙

(5.14a)

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5 Mechanics of Lake Ice

Fig. 5.8 Schematic picture of thermal expansion (modified from Bergdahl, 2002). a After a rapid decrease in air temperature. b Expansion of ice on a sloping shore, in mid-lake, and at a straight shoreline after warming

where α is the coefficient of linear thermal expansion. On the other hand, the relation between strain-rate and stress can be taken as (Bergdahl & Wernersson, 1978; Cox, 1984) ε˙ =

1 σ˙ + Aσ n E

(5.14b)

Equations (5.14a, b) can be combined to solve σ = σ (T ).

5.2.4 Displacements in Ice Cover Occasionally shifts or displacements are observed in a lake ice cover. They result from mechanical forcing by wind, water currents, or lake surface tilt, possibly starting from narrow thermal fractures. These displacements can be detected by surface measurements or high-resolution remote sensing imagery. Airborne drones and satellite SAR interferometry have shown potential for mapping small two-dimensional displacements fields (see Dammert et al., 1998; Vincent et al., 2012). SAR is even able to see through dry snow cover. The main forcing to drive lake ice displacements is due to winds. Strong winds blow in a nearly uniform direction across a lake, and the wind stress τa integrated over a fetch L produces the forcing τa L, where L can reach the cross-section of the lake in this direction. The wind stress depends quadratically on the wind speed, τa = ρa Ca Ua2 , Ca ∼ 1.2 · 10−3 . For a very high wind speed of 20 m s–1 and fetch of 15 km, the wind force about 10 kN m–1 . The strength of an ice slab   becomes is modelled as P = P h; l , where l is the length scale of the loading area. The strength represents the integrated stress through the thickness of ice, and therefore its dimension is force/length. The strength decreases with the length scale: at l ∼ 1

5.2 Ice Cover as a Plate on Water Foundation

173

m we have P ∼ 1 MPa (see Sect. 5.2.1), but at l ∼ 10 km, P ∼ 10 kPa because the probability of defects in the load area increases with the length scale. In general, the ice strength is formulated as    P h; l = P ∗ l h n

(5.15)

The ice cover is stable when the forcing is below the strength, τa L < P ∗ h n . The function P ∗ scales as l−1/2 (Leppäranta, 2011; Sanderson, 1988), and 21 ≤ n ≤ 2 depending on the mode of failure (Ashton, 1986). In the case of crushing and buckling failure, the strength is, respectively, Pc = σc h ∝ h / 3 2

Pb = ρw gL ∝ h , L = 2

4

Eh 3   12ρw g 1 − μ2

(5.16a)

(5.16b)

where σc is the compressive strength and L is the characteristic length of ice plate on water foundation. The crushing and buckling failure criteria become equal at h ≈ 20 cm; for thinner ice buckling is more likely while for thicker ice it is crushing. In coastal landfast ice basins, where the size of open spots corresponds to medium-size lakes, n = 2 has given a good fit possibly due to friction between ice blocks in broken ice (Hopkins, 1994; Leppäranta, 2013). Example 5.3 In sea ice dynamics, the solution has been to take the strength from the ice thickness distribution by the principle ‘thinnest ice breaks first’ (Coon et al., 1974). It has been found that frictional losses in pressure ridging follow the n = 2 power law (Hopkins, 1994), and therefore this power represents small-scale sea ice dynamics. A simple approach with P ∗ = constant ∼ 10 − 100 kPa and n = 1(Hibler, 1979) has been widely used in large-scale sea ice modelling, l ∼  10 − 100 km. The static stability criterion for a lake with L ∼ l is then τa L < P ∗ l h. Since τa /P ∗ ∼ 10−5 , for stable ice cover we have h/L > 10−5 ; e.g., for the lake size of L = 10 km, h > 0.1 m for stability. When thin ice fractures, it is possible that the fractured plates are overridden into a double ice layer. This phenomenon is called rafting. A specific, eye-striking case of rafting is finger rafting, where the interacting ice plates fracture along lines perpendicular to the direction of the fracture and form an interlocked structure with alternate overthrust and underthrust fingers. A model for simple rafting was presented by Parmerter (1975). Two floating ice floes of thickness h are forced together by a force F, and if the edges are not exactly vertical, overriding takes place and develops into rafting. The case can be treated as plates on elastic foundation except where either plate is submerged or lifted with respect to the water surface level. The solution depends on the parameter (ρw − ρi )/ρw , which determines where the lower plate submerges, as well as on the elastic properties of ice. The problem can be formulated using a dimensionless force N and stress Σ:

174

5 Mechanics of Lake Ice

N=

F σt h ,Σ = 2 ρw gλ ρw gλ2

(5.17)

where σt is the tensile strength of ice. The limiting value N ∼ 0.4 is reached when the submerged plate buckles. This force is about 40% of that required in the normal buckling of ice plate on water foundation. Bending of ice during rafting creates tensile stresses. Simple rafting can take place when the stresses are beneath the fracture level, but otherwise blocks break off from the ice to form a rubble field or a ridge (Fig. 5.9). The largest stress experienced by the submerged ice is obtained from an asymptotic limit Σ0 = 0.46 (see Parmerter, 1975). Then we can obtain an estimate for the maximum thickness of rafting ice as  hr f =

1 Σ0

 2 2  3 1 − μ2 σ t ρw g E

(5.18)

The result scales as σt2 /E, and inserting the freshwater ice parameters, we have h r f = 32 cm. This is quite large, and forces over lakes are unlikely to become high enough to cause rafting in this thick ice. Rather, we can deduce that simple rafting is possible in ice-covered lakes but due to limited fetches at thicknesses much lower than the limiting value. Parmerter (1975) examined sea ice, and with the sea ice

Fig. 5.9 Students in a winter lake ice school at an ice ridge in Lake Vanajavesi, Finland in March 2015. Fetches over ice were 5–10 km and the thickness of ice was 26 cm. Photograph by the author

5.3 Bearing Capacity of Ice

175

properties his estimate of the limiting value was h r f = 17 cm which corresponded with observations. If the ice breaks and starts to drift, the next question is that how large lead is formed. Also, the persistence and recurrence of leads is of interest. The location of leads is often at the lee side of lake basins but also across lakes guided by the lake geometry. If leads open up, there will be compression zones elsewhere in the ice cover. In large lakes the unpredictability of sudden displacements in the ice cover endanger ice fishermen. Sometimes in good fishing days, many fishermen have been driven out by the ice and need to be rescued by hydrofoils or helicopters. E.g., on February 9th, 2009, in all 150–300 fishermen experienced this event together in Lake Erie,3 North America. Restriction have been set to ice fishing in Estonia due to ice conditions in Lake Peipsi, a large border lake between Estonia and Russia where thousands of fishermen go on weekends. When ice breakage is common, the ice cover appears as a large set of ice floes, broken ice as usually called in the research of lake ice and river ice. It is a granular medium, where the ‘grains’ are ice floes. Interactions between ice floes contain stress generating mechanisms, and variations in small scale ice properties become less important. When the scale of interest is much more than the size of floes, continuum models are employed. This is the regime of drift ice (Leppäranta, 2011), to be treated in Sect. 5.5 below.

5.3 Bearing Capacity of Ice One of the fundamental practical problems in the research of floating ice is the bearing capacity (see overviews by Masterson, 2009; Goldstein et al., 2017). Ice cover has been used for travel and transportation as long as cold regions have been inhabited. In his book of northern Europe, Gothus (1555) described ice crossing by skiing, skating (metal and bone blades), riding and sledges, and explained techniques in fishing in ice-covered lakes. A dramatic case in the 1200s was a battle between Novgorod and Livonian Knights on the ice of Lake Peipsi/Chudskoe (Fig. 5.10). According to historical sources, the ice broke under the Livonian troops and the battle was then done.4 Winter roads formed a very important traffic network in northern Europe until modern times. The world’s first railway ice crossings were built in the late nineteenth century in Russia. For the first time the track was laid on the ice cover of the Volga River near the city of Kazan. The successful experience contributed to the creation of similar crossings in other regions, particularly on the Volga River near Saratov (1895) and Lake Baikal (1904). 3

This case was reported in many Canadian and U.S. news sites on February 10, 2009; the number of fishermen varied in different sources. 4 The scale of this event is not clear but it is thought to have a true basis. This battle on ice is a scene in the film Alexander Nevsky by Sergey Eisenstein (1938), one of the famous scenes in the film history.

176

5 Mechanics of Lake Ice

Fig. 5.10 Battle on ice in Lake Peipsi/Chudskoe in 1242. A snapshot from the film Alexander Nevsky (premiere in 1938) by Sergey Eisenstein

Substantial progress in the science and technology of lake ice roads was made in Soviet Union during the Siege of Leningrad in 1941–1944. Ice road over Lake Ladoga, known as “The road of life”, was the only way for connections outside the city, and the bearing capacity question was deeply worked on (see Goldstein et al., 2017). The next major progress took place in North America in the 1970s and 1980s in connection with the development of mineral deposits in Northern Quebec, the Canadian Arctic and Northern Alaska (Mesher et al., 2008; Masterson, 2009). The longest winter ice road was 568 km long connecting Tibbitt Lake and Contwoyto Lake and crossing 65 smaller lakes in the Canadian Arctic (Mesher et al., 2008). This road was operated annually from 1979 to 1998 and it provided transportation of cargo to the mines and minerals extracted in the opposite direction. In this road the first attempt was made to use a mobile radar to monitor the thickness of ice. In recent years, interests in ice crossings, roads, and islands to support drilling rigs have risen again in connection with the planned development of oil and gas resources on the continental shelf of the Russian Arctic. Physics The physical basis of the bearing capacity problem was presented by Hertz (1884). Among the first practical results was a special issue of the Proceedings of the Scientific and Technical Committee of the People’s Commissariat of Transport of the USSR (see Goldstein et al., 2017), where there were also links to the papers by Sergeev (1929) and Bernshteyn (1929). The formulation of the theory by Wyman (1950) has been widely used in the western literature as the historical paper. The elastic theory below holds for short-term loading. Consider a one-dimensional ice beam on water foundation under a concentrated load P at the origin. The ice is forced down, while it is supported by the water pressure as ρw gw, where w is the deflection (Fig. 5.11). Thus, a liquid water body

5.3 Bearing Capacity of Ice

177

Fig. 5.11 Schematic picture of the bending of ice under load; h is ice thickness and R is the characteristic length (Goldstein et al., 2017)

acts mathematically in the same manner as an elastic foundation, i.e., the response of the foundation is proportional to the deflection. The deflection from the free surface (w) follows the fourth-order differential equation (e.g., Goldstein et al., 2017).

EI

bh 3 d 4w = −ρ gw, I = w dx4 12

(5.19)

where I is the moment of inertia of the beam, and b is the beam width (here unit width). The point load P acting in the origin is accounted for in the boundary conditions. Under a static load, the solution of Eq. (5.19) is for x ≥ 0 w(x) = −

Pλ −λx e [sin(λx) + cos(λx)], λ = 2ρw g

/ 4

ρw g 4E I

(5.20)

For x ≤ 0, the solution is symmetric. The deflection forms an exponentially damping wave with distance from the origin as specified by one parameter, λ (e.g., Ashton, 1986). Its inverse l1 = λ−1 ∝ h 3/4 is called the characteristic length of a beam on elastic foundation, and the length of the bending wave is 2π L1 . With the elastic properties of lake ice above, L1 = 4.2 m for h = 0.1 m and L1 = 23 m for h = 1.0 m. For h = 0.1 mand 100 kg load (P = 104 Pa), the deflection at the origin is w(0) = 0.72 mm, which is much less than the freeboard of about 10 mm. At the distance of one wavelength, w(2π L1 ) = 1.9 · 10−3 w(0). The maximum tensile bending stress occurs at the bottom surface at the origin. In two dimensions, the general theory of plates on elastic foundation gives the equation of deflection as (Michel, 1978a, 1978b; Wyman, 1950): EI ∇ 4 w = −ρ w gw 1 − μ2

(5.21)

178

5 Mechanics of Lake Ice

This is reduced to the Bessel equation (see Masterson, 2009; Goldstein et al., 2017), and the solution involves special functions. For an infinite plate, the centrally symmetric solution is P K ei w(r ) = 2πρw gL2

/  r Eh 3 4   , L4 = L 12ρw g 1 − μ2

(5.22)

where K ei is the Kelvin function of the 2nd kind, L is the characteristic length of a plate on elastic foundation (as introduced in Eq. 5.16b), and r is the distance from the point of the load application. The maximum deflection occurs at r = 0, and the characteristic length represents the size of the deflection bowl or action radius. The deflection at r = 0 and the stress at r → 0 are P 8ρw gL2

  L 3(1 + μ) P −c log 2 σr = σϕ = 2π h2 r w(0) =

(5.23a) (5.23b)

where c = 0.577216 is Euler’s constant. This stress becomes singular at r = 0 so that a finite size of the loading area needs to be considered. Equation (5.23b) has been the basis of practical formulae of the bearing capacity showing that the stress scales as P/ h 2 , the load over the square of ice thickness. The deflection of the plate and its elastic surface form an exponentially damping wave bowl. At r ≫ L the deflection is practically invisible, and the ice plate can be considered infinite with respect to the vertical load. The maximum tensile stress acts on the lower ice plate surface under the point of the force application. Within the region r < L the upper surface of the ice is pressed down. For practical evaluation of the bowl deflection for freshwater ice, it is convenient to use a simple relation (Gold, 1971): L ≈ 16h 3/4 m1/4 . The deflection bowl is symmetric at the beginning of loading for small deflections. Deformations are changing for large deflections due to formation of radial cracks. Radial cracks occur initially at the lower surface directly under the ice load. The solutions of distributed loads may be obtained by integration of the problem on the action of the concentrated load (see Masterson, 2009; Goldstein et al., 2017). Calculations show a small effect of a distribution of uniform loads on the maximum values of the ice plate deformation parameters. Increasing the radius of a circular loaded area to 10 m reduces the deflection at the center on 12% and at the contour of the area by 3%. The limit load can be written as   −1 b σ f h2 Pcr = 3(1 + μ)C1 L

(5.24)

where σ f is the flexural strength of ice, b is the characteristic size of the loaded area, and C1 = C1 (b/L) is a coefficient (Fig. 5.12). The radius of influence of a point load

5.3 Bearing Capacity of Ice

179

Fig. 5.12 The  coefficient C1 = C1 Lb in limit load on ice plate (Eq. 5.24), b is size of the loaded area and L is the characteristic length of the ice plate (Goldstein et al., 2017)

can be taken as L and, consequently, 2L is the safe distance between loads if the ice thickness is close to the critical level. Example 5.4 The common rule is that ice can bear a mass M < ah 2 , where a ∝ σ f is a parameter, a ≈ 5 kg cm–2 . Equation (5.24) tells that a=

σf 3(1 + μ)C1

With C1 = 0.5 and σ f = 1 MPa, we have Pcr = 5.1h 2 . The value C1 = 0.5 corresponds to the argument b/L = 0.1. Then, with Gold’s (1971) L formula, b = 1.6h 3/4 m1/4 which gives b = 0.3 m for h = 0.1 m and b = 1.0 m for h = 0.5 m, i.e., the load is highly local. Example 5.5 A common practical problem is a crowd of people on lake ice (Nevel & Assur, 1968). Assume that people are evenly spaced at distances of d between, and the mass of each person is m (Fig. 5.13). First, to prevent ice from sinking, the buoyancy condition requires that m ≤ (ρw − ρi )hd 2 Take m = 90 kg. The buoyancy criterion requires that h ≥ 25 cm for d = 2 m. or 25 people in a 100 m2 square. Packing the people much closer requires thick ice,

180

5 Mechanics of Lake Ice

Fig. 5.13 An infantry parade on the ice cover of Lake Eldankajärvi, Viena of Karelia in the Second World War on March 1, 1944. Photograph of SA-kuva archive, Finland

1 m for d = 1 m. Second, to prevent ice from breaking, the number of people, N , is limited by C1 N ∼

σ f h2 3(1 + μ)mg

For h = 25 cm, the right-hand side is 17.8, and L = 5.7 m. Since for a large loading area C1 ∼ 0.1 (Fig. 5.12), we have N > 100 for a 100 m2 square m. Thus, the case is limited by the buoyancy. For a 25 cm thick ice, the bearing capacity for a point load is 3125 kg corresponding to 35 people, so that it is critical how close the people can be together. Interest in dynamical loads originates from practical problems, in particular the problems on loads due to vehicle motion along an ice road. The question became particularly important during the Leningrad Siege in 1941–1944 in the Second World War. The possibility of dangerous resonance phenomena for moving vehicles was expressed by Zubov (1942). This was confirmed by Bregman and Proskuryakov (1943) who proved that there is a certain vehicle speed, which can lead to the fracture of ice cover, based on data analysis of ice cover oscillations in Lake Ladoga. The

5.4 Ice Forces

181

explanation is in the resonance of the vehicle motion and the wave motion in the water caused by the moving pressure (Nevel, √ 1970). The shallow water wave speed is cs = g H , where H is the water depth. If the moving vehicle on ice has the same speed, called as the critical speed, resonance strengthens the bending of ice and, in theory, the bearing capacity disappears (see Ashton, 1986; Masterson, 2009). In practice, it has been observed that the bearing capacity has gone down to around one-third of the static value. When the speed is much less or much more than the critical speed, the resonance disappears. At subcritical speeds, maximal bending moments occur under load, while at supercritical speeds, the peak is in front of the load. In deep water, the deflections are always smaller than in shallow depths because of the effect of water depth on the wave motion.

5.4 Ice Forces 5.4.1 Ice Load Problems Forces due to ice have been investigated for a long time in ice engineering because of practical applications. The maximum loading takes place when the ice fails at a structure, and the loading force depends on ice properties and the failure mode. In lake ice environment, loading cases are concerned with harbour structures, piles, docks, and other coastal constructions, ships, and the natural coastline itself (Fig. 5.14). The failure of ice is not necessarily achieved due to short fetches, and therefore the forces are generally lower than in marine environment. On the other hand, the strength of freshwater ice is higher than the strength of sea ice, and therefore in non-fetch-limited situations loads are higher in lakes. Example 5.6 ‘Ice quakes’. In the beginning of January 2009, a rare ‘ice quake’ event was experienced on the shore of Lake Vesijärvi, Lahti, Finland (Fig. 5.15). The lake ice thickness was about 10 cm, and strong northerly wind over 4–5 km fetch blew onshore. Residents at new, high buildings located about 40 m from the shoreline were alarmed by vibrations of the buildings. Recording of these vibrations were started in three buildings on January 8th. The frequencies of the floors were then 2–10 Hz, around the lowest expected natural frequency of the houses in the so-called pendel vibration mode (Makkonen et al., 2010). All the houses, which felt the vibrations, had been supported by using vertical concrete piles on stiff moraine. The building site itself had been filled in the lake with a 6 m thick filling resting on original quite soft 15 m thick clay/silt layer above the moraine. The in-lake building site was probably the main reason why the vibrations due to cracking of the lake ice could be transported to the buildings via the soil. When the melting of ice began in spring, vibrations started again after a two months quiet period. The highest measured velocities were 1.2 mm s–1 , exceeding the comfort limit of the residents, but no structural damages were expected and indeed not observed.

182

5 Mechanics of Lake Ice

Fig. 5.14 Ice load on shore in Lake Tuusulanjärvi, Finland. Photograph by the author

Fig. 5.15 ‘Ice quakes’ were felt in the houses close to the shore due to pressure from by strong wind in Lahti, Finland. Fetch over the ice was up to 5 km. Photograph by the author

5.4 Ice Forces

183

Example 5.7 Thermal pressure. In Lake Mälaren, Sweden thermal pressure has caused damage in the past (Bergdahl & Wernersson, 1978). In winter 1953 a swing bridge was partly damaged. Air temperature rose by 14 °C in 12 h, and 22 cm thick ice expanded and pressed toward the bridge. The forcing was estimated as 270 kN m–1 . In winter 1976 four beacons mounted on pile groups were destroyed by thermal expansion, with pressures estimated as 100–270 kN m–1 .

5.4.2 Estimation of Ice Loads The load of ice on structures depends on the mode of loading, strength of the structure, and form of the structure. The fundamental method to evaluate ice loads is the Korzhavin (1962) equation for vertical structures: F = I k1 k2 bσ h

(5.25)

where I is the indentation factor, k1 , k2 and b are the contact factor, shape factor and width of the structure, respectively, and σ and h are the strength and thickness of ice. The indentation factor is due to the 3D nature of the stress in ice representing the stress–strain field around the structure and accounting for the difference between the real and the unconfined stress distributions: I = I (δ), δ = b/ h is the aspect ratio, with I ≈ 2.5 for narrow structures and I ≈ 1 for wide structures. The parameters k1 and k2 represent the contact between the structure and ice, k1 for the incomplete contact between the ice edge and structure and k2 for the form of the structure crosssection, k1 ≥ 1/2, k2 ≤ 1. In case of a non-vertical structure, inclination of the loaded surface has also influence. Further simplifications can be derived from the Korzhavin (1962) model. The force on a pier-type structure can be estimated by Ashton (1986) F = K bσc h

(5.26)

where σc is the compressive strength, and K is a scaling coefficient, which depends on the mechanical properties of ice and the geometry of the pile structure. For a narrow structure the aspect ratio is an important parameter, K = 2.5 for δ = 1 and K → 1 for δ ≫ 1. For wide structures and wedge-shaped piles, K = 1 can be taken. For an inclined structure K depends on the slope angle of the structure and ice–structure friction, K < 1. Example 5.8 Consider a 1 m wide pile and freshwater ice with σc = 10 MPa. For thin ice (10 cm), δ = 10 and thus K ≈ 1 and the maximum ice load is F = 1 MN; for thick ice (1 m), δ = 1 and thus K ≈ 2.5 and the maximum ice load is F = 25 MN.

184

5 Mechanics of Lake Ice

When the ice has frozen to the structure, changes in the water level elevation cause vertical forces. The largest forces result when the change in the elevation is fast, since the ice then behaves in elastic manner. Kerr (1975) examined vertical loads on a cylindrical pile. The load increased with the thickness of ice and the aspect ratio. When the ice thickness was 0.5 m, the load magnitude per water level change was 5 · 105 N cm–1 . According to Gamayunov (1960) the ice fails in bending at the cylinder 2 surface with an effective moment M = 16 h σ f . If the ice cover is frozen to vertical walls, uplifting and down dragging forces also result from vertical movement of the water level (Ashton, 1986). Based on the theory of beam on semi-infinite elastic foundation, the load is ρw g P = ∆ξ b L

(5.27)

where ∆ξ is the change in water level elevation, and L is the characteristic length of the ice plate.

5.5 Drift Ice in Lakes 5.5.1 Drift Ice Material The horizontal structure of an ice cover in large lakes is well revealed by optical satellite images (Fig. 5.16). The ice cover is occasionally broken into floes by forcing due to winds and currents, water level variations, and thermal cracking, and the floes may start drifting. In Fig. 5.16, drift ice is seen in Lake Ladoga expect for the landfast ice zone skirting the shoreline. The field of broken ice is a granular medium called drift ice, and here, as usual, for drift ice a continuum approximation is employed. The motion of drift ice can be treated as a two-dimensional phenomenon on the lake surface, and the vertical dimension shows as a material property—the thickness of ice. In sea ice basins, ice drift is a common phenomenon with major short-term and climatological impacts, and the theory and models of sea ice dynamics are applicable to frozen lakes (e.g., Leppäranta, 2011). From the point of view of ice mechanics, ‘large lakes’ are those, in which winddriven mechanical displacements are common all winter. In small and medium-size lakes the ice cover is mostly static, apart from fractures and thermal cracks, but in early and late season the ice cover is weak and mobile. Fetch is a critical factor for the wind forcing, and therefore islands can limit the mobility of ice in large lakes such as Lake Saimaa in Finland. A linear dimension of the order of 100 km can be taken as a scale for lakes where ice displacements in winter are common. In Europe there are several large lakes, where major ice displacements are observed all winter, e.g., Caspian Sea, Lake Ladoga, Lake Vänern, and Lake Balaton (see Burda, 1999; Takács et al., 2018; Wang et al., 2006b). Ice extent in these lakes

5.5 Drift Ice in Lakes

185

Fig. 5.16 Lakes in Finnish and Russian Karelia in Sentinel-3 satellite image on March 6, 2017. Drift ice is seen in Lake Ladoga (area 17,700 km2 , mean depth 51 m), while Lake Onega (area 9700 km2 , mean depth 30 m) shows ice cover with fractures, and Lake Saimaa (area 4400 km2 , mean depth 17 m, complex, mosaic morphology) is fully ice-covered. © European Space Agency

depends on the severity of the winter, and in the Caspian Sea ice formation is limited to the northern part with open boundary in south. As an example, Fig. 5.16 shows the drift ice in Lake Ladoga. Drift ice also forms in the Great Lakes of North America (Wang et al., 2012), and in Asia, wind-driven ice is observed in the Aral Sea (Kouraev et al., 2004) and in Lake Baikal (Semovski et al., 2000). In between large and medium-size lakes, there is a grey area: the ice cover is static in cold winters and may be mobile in mild winters. An example is Lake Lappajärvi, a medium-size lake in Finland. Its surface area is 148 km2 and its mean and maximum depths are 7.4 m and 36 m, respectively. The lake basin depression is a wide meteorite impact crater. It is annually covered by ice, and the maximum annual ice thickness is 50–70 cm. Mechanical ice breakage, displacements and formation of small ridges is a common phenomenon, but the length scale of lateral displacements is limited (Alestalo, 1980). Drift ice physics is conveniently presented in the right-hand horizontal co-ordinate system (x, y) with positive directions east and north, respectively. The drift ice medium is described by the state variables, which contain the properties needed to model the internal stress. The state information is contained in the ice thickness distribution for the ice state (Thorndike et al., 1975), defined for a region Ω by ∏(h) =

1 A(Ω)

 H [h − ζ (ω)]dΩ Ω

(5.28)

186

5 Mechanics of Lake Ice

 where h is the thickness distribution argument, A(Ω) = Ω dΩ is the area of Ω, ζ is the actual thickness at points ω ∈ Ω, and H is the Heaviside function, H (x) = 0(1) for x ≤ 0(x > 0). The value ∏(h) equals the normalized area of ice with thickness less than or equal to h in Ω. Ice state is multi-dimensional of the thickness  function  ˜ distribution, J = J (∏). The two-level ice state is J = A, h , where A = ∏(0) is ice concentration or compactness, and h˜ is the mean ice thickness (Doronin, 1970). For higher resolution of the ice properties, multi-level forms of ice state are needed. For a two-dimensional medium, the stress σ is obtained from the threedimensional stress by integration across the thickness of ice (see, e.g., Leppäranta, 2011). The dimension of the two-dimensional stress is force/length. Drift ice rheology investigations have been made for sea ice and river ice, but the results are applicable for lake ice as well, provided the broken structure of the ice cover prevails. The rheological law of drift ice is in general form ˙ σ = σ (J, ε, ε)

(5.29)

Not including higher-order strain derivatives in Eq. (5.29) means that the system has no memory. The simplest rheology is stress-free ice (σ ≡ 0), the motion called then as free drift. There is no friction between interacting ice floes, and the model is applicable for low ice compactness, A < 0.8, when stress levels are very small and convergence/divergence in ice drift just arranges distances between ice floes. When A ∼ 0.8, collisions and shear friction between ice floes become significant but stresses are still low, and the ice behaves in nonlinear viscous manner (Shen et al., 1986). Compact drift ice (A > 0.9) has a plastic rheology (Coon et al., 1974; Dansereau et al., 2016; Hibler, 1979; Hunke & Dukewicz, 1997). The following general properties describe the ice stress: (1) Strength is sensitive to ice compactness, (2) For A ≈ 1, compressive strength > shear strength ≫ 0, (3) Tensile strength is small, i.e., there is not much friction in driving ice floes further away from each other, and (4) Drift ice is strain hardening in compression. Due to the high yield strength of compact ice, the ice cover in small and medium-size lakes is stable over most of the winter. Strain hardening means that drift ice can be packed to a limited level under a fixed forcing, The original plastic drift ice rheology was the elastic–plastic model developed in the AIDJEX (Arctic Ice Dynamics Joint Experiment) programme (Coon et al., 1974; Pritchard, 1975). A few years later, Hibler (1979) introduced a viscous-plastic rheology, which is more feasible for long-term simulations. These approaches served for long as the basis of plastic drift ice modelling, the main concern was in the shape of the yield curve. Hunke and Dukewitz (1997) developed further a computationally efficient elastic-viscous-plastic model, and recently a Maxwell elasto-brittle rheology has been introduced for sea ice modeling (Dansereau et al., 2016). The strength of the ice depends on the scale and thickness (see Sect. 5.2.4). The basic principle is that thinnest ice breaks first in compression. For the two-level thickness distribution, the formulation of the compressive strength is

5.5 Drift Ice in Lakes

187

P = Pn∗ h n exp[−C(1 − A)]

(5.30)

where Pn and n are the strength parameters of compact ice, and C is the strength reduction for lead opening. In the original version of this form (Hibler, 1979), the power was taken as n = 1, but in general 1/2 ≤ n ≤ 2. The inverse of the parameter C is the e-folding value of strength for change in compactness, and C ≫ 1 due to the high sensitivity when compactness is high. The common value has been C = 20, and the strength of compact ice scales as Pn h n ∼ 25 kPa for h = 1 m. In a research effort of landfast ice in the Baltic Sea, Leppäranta (2013) found n = 2 as a good representative value. The stress depends on the mode of failure in ice and the mode of deformation. In compressive deformation, ice floes may override each other resulting in rafting or break into ice blocks to form pressure ridges (see Sect. 5.2.4). The compressive elastic strength of drift ice is σc / h ≈ 0.1 MN m–2 (Pritchard, 1980). Rafting or piling up of ice blocks may also take place on the windward shore. Then the amount of shore ice accumulation depends on the thickness of ice and shoreline morphology (Bryan & Marcus, 1972). The linear elastic model gives the stress proportional to the strain: σ = σ (ε; k, G) = K (tr ε)I + 2Gε

'

(5.31)

'

where ε = ε − 21 tr εI is the deviatoric strain. The dimension of two-dimensional stress and the moduli K and G is force/length. The magnitude of the drift ice elastic moduli is 10–100 MN m–1 . The original bulk and shear moduli were given as E = 100h MN m–2 and G ≈ 1/2K in the AIDJEX model by Coon et al. (1974). The corresponding Poisson’s ratio is 0.29. The ratio between compressive strength and bulk modulus of ice is σc /E ∼ 10−3 , and therefore elastic strains remain below 10–3 , i.e., less than 1 m across 1 km distances. Pritchard (1980) chose this ratio as σc /E = 2 · 10−3 . The elastic moduli and brittle strength are lower in drift ice mechanics than in small-scale mechanics, since the number of faults increases with length scale, and the scaling has been suggested as σc ∝ L −1/2 (Sanderson, 1988; see also Leppäranta, 2011). Two-dimensional plastic yielding is given with a yield curve F(σ1 , σ2 ) = 0, where σ1 and σ2 are the principal stresses, σ1,2

1 = tr σ ± 2

/

1 tr σ 2

2 − detσ

(5.32)

The plastic failure criterion tells when the ice fails and provides the plastic flow from the equation of motion. Drucker’s postulate for stable materials states that the yield curve serves as the plastic potential, and consequently the failure strain is directed perpendicular to the yield curve, known as the normal, or associated flow rule (e.g., Davis & Selvadurai, 2002). Consequently, the principal strains and stresses are ∂F , where λ is a parameter to be obtained as a part of the solution. parallel and εk = λ ∂σ k

188

5 Mechanics of Lake Ice

Fig. 5.17 Drift ice yield curves in the principal axes space: Mohr–Coulomb or triangular (Coon et al., 1974), teardrop (Coon et al, 1974), and elliptic (Hibler, 1979)

Inside the yield curve, F < 0 and stresses are elastic or viscous, while F > 0 would go outside the yield curve and is not allowed. Since a granular ice cover does not resist tensile stresses, the yield curve is in the quadrant where σ1 , σ2 < 0. Different shapes have been used for the yield curve (Fig. 5.17). The elliptic curve is friendly for a modeller as it provides stress as an explicit function of the strain-rate. Example 5.9 Hibler (1979) presented the elliptic yield curve and formulated the plastic law with strain-rates, σ = σ (˙ε , P, e), where P is the compressive strength and e is the ratio of compressive strength to shear strength. The elliptic yield curve has the advantage that the failure yield can be expressed in closed form. The yield curve is defined as (see Fig. 5.17):  F(σ1 , σ2 ) =

σ1 + P/2 P/2

2

 +

σ2 + P/2 P/2e

2 −1

∂F = The yield equation and the associated flow rule require that F(σ1 , σ2 ) = 0, ∂σ k λ˙εk . This set of equations gives σk = σk (˙ε1 , ε˙ 2 ), and since the principal axes of strain-rate and stress are parallel, the solution is obtained in the original coordinate system as

5.5 Drift Ice in Lakes

σ =

P 2



189

 / ε˙ I 1 − 1 I + 2 (˙ε − ε˙ I I) , ε˙ D = ε˙ 2I + (˙ε I I /e)2 ε˙ D e ε˙ D

where ε˙ I and ε˙ I I are the strain-rate invariants equal to the sum and difference of the principal strains, ε˙ I = ε˙ 1 + ε˙ 2 and˙ε I I = ε˙ 1 − ε˙ 2 . For spherical compressive strain, ε˙ I < 0 and ε˙ I I = 0, and we have, σ = −PI. In the case of pure shear, P ε˙ I = 0, we have σx x = σ yy = − P2 and σx y = σ yx = 2e . Since for drift ice shear strength is significant but less than the compressive strength, we must have 1 < e ≪ ∞; normally e = 2 is chosen. This original model includes an inconsistency of having non-zero stress for an ice field at rest: with σ = −1/2P I for ε˙ = 0. In the case of low, long-term forcing, this stress has an unrealistic consequence in leading to continuous, although very slow, creep. This inconsistency was later removed by letting σ → 0 with ε˙ → 0 (Hibler, 2001). An anisotropic extension was developed by Hibler and Schulson (2000) to examine the dynamics of oriented lead and crack systems. When the yield curve has been fixed, the question is how to deal with stresses below the yield level. In specific analytical modelling, ideal plastic cases can be solved. But in numerical modelling, the medium is taken elastic (Coon et al., 1974), viscous (Hibler, 1979), or elastic-viscous (Hunke & Dukewicz, 1997). The viscous approach is computationally convenient since no reference configuration is needed but only the rate of deformation is relevant (Hibler, 1979). The linear viscous option is given analogously with the elastic model, with strain replaced by strain-rate and the elastic moduli replaced by bulk and shear viscosities ζ and η (compare with 3D Eqs. 5.6 and 5.7). These viscosities have been kept large (~1012 kg s−1 , η ∼ 21 ζ ) to limit viscous deformation and have a good approximation for the plastic flow. Glacier flow obeys a nonlinear viscous law (Paterson, 1999), the linearized viscosity would be ~ 1013 kg s−1 at the strain-rate of 10−8 s−1 .

5.5.2 Equations of Drift Ice Mechanics A drift ice field is described with the thickness of ice, h = h(x, y; t) and velocity u = u(x, y; t), u = ui + vj. A ‘point’ in the drift ice field is a finite cell, where the velocity is represented by the average value, while the material properties of drift ice are represented by the drift ice state J = J (∏). The equation of motion for a continuum is expressed by the Cauchy equation (Hunter, 1976), where the continuum momentum is forced by the external forces and the divergence of the internal stress. The 2D equation of motion of drift ice is derived by integrating across the thickness of the ice and accounting for the Coriolis acceleration (Leppäranta, 2011; Fig. 5.18). The result is   ∂u ˜ + u · ∇u + f k × u = ∇ · σ + τ a + τ w − ρg h∇ξ ρi h˜ ∂t

(5.33)

190

5 Mechanics of Lake Ice

Fig. 5.18 Lake ice drift and the physical problem. Ice is driven by forcing from the atmospheric and lake water body, and the ice responds to forcing through its internal stress field. A typical force diagram is also illustrated

where f = 2Ωsinφ is the Coriolis parameter, Ω = 7.292 · 10−5 s−1 is the angular velocity of the Earth and φ is latitude,5 τ a and τ w are the tangential air–ice and water–ice stresses on the ice top and bottom surfaces, respectively, and ξ is the water level elevation. In very large lakes the Coriolis acceleration becomes significant in ice drift (the derivation is shown, e.g., in Cushman-Roisin, 1994). The external forces consist of the surface pressure gradient and air and water stresses. If the lake water surface is tilting, a pressure gradient force results from the hydrostatic water pressure on ice floes as shown by the last term in Eq. (5.33). The air and water stresses are given by the bulk formulae as discussed in Sect. 4.2 (e.g., Andreas, 1998; McPhee, 2008) τ a = ρa Ca Ua U a τ w = ρw Cw |U w − u|cosθw (U w − u) + sinθw k × (U w − u)

(5.34a) (5.34b)

where Ca and Cw are the air and water drag coefficients, θw is the boundary layer angle in water, and U a and U w are the surface wind and current velocities. The surface air stress is parallel to the wind velocity in the 10 m surface layer. In shallow lakes, the reference depth of the current velocity is taken as 1 m, the vertical mid-depth, or the bottom (where current velocity is zero), while in deep lakes a convenient reference depth is the bottom of the layer of frictional influence of the ice. Representative values of the drag parameters are Ca10 = 1.2·10−3 (10 m reference altitude, Andreas, 1998), Cw1 = 1 · 10–3 (1 m reference depth, Shirasawa et al., 2006). In shallow lakes θw ≈ 0, but flow rotation effects (Coriolis acceleration) need to be considered in deep lakes 5

Latitude is taken positive in the northern hemisphere and negative in the southern hemisphere.

5.5 Drift Ice in Lakes

191

and 0 < |θw | < 25◦ , θw > 0(< 0) in northern (southern) hemisphere (see McPhee, 2008). Figure 5.18 also shows a schematic force diagram of ice drift. Usually, wind is the driving force, balanced by the ice–water drag and the internal friction of the ice (Table 5.3). Internal friction can be strong due to the limited size of lake basins with the presence of land boundary felt all over. Coriolis force is smaller than the three major forces and is always perpendicular to the ice motion, directed to the right (left) in the northern (southern) hemisphere. The other terms are smaller. In rivers, the water current provides the main transport mechanism. For a steep surface slope of 10–5 and ice thickness of 1/2 m, the magnitude of the surface pressure gradient force would be 50 mPa. Thus, hydrostatic pressure, which is a major factor in lake water circulation, is not important in ice dynamics. Inertia becomes significant in very rapid changes. In the lake ice drift problem, there are two principal time scales: local acceleration TI = U/(Cw H ) ∼ 5 min and deformation TD = L/U ∼ 5 d. Where the Coriolis acceleration is significant, the Coriolis period f −1 ∼ 12 h needs to be considered. These time scales are well separated, TI ≪ f −1 ≪ TD . The first-order free drift solution provides a limiting upper scale for the ice velocity (see Leppäranta, 2011), obtained from the steady state equation neglecting the Coriolis and pressure gradient terms: /   ρa Ca cosθ sinθ ≈ 3%, θ ≈ −θ w . (5.35) u=a Ua + U w, a ≈ −sinθ cosθ ρw Cw Table 5.3 Scaling of the equation of motion of drift ice in large lakes Term

Scale

Value mPa

Comments

Local acceleration

ρi HU T −1

0.5

50 for rapid changes (T = 103 s)

Advective acceleration

ρi H (U − Uw )2 L −1

0.1

Usually much less than water stress

Coriolis term

ρi H f U

5

50 for H = 1 and U = 50 cm s–1

Internal friction

P H L −1

0–300

0 for open ice field, limited by forcing

Air stress

ρa Ca Ua2

35

150 for Ua = 10 m s–1

Water stress

ρw Cw (U − Uw )

10

150 for U − Uw = 40 cm s–1

Pressure gradient

ρi H g∇ξ

5

Slopes limited by water dynamics

2

The representative fundamental scales are the following: ice thickness H = 1/2 m, ice velocity U = 10 cm s–1 , ice strength P = 50 kPa, wind velocity Ua = 5 m s–1 , water velocity Uw = 0, lake surface slope ∇ξ = 10−6 , time T = 1 day, and horizontal length L = 50 km

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5 Mechanics of Lake Ice

Fig. 5.19 The free drift solution as the vector sum of wind-driven ice drift and geostrophic current (when the Coriolis acceleration is significant)

where θ is the deviation angle between the directions of wind and ice drift. The θ -matrix rotates the windspeed clockwise by the angle of θ .This solution works in open drift ice, e.g., for the broken state of ice cover in the ice decay period. In shallow water, θ = 0 and u = aU a + U w , while in deep water |θ | ≈ 25◦ (Fig. 5.19). The lower ice velocity limit is zero, when the internal friction is too high to start movement. In compact ice, the forcing needs to be strong enough to break the yield strength. Since water circulation is weak under ice and water flow has a spatially changing direction, the integrated forcing by the ice–water stress does not get as high as by the wind forcing. The role of the internal friction is described by the dimensionless quantity X = σY /F L where F is the forcing and L is the length scale. Ice flows when the forcing overcomes the yield level or X < 1. Ice rheology models have a distinct asymmetry in that during closing the yield strength is very high but during opening it is nearly zero. The wind stress is τa ∼ 0.5 N m–2 for a wind speed of 15 m s–1 . If the fetch L is 10 km, the stress integrates to a force of τa L ∼ 5 kN m–1 at the windward side of the lake, enough to break ice with h ∼ 10 cm by compression. This is the correct magnitude according to observations in ice-covered lakes. In the ice decay period,

5.5 Drift Ice in Lakes

193

ice strength decreases due to deterioration of ice and opening of ice cover, and the conditions approach the free drift state. The lateral boundary of a lake ice field can be simply taken as the shoreline, where the ice velocity is zero. At the land boundary, the ice is allowed to stay or move away into the drift ice basin, but it is not allowed to override the solid boundary out from the basin. Open water is then taken as ‘ice with thickness equal to zero’. In case a lake is partly ice-covered for a longer period, it is often important to track the boundary of ice and open water (Ovsienko, 1976). At this boundary, the ice does not support normal stresses and the motion of ice changes the boundary configuration. As soon as the ice has moved out and lost contact with the shore, the boundary changes to an open water boundary. Toclosethe system, an ice conservation law is needed for the ice state. Taking J = A, h˜ , the conservation law is ∂ h˜ ˜ · u + φh˜ + u · ∇ h˜ = −h∇ ∂t

(5.36a)

∂A + u · ∇ A = −Aψ(˙ε I , ε˙ I I ) + φ A (0 ≤ A ≤ 1) ∂t

(5.36b)

where φh˜ and φ A are the thermal growth rate of mean ice thickness and compactness, respectively, and the function ψ provides the opening and closing of the ice field as a function of the strain-rate. The formulation using ψ is needed, since open water is also created in pure shear. Note that h˜ = Ah˜ F , where h˜ F is the mean thickness of ice floes, i.e., h˜ includes open water patches with h = 0. Thermal change of ice thickness was presented in Chap. 4, and to get the thermal change of mean ice thickness and compactness starts from the ice thickness distribution ∏(h; t). The growth rate of ice, φ(h; t), shifts the distribution in the thickness space, and: ∂∏ ∂∏ ∂ = −φ , φh˜ = ∂t ∂h ∂t φ A ( A) =

∞ h 0

∂∏ dh ∂h

φ(h 0 ) (1 − A) h0

(5.37a)

(5.37b)

Formally, A(t) = 1 − ∏(0; t), but in the two-level ice state the ice concentration is based on the area where ice thickness is less than a demarcation thickness h 0 . Thus A(t) = 1 − ∏(h 0 , t), and Eq. (5.37b) follows for the thermodynamic evolution. When A < 1, lateral melting may take place via absorption of solar radiation in open water patches. This can be added to Eq. (5.37b) by distributing the heat evenly to vertical floe surfaces (see Rothrock, 1986). In a medium-size lake, L ∼ 10 km, the ice cover is stationary for h ∼ 40 cm, but with a major decrease in ice thickness, strength, or increase in wind speed, the ice

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5 Mechanics of Lake Ice

Fig. 5.20 Ice breakage in spring in Lake Päijänne, Finland. Photograph by the author

cover can become mobile. Rare observations of modest ridging exist in Finland to confirm this. In a study of landfast ice in the Baltic Sea, Leppäranta (2013) concluded that in the scales of coastal basins, the power n = 2 for ice thickness in the yield strength (Eq. 5.30) corresponded well to observations. Thus, if h ∼ 30 cm is the critical thickness for lake size L ∼ 50 km, then for h ∼ 0.1 m it would be L ∼ 5 km. In spring, lake ice becomes rotten, and its strength decreases due to internal melting (Fig. 5.20). Then the ice cover may be broken become mobile as sometimes seen in on-shore ride-up or pile-up. However, the period of potential mobility is short and a considerable wind speed is needed for displacements. It is known by experience in Finland that in some years ice cover decays in place and in some years breakage and drift results.

5.5.3 Models of Drift Ice Dynamics A lake ice cover consisting of drift ice can be modelled with continuum models. The conservation equations of momentum and ice (Eqs. 5.33 and 5.36a, b) provide the basis for these models. Despite the important role of ice mechanics in large lakes and during decay in all lakes, only little research has been made with full thermal–mechanical modelling. Continuum models were originally developed for oceanic drift ice in the 1960s (see Leppäranta, 2011 for the history), but for lake ice they followed later. The first efforts were made in Soviet Union for the Caspian Sea (Ovsienko, 1976) and in North America for the Great Lakes (e.g., Wake & Rumer, 1983). Later, sea ice dynamics models progressed in small marine basins (e.g., Kubat et al., 2009; Wang et al., 2003, 2006b), and the results were also utilized for lake

5.5 Drift Ice in Lakes

195

ice modelling in Lake Peipsi (Leppäranta & Wang, 2008; Wang et al., 2006a) and in the Great Lakes (Wang et al., 2010). In long-term modelling, ice dynamics is often neglected. In the early history of research, only thermodynamic models were available for the times of freezing and ice break-up and for the evolution of ice thickness. The objectives of ice drift modelling are basic research of the dynamics of drift ice, ice–land interaction, ice forecasting, and applications for ice engineering. Ice drift has a strong influence on operations on lakes, and changes in ice conditions in large lakes or lake districts are important for weather forecasting over a few days (Yang et al., 2013) or for climate scenarios. Leads may open and close and pressure ridges may build up in a one-day time scale. In the ice decay period, there is a short sub-period when the ice becomes mobile but still has finite strength. Then the lake offshore structures are in danger. It is known in practice in Finnish lakes that small boats, piers etc. need to be taken onshore before freeze-up in fall. The key area of short-term modelling research is the scaling problem, in particular the downscaling of the stress from geophysical to local scale. The free drift solution (Eq. 5.35) gives a scaling of fast drifting ice, but it does not account for the highly important land–ice interaction. One-dimensional Channel Consider a long channel closed at x = L > 0 and covered by compact ice from the origin to L (Fig. 5.21). The thickness of ice is h 0 , the yield strength of ice is σY (h) for compression, and the forcing is by a constant wind Ua > 0. The boundary conditions are u(L) = 0 (solid boundary) and σ = 0 at the ice edge. The one-dimensional channel flow can be treated semi-analytically, and the solution provides a good insight into the role of internal friction. The quasi-steady-state momentum equation is employed, and here A = 0 or 1, i.e., any point is either ice or open water, and we need to consider only mean ice thickness. Thermal growth and decay of ice are neglected in pure dynamics problems, momentum advection and water level tilt are neglected, and current velocity is assumed zero. The system of equations reads: dσ + τa − ρw Cw sgn(u)u 2 = 0 dx Fig. 5.21 The geometry of the channel flow problem. Ice moves with wind toward the solid boundary with eventual deformation to a steady state u ≡ 0

(5.38a)

196

5 Mechanics of Lake Ice

∂h ∂ + (uh) = 0 ∂t ∂x

(5.38b)

In the static case, u ≡ 0 and −σ (x) = τa x ≤ τa L < σY (h 0 ). When τa L > σY (h), the ice starts to move, and the ice cover is packed until the yield stress limits further deformation. At equilibrium, u ≡ 0 and the ice edge has moved from x = 0 to x = x0 > 0, and there are undeformed and deformed zones as: ⎧ ⎪ ⎪ x0 ≤ x ≤ x1 : h = h 0 ⎪ ⎨ x ≤ x ≤ L : τ (x − x ) = σ (h) 1 a 0 Y (5.39) ⎪ L ⎪ ⎪ hd x = h L 0 ⎩ x0

For σY = P1 h, we have 

[ τa (L − x0 ) = P1 h 0 + (L − x0 )h 0 + /

and the solution is x0 = L −

P1 h 0 τa

1 τa 2 P1 (L

2L −

−

P1 h 0 τa

]

τa − x1 ) P1 (L x 1 )2 = h 0 L



, x1 =

P1 h 0 . τa

(5.40)

E.g., for τa ∼ 0.5 N

m and P1 ∼ 50 kN m , L = 100 km and h 0 = 0.5 m, we have x1 = 50 km and x0 = 13.4 km. Thus, the ice cover is shifted by 13.4 km, the undeformed zone is 36.6 km, and the deformation zone is 50 km with ice thickness equal to h L = 0.77 m at the end. When the landfast ice condition is broken, ice drift commences. The theoretical free drift solution is here u F = aUa . Plastic flow begins when the yield strength is reached. The solution of the quasi-steady-state momentum equation is formally: –2

–1

  1 dσ τa + = u 2F − u 2σ ≥ 0 u = ρw Cw dx 2

(5.41)

where u σ is the velocity reduction due to friction. This equation together with the ice conservation law end up finally at the steady state with  x u ≡ 0 and h = h(x). The stress distribution can be integrated from σ (x) = − x0 τa (s)ds, where x0 is the location of the ice edge. There is a rigid zone at |σ (x)| < σY with h = h 0 , the initial x thickness, and then a deformation zone with σ (x) = σY (x) = − x0 τa (s)ds. Example 5.10 Consider ice formation in an open lake with fetch L. Assume that the wind stress is constant across the lake. Due to wind-driven turbulence, frazil ice forms and accumulates into a thickness profile corresponding to the ice strength. In equilibrium, the ice is stationary and covers the lake, and the thickness profile is obtained from P[h(x)] = τa x, where x is the distance from the lee side of the lake. With the linear form P = P1 h, the frazil accumulation has a triangular profile with P1 h(L) = τa L, and thus the frazil volume per unit width is

5.5 Drift Ice in Lakes

197

VF =

1 τa 2 1 h(L)L = · L 2 2 P1

Numerical Modelling A full lake ice model consists of four basic elements: ice state, rheology, equation of motion, and conservation of ice. Any proper ice state has at least two levels, ice compactness and thickness. The primary geophysical parameters are the drag coefficients and the compressive strength of ice. The drag coefficients are used to tune the free drift velocity, while the compressive strength is used to tune the mobility and length scale of motion in the presence of internal friction. There are also secondary rheology parameters connected to ice state and shear deformation. When the plastic flow is approximated by an elastic–plastic (Coon et al., 1974), viscous-plastic (Hibler, 1979) or elastic-viscous-plastic (Hunke & Dukewicz, 1997) rheology, there is an additional rheological formulation for the stresses below the yield strength. The numerical design includes the choice of the grid and, since the system is highly non-linear, the stability of the solution may require tailored techniques. Since the minimum continuum particle size D is fairly large, the grid size can be taken as ∆x ∼ D. The grid size can be down to less than 1 km in basins with small floe size and in the melting season when floes break into small pieces. An elastic–plastic approach was developed in the AIDJEX programme in the 1970s (Coon, 1980). The elastic part necessitates the tracking of the change from the reference configuration, and the basis is applicable for lake ice cover, because often the drift episodes are short events between static situations. However, applications of the AIDJEX model to lake ice are not known to the author. Drift ice dynamics is lateral scale dependent in two aspects. Forcing is limited by the size of basins, the resolution of continuum models is limited by the floe size. In short-term modelling, the time scale of ice cover evolution is 1 h—10 days. In large lakes the scale of the basin size is close to the length scale of ice motion, and therefore a proper treatment of the boundary configuration is critical. Similar models have been also used for large-scale simulations in the polar oceans, semi-enclosed mesoscale marine basins such as the Baltic Sea, Bohai Sea, and Gulf of St. Lawrence, and down to the scale of the Niagara River in North America (Shen et al., 1993). Mechanical displacements are very rare in small or medium-size lakes, apart from freeze-up and decay periods. If the climate becomes warmer, lake ice is expected to become thinner, and lake ice cover is more likely to break all winter in smaller lakes than before. In southern Finland, the maximum annual thickness of lake ice is mostly 30–60 cm, and even with fetches up to L ~ 10 km, the climatological minimum thickness of 25 cm results in a stable ice cover. With this fetch and a wind speed of 15 m s–1 , the ice can be at most about 15 cm thick for any significant breakage to occur. Since 1 °C climate warming would reduce the ice thickness by 5–10 cm (see Chap. 9), ice breakage would become common in South Finland lakes if the winter air temperature increases by 2–3 °C. Example 5.11 Lake Peipsi ice model. Lake Peipsi/Chudskoe is a large (surface area 3555 km2 ) and shallow (7.1 m) lake in Estonia/Russia. A modelling study on ice

198

5 Mechanics of Lake Ice

Table 5.4 The parameterisation of the viscous-plastic three-level lake ice model (Wang et al., 2003, 2006a) I

II

Parameter

Value

Comment

Air–ice drag coefficient

Ca = 1.8 ·

10–3

Surface wind

Ice–water drag coefficient

Cw = 3.5 · 10–3

Model tuning

Ice–water Ekman angle

θw = 15°

Shallow water

Compressive strength

P∗

Yield strength P ∗ h, compact ice

Shear strength

P ∗ /e, e = 2

Elliptic yield curve

e-folding scale for opening

C −1 = 0.05

Yield strength ∝ e−C(1−A)

Maximum creep rate

∆0 = 10−10 s–1

Viscous–plastic transition Open water—ice transition

= 10 kPa

III

Demarcation thickness

h 0 = 10 cm

IV

Spatial grid size

∆x = ∆y = 1.852 km

Much more than floe size

Time step

∆t = 30 min

Stability requirement

Smoothing

Diffusion

Harmonic and bi-harmonic

dynamics was performed by Wang et al. (2006a) for this lake. The model parameters are shown in Table 5.4 based on field data and model experiments in North-European lakes and brackish basins. The model was based on the model for the Gulf of Riga, Baltic Sea (Wang et al., 2003) using the Hibler (1979) viscous-plastic rheology and three-level ice thickness distribution. The area and mean depth of the Gulf of Riga are 17 900 km2 and 23 m. In the Peipsi model, the parameterization was as in the Gulf of Riga model except that the compressive strength of unit ice thickness was decreased from 30 to 10 kPa in the tuning. The free drift wind factor was 2.6%, supported by observations. The transition between viscous and plastic regimes must be very low to avoid production of unrealistic creep of ice; here the creep criterion was taken as ∆0 = 10−10 s–1 , much lower than strain-rates in typical deformation events. The parameter groups are I–atmospheric and oceanic drag parameters, II– rheology parameters (viscous-plastic rheology of Hibler, 1979), III—ice state redistribution parameters, and IV– numerical design parameters. The Lake Peipsi ice dynamics model (Wang et al., 2006a) was validated by comparing the model outcome with a series of MODIS images acquired by the Aqua/Terra satellite. The first image shows that there was an open lead in the northern part of the lake on March 15. The ice then drifted northeast gradually closing the lead and along that more open water appeared in the south and west. Another lead opened in the middle of the lake on March 18 before the ice drifted northwest. In the model simulation, the initial ice thickness and compactness were taken as 40 cm and 99%, respectively, for the ice areas. The winds were taken from the NCAR reanalysis data at 10 m altitude. Northern winds were dominant before March 17 and southern winds thereafter. The highest wind speed was less than 7 m s–1 . The model output and validation are given in Fig. (5.22) showing good agreement. In the first two days,

5.5 Drift Ice in Lakes

199

Fig. 5.22 Day-to-day variation of ice compactness in Lake Peipsi/Chudskoe in March 14–19, 2002. a Result of ice dynamics model simulation. b Validation by daily MODIS images (Wang et al., 2006a)

200

5 Mechanics of Lake Ice

the ice cover was immobile forced by the northern wind, but a lead appeared in the middle part of the lake on March 18. The simulation for the following day shows the ice cover opening towards the south. The openings clearly resulted from wind forcing and the geometry of the lakeshore. When the mechanical forcing reached the yield strength of the ice, the ice cover changed from stable to unstable regime. Lake Peipsi model can be compared with Lake Erie model of Wang et al. (2010). Larger air and water drag coefficients were used in the Lake Erie model than in Lake Peipsi model. The compressive strength of ice of unit thickness was equal to 25 kPa in Lake Erie case, also higher than in Peipsi. The ice was quite thin, less than 10 cm, and the standard strength may result in a too stiff ice cover. Validation of the Lake Erie model was made against ice compactness and ice velocity histograms. There was in general a reasonable agreement. The whole ice season was simulated, and at times the ice velocities were large, 20–60 cm s–1 suggesting that the drift of the ice has an important role in the seasonal ice climatology of the lake.

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Mellor, M. (1986). Mechanical behavior of sea ice. In N. Untersteiner (Ed.), The geophysics of sea ice (pp. 165–281). Plenum Press. Mesher, D. E., Proskin, S. A., & Madsen, E. (2008). Ice road assessment, modeling and management. Proceedings of the Transportation Association of Canada Annual Meeting, 1–20. Metge, M. (1976). Thermal cracks in lake ice. PhD thesis, Queen’s University, Kingston, Ontario. Michel, B. (1978b). Mechanical model of creep of polycrystalline ice. Canadian Geotechnical Journal, 15(2), 155–170. Michel, B. (1978a). Ice mechanics (499 p). Laval University Press. Nevel D. E., & Assur A. (1968). Crowds on ice. CRREL Techn (Report 204). Cold Regions Research and Engineering Laboratory, Hanover, N.H. Nevel, D. E. (1970). Moving loads on a floating ice sheet (CRREL Research report 261). Cold Regions Research and Engineering Laboratory, Hanover, N.H. Ovsienko, S. (1976). Numerical modeling of the drift of ice. Izvestiya, Atmospheric and Oceanic Physics, 12(11), 1201–1206. Palmer, A., & Croasdale, K. (2012). Arctic offshore engineering (p. 327). World Scientific Publishing. Parmerter, R. R. (1975). A model of simple rafting in sea ice. Journal of Geophysical Research, 80(15), 1948–1952. Paterson, W. (1999). Physics of glaciers (3rd ed.) (p. 481). Butterworth-Heinemann. Pritchard, R. S. (1975). An elastic-plastic constitutive law for sea ice. Journal of Applied Mechanics, 42E, 379–384. Pritchard, R. S. (1980). Simulations of nearshore winter ice dynamics in the Beaufort Sea. In R. S. Pritchard (Ed.), Proceedings of ICSI/AIDJEX Symposium on Sea Ice Processes and Models (pp. 49–61). University of Washington, Seattle. Ramseier, R. O. (1971). Mechanical properties of snow ice. Proceedings of POAC, 1, 192–210. Rothrock, D. A. (1986). Ice thickness distribution—measurement and theory. In N. Untersteiner (Ed.), The geophysics of sea ice (pp. 551–575). Plenum Press. Sanderson, T. J. O. (1988). Ice mechanics. Risks to offshore structures (p. 254). Graham & Trotman. Schulson, E. M., & Duval, P. (2009). Creep and fracture of ice (p. 401). Cambridge University Press. Semovski, S. V., Mogilev, N. Y., & Sherstyankin, P. P. (2000). Lake Baikal ice: Analysis of AVHRR imagery and simulation of under-ice phytoplankton bloom. Journal of Marine Systems, 27, 117– 130. Sergeev, B. N. (1929). Creation of winter crossing of wagons on the ice and behavior of a layer of ice under load. Trudy Nauchno-Tekhnicheskogo Komiteta Narodnogo Komissariata Putei Soobshcheniya, 84, 5–35. Shen, H. H., Leppäranta, M., & Hibler, W. D., III. (1986). On applying granular flow theory to a deforming broken ice field. Acta Mechanica, 63, 143–160. Shen, H. T., Chen, Y. C., Wake, A., & Crissman, R. D. (1993). Lagrangian discrete parcel simulation of two-dimensional river ice dynamics. International Journal of Offshore and Polar Engineering, 3(4), 328–332. Shirasawa, K., Leppäranta, M., Kawamura, T., Ishikawa, M., & Takatsuka, T. (2006). Measurements and modelling of the water—ice heat flux in natural waters. In Proceedings of the 18th IAHR International Symposium on Ice (pp. 85–91). Hokkaido University, Sapporo, Japan. Takács, K., Kern, Z., & Pásztor, L. (2018). Long-term ice phenology records from eastern–central Europe. Earth System Science Data, 10, 391–404. Thorndike, A. S., Rothrock, D. A., Maykut, G. A., & Colony, R. (1975). The thickness distribution of sea ice. Journal of Geophysical Research, 80, 4501–4513. Timco, G., & Weeks, W. F. (2010). A review of the engineering properties of sea ice. Cold Regions Science and Technology, 60(2), 107–129. Vincent, C., Descloitres, M., Garambois, S., Legchenko, A., Guyard, H., & Gilbert, A. (2012). Detection of a subglacial lake in Glacier de Tête Rousse (Mont Blanc area). Journal of Glaciology, 58(211). https://doi.org/10.3189/2012JoG11J179

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Wake, A., & Rumer, R. R. (1983). Great lakes ice dynamics simulation. Journal of Waterway, Port, Coastal and Ocean Engineering, 109, 86–102. Wang, K., Leppäranta, M., & Kouts, T. (2003). A model for sea ice dynamics in the Gulf of Riga. Proceedings of Estonian Academy of Sciences. Engineering, 9(2), 107–125. Wang, K., Leppäranta, M., & Reinart, A. (2006a). Modeling ice dynamics in Lake Peipsi. Verh. Internat. Verein. Limnol. 29, 1443–1446. Wang, K., Leppäranta, M., & Kouts, T. (2006b). A study of sea ice dynamic events in a small bay. Cold Regions Science and Technology, 45, 83–94. Wang, J., Bai, X., Hu, H., Clites, A., Holton, M., & Lofgren, B. (2012). Temporal and spatial variability of Great Lakes ice cover, 1973–2010. Journal of Climate, 25, 1318–1329. Wang, J., Hu, H., & Bai, X. (2010). Modeling Lake Erie ice dynamics: Process studies. In Proceedings of the 20th IAHR International Symposium on Ice. University of Helsinki, Finland. Weeks, W.F. (1998). Growth conditions and structure and properties of sea ice. In M. Leppäranta (Ed.), The physics of ice-covered seas (Vol. 1, pp. 25–104). Helsinki University Press, Helsinki. (Available at https://helda.helsinki.fi/handle/10138/39285). Wyman, M. (1950). Deflections of an infinite plate. Canadian Journal of Research, A28, 293–302. Yang, Y., Cheng, B., Kourzeneva, E., Semmler, T., Rontu, L., Leppäranta, M., Shirasawa, K., & Li, Z. (2013). Modelling experiments on air-snow-ice interactions over an Arctic lake. Boreal Environment Research, 18, 341–358. Zubov, N. N. (1942). Basics devices roads ice cover. Gidrometeoizdat, 74 p.

Chapter 6

Proglacial Lakes

Epiglacial lake in New Zealand Southern Alps. Glacial margins can sometimes be easily reached. As such glaciers and epiglacial lakes are beautiful and exciting nature objects, and the exposed land areas at retarding glaciers provide a real-time view into glacial processes creating a new landscape. Here tourists are in a boat excursion in an epiglacial lake in the New Zealand Southern Alps. The edge of this retarding glacier shows up dark due to impurities sedimented from the mountains. Photograph by the author.

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_6

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6.1 Ice Sheets and Glaciers Ice sheets and glaciers—or land ice in general—form in cold regions during long periods from annual snow accumulation exceeding ablation (e.g., Paterson, 1999). Ice sheets are large continental ice masses, also called continental glaciers, with surface area more than 50,000 km2 . In the present Earth only the Antarctic (14 million km2 ) and Greenland (2.2 million km2 ) are covered by ice sheets. Smaller glacial accumulations on land are counted as glaciers. The tongues of ice sheets and glaciers extending over the ocean are called ice shelves. The largest glacier is the Southern Patagonian ice field in Argentina and Chile, covering an area of 13,000 km2 (Rignot et al., 2003), which is 0.6% of the area of Greenland ice sheet. In Europe the largest glaciers are Austfonna, Svalbard (8500 km2 ), and Vatnajökull, Iceland (8100 km2 ) (Fig. 6.1). Glaciers are found at all altitudes in high polar regions but in lower latitudes only in mountains, down to the equator in the Andes in Ecuador and in Kilimanjaro, Tanzania. At the Arctic Circle in northern Europe the glaciation level is at about 1 km altitude, while tropical glaciers are found above 4–5 km altitudes. Our interest in glaciers stems from their acting as freshwater storages and a potential peril of alpine and coastal habitations, as well as from variations of their areal

Fig. 6.1 Landscape photograph showing Vatnajökull, Iceland in summer. This is the largest glacier after Greenland in the Northern Hemisphere. Photograph by the author

6.1 Ice Sheets and Glaciers

207

extent and volume in response to climate changes. Also, the climate history records of glaciers have become extremely valuable in understanding the Earth climate system. The ice sheets of Antarctica and Greenland play a major role in the global climate by regulating the sea level and modifying the atmospheric general circulation. The influence of other glaciers is limited to the local climatic and hydrological conditions. Ice sheets and glaciers are part of the water cycle. They accumulate from snowfall and evolve further from transformation of seasonal snow into firn and ice and from slow flow and deformation. Ablation takes place as sublimation/evaporation, runoff, and calving of icebergs. The equilibrium line separates the accumulation zone from the ablation zone (Fig. 6.2). Glacier impurities are due to enclosure of atmospheric gases and particles at the upper surface, rock particles at the bottom and side boundaries, and meteorites. The ablation zone is located at lower, warmer elevations but not exclusively so. Snow drift and sublimation can create spots of net ablation within the accumulation zone. The surface of these spots is bare ice, as a distinction to the normal surface covered by seasonal snow, and because of their visual appearance, these ablation spots are called blue ice. The local mass balance determines the evolution of the surface topography of a land ice mass and is expressed as dm = P −E + R+Y dt

(6.1)

where m = m(x, y; t) is the ice mass per unit area, P is precipitation, E is evaporation, R is the net runoff, and Y is the net transport due to snowdrift. The dimension of this equation is taken as the rate of change of water equivalent. The annual surface mass balance is positive in the accumulation zone and negative in the ablation zone. At an equilibrium, ice flows from the accumulation zone to the ablation zone to balance the budget and keep the surface topography invariant. Lakes Land ice environments are very cold but several kinds of proglacial lakes1 are connected to them (e.g., Hodgson, 2012; Vincent et al., 2008). These lakes can be seasonal or perennial, and their common characteristic is that their water balance is dominated by glacial meltwater that is available in the ablation zones, in blue ice patches, and at the bottom of ice sheets. Epiglacial lakes are located at the edge of ice sheets and glaciers, supraglacial lakes are found at their top surface, and subglacial lakes exist below them. Integrated over an ice massif, when ∫ A R dA > 0 there is a net liquid water outflow as a potential source for an epiglacial lake (Fig. 6.3; e.g., Menzies, 1995; Vincent et al., 2008; Hodgson, 2012). Within a glacial field in from the edge, epiglacial lakes are found on the surface and at the foot of nunataqs, exposed elements of mountains or ridges from the ice field. Nunataqs have partly bare surface which absorbs solar 1

Proglacial lakes are located at existing glaciers, while glacial lakes refer to the lakes with basins formed by glacial erosion.

208

6 Proglacial Lakes

Fig. 6.2 An alpine glacier in Ötztal, Austrian Tirol in late summer shown from the accumulation area on top down to the outflow stream. Photograph by the author

radiation well, and meltwater is produced from small local glaciers or seasonal snow patches for small ponds (Leppäranta et al., 2013a, 2020). Epiglacial lakes are an important habitat of life in the polar world (Hawes et al., 2008; Kaup, 1994). There are perennial epiglacial ice-covered lakes which are deep enough so that wintertime ice growth does not reach the lake bottom. For example, Lake Untersee is the largest of them in the Dronning Maud Land. Its surface area is 11.4 km2 , maximum depth is 169 m, and it has a permanent 3 m thick ice cover (e.g., Wand et al., 1997). In the same region, there are very shallow seasonal nunataq ponds, with open surface in warm periods (Leppäranta et al., 2020). Epishelf lakes are formed in marine embayments or fjords, dammed by advancing glaciers and ice shelves (Davies et al., 2017). The inflow is meltwater from glaciers and seasonal snow. In some cases, a hydraulic connection persists to the sea under the ice shelf or glacier, sometimes more than 100 m below sea level. Then the epishelf lake becomes meromictic with a saline lower layer.

6.1 Ice Sheets and Glaciers

209

Fig. 6.3 An epiglacial lake at the Russian Novolazarevskaya Station, eastern Dronning Maud Land (70°47' S 11°49' E) in austral summer. Photograph by the author

Due to the low albedo compared with snow, blue ice spots absorb much solar heat resulting in melting in the glacial surface layer, and if the runoff is weak, the meltwater largely stays in place and forms a supraglacial lake (Fig. 6.4). These lakes have very clear water and are extremely low in biota (Keskitalo et al., 2013; Leppäranta et al., 2020) representing one type of extreme habitat. They show high sensitivity to atmospheric conditions and thereby assist detection of regional climate change (e.g., Leppäranta et al., 2013b, 2016; Winther et al., 1996). The most exotic proglacial lakes belong to the category of subglacial lakes. They form from glacial meltwater at the base of land ice due to geothermal heat flux and high pressure of the thick ice. A lake research programme was completed in summers 2002–2015 in western Dronning Maud Land including several supraglacial and epiglacial lakes (Leppäranta et al., 2020. A summary of six lakes is shown in Table 6.1. Supraglacial lakes had low conductivity, almost like distilled water, and nunataq ponds had high conductivity and high pH. There was primary production in all lakes, but it was limited by the phosphorus concentration. Cryoconite holes (MacDonell & Fitzimons, 2008) are tiny, cylindrical supraglacial liquid water pockets in the glacial surface layer (Fig. 6.5). Their diameter is typically 20–30 cm and their depth can be of the order of 1 m. These holes are initiated by windblown material including mineral sediments and dry microbial mats that absorb solar heat to melt their way down into the ice. The subsequent growth of microbial mats can enhance this process and form a self-maintaining ecosystem in

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Fig. 6.4 Supraglacial lake Suvivesi at Basen nunataq is used for the household water source in the Finnish station Aboa in the western Dronning Maud Land (73°03' S 13°29' W). Water is pumped from the lake water body to the Sisu NA-110 bandwagon. Photograph by the author

these holes. They have been proposed as one of the refugia for the eukaryotic biota in the controversial Snowball Earth hypothesis, when the Earth is thought to have undergone extreme freezing-drying events between 750 and 580 million years ago. Proglacial lakes are found at polar ice sheets and as well at mountain glaciers in lower latitudes (Vincent & Laybourn-Parry, 2008). All these lake types exist in Antarctica, where the continent and subantarctic islands form one of the most diverse lake districts on Earth. A debate has been ongoing on the origin of life in the Antarctica (e.g., Gibson et al., 2006), whether the biota is primarily postglacial, or vicariant and have survived glacial advances in lacustrine refuges. The latter case would allow for a greater degree of endemism, and it could be possible that Antarctic lakes contain species that are relicts of Gondwana. There is evidence of zooplankton that has survived an entire glacial cycle, and there is also evidence that species have dispersed to the Antarctic in the present interglacial from the subantarctic islands and from other higher latitude continents of the southern hemisphere.

0.11

0.035 1.0

0.35

1.5

0.75

1.0

2.0

Depth m

9.2

9.2

8.7

6.4

6.5

5.8

pH

103.7

88.0

24.5

26.7

7.8

3.2

EC µS cm–1

159

87

75

87

85

94

O2 %

0.066

0.047

0.002

0.048

0.066

0.010

CDOM a420nm m–1

5

8

8

163

9

44

TP µg L–1

283

156

73

287

38

98

TN µg L–1

For six lakes are shown: type, area, depth, pH, electrical conductivity (EC) at 25 °C, oxygen saturation (O2 ), coloured dissolved organic matter (CDOM) in beam absorption coefficient a, total phosphorus (TP), and total nitrogen (TN) Type: S = supraglacial, E = epiglacial, N = nunataq pond

N

Fossilryggen pond

0.47

E

N

Ringpond

Basen pond

1.5

E

Penaali

361

13.5

S

S

Lake Suvivesi

Area 104 m2

Plogen lake

Type

Lake

Table 6.1 Surface lake study in three summer seasons in 2003–2015, western Dronning Maud Land (Leppäranta et al., 2020)

6.1 Ice Sheets and Glaciers 211

212

6 Proglacial Lakes

Fig. 6.5 A cryoconite hole in the western Dronning Maud Land, initiated by windblown sediment from the Basen nunataq. The vertical meter stick is 2 cm wide with 10 cm long white- and red sections. Photograph by the author

6.2 Supraglacial Lakes 6.2.1 Structure of Supraglacial Lakes Supraglacial lakes are found on the Antarctic and Greenland ice sheets and in many mountain glaciers. Lake water masses may transport latent heat deeper into the ice by convection across the lake water body and by leakage of lake water into fractures (Chikita et al., 1999; Hoffman et al., 2011; Joughin et al., 2008). Lake outflows may form moulins, which extend through the ice transporting heat and reducing the sliding friction at the base. Supraglacial lakes in mountain glaciers possess potential to create floods in the nearby environment (Bajracharya & Mool, 2009). Supraglacial lakes may be seasonal with a recurring annual cycle or they can be interannual growing summer after summer. Supraglacial lakes form in the surface layer of blue ice spots. The necessary condition for the existence of a bare, blue ice surface is negative local mass balance that is possible due to snow drifting and sublimation. Albedo is low for a bare ice (0.4– 0.6) or open water (< 0.1) surface compared to snow cover (around 0.9). Therefore,

6.2 Supraglacial Lakes

213

supraglacial lakes absorb solar energy 5–10 times more than the neighbouring snowcovered ice. The first-order balance between net terrestrial radiation and net solar radiation gives (1 − εa )σ T 4 ∼ (1 − α)Q s

(6.2)

Taking solar radiation Q s = 500 W m–2 and atmospheric emissivity εa = 0.8, we have T ∼ 180 K, 270 K, or 315 K for α = 0.9, 0.5 or 0.1, respectively, illustrating the high sensitivity of surface temperature to albedo. Thus, without extra heat to add to solar radiation, it is difficult to melt snow-covered surface layer of land ice, as can be observed in the Antarctic ice sheet. Solar heating is distributed in the surface layer across the optical depth, which for clean blue ice is 1–2 m (see Sect. 3.4.2). Since conduction of heat in the ice is slow, the solar heat gain is trapped below the blue ice surface and melting of ice may start there. This is one kind of a greenhouse phenomenon. The melt will form a liquid water layer and subsequently grow upward and downward to form a finite water body. Melting has been observed to begin below the surface in the Dronning Maud Land (Leppäranta et al., 2016). The depth of the supraglacial lakes depends on the solar radiation level, and the state of the surface—liquid or solid—depends on the sign of the surface heat balance (Leppäranta et al., 2016). At the surface there are strong losses due to terrestrial radiation balance and evaporation or sublimation. Therefore, the water body may grow in the presence of an ice cover. Figure 6.4 shows a situation when the ice thickness must have been more than 30 cm to carry the five-ton bandwagon in the water pumping operation. The vertical profile is determined by the solar radiation and surface heat balance. The core of the lake consists of a liquid water layer, the surface may be liquid or solid, and the bottom layer is a thick ice–water mixture with ice fraction increasing with depth. Thus, the structure can be presented by three layers: (1) Surface ice lid, eventual; (2) Core; and (3) Ice–water bottom layer. The bottom layer may have a substructure with hard and soft sub-layers. Since the lake bottom is not well-defined, it is convenient to define lake depth based on porosity ν: the lake body extends into the bottom layer to the depth where the porosity satisfies the condition ν = ν(z, t) ≤ ν∗ , where ν∗ is a critical porosity. The porosity level where ice loses its integrity, ν∗ ∼ 1/2 is a natural scale of the critical porosity. The lateral size of supraglacial lakes ranges from small ponds to medium-size shallow lakes of several square kilometres in area. In the Dronning Maud Land, Antarctica there are recurrent summer lakes with the depth scale of 1 m (Leppäranta et al., 2020; Winther et al., 1996). In Greenland they are perennial, much more developed, and have a major impact on the ice mass balance (Hoffman et al., 2011; Liang et al., 2012). Since the main water source is glacial ice, the water quality depends on the geochemistry of the glacial surface layer. The annual life cycle of seasonal supraglacial lakes is a stationary process. It is a reverse process to the freezing cycle of seasonally ice-covered lakes. During the growth of a supraglacial lake, the ice layer on top becomes thinner and albedo decreases that provides positive feedback to the lake growth. In fall, supraglacial

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lakes start to freeze down from the upper surface, as lakes usually do (see Sect. 4.3), and up from the lake bottom. Summer growth of the lake is forced by the solar radiation, which is primarily dependent on the cloudiness and latitude, while winter close-up is primarily dependent on the air temperature. If a supraglacial lake turns perennial, the lake will grow year–by–year and finally may transport latent heat deeper into the ice sheet by convection across the lake water body and penetration of lake water into fractures (Chikita et al., 1999; Hoffman et al., 2011; Joughin et al., 2008). The pressure of a large volume of water may crack the ice sheet, and then the lake water would disappear into the interior of the ice sheet, slowly or catastrophically into moulins. This has been observed in the Greenland ice sheet (Hoffman et al., 2011; Joughin et al., 2008). Meltwater convection is therefore a very important mechanism in the warming and deterioration of ice sheets and glaciers. In a warming climate, ice growth would decrease in winter, and after a certain limit a seasonal lake would turn perennial that would turn retarding of an ice sheet or glacier to a new stage.

6.2.2 Thermodynamics of Supraglacial Lakes The depth scale of supraglacial √ lakes is governed by the light penetration depth κi ≈ 1 m–1 and diffusion length Dτ , D = 0.095 m2 d–1 (Eq. where τ is the time  4.27), √  −1 scale. The active layer can thus be taken as D = 2 κi + Dτ . In winter this layer is frozen,  and the heat content with respect to the 0 °C liquid water reference is  − T f − L f D, where T  is the vertically averaged temperature. The H = ρi ci T heat content changes as (see Sect. 4.1) dH ∂ T = Q 0 + Q sδ + Q b , Q b = −ki dt ∂z z=b

(6.3)

where Q sδ is the solar radiation penetrating into ice, and Q b is the heat flux from the bottom. The heat gain in spring and summer is used to increase the temperature and eventual melting. The main part of this heat gain is due to the penetration of solar radiation into the ice. The surface absorption of solar radiation is to a large degree compensated by heat loss due to terrestrial radiation balance and sublimation, and the bottom heat flux is small and stable, Q b ≈ −2 W m–2 , corresponding to a temperature gradient of 1 °C m–1 in the glacial surface layer. min at the end The temperature of the glacial surface layer reaches its minimum T of winter, the time is then set as t = t0 . A supraglacial lake will form in the following  → T f , or summer if T   min − T f D ∆H = H(t) − H(t0 ) > ρi ci T

(6.4)

6.2 Supraglacial Lakes

215

For example, for a mean Q sδ of 50 W m–2 and D ≈ 10 m, the heat accumulation in min > 30 °C. four months is ∆U = 520 MJ, and the inequality (6.4) is satisfied, if T Example 6.1 In the Aboa station (73°02.5' S 13°44' W) on Basen nunataq, western Dronning Maud Land, the winter temperature of the surface layer of the ice sheet is about –20 °C (Leppäranta et al., 2016). According to observations in four summers, a supraglacial lake formed in summer in a blue ice spot next to the nunataq (altitude 200 m). The formation began in the first week of December at the depth of 0.5– 1 m. During the summer, December–January, the mean solar heating of the surface layer interior was 70 W m–2 . The accumulated heat flux could melt a 1.2 m thick layer of ice, as was approximately observed. The surface heat balance was negative and therefore the lake was covered by a thin ice cover. 200 km further south, in the Svea Research Station (74°35' S, 11°13' W, altitude 1245 m), the annual mean air temperature is about –20 °C (Isaksson & Karlén, 1994), and assuming that the amplitude of the annual cycle is similar there and at Aboa, the minimum surface layer temperature would lie between –30 °C and –25 °C in Svea, still high enough to initiate a supraglacial lake that indeed has been observed. Heat flow in the glacial surface layer follows the heat conduction law with solar radiation as a source term (as discussed for lake ice in Sect. 4.1):

 ∂T ∂ ∂ ki + Q sδ κeκi z (ρi ci T ) = ∂t ∂z ∂z

(6.5)

where the ice surface is at z √ = 0 and the vertical axis is positive upward. The length scale of diffusion in ice is Dt = 0.3 m for t = 1 d, much less than the light penetration depth, and in water the molecular diffusion length scale is one-third of that in the ice. In the spring warming of solid blue ice, we have ∆T = T0 − TH ∼ 10 ◦ C over the depth scale of the thermally active layer of L ∼ 10 m, and the penetration of solar radiation into ice is Q sδ ∼ 50 − 100 W m–2 . The ice temperature may change due to diffusion by D∆T /L 2 ∼ 0.01 °C d–1 , while the vertical mean radiative heating across the 10-m surface layer is 0.2–0.5 °C d–1 . The radiative heating overcomes diffusion down to the depth of about 3.5κi−1 or 3–4 m. The diffusive length scale is 1 m in ice (0.35 m in water) in 10 days. Thus, solar radiation rapidly warms the lake with little diffusive losses. Also surface losses are small in summer compared to the heating period of the lake body. There is positive feedback in that diffusion is slower in water than in ice until convection becomes significant. Once the melting has begun, solar radiation melts the ice vertically by 1–3 cm d–1 . To examine the birth of a supraglacial lake in more detail, the boundary conditions can be taken at the surface as z = 0 : T = T0 ; ki

∂T = Q0 ∂z

(6.6)

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6 Proglacial Lakes

The quasi-steady-state solution of Eqs. (6.5, 6.6) gives the temperature profile in the surface layer and can be applied stepwise for the temperature evolution. The heat flux from the deeper ice is small and ignored here. During the warm-up of ice, the net heating is positive, Q 0 + Q sδ > 0, and the solution reads T (z) = T0 +

 Q 0 + Q sδ Q sδ  1 − eκi z , z ≤ 0 z+ ki ki κi

(6.7)

This consists of a linear profile from the net heating plus a modification by the depth-distributed solar heating. The scale of validity of this solution is up to the length of the steady state conditions; the longer the boundary conditions hold, the deeper the solution goes. For κi → ∞, sunlight is absorbed at the surface and a linear temperature profile is obtained; and for κi → 0, sunlight goes through without any influence. For a finite attenuation coefficient, if Q 0 ≥ 0, the temperature maximum is at the surface. If Q 0 < 0, the temperature increases down until a maximum at the depth h max =

Q sδ 1 log κi Q sδ + Q 0

(6.8)

thereafter decreasing to a linear asymptote. E.g., if Q sδ = 50 W m–2 , Q 0 = −25 W m–2 , and κi = 1 m–1 , we have h max = 0.7 m and T (h max ) − T0 = 3.7 °C. Thus, when the melting stage is achieved inside the surface layer, the surface temperature is still quite low. According to observations in the western Dronning Maud Land, the surface temperature was about –5 °C at the time the lake was born (Leppäranta et al., 2013b). Next, consider the depth scale of the lake in more detail. The distribution of solar energy with respect to depth depends on the transparency of the ice. The solar power absorption is vertically distributed as Q sδ κi eκi z . Furthermore, the evolution of the porosity profile ν = ν(z; t) can be easily obtained from Eq. (6.5) if heat diffusion is ignored. Then, with the lake water body defined by ν(z) > ν∗ , the lake depth scale becomes

 Q sδ κi 1 t (6.9) H = log κi ν∗ ρi L f For κi ∼ 1 m–1 , Q sδ ∼ 100 W m–2 , t ∼ 60 days, and ν∗ ∼ 0.5, we have H ∼ 1.22 m. At ν(z) > ν∗ , , the strength of the ice becomes low and disappears, and the ice matrix changes into slush, as has been observed (Leppäranta et al., 2013b). Thus, the lake depth scales with the light penetration depth. Since light attenuation in ice and water is wavelength-dependent (Sect. 3.4), for a more accurate solution of the light transfer the spectral distribution should be accounted for. However, for simplicity, in the scaling analysis we can take a constant, representative attenuation coefficient across the PAR band and depth.

6.2 Supraglacial Lakes

217

When the lake has formed and the surface heat balance is negative, the lake has an ice cover. Much of the incoming solar radiation penetrates to the lake body, and the surface absorption does not compensate for the losses due to longwave radiation and turbulent transfer. The thin ice cover corresponds to the ‘cold skin phenomenon’ observed in lower latitude lake and sea surfaces in daytime calm summer conditions. This is clear in the early stages of supraglacial lake development when the ice formation starts well beneath the surface. The surface ice layer also becomes thinner by sublimation, which is significant in dry climate regions. Diffusion and possibly convection keep the water temperature at the freezing point. The continuity of heat flux through the surface lid tells that at equilibrium, the surface temperature T0 and surface ice layer thickness h are obtained from −Q 0 (T0 ) = ki

T f − T0 = Q sδ e−κi h h

(6.10)

If Q 0 < 0, a physically realizable solution (T0 ≤ T f , h > 0) does exist. Otherwise, h = 0 and the surface heating is delivered to the lake water to melt more ice (Eq. 6.3). Example 6.2 Take the linear surface heat flux Q 0 (T0 ) = K 0 + K 1 (Ta − T0 ). Then the first and second equation in Eqs. (6.10) give ∆T =

K 0 /K 1 + Ta − T f eκi h Q sδ =− , ki h ki ∆T 1+ K h 1

where the notation ∆T = T0 − T f is for the temperature below the freezing point. To have an ice cover, we must have (i) ∆T < 0 and therefore Ta − T f < −K 0 /K 1 , and (ii) khi Q∆Tsδ < −1 to have a real solution for h. E.g., K 0 /K 1 + Ta − T f ∼ −2 ◦ C and Q sδ ∼ 50 W m–2 give ∆T ∼ −1 ◦ C and h ∼ 5 cm. Recurrent supraglacial lakes close up in winter. The net surface heat flux is negative, and the air temperature is below the freezing point. Lakes start to freeze down from the top surface, as lakes usually do (see Chap. 4), but also a minor growth occurs at the bottom. The surface heat loss is of the order of 50 W m–2 , while the bottom heat loss is around 2 W m–2 . These losses correspond to ice growth rates of 1.4 cm d–1 or 0.06 cm d–1 , respectively. The growth of the top ice layer is obtained from Barnes’ and Zubov’s formula (see Eq. 4.40): h(t) =

/ [h(0) + h a ]2 + a 2 Sa (t) − h a

(6.11)

In the snow-free conditions over supraglacial lakes, variations of the model parameters are small, and we can take a ≈ 3.4 cm (°C d)–1/2 and h a ≈ 10 cm (see Sect. 4.3.2).

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6 Proglacial Lakes

Example 6.3 At Aboa station in the western Dronning Maud Land, in the vicinity of supraglacial lakes the air temperature is continuously below the freezing point and the average is –15 °C in February–November. The freezing-degree-days sum to S = 4500 °C d in ten months, and we have h = 2.18 m growth in the top ice layer. The bottom heat loss adds a little to the growth of ice: the heat loss of 2 W m–2 gives in ten months 18 cm ice growth up from the lake bottom. Here the thickness of ice is not very sensitive to air temperature variations: ± 3 °C overall change in the air temperature would change the ice thickness from 2.18 m by less than 20 cm. The exchange of heat with the underlying ice sheet for the ice growth at the bottom is quite stable. The depth of the lake in summer is around 1 m so that it freezes fully in winter. Recurrent supraglacial lakes undergo annual opening–closing cycles. In a changing climate, these cycles would turn toward a permanent solid blue ice layer or a perennial lake. The lake growth and closing scales are shown by Eqs. (6.9) and 6.11), respectively. For no lake formation, the summer heat gain should be less than heat needed the raise the surface layer temperature to the melting point. If a lake forms, the summer melt and winter ice growth become equal when: /

 2ki   ∆H −TaW tW + h a2 − h a = tS ρi L f ρi L f

(6.12)

aW is the winter mean air temperature, tW and t S are the length of winter and where T summer, respectively, tW + t S = 1 year, and ∆H is the net heat flux in summer. The heat taken out to cool the ice compensates for the heat added for warming the ice in the annual heat balance. These factors are small, since the energy needed to raise the temperature of an ice cube from –15 to 0 °C is only 10% of the energy to melt the ice. With “>” or “ h max A

(6.16)

where A is the lake area and h max is the maximum annual ice thickness. For seasonality, h max ∼ 2 m serves as a good estimate for very cold climate conditions. Example 6.6 For Q n ∼ 100 W m–2 , An ∼ 100 km2 and Tn ∼ 100 days, we have 3 V ∼ 0.28 √km . For H > 10 m, the water body is perennial, and the horizontal scale is L = V /H < 170 m. But for H < 1 m, the water body is seasonal, and the horizontal scale is L > 530 m. There are polar lakes with a perennial ice cover, e.g., in McMurdo Dry Valleys in Antarctica. There Lake Vida has the thickest ice cover, about 20 m (Doran et al., 2004; Mosier et al., 2007). To achieve such large thickness is possible when the summer melting remains small. From ice growth Eq. (6.11) we have an equation for annual growth of multiyear ice after winter n + 1 as (setting h a = 0 for simplicity): / h n+1 =

(h n − ∆h)2 + a 2 Sa

(6.17)

where ∆h is summer melt. Taking the summer melt constant, that is reasonable as it is independent of thickness and only depends on summer climate, and noting that a 2 Sa = h 21 = first-year ice thickness, we can solve for the equilibrium thickness h n = h n+1 = h e after many winters as he =

h 21 − (∆h)2 2∆h

(6.18)

Since h 1 ≈ 2 m, we have h e = 20 m for ∆h = 0.1 m. Chemically, epiglacial lakes range from some of the freshest lakes in the world to hypersaline lakes, which do not freeze over, even in winter when their temperature drops below –10 °C or even much lower. Depending on the degree of chemical influence from the sea by atmospheric transport versus interior climate regimes, lakes are often classified as coastal maritime lakes or coastal continental lakes. The

6.3 Epiglacial Lakes

227

nutrient status of Antarctic lakes is typically oligotrophic, with eutrophy usually restricted to those, which are directly influenced by visiting marine mammals, birds or people. Table 6.1 shows characteristics of the water quality in several epiglacial lakes in the western Dronning Maud Land. Epiglacial lakes are an important habitat of life in the polar world (Kaup, 1994). They contain phytoplankton, few zooplankton species but no fish. Mosses are the highest forms of aquatic plant life on the continent, while higher plants are present in lakes on the peri-antarctic and subantarctic islands (e.g., in South Georgia). However, the bulk of the biomass and primary productivity is benthic, consisting of cyanobacteria, diatoms and green algae, often in luxuriant mats several meters thick (Vincent, 2000; Ellis-Evans, 1996). The mats can be grazed by communities of rotifers, ciliates and crustacea, though usually these zooplankton are present at low densities. Benthic microbial mats are less common in saline lakes, but even the most saline ones contain some biota. Other biological habitats include pockets within the lake ice, the under surface of the lake ice, and the micro-stratified chemoclines of meromictic lakes where gradients of conductivity, oxygen, pH, and sulfur provide numerous niches for photosynthetic sulfur bacteria, other anoxic bacteria, and methanogenic Archaea. With many higher organisms unable to survive there, the Antarctic continent could be seen as a microbial world with the lakes being its principal bastion. Epishelf lakes such as Kakapon Bay in the Bunger Hills can exhibit full marine conditions together with a marine flora and fauna including seal populations.

6.3.2 Thermodynamics of Epiglacial Lakes The thermodynamics of epiglacial lakes is governed by the radiation balance, penetration of solar radiation into the ice, inflow of cold water, and heat storage in the bottom soil or rock (Leppäranta et al., 2020). Ponds on nunataqs are shallow and on ground bottom, which collects radiative heat during the summer. In such environments, the water temperature may vary over a wide range (cf. Quesada et al., 1998), and the stratification can be influenced by dissolved substances in addition to temperature. According to Hawes et al. (2011a, 2011b), small ponds, which are very common in ice-free areas, can possess extreme chemical conditions with high pH levels and dissolved oxygen (DO) concentrations. The thermodynamics of epiglacial lakes follows largely the situation in supraglacial lakes except that the bottom boundary condition can be different (Fig. 6.11). The bottom can be soil or rock, and in deep lakes the lower layers have time scales much more than a year. In shallow lakes solar radiation reaches the bottom and from there warms the bottom water. Then the water temperature can be much above the freezing point.

228

6 Proglacial Lakes

Fig. 6.11 Heat balance of supraglacial lakes (left) and nunataq ponds (right). The numbers provide the scales (in W m–2 ) of net solar radiation, net longwave radiation, turbulent fluxes, conductive flux through surface ice and heat flux to bottom. The circled numbers give the total heat gain by the waterbody, the bottom heat gain depends on the bottom quality and pond depth (Leppäranta et al., 2020)

6.4 Subglacial Lakes Subglacial lakes, rivers and wetlands are very old water bodies at the base of land ice (e.g., Priscu & Foreman, 2009; Siegert et al., 2001). There are such very specific physical and microbiological systems at the bottom of the Antarctic ice sheet (Fig. 6.12), not yet well known. The earliest evidence of Antarctic subglacial lakes was from Russian aircraft pilots flying missions over the Antarctic continent. The presence of the lakes was subsequently verified by seismic investigations and airborne radio–echo sounding during the 1960s and 1970s (Masolov et al., 2006). First was found Lake Vostok by Andrey Kapitsa at the Soviet/Russian Antarctic station Vostok with seismic soundings. The largest subglacial is Lake Vostok (Zotikov, 2006). Its surface lies about 4000 m under the ice sheet. Other lakes have been found in the Dome C and Ridge B regions of eastern Antarctica. More than 150 lakes exist beneath the Antarctic ice sheet, many of which may be connected by large subglacial rivers (Siegert, 2005). Approximately 81% of the detected lakes lie at elevations less than 200 m above the mean sea level, while the majority of the remaining lakes are ‘perched’ at higher elevations. Twothirds of the lakes lie within 50 km of a local ice divide. Recently, subglacial lakes have been found also in Greenland (Palmer et al., 2013) and European Alps (Vincent et al., 2012). There is evidence that subglacial Lake McGregor existed under the Laurentide ice sheet in North America in the last glacial maximum (Munro-Stasiuk, 2003). Interestingly, the presence and research of subglacial lakes on Earth also helps understanding of possibilities to find liquid water in other planets and their moons.

6.4 Subglacial Lakes

229

Fig. 6.12 Schematic picture of a subglacial lake

In particular, a subglacial ocean has been found in the moon Europa of Jupiter (Greenberg, 2005).

6.4.1 Formation and Diversity Subglacial lake environments rest at the intersection of continental ice sheets and the underlying lithosphere. This unique location sets the stage for generating a spectrum of subglacial environments reflective of the complex interplay of ice sheets and the lithosphere. The association of subglacial lakes with local ice divides leads to a fundamental question concerning the evolution of the lake environments whether the location is determined by the ice sheet or by, e.g., basal morphology, geothermal flux, sub-ice aquifers). With the exception of central West Antarctica, where only few lakes exist, only little is known about either the lithospheric character along these catchment boundaries or the history of their migration, given by layering within the ice sheet. There are two factors  that  lead to the formation of subglacial lakes: the presence of geothermal heat flux Q g at the base of the ice sheet and the good insulation capacity of the ice sheet. At the lake surface, the phase equilibrium and the continuity of the heat flow require that T (H ; t) = T f ( p, S)

(6.19a)

230

6 Proglacial Lakes

dH ∂ T = ki ρw L f − Qg dt ∂ z z=−H

(6.19b)

where H is the depth of the lake surface from the top of the ice sheet. Considering a long-term balance, we can assume a linear temperature gradient across the ice sheet. At the steady state, H = constant and the geothermal heat flux is fully conducted through the ice to the atmosphere. Then the left-hand side of Eq. (6.19b) is zero, and we can solve this equation for the depth H:   ki T f − T0 H= Qg

(6.20)

The freezing point decreases with pressure as T f = p × 7.53 · 10−3 °C bar–1 (Eq. 2.8b). Beneath 4 km of ice, the pressure is about 350 bar, and thus the freezing point of fresh water is –2.7 °C. If Q g ∼ 50 mW m–2 and T f − T0 ∼ 50 °C, we have H ∼ 2 km taking the standard conductivity ki = 2.1 W m–1 °C–1 . However, the actual conductivity is higher due to the low temperature and high pressure, so that 4 km is a possible solution. Lake Vostok is weakly saline, around 1‰ (Siegert et al., 2001), that influences the freezing point by less than 0.1 °C. For a physically realizable subglacial lake, the solution for H must be less than the thickness of land ice in the considered location. The time scale of thermal diffusion is L 2 /D (see Sect. 4.3) where L is the length scale and D ∼ 0.1 m2 day–1 is the diffusion coefficient of ice. Thus, for L = 1 km the time scale is 30,000 years. Antarctic subglacial lakes have been categorized into three main types: (1) Lakes in subglacial basins in the ice-sheet interior; (2) Lakes perched on the flanks of subglacial mountains; and (3) Lakes close to the onset of enhanced ice flow. The bedrock topography of the ice-sheet interior involves large subglacial basins separated by mountain ranges. The lakes in the first category are found mostly in and on the margins of subglacial basins. These lakes can be divided into two subgroups in regards with their location: (a) Sites where the subglacial topography is relatively subdued, often toward the centre of subglacial basins; and (b) Sites where significant topographic depressions exist, often closer to subglacial basin margins, but still near the slow-flowing centre of the Antarctic Ice Sheet. Where the bed topography is very subdued, deep subglacial lakes are unlikely to develop. The subglacial lakes in Antarctica have remained unexplored except by using remote sensing technology and recent very few water samples. There are no bottom sediment samples to tell about the history of the ice sheet. Theoretical models reveal that the subglacial environment may hold about 10% of all surface lake water on Earth, enough to cover the whole continent with a uniform water layer with a thickness of about 1 m (see Priscu & Foreman, 2009). These models further reveal that the average water residence time in the subglacial zone is 1000 years. The biology of subglacial lakes is largely an open question. The evidence to date suggests that communities of microorganisms may exist in these

References

231

lakes. Recently, attention has been focused on the possibility that subglacial environments of Antarctica may harbour microbial ecosystems isolated from the atmosphere for as long as the continent has been glaciated (20–25 million years).

6.4.2 Lake Vostok Lake Vostok (Zotikov, 2006) is the largest known subglacial lake. Its surface area is about 14,000 km2 or about the same size as Lake Ladoga, the largest freshwater lake in Europe. The mean depth of Lake Vostok is 400 m, the maximum depth is about 800 m, and the volume is about 5600 km3 . The water residence time has been estimated to be around 50,000 years. It is the only lake that occupies an entire section of a large subglacial trough. An ice core extracted from above Lake Vostok has revealed a diversity of yeasts, Actinomycetes, algae, and spore-forming bacteria. Some of these organisms have remained viable in the ice for 36,000 years. Since the ice above Lake Vostok is over 500,000 years old, the lake could contain microbial genomes, which have been isolated from the rest of the biological world for that period. Ice cores comprised of water from Lake Vostok frozen to the overlying ice sheet have shown the presence, diversity, and metabolic potential of bacteria within the accreted ice overlying the lake water (see Priscu & Foreman, 2009). Estimates of bacterial abundance in the surface waters range from 150 to 460 cells ml–1 and small subunit rDNA gene sequences show low diversity. The sequence data indicate that bacteria in the surface waters of Lake Vostok are similar to present day organisms. This similarity implies that the seed populations for the lake were incorporated into the glacial ice from the atmosphere and were released into the lake water following the downward transport and subsequent melting from the bottom of the ice sheet. Subglacial lakes present a new paradigm for limnology, and once sampled, will produce exciting information on lakes that have been isolated from the atmosphere for more than 10 million years.

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Doran, P. T., Wharton, R. A., Jr., & Lyons, W. B. (2004). Paleolimnology of the McMurdo Dry Valleys, Antarctica. Journal of Paleolimnology, 10(2), 85–114. Ellis-Evans, J.C. (1996). Microbial diversity and function in Antarctic freshwater ecosystems. Biodiversity and Conservation, 5, 1395-1431. Fedorova, I. V., Savatyugin, L. M., Anisimov, M. A., & Azarova, N. S. (2010). Change of the Schirmacher oasis hydrographic net (East Antarctic, Queen Maud Land) under deglaciation conditions. Ice and Glacier, 3(111), 63–70. Moscow, Nauka. Gibson, J. A. E., Wilmotte, A., Taton, A., Van de Vijver, B., Beyens, L., & Dartnall, H. J. G. (2006). Biogeographic trends in Antarctic lake communities. In D. Bergstrom, A. Huiskes, & P. Convey (Eds.), Trends in Antarctic terrestrial and limnetic ecosystems. Kluwer. Greenberg, R. (2005). Europa – The ocean moon. Search for an alien biosphere (p. 395). Springer Praxis Books, Springer. Hawes, I., Howard-Williams, C., & Fountain, A. G. (2008). Ice-based freshwater ecosystems. In W. F. Vincent & J. Laybourn-Parry (Eds.), Polar lakes and rivers: Limnology of Arctic and Antarctic aquatic ecosystems (pp. 103–118). Oxford University Press. Hawes, I., Safi, K., Sorrel, B., Webster-Brown, J., & Arscott, D. (2011a). Summer–winter transition in Antarctic ponds I: The physical environment. Antarctic Science, 23, 235–242. Hawes, I., Safi, K., Webster-Brown, J., Sorrel, B., & Arscott, D. (2011b). Summer–winter transitions in Antarctic ponds II: Biological responses. Antarctic Science, 23, 243–254. Hodgson, D. A. (2012). Antarctic lakes. In L. Bengtsson, R. W. Herschy, & R. W. Fairbridge (Eds.), Encyclopedia of lakes and reservoirs (pp. 26–31). Springer. Hoffman, M. J., Catania, G. A., Neumann, T. A., Andrews, L. C., & Rumrill, J. A. (2011). Links between acceleration, melting, and supraglacial lake drainage of the western Greenland Ice Sheet. Journal of Geophysical Research, 116, F04035. https://doi.org/10.1029/2010JF001934 Huss, M., Bauder, A., Werder, M., Funk, M., & Hock, R. (2007). Glacier-dammed lake outburst events of Gornersee, Switzerland. Journal of Glaciology, 53(181), 189–200. Isaksson, E., & Karlén, W. (1994). Spatial and temporal patterns in snow accumulation, western Dronning Maud Land, Antarctica. Journal of Glaciology, 40(135), 399–409. Joughin, I., Das, S. B., King, M. A., Smith, B. E., Howat, I. M., & Moon, T. (2008). Seasonal speedup along the western flank of the Greenland ice sheet. Science, 320(5277), 781–783. Kaup, E. (1994). Annual primary production of phytoplankton in Lake Verkhneye, Schirmacher Oasis, Antarctica. Polar Biology, 14, 433–439. Kaup, E., Loopmann, A., Klokov, V., Simonov, I., & Haendel, D. (1988). Limnological investigations in the Untersee Oasis (Queen Maud Land, East Antarctica). In J. Martin (Ed.), Limnological studies in Queen Maud Land (East Antarctica) (pp. 6–14). Valgus. Keskitalo, J., Leppäranta, M., & Arvola, L. (2013). First records on primary producers of epiglacial and supraglacial lakes in the western Dronning Maud Land, Antarctica. Polar Biology, 36(10), 1441–1450. https://doi.org/10.1007/s00300-013-1362-0 Leppäranta, M., Järvinen, O., & Lindgren, E. (2013a). Mass and heat balance of snow patches in Basen nunatak, Dronning Maud Land in summer. Journal of Glaciology, 59(218), 1093–1105. Leppäranta, M., Järvinen, O., & Mattila, O.-P. (2013b). Structure and life cycle of supraglacial lakes in the Dronning Maud Land. Antarctic Science, 25(3), 457–467. Leppäranta, M., Luttinen, A., & Arvola, L. (2020). Physics and geochemistry of lakes in Vestfjella, Dronning Maud Land. Antarctic Science, 32(1), 29–42. Leppäranta, M., & Myrberg, K. (2009). Physical oceanography of the Baltic Sea (p. 378). SpringerPraxis. Leppäranta, M., Lindgren, E., & Arvola, L. (2016). Heat balance of supraglacial lakes in the western Dronning Maud Land. Annals of Glaciology, 57(72). Liang, Y.-L., Colgan, W., Lv, Q., Steffen, K., Abdalati, W., Stroeve, J., Gallaher, D., & Bayou, N. (2012). A decadal investigation of supraglacial lakes in West Greenland using a fully automatic detection and tracking algorithm. Remote Sensing of Environment, 123, 127–138. MacDonell, S., & Fitzimons, S. (2008). The formation and hydrological significance of cryoconite holes. Progress in Physical Geography, 32(6), 595–610.

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Masolov, V. N., Popov, S. V., Lukin, V. V., Sheremetyev, A. N., & Popkov, A. M. (2006). Russian geophysical studies of Lake Vostok, Central East Antarctica. Antarctica (pp. 135–140). Springer. Menzies, J. (ed.) (1995) Modern glacial environments. Processes, dynamics and sediments (p. 621). Butterworth–Heinemann, Oxford, UK. Mosier, A. C., Murray, A. E., & Fritsen, C. H. (2007). Microbiota within the perennial ice cover of Lake Vida, Antarctica. FEMS Microbiology Ecology, 59(2), 274–278. Munro-Stasiuk, M. J. (2003). Subglacial Lake McGregor, south-central Alberta, Canada. Sedimentary Geology, 160, 325–350. Palmer, S. J., Dowdeswell, J. A., Christoffersen, P., Young, D. A., Blankenship, D. D., Greenbaum, J. S., Benham, T., Bamber, J., & Siegert, M. J. (2013). Greenland subglacial lakes detected by radar. Geophysical Research Letters, 40, 6154–6159. https://doi.org/10.1002/2013GL058383 Paterson, W. (1999). Physics of glaciers (3rd ed.) (p. 481). Butterworth-Heinemann. Priscu, J. C., & Foreman, C. M. (2009). Lakes of Antarctica. In G. E. Likens (Ed.), Encyclopedia of inland waters (Vol. 2, pp. 555–566). Elsevier. Priscu, J. C. (Ed.) (1998). Ecosystem dynamics in a polar desert: The McMurdo Dry Valleys Antarctica. Research Series vol. 72. American Geophysical Union. Quesada, A., Goff, L., & Karenz, D. (1998). Effects of natural UV radiation on Antarctic cyanobacterial mats. In Proceedings of the NIPR Symposium on Polar Biology (Vol. 11, pp. 98–111). Raj, K. B. G., Kumar, K. V., Mishra, R., & Muneer, A. M. (2014). Remote sensing based assessment of glacial lake growth on Milam Glacier, Goriganga Basin, Kumaon Himalaya. Journal of Geologial Society of India, 83, 385–392. Rignot, E., Rivera, A., & Casassa, G. (2003). Contribution of the Patagonia icefields of South America to sea level rise. Science, 302, 434–437. Siegert, M.J. (2005). Lakes beneath the ice sheet: The occurrence, analysis, and future exploration of Lake Vostok and other Antarctic subglacial lakes. Annual Review of Earth and Planetary Science, 33, 215–245. Siegert, M., Ellis-Evans, J. C., Tranter, M., Mayer, C., Petit, J.-R., Salamatink, A., & Priscu, J. C. (2001). Physical, chemical and biological processes in Lake Vostok and other Antarctic subglacial lakes. Nature, 414, 603–609. Sokratova, I. N. (2011). Hydrological investigations in the Antarctic oases. Russian Meteorology and Hydrology, 36(3), 207–215. Vincent, W.F. (2000). Cyanobacterial dominance in the polar regions. In: Whitton, B.A., & Potts, M. (eds), The ecology of cyanobacteria (pp. 321–340). Kluwer, Dordrecht. Vincent, W. F., Hobbie, J. E., & Laybourn-Parry, J. (2008). Introduction to the limnology of highlatitude lake and river ecosystems. In W. F. Vincent, & J. Laybourn-Parry (Eds.), Polar lakes and rivers. Limnology of Arctic and Antarctic aquatic ecosystems (pp. 1–23). Oxford University Press. Vincent, C., Descloitres, M., Garambois, S., Legchenko, A., Guyard, H., & Gilbert, A. (2012). Detection of a subglacial lake in Glacier de Tête Rousse (Mont Blanc area). Journal of Glaciology, 58(211). https://doi.org/10.3189/2012JoG11J179 Vincent, W. F., & Laybourn-Parry, J. (Eds.). (2008). Polar lakes and rivers (p. 322). Oxford University Press. Wand, U., Schwarz, G., Bruggemann, E., & Brauer, K. (1997). Evidence for physical and chemical stratification in Lake Untersee (central Dronning Maud Land, East Antarctica). Antarctic Science, 9(1), 43–45. Winther, J.-G., Elvehøy, H., Bøggild, C. E., Sand, K., & Liston, G. (1996). Melting, runoff and the formation of frozen lakes in a mixed snow and blue-ice field in Dronning Maud Land, Antarctica. Journal of Glaciology, 42(141), 271–278. Zotikov, I. A. (2006). The Antarctic subglacial Lake Vostok (p. 150). Glaciology, biology and planetology. Springer Praxis Books, Springer.

Chapter 7

Lake Water Body in the Ice Season

Seasonally freezing lakes cool in the autumn and start to freeze from the shore areas. In large or deep lakes, the freeze-up can take a long time and full ice coverage is not necessarily reached. The ice cover introduces major changes to the lake–atmosphere interaction as compared with the open water season. Melting of ice begins from the near-shore areas and progresses with patchy surface characteristics. Photograph from ice decay in May 2018 in Lake Äkäslompolo, northern Finland. Photograph by the author.

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_7

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7 Lake Water Body in the Ice Season

7.1 Ice Formation Ice conditions in a lake develop in close interaction with the water body. Lake water is used for ice growth, and melt water is returned to the water body together with the winter’s precipitation. During the ice season, exchange of heat, momentum, and gases and other substances between atmosphere and lake water takes place through the ice cover. The water body and bottom sediment act as a storage of heat for the ice cover and modify the evolution of the ice season. In most freezing lakes, the ice cover is seasonal. The ice cover buffers the surface water temperature to the freezing point and largely decouples the water body from the atmosphere. Slow heat leakage takes place from the water body by molecular conduction through the ice. Freezing rejects impurities releasing them to surface water, while meltwater of ice is less saline than the lake water (Leppäranta et al., 2003a, 2003b; Yang et al., 2016). In the ice season, inflow and outflow normally decrease since precipitation is mostly solid in the drainage basin. The bottom sediment becomes an important heat source (Falkenmark, 1973). Wind-driven circulation ceases, and weak thermohaline circulation becomes dominant (Huttula et al., 2010; Kirillin et al., 2012). Processes beneath the ice slow down, and lake memory extends over the winter. Oxygen is consumed due to respiration of water organisms, consumption by the lake sediment, and especially at the lake bottom due to bacterial oxidation of organic matter and chemical oxidation of inorganic substances (Golosov et al., 2007). Primary production becomes light-limited, and in polar and snow-covered subpolar frozen lakes it is absent.

7.1.1 Cooling Process In autumn cooling in seasonally freezing lakes, the surface temperature eventually reaches the temperature of maximum density. In freshwater lakes, overturning of the water mass takes place then. At further cooling, the inverse winter stratification develops, and in a calm and cold period the surface may freeze over fast. For example, cooling a 1-m water layer by 50 W m−2 lowers the temperature by 1 °C in one day that can be reached by autumnal terrestrial radiation losses. As phrased by Kirillin et al. (2012), the time of autumn overturning is the beginning of a “pre-winter” in the lake. The quality of the cooling process in seasonally freezing lakes can be classified in terms of the water salinity (Table 7.1). In saline lakes, the seawater formulae (see Eq. 2.8) can be used as approximations for the temperatures of the freezing point T f and maximum density (Tm ). However, especially for hypersaline lakes the exact chemical composition of the lake water is important, and the accuracy of these equations is not good but lake-individual formulae would be preferable. In freshwater and brackish lakes, at T < Tm inverse stratification and wind-mixing events come and go until freezing. The thermal expansion coefficient drops to zero at T = Tm , and near

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237

Table 7.1 Characteristics of cooling process in seasonally freezing lakes Fresh water

Salinity S (‰)

Freezing point T f (◦ C)

Cooling

Turnover

S < 0.5

Tf ≈ 0

T f < Tm

Full

Brackish

0.5 ≤ S < 24.7

−1.3 < T f < 0

T f < Tm

To halocline

Oceanic

24.7 < S < 40

−2.5 < T f < −1.3

T f ≥ Tm

To halocline

Hypersaline

S ≥ 40

T f < −2.5

T f > Tm

To halocline

this temperature the influence of temperature on water density is small. Consequently, quite small differences in salinity may overcome the temperature effects on the density (see Example 2.6). In oceanic and hypersaline salinities, T f > Tm and free convection to the halocline1 or lake bottom continues until ice formation without any pre-winter temperature stratification. In saline lakes, density depends also on the salinity. Then cabbeling effects occur (McDougall, 1987). When two water types with different salinity and temperature are mixed, the mixture has higher density than either of the original types. This results from the nonlinearity of the equation of state ρw = ρw (T , S, p), where, with fixed pressure, the isolines of density are concave up in the T S-plane. In a corresponding manner, thermobaric effects arise in deep lakes from the nonlinear dependence of density on temperature and pressure. Since the influence of temperature on density is at small near the temperature of maximum density, then cabbeling and thermobaric effects are at strongest. Saline lakes, unless very shallow, are normally stratified in salinity. Convective mixing does not reach the lake bottom but only the halocline. In limnology such lakes are called meromictic. In brackish lakes, Tm > T f but the difference  Tm − T f decreases with salinity as well as the density difference, and ρw (Tm )−ρw T f ≤ 0.13 kg m−3 . In the first order, salinity change by 0.1‰ unit influences the density by about 0.08 kg m–3 , independent of the absolute salinity level. Thus, a quite weak halocline can stop convection in cooling near the freezing point. When Tm < T f , convection continues until freezing, and in hypersaline lakes the freezing point can become very low, at extreme about –50 °C in Don Juan Pond in Victoria Land, Antarctica. In a deep lake, the adiabatic temperature change or the adiabatic lapse rate is significant in the vertical stratification (e.g., Curry & Webster, 1999; Imboden & Wüest, 1995). This lapse rate is defined as the rate at which in situ temperature changes with pressure while entropy (and salinity) is held constant. When a water parcel rises, its temperature decreases due to the decreasing pressure, and in sinking down warming takes place in a symmetric manner. The sign of the consequent density change depends on the temperature, whether this is above or below the temperature of maximum density. The potential temperature of a water parcel is defined as the temperature after the parcel has been adiabatically brought to surface. In a neutrally stratified lake, the potential temperature is constant, and in situ temperature increases with depth for T > Tm and decreases with depth for T < Tm . The adiabatic lapse 1

A thin layer with sharp change of salinity (corresponding to thermocline in temperature stratification).

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7 Lake Water Body in the Ice Season

rate can be expressed as (see Gill, 1982):  dT  αgT [≡ = dz ad cp

(7.1)

where α is the coefficient of thermal expansion, and c p is the specific heat at constant pressure. For example, adiabatic effects have a key role in the vertical circulation in the very deep Lake Baikal (Piccolroaz & Toffolon, 2013; Schmid et al., 2008; Shimaraev et al., 2011). Example 7.1 In Lake Baikal thermobaric effects are critically important. The temperature of the maximum density decreases with increasing pressure at the rate of 0.020 °C bar–1 , or about 0.2 °C per 100 m water depth. In Lake Baikal, the equality T = Tm ( p) is observed at H0 ≈ 300 m depth. Deeper the stratification is stable due to thermobaric stability, and therefore seasonal convection is limited to the upper layer H ≤ H0 . Deep waters are renewed due to downwelling from coastal regions. The lake water body is described for the stability of the stratification by the Väisälä frequency or the buoyant frequency N and Richardson number Ri: g ∂ρw ρ0 ∂ z  2  2 ∂u ∂v N2 + Ri = 2 , ε˙ s2 = ε˙ s ∂z ∂z N2 = −

(7.2a)

(7.2b)

where ρ0 is a reference density and ε˙ s is the vertical shear strain-rate. The quantity N gives the highest frequency of free oscillation of a water particle in a fluid with stable stratification, N > 0. If N 2 < 0, the stratification is unstable. The Richardson number equals the ratio between the productions of turbulent energy by buoyancy and vertical shear. In stable (unstable) stratification, Ri > 0(Ri < 0). According to observations, for Ri > 0.2 turbulence is damped by the buoyancy. Example 7.2 In fresh water, the density at 0 °C is lower by 0.13 kg m–3 than at Tm = 3.98 ◦ C. For the temperature change from 0 to 3.98 °C across a 10-m water layer, Väisälä frequency is N = 0.011 s–1 corresponding to oscillation period of 2π N −1 = 9.5 min in the thermocline. To damp the turbulence, the vertical shear must be less than 0.025 s–1 or velocity difference less than 2.5 cm s–1 across 100 cm depth  interval. In brackish water with salinity 10‰, the density difference is ρ(Tm )− ρ T f = 0.045 kg m–3 , and across a 10-m layer this gives N = 0.0066 s–1 . Heat is transferred into/out of a lake by fluxes at the boundaries and by solar radiation into the interior. In three-dimensional analyses it is convenient to fix the zero level of the vertical co-ordinate (z) beneath the lake. The lake bottom and surface are then β = β(x, y) and ξ = ξ (x, y; t), respectively. The heat conservation law is expressed as

7.1 Ice Formation

239

∂T κw Q sδ −κw (ξ −z) + u · ∇T = ∇ · (K w ∇T ) + e ∂t ρw cw

(7.3)

where K w and κw are the thermal diffusion tensor and the light attenuation coefficient of water, respectively, and the heat capacity has been assumed constant, ρw cw ≈ 4.20 MJ m–3 °C–1 . In turbulent conditions, the diffusion tensor is a property of the flow so that the actual water properties appear only in the last term. Turbulent thermal diffusion is anisotropic: K w,x x = K w,yy = K w , K w,zz = K w , and K w, pq = 0 for p /= q. The left-hand side of Eq. (7.3) gives the local rate of change of temperature and advection, and the right-hand side includes diffusion and heating by solar radiation inside the lake. Air–lake and bottom–lake interactions work through the boundary conditions. The surface temperature of a lake decreases together with the ambient atmosphere (Fig. 7.1). The first phase of the winter in seasonally freezing lakes, the ‘pre-winter’, begins at the time the surface temperature passes down below the temperature of maximum density (Kirillin et al., 2012). Surface water experiences a minor supercooling before freezing, depending on the weather. Usually, it is around 0.1 °C but may be up to 1 °C in calm conditions. In pre-winter, the lake has the potential to freeze rapidly, since only a short, calm, and cold event can stabilize a thin surface layer resulting in ice formation. E.g., in winter 2020 in southern Finland the monthly mean temperatures were slightly above the freezing point in all winter months but there were enough cold breaks to freeze all lakes over. In a mild winter without a week or so cold break, the unfrozen, pre-winter phase could persist through the whole winter, but such weather history is quite unlikely. The temperature structure of the water column develops into inverse stratification in pre-winter and sets the initial temperature profile below the ice cover. Surface heat losses can be of the order of 50 W m–2 , while the heat gain from the bottom sediment

Fig. 7.1 Evolution of surface temperature in an arctic tundra lake, Lake Kilpisjärvi in years 2007– 2009 from July to freezing in November. The data source is the open data base Hertta of the Finnish Environment Institute

240

7 Lake Water Body in the Ice Season

Fig. 7.2 Oxygen saturation as a function of temperature

is one order of magnitude lower. The mixing conditions determine the deep-water temperature and the depth of the winter thermocline. If winds are weak, the bulk water temperature remains relatively high, and the thermocline is at a shallow depth. In the course of time, several mixing–re-stratification periods occur. Eventually, a persistent inverse winter thermocline forms, and then freezing can take place fast. Long-lasting, strong mechanically forced mixing deepens the thermocline and cools the water down even close to the freezing point (Homén, 1903). For example, in a large lake Päijänne in central Finland, the maximum depth is 95 m, and in a windy autumn the turnover continued down to 1–2 °C in the observation series of Pulkkanen & Salonen (2013). At the freeze-over, the initial winter stratification is set up. During the pre-winter convection period, the water body is fully saturated with oxygen. Since the saturation level depends on the temperature (Fig. 7.2), the initial oxygen storage for the winter is controlled by the temperature distribution and, consequently, by the cooling process (Terzhevik et al., 2009). Horizontal advection of heat may be significant in the cooling process due to winddriven or thermohaline circulation exchanging water between the central basin and the littoral zone, and in lakes with through flow. In autumn, thermohaline circulation forms with water cooling fast in shallow areas, particularly in the near-shore zone. At T0 > Tm , cooling nearshore waters become denser, sink, and flow toward deeper parts along the sloping bottom. Upwelling follows in the central basin and surface waters flow laterally toward the shore to close the loop. Freezing is delayed in deep parts of the lake because of the large volume of water. Even in a medium-size lake the deep basin may freeze several weeks later than the near-shore zone, depending on the weather conditions (Fig. 7.3). In the progress of cooling, the situation T0 < Tm results first near the shoreline. Then the thermohaline circulation system becomes blocked near the shore (Fig. 7.4), and the blocking 4 °C isotherm, called thermal bar, is shifted further offshore (Zilitinkievich et al., 1992). In principle, a thermal bar is present in all

7.1 Ice Formation

241

Fig. 7.3 Fall cooling at Lake Kesänkijärvi in Lapland with lake smoke over the freezing water. Photograph by the author

lakes but in large lakes the lifetime is long. For example, a thermal bar is regularly observed in Lake Ladoga, the largest lake in Europe, persisting for several weeks (e.g., Rumyantsev et al., 1999). Due to the thermal bar development, freezing of deep waters can be much delayed. Mixing with no freeze-up may continue even through the winter; in limnology such lakes are classified as warm monomictic lakes (Hutchinson, 1957). Fig. 7.4 Thermal bar in Lake Ladoga (redrawn from Melentyev et al., 1997). The 4 °C isotherm separates the circulation from coastal and central parts of the lake for long periods

242

7 Lake Water Body in the Ice Season

Cooling of a lake (Eq. 7.3) can be solved with a prescribed velocity field or simultaneously with a 3-dimensional circulation model. For a small or medium-size lake, this problem is often approximated as a vertical process with one-dimensional models (e.g., Sahlberg, 1983; Mironov et al., 1991; Omstedt, 2011). Modelling the vertical temperature structure can be improved by lateral integration so that instead of temperature T (z, t) the dependent variable is A(z)T (z, t), where A(z) is the crosssectional area of the lake at the level z. These models are further classified into analytic, mixed-layer, and turbulence models. We have, from Eq. (7.3),   ∂ κw Q sδ −κw (ξ −z) ∂T ∂T = Kw + e +F ∂t ∂z ∂z ρw cw

(7.4)

where F accounts formally for the lateral advection and diffusion. Integration this equation over the lateral cross-section is straightforward and arises terms for the profile structure of lake cross-section area. Below, however, we focus on the case A = constant.

7.1.2 Analytic Slab Models Analytic modelling of the cooling process is based on vertical integration of Eq. (7.4). For a one-layer model or the slab model, integration is performed from the level z = b, which represents the lake bottom or the mixed layer bottom, to the surface z = ξ . These levels are assumed fixed, and the water layer depth is thus H = ξ − b. Equation (7.4) becomes: d dt

|ξ b

 |ξ ∂ T z=ξ Q sδ  −κw (ξ −b) T dz = K w + 1−e + Fdz ∂ z z=b ρw cw

(7.5)

b

It is assumed that vertical variations of water temperature are small so that the temperature T represents the whole water layer. The boundary conditions can then be expressed as z = ξ : Kw

∂T = Q 0 = K 0 + K 1 (Ta − T ) ∂z

(7.6a)

∂T = Q b = K b (Tb − T ) ∂z

(7.6b)

z = b : Kw

where Tb and K b are the temperature and the heat transfer coefficient at z = b, respectively. Feasible for analytic modeling, the linearized form (Eq. 4.36) is employed for the surface boundary condition, and a corresponding form is expressed for the heat flux from bottom. Without loss of generality the whole solar flux can be shifted from

7.1 Ice Formation

243

Eq. (7.5) to K 0 in the surface boundary conditions (technically, the surface absorption parameter is then γ = 1). Below, the lateral interaction term F is ignored but it can be easily included into the given solution technique. With the given boundary conditions and F = 0, dividing Eq. (7.5) by H , gives dT = λa (Ta + TR − T ) + λb (Tb − T ) dt λa =

(7.7)

Kb K0 K1 , λb = , TR = ρw cw H ρw cw H K1

Here λa and λb are the inverse response times of the water layer to thermal forcing, and TR comes from the first-order net radiation and latent heat fluxes. Due to solar radiation dependency, in the cooling season TR < 0 ◦ C in polar latitudes and TR > 0 ◦ C below mid-latitudes, where freezing lakes exist in mountains. The solution of Eq. (7.7) is T (t) = T (0)e

−λt

|t +

e−λ(t−τ ) [λa (Ta + TR ) + λb Tb ]dτ

(7.8)

0

where λ = λa + λb . The equilibrium solution and the response time are T =

λa ρw cw λb H (Ta + TR ) + Tb , τ L = λ−1 = λ λ K1 + Kb

(7.9)

  In the cooling period K 1 ∼ 20 W m–2 °C–1 , thus λa ∼ 0.45 d−1 H −1 m . When TR < 0 ◦ C in equilibrium air is a little warmer than the surface to compensate the radiation and evaporation losses, and for TR > 0 ◦ C the situation is reversed. When the lower surface is the lake bottom, the heat flux Q b comes from the sediment. Falkenmark (1973) reported that the sediment heat storage of a shallow Lake Velen (17 m mean depth) in Sweden was at maximum 110 MJ m–2 , which means that the winter heat flux from the sediment is of the order of 5 W m–2 . Maher and Malm (2003) estimated this heat flux as 2–6 W m–2 in early winter decreasing   to 1–2 W m–2 in spring. Thus, K b ∼ 1 W m–2 °C–1 and λb ∼ 0.02 d−1 H −1 m , λb 0. Thus Ta = T f at t = 0, and the water temperature is, from Eq. (7.8),

 λ   α   λa  b Ta + 1 − e−λt 1 − e−λt Tb + T (0)e−λt (7.10) + TR + λ λ λ  Leaving the transient terms out, we have T (t) = λλa Ta + αλ + TR + λλb Tb . Compared with the equilibrium (Eq. 7.9), the falling air temperature leaves the lag αλ−1 ∝ H to the air temperature response. E.g., in North-European autumn, −1 ◦ α ≈ 5 ◦ C month–1 , and then for  ∼ 10 m we have αλ ≈ 4 C. The freeze-up  H time t = t f is obtained from T t f = T f as: T (t) =

tf =

   1 1 λb  + Tb − T f TR + λ α λa

(7.11)

The solution contains a depth-dependent delay λ−1 ∝ H and a shift, which depends on the atmospheric cooling rate. Simojoki (1940) examined the freezing date of lakes in Finland based on field data. He showed that after the air temperature has fallen below the freezing point, the freezing date follows with a delay depending on the lake depth (Fig. 7.5). Shallow lakes (H ≈ 5 m) froze after a 1–2-week cold period but deep lakes needed more than one month. This result can be explained by the analytic solution in Eq. (7.11). Simojoki’s (1940) delay had the slope term proportional to depth but no shift term. At H > 10 m the linear form turned sub-linear corresponding to incomplete mixing in deep lakes. Example 7.4 In the ephemeral lake ice zone, it is of interest to study the full annual cycle. We leave TR = 0 ◦ C and λb = 0. The annual cycle of air temperature is taken as a sine wave Ta = T a + ΔTa sin(ωt), where T a is the mean annual air temperature, 2π is the frequency. Equation (7.8) ΔTa is the air temperature amplitude, and ω = 365d gives the solution ω λ . T = Ta + √ ΔTa sin(ωt − ϕ), ϕ = arctan λ λ2 + ω 2

7.1 Ice Formation

245

Fig. 7.5 Freezing date delay after air temperature downcrossing the freezing point as the function of lake depth. Simojoki (1940) shows data from Finnish lakes, and the lines t f [d] = 2H [m] and t f [d] = 2H [m] + 5d show analytic solutions

When the mixed layer depth is 25 m, we have λ ≈ 0.018d−1 ≈ ω. Then the air temperature cycle is damped by the factor of √12 in the mixed layer and the lag is ϕ = π 4

that accounts for 46 days. The freezing condition is ΔT =

√ λ ΔTa λ2 +ω2

> T a −T f .

If T a ≤ T f , freezing always takes place; if T a > T f and ΔTa > T a − T f , shallow lakes freeze but deep enough lakes do not. For example, Lake Shikotsu in Hokkaido, Japan does not freeze. The mean depth is 265 m, the annual mean and amplitude of air temperature are 5 °C and 14 °C, respectively, and the freezing condition is not satisfied.

7.1.3 Mixed-Layer Models Mixed layer models form an extension of slab models where the depth of the mixed layer is a dependent model variable, H = H (t) (see Niiler & Kraus, 1977). These models do not include horizontal effects. To determine the evolution of the mixed layer depth brings the mechanical energy budget to the model, and then also the water velocity u needs to be solved. The model equations are for heat, momentum, and mechanical energy, respectively:   1  dT (T − TH ) dH + · = Q 0 + Q H + 1 − e−κw H Q sδ dt H dt ρw cw H 1 du (u − u H ) dH + fk×u= · (τ a − τ H ) − dt ρw H H dt

(7.12a) (7.12b)

246

7 Lake Water Body in the Ice Season

H s dH ΔbH 1 − = 2mu 3∗ + [(1 + n)B0 − (1 − n)|B0 |] Ri dt 2      2 2 −κw H + H+ e Q sδ + H− κw κw

(7.12c)

Here f is the Coriolis parameter, τ a and τ H are the shear stresses at the lake surface and mixed layer bottom, respectively, Δb = −g(ρw − ρw H ) is the buoyancy of the mixed layer, Ri = ΔbH |u − u H |−2 is the bulk Richardson number, m, n and s are Q0 is the buoyancy flux at the empirically determined efficiency factors, B0 = gα ρw cw surface, and α is the thermal expansion coefficient. Equations (7.12a, b) are straightforward, but Eq. (7.12c) is the complex heart of mixed-layer models. The right-hand side, say, B = B(H ) is first determined using the forcing data. Then there are two possibilities: (i) B(H ) ≥ 0: There is energy to deepen the mixed layer and the entrainment ≥ 0 can be solved from (7.12c). we = dH dt (ii) B(H ) < 0: The system gains buoyancy, and a new mixed layer thickness is determined from B(H ) = 0. Flake is a widely used mixed-layer–thermocline model, where the variable thermocline depth is determined by the mixing and buoyant flux conditions, and the thermocline is based on the similarity principles (Golosov & Kirillin, 2010; Kirillin, 2010; Kourzeneva et al., 2008; Mironov et al., 2004, 2010). The thermocline extends from the mixed layer to the bottom of the lake described using the assumed self-similar structure of temperature versus depth. The same method has also been used for the temperature structure of the thermally active upper layer of bottom sediments and the ice and snow cover. The result is a computationally efficient bulk model. Flake has been used in several applications, e.g., in numerical weather prediction and climate models to provide a fast and good estimator for the surface temperature of lakes. It works as a stand-alone lake model and as a physical module in aquatic ecosystem models. Barenblatt (1996) defined self-similarity as a time-developing phenomenon where the spatial distributions of its properties at different moments of time can be obtained from one another by a similarity transformation. Self-similarity simplifies computations and the representation of the phenomena under investigation. This approach has been applied to various problems, such as turbulent flows in atmospheric surface layers, wall layers of turbulent shear flows, the dynamics of turbulent spots in fluids with strongly stable stratification, etc. The self-similarity concept was introduced in the marine sciences for the first time by Kitaigorodskii and Miropolski (1970) and later for lake environment (Mironov et al., 1991). The self-similarity for the temperature profile is described for the mixed layer and thermocline as (Mironov et al., 2010)  T =

0≤h≤H T0 , T0 − (T0 − TD )ϕ(ζ ), H ≤ h ≤ D

(7.13a)

7.1 Ice Formation

247

ϕT ≡

h − H (t) T0 (t) − T (h, t) , ζ ≡ T0 (t) − TD (t) D − H (t)

(7.13b)

where D is lake depth, TD is the temperature at the lake bottom, and ϕ(ζ ) is a dimensionless similarity function of the dimensionless depth ζ, 0 ≤ ζ ≤ 1. The selfsimilar structure is restricted to the main thermocline, and the large-scale stratification is assumed stable. An approximate analytical expression for ϕ(ζ ) can be found by using a fourth-order polynomial approximation of the classical type of Karman and Polhausen in boundary layer theory function together with the boundary conditions ϕ(0) = 0, ϕ(1) = 1.

7.1.4 Turbulence Models For a more detailed description of the vertical temperature–velocity structure, advanced turbulence models are employed. A 2nd order closure model of Svensson (1979) has been widely used (Svensson, 1998; Omstedt, 2011). The closure is performed by equations for the turbulent kinetic energy k and the dissipation rate of turbulent kinetic energy ε, and the model belongs to the class of kε-models. The system of equations for the temperature and velocity averaged over turbulent fluctuations is   ∂ Q sδ κw −κw (ξ −z) ∂T ∂T = KT + e (7.14a) ∂t ∂z ∂z ρw cw   ∂ ∂u ∂u = fk×u+ K (7.14b) ∂t ∂z ∂z K = Cμ

k2 K , KT = . ε Pr

(7.14c)

where Cμ = 0.09 is a model constant, Pr = ν/νT = 1.4 is the Prandtl number, ν is the molecular viscosity, and νT is molecular thermal diffusivity. The diffusion coefficients are given by the turbulent kinetic energy and the dissipation rate of turbulent kinetic energy, which are obtained from the 2nd order turbulence closure (Omstedt, 2011; Svensson, 1979)       ∂ u 2 ∂ K ∂k ∂k g ∂ρ   = +K   + · −ε ∂t ∂ z σk ∂ z ∂z ρ0 ∂z        ∂ u 2 ∂ K ∂k ε g ∂ρ ε2 ∂ε = + K c1   + c3 · − c2 ∂t ∂ z σε ∂z k ∂z ρ0 ∂ z k

(7.14d)

(7.14e)

248

7 Lake Water Body in the Ice Season

where the coefficients σk = 1.0, σε = 1.3, c1 = 1.44, c2 = 1.92, and c3 = 0.8 are model constants. The model has six parameters, these five plus Cμ . The most uncertain part of this kε-model is the equation for the rate of dissipation of turbulent kinetic energy, which involves several parameters. The mixed layer physics with its depth comes out quite well in the model but deeper, below the thermocline, horizontal processes become relatively speaking more significant. To reach the observed level of mixing in the lower layer, so-called ‘deep mixing’ functions have been added into the model for tuning. One-dimensional approach thus has limitations in that horizontal advection and diffusion cannot be properly accounted for, and therefore interaction between the coastal zone and central basin is not solved. In the cooling process, particularly advection and sinking of cold water from shallow shore areas to central basins complicates the one-dimensional picture. Based on this 2nd order turbulence closure, a programme package ‘PROBE’ has been constructed to be utilized for several kinds of applications (Omstedt, 2011). Cooling of lake waters in autumn has been widely examined with this model (e.g., Persson et al., 2005; Sahlberg, 1983). In several basins good results have been obtained, and furthermore a group of one-dimensional models has been used as the basis of a connected box-model network. Figure 7.6 shows an example of the model simulation. The whole 50-m water column cools, and at temperatures below 4 °C surface layer starts to stratify and bottom water reaches temperatures down to 2 °C. Heat loss from the lake is almost continuous, and at highest 300 W m–2 . a

b

Fig. 7.6 Lake cooling simulation for Lake Väsman, November 2–December 7, 1981, by PROBE: a the simulated (solid line) and measured (dashed line) temperature profiles at every five days; b net heat loss and wind stress. (Sahlberg, 1983)

7.2 Thermal Structure and Circulation Under Ice

249

7.2 Thermal Structure and Circulation Under Ice 7.2.1 Water Body During the Ice Season The winter of seasonally freezing lakes can be divided into four phases (Kirillin et al., 2012). The pre-winter (Sect. 7.1) forms the 1st phase. The 2nd phase is the freeze-up, then follows the period of complete ice cover (Phase #3), and the fourth one is the ice decay period. The lengths of these phases largely depend on the lake size. In small and medium-size lakes Phase #4 takes normally 2–4 weeks, while the lengths of Phases #2 and #3 depend on the climate zone and can be from zero to more than six months. The physics of lakes is qualitatively different under ice cover as compared with open water season (Fig. 7.7). A complete ice cover is a rigid lid, which almost closes the interaction between the water body and the atmosphere (Table 7.2). Wind stress does not go through the ice, and therefore mechanical forcing is limited to pressure variations. However, thin ice still bends in windy conditions with consequent surface pressure variations. Thermal signals are transmitted across the ice cover. Sunlight penetrates ice or snow surface and reaches the water body, unless there is a thick snow cover, and heat can leak out from the lake slowly by molecular conduction through ice and snow. Exchange of water and other substances between the lake water body and atmosphere does not take place. An ice cover largely isolates the lake water body from the atmosphere with the degree of isolation depending on the ice coverage, ice thickness, and snow thickness. The water body beneath a solid ice cover is cold. In truly fresh water, the temperature is between 0 and 4 °C, but even in weakly saline (order of 1‰) ice-covered shallow lakes the saline bottom layer has reached temperatures up to 10 °C, when dissolution from bottom sediments have produced a thin, dense bottom layer (Leppäranta et al., 2020; Lu et al., 2020). After the ice cover has formed, heat exchange through the surface weakens remarkably, heat flux from the lake bottom becomes significant in comparison with other fluxes, and water flows slow down to a weak thermohaline circulation. Since density changes are very small, even small salinity variations can overcome the influence of temperature in density. Growing ice rejects most of the impurities, but a small fraction remains between ice crystals depending on the ice growth speed. The salinity of lake ice is less than the salinity of the parent water, which may cause weak convection in the surface layer. For salinity less than 1‰ the lake ice captures just a fraction of 0.1 of the salinity of the parent water (Leppäranta et al., 2003b) but at oceanic salinities this fraction is 0.25–0.5. More significant is the melt water release, which is less saline and therefore less dense than the lake water that may result in a thin, cold stable layer under the ice. The lake water body is described for the dynamic state by the Reynolds number: Re =

UL ν

(7.15)

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7 Lake Water Body in the Ice Season

Fig. 7.7 Fieldwork to deploy underice measurement systems in Saroma-ko Lake, Hokkaido. Photograph by the author

Table 7.2 Modification of water body—atmosphere interaction by a complete ice cover

Quantity

Water body in ice season versus open water season

Wind stress

Removed

Surface pressure

Changes due to air pressure variations

Solar radiation

Reduced by more than 50%

Heat transfer to air

Reduced, conductive losses through ice cover

Water balance

Atmospheric exchange removed

Air–lake exchange of matter

Removed

where U and L are the velocity and length scales of water flow, respectively, ν is the kinematic viscosity of water, ν ≈ 1.8·10–6 m2 for cold water. Normally, the flow is laminar for Re < 103 and turbulent for Re > 104 , and in between the flow is in the laminar–turbulent transition regime. In an ice-covered lake, water velocities are typically ~ 1 mm s–1 , and we have Re ∼ 103 for L ∼ 1 m. Therefore, in general in ice-covered lakes the flow may be in any of the three states that brings a large variability in ice–water exchange parameters (Zu et al., 2021). Investigation of the

7.2 Thermal Structure and Circulation Under Ice

251

hydrodynamics beneath ice cover is therefore difficult, either by field observations or by mathematical modelling. In particular, low circulation velocities are difficult to measure (Huttula et al., 2010). Heat transfer in an ice-covered lake follows Eq. (7.3) but now the diffusion tensor depends on the state of the flow and the solar radiation is reduced by attenuation in snow and ice. In laminar flow, ∇ · (K w ∇T ) = Dw ∇ 2 T , where Dw is the thermal (molecular) diffusion coefficient of water. In laminar–turbulent transition, an effective diffusion coefficient K e f f has been employed for vertical diffusion. The magnitudes of the diffusion coefficients vary widely: Dw 0). Karetnikov and Naumenko (2011) studied an ice cover index defined for one winter as t B I =

A(t)dt = ⟨A⟩(t B − t F )

(9.6)

tf

where the operator ⟨·⟩ stands for time averaging. This is a convenient index for the severity of an ice season. Going from the areal averaging to ice volume is limited by the availability of ice thickness data.

320

9 Lake Ice Climatology

9.3 Mathematical Modelling 9.3.1 Linear Surface Heat Flux Analytic modelling provides useful, first-order results to evaluate the climate sensitivity of lake ice seasons (see Sect. 4.4). Since the climate impact comes from the atmosphere and sun, it is convenient to apply a heat flux formula linear in the surface temperature for the forcing, expressed as Q 0 (t, T0 ) = K 0 + K 1 (Ta − T0 )

(9.7)

where the parameters K 0 and K 1 depend on time but do not depend explicitly on T0 . This approach was employed in Sect. 4.3 and can be motivated over synoptic or longer time scales (Karetnikov et al., 2017; Leppäranta, 2014). Equation (9.7) contains the net solar and atmospheric heat flux into the lake, with K 0 dominated by the radiation balance and K 1 by the turbulent exchange. The linear formula is obtained as follows. Solar radiation is absorbed in K 0 . Using the truncated binomial formula, we can write T04 ≈ Ta4 − 4Ta3 (Ta − T0 ), , and the net terrestrial radiation (see Eq. 2.12) is ε0 εa σ Ta4 − ε0 σ T04 ≈ ε0 σ (εa − 1)Ta4 + 4ε0 σ Ta3 (Ta − T0 )

(9.8)

This formula gives an accuracy of about 1% when |Ta − T0 | < 10◦ C and Ta ∼ 0◦ C. There are contributions to both parameters in Eq. (9.7). The bulk formulae (Eqs. 2.16b–c) are used for the turbulent heat fluxes. The sensible heat flux is proportional to Ta − T0 , and thus the proportionality coefficient goes to K 1 . The latent heat exchange is split using the Clausius–Clapeyron equation (e.g., Gill, 1982) for the saturation water vapour pressure es = es (T ), see Annex for the parameterized equation. Assuming that humidity is at the saturation level at the surface, e0 = es (T0 ), and writing ea = Res , where R is the relative humidity, the latent heat flux is expressed as   0.622es (Ta ) LE Q e = −ρa L E C E − T (9.9) (T ) (1 − R) − a 0 Ua pa Rv Ta2 where Rv is the gas constant of dry air. Finally, combining all together, the parameters K 0 and K 1 become (Ta in degrees Kelvin) K 0 = (1 − α)Q s (0) − ε0 (1 − εa )σ Ta4 − ρa L E C E

0.622es (Ta ) (1 − R)Ua pa (9.10a)

9.3 Mathematical Modelling

K 1 = 4ε 0 σ Ta3 + ρ a c p C H Ua + ρa L E C E

321

0.622es (Ta ) L E Ua pa Rv Ta2

(9.10b)

The terms of K 0 contain the radiation balance and latent heat flux for T0 = Ta . The second term of K 1 represents the sensible heat flux, and the first and third terms are the first-order corrections for T0 /= Ta to the terrestrial radiation balance and latent heat flux. The formula has the following interpretation: K 0 can be positive or negative, and K 1 > 4ε0 σ Ta3 > 0 (K 1 > 3 W m−2 ◦ C−1 for Ta > −30 ◦ C ). For K 0 = 0 the surface temperature becomes low pass filtered from the atmospheric temperature. Here we have taken the total net solar radiation into the parameter K 0 , but in applications in Chap. 4 the absorbed solar radiation was partitioned as Q s0 = Q s0 + Q sδ with Q s0 in K 0 and Q sδ as a source term inside the ice. The formulae (9.10a,b) look somewhat complicated, but the resulting climatological values are rather smooth. Cases from the boreal zone and Tibetan Plateau are shown in Table 9.1. Seasonal variation comes mainly from the solar radiation and the latent heat flux. The radiation balance dominates K 0 . In the boreal zone, K 0 < 0 in October–March, while in the Tibetan Plateau K 0 ≈ 50 − 150 W m–2 all year due to the strong solar radiation. The annual averages are 120.5 W m–2 in Ngoring (range 150 W m–2 in monthly means) and 17.8 W m–2 (range 170 W m–2 in monthly means) in Jokioinen. The parameter K 1 is close in both sites, K 1 ≈ 10 − 20 W m–2 °C–1 with contributions more or less evenly from terrestrial radiation, sensible heat flux and latent heat flux. The annual averages are 12.8 W m–2 °C–1 in Ngoring and 18.6 W m–2 °C–1 in Jokioinen. The annual cycle is strong in solar radiation and latent heat flux. The basic analytical models were presented earlier (Karetnikov et al., 2017; Leppäranta, 2014; Leppäranta & Wen, 2022). They include the slab model for lake cooling in Sect. 7.2.1 (Rodhe, 1952), freezing-degree-day model for ice growth in Sect. 4.4.2 (Anderson, 1961; Barnes, 1928; Leppäranta, 1993), and positive-degreeday models for ice melting in Sect. 4.4.2 (Korhonen, 2006; Mueller et al., 2009). The connection of ice melting to air temperature is indirect, and models based on energy balance are more physically based (Ashton, 1980; Zhang & Jeffries, 2000).

9.3.2 Ice Occurrence The slab model (see Sect. 7.1.2 and Eqs. 9.14-15 below) provides a view whether a lake remains ice-free or freezes over, i.e., the fundamental question of ice occurrence. Take a parabolic form of the winter temperature evaluation, expressed as Ta (t) = T a + ct 2 , where T a is the minimum air temperature in winter, placed at time t = 0, and c > 0 is the shape factor of the temperature curve. Using Eq. (9.7) in the slab model (Eq. 9.15), we have the condition for freezing: ∆

τ L2 >

∆

  λb  ρw cw 1

Tc − T a > 0; Tc = T f − Tb − T f − TR , τ L = H (9.11) c λa K1 + Kb ∆

322

9 Lake Ice Climatology

Table 9.1 Linear heat flux parameters in Jokioinen (60°49’N 23°30’°E), southern Finland and Ngoring Lake (34°54’N 97°42’E), Tibetan Plateau (Leppäranta & Wen, 2022) Month

Solar radiation

Terrestrial radiation

Jokioinen

K 0 (W m–2 )

September

90

Latent heat

Total

Terrestrial radiation

Sensible heat

Latent heat

Total

10.5

20.7

K 1 (W m–2 °C–1 )

– 45

– 16

29

5.0

5.2

October

38

– 45

– 9

–16

4.7

5.5

8.2

18.4

November

12

– 43

– 5

–36

4.5

5.8

6.2

16.5

December

3

– 45

– 4

–46

4.3

5.8

4.7

14.8

January

2

– 46

– 5

–49

4.2

5.6

4.0

13.8

February

8

– 48

– 5

–45

4.2

5.5

3.7

13.4

March

30

– 56

– 7

–33

4.4

5.5

5.0

14.9

April

76

– 62

– 10

4

4.7

5.5

7.7

17.9

May

193

– 57

– 36

100

5.0

5.5

11.0

21.5

June

209

– 55

– 40

114

5.2

5.2

13.3

23.7

July

203

– 48

– 41

114

5.4

4.9

14.6

24.9

August

151

– 45

– 29

77

5.3

4.9

13.3

23.5

m–2 )

m–2

°C–1 )

Ngoring

K 0 (W

September

203

– 72

– 19

112

4.5

3.2

5.2

12.9

October

190

– 60

– 11

119

4.2

2.7

2.8

9.8

K 1 (W

November

153

– 53

– 8

92

3.9

3.1

2.1

9.1

December

125

– 50

– 2

73

3.7

2.2

1.0

6.9

January

129

– 50

– 9

70

3.8

3.9

2.1

9.7

February

163

– 53

– 6

104

3.8

3.6

2.0

9.4

March

208

– 56

– 13

139

4.0

4.6

3.4

12.0

April

249

– 64

– 28

157

4.4

4.8

6.3

15.5

May

275

– 73

– 29

173

4.6

4.2

7.4

16.2

June

262

– 78

– 32

152

4.7

4.2

8.2

17.1

July

243

– 79

– 26

138

4.7

4.4

8.9

18.1

August

225

– 82

– 29

114

4.8

3.7

8.7

17.2

The contributions of solar radiation, terrestrial radiation, sensible heat flux and latent heat flux are shown separately, the totals refer the values of the coefficients. Solar radiation is the total net value, including absorption in the near-surface layer and penetration into ice and water. Data from the Finnish Meteorological Institute (Jokioinen) and the Key Laboratory of Land Surface Process and Climate Change in Cold and Arid Region, Lanzhou (Ngoring Lake)

λa =

K1 , ρw cw H

λb =

Kb K0 , TR = ρw cw H K1

where τ L is the response time or the memory of the lake, H is the mixed layer (or lake) depth, and Tb is the temperature of the mixed layer (or lake) bottom. Here

9.3 Mathematical Modelling

323

λa and λb are the relaxation times of the mixed layer for the surface and bottom forcing, and TR is a temperature shift term originating from the radiation balance. Tc is the critical temperature, and a necessary condition for freezing is T a < T c . In addition, the severity of the winter must be strong enough to cool the mixed layer to the freezing point. We can define the ‘cold period’ by T a < Tc , and its length τ f is obtained from cτ 2f = −4Tc using the given temperature curve. Thus, the condition (9.11) becomes simply ∆

∆

τ f > 2τ L

(9.12)

i.e., the length of the cold period must be at least twice the response time of the lake. Very shallow lakes freeze as soon as the air temperature falls below Tc , but deep lakes have long memory and may survive unfrozen even throughout the winter. In geothermal lakes, if the bottom temperature is more than 4 °C, the sediment heat flux forces strong convection with λb >> λa . The heat flux from the sediments is generally much smaller than the surface heat flux in the cooling season, but geothermal lakes may stay ice-free. When Tb > Tw > Tm , the water body is unstable, convective mixing is intensive, and the water temperature is locked to the bottom temperature. E.g., if TR = 0 and λb = 0, for a 5-m deep lake the air temperature must be below the freezing point at least ten days. The temperature condition is external, but the required length of the cold period is lake-specific depending on the mixed layer or lake depth. When the heat flux from bottom is negligible (λb ≈ 0), Eq. (9.11) shows that a necessary condition for freezing is Tˆa − T f < −TR . E.g., TR ≈ −2 ◦ C in Lake Kallavesi (latitude 62°N) in the boreal zone, and TR ≈ 10 ◦ C in Ngoring Lake (latitude 35°N) in the Tibetan Plateau (Leppäranta & Wen, 2022). Thus, in the north radiational and latent heat losses can freeze the surface even when the air temperature is above the freezing point, but in low latitudes solar radiation can compensate for the losses down to quite low temperatures. In northern Poland (around 53°N), lakes do not freeze every winter, and Tˆa ∼ 0 ◦ C and TR ∼ −1 ◦ C.

9.3.3 Freezing and Breakup Dates The slab model can also be employed for the freezing date. The water temperature is assumed vertically uniform, and we have d Q0 + Qb Tw = dt ρw cw H

(9.13)

where Tw is the mixed-layer temperature and H is the mixed-layer depth or lake depth. The freezing date t F is taken as the date when the water temperature reaches the freezing point, Tw (t F ) = T f . The slab model (Eq. 9.13) with the surface heat flux of Eq. (9.7) is first written as

324

9 Lake Ice Climatology

dTw = λa (Ta + TR − Tw )+λb (Tb − Tw ) dt

(9.14)

This equation can be directly integrated: t T (t) =

e−λ(t−τ ) [λa (Ta + TR ) + λb Tb ]dτ

(9.15)

−∞

where λ = λa + λb . Assuming freezing takes place, a linear cooling law is taken to estimate the freezing date: Ta (t) = T f − αt, where α > 0 is the rate of cooling, and t = 0 refers to the time the decreasing air temperature passes the freezing point, Ta (0) = T f . The water temperature is obtained from Eq. (9.15), and the freezing date t F is obtained from the equation T (t F ) = T f : tF = τL +

   1 λb  TR + Tb − T f α λa

(9.16)

The solution shows a delay of freezing after time zero. Also t F < 0 is possible when TR < 0. The local climate impact is included in the time zero, parameter α and the shift TR , while the response time τ L and bottom heating are lake specific. If the heat flux from the sediment is small, the freezing date is delayed behind t = 0 by the lake response time plus the lag ∆t = α −1 TR , t F = τ L + ∆t, from the air temperature. It is interesting to compare the cases of Lake Kilpisjärvi (69°N, H = 19.5 m) in Arctic tundra and Ngoring Lake (35°N, H = 17.6 m) in Tibetan tundra, since their temperature cycles and depths are close (Leppäranta & Wen, 2022). We take τ L ≈ 2H d m−1 as the estimated response time. In Kilpisjärvi we have ∆t = −15 days and t F = 24, 6 day less than in the true climatology, while in Ngoring Lake, we have ∆t = 50 days and t F = 85 , 5 days more than in the true climatology. Thus, the analytic model gives an acceptable estimator, and it is convenient in analysing lake ice climatology across latitude and altitude. Modelling the decay of lake ice is in principle straightforward, since in the melting season ice loss is due to the net gain of energy from the external fluxes, and heat conduction is not significant (Leppäranta et al., 2019). However, the space– time variability of the albedo and light attenuation coefficient of melting ice arise parametrization problems for the solar flux. Ice melting progresses as dh (Q 0 + Q w ) =− ≤0 dt ρi L f

(9.17)

where h is ice thickness, and Q w is the heat flux from water to ice. The breakup date t B is taken from h(t B ) = 0. To predict the breakup date, the thickness of ice at the start of the melting period needs to be known. Ice melting takes place at both boundaries and inside the ice cover. Then ‘ice thickness’ represents the volume of ice per unit area corresponding to the physical

9.3 Mathematical Modelling

325

picture of the decay process. Using the linear surface heat flux (Eq. 9.7), the thickness of ice during the melting period is

hM

1 − h(t) = ρi L f

t



 

K 0 + K 1 Ta − T f + Q w dτ

(9.18)

tM

where h M = h(t M ) is the ice thickness in the beginning of melting. In the melting period, net solar radiation increases fast and therefore we take K 0 = K 0 (t) and thus TR = TR (t). At the start-up time t M , the surface heat flux turns positive after the maximum ice thickness has been reached. This time is approximated from the solution t = t M of the equation

K 0 (t) + K 1 Ta (t) − T f = 0 : Ta (t M ) + TR (t M ) = T f

(9.19)

This is a delicate formula, since in spring both air temperature and solar radiation are increasing, and the temperature TR depends largely on the latitude and altitude. The date of breakup,t B is obtained from the implicit equation h(t B ) = 0. Since melting adds on the incoming heat fluxes, the heat flux from water can be absorbed to the parameter K 0 as is clear from Eq. (9.18). The K 1 -term in Eq. (9.18) gives the positive-degree-day formula. For K 1 = 15Wm−2◦ C−1 , we have the degree-day coefficient ρKi L1 f = 0.42 cm(◦ C d)−1 . When the melting period has begun, the air temperature and the K 0 -term are assumed to increase linearly by the rates β and γ, respectively. Ignoring Q w (or absorbing to K 0 ), we have an equation for the breakup date: 1 (γ + K 1 β)(t B − t M )2 = ρ L f h M 2

(9.20)

The square of the period t B −t M illustrates the rather low sensitivity of the breakup date to ice thickness and the model parameters. The date of ice breakup can be estimated as / 2ρi L f hM tB = tM + (9.21) γ + K1β Any change on the starting points affects as such directly to the breakup date. The length of the decay period can be predicted to the first order, but the start of this period is more difficult to determine. The parameters γ and β tell whether the solar radiation or air temperature has a critical role during melting. Equation (9.21) is expected to be biased upward, since the ice breaks up under its own weight at the critical porosity when the strength of ice has become small. However, the porosity development depends on weather, so that in general there is no strict relation between porosity and ice thickness during the decay period.

326

9 Lake Ice Climatology

Example 9.2. Melting of ice in Kilpisjärvi, Lapland and Ngoring Lake, Tibetan Plateau (Leppäranta & Wen, 2022). The parameters of Eq. (9.20) for these sites were estimated from climate data as: for Ngoring lake, γ ≈ 1 W m–2 d–1 , for Kilpisjärvi −1 –1 ◦ γ ≈ 1.5 W m–2 d–1 , and for both K 1 ≈ 15 W m–2 °C √ and β ≈ 0.17 C d . The A · h M d, A as shown in Eq. date of ice breakup is then estimated as t B ≈ t M + (9.21). In this simple approximation, the length of the melting period depends on the initial ice thickness. Indeed, melt rates usually overlap, and the breakup depends on the start-up time and initial ice thickness. The maximum ice thickness is typically 70 cm in Ngoring Lake and 90 cm in Lake Kilpisjärvi. Furthermore, the estimated time t M is March 15 for Ngoring Lake, and May 10 for Lake Kilpisjärvi, and the observed mean breakup dates are for these sites, respectively, April 30 and June 18. In the observations, melting took 35–45 days from the start in these sites, while the start-up dates ranged over 65 days. In the Tibetan lakes, at the time of ice breakup the air temperature had just reached the freezing point. At the time of start-up, in Kilpisjärvi the air temperature was already about 2 °C higher than the freezing point.

9.3.4 Ice Thickness and Extent To examine ice thickness climate, we take Eq. (4.40) where ice thickness is expressed as a function of K 0 , K 1 and Ta (Andersson, 1961; Barnes, 1928; Leppäranta, 1993). With h(0) = 0, ice thickness is t /   2 2 h(t) = a S a (t) + h a − h a , Sa = max T f − TR − Ta , 0◦ C dτ

(9.22)

0

The model parameters are a 2 ≈ 10 cm2 (◦ C · d)−1 and h a = ki /K 1 ≈ 10 cm. Here T f − T R defines the reference level for the freezing-degree-days Sa , i.e., ice grows only when Ta < T f − T R . Around latitude 60°N, TR ≈ 0, and the usual freezingdegree-days formula is recovered. In mid-latitudes, below 45°N the strong solar radiation gives TR an important role, its value reaching even up to 10 °C (Leppäranta & Wen, 2022). The sensitivity of ice thickness to air temperature is from the total differential formula δh =

a2 δSa 2(h + h a )

(9.23)

The sensitivity is highest at small thickness values. For Sa = 30 ◦ C d, we have h = 10 cm that represents a weak ice cover, and δh = 2.5 cm for δSa = 10 ◦ C d. For Sa = 150 ◦ C d,we have h = 30 cm that represents a stable ice cover in a medium size lake, and δh = 1.25 cm for δSa = 10 ◦ C d.

9.3 Mathematical Modelling

327

The interesting question is that how the freezing-degree-days change with the mean winter temperature. Karetnikov et al. (2017) studied this question with empirical data at Lake Ladoga and obtained fit for the annual freezing the linear regression  degree-days as Sa = 122 − 145T a T f − TR = 0◦ C where Sa is in ◦ Cd and T a is the mean temperature in ◦ C in November–March. The site was Sortavala weather station (61°42’N 30°42’E), and the range of the mean air temperature was from –12 to –2 °C. Another approach is to use the idealized parabolic curve of the winter air temperature, Ta = ct 2 + T a , T a < T f + TR . With fixed shape of the parabola, we ∆

∆

∆

have Sa = − 23 tc T a + T f + TR , where 2 tc is the period when Ta +T f +TR < 0◦ C. The analytic growth and melting models can be combined for the equilibrium multiyear ice thickness. The classical Stefan’s model is used for ice growth, and the summer melting ∆h is over the ‘summer’ defined as the period when the net heat flux to the ice is positive. If not all ice melts in summer, the thickness of ice after the n ' th summer is / (9.24) h n = h 2n−1 + aS − ∆h, n ≥ 1 The growth of ice depends on ice thickness, but we may take ∆h ≈ constant independent of ice thickness. At equilibrium, h n = h n−1 = h e , where h e is the equilibrium thickness, he =

h 21 − (∆h)2 2∆h

(9.25)

The condition of multi-year ice is trivial: h 1 > ∆h. When h 1 − ∆h is h 21 small,h e ≈ h 1 − ∆h, and when h 1 >> ∆h, we have h e = 2∆h . Example 9.3. Multiyear lake ice is found in Antarctic dry valleys. The thickness of ice is 10–20 m. The freezing-degree-days correspond to a first-year ice thickness of around 2 m. For summer melt of 0.1 m, the equilibrium thickness is 20 m. Without melting it would need 100 years to grow the ice, much more with summer melting. The sensitivity of the equilibrium thickness to summer melting is high as seen from Eq. (9.25). Ice extent increases with the frozen area, and in the first-order one can think that freezing front expands depending on the bottom topography of the lake and air temperature. Cooling takes longer when convection reaches deeper, to the bottom of the mixed layer (lake bottom in shallow areas). The lateral growth of ice cover can be formulated as dD dA =− · dt dH



dt F dH

−1

, H < Hmax , t ≤ t1

A ≡ 1, t F (Hmax ) ≤ t ≤ t1

(9.26a) (9.26b)

328

9 Lake Ice Climatology

where D = D(H ) is the hypsographic curve giving the relative area of the lake deeper than H , Hmax is the maximum mixed layer or bottom depth, which is assumed to be constant over time, and t 1 is the end of the growth period. On the right side of Eq. (9.26a), the first factor tells how fast depth increases, and the second factor is the inverse of the rate of freezing date delay when depth increases. When the maximum mixed layer depth is reached in freezing, the whole lake freezes over completely. With linear atmospheric cooling and a linear hypsographic curve, the rate of lateral ice growth is constant. Potential disturbances to this simple model include the thermal bar, ice breakage, and ice drift. The ice cover decay period is straightforward, since the heat gained by ice is used ice for melting. Approximating the rate of heat gain with the linear model, we have   1 dV =− K 0 + K 1 Ta − T f , Ta ≥ T f dt ρi L f

(9.27)

where V = Ah is the ice volume per unit area. The time used to melt all the ice is obtained by integration, and how the ice concentration decreases, depends on the distribution of ice thickness and the patchiness of the ice surface at the beginning of the decay period. At a certain time, when the thickness is still finite, the ice has become so porous that it breaks up and thereafter melts quickly, and the total decay time is less than for an assumed stable ice cover.

9.4 Climate Change Impact on Lake Ice Season 9.4.1 Climate Change The models developed in the previous section can be used for the first-order evaluation of the impact of climate change on the ice season. Climate warming has taken place during the last 30 years as shown in the rise of Northern Hemispheric temperatures (IPCC, 2021). Climate impact on lakes is thoroughly discussed in the book of George et al. (2009). The recent climate warming has affected the ice regime of lakes as shown by numerous publications. A review on the time series work was given by Adrian et al. (2009). The trend comes from the general climate warming during the twentieth century in Eurasia and North America, where most of the lake ice investigations have been made (e.g., Sharma et al., 2019). Aperiodic variations and noise are superposed on the trend toward a milder ice climate. Ice phenology time series predominantly show decrease of the length of the ice season in Eurasian lakes in the last 100 years (e.g., Karetnikov et al., 2017; Kirillin et al., 2012; Korhonen, 2006; Livingstone & Adrian, 2009; Magnuson et al., 2000; Sharma et al., 2019; Yao et al., 2016; ). The shortening is seen more or less equally in the dates of freezing and breakup. Magnuson et al. (2000) reported trends of 5.8 days

9.4 Climate Change Impact on Lake Ice Season

329

later freeze-up dates and 6.5 days earlier break-up dates per 100 years in lake and river ice in the northern hemisphere (1846–1995). In Lake Kilpisjärvi, Arctic tundra in northern Finland, since 1964 the trend has been later freezing by 11.5 days and earlier ice breakup by 5.0 days per 50 years (Lei et al., 2012). Trends of the same order of magnitude were found in Berlin and Brandenburg lakes by Bernhardt et al. (2011). For the irregularly freezing Müggelsee, Berlin, trends were reported for later freezing (5.7 days) and earlier thawing (6.8 days) for the period 1947–2007. Concurrently, the lake revealed increasing number of icefree years, shortening in the total ice duration by 15.6 days, and thinning of the ice cover. In the Tibetan Plateau, shortening of ice seasons is seen in satellite remote sensing data, but, however, some saline lakes show opposite behavior, because their salinity has been decreasing due to the increasing water volume (Cai et al., 2017; Kropacek et al., 2013; Wang et al., 2021; Yao et al., 2016). Notably, trends in the earlier breakup are sometimes reported to be stronger than those in the freezing date (Korhonen, 2006; Jensen et al., 2007), despite ice melting in spring is largely driven by absorption of solar radiation which is affected by the global warming only via the albedo. Prokacheva and Borodulin (1985) examined the variability of lake ice seasons in Northwest Russia, in particular Lake Ladoga. No clear connection was found with sunspots (Wolff index) in Lake Ladoga. Spatial coherence in was seen with large lakes in the region, with Lake Onega, Lake Il’men, Lake Peipsi and Rybinski Reservoir. Thus, climatic factors also other than higher local air temperature in spring affect the ice breakup. They may include decreased ice thickness due to milder autumn/winter conditions, trends in winter precipitation affecting the snow amount, or more cloudy spring conditions directly affecting the radiation input at the lake surface. Apparently, physical mechanisms of the global warming impact on the ice phenology include time-delayed effects and feedbacks not resolved to date. The projected changes in the climate will have a significant effect on qualitative characteristics of the ice and ice season as well as bring quantitative changes. From geographical perspective, the boundaries of ephemeral and seasonal ice zones will move to higher latitudes and altitudes if climate becomes warmer. Also, within the seasonal lake ice zone, there is the boundary of thick ice, referring to ice which is thick to use for on-ice activities, at about 20 cm. With decreasing ice thickness, the period of stable ice cover becomes shorter, and finally the ice cover is unstable during the whole ice season. Today, only very large lakes at around 60ºN in European climate have unstable ice cover but if the ice thickness is reduced to 20–30 cm or less, medium size lakes will also be unstable. Snow accumulation is a critical factor in the ice season that makes climate prediction of ice thickness and breakup difficult. Snow cover is a good insulator, and depending on the snowfall conditions, snow-ice formation may compensate for at least part of the effect of rising air temperatures in ice thickness. Thinner ice also means more breakable ice with more areas of open water, which changes fluxes of heat and moisture between the lake and the atmosphere and may generate frazil ice. Frazil ice formation may, in turn, lead to accumulation of sediments in the ice cover via frazil capturing of particles in the water body and via anchor ice formation. From

330

9 Lake Ice Climatology

the point of view of lake ecology, shorter ice season and presence of openings mean less oxygen deficit problems. But the openings will also create a new type of winter environment in lakes, which have hitherto been completely covered with ice.

9.4.2 Model Prediction Analytic lake ice climate models were presented in Sect. 9.3 for the freezing date and breakup date, ice thickness, and ice extent forced by a linearized surface heat budget, heat flux from below the mixed layer, the mixed layer depth, and freezing point of lake water. The linearized heat budget was expressed as Q 0 = K 0 + K 1 (Ta − T0 ). IPCC climate scenarios provide expected changes for air temperature and precipitation. The former also influences the parameters of the linear surface heat flux formula, K 0 decreasing and K 1 increasing for increasing air temperature (see Eqs. 9.10a,b). However, these changes are small, of order of 1% of the amplitude of K 0 or the value of K 1 , and can be neglected here. These parameters are also affected by humidity and wind speed, but the common climate scenarios do not speculate on them. In addition to air temperature, atmospheric scenarios have been given for precipitation, and changes in snow cover have major impacts on lake ice growth and breakup. For a shift δTa in the annual air temperature, the impact to the possibility of lake freeze-up results from Eq. (9.12). The condition for freeze-up, τ f > 2τ L , says that the cold period Ta ≤ T c must be at least twice the length of the lake memory. With the parabolic air temperature curve Ta (t) = ct 2 + T a , the cold period is changed as ∆

δτ f δT a  , T a < Tc =−

τf 2 T −T ∆

(9.28)

∆

c

a

∆

Example 9.4. In boreal zone, a typical case could be Tc = 0◦ C, T a = −10◦ C and τ f = 100 d. Then in practice all lakes freeze since τ f > 2τ L holds when the lake depth or mixed layer depth is at most 50 m. If air temperature is shifted up by δT a = 2◦ C, τ f drops to 90 days that does not change freezing of lakes. Then, take T a = −4◦ C and τc = 30 d to represent a mild lake ice zone. The condition τ f > 2τ L holds for lakes with depth less than 15 m, and after the warming, τc = 22.5 d, and the freezing lakes are at most 11 m deep. The critical warming δT a = 4◦ C leads to ice-free lakes. ∆

The probability approach can be easily added to the present occurrence of freezing approach. The probability distribution (probability density p) is then needed for the annual minimum winter air temperature  T a . Ice forms annually in very shallow lakes with the probability P T a ≤ Tc , and if we have a climate change by δT a , the 

freezing probability is shifted to P T a + δT a ≤ Tc : ∆

∆

∆

∆

∆

9.4 Climate Change Impact on Lake Ice Season

331 ∆

Tc −∆  Ta



P T a + δT a ≤ Tc = ∆

∆

p(T )dT

(9.29)

−∞

Example 9.5. Take the Normal distribution with mean −5◦ C and standard deviation 5◦ C for the minimum temperature T a , and δT a = 2◦ C. Then, the freezing probability changes as



 • If Tc = −10◦ C, P T a ≤ Tc = 0.16 and P T a + δT a ≤ Tc = 0.08

 

• If Tc = −5◦ C, P T a ≤ Tc = 0.50 and P T a + δT a ≤ Tc = 0.34  



• If Tc = 0◦ C, P T a ≤ Tc = 0.84 and P T a + δT a ≤ Tc = 0.73 ∆

∆

∆

∆

∆

∆

∆

∆

∆

∆

∆

∆

For an even distribution T a ∼ [T1 , T2 ], the probability of freezing is 1 or 1 if Tc > T2 , T1 < Tc < T2 , or Tc < T2 , respectively. P Tˆa ≤ Tc = 0, TT2c −T −T1 ∆

∆

A climate change of δT a shifts the interval by δT a and the probabilities change correspondingly. The maximum change is δT a /(T2 − T1 ). ∆

Assuming now that freezing takes place. The linear cooling law Ta (t) = T f − αt resulted in the freezing date shown in Eq. (9.16). Climate change by δTa shifts the cooling up, and the resulting shift in the freezing date is directly obtained from Eq. (9.16): δt F = α −1 δTa

(9.30)

Thus, the time shift depends on the air temperature increase and the atmospheric cooling rate. E.g., in northern Europe, α ∼ 5 ◦ C month−1 , and the freezing date would change by 6 days for δTa = 1◦ C that agrees quite well with the observed time series data. For faster atmospheric cooling, the time shift is less and vice versa. Taking the parabolic air temperature Ta (t) = ct 2 − T a , the time derivative is dT a /dt = 2ct, i.e., the closer we are to the minimum temperature the less is the rate of change. This indicates that in mild climate the time shift of the freezing date should be more than in cold climate. The thickness of ice depends on the freezing-degree-days Sa that is determined by the climate (Eq. 9.22). The parameter a needs to be considered since in semiempirical models this parameter accounts for the influence of snow and heat flux from water, but the parameter h a is less critical. Thus, the total differential of ice thickness is ∆

δh =

a2 2aSa δSa + δa h + ha h + ha

(9.31)

In the case of climate warming, δSa < 0 but the δa may have either sign. Therefore, past evolution of ice thickness has shown mixed trends (e.g., Karetnikov 2017; Lei,

332

9 Lake Ice Climatology

2012), even though the trends in the length of ice season have been dominantly negative. For bare ice, a ≈ 3.4 cm(◦ C · d)−1/2 , and for strong snow insulation it is reduced by 50%. Therefore 0 ≤ |δa|/a ≤ 0.5.    2 2t The parabolic air temperature can be written as Ta = −T a τc − 1 , where τc ∆

is the length of the period Ta ≤ Tc . Then the total differential of the freezing-degreedays is δS a = −

 2

τc δT a + T a δτc 3 ∆

∆

(9.32) ∆

Thus, when the air temperature curve is shifted up by δT a > 0, then δτc < 0, and the freezing-degree-days decrease, and vice versa. ∆

Example 9.6. Take T a = −10◦ C and, τc = 100 d. Then c = 0.004 ◦ C d−2 , and we have Sa = 667 ◦ C d . For a climate change δ Tˆa = 1 ◦ C, we have / δτc =

−

1 c Tˆa

δ Tˆa = 5 d

and, finally δSa = −100 ◦ C d. Assuming a = 2.5 cm (◦ C d)−1/2 and h a = 10 cm, the thickness of ice changes from 55 to 50 cm. The quality of ice cover is categorized into weak and stable whether the ice is thinner or thicker than 20 cm. If the ice thickness decreases, the period of stable ice becomes shorter or even disappears. To reach the ice thickness of 20 cm, the freezing-degree-days need to sum to at least Sa = 80 ◦ C d for bare ice but with a protective snow cover the required value is Sa = 320 ◦ C d. Usually heat flux from water is small in the ice growth period, and its influence is hidden behind the uncertainties in snow accumulation and metamorphosis. In case the heat flux from water is large, the question is the in the climate sensitivity of the equilibrium thickness. From Eq. (4.39) we have for an air temperature increase by δTa :

ki T f − (Ta + δTa + TR ) he = − ha Qw

(9.33)

Thus, the equilibrium ice thickness is changed by −ki δTa /Q w . For Q w = 10 W m–2 we have δh = 0.05 δTa m ◦ C−1 . Ice extent increases as well in the ice growth period. The freezing front expands depending on the bottom topography and air temperature (see Eq. 9.26a,b). The climate change impact causes change in the timing of the front via δt = α −1 δTa . Therefore Eq. (9.26a,b) does not change but only the initial time when ice starts to form is shifted by δt = α −1 δTa . Then the ice concentration increases as before, and full ice coverage is achieved the time δt later than before. The period of full ice

9.4 Climate Change Impact on Lake Ice Season

333

cover decreases this much from the autumn side. When δTa is large enough, full ice coverage is no more reached. Ice Decay Ice decay was modelled in the previous Section by Eq. (9.18) where melting of ice was predicted from the initial time, ice thickness and the evolution of the heat flux. Solar radiation increases during the melting season, but also air–ice heat fluxes behave so, because the lake’s ice–water bath keeps the surface temperature at the melting point. Melting was solved as t B = t B (t M , h M ; γ , K 1 β) as shown in Eq. (9.20). The new start-up time can be obtained from the equation Q 0 t M , T f = 0 for a climate change δTa as: δt M = −

δTa β

(9.34)

For δTa = 1 ◦ C and β ≈ 1/6 ◦ C d−1 , we have δt M ∼ 6 d. γ /K 1 ≈ 0.1−1 . Another effect to the breakup date √ comes from less ice to melt. The length of the melting period is proportional to h M , and therefore the sensitivity of the breakup date to ice thickness is large for thin ice. From Eq. (9.20) we have ρi L f 1 δh M δt B = − δTa + √ β (γ + β K 1 )h M

(9.35)

Thus, breakup date becomes earlier when air temperature rises, and also due to decreased ice thickness or less ice to melt. Once melting has started, it progresses by the same rate basically independent of the climate within a range of δTa and ceases when all ice is melted. Time series analyses have shown that in the last century in general both freezing date and breakup date have changed about equally much, breakup date even with higher significance level. This is easy to understand in the view of the results of this section, and the higher significance level results from the strong role of solar radiation in the decay period.

9.4.3 Numerical Modelling Numerical models can be used to produce more detailed projections for the future ice seasons. However, the forcing scenarios are largely uncertain, and therefore rather sensitivity analysis is a good way to approach the problem. If air temperature change only is considered, the first order analytic model is usually quite good, but to examine the influence of snow accumulation and qualitative changes, numerical models provide a firmer basis (e.g., Leppäranta, 2009; Yang et al., 2012). An example is shown in Fig. 9.10 for southern Finland, where model simulations for a ‘normal’ winter are compared with those for winters where the mean air temperatures are 1 and 5 °C higher or lower than the present (1980–2010) baseline. In the

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9 Lake Ice Climatology

numerical model, the lake freezes during the first strong cold spell. The observed, present average maximum annual ice thickness of 45 cm decreases by 5 cm for the first 1 °C warming and 20 cm for the 5 °C warming. The ice break-up date also shifts to a much earlier date due to the earlier onset of melting, which reflects the change in the radiation balance and the fact that there is less ice to melt. The resulting ice break-up is 5 days earlier for the 1 °C warming, and 30 days earlier for the 5 °C warming. This illustrates the high instability of the ice season to warming. In a warm climate, ice may form in mid-winter but since it is relatively thin it melts, if the weather becomes warmer, or breaks at significant wind load. The results show that the 1 °C change brings the thickness curve down by 5–10 cm but has no effect on the form of the thickness cycle. With a 5 °C warming there are growth and decay periods through the winter and the maximum ice thickness is about 20 cm. In southern Finland the thickness of 20 cm is enough for a stable ice cover in small lakes, but in large or medium-size lakes stormy winds could break this cover and give rise to areas of open water at any time during the winter. This means a remarkable qualitative change in the lake ice seasons. The length of the ice season is still 3.5 months at 5 °C warming but the period of solid ice with good bearing capacity is almost vanished.

Fig. 9.10 Model sensitivity to the air temperature. The black solid line is the reference of present climate. The grey solid lines show the modelled ice thickness based on air temperature decrease by 5 and 1 °C, and the grey dashed line show modelled ice thickness based on air temperature increase by 5 and 1 °C. The stable ice cover thickness of 20 cm is drawn. (Yang et al., 2012)

References

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Lei, R., Leppäranta, M., Cheng, B., Heil, P., & Li, Z. (2012). Changes in ice-season characteristics of a European Arctic lake from 1964 to 2008. Climatic Change. https://doi.org/10.1007/s10584012-0489-2 Leppäranta, M. (1993). A review of analytical sea ice growth models. Atmosphere-Ocean, 31(1), 123–138. Leppäranta, M. (2009). Modelling of growth and decay of lake ice. In: G George (ed.), Climate change impact on European lakes (pp. 63–83), Springer-Verlag. Leppäranta, M. (2014). Interpretation of statistics of lake ice time series for climate variability. Hydrology Research, 45(4–5), 673–684. Leppäranta, M., & Seinä, A. (1985). Freezing, maximum annual ice thickness and breakup of ice on the finnish coast during 1830–1984. Geophysica, 21(2), 87–104. Leppäranta, M., & Wang, K. (2008). The ice cover on small and large lakes: Scaling analysis and mathematical modelling. Hydrobiologica, 599, 183–189. Leppäranta, M., Lindgren, E., Wen, L., & Kirillin, G. (2019). Ice cover decay and heat balance in Lake Kilpisjärvi in Arctic tundra. Journal of Limnology, 78(2), 163–175. Leppäranta, M., & Wen, L. (2022). Ice phenology in Eurasian lakes over spatial location and altitude. Water, 14, 1037. Livingstone, D. M. (2000). Large scale climatic forcing detected in historical observations of lakeice break-up. Verhandlungen Der Internationalen Vereinigung Für Theoretische Und Angewandte Limnologie, 27(5), 2775–2783. Livingstone, D., & Adrian, R. (2009). Modeling the duration of intermittent ice cover on a lake for climate-change studies. Limnology and Oceanography, 54(5), 1709–1722. Magnuson, J. J., Robertson, D. M., Benson, B. J., Wynne, R. H., Livingstone, D. M., Arai, et al. (2000). Historical trends in lake and river ice cover in the northern hemisphere. Science, 289, 1743–1746 and Errata 2001 Science, 291, 254. Marszelewski, W., & Skowron, R. (2006). Ice cover as an indicator of winter air temperature changes: Case study of the Polish Lowland lakes. Hydrological Sciences Journal, 51, 336–349. Maurer, J. (1924). Severe winters in southern Germany and Switzerland since the year 1400 determined from severe lake freezes. Monthly Weather Review, 52, 222, [Translater from German original in Meteorologische Zeitschrift, 41, 85–86. Mishra, V., Cherkauer, K. A., Bowling, L. C., & Huber, M. (2011). Lake Ice phenology of small lakes: Impacts of climate variability in the Great Lakes region. Global and Planetary Change, 76, 166–185. Mueller, D. R., Van Hove, P., Antoniades, D., Jeffries, M. O., & Vincent, W. F. (2009). High arctic as sentimental ecosystems: Cascading regime shifts in climate, ice cover, and mixing. Limnology and Oceanography, 54(6, part 2), 2371–2385. Prokacheva, V. G., & Borodulin, V. V. (1985). Long-term fluctuations of ice cover in Lake Ladoga. Meteorologiya i Gidrologiya, 10, 86–93. Robertson, D. M., Ragotszkie, R. A., & Magnuson, J. J. (1992). Lake ice records used to detect historical and future climatic change. Climate Change, 21, 407–427. Rodhe, B. (1952). On the relation between air temperature and ice formation in the Baltic. Geografiska Annaler, 1–2, 176–202. Salonen, K., Leppäranta, M., Viljanen, M., & Gulati, R. (2009). Perspectives in winter limnology: Closing the annual cycle of freezing lakes. Aquatic Ecology, 43(3), 609–616. Sharma, S., Blagrave, K., Magnuson, J. J., O’Reilly, C. M., Oliver, S., Batt, R. D., et al. (2019). Widespread loss of lake ice around the Northern Hemisphere in a warming world. Nature Climate Change, 9(3), 227–231. Wang, J., Bai, X., Hu, H., Clites, A., Holton, M., & Lofgren, B. (2012). Temporal and spatial variability of Great Lakes ice cover, 1973–2010. Journal of Climate, 25, 1318–1329. Wang, X., Qiu, Y., Zhang, Y., Lemmetyinen, J., Cheng, B., Liang, E., & Leppäranta, M. (2021). A lake ice phenology dataset for the Northern Hemisphere based on passive microwave remote sensing. Big Earth Data. https://doi.org/10.1080/20964471.2021.1992916

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Chapter 10

Future of Frozen Lakes

Lake ice has changed from a key factor of living conditions in the cold regions to a site for winter activities, such as ice fishing, skiing, and skating, and ice sailing

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9_10

339

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10 Future of Frozen Lakes

10.1 Role of Ice Cover Lake ice forms in cold regions in winter. Ice cover is normally seasonal but in high latitudes and high altitudes permanently ice-covered lakes exist. The presence of seasonal lake ice may last more than half a year, and its thickness can reach up to two meters. In the ephemeral zone, lakes freeze in cold winters but stay open in mild winters. The snow and ice season modifies the hydrological year by its influence on the annual distribution of runoff and by the problems caused by ice to the management of water resources. Ecology in ice-covered lakes must adapt to the long cold, dark and quiet winter period. The research of freezing lakes has experienced a major rise after the year 2000, much due environmental and climate problems (Fig. 10.1). Lake ice is a degree simpler than sea ice and river ice (Fig. 10.2). Lake ice cover is mostly immobile, circulation in the water body beneath the ice cover is weak, and freshwater lake ice is poor in impurities. But lakes with strong throughflow may have river ice type processes (Ferrick & Prowse; 2002; Hicks, 2016; Shen, 2006), the structure of ice in brackish and saline lakes is similar to the structure of sea ice (e.g., Thomas, 2016; Wadhams, 2000; Weeks, 2010), and drift ice occurs in very large lakes (Leppäranta, 2011). Our knowledge of ice in saline waters comes mainly from marine research, and these results are to a large degree applicable for saline lakes.

Fig. 10.1 Students taking water samples from beneath lake ice, Winter limnology field course of the University of Helsinki. Photograph by the author

10.1 Role of Ice Cover

341

Fig. 10.2 Lake Baikal ice season is quite complicated due to its great depth and horizontal scale. In large lakes in general the ice shows significant horizontal displacements due to forcing from winds or water currents. Photograph by Oleg Timoshkin, printed by permission

Hypersaline lakes would rather require specific lake water analyses for ice research rather than extrapolation of the sea ice knowledge. Frozen lakes have belonged to people’s normal life in the cold regions. Solid ice cover has been an excellent base for traffic across lakes and to transport cargo, provided the ice has been thick enough. Indeed, ice cover has isolated island inhabitants from the mainland in periods when the ice has been too weak for on-ice traffic but still limits boating. Until about mid-twentieth century, when refrigerators were not common, the cold content of lake ice was utilized by storing lake ice for summer. Openings were sawed into the lake ice cover for household water and for washing clothes at the site. For ice-covered lakes, special techniques were developed for domestic and commercial winter fishing. More recently, sports and recreation activities on lake have become popular. This book, Freezing of lakes and the evolution of their ice cover, 2nd edition, presents a revision of the first edition (2014) with an up-to-date (year 2022) status of knowledge of the physics of lake ice with applications. It is based on a review of the literature and the lake research of the author since 1990s. The focus is on freshwater lakes, with river ice and sea ice results utilized when helpful. The role of dissolved substances of the parent water is explained here based on the results of sea-ice research.

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The first chapter introduced the topic with a brief historical overview. Chapter 2 presented lakes, their classification, the zones of ice-covered lakes, and remote sensing. Ice formation and the structure and properties of lake ice were treated in Chap. 3, with ice impurities and light transfer in ice. Chapter 4 contained lake ice thermodynamics from the freezing of lakes to ice melting, with thermodynamic models included. Lake ice mechanics was the topic of Chap. 5 including engineering questions such as ice forces and bearing capacity of ice. A section was included on drift ice in large lakes. Glaciers and pro-glacial lakes were treated in Chap. 6. Quite exotic lakes were introduced, particularly lakes at the top and bottom of glaciers and ice sheets. Chapter 7 focused on the water body beneath a lake ice cover with stratification, circulation, and limnology. Environmental applications were treated in Chap. 8, considering the ecology of lake water bodies with limitations brought by the ice, and the life of human settlements at areas with freezing lakes. Chapter 9 discussed lake ice climatology considering time series, mathematical modelling, and climate change impact, and closing words are written in the present in this chapter. References will follow in Chap. 11. Study problems with solutions were given as examples in the text. Useful constants and formulae have been collected in the Annex.

10.2 Science and Technology Scientific research on lake ice was commenced in the 1800s, and at the same time lake ice monitoring was commenced in northern Europe. Ice engineering problems were included in research from the late 1800s, concerning shipping, ice forces on structures, and the bearing capacity of ice. Early literature of the physics of ice-covered lakes is quite sparse. The important books include Shumskii (1956), Pounder (1965), Pivovarov (1973), Michel (1978), and Ashton (1986). Pivovarov (1973) presented a scientific monograph on freezing lakes and rivers with weight on the liquid water body, while Ashton (1986) had a more engineering point of view on this topic, see also Ashton (1980) for an overview paper on lake ice. Also, from the early stage of limnology the ecology of frozen lakes has gained attention, especially in the Central Europe. More recent material is given in the article collections of polar lakes (Vincent & Laybourn-Parry, 2008), limnology (George, 2009), and in the review of the physics (Kirillin et al., 2012) of frozen lakes. Research on winter biology of lakes has largely increased (Salonen et al., 2009; Shuter et al., 2012). Questions of life in extreme conditions have brought in more research of lakes with perennial ice and pro-glacial lakes (Hawes et al., 2011b; Keskitalo et al., 2013; Vincent & Laybourn-Parry, 2008), especially in the Antarctic continent. In large lakes the ice may break and drift considerable distances which influences on the evolution of the winter conditions in lakes and utilization of the lakes (Fig. 10.2). In all, the research of ice-covered lakes turned out to be for long, until about 1970s, scattered and occasional. Thereafter, more effort was put on polar lakes when the access to reach them was better and measurement techniques were feasible.

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343

The leading topics are now the impact of climate change on lakes and the ecology of freezing lakes. Also, together with free access to high or moderate resolution satellite imagery, such as Aqua/Terra and Envisat, monitoring of lake ice cover has expanded to medium-size lakes. The annual ice cycle depends on the atmospheric mass and heat fluxes and is particularly sensitive to air temperature and snowfall. Furthermore, the sensitivity of modelled thickness is in general inversely proportional to the thickness. Therefore, when the ice is thin, a more sophisticated model is needed to simulate the growth of ice. The connection between ice thickness and snow accumulation is quite complicated since the timing of the snowfall is critical (Fig. 10.3). Also, the ice break-up date depends on the structure and thickness of the ice and snow and needs a physically realistic model, with stratigraphy, to explore its variability. The main weakness in the present thermodynamic lake ice models is indeed in the accounting for snow properties and processes. Another question is the ice melting, which takes place at both boundaries and in the interior, the partitioning depending on the ice structure and solar radiation. Fig. 10.3 Snow on lake ice is the main present problem in thermodynamic ice models. The photograph shows strange noodle-like surface structure formed during a snowstorm on an open lake, Lake Suolijärvi in Finland. Photograph by Mr. Vesa Kaloinen, Tuulos, printed with permission

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The modelling problem of lake ice thickness can be approached with semi-analytic or numerical methods. Semi-analytic methods are based on the degree-days and provide the first order approximation of the freezing—ice growth—melting cycle from which simple relations between winter climate and ice season characteristics can be derived. Numerical models are designed to include the full ice physics in the simulations. In these models the stratigraphy of ice, snow physics and atmospheric forcing can be treated in a realistic way, and the evaluation of climate change consequences becomes more detailed than with semi-analytic models. Slush layers are well reproduced, and the physical representation of melting ice is more realistic. In addition, mechanical models are used to examine the breakage of ice, which is a critically important a factor in the character of a lake ice season. Ice drift models have been constructed for marine basins, down to the size of 100 km, but in lakes the corresponding work has been quite limited. The main region of lake ice drift modelling has been the Great Lakes of North America, where shipping has needed information of ice drift and deformation. A major characteristic of lake ice drift is that it consists of short-term displacements between immobile states that require a very close approximation to plastic behaviour in the modelling. The occurrence of ice displacements in any given basin depends on the ice thickness, and therefore climate warming may influence the quality of ice cover. In the lake water body, physical phenomena and processes are very different under ice cover from the open water conditions. The ice cover cuts the transfer of momentum from the wind to the water body that damps turbulence and mixing. The surface water temperature is at the freezing point, and there is very little vertical transfer of heat, apart from geothermal lakes. In all, the temperature structure and circulation are quite stable. In fully ice-covered lakes the water flow under ice can be in laminar or laminar-turbulent transition state that makes the ice–water body interaction a difficult problem for observation techniques and modelling. Even though the water velocities are small, they are stable and result in significant long-term transport of water with its impurities. However, in very large lakes, the ice may experience episodic movements and disturb the water body. In spring, solar radiation provides a strong downward flux of heat, which constitutes the strongest heat flux into fully ice-covered lakes, and the ice melt water with its impurities is released into the water column. Lake ice belongs to the part of cryosphere, which closely interacts with human living conditions. The ice cover has caused problems, but people have been learnt to live with them and to utilize the ice. Especially, in the present time this is true for on-ice traffic and recreation activities. Ice fishing, winter sports such as skiing, skating and ice sailing are now popular winter activities on frozen lakes. In the past, lake ice was used to store perishable products, when it was removed during the winter and stored for the summer. On the negative side, the freezing of lakes makes boating impossible and limits their use for commercial and recreational transport. The bearing capacity of ice is a major safety question. It increases with ice thickness squared so that for a skier or skater 5 cm is enough. When the thickness of the ice is very close to the limiting minimum, special care must be taken because of the natural variability of ice thickness and the resonance effects with shallow water waves for moving loads (Fig. 10.4).

10.3 What Will Be?

345

Fig. 10.4 Travelling on ice is always risky. To prepare for crossing by foot, skiing, or skating, it is good to take a sharp stick to test ice and ice picks to get up after ice breakage, also rope to help others. Photograph by Finnish Swimming and Lifesaving Federation, printed with permission

10.3 What Will Be? Lakes are normally frozen in winter near sea level at latitudes higher than 40–50° and at high altitudes down to the equator. In the northern and southern caps there are open water lakes due to geothermal heat, great depth, or high salinity. Perennial lake ice is rare and found only in specific high-polar and high-altitude environments. The annual course of the thermal structure of lakes is forced by local climatology, and the boundaries of lake ice zones consequently vary along the climate variations. The low latitude limit is the mid-winter 0 °C isothermal where very shallow lakes still may freeze over. The physics of ice-covered lakes is presently rather well understood (Kirillin et al., 2012). The main questions are connected to the melting of ice and to the mechanical displacements and drift of the ice cover. New coupled ice–liquid water body models are needed to improve our understanding of ice season processes and phenomena. The coupling takes place via the solar radiation and heat loss from the lake to the atmosphere, which both depend on the thickness of the ice. Two- and three-dimensional modelling of ice-covered lakes is required more with appropriate process-based links to the seasonal dynamics of the lakes. The climate change question has focused on the physics on the ice phenology, maximum annual ice thickness, and ice extent, as well as implications of these changes to lake ecosystems. The expected changes can be qualitative when the boundaries of perennial ice, seasonal ice and ephemeral zones move with climate variations. In a warming climate, seasonally freezing lakes may change into irregularly freezing lakes, and ice may fully disappear from the ephemeral lake ice zone, as has taken place in Central Europe after the Little Ice Age (Fig. 10.5). The connection between lake ecology and physics is strong in that the ice cover structure and properties provide information about light level in the water, presence of open water spots, and liquid water within the ice cover. Rapid changes bring stresses through light and temperature conditions as well as through budgets of oxygen and nutrients. Climate change issues also necessitate a better understanding of the role

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10 Future of Frozen Lakes

Fig. 10.5 In the Little Ice Age, lakes in Central Europe were covered by ice with bearing capacity to travel and enjoy. Painting by a Flemish artist Jakob Grimmer (1525–1590), from Wikimedia

played by a lake ice cover in the emission of greenhouse gases, especially methane, into the atmosphere, and in the global carbon budget. Climate warming has taken place during the last 30 years as shown in the rise of Northern Hemispheric temperatures (IPCC, 2021). Climate impact on lakes is thoroughly discussed in the book of George (2009). The recent climate warming has affected the ice regime of lakes as shown by numerous publications. A review on the time series work was given by Adrian et al. (2009) and Sharma et al. (2019). Magnuson et al. (2000) reported trends of 5.8 days later freeze-up dates and 6.5 days earlier break-up dates per 100 years in lake and river ice in the northern hemisphere (1846–1995) that has been repeated in more recent works. Trends of the same order of magnitude were found in lakes in Finland (Korhonen, 2006), and Berlin and Brandenburg lakes by Bernhardt et al. (2011). For the irregularly freezing Müggelsee, Berlin, trends were reported for later freezing (5.7 days) and earlier thawing (6.8 days) for the period 1947–2007. In Lake Kilpisjärvi, Arctic tundra in northern Finland, since 1964 the trend has been later freezing by 11.5 days and earlier ice breakup by 5.0 days per 50 years (Lei et al., 2012). Consequently, the implications of a warming climate to the lake ice season are first shorter season and then thinner ice. However, depending on the snowfall conditions, snow-ice formation may increase to counter at least part of the effect of rising air temperatures. Thinner ice also means more breakable ice with more areas of open water, which changes fluxes of heat and moisture between the lake and the atmosphere and may generate frazil ice. Frazil ice formation may, in turn, lead to accumulation of sediments in the ice cover via frazil capturing of particles in the water body and via anchor ice formation. From the point of view of lake ecology, shorter ice season and presence of openings mean less oxygen deficit problems. But the openings will

References

347

also create a new type of winter environment in lakes, which have hitherto been completely covered with ice. The impact of the changes of physical conditions on the ecological status of lakes is less clear and is the topic of future investigations. The key questions are the physical implications of lake ice fracturing and the influence of the projected changes in the ice structure on the transfer of light. When a lake is frozen, the circulation in the water column is weak and driven by solar radiation and heat flux from the lake bottom. These weak circulations are, however, known to be ecologically important since they influence the oxygen budget, regulate the growth of phytoplankton, and have a major effect on the winter survival of lake fish. For human living conditions, shorter ice seasons, while extending the open water season, would severely or even drastically limit the traditional on-ice activities. A key point is the stable ice period when the thickness of the ice is above the critical level of about 20 cm. Then the ice is a stable platform for on-ice traffic, fishing, and recreation. Even a modest warming will have a major impact on the ice season viewed from this practical viewpoint. On the other hand, the lakes would be open for a much longer period and allow the extension of many water-based activities. For the ice cover to be “useful”, it must be thick enough to serve as a solid platform. Lakes in the ephemeral zone, and to a lesser extent those in the unstable zone, are not useful in that sense. A warming by 2–3ºC may reduce the ice thickness by about 20 cm that would mean a level below the critical level in southern Finland and corresponding climatic zones, where presently mean maximum annual ice thickness is 30–40 cm. Even then, the length of the ice season with an unsafe ice cover can still be three months (see Yang et al., 2012). If there were a continuous warming climatic trend, these qualitative changes would become obvious toward the end of the twenty-first century.

References Adrian, R., O’Reilly, C., Zagarese, H., Baines, S. B., Hessen, D. O., Keller, W., et al. (2009). Lakes as sentinels of climate change. Limnology and Oceanography, 54, 2283–2297. https://doi.org/10. 4319/lo.2009.54.6_part_2.2283 Ashton, G. (1980). Freshwater ice growth, motion, and decay. In S. Colbeck (Ed.), Dynamics of snow and ice masses (pp. 261–304). Academic Press. Ashton, G. (Ed.). (1986). River and lake ice engineering (p. 485). Water Resources Publications. Bernhardt, J., Engelhardt, C., Kirillin, G., & Matschullat, J. (2011). Lake ice phenology in BerlinBrandenburg from 1947–2007: Observations and model hindcasts. Climatic Change. https://doi. org/10.1007/s10584-011-0248-9 Ferrick., M. G., & Prowse, T. D. (Eds.). (2002). Special issue: Hydrology of ice-covered rivers and lakes. Hydrological Processes, 16(4), 759–958. George, G. (Ed.). (2009). The impact of climate change on European lakes (p. 607). Springer-Verlag. Hawes, I., Safi, K., Webster-Brown, J., Sorrel, B., & Arscott, D. (2011). Summer–winter transitions in Antarctic ponds II: Biological responses. Antarctic Science, 23, 243–254. Hicks, F. (2016). An introduction to river ice engineering: For civil engineers and geoscientists (p. 170). CreateSpace, Amazon.

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IPCC. (2021). Sixth Assessment Report. Keskitalo, J., Leppäranta, M., & Arvola, L. (2013). First records on primary producers of epiglacial and supraglacial lakes in the western Dronning Maud Land, Antarctica. Polar Biology, 36(10), 1441–1450. https://doi.org/10.1007/s00300-013-1362-0 Kirillin, G., Leppäranta, M., Terzhevik, A., Granin, N., Bernhardt, J., Engelhardt, C., et al. (2012). Physics of seasonally ice-covered lakes: A review. Aquatic Sciences, 74, 659–682. https://doi. org/10.1007/s00027-012-0279-y Korhonen, J. (2006). Long-term changes in lake ice cover in Finland. Nordic Hydrology, 37, 347– 363. Lei, R., Leppäranta, M., Cheng, B., Heil, P., & Li, Z. (2012). Changes in ice-season characteristics of a European Arctic lake from 1964 to 2008. Climatic Change. https://doi.org/10.1007/s10584012-0489-2 Leppäranta, M. (2011). The drift of sea ice (2nd ed.) (p. 347). Heidelberg, Germany: Springer-Praxis. Magnuson, J. J., Robertson, D. M., Benson, B. J., Wynne, R. H., Livingstone, D. M., Arai, T., et al. (2000). Historical trends in lake and river ice cover in the northern hemisphere. Science, 289,1743–1746; Errata. (2001). Science, 291, 254. Michel, B. (1978). Ice mechanics (499 pp.). Laval University Press. Pivovarov, A. A. (1973). Thermal conditions in freezing lakes and rivers (136 pp.). Wiley, New York, N.Y. [Translated from Russian]. Pounder, E. R. (1965). Physics of ice (p. 160). Pergamon Press. Salonen, K., Leppäranta, M., Viljanen, M., & Gulati, R. (2009). Perspectives in winter limnology: Closing the annual cycle of freezing lakes. Aquatic Ecology, 43(3), 609–616. Sharma, S., Blagrave, K., Magnuson, J. J., O’Reilly, C. M., Oliver, S., Batt, R. D., et al. (2019). Widespread loss of lake ice around the Northern Hemisphere in a warming world. Nature Climate Change, 9(3), 227–231. Shen, H. T. (2006). A trip through the life of river ice—research progress and needs. In Proceedings of the 18th IAHR International Symposium on Ice (pp. 85–91). Hokkaido University, Sapporo, Japan. Shumskii, P. A. (1956). Principles of Structural Glaciology [in Russian]. (1964 by Dover Publications (p. 497), Inc. English trans.). Shuter, B. J., Finstad, A. G., Helland, I. P., Zweimüller, I., & Hölker, F. (2012). The role of winter phenology in shaping the ecology of freshwater fish and their sensitivities to climate change. Aquatic Sciences, 74(4), 637–657. Thomas, D. N. (2016). Sea ice (3rd ed) (p. 664). Wiley–Blackwell, Chichester. Vincent, W. F., & Laybourn-Parry, J. (Eds.). (2008). Polar lakes and rivers (p. 322). Oxford University Press. Wadhams, P. (2000). Ice in the ocean (p. 364). Gordon & Breach Science. Weeks, W. F. (2010). On sea ice (p. 664). University of Alaska Press. Yang, Y., Leppäranta, M., Li, Z., & Cheng, B. (2012). An ice model for Lake Vanajavesi, Finland. Tellus A, 64, 17202. https://doi.org/10.3402/tellusa.v64i0.17202

Annex: Ice Properties and Useful Formulae

A.1 Properties of Ice Ih and Liquid Water This section presents a collection of the properties of solid and liquid phases of fresh water. The basic properities of freshwater ice as a function of temperature are shown in Tab. A.1.1, the dependence of freezing point on pressure is given in Tab. A.1.2, and properties of liquid water are given in Tab. A.1.3. The values are based on the books listed in the references. Then, the Unesco equation of state for sea water is presented. It serves here for fresh water and as an approximation for saline lake water.

Table A.1.1 Basic properties of freshwater ice vs. temperature 0 °C

−10 °C

−20 °C

−30 °C

Density (kg m−3 )

916.7

918.7

920.3

921.6

Thermal expansion coefficient, volume (10−6 °C−1 )

159

155

149

143

Adiabatic compressibility (10−5 MPa−1 )

13.0

12.8

12.7

12.5

kg−1

°C−1 )

2.11

2.03

1.96

1.88

Thermal conductivity (W m−1 °C−1 )

2.14

2.3

2.4

2.5

Relative permittivity (dielectric constant)

91.6

94.4

97.5

99.7

333.6

−

−

−

Specific heat (kJ

Latent heat of melting (kJ

kg−1 )

Thermal emissivity

0.96–0.98

Table A.1.2 Dependence of freezing point on pressure Pressure (bar)

1

10

100

1000

Melting point (°C)

0.00

−0.06

−0.74

−8.80

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9

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350

Annex: Ice Properties and Useful Formulae

Table A.1.3 Properties of liquid fresh water (at pressure 1 bar)

Molecular weight (g mol−1 )

18.016

Gas constant for vapour (J °C−1 kg−1 )

461

Temperature of maximum density (°C)

3.98

Boiling point (°C)

100

Triple point

0.1 °C, 611.73 Pa

Thermal emissivity

0.95

Temperature

0 °C

10 °C

Density of water (kg m−3 )

999.84

999.70

Viscosity (N s m−1 )

1.793 ·10−3

1.307 ·10−3

m−1 )

75.64

74.23

Specific heat of water (kJ kg−1 °C−1 )

4.2176

4.1921

Thermal conductivity (W m−1 °C−1 )

0.561

0.580

Relative permittivity

87.90

83.96

Latent heat of evaporation (MJ kg−1 )

2.49

2.47

Surface tension (N

Equation of state of lake water General form (Unesco 1981): ρw (T , S, p) =

ρw (T , 0, 0) − ∆(T , S, 0) . 1 − K (TpS, p)

where ρw is density, T is temperature, S is salinity, p is pressure, ∆ is salinity correction, and K is secant bulk modulus. Pure water: ρw (T , 0, 0) = 999.842594 + 6.793952 · 10−2 T − 9.095290 · 10−3 T 2 + 1.001685 · 10−4 T 3 − 1.120083 · 10−6 T 4 + 6.536332 · 10−9 T 5 Salinity correction (sea water): ∆(T , S, 0) = S(0.824493 − 4.0899 · 10−3 T + 7.6438 · 10−5 T 2 − 8.2467 · 10−7 T 3 + 5.3875 · 10−9 T 4   + S 3/2 −5.72466 · 10−3 + 1.0227 · 10−4 T − 1.6546 · 10−6 T 2 + 4.8314 · 10−4 S 2

Annex: Ice Properties and Useful Formulae

351

Compressibility correction (fresh water and sea water): K (T , 0, 0) = 19652.21 + 148.206T − 2.327105T 2 + 1.360477 · 10−2 T 3 − 5.155288 · 10−5 T 4 K (T , S, 0) = K (T , 0, 0) + S(54.6746−0.603459T + 1.09987 · 10−2 T 2 − 6.1670 · 10−5 T 3 )   + S 3/2 7.944 · 10−2 + 1.6483 · 10−2 T − 5.3009 · 10−4 T 2 K (T , S, p) = K (T , S, 0) + p(3.239908 + 1.43713 · 10−3 T + 1.16092 · 10−4 T 2 − 5.77905 · 10−7 T 3 )   + pS 2.2838 · 10−3 − 1.0981 · 10−5 T − 1.6078 · 10−6 T 2 + 1.91075 pS 3/2   + p 2 8.50935 · 10−5 − 6.12293 · 10−3 T + 5.2787 · 10−8 T 2   + p 2 S −9.9348 · 10−7 + 2.0816 · 10−8 T + 9.1697 · 10−10 T 2

A.2 Physical Information of the Environment This section gives the physics constants referred in text (Table A.2.1), the measures of the Earth (Table A.2.2) General physics constants, method to evaluate incident solar radiation on top of the atmosphere (Table A.2.3), and the atmospheric properties used in the book (Table A.2.4). The Sun–Earth distance r 

AU r

2 = 1.000110 + 0.034221 cos j + 0.001280 sin j + 0.00719 cos 2 j + 0.000077 sin 2 j

Table A.2.1 General physics constants

Gravitational constant

G = 6.67430·10−11 N m2 kg−2

Avogadro’s number

N A = 6.022142·1023

Planck’s constant

h= 6.626068·10−34 J s

Boltzmann’s constant

k B = 1.3807·10−23 J K−1

Stefan-Boltzmann constant

σ = 5.670400·10−8 W m−2 K−4

Velocity of light in vacuum c0 = 2.99792458·108 m s−1 Universal gas constant

R = 8.3145 J K−1 mol−1

Celcius–Kelvin conversion

T [K] = T [°C] + 273.15

352 Table A.2.2 Earth

Table A.2.3 Sun

Annex: Ice Properties and Useful Formulae Equatorial radius

6378.140 km

Polar radius

6356.755 km

Surface area

5.101·1014 m2

Mean density

5515 kg m−3

Standard acceleration due to gravity

g = 9.80665 m s−2 (at 45° latitude)

Period of rotation

23 h 56 min 4.10 s

Rotation rate

 = 0.7292·10−4 s−1

Inclination of rotation axis

ε = 23°27’

Q sc = 1367 W m−2

Solar constant

Sunlight (380—750 nm) fraction γ ≈ 0.48 (range 0.44–0.50)

where AU = 149.60·109 m is the astronomical unit equal to the mean Earth–Sun 2π is the year angle, and J is the day of year (J = 1.0 on distance, j = ( J − 1) 365 January 1st, 00.00 h). The solar zenith angle Z , solar altitude is h = π/2 − Z , cos Z = sin h = sin φ sin δ + cos φ cos δ cos τ   2π sin δ = sin ε sin ( j − 80) 365 τ = τG M T −

+ ∆t 15◦

where φ is the latitude, δ is the declination, τ is the local solar hour angle (zero at solar noon), ε = 23°27' is the inclination of ecliptic, τG M T is hour in GMT (Greenwich Mean Time), is longitude, and ∆t is time correction, |∆t| < 30 min, which arises from the ellipticy of Earth’s orbit around the Sun and is available in astronomical tables. Density Dr y air ρa =

pa Ra T

pa —air pressure. E.g., ρa = 1.293 kg m−3 at 0 °C and pa = p0 .

Annex: Ice Properties and Useful Formulae Table A.2.4 Atmosphere

353

Average molecular mass

28.97 g mol−1

Standard atmospheric pressure

p0 = 1013.25 mbar

Gas constant of dry air

Ra = 287.04 J kg−1 °C−1

Kinematic viscosity (0 °C)

νa = 13.28 · 10−6 m2 s−1

Thermal diffusivity (0 °C)

μa = 19.06 · 10−6 m2 s−1

Specific heat at constant pressure

c p = 1.004 kJ kg−1 °C−1

Specific heat constant volume

cv = 717 J °C−1 kg−1

Thermal conductivity

2.4 ·10−2 W m−1 °C−1 ( p0 , 0 ◦ C)

Atmospheric transmissivity TR ≈ 0.9

Saturation water vapour pressure (mbar) over liquid water surface (ew ) and over ice surface (ei ) as functions of temperature (°C) are given by 0.7859 + 0.03477T 1 + 0.00412T log10 ei (T ) = log10 ew (T ) + 0.00422T (T ≤ 0 ◦ C)

log10 ew (T ) =

The relation between specific humidity (q) and water vapour pressure (e) is q=

0.622e pa

References

Line, D. R. (Ed.) (2001). CRC handbook of chemistry and physics (82nd edn. 2001–2002). CRC Press 2001. Gill, A. (1982). Atmosphere-ocean dynamics. Academic Press. Petrenko, V. F., & Whitworth, R. W. (1999). Physics of ice. Oxford University Press. Unesco. (1981). The practical salinity scale 1978 and the international equation of state of seawater 1980. Technical Papers in Marine Science, 36.

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9

355

Index

A Ablation, 207 Agglomerate ice, 81 Albedo, 98 positive albedo feedback, 219 Antarctica, 37 Dronning Maud Land, 37 dry valleys, 36 Lake Vostok, 38 Archimedes law, 80 Atmosphere adiabatic cooling, 47 air pressure, 45 boundary layer above lake, 117 deposition, 23 potential temperature, 117 wind, 47

limnological, 32 Climate, 39 cold climate zone, 30 El Niño–Southern Oscillation (ENSO), 39 Global Climate Animations, 31 Köppen–Geiger system, 39 North Atlantic Oscillation (NAO), 39 Climate change, 328 climate impact, 328 Coefficient of thermal expansion, 238 Convection, 237 convective layer, 264 Cooling process, 242 weighted air temperature sum, 244 Coriolis, 189 Coriolis acceleration, 257

B Baltic Sea Baltic Ice Lake, 223 Bearing capacity, 175 deflection, 178 dynamical loads, 180 strengthen, 298 Biodiversity, 290 Bottom boundary layer, 283 Bottom sediment, 236, 280

D Density of lake water, 25 Displacements, 172 Dissolved gases, 25 anoxic conditions, 283 diffusion coefficient of DO, 285 dissolved oxygen, 25 oxygen depletion, 282 saturation level, 281 Drainage basin, 236 Drift ice, 184 free drift, 186 ice state, 189 viscous-plastic rheology, 186

C Cabbeling, 237 Classification, 19 ecological, 25 geological, 19 ice season, 33

E Ecosystem, 209

© Springer Nature Switzerland AG 2023 M. Leppäranta, Freezing of Lakes and the Evolution of Their Ice Cover, Springer Praxis Books, https://doi.org/10.1007/978-3-031-25605-9

357

358 carbon dioxide, 288 chlorophyll a, 266 cyanobacteria, 227 diatom, 290 fish, 227 macrophyte, 287 microbial mat, 227, 290 nutrients, 279 pH, 227 photosynthesis, 271 phytoplankton, 227 zooplankton, 227 Elastic, 164 Poisson’s ratio, 165 Young’s modulus, 164 Electromagnetic radiation, 44 Epiglacial lake, 207 Equation of motion, 189 drag coefficient, 190 internal friction, 191 pressure gradient, 190 water–ice stress, 190 wind stress, 190

F Fetch, 184 Fishing, 301 Flexural strength, 168 Flooded water, 258 Fractal, 19 Freeze-up, 249 Freezing point, 236

G Geothermal, 229 Glacier, 206 Great Lakes of North America, 185

H Heat budget, 110 atmospheric thermal radiation, 111 heat flux from the bottom sediments, 111 heat flux from the lake water to ice, 111 latent heat, 111 Heat conduction law, 215 length scale of diffusion, 215 Humidity, 46 absolute, 46 relative humidity, 46 Hydrostatic pressure, 27

Index Hypsographic curve, 18 I Ice compactness, 186 Ice concentration, 186 Ice conservation law, 193 Ice drift model, 195 Ice extent, 308 Ice-free, 308 Ice growth, 124 ate of frazil accumulation, 138 atmospheric surface layer as a buffer, 132 equilibrium thickness, 129 freezing-degree-days, 128 heat diffusion, 127 model for frazil ice, 138 snow as an Insulator, 134 Stefan formula, 145 Stefan’s problem, 127 Ice harvesting, 280 Ice load, 182 indentation factor, 183 Korzhavin, 183 pier, 183 Ice monitoring, 294 Ice occurrence, 313 Ice phenology, 293 Ice plate on water foundation, 173 characteristic length, 173 Ice road, 159 traffic regulations, 297 Ice season, 236 breakup, 308 ice-off, 317 ice-on, 309 stable period, 309 Ice sheet, 206 Ice thickness, 35 Ice thickness distribution, 185 Ice time series, 312 fractile method, 314 Impurities, 24 brine volume, 89 coloured dissolved organic matter (CDOM), 27 ice sediments, 90 Internal waves, 262 Inverse winter stratification, 236 L Lake Baikal rings, 258

Index Lake ice, 69 anchor ice, 81 c-axis, 64 congelation ice, 70 frazil ice, 78 habitat of biota, 292 hermal conductivity, 125 macrocrystal, 69 primary ice, 70 quality of ice, 302 secondary ice, 70 specific heat, 125 stratigraphy, 76 superimposed ice, 70 Lake Ladoga, 18 Lakes brackish, 24 Caspian Sea, 194 dimictic, 32 fladas, 24 geothermal, 30 heat content, 110 hypersaline, 28 Inner Mongolia, 35 Kilpisjärvi, 27 Lake Baikal, 29 Lake Päijänne, 194 Lake Peipsi, 197 Lake Vendyurskoe, 258 meromictic, 32 Pääjärvi, 27 Lake water chlorophyll a, 98 freezing point, 28 salinity, 28 supercoolin, 66 temperature of maximum density, 28 turbidity, 27 Laminar–turbulent transition, 250 Landfast ice, 184 Large lake, 184 Latent heat of freezing, 124 Lead, 175 Light penetration depth, 28 euphotic depth, 28 Secchi depth, 28 Light transfer, 93 apparent optical properties, 95 Beer–Lambert law, 95 diffuse attenuation coefficient, 96 inherent optical properties, 94 irradiance, 95 optically active substances (OAS), 266

359 radiance, 93 suspended matter, 266 Linear surface heat flux, 320 M Mathematical modelling, 309 analytic modelling, 320 Medium-size lake, 197 Melting, 124 candled ice, 142 internal melting, 140 positive-degree-day, 146 Methane, 286 Mixed layer model, 245 Flake, 246 N Near-surface layer, 308 Numerical lake ice models, 147 heat budget model, 147 quasi-steady, 148 time-dependent lake ice model, 149 two-phase lake ice model, 152 Nunataq, 225 O On-ice traffic, 278 P PAR, 266 Perennial ice, 36, 226 Planck’s law, 41 Plastic medium, 165 flow rule, 187 strain hardening, 166 yield strength, 194 yield stress, 165 Polar winter, 280 Porosity, 124 critical porosity, 213 porosity equation, 131 Potential temperature, 237 Prandtl number, 247 Pressure ridge, 187 Pre-winter, 239 Proglacial lakes cryoconite hole, 209 epishelf lake, 208 perennial, 223 recurrent supraglacial lakes, 218 subglacial lake, 207

360 Q Quantum irradiance, 268

R Rafting, 173 Rayleigh number, 252 Remote sensing, 50 drone, 172 radiometry, 56 SAR, 172 thermal infrared, 294 Reynolds number, 249 Rheology, 160 broken ice, 160 compressive strength, 167 continuum, 160 granular medium, 160 internal stress, 161 shear strength, 168 strain, 162 strength, 166 tensile strength, 168 Richardson number, 238 River inflow, 251 Rossby number, 251

S Saline water, 254 Secchi depth, 267 Seiche, 255 Shallow lake, 249 Shallow water wave, 297 Shallow water wave speed, 251 Shoreline, 292 erosion, 293 fringe, 292 ice push, 292 Skating, 301 Slab model, 242 Small lake, 251 Snow, 48 crystal, 48 density of snow, 74 liquid water content, 74 metamorphosis, 74 slush, 79 snow on lake ice, 72 thermal conductivity of snow, 74 Snow accumulation, 21 Snowfall, 253 Stable ice cover, 251

Index Stefan-Boltzmann law, 41 Stratification of lake, 30 halocline, 32 turnover event, 32 Sun, 42 atmospheric transmittance, 112 cloudiness correction, 113 Earth–Sun distance, 112 solar constant, 42 solar elevation, 112 Supraglacial lake, 19

T Temperature of maximum density, 251 Terrestrial radiation, 44 Thermal bar, 241 Thermal crack, 170 Thermal memory, 308 Thermobaric, 237 Thermocline, 246 Thermohaline circulation, 240 Tibetan plateau, 24 Turbulence model, 247 kε-models, 247 PROBE, 248 Turbulent boundary layer, 117 bulk transfer coefficient, 119 eddy covariance, 118 friction velocity, 118 mixing length, 119 Monin–Obukhov length, 120 Reynolds number, 122 Richardson number, 120 roughness length, 119 under-ice boundary layer, 122

V Väisälä frequency, 238 Vatnajökull, 206 Viscous, 165 creep, 169 non-linear viscous, 165 second viscosity, 165 viscosity, 165

W water budget, 20 flushing time, 20 groundwater, 20 inflow, 20 precipitation, 20

Index Water renewal, 252 Water vapour pressure, 21 saturation, 21

361 Wind stress, 249 Winter shipping, 298