Physics of Lakes (eBook)

Preface to the Book Series 7
VOLUME 1: Physics of Lakes – Formulation of the Mathematical and Physical Background 8
VOLUME 2: Physics of Lakes – Lakes as Oscillators 8
VOLUME 3: Physics of Lakes – Methods of Understanding Lakes as Components of the Geophysical Environment 9
Vorwort zur Buchreihe 10
BAND 1: Physik der Seen – Formulierung des mathematischen und physikalischen Hintergrundes 11
BAND 2: Physik der Seen – Seen als Oszillatoren 12
BAND 3: Physik der Seen – Methoden, die Seen als Komponenten des geophysikalischen Umfeldes verstehen 12
.pe..c.o..e . cep.. 14
Acknowledgements 21
References 24
Books, Reports 24
Diploma (M. Sc.) Theses 24
Doctoral Dissertations 25
Habilitation Theses 26
Preface to Volume I 27
Acknowledgements for Copyright Permission 31
Contents 33
Notations 37
Roman Symbols 37
Greek Symbols 42
Miscellaneous Symbols 44
1 Introduction 46
1.1 Motivation 46
1.2 Lakes on Earth 55
1.3 Lakes Characterised by Their Response to the Driving Environment 59
1.3.1 Seasonal Characteristics 59
1.3.2 Characteristics by Mixing 60
1.3.3 Boundary-Related Processes 63
1.3.4 Characterisation by Typical Scales 65
References 67
2 Mathematical Prerequisites 69
2.1 Scalars and Vectors 70
2.2 Tensors 82
2.3 Fields and Their Differentiation 85
2.4 Gradient, Divergence and Rotation of Vector and Tensor Fields 94
2.5 Integral Theorems of Vector Analysis 104
2.5.1 Gauss Theorems 104
2.5.2 Stokes Theorems 106
References 109
3 A Brief Review of the Basic Thermomechanical Laws of Classical Physics 111
3.1 Underlying Fundamentals -- General Balance Laws 111
3.2 Physical Balance Laws 117
3.2.1 Balance of Mass 117
3.2.2 Balance of Linear Momentum 118
3.2.3 Balance of Moment of Momentum 120
3.2.4 Balance of Energy 121
3.2.5 Second Law of Thermodynamics 123
References 126
4 Fundamental Equations of Lake Hydrodynamics 127
4.1 Kinematics 128
4.2 Balance of Mass 144
4.3 Balances of Momentum and Moment of Momentum, Concept of Stress, Hydrostatics 154
4.3.1 Stress Tensor 157
4.3.2 Local Balance Law of Momentum or Newton's Second Law 162
4.3.3 Material Behaviour 167
4.3.4 Hydrostatics 172
4.4 Balance of Energy: First Law of Thermodynamics 180
4.5 Diffusion of Suspended Substances 185
4.6 Summary of Equations 190
4.7 A First Look at the Boussinesq and Shallow-Water Equations 194
References 199
5 Conservation of Angular Momentum--Vorticity 200
5.1 Circulation 200
5.2 Simple Vorticity Theorems 210
5.3 Helmholtz Vorticity Theorem 213
5.4 Potential Vorticity Theorem 220
References 227
6 Turbulence Modelling 228
6.1 A Primer on Turbulent Motions 228
6.1.1 Averages and Fluctuations 228
6.1.2 Filters 230
6.1.3 Isotropic Turbulence 233
6.1.4 Reynolds Versus Favre Averages 235
6.2 Balance Equations for the Averaged Fields 237
6.2.1 Motivation 237
6.2.2 Averaging Procedure 238
6.2.3 Averaged Density Field < .>
6.2.4 Dissipation Rate Density < f>
6.2.5 Reynolds Stress Hypothesis 241
6.2.6 One- and Two-Equation Models 244
6.3 k–e Model for Density-Preserving and Boussinesq Fluids 246
6.3.1 The Balance Equations 246
6.3.2 Closure Relations 247
6.3.3 Summary of (k–e)-Equations 249
6.3.4 Boundary Conditions 250
6.4 Final Remarks 253
6.4.1 Higher Order RANS Models 253
6.4.2 Large Eddy Simulation and Direct Numerical Simulation 254
6.4.3 Early Anisotropic Closure Schemes 255
References 262
7 Introduction to Linear Waves 264
7.1 The Linear Wave Equation and Its Properties 265
7.2 Surface Gravity Waves Without Rotation 277
7.2.1 Short-Wave Approximation 288
7.2.2 Long-Wave Approximation 289
7.2.3 Standing Waves -- Reflection 290
7.3 Free Linear Oscillations in Rectangular Basins of Constant Depth 295
7.4 Concluding Remarks 301
References 304
8 The Role of the Distribution of Mass Within Water Bodies on Earth 305
8.1 Motivation 305
8.2 Processes of Surface Water Penetration to Depth 310
8.3 Homogenisation of Water Masses Requires Energy 316
8.3.1 Constant Density Layers 317
8.3.2 Continuous Density Variation 322
8.3.3 Influence of the Thermal Expansion 325
8.4 Motion of Buoyant Bodies in a Stratified Still Lake 327
8.4.1 Influence of Friction 332
8.5 Internal Oscillations -- The Dynamical Imprintof the Density Structure 336
8.5.1 Fundamental Equations 339
8.5.2 Eigenvalue Problem for the Vertical Mode Structurein Constant Depth Basins 343
8.6 Closure 357
References 359
9 Vertical Structure of Wind-Induced Currentsin Homogeneous and Stratified Waters 361
9.1 Preview and Scope of This Chapter 361
9.2 Hydrodynamic Equations Applied to a Narrow Lake Under Steady Wind 364
9.2.1 Wind-Induced Steady Circulation in a Narrow Homogeneous Lake of Constant Depth 364
9.2.2 Influence of Bottom Slip on the Wind-Induced Circulation 370
9.2.3 Wind-Induced Steady Circulation in a Narrow Lake Stratified in Two Layers 372
9.3 Ekman Theory and Some of Its Extensions 382
9.3.1 Ekman Spiral 383
9.3.2 Steady Wind-Induced Circulation in a Homogeneous Lake on the Rotating Earth 400
9.3.3 Wind-Driven Steady Currents in Lake Erie 406
9.3.4 Time-Dependent Wind-Induced Currents in Shallow Lakes on the Rotating Earth 411
9.3.5 The Dynamical Prediction of Wind Tides on Lake Erie 418
9.4 Final Remarks 426
References 427
10 Phenomenological Coefficients of Water 430
10.1 Density of Water 431
10.1.1 Natural Water and Sea Water 434
10.1.2 Suspended Matter 439
10.2 Specific Heat of Water 440
10.2.1 Specific Heat of Salty Water 440
10.3 Viscosity of Water 445
10.3.1 Pure Water 446
10.3.2 Sea Water 447
10.3.3 Natural Water 450
10.3.4 Suspended Matter 451
10.4 Molecular Heat Conductivity of Water 453
10.4.1 Heat Conductivity of Salt Water 454
10.4.2 Impurities 455
References 457
Name Index 460
Lake Index 464
Subject Index 466