1 Introduction ................................................. 1
1.1 Motivation .............................................. 1
1.2 Lakes on Earth ......................................... 10
1.3 Lakes Characterised by Their Response to the Driving
Environment ............................................ 14
1.3.1 Seasonal Characteristics ........................ 14
1.3.2 Characteristics by Mixing ....................... 15
1.3.3 Boundary-Related Processes ...................... 18
1.3.4 Characterisation by Typical Scales .............. 20
References .................................................. 22
2 Mathematical Prerequisites .................................. 25
2.1 Scalars and Vectors .................................... 26
2.2 Tensors ................................................ 38
2.3 Fields and Their Differentiation ....................... 41
2.4 Gradient, Divergence and Rotation of Vector and
Tensor Fields .......................................... 50
2.5 Integral Theorems of Vector Analysis ................... 60
2.5.1 Gauss Theorems .................................. 60
2.5.2 Stokes Theorems ................................. 62
References .................................................. 65
3 A Brief Review of the Basic Thermomechanical Laws of
Classical Physics ........................................... 67
3.1 Underlying Fundamentals - General Balance Laws ......... 67
3.2 Physical Balance Laws .................................. 73
3.2.1 Balance of Mass ................................. 73
3.2.2 Balance of Linear Momentum ...................... 74
3.2.3 Balance of Moment of Momentum ................... 76
3.2.4 Balance of Energy ............................... 77
3.2.5 Second Law of Thermodynamics .................... 79
References .................................................. 82
4 Fundamental Equations of Lake Hydrodynamics ................. 83
4.1 Kinematics ............................................. 84
4.2 Balance of Mass ....................................... 100
4.3 Balances of Momentum and Moment of Momentum,
Concept of Stress, Hydrostatics ....................... 110
4.3.1 Stress Tensor .................................. 11З
4.3.2 Local Balance Law of Momentum or Newton's
Second Law ..................................... 118
4.3.3 Material Behaviour ............................. 123
4.3.4 Hydrostatics ................................... 128
4.4 Balance of Energy: First Law of Thermodynamics ........ 136
4.5 Diffusion of Suspended Substances ..................... 141
4.6 Summary of Equations .................................. 146
4.7 A First Look at the Boussinesq and Shallow-Water
Equations ............................................. 150
References ................................................. 155
5 Conservation of Angular Momentum-Vorticity ................. 157
5.1 Circulation ........................................... 157
5.2 Simple Vorticity Theorems ............................. 167
5.3 Helmholtz Vorticity Theorem ........................... 170
5.4 Potential Vorticity Theorem ........................... 177
References ................................................. 184
6 Turbulence Modelling ....................................... 185
6.1 A Primer on Turbulent Motions ......................... 185
6.1.1 Averages and Fluctuations ...................... 185
6.1.2 Filters ........................................ 187
6.1.3 Isotropic Turbulence ........................... 190
6.1.4 Reynolds Versus Favre Averages ................. 192
6.2 Balance Equations for the Averaged Fields ............. 194
6.2.1 Motivation ..................................... 194
6.2.2 Averaging Procedure ............................ 195
6.2.3 Averaged Density Field (p) ..................... 197
6.2.4 Dissipation Rate Density {ф) ................... 198
6.2.5 Reynolds Stress Hypothesis ..................... 198
6.2.6 One- and Two-Equation Models ................... 201
6.3 k-e Model for Density-Preserving and Boussinesq
Fluids ................................................ 203
6.3.1 The Balance Equations .......................... 203
6.3.2 Closure Relations .............................. 204
6.3.3 Summary of (k - e)-Equations ................... 206
6.3.4 Boundary Conditions ............................ 207
6.4 Final Remarks ......................................... 210
6.4.1 Higher Order RANS Models ....................... 210
6.4.2 Large Eddy Simulation and Direct Numerical
Simulation ..................................... 211
6.4.3 Early Anisotropic Closure Schemes .............. 212
References ................................................. 219
7 Introduction to Linear Waves ............................... 221
7.1 The Linear Wave Equation and Its Properties ........... 222
7.2 Surface Gravity Waves Without Rotation ................ 234
7.2.1 Short-Wave Approximation ....................... 245
7.2.2 Long-Wave Approximation ........................ 246
7.2.3 Standing Waves - Reflection .................... 247
7.3 Free Linear Oscillations in Rectangular Basins
of Constant Depth ..................................... 252
7.4 Concluding Remarks .................................... 258
References ................................................. 261
8 The Role of the Distribution of Mass Within Water Bodies
on Earth ................................................... 263
8.1 Motivation ............................................ 263
8.2 Processes of Surface Water Penetration to Depth ....... 268
8.3 Homogenisation of Water Masses Requires Energy ........ 274
8.3.1 Constant Density Layers ........................ 275
8.3.2 Continuous Density Variation ................... 280
8.3.3 Influence of the Thermal Expansion ............. 283
8.4 Motion of Buoyant Bodies in a Stratified Still Lake ... 285
8.4.1 Influence of Friction .......................... 290
8.5 Internal Oscillations - The Dynamical Imprint of the
Density Structure ..................................... 294
8.5.1 Fundamental Equations .......................... 297
8.5.2 Eigenvalue Problem for the Vertical Mode
Structure in Constant Depth Basins ............. 301
8.6 Closure ............................................... 315
References ................................................. 317
9 Vertical Structure of Wind-Induced Currents in
Homogeneous and Stratified Waters .......................... 319
9.1 Preview and Scope of This Chapter ..................... 319
9.2 Hydrodynamic Equations Applied to a Narrow Lake
Under Steady Wind ..................................... 322
9.2.1 Wind-Induced Steady Circulation in a Narrow
Homogeneous Lake of Constant Depth ............. 322
9.2.2 Influence of Bottom Slip on the Wind-Induced
Circulation .................................... 328
9.2.3 Wind-Induced Steady Circulation in a Narrow
Lake Stratified in Two Layers .................. 330
9.3 Ekman Theory and Some of Its Extensions ............... 340
9.3.1 Ekman Spiral ................................... 341
9.3.2 Steady Wind-Induced Circulation in a
Homogeneous Lake on the Rotating Earth ......... 358
9.3.3 Wind-Driven Steady Currents in Lake Erie ....... 364
9.3.4 Time-Dependent Wind-Induced Currents in
Shallow Lakes on the Rotating Earth ............ 369
9.3.5 The Dynamical Prediction of Wind Tides on
Lake Erie ...................................... 376
9.4 Final Remarks ......................................... 384
References ................................................. 385
10 Phenomenological Coefficients of Water ..................... 389
10.1 Density of Water ...................................... 390
10.1.1 Natural Water and Sea Water .................... 393
10.1.2 Suspended Matter ............................... 398
10.2 Specific Heat of Water ................................ 399
10.2.1 Specific Heat of Salty Water ................... 399
10.3 Viscosity of Water .................................... 404
10.3.1 Pure Water .................................... 405
10.3.2 Sea Water ...................................... 406
10.3.3 Natural Water .................................. 409
10.3.4 Suspended Matter ............................... 410
10.4 Molecular Heat Conductivity of Water .................. 412
10.4.1 Heat Conductivity of Salt Water ................ 413
10.4.2 Impurities ..................................... 414
References ................................................. 416
Name Index .................................................... 419
Lake Index .................................................... 423
Subject Index ................................................. 425
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