Mathematical Models

Introduction xix
Jean-Michel TANGUY

Chapter 1. Reminders on the Mechanical Properties of Fluids 1
Jacques GEORGE

1.1. Laws of conservation, principles and general theorems 1

1.2. Enthalpy, rotation, mixing, saturation 13

1.3. Thermodynamic relations, relations of state and laws of behavior 20

1.4. Turbulent flow 26

1.5. Dynamics of geophysical fluids 30

Chapter 2. 3D Navier-Stokes Equations 35
Véronique DUCROCQ

2.1. The continuity hypothesis 35

2.2. Lagrangian description/Eulerian description 36

2.3. The continuity equation 37

2.4. The movement quantity assessment equation 38

2.5. The energy balance equation 41

2.6. The equation of state 41

2.7. Navier-Stokes equations for a fluid in rotation 41

Chapter 3. Models of the Atmosphere 43
Jean COIFFIER

3.1. Introduction 43

3.2. The various simplifications and corresponding models 44

3.3. The equations with various systems of coordinates 56

3.4. Some typical conformal projections 61

3.5. The operational models 67

3.6. Bibliography 69

Chapter 4. Hydrogeologic Models 71
Dominique THIÉRY

4.1. Equation of fluid mechanics 71

4.2. Continuity equation in porous media 72

4.3. Navier-Stokes’ equations 74

4.4. Darcy’s law 76

4.5. Calculating mass storage from the equations of state 80

4.6. General equation of hydrodynamics in porous media 82

4.7. Flows in unsaturated media 84

4.8. Bibliography 91

Chapter 5. Fluvial and Maritime Currentology Models 93
Jean-Michel TANGUY

5.1. 3D hydrostatic model 99

5.2. 2D horizontal model for shallow water 107

5.3. 1D models of fluvial flows 119

5.4. Putting 1D models into real time 131

5.5. Bibliography 151

Chapter 6. Urban Hydrology Models 155
Bernard CHOCAT

6.1. Global models and detailed models used in surface flows 156

6.2. Rainfall representation and rainfall-flow transformation 161

6.3. Modeling of the losses into the ground 164

6.4. Transfer function 169

6.5. Modeling of the hydraulic operating conditions of the networks 177

6.6. Production and transport of polluting agents 189

6.7. Conclusion 205

6.8. Bibliography 206

Chapter 7. Tidal Model and Tide Streams 213
Bernard SIMON

7.1. Tidal coefficient 214

7.2. Non-harmonic methods 215

7.3. Compatibilities 216

7.4. Tidal coefficient 222

7.5. Modeling 223

7.6. Tidal currents 226

Chapter 8. Wave Generation and Coastal Current Models 235
Jean-Michel TANGUY, Jean-Michel LEFÈVRE and Philippe SERGENT

8.1. Types of swell models 235

8.2. Spectral approach in high waters 242

8.3. Wave generation models 246

8.4. Wave propagation models 260

8.5. Agitating models within the harbors 266

8.6. Non-linear wave model: Boussinesq model 298

8.7. Coastal current models influenced or created by the swell 320

8.8. Bibliography 325

Chapter 9. Solid Transport Models and Evolution of the Seabed 335
Benoît LE GUENNEC and Jean-Michel TANGUY

9.1. Transport due to the overthrust effect 338

9.2. Total load 344

9.3. Bed forms and roughness 344

9.4. Suspension transport 346

9.5. Evolution model of movable beds 357

9.6. Conclusion 364

9.7. Bibliography 364

Chapter 10. Oil Spill Models 371
Pierre DANIEL

10.1. Behavior of hydrocarbons in marine environment 371

10.2. Oil spill drift models 372

10.3. Example: the MOTHY model 375

10.4. Calculation algorithm of the path of polluting particles 378

10.5. Example of a drift prediction map 379

10.6. Bibliography 379

Chapter 11. Conceptual, Empirical and Other Models 381
Christelle ALOT and Florence HABETS

11.1. Evapotranspiration 382

11.2. Bibliography 394

Chapter 12. Reservoir Models in Hydrology 397
Patrick FOURMIGUÉ and Patrick ARNAUD

12.1. Background 397

12.2. Main principles 399

12.3. Mathematical tools 401

12.4. Forecasting 403

12.5. Integration of the spatial information 405

12.6. Modeling limits 406

12.7. Bibliography 406

Chapter 13. Reservoir Models in Hydrogeology 409
Dominique THIÉRY

13.1. Principles and objectives 409

13.2. Catchment basin 410

13.3. Setting the model up 411

13.4. Data and parameters 412

13.5. Application domains 412

Chapter 14. Artificial Neural Network Models 419
Anne JOHANNET

14.1. Neural networks: a rapidly changing domain 420

14.2. Neuron and architecture models 422

14.3. How to take into account the non-linearity 429

14.4. Case study: identification of the rainfall-runoff relation of a karst 434

14.5. Acknowledgments 441

14.6. Bibliography 441

Chapter 15. Model Coupling 445
Rachid ABABOU, Denis DARTUS and Jean-Michel TANGUY

15.1. Model coupling 446

15.2. Bibliography 488

Chapter 16. A Set of Hydrological Models 493
Charles PERRIN, Claude MICHEL and Vasken ANDRÉASSIAN

16.1. Introduction 493

16.2. Description of the annual GR1A rainfall-runoff model 495

16.3. Description of the monthly GR2M rainfall-runoff model 496

16.4. Description of the daily GR4J rainfall-runoff model 500

16.5. Applications of the models 505

16.6. Conclusions and future work 506

16.7. Bibliography 507

List of Authors 511

Index 515

General Index of Authors 517

Summary of the Other Volumes in the Series 519