Advanced ocean modelling : using open-source software

This book focuses on motions of incompressible ?uids of a freely moving surface being in?uenced by both the Earth's rotation and density strati?cation. In contrast to traditional textbooks in the ?eld of geophysical ?uid dynamics, such as those by by Cushman-Roisin (1994) and Gill (1982), this book uses the method of proce- oriented hydrodynamic modelling to illustrate a rich variety of ?uid phenomena. To this end, the reader can adopt the model codes, found on the Springer server accompanying this book, to reproduce most graphs of this book and, even better, to create animation movies. The reader can also employ the codes as templates for own independent studies. This can be done by a lay person as a hobby activity, undergraduate or postgraduate students as part of their education, or professional scientists as part of research. Exercises of this book are run with open-source software that can be freely downloaded from the Internet. This includes the FORTRAN 95 compiler "G95" used for execution of model simulations, the data visualisation program "SciLab", and "ImageMagick" for the creation of graphs and GIF animations, which can be watched with most Internet browsers.

Author Notes

Jochen Kmpf, a Senior Lecturer in Oceanography at Flinders University, Australia, is known for the discovery of the South Australian Coastal Upwelling System. His research interests cover a broad range of subjects including surface mixed-layer physics, coastal and estuarine circulations, density-driven flows, benthic storms, and more recently dispersal of desalination brine.

1 Introduction	p. 1
1.1 Fundamental Physical Laws	p. 1
1.1.1 Cartesian Coordinates	p. 1
1.1.2 The Navier-Stokes Equations	p. 1
1.1.3 Boundary Fluxes	p. 3
1.1.4 The Hydrostatic Approximation	p. 3
1.1.5 The Stability Frequency	p. 4
1.2 Numerical Methods	p. 4
1.2.1 Finite Differences	p. 4
1.2.2 Requirements for a Finite-Difference Model	p. 5
1.3 Modelling with Fortran 95	p. 5
1.3.1 Writing and Compiling Codes	p. 5
1.3.2 Modular Source Codes	p. 6
1.4 Visualisation with SciLab	p. 6
1.4.1 Writing SciLab Scripts	p. 6
1.4.2 GIF Animations	p. 7
1.5 Organisation of Work	p. 8
1.6 Download of Computer Codes	p. 8
2 1D Models of Ekman Layers	p. 9
2.1 Useful Background Knowledge	p. 9
2.1.1 Inertial Oscillations	p. 9
2.1.2 Semi-implicit Treatment of the Coriolis Force	p. 10
2.2 The Surface Ekman Layer	p. 11
2.2.1 Boundary-Layer Equations	p. 11
2.2.2 Scaling: The Temporal Rossby Number	p. 11
2.2.3 Scaling: The Ekman Number	p. 12
2.2.4 Solutions of the Boundary-Layer Equations	p. 13
2.2.5 Finite-Difference Equations	p. 13
2.2.6 Formulation of Diffusion Terms	p. 14
2.2.7 Stability Criterion for Diffusion Terms	p. 14
2.3 Exercise 1: The Surface Ekman Layer	p. 15
2.3.1 Task Description	p. 15
2.3.2 Results	p. 16
2.3.3 Explanation of the Ekman-Layer Structure	p. 17
2.3.4 Additional Exercises for the Reader	p. 17
2.4 The Bottom Ekman Layer	p. 18
2.4.1 Boundary-Layer Equations	p. 18
2.5 Exercise 2: The Bottom Ekman Layer	p. 18
2.5.1 Task Description	p. 18
2.5.2 Results	p. 18
2.5.3 Additional Exercises for the Reader	p. 19
3 Basics of Nonhydrostatic Modelling	p. 21
3.1 Level Models	p. 21
3.2 2D Vertical-Slice Modelling	p. 22
3.2.1 Configuration	p. 22
3.2.2 The Arakawa C-Grid	p. 23
3.3 Surface Gravity Waves	p. 24
3.3.1 The Governing Equations	p. 24
3.3.2 The Dispersion Relation	p. 24
3.3.3 Orbital Motions of Water Particles and Wave Pressure	p. 26
3.4 Nonhydrostatic Solver	p. 26
3.4.1 Splitting Pressure into Parts	p. 26
3.4.2 Starting as Simple as Possible	p. 27
3.4.3 Finite-Difference Scheme	p. 27
3.4.4 The S.O.R. Method	p. 29
3.4.5 Boundary Conditions for Variable Bathymetry	p. 31
3.4.6 Stability Criterion	p. 31
3.5 Exercise 3: Short Surface Gravity Waves	p. 32
3.5.1 Aim	p. 32
3.5.2 Task Description	p. 32
3.5.3 Results	p. 32
3.5.4 Additional Exercise for the Reader	p. 33
3.5.5 Implementation of Variable Bottom Topography	p. 34
3.5.6 Results	p. 35
3.6 Inclusion of Variable Density	p. 35
3.6.1 The Governing Equations	p. 35
3.6.2 Discretisation of the Advection Terms	p. 36
3.6.3 Stability Criterion for the Advection Equation	p. 38
3.6.4 Implementation of Density Diffusion	p. 38
3.6.5 Required Modifications of the Code	p. 39
3.7 Exercise 4: Density-Driven Flows	p. 40
3.7.1 Aim	p. 40
3.7.2 Task Description	p. 40
3.7.3 Theory	p. 41
3.7.4 Results	p. 41
3.7.5 Can Reduced-Gravity Plumes Jump?	p. 41
3.7.6 Additional Exercise for the Reader	p. 43
3.7.7 The Rigid-Lid Approximation	p. 43
3.8 Internal Waves	p. 44
3.8.1 Theory	p. 44
3.8.2 Normal Wave Modes	p. 45
3.9 Exercise 5: Internal Waves	p. 46
3.9.1 Aim	p. 46
3.9.2 Task Description	p. 46
3.9.3 Results	p. 47
3.9.4 Additional Exercise for the Reader	p. 48
3.10 Mechanical Turbulence	p. 48
3.10.1 Kelvin-Helmholtz Instability	p. 48
3.10.2 Instability of a Stratified Shear Flow	p. 49
3.11 Exercise 6: Kelvin-Helmholtz Instability	p. 50
3.11.1 Aim	p. 50
3.11.2 Task Description	p. 51
3.11.3 Cyclic Boundary Conditions	p. 51
3.11.4 Results	p. 52
3.11.5 Additional Exercise for the Reader	p. 53
3.12 Lee Waves and the Froude Number	p. 53
3.12.1 The Hydraulic Jump	p. 53
3.13 Exercise 7: Lee Waves	p. 54
3.13.1 Task Description	p. 54
3.13.2 Results: Continuous Density Stratification	p. 55
3.13.3 Results: Two-Layer Stratification	p. 56
3.13.4 Additional Exercise for the Reader	p. 57
3.14 Oceanic Convection	p. 57
3.14.1 Background	p. 57
3.14.2 Free Convection	p. 57
3.14.3 The Flux-Rayleigh Number	p. 58
3.14.4 Aspect Ratio of Convection Cells	p. 59
3.14.5 Convective Mixed-Layer Deepening	p. 59
3.15 Exercise 8: Free Convection	p. 60
3.15.1 Aim	p. 60
3.15.2 Task Description	p. 61
3.15.3 A Trick to Avoid Substantial Round-off Errors	p. 62
3.15.4 Inclusion of Momentum Diffusion and Bottom Friction	p. 62
3.15.5 Results	p. 64
3.15.6 Additional Exercise for the Reader	p. 65
3.16 Exercise 9: Convective Entrainment	p. 65
3.16.1 How It Works	p. 65
3.16.2 Entrainment Velocity	p. 65
3.16.3 Task Description	p. 66
3.16.4 Results	p. 66
3.16.5 Additional Exercises for the Reader	p. 67
3.17 Exercise 10: Slope Convection near the Shore	p. 67
3.17.1 Background	p. 67
3.17.2 Implementation of Bottom Friction on a Sloping Terrain	p. 68
3.17.3 Task Description	p. 68
3.17.4 Results	p. 70
3.17.5 Additional Exercise for the Reader	p. 71
3.18 Double Diffusion	p. 72
3.18.1 Background	p. 72
3.18.2 Double-Diffusive Instability	p. 72
3.18.3 Double-Diffusive Layering	p. 73
3.18.4 The Gradient Ratio and the Turner Angle	p. 73
3.19 Exercise 11: Double-Diffusive Instability	p. 74
3.19.1 Aim	p. 74
3.19.2 Task Description	p. 74
3.19.3 Results	p. 75
3.20 Exercise 12: Double-Diffusive Layering	p. 77
3.20.1 Aim	p. 77
3.20.2 Task Description	p. 77
3.20.3 Results	p. 78
3.20.4 Additional Exercises for the Reader	p. 79
3.21 Tilted Coordinate Systems	p. 79
3.21.1 The Governing Equations	p. 79
3.22 Exercise 13: Stratified Flows on a Slope	p. 81
3.22.1 Aim	p. 81
3.22.2 Task Description	p. 81
3.22.3 Results	p. 82
3.22.4 Additional Exercise for the Reader	p. 83
3.23 Estuaries	p. 83
3.23.1 Definition	p. 83
3.23.2 Classification of Estuaries According to Origin	p. 84
3.23.3 The Dynamics of Positive Estuaries	p. 84
3.23.4 Brief Overview of Tides	p. 84
3.23.5 Dynamic Theory of Tides	p. 85
3.23.6 Tides in Estuaries	p. 85
3.23.7 Tidal Patterns	p. 86
3.23.8 Classification of Estuaries According to Stratification and Circulation	p. 86
3.23.9 Transport Timescales in Estuaries	p. 87
3.24 Exercise 14: Positive Estuaries	p. 89
3.24.1 Aim	p. 89
3.24.2 Task Description	p. 89
3.24.3 Implementation of Variable Channel Width	p. 91
3.24.4 Advanced Turbulence Closure	p. 91
3.24.5 Results	p. 92
3.24.6 Additional Exercises for the Reader	p. 93
3.25 Exercise 15: Inverse Estuaries	p. 94
3.25.1 Aim	p. 94
3.25.2 Task Description	p. 94
3.25.3 Results	p. 95
3.25.4 Additional Exercise for the Reader	p. 96
4 2.5D Vertical Slice Modelling	p. 97
4.1 The Basis	p. 97
4.1.1 Adding Another Half Dimension	p. 97
4.1.2 The Geostrophic Balance	p. 97
4.1.3 Scaling	p. 99
4.1.4 Conservation of Potential Vorticity	p. 99
4.1.5 Geostrophic Adjustment	p. 100
4.1.6 The 2.5d Shallow-Water Model	p. 101
4.1.7 Implementation of the Coriolis Force	p. 101
4.1.8 Potential Problems	p. 102
4.2 Exercise 16: Geostrophic Adjustment	p. 103
4.2.1 Aim	p. 103
4.2.2 Task Description	p. 103
4.2.3 Results	p. 104
4.2.4 Additional Exercise for the Reader	p. 105
4.3 Exercise 17: Tidal-Mixing Fronts	p. 106
4.3.1 Background	p. 106
4.3.2 Task Description	p. 106
4.3.3 Results	p. 107
4.3.4 Additional Study	p. 109
4.3.5 Results and Discussion	p. 109
4.3.6 Additional Exercises for the Reader	p. 110
4.4 Coastal Upwelling	p. 110
4.4.1 Background	p. 110
4.4.2 How Does It Work?	p. 111
4.4.3 Partial and Full Upwelling	p. 111
4.4.4 The Upwelling Index	p. 113
4.5 Exercise 18: Coastal Upwelling and Downwelling	p. 113
4.5.1 Aim	p. 113
4.5.2 Task Description	p. 113
4.5.3 Advanced Turbulence Closure	p. 114
4.5.4 Results: Upwelling Scenario	p. 115
4.5.5 Additional Exercise for the Reader	p. 116
4.5.6 Results: Downwelling Scenario	p. 116
4.5.7 Additional Exercise for the Reader	p. 117
4.6 Exercise 19: Ekman Pumping	p. 118
4.6.1 Theoretical Background	p. 118
4.6.2 Aim	p. 118
4.6.3 Task Description	p. 118
4.6.4 Results: Scenario 1	p. 120
4.6.5 Results: Scenario 2	p. 122
4.6.6 Results: Scenario 3	p. 123
4.6.7 Additional Exercises for the Reader	p. 124
5 3D Level Modelling	p. 125
5.1 The Basic Equations	p. 125
5.1.1 The Basics	p. 125
5.1.2 Conservation of Momentum	p. 125
5.1.3 Conservation of Volume	p. 126
5.1.4 Evolution of the Density Field	p. 127
5.2 Numerical Treatment	p. 127
5.2.1 The 3d Arakawa C-grid	p. 127
5.2.2 Treatment of the Advection Terms	p. 128
5.2.3 The Nonhydrostatic Solver of the Momentum Equations	p. 129
5.2.4 Stability Criteria	p. 130
5.3 Exercise 20: Geostrophic Adjustment in 3D	p. 131
5.3.1 Aim	p. 131
5.3.2 Task Description	p. 131
5.3.3 Results	p. 132
5.3.4 Additional Exercise for the Reader	p. 132
5.4 Exercise 21: Eddy Formation in a Strait	p. 133
5.4.1 Background	p. 133
5.4.2 Aim	p. 134
5.4.3 Task Description	p. 134
5.4.4 Creation of Variable Bathymetry	p. 136
5.4.5 Results	p. 136
5.4.6 Bathymetry Creation	p. 137
5.4.7 Additional Exercises for the Reader	p. 137
5.5 Exercise 22: Exchange Flow Through a Strait	p. 137
5.5.1 Aim	p. 137
5.5.2 Mediterranean Seas	p. 138
5.5.3 Task Description	p. 139
5.5.4 Results	p. 140
5.5.5 Additional Exercise for the Reader	p. 142
5.6 Exercise 23: Coastal Upwelling in 3D	p. 142
5.6.1 Aim	p. 142
5.6.2 Task Description	p. 142
5.6.3 Results	p. 143
5.6.4 Additional Exercise for the Reader	p. 145
5.6.5 Time-Splitting Methods	p. 145
5.7 The Thermohaline Circulation	p. 146
5.7.1 The Abyssal Circulation	p. 146
5.7.2 The Stommel-Arons Model	p. 146
5.8 Exercise 24: The Abyssal Circulation	p. 148
5.8.1 Aim	p. 148
5.8.2 Task Description	p. 148
5.8.3 Results	p. 150
5.8.4 Additional Exercise for the Reader	p. 152
5.8.5 Improved Float Tracking	p. 152
5.9 The Equatorial Barrier	p. 155
5.9.1 Inertial Oscillations About the Equator	p. 155
5.9.2 Variation to Exercise 24	p. 156
5.9.3 Results	p. 156
5.9.4 Additional Exercise for the Reader	p. 157
5.10 Equatorial Waves	p. 158
5.10.1 Background	p. 158
5.10.2 Equatorial Kelvin Waves	p. 158
5.10.3 Other Equatorially Trapped Waves	p. 159
5.11 The El-Niño Southern Oscillation	p. 161
5.11.1 Background	p. 161
5.12 Exercise 25: Simulation of an El-Niño Event	p. 162
5.12.1 Aim	p. 162
5.12.2 Task Description	p. 162
5.12.3 The Smagorinsky Turbulence Closure Scheme	p. 164
5.12.4 Warning	p. 164
5.12.5 Results	p. 164
5.12.6 Additional Exercises for the Reader	p. 165
5.13 Advanced Lateral Boundary Conditions	p. 166
5.13.1 Background	p. 166
5.13.2 Consistency	p. 166
5.13.3 Inflow Conditions	p. 166
5.13.4 Outflow Conditions	p. 167
5.13.5 Zero-Gradient Conditions	p. 168
5.13.6 Radiation Conditions	p. 169
5.13.7 Sponge Layers and Low-Pass Grid Filters	p. 170
5.14 Final Remark	p. 171
5.15 Technical Information	p. 171
Bibliography	p. 173
List of Exercises	p. 177
Index	p. 179

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