Advanced Ocean Modelling (eBook)

Using Open-Source Software

Jochen Kämpf (Autor)

eBook Download: PDF

2010 | 2010
XIII, 181 Seiten
Springer Berlin (Verlag)
978-3-642-10610-1 (ISBN)

Infos

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.

Inhalt

Preface 5
Contents 7
to 1 Introduction 14
1.1 Fundamental Physical Laws 14
1.1.1 Cartesian Coordinates 14
1.1.2 The Navier-Stokes Equations 14
1.1.3 Boundary Fluxes 16
1.1.4 The Hydrostatic Approximation 16
1.1.5 The Stability Frequency 17
1.2 Numerical Methods 17
1.2.1 Finite Differences 17
1.2.2 Requirements for a Finite-Difference Model 18
1.3 Modelling with FORTRAN 95 18
1.3.1 Writing and Compiling Codes 18
1.3.2 Modular Source Codes 19
1.4 Visualisation with SciLab 19
1.4.1 Writing SciLab Scripts 19
1.4.2 GIF Animations 20
1.5 Organisation of Work 21
1.6 Download of Computer Codes 21
to 2 1D Models of Ekman Layers 22
2.1 Useful Background Knowledge 22
2.1.1 Inertial Oscillations 22
2.1.2 Semi-implicit Treatment of the Coriolis Force 23
2.2 The Surface Ekman Layer 24
2.2.1 Boundary-Layer Equations 24
2.2.2 Scaling: The Temporal Rossby Number 24
2.2.3 Scaling: The Ekman Number 25
2.2.4 Solutions of the Boundary-Layer Equations 26
2.2.5 Finite-Difference Equations 26
2.2.6 Formulation of Diffusion Terms 27
2.2.7 Stability Criterion for Diffusion Terms 27
2.3 Exercise 1: The Surface Ekman Layer 28
2.3.1 Task Description 28
2.3.2 Results 29
2.3.3 Explanation of the Ekman-Layer Structure 30
2.3.4 Additional Exercises for the Reader 30
2.4 The Bottom Ekman Layer 31
2.4.1 Boundary-Layer Equations 31
2.5 Exercise 2: The Bottom Ekman Layer 31
2.5.1 Task Description 31
2.5.2 Results 31
2.5.3 Additional Exercises for the Reader 32
to 3 Basics of Nonhydrostatic Modelling 33
3.1 Level Models 33
3.2 2D Vertical-Slice Modelling 34
3.2.1 Configuration 34
3.2.2 The Arakawa C-Grid 35
3.3 Surface Gravity Waves 36
3.3.1 The Governing Equations 36
3.3.2 The Dispersion Relation 36
3.3.3 Orbital Motions of Water Particles and Wave Pressure 38
3.4 Nonhydrostatic Solver 38
3.4.1 Splitting Pressure into Parts 38
3.4.2 Starting as Simple as Possible 39
3.4.3 Finite-Difference Scheme 39
3.4.4 The S.O.R. Method 41
3.4.5 Boundary Conditions for Variable Bathymetry 43
3.4.6 Stability Criterion 43
3.5 Exercise 3: Short Surface Gravity Waves 44
3.5.1 Aim 44
3.5.2 Task Description 44
3.5.3 Results 44
3.5.4 Additional Exercise for the Reader 45
3.5.5 Implementation of Variable Bottom Topography 46
3.5.6 Results 47
3.6 Inclusion of Variable Density 47
3.6.1 The Governing Equations 47
3.6.2 Discretisation of the Advection Terms 48
3.6.3 Stability Criterion for the Advection Equation 50
3.6.4 Implementation of Density Diffusion 50
3.6.5 Required Modifications of the Code 51
3.7 Exercise 4: Density-Driven Flows 52
3.7.1 Aim 52
3.7.2 Task Description 52
3.7.3 Theory 53
3.7.4 Results 53
3.7.5 Can Reduced-Gravity Plumes Jump? 53
3.7.6 Additional Exercise for the Reader 55
3.7.7 The Rigid-Lid Approximation 55
3.8 Internal Waves 56
3.8.1 Theory 56
3.8.2 Normal Wave Modes 57
3.9 Exercise 5: Internal Waves 58
3.9.1 Aim 58
3.9.2 Task Description 58
3.9.3 Results 59
3.9.4 Additional Exercise for the Reader 60
3.10 Mechanical Turbulence 60
3.10.1 Kelvin-Helmholtz Instability 60
3.10.2 Instability of a Stratified Shear Flow 61
3.11 Exercise 6: Kelvin-Helmholtz Instability 62
3.11.1 Aim 62
3.11.2 Task Description 63
3.11.3 Cyclic Boundary Conditions 63
3.11.4 Results 64
3.11.5 Additional Exercise for the Reader 65
3.12 Lee Waves and the Froude Number 65
3.12.1 The Hydraulic Jump 65
3.13 Exercise 7: Lee Waves 66
3.13.1 Task Description 66
3.13.2 Results: Continuous Density Stratification 67
3.13.3 Results: Two-Layer Stratification 68
3.13.4 Additional Exercise for the Reader 69
3.14 Oceanic Convection 69
3.14.1 Background 69
3.14.2 Free Convection 69
3.14.3 The Flux-Rayleigh Number 70
3.14.4 Aspect Ratio of Convection Cells 71
3.14.5 Convective Mixed-Layer Deepening 71
3.15 Exercise 8: Free Convection 72
3.15.1 Aim 72
3.15.2 Task Description 73
3.15.3 A Trick to Avoid Substantial Round-off Errors 74
3.15.4 Inclusion of Momentum Diffusion and Bottom Friction 74
3.15.5 Results 76
3.15.6 Additional Exercise for the Reader 77
3.16 Exercise 9: Convective Entrainment 77
3.16.1 How It Works 77
3.16.2 Entrainment Velocity 77
3.16.3 Task Description 78
3.16.4 Results 78
3.16.5 Additional Exercises for the Reader 79
3.17 Exercise 10: Slope Convection near the Shore 79
3.17.1 Background 79
3.17.2 Implementation of Bottom Friction on a Sloping Terrain 80
3.17.3 Task Description 80
3.17.4 Results 82
3.17.5 Additional Exercise for the Reader 83
3.18 Double Diffusion 84
3.18.1 Background 84
3.18.2 Double-Diffusive Instability 84
3.18.3 Double-Diffusive Layering 85
3.18.4 The Gradient Ratio and the Turner Angle 85
3.19 Exercise 11: Double-Diffusive Instability 86
3.19.1 Aim 86
3.19.2 Task Description 86
3.19.3 Results 87
3.20 Exercise 12: Double-Diffusive Layering 89
3.20.1 Aim 89
3.20.2 Task Description 89
3.20.3 Results 90
3.20.4 Additional Exercises for the Reader 91
3.21 Tilted Coordinate Systems 91
3.21.1 The Governing Equations 91
3.22 Exercise 13: Stratified Flows on a Slope 93
3.22.1 Aim 93
3.22.2 Task Description 93
3.22.3 Results 94
3.22.4 Additional Exercise for the Reader 95
3.23 Estuaries 95
3.23.1 Definition 95
3.23.2 Classification of Estuaries According to Origin 96
3.23.3 The Dynamics of Positive Estuaries 96
3.23.4 Brief Overview of Tides 96
3.23.5 Dynamic Theory of Tides 97
3.23.6 Tides in Estuaries 97
3.23.7 Tidal Patterns 98
3.23.8 Classification of Estuaries According to Stratification and Circulation 98
3.23.9 Transport Timescales in Estuaries 99
3.24 Exercise 14: Positive Estuaries 101
3.24.1 Aim 101
3.24.2 Task Description 101
3.24.3 Implementation of Variable Channel Width 103
3.24.4 Advanced Turbulence Closure 103
3.24.5 Results 104
3.24.6 Additional Exercises for the Reader 105
3.25 Exercise 15: Inverse Estuaries 106
3.25.1 Aim 106
3.25.2 Task Description 106
3.25.3 Results 107
3.25.4 Additional Exercise for the Reader 108
to 4 2.5D Vertical Slice Modelling 109
4.1 The Basis 109
4.1.1 Adding Another Half Dimension 109
4.1.2 The Geostrophic Balance 109
4.1.3 Scaling 111
4.1.4 Conservation of Potential Vorticity 111
4.1.5 Geostrophic Adjustment 112
4.1.6 The 2.5d Shallow-Water Model 113
4.1.7 Implementation of the Coriolis Force 113
4.1.8 Potential Problems 114
4.2 Exercise 16: Geostrophic Adjustment 115
4.2.1 Aim 115
4.2.2 Task Description 115
4.2.3 Results 116
4.2.4 Additional Exercise for the Reader 117
4.3 Exercise 17: Tidal-Mixing Fronts 118
4.3.1 Background 118
4.3.2 Task Description 118
4.3.3 Results 119
4.3.4 Additional Study 121
4.3.5 Results and Discussion 121
4.3.6 Additional Exercises for the Reader 122
4.4 Coastal Upwelling 122
4.4.1 Background 122
4.4.2 How Does It Work? 123
4.4.3 Partial and Full Upwelling 123
4.4.4 The Upwelling Index 125
4.5 Exercise 18: Coastal Upwelling and Downwelling 125
4.5.1 Aim 125
4.5.2 Task Description 125
4.5.3 Advanced Turbulence Closure 126
4.5.4 Results: Upwelling Scenario 127
4.5.5 Additional Exercise for the Reader 128
4.5.6 Results: Downwelling Scenario 128
4.5.7 Additional Exercise for the Reader 129
4.6 Exercise 19: Ekman Pumping 130
4.6.1 Theoretical Background 130
4.6.2 Aim 130
4.6.3 Task Description 130
4.6.4 Results: Scenario 1 132
4.6.5 Results: Scenario 2 134
4.6.6 Results: Scenario 3 135
4.6.7 Additional Exercises for the Reader 136
to 5 3D Level Modelling 137
5.1 The Basic Equations 137
5.1.1 The Basics 137
5.1.2 Conservation of Momentum 137
5.1.3 Conservation of Volume 138
5.1.4 Evolution of the Density Field 139
5.2 Numerical Treatment 139
5.2.1 The 3d Arakawa C-grid 139
5.2.2 Treatment of the Advection Terms 140
5.2.3 The Nonhydrostatic Solver of the Momentum Equations 141
5.2.4 Stability Criteria 142
5.3 Exercise 20: Geostrophic Adjustment in 3D 143
5.3.1 Aim 143
5.3.2 Task Description 143
5.3.3 Results 144
5.3.4 Additional Exercise for the Reader 144
5.4 Exercise 21: Eddy Formation in a Strait 145
5.4.1 Background 145
5.4.2 Aim 146
5.4.3 Task Description 146
5.4.4 Creation of Variable Bathymetry 148
5.4.5 Results 148
5.4.6 Bathymetry Creation 149
5.4.7 Additional Exercises for the Reader 149
5.5 Exercise 22: Exchange Flow Through a Strait 149
5.5.1 Aim 149
5.5.2 Mediterranean Seas 150
5.5.3 Task Description 151
5.5.4 Results 152
5.5.5 Additional Exercise for the Reader 154
5.6 Exercise 23: Coastal Upwelling in 3D 154
5.6.1 Aim 154
5.6.2 Task Description 154
5.6.3 Results 155
5.6.4 Additional Exercise for the Reader 157
5.6.5 Time-Splitting Methods 157
5.7 The Thermohaline Circulation 158
5.7.1 The Abyssal Circulation 158
5.7.2 The Stommel-Arons Model 158
5.8 Exercise 24: The Abyssal Circulation 160
5.8.1 Aim 160
5.8.2 Task Description 160
5.8.3 Results 162
5.8.4 Additional Exercise for the Reader 164
5.8.5 Improved Float Tracking 164
5.9 The Equatorial Barrier 167
5.9.1 Inertial Oscillations About the Equator 167
5.9.2 Variation to Exercise 24 168
5.9.3 Results 168
5.9.4 Additional Exercise for the Reader 169
5.10 Equatorial Waves 170
5.10.1 Background 170
5.10.2 Equatorial Kelvin Waves 170
5.10.3 Other Equatorially Trapped Waves 171
5.11 The El-Niño Southern Oscillation 173
5.11.1 Background 173
5.12 Exercise 25: Simulation of an El-Niño Event 174
5.12.1 Aim 174
5.12.2 Task Description 174
5.12.3 The Smagorinsky Turbulence Closure Scheme 176
5.12.4 Warning 176
5.12.5 Results 176
5.12.6 Additional Exercises for the Reader 177
5.13 Advanced Lateral Boundary Conditions 178
5.13.1 Background 178
5.13.2 Consistency 178
5.13.3 Inflow Conditions 178
5.13.4 Outflow Conditions 179
5.13.5 Zero-Gradient Conditions 180
5.13.6 Radiation Conditions 181
5.13.7 Sponge Layers and Low-Pass Grid Filters 182
5.14 Final Remark 183
5.15 Technical Information 183
Bibliography 184
List of Exercises 187
Index 188

Textauszug

"Preface (p. V-VI)

This book focuses on motions of incompressible fluids of a freely moving surface being influenced by both the Earth’s rotation and density stratification. In contrast to traditional textbooks in the field of geophysical fluid dynamics, such as those by by Cushman-Roisin (1994) and Gill (1982), this book uses the method of processoriented hydrodynamic modelling to illustrate a rich variety of fluid 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. Readers new to the subject are advised to read my book “Ocean Modelling for Beginners” (Kämpf, 2009), which gives descriptions, not replicated here, of the basics of geophysical fluid dynamics, finite-difference modelling, and the use of the above software suites.

The latter book deals with so-called layer models, predicting the motions of multiple layers of different densities and freely moving interfaces. Such models are used in practice as forecasting tools for tides and tsunamis. The latter book also contains a detailed description of the Coriolis force. This force contributes to the geostrophic balance that makes the larger-scale oceanic dynamics much more structured and predictable than turbulent motions in a tea cup.

The reader is advised to review the Coriolis force which plays a crucial role in many phenomena described here. This book focuses on so-called level models which, in contrast to layer models, are capable of simulating vertical mixing processes such as density-driven convection or breaking internal waves. I dedicate this book to my doctor father Professor Jan O. Backhaus for his creativity and overwhelming enthusiasm which have been the prime motivation for me to pursue a career in the field of physical oceanography. I particularly remember discussions with Jan on how to include a free sea surface in a nonhydrostatic convection model, a problem that neither of us could resolve at that time.

I have included such a model in this book. Other invaluable sources of motivation behind this work are the classical books of Henry Stommel, namely “An Introduction to the Coriolis Force” published in 1989 and co-authored by Dennis Moore, and “A View of the Sea”, published in 1987. Similar to the approach I take here, Stommel’s work underpins theory with computer programs, written in BASIC, that can be run by the reader for illustration of dynamical processes.

Adelaide, Australia, Jochen

Kämpf October 2009"

Produkt-Details

• Erscheint lt. Verlag: 30.4.2010
• Zusatzinfo: XIII, 181 p.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geografie / Kartografie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Climate Sciences • Computational Modelling • geophysical fluid dynamics • Hydrodynamic Modelling • Marine Engineering • ocean • Oceanography
• ISBN-10: 3-642-10610-2 / 3642106102
• ISBN-13: 978-3-642-10610-1 / 9783642106101