Preface	5
	Contents	7
	The Problem	19
	Physical Interpretation	19
	Example	20
	How to Produce a Simple Graph with SciLab	21
	Finite Time Step and Time Level	22
	Explicit Time-Forward Iteration	22
	Condition of Numerical Stability for Explicit Scheme	22
	Implicit Time-Forward Iteration	23
	Hybrid Schemes	23
	Other Schemes	23
	Condition of Consistency	24
	Condition of Accuracy	24
	Condition of Efficiency	24
	How Model Codes Work	24
	The First FORTRAN Code	25
	How to Compile and Run FORTRAN Codes	25
	A Quick Start to FORTRAN	25
	Aim	27
	Task Description	28
	Instructions	28
	Sample Code	28
	Results	28
	Additional Exercise for the Reader	28
	Detection and Elimination of Errors	29
	Error Messages	29
	Correct Errors One by One	29
	Ignore Error Message Text	29
	Frequent Errors	30
	Trust Your Compiler	30
	Display Warnings	30
	Units	31
	Scalars and Vectors	31
	Difference Between Scalars and Vectors	31
	Contours and Contour Interval	32
	Location and Velocity	32
	Location and Distance	32
	Calculation of Distances with SciLab	33
	Velocity	33
	Types of Motion	34
	Steady-State Motions	34
	Waves	34
	The Sinusoidal Waveform	34
	Visualisation of a Wave Using SciLab	35
	A Simple Wave Made of Vertically Moving Bars	35
	Sample Script	35
	The First SciLab Script	36
	A Quick-Start to SciLab	36
	The First GIF Animation	37
	Modified Animation Script	37
	Creation of an Animated GIF File	38
	Phase Speed	38
	Dispersion Relation	38
	Superposition of Waves	39
	Exercise 2: Wave Interference	39
	Aim	39
	Task Description	39
	Sample Script	40
	A Glimpse of Results	40
	A Rule of Thumb	40
	Forces	41
	What Forces Do	41
	Newton's Laws of Motion	42
	Apparent Forces	42
	Lagrangian Trajectories	42
	Eulerian Frame of Reference and Advection	42
	Interpretation of the Advection Equation	43
	The Nonlinear Terms	43
	Impacts of the Nonlinear Terms	44
	Fundamental Conservation Principles	44
	A List of Principles	44
	Conservation of Momentum	44
	Conservation of Volume -- The Continuity Equation	44
	Vertically Integrated Form of the Continuity Equation	46
	Divergence or Convergence?	46
	The Continuity Equation for Streamflows	47
	Density	48
	The Equation of State for Seawater	48
	Gravity and the Buoyancy Force	48
	Archimedes' Principle	48
	Reduced Gravity	49
	Stability Frequency	49
	Stable, Neutral and Unstable Conditions	50
	Exercise 3: Oscillations of a Buoyant Object	50
	Aim	50
	Task Description	50
	Momentum Equations	50
	Code Structure	51
	Finite-Difference Equations	51
	Initial and Boundary Conditions	51
	Sample Code and Animation Script	52
	Discussion of Results	52
	Analytical Solution	53
	Inclusion of Friction	53
	Additional Exercises for the Reader	55
	The Pressure-Gradient Force	55
	The Hydrostatic Balance	55
	Which Processes are Hydrostatic?	55
	The Hydrostatic Pressure Field in the Ocean	55
	Dynamic Pressure in the Ocean	56
	The Horizontal Pressure-Gradient Force	56
	The Boussinesq Approximation	56
	The Case of Uniform Density	57
	The Coriolis Force	57
	Apparent Forces	57
	The Centripetal Force and the Centrifugal Force	58
	Derivation of the Centripetal Force	59
	The Centrifugal Force in a Rotating Fluid	60
	Motion in a Rotating Fluid as Seen in the Fixed Frameof Reference	61
	Parcel Trajectory	61
	Numerical Code	62
	Analytical Solution	63
	The Coriolis Force	63
	The Coriolis Force on Earth	64
	The Local Vertical	64
	The Coriolis Parameter	65
	The f-Plane Approximation	66
	The Beta-Plane Approximation	66
	Exercise 4: The Coriolis Force in Action	67
	Aim	67
	First Attempt	67
	Improved Scheme 1: the Semi-Implicit Approach	67
	Improved Scheme 2: The Local-Rotation Approach	69
	Yes!	69
	Sample Code and Animation Script	69
	Inertial Oscillations	70
	Sample Code and Animation Script	71
	Turbulence	71
	Laminar and Turbulent Flow	71
	The Reynolds Approach	71
	What Causes Turbulence?	72
	The Richardson Number	72
	Turbulence Closure and Turbulent Diffusion	73
	Prandtl's Mixing Length	73
	Interpretation of the Diffusion Equation	73
	The Navier--Stokes Equations	74
	Complete Set of Equations	74
	Boundary Conditions for Oceanic Applications	75
	Scaling	75
	The Idea	75
	Example of Scaling	76
	More on Finite Differences	78
	Taylor Series	78
	Forward, Backward and Centred Differences	79
	Scheme for the Second Derivative	79
	Truncation Error	80
	Long Surface Gravity Waves	81
	Extraction of Individual Processes	81
	Shallow-Water Processes	81
	The Shallow-Water Model	81
	The Governing Equations	82
	Analytical Wave Solution	82
	Animation Script	83
	Numerical Grid	84
	Finite-Difference Scheme	84
	Stability Criterion	85
	First-Order Shapiro Filter	86
	Land and Coastlines	86
	Lateral Boundary Conditions	86
	Modular FORTRAN Scripting	87
	Structure of the Following FORTRAN Codes	88
	Exercise 5: Long Waves in a Channel	89
	Aim	89
	Instructions	89
	Sample Code and Animation Script	90
	Results	90
	Exercise 6: The Flooding Algorithm	90
	Aim	90
	Redefinition of Wet and Dry	92
	Enabling Flooding of Dry Grid Cells	92
	Flooding of Sloping Beaches	92
	Ultimate Crash Tests	93
	Sample Code and Animation Script	93
	Results	94
	The Multi-Layer Shallow-Water Model	95
	Basics	95
	Exercise 7: Long Waves in a Layered Fluid	97
	Aim	97
	Task Description	97
	Sample Code and Animation Script	98
	Results	98
	Phase Speed of Long Internal Waves	99
	Natural Oscillations in Closed Bodies of Fluid	99
	Merian's Formula	100
	Co-oscillations in Bays	101
	Additional Exercise for the Reader	101
	Long Waves in a Shallow Lake	103
	The 2D Shallow-Water Wave Equations	103
	Arakawa C-grid	103
	Finite-Difference Equations	104
	Inclusion of Land and Coastlines	105
	Stability Criterion	106
	Exercise 8: Long Waves in a Shallow Lake	106
	Aim	106
	Task Description	106
	Sample Code and Animation Script	106
	Snapshot Results	107
	Additional Exercise for the Reader	107
	Exercise 9: Wave Refraction	107
	Aim	107
	Background	107
	Task Description	108
	Lateral Boundary Conditions	108
	Sample Code and Animation Script	110
	Results	110
	Additional Exercise for the Reader	110
	The Wind-Forced Shallow-Water Model	111
	The Governing Equations	111
	Semi-implicit Approach for Bottom Friction	111
	Finite-Difference Equations	112
	Exercise 10: Wind-Driven Flow in a Lake	113
	Aim	113
	Creation of Variable Bathymetry	113
	Sample Code	113
	Task Description	113
	Tricks for Long Model Simulations	114
	Results	114
	Sample Code and Animation Script	115
	Caution	115
	Additional Exercise for the Reader	116
	Movement of Tracers	116
	Lagrangian Versus Eulerian Tracers	116
	A Difficult Task	116
	Eulerian Advection Schemes	116
	Stability Criterion for the Advection Equation	118
	Exercise 11: Eulerian Advection	118
	Aim	118
	Task Description	119
	Results	119
	Recommendation	121
	Sample Code and Animation Script	121
	Exercise 12: Trajectories	121
	Aim	121
	Task Description	121
	Results	121
	Sample Code and Animation Script	123
	Exercise 13: Inclusion of Nonlinear Terms	123
	Aim	123
	Formulation of the Nonlinear Terms	123
	Sample Code	123
	Results	124
	Exercise 14: Island Wakes	124
	Aim	124
	The Reynolds Number	125
	Inclusion of Lateral Friction and Momentum Diffusion	125
	Stability Criterion for Diffusion Terms	127
	Full-Slip, Semi-Slip and No-Slip Conditions	127
	Task Description	128
	Sample Code	129
	Results	129
	Additional Exercises for the Reader	130
	The Complete Shallow-Water Equations	131
	Description	131
	Implementation of the Coriolis Force	131
	Coastal Kelvin Waves	132
	Theory	132
	Exercise 15: Coastal Kelvin Waves	133
	Aim	133
	Task Description	133
	Results	134
	Sample Codes and Animation Script	134
	Additional Exercise for the Reader	134
	Geostrophic Flow	134
	Scaling	134
	The Geostrophic Balance	135
	Geostrophic Equations	135
	Vorticity	136
	Conservation of Potential Vorticity	138
	Topographic Steering	139
	Rossby Waves	139
	Exercise 16: Topographic Steering	141
	Aim	141
	Model Equations	141
	Task Description	141
	Caution	142
	Sample Code	142
	Results	142
	Additional Exercise for the Reader	144
	Instability of Lateral Shear Flows	144
	Theory	144
	Instability to Long Waves	146
	Exercise 17: Barotropic Instability	146
	Aim	146
	Model Equations	146
	Task Description	146
	Results	147
	Sample Code and Animation Script	148
	Additional Exercise for the Reader	149
	The Wind-Driven Circulation of the Ocean	149
	The Dynamical Structure of the Ocean	149
	Steady-State Dynamics and Volume Transport	149
	A Simplified Model of the Wind-driven Circulation	150
	The Surface Ekman Layer	151
	Ekman-layer Transport	151
	Ekman Pumping	152
	The Sverdrup Balance	153
	Interpretation of the Sverdrup Relation	153
	The Bottom Ekman Layer	154
	Western Boundary Currents	155
	The Role of Lateral Momentum Diffusion	156
	Exercise 18: The Wind-Driven Circulation	156
	Aim	156
	Task Description	156
	Results	158
	Sample Code and Animation Script	159
	Additional Exercises for the Reader	159
	Exercise 19: Baroclinic Compensation	160
	Background	160
	Aim	160
	Task Description	160
	Results	161
	Sample Code and Scilab Animation Script	161
	Additional Exercise for the Reader	161
	The Reduced-Gravity Concept	161
	Background	161
	The Rigid-lid Approximation	162
	Geostrophic Adjustment of a Density Front	163
	Background	163
	How Does It Work?	164
	Theory	165
	Exercise 20: Geostrophic Adjustment	167
	Aim	167
	Task Description	167
	Results	168
	Sample Code and Animation Script	169
	Additional Exercise for the Reader	169
	Baroclinic Instability	169
	Brief Description	169
	Exercise 21: Frontal Instability	170
	Aim	170
	Task Description	170
	Results	171
	Sample Code and Animation Script	173
	Additional Exercise for the Reader	174
	Density-Driven Flows	174
	Background	174
	Exercise 22: Reduced-Gravity Plumes	175
	Aim	175
	Task Description	175
	Write a New Simulation Code?	176
	Results	176
	Sample Code and Animation Script	178
	Additional Exercise for the Reader	179
	Technical Information	179