Table of Contents
1 Dynamics of Vortices in Rotating and Stratified Fluids G.J.F. van Heijst 1
1.1 Vortices in Rotating Fluids 1
1.1.1 Basic Equations and Balances 2
1.1.2 How to Create Vortices in the Lab 9
1.1.3 The Ekman Layer 12
1.1.4 Vortex Instability 14
1.1.5 Evolution of Stable Barotropic Vortices 15
1.1.6 Topography Effects 18
1.2 Vortices in Stratified Fluids 20
1.2.1 Basic Properties of Stratified Fluids 20
1.2.2 Generation of Vortices 22
1.2.3 Decay of Vortices 24
1.2.4 Instability and Interactions 30
1.3 Concluding Remarks 33
References 33
2 Stability of Quasi Two-Dimensional Vortices J.-M. Chomaz S. Ortiz F. Gallaire P. Billant 35
2.1 Instabilities of an Isolated Vortex 36
2.1.1 The Shear Instability 37
2.1.2 The Centrifugal Instability 37
2.1.3 Competition Between Centrifugal and Shear Instability 40
2.2 Influence of an Axial Velocity Component 41
2.3 Instabilities of a Strained Vortex 43
2.3.1 The Elliptic Instability 44
2.3.2 The Hyperbolic Instability 46
2.4 The Zigzag Instability 47
2.4.1 The Zigzag Instability in Strongly Stratified Flow Without Rotation 47
2.4.2 The Zigzag Instability in Strongly Stratified Flow with Rotation 50
2.5 Experiment on the Stability of a Columnar Dipole in a Rotating and Stratified Fluid 50
2.5.1 Experimental Setup 50
2.5.2 The State Diagram 51
2.6 Discussion: Instabilities and Turbulence 52
2.7 Appendix: Local Approach Along Trajectories 53
2.7.1 Centrifugal instability 54
2.7.2 Hyperbolic Instability 55
2.7.3 Elliptic Instability 55
2.7.4 Pressureless Instability 56
2.7.5 Small Strain |ε <<1| 56
References 57
3 Oceanic Vortices X. Carton 61
3.1 Observations of Oceanic Vortices 62
3.1.1 Different Types of Oceanic Vortices 62
3.1.2 Generation Mechanisms 67
3.1.3 Vortex Evolution and Decay 70
3.1.4 Submesoscale Structures and Filaments; Biological Activity 72
3.2 Physical and Mathematical Framework for Oceanic Vortex Dynamics 73
3.2.1 Primitive-Equation Model 74
3.2.2 The Shallow-Water Model 76
3.2.3 Frontal Geostrophic Dynamics 86
3.2.4 Quasi-geostrophic Vortices 87
3.2.5 Three-Dimensional, Boussinesq, Non-hydrostatic Models 92
3.3 Process Studies on Vortex Generation, Evolution, and Decay 94
3.3.1 Vortex Generation by Unstable Deep Ocean Jets or of Coastal Currents 94
3.3.2 Vortex Generation by Currents Encountering a Topographic Obstacle 95
3.3.3 Vortex Generation by Currents Changing Direction 96
3.3.4 Beta-Drift of Vortices 98
3.3.5 Interaction Between a Vortex and a Vorticity Front or a Narrow Jet 99
3.3.6 Vortex Decay by Erosion Over Topography 100
3.4 Conclusions 100
References 101
4 Lagrangian Dynamics of Fronts, Vortices and Waves: Understanding the (Semi-)geostrophic Adjustment V. Zeitlin 109
4.1 Introduction: Geostrophic Adjustment in GFD and Related Problems 109
4.2 Fronts, Waves, Vortices and the Adjustment Problem in 1.5d Rotating Shallow Water Model 110
4.2.1 The Plane-Parallel Case 110
4.2.2 Axisymmetric Case 118
4.3 Including Baroclinicity: 2-Layer 1.5d RSW 121
4.3.1 Plane-Parallel Case 121
4.3.2 Axisymmetric Case 127
4.4 Continuously Stratified Rectilinear Fronts 128
4.4.1 Lagrangian Approach in the Case of Continuous Stratification 128
4.4.2 Existence and Uniqueness of the Adjusted State in the Unbounded Domain 130
4.4.3 Trapped Modes and Symmetric Instability in Continuously Stratified Case 133
4.5 Conclusions 136
References 136
5 Wave-Vortex Interactions O. Bühler 139
5.1 Introduction 139
5.2 Lagrangian Mean Flow and Pseudomomentum 142
5.2.1 Lagrangian Averaging 143
5.2.2 Pseudomomentum and the Circulation Theorem 144
5.2.3 Impulse Budget of the GLM Equations 147
5.2.4 Ray Tracing Equations 150
5.2.5 Impulse Plus Pseudomomentum Conservation Law 155
5.3 PV Generation by Wave Breaking and Dissipation 157
5.3.1 Breaking Waves and Vorticity Generation 157
5.3.2 Momentum-Conserving Dissipative Forces 159
5.3.3 A Wavepacket Life Cycle Experiment 160
5.3.4 Wave Dissipation Versus Mean Flow Acceleration 163
5.4 Wave-Driven Vortices on Beaches 165
5.4.1 Impulse for One-Dimensional Topography 166
5.4.2 Wave-Induced Momentum Flux Convergence and Drag 168
5.4.3 Barred Beaches and Current Dislocation 169
5.5 Wave Refraction by Vortices 171
5.5.1 Anatomy of Wave Refraction by the Mean Flow 172
5.5.2 Refraction by Weak Irrotational Basic Flow 173
5.5.3 Bretherton Flow and Remote Recoil 174
5.5.4 Wave Capture of Internal Gravity Waves 177
5.5.5 Impulse Plus Pseudomomentum for Stratified Flow 179
5.5.6 Local Mean Flow Amplitude at the Wavepacket 180
5.5.7 Wave-Vortex Duality and Dissipation 183
5.6 Concluding Comments 184
References 185
Index 189
