IUTAM Symposium on Turbulence in the Atmosphere and Oceans (eBook)

Preface 7
Contents 9
List of Contributors 16
Waves and Imbalance 21
On spontaneous imbalance and ocean turbulence: generalizations of the Paparella--Young epsilon theorem 22
Introduction 22
Spontaneous imbalance 24
Epsilon theorems for realistic ocean models 26
Specific examples 30
Concluding remarks 32
References 33
Inertia-gravity-wave generation: a geometric-optics approach 35
Introduction 35
Geometric-optics approach 36
Applications to simple flows 39
Horizontal strain and vertical shear 39
Elliptical flow 39
Dipole 40
Random-strain models 41
Discussion 42
References 43
Parallels between stratification and rotation in hydrodynamics, and between both of them and external magnetic field in magnetohydrodynamics, with applications to nonlinear waves 45
Introduction 45
Models 46
2D stratified Boussinesq equations 46
2.5D rotating Euler equations 46
2D magnetohydrodynamics 47
Similarity between models I: waves and structures 48
Linear waves 48
Structures: nonlinear waves/vortices 49
Similarity between models II: geometry 50
Hamiltonian structure 50
Geometry of the phase space and nonconstrained dynamical variables 51
Casting PB to the canonical form 51
Triad and quartet wave interactions and wave turbulence (WT) 52
The WT algorithm 52
Known situations leading to get-it-by-hand solutions for stationary energy spectra in WT 53
WT: decay spectra for gravity, gyroscopic and Alfvèn waves 53
WT: non-decay spectra for gravity and gyroscopic waves 54
Conclusions 54
References 55
Generation of an internal tide by surface tide/eddy resonant interactions 56
Introduction 56
Problem definition 57
Governing equations 57
Wave-triad interactions 58
Multiple-scale analysis 59
Numerical simulations 62
Conclusions 65
References 66
Generation of harmonics and sub-harmonics from an internal tide in a uniformly stratified fluid: numerical and laboratory experiments 68
Introduction 68
Experimental set-ups 69
Laboratory experiments 69
Numerical simulations 71
Emission of the wave beam 72
Spatial structure of the wave beam 73
Parametric instability of the wave beam 74
Generation of harmonics 76
Conclusion 76
References 77
Deep ocean mixing by near-inertial waves 80
Introduction 81
Basic Equations and WKB 81
Mixed Bottom Layer 86
Discussion 88
References 89
Turbulence and Convection 91
Eddies and Circulation: Lessons from Oceans and the GFD Lab 92
Introduction 92
Deep pathways in the oceanic overturning circulation 97
Eddies and Rossby waves in the upper ocean 98
Notes from the GFD Lab 104
Conclusion 106
References 107
Observations on Rapidly Rotating Turbulence 110
Introduction 110
How Columnar Eddies Form at Low Ro 112
The Experimental Evidence at Ro1 115
Why Linear Behaviour at Ro1? 116
Why a Cyclone-Anticyclone Asymmetry? 116
The Rate of Energy Decay 117
References 118
Equilibration of Inertial Instability in Rotating Flow 120
Introduction 120
Pure Barotropic Instability 122
Pure Inertial Instability 124
Full 3D Simulation vs. Prediction 125
Discussion 129
References 129
Quasigeostrophic and stratified turbulence in the atmosphere 131
Introduction 131
Divergent and geostrophic modes 134
The numerical configuration 135
Results 137
Conclusions 140
References 142
A Perspective on Submesoscale Geophysical Turbulence 145
The Dynamical Regime of Submesoscale Turbulence 145
The Frontogenetic Route 147
Other Submesoscale Generation Routes 150
Stratified, Non-Rotating Turbulence 153
Summary 154
References 154
Spectra and Distribution Functions of Stably Stratified Turbulence 156
Equations of Motion and their Economical Representation 156
Some Historical Comments 158
More Recent Numerical Results 160
Interpretation of DNS 163
Concluding Comments 164
References 166
Modeling mixing in two-dimensional turbulence and stratified fluids 168
Introduction 168
An analogy between statistical mechanics of 2D flows and density stratified fluids 170
Statistical mechanics of 2D flows 170
Statistical mechanics of stratified fluids 171
Relaxation toward statistical equilibrium 174
Dissipation of density fluctuations by turbulent cascade 175
A simple example: mixing of a two layer stratified fluid 176
Coupling the model with an equation for the kinetic energy 178
Conclusion and perspectives 179
References 180
The solar tachocline: a study in stably stratified MHD turbulence 181
Introduction 181
The Solar Tachocline 182
Properties of the solar tachocline 182
Why is the tachocline there --- and so thin? 184
Simplified models of stratified MHD turbulence 186
The parameter regime 186
A hierarchy of models 186
Formation of jets on a magnetised -plane 187
Future directions 188
References 189
Some Unusual Properties of Turbulent Convection and Dynamos in Rotating Spherical Shells 192
Introduction 192
Mathematical formulation of the problem and methods of solution 193
Convection in rotating spherical shells 196
Chaotic convection 197
Distinct turbulent dynamos at identical parameter values 200
Concluding remarks 203
References 203
Instability and Vortex Dynamics 206
Zigzag instability of the Kármán vortex street in stratified and rotating fluids 207
Introduction 207
Problem formulation 208
Pair of vortices in a stratified and rotating fluid 208
Kármán vortex street in a stratified and rotating fluid 210
Results 212
Conclusion 215
References 215
Instabilities of a columnar vortex in a stratified fluid 217
Introduction 217
A Gaussian vortex in a stratified fluid 218
Instabilities of a tilted vortex 219
Spatial structure of a tilted vortex 219
Tilt-induced instabilities 220
Consequences 222
Radiative instability 222
Linear stability analysis 222
Experimental evidence? 223
Conclusion 224
References 225
Geostrophic vortex alignment in external shear or strain 226
Introduction 226
Physical configuration and model equations 227
Evolution of two point-vortices in external strain and rotation 228
Nonlinear regimes of finite-area vortices with background strain and rotation 232
Conclusions 234
Appendix: Melnikov Theory 235
References 236
Equilibrium States of Quasi-geostrophic Point Vortices 238
Introduction 238
Quasi-geostrophic Approximation and Equations of Motion 239
Equilibrium States of Quasi-geostrophic Point Vortices 241
Maximum Entropy Theory 244
Zero Inverse Temperature State 244
Positive and Negative Temperature States 245
Patch Model 245
Summary 247
References 247
Jets: Formation and Structure 249
The structure of zonal jets in shallow water turbulence on the sphere 250
Introduction 250
Jet undulations 252
The potential vorticity staircase 254
Equatorial superrotation 255
Open questions: the nature of forcing and dissipation 257
References 258
Jet formation in decaying two-dimensional turbulence on a rotating sphere 260
Introduction 261
Parameter sweep experiments (Hayashi et al., 2007) 262
Ensemble experiments (Kitamura and Ishioka, 2007) 265
Summary and Discussion 267
References 269
Triple cascade behaviour in QG and drift turbulence and generation of zonal jets 271
Introduction and the model 271
Charney-Hasegawa-Mima model 273
Conservation of energy and enstrophy 274
Conservation of zonostrophy 274
Triple cascade behaviour 276
Dual cascades in 2D Navier-Stokes turbulence 276
Triple cascades in CHM turbulence 277
Alternative argument for zonation 279
Numerical study 281
Centroids 282
Settings for the weakly nonlinear and the strongly nonlinear runs 283
Weakly nonlinear case 283
Strongly nonlinear case 285
Summary 287
References 292
The HyperCASL algorithm 295
Introduction 295
Brief Description of the Numerical Algorithm 296
Fully Lagrangian Advection 297
Transfer of Diabatic Forcing to Point Vortices 298
An Example: A Diabatically-Forced Jet 299
Conclusions and Future Extensions 302
References 303