Dritschel IUTAM Symposium on Turbulence in the Atmosphere and Oceans

Proceedings of the IUTAM Symposium on Turbulence in the Atmosphere and Oceans, Cambridge, UK, December 8 ? 12, 2008
1. Auflage 2010
ISBN: 978-94-007-0360-5
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 298 Seiten

The text of the Persian poet Rum ¯ ¯ ?, written some eight centuries ago, and reproduced at the beginning of this book is still relevant to many of our pursuits of knowledge, not least of turbulence. The text illustrates the inability people have in seeing the whole thing, the 'big picture'. Everybody looks into the problem from his/her vi- point, and that leads to disagreement and controversy. If we could see the whole thing, our understanding would become complete and there would be no cont- versy. The turbulent motion of the atmosphere and oceans, at the heart of the observed general circulation, is undoubtedly very complex and dif?cult to understand in its entirety. Even 'bare' turbulence, without rotation and strati?cation whose effects are paramount in the atmosphere and oceans, still poses great fundamental ch- lenges for understanding after a century of research. Rotating strati?ed turbulence is a relatively new research topic. It is also far richer, exhibiting a host of distinct wave types interacting in a complicated and often subtle way with long-lived - herent structures such as jets or currents and vortices. All of this is tied together by basic ?uid-dynamical nonlinearity, and this gives rise to a multitude of phen- ena: spontaneous wave emission, wave-induced transport, both direct and inverse energy scale cascades, lateral and vertical anisotropy, fronts and transport barriers, anomalous transport in coherent vortices, and a very wide range of dynamical and thermodynamical instabilities.

Autoren/Hrsg.

Dritschel, David

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;List of Contributors;16
4;Waves and Imbalance;21
4.1;On spontaneous imbalance and ocean turbulence: generalizations of the Paparella--Young epsilon theorem;22
4.1.1;Introduction;22
4.1.2;Spontaneous imbalance;24
4.1.3;Epsilon theorems for realistic ocean models;26
4.1.4;Specific examples;30
4.1.5;Concluding remarks;32
4.1.6;References;33
4.2;Inertia-gravity-wave generation: a geometric-optics approach;35
4.2.1;Introduction;35
4.2.2;Geometric-optics approach;36
4.2.3;Applications to simple flows;39
4.2.3.1;Horizontal strain and vertical shear;39
4.2.3.2;Elliptical flow;39
4.2.3.3;Dipole;40
4.2.3.4;Random-strain models;41
4.2.4;Discussion;42
4.2.5;References;43
4.3;Parallels between stratification and rotation in hydrodynamics, and between both of them and external magnetic field in magnetohydrodynamics, with applications to nonlinear waves;45
4.3.1;Introduction;45
4.3.2;Models;46
4.3.2.1;2D stratified Boussinesq equations;46
4.3.2.2;2.5D rotating Euler equations ;46
4.3.2.3;2D magnetohydrodynamics;47
4.3.3;Similarity between models I: waves and structures;48
4.3.3.1;Linear waves;48
4.3.3.2;Structures: nonlinear waves/vortices;49
4.3.4;Similarity between models II: geometry;50
4.3.4.1;Hamiltonian structure;50
4.3.4.2;Geometry of the phase space and nonconstrained dynamical variables;51
4.3.4.3;Casting PB to the canonical form;51
4.3.5;Triad and quartet wave interactions and wave turbulence (WT);52
4.3.5.1;The WT algorithm;52
4.3.5.2;Known situations leading to get-it-by-hand solutions for stationary energy spectra in WT;53
4.3.5.3;WT: decay spectra for gravity, gyroscopic and Alfvèn waves;53
4.3.5.4;WT: non-decay spectra for gravity and gyroscopic waves;54
4.3.6;Conclusions;54
4.3.7;References;55
4.4;Generation of an internal tide by surface tide/eddy resonant interactions;56
4.4.1;Introduction;56
4.4.2;Problem definition;57
4.4.2.1;Governing equations;57
4.4.2.2;Wave-triad interactions;58
4.4.2.3;Multiple-scale analysis;59
4.4.3;Numerical simulations;62
4.4.4;Conclusions;65
4.4.5;References;66
4.5;Generation of harmonics and sub-harmonics from an internal tide in a uniformly stratified fluid: numerical and laboratory experiments;68
4.5.1;Introduction;68
4.5.2;Experimental set-ups;69
4.5.2.1;Laboratory experiments;69
4.5.2.2;Numerical simulations;71
4.5.3;Emission of the wave beam;72
4.5.4;Spatial structure of the wave beam;73
4.5.5;Parametric instability of the wave beam;74
4.5.6;Generation of harmonics;76
4.5.7;Conclusion;76
4.5.8;References;77
4.6;Deep ocean mixing by near-inertial waves;80
4.6.1;Introduction;81
4.6.2;Basic Equations and WKB;81
4.6.3;Mixed Bottom Layer;86
4.6.4;Discussion;88
4.6.5;References;89
5;Turbulence and Convection;91
5.1;Eddies and Circulation: Lessons from Oceans and the GFD Lab;92
5.1.1;Introduction;92
5.1.2;Deep pathways in the oceanic overturning circulation;97
5.1.3;Eddies and Rossby waves in the upper ocean;98
5.1.4;Notes from the GFD Lab;104
5.1.5;Conclusion;106
5.1.6;References;107
5.2;Observations on Rapidly Rotating Turbulence;110
5.2.1;Introduction;110
5.2.2;How Columnar Eddies Form at Low Ro;112
5.2.3;The Experimental Evidence at Ro1;115
5.2.4;Why Linear Behaviour at Ro1?;116
5.2.5;Why a Cyclone-Anticyclone Asymmetry?;116
5.2.6;The Rate of Energy Decay;117
5.2.7;References;118
5.3;Equilibration of Inertial Instability in Rotating Flow;120
5.3.1;Introduction;120
5.3.2;Pure Barotropic Instability;122
5.3.3;Pure Inertial Instability;124
5.3.4;Full 3D Simulation vs. Prediction;125
5.3.5;Discussion;129
5.3.6;References;129
5.4;Quasigeostrophic and stratified turbulence in the atmosphere;131
5.4.1;Introduction;131
5.4.2;Divergent and geostrophic modes;134
5.4.3;The numerical configuration;135
5.4.4;Results;137
5.4.5;Conclusions;140
5.4.6;References;142
5.5;A Perspective on Submesoscale Geophysical Turbulence;145
5.5.1;The Dynamical Regime of Submesoscale Turbulence;145
5.5.2;The Frontogenetic Route;147
5.5.3;Other Submesoscale Generation Routes;150
5.5.4;Stratified, Non-Rotating Turbulence;153
5.5.5;Summary;154
5.5.6;References;154
5.6;Spectra and Distribution Functions of Stably Stratified Turbulence;156
5.6.1;Equations of Motion and their Economical Representation;156
5.6.2;Some Historical Comments;158
5.6.3;More Recent Numerical Results;160
5.6.4;Interpretation of DNS;163
5.6.5;Concluding Comments;164
5.6.6;References;166
5.7;Modeling mixing in two-dimensional turbulence and stratified fluids;168
5.7.1;Introduction;168
5.7.2;An analogy between statistical mechanics of 2D flows and density stratified fluids;170
5.7.2.1;Statistical mechanics of 2D flows;170
5.7.2.2;Statistical mechanics of stratified fluids;171
5.7.3;Relaxation toward statistical equilibrium;174
5.7.4;Dissipation of density fluctuations by turbulent cascade;175
5.7.5;A simple example: mixing of a two layer stratified fluid;176
5.7.6;Coupling the model with an equation for the kinetic energy;178
5.7.7;Conclusion and perspectives;179
5.7.8;References;180
5.8;The solar tachocline: a study in stably stratified MHD turbulence;181
5.8.1;Introduction;181
5.8.2;The Solar Tachocline;182
5.8.2.1;Properties of the solar tachocline;182
5.8.2.2;Why is the tachocline there --- and so thin?;184
5.8.3;Simplified models of stratified MHD turbulence;186
5.8.3.1;The parameter regime;186
5.8.3.2;A hierarchy of models;186
5.8.3.3;Formation of jets on a magnetised -plane;187
5.8.4;Future directions;188
5.8.5;References;189
5.9;Some Unusual Properties of Turbulent Convection and Dynamos in Rotating Spherical Shells;192
5.9.1;Introduction;192
5.9.2;Mathematical formulation of the problem and methods of solution;193
5.9.3;Convection in rotating spherical shells;196
5.9.4;Chaotic convection;197
5.9.5;Distinct turbulent dynamos at identical parameter values;200
5.9.6;Concluding remarks;203
5.9.7;References;203
6;Instability and Vortex Dynamics;206
6.1;Zigzag instability of the Kármán vortex street in stratified and rotating fluids;207
6.1.1;Introduction;207
6.1.2;Problem formulation;208
6.1.2.1;Pair of vortices in a stratified and rotating fluid;208
6.1.2.2;Kármán vortex street in a stratified and rotating fluid;210
6.1.3;Results;212
6.1.4;Conclusion;215
6.1.5;References;215
6.2;Instabilities of a columnar vortex in a stratified fluid;217
6.2.1;Introduction;217
6.2.2;A Gaussian vortex in a stratified fluid;218
6.2.3;Instabilities of a tilted vortex;219
6.2.3.1;Spatial structure of a tilted vortex;219
6.2.3.2;Tilt-induced instabilities;220
6.2.3.3;Consequences;222
6.2.4;Radiative instability;222
6.2.4.1;Linear stability analysis;222
6.2.4.2;Experimental evidence?;223
6.2.5;Conclusion;224
6.2.6;References;225
6.3;Geostrophic vortex alignment in external shear or strain;226
6.3.1;Introduction;226
6.3.2;Physical configuration and model equations;227
6.3.3;Evolution of two point-vortices in external strain and rotation;228
6.3.4;Nonlinear regimes of finite-area vortices with background strain and rotation;232
6.3.5;Conclusions;234
6.3.6;Appendix: Melnikov Theory;235
6.3.7;References;236
6.4;Equilibrium States of Quasi-geostrophic Point Vortices;238
6.4.1;Introduction;238
6.4.2;Quasi-geostrophic Approximation and Equations of Motion;239
6.4.3;Equilibrium States of Quasi-geostrophic Point Vortices;241
6.4.4;Maximum Entropy Theory;244
6.4.4.1;Zero Inverse Temperature State;244
6.4.4.2;Positive and Negative Temperature States;245
6.4.4.3;Patch Model;245
6.4.5;Summary;247
6.4.6;References;247
7;Jets: Formation and Structure;249
7.1;The structure of zonal jets in shallow water turbulence on the sphere;250
7.1.1;Introduction;250
7.1.2;Jet undulations;252
7.1.3;The potential vorticity staircase;254
7.1.4;Equatorial superrotation;255
7.1.5;Open questions: the nature of forcing and dissipation;257
7.1.6;References;258
7.2;Jet formation in decaying two-dimensional turbulence on a rotating sphere;260
7.2.1;Introduction;261
7.2.2;Parameter sweep experiments (Hayashi et al., 2007);262
7.2.3;Ensemble experiments (Kitamura and Ishioka, 2007);265
7.2.4;Summary and Discussion;267
7.2.5;References;269
7.3;Triple cascade behaviour in QG and drift turbulence and generation of zonal jets;271
7.3.1;Introduction and the model;271
7.3.2;Charney-Hasegawa-Mima model;273
7.3.3;Conservation of energy and enstrophy;274
7.3.4;Conservation of zonostrophy;274
7.3.5;Triple cascade behaviour;276
7.3.5.1;Dual cascades in 2D Navier-Stokes turbulence;276
7.3.5.2;Triple cascades in CHM turbulence;277
7.3.5.3;Alternative argument for zonation;279
7.3.6;Numerical study;281
7.3.6.1;Centroids;282
7.3.6.2;Settings for the weakly nonlinear and the strongly nonlinear runs;283
7.3.6.3;Weakly nonlinear case;283
7.3.6.4;Strongly nonlinear case;285
7.3.7;Summary;287
7.3.8;References;292
7.4;The HyperCASL algorithm;295
7.4.1;Introduction;295
7.4.2;Brief Description of the Numerical Algorithm;296
7.4.2.1;Fully Lagrangian Advection;297
7.4.2.2;Transfer of Diabatic Forcing to Point Vortices;298
7.4.3;An Example: A Diabatically-Forced Jet;299
7.4.4;Conclusions and Future Extensions;302
7.4.5;References;303