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 (IUTAM Bookseries, 28)

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. Preface Contents List of Contributors Waves and Imbalance On spontaneous imbalance and ocean turbulence: generalizations of the Paparella--Young epsilon theorem Introduction Spontaneous imbalance Epsilon theorems for realistic ocean models Specific examples Concluding remarks References Inertia-gravity-wave generation: a geometric-optics approach Introduction Geometric-optics approach Applications to simple flows Horizontal strain and vertical shear Elliptical flow Dipole Random-strain models Discussion References Parallels between stratification and rotation in hydrodynamics, and between both of them and external magnetic field in magnetohydrodynamics, with applications to nonlinear waves Introduction Models 2D stratified Boussinesq equations 2.5D rotating Euler equations 2D magnetohydrodynamics Similarity between models I: waves and structures Linear waves Structures: nonlinear waves/vortices Similarity between models II: geometry Hamiltonian structure Geometry of the phase space and nonconstrained dynamical variables Casting PB to the canonical form Triad and quartet wave interactions and wave turbulence (WT) The WT algorithm Known situations leading to get-it-by-hand solutions for stationary energy spectra in WT WT: decay spectra for gravity, gyroscopic and Alfvèn waves WT: non-decay spectra for gravity and gyroscopic waves Conclusions References Generation of an internal tide by surface tide/eddy resonant interactions Introduction Problem definition Governing equations Wave-triad interactions Multiple-scale analysis Numerical simulations Conclusions References Generation of harmonics and sub-harmonics from an internal tide in a uniformly stratified fluid: numerical and laboratory experiments Introduction Experimental set-ups Laboratory experiments Numerical simulations Emission of the wave beam Spatial structure of the wave beam Parametric instability of the wave beam Generation of harmonics Conclusion References Deep ocean mixing by near-inertial waves Introduction Basic Equations and WKB Mixed Bottom Layer Discussion References Turbulence and Convection Eddies and Circulation: Lessons from Oceans and the GFD Lab Introduction Deep pathways in the oceanic overturning circulation Eddies and Rossby waves in the upper ocean Notes from the GFD Lab Conclusion References Observations on Rapidly Rotating Turbulence Introduction How Columnar Eddies Form at Low Ro The Experimental Evidence at Ro1 Why Linear Behaviour at Ro1? Why a Cyclone-Anticyclone Asymmetry? The Rate of Energy Decay References Equilibration of Inertial Instability in Rotating Flow Introduction Pure Barotropic Instability Pure Inertial Instability Full 3D Simulation vs. Prediction Discussion References Quasigeostrophic and stratified turbulence in the atmosphere Introduction Divergent and geostrophic modes The numerical configuration Results Conclusions References A Perspective on Submesoscale Geophysical Turbulence The Dynamical Regime of Submesoscale Turbulence The Frontogenetic Route Other Submesoscale Generation Routes Stratified, Non-Rotating Turbulence Summary References Spectra and Distribution Functions of Stably Stratified Turbulence Equations of Motion and their Economical Representation Some Historical Comments More Recent Numerical Results Interpretation of DNS Concluding Comments References Modeling mixing in two-dimensional turbulence and stratified fluids Introduction An analogy between statistical mechanics of 2D flows and density stratified fluids Statistical mechanics of 2D flows Statistical mechanics of stratified fluids Relaxation toward statistical equilibrium Dissipation of density fluctuations by turbulent cascade A simple example: mixing of a two layer stratified fluid Coupling the model with an equation for the kinetic energy Conclusion and perspectives References The solar tachocline: a study in stably stratified MHD turbulence Introduction The Solar Tachocline Properties of the solar tachocline Why is the tachocline there --- and so thin? Simplified models of stratified MHD turbulence The parameter regime A hierarchy of models Formation of jets on a magnetised -plane Future directions References Some Unusual Properties of Turbulent Convection and Dynamos in Rotating Spherical Shells Introduction Mathematical formulation of the problem and methods of solution Convection in rotating spherical shells Chaotic convection Distinct turbulent dynamos at identical parameter values Concluding remarks References Instability and Vortex Dynamics Zigzag instability of the Kármán vortex street in stratified and rotating fluids Introduction Problem formulation Pair of vortices in a stratified and rotating fluid Kármán vortex street in a stratified and rotating fluid Results Conclusion References Instabilities of a columnar vortex in a stratified fluid Introduction A Gaussian vortex in a stratified fluid Instabilities of a tilted vortex Spatial structure of a tilted vortex Tilt-induced instabilities Consequences Radiative instability Linear stability analysis Experimental evidence? Conclusion References Geostrophic vortex alignment in external shear or strain Introduction Physical configuration and model equations Evolution of two point-vortices in external strain and rotation Nonlinear regimes of finite-area vortices with background strain and rotation Conclusions Appendix: Melnikov Theory References Equilibrium States of Quasi-geostrophic Point Vortices Introduction Quasi-geostrophic Approximation and Equations of Motion Equilibrium States of Quasi-geostrophic Point Vortices Maximum Entropy Theory Zero Inverse Temperature State Positive and Negative Temperature States Patch Model Summary References Jets: Formation and Structure The structure of zonal jets in shallow water turbulence on the sphere Introduction Jet undulations The potential vorticity staircase Equatorial superrotation Open questions: the nature of forcing and dissipation References Jet formation in decaying two-dimensional turbulence on a rotating sphere Introduction Parameter sweep experiments (Hayashi et al., 2007) Ensemble experiments (Kitamura and Ishioka, 2007) Summary and Discussion References Triple cascade behaviour in QG and drift turbulence and generation of zonal jets Introduction and the model Charney-Hasegawa-Mima model Conservation of energy and enstrophy Conservation of zonostrophy Triple cascade behaviour Dual cascades in 2D Navier-Stokes turbulence Triple cascades in CHM turbulence Alternative argument for zonation Numerical study Centroids Settings for the weakly nonlinear and the strongly nonlinear runs Weakly nonlinear case Strongly nonlinear case Summary References The HyperCASL algorithm Introduction Brief Description of the Numerical Algorithm Fully Lagrangian Advection Transfer of Diabatic Forcing to Point Vortices An Example: A Diabatically-Forced Jet Conclusions and Future Extensions References This book stems from the IUTAM symposium "Rotating Stratified Turbulence and Turbulence in the Atmosphere and Oceans" which took place at the Isaac Newton Institute for Mathematical Sciences in Cambridge from 8 to 12 December 2008, and came at the end of a four-month Programme on "High Reynolds Number Turbulence". This symposium, widely attended by researchers from around the world, aimed to better understand the complex nature of fluid flows found in Nature, particularly in the Earth's atmosphere and oceans, but also in other planetary atmospheres and in the Sun's interior. Talks covered a broad range of subjects, from theoretical studies of fundamental vortex interactions, to laboratory experiments and observations, and to state-of-the-art computational studies. The present volume presents a cross-section of the research presented at the symposium, or in some cases, of the research inspired by it. Herein, 23 peer-reviewed papers are collected (39 talks were presented). The research is loosely divided into four main topics: (1) Waves and Imbalance, (2) Turbulence and Convection, (3) Instability and Vortex Dynamics, and (4) Jets: Formation and Structure