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This page intentionally left blank Non-linear dynamics and statistical theories for basic geophysical flows ANDREW J. MAJDA New York University XIAOMING WANG Florida State University Cambridge University Press Cambridge New York Melbourne Madrid Cape Town Singapore São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521834414 © Cambridge University Press 2006 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2006 ISBN-13 978-0-511-16813-0 eBook (EBL) ISBN-10 0-511-16813-6 eBook (EBL) ISBN-13 978-0-521-83441-4 hardback ISBN-10 0-521-83441-4 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page xi 1 Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction 1 1.1 Introduction 1 1.2 Some special exact solutions 8 1.3 Conserved quantities 33 1.4 Barotropic geophysical flows in a channel domain – an important physical model 44 1.5 Variational derivatives and an optimization principle for elementary geophysical solutions 50 1.6 More equations for geophysical flows 52 References 58 2 The response to large-scale forcing 59 2.1 Introduction 59 2.2 Non-linear stability with Kolmogorov forcing 62 2.3 Stability of flows with generalized Kolmogorov forcing 76 References 79 3 The selective decay principle for basic geophysical flows 80 3.1 Introduction 80 3.2 Selective decay states and their invariance 82 3.3 Mathematical formulation of the selective decay principle 84 3.4 Energy–enstrophy decay 86 3.5 Bounds on the Dirichlet quotient, t 88 3.6 Rigorous theory for selective decay 90 3.7 Numerical experiments demonstrating facets of selective decay 95 References 102 v vi Contents A.1 Stronger controls ont 103 A.2 The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect 107 4 Non-linear stability of steady geophysical flows 115 4.1 Introduction 115 4.2 Stability of simple steady states 116 4.3 Stability for more general steady states 124 4.4 Non-linear stability of zonal flows on the beta-plane 129 4.5 Variational characterization of the steady states 133 References 137 5 Topographic mean flow interaction, non-linear instability, and chaotic dynamics 138 5.1 Introduction 138 5.2 Systems with layered topography 141 5.3 Integrable behavior 145 5.4 A limit regime with chaotic solutions 154 5.5 Numerical experiments 167 References 178 Appendix 1 180 Appendix 2 181 6 Introduction to information theory and empirical statistical theory 183 6.1 Introduction 183 6.2 Information theory and Shannon’s entropy 184 6.3 Most probable states with prior distribution 190 6.4 Entropy for continuous measures on the line 194 6.5 Maximum entropy principle for continuous fields 201 6.6 An application of the maximum entropy principle to geophysical flows with topography 204 6.7 Application of the maximum entropy principle to geophysical flows with topography and mean flow 211 References 218 7 Equilibrium statistical mechanics for systems of ordinary differential equations 219 7.1 Introduction 219 7.2 Introduction to statistical mechanics for ODEs 221 7.3 Statistical mechanics for the truncated Burgers–Hopf equations 229 7.4 The Lorenz 96 model 239 References 255 Contents vii 8 Statistical mechanics for the truncated quasi-geostrophic equations 256 8.1 Introduction 256 8.2 The finite-dimensional truncated quasi-geostrophic equations 258 8.3 The statistical predictions for the truncated systems 262 8.4 Numerical evidence supporting the statistical prediction 264 8.5 The pseudo-energy and equilibrium statistical mechanics for fluctuations about the mean 267 8.6 The continuum limit 270 8.7 The role of statistically relevant and irrelevant conserved quantities _______________________________________last_pages_______________________________ Appendix 2 549 Index advective derivative, 2 auto-correlation function, 252 barotropic beta-plane equations, 19 barotropic one-layer model, 54 barotropic quasi-geostrophic equations, 2, 59 barotropic quasi-geostrophic equations in a channel with topographic stress, 46 barotropic quasi-geostrophic equations on the unit sphere, 483 beta plane effect, 2, 17 beta plane equations with mean shear flow, 28 Casimir invariants, 443 Cauchy–Schwarz inequality, 216, 315 channel geometry, 45, 49 circulation, 48, 50 complete statistical mechanics, 256, 303, 529 conserved quantities, 33, 504 continuously stratified quasi-geostrophic equations, 53 continuum limit, 270 Coriolis force, 2 correlation deficit, 365 correlation function, 25, 238 correlation time, 238 crude closure algorithm, 390 dilute PV theory, 332 Dirichlet quotient, 84 dispersion relation, 20 dispersive wave, 20 dissipation mechanism, 2, 18, 30 EEST, 318 Ekman drag, 5 empirical statistical theory with infinitely many constraints, 309 energy, 22, 33, 42, 61 energy–circulation theory, 292, 537 energy–circulation impulse theory, 297 energy–enstrophy theory, 211, 525 energy spectrum, 435 ensemble prediction, 452, 453 enstrophy, 30, 40, 42, 61 enstrophy identity, 61 EOF basis, 457 equations for large-scale and small-scale flow interaction via topographic stress, 38, 43, 116 ergodicity, 228 ESTMC, 318, 540 ESTP, 318, 536 Euler equations, 6, 12 exact dynamics of the ground state modes and the nth energy shell, 495 flatness, 201 Fourier series representation of barotropic quasi-geostrophic equations, 68 Fourier series tool kit, 10 F-plane equation, 53 free-slip boundary condition, 16 Gamma distribution, 422 Gaussian distribution, 195, 452 generalized enstrophy, 40, 46, 49 geophysical fluid dynamics, 1 geostrophic balance, 4 Gibbs measure, 226 Great Red Spot (GRS), 411 Gronwall inequality, 65 ground state eigenmode, 12 ground state modes dynamics, 492 Hamiltonian system, 442 impulse, 47, 49 information-theoretic entropy, see Shannon entropy invariant measure, 227 isobars, 4 Jacobi identity, 443 Jacobian determinant, 9 kinetic energy identity, 36 Kolmogorov forcing, 17, 60, 76 550 Langevin theory, 332 Laplace–Beltrami operator, 488 large-scale enstrophy, 39, 43, 46, 49 latitude, 4 layered topographic equations, 143 Legendre functions, 489 Limaye profile, 412 Liouville equation, 222, 453 Liouville property, 221 longitude, 4 Lorenz 96 model (L96), 239 maximum entropy principle, 188, 196 maximum relative entropy principle, 192, 196, 204 prior distribution, 203 pseudo-energy, 269 PVST, 318 radiative damping, 5 relative entropy, 192, 195, 204, 452, 455 relative entropy identity, 459 relative vorticity, 2 relaxation, 473 response to large-scale forcing, 67, 511 restricted stability of motion on the first two energy shells, 508 Reynolds number, 97 Rhines measure of anisotropy, 96 Rossby waves, 19, 496 selective decay principle, 81, 85, 518, 519 selective decay state, 81 Shannon entropy, 185, 195, 212, 225 shear flow, 13 signal and dispersion decomposition, 460 sine-bracket truncation, 430 skewness, 201, 421 slaving principle, 78 spectrally truncated quasi-geostrophic equations, 259 spin-up, 394 stability of simple steady states, 116, 505 statistical relevance, 427 statistical solution, 219 statistically sharp, 347 Stokes formula, 488 stratification, 53 stream function, 2 surface spherical coordinates, 482 surface spherical harmonics, 489 swimming pool geometry, 41 swirling eddies, 13 symplectic matrix, 443 tangent plane approximation, 4 Ключевые слова: xj, condition, m t, york, mathematical, forcing, exact, dj, barotropic quasi-geostrophic, flow, solution, ln, xg, quasi, small scale, time, large-scale, von neumann, galileo probe, gas particle, statistical theory, dx, quasi-geostrophic, dipole-vortex street, model, ha, potential, density, poincar map, form, galileo mission, cov, conserved, convective storm, dt dt, assume, math, principle, stability, pure, plane, dissipation, stable, comet shoemaker-levy, field, keixk, theory, vice versa, fj p, vortex, ground, dt, statistical, energy, selective, behavior, divergence free, proposition, equilibrium, subsection, gjhk, limit, proof, stream function, tr dk, prior, kdk, geophysical flow, quasi geostrophic, small, vorticity, constraint, case, linear, equation, term, cec, judicious prior, non-linear, initial, relative, function, geophysical, kj, zn zn, statistical irrelevance, conserved quantity, fact, fourier, moment, potential vorticity, wn, absolutely continuous, velocity, mode, linearly dependent, scale, circulation, prediction, topography, exp, akei, sin, convective tower, dn dm, number, channel, majda, n m, recall, stream, chapter, point, ekman drag, aq dt, pk, northern hemisphere, truncated, special, quantity, shell, maximum, probability, 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