Non-linear dynamics and statistical theories for basic geophysical flows

The general area of geophysical fluid mechanics is truly interdisciplinary. Now ideas from statistical physics are being applied in novel ways to inhomogeneous complex systems such as atmospheres and oceans. In this book, the basic ideas of geophysics, probability theory, information theory, nonlinear dynamics and equilibrium statistical mechanics are introduced and applied to large time-selective decay, the effect of large scale forcing, nonlinear stability, fluid flow on a sphere and Jupiter's Great Red Spot. The book is the first to adopt this approach and it contains many recent ideas and results. Its audience ranges from graduate students and researchers in both applied mathematics and the geophysical sciences. It illustrates the richness of the interplay of mathematical analysis, qualitative models and numerical simulations which combine in the emerging area of computational science.

Preface	p. xi
1 Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction	p. 1
1.1 Introduction	p. 1
1.2 Some special exact solutions	p. 8
1.3 Conserved quantities	p. 33
1.4 Barotropic geophysical flows in a channel domain - an important physical model	p. 44
1.5 Variational derivatives and an optimization principle for elementary geophysical solutions	p. 50
1.6 More equations for geophysical flows	p. 52
References	p. 58
2 The response to large-scale forcing	p. 59
2.1 Introduction	p. 59
2.2 Non-linear stability with Kolomogorov forcing	p. 62
2.3 Stability of flows with generalized Kolmogorov forcing	p. 76
References	p. 79
3 The selective decay principle for basic geophysical flows	p. 80
3.1 Introduction	p. 80
3.2 Selective decay states and their invariance	p. 82
3.3 Mathematical formulation of the selective decay principle	p. 84
3.4 Energy-enstrophy decay	p. 86
3.5 Bounds on the Dirichlet quotient, [Lambda](t)	p. 88
3.6 Rigorous theory for selective decay	p. 90
3.7 Numerical experiments demonstrating facets of selective decay	p. 95
References	p. 102
A.1 Stronger controls on [Lambda](t)	p. 103
A.2 The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect	p. 107
4 Non-linear stability of steady geophysical flows	p. 115
4.1 Introduction	p. 115
4.2 Stability of simple steady states	p. 116
4.3 Stability for more general steady states	p. 124
4.4 Non-linear stability of zonal flows on the beta-plane	p. 129
4.5 Variational characterization of the steady states	p. 133
References	p. 137
5 Topographic mean flow interaction, non-linear instability, and chaotic dynamics	p. 138
5.1 Introduction	p. 138
5.2 Systems with layered topography	p. 141
5.3 Integrable behavior	p. 145
5.4 A limit regime with chaotic solutions	p. 154
5.5 Numerical experiments	p. 167
References	p. 178
Appendix 1	p. 180
Appendix 2	p. 181
6 Introduction to information theory and empirical statistical theory	p. 183
6.1 Introduction	p. 183
6.2 Information theory and Shannon's entropy	p. 184
6.3 Most probable states with prior distribution	p. 190
6.4 Entropy for continuous measures on the line	p. 194
6.5 Maximum entropy principle for continuous fields	p. 201
6.6 An application of the maximum entropy principle to geophysical flows with topography	p. 204
6.7 Application of the maximum entropy principle to geophysical flows with topography and mean flow	p. 211
References	p. 218
7 Equilibrium statistical mechanics for systems of ordinary differential equations	p. 219
7.1 Introduction	p. 219
7.2 Introduction to statistical mechanics for ODEs	p. 221
7.3 Statistical mechanics for the truncated Burgers-Hopf equations	p. 229
7.4 The Lorenz 96 model	p. 239
References	p. 255
8 Statistical mechanics for the truncated quasi-geostrophic equations	p. 256
8.1 Introduction	p. 256
8.2 The finite-dimensional truncated quasi-geostrophic equations	p. 258
8.3 The statistical predictions for the truncated systems	p. 262
8.4 Numerical evidence supporting the statistical prediction	p. 264
8.5 The pseudo-energy and equilibrium statistical mechanics for fluctuations about the mean	p. 267
8.6 The continuum limit	p. 270
8.7 The role of statistically relevant and irrelevant conserved quantities	p. 285
References	p. 285
Appendix 1	p. 286
9 Empirical statistical theories for most probable states	p. 289
9.1 Introduction	p. 289
9.2 Empirical statistical theories with a few constraints	p. 291
9.3 The mean field statistical theory for point vortices	p. 299
9.4 Empirical statistical theories with infinitely many constraints	p. 309
9.5 Non-linear stability for the most probable mean fields	p. 313
References	p. 316
10 Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview	p. 317
10.1 Introduction	p. 317
10.2 Basic issues regarding equilibrium statistical theories for geophysical flows	p. 318
10.3 The central role of equilibrium statistical theories with a judicious prior distribution and a few external constraints	p. 320
10.4 The role of forcing and dissipation	p. 322
10.5 Is there a complete statistical mechanics theory for ESTMC and ESTP?	p. 324
References	p. 326
11 Predictions and comparison of equilibrium statistical theories	p. 328
11.1 Introduction	p. 328
11.2 Predictions of the statistical theory with a judicious prior and a few external constraints for beta-plane channel flow	p. 330
11.3 Statistical sharpness of statistical theories with few constraints	p. 346
11.4 The limit of many-constraint theory (ESTMC) with small amplitude potential vorticity	p. 355
References	p. 360
12 Equilibrium statistical theories and dynamical modeling of flows with forcing and dissipation	p. 361
12.1 Introduction	p. 361
12.2 Meta-stability of equilibrium statistical structures with dissipation and small-scale forcing	p. 362
12.3 Crude closure for two-dimensional flows	p. 385
12.4 Remarks on the mathematical justifications of crude closure	p. 405
References	p. 410
13 Predicting the jets and spots on Jupiter by equilibrium statistical mechanics	p. 411
13.1 Introduction	p. 411
13.2 The quasi-geostrophic model for interpreting observations and predictions for the weather layer of Jupiter	p. 417
13.3 The ESTP with physically motivated prior distribution	p. 419
13.4 Equilibrium statistical predictions for the jets and spots on Jupiter	p. 423
References	p. 426
14 The statistical relevance of additional conserved quantities for truncated geophysical flows	p. 427
14.1 Introduction	p. 427
14.2 A numerical laboratory for the role of higher-order invariants	p. 430
14.3 Comparison with equilibrium statistical predictions with a judicious prior	p. 438
14.4 Statistically relevant conserved quantities for the truncated Burgers-Hopf equation	p. 440
References	p. 442
A.1 Spectral truncations of quasi-geostrophic flow with additional conserved quantities	p. 442
15 A mathematical framework for quantifying predictability utilizing relative entropy	p. 452
15.1 Ensemble prediction and relative entropy as a measure of predictability	p. 452
15.2 Quantifying predictability for a Gaussian prior distribution	p. 459
15.3 Non-Gaussian ensemble predictions in the Lorenz 96 model	p. 466
15.4 Information content beyond the climatology in ensemble predictions for the truncated Burgers-Hopf model	p. 472
15.5 Further developments in ensemble predictions and information theory	p. 478
References	p. 480
16 Barotropic quasi-geostrophic equations on the sphere	p. 482
16.1 Introduction	p. 482
16.2 Exact solutions, conserved quantities, and non-linear stability	p. 490
16.3 The response to large-scale forcing	p. 510
16.4 Selective decay on the sphere	p. 516
16.5 Energy enstrophy statistical theory on the unit sphere	p. 524
16.6 Statistical theories with a few constraints and statistical theories with many constraints on the unit sphere	p. 536
References	p. 542
Appendix 1	p. 542
Appendix 2	p. 546
Index	p. 550

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