| Preface | p. xi |
| Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction | p. 1 |
| Introduction | p. 1 |
| Some special exact solutions | p. 8 |
| Conserved quantities | p. 33 |
| Barotropic geophysical flows in a channel domain - an important physical model | p. 44 |
| Variational derivatives and an optimization principle for elementary geophysical solutions | p. 50 |
| More equations for geophysical flows | p. 52 |
| References | p. 58 |
| The response to large-scale forcing | p. 59 |
| Introduction | p. 59 |
| Non-linear stability with Kolomogorov forcing | p. 62 |
| Stability of flows with generalized Kolmogorov forcing | p. 76 |
| References | p. 79 |
| The selective decay principle for basic geophysical flows | p. 80 |
| Introduction | p. 80 |
| Selective decay states and their invariance | p. 82 |
| Mathematical formulation of the selective decay principle | p. 84 |
| Energy-enstrophy decay | p. 86 |
| Bounds on the Dirichlet quotient, [Lambda](t) | p. 88 |
| Rigorous theory for selective decay | p. 90 |
| Numerical experiments demonstrating facets of selective decay | p. 95 |
| References | p. 102 |
| Stronger controls on [Lambda](t) | p. 103 |
| The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect | p. 107 |
| Non-linear stability of steady geophysical flows | p. 115 |
| Introduction | p. 115 |
| Stability of simple steady states | p. 116 |
| Stability for more general steady states | p. 124 |
| Non-linear stability of zonal flows on the beta-plane | p. 129 |
| Variational characterization of the steady states | p. 133 |
| References | p. 137 |
| Topographic mean flow interaction, non-linear instability, and chaotic dynamics | p. 138 |
| Introduction | p. 138 |
| Systems with layered topography | p. 141 |
| Integrable behavior | p. 145 |
| A limit regime with chaotic solutions | p. 154 |
| Numerical experiments | p. 167 |
| References | p. 178 |
| Appendix 1 | p. 180 |
| Appendix 2 | p. 181 |
| Introduction to information theory and empirical statistical theory | p. 183 |
| Introduction | p. 183 |
| Information theory and Shannon's entropy | p. 184 |
| Most probable states with prior distribution | p. 190 |
| Entropy for continuous measures on the line | p. 194 |
| Maximum entropy principle for continuous fields | p. 201 |
| An application of the maximum entropy principle to geophysical flows with topography | p. 204 |
| Application of the maximum entropy principle to geophysical flows with topography and mean flow | p. 211 |
| References | p. 218 |
| Equilibrium statistical mechanics for systems of ordinary differential equations | p. 219 |
| Introduction | p. 219 |
| Introduction to statistical mechanics for ODEs | p. 221 |
| Statistical mechanics for the truncated Burgers-Hopf equations | p. 229 |
| The Lorenz 96 model | p. 239 |
| References | p. 255 |
| Statistical mechanics for the truncated quasi-geostrophic equations | p. 256 |
| Introduction | p. 256 |
| The finite-dimensional truncated quasi-geostrophic equations | p. 258 |
| The statistical predictions for the truncated systems | p. 262 |
| Numerical evidence supporting the statistical prediction | p. 264 |
| The pseudo-energy and equilibrium statistical mechanics for fluctuations about the mean | p. 267 |
| The continuum limit | p. 270 |
| The role of statistically relevant and irrelevant conserved quantities | p. 285 |
| References | p. 285 |
| Appendix 1 | p. 286 |
| Empirical statistical theories for most probable states | p. 289 |
| Introduction | p. 289 |
| Empirical statistical theories with a few constraints | p. 291 |
| The mean field statistical theory for point vortices | p. 299 |
| Empirical statistical theories with infinitely many constraints | p. 309 |
| Non-linear stability for the most probable mean fields | p. 313 |
| References | p. 316 |
| Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview | p. 317 |
| Introduction | p. 317 |
| Basic issues regarding equilibrium statistical theories for geophysical flows | p. 318 |
| The central role of equilibrium statistical theories with a judicious prior distribution and a few external constraints | p. 320 |
| The role of forcing and dissipation | p. 322 |
| Is there a complete statistical mechanics theory for ESTMC and ESTP? | p. 324 |
| References | p. 326 |
| Predictions and comparison of equilibrium statistical theories | p. 328 |
| Introduction | p. 328 |
| Predictions of the statistical theory with a judicious prior and a few external constraints for beta-plane channel flow | p. 330 |
| Statistical sharpness of statistical theories with few constraints | p. 346 |
| The limit of many-constraint theory (ESTMC) with small amplitude potential vorticity | p. 355 |
| References | p. 360 |
| Equilibrium statistical theories and dynamical modeling of flows with forcing and dissipation | p. 361 |
| Introduction | p. 361 |
| Meta-stability of equilibrium statistical structures with dissipation and small-scale forcing | p. 362 |
| Crude closure for two-dimensional flows | p. 385 |
| Remarks on the mathematical justifications of crude closure | p. 405 |
| References | p. 410 |
| Predicting the jets and spots on Jupiter by equilibrium statistical mechanics | p. 411 |
| Introduction | p. 411 |
| The quasi-geostrophic model for interpreting observations and predictions for the weather layer of Jupiter | p. 417 |
| The ESTP with physically motivated prior distribution | p. 419 |
| Equilibrium statistical predictions for the jets and spots on Jupiter | p. 423 |
| References | p. 426 |
| The statistical relevance of additional conserved quantities for truncated geophysical flows | p. 427 |
| Introduction | p. 427 |
| A numerical laboratory for the role of higher-order invariants | p. 430 |
| Comparison with equilibrium statistical predictions with a judicious prior | p. 438 |
| Statistically relevant conserved quantities for the truncated Burgers-Hopf equation | p. 440 |
| References | p. 442 |
| Spectral truncations of quasi-geostrophic flow with additional conserved quantities | p. 442 |
| A mathematical framework for quantifying predictability utilizing relative entropy | p. 452 |
| Ensemble prediction and relative entropy as a measure of predictability | p. 452 |
| Quantifying predictability for a Gaussian prior distribution | p. 459 |
| Non-Gaussian ensemble predictions in the Lorenz 96 model | p. 466 |
| Information content beyond the climatology in ensemble predictions for the truncated Burgers-Hopf model | p. 472 |
| Further developments in ensemble predictions and information theory | p. 478 |
| References | p. 480 |
| Barotropic quasi-geostrophic equations on the sphere | p. 482 |
| Introduction | p. 482 |
| Exact solutions, conserved quantities, and non-linear stability | p. 490 |
| The response to large-scale forcing | p. 510 |
| Selective decay on the sphere | p. 516 |
| Energy enstrophy statistical theory on the unit sphere | p. 524 |
| 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 |
| Table of Contents provided by Ingram. All Rights Reserved. |
