Note: Cover -- Half-title -- Title -- Copyright -- Contents -- Preface -- 1 Barotropic geophysical flows and two-dimensional fluid flows: elementary introduction -- 1.1 Introduction -- 1.2 Some special exact solutions -- Fourier series tool kit -- 1.2.1 Exact solutions for the Euler equations -- 1.2.2 Exact solutions with beta-effect and generalized Kolmogorov forcing -- 1.2.3 Rossby waves -- 1.2.4 Topographic effect in steady states -- 1.2.5 A dynamical solution with beta-plane and layered topography -- 1.2.6 Beta-plane dynamics with large-scale shear flow -- A test problem for dissipative mechanisms -- 1.3 Conserved quantities -- 1.3.1 Conservation of energy -- Conservation of energy for periodic flows -- 1.3.2 Large-scale and small-scale flow interaction via topography -- 1.3.3 Infinite number of conserved quantities - generalized enstrophy -- 1.3.4 Several conserved quantities -- 1.3.5 Summary of conserved quantities: periodic geometry -- 1.4 Barotropic geophysical flows in a channel domain - an important physical model -- 1.4.1 The impulse and conserved quantities -- 1.4.2 Conservation of circulation -- 1.4.3 Summary of conserved quantities: channel geometry -- 1.5 Variational derivatives and an optimization principle for elementary geophysical solutions -- 1.5.1 Some important variational derivatives -- 1.5.2 An optimization principle for elementary geophysical solutions -- 1.6 More equations for geophysical flows -- 1.6.1 The models -- 1.6.2 Relationships between various models -- Derivation of the barotropic one-layer model from the continuously stratified model -- Derivation of the two-layer model from the continuously stratified model -- Derivation of the one- and one-half-layer model from the two-layer model -- Derivation of the barotropic quasi-geostrophic model from the F-plane model -- References -- 2 The response to large-scale forcing. , 2.1 Introduction -- A remarkable identity -- 2.2 Non-linear stability with Kolmogorov forcing -- 2.2.1 Non-linear stability in restricted sense -- 2.2.2 Finite-dimensional dynamics on the ground modes and non-linear stability -- Fourier representation for the dynamic equations -- 2.2.3 Counter-example of unstable ground state modes dynamics for truncated inviscid flows -- 2.3 Stability of flows with generalized Kolmogorov forcing -- References -- 3 The selective decay principle for basic geophysical flows -- 3.1 Introduction -- 3.2 Selective decay states and their invariance -- 3.3 Mathematical formulation of the selective decay principle -- The Rossby waves degenerate into generalized Taylor vortices in the absence of the geophysical beta-plane effect. -- 3.4 Energy-enstrophy decay -- 3.5 Bounds on the Dirichlet quotient, A (t) -- 3.6 Rigorous theory for selective decay -- 3.6.1 Convergence to an asymptotic state -- 3.6.2 Convergence to the selective decay state -- 3.6.3 Stability of the selective decay states -- 3.6.4 Underlying simplifying mechanisms -- 3.7 Numerical experiments demonstrating facets of selective decay -- 3.7.1 Measure of anisotropy -- 3.7.2 Explicit solutions of the sinh-Poisson equation -- 3.7.3 Numerical examples -- References -- Appendix 1 Stronger controls on A (t) -- Appendix 2 The proof of the mathematical form of the selective decay principle in the presence of the beta-plane effect -- 4 Non-linear stability of steady geophysical flows -- 4.1 Introduction -- 4.2 Stability of simple steady states -- 4.2.1 Non-linear stability and the energy method -- 4.2.2 Simple states with topography, but no mean flow or beta-effect -- 4.2.3 Simple states with topography, mean flow, and beta-effect -- 4.3 Stability for more general steady states -- 4.4 Non-linear stability of zonal flows on the beta-plane. , 4.5 Variational characterization of the steady states -- References -- 5 Topographic mean flow interaction, non-linear instability, and chaotic dynamics -- 5.1 Introduction -- 5.2 Systems with layered topography -- 5.2.1 Hamiltonian structure -- 5.3 Integrable behavior -- 5.3.1 The case h = 0 -- 5.3.2 The case Beta = 0 -- 5.3.3 Single mode topography -- 5.4 A limit regime with chaotic solutions -- 5.4.1 Single mode topography -- 5.4.2 Interaction of non-linear resonances -- 5.4.3 Two modes in the topography: a perturbative Melnikov analysis -- 5.5 Numerical experiments -- 5.5.1 Perturbation of single mode topography -- 5.5.2 Two-mode layered topography and topographic blocking events -- 5.5.3 Random perturbations with multi-mode topography -- 5.5.4 Symmetry breaking perturbations and topographic blocking events -- References -- Appendix 1 -- Appendix 2 -- 6 Introduction to information theory and empirical statistical theory -- 6.1 Introduction -- 6.2 Information theory and Shannon's entropy -- 6.3 Most probable states with prior distribution -- 6.4 Entropy for continuous measures on the line -- 6.4.1 Continuous measure on the line -- 6.4.2 Entropy and maximum entropy principle -- 6.4.3 Coarse graining and loss of information -- 6.4.4 Relative entropy as a "distance" function -- 6.4.5 Information theory and the finite-moment problem for probability measures -- 6.5 Maximum entropy principle for continuous fields -- 6.6.1 The Prior distribution -- 6.6.2 Constraints on the potential vorticity distribution -- 6.6.3 Statistical predictions of the maximum entropy principle -- 6.6.4 Determination of the multipliers and geophysical effect -- 6.7 Application of the maximum entropy principle to geophysical flows with topography and mean flow -- 6.7.1 One-point statistics for potential vorticity and large-scale mean velocity and Shannon entropy. , 6.7.2 The constraints on the one-point statistics -- 6.7.3 Maximum entropy principle and statistical prediction -- 6.7.4 Determination of the multipliers and geophysical effects -- References -- 7 Equilibrium statistical mechanics for systems of ordinary differential equations -- 7.1 Introduction -- 7.2 Introduction to statistical mechanics for ODEs -- 7.2.1 The Liouville property -- 7.2.2 Evolution of probability measures and the Liouville equation -- 7.2.3 Conserved quantities and their ensemble averages -- 7.2.4 Shannon entropy and the maximum entropy principle -- 7.2.5 The most probable state and Gibbs measure -- 7.2.6 Ergodicity and time averaging -- 7.2.7 A simple example violating the Liouville property -- 7.3 Statistical mechanics for the truncated Burgers-Hopf equations -- 7.3.1 The truncated Burgers-Hopf systems and their conserved quantities -- 7.3.2 The Liouville property -- 7.3.3 The Gibbs measure and the prediction of equipartition of energy -- 7.3.4 Numerical evidence of the validity of the statistical theory -- 7.3.5 Truncated Burgers-Hopf equation as a model with statistical features in common with atmosphere -- A scaling theory for temporal correlations -- Numerical evidence for the correlation scaling theory -- 7.4 The Lorenz 96 model -- 7.4.1 Geophysical properties of the Lorenz 96 model -- Rossby waves -- 7.4.2 Equilibrium statistical theory for the undamped unforced L-96 model -- 7.4.3 Statistical properties of the damped forced and undamped unforced L96 models -- Rescaling the damped forced L96 model -- Linear stability of the mean state -- The bulk behavior of the rescaled problem -- The climatology of different forcing regimes in rescaled coordinates -- References -- 8 Statistical mechanics for the truncated quasi-geostrophic equations -- 8.1 Introduction -- 8.2 The finite-dimensional truncated quasi-geostrophic equations. , 8.2.1 The spectrally truncated quasi-geostrophic equations -- 8.2.2 Conserved quantities for the truncated system -- 8.2.3 Non-linear stability of some exact solutions the truncated system -- 8.2.4 The Liouville property -- 8.3 The statistical predictions for the truncated systems -- 8.4 Numerical evidence supporting the statistical prediction -- 8.5 The pseudo-energy and equilibrium statistical mechanics for fluctuations about the mean -- 8.6 The continuum limit -- 8.6.1 The case with a large-scale mean flow -- 8.6.2 The case without large-scale mean flow but with generic topography -- 8.6.3 The case with no geophysical effects -- 8.6.4 The case with no large-scale mean flow but with topography having degenerate spectrum -- High energy subcase -- 8.7 The role of statistically relevant and irrelevant conserved quantities -- References -- Appendix 1 -- 9 Empirical statistical theories for most probable states -- 9.1 Introduction -- 9.2 Empirical statistical theories with a few constraints -- 9.2.1 The energy-circulation empirical theory with a general prior distribution -- 9.2.2 The energy-circulation impulse theory with a general prior distribution -- 9.3 The mean field statistical theory for point vortices -- 9.3.1 Derivation of the mean field point-vortex theory from an empirical statistical theory -- 9.3.2 Complete statistical mechanics for point vortices -- The dynamics of point vortices in the plane -- Liouville property -- The mean field limit equations as N … -- 9.4 Empirical statistical theories with infinitely many constraints -- 9.4.1 Maximum entropy principle incorporating all generalized enstrophies -- 9.4.2 The most probable state and the mean field equation -- 9.5 Non-linear stability for the most probable mean fields -- References. , 10 Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview.