Chapter
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
1.6.2 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
2 The response to large-scale forcing
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
3 The selective decay principle for basic geophysical flows
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
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
4.2 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
5 Topographic mean flow interaction, non-linear instability, and chaotic dynamics
5.2 Systems with layered topography
5.2.1 Hamiltonian structure
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
6 Introduction to information theory and empirical statistical theory
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
7 Equilibrium statistical mechanics for systems of ordinary differential equations
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
7.4.1 Geophysical properties of the Lorenz 96 model
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
8 Statistical mechanics for the truncated quasi-geostrophic equations
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
8.6.1 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
8.7 The role of statistically relevant and irrelevant conserved quantities
9 Empirical statistical theories for most probable states
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
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
10 Assessing the potential applicability of equilibrium statistical theories for geophysical flows: an overview
10.2 Basic issues regarding equilibrium statistical theories for geophysical flows
Some basic applied issues
Some basic theoretical issues
10.3 The central role of equilibrium statistical theories with a judicious prior distribution and a few external constraints
10.4 The role of forcing and dissipation
10.5 Is there a complete statistical mechanics theory for ESTMC and ESTP?
11 Predictions and comparison of equilibrium statistical theories
11.2 Predictions of the statistical theory with a judicious prior and a few external constraints for beta-plane channel flow
11.2.1 Statistical theory
11.2.2 Coherent geophysical vortices (monopoles and dipoles)
Basic solutions with no geophysical effects (V = Beta = 0)
11.2.3 Statistical predictions of generalized Rhines’ scale
Vortex streets in the dilute PV limit
11.3 Statistical sharpness of statistical theories with few constraints
11.3.1 Statistical sharpness and generalized selective decay principle
11.3.2 The statistical sharpness of macrostates from ESTMC
11.3.3 Statistical sharpness of macrostates of ESTP
Energy–enstrophy statistical theories (EEST)
Point-vortex statistical theories (PVST)
11.4 The limit of many-constraint theory (ESTMC) with small amplitude potential vorticity
11.4.1 The asymptotic expansion
11.4.2 Interpretation of the asymptotic equation through renormalized topography
12 Equilibrium statistical theories and dynamical modeling of flows with forcing and dissipation
12.2 Meta-stability of equilibrium statistical structures with dissipation and small-scale forcing
12.2.1 The numerical model
12.2.2 Approximate dynamics for Langevin and dilute PV theory
Algorithm for the approximate dynamics
12.2.3 Statistical consistency of freely decaying vortex states
Free decay of vortex monopoles
Free decay of vortex streets
12.2.4 Statistical consistency of damped and driven vortex states
Inverse cascade from small-scale forcing of single-signed vortices
Maintenance of vortex streets by small-scale, double-signed forcing
Vortex coalescence in strongly forced shear flow
12.3 Crude closure for two-dimensional flows
12.3.1 The equation and the problem
12.3.2 Description of the crude closure dynamics
12.3.3 Numerical results with Newtonian dissipation
Forcing by alternating or opposite signed vortices
12.4 Remarks on the mathematical justifications of crude closure
13 Predicting the jets and spots on Jupiter by equilibrium statistical mechanics
13.1.1 The observational record for Jupiter and the quasi-geostrophic model
13.1.2 Predictions of the ESTP with a suitable prior and the observational record
13.2 The quasi-geostrophic model for interpreting observations and predictions for the weather layer of Jupiter
13.2.1 Fitting the non-dimensional model with the dimensional parameters of Jupiter
13.2.2 Fitting the lower layer topography
13.3 The ESTP with physically motivated prior distribution
13.3.1 The ESTP with a family of prior distributions with anti-cyclonic skewness
13.3.2 The centered Gamma distribution as a family of skewed prior distributions for ESTP
13.4 Equilibrium statistical predictions for the jets and spots on Jupiter
13.4.1 The southern hemisphere domain
13.4.2 The northern hemisphere domain
14 The statistical relevance of additional conserved quantities for truncated geophysical flows
14.1.1 The traditional spectral truncation and equilibrium statistical theory
14.2 A numerical laboratory for the role of higher-order invariants
14.2.1 The spectral truncation with many conserved quantities
14.2.2 Numerical experiments demonstrating the statistical relevance of C3(q) at large scales
14.2.3 Mixing and decay of temporal correlations
14.2.4 The large-scale mean flow
14.2.5 The energy spectrum
14.3 Comparison with equilibrium statistical predictions with a judicious prior
14.3.1 The probability distribution function of potential vorticity
14.3.2 Equilibrium statistical predictions of the non-linear mean state
14.4 Statistically relevant conserved quantities for the truncated Burgers–Hopf equation
Appendix 1 Spectral truncations of quasi-geostrophic flow with additional conserved quantities
A.1.1 Some basic facts about Hamiltonian systems
A.1.2 The equations for Barotropic flow in Fourier space
A.1.3 The sine-bracket truncation with many additional conserved quantities
15 A mathematical framework for quantifying predictability utilizing relative entropy
15.1 Ensemble prediction and relative entropy as a measure of predictability
15.1.1 Practical and mathematical issues for predictability
15.1.2 Gaussian prior distribution for predictability
15.1.3 Invariance of the predictability measure under a general change of coordinates
15.1.4 Canonical form for a Gaussian climate: EOF basis
15.2 Quantifying predictability for a Gaussian prior distribution
15.2.1 The signal and dispersion decomposition for a Gaussian prior
15.2.2 Rigorous lower bounds on predictive information content with a Gaussian prior
15.2.3 Choosing reduced variables to order the predictive information content
15.2.4 The relative entropy and the entropy difference for quantifying predictive information content
15.3 Non-Gaussian ensemble predictions in the Lorenz 96 model
15.4 Information content beyond the climatology in ensemble predictions for the truncated Burgers–Hopf model
15.4.1 Relaxation of ensemble predictions to the climate distribution
15.4.2 The signal, dispersion, and variations in predictive utility
15.5 Further developments in ensemble predictions and information theory
16 Barotropic quasi-geostrophic equations on the sphere
16.1.1 Common differential operators and integration by parts formulas on the sphere
16.1.2 Eigenfunctions of the Laplace operator and the Legendre functions
16.2 Exact solutions, conserved quantities, and non-linear stability
16.2.1 Some special exact solutions
Independent linear dynamics on the ground energy shell
Exact interesting dynamics of the ground state modes and another energy shell
Exact solutions with generalized Kolmogorov forcing
16.2.2 Conserved quantities
Conserved quantities with general topography
Conserved quantities with topography living on the ground energy shell
Conserved quantities in the presence of odd symmetry in z
Conserved quantities in the presence of odd symmetry in z and topography living on the ground energy shell
Summary of conserved quantities
16.2.3 Non-linear stability of exact solutions
Non-linear stability of steady states
Restricted stability of motion on the first two energy shells
16.3 The response to large-scale forcing
16.4 Selective decay on the sphere
16.4.2 Physicist’s selective decay states
16.4.3 Formulation of the selective decay principle
16.4.4 Sketch of the proof
Overview of equilibrium statistical theories for the sphere
16.5 Energy enstrophy statistical theory on the unit sphere
16.5.1 Energy–enstrophy theory with topography – empirical statistical theory
Empirical energy–enstrophy theory with general topography
Empirical energy–enstrophy theory with ground state topography
16.5.2 Complete statistical mechanics.
Conservation of energy and potential enstrophy
Most probable state for the truncated system
Comment on the case with ground state topography
16.6 Statistical theories with a few constraints and statistical theories with many constraints on the unit sphere
A few constraint theory with prior distribution
Energy–circulation theory
16.6.1 Infinitely many constraint statistical theory
Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows
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