Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

Chavanis, Pierre‐Henri

doi:10.1140/epjb/e2009-00196-1

Cited by 46 publications

(139 citation statements)

References 72 publications

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“…We note that the critical points of these different maximization problems (microcanonical, canonical, and grand canonical) are all given by Equation (229). However, the stability of the solutions (whether they are true maxima or saddle points of the thermodynamical potential) may differ because the ensembles can be inequivalent [84,89]. In [89], we have introduced different types of relaxation equations associated with these maximization problems.…”

Section: Application To 2d Turbulencementioning

confidence: 99%

“…However, the stability of the solutions (whether they are true maxima or saddle points of the thermodynamical potential) may differ because the ensembles can be inequivalent [84,89]. In [89], we have introduced different types of relaxation equations associated with these maximization problems. Although being phenomenological, these equations may describe the dynamical evolution of the system towards equilibrium.…”

Section: Application To 2d Turbulencementioning

confidence: 99%

“…This equation increases the generalized grand potential until the equilibrium state is reached (H-theorem). Equation (241) or (242), first introduced in [89], has been recently considered by other authors [90]. It is based on the maximization of a functional G[ω], interpreted as an energy-Casimir functional [89], with no constraint (GCE).…”

Section: Application To 2d Turbulencementioning

confidence: 99%

“…Equation (241) or (242), first introduced in [89], has been recently considered by other authors [90]. It is based on the maximization of a functional G[ω], interpreted as an energy-Casimir functional [89], with no constraint (GCE). However, it may be important to take into account the conservation of energy and circulation, so the relaxation equations (232)-(240) in CE and MCE are also useful.…”

Section: Application To 2d Turbulencementioning

confidence: 99%

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Generalized Stochastic Fokker-Planck Equations

Chavanis

2015

Entropy

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We consider a system of Brownian particles with long-range interactions. We go beyond the mean field approximation and take fluctuations into account. We introduce a new class of stochastic Fokker-Planck equations associated with a generalized thermodynamical formalism. Generalized thermodynamics arises in the case of complex systems experiencing small-scale constraints. In the limit of short-range interactions, we obtain a generalized class of stochastic Cahn-Hilliard equations. Our formalism has application for several systems of physical interest including self-gravitating Brownian particles, colloid particles at a fluid interface, superconductors of type II, nucleation, the chemotaxis of bacterial populations, and two-dimensional turbulence. We also introduce a new type of generalized entropy taking into account anomalous diffusion and exclusion or inclusion constraints.

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Section: Application To 2d Turbulencementioning

confidence: 99%

Section: Application To 2d Turbulencementioning

confidence: 99%

Section: Application To 2d Turbulencementioning

confidence: 99%

Section: Application To 2d Turbulencementioning

confidence: 99%

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Generalized Stochastic Fokker-Planck Equations

Chavanis

2015

Entropy

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“…Thus, for β > γ + 1 , δ 2 J [q 1 , q 2 ] < 0 for every perturbation (δq 1 , δq 2 ). This is the condition of grand-canonical stability, which implies in particular microcanonical stability [49].…”

Section: A Grand-canonical Stabilitymentioning

confidence: 99%

Nonlinear energy transfers and phase diagrams for geostrophically balanced rotating-stratified flows

Herbert

2014

Phys. Rev. E

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Equilibrium statistical mechanics tools have been developed to obtain indications about the natural tendencies of nonlinear energy transfers in two-dimensional and quasi two-dimensional flows like rotating and stratified flows in geostrophic balance. In this article, we consider a simple model of such flows with a non-trivial vertical structure, namely two-layer quasi-geostrophic flows, which remain amenable to analytical study. We obtain the statistical equilibria of the system in the case of a linear vorticity-stream function relation, build the corresponding phase diagram, and discuss the most probable outcome of nonlinear energy transfers, both on the horizontal and on the vertical, in the presence of stratification and rotation.

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Mathematical and physical ideas for climate science

et al. 2014

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The climate is a forced and dissipative nonlinear system featuring nontrivial dynamics on a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the nonequilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate, characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on nonequilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

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Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

Cited by 46 publications

References 72 publications

Generalized Stochastic Fokker-Planck Equations

Generalized Stochastic Fokker-Planck Equations

Nonlinear energy transfers and phase diagrams for geostrophically balanced rotating-stratified flows

Mathematical and physical ideas for climate science

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