Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Chavanis, Pierre‐Henri; Naso, Aurore; Dubrulle, Bérengère

doi:10.1140/epjb/e2010-00264-5

Cited by 11 publications

(37 citation statements)

References 47 publications

(181 reference statements)

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“…and, thanks to the large deviation property [137] (see section 2.3.5) an overwhelming number of potential vorticity fields of the microcanonical ensemble will be close to the maximizer ρ of the variational problem (46).…”

Section: Equilibrium Entropy and Microcanonical Equilibrium Statesmentioning

confidence: 97%

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Statistical mechanics of two-dimensional and geophysical flows

Bouchet¹,

Venaille²

2012

Physics Reports

253

329

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The theoretical study of the self-organization of two-dimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a self-contained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter's troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics.After a brief presentation of the 2D Euler and quasi-geostrophic equations, the specificity of two-dimensional and geophysical turbulence is emphasized. The equilibrium microcanonical measure is built from the Liouville theorem. Important statistical mechanics concepts (large deviations, mean field approach) and thermodynamic concepts (ensemble inequivalence, negative heat capacity) are briefly explained and described.On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of twodimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the GulfStream, and ocean vortices. A detailed comparison between these statistical equilibria and real flow observations is provided.We also present recent results for non-equilibrium situations, for the studies of either the relaxation towards equilibrium or non-equilibrium steady states. In this last case, forces and dissipation are in a statistical balance; fluxes of conserved quantity characterize the system and microcanonical or other equilibrium measures no longer describe the system.

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Section: Equilibrium Entropy and Microcanonical Equilibrium Statesmentioning

confidence: 97%

“…The second route is to try to work directly with the variational problem (46) and to simplify it. In review we will try to rely as much as possible on variational problems only.…”

Section: Equilibrium Entropy and Microcanonical Equilibrium Statesmentioning

confidence: 99%

Statistical mechanics of two-dimensional and geophysical flows

Bouchet¹,

Venaille²

2012

Physics Reports

253

329

View full text Add to dashboard Cite

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“…A statistical theory of 2D turbulence has been developed by Miller [82] and Robert and Sommeria [83] for isolated systems. We consider here the case where the system is forced at small scales and follow the heuristic approach of Ellis et al [84] complemented by Chavanis [85][86][87][88]. We assume that the small-scale forcing is encoded in a prior vorticity distribution χ(σ).…”

Section: Application To 2d Turbulencementioning

confidence: 99%

“…Explicit examples of prior vorticity distributions, and of the corresponding generalized entropies, are given in [85][86][87][88]. For example, when the prior vorticity distribution is a Gaussian, χ(σ) = (2πΩ 2 ) −1/2 exp(−σ 2 /2Ω 2 ), the generalized entropy is proportional to minus the enstrophy S = −(1/2Ω 2 ) ω 2 dr (see [87] and Section 5 of [88]). The functional S[ω] has the status of an entropy in the sense of the theory of large deviations [84].…”

Section: Application To 2d Turbulencementioning

confidence: 99%

“…We note that for a Gaussian prior vorticity distribution, leading to a generalized entropy proportional to minus the enstrophy (C(ω) = ω 2 /2Ω 2 ) [88], the noise term in Equations (243), (245) and (246) is independent on the vorticity since C (ω) = 1/Ω 2 is a constant. When the system possesses different stable equilibrium states for a given entropy S, these stochastic equations describe random transition from one state to the other, similarly to the study of [47].…”

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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Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts

Conti

Badin

2020

J Stat Phys

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We investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or α-models. These models describe both nonlocal and local dynamics, with one example of the latter one given by the Surface Quasi Geostrophy (SQG) model, for which the existence of singularities is still under debate. We aim to study the equilibrium mechanics, using initially a point-vortex approximation and then exploiting the full continuous equations, invoking the maximization of appropriate entropy functionals. The point-vortex approximation highlights an important difference between the 2D turbulence and local dynamics models. In the latter, it is in fact possible to derive a statistical measure only considering two conserved quantities as constraints for the maximization problem, the Hamiltonian and the angular impulse. This result does not hold for 2D turbulence. Both the continuous and the point vortex approximation allow for the derivation of mean field equations that act as constraints for the functional relation between the streamfunction and the active scalar of the model considered. Further, the analysis of the continuous equations suggests the existence of a selective decay principle for the whole family of models. To test these ideas we use numerical simulations of the partial differential equations of the α-models starting from different sets of initial conditions (i.c.s). For random i.c.s, all the solutions tend to a dipolar structure. The functional relation between the active scalar and the streamfunction shows an increase of nonlinearity with a decrease of the locality of the dynamics. We then test the evolution of the specific case of SQG for i.c.s in the form of a hyperbolic saddle, that is a candidate for the possible formation of singularity through a self-similar cascade though secondary instabilities. Results show the presence of a scale dependent selective decay associated to the breaking of the frontal structures emerging from the flow, suggesting a relation with the change of topology of the flow.

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Relaxation equations for two-dimensional turbulent flows with a prior vorticity distribution

Cited by 11 publications

References 47 publications

Statistical mechanics of two-dimensional and geophysical flows

Statistical mechanics of two-dimensional and geophysical flows

Generalized Stochastic Fokker-Planck Equations

Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts

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