The modeling of small scales in two-dimensional turbulent flows: A statistical mechanics approach

Robert, Raoul; Rosier, Carole

doi:10.1007/bf02199111

Cited by 57 publications

(84 citation statements)

References 21 publications

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“…Relaxation equations analogous to equations (31) and (50) have been derived in the context of vortex dynamics [54,12,16,19]. In addition, the problem of "incomplete relaxation" is also encountered in 2D turbulence to explain the confinement of a vortex (e.g., a dipole or a tripole) that forms after a rapid merging [27,23,52]. It has been demonstrated explicitly in two-dimensional turbulence (where the numerical simulations are easier to implement) that the relaxation equations derived from the MEPP and including a space dependant diffusion coefficient related to the fluctuations of the vorticity (analogous to Eq.…”

Section: Resultsmentioning

confidence: 99%

“…However, as we shall see, these difficulties should not throw doubt on the importance of this statistical description. A similar relaxation process is at work in two-dimensional turbulence (described by the 2D Euler equation) and can explain the organization and maintenance of coherent vortices, such as the Great Red Spot of Jupiter, which are common features of large-scale geophysical or astrophysical flows [53,46,23,52,26,6,7]. The mathematical relevance of this statistical description has been given by Robert [50] introducing the concept of Young measures.…”

Section: Introductionmentioning

confidence: 99%

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Statistical Mechanics of Violent Relaxation in Stellar Systems

Chavanis

2002

Multiscale Problems in Science and Technology

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Abstract. We discuss the statistical mechanics of violent relaxation in stellar systems following the pioneering work of Lynden-Bell (1967). The solutions of the gravitational Vlasov-Poisson system develop finer and finer filaments so that a statistical description is appropriate to smooth out the small-scales and describe the "coarse-grained" dynamics. In a coarse-grained sense, the system is expected to reach an equilibrium state of a Fermi-Dirac type within a few dynamical times. We describe in detail the equilibrium phase diagram and the nature of phase transitions which occur in self-gravitating systems. Then, we introduce a small-scale parametrization of the Vlasov equation and propose a set of relaxation equations for the coarse-grained dynamics. These relaxation equations, of a generalized FokkerPlanck type, are derived from a Maximum Entropy Production Principle (MEPP). We make a link with the quasilinear theory of the Vlasov-Poisson system and derive a truncated model appropriate to collisionless systems subject to tidal forces. With the aid of this kinetic theory, we qualitatively discuss the concept of "incomplete relaxation" and the limitations of Lynden-Bell's theory.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Statistical Mechanics of Violent Relaxation in Stellar Systems

Chavanis

2002

Multiscale Problems in Science and Technology

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“…The value of the coefficient v (in (RE2) or (RE3)) is taken as in [17]. Recall that beyond some value (10^2 or 10~3 in our examples), the numerical results for the relaxation equations are quite indistinguishable and very similar to those obtained for the Navier-Stokes equation.…”

Section: Numerical Featuresmentioning

confidence: 65%

“…This equation has not yet been numerically simulated, contrary to (RE3) [17]. In this paper, we first test the validity of this new model by means of numerical simulations compared with those using the NavierStokes equation at high Reynolds number.…”

Section: Introductionmentioning

confidence: 99%

Finite speed propagation in the relaxation of vortex patches

Rosier¹,

Rosier²

2003

Quart. Appl. Math.

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The paper is dedicated to the memory of Louis-Hubert Rosier. Abstract.A degenerate parabolic equation has been proposed by Robert and Sommeria to describe the relaxation towards a statistical equilibrium state for a two-dimensional incompressible perfect fluid with a vortex patch as initial vorticity. In this paper, flows obtained by numerical integration of the Robert-Sommeria equation over a long-time interval are compared with those obtained for the Navier-Stokes equation at high Reynolds number. A finite speed propagation for the extremal values of the vorticity is numerically shown to hold for the Robert-Sommeria equation. A rigorous proof of this (fine) property is also provided.

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“…It describes detailed vortex distribution, where N p , σ i and Ω i (t) denote the number of patches, the vorticity of the i-th patch and the domain of the i-th patch, respectively. Mean field equations for equilibrium and for relaxation time are derived by the principles of maximum entropy [11,12] and maximum entropy production [13,14], respectively. For the latter case, one obtains a system on p ¼ pðx, σ, tÞ,…”

Section: Introductionmentioning

confidence: 99%

Relaxation Theory for Point Vortices

Sawada¹,

Suzuki²

2017

Vortex Structures in Fluid Dynamic Problems

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We study relaxation dynamics of the mean field of many point vortices from quasiequilibrium to equilibrium. Maximum entropy production principle implies four consistent equations concerning relaxation-equilibrium states and patch-point vortex models. Point vortex relaxation equation coincides with Brownian point vortex equation in micro-canonical setting. Mathematical analysis to point vortex relaxation equation is done in accordance with the Smoluchowski-Poisson equation.

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The modeling of small scales in two-dimensional turbulent flows: A statistical mechanics approach

Cited by 57 publications

References 21 publications

Statistical Mechanics of Violent Relaxation in Stellar Systems

Statistical Mechanics of Violent Relaxation in Stellar Systems

Finite speed propagation in the relaxation of vortex patches

Relaxation Theory for Point Vortices

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