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Two-dimensional turbulence and stellar systems

par Pierre-Henri Chavanis - 3 octobre 2006

Two-dimensional flows with high Reynolds numbers have the striking property of organizing spontaneously into large scale coherent vortices. These vortices are common features of geophysical and astrophysical flows with the well-known example of Jupiter’s Great Red Spot, a huge vortex persisting for more than three centuries in a turbulent shear between two zonal jets. Similarly, it is striking to observe that galaxies themselves follow a kind of organization revealed in the Hubble classification or in de Vaucouleurs R^1/4 law for the surface brightness of Ellipticals (NGC3379).

The global organization of these systems can be understood in terms of relatively similar statistical mechanics. This was first investigated by Lynden-Bell [1] in his theory of ``violent relaxation’’ applying to collisionless stellar systems and rediscovered more recently by Kuz’min [2], Miller [3] and Robert & Sommeria [4] in the context of two-dimensional turbulence. A firm mathematical justification of these ideas has been provided by Robert [5] in terms of Young measures. The formal analogy between galaxies and two-dimensional vortices resides in the similar morphology of the Vlasov and the Euler equations (Chavanis [6][7] ; Chavanis et al. [8]). These equations develop a kind of ``mixing process’’ associated with the damped oscillations of a protogalaxy initially far from mechanical equilibrium or with the nonlinear development of the Kelvin-Helmholtz instability in hydrodynamics. The effectiveness of mixing is measured by an entropy functional which tends to be maximum at equilibrium. However, complete mixing, which would give rise to structureless objects, is forbidden by the existence of global constraints like the conservation of energy. Therefore, the star density or the vorticity profiles are obtained by maximizing the entropy while accounting for all the constraints of the Vlasov or the Euler equation. In this point of view, galaxies and two-dimensional vortices can be considered as sort of equilibrium states (in a thermodynamical sense) following a law of chaos : there is a total lack of information at small scales, but the exciting phenomenon is that microscopic disorder leads to macroscopic order. This description, of course, requires a hypothesis of ergodicity which may not be completely fulfilled in practice. The statistical approach provides, however, a systematic and valuable framework to tackle the problem of self-organization in two-dimensional turbulence and for stellar systems.

Because of the long range nature and the singularity at short distances of the gravitational potential, the statistical mechanics of self-gravitating systems makes problem and the notion of equilibrium is not always well-defined [9]. First of all, it is easy to show that there is no entropy maximum if the system is not bounded : its mass would be infinite ! In addition, even if truncated models are introduced or if the system is artificially confined to a box, an equilibrium state does not always exist. Below a minimum energy (or above a maximum radius), there is no critical point of entropy [10] : the system takes a ``core-halo’’ structure and can always increase entropy by making the core denser and denser and hotter and hotter. This ``gravothermal catastrophe’’ is probably related to the negative heat capacities of self-gravitating systems [11]. A priori, the collapse continues up to the formation of a central singularity. Different dynamical models show that the central density becomes infinite in a finite time [12][13][14]. This singularity has been known as ``core collapse’’ and many globular clusters have probably experienced core collapse. In practice, the formation of hard binaries can release sufficient energy to stop the collapse and drive a reexpansion of the system [15]. It has been suggested that a series of oscillations (contraction and reexpansion) should follow [16]. Since collisionless stellar system can reach an isothermal distribution during the process of violent relaxation [1], it is possible that they also undergo the gravothermal catastrophe. However, the core will cease to shrink when it becomes degenerate in Lynden-Bell’s sense. The system will then have a core-halo structure which is a global entropy maximum. This prediction has been fully confirmed by the calculation of Fermi-Dirac spheres by Chavanis \& Sommeria [17].

While stellar systems exhibit a ``core-halo’’ structure with high central densities, two-dimensional vortices have a smoother vorticity profile. However, unlike the stellar case, vorticity can take positive and negative values leading to a wider variety of structures such as monopoles, rotating or translating dipoles, tripoles... In a limit of strong mixing, the maximization of entropy becomes equivalent to a kind of enstrophy minimization and it is possible to obtain a nice classification of the whole ``zoology’’ of coherent vortices usually met in two-dimensional flows [18][19]. Isolated vortices are justified as restricted equilibrium states, introducing the concept of a ``maximum entropy bubble’’ as a heuristic attempt to take into account ``incomplete relaxation’’. Incomplete relaxation is also necessary in the case of stellar systems to avoid the ``infinite mass’’ problem.

To understand more precisely why relaxation does not proceed to completion, we must ressort to kinetic theory and non equilibrium thermodynamics. The route to equilibrium can be described simply in terms of convection-diffusion equations obtained from a general Maximum Entropy Production Principle [20][8][21][22]. It appears that the optimal diffusion current is the sum of two terms : a pure diffusion compensated by an appropriate friction (for stellar systems) or a drift (for two-dimensional vortices) necessary to conserve energy. The diffusion coefficient is related to small-scale fluctuations and can freeze the system in a ``bubble’’ when these fluctuations die away (providing a firmer basis for incomplete relaxation) [23][8]. Finally, a kind of Einstein relation connects the diffusion coefficient to the friction or to the drift, indicating some link with the theory of Brownian motion and Fokker-Planck equations [24][25]. This link is particularly apparent when we consider the more conventional statistical mechanics of Hamiltonian systems made of point vortices [26][27] or point mass stars [28] instead of continuous distributions.

The process of diffusion in two-dimensional turbulence can be analyzed in a completely stochastic framework by studying the statistics of the velocity field arising from a random distribution of point vortices [29]. This problem is intimately related to the statistics of the gravitational field produced by a random distribution of stars [24] and this is another manifestation of the analogy between these two systems. It is possible to derive exact results concerning the probability density function (p.d.f) of the velocity fluctuations, their typical duration and their spatial correlations. The velocity distribution presents a behavior which is intermediate between Gaussian and Levy laws. For that reason, the results are polluted by logarithmic corrections. As an application of these ideas, we can obtain an estimate for the diffusion coefficient of point vortices. When applied to the context of freely decaying turbulence, the diffusion becomes anomalous and we can establish a relation nu=1+xi/2 between the coefficient of anomalous diffusion nu and the exponent xi which characterizes the decay of the vortex density. We have proposed that xi=1, yielding nu=3/2, in the scaling regime of the decay. These results are in agreement with numerical simulations of ``punctuated Hamiltonain dynamics of point vortices’’ performed with a renormalization group procedure [30].Proceedings

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