Whether the two-dimensional Eulerian turbulence evolves to a unique final state

Abstract

The relaxation of two-dimensional Eulerian turbulence to a quasifinal state is studied. We consider the self-organization of localized vortices (in two-dimensional flows) into clusterlike and spiral-like structures and show that quasifinal states do not “forget” conditions of their initial origin. The numerical study confirms a possibility of the “vortex crystals” formation that have been observed in the plasma experiments. We discuss the physical significance of the obtained results.

Quasi-stationary States of Two-Dimensional Electron Plasma Trapped in Magnetic Field

“…In some experiments 14) and simulations, 12,18) it has already been found that the vortex crystal state is sensitive to the microscopic difference in initial states and may lead to completely different quasi-stationary states. In the case where the quasi-stationary state is singly peaked, the quasi-stationary state does not depend on the microscopic details, namely, the position of each electron in the initial state, as long as the configuration gives the same coarse-grained density profile, but we still found that the quasi-stationary state depends on macroscopic differences in the initial state, such as the singleor double-ring state.…”

Evolution of localized vortices in the presence of stochastic perturbations

“…As concluding remarks, let us present some explanations and give useful references (see also Miller et al, 1992;Danilov and Gurari, 2000;Pavlov et al, 2002) concerning some numerical approaches to the problem of the 2-Dturbulence evolution.…”

“…In fact, it is known that one of the difficulties in the description of turbulence is the expansion of the motion at small scales, to scales beyond those of viscous dissipation, when calculations come only from local mean-field at the scale of numerical resolution. Pointin and Lundgren, 1959;Sommeria et al, 1991;Brands et al, 1999;Pavlov et al, 2002 and references therein).…”

Evolution de configurations de tourbillons avec les mêmes invariants globaux

“…D'autre part, on peut supposer, comme cela est postulé dans la Mécanique Statistique (cf. [7][8][9][10][11][12]), que la configuration finale, si elle existe, n'est fixée que par les intégrales globales du système. Parmi ces intégrales, ce sont l'énergie totale du système et les moments, notamment la circulation Γ et l'enstrophie Z 2 , qui jouent évidemment un rôle déterminant.…”

“…A notre connaissance, ces questions n'ont pas encore été abordées dans la littérature scientifique, et ce, probablement, pour des raisons diverses : le système tourbillonnaire est gouverné par des équations d'évolution fortement non linéaires et leurs solutions analytiques ne sont pas envisageables, les approches basées sur les principes variationnels admettent des formulations diverses (cf. [7][8][9][10][11][12]), etc.…”