Coexistence of Two Equilibrium Configurations in 2D turbulence
Résumé
In the past, two families of statistical mechanics approaches have been applied to the two-dimensional Euler equations. The first one is formulated in Fourier space and considers the Galerkin truncated dynamics. The other one is formulated in physical space and considers either point-vortices or coarse-grained vorticity. We show that in a Galerkin truncated system both methods describe a part of the flow. A condensate can be identified assuming that it can be characterized by an unspecified functional relation between the vorticity and the stream-function. It is shown, a posteriori, that this function is a hyperbolic sine relation, as predicted by point-vortex statistical mechanics. The energy spectrum associated with the condensate is well described by an exponential function and the tails of the probability density function of the vorticity are following a power law. Analytical arguments to explain these observations are proposed. After removing the condensate, the remaining field can be described by Fourier-statistical mechanics.
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