Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of “patches” and “points”

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Abstract
Numerical and analytical studies of decaying, two-dimensional Navier–Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Though these predictions might be thought to apply only to the ideal Euler equations, there have been surprising and imperfectly understood correspondences between the long-time computations of decaying states of NS flows and the results of the maximum entropy analyses. Both formulations define an entropy using a somewhat ad hoc discretization of vorticity into “particles.” Point-particle statistical methods are used to define an entropy, before passing to a mean-field approximation. In one case, the particles are delta-function parallel “line” vortices (“points,” in two dimensions), and in the other, they are finite-area, mutually exclusive convected “patches” of vorticity which only in the limit of zero area become “points.” The former are assumed to obey Boltzmann statistics, and the latter, Lynden-Bell statistics. Clearly, there is no unique way to reach a continuous, differentiable vorticity distribution as a mean-field limit by either method. The simplest method of taking equal-strength points and equal-strength, equal-area patches is chosen here, no reason being apparent for attempting anything more complicated. In both cases, a nonlinear partial differential equation results for the stream function of the “most probable,” or maximum entropy, state, compatible with conserved total energy and positive and negative velocity fluxes. These amount to generalizations of the “sinh-Poisson” equation which has become familiar from the “point” formulation. They have many solutions and only one of them maximizes the entropy from which it was derived, globally. These predictions can differ for the point and patch discretizations. The intent here is to use time-dependent, spectral-method direct numerical simulation of the Navier–Stokes equation to see if initial conditions which should relax toward the different late-time states under the two formulations actually do so.
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