Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade

Abstract

We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two-dimensions (2D) in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equations in 2D by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equations clearly show that a formally time-reversible system is able to reproduce the features of a 2D turbulent flow. Statistical quantities based on one-and two-point measurements show an excellent agreement between the two systems, for the inverse-and direct cascade regions. Moreover, we find that the conjecture holds very well for 2D turbulent flows with both conserved energy and enstrophy at finite Reynolds number, which goes beyond the original conjecture for three-dimensional turbulence in the limit of infinite Reynolds number.

Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations

“…Subsequent numerical attempts addressed this conjecture in a few simple systems such as in a version of the 2D NSE truncated to a few modes and with periodic boundary conditions [18], and more recently in [2,19], in the Lorenz-96 model [20] and a shell model of turbulence [21,22]. Recently, an interesting attempt to show equivalence between a reversible 2D NSE, where both energy and enstrophy are kept fixed, with irreversible 2D NSE is presented in [23]. First in [24] and later in [2,25,26] a second conjecture was laid to address the equivalence in the limit of infinite system size at fixed viscosity.…”