Nonequilibrium statistical mechanics of two-dimensional turbulence

Abstract

A complete set of independent real parameters and its dynamic equation are worked out to describe the vorticity dynamics of two-dimensional turbulence. The corresponding Liouville equation is solved by a perturbation method upon the basis of a Langevin–Fokker–Plank Model. The dynamic damping coefficient η of the LFP model is treated as an optimum control parameter to minimize the error of the perturbation solution. Thereby two integral equations, the enstrophy equation and the η equation, are obtained for two unknown functions: the spectrum and the η. The equilibrium spectrum for the inviscid case is obtained as a stationary solution of the enstrophy equation. The nonlocalness of the enstrophy transfer makes the enstrophy equation divergent for a simple power-law spectrum. In order to avoid the divergence problem, a localization factor g is introduced to characterize the actual spectrum. Finally, the localized forms of the two integral equations are numerically solved, leading to the inertial-range spectrum, E(k)=1.82(ln g−1.23)−2/3χ2/3k−3 for g≥10, χ is the dissipation rate of the enstrophy.