The Equilibrium Energy Spectrum of Randomly Forced Two-Dimensional Turbulence

Abstract

From a statistical mechanical treatment of an ensemble of randomly forced two-dimensional flows of a viscous fluid, we derive two independent integral constraints on the form of the equilibrium energy spectrum. With a single hypothesis about the shape-preserving properties of the spectrum, those constraints determine the spectrum to within the value of a single universal dimensionless constant. In all other respects the argument is deductive and does not depend on closure approximations or hypotheses about the process of nonlinear energy transfer. The spectrum exhibits minus third-power dependence for, small scales, but minus first-power dependence for large scales. It is in good agreement with the results of detailed numerical integrations of the Navier-Stokes equations.