On the Clark–α model of turbulence: global regularity and long-time dynamics

Abstract

In this paper we study a well-known three-dimensional turbulence model, the filtered Clark model, or Clark–α model. This is a large eddy simulation (LES) tensor-diffusivity model of turbulent flows with an explicit spatial filter of width α. We show the global well-posedness of this model with constant Navier–Stokes viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an analytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to (L/l d )3, where L is the integral spatial scale and l d is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark–α model has the usual k −5/3 Kolmogorov power law for wave numbers kα ≪ 1 and k −3 decay power law for kα ≫ 1. This is an evidence that the Clark–α model parameterizes efficiently the large wave numbers within the inertial range, kα ≫ 1, so that they contain much less translational kinetic energy than their counterparts in the Navier–Stokes equations.