On the Clark−α Model of Turbulence: Global Regularity and Long–time Dynamics

In this paper we study a well-known three–dimensional turbulence model, the filtered Clark model, or Clark−α model [7]. This is Large Eddy Simulation (LES) tensor-diffusivity model of turbulent flows with an additional spatial filter of width α. We show the global well-posedness of this model with constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an anaytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to (L/ld) , where L is the integral spatial scale and ld 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 semirigorous 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 decay power law for kα ≫ 1. This is 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.