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. 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.

Acknowledgements

This work was supported in part by the US Department of Energy, under contract number W-7405-ENG-36 and by the Office of Science ASCR/AMS/MICS. The work of E. S. T was supported in part by the NSF grant number DMS-0204794, the US CRDF grant number RM1-2343-MO-02 and by the MAOF Fellowship of the Israeli Council of Higher Education.