In this paper we study a well-known three--dimensional turbulence model, the
filtered Clark model, or Clark-alpha model. This is Large Eddy Simulation (LES)
tensor-diffusivity model of turbulent flows with an additional spatial filter
of width alpha (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​)3, 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 semi-rigorous physical arguments we show
that the inertial range of the energy spectrum for the Clark-aËš model has
the usual k−5/3 Kolmogorov power law for wave numbers ka˚≪1 and
k−3 decay power law for ka˚≫1. This is evidence that the
Clark−α model parameterizes efficiently the large wave numbers within
the inertial range, ka˚≫1, so that they contain much less translational
kinetic energy than their counterparts in the Navier-Stokes equations.Comment: 11 pages, no figures, submitted to J of Turbulenc