Length-scale estimates for the 3D simplified Bardina magnetohydrodynamic model

We consider the MHD-α model known as double viscous simplified Bardina MHD (SBMHD) on a 3D periodic box (MHD stands for magnetohydrodynamic). This system is a large eddy simulation model useful to approximate the turbulent behavior of an incompressible homogeneous magnetofluid because of the actual impossibility to handle the MHD model neither analytically nor via direct numerical simulation. In a previous paper (joint work with Secchi), the global existence of strong solutions to the SBMHD has been proved as well as the existence of a global attractor of finite fractal dimension. Upper bounds for such dimension are provided both in terms of the modified Grashof number and the dissipation length associated to the mean rate of energy dissipation. In this paper, we commute the same bound in an estimate in terms of the modified Reynolds number R. This result is useful because the classical Kolmogorov theory of turbulence is expressed using the Reynolds number. The global attractor estimate in terms of R is c...