Mathematical estimates for the attractor dimension in MHD turbulence

The aim of the present work is to derive rigorous estimates for turbulent MHD flow quantities such as the size and anisotropy of the dissipative scales, as well as the transition between 2D and 3D state. To this end, we calculate an upper bound for the attractor dimension of the motion equations, which indicates the number of modes present in the fully developed flow. This method has already been used successfully to derive such estimates for 2D and 3D hydrodynamic turbulence as a function of the L∞ norm of dissipation, as in [5]. We tackle here the problem of a flow periodic in 3 spatial directions (spatial period 2πL), to which a permanent magnetic field is applied. In addition, a detailed study of the dissipation operator provides more indications about the structure of the flow.