Under consideration for publication in J. Fluid Mech. 1 Attractors of lowRm MHD turbulence

Low Rm MHD turbulence is investigated here through estimates of upper bounds for attractor dimension. A ow between two parallel walls with an imposed perpendicular magnetic eld is considered. The ow is dened by its maximum velocity and the intensity of the magnetic eld. Given the corresponding Reynolds and Hartmann numbers, one can derive rigourously an upper bound for the dimension of the turbulent attractor and nd out which modes must be chosen to achieve this bound. This is then used heuristically to estimate the size of the smallest turbulent vortices and the degree of anisotropy of the turbulence. Our upper bound derivation is based on known bounds of the non-linear inertial term, while low Rm Lorentz forces { being linear { can be easily dealt with. The simple conguration considered in this paper allows us to model three impor-tant previously identied transitions. The rst is the transition from classical to MHD turbulence when anisotropy takes the form of a \Joule cone". Second, one can dene the transition from 3D MHD turbulence to quasi-2D MHD turbulence, when all \Som-merfeld modes " disappear and only \Squire modes " are left. Third, Hartmann layers undergo transition to turbulence when more than one \boundary layer mode " exists in the attractor. In addition to this 3D approach, we also determine upper bounds for the dimension of turbulent ows modelled using a 2D MHD equation, which should become physically relevant in the quasi-2D MHD regime. The advantage of this 2D approach is that, while upper bounds are quite loose in three dimensions, optimal upper bounds exist for the nonlinear term. This allows us to derive realistic attractor dimensions for quasi-2D MHD ows. 1