Bounds on the attractor dimension for low-Rm wall-bound MHD turbulence
Résumé
Steady Low Rm MHD turbulence is investigated here through estimates of upper bounds for attractor dimension. A flow between two parallel walls with an imposed perpendicular magnetic field is considered. Given the corresponding Reynolds and Hartmann numbers, one can rigorously derive an upper bound for the dimension of the attractor and find out which modes must be chosen to achieve this bound. The properties of these modes yield quantities which we compare to known heuristics estimates for 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 relatively easily dealt with. The simple configuration considered in this paper allows us to identify some boundaries separating different sets of modes in the space of non-dimensional parameters, which are reminiscent of three important previously identified transitions observed in the real flow. The first boundary separate from classical hydrodynamic sets of modes from MHD sets where anisotropy takes the form of a "Joule cone". In the second, one can define the boundary separating 3D MHD sets from quasi-2D MHD sets. The third separates sets where all the modes exhibit the same boundary layer thickness or so, and sets where many different "boundary layer modes" co-exists in the set. In addition to this 3D approach, we also determine upper bounds for the dimension of forced turbulent flows modelled using a 2D MHD equation.
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