Bounds on the attractor dimension for low-Rm wall-bound magnetohydrodynamic turbulence

International audienceSteady low-Rm magnetohydrodynamic (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. The flow is defined by its maximum velocity and the intensity of the magnetic field. 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 that we compare to known heuristic 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 nonlinear 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 nondimensional parameters, which are reminiscent of three important previously identified transitions observed in the real flow. The first boundary separates 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 three-dimensional (3D) MHD sets from quasi-two-dimensional (2D) MHD sets, when all “Orr-Sommerfeld modes” disappear and only “Squire modes” are left. The third separates sets where all the modes exhibit the same boundary layer thickness or so, and sets where many different “boundary layer modes” coexists in the set. The nondimensional relations defining these boundaries are then compared to the heuristics known for the transition between isotropic and anisotropic MHD turbulence, 3D and quasi-2D MHD turbulence, and that between a turbulent and a laminar Hartmann layer. In addition to this 3D approach, we also determine upper bounds for the dimension of forced turbulent flows modeled 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 2D nonlinear term. This allows us to derive realistic attractor dimensions for quasi-2D MHD flows