  Steady Low $R_m$ 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, we derive an upper bound for the
dimension of the attractor and find out which modes must be chosen to achieve
this bound. Thier, are compared to heuristics estimates for the size of the
smallest scales and anisotropy of the turbulence. We identify 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 separates classical hydrodynamic
sets of modes from MHD sets where anisotropy takes the form of a ``Joule
cone''. The second boundary separates 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. The non-dimensional 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
modelled using a 2D MHD equation, 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.
