  In this paper, we derive estimates for size of the small scales and the
attractor dimension in low $Rm$ magnetohydrodynamic turbulence by deriving a
rigorous upper bound of the dimension of the attractor representing this flow.
To this end, we find an upper bound for the maximum growth rate of any
$n$-dimensional volume of the phase space by the evolution operator associated
to the Navier-Stokes equations. As explained by Constantin et al. (J. Fluid
Mech., 1985), The value of $n$ for which this maximum is zero is an upper bound
for the attractor dimension. In order to use this property in the more precise
case of a 3D periodical domain, we are led to calculate the distribution of $n$
modes which minimises the total (viscous and Joule) dissipation. This set of
modes turns out to exhibit most of the well known properties of MHD turbulence,
previously obtained by heuristic considerations such as the existence of the
Joule cone under strong magnetic field. The sought estimates for the small
scales and attractor dimension are then obtained under no physical assumption
as functions of the Hartmann and the Reynolds numbers and match the Hartmann
number dependency of heuristic results. A necessary condition for the flow to
be tridimensional and anisotropic (as opposed to purely two-dimensional) is
also built.
