  A new model is proposed for low $Rm$ MHD flows which remain turbulent even in
the presence of a magnetic field. These flows minimize the Joule dissipation
because of their tendency to become two-dimensional and, therefore to suppress
all induction effects. However, some small three-dimensional effects, due to
inertia and to the electric coupling between the core flow and the Hartmann
layers, are present even within the core flow. This new model, which may be
seen as an improvement of the Sommeria-Moreau 2D model, introduces this
three-dimensionality as a small perturbation. It yields an equation for the
average velocity over the magnetic field lines, whose solution agrees well with
available measurements performed on isolated vortices.
