  We establish a shallow water model for flows of electrically conducting
fluids in homogeneous static magnetic fields that are confined between two
parallel planes where turbulent Hartmann layers are present. This is achieved
by modelling the wall shear stress in these layers using the Prandtl's mixing
length model, as did the authors of Albousssi\`ere \& Lingwood (Phys. Fluids,
2000). The idea for this new model arose from the failure of previous shallow
water models that assumed a laminar Hartmann layer to recover the correct
amount of dissipation found in some regimes of the MATUR experiment. This
experiment, conducted by the authors of Messadek \& Moreau (J. Fluid Mech.
2002), consisted of a thin layer of mercury electrically driven in differential
rotation in a transverse magnetic field. Numerical Simulations of our new model
in the configuration of this experiment allowed us to recover experimental
values of both the global angular momentum and the local velocity up to a few
percent when the Hartmann layer was in a sufficiently well developed turbulent
state. We thus provide an evidence that the unexplained level of dissipation
observed in MATUR in these specific regimes was caused by turbulence in the
Hartmann layers. A parametric analysis of the flow, made possible by the
simplicity of our model, also revealed that turbulent friction in the Hartmann
layer prevented quasi-2D turbulence from becoming more intense and limited the
size of the large scales.
