Scaling laws of the plasma velocity in visco-resistive magnetohydrodynamic systems

We consider a visco-resistive magnetohydrodynamic modelling of a steady-state incompressible tokamak plasma with a prescribed toroidal current drive, featuring constant resistivity η and viscosity ν. It is shown that the plasma velocity root-mean-square behaves as $\eta f\left(H\right)$ as long as the inertial term remains negligible, where H stands for the Hartmann number $H\equiv {\left(\eta \nu \right)}^{-1/2}$, and that $f\left(H\right)$ exhibits power-law behaviours in the limits $H\ll 1$ and $H\gg 1$. In the latter limit, we establish that $f\left(H\right)$ scales as $H^{1/4}$, which is consistent with numerical results.