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 ηf(H) as long as the inertial term remains negligible, where H stands for the Hartmann number H≡(ην)^−1/2, and that f(H) exhibits power-law behaviours in the limits H≪1 and H≫1. In the latter limit, we establish that f(H) scales as H^1/4, which is consistent with numerical results.