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

We consider a visco-resistive magnetohydrodynamic modeling of a steady-state incompressible tokamak plasma with a prescribed toroidal current drive, featuring constant resistivity η and viscosity ν. We reintroduce in the traditional Grad-Shafranov equation the dissipative viscous term and the non-linear (v · ∇)v term coming from the steady-state Navier-Stokes equation [1, 2, 3, 4, 5, 6]. 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 H¹/⁴, which is consistent with numerical results. These use the finite element method through the open-source platform FreeFem++ for solving partial differential equations [7].