The Magnetic Reynolds Number

The dimensionless ratio of magnetic advection to magnetic diffusion in a conducting flow.

Definition

The magnetic Reynolds number is Rm = U L / eta_m, where U is a characteristic flow speed, L a length scale, and eta_m = eta/mu0 the magnetic diffusivity. It is the magnetic analog of the ordinary Reynolds number, comparing the advection of magnetic field by the flow to its resistive diffusion. It appears directly in the induction equation as the coefficient governing the relative size of the two terms.

The Two Limits

When Rm is much greater than one, advection dominates and the field is effectively frozen into the moving plasma, the ideal-MHD flux-freezing regime. When Rm is much less than one, diffusion dominates and the field slips freely through the medium, decaying as if the flow were absent. Most laboratory and astrophysical plasmas sit deep in the high-Rm regime.

Relation to the Lundquist Number

The Lundquist number is the special case of Rm evaluated with the Alfven speed as the velocity scale, U = v_A. The distinction matters because different processes have different natural velocities: bulk flows for Rm, Alfvenic dynamics for S. Both express the same competition between induction and resistive diffusion but reference different dynamics.

Relevance

High Rm underlies flux conservation, current-profile relaxation timescales, and the slow resistive evolution of tokamak equilibria relative to fast MHD dynamics. It also governs whether externally imposed error fields can penetrate or are screened by plasma rotation. For the Hyperion breeder concept, high Rm justifies treating the bulk plasma as ideal on dynamical timescales while tracking slow resistive current diffusion separately; these are modeling assumptions for a simulated device.