The Magnetic Diffusion Equation

The induction equation in the static-fluid limit, describing how magnetic field decays and penetrates by resistivity.

Derivation

Take the resistive induction equation dB/dt = curl(u x B) + eta_m laplacian B and drop the flow term, either because the fluid is at rest or because diffusion dominates advection (low magnetic Reynolds number). What remains is the magnetic diffusion equation dB/dt = eta_m laplacian B, a vector diffusion equation with magnetic diffusivity eta_m = eta / mu0.

Solutions

Being a diffusion equation, it has the same structure as heat conduction: field gradients smooth out, and a field imposed at a boundary penetrates a conductor over a skin depth that grows as sqrt(eta_m t). The characteristic resistive diffusion time across a system of size L is tau_R = mu0 L^2 / eta = L^2 / eta_m. In a good conductor eta_m is small and tau_R is long, so the field is nearly frozen.

Skin Effect

For a field oscillating at frequency omega, the penetration depth is the skin depth sqrt(2 eta_m / omega). Faster changes are screened to a thinner layer near the surface. This governs how quickly external coil fields, error fields, or feedback fields can reach the plasma interior, and how induced eddy currents in conducting walls shield transient fields.

Relevance

Current-profile evolution in a tokamak is governed by magnetic diffusion of the poloidal field, with the resistive time setting how long a driven current profile takes to relax. Because resistivity falls steeply with temperature, hot plasmas have very long current-relaxation times. For the Hyperion breeder concept, resistive diffusion sets current-drive and profile-control timescales in design-stage modeling of a simulated machine.