Resistive MHD
MHD with finite resistivity, which breaks the frozen-in law and permits reconnection and tearing modes.
Adding resistivity
Ideal MHD forbids field lines from breaking. Real plasmas have finite Spitzer resistivity eta, so Ohm's law becomes E + v x B = eta J. Substituting into Faraday's law gives the resistive induction equation dB/dt = curl(v x B) + (eta/mu0) laplacian B, a competition between advection and magnetic diffusion.
The magnetic Reynolds number
The ratio of advection to diffusion is the Lundquist number S = mu0 L v_A / eta, where v_A is the Alfven speed and L a length scale. In fusion plasmas S is enormous (10^6 to 10^9), so resistivity matters only in thin current layers where gradients are steep.
What it enables
Finite resistivity allows magnetic reconnection: field lines break and reconnect, releasing magnetic energy and changing topology. This underlies tearing modes, neoclassical tearing modes, and the sawtooth crash. These are slower than ideal modes but can still degrade confinement or trigger disruptions.
- Tearing modes grow on the resistive timescale, scaling as S to a fractional power
- Reconnection forms magnetic islands that flatten pressure across them
- Resistive wall modes couple plasma motion to a finite-conductivity wall
How it is solved numerically
Because resistive layers are thin, codes need either adaptive mesh refinement or flux-surface-aligned coordinates that resolve the rational surfaces where q = m/n. Implicit or semi-implicit time stepping is essential: the fast Alfven wave would otherwise force impossibly small explicit time steps. Extended-MHD codes add two-fluid and Hall terms for realistic reconnection rates.
Predicting neoclassical tearing mode onset and island growth (via the Rutherford equation) is part of qualifying any high-performance scenario, including the Hyperion breeder design studies.