The Resistive MHD Equations

Ideal MHD plus finite resistivity, admitting magnetic diffusion, reconnection, and tearing instabilities.

The Modification

Resistive MHD keeps the ideal fluid equations but replaces the ideal Ohm's law with E + u x B = eta J, retaining a finite resistivity eta. Substituting into Faraday's law gives the resistive induction equation dB/dt = curl(u x B) + eta_m laplacian B, where eta_m = eta/mu0 is the magnetic diffusivity. The added diffusion term lets the field slip relative to the plasma.

What Resistivity Enables

In ideal MHD field lines cannot break; topology is conserved. Resistivity relaxes this, permitting magnetic reconnection: field lines can tear and reconnect in thin current layers where gradients are steep. This opens the tearing-mode and resistive-wall-mode instability families, allows the slow ohmic decay of currents, and governs how externally applied fields penetrate the plasma.

Timescale Separation

Because fusion-plasma resistivity is tiny (Spitzer resistivity falls as T^{-3/2}), the resistive diffusion time is far longer than the Alfven time; their ratio is the Lundquist number S, typically very large. Resistive effects therefore act slowly and are localized to thin layers, while the bulk plasma evolves ideally. Resistive-MHD codes must resolve these thin layers, a numerically demanding task.

Relevance

Neoclassical tearing modes, current-profile relaxation, error-field penetration, and disruption dynamics all require resistive MHD. For the Hyperion breeder concept, resistive and extended-MHD modeling assesses tearing stability and disruption avoidance at the design stage; the high plasma temperature implies very small resistivity and very high Lundquist number. These are simulations for a device not yet built.