The Generalized Ohm's Law

The electron momentum balance that connects electric field, current, and the two-fluid and Hall terms.

Beyond E plus v cross B

Ideal MHD uses the simplest Ohm's law, E + v x B = 0. The full generalized Ohm's law comes from the electron momentum equation and contains several additional terms that matter on small scales and short times:

E + v x B = eta J + (1/ne)(J x B) - (1/ne) grad p_e + (m_e/ne^2) dJ/dt

Left to right on the right side: resistive (Spitzer) term, the Hall term, the electron-pressure (thermoelectric/battery) term, and electron inertia. Each dominates in a different regime.

What each term does

• Resistive term: allows reconnection and current diffusion
• Hall term: decouples ion and electron motion at ion-inertial scales, speeds up reconnection
• Electron-pressure term: the Biermann battery, can generate magnetic field from crossed density and temperature gradients
• Electron inertia: matters at electron-inertial scales and very high frequency

Which terms to keep

MHD keeps only the ideal (and sometimes resistive) term. Hall MHD adds the Hall and electron-pressure terms; two-fluid and kinetic models keep everything. The choice depends on the scales of interest: large, slow phenomena need only ideal Ohm's law, while reconnection layers and small-scale turbulence need the extended terms.

How it is used numerically

Extended-MHD codes include the Hall and electron-pressure terms to reproduce realistic, fast reconnection rates that resistive MHD alone gets wrong. The Hall term makes the equations dispersive (whistler waves), which stiffens the numerics and often requires implicit solvers.

Relevance

Accurate reconnection physics, governed by the generalized Ohm's law, is needed to predict sawteeth, tearing modes, and disruptions in tokamaks such as the Hyperion breeder, where the fast crash timescales point beyond simple resistive MHD.