Hall Magnetohydrodynamics

An extension of MHD that retains the Hall term, decoupling ion and electron motion at small scales.

The Hall Term

Hall MHD keeps the J x B / (n e) contribution in the generalized Ohm's law, so E + u x B = eta J + (J x B)/(n e) - grad p_e/(n e). Physically the magnetic field is frozen to the electron fluid rather than to the bulk plasma. Because electrons and ions move differently below the ion inertial scale, the field lines slip relative to the ions.

Characteristic Scale

The Hall term becomes important at the ion inertial length d_i = c / omega_pi, where omega_pi is the ion plasma frequency, and at the ion gyroradius. Above these scales the plasma behaves as a single MHD fluid; below them, dispersion and species decoupling appear. This introduces the whistler and kinetic-Alfven branches into the wave spectrum, which are dispersive unlike the non-dispersive ideal-MHD Alfven wave.

Fast Reconnection

The most cited role of Hall physics is enabling fast magnetic reconnection. In pure resistive MHD, reconnection through a long thin current sheet is far too slow to match observations. The Hall term opens the reconnection geometry into an X-point with an out-of-plane quadrupolar magnetic field, decoupling the electron diffusion region and raising the reconnection rate to a nearly geometry-independent value.

Relevance

Reconnection matters for sawtooth crashes, tearing modes, and disruptions in tokamaks, all of which redistribute energy and can damage confinement. For a spherical tokamak such as the Hyperion breeder concept, low aspect ratio and strong shaping change reconnection geometry, so Hall-resolving extended-MHD simulation is part of the design-stage stability toolset. Any conclusions are computational; no Kronos device has been built.