The fluid theory of plasma behaviour in magnetic fields — governing equilibrium, stability, and the large-scale dynamics of fusion devices.
Magnetohydrodynamics (MHD) treats a plasma as an electrically conducting fluid threaded by magnetic fields. By combining the Navier–Stokes equations of fluid mechanics with Maxwell's equations of electromagnetism, MHD captures the macroscopic behaviour of plasmas on length scales much larger than the ion gyroradius and time scales much longer than the ion cyclotron period. It is the workhorse theory for designing magnetic-confinement configurations, computing equilibria, and assessing gross stability limits.[1]
The ideal-MHD model comprises the continuity equation, the momentum equation (with the J×B Lorentz force and pressure gradient), an energy equation (usually an adiabatic closure), Faraday's law, and Ohm's law in its simplest form E + v×B = 0. This last relation — the frozen-flux condition — states that magnetic field lines move with the plasma, so topology is preserved. When resistivity is retained (resistive MHD), field lines can slip through the plasma and reconnect, enabling an additional class of instabilities.[2]
A fusion plasma in steady state satisfies the Grad–Shafranov equation in axisymmetric geometry or its three-dimensional generalisation (the VMEC code family for stellarators). The solution specifies nested magnetic flux surfaces on which pressure is constant. Equilibrium codes are the starting point for all confinement-device design: they set the safety-factor profile q(ψ), the Shafranov shift, and the shape parameters that determine stability.[1]
MHD stability is assessed through energy principles: if any trial displacement lowers the potential energy of the plasma-plus-field system, the equilibrium is unstable. This variational approach yields sufficient conditions for stability without solving the full time-dependent equations. In practice, numerical codes (DCON, MISHKA, PEST) discretise the energy functional to find growth rates and mode structures for realistic equilibria.[3]
Because MHD averages over particle orbits and velocity distributions, it cannot capture kinetic effects such as Landau damping, trapped-particle dynamics, or finite-Larmor-radius stabilisation. Extended-MHD models add two-fluid corrections (Hall MHD), gyroviscosity, and energetic-particle drive to bridge toward fully kinetic descriptions. For turbulence-driven transport, gyrokinetic theory has largely supplanted MHD, but the equilibrium and large-scale stability framework remains firmly MHD-based.[2]
MHD governs solar flares, stellar interiors, accretion disks, the geodynamo, and industrial processes such as aluminium smelting and electromagnetic casting. The universality of the framework — any conducting fluid in a magnetic field — makes it one of the most widely applied branches of plasma physics.[1]