Magnetohydrodynamic

Magnetohydrodynamics (MHD) is the field that studies the dynamics of electrically conducting fluids in the presence of magnetic fields. It combines fluid dynamics and electromagnetism to describe how magnetic fields influence the motion of plasmas, liquid metals, and salty liquids, and how fluid motion in turn advects and reshapes magnetic fields.

The governing model couples the equations of fluid dynamics with Maxwell's equations under nonrelativistic, quasi-neutral conditions. The momentum balance is ρ(dv/dt) = -∇p + J×B + μ∇^2 v. The induction equation for the magnetic field is ∂B/∂t = ∇×(v×B) + η∇^2B, where η is the magnetic diffusivity (η = 1/(μ0 σ)). The continuity equation ∂ρ/∂t + ∇·(ρ v) = 0 also applies. Ohm's law for a moving conductor relates current to the electric field as J = σ(E + v×B). Maxwell's equations include ∇·B = 0 and ∇×E = -∂B/∂t. In the ideal MHD limit (σ → ∞), E ≈ -v×B and magnetic field lines are frozen into the fluid; finite conductivity yields magnetic diffusion and reconnection.

Applications span astrophysics, including stellar interiors, accretion disks, solar wind, and dynamos; geophysics, such as planetary dynamos; laboratory plasma and fusion research with tokamaks and stellarators; and industrial processes using liquid metals and electromagnetic casting.

Historically, MHD emerged from the work of Hannes Alfvén in the 1940s, who identified Alfvén waves and fundamental coupling between flows and magnetic fields; the field has since become central in plasma physics and astrophysics.