mhd

Magnetohydrodynamics (MHD) is the field that studies the dynamics of electrically conducting fluids in magnetic fields. It encompasses plasmas, liquid metals, and other ionized or conductive media, and is used to model phenomena in astrophysics, geophysics, and engineering.

In single-fluid MHD, fluid motion and magnetic field evolution are described by the conservation laws of mass and momentum together with Maxwell's equations. The momentum equation is ρ dv/dt = -∇p + J × B + ρ g + ∇·τ, where J = ∇ × B / μ0. Ohm's law for a moving conductor is E + v × B = η J, and Ampère's law relates current to the magnetic field by J = ∇ × B / μ0. The magnetic field evolves via the induction equation ∂B/∂t = ∇ × (v × B) + η ∇² B, with the constraint ∇ · B = 0. In ideal MHD, conductivity is infinite (η = 0), and magnetic field lines are frozen into the fluid; non-ideal or resistive MHD allows diffusion and enables magnetic reconnection.

Key concepts include magnetic pressure and tension, Alfvén waves, and the role of diffusion in changing field topology. The magnetic Reynolds number Rm = μ0 σ v L characterizes the relative importance of advection to diffusion, with large Rm implying flux freezing. The Lundquist number and related nondimensional parameters describe wave propagation and diffusion in conductive media.

Applications span astrophysical and space plasmas (solar wind, accretion disks, stellar interiors), planetary and planetary-core dynamics, and laboratory plasmas in fusion devices such as tokamaks. MHD also informs metallurgical processes and liquid-metal technologies.

Historically, MHD was developed in the mid-20th century, with Hannes Alfvén a foundational figure; he received the Nobel Prize in Physics in 1970 for his work in plasma physics and MHD. Limitations include the continuum-fluid assumption and neglect of kinetic effects, with extensions such as resistive, compressible, and relativistic MHD used for more detailed modeling.