= Magnetohydrodynamics
{title2=MHD}
{wiki}
= MHD
{c}
{synonym}
Magnetohydrodynamics couples an electrically conducting <fluid> to a <magnetic field>. The <electric current density> $\mathbf J$ produces a <Lorentz force density> $\mathbf J\times\mathbf B$, and <fluid> motion changes the <magnetic field> through induction.
For constant <magnetic diffusivity> $\eta$, a common nonrelativistic model includes
$$
\rho\frac{D\mathbf u}{Dt}=-\nabla p+\mathbf J\times\mathbf B+\text{other forces},
\qquad
\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\eta\nabla^2\mathbf B,
$$
with $\nabla\cdot\mathbf B=0$. <Ideal magnetohydrodynamics> neglects the diffusion term; <resistive magnetohydrodynamics> retains it. The ratio of induction $UB/L$ to diffusion $\eta B/L^2$ is the <magnetic Reynolds number> $\mathrm{Rm}=UL/\eta$.
<Magnetic tension> supplies a restoring force for an <Alfvén wave>, with speed $v_A=B/\sqrt{\mu_0\rho}$ in a homogeneous medium. <Magnetic pressure> and ordinary <pressure> can both oppose compression. These effects explain why conducting-fluid waves and equilibria differ from those of an unmagnetized gas. <Lecture notes: Astrophysical fluid dynamics by Gordon I. Ogilvie>[Sections 3.1.2–3.1.3, induction, Lorentz force, magnetic pressure/tension and Alfven speed]{ref}
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