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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 includeswith $\nabla \cdot \mathbf{B}=0$. 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. ^[ref]

Resistive magnetohydrodynamics retains finite magnetic diffusivity in the coupled evolution of a conducting fluid and its magnetic field. The resistive induction equation allows field diffusion relative to the fluid, while the magnetohydrodynamic momentum equation includes the Lorentz force density. It contrasts with the zero-diffusivity limit of ideal magnetohydrodynamics.