Magnetostrophic balance ($2\rho \Omega \hat{z}\times u=-\nabla p+f_b+J\times B$)

The leading force balance in a rapidly rotating conducting fluid when inertial and viscous forces are small, while Coriolis force, pressure, buoyancy and Lorentz force density remain significant. The Rossby number and Ekman number measure the omitted inertia and bulk viscosity. Rotation does not balance an arbitrary magnetic torque: impenetrable axisymmetric boundaries impose the Taylor constraint. Magnetic diffusion can remain in the induction equation even when viscosity is neglected in momentum balance.

In inviscid magnetostrophic balance within an impenetrable axisymmetric container, with no azimuthal buoyancy or external mechanical torque, the cylindrically integrated Lorentz force density must vanish on every cylinder coaxial with rotation. The azimuthal pressure derivative integrates to zero, and the cylindrical Coriolis integral is proportional to net radial flux, zero by incompressibility and impermeability. Multiplying by cylinder area and lever arm gives the equivalent zero-torque form. This rotating-fluid condition is distinct from magnetic-helicity relaxation to a force-free plasma state. Its original primary source is Taylor's 1963 rotating-fluid paper.

With small viscosity, the azimuthal cylindrical balance permits Lorentz force torque to be cancelled by viscous stress. For ordinary no-slip rotating boundaries an Ekman layer gives stress of order $\rho \Omega LU\sqrt{E}$, larger than the order-$E$ bulk viscous correction at fixed velocity. Thus bounded small-$E$ flows approach a magnetostrophic Taylor state but need not satisfy its constraint exactly at finite viscosity. An order-one uncompensated magnetic torque would require a divergent flow as $E\to 0$, invalidating the assumed small-inertia balance. Stress-free boundaries remove the leading no-slip friction and require their own torque/regularity conditions.

For an instantaneous magnetostrophic Taylor state, the momentum equation leaves a geostrophic null component $u_g=s\omega (s)\hat{\varphi }$. Decompose the velocity into this component and a particular ageostrophic flow, substitute the induction equation into the displayed derivative constraint, and solve the resulting linear equation for $\omega$ with magnetic boundary conditions imposed. When magnetic surface torque vanishes, the geostrophic part is ${\mu }_0^{-1}{\partial }_s[s^3{\int }_{C_s}{B}_s^{2}d\varphi dz{\omega }^{\prime }]$. Boundary terms generally cannot be discarded for an insulating exterior; the primary analysis is Hardy and collaborators' boundary-compatible geostrophic-flow construction. Regularity and an angular-momentum normalization fix remaining homogeneous freedoms when the magnetic coupling is nondegenerate.

A divergence-free magnetic field satisfying the Taylor constraint on all geostrophic cylinders. It is a compatibility condition for slow rotating-fluid balance, not a local force-free condition. To retain such a state as magnetic induction proceeds, the velocity must also preserve the time derivative of the cylindrical torque constraint.