= Geostrophic flow preserving the Taylor constraint
{title2=$D\mathcal T_B[\nabla\times(u\times B)+\eta\nabla^2B]=0$}
For an instantaneous <magnetostrophic Taylor state>, the momentum equation leaves a geostrophic null component $u_g=s\omega(s)\hat\phi$. 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 $\mu_0^{-1}\partial_s[s^3\int_{C_s}B_s^2d\phi dz\,\omega']$. Boundary terms generally cannot be discarded for an insulating exterior; the primary analysis is https://arxiv.org/abs/1806.06612[Hardy and collaborators' boundary-compatible geostrophic-flow construction]. Regularity and an angular-momentum normalization fix remaining homogeneous freedoms when the magnetic coupling is nondegenerate.
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