= Solution
Start in an inertial frame with the inviscid <magnetohydrodynamic momentum equation>,
$$
\rho D_t\mathbf v=-\nabla P+\mathbf J\times\mathbf B-\rho\nabla\Phi.
$$
Here $\rho$ is <mass density>, $P$ is the isotropic fluid <pressure>, $\mathbf v$ is inertial-frame <velocity>, $\mathbf J$ is <electric current density>, $\mathbf B$ is the <magnetic field>, and $\Phi$ is <gravitational potential> per unit <mass>. In the low-frequency approximation to <Ampère's law>, $\mathbf J=\nabla\times\mathbf B/\mu_0$. The <vector calculus identity>
$$
(\nabla\times\mathbf B)\times\mathbf B=(\mathbf B\cdot\nabla)\mathbf B-\nabla(B^2/2)
$$
separates the <Lorentz force density> into <magnetic tension> and a <magnetic pressure> gradient. Transform the material <acceleration> to a rigid frame rotating with constant <angular velocity> $\boldsymbol\Omega_0$, using the stated frame transformation. The <rotating-frame magnetohydrodynamic momentum equation> is
$$
\boxed{\rho D_t\mathbf u=-\nabla\left(P+\frac{B^2}{2\mu_0}\right)+\frac{(\mathbf B\cdot\nabla)\mathbf B}{\mu_0}-\rho\nabla\Phi-2\rho\boldsymbol\Omega_0\times\mathbf u-\rho\boldsymbol\Omega_0\times(\boldsymbol\Omega_0\times\mathbf r).}
$$
The final terms are the <Coriolis force> and <centrifugal force> densities; $\mathbf u$ is the <velocity> relative to the rotating frame, and $D_t=\partial_t+\mathbf u\cdot\nabla$ uses that frame.
This requires a smooth conducting-fluid continuum with nonrelativistic motion, a constant <magnetic permeability> $\mu_0$, the <solenoidal magnetic-field constraint>, a negligible <displacement current> and negligible net electric-charge force. Viscous forces and anisotropic fluid stresses are omitted. A time-dependent frame would add an <Euler force>. <Incompressible flow> is not needed for this force equation. Finite <electrical resistivity> does not change its displayed instantaneous magnetic force; the later frozen-field calculation additionally uses <ideal magnetohydrodynamics>. A differentially rotating fluid does not define a single rigid frame with spatially varying $\boldsymbol\Omega_0$: choose a fixed local reference rate and retain the fluid's shear in $\mathbf u$.
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