Rotating-frame magnetohydrodynamic momentum equation ($\rho D_t\mathbf{u}=-\nabla (P+B^2/(2{\mu }_0))+(\mathbf{B}\cdot \nabla )\mathbf{B}/{\mu }_0-\rho \nabla \Phi -2\rho \mathbf{\Omega }\times \mathbf{u}-\rho \mathbf{\Omega }\times (\mathbf{\Omega }\times \mathbf{r})$)

For a rigid frame with constant angular velocity $\mathbf{\Omega }$, add the Coriolis force and centrifugal force to the inviscid magnetohydrodynamic momentum equation. A constant magnetic permeability and solenoidal magnetic-field constraint split the Lorentz force density into magnetic pressure and magnetic tension. This nonrelativistic conducting-fluid approximation neglects the displacement current, net electric-charge force and viscosity. Differential rotation belongs to the fluid's velocity field, not to a spatially varying rigid-frame rotation vector. A time-dependent frame requires the additional Euler force.

With aligned uniform magnetic field and rotation, pressure projection couples the two incompressible transverse wave polarization states at rate ${\omega }_I\sim 2\Omega |k_{\parallel }|/k_{\perp }$. Each nonrotating Alfvén wave has rate ${\omega }_A=|k_{\parallel }|v_A$. Their ratio is independent of the anisotropy. Leading finite-timescale decoupling therefore needs $k_{\perp }{v}_A\gg 2\Omega$, with rotational coupling also slow relative to any retained nonlinear evolution. At finite amplitude the usual anisotropic reduction still permits passive advection of pseudo-Alfvénic fluctuations by the Alfvénic subsystem; removing rotational mixing does not eliminate that one-way interaction. Because the two Alfvén polarizations are degenerate, weak coupling can still mix them over the longer time ${\omega }_I^{-1}$.