Magnetohydrodynamic momentum equation ($\rho D_t\mathbf{u}=-\nabla p+\mathbf{J}\times \mathbf{B}+\rho \nu {\nabla }^2\mathbf{u}$)

For constant mass density $\rho$, kinematic viscosity $\nu$ and magnetic permeability ${\mu }_0$, the incompressible momentum equation isIt is the Navier-Stokes equation with Lorentz force density. The magnetic force decomposes into magnetic tension and a magnetic pressure gradient. A uniform streamwise pressure gradient need not make ordinary pressure independent of transverse coordinates.

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 ${\omega }_I^{-1}$.