Rotational coupling of shear and pseudo-Alfvén waves (${\omega }_I/{\omega }_A\sim 2\Omega /(k_{\perp }{v}_A)$)

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