Rapid-rotation slow magneto-Coriolis branch (${\omega }_s^2\sim a^2(K^2{a}^2-H{h}^2)/(4{\Omega }^2{k}_z^2)$)

For fixed wavevector with $k_z\ne 0$, fixed nonzero Alfvén frequency $a$, and $|\Omega |\to \infty$, the two roots in $X={\omega }^2$ of the stratified magneto-Coriolis wave polynomial have $X_f\sim 4{\Omega }^2{k}_z^2/K^2$ and the displayed slow root. This follows from the sum of roots being $4{\Omega }^2{k}_z^2/K^2+O(1)$ and their product being $a^2(K^2{a}^2-H{h}^2)/K^2$. The slow branch is oscillatory for $K^2{a}^2>H{h}^2$ and growing or decaying for the reversed inequality. At equality its frequency vanishes and the original stationary equations must be used.