Stratified magneto-Coriolis wave ($K^2({\omega }^2-a^2{)}^2-4{\Omega }^2{k}_z^2{\omega }^2+H{h}^2({\omega }^2-a^2)=0$)

An incompressible Boussinesq fluid with uniform magnetic field, uniform vertical rotation and uniform temperature gradient supports waves combining magnetic tension, Coriolis force and buoyancy. Put $K=|\mathbf{k}|$, $h^2=K^2-k_z^2$, $a^2=(\mathbf{B}\cdot \mathbf{k}{)}^2/({\mu }_0\rho )$ and $H=g\alpha \beta$, with ${\partial }_z{T}_0=-\beta$. Eliminating pressure and the temperature and magnetic perturbations gives the displayed dispersion relation for nonzero frequencies. Stable thermal stratification has $H<0$. Clearing frequency denominators requires checking any resulting zero roots in the original equations; a stationary magnetostatic-buoyancy balance can exist independently of the propagating branches.

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.