 Pressure equation for inertial oscillations of a barotropic star

In the Cowling approximation, a mode with phase $e^{i\omega t+im\varphi }$ has rotating-frame frequency $\sigma =\omega +m\Omega$. The horizontal momentum matrix has determinant $4{\Omega }^2-{\sigma }^2$. Eliminating velocities yields a pressure-potential equation with vertical coefficient $1-4{\Omega }^2/{\sigma }^2$ and density-gradient term $2m\Omega {\rho }_{R}W/(R\sigma )$. Frequencies zero and plus or minus $2\Omega$ need the original system rather than this inversion.