Uniformly rotating barotropic star ($h+\Phi -{\Omega }^2{R}^2/2=H$)

In a connected inviscid uniformly rotating barotropic fluid, momentum balance makes the displayed sum constant. The pressure term is the barotropic enthalpy function. Perturbations experience Coriolis acceleration, and their rotating-frame frequency depends on the chosen exponential sign convention.

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.

For spherical mass density in the anelastic approximation for a rotating barotropic star, the displayed pressure amplitude gives velocity proportional to $(-iz{R}^{m-1},z{R}^{m-1},i{R}^m)$. This velocity is divergence free and tangent to spherical shells, so its mass-weighted divergence vanishes for every spherical mass density profile. It is an inertial r-mode driven by Coriolis acceleration; with phase $e^{i\omega t+im\varphi }$ its inertial frequency is $\omega =-(m-1)(m+2)\Omega /(m+1)$.

For slow rotation and low-frequency inertial motion, neglect the time derivative of the Eulerian mass density perturbation relative to the mass-weighted velocity divergence. This filters acoustic compressibility while retaining background mass density variation. A spherical background mass density is often sufficient at leading order in rotation.