 Axisymmetric adiabatic displacement operator

In the Cowling approximation, an axisymmetric meridional fluid displacement in a cylindrically rotating barotropic star obeys $-{\omega }^2\xi =\mathcal{F}\xi$, with $\delta \rho =-\nabla \cdot (\rho \xi )$ and $\delta p=-\xi \cdot \nabla p-\gamma p\nabla \cdot \xi$. Conservation of specific angular momentum supplies the epicyclic term, with ${\kappa }^2=R^{-3}d(R^4{\Omega }^2)/dR$. The Cowling energy principle for a rotating barotropic star gives its symmetric quadratic form.