Cowling energy principle for a rotating barotropic star (${\omega }^2\int \rho |\xi {|}^{2}dV=\int [|\delta p{|}^2/(\gamma p)+\rho {\mathcal{N}}^2|\xi \cdot \nabla \Psi {|}^2+\rho {\kappa }^2|{\xi }_R{|}^2]dV$)

With vanishing surface pressure/mass density and regular admissible displacements, integration by parts makes the axisymmetric adiabatic displacement operator symmetric in the mass density-weighted inner product. Completing the pressure square gives the displayed energy form. Nonnegative form for every admissible displacement excludes exponentially growing axisymmetric adiabatic modes in the Cowling approximation. A negative trial Rayleigh quotient proves negative spectrum by the Rayleigh-Ritz variational principle. Nonnegative stratification and epicyclic coefficients give a simple sufficient stability condition.

For barotropic equilibrium $dp/d\Psi =-\rho$, this coefficient equals $-{\rho }^{-1}(dp/d\Psi )[{\gamma }^{-1}d\ln p/d\Psi -d\ln \rho /d\Psi ]$. It weights displacement across effective-potential surfaces in the Cowling energy principle for a rotating barotropic star. Its sign distinguishes stabilizing from destabilizing buoyancy there. It is not itself a squared frequency: the physical local buoyancy-frequency square contains an additional factor $|\nabla \Psi {|}^2$.