Adiabatic buoyancy frequency of an ideal gas ($N^2=(g/T)(dT/dz+g/c_p)$)

For a gas of fixed composition in hydrostatic equilibrium, with $z$ upward and downward gravitational acceleration of magnitude $g$, an adiabatic process gives the displaced element density change $\delta {\rho }_{\mathrm{element}}=\rho {\gamma }^{-1}(d\ln p/dz)\xi$. Comparing with the ambient change $(d\rho /dz)\xi$ yieldsThe ideal gas law $p=\rho \mathcal{R}T$ and $dp/dz=-\rho g$ give the title expression, where $c_p=\gamma \mathcal{R}/(\gamma -1)$ is the constant-pressure specific heat capacity. A temperature gradient less negative than the adiabatic lapse rate gives positive $N^2$; a steeper decline gives convective instability.