= Adiabatic buoyancy frequency of an ideal gas
{title2=$N^2=(g/T)(dT/dz+g/c_p)$}
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 $\delta\rho_{\rm element}=\rho\gamma^{-1}(d\ln p/dz)\xi$. Comparing with the ambient change $(d\rho/dz)\xi$ yields
$$
\ddot\xi=-N^2\xi,\qquad N^2=g\left(\frac1\gamma\frac{d\ln p}{dz}-\frac{d\ln\rho}{dz}\right).
$$
The <ideal gas law> $p=\rho\mathcal RT$ 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>.
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