 Solution

Displace a small parcel upward, with negligible heat exchange, rapid pressure balance, and the same composition as its surroundings. Define the environmental gradient $\nabla =d\log T/d\log P$. An adiabatic ideal gas parcel obeys $P{\rho }^{-\gamma }=\mathrm{constant}$ and hence $T\propto P^{(\gamma -1)/\gamma }$, giving ${\nabla }_{\mathrm{ad}}=(\gamma -1)/\gamma$.

The upward displacement has $\delta \log P<0$. If $\nabla <{\nabla }_{\mathrm{ad}}$, the parcel's temperature changes more negatively than the environment's. At their common pressure it is cooler, denser and driven back down. Therefore the Schwarzschild criterion for stability isThe reversed inequality produces buoyant growth, and equality is neutrally stable in this approximation. A composition gradient requires the Ledoux criterion; pressure equilibrium alone does not ensure equal mean molecular weight.