Schwarzschild criterion for convection

Remarks

Raise a parcel by a small height, where the pressure is lower. Its temperature changes along the adiabat, while the temperature of the surroundings changes along the actual gradient. At equal pressure, the parcel is lighter than its surroundings exactly when it is hotter. This happens when the surroundings cool faster with height than the adiabat, that is, when $\nabla > \nabla_{\mathrm{ad}}$; the parcel then keeps rising. Example: for a fully ionized monatomic ideal gas, $\nabla_{\mathrm{ad}} = 2/5$, so a radiative layer whose radiative gradient exceeds $0.4$ becomes convective. Equivalently, in terms of the Brunt–Väisälä frequency $N$, the layer is unstable when $N^2 < 0$. With a gradient of composition the criterion must be replaced by the Ledoux criterion.

Going deeper

For a student

In stellar models one computes the gradient $\nabla_{\mathrm{rad}}$ that radiation alone would need to carry the luminosity, and compares it with $\nabla_{\mathrm{ad}}$. Where $\nabla_{\mathrm{rad}} > \nabla_{\mathrm{ad}}$, the layer is convective, and the temperature gradient is close to adiabatic. This decides where the convective cores of massive stars and the convective envelopes of cool stars lie. Inside a convection zone the heat flow is usually modelled by mixing length theory.