Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 4 / b

Assume the lower atmosphere is optically thick, hydrogen-helium dominated, convective, and approximately ideal with constant $\gamma =7/5$. The convective temperature gradient is the dry adiabatIf $T_0$ is the temperature at $P_0=0.1\mathrm{bar}$ at the base of the inverted stratosphere, thenIn a radiative lower layer with constant upward flux $F$ and Rosseland opacity ${\kappa }_R$, diffusion would instead giveThe actual profile follows the shallower stable radiative gradient until it exceeds the adiabatic value, where convection begins.