Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S25 / 1 / 1

The tropopause is the transition between the lower, often convecting troposphere and the more stably stratified atmosphere above, where the decrease of temperature with altitude weakens or reverses. Its definition concerns the thermal profile, not a fixed universal altitude. An optical depth is a dimensionless path-integrated extinction, ${\tau }_{\nu }=\int {\kappa }_{\nu }\rho ds$ for mass extinction coefficient ${\kappa }_{\nu }$; without emission into the ray its direct intensity is attenuated by $e^{-{\tau }_{\nu }}$.

In hydrostatic equilibrium, the upward pressure force balances weight:The second equality uses the ideal gas law with specific gas constant $R_a$. For constant $T$ and $g$,For variable temperature the corresponding result is $p(z)=p(z_0)\exp [-{\int }_{z_0}^{z}g/(R_{a}T(z^{\prime }))d{z}^{\prime }]$. The fixed-temperature derivative does not require the entire column to be isothermal.