Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 331 / 4 / d / Solution

At first order, ${\nabla }^2{\psi }_1+R_c{\theta }_{1x}=0$ and ${\nabla }^2{\theta }_1-{\psi }_{1x}=0$. Since ${\nabla }^2{\psi }_1=-2{\pi }^2{\psi }_1$ and $R_c=4{\pi }^2$, the first equation fixes ${\theta }_{1x}={\psi }_1/2$. The homogeneous temperature boundary conditions and the second equation remove any horizontally uniform term, givingAt second order,The forcing is horizontally uniform. Taking no additional critical-mode component, a solution satisfying the homogeneous boundaries isThis mean-temperature correction in weakly nonlinear Darcy convection modifies the vertical temperature gradient and provides the feedback that saturates the convection.