Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 331 / 1 / b / Solution

Write $\mathbf{u}={\mathbf{u}}^{\prime }$ and $\theta =-z+{\theta }^{\prime }$. Dropping quadratic perturbation terms gives the Linearized Boussinesq equationsandThe fixed temperatures give ${\theta }^{\prime }=0$ at $z=0,1$. Impermeable stress-free boundary conditions give