Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 3 / v / Solution

Expand both the forcing and response in the orthogonal vertical normal modes. Projection onto $P_n$ givesIf the horizontal forcing scale obeys $L_h\gg c_n/f_0$, the relative vorticity term ${\nabla }_h^2{\varphi }_n$ is small compared with the stretching term. The resulting long-wave equation isThus each mode communicates the forcing westward at its long Rossby wave speed $C_n$, with the barotropic mode fastest because $c_0$ is largest.