Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 333 / 4 / b / Solution

Put $v=0$ and use amplitudes proportional to $e^{i(kx+mz-\omega t)}$. Zonal momentum giveswhile incompressibility, buoyancy evolution, and hydrostatic balance giveHenceMeridional geostrophic balance requiressoDecay as $|y|\to \infty$, together with $\beta >0$ and $k>0$, requires $\omega >0$. ThereforeThe negative-frequency root makes the Gaussian exponent positive and the solution diverge away from the equator. The acceptable branch is the eastward Equatorial Kelvin wave.