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

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- Solution v

Solution

Expand both the forcing and response in the orthogonal vertical normal modes. Projection onto

P_{n}

gives

\partial_{t} (\nabla_{h}^{2} ϕ_{n} - \frac{f _{0}^{2}}{c _{n}^{2}} ϕ_{n}) + β \partial_{x} ϕ_{n} = F_{n} (x, y) H (t) .

If the horizontal forcing scale obeys

L_{h} ≫ c_{n} / f_{0}

, the relative vorticity term

\nabla_{h}^{2} ϕ_{n}

is small compared with the stretching term. The resulting long-wave equation is

\partial_{t} ϕ_{n} - C_{n} \partial_{x} ϕ_{n} = - \frac{c _{n}^{2}}{f _{0}^{2}} F_{n} (x, y) H (t), C_{n} = \frac{β c _{n}^{2}}{f _{0}^{2}} .

Thus each mode communicates the forcing westward at its long Rossby wave speed

C_{n}

, with the barotropic mode fastest because

c_{0}

is largest.

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