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

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- Solution iv

Solution

Suppose

ϵ = N^{2} H / g ≪ 1

. The smallest root satisfies

μ_{0}^{2} \sim ϵ, c_{0} = \frac{N H}{μ _{0}} \sim g H,

so the leading external gravity wave eigenfunction is depth-independent:

P_{0} (z) \sim 1.

For

n \geq 1

, the roots lie just above

nπ

:

μ_{n} = nπ + \frac{ϵ}{nπ} + O (ϵ^{2}),

and hence

c_{n} = \frac{N H}{nπ} [1 - \frac{ϵ}{n ^{2} π ^{2}} + O (ϵ^{2})] .

In particular, up to an arbitrary normalization and sign,

P_{1} (z) \sim cos [\frac{π ( z + H )}{H}] .

The hierarchy

c_{0} > c_{1} > c_{2} > \dots

separates the fast, nearly barotropic free-surface mode from the internal baroclinic modes.

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