w^{'} (0) = η_{t} .

Hydrostatic pressure and geostrophic-streamfunction normalization give

p^{'} (0) = ρ_{0} f_{0} ψ^{'} (0) = ρ_{0} g η, η = \frac{f _{0}}{g} ψ^{'} (0) .

Combining this with

f_{0} ψ_{z t}^{'} + N^{2} w^{'} = 0

gives

\frac{\partial}{\partial t} (ψ_{z}^{'} + \frac{N ^{2}}{g} ψ^{'}) = 0 (z = 0) .

For an oscillatory disturbance with no time-independent boundary offset, the free-surface boundary condition is therefore

ψ_{z}^{'} + \frac{N ^{2}}{g} ψ^{'} = 0 (z = 0) .

The lower rigid boundary retains

ψ_{z}^{'} (- H) = 0

ii

Solution

For

ψ^{'} = A (x, y, t) cos \frac{nπ z}{H},

the surface value is simply

ψ^{'} (0) = A

. The dynamic free-surface condition from part i gives

η (x, y, t) = \frac{f _{0}}{g} A (x, y, t) .

ξ = - \frac{f _{0}}{N ^{2}} ψ_{z}^{'} .

For the

n

th cosine mode,

ξ = \frac{f _{0} A}{N ^{2}} \frac{nπ}{H} sin \frac{nπ z}{H} .

∣ ξ ∣_{m a x} = \frac{f _{0} ∣ A ∣}{N ^{2}} \frac{nπ}{H} = \frac{g nπ}{N ^{2} H} ∣ η ∣_{m a x} .

For

H = 3000 m

N = 1 0^{- 3} s^{- 1}

, and

∣ η ∣_{m a x} = 1 0^{- 2} m

∣ ξ ∣_{m a x} ≃ \frac{9.81 nπ}{( 1 0 ^{- 3} ) ^{2} ( 3000 )} (1 0^{- 2} m) ≃ 103 n m .

The first baroclinic mode therefore displaces interior density surfaces by about

1 0^{2} m

even though the surface moves only one centimetre.

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