Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 333 / 3 / iv

At the topographic boundary, no normal flow gives

w^{'} = U h_{x}

. Since QG buoyancy is

b^{'} = f_{0} ψ_{z}^{'}

and

U b_{x}^{'} + N^{2} w^{'} = 0

, the linearized lower condition is

f_{0} ψ_{z}^{'} + N^{2} h = 0 (z = 0) .

For the specified ridge

h = h_{0} cos k x

, matching its horizontal structure sets

l = 0

; retaining

K^{2} = k^{2} + l^{2}

also covers a sinusoidal ridge with that horizontal wavevector. When

U > β / K^{2}

γ = \frac{N}{f _{0}} (K^{2} - \frac{β}{U})^{1/2} .

The decaying solution and its boundary amplitude are

ψ (z) = \frac{N h _{0}}{( K ^{2} - β / U ) ^{1/2}} e^{- γ z} .

Increasing

N

raises the streamfunction amplitude in proportion to

N

while reducing the vertical e-folding length

γ^{- 1}

in proportion to

N^{- 1}

: stronger stratification produces a stronger but more shallowly trapped disturbance.

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