Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 333 / 4 / f

Solution

Let the wave-forcing layer have vertical scale

D

and meridional scale

L

f_{0}^{2} \frac{χ ^{*}}{D ^{2}}, N^{2} \frac{χ ^{*}}{L ^{2}} .

Their ratio is controlled by

Λ = \frac{N D}{f _{0} L} .

In the shallow-forcing limit

Λ ≪ 1

, the vertical derivative term dominates:

f_{0}^{2} χ_{zz}^{*} ≃ - f_{0} (\nabla \cdot F)_{z} .

After one vertical integration,

f_{0} \overline{v}_{a}^{*} ≃ - \nabla \cdot F .

\overline{u}_{t} = f_{0} \overline{v}_{a}^{*} + \nabla \cdot F

is small at leading order. The response is primarily an overturning circulation with an associated density tendency.

In the deep-forcing limit

Λ ≫ 1

, the meridional term dominates:

N^{2} χ_{yy}^{*} ≃ - f_{0} (\nabla \cdot F)_{z} .

Now

\frac{f _{0} v _{a}^{*}}{\nabla \cdot F} = O (Λ^{- 2}),

so the residual circulation is too weak to balance the forcing. The dominant momentum response is direct zonal acceleration,

\overline{u}_{t} ≃ \nabla \cdot F,

with a comparatively weak overturning and density response.

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