Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 3 / a

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- Solution a

Solution

Let

W = 2Ω + (\nabla \times u) \cdot e_{z}

be the vertical absolute vorticity. Taking the vertical curl of the momentum equation removes both the tidal potential and the isothermal pressure force, because each is a gradient field. The two-dimensional vorticity equation is

D_{t} W = - W \nabla \cdot u .

The surface-density equation is

D_{t} Σ = - Σ\nabla \cdot u .

The quotient rule now gives

D_{t} (\frac{W}{Σ}) = \frac{D _{t} W}{Σ} - \frac{W}{Σ ^{2}} D_{t} Σ = 0.

Since

ζ = W /Σ

is the vortensity,

D_{t} ζ = 0 .

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