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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-61/2/ii/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 61 / 2 / ii / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

The steady incompressible shearing sheet equation is

u \cdot \nabla u + 2Ω e_{z} \times u = - ρ^{- 1} \nabla p + 2Ω S x e_{x} .

The final term follows from the local tidal potential

- Ω S x^{2}

. For the given linear velocity, the advective acceleration is

- α β (x e_{x} + y e_{y})

, while the Coriolis acceleration is

2Ω β x e_{x} + 2Ω α y e_{y}

. Equating components gives

\frac{p _{x}}{ρ} = [α β + 2Ω (S - β)] x, \frac{p _{y}}{ρ} = α (β - 2Ω) y .

The cross derivatives vanish, so the pressure Hessian of an elliptical shearing-sheet vortex integrates to

\frac{p}{ρ} = \frac{1}{2} A x^{2} + \frac{1}{2} B y^{2} + constant, A = \frac{S ^{2}}{( r - 1 ) ^{2}} - \frac{2Ω S}{r - 1}, B = \frac{S ^{2}}{( r - 1 ) ^{2}} - \frac{2Ω S}{r ( r - 1 )} .

The additive constant is fixed by exterior matching or the pressure reference. No zero-pressure free-boundary condition is imposed on an incompressible vortex patch: it is embedded in a surrounding fluid.

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