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

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

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

The boundary of a steady Kida vortex must be material. Put

E = x^{2} / b^{2} + y^{2} / a^{2}

. Requiring

u \cdot \nabla E = 0

on the ellipse gives

2 x y (\frac{α}{b ^{2}} - \frac{β}{a ^{2}}) = 0, β = r^{2} α .

The relative vorticity is

\partial_{x} u_{y} - \partial_{y} u_{x} = - β - α = - S + ζ_{0}

, so the supplied vorticity relation implies

α + β = S - ζ_{0} = \frac{S ( r ^{2} + 1 )}{r ( r - 1 )} .

Combining these independent conditions yields

α = \frac{S}{r ( r - 1 )}, β = \frac{S r}{r - 1} .

For the usual positive orbital shear both are positive, the flow circulates clockwise and its streamlines have the same aspect ratio as the patch. The velocity coefficients in this question are unrelated to the alpha viscosity parameter in Question 1.

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