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

The boundary of a steady Kida vortex must be material. Put $E=x^2/b^2+y^2/a^2$. Requiring $\mathbf{u}\cdot \nabla E=0$ on the ellipse givesThe relative vorticity is ${\partial }_x{u}_y-{\partial }_y{u}_x=-\beta -\alpha =-S+{\zeta }_0$, so the supplied vorticity relation impliesCombining these independent conditions yieldsFor 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.