= Solution
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 gives
$$
2xy\left(\frac\alpha{b^2}-\frac\beta{a^2}\right)=0,\qquad\beta=r^2\alpha.
$$
The relative <vorticity> is $\partial_xu_y-\partial_yu_x=-\beta-\alpha=-S+\zeta_0$, so the supplied <vorticity> relation implies
$$
\alpha+\beta=S-\zeta_0=\frac{S(r^2+1)}{r(r-1)}.
$$
Combining these independent conditions yields
$$
\boxed{\alpha=\frac S{r(r-1)},\qquad\beta=\frac{Sr}{r-1}.}
$$
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.
Back to article page
