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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.

Write the vortex aspect ratio as $\chi$ to distinguish it from cylindrical radius. The Kida vortex core flow isAs $\chi \to \infty$, $u_x\to 0$ and $u_y\to -(3/2)\Omega x$, recovering Keplerian shear.

For a fluid particle,It therefore circulates around an ellipse with angular frequency $3\Omega /[2(\chi -1)]$ and period

For a perturbation depending only on $z,t$, horizontal pressure gradients vanish. Linearization givesTaking ${\mathbf{u}}^{\prime }\propto e^{-i\omega t}$ yieldsThe first bracket is positive for $\chi >3/2$, while the second is negative for $1<\chi <4$. Their product is therefore negative, so $\omega$ is imaginary and the mode grows precisely when