 Solution

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