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The interface is the material level set $F=r-R(\theta ,t)=0$. The condition $DF/Dt=0$ giveswhich is the stated equation.

Let the perturbation potentials inside and outside beLinearizing the kinematic condition at $r=a$ givesThe linearized interior Euler equations givewhereas the stationary exterior gives $p_o^{\prime }=-\rho \sigma {\varphi }_o$. Since the basic interior pressure has $d{p}_i/dr=\rho {\omega }^{2}r$, pressure continuity on the displaced boundary isEliminating $A$ and $B$ gives the circular vortex-sheet mode relationsoThe $k=1$ mode is a neutral displacement. Every $k>1$ mode has one exponentially growing branch, so the circular interface rolls up through a Kelvin--Helmholtz instability. Its pattern angular velocity is $\omega (k-1)/(2k)$, in the direction of the gyre but slower than the solid-body motion; the disturbances therefore propagate upstream relative to the rotating water.