Polytropic elliptical patch in a shearing sheet

The steady planar velocity $\mathbf{u}=\alpha y{\mathbf{e}}_x-\beta x{\mathbf{e}}_y$ supports a bounded polytropic patch when $\alpha =({\beta }^2-3{\Omega }^2)/\beta$ and $\sqrt{3}\Omega <\beta <2\Omega$. Its polytropic enthalpy is $Q=Q_0-(2\Omega -\beta )(\beta x^2+\alpha y^2)/2$ with $Q_0>0$. Its fluid free boundary is the material ellipse $\beta x^2+\alpha y^2=2Q_0/(2\Omega -\beta )$. No additional self-gravity potential is part of this model.