= Solution
For $\mathbf u=\alpha y\mathbf e_x-\beta x\mathbf e_y$, the velocity divergence is zero and $\mathbf u\cdot\nabla\mathbf u=-\alpha\beta(x\mathbf e_x+y\mathbf e_y)$. Substituting into the steady <Euler equations> yields
$$
Q_x=(\alpha\beta+3\Omega^2-2\Omega\beta)x,\qquad Q_y=\alpha(\beta-2\Omega)y.
$$
Integration gives
$$
Q=Q_0+\frac12(\alpha\beta+3\Omega^2-2\Omega\beta)x^2+\frac12\alpha(\beta-2\Omega)y^2.
$$
The steady enthalpy equation also requires $\mathbf u\cdot\nabla Q=0$. Its $xy$ coefficient is $\alpha(3\Omega^2+\alpha\beta-\beta^2)$, so, since $\alpha>0$,
$$
\boxed{\alpha=\frac{\beta^2-3\Omega^2}{\beta},\qquad Q=Q_0-\frac12(2\Omega-\beta)(\beta x^2+\alpha y^2).}
$$
For a bounded patch with positive <mass density> and a vacuum <fluid free boundary>, choose $Q_0>0$ and require both quadratic coefficients to confine the fluid. With $\Omega>0$ and $\alpha,\beta>0$, this means
$$
\boxed{\sqrt3\,\Omega<\beta<2\Omega.}
$$
The lower bound is imposed by $\alpha>0$; the upper bound by bounded positive $Q$. At $\beta=2\Omega$, $Q$ is constant and has no nontrivial finite vacuum boundary; $\beta\le\sqrt3\Omega$ violates or degenerates the stipulated elliptical flow.
For a <polytropic elliptical patch in a shearing sheet>, the physical <fluid free boundary> is $p=\rho=Q=0$. It is the <ellipse>
$$
\boxed{\beta x^2+\alpha y^2=\frac{2Q_0}{2\Omega-\beta},\qquad \frac{x^2}{a_x^2}+\frac{y^2}{a_y^2}=1,}
$$
where $a_x^2=2Q_0/[\beta(2\Omega-\beta)]$ and $a_y^2=2Q_0/[\alpha(2\Omega-\beta)]$. In the interior $\rho=[Q/((n+1)K)]^n$. The boundary is a material <streamline> because $\mathbf u\cdot\nabla(\beta x^2+\alpha y^2)=0$. Since $\alpha<\beta$, it is elongated in the azimuthal $y$ direction.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-321-elliptical-patch.png]
{title=The polytropic shearing-sheet patch has an elliptical vacuum boundary with tangential anticyclonic flow and positive interior enthalpy}
This is a pressure-supported fluid patch in the supplied tidal model; no additional planetary self-gravity potential was assumed.
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