= Pressure Hessian of an elliptical shearing-sheet vortex
{title2=$A=\alpha\beta+2\Omega(S-\beta),\quad B=\alpha(\beta-2\Omega)$}
For the incompressible velocity $u_x=\alpha y$, $u_y=-\beta x$ in a <shearing sheet>, advection produces $-\alpha\beta(x,y)$ and the <Coriolis acceleration> produces $(2\Omega\beta x,2\Omega\alpha y)$. Including the radial tide $2\Omega Sx$ gives $p/\rho=(Ax^2+By^2)/2+C$. A material ellipse has $\beta=r^2\alpha$. A vortex patch is immersed in surrounding fluid, so this <pressure> need not vanish on its boundary; imposing that condition would describe a different free-boundary problem.
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