Pressure Hessian of an elliptical shearing-sheet vortex ($A=\alpha \beta +2\Omega (S-\beta ),B=\alpha (\beta -2\Omega )$)

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 $p/\rho =(A{x}^2+B{y}^2)/2+C$. 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.