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For $P=K{\rho }^{1+1/n}$, dimensionless pseudo-enthalpy $w=n(\rho /{\rho }_0{)}^{1/n}$, and $\zeta =\Omega z/c_0$, vertical hydrostatic balance and the Poisson equation reduce toThe midplane conditions are $w(0)=n$ and $w^{\prime }(0)=0$, and the first zero gives the disk surface.

Potential-vorticity inversion recovers the balanced quasi-geostrophic streamfunction from interior potential vorticity and suitable boundary buoyancy or pressure data by solving an elliptic boundary-value problem. Boundary anomalies can drive nonzero flow even when the interior potential-vorticity anomaly is zero. Boundary conditions and any pressure gauge or compatibility condition are essential to uniqueness.

Moving a uniform-density boundary produces a delta-function sheet in the Eulerian mass density perturbation. For a slab displaced vertically by ${\eta }_{\pm }$, ${\rho }^{\prime }={\rho }_0{\eta }_{+}\delta (z-H)-{\rho }_0{\eta }_{-}\delta (z+H)$. Poisson equation then gives a continuous gravitational potential with derivative jumps fixed by those sheets.

For a uniform-density sphere with incompressible flow of radius $R$ with one surface spherical harmonic displacement ${\xi }_r(R)=lA{R}^{l-1}{Y}_l^m$, the bulk density perturbation vanishes but the surface contributes $\delta \rho =\rho {\xi }_r(R)\delta (r-R)$. Continuity of the Newtonian gravitational potential and the Poisson equation give $[{\partial }_r\delta \Phi {]}_{\mathrm{out}-\mathrm{in}}=4\pi G\rho {\xi }_r(R)$. The regular interior and decaying exterior solutions areThe coefficient follows because the derivative jump is $-(2l+1)C{R}^{l-1}{Y}_l^m$. This surface source is why neglecting the bulk density perturbation does not justify neglecting the perturbed gravity.