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Let $Q_0>0$ denote the small excess PV in the spherical region, with $Q_0\ll |f|$ for quasi-geostrophic consistency. Let its stretched radius be $R$ and set $r=(X^2+Y^2+Z^2{)}^{1/2}$. Spherical symmetry of the source and uniqueness of the decaying inversion imply a radial streamfunction. Its equations areRegularity at the origin removes the interior $1/r$ term. Decay removes the exterior constant. Integrating gives ${\psi }_{min}=Q_0{r}^2/6+C_0$ and ${\psi }_{mout}=C_1/r$. Since the source has no surface delta function, $\psi$ and ${\psi }_r$ match at $R$. Their matching fixes $C_1=-Q_0{R}^3/3$ and $C_0=-Q_0{R}^2/2$. Therefore the spherical potential-vorticity anomaly hasIts induced horizontal flow isThis flow is azimuthal about the vertical axis, so it is tangent to the PV level surfaces and advects no part of this axisymmetric distribution across its spherical boundary. The example is therefore steady under the quasi-geostrophic PV evolution. In physical coordinates the stretched sphere is the ellipsoid $x^2+y^2+(N^2/f^2)z^2=R^2$, with vertical semiaxis $R|f|/N$.

Relative vertical vorticity is the horizontal Laplacian of the streamfunction, not its full stretched three-dimensional Laplacian. Inside the spherical potential-vorticity anomaly, the quadratic solution givesOutside, in the equatorial plane $Z=0$, let $s=(x^2+y^2{)}^{1/2}>R$ and $C=Q_0{R}^3/3$. The azimuthal velocity is $v_{\theta }={\psi }_s=C/s^2$, soThe opposite signs thus follow directly from solid-body rotation inside and the exterior velocity's rapid radial decay. The azimuthal velocity itself has the same rotational sense on both sides; its radial derivative makes the exterior relative vorticity negative.

There is no contradiction with the zero exterior PV anomaly. At $Z=0$ the exterior solution has ${\psi }_{ZZ}=C/s^3$, so $q_g+{\psi }_{ZZ}=0$. The stratification/stretching contribution compensates the horizontal relative vorticity. Inside it contributes $Q_0/3$, supplementing $2Q_0/3$ to give the prescribed excess $Q_0$.