 Solution

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 $Q_0$.