Spherical potential-vorticity anomaly ($\psi =Q_0(r^2-3R^2)/6(r<R),\psi =-Q_0{R}^3/(3r)(r>R)$)

A uniform positive quasi-geostrophic potential vorticity anomaly in a sphere of stretched radius $R$ has the displayed regular, decaying streamfunction. Both it and its radial derivative match at the sphere. In its equatorial plane the horizontal relative vorticity is $2Q_0/3$ inside and $-Q_0{R}^3/(3r^3)$ outside. The exterior potential-vorticity anomaly remains zero because its vertical stretching contribution compensates that relative vorticity. A stretched sphere corresponds to a flattened ellipsoid in physical coordinates when $N\gg |f|$.