The concept of piecewise constant symmetric vortex in the context of three-dimensional baroclinic balanced geophysical flows is explored. The pressure gradients generated by horizontal cylinders and spherical balls of uniform potential vorticity (PV), or uniform material invariants, are obtained either analytically or numerically, in the general case of Boussinesq and f-plane dynamics as well as under the quasi-geostrophic and semigeostrophic dynamical approximations. Based on the order of magnitude of the different terms in the PV inversion equation, approximated PV equations are deduced. In some of these cases, radial solutions are possible and the interior and exterior solutions are found analytically. In the case of non-radial dependence, exterior solutions can be found numerically. Linear, and upper and lower bound approximations to the full PV inversion equations, and their respective solutions, are also included. However, the general solution for the pressure gradient in the vortex exterior does not have spherical symmetry and remains as an important theoretical challenge. It is suggested that, in order to maintain everywhere the inertial and static stability of the balanced geophysical flows, small balls of finite radius, rather than PV singularities, could become, specially in numerical applications, useful mathematical objects.
Exact solutions of asymmetric baroclinic quasi-geostrophic dipoles with distributed potential vorticity
