Vortex dynamics of exact spatially localized solutions of the (3 + 1)-dimensional Charney–Obukhov equation

New exact spatially localized stationary solutions on the background of a zonal flow are found for the (3 + 1)-dimensional nonlinear non-dissipative quasi-geostrophic potential vorticity equation (Charney–Obukhov equation), which describes Rossby waves and vortices in an exponentially stratified fluid. The three solutions are presented. They include solutions with spherically and cylindrically symmetric cores, as well as partial superposition of these solutions. The nonlinear boundary conditions with a flat bottom and a rigid lid generate an infinite discrete set of baroclinic modes for each solution. For the analytical assessment of the vertical localization of vortices, the condition of maximum total vorticity is used. The solutions show that for a given value of the velocity of stationary motion V, vortices may not exist, or there may be one, two or more vortices localized at different heights and having the same velocity V. All vortices described by the solutions consist of a continuous sequence of the eddy chains, extended along the zonal direction and having an inclination to the horizon in the meridional plane. A pair of vortices in the baroclinic dipole appears and disappears when the velocity V passes through neutral points. In the dipole, the eddy chains belonging to different vortices typically have different inclinations in the meridional plane. The magnitudes of these inclinations can vary from zero to several degrees. The baroclinic dipole generated by the solution with the spherically symmetric core has a “pocket” capable of carrying warm or cold fluid depending on the polarity of the eddies in the dipole.