New exact spatially localized solutions of the (3 + 1)-dimensional Charney–Obukhov equation for the ocean

New exact spatially localized solutions on the background of a zonal flow, propagating along the zonal direction at a constant velocity, are found for the (3 + 1)-dimensional nonlinear Charney–Obukhov equation describing Rossby waves and vortices in ocean. In total, five solutions are presented—two solutions with spherical symmetry and three solutions with cylindrical symmetry. One of the solutions with spherical symmetry is constructed using the Darboux transformation. Visualization of the solutions found shows that, depending on the values of the parameters included in the solutions, they can describe both an irrotational flow and a vortex flow with 1, 2, or more localized vortices. To analytically estimate the vertical localization of vortices, the necessary condition for the instability of the zonal flow and the condition for the maximum total vorticity are used. These estimates are in good agreement with the results of visualization of the vortex flow for the solutions found.