The article is aimed at substantiating theoretically amazing longevity (up to 5 years) of the individual vortices in the World Ocean despite strong fluctuations of the ocean currents and regardless of the Rossby wave dispersion properties. Methods and Results. Evolution of the baroclinic vortices is considered in a hybrid two-layer ocean model over a topographic slope on the beta-plane. In the upper layer with strong potential vorticity anomalies, the currents are assumed to be balanced; in the lower layer at week potential vorticity anomalies, the currents are described in the traditional quasi-geostrophic approximation. Slow evolving almost circular vortices embedded in a vertically sheared current typical of the subtropical part of the ocean are described analytically. The theory shows how a baroclinic vortex is followed by the lee Rossby waves. The vortex drift across the mean current is caused mainly by the baroclinicdipole structure of the represented solution; while the vortex energy loss related to the Rossby wave radiation can be compensated by the energy stored in the mean currents. Conclusions. The constructed model provides reasonable estimates of the energy drift and transfer typical of the ocean vortices with strong anomalies of potential vorticity. Direct support of long-lived vortices by the energy of the baroclinic mean flows irrespective of their stability, is of great importance for better understanding the physical mechanisms relating to significant longevity of the geophysical vortices and their movement.