In the context of hydrodynamic stability of geophysical flows, sufficient stability conditions are obtained for the steady solutions of the quasi geostrophic equation in the equivalent barotropic form. Their application to planetary vortices, to shear flows and to other particular cases is discussed. Our stability conditions are obtained in the framework of the nonlinear stability theory and generalize a classical theorem of Arnold on Eulerian flows. One of these conditions can be regarded as the analog for the equivalent barotropic model of a result by Blumen, valid for a continuously stratified fluid; the other conditions represent new analytical results. Stability is ensured in the nonlinear (i.e. in the rigorous) sense and with respect to small but finite perturbations of any shape.