First- and second-order accurate implicit difference schemes for the numerical solution of the nonlinear generalized Charney–Obukhov and Hasegawa–Mima equations with scalar nonlinearity are constructed. On the basis of numerical calculations accomplished by means of these schemes, the dynamics of two-dimensional nonlinear solitary vortical structures are studied. The problem of stability for the first-order accurate semi-discrete scheme is investigated. The dynamic relation between solutions of the generalized Charney–Obukhov and Hasegawa–Mima equations is established. It is shown that, contrary to existing opinion, the scalar nonlinearity in the case of the generalized Hasegawa–Mima equation develops monopolar anticyclone, while in case of the generalized Charney–Obukhov equation it develops monopolar cyclone.