Investigation and numerical resolution of initial-boundary value problem for the generalized Charney-Obukhov and Hasagewa-Mima equations

In the present work, first-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 a two-dimensional nonlinear solitary vortical structures is studied. For the considered equations the initial-boundary value problem is set when at the initial moment the solution in the form of solitary dipole structure is taken. For this problem uniqueness of the solution in case of periodic boundary conditions is proved. The dynamic relation between solutions of the generalized Charney-Obukhov and Hasegawa-Mima equations is established. It is shown that, in spite of the existing opinion, the scalar nonlinearity in case of the generalized Hasegawa-Mima equation develops monopolar anticyclone, while in case of the generalized Charney-Obukhov equation develops monopolar cyclone.