By using a perturbation method, a forced mKdV-Burgers equation is derived from the geostrophic potential vorticity equation including dissipation and topography. An approximate analytic solution of the mKdV-Burgers equation is obtained for the case with a small dissipation. The time evolution of mass and energy of the solitary waves is analyzed, and finally the numerical solutions of the forced mKdV-Burgers equation with a small dissipation are given for a localized topographic forcing by using the pseudo-spectral method. The numerical results show that the presence of small dissipation causes a slow decrease of the amplitude and the propagation speed of the solitary waves and slow increase of the solitary wave width. In the nonlinear system with dissipation and topographic forcing, the dissipation factor forces a moving solitary wave to oscillate in the forcing region during the interaction between the solitary wave and the topographic forcing, and it is advantageous to form large amplitude disturbances.

An inhomogeneous Benjamin-Davis-Ono-Burgers equation including topographic forcing and turbulent dissipation is derived in terms of the quasi-geostrophic vorticity equation, an approximate analytic solution of the BDO-Burgers equation with a small dissipation is obtained, the time variations of mass and energy of the algebraic solitary waves are discussed, and finally, the forced BDO-Burgers equation is integrated numerically and the numerical solutions are given for a given basic flow with a weak shear and a localized topographic forcing.

By using a perturbation method, a forced mKdV equation is derived from the so-called quasi-geostrophic vorticity equation, and time evolutions of mass and energy of the mKdV solitary waves are discussed. Finally, numerical solutions of the forced mKdV equation are obtained by using the pseudo-spectral method. The calculation results show that the features of the mKdV solitary waves excited by localized external source are closely related to the detuning parameter α and the strength of the external source. The external forcing source in a forced mKdV system can excite larger amplitude and more instable disturbances than those in a forced KdV system.

It generalizes the theory developed by Helfrich and Pedlosky^[1] for time-dependent coherent structures in a marginally stable zonal flow by including forcing.Such forcing could be due to topography or to an external source.By using a perturbation method,the nonlinear differential equation is obtained for governing the evolution of a disturbance excited by those forcings.Some general features of the excited disturbance are given by numerically solving the governing equation.It further studies the interaction between solitary wave and topography and reveals that the solitary wave can always climb over the topography,but depending on the initial conditions of solitary wave and the height of topography,the initial solitary wave could keep most of its mass or be fissioned into two solitary waves traveling in opposite directions.