Vortices and internal waves in a stratified fluid

Equations describing the evolution of potential vortices and internal waves in a stably stratified fluid which fills a half-space are derived in Euler-Lagrange variables. Asymptotic series in a small parameter are constructed which give approximate solutions of the non-linear problem. The equations of the linear approximation for a potential vortex and internal waves are independent, while the equations of the higher-order approximations describe the interaction between the potential vortices and the internal waves. It is shown that distributed sources (which can be interpreted as intense atmospheric rainfall) cause an exponential increase in the potential vorticity, which in turn may lead to a considerable increase in the amplitudes of the internal waves. Asymptotic forms for the field of the internal waves in different space-time regions are constructed for the case of an exponential stratification.