The non-linear interaction of a gravity wave with the vortical modes

The wave-wave interaction theory has been used successfully in describing one class of weakly non-linear wave phenomena. The application of this theory to the atmosphere shows the possibilities of energy and momentum transfer among three interacting gravity waves, as well as from the gravity wave to the other modes of motion. It has been found that the non-resonant interaction of a gravity wave with two vortical modes can proceed at a reasonably rapid rate. With the gravity wave viewed as the primary wave and the two vortical modes as the secondary waves, the interaction equation can be linearized and solved. The resulting analytic formula gives the growth rate of the interaction. In the absence of the Earth's rotation, the growth is limited to a threshold effect. The theory shows that whenever the horizontal air parcel velocity of a gravity wave exceeds a factor of √2 times the horizontal trace velocity of the wave, energy and momentum transfer from the gravity wave to the vortical modes can proceed. The rotation of the Earth will blur this threshold effect by making the interaction more likely to occur. Thus, through this mechanism, a gravity wave can transfer its energy and momentum to the horizontal velocity field in the vortical mode. In this sense, the small scale vortical motions would serve as the sink of both energy and momentum of a propagating gravity wave. When scales of vortical modes reach sufficiently small values, dissipation through viscosity becomes important. At this scale and smaller, the vortical modes are damped out quickly and its energy spectrum must exhibit a sharp decay.