Hayashi, Y., and D. G. Golder, 1987: Effects of wave-wave and wave-mean flow interactions on the growth and maintenance of transient planetary waves in the presence of a mean thermal restoring force. Journal of the Atmospheric Sciences, 44(22), 3392-3401.

Abstract: In order to clarify the effects of wave-wave and wave-mean flow interactions on the growth and maintenance of extratropical tropospheric transient waves in the presence of a mean thermal restoring force, numerical experiments are conducted with the use of a dry general circulation model having a zonally uniform ocean surface. After the model has reached its steady state in the absence or presence of eddies, waves are allowed to grow from small disturbances by including all or some of the zonal wavenumber components.

In the presence of all wavenumbers (1-21), ultralong waves (wavenumber 1-3) and cyclone-scale waves (wavenumber 4-9) initially grow as fast as short-scale waves (wavenumber 10-21), whereas ultralong waves do not initially grow as fast in the absence of wave-wave interactions. However, in the mature stage, ultralong waves attain a smaller amplitude in the presence of higher wavenumber components than they do in the absence of these components. This smaller amplitude is due to the fact that the mean baroclinicity is reduced by ultralong waves together with the higher wavenumber components to maintain equilibrium.

It is found that wave-wave interactions energetically play a more important role in the growth of ultralong waves than in their maintenance, being consistent with their nonlinear growth. This implies that the wave-wave energy transfer is sensitive to phase relations and is more efficient in the growing stage. It is also found that the ratio between the kinetic and available potential energies of ultralong waves is increased in the presence of wave-wave interactions. This implies that ultralong waves become more barotropic due to the nonlinear growth of external Rossby waves.