Weakly nonlinear internal gravity wavepackets

Horizontally periodic, vertically localized internal wavepackets evolve nonlinearly due only to interactions between the waves and their wave-induced mean flow. The corresponding weakly nonlinear equation that describes the evolution of the amplitude envelope before the onset of parametric subharmonic instability is examined. The results are compared with fully nonlinear numerical simulations and are shown to lie in excellent agreement for over 15 buoyancy periods. Analysis of the equation shows that the evolution is modulationally unstable if the wave frequency exceeds that of waves with the fastest vertical group speed and if the amplitude is sufficiently large. Waves that move close to the fastest vertical group speed are unstable even if their relative amplitude is a tiny fraction of the inverse relative vertical extent of the wavepacket. At late times in the evolution of an unstable wavepacket third-order dispersion terms become non-negligible and act in conjunction with weakly nonlinear effects to retard the vertical advance of the wavepacket as a whole.