Weakly nonlinear internal gravity wavepackets

Abstract

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.

Dispersive corrections to a modulation theory for stratified gravity waves

“…The amplitude equation then reduces to the form (5.1). This re-affirms the quasi-monochromatic analysis for gravity waves of Sutherland [33] and Tabaei and Akylas [34]. It is noteworthy that the sign of the dispersion term in (5.9) is controlled by C 0 -this is precisely the distinguishing ingredient in the leading versus trailing oscillations seen in run1 and run2 (Figs.…”

“…We also simplify by assuming a uniform mean flow, and by changing to a co-moving frame, we can take u ¼ 0. We remark that the above treatment of the mean flow is not permissible in the nonlinear case because of mean flow coupling [33,34]. The resulting constant coefficient problem (2.1) now permits an exact gravity wave dispersion relation…”

Beyond ray tracing for internal waves. II. Finite-amplitude effects

“…This was confirmed by the derivation and analysis of a nonlinear Schrödinger equation governing the evolution of horizontally periodic internal wavepackets. 11,12 The equation, which filtered the wave-wave interactions associated with parametric subharmonic instability, captured well the simulated fully nonlinear evolution of the waves for over 15 buoyancy periods. 10,12 If of sufficiently large amplitude, the wave-induced mean flow significantly Doppler shifted the waves, changing their structure and consequently changing the response of the wave-induced mean flow.…”

“…Even if condition ͑7͒ is not met, weakly nonlinear interactions between the waves and the wave-induced mean flow may nonetheless dominate the dynamics governing the evolution of a vertically compact wavepacket. 12 Subharmonic parametric instability, in which the primary waves interact nonlinearly with subharmonically excited waves, dominates the initial evolution of the waves only if they are both vertically and horizontally periodic. 20 One consequence of the weakly nonlinear dynamics is that the amplitude envelope of vertically compact wavepackets grows faster than predicted by linear theory and the vertical group velocity slows if ͉⌰͉ Շ 35.3°͉͑m͉ Շ 0.71͉k͉͒ provided the amplitude is sufficiently large.…”

2013 program of study : buoyancy-driven flows

“…As for surface gravity waves (see for example Dysthe (1979)), the interaction with the horizontal induced mean flow of horizontally periodic but vertically compact wave groups is wellcaptured by non-linear Schrödinger type evolution equations subject to an instability of the Benjamin-Feir type (Benjamin & Feir (1967)). For Boussinesq, non-Boussinesq and anelastic horizontally periodic wave packets Sutherland (2006) and Dosser & Sutherland (2011a, 2011b have derived such equations and compared their predictive ability to fully non-linear simulations.…”