Finite Amplitude Gravity Waves: Harmonics, Advective Steepening and Saturation

Abstract

Steepening, harmonics generation, and saturation of a finite amplitude gravity wave are calculated in an approximate fashion by means of a new and elementary approach. The mechanism for these phenomena is nonlinear advection. For simplicity, attention is confined to a single wave or wave packet in a quasi-stationary background. A principal result of the calculation is that harmonics are generated which cause the wave velocity fluctuation to steepen; harmonics of the density fluctuation are neglected. The wave velocity steepens more and more with increasing amplitude until, at a particular amplitude, it breaks and saturates. Energy flows from the primary wave into harmonies down to turbulence. This steepening and saturation resembles the shoaling of an ocean wave. In both cases, steepening of the velocity fluctuation is caused by the advective, v? · ?v? term in the Navier-Stokes equation. Of special interest is that the lapse rate becomes marginally stable at about the same time that the wave velocity begins to break. This conforms to observations that saturation of velocity growth is accompanied by a near adiabatic lapse rate and turbulence. However, here, the adiabatic lapse rate is not utilized to saturate the wave; i.e. unstable lapse rate and saturation are more nearly concurrent than sequential. Turbulence is necessarily produced?by steepened velocity gradient (shear) as well as by unstable lapse rate?but eddy diffusion is not invoked for saturation. The goal is to show that wave-wave interactions (energy cascade) have the possibility of causing saturation at the wave amplitudes observed. Generally speaking, saturation by turbulence is viewed as competitive with wave-wave interactions, with their relative importance depending on wave frequency. In all cases, the superadiabatic lapse rate is a ?signature? of saturation. These conclusions pertain to the idealized case considered and can be altered by the presence of mean shears or more than one primary wave. The most serious approximation is neglect of density fluctuation harmonics.