Finite Amplitude Gravity Waves: Harmonics, Advective Steepening and Saturation

Abstract

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 velo...