Effect of Nonlinearity on Atmospheric Gravity Waves

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Abstract
The weakly nonlinear limit of two-dimensional gravity waves in an incompressible, inviscid and stably stratified atmosphere is studied. The three-wave resonant interaction theory indicates an energy cascade from a vertically propagating wave (primary wave) to waves having smaller absolute values of Doppler-shifted frequency (Ω). When the nondimensional parameter |Ω|/N is in the range 1 to ∼0.8 (N being the Brunt-Väisälä frequency), the energy of the primary wave is transferred to bands of small amplitude waves. The triads in these bands include a member with the same vertical group velocity as the primary wave while other triads contain a member with larger vertical group velocity. The band widths approach zero as the primary wave amplitude is reduced. The analysis suggests that the so-called induced diffusion, and parametric subharmonic instability can lead to gravity wave saturation. The latter, when |Ω|/N is near one, indicates effective drag force on the horizontal long waves. This study is only valid for waves having vertical scale smaller than the scale height of the atmosphere.