Effect of Nonlinearity on Atmospheric Gravity Waves

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  • 1 Department of Meteorology, King Abdulaziz University, Jeddah-21413, Saudi Arabia
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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.

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.

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