Effect of Nonlinear Instability on Gravity-Wave Momentum Transport

View More View Less
  • 1 Northwest Research Associates, Inc., Bellevue, WA 98009
© Get Permissions
Full access

Abstract

Internal gravity waves, and the stress divergence and turbulence induced by them, are essential components of the atmospheric and oceanic general circulations. Theoretical studies have not yet reached a consensus as to how gravity waves transport and deposit momentum. The two best-known theories, resonant interaction and Eikonal saturation, yield contradictory answers to this question. In resonant interaction theory, an energetic, high-frequency, low-wavenumber wave is unstable to two waves of approximately half the frequency and is backscattered by a low-frequency wave or mean finestructure of twice the vertical wavenumber. By contrast, the Eikonal saturation model, as it is commonly used, ignores reflection by assuming a slowly varying basic state and does not question the longevity of the primary wave in the presence of local Kelvin–Helmboltz or convective instabilities. The resonant interaction formalism demands that the interactions be weakly nonlinear. The Eikonal saturation model allows strong, “saturated” waves but ignores reflection and eliminates nonlinear instability with respect to other horizontal wavenumbers by invoking the linear or quasi-linear assumption.

To help bridge the gap between the two theories, results from prototype, nonlinear numerical simulations are presented. Attention is directed at the nonlinear instability of gravity waves in a slowly varying basic state. Parametric instability theory yields a group trajectory length scale for the primary wave expressed in terms of the dominant vertical wavelength and degree of convective saturation. This result delimits the range of validity for the Eikonal saturation model: a low-amplitude wave introduced into an undisturbed slowly varying basic state easily traverses many vertical wavelengths; conversely, a convectively neutral wave soon undergoes decay through nonlinear instability provided that some noise is present initially or created in situ by off-resonant interactions.

The numerical results establish the existence of a cascade in wavenumber space, which for hydrostatic waves proceeds toward both higher and lower horizontal wavenumbers, in accord with theory. Substantial reductions in momentum flux are found relative to the linear values.

Abstract

Internal gravity waves, and the stress divergence and turbulence induced by them, are essential components of the atmospheric and oceanic general circulations. Theoretical studies have not yet reached a consensus as to how gravity waves transport and deposit momentum. The two best-known theories, resonant interaction and Eikonal saturation, yield contradictory answers to this question. In resonant interaction theory, an energetic, high-frequency, low-wavenumber wave is unstable to two waves of approximately half the frequency and is backscattered by a low-frequency wave or mean finestructure of twice the vertical wavenumber. By contrast, the Eikonal saturation model, as it is commonly used, ignores reflection by assuming a slowly varying basic state and does not question the longevity of the primary wave in the presence of local Kelvin–Helmboltz or convective instabilities. The resonant interaction formalism demands that the interactions be weakly nonlinear. The Eikonal saturation model allows strong, “saturated” waves but ignores reflection and eliminates nonlinear instability with respect to other horizontal wavenumbers by invoking the linear or quasi-linear assumption.

To help bridge the gap between the two theories, results from prototype, nonlinear numerical simulations are presented. Attention is directed at the nonlinear instability of gravity waves in a slowly varying basic state. Parametric instability theory yields a group trajectory length scale for the primary wave expressed in terms of the dominant vertical wavelength and degree of convective saturation. This result delimits the range of validity for the Eikonal saturation model: a low-amplitude wave introduced into an undisturbed slowly varying basic state easily traverses many vertical wavelengths; conversely, a convectively neutral wave soon undergoes decay through nonlinear instability provided that some noise is present initially or created in situ by off-resonant interactions.

The numerical results establish the existence of a cascade in wavenumber space, which for hydrostatic waves proceeds toward both higher and lower horizontal wavenumbers, in accord with theory. Substantial reductions in momentum flux are found relative to the linear values.

Save