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The k−3 and k−5/3 Energy Spectrum of Atmospheric Turbulence: Quasigeostrophic Two-Level Model Simulation

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  • 1 Department of Applied Mathematics, University of Washington, Seattle, Washington
  • | 2 Colorado Research Associates Division, Northwest Research Associates, Inc., Boulder, Colorado
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Abstract

The Nastrom–Gage energy spectrum of atmospheric turbulence as a function of wavelength is simulated here with a two-level quasigeostrophic (QG) model. This simple model has no topography, no direct wave forcing, and no small-scale forcing, nor any kind of gravity wave generation. The two-level model does, however, allow for the simple mechanism of baroclinic energy injection at the large (synoptic) scales as the model atmosphere relaxes to a specified north–south “radiative equilibrium” temperature gradient. It also has a small sink of energy at the small scales due to subgrid hyperdiffusion; this attempts to model the small-scale sink not resolved by the two-level QG model, in particular, enhanced viscous dissipation in atmospheric fronts. The magnitude and shape of the observed energy spectrum, with its characteristic k−3 power-law behavior in the synoptic and subsynoptic scales (from several thousand to about eight hundred kilometers) and the characteristic k−5/3 behavior in the mesoscales (less than about six hundred kilometers), are reproduced convincingly in the model.

The picture that emerges for the energy spectrum of atmospheric turbulence from a few kilometers to tens of thousands of kilometers is actually quite simple. The potential energy of the mean flow, which is derived from solar heating with no scale dependence, is transferred selectively to the long synoptic scales of motion via the mechanism of (nonlinear) baroclinic instability. The injected energy moves both upscale, to the planetary waves where it is damped by Ekman damping, and also downscale, through the short synoptic waves, through the mesoscales, to the short mesoscales, where it can be damped by viscous dissipation. There is no need for dynamics other than QG to produce the spectrum. (However, the present work cannot be used to rule out other explanations, such as gravity wave generation, or a separate energy source at the small scales.)

Nee Wendell Tyler Welch

Corresponding author address: K. K. Tung, Department of Applied Mathematics, University of Washington, P.O. Box 352420, Seattle, WA 98195-2420. Email: tung@amath.washington.edu

Abstract

The Nastrom–Gage energy spectrum of atmospheric turbulence as a function of wavelength is simulated here with a two-level quasigeostrophic (QG) model. This simple model has no topography, no direct wave forcing, and no small-scale forcing, nor any kind of gravity wave generation. The two-level model does, however, allow for the simple mechanism of baroclinic energy injection at the large (synoptic) scales as the model atmosphere relaxes to a specified north–south “radiative equilibrium” temperature gradient. It also has a small sink of energy at the small scales due to subgrid hyperdiffusion; this attempts to model the small-scale sink not resolved by the two-level QG model, in particular, enhanced viscous dissipation in atmospheric fronts. The magnitude and shape of the observed energy spectrum, with its characteristic k−3 power-law behavior in the synoptic and subsynoptic scales (from several thousand to about eight hundred kilometers) and the characteristic k−5/3 behavior in the mesoscales (less than about six hundred kilometers), are reproduced convincingly in the model.

The picture that emerges for the energy spectrum of atmospheric turbulence from a few kilometers to tens of thousands of kilometers is actually quite simple. The potential energy of the mean flow, which is derived from solar heating with no scale dependence, is transferred selectively to the long synoptic scales of motion via the mechanism of (nonlinear) baroclinic instability. The injected energy moves both upscale, to the planetary waves where it is damped by Ekman damping, and also downscale, through the short synoptic waves, through the mesoscales, to the short mesoscales, where it can be damped by viscous dissipation. There is no need for dynamics other than QG to produce the spectrum. (However, the present work cannot be used to rule out other explanations, such as gravity wave generation, or a separate energy source at the small scales.)

Nee Wendell Tyler Welch

Corresponding author address: K. K. Tung, Department of Applied Mathematics, University of Washington, P.O. Box 352420, Seattle, WA 98195-2420. Email: tung@amath.washington.edu

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