An Analysis of Wave-Turbulence Interaction

D. Fua Cooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado/NOAA, Boulder 80309 and IFA/CNR, 00044 Frascati, Italy

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G. Chimonas School of Geophysical Sciences, Georgia Institute of Technology, Atlanta 30332

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F. Einaudi School of Geophysical Sciences, Georgia Institute of Technology, Atlanta 30332

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O. Zeman Chesapeake Bay Institute and The Earth and Planetary Science Department, The Johns Hopkins University, Baltimore, MD 21218

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Abstract

We present the results of an analytical and numerical calculation of the interaction between an internal gravity wave and a wave-induced turbulence. The initial atmospheric state, assumed horizontally homogeneous, is statically and dynamically stable with the background Richardson number Ri0 approaching ¼ over some height regions. An initial non-singular neutral gravity wave propagates through such a system and modifies the Richardson number. The new Richardson number Ri may become smaller than ¼ and turbulence may develop. Using a “1½th order” scheme for the turbulence, we calculate the mean and the fluctuating part of the eddy diffusion coefficient. We show that the fluctuating part of the diffusion coefficient, because of its amplitude and phase, may overcome the damping effect of its mean part and force the original wave to grow in time. As the wave grows, it may further lower the Richardson number, increase the intensity of the turbulence, and further strengthen its interaction with it. At least in its initial stages, wave-induced turbulence appears to be an effective mechanism for transfer of energy from the background state into the wave. By showing that the early stages of the wave-induced turbulence interaction can lead to energy being transferred into the wave, we strengthen the case for gravity waves as important elements in the generation of turbulence in the atmosphere. The values we obtain for the eddy diffusion coefficients suggest that the process is quite capable of producing the empirically observed mixing rates at substantial heights above the ground. While the present calculations cannot describe the long-time limit of the wave-turbulence system, one may suggest that the often observed atmospheric conditions in which turbulence and waves appear to co-exist for several hours may result from a sort of equilibrium between the roles of the mean and the fluctuating parts of the eddy diffusion coefficient in taking away from and feeding energy into the wave.

Abstract

We present the results of an analytical and numerical calculation of the interaction between an internal gravity wave and a wave-induced turbulence. The initial atmospheric state, assumed horizontally homogeneous, is statically and dynamically stable with the background Richardson number Ri0 approaching ¼ over some height regions. An initial non-singular neutral gravity wave propagates through such a system and modifies the Richardson number. The new Richardson number Ri may become smaller than ¼ and turbulence may develop. Using a “1½th order” scheme for the turbulence, we calculate the mean and the fluctuating part of the eddy diffusion coefficient. We show that the fluctuating part of the diffusion coefficient, because of its amplitude and phase, may overcome the damping effect of its mean part and force the original wave to grow in time. As the wave grows, it may further lower the Richardson number, increase the intensity of the turbulence, and further strengthen its interaction with it. At least in its initial stages, wave-induced turbulence appears to be an effective mechanism for transfer of energy from the background state into the wave. By showing that the early stages of the wave-induced turbulence interaction can lead to energy being transferred into the wave, we strengthen the case for gravity waves as important elements in the generation of turbulence in the atmosphere. The values we obtain for the eddy diffusion coefficients suggest that the process is quite capable of producing the empirically observed mixing rates at substantial heights above the ground. While the present calculations cannot describe the long-time limit of the wave-turbulence system, one may suggest that the often observed atmospheric conditions in which turbulence and waves appear to co-exist for several hours may result from a sort of equilibrium between the roles of the mean and the fluctuating parts of the eddy diffusion coefficient in taking away from and feeding energy into the wave.

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