Eddy diffusion coefficients and their upper limits based on application of the similarity theory

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Abstract

The equation for the diffusion velocity in the mesosphere and the lower thermosphere (MLT) includes the terms for molecular and eddy diffusion. These terms are very similar. For the first time, we show that, by using the similarity theory, the same formula can be obtained for the eddy diffusion coefficient as the commonly used formula derived by Weinstock (1981). The latter was obtained by taking, as a basis, the integral function for diffusion derived by Taylor (1921) and the three-dimensional Kolmogorov kinetic energy spectrum. The exact identity of both formulas means that the eddy diffusion and heat transport coefficients used in the equations, both for diffusion and thermal conductivity, must meet a criterion that restricts the outer eddy scale to being much less than the scale height of the atmosphere. This requirement is the same as the requirement that the free path of molecules must be much smaller than the scale height of the atmosphere. A further result of this criterion is that the eddy diffusion coefficients K<inf>ed</inf>, inferred from measurements of energy dissipation rates, cannot exceed the maximum value of 3.2 × 10<sup>6</sup> cm<sup>2</sup> s<sup>-1</sup> for the maximum value of the energy dissipation rate of 2 W kg<sup>-1</sup> measured in the mesosphere and the lower thermosphere (MLT). This means that eddy diffusion coefficients larger than the maximum value correspond to eddies with outer scales so large that it is impossible to use these coefficients in eddy diffusion and eddy heat transport equations. The application of this criterion to the different experimental data shows that some reported eddy diffusion coefficients do not meet this criterion. For example, the large values of these coefficients (1 × 10<sup>7</sup> cm<sup>2</sup> s<sup>-1</sup>) estimated in the Turbulent Oxygen Mixing Experiment (TOMEX) do not correspond to this criterion. The K<inf>ed</inf> values inferred at high latitudes by Lübken (1997) meet this criterion for summer and winter polar data, but the K<inf>ed</inf> values for summer at low latitudes are larger than the K<inf>ed</inf> maximum value corresponding to the criterion. Analysis of the experimental data on meteor train observations shows that energy dissipation with a small rate of about 0.2 W kg<sup>-1</sup> sometimes can induce turbulence with eddy scales very close to the scale height of the atmosphere. Our results also explain the discrepancy between the large cooling rates calculated by Vlasov and Kelley (2014) and the temperatures given by the MSIS-E-90 model because, in these cases, the measured eddy diffusion coefficients used in calculating the cooling rates are larger than the maximum value presented above.

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Vlasov, M. N., & Kelley, M. C. (2015). Eddy diffusion coefficients and their upper limits based on application of the similarity theory. Annales Geophysicae, 33(7), 857–864. https://doi.org/10.5194/angeo-33-857-2015

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