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Statistical Distribution of Atmospheric Dust Devils on Earth and Mars

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Abstract

The theoretical foundations of the exponential and power-law analytical formulations for the size–frequency and intensity–frequency distributions of the convective vortices, including dust devils, are re-examined. Jaynes’ general statistical arguments based on Shannon’s entropy maximum principle leading to an exponential distribution are supplemented by Rényi’s maximum entropy principle which is shown to lead to a power-law distribution. In both cases, a key ingredient of the theory is the a priori knowledge of a first finite moment of the distribution. Applications to statistics of convective vortices, including dust devils, on Earth and Mars are discussed. The existence of a finite expectation value of the vortex diameter related to the absolute value of the Obukhov length scale in the atmospheric boundary layer allows a quantitative explanation of a burst of convective vortex activity observed at the InSight landing site in northern autumn on Mars.

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This paper explains the used theoretical methods; it uses published observational data for illustration of the obtained theoretical results.

Notes

  1. Equation 6 has real positive solutions \(\lambda_{1}\) for \({{\overline{x}} \mathord{\left/ {\vphantom {{\overline{x}} {x_{\max } < \tfrac{1}{2}}}} \right. \kern-\nulldelimiterspace} {x_{\max } < \tfrac{1}{2}}}\) where the upper limit value \({{\overline{x}} \mathord{\left/ {\vphantom {{\overline{x}} {x_{\max } = \tfrac{1}{2}}}} \right. \kern-\nulldelimiterspace} {x_{\max } = \tfrac{1}{2}}}\) corresponds to the uniform distribution \(p\left( x \right) = x_{\max }^{ - 1}\) in the interval \(\left[ {0,x_{\max } } \right]\).

  2. Similar arguments, not detailed here, concern Rényi entropy maximization (see below), if infinity is replaced by \(x_{\max }\) in (2), (3), and (7) in the upper limit of integration. In particular, an estimate for the Insight convective vortices shows that such a replacement leads to a relative error of about 0.05%, and for the power-law distribution (with \(m = 6\)) of the Spirit dust devils is about 0.15% (see Sect. 3).

  3. A slight difference between the number of pressure drops in Table 1 and in the text is explained by the fact that when using a catalogue of pressure drops presented as supplementary material in Spiga et al. (2021), we calculated the number of vortices with a pressure drop strictly above the given value \(\Delta P\), which coincides with the definition of CCDF, while Spiga et al. (2021) apparently calculated the number of vortices with pressure drops greater than or equal to \(\Delta P\).

  4. This goodness-of-fit is supported by the following simple, but quite convincing argument. Using the exponential fit, we estimate the median value \(M\) of the diameter \(x \equiv D\) for the population of Spirit dust devils (Greeley et al. 2006) from the equation \(Pr \left( { > M} \right) \equiv \exp \left( { - {M \mathord{\left/ {\vphantom {M {\overline{x}}}} \right. \kern-\nulldelimiterspace} {\overline{x}}}} \right) = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}\). It gives \(M \approx 27.97\,{\text{m}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \times {\kern 1pt} {\kern 1pt} \ln 2 \approx 19.39\,{\text{m}}\), which is quite close to the median diameter value of \(19\,{\text{m}}\) for Spirit dust devils analysed in Greeley et al. (2006) (D. Waller, personal communication).

  5. In these arguments, the exact value of the multiplier \(\gamma\) in the relation \(\overline{D} = \gamma \left| L \right|\) is not important because \(\gamma\) cancels out in (23).

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Acknowledgements

I am thankful to two anonymous reviewers whose critical review and useful suggestions contributed greatly to improvements in the content and in the style of this article. This work was supported by the Russian Science Foundation (Grant No. 18-77-10076).

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  1. A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevsky 3, Moscow, Russian Federation, 119017

    Michael V. Kurgansky

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Kurgansky, M.V. Statistical Distribution of Atmospheric Dust Devils on Earth and Mars. Boundary-Layer Meteorol 184, 381–400 (2022). https://doi.org/10.1007/s10546-022-00713-w

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