Icarus 203 (2009) 683–684
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Power law of dust devil diameters on Mars and Earth Ralph D. Lorenz * Johns Hopkins University, Applied Physics Laboratory, Laurel, MD 20723, USA
a r t i c l e
i n f o
Article history: Received 29 April 2009 Revised 24 June 2009 Accepted 29 June 2009 Available online 5 July 2009
a b s t r a c t Estimates from visual surveys of the frequency of dust devils, even at terrestrial localities known for their abundance, vary by some four orders of magnitude, making a quantitative hazard assessment difﬁcult. Here I show (1) that new high-quality observations from Mars ﬁt a power law size distribution, (2) that such a power law population can unify the discrepant terrestrial surveys, and (3) that the populations on the two planets appear similar. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Mars Atmosphere Meteorology Earth
1. Introduction Dust devils (e.g. see review by Balme and Greeley (2006)) are dry convective vortices that loft dust into the air, often a nuisance for outdoor activities and occasionally responsible for structural damage and fatal aircraft accidents. They are also the most prominent dynamic phenomena observed on the surface of Mars, where they inﬂuence the climate by acting as the principal mechanism of dust-raising. Atmospheric dust, and its removal by dust devils from solar panels, can also significantly impact the operation of spacecraft on the martian surface. Visual census data generally agree that dust devil activity peaks when the convective heat ﬂux driving the vortices is strongest, that is, in early afternoon during the summer (e.g. Sinclair, 1969; Ryan and Carroll, 1970; Fitzjarrald, 1973; Snow and McClelland, 1990; Oke et al., 2007). Air accident statistics (Lorenz and Myers, 2005), imply a similar climatology. However, the overall reported populations in terms of dust devils per unit area per unit time are vastly different (see Table 1). This is a puzzle: while there is evident spatial variation in dust devil frequency owing to dust availability and vorticity generation by terrain (e.g. local counts vary between 1 and 80 per square mile in Sinclair’s (1969) Avra Valley survey, with dry river beds having the highest observed density), the surveys are by deﬁnition performed in areas known to have frequent dust devils and the populations reﬂect roughly the same summer clear-sky absorbed shortwave heat ﬂux and so, averaged over long periods, should be similar. The long period is important, since synoptic scale weather systems may introduce mechanically-forced turbulent convection of heat which might otherwise manifest as free convection, including dust devils (or, put simply, windy days may have few dust devils). 2. Measured size distributions Because of the difﬁculty of measuring size in real time in the ﬁeld, visual surveys have tended to estimate dust devil diameters with coarse size categories. However, more quantitative analysis tools – and, indeed, completely automated analysis, Castano et al. (2008) – can be applied at leisure to digitally-recorded images, such as those obtained from landed spacecraft on Mars.
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Lengthy observations by the Mars Exploration Rover Spirit recorded some 533 dust devils (Greeley et al., 2006) whose sizes were well-determined in 10 m size bins. These data are shown on logarithmic axes in Fig. 1, and the linear trend is consistent with a power law with an exponent of 2 (except in the smallest size bin, not shown, which is depleted due to the existence of a threshold size of either the phenomenon, its detection, or both, which truncates the distribution). An exponential function had been proposed (Kurgansky, 2006) on informationtheoretic grounds to ﬁt the coarsely-binned diameter statistics of the terrestrial surveys. However the better-quality Mars data (Greeley et al., 2006) show that an exponential distribution falls off too quickly at large sizes (i.e. its tail is too small: an exponential ﬁt has a correlation coefﬁcient R2 = 0.84, while the power law yields a much better R2 = 0.94). A power law, a common distribution for geophysical phenomena, also suggests a clue by which the discrepant terrestrial data can be reconciled. 3. Reconciling discrepant surveys The surveys span a wide range of sample area, and although detection thresholds are not reported, it is obvious that small dust devils will be difﬁcult to see at long distance. Thus (as noted by Balme and Greeley (2006)) large area surveys will be less efﬁcient at detecting the small, more abundant features. Simplistically, for a constant density with a 2 power law differential size distribution (dN/dx = kx2, where N is the number of devils of size x, with k a constant), the cumulative size distribution will be N(>d) = kd1, where d is the minimum detectable diameter. The cumulative number density of detections exceeding a ﬁxed angular span threshold will fall off inversely with detection range R, or equivalently with the square root of survey area A R2 and so Nobs 1/A0.5. Other factors, including the suppression of distant optical contrast by atmospheric scattering, can cause the observed density to fall off more steeply. More particularly, if dust devil height and width d are correlated, and the detection threshold relates to the solid angle subtended by the feature d2 rather than merely its diameter d, then the number density of detections will fall off as the inverse square of range R, or inversely with survey area A, thus Nobs 1/A. This is in fact what is seen (Fig. 2) which shows that all the surveys from Earth and Mars follow the same relationship of observed density Nobs (devils km2 day1) = 50/A (km2). For the smallest study area of 0.1 km2, the relevant size threshold (taking into account the angular resolution of the naked eye) will be 0.5 m. Adopting this as a general minimum diameter implies a maximum occurrence rate of 500 devils km2 day1 at 0.5 m and above, with e.g. 50 dev-
Note / Icarus 203 (2009) 683–684
Table 1 Chronological summary of published visual dust devil surveys. IMP is the Imager for Mars Pathﬁnder. Note the trend of larger areas having smaller densities, even for surveys at the same location. References: (1) Sinclair (1969) (2) Ryan and Carroll (1970) (3) Fitzjarrald (1973) as cited by Balme and Greeley (2006) (4) Snow and McClelland (1990) (5) Ferri et al. (2003) (6) Greeley et al. (2006) (7) Oke et al. (2007). Ref.
Density (km2 day1)
1 1 2 3 4 4 5 6 6 7
Tucson, Arizona Avra Valley, Arizona Mojave Desert, California Mojave Desert, California White Sands, New Mexico White Sands, New Mexico Ares Vallis, Mars Gusev crater, Mars Gusev crater, Mars Fowlers Gap, Australia
Visual Visual Visual Visual Visual Visual IMP Survey Navcam Survey Navcam Stare Visual
11 22 10 12 61 36 0.25 270 2 20
500 388 0.15 0.15 64.5 33.8 900 500 4.9 33.8
610 1663 1151 156 2117 1017 16 533 351 557
0.11 0.19 767.3 86.7 0.54 0.84 0.07 0.05 50 0.84
Sols on Mars. Mars numbers are approximate, since observations were not continuous.
ils km2 day1 at a diameter of 5 m and above, per the 2 differential power law above. This power law perspective permits more conﬁdent quantitative investigation of dust devil properties and effects, in that observations at different scales can be seen as different parts of the same population. 4. Conclusions and discussion For a ﬁnite population, a power law must be truncated at small and large sizes. It is impossible to say without better (digital) terrestrial data whether the same detection thresholds apply for surveys on Mars and Earth, and thus whether the same intrinsic minimum size applies on the two worlds (the minimum size may relate to the Obhukov length scale – Kurgansky, 2006). It is noteworthy, however, that the martian and terrestrial surveys fall on the same line, suggesting that the underlying populations may be similar, despite the very different atmospheric densities on Earth and Mars. It should be relatively straightforward, using timelapse cameras deployed in the ﬁeld, to acquire statistics of dust devil diameters on Earth of comparable quality to those from Mars: efforts are presently underway in this direction. Acknowledgment
This work was supported by the NASA Applied Information Systems Research program. p Fig. 1. Martian dust devil counts from Greeley et al. (2006) as ﬁlled circles with N error bars in 10 m size bins, except the last 2. Dotted curve is an exponential ﬁt, which is too convex to correctly represent the data: solid line is a 2 power law which follows the trend very well (formal best ﬁt has an exponent of 1.92).
Fig. 2. Observed dust devil frequency from surveys on Earth (gray diamonds) and Mars (open circles) – see Table 1 – as a function of survey area. Solid line is a simple model with density (km2 day1) = 50/A (km2), suggesting a solid angle detection threshold and a 2 power law diameter distribution.
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