Abstract

A phenomenological model is developed wherein vortices are introduced at random into a virtual arena with specified distributions of diameter, core pressure drop, longevity, and translation speed, and the pressure history at a fixed station is generated using an analytic model of vortex structure. Only a subset of the vortices present are detected as temporary pressure drops, and the observed peak pressure-drop distribution has a shallower slope than the vortex-core pressure drops. Field studies indicate a detection rate of about two vortex events per day under favorable conditions for a threshold of 0.2 mb (1 mb = 1 hPa): this encounter rate and the observed falloff of events with increasing pressure drop can be reproduced in the model with approximately 300 vortices per square kilometer per day—rather more than the highest visual dust devil counts of approximately 100 devils per square kilometer per day. This difference can be reconciled if dust lifting typically only occurs in the field above a threshold core pressure drop of about 0.3 mb, consistent with observed laboratory pressure thresholds. The vortex population modeled to reproduce field results is concordant with recent high-resolution large-eddy simulations, which produce some thousands of 0.04–0.1-mb vortices per square kilometer per day, suggesting that these accurately reproduce the character of the strongly heated desert boundary layer. The amplitude and duration statistics of observed pressure drops suggest large dust devils may preferentially be associated with low winds.

1. Introduction

Dust devils (e.g., Balme and Greeley 2006) are dry convective vortices that are a source of enduring meteorological fascination, having many of the dynamical features of tornadoes, yet being rather easier and less dangerous to study. Desert whirlwinds are nonetheless responsible for disruption of picnics and sports activities, structural damage, and even an occasional fatality; thus, it is important that the frequency of occurrence be evaluated. Dust devil vortices generate a surface pressure signal that acts as an enhanced noise background in microbarograph applications such as nuclear test ban monitoring. Dust devils may also be an agent of dust lifting that could influence air quality, and the correct estimation of their contribution requires that the dust devil population be understood.

Under conditions of strong surface heating with modest ambient wind, columnar vortices appear in large-eddy simulations (LESs) of convecting boundary layers (e.g., Kanak 2005; Gheynani and Taylor 2011), suggesting many may be present in nature. Such features may or may not be visible as dust devils, depending on whether they are strong enough to loft dust—occasionally, vortices can be rendered visible by leaves, snow, hay, and even tumbleweed plants. Hence, it is useful to reconcile the occurrence of vortices with observed dust devils and, thereby, infer the fraction of “dustless devils.” These same questions pertain to the planet Mars, where dust devils are perhaps the most dynamic observable feature of the near-surface environment and may be a dominant agent of dust transport (e.g., Newman et al. 2002; Michaels and Rafkin 2004).

In this paper, we review visual observations of terrestrial dust devils and estimate the formation rate taking into account observing biases. We underscore that no report of dust devil density, or occurrence frequency, is useful without an accompanying statement of observation threshold. We then note recent surveys of the encounter rate of vortices detected by field barometric measurements, where vortex passage manifests as a local drop in pressure. We then construct a model of vortex migration to relate the areal formation rate of vortices with point observations of pressure drops and, thereby, derive the properties of the underlying vortex population. We deliberately use the word “population” to underscore that observations provide only an incomplete sample of the full set of vortices; in effect, dust devil surveys are performing a census, and statistical methods must be applied to estimate the full population from incomplete data. The problem is very similar to that confronted in ecology, where wildlife populations must be estimated by catching a small number or observing a small area: the detection efficiency must be explicitly determined or modeled. The problem may in a sense be expressed as a parameter estimation exercise, where the parameters describe the distribution of diameters, core pressure drops, dust loading, etc. of the population. These distributions may be independent, or they may be correlated (e.g., only large diameter devils having large pressure drops).

2. Observations

a. Visual dust devil counts

As noted by Balme and Greeley (2006), the observed formation rate of dust devils varies with the area surveyed. Lorenz (2009) showed that this variation is systematic, with detected rates N (devils per square kilometer per day) being inversely proportional to survey area A (km2), with N ~ 50/A, and suggested that this could be explained by a power-law diameter distribution and a detection threshold that varied with distance (and thus survey area). The detection efficiency of small and/or optically thin devils is essential to take into account—for example, in comparing populations observed at different sites—and to understand the shape of the actual dust devil population function (e.g., Lorenz 2011; Kurgansky 2012).

In this paradigm, clearly the smallest areas will have the highest detection rates as 100% of even the smallest devils will be detected and the inventory can be considered “complete.”1 The highest rates reported in the field are those by Carroll and Ryan (1970), who encountered some 1151 devils in 10 days in a 0.15-km2 area or 767 devils per square kilometer per day. This count is especially high for a couple of reasons. First, the site was completely clear of vegetation and they “groomed” the desert surface daily to maximize the availability of dust. This likely reduced the threshold wind speed or pressure drop required for dust lifting (see later). Second, counts in very small study areas are inflated because dust devils that form elsewhere but are advected into the site are counted—for the 0.15-km2 area and the typical sizes (and thus longevities) and wind speeds encountered, the rate at which vortices form in the study area is probably lower by a factor of 2–3 [see Fig. 6 of Lorenz (2013a,b)]. Hence, we consider their corrected result as a formation rate of approximately 300 devils per square kilometer per day, noting that this occurred with an artificially modified surface.

The highest count in more typical field conditions (indeed, at El Dorado Playa in Nevada, where the pressure-drop data in the next section was obtained) is that by Pathare et al. (2010) of 528 devils in 0.55 km2 in 9 days. However, this period included some less favorable days, with daily counts varying between 18 and 83. Adopting the highest number yields N = 151 devils observed per square kilometer per day, although again this should be reduced by a factor relating to the advection speed, so 100 devils per square kilometer per day is probably about right for the formation rate and certainly greater than 75 devils per square kilometer per day. Other surveys (Lorenz 2011; Lorenz and Lanagan 2014) have lower counts. Snow and McClelland (1990) report mean counts of about 0.5 devils per square kilometer per day at White Sands Missile Range, but close inspection of their Fig. 3 shows that these were very unevenly distributed in the survey area, and one 0.5 × 0.5 km2 survey plot saw 151 devils in 41 days or 14 devils per square kilometer per day. This plot was some kilometers from the observer location, suggesting that this number is rather incomplete, and they note that “it is likely that less than 10% of whirlwinds in the small category … were counted, greater than 80% of dust devils in the medium category and all those in the large category … were counted by the observer” (this is the only completeness estimation known to this author for terrestrial visual surveys). Hence, the overall dust devil rate was probably a factor of a few higher than the 14 devils per square kilometer per day above and seems consistent with the approximately 100 devils per square kilometer per day indicated by Pathare et al. (2010).

Hence, we adopt approximately 100 devils per square kilometer per day as the likely formation rate of visible dust devils under favorable meteorological conditions with favorable but “natural” dust availability and note that surface conditioning may increase this rate by a factor of about 3. Of course, many values lower than this can be derived from surveys in the literature, resulting either from poorer vortex formation conditions, lower detection efficiency, or both.

b. Vortex encounters from in situ meteorological observations

Dust devil passage manifests itself in meteorological data by several clear signatures (e.g., Sinclair 1973; Ryan and Lucich 1983; Ringrose et al. 2003, 2007; Tratt et al. 2003)—most notably a pressure drop lasting from a few seconds to hundreds of seconds and a rotation of the wind direction. Wyett (1954) noted a dust devil passage in a recording barograph at an airport meteorology station, and Lambeth (1966) documented about 20 pressure drops.

The shapes of these signatures depend on the miss distance of the devil from the meteorological station and the diameter of the devil: usually the time base of the recorded signatures is mapped into a spatial coordinate by an assumed constant translational velocity that allows the signature to be normalized by the diameter of the devil, although in many instances a dust devil’s path may be curved or even cycloidal (Lorenz 2013a), resulting in a complex mapping of distance to time.

On Earth, despite the existence of thousands of meteorological stations worldwide, few are monitored with the high time resolution (typically ~1 s) required to detect dust devils. Lambeth (1966) installed an array of six stations at White Sands, New Mexico. These stations recorded wind, pressure, and temperature data: the stations, each with instruments in a small shelter and on a mast, were spread in a line about 16 m wide orthogonal to the prevailing wind direction, and data were acquired by chart recorder in a trailer van some 35 m away. Although the site was known to experience “numerous” dust devils, over a period of some 4 months (15 May–15 September 1959) only 21 close encounters were made: pressure drops were recorded in only 19 cases. As discussed in Lorenz (2012a), the detection efficiency here seems low.

Ryan and Carroll (1970) set up a meteorological mast in the Mojave Desert and acquired an extensive set of in situ data over a 30-day field period: these data so far are the best for correlating some observed quantities such as diameter and wind speed. Some 1119 devils were observed in a 300 × 500 m2 study area, only about 80 encounters with the pole of devils larger than 1 m were observed. It was further observed that of the 30-day study, only 11 days were conducive to dust devil formation. Unfortunately, pressure data were not recorded. However, one can use the observed maximum wind velocity V as a proxy for the core pressure drop ΔP, invoking (ΔP ~ ρV2) as an estimate (which follows from assuming cyclostrophic balance, with the air density ρ): examining their data yields 17 dust devils where ρV2 exceeded 0.3 mb (1 mb = 1 hPa), about 20%–25% of the total, when allowance for missing data is made. Discounting “unfavorable” days then gives an encounter rate of about 2 day−1 for ΔP > ~0.3-mb devils.

As discussed in Lorenz (2011), a larger database of terrestrial encounters is needed for robust comparisons with Mars. Therefore, after some initial experiments [yielding a 0.6-mb devil in a 3-day survey reported in Lorenz (2010)], new experiments have been made with small pressure loggers (Lorenz 2012b, 2013a). A preliminary inspection of data from a trial field campaign at El Dorado Playa in Nevada (Lorenz 2013b) suggested detectable encounter rates (>0.3 mb) on average of 1.3–2.6 day−1, although this value of course is subject to stochastic variation and (as Fig. 1 implies) is a strong function of the detection threshold. A more systematic evaluation recording over a hundred candidate events (Lorenz and Lanagan 2014) has been conducted: the population of pressure drops recorded at one station is shown in Fig. 2. A useful additional result (see later) is the distribution of magnitude of the pressure drop versus the duration of the drop, and these data are shown in Fig. 3.

Fig. 1.

A fairly deep (~0.6 mb) pressure drop recorded on El Dorado Playa (Lorenz and Lanagan 2014). 

Fig. 1.

A fairly deep (~0.6 mb) pressure drop recorded on El Dorado Playa (Lorenz and Lanagan 2014). 

Fig. 2.

The cumulative number of recorded pressure drops (diamonds) at a fixed station on El Dorado Playa in Nevada [here P28 from Lorenz and Lanagan (2014)] follows a broken power law—the major segment has a cumulative slope of about 1.8, suggesting a differential power law of −2.8. Note that the only other dataset, from Lambeth (1966), has poor statistics and a low detection efficiency but a consistent distribution shape. The solid circle marked Avra Valley is the detection by Lorenz (2010).

Fig. 2.

The cumulative number of recorded pressure drops (diamonds) at a fixed station on El Dorado Playa in Nevada [here P28 from Lorenz and Lanagan (2014)] follows a broken power law—the major segment has a cumulative slope of about 1.8, suggesting a differential power law of −2.8. Note that the only other dataset, from Lambeth (1966), has poor statistics and a low detection efficiency but a consistent distribution shape. The solid circle marked Avra Valley is the detection by Lorenz (2010).

Fig. 3.

While most events are small and short in duration, a number of short–large and long–small events are detected. There are few long–large events.

Fig. 3.

While most events are small and short in duration, a number of short–large and long–small events are detected. There are few long–large events.

3. Relating observations to the underlying population

A very simple thought experiment shows that the observed pressure-drop distribution will have a different slope than the underlying vortex population. Consider the case of a flat distribution, wherein all vortices have a central pressure drop of, say, 5 mb. A vortex that traverses the station will have a 5-mb drop recorded, but many more will give near misses, yielding smaller observed pressure drops. For a fixed wind speed, size, and duration, one can simply distribute miss distances uniformly and estimate the relative number of recorded drops. However, the situation is in fact much more complicated as the detection efficiency for large devils is much larger even when the core pressure drop remains fixed. This is for two reasons: large dust devils cause a given pressure drop over a wider area as they traverse the landscape but also empirically last longer (e.g., Lorenz 2013b) and so have longer traverses. Figure 4 shows schematically the effects at work. Since the largest pressure drop measured at a fixed point may be determined by a single “lucky” near miss, rather than being representative of any sort of “average” pressure drop, it is difficult to evaluate the population analytically. Indeed, the notion of average in a population that has highly skewed properties requires considerable care (e.g., Lorenz 2011, 2012a).

Fig. 4.

Schematic of the area swept by a visible dust devil, and the corresponding area that sees a pressure perturbation of some given threshold (e.g., 0.1 mb). Larger-diameter devils have both a larger cross section and last longer, resulting in longer paths for a given wind speed. Larger central pressure drops push the 0.1-mb contour farther out. Weaker winds lead to shorter migration paths. Note that diameter (the visible wall, where maximum wind speed occurs) is specified entirely independently of pressure drop: the area swept by the visible devil may not correspond exactly to the area where dust lifting occurs.

Fig. 4.

Schematic of the area swept by a visible dust devil, and the corresponding area that sees a pressure perturbation of some given threshold (e.g., 0.1 mb). Larger-diameter devils have both a larger cross section and last longer, resulting in longer paths for a given wind speed. Larger central pressure drops push the 0.1-mb contour farther out. Weaker winds lead to shorter migration paths. Note that diameter (the visible wall, where maximum wind speed occurs) is specified entirely independently of pressure drop: the area swept by the visible devil may not correspond exactly to the area where dust lifting occurs.

4. Monte Carlo model

It follows, then, that an evaluation of the population properties is best performed by numerical simulation. Our model is named Vortex Evolution Model with Observation Operations Simulated Explicitly (VEMOOSE). The model is coded in the Interactive Data Language (IDL). Essentially, vortices are introduced into a virtual arena (2 × 2 km2) one at a time with a random starting position. The vortex is assigned a wall diameter and a core pressure drop and is advected at a specified speed and direction for a given vortex lifetime. For the present study, the lifetime is simply an explicit function of diameter; specifically duration in seconds is 40d0.66 (Lorenz 2013b) with diameter d in meters, although in principle some random variation on this could be superposed.

The arena size was set to accommodate typical traverse paths, although a periodic boundary condition is applied such that devils that migrate out of the arena on one side are replaced at the other. A flag is set in the model such that a given devil is not detected more than once. Straight-line constant-speed advection is assumed.

The key parameters for the model are the distributions of core pressure drop, diameter, and advection speed. These can be independent or coupled. Various observation processes could be simulated (e.g., whether a vortex would subtend a given angle as seen from a fixed point, simulating optical detection by visual survey). For the present application, we consider pressure sensing at a point.

The pressure perturbation at each time step (nominally 1 s) that would be measured at a fixed station in the center of the arena is computed (see next section) and a detection criterion applied, such as the exceedance of a given pressure perturbation threshold. If this criterion is satisfied, the peak pressure drop is measured, and the duration of the threshold exceedance is recorded. It was found in general that to obtain reasonable statistics (>100 detections) that around 200 000 devils needed to be introduced into the 2 × 2 km2 arena. Results of the modeling are presented in section 5. First, we discuss the input parameter distributions and the important specification of sensed pressure.

a. Pressure drop versus distance

The pressure drop and wind speed in dust devils have traditionally been modeled with the classic 1880s Rankine vortex model, which captures the overall structure well, specifically with a “solid body” rotation in the core; that is, Vt(x) = Vr(2x/d), and a tangential speed outside the core that falls off as 1/x; that is, Vt(x) = Vr(d/2x), where Vr is the tangential velocity at the “wall” of the vortex and x = (d/2), d being the diameter.

This idealized model, however, introduces an undifferentiable singularity in the radial profiles that, in real vortices, is smoothed out by friction. Various analytical descriptions of more physical vortex functions are described in the literature (e.g., Burgers and Rott) that have a less-peaked structure; see also work on laboratory vortices by Snow et al. (1980). A particularly succinct model by Vatistas et al. (1991) gives excellent agreement with laboratory data and has the following functions of tangential velocity and pressure as a function of normalized distance r (r = 2x/d):

 
formula
 
formula

We show here that the velocity structure specified in this model agrees well with those measured for dust devils in the field. Tratt et al. (2003) obtained a velocity profile as a function of distance with a 7-m-diameter dust devil in the Mars Atmosphere and Dust in the Optical and Radio (MATADOR) field campaign and noted that while a Rankine model was a reasonable description of the central velocity profile, the speed fell off with distance outside the wall more shallowly than the Rankine model would predict, specifically approximately as r−0.5 instead of r−1. In the near field, the Vatistas model has behavior very close to this, as shown in Fig. 5. (The Vatistas model has asymptotic Rankine-like r−1 behavior in the extreme far field.)

Fig. 5.

Field data (circles) of an Arizona dust devil recorded by Tratt et al. (2003) compared with the model by Vatistas et al. (1991) described in the text. The idealized (inviscid) Rankine vortex has too steep a falloff of speed with distance outside the devil.

Fig. 5.

Field data (circles) of an Arizona dust devil recorded by Tratt et al. (2003) compared with the model by Vatistas et al. (1991) described in the text. The idealized (inviscid) Rankine vortex has too steep a falloff of speed with distance outside the devil.

Ellehoj et al. (2010) empirically fit the shape of pressure drops caused by dust devils passing near the Phoenix lander on Mars with a Lorentzian function—namely,

 
formula

Figure 5 shows that this empirical expression is very nearly equivalent to the Vatistas one—both being within 20% of each other, and both indicating that the pressure drop at the wall (r = 1) is 50% of that at the center (r = 0). Since the Vatistas expression has a more physical basis, being derived in part from conservation laws, we adopt it in further work. As Fig. 6 shows, even in the far field, the two expressions are only 20% different, so high quality data would be required to discriminate the two functions even in the case of a perfect single vortex following a linear path.

Fig. 6.

Both a Lorentzian function [Eq. (3)] and the Vatistas expression [Eq. (2)] have similar radial pressure profiles. In both, the pressure drop at the wall is 0.5 times the central drop, and the drop decays to a tenth at 2.5–3 radii.

Fig. 6.

Both a Lorentzian function [Eq. (3)] and the Vatistas expression [Eq. (2)] have similar radial pressure profiles. In both, the pressure drop at the wall is 0.5 times the central drop, and the drop decays to a tenth at 2.5–3 radii.

b. Diameter distribution

It is well known that there are typically many more small dust devils seen than large ones (e.g., Lorenz 2011). There is generally considered to be a minimum size for dust devils [typically associated with the Obukhov length; e.g., Hess and Spillane (1990) and Kurgansky et al. (2010)], although it is conceivable that if small devils tend to be optically thin and/or short lived, the minimum size may not be well characterized. Similarly, a likely upper limit on dust devil height is associated with the thickness of the planetary boundary layer, and since dust devils tend to have an aspect ratio of about 5 [and rarely more than 10; e.g., Lorenz (2013b)], this in turn puts an upper limit on diameter. Within these bounds, opinions differ as to whether the observed population is best fit with a (truncated) power law (e.g., Lorenz 2009) or an exponential (e.g., Kurgansky 2006; Pathare et al. 2010; Kurgansky 2012). In fact, other functions such as lognormal or Rayleigh might also fit the present coarsely binned data just as well—the data are not good enough to robustly discriminate.

For numerical convenience, we adopt a power-law diameter distribution with a −1.8 differential exponent. The distribution is truncated at 1-m minimum diameter and 200-m maximum diameter.

c. Core pressure-drop distribution

Since observed pressure drops approximately follow a truncated power law (Lorenz 2012a), we similarly adopt this formulation as our default, although we have also trialed drops proportional to diameter. We use a minimum drop of 0.2 mb for practical reasons—much smaller pressure drops are not detectable against instrument noise and turbulent fluctuations in pressure (Lorenz and Lanagan 2014). We consider the slope of the power law a free parameter which can be tuned to yield the observed distribution. An upper limit of 10 mb for the drops is sometimes used (in which case the random number generator is resampled to choose a drop from the power-law distribution) but in practice this upper limit rarely makes a difference since the power-law slope requires so few devils with such a high drop anyway.

d. Advection speed distribution

It is typically considered that a dust devil will migrate with a speed and direction close to that of the near-surface wind (e.g., Balme et al. 2012). However, it is generally considered that modest wind speeds (a few meters per second) are particularly conducive to dust devil formation, with high winds generally unfavorable.

Initially, we considered the results of Oke et al. (2007), who documented the wind speed at which some 557 dust devils were observed at a field site in Australia. Dust devils were not seen for winds of 1 m s−1or less or above 8 m s−1. Their observed relative frequencies could be described with a Gaussian with a mean of 4.5 m s−1 and a standard deviation of 1.3 m s−1.

On encountering difficulty in reproducing duration statistics, we revisited this parameterization, and note rather wider ranges of observations in the literature. Sinclair (1973) finds dust devils occurring up to about 8 m s−1, with frequency declining above that speed. Kurgansky et al. (2010) found that dust devils formed at arbitrarily low wind speeds but did not observe any at speeds above 8 m s−1. However, Snow and McClelland (1990) present a summary of advection distances and observed lifetimes (see their Table 5) of their extensive dataset of more than 2000 dust devils. Reexpressing these data as a translation speed, we find that 3–4 m s−1 is particularly favorable but note that devils were encountered for speeds of less than 1 m s−1 and a number at 10 m s−1 or above.

Given these somewhat disparate data, we evaluate a number of advection speed distributions but note that a peak around 3–5 m s−1, and a decline toward lower and higher speeds, is indicated. For aesthetic purposes (Fig. 7), we impose a small variation in advection direction, but the symmetry of the present problem with a single measurement station makes this unimportant.

Fig. 7.

Model parameters and results. (a) A subset of 100 tracks of the 200 000 in the run. (b) The diameter distribution specified as a power law. Crosses are error bars of observations by Pathare et al. (2010). (c) A Gaussian wind speed distribution. (d) The observed duration and magnitude of pressure dips exceeding a measurement threshold of 0.1 mb. (e) The model core pressure dip population (dashed line) specified as a power law above 0.2 mb, and the measured pressure dips (solid line). (f) A zoom of the measured pressure dips compared with the observed pressure dips with 1σ error bars at El Dorado station P28 (Lorenz and Lanagan 2014) scaled to show the shape of the distribution function (absolute number is not meaningful).

Fig. 7.

Model parameters and results. (a) A subset of 100 tracks of the 200 000 in the run. (b) The diameter distribution specified as a power law. Crosses are error bars of observations by Pathare et al. (2010). (c) A Gaussian wind speed distribution. (d) The observed duration and magnitude of pressure dips exceeding a measurement threshold of 0.1 mb. (e) The model core pressure dip population (dashed line) specified as a power law above 0.2 mb, and the measured pressure dips (solid line). (f) A zoom of the measured pressure dips compared with the observed pressure dips with 1σ error bars at El Dorado station P28 (Lorenz and Lanagan 2014) scaled to show the shape of the distribution function (absolute number is not meaningful).

5. Results

After some minimal experimentation with the model, rather satisfactory results are obtained (e.g., Fig. 7). As expected, the core pressure-drop power law must be steeper than the observed dips because of the distribution of miss distances—we find that a differential population exponent of about −2.8 is needed to reproduce the observed slope of about −2.

A key result is the overall “efficiency: of detection—in effect, a measure of the detection footprint of a given dust devil subject to the pressure detection threshold. With the default model parameters and a 0.1-mb detection threshold, we find about 400 detections out of 200 000 devils introduced into the arena. In other words, with a −2.8 pressure power law and the Oke wind speed distribution, the 400 detections correspond to a density of 50 000 devils per square kilometer. In nature, such a number would require many tens of days—see next section.

Although we caution against considering a “typical” dust devil, it is a worthwhile “reality check” to evaluate the effective detection footprint over the population. Dividing 400 by the 50 000 devils per square kilometer yields a footprint of 0.008 km2. A 5-m dust devil would last around 115 s, and so with a 4 m s−1 wind would be advected about 500 m, and so the dust devil itself would sweep 0.0025 km2. Thus, the pressure-sensed footprint of the average over the population is about 3 times larger than the optical footprint of a 5-m dust devil (see, e.g., Figure 4). While not in itself a quantitatively meaningful comparison without a definition of pressure footprint, it serves to illustrate the general reasonableness of the model results.

Since the model runs in only a few minutes, it is of course possible to simulate a range of scenarios. Figure 8 shows the effect of different wind speed and pressure distributions, for example. Increasing the wind speed raises the cross section and thus realizes more detections—in this case, around 700 of the 200 000 are detected.

Fig. 8.

As in Fig. 7, using the same diameter distribution, but with a shallower core pressure drop slope (−1.5), one finds (f) a shallower measured pressure drop slope than observations. Introducing a wider speed range causes (d) the events of longer duration to be detected and (a) the paths to be longer, but the scatterplot in (d) is still too concentrated at the short-duration end.

Fig. 8.

As in Fig. 7, using the same diameter distribution, but with a shallower core pressure drop slope (−1.5), one finds (f) a shallower measured pressure drop slope than observations. Introducing a wider speed range causes (d) the events of longer duration to be detected and (a) the paths to be longer, but the scatterplot in (d) is still too concentrated at the short-duration end.

While the shape of the detected pressure-drop distribution is fairly easy to recover with a range of combinations of model parameters, trial and error revealed that the drop–duration scatterplot (Fig. 3) is in fact rather more challenging to reproduce. Specifically, the number of long pressure drops is difficult to recover: if wind speeds are reduced to extend the duration of a given miss distance, then the detection efficiency goes down and far too many vortices are needed. On the other hand, observations prevent the proportion of large-diameter vortices being too high. It seems possible that correlation between two or more variables may be required—we offer one such model in section 5c.

a. Reconciling devils and vortices—Dust-lifting threshold

One way to reconcile the difference between the visual dust devil count and the pressure-drop statistics is to posit that only those vortices with a core pressure drop above some dust-lifting threshold (which is higher than the vortex detection threshold) become visible. The physics of dust lifting are in fact rather complex (e.g., Kok 2011; Kok et al. 2012), involving breakup of dust aggregates, hysteresis effects, and possibly “sand blasting,” launching of dust by more easily saltated sand grains, etc. For the present purpose, we posit that a given surface has a single threshold and that this threshold can be expressed (whether due to tangential winds, radial winds, pressure drop, or some combination of these) as an equivalent pressure drop.

Conventional expressions for particle-lifting threshold speeds in boundary layer flow fail to predict dust lifting for the wind speeds typically encountered in dust devils, suggesting that additional mechanisms prevail (e.g., Greeley et al. 2003). Notably, whereas friction velocities exceeding 1 m s−1 (i.e., freestream speeds of about 10–20 m s−1) would be predicted to be required lift particles of about 2 μm, the observed lifting threshold in laboratory vortex experiments is only about ~ 0.2 m s−1.

Balme et al. (2003) report peak friction speeds inside dust devils in the field of between 0.9 and 2.4 m s−1. It should be borne in mind that these were of dust devils large enough, optically thick enough, and long-lived enough to race ahead of and deposit an instrumentation array into [like the Lambeth (1966) survey, there is in effect a large threshold to trigger the observation process]. Thus the typical field threshold for dust lifting must be rather less than 0.9 m s−1. On the other hand, the ambient friction speeds measured by Balme et al. (2003) outside the dust devil (which therefore must be lower than the threshold) were measured between 0.1 and 0.65 m s−1 (mean = 0.32 m s−1). Hence the dust-lifting threshold friction speed in field conditions appears to be above 0.3 m s−1, and in some cases is around 0.7–0.8 m s−1, or 3–4 times the laboratory threshold.

We can equate this friction speed to an approximate pressure drop, in that by definition the freestream velocity V is () times the friction speed , or V ~ (10 to 20) for typical land surfaces, where the drag coefficient Cd ~ 0.0025–0.01 (e.g., Dong et al. 2001). We then assume that the threshold velocity is mostly due to Vt, which can be related to the central pressure drop via cyclostrophic balance as , or ~100–400 . At El Dorado (elevation ~ 800 m, with ρ ~ 1.1 kg m−3), where Balme et al. (2003) found that V ~ 12 in field measurements in dust devils, we obtain and, hence, the field threshold of approximately 0.75 m s−1discussed above implies ΔPt ~ 85 Pa or 0.8 mb. On the other hand, if the dust (and sometimes sand)-laden flow within the dust devil in effect causes a higher local drag, such that V ~ 8, then ΔPt would be lower.

Neakrase et al. (2006) evaluate dust-lifting fluxes in laboratory vortices, measuring ΔP directly. For terrestrial conditions, they find 2-μm clay dust is not lifted when ΔP is less than 0.1 mb, and the dust flux increases by an order of magnitude over the range 0.15–0.3 mb. Whether the dust-lifting process invokes a pressure drop directly (sucking dust off the ground or, more formally, lifting dust via vertical flow from exhausting the pore space in the ground and/or by a vertical pressure gradient across the grain) in addition to the aerodynamic lift and drag due to the rotating flow is open to discussion [Balme and Hagermann (2006) suggest that the suction or “ΔP effect” is indeed important] but, in any case, it seems that there is broad consistency between laboratory measurements of ΔP and dust lifting and the ΔP corresponding to the dynamic pressure associated with the measured threshold friction speeds. We are thus comfortable generalizing the approach to derive ΔPt ~ 0.8 mb to make dust devils visible in some field conditions field or, perhaps, as low as about 0.2 mb in the most dust-available areas with ideal (near laboratory) conditions. Further, since ΔP in the field is straightforward to measure, we retain this parameter as the focus for discussion.

Considering El Dorado Playa, we see in Fig. 2 that a 0.3-mb threshold is encountered about once per day, compared to 0.1-mb detections about six times per day. Thus, the 50 000 devils per square kilometer in our model yielding about 400 detections at 0.1 mb correspond to about 65 days of observation: in other words, there are about 750 0.1-mb vortices per square kilometer per day and about 120 vortices per square kilometer per day with core pressure drops of 0.3 mb. That is to say, the number of vortices with pressure drops of 0.3 mb appears to correspond with the number of dust devils observed visually (~100 vortices per square kilometer per day; see section 2a).

This population-averaged threshold appears to be lower than the thresholds measured by Balme et al. (2003). One or more of their handful of encounters may be spurious, there may be a detection effect (largest, slowest devils being more likely to be sampled), or perhaps a feedback wherein once dust is lofted, the circulation becomes more intense. A mechanism for such feedback is that lofted dust introduces solar heating into the vortex itself (Lorenz 2005; Fuerstenau 2006). The impression of intensification by dust is sometimes given from field observation: the question deserves further study.

We have assumed in this discussion that a single threshold prevails. In all likelihood in the field there are patches where dust is readily lifted and areas where it is not, and if feedbacks introduce hysteresis, then the patches of easy lifting may dominate the overall characteristics of the population. Kok et al. (2012) note the role of hysteresis in saltation of sand. Stochastic thresholds could be applied in future development of the model, much as employed in some recent LESs (e.g., Klose and Shao 2013).

b. Comparison of vortex population with large-eddy simulations

Growing computational capabilities now make it feasible to explicitly model dynamics in the convecting atmospheric boundary layer with sufficient spatial resolution to resolve the majority of observed dust devils (i.e., a few meters) (e.g., Kanak 2005). Vertical vortices tend to form at the upwelling vertices and sometimes edges of the convection cells in the convective boundary layer (CBL). If these LESs are correctly representing the boundary layer in which dust devils are observed to form, then the characteristics of vertical vortices in these simulations should be the same as those measured in nature. Unfortunately, LES modelers rarely report results in a manner that permits robust quantitative comparison with field results, where diameter (Lorenz 2011) and pressure-drop (Lorenz 2012a,b) statistics show distinctive functions.

However, some workers have reported results which allow deduction of the number density of dust devil–like vortices. Two recent papers in particular are of note in this connection. First, Ohno and Takemi (2010) report vortex formation and evolution in an LES with a 1 × 1 km2 domain and a horizontal resolution of 3 m, with a CBL height of 800 m and a surface heat forcing of 0.24 K m s−1 (i.e., a sensible heat flux of about 290 W m−2, typical of desert conditions on Earth). They find after model convergence that 225 vortices were detected (with a 10-Pa pressure-drop threshold and a vorticity threshold ζ of 0.15 s−1) in a 2000-s period. If we extrapolate to a typical sunny day (i.e., ~4 h or 15 000 s of such strong heating), then this corresponds to 1600 vortices per square kilometer per day.

Second, Raasch and Franke (2011) show results for a 4 × 4 km2 domain and 2-m resolution (as well as a smaller run at 1-m resolution) with a CBL of 700 m and 0.24 K m s−1 surface sensible heat flux. They used a vortex detection threshold of ζ > 1 s−1 and ΔP > 4 Pa and found 25 000 vortex tracks in a 5400-s run. This corresponds, given their 16-km2 model domain, to about 1500 tracks per square kilometer in the run or about 4500 vortices per square kilometer per day.

Although conducted with entirely independent simulation codes and using different detection criteria, these models (with the same basic parameters) are in fact quite consistent. Since natural pressure drops have an approximately 2–2.5-power-law differential distribution, or 1–1.5 cumulative, it follows that 4-Pa pressure drops should occur (10/4)(1 to 1.5) = 2.5–4 times as often as 10-Pa drops. This agrees nicely with the ratio of the two results above of 2.7.

These results are furthermore reassuringly consistent with field measurements. Although turbulent fluctuations and instrument noise make it difficult to detect vortices by pressure drops alone for drops as small as those in the LES, the population statistics implied in this paper for 30-Pa vortices of about 300 km−2 day−1 imply that 10-Pa vortices should occur 3–5 times as often (i.e., 900–1500 vortices per square kilometer per day), in reasonable agreement with the Ohno and Takemi (2010) result and, by implication, also with the work of Raasch and Franke (2011).

Although the two LES studies noted above appear to have consistent pressure drop statistics, they did use different vorticity criteria. It would be useful to understand the detection efficiency/false alarm rates using different criteria so that different studies can be more accurately compared. We hope the discussion in the present paper may encourage LES modelers to report vortex statistics in their simulations in greater detail to permit more extensive comparison with the improving field data. In particular, plotted or tabular data exploring the relationship of detected vortex parameters (e.g., scatterplot of diameter versus core pressure drop, advection speed versus pressure drop) would be useful, as would parameters of the overall population. Time histories of the evolution of vortex parameters (e.g., do dust devils grow and shrink in diameter, or do they vary intensity with a fixed diameter, and over what time scale?) would similarly be interesting—such evolution is very difficult to obtain in the field. Study of dust devils with radiatively active dust [which may introduce heat into the core and thereby intensify the vortex (Lorenz 2005; Fuerstenau 2006)] would also be important.

c. Event durations—A wind speed–diameter relationship?

As noted earlier, the pressure-drop amplitude–duration scatter, with a short–deep arm and a long–shallow arm, but with few long–deep drops, is difficult to reproduce. One solution is to posit that large dust devils move more slowly (or equivalently, that low wind speeds favor the formation of larger dust devils). Such a scenario—we can modify the random advection speed by a factor of approximately 10/(5 + d) or 2/(2 + ΔPc), where ΔPc is the core pressure drop (mb)—yields the observed amplitude–duration distribution much better (Fig. 9) than independent speeds, diameters, and pressure drops, but a formal fit exercise is probably not justified without a larger field dataset. There is some indication in visual observations of Mars devils (Greeley et al. 2010, their Fig. 13) that larger dust devils appear to be associated with slower wind speeds. Such a speed–diameter–intensity relationship should be manifest in the statistics of tracks formed on the surface.

Fig. 9.

As in Fig. 7, except the (a) observed diameter distribution, (d) pressure drop distribution, and (c) amplitude–duration statistics can be reproduced by forcing larger devils to be associated with slower winds. Note that the upper-right corner of the scatterplot in (d) is unpopulated, as observed.

Fig. 9.

As in Fig. 7, except the (a) observed diameter distribution, (d) pressure drop distribution, and (c) amplitude–duration statistics can be reproduced by forcing larger devils to be associated with slower winds. Note that the upper-right corner of the scatterplot in (d) is unpopulated, as observed.

6. Conclusions and recommendations for future work

A modeling approach such as that described here is necessary to account for various observational effects and, in particular, for the very nonlinear dependence of pressure sensed as a function of distance from the center of a dust devil.

The statistical modeling approach here exposes interdependencies between dust devil properties. Specifically, the amplitude–duration characteristics of single-station measurements suggest a possible correlation between wind speed and diameter. LES investigation might be usefully able to explore whether such a correlation emerges in dynamic simulations.

Visual dust devil counts and measured vortex encounters can be reconciled if only a fraction of vortices are rendered visible by lofted dust. The observed fraction on Earth (about 1/6 of 0.1-mb vortices) could be explained if only devils with central pressure drops exceeding a threshold ΔPt of about 0.3 mb are rendered visible. Field measurements of the friction speed in dust devils suggest a possibly larger core pressure drop: whether this is a selection effect or a feedback (once dust lifting starts, the devil becomes intensified) requires further investigation.

It is straightforward to extend this model to simulate other observation processes, such as passing within some region of study (as might be observed by a time-lapse camera, for example). Additionally, the Vatistas model functions can be used to simulate the wind speed and direction that might be measured at a fixed point. Furthermore, parameters measured at several fixed points, by a distributed array of pressure sensors, for example, can be simulated to assess how well miss distance, core pressure drop, and diameter can be independently estimated.

In future work we will also apply this model to Mars, where there exists a handful of barometric surveys from landers and dust devil counts from various spacecraft, as well as LESs. Furthermore, an additional insight into the vortex population is the formation of persistent trails on the surface. Such tracks likely have a threshold condition for their formation that is more rarely exceeded than that for forming a visible dust devil.

Acknowledgments

This work was funded by NASA through the Mars Fundamental Research Program Grant NNX12AI04G.

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Footnotes

1

Complete in the sense of detecting all devils present in the survey area. Surveys in small areas may statistically encounter less than one devil for large sizes and so are not complete in the sense of fully characterizing the population unless they are conducted for a long enough period—see Lorenz (2011).