1. Introduction
Dust devils are small-scale vertical vortices often observed over deserts and bare land in the early afternoon under fair weather conditions. Previous studies (e.g., Williams 1948; Sinclair 1965; Snow and McClelland 1990; Greeley et al. 2006; Oke et al. 2007; Kurgansky et al. 2011) have shown that dust devils are ubiquitous in convective mixed layers. In fact, we know that dust devils also occur in Mars’s atmosphere during the daytime (e.g., Cantor et al. 2006; Greeley et al. 2006; Balme and Greeley 2006) and that they may contribute to global warming on Mars (Fenton et al. 2007). These vortices are believed to occur even when dust particles to be picked up from the bottom surface of the vortices are not available: a feedback of the dust particles is not essential for the vortices (Sinclair 1969; Gheynani and Taylor 2010b). Indeed, Fujiwara et al. (2011, 2012) detected a number of invisible vertical vortices in convective mixed layers by a Doppler lidar. Experimental (e.g., Willis and Deardorff 1974) and numerical (e.g., Kanak et al. 2000; Toigo et al. 2003; Kanak 2005, 2006; Ito et al. 2010; Gheynani and Taylor 2010a; Raasch and Franke 2011) studies have succeeded in simulating dust devil–like vortices (DDVs) in convective mixed layers.
A natural question to be raised then is how such strong vortices are formed in convective mixed layers. The preceding numerical studies, which have demonstrated formation of DDVs in convective mixed layers in the absence of background winds, shed some light on this question (Kanak et al. 2000; Toigo et al. 2003; Kanak 2005; Ohno and Takemi 2010; Raasch and Franke 2011). However, the source of the vertical vorticity and the formation mechanism of the vortices remain controversial, and no quantitative study on the source of the vertical vorticity has been performed.
For DDVs to form, there must be a source of vertical vorticity that is stretched by an updraft caused by air with a large buoyancy originating from the surface layer subject to a superadiabatic lapse rate. Several hypotheses have been proposed for the source of the vertical vorticity in DDVs: 1) mechanism A (mech. A): effects of topography or even a small animal (Williams 1948; Barcilon and Drazin 1972); 2) mechanism B (mech. B): tilting of horizontal vorticity, associated with vertical shear of a background wind, by the updraft of the DDV (Maxworthy 1973); 3) mechanism C (mech. C): tilting of horizontal vorticity, associated with a convective cell within the convective mixed layer, by the updraft of the convective cell (Willis and Deardorff 1979; Hess et al. 1988; Cortese and Balachandar 1993); and 4) mechanism D (mech. D): nonuniform convergence in convection cells and associated horizontal shear near the ground (Carroll and Ryan 1970; Cortese and Balachandar 1993). Recently, numerical studies have started to provide important suggestions for the generation mechanism of DDVs. Kanak et al. (2000) mentioned that mech. C occurred in their large-eddy simulation (LES) while mech. A and mech. B did not. Kanak (2005) later suggested that mech. D occurs in low-level convergence zones associated with convective cells, while mech. C occurs at convective cell vertices. Raasch and Franke (2011) performed an LES with an extraordinarily fine mesh, and suggested that mech. C is significant for the formation of DDVs while mech. D can operate only in short-lived vertical vortices.
This paper attempts to quantitatively examine the ways in which vertical vorticity of DDVs is generated in the convective mixed layer without background wind. For this purpose, a backward-trajectory analysis may provide a useful approach. Using a backward-trajectory analysis, Markowski and Hannon (2006) examined the vorticity budgets of miso-scale vortices that occur in convective mixed layers and have a horizontal scale larger than that of dust devils. However, a disadvantage of the vorticity analysis is that vorticity is not conserved, even in the absence of turbulent mixing or baroclinic production, when stretching and compression are present. Thus, we will instead examine a circulation, which is defined as an area integral of vorticity or a line integral of velocity along a closed curve and is a conserved quantity in the absence of turbulent mixing and baroclinic production. A similar analysis using a circulation was performed by Mashiko et al. (2009), who studied the mechanism of a tornadogenesis. While they calculated the circulation using a line integral, we calculated the circulation using an area integral of vorticity over a material surface (MS), which has an advantage of giving additional information about how tilting and stretching of vorticity take place.
The next section describes a numerical simulation of DDVs in a convective mixed layer, performed at a very fine resolution. Section 3 describes results of the backward-trajectory analysis on the circulation. The results are discussed in section 4, and conclusions are given in section 5.
2. Numerical methodology
a. Model description
The LES model used in the present study is the same as that described by Nakanishi (2000) and Ito et al. (2010), except that the bottom boundary condition is changed to free slip. A brief description of the model is given in this subsection.
Spatial derivatives are approximated by second-order centered differences on a staggered grid system. Time integration is performed by a second-order Adams–Bashforth scheme with a time step of 0.2 s. A predictor–corrector scheme is used for the momentum equations to ensure incompressibility. The Poisson equation for pressure is solved by applying the Fourier transformation in the horizontal directions and finite differencing in the vertical direction. To avoid reflections of gravity waves from the upper boundary, a Rayleigh friction term is added to all the prognostic equations in the upper 10 layers.
b. Experimental design
The experimental design is nearly the same as that described by Ito et al. (2010) except for smaller domain size, finer resolution, and the bottom boundary condition for momentum. Table 1 summarizes the experimental settings.
Summary of the experimental design.
Backward trajectories of air parcels located in the updraft region of a DDV tend to pass near the ground, and the nonslip condition makes the calculation of the backward trajectories extremely complex. The present LES assumes zero surface friction, and starts from a horizontally uniform initial state. Thus, there is no mean vertical shear. This configuration completely excludes the formation of DDVs by mech. A or mech. B.
To perform accurate backward-trajectory analyses, a fine grid size is desired; in the present LES, the grid size is set to 5 m. The surface heat flux Q is prescribed by a sinusoidal function, Q = Qmax × sin[π(t − 7)/11], where t is the local standard time (LST) in hours and Qmax is set to 0.24 K m s−1. The initial potential temperature increases linearly with height at a constant rate of 4.0 K km−1 from the surface to the top of the computational domain (Fig. 1). Time integration is started at 0700 LST. To reduce computational cost, the size of the calculation domain is set to 1.8 km in the horizontal directions and 1.6 km in the vertical direction. Although this domain is not as large as that used by Ito et al. (2010), development of the convective mixed layer (Fig. 1) and cellular convection with DDVs are reasonably similar to those in Ito et al. (2010) (Fig. 2). Note that the top of the convective mixed layer does not reach the top of calculation domain until 1230 LST.
Vertical profiles of horizontally averaged potential temperatures at 1-h time increments from 0700 to 1200 LST.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
Iso-surface of vertical vorticity of 0.25 (red) and −0.25 s−1 (green) at 1210 LST. The lower part of the calculation domain at heights < 100 m is displayed. The color shading shows the vertical velocity at the lowest model level (m s−1). The DDV to be examined in detail in this study is indicated by the white arrow.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
The trajectory analyses are performed using simulation data stored at each time step (0.2 s) between 1200 and 1210 LST. A number of clockwise and anticlockwise vertical vortices with absolute vorticity exceeding 1.0 s−1 can be observed during this period. Note that vertical vorticity in the background is on the order of 10−3 s−1, which is two orders of magnitude smaller than that of DDVs. Maximum absolute values of the vertical vorticity and the convection pattern observed in the present simulation appear to be similar to a simulation with surface friction (not shown), suggesting that the effects of surface friction are not essential for the formation of DDVs.
3. Analysis of the generation mechanism of DDVs
a. Tracking of a material surface
Hereafter, the analysis will concentrate on a strong DDV at 1210 LST, indicated by the white arrow in Fig. 2. This DDV is a one-celled vortex with an updraft occupying its center (Fig. 3). It has the largest vertical vorticity (0.8 s−1) and pressure depression (30 Pa) among vortices in the simulated domain at this moment. Note also that the radius of the maximum wind of the vortex is approximately 20 m. Figure 4 shows vertical distribution of maximum vertical vorticity over each horizontal cross section, which gives a typical vertical profile of vertical vorticity in a DDV. The vertical vorticity in a DDV generally increases toward the ground (Fig. 4). Hence, we examine the generation mechanism of the vertical vorticity in a DDV at lower levels.
Horizontal distribution of vertical velocity (gray scale) and the horizontal component of wind vectors (arrows) at 1210 LST at the lowest layer around the DDV for which backward trajectories were obtained.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
Vertical profile of maximum vertical vorticity in the domain at 1210 LST.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
Backward trajectories of fluid particles are obtained from the LES data recorded at each time step. Backward time integration is performed by a second-order Adams–Bashforth scheme. Figure 5 shows a backward trajectory of an air parcel that is placed in the core of the DDV at a height of z = 7.5 m at 1210 LST, and is tracked for 128 s. At an early stage of the backward tracking, the parcel exhibits a downward helical movement, whereas at a later stage, it shows a weak upward motion, indicating that the parcel is in a downdraft region.
Example of a backward trajectory of an air parcel initially located in the core region of a DDV.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
In the present study, multiple trajectories are used to track an MS. The procedure for tracking the MS follows that of Goto and Kida (2007). An MS is constructed by a set of small triangles, whose vertices are tracked as three backward trajectories.
The tracked MS is initially (at time t = 0) a horizontal square of 20 m × 20 m at z = 7.5 m (Fig. 6a). The MS is divided into about 40 000 triangular patches, whose vertices are tracked with about 20 000 backward trajectories. It turns out that some of the patches experience enormous expansions, which cause a significant reduction in the accuracy of the analysis. To circumvent this problem, we follow Goto and Kida (2007): if the side length of each patch becomes longer than 9.5 m, the triangle is divided into two. In this way, the area of patches is kept small throughout the tracking; otherwise, the number of the patches increases by a factor of hundreds, between t = 0 and t = −120 s, and the number of patches continues to grow exponentially when we track the patches further; therefore, we stopped the tracking at t = −128 s. Tracking for times prior to t = −128 s or from various other positions or instances may seem desirable, but the associated computational costs are large. Tracking is performed based on the resolved-scale velocity. In the analysis of circulation, however, we also take into account contributions from the modeled SGS motions, as described later [see FT in Eq. (5)].
Time–space evolution of an MS. The color shading shows the height of each point on the MS (m).
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
The time evolution of the tracked MS, together with the heights of its parts, is displayed in Figs. 6b–e. As the tracking proceeds, the MS moves downward while rotating around the core of the DDV (Fig. 6b). It then expands its size horizontally near the ground (Fig. 6c). The outer edge of the MS eventually begins to move upward as it expands into downdraft regions (Fig. 6e).
b. Circulation associated with the MS
The solid line in Fig. 7a shows the time evolution of Γ on the MS. The circulation is roughly conserved between −128 and −55 s, increases slightly between −55 and −40 s, and then decreases between −40 and 0 s.
Time series of (a) the circulation Γ, its horizontal components Γx and Γy, and vertical component Γz [Eq. (7)]; (b) the turbulent transport FT and its horizontal and vertical components; (c) the area of the MS and its projections on the y–z plane Syz, x–z plane Sxz, and x–y plane Sxy; and (d) mean vorticity ωm ≡ Γ/S over the MS.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
The causes for the circulation change are examined on the basis of Eq. (5). Figure 7b shows the time series of FT together with its horizontal and vertical components. The magnitude |B| is found to be always less than 0.2 m2 s−2, and does not contribute to the circulation change (not shown). These results indicate that horizontal turbulent transport is mainly responsible for the changes in circulation between −55 and 0 s, and that baroclinic generation is negligible throughout the tracking.
4. Discussion
a. Contraction of the MS to form a DDV
After t = −55 s, the MS approaches the DDV, and both stretching and considerable turbulent transport mixing begin to occur (Figs. 7b,c). Circulation increases between t = −55 and −40 s (Fig. 7a) when the major part of the MS is located outside of the core region of the DDV. This is caused by horizontal turbulent transport of angular momentum1 from the core region of the DDV. After t = −40 s, the circulation begins to decrease (Fig. 7a) because of the horizontal turbulent transport of angular momentum (Fig. 7b). Figure 8 shows the distribution of turbulent transport on the MS, which implies outward turbulent transport of angular momentum. However, the circulation redistributed by the turbulent mixing is considerably smaller than the magnitude of the initial circulation in the MS. Figure 7d shows the evolution of mean vorticity in the MS ωm, defined as Γ/S, where S is the area of the MS. It is seen that the vorticity is continuously amplified by stretching of the MS, even in the presence of outward horizontal turbulent transport of angular momentum (Figs. 7b,c).
Turbulent transport 
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
When a MS is placed slightly higher (z = 17.5 m) and is tracked backward (Fig. 9), the MS simply passes through z = 7.5 m in the DDV, and shows a nearly similar evolution of the circulation to Fig. 7b.
As in Fig. 7a, except that the initial MS is placed at a height of z = 17.5 m.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
b. Circulation around simulated DDVs
Section 4a examined the formation mechanism of a particular DDV. Hereafter, we attempt to generalize our discussion of the formation of the large vertical vorticity in DDVs in convective mixed layers. Because the computational cost of MS tracking is extremely large, we need to employ a reasonable alternative to the backward-trajectory analysis when analyzing a number of DDVs. To this end, we consider circulation within horizontal circles, which is centered at each grid point and has a radius of 300 m. This choice of the radius is based on Fig. 7c, which shows that the MS at t = −120 s is nearly horizontal and has its areal size of 106 m2.
Figure 10 shows circulations calculated for a circle of 300-m radius around each grid point at z = 2.5 m from 1200 to 1214 LST together with the distribution of DDVs. To exclude the contribution of DDVs themselves to the circulations, vertical vorticity only in downdraft regions is integrated. Although the calculations do not exclude a possibility that circulation in downdraft regions in the core of two-celled vortices could contribute to the circulation, its contribution is believed to be small as compared with the total circulation over a circular region of radius 300 m. Figure 10 shows that downdraft regions have significant circulation—for example, a region of negative circulation near the southeast corner at 1204 LST, a region of positive circulation slightly southwest of the domain center at 1206 LST, and so on. The circulation in these downdraft regions is a possible source for the formation of DDVs if it is nearly conserved when the air converges to narrow updraft regions. The DDV, which is tracked in section 3, is indicated by the arrow labeled “A” in Fig. 10c. Figures 10c and 10d show that the DDV A is accompanied by positive circulation in surrounding downdraft regions. In the subsequent evolution of the DDV (Fig. 10f; 1210 LST), the significant positive circulation around DDV A disappears (Fig. 10g), and the DDV starts to dissipate (Fig. 10h).
Time series of vertical vorticity iso-surfaces of 0.25 (red) and −0.25 s−1 (green) from (a) 1200 to (h) 1214 LST, at 2-min intervals. The lower domain at heights < 100 m is displayed. Circulation, calculated only for downdraft regions within a horizontal circle of radius 300 m at each grid point at z = 2.5 m, is shown by color shading ( m2 s−1). Regions where vertical velocities are > 0.1 m s−1 at the lowest model level are shown by gray isolines.
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
We should mention that the data in Fig. 10 are not always explained by such behavior: for example, DDV A seems to be affected by the negative circulation around it at 1208 LST. A part of the problem results from the crude assumption that the low-level air converges from circles of 300-m radius (cf. Fig. 6).
A series of similar life cycles is also observed for other DDVs: for example, DDV “B,” which displays negative vertical vorticity, as indicated in Fig. 10f, and DDV “C,” which displays positive vertical vorticity, as indicated in Fig. 10g. The typical time scale for a development of circulation seems to be several minutes, which is of the same order of magnitude as the duration of dust devils (cf. Oke et al. 2007). The relationship between DDVs and surrounding circulation, such as the one found for DDV A and other DDVs, implies that stretching of vertical vorticity occurs through convergence from a wide horizontal area to the updraft region, causing the DDV within several minutes (Fig. 7c).
Figure 10 suggests that the circulation necessary for forming DDVs commonly exists in the downdraft regions of the convective mixed layer. If there is convergence near the surface, DDVs would form. Stronger DDVs are likely to form at the vertices of convective cells, where the updrafts are stronger than those at their edges (Williams and Hacker 1993).
c. Characteristics of circulation generated in a convective mixed layer
The previous subsection shows that DDVs are formed by converging air, which possesses a significant z component of circulation (Γz), and flows from downdraft regions toward updraft regions. It is of interest to know where and how the circulation is generated in the convective mixed layer, which is statistically homogeneous in the horizontal directions. Since the probability that the circulation has either a positive or negative sign is equal, and that its horizontal average at each height is approximately zero, we employ a standard deviation as a representative statistical measure of the circulations. Figure 11 displays standard deviations of circulations in the horizontal plane [σ(Γz)] near the surface (z = 2.5 m), at five elapsed times, where the abscissa is the radius of the horizontal circle over which the circulation is calculated. The standard deviation increases with time as the convective mixed layer grows, until 1100 LST, but seems to attain a steady state by 1200 LST. Given a certain amount of circulation, as suggested by Fig. 11a, a DDV would form in regions where convergence toward an updraft exists.
Time evolution of (a) standard deviations of circulation σ(Γz), at z = 2.5 m, where the abscissa is the radius over which the circulation is calculated; (b) as in (a), except that σ(Γz) is scaled by h × w*; and (c) σ(Γz) at 1200 LST (solid line) together with σ(Γz) for six different realizations of randomly distributed vorticity (dotted and dashed lines).
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
We examined whether the standard deviation of the circulation is subject to convective scaling. Figure 11b shows σ(Γz) scaled by h × w*, where h is the depth of the mixed layer and w* ≡ (gQh/θ0)1/3 is the convective velocity scale (Deardorff 1970). The data show that σ(Γz) is well scaled by h × w* after 1000 LST, although some deviations exist at earlier times. This implies that σ(Γz) is produced by the organized structure of convection in convective mixed layers. In other words, convective mixed layers are inherently accompanied by σ(Γz), which can be a source of rotation for DDVs.
It is possible that random fluctuations of vertical vorticity having a normal distribution could induce a finite σ(Γz). Figure 11c shows σ(Γz) obtained from the LES at 1200 LST (solid line), together with σ(Γz) values derived from artificial random distributions of vertical vorticity (dotted and dashed lines). In the latter, values of vertical vorticity at all grid points at the lowest level are randomly redistributed so that systematic spatial correlations of vertical vorticity due to convection do not exist. Figure 11c shows that σ(Γz) in the convective mixed layer is not a simple product of random fluctuation, but is coherently produced by convection. These observations indicate that convective circulation is important for producing DDVs.
The present simulation begins with zero initial circulation over whole domain and remains so when integrated over the horizontal plane, yet it generates DDVs. This may be possible because convection is always associated with baroclinically generated horizontal vorticity, and tilting of the horizontal vorticity by differential vertical motions of convection produces vertical vorticity.
Figure 12a shows σ(Γz) at various heights, where time averaging over 30 min is performed to clarify the differences between different heights. It shows that the standard deviations of the circulations are largest at midlevels of the convective mixed layer, implying that the circulation is produced at midlevels and is advected toward the bottom and top of the convective mixed layer.
Time evolution of (a) σ(Γz) at different heights obtained by averaging data values at 1-min intervals between 1200 and 1230 LST; (b) as in (a), except that the standard deviation of the tilting term is integrated over horizontal circles of different radii; and (c) as in Fig. 11c, representing the standard deviation of the tilting term at z = 0.25h (solid line), and that for six realizations of a randomly redistributed tilting term (dotted and dashed lines).
Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-085.1
The magnitude of σ(Γz) decreases as the radius of the area integration exceeds 800 m (Fig. 11), and approaches zero as the radius approaches the size of the calculation domain (not shown). This demonstrates that the probability of generating positive and negative vertical vorticities is equal.
d. Generation mechanism of DDVs
The above analyses show that DDVs are formed because turbulent convection produces some circulation, mainly at the midlevels of convection cells, which is transported close to the surface via downdrafts and then to regions of strong updrafts, which amplifies vorticity through stretching. Because tilting does not seem to be significant in updraft regions near the surface, mech. C may not be important for the formation of DDVs, even for those located at the vertices of convection cells. In contrast, mech. D may be possible if a convergent flow transports sufficient circulation to form DDVs. Merging of vertical vortices, as suggested by Ohno and Takemi (2010), can also occur, although it does not explain the ultimate source of rotation for DDVs.
The present result that circulation in the lower horizontal plane is important for DDV formation is also implied by results of our recent study (Ito et al. 2011) that examined the effects of ambient rotation on DDVs by means of an LES. While the LES without ambient rotation shows no difference in the magnitudes of positive and negative vertical vorticity in DDVs, the planetary vorticity on the order of 10−4 s−1 causes a difference between the negative and positive vertical vorticity on the order of 10−2 s−1, without appreciably changing the structure of the convective mixed layer. This suggests that convergence into convective updrafts stretches both planetary vorticity and the circulation created within the convective turbulence by 100 times. As typical vertical vorticity in a DDV is of the order of 10−1 s−1 in an LES with a grid spacing on 50 m and a prevailing vertical vorticity of 10−3 s−1, an estimated convergence of air with a circulation of 250 m2 s−1 from a square area 500 m × 500 m to a grid point can be inferred to create a DDV. This estimate is consistent with what is observed in the present study.
The present idealized LES model does not consider the possibilities of DDV formation by mech. A or mech. B. However, stronger DDVs are likely to form in the presence of a source of vertical vorticity associated with an externally forced horizontal shear (Ito et al. 2011). A number of causes of externally forced horizontal shears in convective mixed layers exist in the atmosphere. The frequency of actual dust devils appears to be less than that of simulated DDVs in the LES, and actual dust devils may form with the aid of circulation induced by external forcings. On the other hand, mech. B is not likely to be important for the formation of DDVs, since an LES study by Ito et al. (2010) has shown that DDVs do not favor strong vertical shear. However, Raasch and Franke (2011) showed that moderate vertical shear result in their longer lifetime. Furthermore, Zhao et al. (2004) showed that surface friction has a strong influence on the structure of DDVs and enhances their tangential velocities. To clarify the dynamics of real dust devils, further observational and numerical studies are desired: observations should focus not only on dust devils themselves but also on circulation at the scale of convective cells, which may be observed by recent remote sensing devices such as Doppler lidar (Fujiwara et al. 2011). In addition, high-resolution numerical simulations using more sophisticated boundary conditions for momentum and heat fluxes and SGS parameterizations near the surface should be developed.
5. Conclusions
To investigate the formation mechanism of dust devil-like vortices (DDVs) in a convective mixed layer, we have performed an LES using a grid size of 5 m. The simulations have reproduced DDVs in the convective mixed layer with maximum vertical vorticities of approximately 1.0 s−1. Analyses of circulation associated with an MS, which is initially a horizontal square placed in a strong DDV and is tracked backward, has been made to quantitatively clarify the formation mechanism of DDVs.
The MS that eventually flows into a DDV originates in downdraft regions of convective cells, and converges toward the DDV while reasonably conserving circulation. Statistical quantities in the convective mixed layer show that the standard deviation of the circulation is scaled by h × w*, where h is the depth of the convective mixed layer and w* is the convective velocity scale, demonstrating that the circulation is intrinsic to the convective mixed layer. When the MS approaches the DDV, the circulation increases because of turbulent transport from the core region of the DDV. When it eventually enters the core region, the circulation decreases by the outward turbulent transport of angular momentum. This manner of DDV enhancement through conservation of circulation as the MS is contracted is also supported by the results of a numerical study (Ito et al. 2011), which clarifies the effects of ambient rotation on the formation of DDVs.
The ultimate source of the circulation available for generating DDVs appears to be produced through tilting of horizontal vorticity at the midlevels of the convective mixed layer. The circulation generated in this way is likely to be advected to the bottom of the convective mixed layer by downdrafts and then to the vertices of the updrafts to form DDVs.
Acknowledgments
We thank Dr. Takashi Noguchi and Prof. Keita Iga for their useful suggestions. This work was supported in part by a Grant-in-Aid for Scientific Research (B)(2) (21340134), the Japan Society for the Promotion of Science Projection of Planet Earth Variations for Mitigating Natural Disasters (Field 3), the Strategic Programs for Innovative Research in Establishment of the Research System for Computational Science, and Ministry of Education, Culture, Sports, Science, and Technology in Japan.
APPENDIX
SGS Model
REFERENCES
Balme, M., and R. Greeley, 2006: Dust devils on Earth and Mars. Rev. Geophys., 44, RG3003, doi:10.1029/2005RG000188.
Barcilon, A., and P. Drazin, 1972: Dust devil formation. Geophys. Fluid Dyn., 4, 147–158.
Cantor, B. A., K. M. Kanak, and K. S. Edgett, 2006: Mars Orbiter Camera observations of Martian dust devils and their tracks (September 1997 to January 2006) and evaluation of theoretical vortex models. J. Geophys. Res., 111, E12002, doi:10.1029/2006JE002700.
Carroll, J. J., and J. A. Ryan, 1970: Atmospheric vorticity and dust devil rotation. J. Geophys. Res., 75, 5179–5184.
Cortese, T., and S. Balachandar, 1993: Vortical nature of thermal plumes in turbulent convection. Phys. Fluids, A5, 3226–3232.
Deardorff, J. W., 1970: Convective velocity and temperature scale for the unstable planetary boundary layer and for Rayleigh convection. J. Atmos. Sci., 27, 1211–1213.
Fenton, L. K., P. E. Geissler, and R. M. Haberle, 2007: Global warming and climate forcing by recent albedo changes on Mars. Nature, 446, 646–649.
Fujiwara, C., K. Yamashita, M. Nakanishi, and Y. Fujiyoshi, 2011: Dust devil-like vortices in an urban area detected by a 3D scanning Doppler lidar. J. Appl. Meteor. Climatol., 50, 534–547.
Fujiwara, C., K. Yamashita, and Y. Fujiyoshi, 2012: Observed effect of mesoscale vertical vorticity on rotation sense of dust devil-like vortices in an urban area. SOLA, 8, 25–28.
Gheynani, B. T., and P. A. Taylor, 2010a: Large-eddy simulations of vertical vortex formation in the terrestrial and Martian convective boundary layers. Bound.-Layer Meteor., 137, 223–235.
Gheynani, B. T., and P. A. Taylor, 2010b: Large eddy simulation of typical dust devil-like vortices in highly convective Martian boundary layers at the Phoenix lander site. Planet. Space Sci., 59, 43–50.
Goto, S., and S. Kida, 2007: Reynolds-number dependence of line and surface stretching in turbulence: Folding effects. J. Fluid Mech., 586, 59–81.
Greeley, R., and Coauthors, 2006: Active dust devils in Gusev crater, Mars: Observations from the Mars Exploration Rover Spirit. J. Geophys. Res., 111, E12S09, doi:10.1029/2006JE002743.
Hess, G. D., K. T. Spillane, and R. S. Lourensz, 1988: Atmospheric vortices in shallow convection. J. Appl. Meteor., 27, 305–317.
Ito, J., H. Niino, and M. Nakanishi, 2010: Large eddy simulation of dust devils in a diurnally-evolving convective mixed layer. J. Meteor. Soc. Japan, 88, 64–77.
Ito, J., H. Niino, and M. Nakanishi, 2011: Effects of ambient rotation on dust devils. SOLA, 7, 165–168.
Kanak, K. M., 2005: Numerical simulation of dust devil-scale vortices. Quart. J. Roy. Meteor. Soc., 131, 1271–1292.
Kanak, K. M., 2006: On the numerical simulation of dust devil-like vortices in terrestrial and Martian convective boundary layers. Geophys. Res. Lett., 33, L19S05, doi:10.1029/2006GL026207.
Kanak, K. M., D. K. Lilly, and J. T. Snow, 2000: The formation of vertical vortices in the convective boundary layer. Quart. J. Roy. Meteor. Soc., 126, 2789–2810.
Kurgansky, M. V., A. Montecinos, V. Villagran, and S. M. Metzger, 2011: Micrometeorological conditions for dust-devil occurrence in the Atacama Desert. Bound.-Layer Meteor., 138, 285–298.
Lilly, D. K., 1966: On the application of the eddy-viscosity concept in the inertial subrange of turbulence. National Center for Atmospheric Research Manuscript 123, 19 pp.
Markowski, P., and C. Hannon, 2006: Multiple-Doppler radar observation of the evolution of vorticity extrema in a convective boundary layer. Mon. Wea. Rev., 134, 355–374.
Mashiko, W., H. Niino, and T. Kato, 2009: Numerical simulation of tornadogenesis in an outer rainband mini-super-cell of Typhoon Shanshan on 17 September 2006. Mon. Wea. Rev., 137, 4238–4260.
Maxworthy, T., 1973: A vorticity source for large scale dust devils and other comments on naturally occurring columnar vortices. J. Atmos. Sci., 30, 1717–1722.
Nakanishi, M., 2000: Large-eddy simulation of radiation fog. Bound.-Layer Meteor., 94, 461–493.
Ohno, H., and T. Takemi, 2010: Mechanisms for intensification and maintenance of numerically simulated dust devils. Atmos. Sci. Lett., 11, 27–32.
Oke, A. M. C., N. J. Tapper, and D. Dunkerley, 2007: Willy-willies in the Australian landscape: The role of key meteorological variables and surface conditions in defining frequency and spatial characteristics. J. Arid Environ., 71, 201–215.
Raasch, S. and T. Franke, 2011: Structure and formation of dust devil–like vortices in the atmospheric boundary layer: A high-resolution numerical study. J. Geophys. Res., 116, D16120, doi:10.1029/2011JD016010.
Sinclair, P. C., 1965: On the rotation of dust devils. Bull. Amer. Meteor. Soc., 46, 388–391.
Sinclair, P. C., 1969: General characteristics of dust devils. J. Appl. Meteor., 8, 32–45.
Smagorinsky, J., 1963: General circulation experiments with the primitive equations. I. The basic experiment. Mon. Wea. Rev., 91, 99–164.
Snow, J. T., and T. M. McClelland, 1990: Dust devils at White Sands Missile Range, New Mexico: 1. Temporal and spatial distributions. J. Geophys. Res., 95 (D9), 13 707–13 721.
Sullivan, P. P., J. C. McWilliams, and C. H. Moeng, 1994: A subgrid-scale model for large-eddy simulation of planetary boundary layer flows. Bound.-Layer Meteor., 71, 247–276.
Toigo, A. D., M. I. Richardson, S. P. Ewald, and P. J. Gierasch, 2003: Numerical simulation of Martian dust devils. J. Geophys. Res., 108, 5047, doi:10.1029/2002JE002002.
Williams, A. G., and J. M. Hacker, 1993: Interaction between coherent eddies in the lower convective boundary layer. Bound.-Layer Meteor., 64, 55–74.
Williams, N. R., 1948: Development of dust whirls and similar small scale vortices. Bull. Amer. Meteor. Soc., 29, 106–117.
Willis, G. E., and J. W. Deardorff, 1974: A laboratory model of the unstable planetary boundary layer. J. Atmos. Sci., 31, 1297–1307.
Willis, G. E., and J. W. Deardorff, 1979: Laboratory observations of turbulent penetrative-convection planforms. J. Geophys. Res., 84 (C1), 295–302.
Zhao, Y. Z., Z. L. Gu, Y. Z. Yu, Y. Ge, Y. Li, and X. Feng, 2004: Mechanism and large eddy simulation of dust devils. Atmos.–Ocean, 42, 61–84.
Angular momentum is the same measure as circulation on the MS if changes of the air density are negligible. We refer to “angular momentum” here instead of circulation because, although circulation is always associated with an MS, angular momentum can be defined locally. What is transported by turbulence through an MS, and results in circulation changes in an MS, is the angular momentum.
