Formation Mechanism of Dust Devil–Like Vortices in Idealized Convective Mixed Layers

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JUNSHI ITO AND HIROSHI NIINO Atmosphere and Ocean Research Institute, The University of Tokyo, Kashiwa, Japan

MIKIO NAKANISHI National Defense Academy, Yokosuka, Japan

(Manuscript received 5 March 2012, in ﬁnal form 9 November 2012)

ABSTRACT

Dust devils are small-scale vertical vortices often observed over deserts or bare land during the daytime under fair weather conditions. Previous numerical studies have demonstrated that dust devil–like vertical vortices can be simulated in idealized convective mixed layers in the absence of background winds or envi- ronmental shear. Their formation mechanism, however, has not been completely clariﬁed. In this paper, the authors attempt to clarify the vorticity source of a dust devil–like vortex by means of a large-eddy simulation, in which a material surface initially placed in the vortex is tracked backward and the circulation on the material surface is examined. The material surface is found to originate from downdrafts, which already have sufﬁcient circulation. As the material surface converges toward the vortex, the vorticity is increased because of conservation of circulation. It is shown that a convective mixed layer is inherently accompanied by circulation, which is scaled by a product of the convective velocity scale and the depth of the convective mixed layer. This circulation is considered to be originally generated by tilting of baroclinically generated horizontal vorticity principally at middepths of the convective mixed layer.

1. Introduction Taylor 2010b). Indeed, Fujiwara et al. (2011, 2012) detected a number of invisible vertical vortices in con- Dust devils are small-scale vertical vortices often ob- vective mixed layers by a Doppler lidar. Experimental served over deserts and bare land in the early afternoon (e.g., Willis and Deardorff 1974) and numerical (e.g., under fair weather conditions. Previous studies (e.g., Kanak et al. 2000; Toigo et al. 2003; Kanak 2005, 2006; Williams 1948; Sinclair 1965; Snow and McClelland Ito et al. 2010; Gheynani and Taylor 2010a; Raasch and 1990; Greeley et al. 2006; Oke et al. 2007; Kurgansky Franke 2011) studies have succeeded in simulating dust et al. 2011) have shown that dust devils are ubiquitous devil–like vortices (DDVs) in convective mixed layers. in convective mixed layers. In fact, we know that dust A natural question to be raised then is how such strong devils also occur in Mars’s atmosphere during the day- vortices are formed in convective mixed layers. The time (e.g., Cantor et al. 2006; Greeley et al. 2006; Balme preceding numerical studies, which have demonstrated and Greeley 2006) and that they may contribute to formation of DDVs in convective mixed layers in the global warming on Mars (Fenton et al. 2007). These absence of background winds, shed some light on this vortices are believed to occur even when dust particles question (Kanak et al. 2000; Toigo et al. 2003; Kanak to be picked up from the bottom surface of the vortices 2005; Ohno and Takemi 2010; Raasch and Franke 2011). are not available: a feedback of the dust particles is not However, the source of the vertical vorticity and the essential for the vortices (Sinclair 1969; Gheynani and formation mechanism of the vortices remain contro- versial, and no quantitative study on the source of the vertical vorticity has been performed. Corresponding author address: Junshi Ito, Atmosphere and Ocean Research Institute, The University of Tokyo, 5-1-5 Ka- For DDVs to form, there must be a source of vertical shiwanoha, Kashiwa, Chiba 277-8564, Japan. vorticity that is stretched by an updraft caused by air E-mail: [email protected] with a large buoyancy originating from the surface layer

DOI: 10.1175/JAS-D-12-085.1

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TABLE 1. Summary of the experimental design.

Domain (km) 1.8 3 1.8 3 1.6 Grid points 362 3 362 3 320 Grid spacing (m) 5 2 Surface heat flux Q (K m s 1) 0.24 3 sin[p(t 2 7)/11] subject to a superadiabatic lapse rate. Several hypothe- ses 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 FIG. 1. Vertical profiles of horizontally averaged potential D (mech. D): nonuniform convergence in convection temperatures at 1-h time increments from 0700 to 1200 LST. cells and associated horizontal shear near the ground (Carroll and Ryan 1970; Cortese and Balachandar 1993). circulation using an area integral of vorticity over a ma- Recently, numerical studies have started to provide im- terial surface (MS), which has an advantage of giving portant suggestions for the generation mechanism of additional information about how tilting and stretching DDVs. Kanak et al. (2000) mentioned that mech. C oc- of vorticity take place. curred in their large-eddy simulation (LES) while mech. The next section describes a numerical simulation A and mech. B did not. Kanak (2005) later suggested of DDVs in a convective mixed layer, performed at that mech. D occurs in low-level convergence zones a very fine resolution. Section 3 describes results of the associated with convective cells, while mech. C occurs backward-trajectory analysis on the circulation. The at convective cell vertices. Raasch and Franke (2011) results are discussed in section 4, and conclusions are performed an LES with an extraordinarily fine mesh, given in section 5. and suggested that mech. C is significant for the formation of DDVs while mech. D can operate only in short- 2. Numerical methodology lived vertical vortices. This paper attempts to quantitatively examine the a. Model description ways in which vertical vorticity of DDVs is generated in The LES model used in the present study is the same the convective mixed layer without background wind. as that described by Nakanishi (2000) and Ito et al. For this purpose, a backward-trajectory analysis may (2010), except that the bottom boundary condition is provide a useful approach. Using a backward-trajectory changed to free slip. A brief description of the model is analysis, Markowski and Hannon (2006) examined the given in this subsection. vorticity budgets of miso-scale vortices that occur in The resolved-scale momentum equation, the ther- convective mixed layers and have a horizontal scale modynamic equation, and the continuity equation un- larger than that of dust devils. However, a disadvantage der the Boussinesq approximation are, respectively, of the vorticity analysis is that vorticity is not conserved, expressed as even in the absence of turbulent mixing or baroclinic production, when stretching and compression are pres- ›u ›u u 1 ›p ›t g i 1 j i 52 2 ij 1 (u 2 u )d , (1) ent. Thus, we will instead examine a circulation, which is ›t ›x r ›x ›x u 0 i3 defined as an area integral of vorticity or a line integral j 0 i j 0 of velocity along a closed curve and is a conserved ›u ›u u ›t quantity in the absence of turbulent mixing and baro- 1 j 52 uj › › › , and (2) clinic production. A similar analysis using a circulation t xj xj was performed by Mashiko et al. (2009), who studied the ›u mechanism of a tornadogenesis. While they calculated i 5 › 0, (3) the circulation using a line integral, we calculated the xi

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21 FIG. 2. Iso-surface of vertical vorticity of 0.25 (red) and 20.25 s (green) at 1210 LST. The lower part of the calculation domain at heights , 100 m is displayed. The color shading shows 2 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. where the overbars denote resolved-scale variables; scale (SGS) motions; dij is the Kronecker delta; g is the ui (i 5 1, 2, 3) the velocity components (u, y, and w)in gravitational acceleration; u is the potential tempera- the x, y, and z directions, respectively; p is the pertur- ture; u0 is the basic potential temperature; r0 is the air bation pressure; tij is the stress tensor due to subgrid- density; and tuj is the SGS heat ﬂux. The Coriolis force is

FIG. 3. 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.

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FIG. 5. Example of a backward trajectory of an air parcel initially located in the core region of a DDV.

is set to 5 m. The surface heat flux Q is prescribed by FIG. 4. Vertical profile of maximum vertical vorticity in the domain 5 3 p 2 at 1210 LST. a sinusoidal function, Q Qmax sin[ (t 7)/11], where t is the local standard time (LST) in hours and 21 Qmax is set to 0.24 K m s . The initial potential temperature increases linearly with height at a constant rate not considered. SGS fluxes t and tu in Eqs. (1) and (2) 2 ij j of 4.0 K km 1 from the surface to the top of the com- are modeled after Smagorinsky (1963) and Lilly (1966) putational domain (Fig. 1). Time integration is started at (see appendix). At the ground surface, w is set equal to 0. 0700 LST. To reduce computational cost, the size of the Lateral boundary conditions are cyclic, and the upper calculation domain is set to 1.8 km in the horizontal boundary conditions are w 5 0, free slip for u and y, and directions and 1.6 km in the vertical direction. Although adiabatic for u. this domain is not as large as that used by Ito et al. Spatial derivatives are approximated by second-order (2010), development of the convective mixed layer (Fig. centered differences on a staggered grid system. Time 1) and cellular convection with DDVs are reasonably integration is performed by a second-order Adams– similar to those in Ito et al. (2010) (Fig. 2). Note that the Bashforth scheme with a time step of 0.2 s. A predictor– top of the convective mixed layer does not reach the top corrector scheme is used for the momentum equations of calculation domain until 1230 LST. to ensure incompressibility. The Poisson equation for The trajectory analyses are performed using simula- pressure is solved by applying the Fourier transforma- tion data stored at each time step (0.2 s) between 1200 tion in the horizontal directions and finite differencing and 1210 LST. A number of clockwise and anticlockwise in the vertical direction. To avoid reflections of gravity 2 vertical vortices with absolute vorticity exceeding 1.0 s 1 waves from the upper boundary, a Rayleigh friction can be observed during this period. Note that vertical term is added to all the prognostic equations in the up- 2 2 vorticity in the background is on the order of 10 3 s 1, per 10 layers. which is two orders of magnitude smaller than that of b. Experimental design DDVs. Maximum absolute values of the vertical vorticity and the convection pattern observed in the present The experimental design is nearly the same as that simulation appear to be similar to a simulation with described by Ito et al. (2010) except for smaller domain surface friction (not shown), suggesting that the effects size, finer resolution, and the bottom boundary condi- of surface friction are not essential for the formation of tion for momentum. Table 1 summarizes the experi- DDVs. mental settings. Backward trajectories of air parcels located in the updraft region of a DDV tend to pass near the ground, 3. Analysis of the generation mechanism of DDVs and the nonslip condition makes the calculation of the a. Tracking of a material surface backward trajectories extremely complex. The present LES assumes zero surface friction, and starts from a Hereafter, the analysis will concentrate on a strong horizontally uniform initial state. Thus, there is no mean DDV at 1210 LST, indicated by the white arrow in Fig. 2. vertical shear. This configuration completely excludes This DDV is a one-celled vortex with an updraft occu- the formation of DDVs by mech. A or mech. B. pying its center (Fig. 3). It has the largest vertical vor- 2 To perform accurate backward-trajectory analyses, ticity (0.8 s 1) and pressure depression (30 Pa) among a fine grid size is desired; in the present LES, the grid size vortices in the simulated domain at this moment. Note

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FIG. 6. Time–space evolution of an MS. The color shading shows the height of each point on the MS (m). also that the radius of the maximum wind of the vortex is The tracked MS is initially (at time t 5 0) a horizontal approximately 20 m. Figure 4 shows vertical distribution square of 20 m 3 20 m at z 5 7.5 m (Fig. 6a). The MS is of maximum vertical vorticity over each horizontal cross divided into about 40 000 triangular patches, whose ver- section, which gives a typical vertical profile of vertical tices are tracked with about 20 000 backward trajectories. vorticity in a DDV. The vertical vorticity in a DDV It turns out that some of the patches experience enor- generally increases toward the ground (Fig. 4). Hence, mous expansions, which cause a significant reduction in we examine the generation mechanism of the vertical the accuracy of the analysis. To circumvent this prob- vorticity in a DDV at lower levels. lem, we follow Goto and Kida (2007): if the side length Backward trajectories of fluid particles are obtained of each patch becomes longer than 9.5 m, the triangle is from the LES data recorded at each time step. Back- divided into two. In this way, the area of patches is kept ward time integration is performed by a second-order small throughout the tracking; otherwise, the number of Adams–Bashforth scheme. Figure 5 shows a backward the patches increases by a factor of hundreds, between trajectory of an air parcel that is placed in the core of t 5 0 and t 52120 s, and the number of patches con- the DDV at a height of z 5 7.5 m at 1210 LST, and is tinues to grow exponentially when we track the patches tracked for 128 s. At an early stage of the backward further; therefore, we stopped the tracking at t 5 tracking, the parcel exhibits a downward helical move- 2128 s. Tracking for times prior to t 52128 s or from ment, whereas at a later stage, it shows a weak upward various other positions or instances may seem desirable, motion, indicating that the parcel is in a downdraft but the associated computational costs are large. Track- region. ing is performed based on the resolved-scale velocity. In In the present study, multiple trajectories are used to the analysis of circulation, however, we also take into track an MS. The procedure for tracking the MS follows account contributions from the modeled SGS motions, that of Goto and Kida (2007). An MS is constructed by as described later [see FT in Eq. (5)]. a set of small triangles, whose vertices are tracked as The time evolution of the tracked MS, together with three backward trajectories. the heights of its parts, is displayed in Figs. 6b–e. As the

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FIG. 7. Time series of (a) the circulation G, its horizontal components Gx and Gy, and vertical component Gz [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,andx–y plane Sxy;and(d)meanvorticityvm [G/S over the MS.

tracking proceeds, the MS moves downward while ro- under the Boussinesq approximation. We can write FT tating around the core of the DDV (Fig. 6b). It then and B as expands its size horizontally near the ground (Fig. 6c). ð dG › › The outer edge of the MS eventually begins to move 5 F 1 B 5 t dS T kli› › ij k upward as it expands into downdraft regions (Fig. 6e). dt S x x ð l j › g b. Circulation associated with the MS 1 u 2 u d dS kli› u ( 0) i3 k , (5) S x A circulation G on the MS, denoted by S, is deﬁned as l 0 the inner product of a vorticity vector v 5 (j, h, z) [ where kli is a permutation symbol. (›w/›y 2 ›y/›z, ›u/›z 2 ›w/›x, ›y/›x 2 ›u/›y) and a sur- The solid line in Fig. 7a shows the time evolution of face element vector S normal to the MS: G on the MS. The circulation is roughly conserved be- ð tween 2128 and 255 s, increases slightly between 255 2 2 G5 v dS. (4) and 40 s, and then decreases between 40 and 0 s. S The causes for the circulation change are examined on the basis of Eq. (5). Figure 7b shows the time series

A summation of vp dSp for all the triangular patches of FT together with its horizontal and vertical compo- p is used to approximate the right-hand side (RHS) of nents. The magnitude jBj is found to be always less 2 22 Eq. (4), where vp is the vorticity vector at the center of than 0.2 m s , and does not contribute to the circu- gravity of triangular patch p and Sp is the area of the lation change (not shown). These results indicate that patch. The circulation is conserved through advection, horizontal turbulent transport is mainly responsible for although it is varied by turbulent transport FT and baro- the changes in circulation between 255 and 0 s, and clinic generation of horizontal vorticity B. Note that that baroclinic generation is negligible throughout the vertical vorticity is not generated by baroclinic processes tracking.

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FIG. 8. Turbulent transport kli(›/›xl)(›/›xj)tijnk at each point on the MS at t 5240 s, where nk is the k component of the normal vector of MS. Color shading represents the magnitude 2 of turbulent transport (s 2).

4. Discussion regions (Fig. 6e). The impact of the tilting on circulation can be evaluated by assessing the x, y, and z components, a. Contraction of the MS to form a DDV according to Deformation of the MS gives an interpretation of ð ð ð information obtained from the conventional vorticity G5G 1G 1G 5 j 1 h 1 z x y z dSx dSy dSz , (7) budget analysis. The vertical vorticity equation is S S S

Dz ›w ›w ›w where dS , dS , and dS are, respectively, the x, y, and z 5 j 1 h 1 z 1 f , (6) x y z Dt ›x ›y ›z d components of the surface element vector of the MS. Time series of Gx, Gy, and Gz are displayed in Fig. 7a. 52 G where fd denotes the contribution of turbulent mixing. Before t 90 s, values of x are negative. Tilting of The ﬁrst and second terms in the RHS of Eq. (6) are the tilting of horizontal vorticity and the third term is the stretching of vertical vorticity. Contraction of the MS implies stretching of vorticity, whereas tilting of the MS corresponds to tilting of vorticity. Figure 7c shows the time series of the area of the MS together with its projections on the y–z plane Syz, the x–z plane Sxz, and the x–y plane Sxy. It is seen that the change of the area of the MS is mainly contributed by a decrease of Sxy. Because circulation is roughly conserved except for stages at which most parts of the MS are located in the core of the DDV (as shown in Fig. 7a), stretching of vertical vorticity is the major cause of vertical vorticity ampliﬁca- tions in the DDV.

As the MS is horizontal at t 5 0, both Syz and Sxz are initially 0. However, Syz and Sxz begin to have nonzero values as the backward tracking proceeds. Tilting of the FIG. 9. As in Fig. 7a, except that the initial MS is placed at a height MS occurs at its outer edges when it reaches downdraft of z 5 17.5 m.

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21 FIG. 10. Time series of vertical vorticity iso-surfaces of 0.25 (red) and 20.25 s (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 2 2 of radius 300 m at each grid point at z 5 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.

vorticity in the MS from horizontal (j) to vertical (z) of angular momentum (Fig. 7b). Figure 8 shows the is occurring during this period, but this has a negative distribution of turbulent transport on the MS, which contribution to z. implies outward turbulent transport of angular mo- After t 5255 s, the MS approaches the DDV, and mentum. However, the circulation redistributed by the both stretching and considerable turbulent transport turbulent mixing is considerably smaller than the mag- mixing begin to occur (Figs. 7b,c). Circulation increases nitude of the initial circulation in the MS. Figure 7d between t 5255 and 240 s (Fig. 7a) when the major shows the evolution of mean vorticity in the MS vm, part of the MS is located outside of the core region of defined as G/S, where S is the area of the MS. It is seen the DDV. This is caused by horizontal turbulent trans- that the vorticity is continuously amplified by stretch- port of angular momentum1 from the core region of the ing of the MS, even in the presence of outward hori- DDV. After t 5240 s, the circulation begins to decrease zontal turbulent transport of angular momentum (Figs. (Fig. 7a) because of the horizontal turbulent transport 7b,c). When a MS is placed slightly higher (z 5 17.5 m) and is tracked backward (Fig. 9), the MS simply passes 1 Angular momentum is the same measure as circulation on through z 5 7.5 m in the DDV, and shows a nearly the MS if changes of the air density are negligible. We refer to similar evolution of the circulation to Fig. 7b. ‘‘angular momentum’’ here instead of circulation because, although circulation is always associated with an MS, angular mo- b. Circulation around simulated DDVs mentum can be defined locally. What is transported by turbulence through an MS, and results in circulation changes in an MS, is the Section 4a examined the formation mechanism of angular momentum. a particular DDV. Hereafter, we attempt to generalize

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FIG. 10. (Continued) our discussion of the formation of the large vertical example, a region of negative circulation near the vorticity in DDVs in convective mixed layers. Because southeast corner at 1204 LST, a region of positive the computational cost of MS tracking is extremely circulation slightly southwest of the domain center at large, we need to employ a reasonable alternative to 1206 LST, and so on. The circulation in these down- the backward-trajectory analysis when analyzing a draft regions is a possible source for the formation number of DDVs. To this end, we consider circulation of DDVs if it is nearly conserved when the air con- within horizontal circles, which is centered at each verges to narrow updraft regions. The DDV, which grid point and has a radius of 300 m. This choice of the is tracked in section 3, is indicated by the arrow la- radiusisbasedonFig.7c,whichshowsthattheMSat beled ‘‘A’’ in Fig. 10c. Figures 10c and 10d show that t 52120 s is nearly horizontal and has its areal size the DDV A is accompanied by positive circulation of 106 m2. in surrounding downdraft regions. In the subsequent Figure 10 shows circulations calculated for a circle evolution of the DDV (Fig. 10f; 1210 LST), the sig- of 300-m radius around each grid point at z 5 2.5 m niﬁcant positive circulation around DDV A disap- from 1200 to 1214 LST together with the distribution of pears (Fig. 10g), and the DDV starts to dissipate (Fig. DDVs. To exclude the contribution of DDVs them- 10h). selves to the circulations, vertical vorticity only in down- We should mention that the data in Fig. 10 are not draft regions is integrated. Although the calculations do always explained by such behavior: for example, DDV not exclude a possibility that circulation in downdraft A seems to be affected by the negative circulation regions in the core of two-celled vortices could con- around it at 1208 LST. A part of the problem results tribute to the circulation, its contribution is believed to from the crude assumption that the low-level air con- be small as compared with the total circulation over verges from circles of 300-m radius (cf. Fig. 6). a circular region of radius 300 m. Figure 10 shows that A series of similar life cycles is also observed for other downdraft regions have signiﬁcant circulation—for DDVs: for example, DDV ‘‘B,’’ which displays negative

Unauthenticated | Downloaded 09/30/21 09:10 AM UTC 1182 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 70 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 (Gz), 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 [s(Gz)] near the surface (z 5 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. We examined whether the standard deviation of the FIG. 11. Time evolution of (a) standard deviations of circulation s G 5 circulation is subject to convective scaling. Figure 11b ( z), at z 2.5 m, where the abscissa is the radius over which the circulation is calculated; (b) as in (a), except that s(G ) is scaled by shows s(Gz) scaled by h 3 w , where h is the depth of z * h 3 w s G s G the mixed layer and w [ (gQh/u )1/3 is the convective *; and (c) ( z) at 1200 LST (solid line) together with ( z) * 0 for six different realizations of randomly distributed vorticity velocity scale (Deardorff 1970). The data show that (dotted and dashed lines). s G 3 ( z) is well scaled by h w* after 1000 LST, although some deviations exist at earlier times. This implies that s(Gz) is produced by the organized structure of It is possible that random fluctuations of vertical convection in convective mixed layers. In other words, vorticity having a normal distribution could induce a fi- convective mixed layers are inherently accompanied by nite s(Gz). Figure 11c shows s(Gz) obtained from the s(Gz), which can be a source of rotation for DDVs. LES at 1200 LST (solid line), together with s(Gz) values

Unauthenticated | Downloaded 09/30/21 09:10 AM UTC APRIL 2013 I T O E T A L . 1183 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 s(Gz) in the convective mixed layer is not a simple product of random ﬂuctua- tion, 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 s(Gz) 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. In the present convective mixed layer, tilting of horizontal vorticity is the only way to yield Gz composed only of vertical vorticity. The contribution of the tilting of horizontal vorticity to the circulation Gz in a horizontal plane Sh at a given height, denoted by Gt, is given by ð ð ›w ›w G 5 j dS 1 h dS. (8) t ›x ›y Sh Sh

Figure 12b shows standard deviations of Gt, s(Gt). The strongest tilting occurs at midlevels in the convective mixed layer, where the magnitudes of convective-scale horizontal vorticity and the velocities of updrafts and downdrafts are largest. Figure 12c shows a plot similar to that in Fig. 11c, except for the standard deviation of the tilting term. The random spatial distribution of the tilting term does not produce a signiﬁcant standard deviation of the tilting term when integrated over FIG. 12. Time evolution of (a) s(Gz) at different heights obtained a horizontal circle of a certain radius. The contribution by averaging data values at 1-min intervals between 1200 and of tilting to the circulation is likely to become large, 1230 LST; (b) as in (a), except that the standard deviation of the presumably because an organized structure of con- tilting term is integrated over horizontal circles of different radii; and (c) as in Fig. 11c, representing the standard deviation of the vective motions causes spatial correlation of tilting tilting term at z 5 0.25h (solid line), and that for six realizations terms. To clarify how these structures produce a sig- of a randomly redistributed tilting term (dotted and dashed lines). niﬁcant standard deviation of area-integrated tilting terms and resulting circulation, however, is left for fu- approaches zero as the radius approaches the size of ture study. the calculation domain (not shown). This demonstrates Themagnitudeofs(Gz) decreases as the radius that the probability of generating positive and negative of the area integration exceeds 800 m (Fig. 11), and vertical vorticities is equal.

Unauthenticated | Downloaded 09/30/21 09:10 AM UTC 1184 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 70 d. Generation mechanism of DDVs (2004) showed that surface friction has a strong in- fluence on the structure of DDVs and enhances their The above analyses show that DDVs are formed be- tangential velocities. To clarify the dynamics of real dust cause turbulent convection produces some circulation, devils, further observational and numerical studies are mainly at the midlevels of convection cells, which is desired: observations should focus not only on dust transported close to the surface via downdrafts and then devils themselves but also on circulation at the scale of to regions of strong updrafts, which amplifies vorticity convective cells, which may be observed by recent re- through stretching. Because tilting does not seem to mote sensing devices such as Doppler lidar (Fujiwara be significant in updraft regions near the surface, mech. et al. 2011). In addition, high-resolution numerical C may not be important for the formation of DDVs, simulations using more sophisticated boundary condi- even for those located at the vertices of convection cells. tions for momentum and heat fluxes and SGS parame- In contrast, mech. D may be possible if a convergent terizations near the surface should be developed. flow transports sufficient circulation to form DDVs. Merging of vertical vortices, as suggested by Ohno and 5. Conclusions Takemi (2010), can also occur, although it does not ex- plain the ultimate source of rotation for DDVs. To investigate the formation mechanism of dust devil- The present result that circulation in the lower hori- like vortices (DDVs) in a convective mixed layer, we zontal plane is important for DDV formation is also have performed an LES using a grid size of 5 m. The implied by results of our recent study (Ito et al. 2011) simulations have reproduced DDVs in the convective that examined the effects of ambient rotation on DDVs mixed layer with maximum vertical vorticities of ap- 2 by means of an LES. While the LES without ambient proximately 1.0 s 1. Analyses of circulation associated rotation shows no difference in the magnitudes of pos- with an MS, which is initially a horizontal square placed itive and negative vertical vorticity in DDVs, the in a strong DDV and is tracked backward, has been 2 2 planetary vorticity on the order of 10 4 s 1 causes a made to quantitatively clarify the formation mechanism difference between the negative and positive vertical of DDVs. 2 2 vorticity on the order of 10 2 s 1, without appreciably The MS that eventually flows into a DDV originates in changing the structure of the convective mixed layer. downdraft regions of convective cells, and converges This suggests that convergence into convective updrafts toward the DDV while reasonably conserving circula- stretches both planetary vorticity and the circulation tion. Statistical quantities in the convective mixed layer created within the convective turbulence by 100 times. show that the standard deviation of the circulation is As typical vertical vorticity in a DDV is of the order of scaled by h 3 w , where h is the depth of the convective 2 2 * 10 1 s 1 in an LES with a grid spacing on 50 m and mixed layer and w is the convective velocity scale, 2 2 * a prevailing vertical vorticity of 10 3 s 1, an estimated demonstrating that the circulation is intrinsic to the 2 convergence of air with a circulation of 250 m2 s 1 from convective mixed layer. When the MS approaches the a square area 500 m 3 500 m to a grid point can be in- DDV, the circulation increases because of turbulent ferred to create a DDV. This estimate is consistent with transport from the core region of the DDV. When it what is observed in the present study. eventually enters the core region, the circulation de- The present idealized LES model does not consider creases by the outward turbulent transport of angular the possibilities of DDV formation by mech. A or mech. momentum. This manner of DDV enhancement through B. However, stronger DDVs are likely to form in the conservation of circulation as the MS is contracted is presence of a source of vertical vorticity associated with also supported by the results of a numerical study (Ito an externally forced horizontal shear (Ito et al. 2011). A et al. 2011), which clarifies the effects of ambient rota- number of causes of externally forced horizontal shears tion on the formation of DDVs. in convective mixed layers exist in the atmosphere. The The ultimate source of the circulation available for frequency of actual dust devils appears to be less than generating DDVs appears to be produced through tilt- that of simulated DDVs in the LES, and actual dust ing of horizontal vorticity at the midlevels of the con- devils may form with the aid of circulation induced by vective mixed layer. The circulation generated in this external forcings. On the other hand, mech. B is not way is likely to be advected to the bottom of the con- likely to be important for the formation of DDVs, since vective mixed layer by downdrafts and then to the ver- an LES study by Ito et al. (2010) has shown that DDVs tices of the updrafts to form DDVs. do not favor strong vertical shear. However, Raasch and Franke (2011) showed that moderate vertical shear re- Acknowledgments. We thank Dr. Takashi Noguchi sult in their longer lifetime. Furthermore, Zhao et al. and Prof. Keita Iga for their useful suggestions. This

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