The need for downdrafts

Carroll and Ryan (1970) were perhaps the first to speculate that downdrafts are ultimately responsible for the production of surface vertical vorticity in the convective boundary layer. The requirement for downdrafts in producing ground-level rotation in the absence of preexisting vertical vorticity has been highlighted in the field of tornado research (e.g., Davies-Jones 1982; Davies-Jones and Brooks 1993; Walko 1993; Parker and Dahl 2015; Fischer et al. 2024). The main idea is that appreciable vertical vorticity cannot arise at the ground by updraft gradients tilting horizontal vortex lines vertically: As the vorticity is being reoriented vertically, the air rises away from the ground, and consequently significant vertical vorticity only develops some distance above the ground (see also Markowski and Richardson 2010, p. 277; Markowski et al. 2014). While the mechanism by which surface vertical vorticity develops in supercell thunderstorms prior to tornadogenesis is reasonably well understood (Davies-Jones and Brooks 1993; see also Dahl et al. 2014; Markowski and Richardson 2014; Markowski et al. 2014; Parker and Dahl 2015; Fischer et al. 2024), such a mechanism does not seem to have been discussed in the context of boundary layer vortices.

The purpose of this study is thus to revisit the question of where the rotation in numerically simulated dust-devil-like vortices (DDLVs) originates from, focusing on the period prior to the development of fully developed DDLVs. By beginning the analysis when the atmosphere is still at rest, 3D vorticity dynamics along air parcel trajectories can be applied to identify the origin of vertical vorticity at the bottom of the CBL.

a. Development of the convective boundary layer and vorticity noise

To trigger convective mixing in the boundary layer, potential temperature (θ) perturbations with a maximum amplitude of 0.1 K were added in the entire domain at the first time step of the simulation. Without such perturbations, no mixing ensues even in the presence of superadiabatic near-surface temperature lapse rates. Interestingly, weak vertical vorticity at the lowest scalar model level (2.5 m AGL) already appears a few seconds after the simulation has started. This development is related to the θ perturbations, which are associated with weak buoyancy perturbations, which result in weak horizontal baroclinically produced circulations. The weak horizontal vorticity thus produced is reoriented and leads to extrema of vertical vorticity on the order of 10⁻⁴ s⁻¹ throughout the domain. The resulting vertical vorticity is an order of magnitude weaker than the vorticity that develops around 400 s into the simulation when convective mixing due to the developing superadiabatic surface layer sets in. Indeed, Fig. 1 shows that the perturbations associated with the convection driven by surface heating is much stronger than the random perturbations, which are visible as weak noise throughout the domain. In other words, the weak initial vertical vorticity is associated with the potential-temperature noise and is not related to the developing convective boundary layer.

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(a) Vertical cross section (y = 0.495 km) of the potential-temperature perturbation (shaded) and vertical velocity [black contours, (−1, −0.5, 0.5, 1) m s⁻¹] at 400 s. (b) Vertical vorticity (shaded) and horizontal vorticity magnitude [black contours, (−0.1, −0.05, 0.05, 0.1) s⁻¹] at 400 s.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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b. Early vertical vorticity evolution

The convective structures that develop about 400 s into the simulation are larger and more coherent compared to the random noise (Fig. 1). At that time, a cellular structure appears also in the horizontal plane, and vertical vorticity extrema that mirror this structure emerge (Fig. 2). It is this initial vertical vorticity whose origin is of interest. At this point, the CBL is still in its infancy: Mixing is weak (peak vertical velocities on the order of 0.5 m s⁻¹) and shallow (about 100 m).

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Vertical vorticity (shaded) and horizontal wind vectors (arrows) at the lowest model level (2.5 m AGL) at 600 s.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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To understand this initial development of surface vertical vorticity, 510 050 trajectories were launched at 300-s simulation time in a roughly 500 × 500 × 250 m³ box in the middle of the domain with an initial horizontal and vertical spacing of 5 m. Only those trajectories were considered that terminated close to the surface (between 2.5 and 5 m AGL) between 599 and 600 s and within a region of weak positive vertical vorticity (ζ ≥ 0.015 s⁻¹). These considerations reduced the number of useful parcels to n = 584. These trajectories are shown in Fig. 3. The parcels initially rest in the air above the growing CBL at roughly 60 m AGL and at some point are entrained into the CBL, with most parcels simply descending to the ground (others first rise and then descend). To analyze vorticity budgets, parcels were discarded if they descended below the lowest scalar model level, as the horizontal winds are not defined there (e.g., Vande Guchte and Dahl 2018). Moreover, the integrated vorticity budgets had to match the interpolated vorticity reasonably well; this determination was made subjectively (but in an automated manner, by filtering trajectories based on the accumulated absolute error of the integrated vorticity). Only about a dozen parcels remained after this procedure (n = 11), which are shown in black in Fig. 3.

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Trajectories (n = 584) that acquire weak near-surface vertical vorticity are shown as red line segments. The plan view also shows horizontal velocity vectors at the lowest model level (2.5 m AGL), as well as the magnitude of horizontal baroclinic vorticity production (shaded) and downdraft speed at 12.5 m AGL (contoured, w = −0.5 m s⁻¹) at 600 s. The black line segments represent the 11 trajectories analyzed in detail. In this plot, the baroclinic vorticity production strength is approximated as |∇_hB|, where B is the buoyancy.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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Figure 4a shows the different integrated terms of Eq. (2) for the average parcel. Perhaps not surprisingly, baroclinic production is the dominant contribution in the horizontal vorticity budget. Moreover, the parcels are descending while acquiring their horizontal vorticity (Fig. 4b). This baroclinically produced vorticity is associated with the convective circulations that make up the CBL. Horizontal stretching of the baroclinically produced vorticity occurs as the horizontal flow diverges and accelerated away from the downdraft center. Turbulent mixing and numerical diffusion contribute slightly negatively close to the ground.

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(a) Integrated budgets of the horizontal vorticity magnitude of the average trajectory (n = 11). The shaded areas show the 10th–90th percentiles of the integrals. (b) As in (a), but for the vertical vorticity component. The average parcel height is shown as the black dashed line (the heights are shown on the axis on the right side of the panel).

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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The vertical vorticity budgets are shown in Fig. 4b. Aside from relatively small contributions from SGS mixing and implicit diffusion, positive vertical vorticity arises from upward tilting of the baroclinically produced horizontal vorticity while the parcels are descending. The effect of the stretching term is negative because of horizontal divergence at the base of the downdraft. This vertical vorticity acquisition is visualized in Fig. 5, which shows the vorticity vectors along the average trajectory in the (x, z) plane. The vorticity vectors are initially negligibly small in magnitude and subsequently attain a nonzero horizontal component due to baroclinic production. As the trajectory approaches the ground, the vorticity is tilted upward (using all 584 parcels to calculate the average vorticity evolution reveals the same general picture of upward tilting during descent).

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Projection of the 3D average trajectory (n = 11) and its vorticity vectors onto the (x, z) plane.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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To demonstrate the general principle of how the horizontal geometry of downdrafts affects vertical-vorticity production, two separate, idealized downdraft simulations were run also using the CM1 model, but in a different configuration than the CBL simulation. The simulations feature an artificial heat sink, which triggers and maintains a downdraft. The model settings are essentially the same as in Parker and Dahl (2015) and Fischer and Dahl (2020) and are summarized in the appendix. For these simulations, a horizontal grid spacing of 50 m was used. The heat sink had a circular horizontal footprint in one simulation and an elliptic footprint in the other.

In the simulation with the circular footprint, surface outflow spreads out radially, implying that the baroclinically produced vortex lines likewise are circles, and consequently they do not intersect the updraft contours as shown in Fig. 6a. As a result, the vertical vorticity at the surface remains practically zero, as seen in Fig. 6b. The reason that the outflow spreads out in an axisymmetric manner is that the region of high pressure, which drives the horizontal outflow, likewise is circular at low levels (red contours in Fig. 6c). This behavior is ultimately tied to the fact that the Laplacian of the pressure field is approximately determined by velocity and buoyancy gradients (e.g., Rotunno and Klemp 1982). In the Boussinesq limit, this may be written as

$-\frac{1}{{\rho }_0}{\nabla }^2{p}^{\prime }=\left[{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial \upsilon }{\partial y}\right)}^2+{\left(\frac{\partial w}{\partial z}\right)}^2\right]+2\left(\frac{\partial u}{\partial y}\frac{\partial \upsilon }{\partial x}+\frac{\partial u}{\partial z}\frac{\partial w}{\partial x}+\frac{\partial \upsilon }{\partial z}\frac{\partial w}{\partial y}\right)-\frac{\partial B}{\partial z},$(4)

where ρ₀ is the base-state density and p′ is the perturbation pressure [see, e.g., Markowski and Richardson (2010, p. 27) for a derivation and interpretation of this equation]. The terms in the first square bracket are referred to as the fluid-extension (FE) terms, and the second bracket contains the shear terms; the last term describes buoyancy (B) forcing. This equation may be used to attribute the pressure field to flow features (e.g., a stagnation point is associated with pressure maximum). In the downdraft simulations, the largest forcing comes from the fluid-extension terms, which are also shown in Fig. 6c for the circular downdraft geometry. Indeed, the perturbation pressure forcing is positive (implying positive perturbation pressure), and it is axisymmetric, thus driving radial outflow (Fig. 6c).

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(a) Vertical velocity [shaded and contoured, (−5, −3, −2, −1) s⁻¹], horizontal vorticity ${\mathit{\omega }}_h^{\text{shr}}$ (arrows), and the tilting term of the vertical vorticity equation (contoured in red, −2, −1, 1, 2 × 10⁻⁴ s⁻²) at 142 m AGL. The tilting term is theoretically zero but attains small values due to numerical noise. (b) Vertical velocity at 142 m AGL [contoured, (−5, −2, −1, 1, 2, 5) m s⁻¹] and 2.5 m AGL wind vectors (arrows) as well as 2.5 m AGL vertical vorticity (shaded). (c) FE terms (shaded), perturbation pressure [red contours, (0.5, 1.0, 1.5) hPa] at 2.5 m AGL, wind vectors at 2.5 m AGL, and vertical velocity at 142 m AGL [black contours, (−5, −2, −1, 1, 2, 5) m s⁻¹]. The time is 360 s for all panels.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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If the horizontal footprint of the downdraft is elliptic, the symmetry is broken and ${\mathit{\omega }}_h^{\text{shr}}$ does intersect the w contours as depicted in Fig. 7a. The tilting term exhibits a quadrupole pattern, resulting in a cloverleaf-like distribution of vertical vorticity at the lowest model level (Fig. 7b).³ As shown in Fig. 7c, the fluid-extension terms are maximized beneath the downdraft, and they exhibit nearly the same ellipticity as the downdraft contours. The pressure field itself, however, is more circular than the downdraft contours because the solution of a Poisson equation tends to be more “spread out” than the forcing (e.g., Shapiro and Kanak 2002). Since the outflow is driven by the horizontal perturbation pressure gradient, v_h is nonorthogonal to the w contours, and ${\mathit{\omega }}_h^{\text{shr}}$ is nonparallel to the w contours. The result is upward/downward tilting of the baroclinically produced horizontal vorticity in descending air. Shapiro and Kanak (2002) discuss a similar scenario in the context of an ellipsoidal thermal, which results in a quadrupole-like distribution of vertical vorticity within the thermal [in their case, the perturbation pressure field is forced by the buoyancy term in Eq. (4), however].

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(a) Vertical velocity [shaded and contoured, (−5, −3, −2, −1) × m s⁻¹], horizontal vorticity ${\mathit{\omega }}_h^{\text{shr}}$ (arrows), and the tilting term of the vertical vorticity equation (contoured in red, −2, −1, 1, 2 × 10⁻⁴ s⁻²) at 142 m AGL. (b) Vertical velocity at 142 m AGL [contoured, (−5, −2, −1, 1, 2, 5) m s⁻¹] and 2.5 m AGL wind vectors (arrows) as well as 2.5 m AGL vertical vorticity (shaded). (c) FE terms (shaded), perturbation pressure [red contours, (0.5, 1.0, 1.5) hPa] at 2.5 m AGL, wind vectors at 2.5 m AGL, and vertical velocity at 142 m AGL [black contours, (−5, −2, −1, 1, 2, 5) m s⁻¹]. The time is 360 s for all panels.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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Indeed, 1270 s into the present CBL simulation, a downdraft loosely featuring a cloverleaf-like vertical-vorticity field is present as shown in Fig. 8 (most downdrafts produce less organized vertical vorticity patterns because they exhibit seemingly arbitrary nonaxisymmetric geometries).

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Vertical vorticity (shaded; plus and minus signs show the signs of vertical vorticity extrema produced by the downdraft) at the lowest model level (2.5 m AGL) at 1270 s. Also shown is the vertical velocity at 22.5 m AGL (black contours; m s⁻¹) and horizontal convergence at 2.5 m AGL (red contour; 0.01 s⁻¹).

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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The cellular vertical-vorticity pattern at the lowest model level gradually widens and intensifies as the CBL deepens [as also found by Kanak et al. (2000)]. However, the vertical vorticity at the bottom of the CBL is still maintained primarily by the same baroclinic mechanism just described. Indeed, Fig. 9 shows that the horizontal vorticity vectors are mostly parallel to the potential-temperature contours at 1420 s, implying that the vorticity is predominantly of baroclinic origin. The tilting term is also shown in Fig. 9, but to reduce clutter, only the positive values are considered; also, the tilting term is only plotted where it coincides with downdraft to highlight regions where vertical vorticity arises in descending air (black contours in Fig. 9). The red contours show the resulting positive vertical vorticity, which tends to be located close to the regions of positive tilting, though there is some offset because vertical vorticity is maximized where the most vertical stretching has subsequently been accumulated, which is in regions of horizontal convergence at the periphery of the downdraft cells. Negative vertical vorticity contours are suppressed in Fig. 9 for better visibility, but the downdraft mechanism produces positive and negative vertical vorticity with the average vertical vorticity in the domain being practically zero.

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Horizontal vorticity vectors (arrows), potential temperature perturbations (shaded), and tilting term [black contours, (0.01, 0.05, 0.1) × 10⁻² s⁻²] shown only where w < 0, at 12.5 m AGL and 1420 s. Vertical positive vorticity at 12.5 m AGL (s⁻¹) is shown in red contours.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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To summarize, it is the nonaxisymmetric horizontal footprint of the downdraft cells that allows for the baroclinically produced horizontal vorticity to be tilted into the vertical within the downdrafts. In this simulation, this mechanism is the origin of vertical vorticity at the lowest model level.

As the CBL grows, an interesting aspect becomes evident: After an initial increase, the average vertical vorticity magnitude at the lowest model level decreases in time (Fig. 10b), while the maximum increases (Fig. 10c). The reason is that the regions of vertical vorticity accumulate beneath the horizontally convergent updraft regions of the convective cells. As the diameter of the cells increases, there are larger “voids” of vertical vorticity while the intensity of the vorticity increases.

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Time series of (a) sensible heat flux, $\overline{{\theta }^{\prime }{w}^{\prime }}$, (b) average vertical vorticity magnitude at the lowest model level, (c) maximum vertical vorticity magnitude at the lowest model level (red line) and domain-wide maximum horizontal vorticity (green line), (d) maximum deformation magnitude at the lowest model level, and (e) maximum Okubo–Weiss number at the lowest model level.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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c. Vortex formation

The presence of vertical vorticity at the lowest model level does not imply the presence of vortices (e.g., Jeong and Hussain 1995). The reason is that aside from vorticity, there usually is also nonzero straining motion present in the flow. If, e.g., there is deformation and vorticity,⁴ and each has the same magnitude, the result is simply a sheared, unidirectional flow within the patch of vorticity (Fig. 11a; see also Batchelor 2002, p. 83). Only if the vorticity is much stronger than the straining motion, the flow exhibits a spinning motion and a vortex is present (Fig. 11b). In the ideal case of pure, solid-body-like rotation, the straining motion is zero.

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(a),(b) Example of the linear superposition of a flow field characterized by stretching deformation, and the flow associated with a circular uniform vortex patch (i.e., a Rankine vortex profile). In (a), the deformation strength is the same as the rotation strength (0.2 s⁻¹), and the result is a sheared, rectilinear flow in the vortex patch (black circle). In (b), the rotation magnitude is 5 times that of the deformation, and a vortex is present.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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In the present simulation, the straining motion is associated with the convective cells. Deformation, like horizontal convergence, is maximized beneath the low-level updraft regions, serving as conduits where the vortex patches are collected and along which the vortex patches propagate (Fig. 12). The maximum deformation magnitude at the lowest model level increases as the CBL grows (Fig. 10d). Despite this increase, Fig. 10e shows that the maximum OW still grows as the simulation progresses, implying that vorticity progressively dominates over deformation and that vortex patches increasingly become associated with vortices (e.g., closed streamlines and low pressure centers). Why does the vorticity become dominant relative to the deformation?

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Deformation magnitude ($\text{Def}=\sqrt{D_1^2+D_2^2}$; shaded), and vertical vorticity at the lowest model level [contoured, ζ = (−0.05, 0.05) s⁻¹] at 2860 s.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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The initial vortex patches are small in size and weak in magnitude and seem to exhibit behavior similar to that shown in Fig. 11a, which is a linear superposition of the deformation and rotation parts with no closed streamline patterns. Hence, their small OW numbers are as seen in Fig. 13a. A possible explanation for the subsequent increase in OW is that horizontal convergence strongly affects the vertical vorticity due to conservation of angular momentum, while the deformation magnitude appears not to be affected as much by horizontal convergence. The result is that vorticity increases beyond the deformation magnitude.

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Okubo–Weiss number (shaded), vertical vorticity [contoured, ζ = (−0.3, −0.1, −0.05, 0.05, 0.1, 0.3) s⁻¹], and horizontal wind vectors at the lowest model level, at (a) 1070 and (b) 2860 s.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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Moreover, aside from an increase of vorticity, the deformation magnitude tends to decrease within the developing vortex cores (Fig. 14). The reason for the decrease of the deformation is not obvious but may be related to mergers of individually weak, like-signed vortex patches, also visible in Fig. 14. These mergers are mostly a stochastic process, depending on the supply of like-signed vortex patches from the downdraft regions. Following these mergers, the resulting vortex patch increases in size, and at that point, there appears to be a significant amount of interaction between the deformation and rotation parts of the flow, and the deformation decreases in the vortex core (and it increases outside the vortex core, which may contribute to the increase of deformation in Fig. 10d). A similar behavior was observed in the idealized 2D simulations by Dahl (2020), who showed that during vortex formation, while the vorticity remained constant, the ratio between vorticity and deformation increased (implying a decrease in deformation in the developing vortex core). It seems that both of these effects, the increase in vorticity via stretching and the decrease in deformation within the vortex patches, lead to an increase in the OW number and hence vortex formation.

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Deformation magnitude (shaded), horizontal wind vectors, and vertical vorticity [contoured, ζ = (−0.5, −0.3, −0.1, 0.1, 0.3, 0.5) s⁻¹] at the lowest model level: (a) 2730 and (b) 2850 s. The labels A–D refer to individual cyclonic vortex patches.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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d. Vorticity dynamics of a maturing vortex

Although the dynamics of the mature vortices are not the focus of this study due to their somewhat poor representation given the relatively coarse resolution of the simulation, a few insights into the structure and dynamics of these vortices may still be gleaned.

Notably, as circular streamlines are attained, the flow interacts with the lower boundary, and the structure of the vortices in the present simulation is broadly consistent with seminal early work that has focused on the interaction of a vortex with the lower boundary, e.g., Barcilon (1967), Burggraf et al. (1971), and Rotunno (1980). That work has more recently been reviewed and expanded upon by Rotunno (2013, 2014). The main idea is that a vortex boundary layer develops, in which, due to the frictional retardation of the near-ground azimuthal flow, radial vorticity arises (i.e., the horizontal near-ground vorticity vectors point toward the vortex axis in a cyclonic vortex). In addition, the air in the vortex boundary layer accelerates toward the vortex center because of the imbalance between the pressure-gradient force and the centrifugal force in that layer (e.g., Rotunno 2013). As a consequence, the boundary layer air converges toward the axis of rotation where this air is forced to rise, thereby abruptly tilting the radial vorticity into the vertical in the vortex center. The result is a swirling upward-directed jet, a configuration referred to as an endwall vortex because the jet consists of the endwall boundary layer air of the parent vortex (Rotunno 2013). As shown in Fig. 15a, there is indeed radial vorticity at the base of the vortices shown. Now, because the 3D vorticity vector field is solenoidal, i.e., ∇ ⋅ ω = 0, it follows that

${\nabla }_h\cdot {\mathit{\omega }}_h=-\frac{\partial \zeta }{\partial z}.$(6)

That is, in a region where the horizontal vortex lines converge near the surface, they must bend upward, as supported by Fig. 15b (The vorticity vectors point away from the vortex center in anticyclonic vortices, as seen in Fig. 15a). These results are consistent with the analysis by Simpson and Glezer (2016), who likewise found that it was the near-ground horizontal vorticity associated with surface drag that is tilted upward at the base of their modeled vortex. In the present simulation, the radial inflow into the vortex is comparatively weak, probably because the vortex boundary layer is poorly resolved. However, the tilting term is indeed positive at the base of the vortex (but exhibits a wavenumber m = 2 asymmetry), as seen in Fig. 16.

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(a) Horizontal divergence of the horizontal vorticity vectors (shaded; m⁻¹ s⁻¹), horizontal vorticity vectors (arrows), and vertical vorticity [red contours, (−0.3, −0.1, 0.1, 0.3) × s⁻¹] at z = 2.5 m AGL and 2780 s. (b) Vertical cross section of vertical vorticity (shaded) and vorticity vectors in the (x, z) plane (arrows) at y = 2885 m.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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Tilting term of the vertical vorticity equation (shaded), horizontal wind vectors, and vertical vorticity [contoured, ζ = (−0.5, −0.3, −0.1, 0.1, 0.3, 0.5) s⁻¹] at the lowest model level at 2950 s.

Citation: Journal of the Atmospheric Sciences 81, 11; 10.1175/JAS-D-23-0228.1

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This brief analysis is consistent with the notion that, as the vortex matures, it is mainly radial vorticity due to surface drag that feeds into the vortex base—in contrast to the early stages, where the main vorticity source is baroclinic production. Similar dynamics were recently identified in idealized simulations of tornado-like vortices (Fischer and Dahl 2022) and are also implied by the conceptual model of Raasch and Franke (2011; their Fig. 15).

a. Relation to previous vorticity analyses

The results presented here are consistent with previous studies. Importantly, horizontal baroclinic vorticity production and subsequent tilting has long been considered a viable source of vertical vorticity (e.g., Carroll and Ryan 1970; Kanak et al. 2000; Fiedler and Kanak 2001; Renno et al. 2004; Ito et al. 2010, among others). A new aspect in the present study is the identification of the baroclinic downdraft mechanism that leads to vertical vorticity at the bottom of the CBL, which is the ultimate origin of DDLVs in the present simulation. Moreover, studies considering fully developed vortices have consistently found that tilting and stretching is large at the vortex base (e.g., Ohno and Takemi 2010; Raasch and Franke 2011; Ito et al. 2010; Simpson and Glezer 2016). These findings are consistent with the abrupt upward tilting of horizontal vorticity generated at the lower boundary found in the present analysis after the vortex has formed.

When considering the origin of the vortex, the vorticity prior to its development is most relevant; once a fully developed vortex is present, its interaction with the lower boundary typically reveals large tilting and stretching of the vorticity present in the vortex boundary layer. While this aspect is relevant for the maintenance (and perhaps, decay) of the vortex, the vorticity budgets at that point usually no longer reveal why the vortex formed in the first place.

b. Potential additional vorticity contributions

One possible origin of vorticity was excluded from this study by using a resting initial state. Real-world CBLs are usually sheared given nonzero horizontal background flow. In sheared CBLs, the horizontal vorticity associated with the background flow may be tilted into the vertical and potentially contribute to weak vertical vorticity. However, it seems that fundamentally the background wind shear is not needed because boundary layer vortices also form without its presence. Indeed, too strong a background flow seems to be detrimental to dust-devil formation (e.g., Ito et al. 2010), probably because the superadiabatic surface layer is weakened due to turbulence in strong-wind conditions (Rafkin et al. 2016). Perhaps the elongated downdraft geometry in sheared flows also affects the surface vertical vorticity production (horizontal rolls may be less efficient in delivering vertical vorticity at the ground).

c. Possible alternative view on rapid vortex spin-up

Visually, it often appears that a dust devil is the result of a particularly buoyant thermal that suddenly rises, thereby rapidly concentrating the available surface vertical vorticity via vertical stretching. However, based on the argument in the previous section, it also seems possible that the rapid vortex intensification may be dominated by the development of the radial inflow associated with the secondary circulation in the vortex boundary layer. In this case, one might likewise observe a sudden onset of rotation and near-surface upward motion. As this intensification does not require buoyancy forces (e.g., Rotunno 2014), the upward acceleration would mainly be driven by the dynamic vertical perturbation pressure gradient force.

d. Requirement for vertical-vorticity production at the downdraft base

The present analysis suggests that negatively buoyant downdrafts tend to produce surface vertical vorticity as long as their horizontal geometry is noncircular. Previous analyses identified nonzero downdraft-relative flow as requirement for vertical-vorticity production in downdrafts (e.g., Davies-Jones 2002; Straka et al. 2007; Parker and Dahl 2015). In Parker and Dahl’s (2015) analysis, the through-flow was needed because the heat sink was circular, so the only way of creating nonaxisymmetric downdrafts was the ambient wind. This condition is not necessary, however, as shown herein. That is, even without background flow, the downdraft structures tend to be nonaxisymmetric. This has important ramifications for precipitating deep convection: Whenever the downdraft footprint is noncircular (i.e., in practically all cases), one should expect regions of nonzero vertical vorticity within the storm’s downdraft. While this vertical vorticity in most instances cannot easily be concentrated, it suggests that practically all precipitating convective clouds likely produce at least some weak vertical vorticity within their outflow.

e. Parallels to the vorticity dynamics of supercell tornadoes

The chain of events presented here that lead to the formation of dust-devil-like boundary layer vortices in many respects resembles the mechanisms by which tornadoes form in supercells in the absence of preexisting vertical vorticity (e.g., Fischer et al. 2024). In supercells, the negatively buoyant main downdraft northeast of the mesocyclone produces horizontal vorticity baroclinically at its periphery (e.g., Rotunno and Klemp 1985; Markowski and Richardson 2014; Dahl et al. 2014; Parker and Dahl 2015). It is this initially horizontal vorticity that is tilted into the vertical while the air is descending, resulting in regions of vertical vorticity in the storm’s outflow (e.g., Davies-Jones and Brooks 1993; Markowski and Richardson 2014; Fischer and Dahl 2020). This baroclinic downdraft mechanism, at a smaller scale, is also responsible for the vertical vorticity at the bottom of the CBL. In supercells, this vorticity subsequently intensifies due to horizontal convergence along the leading edge of, as well as along internal boundaries within, the storm’s outflow. As in the CBL, the resulting vertical vorticity maxima in supercells organize and merge (Dahl 2020; Parker 2023), eventually leading to pressure deficits and circular streamlines, setting the stage for vortex intensification as the secondary circulation in the vortex boundary layer develops (e.g., Fischer et al. 2024). From a pure vorticity dynamics perspective, tornado formation in a supercell and dust-devil formation thus seem rather similar.