Tilting of Horizontal Shear Vorticity and the Development of Updraft Rotation in Supercell Thunderstorms

Johannes M. L. Dahl Atmospheric Science Group, Department of Geosciences, Texas Tech University, Lubbock, Texas

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Abstract

The question of how rotation arises in sheared updrafts is analyzed using the shear and curvature vorticity framework. Local rotation exists where the shear and curvature vorticity have a similar magnitude and the same sign, such that parcels are in near-solid-body rotation. It is shown that the tilting terms of the vertical vorticity equation cannot explain the development of local rotation in the canonical cases where the horizontal vorticity is either purely streamwise or purely crosswise. Rather, vertical shear vorticity develops if crosswise vorticity is tilted, and vertical curvature vorticity develops if streamwise vorticity is tilted. To analyze how local rotation develops, two simulations of updrafts in an environment with crosswise and mostly streamwise vorticity, respectively, are discussed. A trajectory analysis is performed and shear and curvature vorticity budgets are analyzed. It is found that much of the horizontal vorticity near the updraft becomes streamwise, which results from pressure gradient accelerations in the vicinity of the updraft. Consequently, in the analyzed scenarios, the tilting mechanism results primarily in vertical curvature vorticity. Local rotation is achieved via an interchange process that facilitates a partial conversion of vertical curvature vorticity to vertical shear vorticity. Updraft rotation in supercells thus does not result from tilting of horizontal vorticity alone, but partial conversion of curvature to shear vorticity is also required.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. Johannes Dahl, johannes.dahl@ttu.edu

Abstract

The question of how rotation arises in sheared updrafts is analyzed using the shear and curvature vorticity framework. Local rotation exists where the shear and curvature vorticity have a similar magnitude and the same sign, such that parcels are in near-solid-body rotation. It is shown that the tilting terms of the vertical vorticity equation cannot explain the development of local rotation in the canonical cases where the horizontal vorticity is either purely streamwise or purely crosswise. Rather, vertical shear vorticity develops if crosswise vorticity is tilted, and vertical curvature vorticity develops if streamwise vorticity is tilted. To analyze how local rotation develops, two simulations of updrafts in an environment with crosswise and mostly streamwise vorticity, respectively, are discussed. A trajectory analysis is performed and shear and curvature vorticity budgets are analyzed. It is found that much of the horizontal vorticity near the updraft becomes streamwise, which results from pressure gradient accelerations in the vicinity of the updraft. Consequently, in the analyzed scenarios, the tilting mechanism results primarily in vertical curvature vorticity. Local rotation is achieved via an interchange process that facilitates a partial conversion of vertical curvature vorticity to vertical shear vorticity. Updraft rotation in supercells thus does not result from tilting of horizontal vorticity alone, but partial conversion of curvature to shear vorticity is also required.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Dr. Johannes Dahl, johannes.dahl@ttu.edu

1. Introduction

Supercell thunderstorms develop in environments with strong vertical wind shear and their defining—and arguably most spectacular—characteristic is their persistent, rotating updraft (Doswell and Burgess 1993). Although the association between updraft rotation and ambient wind shear has been recognized in some of the earliest formal documentations of rotating thunderstorms (Wegener 1917, 1918; Markgraf 1928), it was not until the 1960s that Ludlam (1963, p. 24) and Browning and Landry (1963) speculated that thunderstorm rotation may result from tilting of ambient shear vorticity. Barnes (1970) subsequently explored observational data, which supported the viability of this mechanism and it was around that time that these storms were christened “supercells” (Browning 1962, 1964). Early numerical modeling efforts further bolstered the notion of updraft rotation being a result of upward tilting of horizontal shear vorticity (Schlesinger 1975; Wilhelmson and Klemp 1978), which was put on strong theoretical footing by Rotunno (1981), Lilly (1982), Rotunno and Klemp (1982), Davies-Jones (1984), and Rotunno and Klemp (1985).1

The process is most succinctly conceptualized using horizontal vortex lines associated with the ambient sheared flow (e.g., Davies-Jones 1984; Klemp 1987). These vortex lines are materially reoriented by the updraft, thus explaining how vertical vorticity arises (Fig. 1). However, whether this vertical vorticity is associated with a vortex (section 3a) cannot be inferred from this framework. The fact that visualizations typically include horizontal roll vortices surrounding isolated horizontal vortex lines seems to imply that the tilting mechanism directly leads to a vertically oriented vortex. However, since no mesocyclone-scale horizontal roll vortex is observed in the environment of supercells, it is not clear how the resulting vertical vorticity manifests itself. Moreover, tilting of a horizontal roll vortex into the vertical at the updraft edge would probably not explain the observed vorticity structure in a supercell.2 Smaller-scale horizontal circulations [horizontal convective rolls (HCRs)] may exist in a supercell’s environment (Nowotarski et al. 2015), but the development of updraft rotation does not require HCRs. Rather, numerical simulations reliably produce supercells in horizontally homogeneous environments (e.g., Rotunno and Klemp 1985; Wicker and Wilhelmson 1995; Dahl et al. 2012; among many others). Moreover, the horizontal scale of the vertical vorticity extrema related to the interaction of HCRs with supercells is much smaller than the scale of the mesocyclone (Nowotarski et al. 2015).

Fig. 1.
Fig. 1.

Conceptual model of the tilting mechanism that leads to updraft rotation. The pink ribbons represent the vortical flow associated with the vortex lines, which are depicted as black line segments. The black arrows represent the ambient vertical wind profile. Adapted from Markowski and Richardson (2010, their Fig. 8.35a), based on Klemp (1987).

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

The question is thus how—and whether—the interaction of horizontal shear vorticity with an updraft results in a vertically oriented vortex. As will be shown, tilting of horizontal shear vorticity into the vertical may be understood in terms of differential vertical advection of horizontal momentum, and the resulting flow field is not a vortex in general.

A robust understanding of vortex formation in a convective updraft is desirable because of the concomitant pressure perturbations, which may facilitate updraft propagation (Rotunno and Klemp 1982; Weisman and Rotunno 2000; Davies-Jones 2002) and intensification (Brooks and Wilhelmson 1993; Weisman and Rotunno 2000). In fact, the pressure field due to the vorticity in a supercell’s updraft is the defining dynamical characteristic of a supercell that sets it apart from nonsupercellular storms, and it plays an important role in the supercell’s vigor and longevity. Moreover, hydrometeor pseudotrajectories strongly depend on the horizontal flow structure in the updraft (Dennis and Kumjian 2017), and an understanding of how the ambient wind profiles affect this structure may be helpful in developing more detailed knowledge about the growth mechanisms of large hailstones.

The purpose of this paper is to

  • clarify the mechanism behind the tilting term when horizontal shear vorticity is reoriented into the vertical, and

  • present the mechanisms that lead to updraft rotation.

The analysis begins with simple conceptualizations using canonical hodographs, which will be presented in section 2. A more formal approach, including shear- and curvature-vorticity equations as well as numerical simulations will be introduced in sections 3 and 4, and the results will be presented in section 5. A discussion is offered in section 6, followed by conclusions in section 7.

2. Conceptualization of the tilting term

The tilting term in the vertical vorticity equation arises from the curl of the advection term in the equation of motion (e.g., Markowski and Richardson 2010, p. 21), such that
e1
Here, is the vertical unit vector, is the 3D velocity, is the horizontal velocity, is the horizontal vorticity, ζ is the vertical vorticity, w is the vertical velocity, and is the horizontal gradient operator. Aside from the tilting term, which is the second term on the rhs of Eq. (1), the curl of momentum advection also yields the vorticity advection and divergence terms (first and last term, respectively). The tilting term incorporates the effect of materially rearranging vortex lines such that their vertical component changes, which, per Eq. (1), is equivalent to differential advection of momentum.3 Although vortex lines are an extremely powerful tool, some details of the flow remain hidden (e.g., a vortex line could represent either a vortex or a sheared flow). It thus seems most instructive to interpret the tilting term from the momentum perspective. Instead of considering the configuration of the curl of the velocity field, the velocity field itself will be analyzed.

To gain a qualitative understanding of the mechanism underlying the reorientation of horizontal shear vorticity into the vertical, a heuristic model using material surfaces will be used (Dahl 2006). Material surfaces may conveniently be defined by the common height of parcels at an arbitrary reference time (e.g., Davies-Jones 2015). These surfaces are Lagrangian surfaces (Z surfaces in the following) that become deformed as they materially follow the flow. Assuming a stationary updraft–downdraft couplet with nonzero ambient flow (and no critical level where the flow magnitude is zero), a Lagrangian surface will exhibit a bulge that lags the vertical-motion field by a quarter of a wavelength in the downstream direction [much like discussed by Holton and Hakim (2013, chapter 9) in the context of a flow over topography], consistent with the rendering in Fig. 2. The reason is that parcels accumulate vertical displacement while they are residing within an updraft. The Lagrangian surface will thus exhibit maximum vertical displacement just downstream of the updraft. In this kinematic approach no direct information is included—or needed—about the nature of the upward accelerations. However, an important assumption in this model is that there are no material horizontal accelerations. In other words, the flow at a given location in Eulerian space can only change as a result of vertical advection of horizontal momentum. This implies that the horizontal flow remains unchanged on the (initially horizontal) Lagrangian surfaces, even as they are deformed by the flow. This assumption is made to isolate the effects of differential momentum advection, which facilitates the interpretation of the tilting term.

Fig. 2.
Fig. 2.

Three-dimensional rendering of an upward bulging Z surface with streamlines. The horizontal motion (indicated by the gray arrow) remains unchanged as the flow crosses the hump.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Instead of considering a single Z surface, one might consider a number of vertically stacked Z surfaces, with flow speed and/or direction varying on each surface, as shown in Fig. 3. Each individual surface will be displaced upward (downward) within the updraft (downdraft), whereupon a stack of upward-bulging Z surfaces arises (Fig. 4a). Now consider a horizontal slice through this configuration at a given height and time t. The result is the horizontal Z field at time (Fig. 4b) at the height of interest. This field exhibits a minimum downstream of the updraft (or weaker updraft). The flank at which the updraft occurs thus depends on the horizontal wind direction on a given surface, and the Z minimum corresponds to a maximum in vertical displacement (at a given level, those parcels that have been displaced vertically the most had the smallest initial height).

Fig. 3.
Fig. 3.

Vertically stacked, horizontally oriented Z surfaces in the unperturbed environment. The arrows represent the wind vectors on each surface.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Fig. 4.
Fig. 4.

(a) Vertical cross section through the deformed stacked Z surfaces. An updraft is located at the upstream side of the hump. The red dashed line marks the location of the horizontal slice through the configuration. (b) Horizontal Z distribution at the height of the horizontal slice.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

As the horizontal flow on each Z surface is homogeneous and constant in time, the horizontal wind in Eulerian space is obtained simply from the initial vertical wind profile and the configuration of the time-dependent Z surfaces. The wind along each Z contour on a horizontal plane corresponds to the wind at the height where the respective Z surface originated. As a final assumption, the Z field is taken to be axisymmetric, which allows for an easy interpretation of the result, but which affects the shape of the up- and downdraft for a given wind profile. However, this assumption has no effect on the interpretation of the heuristic results.

a. Crosswise horizontal vorticity

Consider the case of vertically sheared, unidirectional flow, as shown in Fig. 3. If we assume the updraft to be stationary, this corresponds to the canonical crosswise vorticity scenario. The vertical cross section parallel to the flow through the stack of Z surfaces corresponds to the configuration depicted in Fig. 4a. The updraft is found at all levels to be on the left (upstream) side of the Z hills because the flow is unidirectional. Using six Z surfaces and allowing the horizontal flow to be preserved on these surfaces, a flow pattern shown in Fig. 5 arises. An updraft (downdraft) is present where the arrows have warm (cool) colors. As can be seen, differential momentum advection in this case leads to unidirectional flow within the updraft, which is not surprising because the ambient flow is also unidirectional at all levels. Momentum advection thus cannot engender changes in the flow direction. A couplet of vertical shear vorticity emerges, which is maximized on the downstream flanks of the updraft. This vorticity pattern is fully consistent with the models by Lilly (1982) and Davies-Jones (1984). However, no vortex is apparent, and the first conclusion—to be built on more rigorous grounds in the next section—is that if crosswise horizontal vorticity is tilted upward, vertical shear vorticity arises.

Fig. 5.
Fig. 5.

Horizontal velocity vectors at a given height that is penetrated by upward-bulging Z surfaces. The length of the arrows is proportional to the wind speed, and the gray circles represent the horizontal Z field. The normalized vertical speed W is coded according to the colorbar. The thin black line represents the 0 m s−1 updraft contour. Shear vorticity extrema are outlined with thick black contours [positive values (solid); negative values (dashed)]. A hodograph is plotted at the bottom-left corner, with updraft motion lying in the origin. The flow at each height mark corresponds to the flow on the respective Z contour in the horizontal cross section.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

b. Streamwise horizontal vorticity

As another canonical wind profile, consider a semicircular hodograph with veering shear in a reference frame in which the updraft–downdraft couplet is stationary and located at the hodograph’s center of curvature. The flow magnitude does not change with height, and, as in the previous example, the updraft is found on the local upstream side of the Z minimum. The result is shown in Fig. 6. The updraft spirals toward the center of the Z minimum, and it exhibits a cyclonically curved flow. Anticyclonic curvature is present in the downdraft (blue arrows in Fig. 6). There are no variations in flow magnitude within the updraft or downdraft because the wind speed is constant in the ambient wind profile.

Fig. 6.
Fig. 6.

As in Fig. 5, but for the streamwise vorticity scenario, and the thick black contours now outline extrema of curvature vorticity [positive values (solid); negative values (dashed)].

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

The implication is that if streamwise horizontal vorticity is tilted into the vertical, vertical curvature vorticity arises. Importantly, the cyclonic vorticity is located amid the updraft (rather than at its flanks), consistent with Lilly (1982) and Davies-Jones (1984).

This scenario highlights a curious aspect of the flow: the upward-bending streamlines in 3D space are straight lines if projected onto the horizontal plane. However, at a given level, the horizontal wind vectors associated with these 3D streamlines, which intersect that level, may result in a curved flow pattern if the wind is turning with height. In fact, the 3D streamlines may even be curved anticyclonically, and they may still result in cyclonic curvature on a 2D plane (Fig. 7), as was observed by Klemp and Wilhelmson (1981) and Lilly (1986).

Fig. 7.
Fig. 7.

Example of how anticyclonic 3D streamlines may result in cyclonic horizontal flow at a given level. The colored, directed line segments represent projections of ascending, anticyclonically curved streamlines. As they cross the level of interest, the horizontal velocity components yield a cyclonically curved flow, highlighted by the arrows. The black contour outlines the updraft.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

c. Intermediate horizontal vorticity

In the more common case where the ambient flow is characterized by both streamwise and crosswise vorticity, a pattern that combines both previous cases is obtained (Fig. 8). Here, the angle between the ambient vorticity and wind vectors varies between 10° and 35° with height. As shown in Fig. 8, the curvature vorticity tends to be maximized upstream of the Z minimum within the updraft, which is to be expected as the tilting of streamwise vorticity occurs upstream of the updraft (the streamwise vorticity component is parallel to the velocity vector). Shear vorticity, on the other hand, is accumulated at the lateral flanks of the Z minimum relative to the local flow, consistent with the pure-crosswise-vorticity scenario. The rule for intermediate shear profiles is thus that the streamwise horizontal vorticity component contributes to vertical curvature vorticity within the updraft, while the horizontal crosswise vorticity component results in vertical shear vorticity at the flanks of the updraft.

Fig. 8.
Fig. 8.

As in Fig. 5, but for a scenario where streamwise and crosswise vorticity are present. Shear vorticity extrema are outlined by the thick black contours [positive values (solid); negative values (dashed)], and curvature vorticity extrema are outlined by the thick green contours [positive values (solid); negative values (dashed)].

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Using (for now) the intuitive notion of a vortex as being characterized by a circular, closed streamline pattern, it is apparent that tilting of horizontal shear vorticity into the vertical alone generally cannot explain vortex formation in the updraft. In the following section, the above heuristic arguments will be approached from a more formal perspective.

3. Definition of rotation and the shear and curvature vorticity equations

a. Local rotation

Although a number of rather sophisticated criteria for identifying 3D vortex structures have been developed (e.g., Jeong and Hussain 1995; Haller 2005; Wu et al. 2006, p. 310 ff), on a 2D plane most of these vortex definitions are equivalent. An example of a vortex criterion is the Okubo–Weiss (OW) number (Wu et al. 2006; Markowski et al. 2011), which highlights flow regimes in which rotation dominates over strain (corresponding to a negative OW number). On a 2D plane, the OW number is related to the right-hand side of the Poisson pressure equation (Bradshaw and Koh 1981), which diagnoses a pressure minimum within and surrounding regions dominated by rotation. Thus, a negative OW number highlights vortices consistent with the intuitive measure of a vortex as a local pressure minimum (Wu et al. 2006). If inclusion of the vortex intensity is not desired in the definition of a vortex, one might use the kinematical vorticity number, which is the ratio of the rotation and strain rates (Truesdell 1954; Schielicke et al. 2016). While these measures are valuable in identifying vortex structures, they do not provide a useful dynamical framework for understanding vortex genesis. The most powerful tool to analyze vortex dynamics thus remains the vorticity framework, although it provides merely a local measure of rotation, and hence it does not unambiguously reveal vortex formation. In this work, the vorticity concept, refined by considering curvature and shear vorticity, will be applied. The following considerations connect the vorticity framework with the nonlocal definitions of a vortex:

  • A sheared flow is characterized by zero curvature vorticity and equal amounts of rotation and strain (e.g., Batchelor 2002, p. 83), which leads to zero dynamic pressure deficits and zero OW number, consistent with the intuitive notion that no vortex is present despite the nonzero vorticity. The implication is that a region of pure shear vorticity cannot contain a vortex.

  • Solid-body rotation, which is characterized by zero strain and pure rotation, exhibits equal amounts of shear and curvature vorticity (this is readily shown by writing ζ in polar coordinates). Given the absence of straining motion, there is a pressure deficit in the vortex center, and the OW number is negative, consistent with closed streamlines. Equal amounts of shear and curvature vorticity of like sign are thus required for solid-body rotation.

  • Finally, if a flow is characterized only by curvature vorticity and no shear, a closed streamline pattern cannot be achieved, and hence no vortex could be present.

Based on these considerations, a local vortex criterion is that shear and curvature vorticity ought to have the same sign and magnitude (solid-body rotation). However, since most real-world vortex cores are not in perfect solid-body rotation, the vortex criterion used herein is that shear and curvature vorticity need to have similar magnitudes but the same sign. Such a flow is in local solid-body-like rotation, which in the following will be referred to simply as “local rotation.” It is clear that no vortex, or local rotation, can exist if only one flavor of vorticity, curvature, or shear, is present. Also, consistent with the heuristic model, upward tilting of either streamwise or crosswise horizontal vorticity alone cannot result in local rotation. Note, however, that in intermediate cases such as discussed in section 2c, some parts of the inflow may contain suitably oriented streamwise and crosswise vorticity, which may in principle be tilted upward to result in local rotation. This point will briefly be addressed again at the end of section 5a.

b. Vertical shear and curvature vorticity equations

To analyze the problem from a more formal perspective, the vertical curvature and vertical shear vorticity equations are used. In the following, the qualifier “vertical” will be omitted for shear and curvature vorticity, and the qualifier “horizontal” will be omitted for streamwise and crosswise vorticity. The curvature and shear vorticity equations (Hollman 1958; Bleck 1991; Viúdez and Haney 1996) have been applied to synoptic-scale flows (Pichler and Steinacker 1987; Bell and Keyser 1993) but to the author’s knowledge not to supercells. Here we use the formulation by Viúdez and Haney (1996), but the equations are cast in height coordinates. Furthermore, the Coriolis force is neglected. As production of vertical vorticity by baroclinity and viscosity may be neglected at the scale of the mesocyclone, these equations are given by
e2
and
e3
The main steps of the derivation of these equations, and the definition of the symbols, are provided in the appendix. Equation (2) is the curvature vorticity equation. The physical content of this equation is that vertical curvature vorticity increases when (i) there is horizontal convergence acting on existing curvature vorticity, (ii) streamwise vorticity is tilted into the vertical, and (iii) the pressure field is distributed in a way that allows for an exchange between shear and curvature vorticity, while the total vorticity remains unchanged. The terms on the rhs of the shear vorticity equation [Eq. (3)] have an analogous interpretation, except that shear vorticity results from tilting of crosswise vorticity. Importantly, the tilting terms of these equations are fully consistent with the heuristic results obtained in section 2.

Equipped with this background, the implication is that tilting of horizontal streamwise or crosswise shear vorticity alone—which represent the two commonly discussed scenarios in supercell dynamics—cannot explain updraft rotation in the supercell. If shear (curvature) vorticity develops in the updraft because of the tilting and stretching terms, partial conversion to curvature (shear) is required for local rotation. In the remainder of the paper, the details of this mechanism will be explored.

4. Modeling framework

To address vortex formation in sheared updrafts, results from two numerical simulations using the Bryan Cloud Model 1 [CM1; Bryan and Fritsch (2002) and the appendix of Bryan and Morrison (2012)], release 17, will be presented. The horizontal grid spacing in both simulations is 250 m, and the vertical grid spacing increases from 100 m near the ground to 250 m at the top of the domain, which is at 20 km AGL. The horizontal domain size is about 120 km in each direction. A sponge layer is employed in the upper 4 km of the domain to limit reflection of vertically propagating gravity waves at the upper boundary. The lateral boundary conditions are open, and the top and bottom boundary conditions are free slip. The single-moment Gilmore et al. (2004) microphysics scheme is used (with a rain intercept parameter of 106 m−4). The thermodynamic profile in all simulations is taken from Weisman and Klemp (1982) (using a surface potential temperature of 300 K and a surface water vapor mixing ratio of 14 g kg−1).

To initiate the updrafts, convergence forcing (Loftus et al. 2008) as well as “warm bubble” forcing were used. Although the cells develop faster if the bubble forcing is utilized, the main results are insensitive to the forcing. However, while the convergence forcing is active, the artificial nature of this forcing would render budget calculations of shear and curvature vorticity unreliable. For this reason, the results from the experiments using the warm bubble will be discussed.

To apply the shear and curvature vorticity equations, a trajectory analysis was performed. Trajectories are calculated online within CM1 and obtained using a fourth-order Runge–Kutta scheme and trilinear spatial interpolation on every large model time step (2.0 s). The terms on the rhs of the shear and curvature vorticity equations are cast in Cartesian form (see appendix), approximated to second-order accuracy on staggered grids, and then interpolated to the parcels’ locations. Parcel data are stored every 10 s. Finally, the individual terms on the rhs of the vorticity equations are integrated over time using a trapezoidal scheme with a 10-s time step.4

The goal was to present simulations of updrafts in environments with purely crosswise and purely streamwise vorticity. The straightforward approach is to initialize updrafts in vertical wind profiles featuring straight and circular hodographs. In the straight-line hodograph case with zonal flow increasing linearly from −15 m s−1 at the ground to +15 m s−1 at 7.5 km AGL, at least initially the base-state vorticity is purely crosswise, rendering this case suitable for analysis. Unfortunately, the circular hodographs (using a full- and semicircle hodograph with a constant wind speed of 10 m s−1 and a veering rate of 60° km−1) do not result in streamwise updraft-relative base-state vorticity because the motion vector of the nascent updraft does not lie in the center of the hodograph. One reason is that the full-physics simulations include buoyancy, which is zero in the steady-state Beltrami solutions where updraft motion does fall on the center of the hodograph. However, almost perfectly streamwise vorticity in the inflow was attained by a mature supercell using the quarter-circle hodograph of Weisman and Rotunno (2000). While these two scenarios represent different stages in the storm’s life cycles, they do highlight the similarities and differences between the straight- and curved-hodograph cases. Also, the maturity of the vortex may not be as important because of the Lagrangian nature of the analysis.

The different flavors of vorticity are not Galilean invariant (Viúdez and Haney 1996), so a frame of reference must be chosen for the analysis. As pointed out by Davies-Jones (1984), the relevant frame in supercell dynamics is the updraft-relative frame. The ensuing analysis is thus performed in that frame, which is achieved by using a grid motion that matches the storm motion.

Conversion terms

When writing out the conversion terms [Eq. (A19)] in Cartesian coordinates, they expand into a collection of terms that involve multiple products of first- and second-order derivatives. Unfortunately, numerical truncation errors escalate if products of second-order derivatives are involved, and the directly integrated shear and curvature vorticity thus tend to drift away from the known interpolated vorticity values along the trajectories.

However, the following technique allows for an accurate estimation of the conversion terms despite these complications as long as it can be established that the main source of error indeed comes from the conversion terms. To make this assertion, the following tests were performed: (i) the total vertical vorticity accurately matches the integrals of the tilting and stretching terms; (ii) when adding the integrals of the tilting and stretching terms of the shear and curvature vorticity equations, respectively, the total vertical vorticity is accurately reproduced (which will be demonstrated in section 5); and (iii) the sum of the tilting (divergence) terms of the shear and curvature equations add up to the tilting (divergence) term of the total vertical vorticity equation. These conditions are fulfilled for the analysis presented here, and it is thus permissible to conclude that indeed the main errors arise from the conversion terms. Importantly, this allows us to treat the conversion terms as a residual. We may write
e4
or
e5
where is the residual (integral of conversion from to ), is the shear vorticity due to the accumulated stretching term, and is the shear vorticity due to the accumulated tilting term. Likewise,
e6
where now is the residual representing the vorticity due to conversion of shear vorticity, is the curvature vorticity due to the accumulated stretching term, and is the curvature vorticity due to the accumulated tilting term. The advantage of this approach is that the complicated explicit formulation of the conversion terms can be avoided. Reassuringly, both approaches yield the same qualitative results, as will be shown in the next section. Since the residual appears to be the more accurate estimate of the conversion terms, the budget calculations in the next section are based on this residual.

5. Simulation results

First, we will focus on the initial vortex development within a convective updraft developing in a kinematic environment characterized by purely crosswise base-state vorticity. These early vortices in this simulation are related to the first updraft pulse associated with the warm bubble and most closely represent the theoretical models describing the onset of updraft rotation (Lilly 1982; Davies-Jones 1984). Subsequently, an analysis of a mature supercell will be presented.

a. Crosswise vorticity scenario

The first appearance of horizontally perturbed flow associated with the updraft, which is characterized by a combination of sheared and curved flow, corresponds to a weak vertical vorticity dipole straddling the updraft (Fig. 9a). Within a few minutes, the flow develops a couplet of coherent vortices, consistent with regions of negative Okubo–Weiss numbers (Figs. 9b,d) and negative pressure perturbations. With some delay, the same development can be observed at successively higher levels as the updraft grows upward (not shown).

Fig. 9.
Fig. 9.

Vertical vorticity (shaded), updraft (black contours; m s−1), horizontal wind vectors (scale in the middle of the figure), and pressure perturbations (hPa) at about 2000 m AGL and (a) 210 s and (b) 510 s. (c),(d) As in (a),(b) but the OW number is displayed instead of vertical vorticity.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

The pressure field, which will play an important role in the interpretation of the results, initially appears to be related to the linear interaction between the updraft and the ambient wind shear (Fig. 9a), leading to high (low) perturbation pressure on the upshear (downshear) side of the updraft (Rotunno and Klemp 1982; Markowski and Richardson 2010, p. 31). However, the pressure field quickly becomes dominated by two minima centered on the developing vortices (Rotunno and Klemp 1982, 1985). During vortex formation, the flow in the updraft center reverses direction against the pressure gradient acceleration (which points to the east in Fig. 9a).

To gain an understanding of this development, a total of 77 775 forward trajectories was seeded in a box surrounding the updraft ( km3) 30 s after the bubble was inserted into the base state. The initial spacing of the parcels in all three Cartesian directions was 200 m. To find suitable parcels that became part of the rotating updraft, a filter was applied to all parcels, keeping those whose vertical vorticity exceeded 0.01 s−1 in an at least 5 m s−1 updraft between 3000 and 4900 m, and between 530 and 600 s. The focus here is on the cyclonic vortex, but the anticyclonic vortex evolves in a mirror-symmetric fashion. These values were chosen based on the onset of rotation surrounding this space–time window. Altogether, 1111 parcels were identified, which are shown in Fig. 10.

Fig. 10.
Fig. 10.

Overview of all 1111 trajectories, broken up in trajectories originating (a) below 3 km AGL and (b) above 3 km AGL. In (a) only every fourth of a total of 830 trajectories is plotted for better visibility. In (b) all remaining 281 trajectories are shown. The red (blue) contours in both panels denote cyclonic (anticyclonic) vertical vorticity [shown are the s−1 contours], and the black contours represent updraft speed (m s−1) at 4125 m AGL and 540 s. The arrows represent horizontal wind vectors according to the scale at the top-right corner of the figure, which pertains to (a) and (b).

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

There appear to be two regimes of trajectories, those below about 3 km AGL (Fig. 10a), and those originating above that level (Fig. 10b). In most real-world severe thunderstorm environments, the trajectories originating from higher than about 3 km would be located within the elevated mixed layer (Carlson et al. 1983), which is characterized by dry, and at best weakly buoyant air. It is thus unlikely that these parcels contribute to updrafts in typical storm environments. The fact that they are identified here could be related to the CAPE profile of the Weisman–Klemp sounding, which is uncharacteristically moist through midlevels. Since these parcels do contribute in this simulation, however, they are retained in the analysis. Although the details of vorticity evolution among individual trajectories vary, the vast majority share a common basic behavior (including trajectories from above the 3-km level), an example of which will be shown in detail in the next subsection.

Trajectory analysis

A representative member of the low-level trajectories is parcel number 150. The parcel enters the storm from the southeast and rises from about 600 to 1900 m AGL by 530 s, while it becomes part of the nascent cyclonic vortex at the southern flank of the updraft (Fig. 11). To check whether the integration of the tilting and stretching terms is accurate (and, consequently, whether the conversion terms may be treated as residual), the integrated tilting and stretching terms of the shear and curvature vorticity equations were summed up and compared to the interpolated total vertical vorticity (black line in Fig. 12). The integral matches the interpolated vorticity, such that the conversion terms may be inferred from the residuals. The time series also shows that the parcel acquires similar amounts of shear and curvature vorticity, suggesting that it is in a state of local rotation. To understand this development, Fig. 13a shows the different contributions to changes in shear vorticity along the parcel trajectory. While the effect of upward tilting of crosswise vorticity is positive through much of the period, it is much weaker than the action of the conversion and stretching terms. From this analysis, it cannot be inferred whether the stretching term acts on shear vorticity that resulted from tilting or from conversion, but it is clear that the conversion terms are at least as important as the combined effects of tilting and stretching. Shear vorticity in the updraft thus does not primarily result from tilting of crosswise vorticity but, to a significant portion, from conversion of curvature vorticity. The processes by which curvature vorticity arises are shown in Fig. 13b. The increase in curvature vorticity is dominated by the tilting and stretching terms. The conversion terms predominantly deplete curvature vorticity to the benefit of shear vorticity, especially after about 400 s. In light of the discussion in section 2, these results are perhaps surprising, because one might have expected that in the crosswise vorticity case the shear vorticity would result mainly from the tilting term (and subsequent stretching). The implication of this result is that, although the base-state vorticity is crosswise, it is mostly streamwise vorticity that is being tilted upward.

Fig. 11.
Fig. 11.

Shown is trajectory number 150 (black line segment; the parcel location at 540 s is depicted as a red circle) as well as curvature vorticity (positive values 10−2 s−1; red contours) and shear vorticity (negative values 10−2 s−1; blue contours). Since the storm is mirror symmetric, the magnitude of shear and curvature vorticity pertains to both vortices. Total vertical vorticity is shaded, and the wind vectors are represented by the arrows (according to the scale at the top right). The fields are displayed at about 2017 m AGL and 540 s.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Fig. 12.
Fig. 12.

Time series along the trajectory portrayed in Fig. 11 of interpolated vertical vorticity (black line), integrated vertical vorticity, obtained by summing the integrated tilting and stretching terms of the shear and curvature vorticity equations (black dashed line, which nearly coincides with the solid black line), curvature vorticity (solid green line), shear vorticity (dashed green line), and integrated conversion terms. The blue line represents the residual, and the blue-shaded interval represents the difference between the integration of the actual conversion terms and the residual.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Fig. 13.
Fig. 13.

Time series along the trajectory portrayed in Fig. 11 of (a) shear vorticity (black contour; the shaded interval represents the difference between the interpolated and integrated values), integrated tilting (green contour), integrated stretching (blue contour), and integrated conversion from curvature to shear (red contour). (b) As in (a), but for curvature vorticity, and the conversion is from shear to curvature.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Indeed, as shown in Fig. 14, the streamwise vorticity of the parcel increases from near zero to about s−1 as the parcel reaches the updraft (at about 250 s). This streamwise vorticity is sufficient to result in appreciable vertical curvature vorticity upon interaction with the updraft. To understand this development, the inviscid Boussinesq version of the streamwise-vorticity equation may be utilized (Adlerman et al. 1999), which is given by
e7
where ϕ is the angle between the horizontal velocity vector and some fixed horizontal axis, is the 3D vorticity vector, and B is buoyancy. The first term in this equation will be referred to as “curvature term” and will be discussed below. The second term combines the effects of tilting of the 3D vorticity vector into the streamwise direction as well as stretching in this direction. The last term is the baroclinic production term.
Fig. 14.
Fig. 14.

Time series along the trajectory portrayed in Fig. 11 showing the streamwise vorticity (black contour; the shaded interval represents the difference between the integrated and interpolated values), the integrated curvature term (red), integrated stretching/tilting (green), and integrated baroclinic production of streamwise vorticity (blue).

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

The main contributor to the increase of streamwise vorticity is the curvature term, as shown in Fig. 14. This term engenders the development of streamwise vorticity in curved deformation flows if crosswise vorticity is present [see Shapiro (1969) and Scorer (1978, p. 84 ff.); this phenomenon is often referred to as the river-bend effect]. Since this term depends on the crosswise vorticity, which may change via, for example, stretching, the curvature term is not in general an exchange term (i.e., the crosswise vorticity is not depleted at the same rate as streamwise vorticity is gained via the curvature terms). Tilting and stretching of the 3D vorticity vector into the streamwise direction contribute comparatively less to the increase of streamwise vorticity and appear to be related mainly to downstream acceleration and stretching. The baroclinic production is comparatively weak and antistreamwise before the parcel starts rising around 250 s. The curvature effect that leads to an increase of streamwise vorticity is revealed in Fig. 15, depicting the horizontal velocity and vorticity vectors, as well as the angle between these vectors. Because there is a pressure deficit beneath and downshear of the buoyant updraft, the air accelerates toward the low pressure center. The result is that low-level parcels at the southern (northern) periphery of the updraft turn anticyclonically (cyclonically) toward the updraft (see the dip in curvature vorticity during the first 250 s in Fig. 13b). This irrotational turn is associated with an increase of streamwise vorticity via the river-bend effect. Also, Fig. 15 shows that the velocity vectors are nearly orthogonal to the updraft contours, consistent with the observation that it is mainly streamwise vorticity that is tilted upward.

Fig. 15.
Fig. 15.

Perturbation pressure field (hPa; shaded), angles between the horizontal velocity and vorticity vectors in degrees (blue contours for angles < 90° and red contours for angles > 90°) at about 650 m AGL and 270 s. The black contours represent updraft speed (m s−1). The scales for the horizontal vorticity and velocity vectors are located at the bottom of the figure.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Although not presented in detail, parcels that originate from above the critical level (where the storm relative flow is zero) approach the updraft from the west and undergo a cyclonic curve instead of an anticyclonic one; during this turn, their horizontal vorticity is likewise converted to streamwise vorticity, such that their acquisition of shear and curvature vorticity proceeds in the same manner as for the parcel just analyzed. Trajectories initially above about 3 km (Fig. 10b) descend slightly as they curve around the updraft and acquire streamwise vorticity via the curvature effect and baroclinic production at the edge of the updraft. It is again predominantly the streamwise vorticity that is tilted into the vertical leading to curvature vorticity, while shear vorticity is attained via the conversion of curvature vorticity.

To gain a sense of the generality of these results, the bulk behavior of all 1111 parcels is shown in Fig. 16. The filled intervals show the interquartile range of the integrated variables, and the lines show the median values. Figure 16a reveals that the conversion term plays the dominant role in explaining the increase of shear vorticity. Although the strength (and even the sign) of the integrated tilting/stretching and conversion terms varies quite a bit, the bulk behavior highlights the importance of conversion from curvature to shear vorticity. The tilting/stretching contribution to shear vorticity is dominated by the stretching term (the net effect of tilting alone is slightly negative; not shown). The median curvature vorticity primarily grows because of tilting of streamwise vorticity and stretching, with some of this vorticity being converted to shear (Fig. 16b).

Fig. 16.
Fig. 16.

Time series of the contribution to shear and curvature vorticity of the median parcel. The shaded intervals show the interquartile ranges of the integrated quantities. (a) The black contours represent the shear vorticity [interpolated (solid); integrated (dashed)], the green contour represents the integrated effect of stretching/tilting, and the red contour depicts the integral of the conversion terms. (b) As in (a), but for curvature vorticity. The number of parcels in this sample is n = 1111.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

The main result of this experiment is that, although the far-field horizontal vorticity is crosswise, the near-storm vorticity at the southern flank of the updraft tends to be streamwise because of the horizontal accelerations associated with the perturbation pressure field. It is this streamwise vorticity that is tilted into the vertical, resulting in cyclonic curvature vorticity as the parcels rise. Shear vorticity develops via conversion at the expense of curvature vorticity. Interestingly, while in principle it would be possible for the updraft to attain local rotation directly via upward tilting of suitably oriented streamwise and crosswise vorticity (section 3a), the majority of the parcels do not follow that route (mainly because the velocity vectors are parallel to the horizontal updraft gradients, such that the crosswise vorticity cannot be tilted upward).

b. Streamwise vorticity scenario

The other simulation, using the quarter-circle hodograph, was carried out until a persistent supercell structure developed. During the mature stage of the storm (beginning about 3000 s into the simulation), the updraft is characterized by a broad, curved flow that does not exhibit closed streamlines in the storm-relative frame, however (Fig. 17). Rather, the pattern resembles a segment of a vortex in solid-body-like rotation, suggested by the fact that the shear and curvature vorticity have the same sign and roughly the same magnitude in the updraft and by the negative OW number (Fig. 17). The western edge of this “open” vortex is characterized by a stagnation zone, which usually gives rise to hydrometeor accumulation and hook-echo formation (Byko et al. 2009). It is not uncommon for supercell updrafts to lack a complete updraft-scale vortex at midlevels (e.g., Lemon and Doswell 1979; Bluestein and Gaddy 2001). However, transient, compact vortices often appear at the southern periphery of the updraft and are swept to its western flank (one such vortex is visible in Fig. 17, which will be addressed again in section 6).

Fig. 17.
Fig. 17.

Horizontal cross section through the supercell at 5125 m AGL and 3300 s. (a) Vertical vorticity (10−2 s−1; shaded), storm-relative streamlines, and updraft speed (black contours; m s−1). (b) As in (a), but showing the OW number instead of vertical vorticity. (c) As in (a), but showing curvature vorticity. (d) As in (a), but showing shear vorticity. The colorbar at the bottom right pertains to ζ, , and .

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

To see how local rotation arises in this case, a total of 43 911 trajectories was initialized at 2700 s in a box given by km3. The initial spacing of the parcels along the Cartesian directions was 300 m. Using the criteria that parcels needed to exceed a vertical vorticity threshold of 0.01 s−1 in at least 5 m s−1 updraft between 3240 and 3300 s at 6000 ± 100 m AGL, 460 parcels were identified. These parcels primarily contribute to the ribbon of large cyclonic vorticity at the southern periphery of the updraft (Fig. 17) and are shown in Fig. 18. The bulk development of the shear and curvature vorticity parts is shown in Fig. 19. Although there is again appreciable variability between the different parcels, the overall tendency is that shear vorticity arises mainly via partial conversion of curvature (Fig. 19a) and that the curvature vorticity to a large part originates from tilting of streamwise vorticity into the vertical and subsequent stretching (Fig. 19b). Indeed, the near-storm inflow below about 4-km altitude is nearly helical, as shown in Fig. 20, suggesting that barely any shear vorticity can develop via tilting. This enhancement of streamwise vorticity in the near-storm environment due to pressure gradient accelerations is consistent with, for example, Brooks and Wilhelmson (1993), Potvin et al. (2010), or Parker (2014).

Fig. 18.
Fig. 18.

Overview of the 460 analyzed trajectories, with their initial heights (km) color coded according to the colorbar. The gray arrows represent the horizontal wind vectors (scale vector at the top right of the figure), and the black contours represent updraft speed (m s−1) at 6125 m AGL and 3300 s.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Fig. 19.
Fig. 19.

As in Fig. 16, but for the mature supercell. Here, n = 460 parcels.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Fig. 20.
Fig. 20.

Angle in degrees between the horizontal wind and velocity vectors (shaded), as well as wind vectors (gray arrows) and horizontal vorticity vectors (orange arrows), shown at about 2000 m AGL and 3000 s. The updraft is represented by the white contours (m s−1). The scale vectors for the vorticity and velocity vectors are located at the bottom of the figure.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

6. Discussion

a. Physical interpretation of the conversion terms

The conversion (or interchange) terms in the shear and curvature vorticity equations are related to the perturbation pressure gradient force (PGF) field, which in the present application is practically conservative and thus cannot create vertical vorticity (i.e., vertical baroclinic torques are zero). However, the PGF may create shear vorticity at the same rate that curvature vorticity is depleted, and vice versa. One way this conversion (i.e., the net effect of both conversion terms) may be conceptualized is to consider a simple 2D framework with two point masses that pass through the periphery of a potential well, as depicted in Fig. 21. The trajectories of the two masses were obtained numerically assuming a Gaussian potential well. Each mass has the same initial velocity. Both masses acquire, and then lose, cyclonic curvature during their journey through the well. At the same time, the mass closer to the center of the well experiences a greater longitudinal acceleration than the mass farther from the center, thus creating a speed variation in the direction normal to the direction of motion. If these two masses were to represent two fluid parcels, this differential acceleration would correspond to the production of anticyclonic shear. As the total acceleration is irrotational, the total vorticity remains constant (zero in this case) while the changes in shear and curvature vorticity are equal in magnitude but opposite in sign. Indeed, the parcels losing curvature vorticity in the updraft are located on the downstream side of the low center, as shown in Fig. 22, and the dipole-like distribution of the conversion term implied in Fig. 21 is evident. The conversion requires the flow to have a component normal to the pressure contours. The reason that these contours are not parallel to the flow is that the pressure field is not only determined by the rotational part of the flow, but also by deformation (in particular, by the linear interaction of the updraft with the base-state shear). Also note in Fig. 22 that the cyclonic updraft cores are dominated by negative values of the conversion terms, consistent with the trajectory analyses.

Fig. 21.
Fig. 21.

Trajectories of two point masses passing through the periphery of a potential well (black contours). The normalized speed V of the masses is color coded along the trajectories, and each arrow represents the progression of a constant time interval. The length of the arrows is proportional to the velocity of the respective point mass.

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

Fig. 22.
Fig. 22.

(a) Conversion terms C (shaded) for the nascent updraft at 540 s and 2017 m AGL in the crosswise vorticity case along with trajectory number 150 (thick, black line segment), updraft speed (black contour; m s−1), perturbation pressure (hPa; red contours, only negative values are shown), and wind vectors (scale vector at the top right of the panel). (b) Conversion terms (shaded) in the supercell simulation, shown at 4125 m AGL and 3240 s. The red contours represent negative pressure perturbations (hPa), the black contours show updraft speed (m s−1), and the arrows depict the wind vectors (scale vector at the bottom right of the panel).

Citation: Journal of the Atmospheric Sciences 74, 9; 10.1175/JAS-D-17-0091.1

b. Connection to established concepts

Some of the results presented here seem to be at odds with the analysis by Lilly (1982) and Davies-Jones (1984), who predicted that updrafts in an environment with streamwise vorticity exhibit net cyclonic rotation (i.e., the updraft center coincides with the vertical vorticity center). Here it was found that, although mostly streamwise vorticity was tilted by the updraft discussed in section 5a, two counterrotating vortices (i.e., no net rotation) developed within the updraft. The cause of this discrepancy is the linear nature of the models by Lilly (1982) and Davies-Jones (1984). In these models, the base-state flow is not affected by the perturbation flow, and the inflow into the updraft is simply the base-state flow. In the full-physics numerical model, the inflow is altered by the perturbation pressure field surrounding the updraft. These nonlinear effects do not change the basic results of Davies-Jones (1984), because tilting of horizontal vorticity still occurs symmetrically at the southern and northern flanks of the updraft (i.e., in inflow contains streamwise vorticity at the southern flank of the updraft and antistreamwise vorticity at its northern flank).

c. Limitations of the shear and curvature vorticity approach

Because vorticity and its shear/curvature parts are a local measure of rotation, the analysis only reveals a necessary but insufficient condition for the presence of a vortex. In particular, only the local requirement for solid-body-like rotation () was analyzed. With this approach, a flow exhibiting a complete vortex (closed streamlines) cannot be distinguished from a flow characterized by only a “segment” of a vortex (local rotation without closed streamlines). However, established criteria such as the Okubo–Weiss number (or the pressure criterion or the kinematical vorticity number) suffer from the same shortcoming (see the minimum of the OW number in Fig. 17b). These problems reflect the general difficulty of diagnosing vortex structures formally (Haller 2005).

d. Mesocyclone structure

The midlevel mesocyclone of the mature supercell only contains a segment of a vortex (there are no closed streamlines within the updraft). This behavior can be understood based on the model developed in section 2. Near the mature supercell, the vorticity was nearly streamwise. In this case, the storm-relative winds veered by nearly 180° in the inflow layer, which is typical for most supercell environments. Accordingly, tilting contributes to a flow that describes roughly a cyclonic 180° turn within the updraft (cf. Figs. 17, 6). Part of is converted to , such that an open vortex in near-solid-body rotation arises.

However, this does not preclude the possibility of a deep coherent vortex to form, and ongoing research suggests that in these cases the vortex is made up of parcels carrying baroclinically augmented vorticity (J. M. L. Dahl 2015, oral presentation). These baroclinically augmented vortices tend to be rather compact and unsteady (such as those described by French et al. 2013). “Updraft rotation” thus involves additional processes not included in the results presented in this paper. The completion of the vortex within a supercell by what appears to be baroclinic vorticity is reminiscent of the divided mesocyclone structure introduced by Lemon and Doswell (1979).

The updraft-scale vortex segment may be referred to as “barotropic mesocyclone,” because it is fed by ambient (barotropic) vorticity [see e.g., Davies-Jones (2000) and Dahl et al. (2014)]. The trajectories in this flow regime do not exhibit a “swirling” pattern; rather, in environments with large streamwise vorticity they tend to be slightly cyclonically curved (Fig. 18) up until midlevels, where the mesocyclone typically is strongest [this still allows for an anticyclonic turn farther aloft as the parcels exit the updraft toward easterly directions, as suggested by Klemp and Wilhelmson (1981) and Lilly (1986)]. Swirling trajectories do appear in numerical simulations (e.g., Coffer and Parker 2017), but these seem to be associated with baroclinically augmented vorticity.

7. Summary and conclusions

In this study, the development of rotation in updrafts immersed either in crosswise or streamwise environmental horizontal vorticity was analyzed employing heuristic arguments, theory, and numerical simulations using CM1. Here, rotation is defined locally as vertical shear vorticity having the same sign and a similar magnitude as vertical curvature vorticity. Trajectory analyses were performed, quantifying the different mechanisms by which vertical shear and curvature vorticity change as parcels are rising in the updraft. This analysis leads to the following conclusions:

  • In canonical supercell environments, upward tilting of environmental horizontal vorticity alone cannot explain the onset of local rotation in updrafts. Rather, the tilting mechanism results in vertical shear vorticity if purely crosswise vorticity is tilted and in vertical curvature vorticity if purely streamwise vorticity is tilted.

  • The pressure gradient field surrounding sheared buoyant updrafts accelerates inflow air such that the horizontal vorticity attains a significant streamwise component even if the far-field horizontal vorticity is purely crosswise. The vertical vorticity developing in the updraft via tilting of this near-storm streamwise vorticity is thus manifest mainly as curvature.

  • For completion of local rotation, a mechanism not previously considered in supercell dynamics was found to be relevant: partial conversion from vertical curvature vorticity to vertical shear vorticity. This mechanism is tied to the perturbation pressure field and a necessary step in the development of local rotation.

  • A simulated mature supercell was found to ingest almost perfectly streamwise vorticity and consequently the tilting mechanism contributed to vertical curvature vorticity within the updraft. The horizontal flow structure in the updraft was characterized only by a segment of a vortex (i.e., curved flow in solid-body-like rotation), but without closed streamlines. This is consistent with a heuristic model that suggests tilting of streamwise vorticity to result in a broad, curved flow in the updraft. The change of direction within this curved flow appears to be directly related to the degree of veering in the vertical wind profile in the storm inflow. Partial conversion of this curvature to shear maintains (local) solid-body-like rotation.

This study was concerned only with the interaction of an updraft with ambient horizontal vorticity. Further study is warranted to analyze the role of baroclinically-augmented vorticity. Preliminary results suggest that this vorticity may augment the broad mesocyclonic flow such that compact vortices form. This seems particularly likely in intense supercells that are able to lift parcels with large baroclinic vorticity out of the cold pool.

Finally, the results presented here are fully consistent with well-established, previous findings regarding the reorientation of ambient vortex lines by convective updrafts, as well as the spatial correlation between vertical velocity and vertical vorticity extrema. However, the results add more detail about the flavor of vorticity that is produced via the tilting mechanism and emphasize the requirement for curvature-to-shear conversion to accomplish rotation in sheared updrafts.

Acknowledgments

Numerous beneficial discussions in the early 2000s with Drs. Chuck Doswell, Bob Davies-Jones, Bernold Feuerstein, and more recently with Drs. Matt Parker, Paul Markowski, and Fazle Hussain are gratefully acknowledged. Dr. George Bryan generously maintains and supports CM1, without which this research would not have been possible. Dr. Rich Rotunno, Brice Coffer, and two anonymous reviewers provided insightful comments that helped improve the initial version of the manuscript. The author fondly remembers numerous exchanges about severe storms with the late Prof. Werner Wehry of the Free University of Berlin, to whom this paper is dedicated.

APPENDIX

The Shear and Curvature Vorticity Equations

The derivation of the shear and curvature vorticity equations closely follows Viúdez and Haney (1996). The only difference is that the equations are formulated in height coordinates and that the Coriolis parameter is set to zero. The equations obtained here correspond to Eqs. (38) and (39) of Viúdez and Haney (1996).

The main steps in the derivation involve application of the material derivative operator to the expressions of shear and curvature vorticity in natural coordinates, a few vector identities and commutation rules of differential operators, and use of the Euler equations. The vector identities are most easily proven in index notation. In the following, Latin indices refer to horizontal dimensions while Greek indices run from one to three, and the summation convention is in force.

The commutator of the material derivative operator and the gradient operator may be found via
ea1
where is the (3D) material derivative operator and a is an arbitrary scalar. In vector notation we may write
ea2
Similarly,
ea3
where is an arbitrary horizontal vector and the colon symbol signifies the contraction of both indices of each tensor ().
Replacing with in the last expression, and using the fact that
ea4
we obtain
ea5
Finally, we will use that
ea6
Here, is the vector normal to the streamline.5 The expressions for shear and curvature vorticity are obtained by taking the vertical component of the curl of the horizontal velocity field, which is given by , where is the tangent vector and V is the wind speed. Then,
ea7
Upon taking the material derivative and applying Eqs. (A5) and (A6), one obtains
ea8
eq1
where includes all terms with material derivatives.
A preliminary version of the shear vorticity equation is obtained by applying the material derivative operator to and using Eq. (A2) as well as .6 Then,
ea9
The terms involving time derivatives of V and ( and ) may be simplified by applying the Boussinesq approximation, which implies that the horizontal pressure gradient force has a potential, given by ( is the specific heat at constant pressure, is the base-state potential temperature, and π is the Exner function). Then,
ea10
where is the horizontal acceleration. To simplify , we need to find the commutator of the directional (or intrinsic) derivative operators, and . Using , it follows that
ea11
After a few steps, one finds that
ea12
The Euler equations in an inertial reference frame are, in natural coordinates, given by
ea13
and
ea14
Using these equations and Eq. (A12), we find that
ea15
The final form of Eq. (A8) is obtained by noting that (ϕ is the angle between the streamline and a fixed horizontal axis) and recalling that the streamwise vorticity is given by .7 Then,
ea16
Likewise, we insert Eqs. (A13) and (A14) in the expression for and directly find that
ea17
so that Eq. (A9) becomes
ea18
By convention, the conversion terms, bundled together into C, are positive if curvature vorticity increases, such that
ea19
To implement the shear and curvature vorticity equations, they were transformed to Cartesian coordinates [see also the appendix of Bell and Keyser (1993)] using
ea20
and
ea21
as well as
ea22
and
ea23

Inserting these expressions into the shear and curvature equations, after some algebra, leads to the desired Cartesian formulations.

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1

Rotation near the surface is most likely accomplished by a different mechanism than upward tilting of ambient vortex lines by an updraft (Dahl et al. 2014; Dahl 2015) and is not considered in this paper.

2

The initially horizontal vortex would wrap the ring-shaped vortex lines surrounding the updraft around itself, which would ultimately lead to a sheath of anticyclonic (cyclonic) vorticity enshrouding the cyclonic (anticyclonic) vortex that arises during tilting as discussed by Kida et al. (1995).

3

It is not uncommon for the tilting term to be visualized as a rigid, spinning object like a cylinder that is rotated in space [e.g., Dutton (1976, p. 341) or Holton and Hakim (2013, p. 105)]. However, such a rigid cylinder would precess if a torque acted on it, because unlike fluids it cannot be deformed. Moreover, per Eq. (1), no torque is involved during vortex-line reorientation.

4

Other numerical schemes as well as different output intervals were tested but yielded the same results.

5

To verify Eq. (A6), carry out the operations on each side and use and , as well as .

6

Upon writing the lhs of this expression in index notation, one finds that . Noting again that , it can be seen that . Here, is the permutation symbol.

7

To verify, note that . Using and , one readily obtains , where and are the streamwise and crosswise vorticity vectors, respectively.

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