1. Introduction
One of the outstanding questions in tornado research remains the origin of “seed” ground-level rotation1 that horizontal convergence can act upon to realize a compact vortex (e.g., Davies-Jones et al. 2001). It has been established that the onset of this initial rotation is due to the rearrangement of initially horizontal vortex lines within downdrafts (Davies-Jones and Brooks 1993; Walko 1993; Wicker and Wilhelmson 1995; Adlerman et al. 1999; Davies-Jones et al. 2001; Davies-Jones and Markowski 2013; Dahl et al. 2014; Markowski et al. 2014; Schenkman et al. 2014; Parker and Dahl 2015), assuming negligible preexisting vertical vorticity in the storm’s environment. In general terms, these horizontal vortex lines may either be generated within the storm by buoyant or frictional torques, or the horizontal vorticity may be imported into the downdraft from the environment. This study is part of an ongoing effort to explore which of these contributions dominate in what situation, which is crucial for a complete understanding of tornado dynamics. In an attempt to tackle this problem, Dahl et al. (2014) applied a vorticity decomposition technique to quantify the roles of storm-generated and imported vorticity in an idealized free-slip simulation of the Del City, Oklahoma, supercell. They found that the imported ambient vorticity (treated as “barotropic vorticity”) did not contribute much to the development of vertical-vorticity maxima at the lowest model level. Rather, the near-surface vertical vorticity originated primarily from horizontal buoyancy torques within the storm. This behavior is due to the orientation of the ambient vorticity vector relative to the storm-relative flow: if this barotropic vorticity is mostly streamwise it will tend to remain aligned with the velocity vectors. Hence, as the trajectories turn on the horizontal plane near the ground, the barotropic vorticity likewise turns horizontally and thus cannot contribute to vertical near-ground vorticity, as already anticipated by Davies-Jones and Brooks (1993) and Davies-Jones et al. (2001). However, if the ambient vorticity has a crosswise component, it is generally possible for this vorticity to contribute to, and perhaps dominate, the vertical vorticity at the downdraft base. Rotunno and Klemp (1985) simulated a supercell in an environment with crosswise vorticity. From their work [Fig. 12 in Rotunno and Klemp (1985)] it may be inferred that in their simulation the barotropic vertical vorticity is negative near the ground and that it is dominated by the positive baroclinic vertical vorticity.2 However, Rotunno and Klemp (1985) and other studies demonstrating the dominance of the baroclinic mechanism (Davies-Jones and Brooks 1993; Wicker and Wilhelmson 1995; Adlerman et al. 1999) used a warm-rain Kessler microphysics parameterization, which has long been known to overestimate low-level baroclinity (e.g., Markowski 2002). The question is thus whether the prevalence of the baroclinic mechanism is merely an artifact of the microphysics parameterization.
The purpose of this study is to analyze the relative importance of the baroclinic and barotropic mechanism in the presence of crosswise vorticity employing the vorticity decomposition approach and to test the sensitivity of the results by using two different microphysics schemes. Also, a scaling argument is offered as explanation for the results.
This study is focused on the initial development of tornado–cyclone-scale vorticity maxima at the lowest model level. That is, only the seed near-ground rotation for possible tornadogenesis is addressed herein. Whether or not this vorticity is actually concentrated into a strong tornado-like vortex by vertical stretching is a separate problem not considered in this study [but it is discussed elsewhere, e.g., by Markowski and Richardson (2014)].
2. Methods
a. Experimental design
The goal is to produce simulations of a supercell that develops vertical vorticity ζ at the lowest model level while ingesting appreciable crosswise storm-relative vorticity. The most straightforward way to accomplish this goal is to use a unidirectionally sheared base-state flow (e.g., Rotunno and Klemp 1985), as detailed below. The simulations were carried out with the Bryan cloud model, version 1 [CM1; Bryan and Fritsch (2002)], release 17. The forward trajectory calculations (see section 2c) within the model were modified, using a fourth-order Runge–Kutta time integration and Lagrange polynomials for the spatial interpolation to obtain the velocities at the parcels’ locations (rather than the trilinear interpolation in the standard distribution of CM1). The horizontal model domain (~125 × 125 km2) has a grid spacing of 250 m. The vertical grid spacing varies from 100 m near the ground to 250 m at the domain top, which is at 20 km AGL. The lowest scalar model level is at 50 m AGL. A sponge layer is employed in the uppermost 6 km and the lateral boundary conditions are open while the lower and top boundaries are free slip, and the Coriolis parameter is set to zero. One of the simulations uses a single-moment Lin-type microphysics parameterization (Gilmore et al. 2004), in which the rain-intercept parameter was reduced to 106 m−4 to prevent overly cold outflow (Dawson et al. 2010). The other simulation utilizes the double-moment Morrison microphysics scheme (Morrison et al. 2009) using the “hail-like” graupel option.
Guided by Rotunno and Klemp (1985), the base state is given by a unidirectional wind profile, where the x component of the base-state flow, 
b. Vorticity decomposition
In this study the vorticity separation technique described by Dahl et al. (2014) is used. In general terms, the total vorticity at a given time t may be decomposed into two parts: a barotropic part, which is due to the rearrangement of vorticity present at an arbitrary initial time, and a nonbarotropic part, which is due to production of vorticity by pressure (and frictional/diffusional) torques, and subsequent reconfiguration (e.g., Dahl et al. 2014 and the references therein). The barotropic vorticity may be determined by calculating the deformation gradient of a fluid volume along its trajectory, which is done by tracking the relative displacements of parcels within “Lagrangian stencils” (i.e., sets of six parcels that are each centered around the parcel of interest and initially aligned along the three Cartesian axes). Once the barotropic vorticity is determined, the nonbarotropic vorticity may simply be inferred from the difference between the known total vorticity and the barotropic vorticity along the trajectories of interest. Herein the barotropic vorticity represents the ambient vorticity, which characterizes the kinematic environment of the storm. The nonbarotropic vorticity then represents the storm-generated vorticity. This interpretation requires that the forward trajectories are launched far enough away from the storm such that the initial vorticity is not contaminated by baroclinic production in the storm’s far field. The procedure to obtain suitable trajectories is described next.
c. Parcel trajectories
The objective is to obtain highly accurate forward trajectories calculated on the large time step (2.0 s) within CM1 and to analyze those parcels that acquire positive vertical vorticity while descending through the lowest model level. This criterion is consistent with the notion that the initial vertical vorticity in supercells is generated in downdrafts (Davies-Jones 1982; Davies-Jones and Brooks 1993; Davies-Jones 2000; Davies-Jones and Markowski 2013; Dahl et al. 2014; Parker and Dahl 2015).
Dahl et al. (2012) suggested that forward trajectories near vorticity extrema are more accurate than backward trajectories. Moreover, forward trajectories can be calculated within CM1 on every large model time step without the need to store such high-resolution output. The disadvantage of forward trajectories is that the initial locations of the trajectories ending up in a certain region of interest, are unknown. In contrast to the approach by Dahl et al. (2014), an iterative technique using backward trajectories was used to identify the source regions of relevant parcels. First, a dense cloud (~2020 parcels km−3) of forward trajectories was seeded at 3000 s in a 3-km-deep, 20 × 20 km2 box surrounding the main downdraft cores that produce vertical vorticity at their bases. Only those trajectories were captured with 
Once parcels of interest were identified, backward trajectories were calculated for an interval of 30 min, using history files every 30 s and a second-order Runge–Kutta scheme with a time step of 2.0 s. This 30-min time interval was found to be necessary to ensure that the parcels are far enough away from the storm such that their initial vorticity is sufficiently unperturbed, as detailed below. These comparatively coarse backward trajectories merely served as guidance for where to seed the (~30 min) forward trajectories in a restart run. The initial positions at t = 1200 s of these long-history trajectories were located in the subdomain 
For the single-moment run, the same filter criteria as for the initial set of forward trajectories were applied, yielding 3481 parcels of interest. For the initial (barotropic) vorticity to represent the ambient vorticity as accurately as possible, another filter was applied to keep only those parcels whose initial vertical vorticity was 0.0005 s−1 or less, and whose horizontal vorticity was perturbed by less than 10% from the base-state value, which yielded 1846 parcels. Finally, to obtain the barotropic vorticity, another restart run is necessary, this time including the Lagrangian stencils surrounding each of the 1846 parcels, analogous to the description by Dahl et al. (2014).
For the double-moment simulation, the same initial conditions and filtering criteria of the trajectories in the downdraft were used, except that 
3. Results
a. Single-moment microphysics simulation
An overview of the storm including the analyzed trajectories is shown in Fig. 1. The trajectories, as in previous simulations (e.g., Adlerman et al. 1999; Dahl et al. 2012, 2014; Markowski and Richardson 2014; Parker and Dahl 2015), originate from the lowest few kilometers above the ground. This trajectory sample includes several parcels that start out near the ground, but then rise along the left-flank convergence boundary (Beck and Weiss 2013) and subsequently descend within the main downdraft north of the mesocyclone. The trajectories reach the lowest model level at different times [at the time shown, some of the parcels are already rising in the main updraft near 
Shown are the 1695 trajectories at 3780 s that contribute to vertical vorticity at the lowest model level, color coded based on their initial altitude (see color bar). Shown are (top) the projection onto the 
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
The result of the vorticity decomposition applied to all these trajectories is shown in Fig. 2, displaying the average over all 1695 trajectories of horizontal barotropic and nonbarotropic vorticity. To obtain the average, the trajectories were transformed to a common time frame defined with respect to the time when the parcels reach the lowest model level. The initial barotropic vorticity of the parcels is determined by the base state and keeps pointing northward throughout the analysis period, but undergoes substantial horizontal stretching before reaching the base of the downdraft. The nonbarotropic vorticity is due primarily to southward horizontal buoyancy torques, consistent with Fig. 3, and subsequent horizontal stretching.4
Horizontal projection of the average nonbarotropic (blue) and barotropic (red) vorticity vectors along the average trajectory (
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
Shown is every 10th trajectory of the set containing 1695 parcels. The buoyancy field (including hydrometeor load) is shaded and the direction of the baroclinic vorticity production is shown by the arrows. The magnitude of baroclinic production is proportional to the horizontal buoyancy gradients: (top) at 1818 m AGL and 3000 s and (bottom) 265 m AGL and 4200 s.
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
The baroclinic and barotropic vorticity parts along the averaged trajectory in the 
Average baroclinic (blue) and barotropic (red) vorticity vectors in the 
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
The distributions of total, barotropic, and baroclinic vertical vorticity of all parcels as they are descending through 50 m AGL are shown for two deformation-magnitude thresholds in Fig. 5a (
Box-and-whisker plots of the total, baroclinic, and barotropic vertical-vorticity distribution of the parcels as they reach the lowest model level (50 m AGL). (a) Parcels with 
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
Several additional tests were performed to evaluate the robustness of the results: (i) to assess the technique used to obtain the barotropic vorticity, this vorticity was calculated by numerically integrating the 3D vorticity equation but omitting baroclinic- and frictional-production terms (this integration was performed using updates of the forcing function every 10 s and an integration step of 0.1 s); (ii) to assess the sensitivity of the results to errors in the initial orientation and magnitude of the vorticity (cf. the base-state value), the analysis was repeated using the assumption that the initial barotropic vorticity of each of the 3481 parcels was given precisely by the base-state vorticity 
Sample averages of barotropic and baroclinic vertical vorticity at the downdraft base for the single-moment simulation. Shown are the results for different methods, filter thresholds, and model runs. The method of obtaining the barotropic vorticity is either via Cauchy’s formula (the “Lagrangian stencil technique”) or RK2 integration. The filter based on initial perturbation is referred to as “IC filter” and is described in the text. The F filter pertains to the maximum allowed deformation magnitude, as also described in the text. The symbol 
b. Double-moment microphysics simulation
Now that the dominance of the baroclinic vorticity within the near-ground vorticity maxima has been established for the single-moment run, we turn to the sensitivity of this result using a double-moment scheme. Overall, the simulation with the double-moment microphysics parameterization evolves similarly to the single-moment simulation. The 330 trajectories that contribute to near-ground vorticity maxima are shown in Fig. 6 and generally originate from lower altitudes than those in the single-moment run. Strikingly, in the double-moment simulation there are fewer downdrafts and less frequent downdraft surges, and thus fewer ζ extrema in the cold pool (Fig. 7). Moreover, the horizontal baroclinic vorticity production is weaker overall than in the single-moment run (Figs. 7b,d). However, as also shown in Figs. 7b and 7d the ζ maxima in each simulation emanate from the most intense downdraft cores and conspicuously emerge from concentrated regions of large horizontal buoyancy gradients (see also the animated version of Fig. 7, available in the online supplemental material). To better understand the relative importance of baroclinic and barotropic contributions in this case, the horizontal projection of the averaged two vorticity parts is shown in Fig. 8. The vorticity parts evolve qualitatively identical to those in the single-moment run, and again the barotropic negative vertical vorticity is overwhelmed by the cyclonic baroclinic contribution (Fig. 9). This experiment demonstrates that also with a double-moment microphysics scheme the baroclinic contribution dominates.
This figure displays the 330 trajectories for the double-moment simulation at 4200 s that contribute to vertical vorticity at the lowest model level, color coded based on their initial altitude (see color bar; note the different scale compared to Fig. 1). Shown are (top) the projection onto the 
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
A snapshot of the simulation using (a),(b) the single-moment microphysics scheme and (c),(d) the double-moment microphysics scheme. (a) Buoyancy (including hydrometeor load; shaded), vertical velocity (−2 m s−1; black contours) at 265 m AGL, and positive ζ at the lowest model level (contoured for 0.005 and 0.01 s−1) at 4500 s. (b) As in (a), but that the shaded field is the magnitude of the buoyancy torque. (c),(d) As in (a),(b), but for the double-moment simulation and at 4710 s.
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
Horizontal projection of the two vorticity parts for the average trajectory in the double-moment simulation. The nonbarotropic (blue) and barotropic (red) vorticity vectors are plotted along the average trajectory (
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
Average baroclinic (blue) and barotropic (red) vorticity vectors in the 
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
Summarizing the results so far, the warm-rain Kessler scheme (Rotunno and Klemp 1985), as well as the Lin-type single-moment scheme and the Morrison scheme presented herein, each exhibiting different outflow characteristics, favor the baroclinic mechanism. A cartoon of the general behavior of the vorticity in these simulations is shown in Fig. 10.
Conceptual model of the vorticity evolution along a typical trajectory (black line) within the simulations analyzed herein. Red arrows represent barotropic vorticity and blue arrows represent baroclinic vorticity.
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
4. Discussion
a. Why does the baroclinic mechanism seem to dominate?
It is intriguing that the above results and a large number of previous studies analyzing observed storms and idealized simulations consistently find that downdraft production of vertical vorticity near the ground is due primarily to the baroclinic mechanism.5 This implies that the barotropic mechanism is ineffective for a wide range of representations of cloud microphysics ranging from warm-rain (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Wicker and Wilhelmson 1995; Adlerman et al. 1999) to Lin-type (Dahl et al. 2014 and this study) to double-moment (this study) to idealized heat sink (Markowski and Richardson 2014; Parker and Dahl 2015) parameterizations, as well as observed cases (Markowski et al. 2008, 2012). It is, thus, tempting to speculate that there is a fundamental reason that leads to this dominance of baroclinic vorticity.6 The leading-order effect is most likely that tornadic environments tend to be dominated by streamwise ambient storm-relative vorticity, implying that in such cases the ambient vorticity does not contribute to ground-level ζ as discussed in section 1. But, why does the baroclinic vorticity also seem to dominate in cases where the ambient storm-relative vorticity has an appreciable crosswise component?
If we assume a constant characteristic time scale, the implication is that 
The curves describe the surface vertical-vorticity parts [baroclinic (blue) and barotropic (red)] at the downdraft base as a function of downdraft intensity based on the scaling argument. The vertical dashed line marks the downdraft intensity above which the baroclinic vertical vorticity dominates. The horizontal barotropic vorticity is assumed to be constant.
Citation: Monthly Weather Review 143, 12; 10.1175/MWR-D-15-0115.1
For a storm exhibiting a variety of downdraft intensities, the argument predicts that weaker downdrafts will tend to deliver less vertical vorticity at its base than stronger downdrafts. However, the important finding is that the barotropic vorticity dominates only within those weaker downdrafts. This corresponds to the region left of the vertical line in Fig. 11. While the stronger downdrafts more effectively tilt horizontal barotropic vorticity (
Based on the above argument it may be speculated why strong surface ζ extrema dominated by barotropic vorticity seem to be rare: downward tilting of horizontal vortex lines (and downward advection of the resulting vertical vorticity) is most effective in strong downdrafts. Weaker downdrafts on the other hand, are ineffective at tilting horizontal vorticity into the vertical and transporting the vertical vorticity to the ground. However, the barotropic mechanism only dominates in this weak-downdraft regime. The resulting weak barotropic vertical surface vorticity may take too long to be stretched into an intense vortex within time scales that parcels typically spend in horizontally convergent flow. Put another way, the scaling argument predicts that as downdraft strength increases not only does the total vertical vorticity delivered at the downdraft base increase, but also that this vorticity is increasingly dominated by baroclinic vorticity.
Near-ground rotation in axisymmetric simulations (Markowski et al. 2003; Davies-Jones 2008; Parker 2012) is dominated by barotropic vortex-line reconfiguration, because the azimuthal baroclinic vorticity cannot be tilted into the vertical. However, the horizontal flow field in which the downdraft is embedded in axisymmetric simulations is not particularly representative of sheared, 3D convective storms. Thus, in the above argument a more realistic setting was assumed with nonaxisymmetric horizontal flow through the downdraft, which was found to be the basic requirement for the onset of near-ground rotation by Davies-Jones (2000) and Parker and Dahl (2015).
b. The role of surface friction
In this study surface friction is neglected and the focus is strictly on the relative roles of ambient and storm-generated vorticity at the base of downdrafts. At this stage of vorticity acquisition, surface friction should play a rather small role as the horizontal-velocity profile of air reaching the surface during descent has not yet adjusted to surface friction [Letchford et al. 2002; a sample of measured wind profiles within thunderstorm outflows can be found in Gunter and Schroeder (2015)]. Parker and Dahl (2015) found no appreciable difference between their simulations with and without surface friction. This is in contrast to Schenkman et al. (2014), who did find that frictional torques at the base of downdrafts were the dominant source of horizontal vorticity for some of the parcels they analyzed. More research is needed to explain this discrepancy, but it is likely that the completion of tornadogenesis as well as tornado maintenance rely on processes beyond the barotropic and baroclinic mechanisms discussed herein.
The idealized base-state shear profile used herein could not be maintained in the presence of surface friction, which would alter the low-level ambient vorticity. The orientation and perhaps the magnitude of the barotropic vorticity of near-ground parcels riding up the left-flank boundary would thus be expected to vary from the results presented herein. The explicit effect of surface friction in the context of vorticity decomposition is left for future research.
5. Conclusions
In this study the relative importance of ambient crosswise (barotropic) vorticity and storm-generated (baroclinic) vorticity in producing vertical-vorticity maxima at the base of downdrafts in supercells was investigated. The goal was to analyze how robust the baroclinic mechanism is. Two supercells in unidirectional shear were simulated, using a single-moment and a double-moment microphysics parameterization, respectively. A large number of forward trajectories that contribute to cyclonic vorticity at the base of downdrafts was analyzed for a time period of about 30 min and the vorticity was decomposed into barotropic and baroclinic parts. Independent of the microphysics parameterization, the barotropic vorticity remains weaker than the baroclinic vorticity and is tilted downward within downdrafts, while the baroclinic vorticity has a much larger magnitude and is tilted upward.
The observation based on this study and previous work that the dominance of the baroclinic mechanism seems rather insensitive to the microphysics parameterizations (and the shear profiles) may be related to the following factors: (i) in cases with streamwise ambient vorticity, the barotropic contribution to near-ground rotation is small because streamwise vorticity becomes horizontal along trajectories near the surface; and (ii) in cases with crosswise ambient vorticity, a scaling argument predicts that the baroclinic vertical vorticity becomes increasingly dominant as downdraft strength increases. That is, the imported barotropic vorticity tends to be overwhelmed by baroclinic vorticity except in the weakest downdrafts, which, however, do not yield much vertical vorticity altogether at their base. This mostly barotropic vorticity may be too weak to be concentrated effectively by horizontal convergence.
Acknowledgments
I would like to thank Drs. Matt Parker, Paul Markowski, Lou Wicker, George Bryan, Yvette Richardson, Bob Davies-Jones, Dan Dawson, Marcus Büker, and Scott Gunter for insightful discussions. George Bryan is gratefully acknowledged for maintaining the CM1 model and for implementing the Lagrange polynomials in the parcel interpolation routine. I also thank the students in the Atmospheric Science Group at TTU for comments on an early draft of the manuscript. Reviews by Drs. Rich Rotunno, Alex Schenkman, and an anonymous reviewer contributed insightful comments that led to additional analysis and improved the overall presentation.
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In this study, “ground level” rotation or “near ground” rotation refer to rotation of air about a vertical axis an arbitrarily small distance above the lower boundary.
The material circuit analyzed by Rotunno and Klemp (1985) was located in a regime of appreciable baroclinity at the initial time, such that the association of the initial circulation with the ambient contribution is somewhat uncertain. However, because the baroclinic production (see their Fig. 12) was positive at the initial time while the circulation was negative, it seems likely that the ambient circulation indeed was negative.
Isolating thousands of trajectories out of a set of several million parcels is computationally quite expensive. Increasing the upper boundary of the ζ interval would have increased the number of parcels.
Since baroclinic vorticity production is rather inhomogeneous and unsteady, it is impossible to display the buoyancy field at a single time and height representative of the baroclinic vorticity generation for all parcels. However, the results for individual parcels were carefully checked and the inferred baroclinic vorticity is consistent with the horizontal buoyancy gradients within the storm. As in the Del City simulation (Dahl et al. 2014), the dominant vorticity production occurs in the lowest few 100 m above the ground within strong horizontal buoyancy gradients at the periphery of the main downdraft north of the mesocyclone. It thus seems justified to refer to the nonbarotropic vorticity as baroclinic vorticity.
The author is aware of only one study that suggests that ambient vorticity is the dominant contributor to an intense near-ground vortex in a supercell (Mashiko et al. 2009). However, these authors calculated parcel histories of only about 5 min prior to the parcels entering the vortex, which makes it rather unlikely that the initial (barotropic) vorticity corresponded to the ambient vorticity.
The basic downdraft processes simulated by Parker and Dahl (2015) were not changed in important ways when surface friction was included in their simulations, implying that at least for the onset of near-ground rotation, surface friction is not the dominant contributor. This point will be addressed again at the end of this section.
Dynamically forced downdrafts, such as the occlusion downdraft (Klemp and Wilhelmson 1983; Markowski 2002) are approximately irrotational and, hence, cannot directly produce horizontal vorticity baroclinically. The argument herein pertains only to downdrafts produced by negative buoyancy.
An assumption implicit in this argument is that the orientation of the horizontal vorticity parts does not depend on the accumulated baroclinity (~downdraft strength). The barotropic horizontal vorticity is assumed to be given by the base state and, hence, does not vary for a given storm. That the baroclinic vorticity does not change its horizontal orientation appreciably as downdraft intensity is increased was confirmed by idealized downdraft simulations such as those in Parker and Dahl (2015; not shown).
In the simulations discussed herein, there is a considerable amount of variation among the parcels regarding residence time within downdrafts and, hence, accumulated baroclinic vorticity, as well as horizontal deformation, which in turn yields a large range of horizontal barotropic vorticity magnitudes among the parcels, making it difficult to test the prediction of the above argument. Instead, one probably would have to perform highly controlled experiments that only vary downdraft strength/baroclinity [perhaps similar to those by Parker and Dahl (2015)].
