1. Introduction
Recent work on the vorticity dynamics of supercell tornadoes has emphasized the transition of the dominant processes by which parcels acquire nearsurface vertical vorticity during tornadogenesis. First, regions of surface vertical vorticity (vortex patches) appear within outflow air of the supercell. The production of these vortex patches relies on negatively buoyant downdrafts (e.g., DaviesJones and Brooks 1993; Walko 1993; Dahl et al. 2014; Parker and Dahl 2015). These seed vortex patches tend to axisymmetrize while also amplifying (Dahl 2020), whereupon horizontal vorticity can be tilted upward abruptly within the corner flow of the vortex (Fischer and Dahl 2022, and the references therein).
Since parcels in a supercell’s environment carry abundant horizontal vorticity, a question that arises is whether this vorticity can directly feed the tornado (even if a detour through a downdraft may occur). Although the association of tornadoes with strong lowlevel wind shear has been known for some time (e.g., Rasmussen and Blanchard 1998), so far only indirect effects have been identified by which the ambient vorticity is beneficial to tornadoes. For instance, Markowski and Richardson (2014) and Coffer and Parker (2017) have shown that the more intense and streamwise the ambient vorticity is, the more intense and better organized is the mesocyclone, and the lower its base, implying stronger upward pressuregradient accelerations. These help lift the verticalvorticitybearing outflow air and thus support tornadogenesis and tornado maintenance.
One challenge in analyzing the direct role of ambient vorticity in maintaining tornadoes is that the integral of the traditional vorticity equation (i.e., integrating each term of the vorticity equation separately) merely reveals that, e.g., tilting and stretching dominates, but the origin of the vorticity that is being tilted and stretched cannot be identified. DaviesJones (2000) pioneered application of a different formulation of the integral of the vorticity equation, often referred to as vorticity decomposition, to idealized supercelllike flows.^{1} This vorticity source decomposition allows one to determine the effect of tilting and stretching of the part of the vorticity that has (i) existed initially and (ii) that has subsequently been produced via baroclinic or mixing effects. Dahl et al. (2014) and Dahl (2015) applied a simplified version of this decomposition to the development of vortex patches that serve as seeds for tornadogenesis within the storm’s outflow. They found that baroclinically produced and subsequently reoriented vorticity (“baroclinic vorticity”) dominates in these vortex patches, and that the average effect of vorticity imported from the environment at best is neutral (Dahl et al. 2014), or in some cases contributes negatively (Dahl 2015). Markowski (2016) applied the vorticity source decomposition to idealized “pseudosupercell” simulations and found that as the storm matured the main vorticity source of the tornadolike vortices (TLVs) that developed in the simulation was baroclinic vorticity, though the imported (“barotropic”) vorticity also contributed. Due to limitations of the numerical integration of the vorticity parts, the analysis by Markowski (2016) focused on vortices relatively early in the simulation, rather than what may be considered mature TLVs. In this paper, the problem is thus revisited by considering a mature, simulated tornadolike vortex in a fullphysics simulation, and by using the complete vorticity source decomposition (including numerical and subgridscale mixing terms) along 30min forwardintegrated trajectories. We will also contrast the vorticity source decomposition with the traditional integral of the vorticity equation.
In the next section, the simulation will be introduced. In section 3 the vorticity equations, their integrals, and their utility will be reviewed. Further, a derivation of Truesdell’s basic vorticity formula utilizing tools from differential geometry will be offered. In section 4 trajectories will be analyzed, and the results will be discussed in section 5. Finally, conclusions are offered in section 6.
2. Model setting
We used the Bryan Cloud Model 1 (CM1; Bryan and Fritsch 2002), release 19, to which parcel diagnostics have been added, such that the velocity gradient, the curl of the pressuregradient force and of numerical and subgridscale (SGS) mixing, as well as the curl of the Coriolis force, are available along the parcel trajectories. These contributions were taken directly from the diagnostics available within CM1, and calculated on the respective staggered grids. The simulation analyzed here is the same as the fullphysics simulation of the mature and intense tornadolike vortex analyzed by Fischer and Dahl (2022). The horizontal domain has a size of 200 × 200 km^{2}, with a constant horizontal grid spacing of 200 m in the inner portion of the domain (50 < x < 150 km), stretched to circa 1.8 km toward the domain boundary. The vertical grid spacing increases from 20 m near the ground to 280 m at the domain top, which is located at about 18 km AGL. Cloud and precipitation processes are parameterized using the doublemoment NSSL scheme. The VORTEX2 tornadic composite sounding obtained by Parker (2014) served as base state, and the storm was initiated using the updraft nudging technique as discussed by Fischer and Dahl (2022). The lower boundary condition is semislip, and the lateral boundary conditions are openradiative. A fifthorder horizontal and vertical advection scheme is employed.
When a semislip boundary condition is used at the lower boundary, a pseudo pressure gradient force needs to be present in order to maintain a quasisteady environment (without this, the surface flow will continuously lose momentum). One common approach, which is also adopted herein, is to include the Coriolis force, but only allow it to act on the perturbation winds. In the present case this led to minor adjustment of the flow field early in the simulation. DaviesJones (2021) has warned of some pitfalls in this technique. In this approach, it is implicitly assumed that the base state is in thermal wind balance. Consequently, continued circulation production is implied in a vertical plane due to the baroclinity associated with the thermal wind shear, but this circulation production is balanced by the tilting of planetary vorticity into the horizontal (DaviesJones 1991). The Coriolis force thus has an effect only on the unbalanced (i.e., ageostrophic) part of the flow. However, since the real wind profile most likely is not geostrophically balanced (due, e.g., to isallobaric effects and surface drag), the effect of the Coriolis force may be erroneous. However, in the traditional integral of the vorticity equation, this effect is negligible (the Coriolis forcing is practically zero for all parcels we investigated). For the vorticity source decomposition (section 3b), the Coriolis effect is nonnegligible but it does not contribute meaningfully to the vorticity budgets as shown in section 4, so this error is unlikely to affect the main results significantly.
A total of 5.4 million trajectories were launched at 2400 s simulation time covering the domain [100, 145] × [90, 120] × [0.01, 1.8] km^{3}. These initial locations were found iteratively as described by Fischer and Dahl (2022). The trajectories were then integrated forward for 30 min. Such long histories are needed if the intention is to decompose the vorticity into parts that stem from (approximately) the environment as well as stormgenerated contributions. That is, the parcels ought to begin their journey in the unperturbed environment upstream of the storm. Only parcels that became part of the TLV, i.e., attained vertical vorticity in excess of 0.1 s^{−1} between 20 and 70 m AGL and between 4135 and 4145 s were considered, giving 95 parcels. Of those parcels, the majority dropped below the lowest scalar model level (10 m AGL), rendering them unsuitable for analysis (Vande Guchte and Dahl 2018). This is a rather consequential criterion, because tornadoes are characterized by strong upward motion very close to the ground. This means that there is upward motion at the lowest model level, implying that parcels generally enter the vortex base from below the lowest model level. Looking higher above the ground to identify parcels within the vortex reduces this problem, but these parcels may no longer capture the “nearsurface” dynamics. Parcels that enter the vortex in the lowest ∼50 m AGL and at the same time have not dipped below the lowest scalar model level are thus very rare.
Moreover, the initial horizontal vorticity of the parcels, rounded to one hundredth, was not allowed to differ from the basestate horizontal vorticity rounded to one hundredth. This was done to ensure that the parcels approximately had the same vorticity as the base state when starting the integration. For most parcels, this meant
3. Vorticity equations and their integrals
a. Traditional integration of the vorticity equation
b. Vorticity source decomposition: Truesdell’s basic vorticity formula
An alternative way of decomposing the vorticity vector is based on Truesdell’s “basic vorticity formula” (Truesdell 1954, p. 154). An elegant way of obtaining it is afforded by the use of the material (Lagrangian) coordinate system. This system is made up of a coordinate grid that moves with the flow and is deformed along the way (e.g., Salmon 1998, p. 5). The derivation here utilizes classical tensor calculus following, e.g., Truesdell (1954) and Dutton (1976), and it extends the treatment by Dahl et al. (2014, their appendix A) by leveraging the material description of vorticity. The reader is referred to the extensive work by DaviesJones (2000, 2006, 2015, 2022) for alternative derivations and formulations applicable to specific flows and configurations with an eye to supercelllike flows, and Epifanio and Durran (2002) for applications to orographic vortices.
To begin, we introduce the fixed spatial (Eulerian) coordinates x^{i} and the timedependent material (Lagrangian) coordinates ξ^{α}. In addition, in classical tensor analysis there always exist background Cartesian coordinates y^{j}, relative to which the other coordinate systems and their coordinate basis vectors are defined (e.g., Marsden and Hughes 1983, p. 44).
All indices run from 1 to 3, where the spatial components have Roman letters as indices, and the material components are represented by Greek letters. That is, the components a^{i} of a vector a pertain to the spatial basis, and the components a^{α} pertain to the material basis. The fixed spatial coordinates are assumed to be nonCartesian to keep the analysis as general as possible, though practically the spatial coordinates are often taken to be Cartesian. The material coordinates, in the present interpretation, remain attached to each fluid parcel and are given by the spatial coordinates that the parcels occupy at some arbitrary reference time t_{0}. Throughout, Einstein’s summation convention is used. The reader is referred to Simmonds (1994) for some background on classical tensor calculus in curvilinear coordinates. The advantage of using material coordinates is that the vorticity equation and its integral are comparatively simple in these coordinates. DaviesJones (2022) leveraged this observation by writing the evolution equation of vorticity in material coordinates, which allowed for a straightforward integration. The starting point of our approach is the expression for vorticity itself and subsequent use of a simple integral of the momentum equation to obtain the desired vorticity formulas.
The second part of Eq. (26) describes the effect of the torques due to nonconservative forces acting on the flow, taking into account the incremental expansion/compression effect. In essence, this part is the integral of the material torque. In this analysis, the torque includes the baroclinic vorticity production, vorticity generated via SGS and numerical mixing, as well as curl of the Coriolis force.
Implementation of the vorticity source decomposition
The vorticity source decomposition offers different insights into the vorticity dynamics than the traditional formulation. Aside from the effect just described, whereby a small amount of vorticity production may be identified as dynamically important, the impact of the initial, or preexisting, vorticity can be quantified. If the initial vorticity closely corresponds to the basestate vorticity, the direct contribution of imported ambient vorticity to the rotation of the tornado can be assessed. A practical disadvantage of this vorticity source decomposition is that the vorticity parts in the integrands are not known along the trajectory and they must be calculated using an iterative or an implicit scheme (appendix B). Unfortunately, this integration is prone to error growth, and matching vorticity budgets are less likely to be achieved than for traditional budgets. While Markowski (2016) attributed most of the error growth to the nonlinear advection, in our case this is related to the fact that the tilting and stretching terms are calculated using the partial vorticity obtained from the previous integration steps (in the “traditional” integration, one uses the known interpolated vorticity components, so the problem is not as severe). This effect is aggravated near the vortex where large velocity gradients exist, so that the errors rapidly escalate. To alleviate these errors, we used trajectory data output on every large model time step (0.5 s). The trajectory data were calculated during runtime within CM1 on their respective staggered grids, and then interpolated trilinearly to the parcel locations. When integrating the vorticity budgets several solvers were tested to mitigate error growth (including those available in the Python library solve_ivp as well as manually implemented Runge–Kutta and implicit schemes). While all solution methods yielded the same results, the implicit scheme (appendix B) was most efficient computationally.^{6} Despite the efforts to maximize trajectory accuracy, as well as accuracy when calculating the forcing terms, only for a small number of parcels (n = 7; see section 2) were the integrals of sufficient quality to warrant further analysis.
4. Results
Figure 2 shows the tornadolike vortex at the lowest scalar model level (Fig. 2a), along with the vertical motion field at 250 m AGL. Besides the main cyclonic vortex, a weaker anticyclonic vortex is also present, and there is a downdraft between the vortices and southwest of the main (cyclonic) vortex. These downdrafts are likely forced by dynamic pressure gradient accelerations (e.g., Klemp and Rotunno 1983; Schenkman et al. 2016). Northwest of the main vortex there is a rather stationary internal boundary as revealed by the horizontal windvector convergence, which is accompanied by a tongue of rising motion northwest of the TLV. This boundary and the associated region of upward motion persisted throughout the TLV’s lifetime. As will be seen, these vertical motion regimes are critical in shaping the parcel trajectories as they approach the main vortex.
Figure 2b shows a vertical (south–north) cross section through the center of the cyclonic vortex. Although the vertical vorticity in the vortex core does not exhibit an annular structure, a twocelled structure is apparent in the flow field, with downflow in the central vortex portions and an annular cornerlike flow surrounding it. The upward velocity at the southern vortex periphery exceeds 40 m s^{−1} at about 1 km AGL. The vortex formed around 3000 s simulation time and persisted through the end of the simulation at 5400 s.
a. Trajectory overview
The seven trajectories based on the stormrelative wind are shown as black line segments in Fig. 3 and are seen to enter the TLV. The vast majority of the trajectories originating from the north/northeast of the vortex descend below the lowest scalar model level or had poorly matching budgets and were not considered. The focus is thus on the airstream originating east of the mesocyclone. While parcels with similar trajectories have been identified in previous studies (e.g., Wicker and Wilhelmson 1995; Markowski and Richardson 2014; Dahl et al. 2014; Mashiko 2016; Boyer and Dahl 2020), in the present case the descent in the forward flank region is rather gentle, and the parcels do not yet reach the surface. Instead, they rise again as they traverse the tongue of rising motion northwest of the main TLV discussed in the previous paragraph. The parcels subsequently descend comparatively quickly close to the surface in the occlusion downdraft before they enter the TLV [as was also observed by, e.g., Schenkman et al. (2014) and Mashiko (2016)].
Before analyzing the vorticity dynamics in detail, the vertical vorticity parts of the parcels as they enter the base of the vortex, obtained by integrating Eqs. (41)–(45) along each of the seven trajectories, are shown in Fig. 4. Parcel A is represented in blue, and parcel B in red. Notably, for all parcels the vertical vorticity is dominated by the SGS part caused mainly by surface drag [even for parcel A (blue dots), ζ_{SGS} becomes slightly positive as the parcel rises in the vortex]. The ζ_{P} and ζ_{BC} parts contribute positively only for some parcels. Interestingly, for the parcels analyzed here, when the imported part is positive, the baroclinic part is negative, and vice versa. The part due to numerical mixing ζ_{DIFF} also makes a net positive contribution to the vorticity budgets and will be analyzed separately in section 5b. Finally, the contribution due to the Coriolis force (rearrangement of planetary vorticity) is generally small and will not be discussed in the subsequent analysis. In the following we will consider two representative parcels for which either the imported or the baroclinic vertical vorticity dominates at the base of the TLV.
b. Parcel A
First, we consider parcel A (Fig. 3). Before presenting the vorticity source decomposition, the results of the traditional integration of the vorticity equation, Eqs. (2)–(6), will be shown separately for the horizontal vorticity magnitude and the vertical vorticity component. The contributions to the horizontal vorticity magnitude are displayed in two different ways: (i) by calculating the magnitudes due to each contribution separately (e.g.,
The analysis reveals that the horizontal vorticity magnitude is dominated by horizontal stretching as the parcels accelerate toward the vortex (Fig. 5a). There are relatively large contributions from the SGS mixing terms after about 4000 s, which however are oriented such that the horizontal vorticity magnitude decreases somewhat. This is seen in Fig. 5b, which shows the projection of each integrated horizontal vorticity contribution onto the horizontal vorticity vector.
Notably, there is practically (but not perfectly) zero baroclinic production of horizontal vorticity along the parcel’s 30 min history. What little baroclinic vorticity production the parcel experiences occurs in the forward flank region of the storm, where there are weak horizontal buoyancy gradients (Fig. 6). Notably, but perhaps not unexpectedly, no appreciable horizontal vorticity is generated baroclinically as the parcels descend through the occlusion downdraft.^{7}
The vertical vorticity budget, shown in Fig. 7 is dominated by the tilting and stretching terms, as well as by implicit diffusion. It is seen that the tilting and numerical diffusion contributions increase as the parcel reaches its nadir (at about 13 m AGL), leading to an increase in vertical vorticity. Subsequently, the stretching term, acting on that vertical vorticity, leads to a rapid increase of vertical vorticity as the parcel enters, and rises within, the vortex. This progression is consistent with the “inandup” mechanism (Boyer and Dahl 2020; Fischer and Dahl 2022) augmented by diffusion.
Perhaps surprisingly, when considering the vorticity source decomposition (Fig. 8) the seemingly negligible amount of baroclinically produced vorticity is subject to significant stretching, such that the baroclinic vorticity becomes a dominant source in terms of vorticity magnitude. Another significant contribution comes from the imported vorticity. However, as seen in Figs. 8a and 8b, the two vorticity parts point in opposite directions. The ω_{P} part contributes positively to the vertical vorticity, while the baroclinic part contributes negatively. That is, the imported vorticity directly helps maintain the rotation of the TLV via abrupt upward tilting as the parcel enters the TLV. The vorticity due to SGS mixing remains horizontal as the parcel enters the vortex, and thus does not contribute to the vertical vorticity of the parcel (only a bit later the SGS vorticity attains a positive vertical component as the air continues to rise). The diffusive vertical vorticity contributes the most, but as will be discussed in section 5b, this effect does not contribute to vortex maintenance. Figure 8 only affords a qualitative picture of the relative magnitudes and general orientations of the different vorticity parts. To quantify the different vorticity parts, Fig. 9 shows the individual horizontal vorticity magnitudes as well as the vertical vorticity, highlighting the diffusive and preexisting (imported) parts as the dominant contributors.
c. Parcel B
The traditional integrals for parcel B (Fig. 3) reveal a similar evolution as was found for parcel A. When considering the magnitudes of the horizontal vorticity contributions using the traditional integral, shown in Fig. 10a, the dominant terms are again the tilting/stretching terms as the parcel accelerates toward the vortex, entering it about 20 m AGL. The magnitude of the integrated baroclinic production is nonzero, but this contribution has a component in the opposite direction of the horizontal vorticity vector, implying that it tends to reduce the horizontal vorticity magnitude. This is inferred from the projections of the horizontal vorticity contributions onto the horizontal vorticity vector, shown in Fig. 10b. Likewise, implicit mixing again contributes negatively, but in contrast to parcel A, SGS mixing leads to an increase of horizontal vorticity magnitude due to surface drag.
The vertical vorticity budget, shown in Fig. 11, likewise resembles that of parcel A. First the vertical vorticity due to tilting increases, and this vorticity is subsequently stretched vertically as the parcel rises in the updraft. The effect of implicit diffusion also contributes to an increase in vertical vorticity, as for the other parcel. In short, the traditional vorticity budgets are quite similar for both parcels.
The source decomposition reveals rather large differences in the vorticity dynamics of parcel B compared to parcel A. As shown in Fig. 12, both the imported and baroclinic vorticity vectors along trajectory B become oriented oppositely to the corresponding vectors along trajectory A. That is, the imported preexisting vorticity attains a downward component, while the baroclinic vorticity points upward, which thereby contributes positively to the rotation of the TLV. This is also seen in the time evolution of the different vorticity parts displayed in Fig. 13. Moreover, the SGS vorticity, caused mainly by surface drag is abruptly tilted upward at the base of the vortex, as also recognized by Schenkman et al. (2014), Roberts et al. (2016), and Roberts and Xue (2017).
In summary, while the analysis of the traditional vorticity equation suggests that both parcels behave rather similarly to one another, the vorticity source decomposition reveals stark differences in the relative contributions of imported and baroclinic vorticity. These differences are explored in the next section.
5. Discussion
a. Difference in behavior of imported and baroclinic vorticity between trajectories A and B
An interesting observation in the analysis is that, while the traditional vorticity budgets pointed to rather similar vorticity dynamics, the vorticity source decomposition yielded large differences between parcels A and B in the configuration of both the baroclinic and imported vorticity vectors. While the vorticity vectors (either imported or baroclinic) are initially aligned similarly, they are imbedded in different strain fields, which is in part related to the inhomogeneous downdraft field these parcels pass through (Fig. 14). To see the differences in the strain field along the trajectories, the principal axes of the local strain tensor corresponding to the intermediate eigenvalues are shown in Fig. 15, along with the imported vorticity. The eigenvectors were calculated following Kundu and Cohen (2008, p. 41). The orientation of the eigenvectors corresponds to the axes of deformation. Although one might expect the vorticity vectors to become aligned with the eigenvector associated with the largest positive eigenvalue, which signifies the direction of maximum stretching (Hamlington et al. 2008), it has long been recognized that the vorticity vectors tend to become aligned with the intermediate axis of deformation of the rateofstrain tensor (e.g., Hamlington et al. 2008; Coriton and Frank 2016). Hamlington et al. (2008) explain this behavior with the strain induced by the local vortex structure itself, which often exceeds the background strain that reoriented and stretched the vortex in the first place. As can be seen in Fig. 15, the imported vorticity has a different alignment for each parcel relative to the deformation axis, indicating that the vorticity vectors along each trajectory will be reoriented in a different fashion, ultimately resulting in opposite orientations of the vectors for each parcel. The final alignment approaches the intermediate deformation axis (Hamlington et al. 2008).
The result is that all vorticity parts tend to become oriented approximately parallel or antiparallel to one another, as is also apparent in the analysis by Dahl et al. (2014) and Dahl (2015). This implies that even in the presence of the source terms, influencing the vorticity evolution for all but the imported vorticity parts, the deformation effect (tilting/stretching) dominates over those sources. That is, the tendency to become aligned with the intermediate deformation axis is not affected much by the production of new vorticity, which would not in general be aligned with the intermediate deformation axis.
b. Role of implicit diffusion
The terms in Eq. (52) may be integrated numerically to see how parcels acquire their vertical vorticity as they enter the vortex. Figure 16 shows that parcels first gain vertical vorticity via diffusion, whereupon this vorticity is strongly stretched vertically. This increase of vertical vorticity via diffusion is also evident in the vorticity budgets discussed above (Figs. 7 and 11). Because all parcels in the Burgers–Rott solution acquire their vertical vorticity via diffusion, one might conclude that the vortex core is maintained by diffusion. However, to obtain the steady vortex solution some initial vertical vorticity must have been present that subsequently has been concentrated. In the steady state, this initial vorticity resides at r = 0. Given the freeslip lower boundary in the Burgers–Rott model, the parcels initially at z = 0 remain there while parcels at z > 0 rise. The result is continued vertical stretching of their initial vertical vorticity at r = 0. It is this stretched initial vorticity that maintains the vortex. This stretching effect alone would result in unbounded growth, but it is offset by diffusion. The relevant vorticity that maintains the vortex is thus infinitely thin and located at the symmetry axis r = 0. All parcels outside the symmetry axis acquire their vorticity via diffusion. The vertical vorticity is never exhausted because the vortex line terminates at z = 0 where the vertical velocity is zero. In realworld vortices, the vorticity in the inner core must be supplied continuously, in the case of a mature tornado mainly via upward tilting of horizontal vorticity at the base of the vortex (e.g., Fischer and Dahl 2022). Incidentally, drain vortices likewise need a continuous source of vertical vorticity to persist (Yokoyama et al. 2012).
The implication is that parcels that acquire vertical vorticity via diffusion do not maintain the core vorticity and hence do not actively partake in vortex maintenance—these parcels merely benefit from large vorticity in the central core. Analogous arguments apply to the horizontal vorticity: Another process must have created large horizontal vorticity gradients in order for parcels to acquire horizontal vorticity via diffusive processes. Diffusion can thus not be invoked to explain vortex maintenance.
c. Connection to angular momentum conservation
At first sight, an easier perspective of vortex maintenance, rather than using vorticity dynamics, may be to consider angular momentum (or perhaps more appropriately, circulation) conservation. Here, the air outside the vortex core has approximately zero vertical vorticity while the swirl velocity of air parcels increases as they converge toward the vortex center. In other words, the airflow just outside the vortex core, where still much of the damage associated with a tornado occurs, has no vertical vorticity. One might wonder, then, why the acquisition of vertical vorticity matters. The reason is that the air in the far field has positive circulation only because there is a vortex core—without its presence the circulation would be zero, because circulation is determined by the vorticity vectors piercing the surface around which circulation is calculated. Consequently, the average swirl velocity along the boundary of this surface would be zero if the circulation was zero (and conservation of zero angular momentum would not result in increasing swirl velocities upon convergence). The implication is that the angular momentum explanation of vortex maintenance ultimately relies on the mechanisms that maintain the vortex core.
d. Limitations
While the analysis suggests that there are rather diverse pathways by which parcels may acquire large nearground vertical vorticity as they enter a tornado, a shortcoming of this study is the relatively small number of parcels considered. Aside from the problem of parcels descending beneath the lowest scalar model level, the main culprit is the fact that the parcels are subject to strong velocity gradients, potentially rendering (i) the trajectories themselves and (ii) the vorticity integration, unreliable. Problems such as this one, requiring very small time steps for solutions to converge, are sometimes referred to as stiff (e.g., Shampine et al. 2003) and they are inherently difficult to solve. As a result, while meaningful solutions could be found for only a few parcels, the vast majority of the trajectories could not be used for the analysis. While in the small sample shown in Fig. 4 the vorticity related to surface drag (SGS mixing) dominates, it is not clear how representative this sample is. However, the importance of ω_{SGS} in mature TLVs has been highlighted also in previous studies (e.g., Schenkman et al. 2014; Roberts et al. 2016; Roberts and Xue 2017; Fischer and Dahl 2022). Moreover, for our parcels to acquire large horizontal vorticity due to surface drag, they need to be close to the lowest scalar model level (only at the lowest scalar model level the effect of surface drag is included). Unless parcels reached heights of ∼10 m AGL, they experience less vorticity production due to surface drag than realworld “nearsurface” parcels would experience. Such parcels enter the vortex from within the boundary layer of the tornado, such that they most likely experience significant horizontal vorticity production due to surface drag. This implies that in the present sample the importance of surface drag may still be underestimated.
An alternative to the integration method used here could be to use the material stencils as implemented by Dahl et al. (2014), but the large deformation near the vortex, as well as the requirement that all stencil parcels remain above the lowest scalar model level rendered this technique not as effective for analyzing an intense vortex. For this reason, Dahl et al. (2014) and Dahl (2015) avoided considering mature vortices, and instead only focused on the weak nearground “seed” vortex patches, such that the velocity gradients still had relatively small magnitudes and the stencils remained relatively wellbehaved. In those settings, a larger number of parcels could be considered, and for these seed vortex patches the robustness of the baroclinic downdraft mechanism could be affirmed (Dahl 2015).
Another possible solution to the small yield of the vorticity source decomposition technique as applied here is to consider the circulation evolution of material circuits. Potential disadvantages are that these circuits tend to become extremely contorted (e.g., Mashiko 2016), possibly rendering the calculations inaccurate, and that only the average partial vorticity component normal to the surface can be identified.
An error source potentially affecting the interpretation of the vorticity source decomposition involves the initial condition. To be able to associate the initial vorticity with the ambient vorticity, long parcel histories are necessary to ensure that the parcels are initially in the unperturbed environment. For that purpose, rather long parcel histories were considered (30 min), but even with such long histories, there were still minor perturbations at the parcels’ initial locations (some parcels may have experienced, e.g., small amounts of baroclinic production prior to the beginning of the analysis). The initial vorticity thus did not always perfectly correspond to the base state. Practically, the consequences may be minor, however, given that actual environments are not uniform in the first place, and it is not trivial to even define what constitutes a storm’s environment (e.g., Markowski et al. 1998).
These limitations highlight the challenge of obtaining accurate information about vorticity and circulation budgets for concentrated vortices.
6. Conclusions
Using the CM1 model to simulate an idealized supercell that produced a longlived tornadolike vortex, we presented the integrals of the traditional vorticity equation along seven trajectories. In addition, the vorticity vector was decomposed into parts that arise from the rearrangement of (i) preexisting (ambient) vorticity, and (ii) vorticity that has subsequently been generated within the storm. The main findings may be summarized as follows:

The most reliable dynamical source contributing to the rotation of the mature tornadolike vortex is surface drag (ω_{SGS}), which for all analyzed parcels contributed either positively or, in one case, negligibly, to the vertical vorticity at the base of the vortex (i.e., ω_{SGS} never contributed negatively, unlike the other vorticity parts). Baroclinic vorticity contributed positively for the majority, although not all of the parcels. While not the dominant source across the sampled parcels, the preexisting vorticity did directly contribute to the vortex for some parcels. However, given the small sample size (n = 7), it cannot be established how representative these findings are.

Although the integrated baroclinic vorticity production may seem negligible along a trajectory using the traditional vorticity equation, the subsequent stretching of this initially weak vorticity may lead to large baroclinic vorticity near the vortex as revealed by the vorticity source decomposition. The traditional vorticity analysis in this case emphasizes the importance of tilting and stretching of vorticity, but deemphasizes the importance of small amounts of vorticity production that may end up having a large effect.

Although numerical diffusion is found to contribute positively to the parcels’ vertical vorticity on average, it is shown that diffusion does not aid in maintaining the vortex using the Burgers–Rott model.

The parcels experience the strongest descent in the occlusion downdraft close to the vortex, rather than in the main downdraft northeast of the mesocyclone. Previous studies have shown that trajectories early in a vortex patch’s lifetime are maintained by the downdraft mechanism during which time the contributing parcels descend to the ground in the main downdraft and acquire baroclinically generated vorticity. In contrast, the present analysis suggests that once the vortex matures and the vortex is maintained via abrupt upward tilting of horizontal vorticity at the vortex base, the trajectory topology likewise changes, featuring strong descent mainly in the occlusion downdraft.
Further research is warranted to improve the management of the challenges posed by parcels that descend below the lowest scalar model level, and to increase the accuracy of the integration schemes used to solve the vorticity equations.
Here we refer to this technique as vorticity source decomposition, because the traditional integral of the vorticity equation, discussed in section 3 also constitutes a “decomposition” of the vorticity vector.
If the individual components are written out, the traditional “tilting” and “stretching” terms can be isolated; e.g.,
If
The matrix
Consider an inviscid fluid, in which one may always define a preexisting vorticity part that behaves as if the flow were barotropic, hence the designation “barotropic vorticity.” The remaining vorticity part is newly generated via baroclinity and then rearranged, which is the baroclinic vorticity. However, in barotropic flows with nonzero (molecular, eddy, or computational) mixing, the designation is perhaps misleading because now there is a barotropic part (preexisting vorticity being rearranged) and a “nonbarotropic” part that is caused by viscous effects. There would thus be nonbarotropic vorticity even though the flow is purely barotropic. Moreover, this nomenclature has led several authors to refer to the rearrangement of preexisting vortex lines as a “barotropic” process (e.g., DaviesJones 2008; Markowski 2016; Dahl 2017), which in most cases is misleading because horizontal gradients in vertical motion in convective storms are often associated with horizontal buoyancy gradients, i.e., baroclinity.
Note that the data required along the trajectory are the same for the traditional integration as for the vorticity source decomposition.
The perturbation pressure gradient accelerations are approximately irrotational, and consequently cannot generate vorticity.
Acknowledgments.
Discussions about the numerical solvers with Drs. Eric Bruning and Lou Wicker are gratefully acknowledged, as are exchanges with Drs. Paul Markowski and Matt Parker about various aspects about this research. Thanks are extended also to Dr. George Bryan for his ongoing support of the CM1 model and Dr. Brice Coffer for providing the base state for the simulation. Three anonymous reviewers provided valuable comments that improved the presentation. The figures were created using the Python Matplotlib package, as well as the Grid Analysis and Display System (GrADS). Financial support for this study was provided by NSF under Grants AGS1651786 and AGS2152537.
Data availability statement.
The namelist settings of the CM1 model as well as analysis software are available from the corresponding author upon request.
APPENDIX A
Practical Form of the Vorticity Source Decomposition
In Truesdell’s formulation of the vorticity, Eq. (29), the transformation matrices (in this context also referred to as the deformation gradients) need to be known to obtain a solution, but these are not easily available along trajectories obtained from model or radar data. Also, near the vortex the deformation becomes very large, rendering explicit calculation of the deformation gradient (as was done by, e.g., Dahl et al. 2014) inaccurate. As an alternative, Eq. (29) may be differentiated with respect to time, which gives the evolution equation of the individual vorticity parts. The required steps are demonstrated in this appendix. The resulting equations can then be integrated without the need to calculate the transformation matrices.
APPENDIX B
Numerical Integration of the Vorticity Equations
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