1. Introduction
Tornadoes have great impact on society, and thus, the mysteries of tornadogenesis have been investigated extensively. Most studies have focused on tornadogenesis in conjunction with the mesocyclones of supercells, which produce the vast majority of significant tornadoes (Smith et al. 2012). The pathway to mesocyclonic tornadogenesis has been conceptualized in three stages (Davies-Jones et al. 2001; Davies-Jones 2015):
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the development of a mesocyclone aloft via tilting of ambient horizontal vorticity,
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the development of initial pretornadic vertical vorticity (ζ) at the ground, and
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the amplification of this ζ into a tornado via convergence and stretching.
A great deal of effort has been devoted to understanding stage 2; this is the “origins of rotation” problem. The leading candidates for stage 2 mechanisms1 include the baroclinic production and subsequent tilting of horizontal vorticity in downdrafts (e.g., Davies-Jones and Brooks 1993; Markowski et al. 2008, 2012b; Dahl et al. 2014; Markowski and Richardson 2014) and the frictional generation and subsequent tilting of horizontal vorticity (e.g., Schenkman et al. 2014; Markowski 2016; Roberts et al. 2016; Yokota et al. 2018; Tao and Tamura 2020). In this article, we refer to such vorticity sources as “coherent,” meaning that they produce surface ζ that is more structured than random noise.
Even after stage 2 is accomplished, there are known failure points that can prevent the completion of stage 3; indeed, the majority of supercells are nontornadic (Trapp et al. 2005). The pretornadic ζ from stage 2 is generally found within supercells’ outflow air (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Wicker and Wilhelmson 1995; Markowski et al. 2008, 2012a, 2014; Dahl et al. 2012, 2014; Coffer and Parker 2017; Orf et al. 2017). Relatively cold outflow may be too negatively buoyant to be stretched by the parent storm’s updraft, preventing stage 3 (Markowski et al. 2002; Grzych et al. 2007; Lee et al. 2012; Markowski and Richardson 2014). It is also possible that fast moving outflows may prevent a sustained vertical superposition between pretornadic surface ζ and the strong, deep vertical accelerations of a supercell’s parent updraft (Brooks et al. 1994; Snook and Xue 2008; Guarriello et al. 2018; Brown and Nowotarski 2019; Fischer and Dahl 2023).
Beyond these stage 3 failure points, Coffer and Parker (2017), Coffer et al. (2017), and Coffer and Parker (2018) found that the most important factor in successful tornadogenesis was an intense zone of dynamic lifting that resulted from a stronger, steadier low-level mesocyclone. This finding mirrored the earlier work of Wicker and Wilhelmson (1995), Markowski et al. (2012b), Markowski and Richardson (2014), Skinner et al. (2014), and Coffer and Parker (2015), and has since been reflected in the studies of Fischer and Dahl (2020), Murdzek et al. (2020), and Wade and Parker (2021). In this line of thinking, another likely failure point for supercell tornadogenesis is the absence of necessary dynamic upward accelerations produced by the low-level mesocyclone. Ancillary evidence for the importance of stage 3 dynamic lifting includes the high operational skill of the environmental storm relative helicity in predicting tornadic supercells, particularly in layers very near the ground (0–1 km AGL, or even moreso 0–0.5 km AGL; Rasmussen 2003; Markowski et al. 2003a; Esterheld and Giuliano 2008; Nowotarski and Jensen 2013; Parker 2014; Coffer et al. 2019, 2020); this parameter has been shown to control the strength of the low-level mesocyclone and its associated upward dynamic accelerations (Markowski and Richardson 2014; Coffer and Parker 2015, 2017; Goldacker and Parker 2021).
The development of this line of research, emphasizing the ingredients for success or failure of stage 3, has led to questions about the comparative importance of the origins of surface ζ (stage 2) in the tornadogenesis process (e.g., as discussed by Fischer and Dahl 2020). Clearly, the source of ζ for tornadoes (stage 2) remains an important fundamental question, and it will continue to fuel scientists’ curiosity. But, however stage 2 occurs, it appears that all mature, surface-based supercells have appreciable pockets of ζ at the bottom data levels of simulations (e.g., Fig. 1; Yokota et al. 2018; Markowski 2020; Roberts et al. 2020; Fischer and Dahl 2022) or dual-Doppler wind observations (e.g., Wurman et al. 2007, 2010; Markowski et al. 2011; Kosiba et al. 2013; Murdzek et al. 2020), and that surface vorticity is rather easy to produce (Markowski and Richardson 2014; Parker and Dahl 2015; Fischer and Dahl 2020). This ubiquity of surface ζ, in conjunction with the importance of mesocyclone-associated upward accelerations (stage 3), led Coffer and Parker (2018) to argue that the specific origins of surface ζ (stage 2) are of minimal consequence to whether a supercell produces a tornado or not.
Plan view from Coffer et al. (2017, their Fig. 12) of vertical vorticity at the bottom model level (s−1, shaded) for an ensemble of 15 simulated supercells at the key time period of tornadogenesis or tornadogenesis failure within the nontornadic VORTEX2 composite environment of Parker (2014). The 10-dBZ reflectivity contour (black) for the respective ensemble members is shown at the bottom model level.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
The processes responsible for stages 1, 2, and 3 are not completely independent of one another; for example, the downdrafts that may accomplish stage 2 are linked to the outflow temperature, which may inhibit stage 3. A number of scientists have therefore pursued simpler numerical approaches that at least partly separate the associated processes (Walko 1993; Markowski et al. 2003b; Straka et al. 2007; Davies-Jones 2008; Parker 2012; Davies-Jones and Markowski 2013; Markowski and Richardson 2014; Parker and Dahl 2015; Houston 2016; Markowski 2016; Houston 2017; Markowski and Richardson 2017; Rotunno et al. 2017; Dahl 2020; Fischer and Dahl 2020; Goldacker and Parker 2021). Whereas Goldacker and Parker (2021) specifically sought to isolate the development of large vertical accelerations in environments without surface ζ, the present study takes the complementary approach of examining the fate of preexisting surface ζ in the presence of an updraft already possessing large vertical accelerations. Our research question is, If an updraft has vertical velocities comparable to those in tornadic supercells, does it matter whether the available surface vorticity is well organized? This is relevant given some of the extremely fine-scale structures seen within supercells’ surface ζ fields (e.g., Fig. 1; also seen in studies such as Orf et al. 2017; Markowski 2020), which may not always be statistically distinguishable from random noise.
The present experimental design therefore hinges on the fate of random “vorticity noise.” To make sense of this noise, the regional integral of ζ represented by circulation,
Our working hypothesis is that quasi-random nonzero surface ζ is sufficient for tornadogenesis within a low-level updraft that is characteristic of tornadic supercells. Rejection of the hypothesis would mean that a process for producing well-organized surface ζ (stage 2) is important to whether or not a supercell produces a tornado. Alternatively, if the hypothesis survives testing, this could mean that the final stretching process (stage 3) is the dominant consideration in mesocyclonic tornadogenesis, since all mature supercells appear to have nonzero surface ζ. In pursuit of these aims, the experiments also address whether the horizontal scale of the quasi-random ζ maxima is consequential, whether the updraft-relative speed of the outflow is consequential, and whether there is more impact from the peak in updraft vertical velocity or from its vertical gradient. Methods are described next, followed by results and conclusions.
2. Methods
As explained in section 1, the philosophy of these experiments is to ignore the chain of events that lead to the production of surface ζ. Instead, the focus is on the fate of preexisting, quasi-random surface ζ when subjected to an updraft that is typical for tornadic supercells. This study therefore employs a highly idealized framework for numerical simulations to isolate this specific interaction.
a. Numerical model
The simulations for this study were performed using Cloud Model 1 (CM1; Bryan and Fritsch 2002; Bryan and Morrison 2012), release 20.3. The model simulations excluded moisture, surface drag, radiation, and Coriolis accelerations. The lack of surface drag would be considered an impediment to realistic simulation of mature tornado structure, but seems not to hinder the initial formation of a tornado-like vortex (TLV); as will be shown, the vortices produced in these simulations are still strongly convergent at the bottom model level (i.e., the free-slip analog to a corner flow is established). Because tornadogenesis is typically a rapid, fine-scale process, this study focused on a small horizontal area, close to the surface, for a short period of time. The model grid has an isotropic 50 m grid spacing, providing reasonable resolution of developing TLVs, but not of sub-vortex-scale details. With this grid spacing, CM1 is run in LES mode with the Deardorff (1980) TKE-based subgrid-scale parameterization. The domain extends 20 km in x and y with open lateral boundary conditions, and extends to 10 km in z, with rigid flat upper and lower boundaries and a Rayleigh sponge layer above 7 km AGL. The Rayleigh sponge does directly damp the uppermost part of the simulated updraft, but this had little effect on the near-surface processes being studied here during the period of simulation.
The model simulations were run for 8 min, which is sufficient for development of TLVs given the preexistence of surface ζ and a strong updraft (details of which follow below). Based on previous studies, both the duration of steady dynamic lifting by the low-level mesocyclone and the time scale for tornadogenesis itself seem to be roughly 10 min or less (e.g., Davies-Jones et al. 2001, p. 190). Output was examined and statistics were recorded at 15 s intervals. For each simulation, vortices were identified as extrema in the Okubo–Weiss parameter (Okubo 1970; Weiss 1991), and the strongest vortex in terms of υtan is reported in Tables 1 and 2. If a surface vortex was continuing to strengthen at the 8 min end time, the ending value is reported (but flagged with a dagger in Tables 1 and 2). The stated motivation of the experiments is to examine the interaction of a quasi-random surface ζ field with an intense low-level updraft. Within the simplified moisture-free model configuration, an environment of constant potential temperature (i.e., neutral static stability) is therefore subjected to a localized artificial vertical velocity tendency, as detailed next.
Summary of primary experiments and results. For each simulation, the wavelength (λ) and number used to seed the random number generator in the Perlin noise routine are reported. Circulation is reported both for the outer perimeter of air parcels joining the updraft by 8 min, Ci(outer), and for the maximum value of circulation within that radius, max(Ci), as explained in the text. Each simulation’s peak intensity is reported in terms of maximal tangential velocity at the bottom model level (υtan), maximal vertical vorticity at the bottom model level associated with that time, ζx(υtan), and minimal pressure deficit at the bottom model level associated with that time p′(υtan). Finally, the time corresponding to the maximal υtan is reported, along with the duration (computed as the contiguous time, in 15 s increments, during which υtan was ≥20 m s−1), where a dagger (†) indicates that the duration is likely underreported because the criteria continued to be met at the ending time of the simulation (t = 480 s).
Summary of special experiments and results. Most columns are identical to those in Table 1, with the following exceptions. The updraft nudging (
b. Updraft forcing
The updraft of a potentially tornadic supercell was emulated using the updraft nudging technique of Naylor and Gilmore (2012), with an implementation closely mirroring that in Goldacker and Parker (2021). Specifically, the spatial structure and magnitude of the updraft nudging in the present simulations were tuned to match as closely as possible the updraft of the tornadic supercell simulated by Coffer and Parker (2017) and the four updrafts with the strongest dynamical feedbacks2 simulated by Goldacker and Parker (2021). The nudging was applied within a 3 km radius sphere, centered in the domain and at a height (zc) of 2 km AGL, having a peak (wmax) value of 35 m s−1 at its center, and using a nudging coefficient of 0.5 s−1. The nudging is positive definite and applied at every time step throughout the full 8 min simulation, producing vertical velocities as shown in Fig. 2a for a quiescent model environment. For reference, the quasi-steady updraft velocity at 1 km AGL in this simulation is 26.1 m s−1. To isolate the roles of the peak updraft speed (wmax) versus the near-ground updraft speed, several simulations were rerun with wmax halved (suffix _half in Table 2), and with zc raised to 3 km (suffix _raise in Table 2). The resulting updrafts for these experiments are shown in Figs. 2b and 2c. For reference, the quasi-steady updraft velocities at 1 km AGL in these simulations are 14.5 m s−1 (_half) and 9.3 m s−1 (_raise), respectively. While these updrafts speeds are designed specifically to represent tornadic (mesocyclonic) supercells, the updraft itself is initially nonrotating; thus, this experimental design is equally applicable to nonsupercellular (nonmesocyclonic) tornadoes to the extent that they represent simple interactions between updrafts and surface ζ. We discuss this point further in section 4b.
Vertical cross sections of vertical velocity (m s−1, shaded) averaged from t = 1 to 8 min for the three updraft nudging profiles used in this study (as explained in the text): (a) control nudging (used for most experiments); (b) “half” nudging profile (wmax = 17.5 m s−1); (c) “raise” nudging profile (zc = 3.0 km); (d) control nudging profile including a 10 m s−1 easterly mean wind (“M10” experiments). In each panel the contour of ∂w/∂z = 0.01 s−1 is plotted in black, and the maximum value of ∂w/∂z at the bottom model level is reported.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
Notably, the Naylor and Gilmore (2012) nudging approach is different from the updraft tendencies (applied either via heating or directly to w) that have been used in some other recent studies. The updraft forms more quickly as a result of nudging, such that the resulting updraft is quasi steady in the lowest 2 km AGL after approximately 1 min of simulation time. This made the high-resolution 3D simulations affordable, by allowing immediate interaction with the local ζ noise that was initially imposed beneath the updraft. Interactions of the updraft with the ζ field may cause upward velocities in excess of what is provided by the nudging. However, the nudging approach largely opposes the development of downdrafts from adverse vertical perturbation pressure gradient accelerations, such as supercells’ occlusion downdrafts (e.g., Klemp and Rotunno 1983; Markowski 2002) or the axial downdrafts in high-swirl tornadoes (e.g., Rotunno 1977). The absence of such features in these simulations is consistent with the fact that interactions with the background supercell’s structure have already been omitted. The benefit of nudging is that it ensures a very consistent updraft across all simulations, enabling a truer head-to-head comparison across many realizations of noisy surface ζ. Next, we describe the specification of the initial ζ field with which the updraft interacts.
c. Initial vertical vorticity field
A quasi-random surface ζ field was generated through the use of “Perlin noise” (Perlin 1985), which was implemented via the Python package Perlin-noise (https://pypi.org/project/perlin-noise). This approach is widely used in computer graphics rendering and was originally created to produce “naturalistic visual complexity” replicating highly textured phenomena (e.g., clouds, fires, crystals; Perlin 1985). The most desirable trait for the present application is the ability to specify the wavelength of the noise being created. Whereas Perlin (1985) built up visual complexity by superposing noise seeds with differing wavelengths, almost all of the simulations in this study used only a single wavelength that could be varied among experiments (as a measure of how well the noise is organized). The use of a single wavelength did lead to some modest visible repetition of horizontal structures within the model domain in the initial conditions. But these structures were different within each individual model realization, and any repetition proved inconsequential to the regional integral of ζ represented by circulation.
To create the model initial conditions, the Perlin-noise package was first used to generate quasi-random values of surface ζ. The user-selected parameters describing the Perlin noise for each run included a dominant horizontal wavelength for the noise (λ in Tables 1 and 2) and an initial seed value for the random number generator (“seed” in Tables 1 and 2). At the end of the Perlin noise routine, a small constant was uniformly added or subtracted across the entire domain to zero the mean ζ value. The initial conditions in CM1 were specified in terms of horizontal velocity components, which were recovered from the ζ fields using the induced flow solver3 of Dahl (2020). The surface ζ (and thus velocity) values were assumed to change linearly from their surface values to zero at a height of 1 km AGL. The decline in horizontal velocities up to 1 km implies the existence of horizontal vorticity at individual grid points, but just like the initial ζ, this horizontal vorticity is noisy and has a near-zero domainwide average.
As a baseline, the experimental design began with an initial ζ noise amplitude of
We use the circulation of the initial condition, Ci, to distill the availability of ζ in each simulation. Passive trajectories were initialized on every surface grid point; those that ascended past 500 m AGL by the 8 min simulation end were used to define the footprint of air at the surface that was ingested by the updraft. For a reference simulation without any initial flow disturbances, this footprint was a circular area with a radius of 4.5 km. In this article, Ci is reported for the circuit corresponding to the outer perimeter of this surface parcel source region, Ci(outer), as well as the peak magnitude6 for any smaller concentric circuit contained therein, max(Ci), determined by proportionally shrinking the outer perimeter in increments of 1%. Occasionally, Ci(outer) and max(Ci) are the same value (Tables 1 and 2; this can also be seen by looking ahead to Fig. 4). Experimentation revealed that the peak vortex intensity in the simulations was more strongly related to max(Ci) than to Ci(outer), so that is the value discussed most in this article.
The integers 1–5000 were used as random number seeds to produce an ensemble of initial ζ fields for each λ. The associated distributions of max(Ci) are presented for λ = 4, 2, 1, and 0.5 km in Fig. 3. Notably, as the wavelength of the noise is increased, the initial ζ field becomes smoother and it becomes more likely that ζ of one sign will predominate across the entirety of the circuit area (blue curves in Fig. 4). In contrast, as the wavelength is decreased, the initial ζ field becomes less smooth, and it becomes more likely that at least one substantial peak of Ci is found somewhere within the circuit area but less likely that ζ of one sign will predominate throughout the entire area (red curves in Fig. 4). As a result of these effects, the initial distribution of max(Ci) is considerably wider (with far larger extrema) for the longer wavelengths, as shown by the blue bars in Fig. 3. Shorter wavelengths are more likely to have at least moderate nonzero max(Ci), but much less likely to have large nonzero max(Ci), as shown by the red bars in Fig. 3.
Distributions of max(Ci) for 5000 realizations (i.e., unique random number seeds) of Perlin noise for each of the four wavelengths considered in this study (colored as in the key). The count shown by each vertical bar represents a bin width of 5000 m2 s−1.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
Initial circulation at the bottom model level (Ci, m2 s−1) as a function of radius for a circuit centered on the domain’s midpoint for the simulations listed in Table 1. The corresponding wavelengths of the Perlin noise are colored as in the key.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
Full model simulations were not possible for all 5000 ensemble members for each of the four wavelengths. This study investigates a subset of selected percentiles from the distributions of the longest (4.0 km) and shortest (0.5 km) wavelengths, as summarized in Table 1. For each wavelength, the median max(Ci) is near 0 m2 s−1 (i.e., half of the randomly created values are negative; Fig. 3). Because the vast majority of observed tornadoes in the Northern Hemisphere are cyclonic, this study focused only on the subset of initial ζ values where max(Ci) is positive. The selected initial conditions were for the realizations with max(Ci) at the 60th, 70th, 80th, 90th, and 99th percentiles (indicated in Table 1 by the suffixes _60, _70, _80, _90, and _99). In addition, one simulation is performed for the 4.0 and 0.5 km realizations whose Ci fell closest to, but was greater than, 0 m2 s−1 (indicated in Table 1 by the suffixes _min). Examples of these initial noisy ζ fields are provided in Fig. 5. This suite of 4.0 and 0.5 km runs constitute the primary experiment in this study (Table 1). Based upon experimentation, a value of max(Ci) = 10 000 m2 s−1 appears to be meaningful for testing the overall sensitivities to the wavelength of the Perlin noise (the motivation for using this value is explained further in section 3c). Thus, additional simulations were carried out using the realization of noise falling closest to max(Ci) = 10 000 m2 s−1 for all four values of λ; these simulations have the suffix _10K in Table 2. Because real-world fields constitute a spectrum of wavelengths, this study also includes several simulations with combinations of noise at both the 4.0 and 0.5 km wavelengths (the simulations with prefix C_ in Table 2).
Initial values of vertical vorticity (shaded, s−1) at the bottom model level for (a)–(d) four simulations using λ = 4 km and (f)–(i) four simulations using λ = 0.5 km. The area used to compute Ci(outer) is enclosed by a black circle in each panel. (e),(j) Zoomed depiction for the square areas outlined with white boxes in (d) and (i), and also including horizontal wind vectors on the bottom model level (m s−1, scaled as shown, plotted at every fourth grid point). The shading scales differ by an order of magnitude between the top and bottom rows, with corresponding shading scales given on the far right of each row.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
There was no mean updraft-relative flow in the control experiments. It has been argued by Fischer and Dahl (2023) that the speed of the outflow carrying surface ζ past the updraft can have an important influence on vortex development by altering the time scale on which surface ζ resides beneath the updraft. To evaluate whether this would alter the present results, this study includes additional experiments in which the initial ζ field remains fixed but the model includes an added mean wind from the east at 10 m s−1, indicated in Table 2 by the suffixes _M10. Among other effects, the relative flow produced updrafts (Fig. 2d) that were slightly elongated in the downstream direction. As noted previously, Ci is computed for the footprint of surface air parcels that join the main updraft within the 8 min study period. For the _M10 runs, this area becomes elliptical (elongated in the direction of the flow, narrowed in the direction across the flow, and shifted upstream, as shown in the online supplemental Fig. 1).
d. Nomenclature for simulations
In Tables 1 and 2 and throughout the text, we refer to the simulations described above using a naming convention of the form W_P_T, where W is the wavelength of the random noise (or else “C” for a combination of wavelengths), P is the percentile of max(Ci) from the noise distribution for that wavelength (or else “min” for the smallest positive value, or “10K” for the value closest to 10 000 m2 s−1), and when present T is the special treatment in the simulation, if any (e.g., the “half” or “raise” updraft profile, or the “M10” added mean wind). So, for example, “0.5_80_M10” refers to an initial condition having 0.5 km wavelength noise whose max(Ci) fell at the 80th percentile of the distribution, within a simulation having a 10 m s−1 added easterly mean wind.
3. Results
For the purposes of this discussion, a TLV is defined as one with υtan of at least 20 m s−1, following the definition of Alexander and Wurman (2008) and Wurman and Kosiba (2013). All of the present simulations exceeding the 20 m s−1 threshold possessed surface ζ of at least 0.69 s−1 [well above the Coffer et al. (2017) TLV threshold of 0.3 s−1], and the vast majority (15 of the 19) had pressure deficits of at least 10 hPa [the Coffer et al. (2017) TLV threshold], including all of the simulations with υtan > 26 m s−1. The vast majority of the qualifying TLVs (again, 15 of the 19) also had durations of at least 1 min, including all of the simulations with υtan > 23 m s−1. Examples of developing TLVs are shown in Fig. 6 (further commentary is provided in the following subsections), and supplemental Figs. 2–7 provide animated versions of Fig. 6 for every simulation in this article. Cross sections reveal free-slip TLV structures (Fig. 7), lacking a true corner flow but having both radial inflow and the strongest tangential velocities at the bottom model level, along with a columnar pressure deficit tube. Inspection of Tables 1 and 2 and Figs. 6 and 7 (plus animations in the supplemental Figs. 2–7) reveals that a noisy initial ζ field is in many cases sufficient to support development of a TLV beneath an updraft typifying a tornadic supercell. The following subsections distill the results of the individual experiments by group.
Plan views depicting minute-by-minute evolution of vertical vorticity on the bottom model level (s−1, shaded) for selected simulations in Table 1: (a)–(e) simulation 4.0_70; (f)–(j) simulation 4.0_90; (k)–(o) simulation 0.5_70; (p)–(t) simulation 0.5_90. Horizontal velocity vectors on the bottom model level are shown using the scale vector at bottom right. Times that satisfy the study’s 20 m s−1 criterion for a tornado-like vortex (TLV) are labeled in the upper right of the corresponding panel.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
Azimuthally averaged vertical cross sections through vortices from selected simulations in Table 1: tangential velocity (m s−1, shaded), radial and vertical velocities (m s−1, vectors using the scale at bottom right), and pressure perturbations (contoured in white every 5 hPa, with negative values dashed). Each panel is created at the time of maximal tangential velocity for that simulation, as labeled.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
a. 4.0 km noise
All of the control simulations with a noise wavelength of 4.0 km produce a TLV with the exception of 4.0_min (Table 1, crosses in Fig. 8). In these simulations, positive C associated with weak, but on average positive, ζ is advected inward and stretched (e.g., Figs. 6a–j, animations in supplemental Fig. 2) to TLV strength. Provided it is associated with at least moderate nonzero Ci (all of the 4.0 km simulations except for 4.0_min), a noisy initial ζ field will indeed support TLV development. Thus, our working hypothesis survives the initial longer-wavelength test. The continued influx of net positive C leads to TLVs that persist for several minutes; in the two simulations with the highest max(Ci), the TLV lifetimes exceed 5 min (Table 1). The TLVs that decay before the end of the simulation (all but 4.0_90) are associated with noise patterns that have large C somewhat close to the updraft axis, which is eventually exhausted and replaced by air with negative C thereafter. Such simulations can be identified from Table 1 as those where max(Ci) ≫ Ci(outer), which is a rather common configuration under random initial ζ. Among all relationships examined among variables (not shown), peak tangential winds (υtan) and max(Ci) exhibited the highest correlation, which can be seen for the 4.0 km simulations via the crosses in Fig. 8. This could be anticipated, as in the discussion of circulation from section 1. We next ask whether these findings hold for much finer-scale noise.
Scatterplot of peak υtan for each simulation in Table 1 vs max(Ci). Wavelengths of the initial Perlin noise in the vorticity field are indicated by crosses and circles, as shown in the key. All simulations plotted here used the control updraft nudging. A best-fit line (and its associated correlation coefficient) are superimposed, as well as the 20 m s−1 threshold (dashed) used to identify tornado-like vortices in this study.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
b. 0.5 km noise
As described earlier (recall Figs. 3 and 4), when the initial ζ noise has a shorter wavelength, it is less likely for ζ of one sign to dominate within a given area. Thus, for each of the 60th–99th percentiles, the Ci values are correspondingly smaller than for the 4.0 km noise fields (Table 1). When compared to the 4.0 km noise simulations, the 0.5 km noise simulations are slower to develop their peak in surface rotation, and the TLVs that do occur are shorter lived (Table 1). The evolution underpinning these differences is rather clear in Figs. 6k–t (see also the animations in supplemental Fig. 3); the sign of the inflowing ζ is less uniform in the 0.5 km runs, and thus, the features are more transient because of the interspersed times when the converged C changes sign. Interestingly, however, the peak υtan in the 0.5 km noise simulations is still well described by the same linear trend line with max(Ci) that emerged in the 4.0 km noise simulations (circles in Fig. 8). In other words, the development and ultimate intensity of TLVs in both the 0.5 and 4.0 km noise simulations is largely determined by the available circulation in proximity to the updraft. Even short wavelength noise can possess sufficient Ci to support TLV formation. The differences in intensity of the vortices between the 0.5 and 4.0 km noise experiments are attributable to the shift in the distribution of Ci values that occurs as the wavelength of the noise is changed (e.g., Fig. 3). To add confidence to this interpretation we next ask whether, for a given Ci, the wavelength matters at all (section 3c), and whether the addition of 0.5 km noise modifies the properties of simulations with 4.0 km noise (and thus larger Ci; section 3d).
c. Fixed circulation with varying wavelength
As can be verified in Table 1, the particular values of max(Ci) in the various 4.0 and 0.5 km noise experiments (described above) do not necessarily overlap. To isolate the possible roles of max(Ci) versus wavelength, an example of initial noise is selected from four different wavelengths (4.0, 2.0, 1.0, and 0.5 km) having roughly equivalent max(Ci). Given that the trend line in Fig. 8 crosses the minimal threshold for a TLV near max(Ci) = 9000 m2 s−1, a control value of max(Ci) = 10 000 m2 s−1 serves well for this direct comparison. This value represents a moderate percentile from each wavelength’s distribution (58th, 56th, 57th, and 63rd percentiles for 4.0 through 0.5 km, respectively). From the 5000 realizations of random noise at each wavelength (summarized in Fig. 3), the member with max(Ci) closest to 10 000 m2 s−1 serves as the chosen initial condition. The details of these members are shown in Table 2 (labels with the suffix _10K).
Although the 4.0_10K simulation produces the largest value of υtan among the four, the simulations are not ordered by wavelength in terms of any of the summary values (υtan, ζx, p′, and duration in Table 2). The 1.0_10K simulation produces the largest pressure deficit and develops its TLV more quickly, whereas the 2.0_10K simulation produces the weakest rotation across all summary values. The primary finding is that the simulations with nearly identical max(Ci) produce a rather narrow range of results, with υtan ranging from 17 to 26 m s−1 (and similarly narrow ranges in ζx and p′). The spread among the runs seems to be an inherent feature of the arbitrary structures present in the initial noisy ζ fields, and animations reveal periods where negative ζ dominates beneath the updraft (animations in supplemental Fig. 4). But all four simulations still ultimately fall rather close to the original υtan versus max(Ci) trend line as shown in Fig. 9 (the roughly vertical stack of blue cross, “1,” circle, and “2”). This result lends support to the previous claim that differences between the 0.5 and 4.0 km noise experiments stem largely from the upward shift in Ci as the wavelength of the noise increases (as previously documented in Figs. 3 and 4 and Table 1).
Scatterplot of peak υtan for each simulation in Table 2 vs max(Ci). Wavelengths of the initial Perlin noise in the vorticity field for each simulation are shown by the key. The color of each symbol indicates the associated model configuration, as shown by the key. The best-fit line from Fig. 8 is superimposed, as well as the 20 m s−1 threshold (dashed) used to identify tornado-like vortices in this study.
Citation: Journal of the Atmospheric Sciences 80, 6; 10.1175/JAS-D-22-0195.1
d. Combination of wavelengths
To this point, we have emphasized the primacy of max(Ci) associated with the initial noisy ζ fields. Given that the 0.5 km noise simulations appear to have more transient features (e.g., Fig. 6 and the durations in Table 1), we next ask whether the addition of 0.5 km noise will act to disrupt the steadier features that emerge in the 4.0 km noise simulations. In other words, will initial random ζ with comparatively large values of max(Ci), i.e., those from the 4.0 km noise simulations that produced TLVs, be interrupted by the finer-scale features that induce more unsteadiness in the 0.5 km noise simulations? To address this, the initial ζ fields from the 4.0_min, 4.0_60, and 4.0_70 simulations are summed with the ζ field from the 0.5_min simulation (i.e., fine-scale noise that will add very little to the 4.0 km values of Ci); the simulations using this combined initial noise are denoted by the prefix C_ in Table 2. The procedure for retrieving induced velocities (as outlined in section 2c) is undertaken after the two ζ fields have been combined. This means that the computed values for max(Ci) for the combination simulations are not perfectly identical to the summed max(Ci) from the 4.0 and 0.5 km simulations, although they are reasonably close.
A comparison of 4.0_min, 4.0_60, and 4.0_70 (Table 1) to C_min_min, C_60_min, and C_70_min (Table 2), respectively, reveals that the addition of the 0.5 km noise in the C_ simulations changes the outcomes of the original 4.0_ simulations very little (this can also be seen by comparing the animations in supplemental Figs. 2 and 5). This lack of sensitivity is reinforced by how closely the combination simulations fall to the trend line from the original experiments (“C” symbols in Fig. 9, which can also be directly compared to the locations of the three leftmost cross symbols in Fig. 8). This outcome can be taken to mean that the fine-scale noise structures do not interrupt the overarching sensitivity of υtan to max(Ci). The experimental design and results of the C_ simulations also fortify the previously stated conclusion that the horizontal scale of the ζ noise largely impacts the problem through its influence on the distribution of max(Ci), again with larger wavelengths more likely to support higher values. Having largely established this primary relationship, we next turn to other possible impacts upon TLV development (the vertical profile of the updraft, and the potential for updraft-relative flow).
e. Variations in updraft profile
As reviewed in the Introduction, the tornadogenesis process requires the development of near-ground ζ (“stage 2”), followed by the vertical stretching of that ζ (“stage 3”). In the present study, the nonzero random surface ζ is present in the initial condition (i.e., stage 2 is treated as having been accomplished). This prescribed ζ is then subjected to an updraft typifying a tornadic supercell to evaluate whether a persistent TLV might form. In the introduction we suggested that if quasi-random nonzero surface ζ is sufficient for tornadogenesis, that could mean that the final stretching process (stage 3) is the dominant consideration. To put a finer point on this, we return to the 4.0_60, 4.0_70, and 4.0_80 simulations, each of which produced a TLV in the original experiments. These simulations are rerun with unchanged initial conditions, but with a profile of updraft nudging that is either halved in magnitude (as shown in Fig. 2b, indicated by labels with suffix _half in Table 2) or whose center is raised in height by 1 km (as shown in Fig. 2c, indicated by labels with suffix _raise in Table 2).
The _half updraft profile would correspond to a much weaker midlevel updraft, but it retains a moderate vertical gradient in upward velocity (∂w/∂z) near the surface (Fig. 2b). Such a profile might in nature be associated with lower total CAPE but appreciable dynamical upward accelerations below the LFC. In contrast, the _raise updraft profile would correspond to roughly the same peak updraft speed in the middle troposphere, but it lacks appreciable ∂w/∂z near the surface (Fig. 2c). Such a profile might in nature be associated with comparable total CAPE but a lack of dynamical upward accelerations below the LFC. This comparison is of interest in light of several recent studies speculating that middle–upper-tropospheric updraft speeds or structures may be relevant to the likelihood or intensity of tornadoes (e.g., Peters et al. 2019; Marion et al. 2019; Sandmæl et al. 2019).
Perhaps surprisingly, halving the updraft magnitude decreases the intensity of the resulting TLVs by only 10%–15% (cf. Table 1 versus Table 2; this can also be seen by comparing the animations in supplemental Figs. 2 and 6); the respective values of υtan decline from 22 to 20 m s−1 for 4.0_60, from 33 to 29 m s−1 for 4.0_70, and from 42 to 37 m s−1 for 4.0_80. The values of peak ζ and pressure deficit are also proportionally weaker, but perhaps the most pronounced difference is that the TLVs take longer to develop in the _half simulations. This delay in onset occurs because the inflow and stretching of surface circulation happens more slowly with the weakened updraft profile. The value of ∂w/∂z is decreased by 40%–45% in the _half updraft (as measured over either the surface–0.5 km or surface–1.0 km layers), but this is still able to intensify the noisy surface ζ to comparable strength; it simply takes longer for the circuit possessing max(Ci) to be converged. The implication is that, given sufficient stretching, the peak in vortex intensity is still largely described by Ci, a point that is visualized by the close correspondence of υtan from the _half simulations to the trend line from the original experiments (green crosses in Fig. 9).
In sharp contrast to the _half simulations, when the original updraft nudging is held at fixed intensity but raised in height by 1 km (the _raise simulations in Table 2), no TLVs form. This is true even within the 4.0_80 initial noise profile, which produced υtan of 42 m s−1 in the control profile but only 6 m s−1 in the _raise profile. Although the midlevel peak in w is essentially unchanged, the near-surface ∂w/∂z declines dramatically (Fig. 2); this results in a dearth of stretching for the available surface ζ. Instead, the peak in horizontal convergence occurs farther aloft, above the layer where nonzero ζ has been seeded. As can be readily seen in Fig. 9 (yellow crosses), the _raise simulations produce the least intensification among any of the realizations in this study, falling well below the trend line from the original experiments and also well below other experiments with comparable values of Ci.
The picture that emerges from the _half and _raise experiments is that both sufficient surface circulation and sufficient ∂w/∂z must coexist in order for TLV formation to occur. The importance of near-ground stretching to TLV formation is probably unsurprising, but it highlights the key role of processes that support upward accelerations at altitudes that would almost always be below the LFC in real storms. Following the conceptual model of dynamic lifting reviewed in section 1, and as explained in studies such as Markowski and Richardson (2014), Coffer and Parker (2017), and Goldacker and Parker (2021), in nature large near-ground ∂w/∂z is almost invariably associated with pressure falls within an intense low-level mesocyclone. This supports the contention of Coffer et al. (2019) that proxies for potential low-level mesocyclone strength (e.g., lower-tropospheric storm-relative helicity) may be more meaningful to the probability of tornadogenesis in supercells than proxies for peak updraft strength (e.g., CAPE).
f. Addition of updraft-relative flow
This study has simplified a number of aspects of the tornadogenesis problem in a real storm in order to exercise control over the direct interaction between an updraft and random surface ζ. Among the simplifications that have not yet been examined, the lack of mean flow in the preceding experiments would be rather unlike the outflows of supercells within which rivers of ζ maxima and minima flow past the parent updraft (e.g., Dahl et al. 2014). As discussed by Fischer and Dahl (2023), faster outflows in an updraft-relative reference frame provide a shorter period within which individual elements of surface ζ can be stretched. Thus, a final experiment uses both the 4.0 and 0.5 km wavelengths with an added 10 m s−1 easterly mean wind (indicated by the suffix _M10 in Table 2); the original control updraft profile (Fig. 2a) is reinstated for these simulations. The _min, _60, and _80 ζ percentiles are again utilized, but the selection of initial random number seeds differs from the original experiments because the area over which the initial circulation must be calculated (i.e., the footprint of inflow into the updraft) is now elongated and shifted upstream, as explained in section 2c and shown in supplemental Fig. 1. This means that the _M10 simulations are not quite as directly comparable to the original simulations; nevertheless, they are illustrative.
The evolution of the _M10 simulations differs from all those previously discussed because the mean flow advects the vortices westward as they are developing (see animations in supplemental Fig. 7). This effect appears to add spread to the experimental results, as can be seen in Table 2 and from the magenta markers in Fig. 9. The simulated υtan for 4.0_80_M10 (rightmost magenta cross in Fig. 9) and 0.5_min_M10 (leftmost magenta circle in Fig. 9) fall close to the trend line from the original experiments. However, the other four experiments are rather far from it. The 4.0_min_M10 simulation produces an unexpectedly strong TLV and the 4.0_60_M10 simulation produces the second-strongest TLV across the entire study (Table 2 and two leftmost magenta crosses in Fig. 9). In contrast, the 0.5_60_M10 and 0.5_80_M10 simulations fail to produce TLVs with starting values of Ci that might otherwise seem supportive. These discrepancies are probably attributable in part to the greater difficulty of computing a single representative value of max(Ci) for each initial condition; as an _M10 simulation proceeds in time, the elliptical source region of updraft air parcels not only enlarges but also shifts progressively farther upstream. Given such limitations, we cannot definitively conclude from this small set of simulations that the addition of updraft-relative motion is either harmful or beneficial to tornadogenesis. To the extent that the relationship between peak υtan and max(Ci) is in fact disrupted, it does appear that the simulations with longer-wavelength noise fare better than those with shorter-wavelength noise. As shown in the animations in supplemental Fig. 7, once a TLV is established (in the longer-wavelength scenarios), it seems to be more resistant to the disruptive effect of the lateral advection. This might mean that, as the speed of the ζ-bearing outflow increases relative to the updraft, the reduced time scale for stretching (e.g., Fischer and Dahl 2023) can produce and sustain a TLV only when combined with the more horizontally extensive patches of nonzero surface ζ.
4. Conclusions
This study used idealized numerical simulations with initial surface vertical vorticity (ζ) fields defined by random noise in combination with an updraft designed to typify those found in tornadic supercells. In response to the question of whether quasi-random nonzero surface ζ is sufficient for the production of a tornado-like vortex (TLV) in such updrafts, we can answer with a qualified yes; TLVs occurred in 19 of the 31 simulations presented here. This might mean that, for tornadogenesis in nature, producing well-organized surface ζ matters less than having sufficient stretching from the storm’s low-level updraft. We find support for this contention from the following main results.
a. Primary findings
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The most explanatory predictor of whether a TLV would form (and how strong it would become) was the maximal value of initial surface circulation (Ci) found near the updraft. Even quasi-random ζ fields are still often associated with pockets of rather large Ci.
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Because Ci represents the areal integral of surface ζ, longer-wavelength noise is more likely to be associated with substantial values of Ci. That said, for a fixed value of Ci, the sensitivity to the wavelength of the noise was small. The addition of shorter-wavelength noise also did not appear to disrupt the development of TLVs in simulations that already possessed sufficient Ci.
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Given sufficient Ci in the random ζ field, the other key ingredient is a large near-ground vertical gradient in vertical velocity (∂w/∂z) to promote stretching. Decreasing the speed of the midlevel updraft (at altitudes that would be well above the LFC in nature) had a minimal effect on the resultant TLVs, whereas raising the center of the updraft (significantly diminishing ∂w/∂z at altitudes that would be below the LFC in nature) prevented TLV formation.
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The addition of updraft-relative motion in the surface ζ field (such as would be found in the outflow of a natural supercell) leads to more uncertain findings. Such motion seems to have a more negative (or less positive) effect on developing vortices in shorter-wavelength noise compared to those in longer-wavelength noise.
b. Context, caveats, and future work
The title of this article asks how well organized surface vorticity must be for tornadogenesis. The most direct answer is that statistically random vorticity structures can suffice. However, these experiments do reveal that longer wavelengths provide a higher likelihood of large Ci. Thus, the importance of more coherent sources of ζ may be in their creation of structures with larger horizontal scales. While the origins of nonzero surface ζ along individual air parcel trajectories can be meaningful, future research might profit from focusing on the origins of larger pools of surface circulation. In this regard, it is interesting to note that some of the vorticity noise in Fig. 1 is rather banded in nature; perhaps the elongation (e.g., in the along-flow direction) of noisy structures could provide an important bias to the sign of circulation flowing into the updraft prior to tornadogenesis.
In these simulations, most of the vortices in the 0.5 km simulations were weaker and shorter lived than those in the 4.0 km simulations. This might be taken as evidence that a coherent vorticity source is also important to tornado intensity and longevity. However, the degree of idealization7 in the present experiments suggests that we should be cautious about direct inferences of the traits of mature tornadoes in nature. Perhaps the processes depicted here are most relevant to the early moments of tornadogenesis during which surface pressure falls begin to facilitate the development of a corner flow and a transition to the “in and up” (tilting-dominated) vortex trajectories highlighted by Fischer and Dahl (2022). Future experiments with surface drag and more representative environmental wind profiles could begin to bridge the gaps among the various idealized studies in the extant literature.
Among other curiosities, the present simplified approach (which omits rotation from the main updraft) also raises the question of whether or not the parent storm even needs to be a supercell. It is already well-known that nonsupercell tornadoes occur in nature when preexisting surface ζ is combined with strong low-level stretching (Wakimoto and Wilson 1989). Nevertheless, most significant tornadoes are associated with supercells (e.g., Smith et al. 2012). Supercells are rather unique in their capacity to produce steady, intense near-ground ∂w/∂z within air that is below the LFC (and hence neutrally or negatively buoyant), as a consequence of the dynamically induced vertical accelerations that result from pressure falls in the low-level mesocyclone (e.g., Markowski and Richardson 2014; Coffer and Parker 2017; Goldacker and Parker 2021). Updraft nudging in the present experiment mimics this effect, and interactions with a supercell’s internal vorticity field can further intensify this dynamic lifting (e.g., Roberts and Xue 2017; Orf et al. 2017). In contrast, such large values of near-ground ∂w/∂z would be less common in nonsupercellular storms. In addition, whereas the nonsupercell mechanism would operate equally well on pools of either positive or negative surface C, the predominance of cyclonically rotating mesocyclones in (right-moving) supercells normally requires same-signed surface ζ for tornadogenesis (e.g., Markowski et al. 2008; Markowski and Richardson 2014). This may be another juncture at which coherent origins of vorticity emerge as important. Supercells’ unique geometry, which commonly includes flow from the forward flank that passes through a baroclinic zone as it descends, can produce a bias toward positive ζ among outflow parcels approaching the updraft (e.g., Davies-Jones and Brooks 1993; Parker and Dahl 2015); this fits nicely with the need for pools of positive C beneath the updraft in real supercells. There appear to be many motivating questions for the future of supercell process studies.
Finally, the present study is agnostic to the origins of the surface ζ field, but the conclusions logically prompt the following question: What is the meaning of “vorticity noise” in real supercells? Presumably, convective-scale heterogeneity in the vertical drafts and precipitation structures of parent storms is sufficient to provide a rather noisy within-storm ζ field, as can be seen even in idealized supercell simulations (Fig. 1). Could such random noise also emerge naturally from external sources such as variable land cover or terrain, which have been cited as possibly influencing tornadoes (e.g., Lyza and Knupp 2018; Houser et al. 2020)? The demonstrations provided by Coleman et al. (2021) suggest that the answer is very likely yes. However, given that supercell simulations in horizontally homogeneous environments already exhibit well-developed noise within their surface ζ fields (Fig. 1), these external factors may only be impactful to the extent that they bias the sign of ζ (and hence C) in one direction. It is also worth noting that, while the present simulations have quasi-random ζ fields, the initial flows are purely horizontal and are thus in some sense laminar. Real storms (and storm environments) include 3D turbulence, which might further modify the time scales and magnitudes of the noise and resultant vortices, although ζ-bearing outflows often have at least moderate static stability (which would damp turbulent overturning). The work of Markowski (2020) suggests that the net effects of turbulent perturbations are not easily predictable. Future work on the origins, structures, natural scales, and impacts of vorticity noise within (and near) supercells would therefore be of interest.
This article does not recapitulate the currently proposed mechanisms in great detail because the present study is purposefully agnostic to how stage 2 is accomplished. Interested readers can consult with the citations given in the referring sentence.
These four simulations produced the largest 0.5 km AGL dynamical vertical pressure gradient accelerations (and vertical velocities) in response to the environmental wind profiles from supercell cases described by Goldacker and Parker (2021); the four simulations were cases C, D, H, and L as seen in Figs. 2 and 11 of Goldacker and Parker (2021).
This solver returns the nondivergent velocity vectors associated with the ζ field. The pressure field is not modified in this process, but is adjusted quickly by acoustic waves upon model startup. The values of Ci reported in the tables and plotted in the figures of this article are computed from the retrieved horizontal velocity vectors in the CM1 initial condition, not the raw ζ field itself.
The retrieved values for this maximum initial wind perturbation,
Notably, even for more coherent (larger-scale) surface vorticity structures, in real storms many fine-scale structures are likely to emerge due to turbulence within the flow. For reference, a 2D fast Fourier transform in space was performed for the surface vorticity field from 10 of the output times shown in Fig. 1, as well as for a comparable unpublished simulation that used a horizontal grid spacing of 80 m (instead of the 125 m grid spacing depicted in Fig. 1). This analysis revealed that roughly equal amplitude was present at all wavelengths greater than 1.0 km in the simulated supercells’ surface vorticity field. In the simulations with 125 m grid spacing, the 0.5 km wavelength had 31% of the amplitude of the 4.0 km wavelength, but such waves are of 4Δx length, and thus fall below the effective resolution of the model (roughly 6Δx; e.g., Bryan et al. 2003; Bryan 2005). In the simulation with 80 m grid spacing, the 0.5 km wavelength had 57% of the amplitude of the 4.0 km wavelength. In short, the wavelengths used in this study occur routinely in the surface vorticity fields of simulated supercells, even without external sources of noise (e.g., a convective boundary layer, surface heterogeneity).
In reporting max(Ci), we use the largest absolute value,
The bottom boundary is free slip, the forced updraft is steady and does not acquire rotation from ambient streamwise horizontal vorticity, and the simulations are ended at 8 min when some of the TLVs still persist.
Acknowledgments.
George Bryan is gratefully acknowledged for providing and supporting the CM1 model. Johannes Dahl generously shared his induced flow solver, which was used to create the initial conditions, and Brice Coffer provided access to the simulations shown in Fig. 1 and described in footnote 5. This work was supported by the National Science Foundation under Grants AGS-1748715 and AGS-2130936. Simulations were accomplished with high-performance computing support from Cheyenne (DOI:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. The author thanks the members of the NCSU Convective Storms Group for their beneficial comments.
Data availability statement.
The CM1 model code is available from https://www2.mmm.ucar.edu/people/bryan/cm1/. All other namelists, input files, and previously unpublished analysis scripts with nontrivial computations are available from https://doi.org/10.5061/dryad.mcvdnck4n.
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