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    (a) Vertical profile of environmental potential temperature . (b) Hodographs in the strong-shear (m = 8) and weak-shear (m = 2) environments. Labels along the hodographs are select altitudes (km). Storm-relative helicity (SRH) in the 0–1- and 0–3-km layers also is indicated.

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    Three-dimensional structure of the heat source (red) and heat sink (blue) as viewed from (a) above and (b) the southeast in the subdomain −10 ≤ x ≤ 10 km and −10 ≤ y ≤ 10 km, within which the horizontal grid spacing is 100 m. Axes indicate model coordinates (km). The black contours are schematic isochrones, at select times, of the leading edge of the cool outflow that emanates from the heat sink. The heat sink is activated at 900 s (15 min). The small circular arrows near the base of the heat source in (b) indicate where a strong cyclonic vortex develops at the surface in some simulations.

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    West–east-oriented vertical cross sections through the heat-source-driven updraft at 900 s (i.e., when the updraft is in roughly a steady state, immediately prior to the activation of the heat sink) in (a),(c) the simulations initialized with relatively weak low-level shear (m = 2) and (b),(d) relatively strong low-level shear (m = 8). The vertical cross sections are at y = 0. The vertical velocity (w) field (color shaded; see legend) is shown in all four panels. In (a),(b), potential temperature perturbations (θ′) are overlaid on the vertical velocity field (black contours every 3 K; dashed contours indicate negative values). In (c),(d), the vertical vorticity (ζ) field is overlaid (black contours every 0.005 s−1 for ζ ≥ 0.005 s−1).

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    Horizontal cross sections of 0–1-km-mean (a),(b) dynamic VPPGF , (c),(d) linear dynamic VPPGF , and (e),(f) nonlinear dynamic VPPGF in the (a),(c),(e) weak low-level shear (m = 2) and (b),(d),(f) strong low-level shear (m = 8) simulations at 900 s (i.e., when the updraft is in roughly a steady state, immediately prior to the activation of the heat sink); π′ is the nondimensional pressure perturbation, and the subscripts D, DL, and DNL refer to dynamic, linear dynamic, and nonlinear dynamic nondimensional pressure perturbations . The VPPGF contours are black and drawn every 0.003 m s−2 (negative isopleths are dashed). In each panel, the vertical velocity (w) at z = 1 km (m s−1) also is shown (color shading).

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    Evolution of near-surface θ′, ζ, w, and horizontal wind fields in the Sc8m8 simulation. The fields are shown (a) 10 and (b) 5 min prior to the time that cyclonic vorticity is a maximum (t − 10 min and t − 5 min, respectively); (c) at the time of maximum cyclonic vorticity (t − 0 min); and (d) 5 min after the time of maximum cyclonic vorticity (t + 5 min). The θ′ field at z = 50 m is shaded (see legend). Isovorts of ζ at z = 50 m (interpolated to the scalar grid points) are drawn using thin red contours for ζ = ±0.05, 0.15, 0.25 s−1, etc.; dashed contours enclose anticyclonic vorticity. Horizontal wind vectors at z = 50 m are plotted at every fifth grid point. The w = 4 m s−1 isotach at z = 450 m is indicated with thick black contours. In (a), the outlines of the heat source and heat sink are indicated with red and blue dashed rings, respectively. The cyclonic and anticyclonic vorticity maxima discussed in the text are indicated with thick arrows in (c). The gray ring in (c) is the material circuit analyzed in Fig. 10. In (b),(c), the small magenta disks with white and black borders indicate, respectively, the horizontal positions of the parcels analyzed in Figs. 11 and 12 (i.e., parcels that acquire cyclonic and anticyclonic ζ as they near the surface).

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    Three-dimensional view of the midlevel updraft, near-surface θ′ field, and key vortex lines and trajectories at t − 10 min (2160 s) in the Sc8m8 simulation. The view is from the south–southeast. Axes are in kilometers. The w = 15 m s−1 isosurface is gray. The near-surface θ′ field is color shaded (see legend). The yellow lines are vortex lines (their direction is indicated with arrows) that pass within 200 m of ζmax at the lowest scalar level (z = 50 m). The blue lines are vortex lines that pass within 500 m of ζmax at z = 3 km. The white lines are forward-integrated trajectories that pass within 500 m of ζmax at t − 10 min and have ζ ≥ 0.008 s−1 in the lowest 75 m. The trajectory that passes nearest to the cyclonic vorticity maximum at t − 10 min is red. Arrows are placed along it at 5-min intervals.

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    (a) As in Fig. 6, but at t − 5 min (2460 s) in the Sc8m8 simulation. Trajectories have been omitted. The black and white lines are additional vortex lines; the black one originates in the environment and descends through anticyclonic vorticity in the cold pool. The purple vortex lines originate within the low-level cyclonic vortex and arches toward the anticyclonic vorticity before turning upright and passing into the midlevel updraft and mesocyclone. The dotted white and black lines suggest, respectively, plausible connections between the anticyclonic and cyclonic vortices in the cold pool and between the environmental vortex lines and vortex lines entering the midlevel mesocyclone. See text for further details. (b) Schematic evolution (left to right) of “vortex-line surgery” that joins a vortex line arching upward out of the cold pool with an environmental vortex line that enters the midlevel mesocyclone. The view is from the east–northeast.

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    As in Fig. 6, but at t − 0 min (2760 s) in the Sc8m8 simulation. The trajectories pass within 200 m of ζmax, within 75 m of the surface, at t − 0 min. The trajectory that passes nearest to cyclonic vorticity maximum at t − 0 min is red. The magenta trajectory nears the lowest scalar level approximately 1 km west of ζmax at t − 0 [see section 3a(3) and Fig. 11 for details]. The view is from the south–southeast.

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    As in Fig. 8, but the yellow and white lines are vortex lines and trajectories, respectively, that pass within 200 m of ζmin [at the lowest scalar level (z = 50 m) in the case of the vortex lines and within 75 m of the surface in the case of the trajectories]. The trajectory that passes nearest to the anticyclonic vorticity maximum at t − 0 min is red. The magenta trajectory nears the lowest scalar level near ζmin at t − 0 [see section 3a(2) and Fig. 12 for details]. The view is from the southwest.

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    (a) View from the southwest and from above (inset) of the material circuit in the Sc8m8 simulation at t − 20, t − 10, and t − 0 min. The circuit is a 1-km-radius circle centered on the cyclonic vorticity maximum at z = 250 m at t − 0 min (this is the starting point of the backward trajectory calculations for the parcels comprising the circuit; the circuit is indicated with a gray ring in Fig. 5c). Six select parcels within the circuit are shown at each time using orange, yellow, gray, green, cyan, and purple markers. The perturbation potential temperature field (θ′) at z = 50 m at t − 0 min also is shown (legend). The vertical scale in the view from the southwest is exaggerated by approximately a factor of 2. (b) Circulation (C; black curve) and solenoidal generation of circulation (; blue shading) about the circuit as a function of time; also is shown (gray curve) for comparison, where C0 is the constant that makes match the circulation about the circuit at t − 0 min.

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    Vorticity and vertical velocity forcings along the trajectory discussed in section 3a(3) (the magenta trajectory in Fig. 8) that nears the lowest scalar level, 1 km northwest of ζmax at t − 0. (a) The vorticity vector (ω; black) is plotted every 2.5 min along the trajectory (magenta line) in the sz plane. Axes are in kilometers. The projections of the integrated barotropic, baroclinic, and turbulent forcings in the intervening 2.5-min periods are shown with green, blue, and brown vectors, respectively (s−1). The integrated forcing from crosswise–streamwise exchange is shown with orange vectors (s is the unit vector that points in the streamwise direction). The scale of all of the vorticity forcing vectors matches the scale of the vorticity vectors (see legend in panel) and the vectors are scaled differently in the horizontal and vertical directions. The vertical components of the vectors are exaggerated by the same degree to which the vertical scale of the viewing window is exaggerated. (b) As in (a), but in the horizontal plane (there is no crosswise–streamwise exchange in the xy plane); blue (red) shading indicates the horizontal extent of the heat sink (source). (c) Altitude (zp) and ζ of the parcel during its ascent and subsequent descent (t − 20 to t − 1). (d) Vertical inclination angle δ between the trajectory and ω during the same time period as (c).

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    As in Fig. 11, but for the trajectory discussed in section 3a(3) (the magenta trajectory in Fig. 9) that nears the lowest scalar level near ζmin at t − 0.

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    Horizontal distribution of circulation (C) at z = 50 m computed over 1-km-radius rings (color shading) at (a) t − 10, (b) t − 5, and (c) t − 0 min in the Sc8m8 simulation. The dynamic VPPGF field vertically averaged over the 0–1-km layer is overlaid with gray contours (0.05 m s−2 interval, dashed contours indicate negative values). The plus and minus signs in (c) indicate the cyclonic and anticyclonic vorticity maxima, respectively, at z = 50 m. The gray rings are centered on the maximum 0–1-km mean dynamic VPPGF and have radii of 2 km. (d) The potential temperature perturbation averaged within this ring (〈θ′〉; cf. Figure 5) and C about this ring, both at z = 50 m, are shown from t − 20 to t + 5 min.

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    As in Fig. 5, but for the Sc16m8 simulation. (c) The cyclonic vorticity maximum discussed in the text is indicated with a thick arrow.

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    As in Fig. 6, but for the Sc16m8 simulation at t − 0 min (2220 s). The trajectories pass within 200 m of ζmax, within 75 m of the surface, at t − 0 min. The trajectory that passes nearest to the cyclonic vorticity maximum at t − 0 min is red. The view is from the south–southeast.

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    As in Figs. 13c,d, but for the Sc16m8 simulation. (a) The plus sign indicates the cyclonic vorticity maximum at z = 50 m. (b) The blue lines are for the Sc8m8 simulation (i.e., they are traced from Fig. 13d).

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    As in Fig. 5, but for the Sc8m2 simulation. (a) The outline of the heat source and heat sink are indicated with dashed red and blue rings, respectively. (c) The cyclonic vorticity maximum discussed in the text is indicated with a thick arrow.

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    As in Fig. 6, but for the Sc8m2 simulation at t − 0 min (2640 s). The view is from the southeast.

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    As in Fig. 16, but for the Sc8m2 simulation.

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    As in Fig. 17, but for the Sc4m8 simulation. (d) The fields are shown 14 min (not 5 min) after the time of maximum cyclonic vorticity (t + 14 min), at which time a strong anticyclonic vortex is present at the surface at the location indicated by the thick arrow.

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    As in Fig. 6, but for the Sc4m8 simulation at 3660 s. The view is from the southeast.

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    As in Fig. 16, but for the Sc4m8 simulation.

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    As in Fig. 6, but for the Sc4m8 simulation at 4380 s (here t − 0 min refers to the time relative to the maximum anticyclonic vorticity, rather than maximum cyclonic vorticity, as in Figs. 20 and 21). The trajectories pass within 200 m of ζmin, within 75 m of the surface, at t − 0 min. The trajectory that passes nearest to anticyclonic vorticity maximum at t − 0 min is red. The view is from the southeast.

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    Magnitude of the cyclonic vorticity maximum as a function of the horizontal position of the heat sink in the (a) Sc8m8 (baseline), (b) Sc16m8, (c) Sc8m2, and (d) Sc4m8 experiments. The x and y axes indicate the heat sink coordinates (xc, yc) (km), which are each shifted up to ±2 km from the locations (indicated with unshaded circles at the center of each domain) of the heat sinks in the experiments presented in section 3.

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    Schematic summarizing the simulation outcomes: (a) the baseline simulation in which a strong cyclonic vortex develops (Sc8m8), the simulations in which (b) the heat sink is either too strong (results in colder outflow, i.e., Sc16m8) or (c) the environmental wind shear is too weak (results in a weaker dynamic VPPGF, i.e., Sc8m2), and (d) the simulation in which the heat sink is excessively weak (i.e., Sc4m8). A schematic trajectory bound for ζmin (and evolution of ω along this trajectory) is shown in (d) only (the anticyclonic vortex is dominant in this simulation), but the generalized trajectory and vorticity evolution also applies to trajectories approaching ζmin in (a)–(c).

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The Influence of Environmental Low-Level Shear and Cold Pools on Tornadogenesis: Insights from Idealized Simulations

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

Idealized, dry simulations are used to investigate the roles of environmental vertical wind shear and baroclinic vorticity generation in the development of near-surface vortices in supercell-like “pseudostorms.” A cyclonically rotating updraft is produced by a stationary, cylindrical heat source imposed within a horizontally homogeneous environment containing streamwise vorticity. Once a nearly steady state is achieved, a heat sink, which emulates the effects of latent cooling associated with precipitation, is activated on the northeastern flank of the updraft at low levels. Cool outflow emanating from the heat sink spreads beneath the updraft and leads to the development of near-surface vertical vorticity via the “baroclinic mechanism,” as has been diagnosed or inferred in actual supercells that have been simulated and observed.

An intense cyclonic vortex forms in the simulations in which the environmental low-level wind shear is strong and the heat sink is of intermediate strength relative to the other heat sinks tested. Intermediate heat sinks result in the development (baroclinically) of substantial near-surface circulation, yet the cold pools are not excessively strong. Moreover, the strong environmental low-level shear lowers the base of the midlevel mesocyclone, which promotes strong dynamic lifting of near-surface air that previously resided in the heat sink. The superpositioning of the dynamic lifting and circulation-rich, near-surface air having only weak negative buoyancy facilitates near-surface vorticity stretching and vortex genesis. An intense cyclonic vortex fails to form in simulations in which the heat sink is excessively strong or weak or if the low-level environmental shear is weak.

Corresponding author address: Dr. Paul M. Markowski, Department of Meteorology, The Pennsylvania State University, 503 Walker Building, University Park, PA 16802. E-mail: pmarkowski@psu.edu

Abstract

Idealized, dry simulations are used to investigate the roles of environmental vertical wind shear and baroclinic vorticity generation in the development of near-surface vortices in supercell-like “pseudostorms.” A cyclonically rotating updraft is produced by a stationary, cylindrical heat source imposed within a horizontally homogeneous environment containing streamwise vorticity. Once a nearly steady state is achieved, a heat sink, which emulates the effects of latent cooling associated with precipitation, is activated on the northeastern flank of the updraft at low levels. Cool outflow emanating from the heat sink spreads beneath the updraft and leads to the development of near-surface vertical vorticity via the “baroclinic mechanism,” as has been diagnosed or inferred in actual supercells that have been simulated and observed.

An intense cyclonic vortex forms in the simulations in which the environmental low-level wind shear is strong and the heat sink is of intermediate strength relative to the other heat sinks tested. Intermediate heat sinks result in the development (baroclinically) of substantial near-surface circulation, yet the cold pools are not excessively strong. Moreover, the strong environmental low-level shear lowers the base of the midlevel mesocyclone, which promotes strong dynamic lifting of near-surface air that previously resided in the heat sink. The superpositioning of the dynamic lifting and circulation-rich, near-surface air having only weak negative buoyancy facilitates near-surface vorticity stretching and vortex genesis. An intense cyclonic vortex fails to form in simulations in which the heat sink is excessively strong or weak or if the low-level environmental shear is weak.

Corresponding author address: Dr. Paul M. Markowski, Department of Meteorology, The Pennsylvania State University, 503 Walker Building, University Park, PA 16802. E-mail: pmarkowski@psu.edu

1. Introduction

Among the outstanding questions related to tornadogenesis in supercell thunderstorms are the relative roles of environmental horizontal vorticity associated with the ambient vertical wind shear versus the horizontal vorticity that is produced baroclinically by storm-generated horizontal buoyancy gradients, particularly those associated with precipitation regions. Climatological studies derived from proximity soundings repeatedly find that tornadoes are more probable as the environmental horizontal vorticity increases, particularly at low levels (nominally in the lowest 1 km) (e.g., Rasmussen 2003; Thompson et al. 2003; Markowski et al. 2003a; Craven and Brooks 2004). On the other hand, analyses of circulation tendencies following material circuits derived from both numerical simulations (Rotunno and Klemp 1985, hereafter RK85; Davies-Jones and Brooks 1993, hereafter DJB93) and dual-Doppler observations of supercells (Markowski et al. 2012b) reveal that the vast majority of the low-level mesocyclone’s circulation is attributable to baroclinic vorticity generation rather than environmental vorticity. Moreover, the configuration of vortex lines in the low-level mesocyclone region of simulated (e.g., RK85) and observed (e.g., Straka et al. 2007; Markowski et al. 2008, 2011, 2012a; Marquis et al. 2012; Kosiba et al. 2013) supercells strongly suggests a leading role for baroclinic vorticity generation, in that there is a strong tendency for the vortex lines to form arches that join the low-level mesocyclone with a low-level mesoanticyclone on the trailing flank of the hook echo and rear-flank downdraft. These vortex lines have a horizontal projection that is roughly 90° to the right of the observed horizontal buoyancy gradient (or horizontal buoyancy gradient that might be inferred from the reflectivity gradient, if the former is not observed) within the relatively cool outflow; this orientation is often nearly 180° different from the orientation of the environmental vortex lines.

In summary, large environmental vorticity at low levels favors tornadic supercells, but studies of the dynamical origins of low-level rotation in supercells implicate baroclinically generated vorticity rather than environmental vorticity. To make matters even more confusing, tornadogenesis is more likely to be observed in supercells that have cold pools characterized by only intermediate negative buoyancy1 rather than strong cold pools (Markowski et al. 2002; Grzych et al. 2007; Hirth et al. 2008), but the potential for significant baroclinic vorticity generation would seem to decrease as cold pools become weaker.

Why is strong low-level vertical shear—especially when combined with a low lifting condensation level (LCL)—so favorable for tornadoes in proximity sounding studies (e.g., Thompson et al. 2003; Craven and Brooks 2004)? Why does tornado likelihood decrease as the negative buoyancy of the outflow increases (e.g., Markowski et al. 2002)? Why does the low-level cyclonic vortex typically become so much stronger than the anticyclonic vortex? We have been tackling these questions by way of idealized numerical simulations designed to emulate what we believe to be some of the salient aspects of the processes by which strong vortices are produced at the surface in supercell thunderstorms. The simulations are dry but include the generation of near-surface2 rotation beneath supercell-like (i.e., helical) updrafts in a way consistent with our present understanding of the importance of a downdraft in environments in which vertical vorticity is initially absent at the surface (Davies-Jones et al. 2001).

A stationary, cylindrical heat source is imposed within a horizontally homogeneous environmental wind field containing vertical shear. The vertical wind profile is described by a semicircular hodograph; that is, the horizontal vorticity is purely streamwise for a storm motion at the origin.3 The interaction of the heat source and wind field results in a cyclonically spinning updraft with maximum rotation at midlevels. The rotation is small at the lowest grid level; ideally, in the absence of any downdrafts, it would vanish near the surface because parcels of air only acquire vertical vorticity from the tilting of environmental horizontal vorticity as they rise away from the surface (Davies-Jones 1982a,b). Once a nearly steady state is achieved, a heat sink is imposed on the northeastern flank of the updraft at low levels. The heat sink produces a downdraft and low-level outflow, and the horizontal temperature gradients that accompany the downdraft and outflow baroclinically generate horizontal vorticity. The outflow from the heat sink eventually spreads beneath the updraft, and under the right conditions, the interaction between the overlying updraft and the baroclinically generated vorticity that accompanies the outflow can result in the formation of strong surface vortices (Fig. 2). The simulations are not as realistic as those that incorporate more physical processes (e.g., latent heating and microphysics), but we believe their simplicity makes it easier to isolate key dynamical processes and assign cause and effect. The simulations emulate the salient characteristics of an actual supercell (e.g., the hook echo, forward-flank downdraft, rear-flank downdraft, occlusion downdraft, and cyclic mesocyclogenesis) with just a couple of tunable parameters, that is, by adjusting the strength and location of a single heat source and heat sink, as opposed to having a dozen parameters that might have to be tuned in a microphysics parameterization alone in a more complicated simulation.

The methodology is explained in more detail in section 2. The results are presented in section 3 and are discussed in section 4. A summary and conclusions are presented in section 5.

2. Methodology

The dry version of the Bryan cloud model version 1 (CM1), release 16, is used (Bryan and Fritsch 2002). A fifth-order advection scheme is used, which has implicit diffusion, so no additional artificial diffusion is included. There are no surface fluxes, Coriolis force, or radiative transfer. The domain is 100 km × 100 km × 18 km, with a rigid top and bottom boundary and open lateral boundaries. The horizontal grid spacing is 100 m within a 20 km × 20 km region centered in the domain and gradually increases to 3.9 km from the edge of this inner region to the lateral boundaries via the function given by Wilhelmson and Chen (1982). The vertical grid spacing varies from 100 m in the lowest 1 km to 400 m at the top of the domain, following the same Wilhelmson and Chen function. The large (small) time step is 1 s (0.1 s). The simulations are run for 1.25 h. Owing to the horizontal and vertical grid spacing, one cannot expect to resolve tornadoes. The free-slip lower boundary condition also precludes the development of vortices that could be regarded as tornadoes. Thus, the intense, submesocyclone-scale vortices that develop in some of the simulations should be regarded as no more than “tornado like” vortices.

The initial environmental potential temperature field is horizontally homogeneous and follows that of Walko’s (1993) idealized simulations in the vertical. At the surface, ; increases with height at 1 K km−1 in the lowest 10 km AGL, and 10 K km−1 above 10 km AGL (Fig. 1a). The environmental wind profile (Fig. 1b) is initialized using the formulation for a semicircular hodograph given by McCaul and Weisman (2001) that has a vertically varying wind shear magnitude, such that
e1
and
e2
where and are the zonal and meridional environmental wind components, z is the vertical coordinate, A is the hodograph radius, m is the profile “compression parameter,” H is the vertical scale, and z0 is the height where is a maximum. The simulations use A = 8 m s−1, H = 6 km, and m = 2, 8. The m = 2 (m = 8) simulations are referred to as weak-shear (strong-shear) simulations, in reference to the magnitude of the low-level shear (the environments have a similar mean shear over the depth of the domain). The 0–1 km storm-relative helicity is 50 m2 s−2 (120 m2 s−2) in the weak-shear (strong-shear) simulations. Though low-level hodographs in actual supercell environments are commonly characterized by large curvature and streamwise vorticity, hodographs are not typically semicircles over the depth of the troposphere (e.g., Markowski et al. 2003a). The use of a semicircle hodograph is largely a matter of convenience; in such an environment, the updraft is very helical (Davies-Jones 2002) and its motion is near the center of curvature. The results are insensitive to the mid- to upper-level hodograph traits in the simulations—probably more so than we would expect actual storms to be—because the heat sink’s characteristics (to be described below) are user controlled rather than being governed by the microphysics parameterization and hydrometeor trajectories.
Fig. 1.
Fig. 1.

(a) Vertical profile of environmental potential temperature . (b) Hodographs in the strong-shear (m = 8) and weak-shear (m = 2) environments. Labels along the hodographs are select altitudes (km). Storm-relative helicity (SRH) in the 0–1- and 0–3-km layers also is indicated.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

A heat source (Sw) and heat sink (Sc) were added to CM1’s potential temperature tendency equation. They are specified by
e3
where φ is either w or c to indicate a heat source or sink, respectively, Sφ0 is the heat source–sink amplitude, and
e4
and
e5
where Rφ is the horizontal radius of the heat source–sink, Zφ is the half depth of the heat source–sink, r2 = (xxφ)2 + (yyφ)2, and the heat source–sink is centered at (xφ, yφ, zφ) (Fig. 2).
Fig. 2.
Fig. 2.

Three-dimensional structure of the heat source (red) and heat sink (blue) as viewed from (a) above and (b) the southeast in the subdomain −10 ≤ x ≤ 10 km and −10 ≤ y ≤ 10 km, within which the horizontal grid spacing is 100 m. Axes indicate model coordinates (km). The black contours are schematic isochrones, at select times, of the leading edge of the cool outflow that emanates from the heat sink. The heat sink is activated at 900 s (15 min). The small circular arrows near the base of the heat source in (b) indicate where a strong cyclonic vortex develops at the surface in some simulations.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

The heat source has an amplitude of Sw0 = 0.070 K s−1, is centered above the origin [(xw, yw, zw) = (0, 0, 5.25) km], and has dimensions given by Rw = 3 km and Zw = 4.75 km. The heat source emulates latent heat release; from the difference of zw and Zw, the LCL is effectively at z = 0.5 km. The lowest level at which positive buoyancy is produced by the heat source (the level of free convection) is approximately z = 1.35 km (it varies slightly with m). The heat source is present throughout the simulation. It produces an approximately steady, cyclonically rotating updraft by 900 s, at which time the updraft in the m = 8 (m = 2) simulation has a maximum potential temperature excess of 10.2 K (9.9 K), a maximum vertical velocity of 53 m s−1 (57 m s−1), and a maximum vertical vorticity of 0.026 s−1 (0.013 s−1) (Fig. 3). As a result of the stronger low-level shear, the upward-directed dynamic vertical perturbation pressure gradient force (VPPGF; Rotunno and Klemp 1982, hereafter RK82) is more than 5 times larger in the m = 8 simulation than in the m = 2 simulation in the 0–1-km layer (Figs. 4a,b). A decomposition of the dynamic VPPGF into linear and nonlinear parts, following RK82, reveals that the large difference in the dynamic VPPGF is primarily attributable to a large difference in the nonlinear dynamic VPPGF (Figs. 4c–f), which is itself largely attributable to the large difference in the vertical gradient of ζ2 (where ζ is the vertical vorticity) between the updrafts driven in the weak-shear and strong-shear environments. In other words, the base of the midlevel mesocyclone is lowered as the environmental low-level shear increases, which increases the dynamically induced suction (Lilly 1986) at low levels. The implications of the differences in low-level dynamic lifting will be apparent in section 3.

Fig. 3.
Fig. 3.

West–east-oriented vertical cross sections through the heat-source-driven updraft at 900 s (i.e., when the updraft is in roughly a steady state, immediately prior to the activation of the heat sink) in (a),(c) the simulations initialized with relatively weak low-level shear (m = 2) and (b),(d) relatively strong low-level shear (m = 8). The vertical cross sections are at y = 0. The vertical velocity (w) field (color shaded; see legend) is shown in all four panels. In (a),(b), potential temperature perturbations (θ′) are overlaid on the vertical velocity field (black contours every 3 K; dashed contours indicate negative values). In (c),(d), the vertical vorticity (ζ) field is overlaid (black contours every 0.005 s−1 for ζ ≥ 0.005 s−1).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Fig. 4.
Fig. 4.

Horizontal cross sections of 0–1-km-mean (a),(b) dynamic VPPGF , (c),(d) linear dynamic VPPGF , and (e),(f) nonlinear dynamic VPPGF in the (a),(c),(e) weak low-level shear (m = 2) and (b),(d),(f) strong low-level shear (m = 8) simulations at 900 s (i.e., when the updraft is in roughly a steady state, immediately prior to the activation of the heat sink); π′ is the nondimensional pressure perturbation, and the subscripts D, DL, and DNL refer to dynamic, linear dynamic, and nonlinear dynamic nondimensional pressure perturbations . The VPPGF contours are black and drawn every 0.003 m s−2 (negative isopleths are dashed). In each panel, the vertical velocity (w) at z = 1 km (m s−1) also is shown (color shading).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

The heat sink is activated at 900 s at low levels to the north-northeast of the heat source (Fig. 2) in order to emulate the evaporation, melting, and sublimation of hydrometeors in the region where precipitation would be expected to fall relative to the updraft for the specified environmental wind profile. The heat sink is present for the remainder of the simulation. Its amplitude (Sc0) is varied from −0.004 to −0.016 K s−1, and it is centered at (xc, yc, zc) = (2, 4, 0) km. The dimensions of the heat sink are given by Rc = 3 km and Zc = 5 km. The location and dimensions of the heat sink were determined after extensive testing in order to determine which heat sink characteristics result in a low-level temperature field that best emulates the low-level buoyancy field in supercell thunderstorms (e.g., an actual supercell typically has regions of negative buoyancy at low levels on its forward and rear flanks, which are separated from the environmental air by gust fronts). In addition to the development of near-surface (z = 50 m) vertical vorticity being sensitive to the amplitude of the heat sink (one of the primary objectives of the paper is to investigate this sensitivity), there is considerable sensitivity to the location of the heat sink. The sensitivity of the evolution to the location of the heat sink is discussed in section 4.

Though the use of a stationary heat source and sink affords us considerable control over the experiments, there are some limitations. For example, a stationary heat source precludes cold pool–driven storm propagation. Though supercell propagation is largely independent of the cold pool properties (e.g., RK82; RK85; Bunkers et al. 2000), the propagation of some high-precipitation (HP) supercells (Moller et al. 1990; Doswell and Burgess 1993) occasionally can be influenced by their strong cold pools (e.g., Ziegler et al. 2010). Regarding the use of a stationary heat sink, in an actual supercell, evaporation and melting of hydrometeors in the hook echo can alter the buoyancy field there. However, in these idealized simulations, parcels conserve their potential temperature (in the adiabatic limit) once they leave the heat sink to the northeast of the updraft. Despite the simplifications made in our modeling approach, it will be evident in section 3 that the idealized storms have many similarities with actual supercells. Our ability to replicate so many supercell properties using only a stationary heat source and sink might be due to the relative steadiness of supercells.

The simulations have many similarities with Walko’s (1993) idealized simulations, though there are some differences worth noting. Walko’s hodograph was straight and passed through the origin, such that the environmental vorticity was crosswise at all levels for a stationary storm. Because of this, Walko’s environment and stationary heat source are inconsistent with expected storm behavior. In reality, the tilting of crosswise vorticity results in a pair of mesolows that straddle the midlevel updraft. These lows are associated with an upward-directed dynamic VPPGF on the flanks of the updraft, which promotes storm splitting and updraft propagation in actual supercells (e.g., RK82). The resulting pair of propagating updrafts cannot be represented by a single stationary heat source. The stationary heat source in Walko’s environment also does not represent either individual member of the pair because the real cyclonic (anticyclonic) member of the pair ingests significant streamwise (antistreamwise) vorticity owing to the rightward (leftward) updraft motion with respect to the direction of the shear vector (Davies-Jones 1984). In contrast, the stationary heat source used in this study produces an updraft with a motion that is consistent with a right-moving, cyclonically rotating updraft developing in the environment characterized by the environmental hodographs in Fig. 1b. That is, the mean wind and deviant (southward) propagation [attributable to the linear dynamic VPPGF (Davies-Jones 2002)] would be expected to yield a nearly stationary motion.

Near-surface vortices form as a result of a barotropic redistribution of initial vorticity in the idealized simulations of Davies-Jones (2000a, 2008), Markowski et al. (2003b), and Parker (2012)4; however, the axisymmetry of their simulations precludes them from addressing the questions posed in section 1 (e.g., baroclinically generated vorticity cannot be tilted in an axisymmetric model). The idealized simulations herein also share some similarities with those of Straka et al. (2007). A key difference is that no vertical shear is present in their simulations.

3. Simulation results and analysis

a. Baseline pseudostorm simulation in which a strong vortex develops (Sc8m8)

1) Overview

The baseline simulation in the strong-shear environment (m = 8) with a moderately strong heat sink (Sc0 = −0.008 K s−1)—also referred to as the Sc8m8 simulation—is described first because it develops the most intense near-surface vortex. Simulations with less favorable environments and heat sinks will be compared to this benchmark case in subsequent subsections.

Following the activation of the heat sink, cool air gradually spreads southward beneath the midlevel updraft. The leading southern edge of the cool outflow develops an inflection once it reaches the region of broad convergence beneath the midlevel updraft. The inflection is associated with low-level cyclonic vorticity, and the cyclonic vorticity rapidly intensifies in the 2160–2760-s time period (Fig. 5). The vortex attains a maximum cyclonic vorticity, pressure drop, and tangential wind of 0.83 s−1, 13.6 hPa, and 31 m s−1, respectively, and persists for nearly 10 min (|ζ| > 0.2 s−1 for 565 s). Hereafter, the time of maximum near-surface cyclonic vorticity (2760 s in this simulation) is referred to as t − 0. Similarly, t ± X refers to X min before or after the time of maximum near-surface cyclonic vorticity (Fig. 5).

Fig. 5.
Fig. 5.

Evolution of near-surface θ′, ζ, w, and horizontal wind fields in the Sc8m8 simulation. The fields are shown (a) 10 and (b) 5 min prior to the time that cyclonic vorticity is a maximum (t − 10 min and t − 5 min, respectively); (c) at the time of maximum cyclonic vorticity (t − 0 min); and (d) 5 min after the time of maximum cyclonic vorticity (t + 5 min). The θ′ field at z = 50 m is shaded (see legend). Isovorts of ζ at z = 50 m (interpolated to the scalar grid points) are drawn using thin red contours for ζ = ±0.05, 0.15, 0.25 s−1, etc.; dashed contours enclose anticyclonic vorticity. Horizontal wind vectors at z = 50 m are plotted at every fifth grid point. The w = 4 m s−1 isotach at z = 450 m is indicated with thick black contours. In (a), the outlines of the heat source and heat sink are indicated with red and blue dashed rings, respectively. The cyclonic and anticyclonic vorticity maxima discussed in the text are indicated with thick arrows in (c). The gray ring in (c) is the material circuit analyzed in Fig. 10. In (b),(c), the small magenta disks with white and black borders indicate, respectively, the horizontal positions of the parcels analyzed in Figs. 11 and 12 (i.e., parcels that acquire cyclonic and anticyclonic ζ as they near the surface).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Despite the simplicity of the model configuration, the structure and evolution of the simulated pseudostorm—a term borrowed from Trapp and Fiedler (1995)—are strikingly similar to long-standing conceptual models of supercells derived from observations and simulations (e.g., Lemon and Doswell 1979; Doswell and Burgess 1993; Davies Jones et al. 2001) using more complicated model configurations. For example, the boundary between the ambient air and air cooled by the heat sink resembles the gust front structure of an actual supercell, with both a forward-flank and rear-flank gust front obvious in the pseudostorm (Fig. 5). In the early stages of the development of low-level cyclonic rotation, a cyclonic–anticyclonic vorticity couplet develops within the cool air. The couplet is joined by arching vortex lines (yellow lines in Fig. 6), similar to those documented by Straka et al. (2007), Markowski et al. (2008, 2011, 2012a), Marquis et al. (2012), and Kosiba et al. (2013) in observed supercells.

Fig. 6.
Fig. 6.

Three-dimensional view of the midlevel updraft, near-surface θ′ field, and key vortex lines and trajectories at t − 10 min (2160 s) in the Sc8m8 simulation. The view is from the south–southeast. Axes are in kilometers. The w = 15 m s−1 isosurface is gray. The near-surface θ′ field is color shaded (see legend). The yellow lines are vortex lines (their direction is indicated with arrows) that pass within 200 m of ζmax at the lowest scalar level (z = 50 m). The blue lines are vortex lines that pass within 500 m of ζmax at z = 3 km. The white lines are forward-integrated trajectories that pass within 500 m of ζmax at t − 10 min and have ζ ≥ 0.008 s−1 in the lowest 75 m. The trajectory that passes nearest to the cyclonic vorticity maximum at t − 10 min is red. Arrows are placed along it at 5-min intervals.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

During the intensification of cyclonic low-level rotation, the low-level updraft takes on the classic horseshoe-shaped appearance (Lemon and Doswell 1979), the rear-flank gust front wraps up and occludes the near-surface mesocyclone (i.e., the vortex is surrounded by outflow air), an occlusion downdraft develops (Klemp and Rotunno 1983), and the low-level θ field develops a structure similar to the hook echo that would be observed in a radar reflectivity image [Fig. 5c; cf. Fig. 14 of Klemp and Rotunno (1983)]. The arching vortex lines that earlier emanated from the near-surface mesocyclone (Fig. 6) are transformed into approximately vertical vortex lines that extend to the updraft summit (yellow lines in Figs. 7a and 8), similar to the evolution of the Goshen County storm (Markowski et al. 2012b) observed by the Second Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX2; Wurman et al. 2012). Cyclic low-level mesocyclogenesis (Burgess et al. 1977; Adlerman et al. 1999; Dowell and Bluestein 2002) is observed at later times (not shown). The anticyclonic vorticity found on the trailing flank of the pseudostorm’s hook echo remains weak throughout the simulation.

Fig. 7.
Fig. 7.

(a) As in Fig. 6, but at t − 5 min (2460 s) in the Sc8m8 simulation. Trajectories have been omitted. The black and white lines are additional vortex lines; the black one originates in the environment and descends through anticyclonic vorticity in the cold pool. The purple vortex lines originate within the low-level cyclonic vortex and arches toward the anticyclonic vorticity before turning upright and passing into the midlevel updraft and mesocyclone. The dotted white and black lines suggest, respectively, plausible connections between the anticyclonic and cyclonic vortices in the cold pool and between the environmental vortex lines and vortex lines entering the midlevel mesocyclone. See text for further details. (b) Schematic evolution (left to right) of “vortex-line surgery” that joins a vortex line arching upward out of the cold pool with an environmental vortex line that enters the midlevel mesocyclone. The view is from the east–northeast.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Fig. 8.
Fig. 8.

As in Fig. 6, but at t − 0 min (2760 s) in the Sc8m8 simulation. The trajectories pass within 200 m of ζmax, within 75 m of the surface, at t − 0 min. The trajectory that passes nearest to cyclonic vorticity maximum at t − 0 min is red. The magenta trajectory nears the lowest scalar level approximately 1 km west of ζmax at t − 0 [see section 3a(3) and Fig. 11 for details]. The view is from the south–southeast.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

2) Evolution of vortex lines

A few additional comments are warranted regarding the evolution of the vortex lines in the vicinity of the near-surface cyclonic vorticity maximum (ζmax) from arches to lines that extend to the summit of the pseudostorm (Figs. 68). Although the lifting of air parcels out of the cold pool (note the trajectories in Fig. 6) contributes to the arching of the vortex lines, the rapid transformation of the arching vortex lines to tall, nearly vertical lines cannot be explained solely in terms of parcel trajectories (the vortex lines are not material lines, especially in this part of the pseudostorm, where baroclinity and turbulence are significant). The transformation is partly the result of the vertical vorticity component becoming very large, but interactions between the arching vortex lines and environmental vortex lines may also play a role. As noted by Morton (1984), vortex lines can be severed and reattached to other lines instantaneously as long as there are no loose ends at any time. “Surgery” occurs when two oppositely directed lines come close together [see also Markowski et al. (2008), section 4e].

This vorticity cross-diffusion effect is one possible explanation for the aforementioned transformation of vortex lines from arches to more vertical lines that extend to high elevations. In Fig. 7a, in addition to the vortex lines that pass near ζmax, two additional vortex lines are shown. The black vortex line originates in the environment and descends through anticyclonic vorticity in the cold pool, and the white vortex line originates within the low-level cyclonic vortex and arches toward the anticyclonic vorticity before turning upright and passing into the midlevel updraft and mesocyclone. The dotted white and black lines suggest, respectively, plausible prior connections between the anticyclonic and cyclonic vortices in the cold pool and between the environmental vortex lines and vortex lines entering the midlevel mesocyclone. The possible process leading to the final configuration is illustrated schematically in Fig. 7b [cf. Figs. 10i and 10iii in Morton (1984)].

The vortex lines that pass near the near-surface anticyclonic vorticity maximum (ζmin) at t − 0 are arches (Fig. 9); that is, even though the vortex lines through ζmax are no longer arches at t − 0, we can still find arching vortex lines by looking farther away from ζmax. The arching vortex lines are highly contorted owing to the influence of the stronger, proximate cyclonic vorticity maximum to the northeast. They ultimately extend into the pseudostorm’s forward flank.

Fig. 9.
Fig. 9.

As in Fig. 8, but the yellow and white lines are vortex lines and trajectories, respectively, that pass within 200 m of ζmin [at the lowest scalar level (z = 50 m) in the case of the vortex lines and within 75 m of the surface in the case of the trajectories]. The trajectory that passes nearest to the anticyclonic vorticity maximum at t − 0 min is red. The magenta trajectory nears the lowest scalar level near ζmin at t − 0 [see section 3a(2) and Fig. 12 for details]. The view is from the southwest.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

3) Development of near-surface circulation

In the Sc8m8 simulation, as well as the others described in sections 3bd, near-surface cyclonic rotation develops via the “baroclinic mechanism” described by Davies-Jones (2000b) and Davies-Jones et al. (2001), which has been found to be operating in the supercells simulated by RK85, DJB93, Adlerman et al. (1999), and Dahl et al. (2013, manuscript submitted to J. Atmos. Sci.) and observed by Markowski et al. (2012b). One way of demonstrating this is by analyzing the development of circulation about a material circuit (Figs. 5c, 10). In the inviscid, Boussinesq limit, , where D/Dt is a material derivative, B is the buoyancy, and is the solenoidal generation. The approach of RK85, among others, is followed, whereby a ring of parcels is placed around the vertical vorticity maximum and subsequently tracked backward in time. The backward trajectories are computed using velocity data saved every 10 s. A fourth-order Runge–Kutta scheme is used with a time step of 5 s. Dahl et al. (2012) have shown that backward trajectories computed from regions of strongly confluent flow (e.g., the near environs of intense near-surface vortices) are less reliable than forward trajectories, but by tracking a broader material circuit, parcels avoid the most severe accelerations present in the immediate vicinity of the vortices.

Fig. 10.
Fig. 10.

(a) View from the southwest and from above (inset) of the material circuit in the Sc8m8 simulation at t − 20, t − 10, and t − 0 min. The circuit is a 1-km-radius circle centered on the cyclonic vorticity maximum at z = 250 m at t − 0 min (this is the starting point of the backward trajectory calculations for the parcels comprising the circuit; the circuit is indicated with a gray ring in Fig. 5c). Six select parcels within the circuit are shown at each time using orange, yellow, gray, green, cyan, and purple markers. The perturbation potential temperature field (θ′) at z = 50 m at t − 0 min also is shown (legend). The vertical scale in the view from the southwest is exaggerated by approximately a factor of 2. (b) Circulation (C; black curve) and solenoidal generation of circulation (; blue shading) about the circuit as a function of time; also is shown (gray curve) for comparison, where C0 is the constant that makes match the circulation about the circuit at t − 0 min.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

The material circuit is initially 2 km in diameter, is centered on the vertical vorticity maximum at z = 250 m at t − 0, and initially comprises 500 parcels (the initial spacing between parcels is ~10 m). Horizontal velocity components from the lowest scalar level are extrapolated to parcels below the lowest scalar level in accordance with the free-slip lower-boundary condition. However, we have found that DC/Dt better matches when the extrapolation of the circuit is limited. Thus, the circuit is introduced at z = 250 m rather than z = 50 m in order to limit the time that portions of the circuit spend below the lowest scalar level. Moreover, because the vdl calculations are unreliable for large spacings between adjacent parcels within the circuits, additional parcels are added to the circuits as needed in order to ensure that adjacent parcels in the circuit are never more than 100 m apart. This is important for circuits that unavoidably pass through the wind-shift line between the cool air and ambient air, where parcels on opposite sides of the line diverge as the trajectories are integrated backward in time. After just 15 min of backward integration, gaps between adjacent parcels as large as 5 km develop within parts of the material circuits if additional parcels are not added.

The circuit’s circulation is near its maximum at t − 0, when C = 7.9 × 104 m2 s−1 (Fig. 10b). Owing to the complexity of the wind fields, the circuit develops numerous folds as it is tracked backward in time (Fig. 10a), making its evolution more complicated than in previous studies that have analyzed material circuits using wind fields that were considerably smoother and available less frequently (e.g., RK85; DJB93; Markowski et al. 2012b). Complexity aside, the gross evolution of the circuit and its circulation is similar to the evolution in previous supercell studies. For example, the circuit acquires a vertical projection as it is stepped backward into the forward-flank region of the pseudostorm, as it must, given that vortex lines are nearly horizontal away from the vorticity maximum (circulation is proportional to the flux of vortex lines through a surface bounded by the circuit).

The circuit acquires significant circulation northeast of the updraft between t − 28 and t − 8 as a result of solenoidal generation (Fig. 10b). The circulation trends in Fig. 10b are believed to be credible given the relatively good agreement with . The quantitative agreement is poorest when segments of the circuits experience large accelerations (i.e., when trajectory errors are largest) and when significant segments are below the lowest scalar level. The vast majority of the circulation associated with the near-surface mesocyclone develops after the heat sink’s activation at t − 31 (900 s) and is the result of baroclinic vorticity generation associated with the heat sink. The environmental vorticity does not contribute significantly to the circulation of the near-surface mesocyclone in this case, despite the presence of large environmental streamwise vorticity. This result is consistent with Dahl et al.’s (2013, manuscript submitted to J. Atmos. Sci.) finding that the vertical vorticity of parcels reaching the near-surface mesocyclone of a simulated supercell is virtually entirely attributable to vorticity generated baroclinically (i.e., baroclinic vorticity) within the storm rather than preexisting vorticity associated with the environmental wind shear that is reoriented and stretched within the storm. These parcels, though not originating entirely outside the storm, were chosen such that their initial vorticity (i.e., barotropic vorticity) was close to that of the environment.

Though we have not diagnosed baroclinic and barotropic vorticity along individual trajectories, the Lagrangian material circuit analysis is complemented by a Lagrangian analysis of the three-dimensional vorticity along a representative trajectory that enters the near-surface mesocyclone (Figs. 8, 11). At t − 0, the parcel is 1 km northwest of ζmax at an altitude of 70 m, which puts it just below the aforementioned material circuit (Fig. 5c). The trajectory is a forward trajectory computed during model run time using the velocities obtained from the staggered C grid (Arakawa and Lamb 1977) at each large time step.5 The trajectory remains above the lowest scalar level at all times; thus, extrapolation of velocity and vorticity components and vorticity forcings is avoided.6

Fig. 11.
Fig. 11.

Vorticity and vertical velocity forcings along the trajectory discussed in section 3a(3) (the magenta trajectory in Fig. 8) that nears the lowest scalar level, 1 km northwest of ζmax at t − 0. (a) The vorticity vector (ω; black) is plotted every 2.5 min along the trajectory (magenta line) in the sz plane. Axes are in kilometers. The projections of the integrated barotropic, baroclinic, and turbulent forcings in the intervening 2.5-min periods are shown with green, blue, and brown vectors, respectively (s−1). The integrated forcing from crosswise–streamwise exchange is shown with orange vectors (s is the unit vector that points in the streamwise direction). The scale of all of the vorticity forcing vectors matches the scale of the vorticity vectors (see legend in panel) and the vectors are scaled differently in the horizontal and vertical directions. The vertical components of the vectors are exaggerated by the same degree to which the vertical scale of the viewing window is exaggerated. (b) As in (a), but in the horizontal plane (there is no crosswise–streamwise exchange in the xy plane); blue (red) shading indicates the horizontal extent of the heat sink (source). (c) Altitude (zp) and ζ of the parcel during its ascent and subsequent descent (t − 20 to t − 1). (d) Vertical inclination angle δ between the trajectory and ω during the same time period as (c).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Figure 11b displays the horizontal projections of the vorticity vector [ω = (ξ, η, ζ)] and its forcings. The vorticity vector is plotted along the trajectory every 2.5 min. The projections of the vorticity forcings over the intervening 2.5-min periods also are shown and are related to the change in ω via
e6
where v = (u, υ, w) is the velocity vector, cp is the specific heat at constant pressure, π is nondimensional pressure, and F is the forcing for Dv/Dt from subgrid-scale turbulence. The , , and terms represent the mean barotropic, baroclinic, and turbulent forcings over the time interval Δt (=2.5 min). To a good approximation, −cpθ × π × Bk.
Figure 11a shows, for the same trajectory, the projections of the vorticity vector and its forcings onto the sz plane, where s is the streamwise direction. Changes in the streamwise vorticity (ωs; i.e., the horizontal vorticity component lying in the changing vertical plane) are governed by
e7
where (⋅)s indicates the projection of a vector forcing in the streamwise direction and the last term on the rhs represents the conversion of crosswise vorticity (ωc) to streamwise vorticity as a parcel changes direction [ψ = tan−1(υ/u)] in the horizontal plane (Scorer 1978, p. 88; Adlerman et al. 1999; Davies-Jones et al. 2001). The vorticity tendencies evident in Figs. 11a and 11b are in good qualitative agreement with the forcings (i.e., the vector change in ω over each interval approximately matches the vector sum of the mean forcings).

The trajectory approaches the pseudostorm from the southeast and reaches the heat sink at approximately t − 13, at which point the parcel begins experiencing southwestward baroclinic vorticity generation (Fig. 11b). The baroclinic generation is initially crosswise but becomes increasingly streamwise as the trajectory is bent to the left between t − 13 and t − 5 by the horizontal pressure gradient force between the heat sink’s high pressure and the mesocyclone’s low pressure. The vorticity vector itself remains highly streamwise during this time, though along other trajectories (e.g., the white and red trajectories in Figs. 6 and 8), the vorticity vector develops a significant crosswise component, with the crosswise vorticity subsequently becoming streamwise owing to crosswise–streamwise exchange (not shown). Early crosswise baroclinic vorticity generation and conversion of crosswise vorticity to streamwise vorticity also was identified along trajectories in the RK85 and Adlerman et al. (1999) simulations and suggested by Markowski et al.’s (2012b) analysis of the Goshen County storm.

Cyclonic vorticity is initially acquired by the parcel between t − 20 and t − 11 owing to upward tilting as the parcel rises over the forward-flank gust front a few minutes before the parcel reaches the heat sink (Figs. 11a,c). The aforementioned leftward turning of the trajectory occurs shortly thereafter and is accompanied by rapid descent (~1 km between t − 11 and t − 5) owing to negative buoyancy. The vorticity vector is initially tipped downward (i.e., anticyclonic vorticity is acquired) once descent commences, but the vorticity vector gradually becomes inclined upward relative to the descending trajectory, such that the parcel develops cyclonic vorticity by the time it is within 100 m of the surface (Figs. 11a,c). The evolution is similar to what previously has been described by DJB93 and Davies-Jones (2000b), who showed that cyclonic vorticity can arise along a trajectory that passes through the left side of a cool downdraft, which is analogous to the heat sink herein. In the words of DJB93, the baroclinicity introduces “slippage between the vortex lines and trajectories” so that the vorticity vector can have a vertical component next to the surface despite the trajectory turning horizontal there (see DJB93’s Fig. 9). This basic mechanism was also found to be the means by which near-surface cyclonic vorticity developed in the Adlerman et al. (1999) simulation, and seemingly in the RK85 simulation as well (their Fig. 8 shows the growth of cyclonic vorticity along a descending trajectory, along which the upstream vorticity is highly streamwise).

Figure 11d shows the evolution of the vertical inclination angle between the vorticity vector and trajectory δ (positive δ implies the vorticity vector is angled upward relative to the trajectory). The vorticity vector is temporarily tipped below the inclination of the trajectory as the trajectory rises over the gust front (δ is a minimum at t − 15). During the descent that follows, the vorticity vector is tilted upward relative to the trajectory, with the positive inclination reaching its maximum (δ ~ 18°) roughly halfway to the surface at t − 8. Throughout most of the descent the vorticity is anticyclonic, however (Fig. 11c). The angle δ is notably small, though still positive, as the trajectory reaches its nadir (Fig. 11d). The small cyclonic vorticity acquired as the trajectory nears the surface is amplified—primarily via stretching—to mesocyclone strength [O(10−2 s−1)] as the trajectory subsequently travels quasi horizontally ~70 m above the surface between t − 3 and t − 0 (Fig. 11a).

A similar analysis was carried out for a trajectory that approaches the near-surface anticyclonic vorticity maximum (Fig. 12) that develops to the southwest of the cyclonic vortex (Figs. 5b,c and 9). The trajectory follows a path a few kilometers to the west of the trajectory that passes nearest to ζmax (cf. Figs. 11b and 12b), such that the trajectory descends on the western flank of the cold pool (Figs. 5b,c). In doing so, the trajectory experiences northward-pointing baroclinic vorticity generation from t − 8 to t − 0, in contrast to the southward-pointing baroclinic generation experienced by the parcel that approaches ζmax (Fig. 12b). The combination of descent, northward baroclinic generation, and conversion of anticrosswise vorticity to antistreamwise vorticity promotes negative δ; that is, the vorticity vector is tipped below the trajectory (Figs. 12a,d). Though the parcel develops significant anticyclonic vorticity early in its descent (say, by t − 10; Figs. 12a,c), as was the case for the trajectory that approaches ζmax (Figs. 11a,c), the trajectory that approaches ζmin retains its anticyclonic vorticity as it nears the surface as a result of the negative δ (Figs. 12a,c,d). The rapid fluctuations in ζ between t − 5 and t − 1 (Fig. 12c) are the result of turbulent mixing. The effects of the turbulent mixing on the trailing cold pool also are evident in Figs. 5b and 5c, which shows that the cold air has been eroded on its western flank.

Fig. 12.
Fig. 12.

As in Fig. 11, but for the trajectory discussed in section 3a(3) (the magenta trajectory in Fig. 9) that nears the lowest scalar level near ζmin at t − 0.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

4) The juxtaposition of near-surface circulation and dynamic lifting

Though the Lagrangian perspective provided in section 3a(3) is enlightening, it is incomplete because it fails to reveal where circulation is available. Because the Lagrangian analyses focus on parcels or rings of parcels in close proximity to ζmax, they do not allow us to assess whether there might be larger circulation away from ζmax that simply does not experience sufficient convergence to yield large vertical vorticity. In this subsection, the Lagrangian perspective is complemented by an Eulerian perspective in order to examine the spatial relationship between near-surface circulation and the dynamic lifting.

In Figs. 13a–c, circulation is computed about 1-km-radius rings (the radii match the initial radius of the material circuit introduced at t − 0) centered on each grid point at z = 50 m, and the field is subsequently contoured to reveal the horizontal distribution of circulation. It is evident that the maximum cyclonic vorticity develops near (though not exactly at) the location of maximum circulation (this is not the case for all of the simulations described in section 3). Near-surface circulation reaches a maximum at approximately the same time that the maximum near-surface cyclonic vorticity is observed, that is, t − 0 (Fig. 13c). This is also true of the circulation about a broad 2-km-radius ring centered on the maximum 0–1-km-mean dynamic VPPGF (Fig. 13d).

Fig. 13.
Fig. 13.

Horizontal distribution of circulation (C) at z = 50 m computed over 1-km-radius rings (color shading) at (a) t − 10, (b) t − 5, and (c) t − 0 min in the Sc8m8 simulation. The dynamic VPPGF field vertically averaged over the 0–1-km layer is overlaid with gray contours (0.05 m s−2 interval, dashed contours indicate negative values). The plus and minus signs in (c) indicate the cyclonic and anticyclonic vorticity maxima, respectively, at z = 50 m. The gray rings are centered on the maximum 0–1-km mean dynamic VPPGF and have radii of 2 km. (d) The potential temperature perturbation averaged within this ring (〈θ′〉; cf. Figure 5) and C about this ring, both at z = 50 m, are shown from t − 20 to t + 5 min.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Not only does the near-surface circulation arise in relatively close proximity to the strongest dynamic VPPGF, but the dynamic VPPGF itself intensifies as near-surface circulation and cyclonic vorticity increase. The maximum dynamic VPPGF more than doubles between t − 10 and t − 0 (Figs. 13a–c) owing to a positive feedback involving the intensifying low-level rotation, low-level updraft, and low-level dynamic VPPGF, as has been noted in other simulations as well (e.g., Wicker and Wilhelmson 1995; Davies-Jones 2008). Within a 2-km radius of the maximum dynamic VPPGF, the mean θ′ at z = 50 m ranges from −1 to −2 K from t − 20 to t + 5 (Fig. 13d).

One of the goals expressed in section 1 is to investigate why the anticyclonic vorticity maximum commonly observed in supercells tends to be weaker than the cyclonic vorticity maximum. The near-surface anticyclonic vorticity is considerably weaker than the cyclonic vorticity in the pseudostorm as well (e.g., Fig. 5c). The reasons are twofold. The first is that the anticyclonic near-surface circulation is less than the cyclonic near-surface circulation (Figs. 13a–c) owing to less (in magnitude) solenoidal generation experienced by circuits converging upon ζmin than circuits converging upon ζmax, per additional material circuit analyses (not shown). The differences in solenoidal generation appear to be due to weaker baroclinicity on the trailing flank of the cold pool than on the leading flank (Fig. 5). The forward (eastern) flank of the pseudostorm is associated with stronger frontogenesis. The second reason for the relative weakness of the anticyclonic vorticity maximum is that the anticyclonic near-surface circulation develops farther from the region of maximum dynamic lifting than the cyclonic near-surface circulation (Fig. 13c; also see Figs. 11b, 12b), as presumed by Davies-Jones et al. (2001). The differences in dynamic lifting also are evident in the differences in the trajectories between t − 0 and t + 5 (cf. Figs. 8 and 9). Despite the parcels associated with ζmin having less negative buoyancy than those associated with ζmax (Fig. 5c), the trajectories associated with ζmax rise abruptly and experience considerably more vertical vorticity stretching than the trajectories associated with ζmin.

In summary, the intense cyclonic vortex develops within a region of relatively large near-surface circulation—circulation that arises within outflow possessing relatively small negative buoyancy, with the circulation-rich air also being within a region of strong dynamic lifting. The buoyancy of the circulation-rich air, the location where significant near-surface circulation develops relative to the region of dynamic lifting, and/or strength of the dynamic lifting are decidedly less favorable in the other simulations of the parameter space.

b. Pseudostorm with a stronger cold pool relative to the baseline simulation (Sc16m8)

This simulation (Sc16m8) is initialized with the same environmental hodograph (m = 8) as the baseline simulation analyzed in section 3a, but the heat sink is twice as strong (Sc0 = −0.016 K s−1). The minimum θ′ in the cold pool approaches −8 K (Fig. 14). The stronger cold pool spreads beneath the midlevel updraft faster than in the Sc8m8 simulation. A near-surface cyclonic vortex develops at 2220 s, but it is shorter-lived and weaker than the vortex in the Sc8m8 simulation (Table 1; Fig. 14). The pseudostorm’s gust front structure and low-level θ′ field at times resemble those expected in an HP supercell (e.g., Fig. 14d).

Fig. 14.
Fig. 14.

As in Fig. 5, but for the Sc16m8 simulation. (c) The cyclonic vorticity maximum discussed in the text is indicated with a thick arrow.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Table 1.

Summary of the characteristics of the vortices that develop in the simulations. The maximum vertical vorticity magnitude, minimum pressure perturbation, maximum tangential wind, and radius of maximum tangential wind are ζmax, , , and rmax, respectively (all are valid at the lowest scalar level, i.e., z = 50 m). The duration of the vortex is arbitrarily defined as the time period over which |ζ| > 0.2 s−1. For the Sc4m8 simulation, the cyclonic vertical vorticity maximum does not develop into a vortex (for this reason, several columns show N/A for “not applicable”). In the same simulation, a strong anticyclonic vortex develops. Its characteristics (denoted by the letter A) appear in the fifth row.

Table 1.

The three-dimensional perspective of Fig. 15 suggests a midlevel updraft that has been undercut by excessively cold outflow. The near-surface cyclonic vortex is decoupled from the midlevel mesocyclone and develops several kilometers south of where the vortex develops in the Sc8m8 simulation. The trajectories that reach the surface within the cold pool originate from a higher altitude than in the Sc8m8 simulation, on average. Those that pass through the near-surface cyclonic vortex simply slide beneath the midlevel updraft without ever ascending into it. Vortex lines arch only a short distance upward out of the near-surface vortex before turning southwestward and paralleling the gust front; they eventually descend into a region of weak anticyclonic vorticity 5–8 km southwest of the cyclonic vortex. The trajectories and vortex lines resemble those that have been observed in nontornadic low-level mesocyclones (Markowski et al. 2011).

Fig. 15.
Fig. 15.

As in Fig. 6, but for the Sc16m8 simulation at t − 0 min (2220 s). The trajectories pass within 200 m of ζmax, within 75 m of the surface, at t − 0 min. The trajectory that passes nearest to the cyclonic vorticity maximum at t − 0 min is red. The view is from the south–southeast.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Though the maximum near-surface circulation is larger than in the Sc8m8 simulation (cf. Figs. 13c and 16a), the circulation-rich air is shunted well to the south of the strongest dynamic forcing as a result of the stronger cold pool (Fig. 16a). As a result, the dynamic VPPGF is weaker along the trajectories that pass near ζmax in the Sc16m8 simulation than in the Sc8m8 simulation, despite both simulations having identical environmental wind profiles. Moreover, the enhancement of the dynamic VPPGF that accompanies the development of the intense vortex in the Sc8m8 simulation is absent in the Sc16m8 simulation (cf. Figs. 13c and 16a). Within a 2-km radius of the maximum dynamic VPPGF in the Sc16m8 simulation, the near-surface air mass is colder (e.g., at t − 0, the mean θ′ is −4 K at z = 50 m) and possesses less circulation (e.g., at t − 0, C is 1.1 × 104 m2 s−1 at z = 50 m about the 2-km-radius ring surrounding the maximum dynamic VPPGF) than in the Sc8m8 simulation (Fig. 16b). By comparison, in the Sc8m8 simulation the mean θ′ is −2 K within 2 km of the maximum dynamic lifting at t − 0, and the C about the same broad ring is 3.4 × 104 m2 s−1 (Fig. 13d).

Fig. 16.
Fig. 16.

As in Figs. 13c,d, but for the Sc16m8 simulation. (a) The plus sign indicates the cyclonic vorticity maximum at z = 50 m. (b) The blue lines are for the Sc8m8 simulation (i.e., they are traced from Fig. 13d).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

In summary, the excessively cold outflow is detrimental to the intensification of near-surface vorticity in the Sc16m8 simulation for two reasons: 1) the excessive negative buoyancy impedes vertical accelerations and 2) the large (southward) horizontal accelerations associated with the strong cold pool displace the developing near-surface vortex from the strongest dynamic lifting. With regards to the second reason, this detriment might have been mitigated somewhat if the updraft had been free to propagate with the gust front.

c. Pseudostorm in weaker environmental low-level shear than the baseline pseudostorm (Sc8m2)

In this simulation (Sc8m2) the heat sink has the same strength as in the baseline Sc8m8 simulation (Sc0 = −0.008 K s−1), but the low-level environmental shear is weaker (m = 2). A near-surface vortex also develops in this simulation, but it is again shorter-lived and weaker than the vortex in the Sc8m8 simulation (Table 1; Fig. 17). As was the case for the Sc16m8 simulation, the structure of the Sc8m2 pseudostorm’s gust fronts and low-level θ′ field also occasionally resemble those of an HP supercell.

Fig. 17.
Fig. 17.

As in Fig. 5, but for the Sc8m2 simulation. (a) The outline of the heat source and heat sink are indicated with dashed red and blue rings, respectively. (c) The cyclonic vorticity maximum discussed in the text is indicated with a thick arrow.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Similar to the Sc16m8 simulation, the intensification of near-surface vertical vorticity is inhibited by the undercutting of midlevel heat source and updraft (Fig. 17), which occurs despite the Sc8m2 simulation having a weaker cold pool than the Sc16m8 simulation. The undercutting could be due to a number of factors, though it is difficult to quantify their relative influences on the motion of the outflow: 1) weaker dynamic lifting at low levels owing to the weaker environmental shear (the degree to which cool air is drawn upward upon sliding beneath the overlying updraft is a function of the relative magnitudes of the VPPGF and negative buoyancy of the outflow), 2) differences in the outflow-relative headwinds between the m = 2 and m = 8 environments, and 3) differences in the gust front normal component of the low-level vertical wind shear between the two environments. Regarding the first factor, the differences in the low-level dynamic lifting are evident in the steady-state (900 s) dynamic VPPGF fields (Fig. 4), as well as in the three-dimensional perspective of Fig. 18 (note the differences in the height of the base of the w = 15 m s−1 isosurface at t − 0 in Figs. 8 and 18). The vortex lines that pass near ζmax display the classic arching structure (Fig. 18), but at no time are they nearly vertical lines extending to the summit of the pseudostorm, as is the case in the Sc8m8 simulation as cyclonic vorticity intensifies.

Fig. 18.
Fig. 18.

As in Fig. 6, but for the Sc8m2 simulation at t − 0 min (2640 s). The view is from the southeast.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

As is the case for the Sc16m8 simulation and observed nontornadic low-level mesocyclones (Markowski et al. 2011), the trajectories that acquire significant cyclonic vertical vorticity near the surface fail to ascend and participate in the midlevel updraft after entering the near-surface cyclonic vortex (Fig. 18). The dynamic VPPGF field (Fig. 19a) is substantially weaker in the weak-shear Sc8m2 environment compared with the strong-shear Sc8m8 and Sc16m8 environments (cf. Figs. 16a and 19a). The circulation-rich air also is more negatively buoyant in the Sc8m2 simulation (Fig. 19b) than in the Sc8m8 simulation from t − 6 to t + 2. The difference can only be due to longer parcel residence times in the Sc8m2 case, given that the Sc8m8 and Sc8m2 heat sinks are identical.

Fig. 19.
Fig. 19.

As in Fig. 16, but for the Sc8m2 simulation.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Though the maximum near-surface circulation is less than in the Sc16m8 simulation (cf. Figs. 16a and 19a), the circulation about a broad 2-km-radius ring centered on the maximum dynamic lifting is larger in the Sc8m2 simulation (cf. Figs. 16b and 19b). The near-surface cyclonic vorticity maximum develops within a region of the outflow having considerably less circulation than the maximum circulation; the maximum circulation is present ~4 km southwest of the cyclonic vorticity maximum (Fig. 19a).

d. Pseudostorm with a weaker cold pool relative to the baseline simulation (Sc4m8)

Last, we present a simulation (Sc4m8) in which the heat sink is half as strong as in the Sc8m8 simulation (i.e., Sc0 = −0.004 K s−1), but the same environmental wind profile is used as in the Sc8m8 and Sc16m8 simulations (i.e., m = 8). The weaker heat sink results in a cold pool having near-surface θ deficits <2.5 K (Figs. 20, 21). The pseudostorm, which in some ways resembles a low-precipitation (LP) supercell (Bluestein and Parks 1983; Doswell and Burgess 1993), never develops a strong near-surface vortex.

Fig. 20.
Fig. 20.

As in Fig. 17, but for the Sc4m8 simulation. (d) The fields are shown 14 min (not 5 min) after the time of maximum cyclonic vorticity (t + 14 min), at which time a strong anticyclonic vortex is present at the surface at the location indicated by the thick arrow.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Fig. 21.
Fig. 21.

As in Fig. 6, but for the Sc4m8 simulation at 3660 s. The view is from the southeast.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

The peak near-surface vorticity is 0.13 s−1, but this occurs far from where strong cyclonic vortices typically develop in actual supercells or in the other pseudostorm simulations. The maximum vorticity is associated with a shallow, transient vortex (the pseudostorm’s version of a gustnado) that develops at (x, y) = (−0.6, 9.3) km, which is along the northern fringe of the cool outflow, well north of the region displayed in Fig. 20. The maximum vorticity indicated in Table 1 (0.03 s−1) is the maximum that develops in the traditional location (at 3660 s), that is, at the inflection in the gust fronts where the rear-flank gust front intersects the forward-flank gust front [(x, y) = (1.4, −1.5) km in Fig. 20c]. We deliberately do not refer to the cyclonic vorticity maximum as a cyclonic vortex. At the location of ζmax, the magnitude of the deformation exceeds the magnitude of the vertical vorticity, such that a relative maximum (not minimum) in pressure coincides with ζmax (not shown).

The cyclonic vorticity maximum in the Sc4m8 simulation is associated with the smallest positive circulation of the four simulations presented (Fig. 22). Material circuit analyses for this simulation (not shown) reveal that the solenoidal generation is also the smallest among the simulations, which is unsurprising given the weak cold pool and the related fact that the downward vertical excursions of parcels are smallest in this simulation (Fig. 21).

Fig. 22.
Fig. 22.

As in Fig. 16, but for the Sc4m8 simulation.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

Some of the vortex lines near ζmax at t − 0 form arches, whereas others (only one is shown in Fig. 21) extend to the summit of the pseudostorm. Although vortex lines that are nearly vertical and extend to high altitudes often imply the presence of a deep, intense vortex, nearly vertical vortex lines that extend to great heights also can be associated with relatively weak vorticity (the path a vortex line takes depends only on the orientation of the vorticity, not its magnitude, though the vertical orientation does imply that ζ ≫ |ωh|, where ωh is the horizontal vorticity vector). The trajectory that passes nearest to ζmax also rises abruptly upward (Fig. 21). Other trajectories that pass near ζmax, however, fail to rise as abruptly owing to the influence of an intensifying anticyclonic vortex to the southwest that trails the pseudostorm’s hook echo (Figs. 20b–d). These trajectories are accelerated more southwestward than upward (Fig. 21).

As alluded to above, a significant, long-lived anticyclonic vortex develops in this simulation (Fig. 20d). Though the cyclonic member of the cyclonic–anticyclonic vorticity couplet that typifies the rear-flank outflow of supercells usually becomes the most intense (Markowski 2002), occasionally a strong, even dominant, anticyclonic vortex develops (e.g., Fujita and Wakimoto 1982). The anticyclonically bending hook echo (implied by the near-surface θ′ field) in Fig. 20 resembles hook echoes in some observed supercells as well [e.g., the 9 June 1998 nontornadic supercell in Fig. 1 of Markowski et al. (2002)].

The anticyclonic vortex attains a peak intensity of ζ = −0.43 s−1 (Table 1) at 4500 s, or 14 min after the maximum cyclonic vorticity is attained. The trajectories that pass near ζmin (Fig. 23) and evolution of vorticity along them (not shown) are qualitatively similar to what was analyzed for the anticyclonic vorticity maximum in the Sc8m8 simulation (Figs. 9, 12). The vortex lines that pass near ζmin descend from high aloft in the pseudostorm, but they ultimately all can be traced back into the environment to the south, as opposed to joining the anticyclonic vorticity with a region of low-level cyclonic vorticity to the northeast within the cold pool (cf. Figs. 9 and 23). The vortex lines likely obtain this configuration via the process illustrated in Fig. 7b.

Fig. 23.
Fig. 23.

As in Fig. 6, but for the Sc4m8 simulation at 4380 s (here t − 0 min refers to the time relative to the maximum anticyclonic vorticity, rather than maximum cyclonic vorticity, as in Figs. 20 and 21). The trajectories pass within 200 m of ζmin, within 75 m of the surface, at t − 0 min. The trajectory that passes nearest to anticyclonic vorticity maximum at t − 0 min is red. The view is from the southeast.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

The anticyclonic vortex is associated with a considerably larger magnitude of circulation than the cyclonic vortex (Fig. 22a). In fact, this is the only simulation in which the near-surface anticyclonic circulation magnitude exceeds the near-surface cyclonic circulation magnitude (cf. Figs. 13a, 16a, 19a, and 22a). That the θ′ fields evident in Figs. 20c and 20d would promote larger negative solenoidal generation of circulation than positive generation would not be easy to anticipate. Though strong baroclinicity is present on the western flank of the hook echo (Figs. 20c,d), and the material circuits (not shown) that converge upon ζmin pass through this baroclinicity, the rear-flank baroclinicity is not obviously stronger than the forward-flank baroclinity through which a circuit passes en route to ζmax. The differences in solenoidal generation must be due to details in the trajectories and their residence times in the baroclinicity.

Another factor in the development of the strong anticyclonic vortex is the overall weakness of the cold pool. Not only do the parcels that feed the anticyclonic vortex have small negative buoyancy, but the cold pool does not surge nearly as far southward in the Sc4m8 simulation as in the other simulations. Thus, the developing anticyclonic vortex is in a more favorable region (i.e., beneath the midlevel heat source; Fig. 20d) than the anticyclonic vorticity maxima in the other simulations. Although Fig. 22a only shows the horizontal distribution of circulation at t − 0 (i.e., the time of maximum near-surface cyclonic vorticity), it is evident that the region of strong anticyclonic circulation is much nearer to the strong dynamic lifting than in the Sc8m8, Sc16m8, and Sc8m2 simulations (cf. Figs. 13c, 16a, 19a, and 22a). The relative warmth of the outflow in proximity to the dynamic lifting is evident in Fig. 22b, with mean θ deficits within 2 km of the maximum dynamic VPPGF being less than 1 K through 3660 s (t − 0), and through 4500 s as well (not shown).

4. Discussion

Additional simulations were performed with weak low-level shear (m = 2), but are not summarized herein. They all resulted in weaker low-level vortices than in strong-shear (m = 8) simulations with the same heat sink strength. Simulations also were performed in which the hodograph was straight and the environmental vorticity had a large crosswise component; unlike Walko’s (1993) environment, this environment also contained streamwise vorticity because a southerly wind was imposed (i.e., the hodograph was shifted north of the origin rather than passing through the origin, to be consistent with the stationary heat source). These pseudostorms developed weaker vortices as well. The analysis of these simulations is beyond the scope of the present paper and does not affect its conclusions.

The results of section 3 are consistent with the axisymmetric simulations of Markowski et al. (2003b). In those simulations, the strongest tornado-like vortex developed in the case with the strongest midlevel updraft and a precipitation-induced downdraft associated with relatively small negative buoyancy. This combination was favorable because significant circulation reached the surface, and the relatively small negative buoyancy of the circulation-rich air, combined with the strong midlevel updraft, facilitated intense low-level vorticity stretching. Trajectories rose to great heights after passing through the developing near-surface vortices in the strong-vortex cases; in the weak-vortex cases, trajectories ascended only a short distance before turning horizontally. The Markowski et al. (2003b) simulations, owing to the axisymmetry, could not reveal the additional importance of the horizontal displacement of near-surface circulation (by excessively cold outflow) relative to the region where dynamic lifting is strongest.

The results also are similar to those reported by Snook and Xue (2008), who performed a series of supercell simulations (also using 100-m horizontal grid spacing) in which the environment was held fixed but the microphysical characteristics of the storms were varied. Intense vortices only developed in the simulations in which the outflow was relatively mild. In the simulations in which tornado-like vortices failed to develop, the outflow was cold and the near-surface cyclonic vorticity maxima were displaced far from the dynamic forcing associated with the midlevel updraft and mesocyclone. In the words of Snook and Xue (2008, p. 1), “When the cold pool is strong, the updraft is tilted rearward by the strong, surging gust front, causing a disconnect between low-level circulation centers near gust front and the mid-level mesocyclone. Weaker cold pool cases have strong, sustained, vertical updrafts positioned near and above the low-level circulation centers, providing strong dynamic lifting and vertical stretching to the low-level parcels and favoring tornadogenesis.”

We believe our simulations provide a plausible explanation for why tornadic supercells are favored in environments that have low LCLs and contain strong low-level wind shear (e.g., Thompson et al. 2003; Craven and Brooks 2004). Given that most supercells (those with at least some outflow) seem to develop near-surface vertical vorticity (including all of the pseudostorms), the development of an intense vortex requires large near-surface stretching. Large vertical stretching is associated with a strong net upward force, or, equivalently, small negative buoyancy and a large upward VPPGF. Supercell cold pools tend to be suppressed (i.e., have small negative buoyancy) in environments with low LCLs (Markowski et al. 2002; Shabbott and Markowski 2006). The low-level dynamic VPPGF experienced by parcels in proximity to ζmax increases as the low-level environmental shear increases (strong low-level shear has been found to favor tornadic supercells in the aforementioned tornado environment studies), and as the negative buoyancy decreases, as suggested by the simulations herein, because a strong cold pool tends to displace the developing near-surface vortex far from the dynamic lifting, which is maximized beneath the midlevel mesocyclone. We are unable to refute the hypothesis of Markowski et al. (2012b) that the role of strong low-level environmental shear is to lower the base of the midlevel mesocyclone (tilting of horizontal vorticity and stretching of vertical vorticity increase with increasing environmental shear, which reduces the altitude at which significant vertical vorticity arises), thereby enhancing the dynamic VPPGF and stretching of negatively buoyant outflow parcels that possess cyclonic vorticity next to the ground.

Not only is the intensification of near-surface vertical vorticity sensitive to the heat sink’s strength, but intensification also is sensitive to the location and dimensions of the heat sink. Figure 24 reveals the sensitivity of ζmax to the coordinates of the heat sink. For the Sc8m8 configuration, a horizontal shift in the heat sink’s position of just 1 km can result in changes in ζmax by 100% (Fig. 24a). The results are similarly sensitive to the dimensions of the heat sink (not shown). Nevertheless, for fixed heat sink properties, the results herein are robust; that is, for a heat sink fixed at a particular location with particular dimensions, strong cyclonic vortices are most likely to develop in simulations with intermediate heat sinks.

Fig. 24.
Fig. 24.

Magnitude of the cyclonic vorticity maximum as a function of the horizontal position of the heat sink in the (a) Sc8m8 (baseline), (b) Sc16m8, (c) Sc8m2, and (d) Sc4m8 experiments. The x and y axes indicate the heat sink coordinates (xc, yc) (km), which are each shifted up to ±2 km from the locations (indicated with unshaded circles at the center of each domain) of the heat sinks in the experiments presented in section 3.

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

One could interpret the sensitivity as implying that outflow with relatively small negative buoyancy and strong low-level dynamic upward forcing are necessary but insufficient conditions for the development of intense near-surface vertical vorticity, with the formation of an intense near-surface vortex also requiring the development of large near-surface circulation (within the outflow) in a region of strong dynamic lifting. (In actual supercell environments, low LCLs and strong low-level shear are, of course, insufficient conditions for tornadogenesis as well.) Though the amplitude of the negative buoyancy and the strength of the low-level dynamic upward forcing are relatively easily controlled by the heat sink strength and environmental low-level vertical shear, respectively, the magnitude of the near-surface circulation and the potential colocation of circulation-rich air with strong dynamic upward forcing (and possible enhancement of the dynamic forcing that accompanies the intensifying cyclonic vortex, as is evident in the baseline simulation; Fig. 13c) are extremely sensitive to the details of the buoyancy field, its gradients, and trajectories, all of which are very sensitive to the location and three-dimensional structure of the heat sink. For example, the Sc8m8 simulation has warmer outflow than the Sc8m2 simulation despite the simulations having identical heat sinks. The environmental wind directions through the heat sink differ in the two simulations. Parcel residence times in the heat sink must be longer in the Sc8m2 simulation, but more (not less) near-surface circulation develops in the Sc8m8 simulation. This implies that there is either less solenoidal generation in the Sc8m2 simulation despite the longer residence times of parcels in the heat sink, or that there are differences in the tilting of horizontal vorticity into the vertical associated with the “flattening out” of material circuits as they near the surface.7 Material circuits that converge upon ζmax in the Sc8m2 simulation (not shown) indeed experience less solenoidal generation. Moreover, as evidenced by the Sc4m8 results, if the heat sink is too weak, then regardless of the details of the resultant low-level buoyancy field and trajectories, the development of near-surface circulation is limited (Fig. 24d). It seems plausible that LP supercells are rarely tornadic because their cold pools are too weak and the resultant near-surface circulation is too small.8

The sensitivity of vortex development to the heat sink is one reason we opted for the idealized simulation approach. We can control for the sensitivities to the various characteristics of the heat sink, which cannot be done in a more realistic simulation of a supercell with moist processes. In an actual supercell, the location, size, and amplitude of the “heat sink” (i.e., the distribution of negative buoyancy) is affected by the deep-layer wind shear, storm-relative flow, hydrometeor fall speeds, and hydrometeor species. The effects of hydrometeors on the buoyancy field cannot be expected to be handled well by numerical models. The sensitivity of ζmax to the heat sink properties in the simple simulations herein presents a challenge for initiatives like Warn on Forecast (Stensrud et al. 2009, 2013; Potvin and Wicker 2013), which aspires to use short-term numerical simulations in an operational setting to predict, among other things, the location, magnitude, and timing of low-level vertical vorticity development in observed supercells.

5. Summary and conclusions

Idealized simulations were used to investigate the formation of strong near-surface vortices in supercell-like pseudostorms. Of greatest interest was the sensitivity of the intensification of near-surface rotation to the characteristics of an artificially generated cold pool and the relative roles of baroclinic vorticity generation and environmental wind shear in vortexgenesis.

Even though the environment has large streamwise vorticity, baroclinic vorticity generation and downdrafts are crucial for the development of significant circulation next to the ground in the simulations. Along trajectories bound for ζmax, baroclinic vorticity generation associated with the eastern flank of the pseudostorm’s cold pool rotates the vorticity vector 90°–180° to the left of its orientation in the environment (Fig. 25). Along trajectories bound for ζmin, baroclinic generation encountered on the eastern flank of the cold pool initially rotates the vorticity vector to the left of its orientation in the environment as for the trajectories bound for ζmax, but subsequent baroclinic generation encountered on the western flank of the cold pool rotates the vorticity vector back to the right (north) (Fig. 25d). Moreover, the baroclinicity coupled with descent causes the vorticity vector to become tipped upward (downward) relative to the trajectories bound for ζmax (ζmin).

Fig. 25.
Fig. 25.

Schematic summarizing the simulation outcomes: (a) the baseline simulation in which a strong cyclonic vortex develops (Sc8m8), the simulations in which (b) the heat sink is either too strong (results in colder outflow, i.e., Sc16m8) or (c) the environmental wind shear is too weak (results in a weaker dynamic VPPGF, i.e., Sc8m2), and (d) the simulation in which the heat sink is excessively weak (i.e., Sc4m8). A schematic trajectory bound for ζmin (and evolution of ω along this trajectory) is shown in (d) only (the anticyclonic vortex is dominant in this simulation), but the generalized trajectory and vorticity evolution also applies to trajectories approaching ζmin in (a)–(c).

Citation: Journal of the Atmospheric Sciences 71, 1; 10.1175/JAS-D-13-0159.1

In the case of strong low-level environmental wind shear and a heat sink of intermediate strength (Fig. 25a), such as in the baseline simulation (Sc8m8), significant near-surface cyclonic circulation develops in close proximity to strong dynamic lifting, which is itself attributable to the larger environmental wind shear, mainly by way of large ∂ζ2/∂z. Moreover, the circulation-rich air is associated with relatively small negative buoyancy. The combination of large circulation, relatively small negative buoyancy, and strong dynamic lifting leads to the formation of an intense cyclonic vortex. Anticyclonic near-surface circulation develops on the opposite side of the buoyancy minimum as the cyclonic circulation, but the displacement of the circulation from the strong dynamic uplifting prevents the formation of an intense anticyclonic vortex. The anticyclonic circulation also is weaker than the cyclonic circulation (this is the case for most of the simulations). The outflow from the intermediate heat sink in the baseline (Sc8m8) simulation, though colder than that of the Sc4m8 simulation, would be regarded as relatively weak by most standards, as the potential temperature perturbations within a few kilometers of ζmax are only ~2 K on average, and there was no contribution to negative buoyancy from hydrometeors as there would be in actual storms.

In the case of strong low-level environmental wind shear and an excessively strong heat sink (Fig. 25b), such as in the Sc16m8 simulation, slightly larger near-surface cyclonic circulation develops relative to the baseline simulation, but an intense cyclonic vortex fails to form because the circulation is associated with significant negative buoyancy and is displaced far from the strong dynamic lifting. In other words, cyclonic vorticity develops along trajectories that descend through the heat sink, but weak vertical stretching after trajectories reach their nadirs precludes the formation of an intense cyclonic vortex (the outflow air also fails to be drawn upward as it spreads beneath the overlying updraft). The maximum near-surface anticyclonic vorticity is less than the maximum near-surface cyclonic vorticity owing to the even greater displacement of the anticyclonic circulation from the region of strong dynamic lifting.

In the case of weak low-level environmental wind shear and a heat sink of intermediate strength (Fig. 25c), such as in the Sc8m2 simulation, the near-surface cyclonic circulation is weaker than in the Sc8m8 or Sc16m8 simulations (less solenoidal generation of circulation occurs than in the Sc8m8 or Sc16m8 simulations). Though significant near-surface cyclonic circulation develops not far from the maximum dynamic lifting, the dynamic lifting is weak relative to the simulations in strong-shear environments. Moreover, the near-surface cyclonic circulation is associated with larger negative buoyancy than the circulation in the Sc8m8 simulation. An intense cyclonic vortex fails to develop, and the maximum anticyclonic vorticity is weak as well for the same reasons it is weak in the Sc8m8 and Sc16m8 simulations.

If the low-level environmental wind shear is strong, but the heat sink is weak, such as in the Sc4m8 simulation (Fig. 25d), cyclonic near-surface circulation develops in close proximity to strong dynamic lifting, and the circulation is associated with small negative buoyancy; however, the cyclonic circulation is weak owing to limited positive solenoidal generation. Though an intense cyclonic vortex fails to form, an intense anticyclonic vortex develops on the trailing flank of the pseudostorm’s hook echo, where more significant near-surface anticyclonic circulation develops, and in sufficiently close proximity to the region of strong dynamic lifting.

We believe the simulations provide a plausible explanation for why tornadic supercells are favored in environments that limit cold pool production (i.e., environments that have a high boundary layer relative humidity) and contain large low-level wind shear. Future experiments will investigate the effects of surface drag on the evolution of vorticity along trajectories and development of near-surface circulation. Future work also likely will extend the idealized modeling approach to studies of tornado maintenance.

Acknowledgments

We are grateful for the discussions with our collaborators, especially Drs. George Bryan, Johannes Dahl, Bob Davies-Jones, David Dowell, Ryan Hastings, Karen Kosiba, Mario Majcen, Jim Marquis, Chris Nowotarski, Matt Parker, Erik Rasmussen, Jerry Straka, Lou Wicker, and Josh Wurman. We thank Dr. Johannes Dahl for sharing some of his analysis code, Dr. George Bryan for his ongoing support of CM1, and three anonymous reviewers. Support from NSF Grants ATM-0801035 and AGS-1157646 also is acknowledged.

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1

By intermediate, we are referring to density potential temperature deficits in the near-surface mesocyclone region of, very roughly, only 1–4 K, which most would probably regard as relatively mild outflow.

2

Throughout the paper, near surface refers to the lowest model grid level for horizontal winds.

3

The streamwise (antistreamwise) vorticity is the horizontal vorticity component aligned with (pointed in the opposite direction as) the updraft-relative wind. The crosswise (anticrosswise) vorticity is the horizontal vorticity component 90° to left (right) of the updraft-relative wind.

4

We deliberately refer to this vorticity as initial vorticity rather than environmental vorticity because the initial conditions of these simulations include a vertical vortex that resembles a midlevel mesocyclone.

5

The forward trajectories are computed via the first-order Euler scheme. Though it is not as accurate as higher-order schemes, the large model time step of 1 s results in sufficiently accurate trajectories. Trajectories computed using a time step of 0.1 s are virtually indistinguishable from the 1-s trajectories.

6

In contrast, virtually all trajectories closer to ζmax drop below the lowest scalar for significant periods of time (e.g., the trajectory that passes nearest to ζmax drops below z = 50 m nearly 5 km upstream of ζmax). Though there is justification for extrapolating horizontal velocity components to parcels below the lowest scalar level when the lower boundary is free slip, the extrapolation of vorticity forcings is problematic.

7

The circulation of a material circuit, following the circuit, is unaffected by the reorientation of vortex lines that thread a surface bounded by the circuit. However, the circulation about fixed points in a horizontal plane (such as in Figs. 13a–c, 16a, 19a, and 22a) also depends on the degree to which baroclinically generated horizontal vorticity is tilted into the vertical.

8

Though there is a low-LCL cutoff for tornadoes at approximately 500 m in Craven and Brooks’s (2004) dataset [see Fig. 10.13 in Markowski and Richardson (2010), which is an updated version of Craven and Brooks’s Fig. 12, which shows a scatterplot of tornadic versus nontornadic supercell environments as a function of low-level shear and LCL], we suspect that the tornado-scarce low-LCL regime probably represents a regime in which convective available potential energy (CAPE) is rarely present rather than an LP supercell regime in which cold pools are very weak.

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