Editorial Type:
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Article Type:
Research Article

An Idealized Numerical Simulation Investigation of the Effects of Surface Drag on the Development of Near-Surface Vertical Vorticity in Supercell Thunderstorms

Paul M. Markowski Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania

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Print Publication:
01 Nov 2016
DOI:
https://doi.org/10.1175/JAS-D-16-0150.1
Page(s):
4349–4385
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Abstract

Idealized simulations are used to investigate the contributions of frictionally generated horizontal vorticity to the development of near-surface vertical vorticity in supercell storms. Of interest is the relative importance of barotropic vorticity (vorticity present in the prestorm environment), baroclinic vorticity (vorticity that is principally generated by horizontal buoyancy gradients), and viscous vorticity (vorticity that originates from the subgrid-scale turbulence parameterization, wherein the effects of surface drag reside), all of which can be advected, tilted, and stretched. Equations for the three partial vorticities are integrated in parallel with the model. The partial vorticity calculations are complemented by analyses of circulation following material circuits, which are often able to be carried out further in time because they are less susceptible to explosive error growth.

Near-surface mesocyclones that develop prior to cold-pool formation (this only happens when the environmental vorticity is crosswise near the surface) are dominated by only barotropic vertical vorticity when the lower boundary is free slip, but both barotropic and viscous vertical vorticity when surface drag is included. Baroclinic vertical vorticity grows large once a cold pool is established, regardless of the lower boundary condition and, in fact, dominates at the time the vortices are most intense in all but one simulation (a simulation dominated early by a barotropic mode of vortex genesis that may not be relevant to real convective storms).

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

Abstract

Idealized simulations are used to investigate the contributions of frictionally generated horizontal vorticity to the development of near-surface vertical vorticity in supercell storms. Of interest is the relative importance of barotropic vorticity (vorticity present in the prestorm environment), baroclinic vorticity (vorticity that is principally generated by horizontal buoyancy gradients), and viscous vorticity (vorticity that originates from the subgrid-scale turbulence parameterization, wherein the effects of surface drag reside), all of which can be advected, tilted, and stretched. Equations for the three partial vorticities are integrated in parallel with the model. The partial vorticity calculations are complemented by analyses of circulation following material circuits, which are often able to be carried out further in time because they are less susceptible to explosive error growth.

Near-surface mesocyclones that develop prior to cold-pool formation (this only happens when the environmental vorticity is crosswise near the surface) are dominated by only barotropic vertical vorticity when the lower boundary is free slip, but both barotropic and viscous vertical vorticity when surface drag is included. Baroclinic vertical vorticity grows large once a cold pool is established, regardless of the lower boundary condition and, in fact, dominates at the time the vortices are most intense in all but one simulation (a simulation dominated early by a barotropic mode of vortex genesis that may not be relevant to real convective storms).

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

1. Introduction

a. Simulations of supercell storms employing a free-slip lower boundary condition

Numerical simulations of supercell thunderstorms, the storms responsible for most significant tornadoes, repeatedly have shown that vertical vorticity can develop next to the surface within air parcels that have descended through a downdraft and accompanying baroclinic zone (e.g., Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Adlerman et al. 1999; Davies-Jones et al. 2001; Dahl et al. 2014; Markowski and Richardson 2014, hereafter MR14). A tornado or tornadolike vortex1 can develop if this vertical vorticity is subsequently stretched. Tornadogenesis also has been linked to downdrafts in numerous observational studies [see the review by Markowski (2002)]. Moreover, in the subset of observational studies in which a volume of dual-Doppler observations is available, the configuration of the vortex lines in the tornadic region suggests a strong influence of baroclinity (e.g., Straka et al. 2007; Markowski et al. 2008, 2012a,b; Marquis et al. 2012).

In the above-referenced numerical simulation studies, a free-slip lower boundary condition was used. A free-slip lower boundary condition has been a popular choice for idealized convective storm simulations ever since the pioneering papers on storm modeling by Schlesinger (1975) and Klemp and Wilhelmson (1978). Convective storm simulations commonly are initialized with horizontally homogeneous environments, particularly idealized studies attempting to relate storm characteristics and behavior to a specific storm environment (e.g., Rotunno and Klemp 1982; Weisman and Klemp 1982, 1984; McCaul and Weisman 1996, 2001; McCaul and Cohen 2002; McCaul et al. 2005; Kirkpatrick et al. 2007, 2009, 2011). Such investigations are more difficult if the environment is horizontally heterogeneous and unsteady. The use of a free-slip lower boundary condition, a horizontally homogeneous environment (i.e., there is no large-scale horizontal pressure-gradient force), and neglecting the Coriolis force (another popular choice in supercell studies, owing to the dominance of vertical vorticity production by tilting) allows the environmental wind profile to remain steady over the course of the simulation.

The vertical wind profile used to initialize the horizontally homogeneous environment often is taken from an actual storm environment (in which the Coriolis force and surface drag are present), or at least resembles the wind profile that might be found in an actual storm environment. Thus, though Coriolis-free, free-slip storm simulations neglect the effects of the Coriolis force and drag on internal storm dynamics, these simulations often implicitly include the effects of the Coriolis force and drag on the storm environment.

b. Influences of a nonfree-slip boundary condition on supercell storms

Although the realism of storms simulated using the above approach has been compelling, many of the small-scale details are undoubtedly sensitive to the lower boundary condition. For example, it is well known that surface drag can intensify vertical vortices by inducing radial inflow within the boundary layer, thereby promoting the convergence of angular momentum toward the axis of rotation (e.g., Rotunno 1979; Howells et al. 1988; Lewellen 1993; Davies-Jones 2015): that is, vorticity stretching. However, in recent supercell simulations by Schenkman et al. (2014, hereafter SXH14) and Roberts et al. (2016, hereafter RXSD16), the authors concluded that surface drag was a crucial source of vorticity for tornadogenesis. In other words, drag did not simply promote the stretching of existing vertical vorticity near the surface. Instead, SXH14 and RXSD16 concluded that drag was an important source of near-surface horizontal vorticity, and ultimately, as a result of vorticity tilting, vertical vorticity and angular momentum. [The vorticity tendency from drag in both of these studies is really the vorticity tendency attributable to the curl of the subgrid-scale (SGS) momentum tendency , which, in general, is nonzero within most of a storm even in simulations using a free-slip lower boundary condition. Hereinafter, “viscous effects” (or similar wording) refers to the influence of both a nonfree-slip lower boundary condition, if applicable, and SGS turbulence away from the lower boundary.] Intense vertical vortices like tornadoes are highly helical (i.e., the most intense cyclonic vertical vorticity is within rising air). To contribute to such a helical flow, the horizontal vorticity generated by viscous effects must develop a streamwise component (Davies-Jones 1984), where streamwise refers to the alignment of the horizontal vorticity with the storm-relative wind .

Adlerman and Droegemeier (2002) also explored the effects of surface drag on low-level mesocyclone development in simulated supercells, though all of the drag coefficients were relatively small. They found that low-level mesocyclones were weakened (relative to a free-slip control simulation) for 0.001 (where is the drag coefficient), enhanced for 0.0005, and weakened for 0.000 25. Changes in the drag coefficient also affected the speed at which new low-level mesocyclones developed (i.e., the cycling frequency). It is not known whether the differences in low-level mesocyclone strength were due to differences in the convergence of angular momentum (à la Howells et al. 1988) or differences in angular momentum generation (à la SXH14). Additional supercell simulations have included surface drag, but the contribution of viscous vorticity to the low-level mesocyclone’s rotation was either not the focus (Frame and Markowski 2010, 2013; Nowotarski et al. 2015; Nowotarski and Markowski 2016) or was not found to be significant (Mashiko et al. 2009).2

In two additional recent papers, surface drag also has been concluded to be important for the development of intense surface vortices within mesoscale convective systems (MCSs). Schenkman et al. (2012) found that surface drag promoted the generation of a horizontal rotor within a simulated MCS, and the upward branch of the rotor was found to be responsible for the development of an intense surface vortex by enhancing vertical vorticity stretching. Xu et al. (2015) concluded that surface drag was an important source of circulation in the genesis of intense surface vortices in a simulated bow echo, although the contribution was obtained as a residual. Analysis errors no doubt contributed to part of the residual, though it is not known how much.

Although it is unclear how one would use observations to directly measure the contribution of viscosity to the vorticity of a supercell’s low-level mesocyclone, in their analysis of the 5 June 2009 tornadic supercell intercepted by VORTEX2 (Wurman et al. 2012), Markowski et al. (2012b) found that the diagnosed circulation growth about a material circuit was larger than predicted by Bjerknes’ theorem. Though the available thermodynamic observations were limited and some error sources in dual-Doppler wind retrievals are not easily quantified, the authors acknowledged (p. 2931) that they could not “exclude the possibility that surface drag contributed positively to the circulation tendency.”

c. Unanswered questions

This study further investigates the importance of surface drag to the development of near-surface vertical vorticity in supercell storms. More precisely, this study seeks to compare the viscous vorticity—that is, vorticity that originates from the SGS turbulence parameterization (wherein the effects of surface drag reside)—to the barotopic vorticity and baroclinic vorticity, which are, respectively, the part of the vorticity that behaves as though the flow is inviscid and barotropic (commonly regarded as the vorticity present in the prestorm environment) and vorticity generated by horizontal buoyancy gradients (Davies-Jones 2000; Davies-Jones et al. 2001). All three partial vorticities may be modified by tilting and stretching. As explained in section 1a, baroclinically generated vorticity has been found to be the dominant vorticity source for near-surface mesocyclones in free-slip simulation studies, even though supercell environments are characterized by large barotropic vorticity owing to the strong environmental vertical wind shear (the barotropic vorticity is mostly horizontal).

This paper is motivated by the following outstanding questions:

  • What are the relative contributions of barotropic vorticity, baroclinic vorticity, and viscous vorticity within the near-surface mesocyclone of supercell storms when surface drag is present?

  • How does the environmental vertical wind profile affect the relative contributions?

  • How does the viscous horizontal vorticity acquire a streamwise component so that it can contribute to cyclonic vorticity within a rising airstream?

Numerical models are essential for this investigation, given that the effects of surface drag are generally unobservable (mobile radars used in field projects typically cannot observe winds below 100-m altitude except very near the radar), and there is no way to know from observations alone how a storm would have evolved in the absence of drag. The “toy model” approach of MR14 is employed, in which supercell-like “pseudostorms” are produced via a heat source and heat sink that, respectively, drive an updraft and downdraft (the updraft rotates cyclonically owing to the vertical shear in the environmental wind profile). As discussed in MR14, the simplicity of the simulations makes it considerably easier to isolate key dynamical processes and untangle complicated cause-and-effect relationships. In the MR14 simulations, the development of an intense cyclonic vortex at the lowest model level is the result of circulation-rich, near-surface air being associated with weak negative buoyancy and also experiencing a large upward-directed vertical perturbation pressure-gradient force (VPPGF) owing to its proximity to the midlevel mesocyclone (Fig. 1). The MR14 simulations used a semicircular hodograph with a free-slip lower boundary condition. In the simulations herein, the near-surface vertical wind profile is modified by surface drag.

Fig. 1.
Fig. 1.

Summary of the MR14 simulation with strong low-level environmental shear and a moderately strong heat sink, which resulted in the development of a tornadolike vortex (this was MR14’s Sc8m8 simulation). The Sc8m8 simulation was rerun on the grid used for the simulations in the present paper, which has a finer vertical grid spacing near the surface than the original MR14 simulations (see section 2). (a) Three-dimensional structure of the heat source (red) and heat sink (blue) as viewed from the southeast in the subdomain km and km, within which the horizontal grid spacing is 100 m (adapted from MR14’s Fig. 2). Axis labels indicate model coordinates (km). The heat sink is activated 900 s (15 min) after the heat source is activated. (b) Potential temperature perturbation (shaded) at 10 m, vertical vorticity ζ at 10 m (thin red contours of 0.05, 0.10, 0.15 s−1, etc.), vertical velocity w at z = 402 m (thick black contours of 4, 8, 12 m s−1, etc.), and horizontal wind vectors at 10 m (plotted every second grid point) in the Sc8m8 simulation at the time the vortex is most intense (2600 s) (cf. MR14’s Fig. 5c). (c) Time series of maximum vertical vorticity at 10 m in the Sc8m8 simulation. The black arrow indicates the point in the time series for which the fields in (b) are shown. (d) Schematic summarizing MR14’s simulation Sc8m8. See MR14 for further details.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Additional details of the methodology are explained in section 2. The results are presented in sections 3 and 4, and a discussion follows in section 5. A summary and conclusions are presented in section 6.

2. Methods

a. Numerical model configuration

The numerical modeling approach is almost identical to that of MR14 (see their section 2). Cloud Model, version 1 [CM1; see Bryan and Fritsch (2002) and the appendix of Bryan and Morrison (2012)], release 18, is used for the simulations herein. As was the case in MR14, a fifth-order advection scheme with implicit diffusion is used; no additional artificial diffusion is included. SGS turbulence is parameterized using a turbulent kinetic energy (TKE) scheme similar to that of Deardorff (1980); that is, SGS turbulence is parameterized as it often is in large-eddy simulations (LES).

The domain is 100 km × 100 km × 18 km, with rigid top and bottom boundaries and periodic 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 20 m in the lowest 150 m (the lowest scalar level is at 10 m) to 380 m at the top of the domain, following the same Wilhelmson and Chen function. The increased vertical resolution near the surface, relative to MR14, is used so that the influence of surface drag on the near-surface vertical wind profile is better resolved. The large (small) time step is 1 (0.1) s.

The heat source and heat sink dimensions and locations are the same as in MR14. The heat sink amplitude and initial low-level environmental shear match those in MR14’s Sc8m8 simulation. This is the simulation with strong low-level environmental shear and a moderately strong heat sink, which resulted in the most intense vortex (Fig. 1); that is, the heat sink amplitude is K s−1, and the low-level shear parameter is 8, using the symbology of MR14.

Four simulations are conducted (Table 1). Two of the simulations utilize a free-slip lower boundary condition. The SGS stresses and (where is the SGS stress tensor) need to be specified at the lower boundary [CM1 uses the Arakawa and Lamb (1977) C grid]. The free-slip condition in CM1 is that and ; thus, there is no SGS-related vertical momentum flux divergence at the lowest level at which horizontal velocities are computed (). In the other two simulations, a semislip lower boundary condition is used,3 in which the shear stresses and at the lower boundary are
e1
e2
where and are the filtered (resolved) horizontal winds at height 10 m, which is the first horizontal wind level above the surface, is the friction velocity, and is the drag coefficient. A roughness length of 12 cm is used, which yields 0.008, where k is the von Kármán constant. This roughness length is appropriate for grasslands like those in the central United States. Though a grid aspect ratio near unity is desirable when using LES turbulence closures (e.g., Brasseur and Wei 2010), it is more appropriate to use a large aspect ratio next to the surface (i.e., ) when the SGS stress at the boundary is diagnosed from resolvable-scale variables through surface-exchange coefficients (Wyngaard et al. 1998). In the simulations herein, within the central 20 km × 20 km fine mesh, the grid aspect ratio is 10 at z = 0 and 0.9–2.6 for z = 200–1000 m.
Table 1.

Summary of the numerical simulations and key characteristics: bottom boundary condition (BC), orientation of near-surface , , 0–1-km SRH, and near-surface . The parameters characterize the pseudostorm environment at the time the heat source is activated.

Table 1.

b. Pseudostorm environments

The environment is horizontally homogeneous; that is, there is no large-scale horizontal pressure-gradient force. Moreover, the Coriolis force is excluded. Thus, once each of the aforementioned simulations with surface drag commences, the near-surface portion of the hodograph evolves owing to a force imbalance. The environmental hodograph would evolve forever without intervention. Instead, after 1 h elapses, an additional term is added to the horizontal equations for u and υ:
e3
e4
Here (, ) is the horizontal average of (u, υ), (, ) is a reference wind profile, and is a time constant. Gibbs et al. (2011) used a similar approach and referred to these terms as “force–restore” terms. In the present work, these terms crudely mimic the influence of a large-scale horizontal pressure-gradient force and Coriolis force. The artificial forcing minimizes the evolution of the base state far away from the pseudostorms without applying so much nudging that the simulations have no hope of evolving naturally within the pseudostorms themselves. Setting (, ) to be the wind profile at 1 h and setting 1800 s yields a steady wind profile by 2 h, at which time the heat source is activated. As in MR14, the heat sink is activated 15 min after the heat source.

The two simulations with surface drag use different ground-relative wind profiles and heat source/sink motions. In one simulation, a stationary heat source (centered on the origin) and sink (centered 4 km north and 2 km east of the origin) are used, and the initial semicircular hodograph is centered on the origin, as in MR14 (Fig. 2b). The steady-state hodograph (i.e., the hodograph that is obtained after 2 h of evolution) is characterized by near-surface crosswise vorticity. Hereinafter, this simulation is referred to as the DRAG-CROSSWISE simulation. In a second simulation with surface drag, the heat source and sink move northeastward at a velocity of (6.0, 5.3) m s−1, and the hodograph is shifted accordingly (Fig. 2c).4 After 2 h of evolution, the steady-state hodograph is characterized by near-surface streamwise vorticity. Hereinafter, this simulation is referred to as the DRAG-STREAMWISE simulation. The components of the pseudostorm motion needed to produce near-surface streamwise vorticity were determined via trial and error. The pseudostorm motion also was specified so that the ground-relative wind speed at the lowest grid level (and therefore surface shear stresses and ) would be identical in the DRAG-CROSSWISE and DRAG-STREAMWISE simulations (Table 1).

Fig. 2.
Fig. 2.

(a) Vertical profile of . (b) The gray hodograph depicts the initial environmental vertical wind profile in the CROSSWISE experiments (and also in the MR14 Sc8m8 simulation). The red hodograph depicts the steady-state environmental vertical wind profile after 2 h. Both the FREESLIP-CROSSWISE and DRAG-CROSSWISE pseudostorm simulations are run with the red environmental hodograph. Labels along the hodographs are select altitudes (km), with the exception of the lowest marker (altitude in m). The pseudostorm motion is indicated with the black dot, and vectors at 50 m are indicated with the green and blue vectors, respectively, and SRH in the 0–1-km layer also is indicated for each hodograph. (c) As in (b), but for the STREAMWISE experiments.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

The pair of DRAG simulations is similar to RXSD16’s “full-wind friction” (FWFRIC) simulations. RXSD16 started with a hodograph obtained from the environment of the 3 May 1999 Moore, Oklahoma, tornadic supercell. They ran the model for 48 h before triggering a storm, during which time surface drag acted on the total wind and the Coriolis force acted on the wind perturbation (i.e., the departure of the wind from the initial hodograph). (If the initial hodograph can be regarded as being in geostrophic balance, then the effects of a horizontal pressure-gradient force can be included implicitly by applying the Coriolis force only to the wind perturbation.) The resulting modified wind profile was quasi steady, with a three-way force balance between the surface drag, horizontal pressure-gradient force, and Coriolis force. A storm was then triggered with a warm bubble. This methodology was not appropriate for the present study, intended to be an extension of MR14, because the MR14 wind profile could not be regarded as being in geostrophic balance [geostrophic wind hodographs tend to be much straighter (Banacos and Bluestein 2004)]. The semicircular hodograph used by MR14 might best be viewed as an idealization of a hodograph that already has been influenced to some extent by surface drag. The drag-influenced hodographs used herein, however, have more realistic shapes in the lowest several hundred meters.

In the remaining two simulations, the simulations are initialized with the steady-state hodographs obtained at 2 h in the aforementioned simulations with surface drag, but the lower boundary condition is free slip. These two simulations are referred to as FREESLIP-CROSSWISE and FREESLIP-STREAMWISE. Their purpose is to expose whether differences between the DRAG simulations and MR14’s Sc8m8 simulation are due to differences in the lower boundary condition itself as it acts on the pseudostorm, or to changes to the shape of the environmental hodograph brought about by surface drag acting on the base-state wind profile. For the sake of consistency, the force-restore forcing also is included in the FREESLIP simulations. The FREESLIP simulations are similar to RXSD16’s EnvFRIC simulation, in which surface drag operates only on the environmental wind components.

c. Partial vorticity calculations

Throughout the simulations, equations for the barotropic, baroclinic, and viscous vorticity (, , and , respectively) are integrated in parallel, as in Epifanio and Durran (2002, hereafter ED2002), where
e5
e6
e7
The components of each vorticity vector are (ξ, η, ζ), the three-dimensional velocity vector is (u, υ, w), is the specific heat at constant pressure, θ is potential temperature, π is the Exner function, and the SGS momentum tendency is , where and is the SGS eddy viscosity, which is parameterized from the SGS TKE. To a good approximation, , where B is the buoyancy [because these are dry simulations, , where ]. The force-restore terms described in section 2b, which are a small source of horizontal vorticity, are excluded from both the partial vorticity calculations and partial circulation calculations (to be described in section 3d). Their influence on the calculations is dwarfed by the evolution of the barotropic, baroclinic, and viscous vorticity for the time scales analyzed herein.

At the time the heat source is activated ( 0 in the FREESLIP simulations and 2 h in the DRAG simulations), the barotropic vorticity is set to the environmental vorticity: that is, (, , ) (ξ, η, 0). (Given the horizontally homogeneous environment, 0 before each pseudostorm is triggered.) The baroclinic and viscous vorticity both are defined to be zero at this time: that is, (, , ) and (, , ) .

The partial vorticity components are solved at the thermodynamic grid points every large time step via a two-step, third-order, Adams–Bashforth–Moulton technique (Wicker 2009).5 The partial vorticity integrations are decoupled from the integration of CM1’s prognostic equations (i.e., the simulations evolve independently of the partial vorticity calculations). Errors owing to interpolation are unavoidable, especially in the tilting and stretching calculations, which require more interpolation than advection calculations on a C grid. These errors can be quantified by comparing the vorticity computed from the predicted model velocity fields with the summed barotropic, baroclinic, and viscous partial vorticities. A sixth-order computational diffusion term is added to (5)–(7) to suppress the growth of these errors and keep the solution stable. Without it, explosive error growth occurs owing to nonlinear computational instability [i.e., instability not related to the time step size (Phillips 1959)] within 600–900 s of integration.

Even with the artificial diffusion, errors still eventually grow large as vorticity intensifies in the pseudostorm simulations. To mitigate this error growth, additional smoothing is applied via a Shapiro (1975) digital filter (Shapiro’s filter is used) once a partial vertical vorticity component exceeds 0.2 s−1 (this threshold was identified via trial and error). The filter has a response of 0.00, 0.44, 0.75, and 0.94 for wavelengths of 2, 3, 4, and 6, respectively. This filter extends the usefulness of the prognosed partial vorticity fields to roughly 1500–2500 s of simulation time (it varies by experiment). It is speculated that ED2002 did not encounter this problem because the flow they studied (stratified flow over an isolated hill) possessed weaker velocity gradients and was much more steady than the wind field of a convective storm.

Dahl et al. (2014) obtained the barotropic vorticity via a Lagrangian technique in which material fluid volume elements were tracked so that the rearrangement of ambient vortex-line segments could be analyzed. The residual between the vorticity field obtained from the velocity fields predicted by the model and the diagnosed barotropic vorticity was regarded as the nonbarotropic vorticity (alternatively, storm-generated vorticity). The approach used herein has the advantages of not relying on residuals and separately providing both the baroclinic and viscous partial vorticity.

d. Lagrangian circulation analyses

A second tool for quantifying the contributions of barotropic, baroclinic, and viscous vorticity to the near-surface mesocyclones is the analysis of circulation about material circuits (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; MR14). Although these Lagrangian circulation analyses tell a similar tale as the partial vorticity decompositions, the circulation analyses frequently maintain their reliability beyond the time when errors in the partial vorticity fields become unacceptably large, at least if two conditions are met. The first is that the circuits, where introduced, should be broader than the vortices they surround (herein, 1-km-radius rings of air parcels that surround the ζ maxima are introduced) so that the air parcels comprising the circuits avoid extreme accelerations. The second condition is that the circuit must be well represented at all times: that is, adjacent parcels within the circuit must not be allowed to drift too far apart lest the numerical calculation of the circulation , where is a vector line element of the circuit, becomes error prone. Parcels unavoidably diverge from one another as they are tracked backward in time (the vortices encircled by the circuits form within strongly confluent horizontal flow). The extended reliability of the circulation analyses relative to the partial vorticity calculations is probably due to the absence of volatile vorticity stretching terms.

The backward trajectories of the air parcels comprising the material circuits are computed using velocity data saved every 2 s. A fourth-order Runge–Kutta scheme is used with a time step of 1 s. Additional parcels are inserted each time step, as needed, in order to maintain <50 m of separation between adjacent parcels.6 In the FREESLIP simulations, second-order extrapolation is used to assign horizontal wind components to parcels that pass below the lowest scalar level. In the DRAG simulations, parcels that pass below the lowest scalar level are assigned horizontal velocities consistent with the semislip lower boundary condition (i.e., a log wind profile is assumed from to 10 m).

To a good approximation, the circulation about each material circuit is governed by
e8
and the circulation at any time is
e9
where is the circulation at 0, and , , and are referred to as the barotropic, baroclinic, and viscous circulation, respectively.

3. Pseudostorm simulations with near-surface crosswise vorticity (FREESLIP-CROSSWISE and DRAG-CROSSWISE)

a. Overview

Significant near-surface cyclonic vorticity develops early in the FREESLIP-CROSSWISE simulation: that is, before the arrival of cool outflow from the heat sink. By 900 s (recall that the heat sink is only activated at this time), 10-m ζ exceeds “mesocyclone strength” (0.01 s−1) (Figs. 3a,e). The cyclonic vortex rapidly intensifies after 1150 s (Figs. 3b–e), once cool air from the heat sink reaches it; the maximum 10-m ζ (0.57 s−1) is attained at 2550 s (Fig. 3d). The vortex gradually weakens thereafter as cold air shunts it southward (northerly winds in the outflow exceed 20 m s−1 at 10 m), away from the updraft forcing associated with the elevated heat source (note that the vortex is 3 km farther south in Fig. 3d than in Fig. 3b).

Fig. 3.
Fig. 3.

(a)–(d) Evolution of the , ζ, and horizontal wind fields at 10 m, and w at z = 402 m, at select times in the FREESLIP-CROSSWISE simulation. The field is shaded (see legend). Isovorts of cyclonic ζ are drawn using thin red contours every 0.02 s−1 for 0.01 s−1 (i.e., 0.01, 0.03, 0.05 s−1, etc.). Isotachs of w are drawn using black contours every 4 m s−1 for 4 m s−1 (i.e., 4, 8, 12 m s−1, etc.). Horizontal storm-relative wind vectors are plotted at every second grid point. The rings in (a) and (d) indicate the starting positions of the material circuits analyzed in Fig. 12. (e) Time series of at 10 m in the FREESLIP-CROSSWISE simulation (black). The markers (a)–(d) indicate the times shown in (a)–(d). The time series of in MR14’s Sc8m8 simulation (free-slip lower boundary condition, semicircular hodograph) shown in Fig. 1c is shown in the background (cyan).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Even stronger near-surface cyclonic vorticity develops prior to cold-pool development in the DRAG-CROSSWISE simulation. By 900 s (from now on, times given in the description of the DRAG simulations refer to the time elapsed since the activation of the heat source at 2 h), 10-m ζ reaches 0.20 s−1 (Figs. 4a,e). The vortex reaches peak intensity at 1400 s, shortly after the arrival of cool outflow from the heat sink (Fig. 4b), at which time 10-m ζ is 0.31 s−1. The vortex is engulfed in outflow by 2100 s (Fig. 4c), though the vortex is not shunted southward as quickly as in the FREESLIP-CROSSWISE simulation. Despite an identical heat sink amplitude and heat sink–relative flow as in the FREESLIP-CROSSWISE simulation, the outflow is not as strong in the DRAG-CROSSWISE simulation, probably owing to stronger near-surface turbulent mixing within the outflow (cf. Figs. 3c,d and 4c,d). In spite of being closer to the elevated updraft forcing and in less negatively buoyant outflow than in the FREESLIP-CROSSWISE simulation, the vortex in the DRAG-CROSSWISE simulation gradually weakens after 1400 s and has a peak intensity much less than that at 2550 s in the FREESLIP-CROSSWISE simulation (cf. Figs. 3d and 4d).

Fig. 4.
Fig. 4.

As in Fig. 3, but for the DRAG-CROSSWISE simulation. (a),(b) Rings indicate the starting positions of the material circuits analyzed in Fig. 13. (e) The time series of are for the DRAG-CROSSWISE simulation (black) and MR14’s Sc8m8 simulation (free-slip lower boundary condition, semicircular hodograph; cyan).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

b. Origins of near-surface cyclonic vorticity prior to cold-pool development

The early development of near-surface cyclonic vorticity in each of the CROSSWISE simulations is the result of a downward displacement of vortex lines by dry, dynamically driven descent, which is strongest on the northern and northeastern flanks of the updraft (Fig. 5). In other words, vortex formation is via the so-called barotropic mechanism [see Fig. 16b of Markowski et al. (2008) and Fig. 2e of Markowski and Richardson (2009)]. Markowski et al. (2003a), Davies-Jones (2008), and Parker (2012) have demonstrated this mechanism in axisymmetric models, though only in Parker’s simulation was the descent forced without imposing negative buoyancy. The parcels entering the vortex experience relatively shallow downward displacements of generally less than 100 m (Fig. 6), in what might best be regarded as compensating subsidence.7 This appears to be the same mechanism by which early vortex formation begins in the idealized supercell simulation of RXSD16 and is probably a three-dimensional version of the mechanism at work in Parker’s (2012) simulations.

Fig. 5.
Fig. 5.

Vortex lines at (a) 300, (b) 600, and (c) 900 s in the FREESLIP-CROSSWISE simulation. The 0.5 m s−1 (gray), 5 m s−1 (red), and 0.02 s−1 (yellow) isosurfaces are also shown. The view is from the northwest. The vortex lines are at z =100 m in the far field north of the pseudostorm. The broad gray arrow overlaid in (c) locates the subsiding airstream, which, in conjunction with the updraft, led to the development of significant near-surface vertical vorticity on the updraft’s north flank early in the evolution of the pseudostorm. The evolution of the drafts and vortex lines in the DRAG-CROSSWISE simulation is qualitatively similar.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Fig. 6.
Fig. 6.

Horizontal cross section of pressure perturbation (see legend) at 10 m at 600 s in the FREESLIP-CROSSWISE simulation. Vertical velocity w isotachs of 2, 4, 6, and 8 m s−1 at 1.03 km (white contours) are overlaid. Twenty trajectories computed backward in time from 900 s to the start of the simulation also are overlaid (black). At 900 s, the air parcels are all at 30 m and compose a 1-km-radius ring surrounding the developing near-surface cyclonic vorticity maximum. The altitudes of the air parcels at the start of the simulation (m) are shown beside each parcel’s position at 0 s (black dots).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

The vortex lines that are tilted in the FREESLIP-CROSSWISE simulation are environmental vortex lines: that is, vortex lines associated with the vertical wind shear (horizontal vorticity) present in the initial vertical wind profile. According to Helmholtz’s vorticity theorem, vortex lines are material lines in the barotropic, inviscid limit. Thus, to a good approximation, the vortex lines shown in Fig. 5 move as material lines, especially in the first 600 s of the simulation (Figs. 5a,b), when the vortex lines shown are mostly below the altitude of the elevated heat source (baroclinity is present on the periphery of the heat source) and the flow is still highly laminar.

Not surprisingly, the ζ decomposition (Fig. 7) indicates a dominance of ; is negligible,8 and is negative (though small) owing to the effects of SGS turbulent diffusion. [Note that the sum of the partial vorticities, (Fig. 7b), is in decent agreement with the ζ obtained directly from the model’s velocity fields (Fig. 7a); partial vorticities are shown at 30 m rather than 10 m because of larger partial vorticity errors at the first interior grid point ( 10 m).] What is perhaps unexpected, however, is that the ζ maximum (and therefore maximum) very nearly coincides with the w maximum. This is the result of tilting streamwise rather than crosswise horizontal vorticity (Davies-Jones 1984). Streamwise characterizes the trajectories entering the vortex from the southwest through northwest through northeast, the sector through which most of the air enters the vortex (Fig. 8; note especially the heavy dashed line, northwest of which a large streamwise component is present). The horizontal vorticity is dominated by ; is negligible, and is generally antistreamwise and small.

Fig. 7.
Fig. 7.

Horizontal cross sections of (a) ζ obtained from the model’s velocity fields, (b) the sum of the partial vertical vorticities (), (c) barotropic vertical vorticity , (d) baroclinic vertical vorticity , and (e) viscous vertical vorticity at 30 m in the FREESLIP-CROSSWISE simulation at 900 s (black contours). The contour interval is 0.01 s−1; negative isopleths are dashed. The contours are overlaid on the w field at 30 m (color shading).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Fig. 8.
Fig. 8.

Horizontal cross sections of the magnitudes of the (a) total horizontal vorticity obtained from the model’s velocity fields, (b) barotropic horizontal vorticity , (c) baroclinic horizontal vorticity , and (d) viscous horizontal vorticity at 30 m in the FREESLIP-CROSSWISE simulation at 900 s (color shading). Horizontal vorticity vectors (white) and horizontal storm-relative velocity vectors (black) also are overlaid (see legend). The 0.01- and 0.05-s−1 isopleths are overlaid (thin white contours) in order to facilitate comparisons with Fig. 7. The white dashed lines enclose regions (labeled with an “S”) in which the streamwise component of the horizontal vorticity (either the total horizontal vorticity or partial horizontal vorticities) exceeds 0.04 s−1 (i.e., where a significant streamwise component exists).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Although near-surface is crosswise in the environment (Fig. 2b), becomes highly streamwise along trajectories approaching the near-surface ζ maximum owing to crosswise-to-streamwise exchange, also known as the “riverbend effect” (Scorer 1978, p. 88 ff.; Davies-Jones et al. 2001). The leftward bending of trajectories bound for the near-surface cyclonic vortex is apparent in Fig. 6, and it is this bending that is associated with large conversion of crosswise to streamwise . The pressure field in Fig. 6 is ultimately responsible for the bending trajectories. The precold-pool pressure field closely resembles the pressure field in a mature supercell with a substantial cold pool (see Fig. 2 in Markowski et al. 2012b). The implication is that the pressure field resulting from just the interaction between an updraft and environmental shear can promote crosswise-to-streamwise exchange. A cold pool apparently is not critical, though most mature supercells likely possess a horizontal precipitation distribution (which ultimately is dictated by the wind field through which hydrometeors grow and fall) that favors a similar pressure field. Crosswise-to-streamwise exchange also has been found to be significant in several (free-slip) simulations of supercells in which baroclinically generated horizontal vorticity was initially crosswise but subsequently converted to streamwise vorticity as parcels that were initially traveling westward through the supercell’s forward-flank precipitation region turned southward/leftward as they approached the low-level mesocyclone (Adlerman et al. 1999; MR14; Dahl et al. 2014).

Although the fundamental means of precold-pool ζ development is similar in the DRAG-CROSSWISE simulation (i.e., the trajectories, pressure field, development of vertical drafts, and evolution of the vortex lines are qualitatively similar to the FREESLIP-CROSSWISE simulation), in contrast to the FREESLIP-CROSSWISE simulation, both and are large and positive during the precold-pool ζ development (Figs. 9c,e). The maximum, which is roughly twice as strong as the maximum (the total ζ is roughly 3 times stronger at 900 s in the DRAG-CROSSWISE simulation than in the FREESLIP-CROSSWISE simulation), resides on the eastern flank of the low-level updraft (Fig. 9e). Anticyclonic is present on the western through northern flank of the low-level updraft.

Fig. 9.
Fig. 9.

As in Fig. 7, but for the DRAG-CROSSWISE simulation at 900 s. The contour interval is the same as in Fig. 7 (i.e., 0.01 s−1, and negative isopleths are dashed).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

As is the case in the FREESLIP-CROSSWISE simulation, the vortex in the DRAG-CROSSWISE simulation ingests considerable streamwise despite near-surface crosswise in the far-field environment (Fig. 10b). Of greater interest is how large cyclonic develops within updraft (albeit, on the eastern flank of the updraft) in the DRAG-CROSSWISE simulation. As in the FREESLIP-CROSSWISE simulation (Fig. 8d), is generally antistreamwise (Fig. 10d). However, in two narrow swaths on the southern and southeastern flanks of the low-level updraft, acquires a large (≥0.04 s−1) streamwise component. Trajectories bound for the eastern flank of the updraft and vortex experience the largest amount of leftward bending (cf. the yellow and red trajectories in Fig. 11) and crosswise-to-streamwise exchange. The largest generation of occurs on the western flank of the vortex, but this generation is antistreamwise (Fig. 11) and is the result of large turbulent diffusion, which weakens vorticity extrema ( is a maximum here; Fig. 10a). Streamwise arises, instead, from the combination of approximately crosswise generation on the southern flank of the updraft [a relative maximum is at (x, y) (0.8, 0.5) in Fig. 11] and simultaneous crosswise-to-streamwise conversion as trajectories bend leftward into the eastern flank of the updraft (note the black contours in Fig. 11). Cyclonic arises via tilting where the trajectory enters the eastern flank of the updraft. This appears to be the same mechanism by which parcels acquire streamwise vorticity in the simulation of SXH14, though crosswise and streamwise are defined with respect to the ground-relative wind in their analysis, and is unavailable. This also seems to be what is described by Rotunno (1980) in his discussion of the vorticity dynamics of a convective swirling boundary layer (see section 3a of that paper).

Fig. 10.
Fig. 10.

As in Fig. 8, but for the DRAG-CROSSWISE simulation at 900 s. The 0.01- and 0.05-s−1 isopleths are overlaid (thin white contours) in order to facilitate comparisons with Fig. 9.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Fig. 11.
Fig. 11.

Magnitude of horizontal vorticity generation by viscosity (; shaded) at 30 m at 900 s in the DRAG-CROSSWISE simulation. Generation vectors (white; see legend) are overlaid at every other grid point in regions where their magnitude exceeds approximately 2 × 10−4 s−2. The 0.01- and 0.05-s−1 isopleths are overlaid (white contours) in order to facilitate comparisons with Figs. 9 and 10. Regions of large crosswise-to-streamwise exchange of viscous horizontal vorticity are indicated with black contours, which enclose exchange rates of 6, 12, 18, and 24 × 10−4 s−2 (cf. with the region enclosed by the heavy dashed line in Fig. 10d, which is where the viscous horizontal vorticity has a significant streamwise component). The crosswise-to-streamwise exchange is , where is the crosswise vorticity component and ψ is the wind direction. The yellow trajectory enters the maximum; the red trajectory enters the maximum (see Fig. 9).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Most of the trajectories that enter the near-surface vortex (approximately 70% of the region in which > 0.01 s−1 is on the antistreamwise side of the heavy dashed line in Fig. 10d) possess antistreamwise that contributes to anticyclonic (Fig. 9e); the relative few that enter the eastern flank of the vortex with streamwise are much more strongly vertically stretched owing to their closer proximity to the w maximum and thus contribute to very large cyclonic (>0.1 s−1). The potential exists for a positive feedback, in which the stronger a vortex becomes, the more trajectories are bent cyclonically, thus increasing the possibility that can develop a streamwise component and further intensify a cyclonic vortex once the streamwise is tilted into the vertical.

Circulation and its forcings are evaluated about material circuits that reach the cyclonic vorticity maxima at 900 s in both the FREESLIP-CROSSWISE (Figs. 12a,b) and DRAG-CROSSWISE (Figs. 13a,b) simulations. At 900 s, the circuits are 1-km-radius rings centered on the cyclonic vorticity maxima at 30 m (the locations of the circuits are shown in Figs. 3a and 4a). The parcels that comprise the circuits are followed backward in time to the start of the simulations following the methodology described in section 2d.

Fig. 12.
Fig. 12.

(a),(c) Total circulation C (black) and partial circulations , , and (green, blue, and red, respectively) about material circuits followed backward in time to 0 s from 900 and 2550 s, respectively, in the FREESLIP-CROSSWISE simulation. The black dashed curves represent the total circulations obtained from the sums of the diagnosed partial circulations [in (a), the dashed black curve lies nearly on top of the solid black curve]. In (c), the gray dashed curve is the diagnosed circulation if the effects of implicit diffusion are included. The positions of the circuits at 900 and 2550 s are indicated in Figs. 3a and 3d (the circuits are centered at 30 m at these times). The colored stars along the right margins of (a) and (c) indicate the values of the respective area averages of partial ζ at 900 and 2550 s times , where r (=1 km) is the radius of the material circuit at 900 and 2550 s, respectively. For example, the green stars represent . (b),(d) Forcings for the baroclinic circulation (blue) and viscous circulation (red). Note that the scale of (d) differs from that of (b).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Fig. 13.
Fig. 13.

As in Fig. 12, but for the DRAG-CROSSWISE simulation. The circuits are followed backward in time to 0 s from 900 and 1400 s, respectively. The positions of the circuits at those times are indicated in Figs. 4a and 4b (the circuits are at 30 m).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Circulation is approximately conserved following the material circuit from 0 to 900 s in the FREESLIP-CROSSWISE simulation (Fig. 12a), though a small loss to viscosity is evident (Fig. 12b). In the DRAG-CROSSWISE simulation, half of the total circulation comes from viscous generation (Figs. 13a,b). The growth of is largest when the circuit has a significant vertical projection (not shown): that is, when part of the circuit is next to the surface and part is well above the surface (this is true at other times as well and also in the DRAG-STREAMWISE simulation). Interestingly, the circuit analyzed in the FREESLIP-CROSSWISE simulation possesses approximately 50% larger than the circuit analyzed in the DRAG-CROSSWISE simulation (cf. Figs. 12a and 13a). Apparently there are significant differences in the degree to which environmental vortex lines thread the surfaces defined by the circuits, which inevitably develop complex geometries after being tracked backward for 900 s.

For both circulation analyses, there is excellent agreement between C and . Moreover, each partial C at 900 s nearly matches times its corresponding partial ζ averaged over the circular area spanned by the circuit at that time, where 1 km is the radius of the circuit when it encircles the ζ maximum. Note the colored stars in Figs. 12a and 13a. From Stokes’ theorem, the mean partial ζ should equal the partial C divided by the area spanned by the circuit. Thus, in the case of perfect partial C and partial ζ calculations, the colored stars should match their respective partial C values at 900 s. Differences are the result of both partial C and partial ζ errors.

c. Vortex maintenance once cool outflow develops in the pseudostorms

As summarized in section 3a, low-level rotation (both ζ and C) continues intensifying in the FREESLIP-CROSSWISE simulation as cool outflow is entrained into the low-level mesocyclone (Figs. 3d,e). Although the agreement between ζ and is not as good at 2550 s (the time of maximum 10-m ζ) as at 900 s, the agreement is arguably good enough for some qualitative inferences to be made. At 2550 s, dominates the total ζ within the vortex (Fig. 14d), being roughly 300% larger than (Fig. 14c). Consistent with the dominance of , the total C is dominated by in the complementary Lagrangian circulation analysis (Fig. 12c). Moreover, much larger total C is acquired by the circuit tracked from 0 to 2550 s than for the circuit tracked from 0 to 900 s (cf. Figs. 12a and 12c). Regarding the viscous effects, both and indicate that turbulent diffusion is limiting the intensity of the vortex; is negative within the vortex (Fig. 14e), and negative is diagnosed about the material circuit (Fig. 12c). [The agreement between C and is not as good in Fig. 12c as for the other Lagrangian circulation analyses. The relatively large discrepancy appears to be attributable to the effects of artificial, implicit diffusion resulting from the use of a fifth-order advection scheme (the implicit diffusion has a greater influence on this circuit than on the other circuits analyzed in the paper). When the effects of this diffusion are included, there is much better agreement between the measured and diagnosed C tendencies (see gray dashed line in Fig. 12c).]

Fig. 14.
Fig. 14.

As in Fig. 7, but at 2550 s. The contour interval is 0.03 s−1, and negative isopleths are dashed.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

In the DRAG-CROSSWISE simulation, at 1400 s (the time of maximum 10-m ζ), and ( and ) continue to dominate the total ζ (C) (Figs. 13c and 15); C is twice as large as at 900 s (Fig. 13c). The maximum remains situated on the eastern flank of the ζ maximum, where crosswise-to-streamwise exchange is maximized (not shown, but qualitatively similar to Fig. 11). Baroclinically generated vorticity gradually gains importance as the simulation progresses. The partial ζ fields at 2100 s reveal a continued importance of but a growing importance of and declining importance of (Fig. 16). Unfortunately, both the partial ζ calculations and Lagrangian circulation analyses in the DRAG-CROSSWISE simulation have large errors at later times (not shown). It is tempting to speculate that baroclinic vorticity might continue to gradually grow in importance beyond 2100 s, but the issue is also somewhat moot given that the vortex is steadily weakening following its peak intensity at 1400 s.

Fig. 15.
Fig. 15.

As in Fig. 7, but for the DRAG-CROSSWISE simulation at 1400 s. The contour interval is 0.02 s−1, and negative isopleths are dashed.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Fig. 16.
Fig. 16.

As in Fig. 7, but for the DRAG-CROSSWISE simulation at 2100 s. The contour interval is 0.02 s−1, and negative isopleths are dashed.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

d. Comparison to the MR14 simulation

MR14’s Sc8m8 simulation was rerun on the grid used for the FREESLIP-CROSSWISE and DRAG-CROSSWISE simulations. The evolution is not significantly different from that in the original MR14 Sc8m8 simulation (cf. Fig. 1 and MR14’s Fig. 5). The time series of 10-m ζ in the MR14 simulation is overlaid in Figs. 3e and 4e. The early development of ζ is the most obvious difference between the two CROSSWISE simulations and the MR14 simulation. The early ζ development is stronger in the DRAG-CROSSWISE simulation than in the FREESLIP-CROSSWISE simulation, but early development is present in both CROSSWISE simulations, unlike the MR14 simulation, in which the near-surface is streamwise. In the MR14 simulation, near-surface ζ does not develop until a cold pool is well established (i.e., after 1800 s). The material circuit analysis of MR14 indicates that the near-surface vertical vorticity maximum develops baroclinically in their simulation (see their Fig. 10).

The evolution of the FREESLIP-CROSSWISE simulation from 1800 to 3600 s shares some similarities with the MR14 simulation (Fig. 3e) in the sense that the 10-m ζ also reaches its maximum only after the cold pool is well established and that baroclinic vorticity dominates the near-surface mesocyclone at this time. Although the maximum 10-m ζ in the FREESLIP-CROSSWISE simulation (0.57 s−1 at 2550 s) is nearly twice as strong as in the DRAG-CROSSWISE simulation (0.34 s−1 at 1400 s), it is considerably weaker than in the MR14 simulation (0.85 s−1 at 2600 s). Two likely explanations for the weaker vortex in the FREESLIP-CROSSWISE simulation relative to the MR14 vortex are the greater degree of undercutting of the elevated updraft forcing by the cool outflow in the FREESLIP-CROSSWISE simulation (note the differences in the 10-m fields displayed in Figs. 1b and 3d; in the latter figure, the outflow boundary has pushed south of the southern boundary of the plotting region) and the lesser circulation (about a vertical axis) that develops in the FREESLIP-CROSSWISE simulation relative to the MR14 simulation (not shown). The cool outflow spreads south faster in the FREESLIP-CROSSWISE simulation than in the MR14 simulation presumably because the outflow is colder in the former simulation (the DRAG-CROSSWISE outflow also is colder, overall, than the MR14 outflow, though one must look farther north than the regions shown in Figs. 4c,d). Although the heat sinks are identical, the heat sink–relative flow is weaker in both CROSSWISE simulations than in the MR14 simulation; thus, parcels experience greater cooling in the CROSSWISE simulations than in the MR14 simulation (note the differences between the gray and red hodographs in Fig. 2b).

There are likely additional factors responsible for the differences in the evolution of the cold pools and ζ development between the MR14 and CROSSWISE simulations, such as differences in storm-relative helicity (SRH), dynamic VPPGF, and low-level updraft strength (low-level w differences are obvious in comparing Figs. 1b, 3, and 4). These are beyond the scope of the present paper but are worthy of investigation in a future study (additional discussion on this topic appears in section 5).

4. Pseudostorm simulations with near-surface streamwise vorticity (FREESLIP-STREAMWISE and DRAG-STREAMWISE)

a. Overview

In contrast to the CROSSWISE simulations, near-surface cyclonic vortices do not develop in the STREAMWISE simulations until well after the heat sink is activated (Figs. 17 and 18). In this regard, both STREAMWISE simulations are similar to the MR14 simulation, although the maximum 10-m ζ in the MR14 simulation (0.85 s−1; Fig. 1c) is considerably greater than the maximum in either STREAMWISE simulation. In the FREESLIP-STREAMWISE simulation, 10-m ζ reaches a maximum of 0.38 s−1 at 2410 s (Figs. 17d,e), which is approximately 30% less than the maximum in the FREESLIP-CROSSWISE simulation. In the DRAG-STREAMWISE simulation, 10-m ζ also reaches a maximum of 0.38 s−1, which is similar to the maximum in the DRAG-CROSSWISE simulation, though it occurs later in the DRAG-STREAMWISE simulation at 3300 s (Figs. 18d,e).

Fig. 17.
Fig. 17.

As in Fig. 3, but for the FREESLIP-STREAMWISE simulation. (a)–(c) The dashed red isovorts are 0.0025, 0.005, and 0.0075 s−1. (d) The ring indicates the starting position of the material circuit analyzed in Fig. 20. (e) The time series of are for the FREESLIP-STREAMWISE simulation (black), the Sc8m8 simulation (free-slip lower boundary condition, semicircular hodograph; cyan), and the FREESLIP-CROSSWISE simulation (gray; only for the first ~2100 s, for the sake of clarity).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Fig. 18.
Fig. 18.

As in Fig. 3, but for the DRAG-STREAMWISE simulation. (a) The dashed red isovorts are 0.0025, 0.005, and 0.0075 s−1. (d) The ring indicates the starting position of the material circuit analyzed in Fig. 23. (e) The time series of are for the DRAG-STREAMWISE simulation (black), the Sc8m8 simulation (free-slip lower boundary condition, semicircular hodograph; cyan), and the DRAG-CROSSWISE simulation (gray; only for the first ~2100 s, for the sake of clarity).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

At the time of the formation of the intense vertical vortices, vortex lines, trajectories, and the evolution of the vorticity vector along the trajectories are similar to what is depicted in the schematic summary of the MR14 simulation (Fig. 1c). Although there are no doubt many other nuances of the FREESLIP-STREAMWISE and DRAG-STREAMWISE simulations that could be contrasted with the each other, as well as with the MR14 simulation, the focus below is on the relative contributions of the barotropic, baroclinic, and viscous vorticity to the development of the vertical vortices.

b. Origins of near-surface vertical vorticity

In the FREESLIP-STREAMWISE simulation, partial ζ calculations are sufficiently reliable until roughly 2100 s, which is 5 min prior to the time of maximum 10-m ζ (Fig. 19). At this time, dominates the total ζ at the location of the ζ maximum, with the maximum in being 0.15 s−1 (Fig. 19d); also contributes to the ζ maximum, though to a lesser extent, with the maximum in being 0.006 s−1 (Fig. 19c). Positive and are both highly correlated with updraft, which is consistent with and both being highly streamwise within the airstream entering the ζ maximum from the north-northeast (not shown). Weakly positive is present within much of the region of positive total ζ, particularly the southern flank (Fig. 19e). This appears to be the result of downward turbulent diffusion of ζ, which increases rapidly with height at this time within the developing near-surface mesocyclone.

Fig. 19.
Fig. 19.

As in Fig. 7, but for the FREESLIP-STREAMWISE simulation at 2100 s. The contour interval is 0.005 s−1, and negative isopleths are dashed.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Although partial ζ fields are unreliable by the time the vortex attains its peak intensity in the FREESLIP-STREAMWISE simulation (2410 s), a Lagrangian circulation analysis reveals a dominant baroclinic contribution to the circulation (Fig. 20), with being 300% larger than . Moreover, the total C of the material circuit at 2410 s is only ~30% of the C of the circuit surrounding the ζ maximum in the FREESLIP-CROSSWISE simulation (cf. Figs. 12c and 20a). This unexpected result is revisited in section 5.

Fig. 20.
Fig. 20.

As in Fig. 12, but for the FREESLIP-STREAMWISE simulation. The circuit is followed backward in time to 0 s from 2410 s. The position of the circuit at 2410 s is indicated in Fig. 17d (the circuit is at 30 m).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

In the DRAG-STREAMWISE simulation, errors begin dominating the partial ζ fields much earlier than in the FREESLIP-STREAMWISE simulation. At 1500 s, which is unfortunately near the end of the limited window in which the partial ζ fields are trustworthy (and is still a half-hour before the time the vortex reaches its peak intensity), the total ζ field is relatively disorganized (i.e., there are multiple, elongated patches of positive ζ; Figs. 18b and 21a). At this time, dominates the total ζ field (Fig. 21e). It should not be surprising that (Fig. 21d) would be insignificant at 1500 s, given its insignificance at similarly early times in the FREESLIP-STREAMWISE simulation and in both CROSSWISE simulations. It is surprising, however, that would be so inconsequential (Fig. 21c), given its magnitude at 2100 s in the FREESLIP-STREAMWISE simulation (Fig. 19c) and in both CROSSWISE simulations (Figs. 14c and 16c).

Fig. 21.
Fig. 21.

As in Fig. 7, but for the DRAG-STREAMWISE simulation at 1500 s. The contour interval is 0.01 s−1, and negative isopleths are dashed.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

The horizontal vorticity field also is dominated by (Fig. 22). As is the case in the DRAG-CROSSWISE simulation, has a streamwise component in relatively limited areas. One area is on the eastern flank of the ζ maximum near (0.0, −2.0). A second area is on the western flank of the ζ maximum near (−0.3, −0.8). Both are locations where significant crosswise-to-streamwise exchange is present (not shown).

Fig. 22.
Fig. 22.

As in Fig. 8, but for the DRAG-STREAMWISE simulation at 1500 s. The 0.01- and 0.03-s−1 isopleths are overlaid (thin white contours) in order to facilitate comparisons with Fig. 21.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Despite the inability to obtain partial ζ fields closer to the time of maximum 10-m ζ, a Lagrangian circulation analysis is fortunately more successful. The circulation about a material circuit tracked backward to the start of the simulation from 3300 s is in good agreement with the integrated circulation forcings (Fig. 23). Although the contribution to total C from is positive, is the dominant contributor to the total C. At 3300 s, is 300% larger than (Fig. 23a), although might have rivaled if not for the abrupt loss of from 1500 to 1800 s, when a portion of the circuit is “caught up” within strong velocity gradients along the updraft flank and experiences large circulation loss owing to turbulent diffusion (Fig. 23b). There is also nonzero baroclinic generation prior to heat sink activation (negative from 0 to 400 s and positive from 400 to 900 s), resulting in small positive by 900 s; this also implies that some parcels in the material circuit were at some point influenced by the updraft. Even if only a small fraction of the parcels that surround the cyclonic vorticity maximum at 3300 s have complicated histories (e.g., updraft parcels that are dynamically forced to descend within a strong downward-directed VPPGF), the material circuits acquire horribly complex geometries. It is not known how common it is for material circuits to be influenced by the updraft in analyses in which the circuits are tracked so far backward in time.9 Last, as is the case for the FREESLIP-STREAMWISE simulation, the total C in the DRAG-STREAMWISE simulation also is considerably less than the total C in the DRAG-CROSSWISE simulation.

Fig. 23.
Fig. 23.

As in Fig. 12, but for the DRAG-STREAMWISE simulation. The circuit is followed backward in time from 3300 to 0 s. The position of the circuit at 3300 s is indicated in Fig. 18d (the circuit is at 30 m).

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

5. Discussion

a. Attribution challenges

The peak 10-m ζ in the DRAG-CROSSWISE simulation is weaker than in the FREESLIP-CROSSWISE simulation. The peak 10-m ζ is similar in the DRAG-STREAMWISE and FREESLIP-STREAMWISE simulations, but there are almost countless differences in other aspects of the simulations. Moreover, all of the cyclonic vortices that develop in the simulations presented in this paper are weaker than the vortex in MR14’s Sc8m8 simulation. Although one can conclude that surface drag is in some way responsible for all of these differences (recall that even the FREESLIP hodographs were obtained by allowing surface drag to act on the MR14 hodograph for 2 h), it is difficult to say exactly how. The possible effects of surface drag are legion.

A comparison of the w, , ζ, and fields in a FREESLIP simulation and its complementary DRAG simulation (e.g., cf. Figs. 3d and 4d) reveals a multitude of differences. In addition to the modification of the horizontal vorticity of air parcels that approach the mesocyclone in the DRAG simulations, both the motion of the cool outflow and the vertical mixing within it are affected by surface drag, which potentially affect the buoyancy field and its horizontal gradients. Moreover, it is impossible to simultaneously control for near-surface , which is related to , and in modifying the near-surface (Figs. 2b,c). Differences in between the CROSSWISE and STREAMWISE hodographs, and also in comparison to the MR14 hodograph, imply differences in the headwinds encountered by the outflow (thus, outflow motion), parcel residence times within the heat sink (thus, the buoyancy field, which also influences the outflow motion, and baroclinic vorticity generation), and the extent to which baroclinic vorticity can be tilted [an optimal heat sink–relative wind speed exists; Parker and Dahl (2015)]. Differences in the hodograph shapes also imply differences in the dynamic VPPGF among the MR14, CROSSWISE, and STREAMWISE simulations.

In the MR14 simulation, the heat sink was placed where it would produce the most intense vortex. MR14’s Fig. 24a reveals that shifting the heat sink away from its default position, 2 km east and 4 km north of the heat source (the same position used in this study), results in weaker vortices relative to the optimal position of the heat sink. It is possible that the MR14 heat sink location is not optimized for the modified wind profiles used in the present study. Thus, the present results are not an indication that surface friction always weakens vortices. All that can be said is that weaker vortices develop relative to MR14’s free-slip, semicircle hodograph simulation when surface drag is included and the heat sink is configured identically. A wide range of heat sink positions and amplitudes were not explored in the present paper as in MR14. The purpose of the paper was not to make the most intense vortex but to understand how surface drag contributes to the vorticity of a strong near-surface vortex (all of the vortices in this paper should be regarded as strong—all attain maximum 10-m ζ exceeding 0.3 s−1).

It is also worth pointing out that the vorticity decomposition does not isolate the individual vorticity components from the effects of the others on the flow, as all three partial vorticities contribute to the induction of the common velocity field used to advect, tilt, and stretch vorticity; to truly isolate the effects of each component one would have to follow the procedures described in Epifanio and Rotunno (2005).

b. The importance of viscous vorticity in the development of near-surface vertical vorticity

As explained in section 1, this study was motivated by recent findings that frictionally generated vorticity is the dominant contributor to the vorticity of simulated tornadolike vortices. In past studies, however, the viscous vorticity has not been assessed. Viscous vorticity generation has been assessed along trajectories (e.g., SXH14; RXSD16), and it has been inferred that viscous vorticity contributed to the vertical vorticity owing to diagnoses of reorientation of horizontal vorticity into the streamwise and vertical directions. But the contribution of viscous vertical vorticity to the tornadolike vortices could not be quantified because the tilting, stretching, and exchange terms in the Lagrangian vorticity budgets acted on the total rather than partial vorticity.

Another complication in some past studies has been the limited extent to which backward trajectories could be computed. For example, in SXH14, vorticity budget calculations for the first tornado (see their Figs. 12–14) were limited to the last 7 min of the parcels’ approach to the ζ maximum (longer-period calculations were possible for the second tornado, however), during which time viscous generation was large and additional baroclinic generation was negligible (the “additional” qualifier is used because there may have been previously generated baroclinic vorticity in the tilting, stretching, and exchange terms in their budgets).10 In the DRAG-STREAMWISE simulation, which is believed to most closely resemble the SXH14 simulation (i.e., in the storm-relative reference frame, the environmental hodograph in SXH14 has predominantly near-surface streamwise vorticity; see their Fig. 4), the material circuit analysis reveals that viscous generation also is dominant—and baroclinic generation is negligible—in the last 7 min of the circuit’s approach to the near-surface mesocyclone (Fig. 23; note the differences in the trends of and from 2880 to 3300 s).

However, a different picture emerges if a longer history is considered. Over the course of 3300 s, exceeds , though is still significant (Fig. 23a). Put another way, C would have remained negative in the absence of baroclinic generation. In the Xu et al. (2015) study, the net contribution of to the total C is less impressive—approximately zero—owing to a negative viscous circulation tendency over roughly half of the time period analyzed (this can be inferred from their Figs. 10d and 13d). Though the details of the circulation analyses (and partial vorticity fields, if available) are no doubt sensitive to the exact time period considered, circuit placement, and small changes in the environment (and the many tunable parameters in less idealized simulations), the safest conclusion is that baroclinically generated vorticity becomes increasingly important as time elapses, as predicted by Dahl (2015).

In neither the DRAG-STREAMWISE nor DRAG-CROSSWISE simulation is there an indication that baroclinic generation is unimportant after ~30 min of simulation time (or ~15 min after the heat sink is activated). However, one could perhaps imagine a scenario in which surface drag reduces the contribution of baroclinic vorticity to the mesocyclone’s rotation, compared with a free-slip simulation, by virtue of enhanced turbulent mixing near the surface (owing to larger near-surface vertical shear) and the weakening of horizontal buoyancy gradients. The outflow in both FREESLIP simulations is colder than in their complementary DRAG simulations despite identical heat sink–relative wind speeds and therefore similar residence times in the heat sink.

As mentioned in section 1b, the viscous horizontal vorticity must have a streamwise component in order to contribute to cyclonic vorticity within a rising airstream (intense vertical vortices are highly helical and within rising air). However, as shown in sections 3 and 4, there is a strong tendency for to have an antistreamwise component and therefore contribute to anticyclonic . This is true of not only the FREESLIP simulations (e.g., Fig. 8d), but also the DRAG simulations (e.g., Fig. 10d). Only in localized regions in the DRAG simulations, where strong leftward bending of trajectories implies large crosswise-to-streamwise exchange (typically southeast of a vortex; e.g., Fig. 11), does acquire a streamwise component and ultimately yield large positive .

The explanation for the general tendency for to have an antistreamwise component is as follows. Turbulent diffusion tends to reduce vorticity extrema. Thus, where is a relative maximum and has a streamwise component, as is commonly the case in free-slip supercell simulations within the airstream that feeds the low-level mesocyclone from the storm’s precipitation region to the north [assuming a Northern Hemisphere supercell in predominantly westerly shear (e.g., Rotunno and Klemp 1985); also note the red trajectory in Fig. 1d], generated by has a strong tendency to be antistreamwise (Fig. 24a). On the other hand, there may be a lesser tendency for to point in the opposite direction as if surface drag is present. Most idealized severe storms modeling studies are designed so that the environmental wind profile remains steady throughout the simulation (see section 1a), either because there are no horizontal forces in environment (as is the case in Coriolis-free simulations using a free-slip lower boundary condition and horizontally homogeneous environment) or the horizontal forces are balanced. In such applications, as is the case in this study, generation owing to surface drag is directed to the left of the ground-relative wind perturbation , where the prime indicates a departure of the ground-relative wind from the initial (base state) environmental wind, rather than to the left of the ground-relative wind itself. In the case of , , and pointing toward the south, as is often the case within the aforementioned southbound airstream that feeds a low-level mesocyclone, can develop both a crosswise and antistreamwise component owing to the combined influences of surface drag (tends to promote pointing to the left of ) and turbulent diffusion (tends to promote pointing opposite ) (Fig. 24b). One of the representative parcels analyzed in the simulation of RXSD16 (see their Fig. 12) possesses both crosswise and antistreamwise viscous vorticity generation, similar to that depicted in Fig. 24b.

Fig. 24.
Fig. 24.

Schematic illustration of the relationship among the horizontal vorticity , viscous vorticity tendency , viscous horizontal vorticity , storm-relative wind , and ground-relative wind perturbation in the x–y plane, just above the surface, for the case of (a) a free-slip lower boundary and (b),(c) surface drag. In (b), is aligned with and . In (c), points 90° to the right of and . See text for further details.

Citation: Journal of the Atmospheric Sciences 73, 11; 10.1175/JAS-D-16-0150.1

Although not impossible, it generally would seem to be difficult for to acquire a streamwise component within the airstream described above, though Fig. 24c illustrates one possible way. The magnitude of generation owing to surface drag exceeds the magnitude of the tendency attributable to turbulent diffusion (note that is larger in Fig. 24c than in Fig. 24b), and points 90 to the right of . If is from north (i.e., if it is aligned with southward-pointing ), then would be streamwise.

Given the surmised difficulty in the “direct” generation of a streamwise component, crosswise-to-streamwise exchange seems likely to be of fundamental importance in the acquisition of a streamwise component and ultimately cyclonic . Crosswise-to-streamwise exchange of also was inferred by SXH14 as the mechanism by which ultimately could contribute to cyclonic vertical vorticity, but the interpretation of their results is complicated by the fact that crosswise and streamwise vorticity were not defined in a storm-relative reference frame. Thus, the relevance to producing a collocated cyclonic vortex and updraft is unclear. Moreover, only modifications to the crosswise or streamwise component of the total from tilting, stretching, and exchange were diagnosed, as opposed to modifications to .

Although the focus of this paper is on the surface drag as a source of near-surface mesocyclone vorticity, drag weakens the low-level updraft in the DRAG-STREAMWISE simulation relative to the FREESLIP-STREAMWISE simulation (cf. Figs. 17a–c and 18a–c), although the maximum 10-m ζ is similar in the two simulations. Conversely, low-level updraft strength is similar in both CROSSWISE simulations (cf. Figs. 3a–c and 4a–c). The influence of the lower boundary condition on the low-level updraft might be worth exploring in future work; however, in these simulations, the low-level updraft is even more sensitive to the orientation of relative to near the surface (see section 5d).

c. Issues with running the simulations as LES

Simulations of convective storms are almost always run as LES, in which it is assumed that the most important scales of motion (the energy-containing, or “large” eddies) are well resolved on the grid. Markowski and Bryan (2016; hereafter MB16) have shown that the use of LES when the flow is insufficiently turbulent (i.e., eddy-less LES), particularly when a nonfree-slip lower boundary condition is used, leads to the development of unrealistically strong vertical wind shear (i.e., horizontal vorticity) near the surface (>1000% too strong in the example presented in their paper). Unfortunately, most prior studies investigating the effects of surface drag on storms (see section 1b) have likely suffered to some extent from this problem, either because the environment was horizontally homogeneous or the grid spacing was too coarse (both ultimately lead to environments that are too laminar).

In some studies (Frame and Markowski 2010, 2013; SXH14), the mixing length is lengthened in the neutral or slightly unstable boundary layer surrounding the storm, which probably mitigates the development of unrealistic near-surface vertical shear, at least in the environment, though it is difficult to specify how much. It is also possible that the problem exposed by MB16 might not be as severe within the outflow of storms, where most of the low-level mesocyclone’s vorticity is generated anyway (regardless of whether or dominates), because the outflow tends to be turbulent even when the environment is fairly laminar, assuming sufficiently high resolution is used (see MB16’s Fig. 1). It is unclear precisely what resolution would be needed and whether the flow within the outflow would become sufficiently turbulent naturally (i.e., without imposing random perturbations on temperature, wind, etc.). Moreover, the development of unrealistically large vertical shear in the environment could still be problematic, because the interaction of the storm’s main updraft with the environmental shear affects the updraft’s dynamic VPPGF, which is the means by which negatively buoyant air parcels within the outflow can be accelerated upward. Thus, the vorticity stretching experienced by parcels within the ζ-rich outflow could be affected by unrealistic shear in the far-field environment. In RXSD16’s simulation with surface drag, the intense vertical vortex forms prior to the development of outflow anyway, when the flow is still highly laminar in the region of developing rotation.

In addition to the problem with eddy-less LES, there are other issues that likely lead to unrealistic near-surface vertical shear. One is the so-called law-of-the-wall problem, which refers to the development of excessive vertical wind shear near the surface (or, in general, a wall) owing to turbulent eddies becoming inadequately resolved as the surface is neared (eddy size scales with the distance from the surface). The issue is reviewed at length by Mason and Thomson (1992), Sullivan et al. (1994), Chow et al. (2005), Brasseur and Wei (2010), and Moeng and Sullivan (2014), among others. Another uncertainty arises from the formulation of the lower boundary condition itself. In severe storms modeling in which surface drag is included, the most common parameterization (used in this study as well; see section 2a) specifies surface stress at each grid point based on the assumption that the wind profile obeys the log law below the lowest scalar grid level. However, this approach is not strictly consistent with the fact that log laws were derived for average quantities and are not generally valid instantaneously. Some boundary conditions consider average rather than instantaneous quantities (e.g., Schumann 1975; Moeng 1984; Grötzbach 1987; Hultmark et al. 2013), and even more sophisticated ones consider the vertical velocity near the surface (e.g., Piomelli et al. 1989). Others are still under development (J. Brasseur 2015, personal communication). The wind fields of supercells are very different from those studied in the boundary layer community (e.g., parcels can suddenly find themselves adjacent to the surface after violently descending from several kilometers aloft, and having a very different momentum and temperature than any other parcels in the boundary layer), which might make the lower boundary condition even more uncertain.

In spite of the potential problems with prior severe storms simulations that have included surface drag, in the simulations herein, the LES approach used in most previous studies was followed for two reasons. The first is that comparisons to these simulations would be more challenging if an entirely new approach were utilized. Different approaches to handling surface drag will be explored in a future paper. The second reason is that most, if not all of the aforementioned issues in the prediction of near-surface vertical wind shear in LES seem to result in unrealistically large shear. It might be worth knowing the “worst case” influence of surface drag on the development of vertical vortices within storms, and it is believed that the results obtained in this paper might best be regarded as such.

d. The influence of low-level hodograph curvature independent of the lower boundary condition

The simulations expose two modes of vortex genesis. A barotropic mode is observed early in the evolution, prior to cold-pool development, when near-surface environmental is crosswise (Fig. 2b). Surface drag is not necessary, but it enhances the process by augmenting the that is tilted. To the extent that numerical simulations can be regarded as an extension of theory, this mode at least appears theoretically possible even though, to the author’s knowledge, it has not been shown to operate in real convective storms. Some possible candidate storms, in the sense that tornadoes were observed relatively early in the storm’s life (though not within 15–20 min of updraft initiation, as observed herein), might be the 27 May 1997 Jarrell, Texas, tornado; the 10 May 2010 Norman, Oklahoma, tornado [this tornado was observed by scientists at the National Weather Center and by the University of Oklahoma Prime radar]; and the 19 May 2010 Leedey, Oklahoma, tornado (the latter tornado was observed visually by VORTEX2 scientists). However, without high-resolution thermodynamic observations near these developing tornadoes, it would be difficult to evaluate whether the barotropic mode might have been operating.

The barotropic mode is absent in the simulations in which the near-surface environmental is streamwise (Fig. 2c). Instead, these simulations exhibit a baroclinic mode of vortex genesis: that is, the vertical vorticity originates from baroclinic (horizontal) generation by horizontal buoyancy gradients associated with cool outflow and downdrafts and subsequent tilting. As reviewed in section 1a, the development of near-surface vertical vorticity in prior supercell simulations has been attributed to this mechanism, and past observations, where available, also seem to implicate baroclinity. Baroclinic ζ also dominates late in the FREESLIP-CROSSWISE simulation (2550 s) and is gaining importance late in the DRAG-CROSSWISE simulation (2100 s).

Another important aspect of orientation of the near-surface is its influence on both the low-level updraft and near-surface C. Low-level updrafts are considerably stronger in the STREAMWISE simulations than in the CROSSWISE simulations at most times (cf. Figs. 3, 4 and Figs. 17, 18). On the other hand, significantly less near-surface C characterizes the near-surface mesocyclones that develop in the STREAMWISE simulations compared with the CROSSWISE simulations (cf. Figs. 12c, 13c and Figs. 20a, 23a). This finding is surprising, given that proximity sounding climatologies have found tornadic supercell environments to have greater low-level streamwise than nontornadic supercell environments (e.g., Markowski et al. 2003b; Rasmussen 2003; Thompson et al. 2003; Nowotarski and Jensen 2013; Parker 2014). Moreover, some less idealized supercell simulations have demonstrated that environments with near-surface streamwise are more favorable than environments with near-surface crosswise for the development of near-surface ζ and C (e.g., Wicker 1996; Webster et al. 2014), although these simulations have investigated only a limited parameter space. Perhaps the main benefit of streamwise vorticity in the low-level hodograph is through its influence on the low-level updraft rather than its influence on the development of near-surface C. Past parameter space studies have tended to focus on the maximum updraft speed (which is typically found at middle or upper levels) and maximum vertical vorticity (which is more volatile than circulation). The simulations presented in this paper expose a gap in our knowledge, probably worthy of future study, of how the low-level updraft and development of near-surface C are influenced by the near-surface environmental hodograph.

6. Summary and conclusions

It has long been known that surface drag can intensify vertical vortices by preventing cyclostrophic balance and promoting radial inflow, thereby promoting the stretching of vertical vorticity. However, a few recent simulation studies have concluded that surface drag is a crucial vorticity source for intense vertical vortices. In this study, the “toy model” approach of MR14 was used to further investigate the role of surface drag on the development of near-surface vertical vorticity in supercell storms.

Two pairs of pseudostorm simulations (four total) were presented. In one pair, the environmental vorticity was crosswise just above the surface; in the other pair, the environmental vorticity was streamwise. In contrast to the original MR14 hodographs, which were semicircular, both boundary layer wind profiles used herein resemble those that might be observed in the presence of surface drag and a Coriolis force. One simulation in each pair used a free-slip lower boundary condition; the other used a semislip lower boundary condition. Following the approach of ED2002, the relative contributions to the vertical vorticity and circulation from barotropic, baroclinic, and viscous vorticity were evaluated. The partial vorticity analyses were complemented by analyses of circulation following material circuits.

Intense cyclonic vortices developed in all four simulations, although all four were weaker than the baseline vortex that developed in the MR14 study. In the pair of simulations initialized with near-surface crosswise environmental vorticity, an intense near-surface vortex developed early in the simulation, prior to the development of a cold pool. In this barotropic mode of vortex genesis, environmental vortex lines (augmented by viscous vorticity when surface drag was included) were displaced downward toward the surface by a dynamically driven downdraft. The downdraft was maximized on the northern and northeastern flank of the updraft. This mode was absent in the simulations in which the near-surface environmental vorticity was streamwise.

In simulations in which the lower boundary condition was free slip, baroclinically generated vorticity dominated the near-surface cyclonic vortices when they were at their maximum intensity, as has been found in prior numerical simulations and inferred from some past observations. In the simulations in which surface drag was present, viscous vertical vorticity dominated the vortices early in the simulations (prior to and during the early stages of cold-pool development), but baroclinically generated vorticity grew in importance as the simulations advanced (i.e., once a mature cold pool was established).

The highly idealized simulations herein suggest that conclusions from prior simulation studies about the importance of frictionally generated vorticity in tornadogenesis might be skewed by not being able to examine sufficiently long parcel histories or by analyzing the development of strong near-surface cyclonic vorticity prior to the development of a significant cold pool (at best, such early vortex formation is atypical). Whether or not viscous vertical vorticity contributes positively to a cyclonic vortex also is sensitive to the trajectories (or flank of the vortex) considered. There also is large uncertainty in the influence of surface drag on the simulated pseudostorms, owing to uncertainties in the formulation of the lower boundary condition, the intrinsic limitations of LES near boundaries, and even the misuse of LES (the lack of turbulent eddies). The influence of surface drag in these simulations might represent a “worst case” scenario. I hope to address several of the aforementioned issues in future publications.

Acknowledgments

I am grateful for numerous constructive discussions on this topic with many colleagues: Elie Bou-Zeid, Jim Brasseur, George Bryan, Marcelo Chamecki, Tina Chow, Johannes Dahl, Evgeni Fedorovich, Jeff Mirocha, Matt Parker, Yvette Richardson, Rich Rotunno, Alex Schenkman, Peter Sullivan, Lou Wicker, and John Wyngaard. I am particularly indebted to Yvette Richardson and George Bryan for their detailed reviews of an earlier version of this manuscript, in addition to George Bryan’s seemingly boundless, generous support of CM1. I also thank Matt Parker, Rich Rotunno, Alex Schenkman, and an anonymous reviewer for their critical evaluations of the submitted manuscript. This work would not have been possible without the generous support of the National Science Foundation (Grants AGS-1157646 and AGS-1536460).

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