1. Introduction and motivation
The large number of hailstorms that occur each year in the United States make these severe convective storms collectively one of the costliest hazards in terms of insured losses to homes, businesses, automobiles, aircraft, and agriculture. Annual insured losses in the United States from hailstorms exceed $1 billion (Munich Re 2013). Moreover, individual hailstorms can produce hundreds of millions of dollars to over $1 billion in damage (e.g., Picca and Ryzhkov 2012), including the San Antonio 12 April 2016 storm, which is estimated to have produced $1.36 billion in damage (NWS–San Antonio 2016). Other parts of the world are affected as well. Recently, a series of hailstorms in Germany unexpectedly led to losses in excess of EUR 3 billion (Munich Re 2014). Thus, understanding processes contributing to hail formation, hail growth, and the environmental controls over hail production are important for anticipating hail threats and potentially mitigating the risk (e.g., moving automobiles and aircraft to cover).
A large thrust in hail-related research occurred during the 1970s as a part of the National Hail Research Experiment (NHRE), a large multidisciplinary project with an ultimate aim of weather modification and hail suppression (e.g., Foote and Knight 1979). Regarding hail suppression, the results showed little hope: Knight et al. (1979, p. 1630) summarize the project’s findings by stating “no influence of seeding on hailfall has been detected.” Funding and interest in hail suppression research faded after this project (C. Knight 2014, personal communication). However, several of the well-sampled storms during NHRE and other campaigns were the subject of further research exploring the mechanisms of hail formation and growth in convective storms. Many found that the storm’s airflow patterns were just as important for hail production (if not more important) than its maximum updraft speed. Several of these studies have shown that hail growth can occur from numerous initial conditions, but the favorable initial region for hail embryos is restricted (e.g., Foote 1984). Growth of embryos into large hailstones depends on the trajectories they take through the storm’s updraft. For supercells in particular, three main embryo trajectories have been proposed (e.g., Browning and Foote 1976, their Fig. 19; Conway and Zrnić 1993): 1) embryos enter a region of strong updraft and are ejected through the forward anvil before significant growth can occur; 2) embryos enter the less-intense western edge of the updraft, grow at moderate rates in ascent, and cycle into the reflectivity overhang (known as the “embryo curtain”); and 3) embryos from 2) fall cyclonically around the updraft and recycle at the bottom of the embryo curtain into a region of the updraft where a balance is achieved between updraft velocity and hailstone fall velocity. In this crucial suspended state, hailstones can experience prolonged exposure to regions where optimal growth conditions may exist.
Numerous studies point out that this balance is a key condition for optimum hail growth (e.g., Browning 1964; Foote 1984; Heymsfield 1983; Nelson 1983; Miller and Fankhauser 1983; Musil et al. 1986; Miller et al. 1990; Conway and Zrnić 1993; Kennedy and Detwiler 2003; Grant and van den Heever 2014, hereafter GvdH14). If the updraft is too strong, small particles will be ejected out of the top of the growth region too quickly to experience significant growth; if it is too weak, embryos simply sediment out before substantial growth can occur (e.g., Browning and Foote 1976; Nelson 1983; Foote 1984; Musil et al. 1986; Rasmussen and Heymsfield 1987). Though the maximum updraft intensity does place an upper limit on hailstone size, high-speed updrafts are insufficient to ensure that large stones are grown; stones must also reside in the updraft for enough time, and this is largely governed by the horizontal path growing hailstones take through the updraft (Nelson 1983; Miller et al. 1990). In part, this depends on the width of the updraft, with broader updrafts favored for enhanced hail growth (e.g., Nelson 1983; Foote 1984).
Foote (1984) shows embryos that have grown by some mechanism to millimeter-sized diameters before entering the updraft are most likely to achieve the required balance between fall speed and updraft speed. This balance must occur in a region with sufficient supercooled liquid water (cloud droplets and raindrops) and an appropriate temperature for efficient growth. For large-hail growth, the hailstones/embryos must be balanced such that they experience prolonged exposure to the moisture-rich updraft at temperatures between −10° and −25°C (e.g., Nelson 1983; Ziegler et al. 1983; Foote 1984; Miller et al. 1988). However, competition among embryos for supercooled liquid water has also been cited as a limiting factor for hail growth (e.g., Heymsfield 1983).
To summarize, optimal hail growth is dependent on five key characteristics: 1) appropriate updraft strength and breadth to facilitate the suspension of hailstones/embryos, 2) sufficient amounts of supercooled liquid water, 3) temperatures conducive to hail growth, 4) the size and number of embryos such that the particles are balanced in the updraft and competition does not deplete the total water content, and 5) trajectories that take these embryos through the optimal growth environments. Regarding the latter, Nelson (1987) argues that storm airflow patterns are “critically important” for controlling severe hail events, a conclusion echoed by numerous other studies cited above. However, none of these studies systematically investigated how environmental factors alter supercell airflow patterns and thus hail production.
The structure and intensity of supercell storms has been explored extensively in the last half century. Numerical simulations examining the effects of varying unidirectional vertical wind shear and buoyancy (CAPE) on idealized convective storms were carried out by Weisman and Klemp (1982). They found a spectrum of storm behaviors similar to observed storms, including short-lived single-cell storms, multicell storms, and supercells. For fixed CAPE, they found increasing shear decreases maximum updraft intensity and increases the updraft tilt. A follow-up study (Weisman and Klemp 1984) tested the influence of directionally varying wind shear on simulated convective storms. Their results were qualitatively similar to their previous work but added insights into how clockwise-turning hodographs favor cyclonically rotating supercells. Varying only midlevel shear, Brooks et al. (1994) found differences in the development, maintenance, and intensity of low-level mesocyclones, identifying the strength of the midlevel storm-relative flow as being critical for the simulated supercell precipitation structure. Weisman and Rotunno (2000) simulated supercells in environments with straight through circular hodographs to clarify mechanisms by which convective storms acquire rotation and propagate across the environmental vertical shear. McCaul and Weisman (2001) found that stratification of buoyancy and shear and shapes of their profiles cause substantial impacts on updraft intensity, cold pool intensity, and midlevel and surface vorticity.
Though these studies have greatly advanced our understanding of supercell storm structure and dynamics and how these vary with environmental parameters, less attention has been paid to how these variations alter microphysics and, specifically, hail formation and growth. Van den Heever and Cotton (2004) analyzed the resulting changes to storm structure and evolution by modifying hail size distribution parameters in a single-moment microphysics scheme. However, these changes to the hail size distribution were made a priori, without explanation of how these may arise; thus, the authors call for a better understanding of how environmental factors (including aerosol concentrations, the presence of dry layers, and vertical wind shear) influence the hail size distribution.
More recently, GvdH14 analyzed the impact of varying midlevel relative humidity on resulting simulated supercell structure and hail production. They found that changes in storm-relative winds led to different hail growth mechanisms in “low precipitation” supercells versus “classic” supercells. Classic supercells had the largest riming rates on the western side of the updraft, whereas low-precipitation supercells had the largest riming rates on the north and northeast sides of the updraft. Whereas the observational study of Rasmussen and Straka (1998) suggested anvil-level storm-relative winds could be a factor in determining supercell mode (with low-precipitation supercells favored in environments with larger anvil-level storm-relative winds), the differences in GvdH14 arose from changes in storm motion and storm-scale-generated flow in each simulation, in which midlevel relative humidity was changed but environmental wind profiles were fixed.
In contrast, this study aims to investigate the role of environmental wind profiles on supercell structure and subsequent hail growth. The next section describes the methods used and the numerical simulations performed. Results are presented in section 3. Section 4 provides a discussion of the results and outlines the main conclusions of this study.
2. Methods
To investigate the impact of environmental shear on hail growth in idealized supercell storms, a series of idealized simulations is performed. Cloud Model 1 (CM1; Bryan and Fritsch 2002), version 17, is used for these simulations, with a 240 × 240 × 80 grid using 500-m horizontal grid lengths and 250-m vertical grid lengths. In each simulation, the thermodynamic environment is fixed (CAPE = 2200 J kg−1), using the idealized supercell sounding from Weisman and Rotunno (2000). The simulations were run for 2 h using a 1.5-s time step, with output files written every 5 min. Lateral boundaries are gravity wave radiating, similar to Klemp and Lilly (1978) and Klemp and Wilhelmson (1978), but with a modification to prevent mass leakage from the domain (G. Bryan 2016, personal communication). The lower boundary is free slip with no fluxes considered. The domain is translated with storm motion to keep the storm centered in the domain. Storm motion was determined by following the midlevel updraft core in a simulation with a stationary domain. The average motion over the simulation period was applied. This was done for each of the simulations herein, resulting in different translation speeds for each storm. Though admittedly redundant, this allowed for more accurate storm motion estimates than those based on the hodograph alone. This observed storm motion was used to compute storm-relative helicity from the prescribed hodographs.
A Rayleigh dampening layer is applied in the stratosphere, in the upper 5 km of the domain. Storms are initiated with a warm (1-K potential temperature perturbation) bubble. The two-moment Morrison microphysics scheme (Morrison et al. 2005, 2009) is used, with the rimed ice category set to the high-density option (“hail”). In the standard version of the two-moment Morrison scheme, cloud droplet number concentration is held constant at 250 cm−3. Assuming a fixed cloud liquid mass mixing ratio, changes in this droplet number concentration would alter droplet sizes. Although this effect can be important for hail production and other microphysical process rates (e.g., Lebo 2014; Kalina et al. 2014; Loftus and Cotton 2014), this and other aerosol effects are not accounted for in the present study.
In the control run, umax1 = 7.0 m s−1, umax2 = 31 m s−1, υmax = 7 m s−1, h1 = 2 km, and h2 = 6 km. (These are the parameter settings in the standard CM1 package quarter-circle hodograph supercell case.) The main parameters altered in this study are 
Simulations performed for this study. Naming convention includes the value of the umax2 and υmax parameters. If no υmax value is listed, the default value of 7 m s−1 is used. Also listed are the total (0–6 km) hodograph length and 0–3-km storm-relative helicity.
Of relevance for storm structure and microphysics is the storm-relative flow. Although vertical wind shear is Galilean invariant, how it is partitioned into u and υ components may change depending on storm motion. For typical supercell environments, most turning of the hodograph occurs in the lowest few kilometers, followed by roughly unidirectional shear aloft, which tends to be zonal on average in North America (e.g., Maddox 1976; Weisman and Rotunno 2000; Markowski et al. 2003; Markowski and Richardson 2010). The analytic quarter-circle hodograph used herein has deep-layer unidirectional shear in the east–west direction (u component), whereas the hodograph turning at low levels is associated with north–south (υ component) shear. Clearly, other cases (e.g., in a northwest flow regime) would not conform well to such a u and υ partitioning. However, for simplicity we adopt the u- and υ-component terminology for this study.
A detailed understanding of how hail sizes change with environmental factors requires the use of sophisticated bin microphysical models (e.g., Lebo et al. 2012). This is because bulk schemes do not adequately treat processes like size sorting that can affect the tails of the particle size distributions (e.g., Milbrandt and Yau 2005a; Wacker and Seifert 2001; Milbrandt and McTaggart-Cowan 2010; Dawson et al. 2010; Kumjian and Ryzhkov 2012). Though some three-moment schemes have been developed and/or used to mitigate many of these concerns (e.g., Milbrandt and Yau 2005b; Kumjian and Ryzhkov 2012; Dawson et al. 2014; Loftus et al. 2014), these are not widely available in CM1 or WRF and thus are not used in the present study. As such, we focus on hail production in terms of the total mass mixing ratio and refrain from much discussion of hailstone sizes. Future work focusing specifically on hail sizes should use more detailed bin microphysical models that most realistically treat the evolution of the particle size spectrum.
3. Analysis and results
The simulation results are presented in this section and are divided into subsets to focus on the impacts of changing individual parts of the hodograph at a time.
a. The u-component (deep layer) shear
The first subset of simulations includes those for which only the 
Storm-relative hodographs for the subset of supercell simulations in which only the deep-layer east–west (u) component of the shear is varied. The naming convention for each storm is indicated in the upper left of each panel; it refers to the umax2 value (m s−1) from (2) and (3) (see Table 1). The control run is the standard CM1 supercell case and is equivalent to umax31. Each of these storms has υmax = 7 m s−1. Winds in the lowest 3 km are shown in black, 3–6-km winds in blue, and >6-km winds in magenta. Color frames around each panel are consistent with subsequent figures.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Modifying the environmental shear is expected to have an impact on the subsequent storm structure and evolution. To assess any changes in updraft speed, time series of 1-min maximum vertical velocity for each of the simulations are shown in Fig. 2. From the figure, it is evident that maximum updraft speeds were slightly different among the different simulations. When averaged over the last 40 min, the highest and lowest average maximum updraft speed did not differ by more than 7 m s−1. There is, however, an inverse relationship between the average maximum updraft strength and u-component shear: the highest-shear case (umax50) has the lowest average maximum updraft strength over the last 40 min of the simulation, whereas the lowest-shear case (umax25) has the strongest updraft. Thus, stronger east–west (u component) shear leads to slower-developing storms and slightly reduced wmax, consistent with the findings of Weisman and Klemp (1982). The reduced wmax for a given amount of CAPE may be related to the effects of the downward-directed perturbation pressure gradient force arising from buoyancy gradients atop updrafts. The magnitude of this perturbation pressure gradient force has been shown to increase for broader (e.g., Markowski and Richardson 2010; Morrison 2016a,b) and tilted (Parker 2010) updrafts. As shown below, updrafts in the higher-shear cases indeed are broader.
Time series of the maximum vertical velocity wmax (m s−1) for the five umax storms.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Simulated hail swaths of maximum hail mass mixing ratio at the lowest model grid level are shown in Fig. 3, overlaid with the 30 m s−1 5-km-AGL updraft contour (magenta line). Each reveals an elongated, ~10-km-wide swath of lower values (<2 × 10−1 kg kg−1), within which are embedded localized maxima that tend to favor the southern edge of the swath. However, differences exist between the simulations. For example, the high-shear cases (umax41 and umax50) appear to have larger values in these embedded maxima; peak values approach 1.0 × 10−3 kg kg−1 in the umax50 case. Additionally, higher-shear cases feature less hail mass directly beneath the midlevel updraft core. Given the spatiotemporal variability of hailfall characteristics in these storms, robust comparisons of the storm’s structure are desired. To do so, we construct composites for each storm to determine the average structural features during their mature phase.
Time-integrated maximum hail mass mixing ratio (shaded according to scale; kg kg−1) at the lowest model grid level with a moving domain throughout the 2-h storm simulation for each of the umax storms. Overlaid is the trace of the 5-km-AGL updraft 30 m s−1 contour (magenta).
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Composites of each simulated storm are produced following methods similar to those used in GvdH14. First, the right-moving supercell’s midlevel (6.125 km AGL) updraft maximum is found for each output time during the last hour of simulation. The data are recentered on this midlevel updraft maximum, and the domain is constricted to ±25 km in the x and y directions from this center. Then, each field on the 50 km × 50 km recentered grid from each time is averaged.
Figure 4 shows vertical profiles of the average hail mass mixing ratio qh computed from the composites for each of the u-component shear subset of cases. To assess the statistical significance of differences in the average qh at a given altitude, we used a bias-corrected-and-accelerated bootstrapping technique (e.g., Efron and Tibshirani 1993) with 1000 resamples to estimate the 95% confidence intervals about the average 
Vertical profiles of the 95% confidence intervals about the average hail mass mixing ratio qh for each storm, computed from the composites.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
So, changing the u-component (deep layer) shear via the 
To determine the underlying reason for the differences in hail production aloft, we first investigate whether certain microphysical processes are acting at higher rates in the high-shear cases. In the Morrison two-moment microphysics scheme, there are several microphysical processes relevant for hail formation and growth (Table 2). These can be broadly categorized into conversion processes, in which the particles are converted from some other hydrometeor category to hail, or growth processes, in which mass is added to existing hail. After consulting with the parameterization scheme’s architect (H. Morrison 2014, personal communication), we have combined these process rates into four broad categories: conversion to hail by freezing, which includes immersion freezing of raindrops and contact freezing of raindrops resulting from collisions with cloud ice; conversion to hail by riming, which includes riming of snow by raindrops and cloud droplets; growth of hail by riming, which includes collection of raindrops and cloud droplets; and vapor depositional growth of hail. For brevity, hereinafter these are referred to as freeze conversion, rime conversion, rime growth, and deposition growth, respectively (Table 2). In the Morrison two-moment scheme, only a portion of the mass acquired from riming is converted to hail, following Reisner et al. (1998). Additionally, mass mixing ratios of the species involved with riming must exceed 0.1 g kg−1 for conversion to be active (Morrison and Grabowski 2008). Deposition growth is not included in the subsequent analyses because its rate was at least an order of magnitude less than the other categories in our simulations.
Microphysical processes in the Morrison two-moment scheme relevant to hail formation and growth. Processes are combined into categories discussed in this study.
For each of the simulations, vertical profiles of the processes rates are conditionally averaged for the composites over the last hour of the simulations, computed along with their 95% confidence interval about that mean at each height level. Grid boxes were only included in the averages if the process rates exceeded 1 × 10−5 kg kg−1 s−1 for rime growth and freeze conversion, and 1 × 10−6 kg kg−1 s−1 for rime conversion. These thresholds were tested over a range of orders of magnitudes, and the qualitative results were not affected. The statistics were again estimated using the bootstrapping technique (e.g., Efron and Tibshirani 1993) with 1000 resamples. The 95% confidence intervals about the average vertical profiles of the process rates are shown in Fig. 5. Though differences between the various simulations do exist, many of the confidence intervals indeed overlap. Further, there is no clear relationship between the magnitude of the process rates and hail production. For example, much of the domain below 8 km AGL reveals lower rime growth rates in the umax50 storm (Fig. 5a), which had the largest surface hail mass mixing ratio. Even above 8 km AGL, the umax50 storm does not exhibit larger rime growth rates than the other storms, although it does feature enhanced rime conversion between 6 and 8 km that may contribute to increasing hail mass. However, given these rates are an order of magnitude less than the rime growth and that it is not present in other storms, it is likely a secondary effect. Similarly, each storm’s conversion rates overlap (Figs. 5b,c). Thus, it seems unlikely that differences in the average microphysical process rates are a major contributing factor to the observed differences in hail formation and growth aloft.
Vertical profiles of the 95% confidence intervals about the conditionally averaged microphysical process rates for each umax subset storm, computed from the composites: (a) rime growth, (b) rime conversion, and (c) freeze conversion. Note the different ranges of values on the abscissa.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
If the magnitudes of the process rates are not the main cause of hail production differences, then it could be the differences in updraft volume or hailstone residence time within the updraft; even if the average conversion rate and growth rate values are the same among the different storms, a larger growth region could lead to more hail production as these process rates act over a greater volume of the storm, and/or hailstones remaining in the growth region for longer periods of time could lead to larger hail mass. Recall that previous observational and modeling studies suggest that broader updrafts are more conducive to hail growth (e.g., Nelson 1983; Foote 1984; Picca and Ryzhkov 2012). Specifically, a broad region of moderate updraft allows hailstones to remain balanced in a region of efficient growth for a longer amount of time.
To investigate these differences, we first inspect the three-dimensional storm structures. Figure 6 shows a horizontal cross section of simulated reflectivity factor ZH (a direct model output variable, based on the Rayleigh approximation to electromagnetic scattering) through the composite umax25 and umax50 storms at 5.125 km AGL. These storms are chosen to illustrate the two extremes of the subset of storms considered thus far. A semicircular bounded weak echo region (BWER) is evident in the umax25 storm (Fig. 6a), with a high-ZH core to its northwest. Vertical cross sections are taken through the center of the composite (updraft maximum at 6.125 km AGL) to explore the vertical structure of the storm (Figs. 7a,b). The storm exhibits a robust, vertically erect updraft. Below 8 km, substantial liquid mass mixing ratio is evident in a column on the western flank of the storm, as are maxima in the process rates (Fig. 7a, just west of 0 km). This is consistent with the findings of GvdH14 for “classic” supercells. Above this level, a maximum in freeze conversion caps the liquid mass mixing ratio contours, and rime growth contours are found throughout the updraft. Above 6 km, the process rates tend to favor the south flank of the updraft, just inside the ZH echo overhang (Fig. 7b).
Horizontal cross sections of simulated reflectivity factor (dBz; shaded according to scale) at 5.125 km AGL for the composites of the (a) umax25 and (b) umax50 storms. The black lines indicate where vertical cross sections are taken in subsequent figures.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Vertical cross section through the composite umax25 storm along the (a) east–west and (b) north–south directions. (c),(d) As in (a) and (b), respectively, but for the umax50 case. Grayscale shading is simulated reflectivity factor (dBz); white vectors are the two-dimensional winds in the plane of the cross section, with vector maxima in the lower left of each panel; blue contours are liquid (cloud and rain) drop mass mixing ratio (0.001, 0.005, and 0.01 kg kg−1); blood red is hail mass mixing ratio (0.001, 0.005, and 0.01 kg kg−1); chartreuse is rime growth (0.1, 1.0, and 2.0 × 10−4 kg kg−1 s−1); cyan is rime conversion (0.1, 1.0, and 2.0 × 10−5 kg kg−1 s−1); magenta is freeze conversion (0.1, 1.0, and 2.0 × 10−4 kg kg−1 s−1); and updraft (20, 30, and 40 m s−1) contours are goldenrod.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
In contrast, the umax50 storm exhibits an elongated, well-defined BWER at midlevels (Fig. 6b), with an enclosed region of ZH < 30 dBz stretching more than 5 km in the east–west direction. Vertical cross sections through this storm (Figs. 7c,d) exhibit similar structures to the umax25 storm, especially in the north–south plane. However, the most striking difference is the much larger horizontal extent of the updraft region in the umax50 storm in the east–west direction (Fig. 7c). For example, at 5.125 km AGL, the umax25 storm’s updraft region bounded by the 20 m s−1 contour is approximately 5 (6.5) km wide in the east–west (north–south) direction, whereas the umax50 storm’s is 11 (7) km wide. Clearly, the greater shear promotes a broader updraft in the direction of the enhanced shear vector. Rather than being tilted more downshear, the umax50 storm’s updraft appears to be vertically erect, consistent with the discussion in Markowski and Richardson (2010) of how large vertical shear can promote upright updrafts (see also Rotunno et al. 1988). This elongated updraft also potentially serves to increase the volume over which microphysical processes act to increase hail mass and number concentration.
To quantify the impact of this larger updraft on process rates, we look at volumes in the domain (for the right-moving supercell only) in which process rates exceed certain thresholds: 6 × 10−5 kg kg−1 s−1 for rime growth and freeze conversion, and 6 × 10−6 kg kg−1 s−1 for rime conversion. These thresholds were chosen to ensure that smaller values associated with secondary convection or the left-moving supercell, which could spuriously increase volumes, are not included. Figure 8 shows the time series of these volumes for the simulated storms. For rime growth (Fig. 8a), the storms exhibit a clear separation, with high-shear storms displaying larger volumes throughout the time period shown. By the end of the simulation, the umax50 storm exhibits nearly twice the volume of rime growth exceeding the chosen threshold than the umax25 storm. The relationship for rime conversion (Fig. 8b) is not as clear, though for much of the time period the general trend is for the higher-shear cases to have larger volumes than the lower-shear cases. Freeze conversion (Fig. 8c) again exhibits a clear separation throughout most of the time period, with umax50 exhibiting more than twice the volume exceeding the chosen threshold as umax25 by the end of the simulation.
Volumes of different process rates exceeding specified thresholds for the umax storms. (a) Rime growth exceeding 6 × 10−5 kg kg−1 s−1, (b) rime conversion exceeding 6 × 10−6 kg kg−1 s−1, and (c) freeze conversion exceeding 6 × 10−5 kg kg−1 s−1. Colors indicate simulations as in previous figures. Note the different ordinate axes used in each panel.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
These results, when considering the similar overall magnitudes of the process rates, imply that increased u-component deep-layer shear causes broader updrafts (elongated in the west–east direction), in which larger volumes of optimal growth conditions occur. It is not the magnitudes of the process rates or the updraft speed that seems to be most important for hail growth in this subset of simulations but, rather, the volume in which optimal growth conditions exist. This is consistent with previous studies that suggested broad updrafts are conducive to hail growth (e.g., Nelson 1983).
Investigating hailstone residence time within the prime hail growth region is more difficult, as the bulk microphysics model does not explicitly predict individual hailstones or their trajectories through the storm. Instead, we take the distribution of storm-relative u and υ winds in the hail growth region to determine if hailstones would spend more time in the updraft. Distributions are constructed for values within updraft regions >20 m s−1 between 5.125 and 8.625 km AGL, which corresponds to environmental temperatures from −9.5° to −33°C. Figures 9 and 10 show these distributions. Although increasing 
Box-and-whisker plots showing distributions of storm-relative u-wind components in the hail growth region (w > 20 m s−1; 5.125–8.625 km AGL) from the composites of each storm. The blue circle is the median value, the colored box is the 25th–75th-percentile (interquartile) range, the whiskers extend from the 10th- to 90th-percentile range, and the red crosses indicate the maximum and minimum values.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
As in Fig. 9, but for the storm-relative υ component.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
In summary, this subset of simulations provides strong evidence that environmental wind shear plays a significant role in modulating the hail production in supercell storms. In particular, enhanced u-component (deep layer) shear leads to updrafts elongated in the direction of the shear vector. Maximum updraft speeds are slightly less for increased deep-layer shear, which, when combined with these larger volumes and only slight increases in u and υ storm-relative winds within the updraft, likely lead to longer residence times. The broader updrafts provide more volume over which hailstones may form (predominantly by freeze conversion), and, when combined with longer residence times, provide more volume and time in which hailstones may grow (predominantly by accretion of supercooled cloud and rainwater).
b. The υ-component cases
In this subsection, we present a subset of simulations in which the υ component of shear is altered. Recall that the υ shear occurs over the lowest 3 km of the troposphere for the idealized hodographs used herein. Enhancements in low-level north–south shear may be associated with a nocturnal low-level jet, for example. In typical supercell environments, the magnitude of this north–south shear is much less than the east–west shear, given the predominantly westerly flow across midlatitudes.
These simulations vary the 
As in Fig. 1, but for the υmax cases.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Vertical profiles of average 
The 95% confidence intervals about the average qh at the lowest grid level for each composited storm. Each group of umax storms is color coded as before; υmax storms within each umax group are indicated by different dashed lines.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Similar to the 
To investigate the impact of the low-level υ-shear changes on the updraft structure, we construct horizontal cross sections at midlevels (5.625 km) for the subset of υ-shear cases (Fig. 13). The control run (Fig. 13a) features a classic crescent-shaped updraft region and a well-defined BWER. The maxima in process rates are confined to the western edge of the updraft at this height. Similarly to the 
Horizontal cross sections at 5.625 km AGL for the (a) control, (b) umax31vmax10, (c) umax31vmax13, and (d) umax31vmax16 cases. Grayscale shading is simulated reflectivity factor (dBz); white vectors are the two-dimensional winds within the plane of the cross section; blue contours are liquid (cloud and rain) drop mass mixing ratio (0.001, 0.005, and 0.01 kg kg−1); chartreuse is rime growth (0.1, 1.0, and 2.0 × 10−4 kg kg−1 s−1); magenta is freeze conversion (0.1, 1.0, and 2.0 × 10−4 kg kg−1 s−1); and updraft (20, 30, and 40 m s−1) contours are goldenrod.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
Similar to the 
As in Fig. 8, but for the υmax storms. Gray curves indicate the control run, blue curves indicate vmax10, green curves indicate vmax13, and goldenrod curve indicate vmax16.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
c. Hailstone pseudotrajectories
As discussed in GvdH14, computing hailstone growth trajectories is not possible in bulk microphysics schemes. This is largely because bulk schemes predict only integrated moments of the distribution instead of individual sizes, and these moments can have different weighted fall speeds. Nonetheless, we can construct “pseudotrajectories” that provide some insights into plausible hail pathways through the storm using the composited fields (i.e., assuming steady-state average storm structure for these calculations). We prescribe embryo locations and sizes to initialize a pseudotrajectory. Then, using a 1-s time step, these embryos are advected with the composited storm-relative u and υ fields. Vertical motion is given by w minus the particle’s fall speed, which is determined using the hail fall-speed relationship used in the Morrison two-moment scheme that is based on Matson and Huggins (1980). As the particles enter a region of hailstone growth (as determined by composited microphysical process rate fields), their growth is computed from the composited rime growth rate at that grid box. Throughout the growth, number mixing ratio is held constant at 10 kg−1; though results are quantitatively sensitive to this choice, the qualitative trajectories and insights gained from them are the same. Similarly, though quantitative sensitivity was found for initial embryo sizes (which are arbitrarily chosen based on realistic observed embryo sizes), the qualitative results remain unchanged. Hailstone sizes predicted by these pseudotrajectories should not be taken literally but do show which trajectories encounter larger growth rates, which we can take as a proxy for larger sizes. The calculations are terminated when the hailstones reach the lowest grid level or exit through other domain boundaries. No melting is considered in these calculations.
Though these are an approximation for real hailstone growth trajectories, some confidence in their plausibility is given by their similarity to those found in previous studies (e.g., Nelson 1983; Foote 1984; Miller et al. 1990). Further, the trajectories reach the ground within reflectivity contours and regions of nonzero 
Simulated hailstone pseudotrajectory (blue line) initialized as a 2-mm embryo (green circle). Updraft 20 m s−1 (red) and simulated reflectivity factor 45-dBz (yellow) isosurfaces are shown. White contours at the surface show 30-, 40-, and 50-dBz reflectivity factor at the lowest model grid level.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
For each composited storm, we seed 2-mm embryos at every grid box within the domain for heights from 4.875 to 11.125 km AGL. Though not each of these initial locations is realistic, the resulting calculations show which of these locations result in significant hailstone growth. These potential embryo initial locations can then be compared to the actual locations of hydrometeors in the simulated storms to assess whether the embryos could be realized (i.e., if hydrometeors are unavailable in a region favorable for embryos to grow, the growth is not realized for that case).
Figure 16 shows examples of such calculations for the control simulation. Each panel shows embryos initialized at a particular height. At each height, the embryo initial locations are shaded by the final size attained by that embryo at the lowest grid level (recall that melting is not accounted for in these calculations). The darker-shaded regions indicate that embryos initialized at those locations undergo significant growth during their path through the storm. Figure 16 reveals several important embryo source regions that lead to significant hail growth. First, at lower levels, the right (southern) flank of the storm reveals a region for preferred growth that shifts rearward to the western flank of the storm with increasing height. This is the embryo curtain discussed in numerous previous studies (e.g., Browning and Foote 1976). Its consistency with previous work provides confidence in our pseudotrajectories. Aloft, there is a narrow band of locations west of the updraft that also results in substantial hail growth. Finally, embryos within the updraft below 8 km (Figs. 16a–c) are in favorable locations for significant growth. In real supercells, such embryos would be frozen raindrops, supplied from below in the region observed with polarimetric radar called the “
Initial embryo locations at different heights, shaded by the final hailstone diameter at the surface (according to scale; mm) for the control simulation. Simulated reflectivity factor −20-, −10-, and 0-dBz contours at that height are overlaid (black), as are 10, 20, 30, and 40 m s−1 updraft contours (goldenrod). Shading outside the black (reflectivity) contours indicates that simulated hydrometeors are unavailable at those locations, and thus the initial embryo locations favorable for growth are not realized.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
The embryo curtain or corridor on the south flank of the updraft is partially located within simulated reflectivity contours, suggesting that hydrometeors are present at these locations and may participate in hail growth. However, portions of the embryo corridor (especially aloft) are found outside these reflectivity contours. This implies that these potential embryo pathways are not realized by the simulated storm because there are no hydrometeors present there.
Figure 17 shows the same type of calculations for the umax50 storm. The three main starting locations for embryos are evident in this storm, similar to those in the control run. However, the embryo corridor region is elongated in the direction of the shear vector, similar to the updraft. This provides a broader region of initial embryo locations that can result in significant growth. For example, the umax50 storm has nearly 20% more grid boxes of initial embryo locations that start in regions of ZH > 0 dBz that result in final hailstone sizes >10 mm. Thus, in addition to broader updrafts, enhanced zonal (deep layer) shear results in a larger source volume for hailstone embryos.
As in Fig. 16, but for the umax50 storm. Note the expanded axes.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
In contrast, enhancing the low-level meridional shear for the umax50vmax16 actually decreases the source region for embryos compared to the umax50 case (Fig. 18). The stronger southerly storm-relative winds serve to decrease the hail production by two combined effects. First, hydrometeors that could serve as potential embryos are advected farther north by the stronger southerly storm-relative winds, resulting in a lack of embryos in the potentially favorable region (i.e., notice the ~5-km northward shift of the simulated reflectivity contours from umax50 to umax50vmax16; cf. Figs. 17 and 18). Second, the potentially favorable embryo region is shifted southward, as embryos now take longer northward paths before entering the updraft (cf. Figs. 17a and 18a). In this way, the region of potentially favorable embryo initial locations is separated from the supply region of actual hydrometeors. Thus, increased low-level north–south shear reduces the potential embryo source regions in the embryo corridor.
As in Fig. 17, but for the umax50vmax16 storm.
Citation: Journal of the Atmospheric Sciences 74, 3; 10.1175/JAS-D-16-0066.1
4. Discussion and summary
To investigate the impact of vertical wind shear on hail growth in supercell storms, a series of idealized supercell simulations is performed in which the thermodynamic profile is held constant and the hodograph is systematically altered. Our results show that changes in the environmental hodograph affect the shape, size, and strength of the updraft in simulated supercell storms. This has significant impacts on the hail produced in a given idealized supercell. Composite fields over the last hour of simulation time for each case were constructed to robustly compare structural differences among the different storms.
Increasing the u-component shear in the hodograph, which is equivalent to increasing deep-layer shear, leads to an elongated updraft in the east–west direction. This leads to statistically significant increases in hail mass mixing ratio with increasing shear. Though the magnitudes of the Morrison two-moment microphysical scheme process rates relevant to hail production (grouped into conversion by freezing, conversion by riming, and riming growth) are not significantly different among the different simulations, the increased volume of updrafts over which these process rates contribute to hail growth leads to the observed correlation between hail production and deep-layer zonal shear. As a secondary effect, the broader, weaker updrafts and insignificantly changed horizontal flow within the updrafts may contribute to larger residence times for hailstones. Increasing the υ-component shear, which is the north–south shear in the lowest 3 km AGL, similarly leads to an elongation of the updraft in the north–south direction. However, unlike the u-component shear cases, increasing the low-level meridional shear for a given amount of zonal shear tends to decrease the hail mass mixing ratio.
To determine why, hailstone growth pseudotrajectories were constructed from the composited fields. These pseudotrajectories reveal three main embryo source regions that are consistent with prior studies: (i) the right (south) flank of the updraft, a region known as the embryo corridor; (ii) a narrow ribbon along the rear (west) flank of the updraft aloft; and (iii) within the updraft below 8 km AGL. Increased deep-layer (west–east) shear broadened the embryo source regions (particularly the embryo corridor and the updraft), resulting in a larger volume of the storm over which hydrometeors could start paths that result in substantial growth. However, increased low-level (south–north) shear reduces embryo sources by shifting the embryo corridor southward and shifting hydrometeors that can seed the embryo corridor northward, resulting in minimal overlap between the supply region of actual hydrometeors and the potential initial locations that would lead to significant growth.
In both sets of simulations, increasing shear leads to an elongation of the updraft along the direction of the shear vector. Though our model output fields are insufficient to diagnose the reasons for this elongation, we speculate on two possibilities. First, because the horizontal gradients of vertical velocity on the upshear side of the storm are not significantly different among the different simulations, increased shear would promote larger perturbation pressure gradients across the updraft in the direction of the shear vector (e.g., Rotunno and Klemp 1982, 1985; Markowski and Richardson 2010). This could lead to larger horizontal accelerations for parcels ascending in the main updraft. Second, the enhanced shear case has larger horizontal storm-relative winds within the updraft (Figs. 9, 10), leading to more lateral displacement as parcels ascend. Additional idealized modeling studies in a simplified framework (e.g., dry convection) could provide a more rigorous explanation for this observed behavior.
Clearly, the simulations used herein are highly idealized, and real cases may have substantially different hodograph shapes. However, we opted for the simple quarter-circle hodograph to facilitate a systematic exploration of the parameter space. Thus, a natural question arises: how representative are these results of real cases in which hodographs may be more complex? Though observational studies are limited and beyond the scope of this paper, a recent study by Johnson and Sugden (2014) shows composited hodographs for different size classes of surface hail reports. Based on their cases, a trend can be seen for larger hail reports to be associated with hodographs elongated in the east–west direction. In other words, deep-layer u-component shear was correlated with reports of larger hail size. This observational study thus appears to be broadly consistent with our idealized modeling results.
The results of our study open an avenue for future research that could potentially lend skill to operational severe weather forecasting. Identifying environmental controls favorable for large-hail production given an environment otherwise favorable for convective storms could allow for improved assessment of the conditional large-hail threat. However, spectral bin microphysics treatments are necessary to understand environmental impacts on the surface hail size distribution, which is of chief operational concern. Although suppression is not a major focus of research today, the ability to forecast hail events can allow movement of assets (e.g., cars, aircraft, and people) in advance of storms, potentially reducing damage, and thus reducing cost. Also, anticipation of major hail days will allow insurance companies to staff adequate numbers of claims adjusters.
Some studies (e.g., Nelson 1987; Kumjian and Ryzhkov 2008) have suggested that nontornadic supercell storms are more prolific large-hail producers than tornadic ones. Our finding that increasing υ-component (low level) shear for a given amount of deep-layer u-component shear tends to decrease hail production may be consistent with these suggestions; increased υ-component shear at low levels is associated with increases in storm-relative helicity. Further, for the hodographs considered in this study with similar lengths, the one with larger 0–3-km storm-relative helicity tends to have less hail at the surface. All else being equal, hodographs with greater low-level curvature and/or larger low-level storm-relative helicity would tend to be more favorable for tornadogenesis (e.g., Rasmussen and Blanchard 1998; Thompson et al. 2003, 2007). We speculate that hodographs supportive of tornadic supercells are less supportive for large-hail production because of the changes in storm structure and embryo source regions identified in this study. Thus, it is not simply the total hodograph length (or total shear) that is important for hail production, but rather how it is partitioned, reminiscent of the findings of McCaul and Weisman (2001) for supercell intensity and structure.
The idealized simulations performed herein only occupy a portion of the possible parameter space. As such, this study should be seen as a first step toward exploring the role vertical wind shear plays on controlling hail growth. A remaining question is how melting responds to these environmental changes, given that melting plays a large role in modulating the hail size distribution reaching the surface. Limitations of two-moment schemes like the one used herein prevent a rigorous treatment of this question; future work involving more sophisticated bin microphysics is needed to adequately address the problem of melting and the resultant surface hail size distributions. Future efforts should also expand on the parameter space, using more realistic hodograph shapes and/or hodographs from real cases, as well as investigating how different combinations of CAPE and shear affect hail production. For example, the results found herein should be validated for different thermodynamic environments. In addition to providing insights into controls on hail growth, such studies can improve our basic understanding of how environmental factors affect updraft size. Future studies should use more detailed microphysical treatments (e.g., spectral bin models) to explore the impacts of these environmental factors on hailstone sizes and size distributions reaching the surface to better understand the nature of a given event’s potential hail risk.
Acknowledgments
Support for the authors came from an award from the Insurance Institute for Business and Home Safety (IBHS) to coauthor Kumjian while the first author was an undergraduate student at Penn State University. We thank Drs. Tanya Brown-Giammanco and Ian Giammanco (IBHS) and Yvette Richardson and Paul Markowski (Penn State University) for numerous, beneficial discussions. Dr. Charlie Knight (NCAR-MMM) is thanked for a useful discussion as well. We are grateful to Dr. George Bryan (NCAR-MMM) for his expertise and helpful insights about CM1 and to Dr. Hugh Morrison (NCAR-MMM) for useful information about the microphysics scheme. Three anonymous reviewers provided extremely thorough and helpful reviews that substantially improved the manuscript.
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