Abstract

The HAILCAST hail growth model has been integrated into the Advanced Research version of the Weather Research and Forecasting (WRF-ARW) Model to predict hail size at the ground. Significant updates to the physics of the hail growth model are added, including variable hail density for both wet and dry growth regimes, an updraft multiplier that parameterizes advection of the hail embryo across an updraft, temperature-dependent ice collection efficiency, mass growth by vapor deposition or condensation, and an improved liquid water shedding threshold. Sample hail trajectories from three different updrafts are presented showing the effects of these physical updates. The updraft multiplier in particular improves the representation of the hail growth by not requiring a hail embryo to be locked in the center of an updraft until it grows large enough to fall. Five weeks of hail diameter forecasts are verified using a maximum expected size of hail (MESH) product. At points where WRF successfully forecasts convection, the forecasted hail size is within 0.5 in. 66% of the time.

1. Introduction

Hailstorms are a significant convective storm hazard in the United States, often causing property and crop damage of over $1 billion (U.S. dollars) annually (Herndon 2007, 2010). Aircraft are particularly sensitive to this hazard. For example, 60 airline passengers were killed in 1977 when a DC-9 ingested hail into its engines and crashed near New Hope, Georgia (NSSL 2014). Even hailstones no larger than 10 mm can seriously damage an aircraft in only a few seconds [(Federal Aviation Administration) FAA 1975]. Despite its frequent occurrence, hail still remains a difficult phenomenon to model or forecast. Significant complexities in hailstone growth have been highlighted in the literature, including how hailstones tumble while falling (Knight and Knight 1970), how water is retained on the hailstone surface and when that water is shed (Chong and Chen 1974; Rasmussen and Heymsfield 1987; Miller et al. 1988; Garcia-Garcia and List 1992; Phillips et al. 2014), how transitions from wet to dry growth regimes occur (Lesins and List 1986; Garcia-Garcia and List 1992), and how their density varies (Heymsfield 1978; Knight and Heymsfield 1983; Ziegler et al. 1983; Gilmore et al. 2004; Knight et al. 2008). These significant complexities ensure that understanding, and hence modeling or forecasting, hailstone growth is a difficult proposition.

Current warning methods involve inferences from radar data of the 50-dBZ echo and melting level heights (Donavon and Jungbluth 2007) or vertically integrated liquid (VIL; Edwards and Thompson 1998). However, such methods can only be applied to a storm that already exists and has been detected by radar, and therefore are more of a “nowcast” than a forecast. More recently the HAILCAST one-dimensional coupled cloud and hail model was developed by Poolman (1992) and improved upon by Brimelow et al. (2002). HAILCAST forecasts the maximum expected hail diameter at the surface using a profile of nearby atmospheric temperature, moisture, and winds. A version of the model updated by Jewell and Brimelow (2009) has been in use at the Storm Prediction Center for over 10 years. However, with the advent of “convection-allowing” models (CAMs), the cloud model portion of HAILCAST can be replaced by the prognostic variables from these models. In this work, the HAILCAST hail model has been significantly updated and one-way coupled directly with a widely used CAM, the Advanced Research version of the Weather Research and Forecasting (WRF) Model (WRF-ARW; Skamarock et al. 2008). Such a coupling uses the more physically realistic updraft and microphysical information produced by the three-dimensional WRF, as compared to the 1D cloud model originally used in HAILCAST. Coupling HAILCAST with WRF also produces hail forecasts along model simulated storm tracks at finer temporal and spatial scales, allowing comparison of forecast hail sizes among individual convective systems, or evaluation of forecast hail size over time throughout a convective life cycle.

Section 2 describes the original HAILCAST model. Section 3 details the updates made to the HAILCAST hail model and its implementation within WRF. Section 4 presents WRF-HAILCAST hailstone trajectories through three different updrafts, and section 5 presents a case study forecast and some sensitivity tests. Section 6 discusses verification statistics of experimental HAILCAST forecasts during two multiweek periods in 2014 and 2015.

2. Original HAILCAST model

HAILCAST consists of a 1D, steady-state cloud model coupled with a time-dependent hail growth model (Brimelow et al. 2002). A profile of atmospheric temperature, moisture, and winds is used to drive the cloud model, which produces simulated vertical profiles of vertical velocity, liquid and ice water content, and temperature associated with a cloud. Allowance is made for the impacts of water loading and entrainment on the updraft. These profiles are assumed to be steady state when passed to the hail growth model. Coupling a steady-state cloud model with a time-dependent hail model is complicated as the steady-state updraft neither initiates nor dissipates. Brimelow et al. (2002) chose to limit the lifetime of the updraft based on the energy shear index (ESI), the product of the surface-based CAPE and 850 hPa–6 km AGL wind shear. Larger values of the index allow a longer-lasting updraft. Updraft lifetimes are bounded between different intervals among different versions of HAILCAST. The version used in this study was obtained from the Storm Prediction Center (SPC), and updraft lifetimes are bounded between 10 and 60 min.

The hail growth model inserts a liquid embryo at the cloud base. The embryo is given initial upward motion of 4 m s−1, and at each time step its vertical position is calculated based on its terminal velocity and the updraft vertical velocity interpolated to that height. As time advances, the embryo travels upward in the updraft, accreting liquid water as it rises. When it passes above −8°C, it automatically freezes. The mass budget of the hailstone is then tracked and the heat balance equation solved at every time step, making the hailstone temperature a time-dependent variable. (Hailstone here is simply defined as a spherical ice particle.) The hailstone enters wet or dry growth regimes (as appropriate) depending on its internal temperature and the in-cloud temperature. If the hailstone grows larger than 9 mm in diameter, any liquid water on its surface above a mass threshold is shed. Hailstone terminal fall speed is calculated using the method of Rasmussen and Heymsfield (1987), incorporating the Reynolds number and ventilation coefficients. Once the hailstone becomes too large to be sustained aloft by the updraft, or if the updraft is shut off, it starts its descent. When it falls below the cloud base, the cloud moisture profile is set to 0. In this version of HAILCAST, for input soundings designed as “elevated” the hailstone melting rate is calculated using the mean wet bulb temperature in the subcloud layer.

Key assumptions, in addition to the steady-state updraft, include a constant ice density of 900 kg m−3, a spherical hailstone, and a 300-μm initial liquid embryo diameter. Cloud ice also only exists between −20° and −40°C, and its amount is determined by an exponential relation from Vali and Stansbury (1965). Further details of the HAILCAST cloud and hail models can be found in Poolman (1992), Brimelow et al. (2002), and Jewell and Brimelow (2009).

3. New WRF-HAILCAST model

a. Implementation in WRF

One of the largest assumptions made in HAILCAST is the use of the cloud model to simulate a thunderstorm updraft. The WRF Model, when run at a horizontal grid spacing of 4 km or finer, can reproduce the dominant, larger-scale circulations and hydrometeor fields associated with organized storms and convective systems (Weisman et al. 1997; Kain et al. 2006). While Bryan et al. (2003) found that deep moist convective processes such as entrainment or overturning are only accurately resolved at very fine grid spacings on the order of 100 m, they also noted that 1-km simulations are able to reproduce the basic storm-scale structure of a cumulonimbus cloud. The array of operational WRF simulations currently running at horizontal convection-allowing grid spacings of 1–4 km contains more valuable physical information about convective updrafts than the 1D cloud model updrafts simulated by the original HAILCAST, which are based on fields of mesoscale storm updraft ingredients rather than actual predicted storms. Fields of updraft helicity produced by WRF forecasts of 4-km grid spacing have been shown to correlate well with observed tornado tracks (Sobash et al. 2011; Clark et al. 2012; Gallo et al. 2016), indicating the skill of WRF in simulating storm-scale convective circulations and updrafts. While the results shown in sections 4, 5, and 6 are from model runs at 4-km horizontal grid spacing, WRF-HAILCAST has been tested on grid spacings as fine as 1 km (not shown) and it produces realistic forecasts without modification.

The time-dependent hail model would ideally be incorporated to run inline within WRF, allowing the embryo to move in three dimensions through the storm. However, that could involve incorporating up to millions of passive tracers into a simulation, a highly computationally intensive endeavor. Instead, the vertical updraft, liquid and ice water content, and temperature profiles from a given WRF time step and grid column are passed to the time-dependent WRF-HAILCAST hail model (Fig. 1). Embryos are inserted into the updraft and tracked as they rise, grow, and fall. The resulting hail forecast is passed back to WRF, which stores that data and then moves to its next time step. The only data that are presently passed from WRF-HAILCAST back to WRF is hail size information at the surface.

Fig. 1.

Conceptual model of WRF-HAILCAST processing; A is the term in brackets in (1).

Fig. 1.

Conceptual model of WRF-HAILCAST processing; A is the term in brackets in (1).

Ice water content is determined using the cloud ice and snow mixing ratios from the WRF Model. The cloud water mixing ratio profile from WRF, however, has proved to be an inadequate representation of the supercooled water profile available for hailstone growth by riming, because it has already been depleted by collisions with graupel and autoconversion to rain within the microphysical parameterization. The spatially rather smooth horizontal profiles of CAM updraft and cloud fields are unable to resolve the localized bounded weak echo region (BWER) with its reduced precipitation content and scavenging and adiabatic cloud water profiles that are ubiquitous features of observed hailstorm updraft cores. To determine the vertical profile of the maximum cloud liquid water content that could be maintained within the column in the absence of the latter depletion processes, the difference between the water vapor mixing ratio at cloud base and the saturation vapor mixing ratio throughout the cloud column (the latter dependent on the model temperature and pressure) is integrated over the depth of the cloud (Kessler 1969). The new adiabatic profile of cloud water mixing ratio is assumed to be glaciated using a linear function decreasing from 1 to 0 between −31° and −38°C, following observations from Rosenfeld and Woodley (2000).

Hailstone growth by collision with precipitation-sized particles (i.e., graupel or rain) in WRF-HAILCAST is ignored. While these collisions can occur, they are rare (Knight and Knight 1970; Knight et al. 2008) due to the low concentrations of precipitation-sized particles compared to cloud particles. If a collision does occur, a graupel particle is likely to bounce off a hailstone (e.g., Knight et al. 2008, their Fig. 11 and discussion). Thus, continuous collection in the presence of rain or graupel mixing ratios is excluded. However, continuous collection in the presence of snow mixing ratios is included, due to the difficulty of determining when to convert cloud ice to snow within WRF microphysical parameterizations (Morrison and Milbrandt 2015).

In contrast with the original HAILCAST model, hailstone growth in WRF-HAILCAST is not assumed to be confined within the 1D updraft core until the hailstone grows large enough to overcome the updraft and fall to the ground. Previous observational and modeling studies have found that hailstone embryos are typically located on the edge of the updraft in the upshear portion of the storm or a feeder cell (Ludlam 1958; English 1973; Heymsfield et al. 1980; Heymsfield 1982, 1983a,b; Heymsfield and Musil 1982; Ziegler et al. 1983; Foote 1984). The embryo is advected horizontally into the main portion of the updraft. It increases its mass via accretion riming of supercooled water and aggregation of snow and ice crystals, and traverses the updraft horizontally while simultaneously increasing its fall speed and moving vertically in the updraft (Heymsfield et al. 1980; Heymsfield 1982; Heymsfield and Musil 1982; Nelson 1983; Heymsfield 1983a,b; Ziegler et al. 1983; Foote 1984; Miller et al. 1990). The horizontal motion of hail in 3D flows can result from convergent (divergent) flow below (above) the updraft maximum, a tilted updraft, or a mesocyclonic or mesoanticyclonic updraft (e.g., Ziegler et al. 1983; Miller et al. 1990). To approximately simulate the embryo’s horizontal motion relative to the updraft in a Lagrangian (particle following) sense, a time-dependent multiplier of the following form:

 
formula

is used to calculate the hailstone-relative updraft speed at any level z for any internal WRF-HAILCAST time τ, where is the updraft speed encountered by the hail particle, is the updraft profile from WRF, and is the updraft duration time limit. The particular chosen form of the sinusoidal weighting term in (1) provides an initially weaker updraft to help sustain early embryo growth following the aforementioned hail growth studies.

To determine the updraft duration in (1) a new WRF diagnostic WDUR (updraft duration) is introduced. At each model time step, the updraft fields at the current and previous time steps are compared. If either a grid column or any of its adjacent grid columns has a maximum updraft exceeding 10 m s−1 at both times, the updraft duration field in that column is incremented by one WRF time step. In this way, updraft objects and their lifetimes are tracked as they move across the model grid, similar to the methods of Clark et al. (2012). This derived updraft object duration field could be useful for convective algorithms generally. In WRF-HAILCAST, the updraft duration variable is used to prescribe the updraft time limit in the hail model. At coarser CAM grid spacings (e.g., 4 km), this tracking method identifies convective cells or organized convective systems rather than individual convective updrafts. “Updraft durations” of 3–4 h are common as the algorithm tracks long-lived observed supercells or mesoscale convective systems instead of CAM-simulated updrafts. Having a hailstone stay in an updraft for 3–4 h is not physically realistic, so in (1) is capped at a maximum value of 2000 s in WRF-HAILCAST. Such a limit is consistent with previous observational studies of updraft and hail growth lifetimes in multicell and supercell storms of up to about 30 min (Dennis et al. 1970; Chalon et al. 1976; Fankhauser et al. 1982; Knight et al. 1982).

The computational aspects of the WRF-HAILCAST model are as follows (Fig. 1). At each WRF time step and within any grid column that contains an updraft of 10 m s−1 or larger, the updraft duration field must be 15 min or longer (i.e., the updraft must have existed for at least 15 min, but not necessarily at that grid cell). If both these tests are passed, then the vertical velocity, temperature, and cloud water, cloud ice, and snow mixing ratio profiles from that grid column are passed to the WRF-HAILCAST hail model. Five embryos are inserted at either −8° or −13°C. These temperatures were chosen to correspond with the location of initial radar echoes reported by Chalon et al. (1976) and Foote and Wade (1982), as well as the findings of Heymsfield et al. (1980) that embryos located in the dendritic growth temperature range are frequent hail producers. The updraft profile then varies with respect to time as described in (1). The temperature and cloud liquid and ice water profiles are considered to be in locally steady state relative to the growing hailstones until is reached. At that point the hailstones are assumed to have exited the cloud, the updraft speed is set to zero following (1), and the liquid and ice water profiles are set to zero. Once the hailstones reach the surface, their diameters are passed back to WRF. Then WRF moves to its next time step, and is incremented. The new updraft properties are passed to WRF-HAILCAST, five new embryos are tracked as they travel through the new updraft, and another hail size forecast is produced. WRF stores the maximum hail diameter returned from WRF-HAILCAST in each grid column. This process continues until any given updraft column no longer contains a vertical motion of 10 m s−1 or larger and a WDUR of at least 15 min.

The HAILCAST hail model is executed once for each of five different embryos at each WRF time step. Ziegler et al. (1983) suggested that employing a range of initial embryo sizes would better estimate the spectrum of hail sizes produced by a storm. Extensive testing of WRF-HAILCAST confirmed a strong sensitivity of forecast hail diameter to the initial embryo size and the initial temperature at which the embryo was inserted (shown in section 4a). Five specific combinations of initial embryo size and temperature were chosen after evaluating the performance of hail forecasts produced by a range of size and temperature combinations during the five-week period described in section 6. Embryos of sizes 5 and 7.5 mm are inserted at −8°C, and sizes 5, 7.5, and 10 mm are inserted at −13°C. These embryos sizes coincide with observations of hailstone embryos. Magono and Nakamura (1965) observed aggregates of up to 15–20 mm serving as hailstone embryos, although Heymsfield (1982) found that aggregates of 5–7 mm were a more preferred size as they needed less time to develop than the larger-sized aggregates. Heymsfield and Musil (1982) observed particles of up to 10 mm on the edge of the updraft, and even some particles within the updraft core of no more than 2–3 mm. The modeling study of Heymsfield (1982) found that aggregates of up to 10 mm, and crystals, graupel, and frozen drops of approximately 1 mm, can all serve as effective hailstone embryos. Analyses of in situ hail samples obtained using thin-sectioning and deuterium analysis have implied the existence of plentiful millimetric or larger frozen drop and graupel embryos (Knight 1981) that form at a range of subfreezing in-cloud temperatures (Macklin et al. 1977; Ziegler et al. 1983; Nelson and Knight 1983). A wider range of initial embryo sizes will be shown in the sensitivity tests in the next section.

When originally designing WRF-HAILCAST, percentile points along the graupel size distribution from the WRF microphysical parameterization were used to determine the initial embryo size. Unfortunately, the graupel size distributions varied widely among various existing microphysical parameterizations. As an example, the wide variation in the lower 20th percentile of the graupel size distribution from three different microphysical parameterizations is evident in a histogram of initial embryo sizes from a WRF run (Fig. 2; WRF run described in section 4). Because of these large differences in retrieved initial embryo sizes among parameterizations, the resulting WRF-HAILCAST hail forecasts were highly dependent upon the chosen microphysical parameterization. Thus, a constant set of initial embryo sizes were chosen from the observations as previously described.

Fig. 2.

Histogram of initial embryo size, as retrieved from the 20th percentile of the graupel size distribution for the Thompson (light gray), Morrison (red), and WDM6 (blue). Results are from the 27–28 Apr 2011 WRF simulation described in section 5.

Fig. 2.

Histogram of initial embryo size, as retrieved from the 20th percentile of the graupel size distribution for the Thompson (light gray), Morrison (red), and WDM6 (blue). Results are from the 27–28 Apr 2011 WRF simulation described in section 5.

WRF-HAILCAST outputs maximum, mean, and standard deviation hail diameters for the previously described set of five embryos. In the verification analyses presented in section 6, the maximum hail diameter is used. The mean and standard deviation are provided so that information is available about the uncertainty inherent in the hail forecast. The largest values of the maximum and mean hail diameters (and associated standard deviation) produced at each grid point are tracked and output in each WRF history output file, then reset after the output file is generated, similar to the “hourly maximum fields” described by Kain et al. (2010).

An internal time step for WRF-HAILCAST of 5 s was chosen after experimentation. This value provided a balance between the desires of fast processing and accurate tracking of the hailstone through the updraft. A time-differencing method was added to the calculation of the hailstone temperature to avoid instabilities at small hailstone sizes.

b. Modifications to the WRF-HAILCAST hail model

1) Variable hailstone density

The importance of variable rime layer and hailstone density when simulating the accretional growth of ice particles in convective storms has been noted in several modeling studies that either simulate individual Lagrangian hail growth trajectories in radar-observed storms (Ziegler et al. 1983; Foote 1984; Miller et al. 1988, 1990), or simulate storms using multidimensional cloud-mesoscale models (Milbrandt and Morrison 2013; Morrison and Milbrandt 2015). In WRF-HAILCAST, the density of the frozen hailstone is varied similarly to Ziegler et al. (1983), which uses the basic form of rime layer density originally provided by Macklin (1962). The initial embryo density in WRF-HAILCAST is now set to 500 kg m−3. If the growing hailstone subsequently enters a wet growth regime, any unfrozen accreted water is assumed to soak into the hailstone if its density is below 900 kg m−3. An ice layer that forms during wet growth is assumed to have a density of 900 kg m−3.

During the dry growth regime, the density of a new ice layer that forms by accretion of supercooled water is determined by Eqs. (15) and (16) from Heymsfield and Pflaum (1985):

 
formula
 
formula

Here is the hailstone rime layer density (kg m−3), r is the mean cloud droplet radius (μm), is the cooling of the hailstone surface below 0°C, and is the impact velocity of the cloud droplet on the hailstone, calculated using the method of Rasmussen and Heymsfield (1985). The cloud droplet radius r is calculated using the adiabatic cloud water mixing ratio and an assumed cloud droplet concentration of 300 cm−3, consistent with continental regimes (Thompson et al. 2004).

During periods in which the new ice layer is formed by collision and aggregation with cloud ice or snow, the new layer density is assumed to be 700 kg m−3 following observations of cloud ice density from Heymsfield (1972). Other densities were tested, but WRF-HAILCAST was not sensitive to the assumed cloud ice density as most hailstone growth occurs by riming in regions of liquid water.

2) Hailstone vapor growth and ice collection efficiency

Within the previous HAILCAST hail model, the ice collection efficiency was a step function, equal to 1 if the environmental temperature was warmer than −5°C, and 0.21 if colder. The ice collection efficiency was changed to a linear function that varies from 1 at 0°C to 0 at −40°C, similar to the method of Ziegler et al. (1983). The liquid water collision/collection efficiency is 1 for all temperatures. The temperature dependence of the ice collection efficiency was added because of the increased ability of an impacting ice particle to stick to the hailstone at warmer temperatures, particularly if the hailstone has a liquid layer on its surface caused by collected cloud droplets in the process of freezing. Ice particles are also more likely to be larger and, therefore, more likely to undergo collisions with hail at relatively warmer temperatures, particularly in the dendritic zone (Pruppacher and Klett 1997). Figure 3 compares the previous and new ice collection efficiency functions.

Fig. 3.

Original (dashed) and new (solid) ice collection efficiency functions.

Fig. 3.

Original (dashed) and new (solid) ice collection efficiency functions.

WRF-HAILCAST also now includes mass growth by vapor deposition or condensation following this expression:

 
formula

where is hail diameter; is the water vapor ventilation coefficient; is diffusivity of water vapor in air; is the gas constant for water vapor; and and T are, respectively, the saturation vapor pressure and temperature in the environment () and at the surface of the stone (). The parameters and are defined as in Rasmussen and Heymsfield (1987) and Pruppacher and Klett (1997).

3) Hailstone melting and shedding

The previous version of HAILCAST allowed water to be retained on the surface of the hailstone during wet growth if the hailstone diameter was smaller than 9 mm or if the water mass did not exceed a critical limit that depended on the hailstone ice mass (Brimelow et al. 2002). These threshold values for retained surface water layer thickness and mass were determined by Rasmussen and Heymsfield (1987) to be valid for nontumbling hailstones. However, falling hailstones are likely to tumble, even at sizes smaller than 9 mm (Knight and Knight 1970; Heymsfield and Hjelmfelt 1984; Lesins and List 1986; Phillips et al. 2014). Hail in WRF-HAILCAST is assumed to tumble and efficiently shed excess water in the process, thus maintaining no more than a thin liquid surface layer both during wet growth and melting (List and Dussasult 1967; Lesins and List 1986; List 1990; Garcia-Garcia and List 1992). Although unfrozen water is retained on the hailstone surface, excess water is assumed to be shed when it reaches above a threshold mass of 2 × 10−4 kg. A constant mass threshold was chosen as Chong and Chen (1974) found that thinner layers of water can be retained on larger hailstones. Furthermore, while Lesins and List (1986) found that the amount of liquid water retained on the hailstone does vary with respect to the hailstone mass, the amount retained was more highly sensitive to the tumbling rate of the hailstone, the calculation of which is beyond the scope of this hail model.

Finally, WRF-HAILCAST now contains an enhanced melting rate when the hailstone collides with liquid water below the freezing level, taken from Eqs. (16)–(40) and (16)–(80) in Pruppacher and Klett (1997).

c. Processing time considerations

Because WRF-HAILCAST only runs at grid points with updrafts of at least 10 m s−1, the resulting increase in processing time required for a simulation varies with the time-varying coverage of deep convection. The ratio between the processing times needed for each WRF time step in comparative runs with or without WRF-HAILCAST over the domain discussed in section 6 is slightly larger than unity during the periods 0000–0600 UTC 12–13 May (Fig. 4a) and 13–14 May 2014 (Fig. 4b). These domains covered the full continental United States, and the days chosen for simulation were heavily convective, representing the upper end of required extra processing time. The total increase in processing time required for the simulation was 3% during 12–13 May and 2% during 13–14 May 2014. Further decreases in processing time could be obtained by calling WRF-HAILCAST at regular subintervals within WRF (e.g., every 10 time steps) without degradation of the hail forecast.

Fig. 4.

Ratio between processing time required for each WRF time step for a run with WRF-HAILCAST and a run without: (a) 0000 UTC 12 May–0600 UTC 13 May and (b) 0000 UTC 13 May–0600 UTC 14 May 2014. Ratio of total processing required for each simulation given in text in each figure.

Fig. 4.

Ratio between processing time required for each WRF time step for a run with WRF-HAILCAST and a run without: (a) 0000 UTC 12 May–0600 UTC 13 May and (b) 0000 UTC 13 May–0600 UTC 14 May 2014. Ratio of total processing required for each simulation given in text in each figure.

4. Sensitivity tests

The characteristics of three updrafts from selected WRF grid columns in the case study simulation discussed in the next section are shown in Fig. 5. Even though those example updrafts represent only a small fraction of the total parameter space encompassing potential hail-bearing updrafts in the CAM simulations, they usefully depict some broad characteristics of WRF-HAILCAST outputs. Hailstone trajectories through these updrafts have been obtained in several sensitivity tests of the WRF-HAILCAST hail model. The model described in the previous section corresponds to the control (CONTROL), while the sensitivity tests are listed in Table 1. The strongest updraft (Fig. 5a) contains peak speeds of almost 25 m s−1 and substantial supercooled liquid water; conditions that will be shown to produce a maximum hail diameter of 31.2 mm in the CONTROL model. A “dry” updraft (Fig. 5b) consists of an updraft with peak magnitudes on the same order as the strong updraft, but with a higher cloud base and less supercooled water. The dry updraft produces a maximum hail diameter of 36.7 mm. The weakest updraft (Fig. 5c) minimally exceeds the 10 m s−1 updraft threshold for WRF-HAILCAST, has a high cloud base above the freezing level, and produces no hail. The ratio of the peak values of WRF-simulated cloud liquid water to the adiabatic value at the same level are about 1:3 in the strong and weak updraft cases, and about 1:1 below 7 km in the dry updraft case.

Fig. 5.

Updrafts characteristics from three different grid columns within a WRF simulation discussed in section 4. The teal line is the vertical velocity (m s−1), the light gray line is the WRF cloud water mixing ratio (g kg−1), the dashed light gray line is the adiabatic cloud water mixing ratio (g kg−1), and the dark gray line is the combined snow and cloud ice mixing ratio (g kg−1). (a) The “strong” updraft, (b) the “dry” updraft, and (c) the “weak” updraft.

Fig. 5.

Updrafts characteristics from three different grid columns within a WRF simulation discussed in section 4. The teal line is the vertical velocity (m s−1), the light gray line is the WRF cloud water mixing ratio (g kg−1), the dashed light gray line is the adiabatic cloud water mixing ratio (g kg−1), and the dark gray line is the combined snow and cloud ice mixing ratio (g kg−1). (a) The “strong” updraft, (b) the “dry” updraft, and (c) the “weak” updraft.

Table 1.

WRF-HAILCAST model sensitivity test configurations.

WRF-HAILCAST model sensitivity test configurations.
WRF-HAILCAST model sensitivity test configurations.

a. Initial embryo size and temperature

Figure 6 shows hailstone trajectories in the CONTROL configuration of WRF-HAILCAST, through three updrafts. It also shows the trajectory of two hailstones with a smaller initial embryo size, 0.9 mm, inserted at −8° and −13°C for comparison. The seven different hailstone trajectories from the strong updraft (Fig. 6a) display two different scenarios. A large amount of supercooled water is available at both insertion temperatures (Fig. 5a). Because of the time-dependent modifier applied to the updraft strength, all five embryos have some time to grow before they are assumed to enter the updraft core. However, the 0.9-mm embryos are quickly lofted above the supercooled liquid water to heights mainly containing cloud ice. Because the ice collection efficiency decreases linearly with temperature the hailstone grows little, and quickly melts below the freezing level. Conversely, the larger 5-, 7.5-, and 10-mm embryos are able to grow large enough to hover in the supercooled water layer even after entering the updraft. While in the supercooled water region the hailstones transition from a rather brief initial period of dry growth into a wet growth regime and subsequently become very dense. They grow quickly, eventually reaching diameters between 29.7 and 31.2 mm.

Fig. 6.

Hailstone trajectories from hailstones in (a) the strong updraft, (b) the dry updraft, and (c) the weak updraft. (from top to bottom) Height (km), density (kg m−3), diameter (mm), and the hailstone temperature (K). The six colors correspond to four different initial embryo sizes, 0.9, 5, 7.5, and 10 mm, and two insertion temperatures of −8° and −13°C. The 0.9-mm embryo lines are dotted and dashed as that embryo is not included in the CONTROL model.

Fig. 6.

Hailstone trajectories from hailstones in (a) the strong updraft, (b) the dry updraft, and (c) the weak updraft. (from top to bottom) Height (km), density (kg m−3), diameter (mm), and the hailstone temperature (K). The six colors correspond to four different initial embryo sizes, 0.9, 5, 7.5, and 10 mm, and two insertion temperatures of −8° and −13°C. The 0.9-mm embryo lines are dotted and dashed as that embryo is not included in the CONTROL model.

In the dry updraft, the embryos are inserted near the height of the peak updraft as in the strong updraft. However, the hailstones spend a longer period in lower density dry growth regimes (Fig. 6b) because of the lower cloud liquid water contents (Fig. 5b). The 5-mm embryo inserted at −8°C (red line in Fig. 6b) in particular spends approximately 500 s in a low-density dry growth regime before entering a wet growth regime above 8 km where liquid water contents are slightly higher. At that point its density quickly increases due to rime soaking. Both 5-mm embryos, and the 7.5-mm embryo inserted at −8°C, are able to grow to a slightly larger size than corresponding embryos in the strong updraft, because their initial low-density growth means the same amount of mass accumulation results in larger diameter growth. Conversely, the 0.9-mm embryos follow very similar trajectories as in the strong updraft, being quickly lofted above the supercooled water layer and growing little.

Interestingly, only the 0.9-mm embryos produce nonzero size hailstones in the weak updraft (Fig. 6c). WRF-HAILCAST requires that hailstones either be lofted above their insertion point or remain aloft for at least 15 min to ensure, for example, that the 10-mm embryo does not simply fall to the surface and be forecast as a 10-mm hailstone. Thus the 5-, 7.5-, and 10-mm embryo hailstone diameters are set to zero once they reach the surface. These larger embryos are unable to remain aloft in the weak updraft, which additionally contains little liquid water. However, the two 0.9-mm embryos are small enough that they can hover around 6 km, accreting a mixture of supercooled water and snow and ice crystals (Fig. 5c). Because of the lesser cloud liquid water amounts, the hailstones remain in dry growth with a lower bulk density until the updraft multiplier increases enough to loft them into a layer with larger cloud water mixing ratios. At that point the hailstones enter a wet growth regime and quickly undergo rime soaking, subsequently increasing their density to 900 kg m−3. The 0.9-mm embryo inserted at −13°C is able to ascend farther because of its higher starting point, affording it more time for accretion and growth. It reaches a final hail diameter of 15.4 mm, compared to the final hail diameter of 10.2 mm for the embryo inserted at −8°C.

b. Constant hail density

Figure 7 shows two sets of three different hailstone trajectories through the strong updraft. The left column compares three initial embryo sizes (2, 7.5, and 10 mm) in the CONTROL run to a sensitivity test with a constant hailstone bulk and layer density of 900 kg m−3 (run CONSDENSE900). The initial embryo density is 500 kg m−3 in CONTROL, so the same diameter embryo in the CONSDENSE900 test with its 900 kg m−3 density would have more mass and, therefore, initially fall slightly faster. The 7.5- and 10-mm embryos follow similar paths in CONTROL and CONSDENSE900. In the control runs these hailstones quickly reach a density of 900 kg m−3 due to rime soaking, so assuming a constant density of 900 kg m−3 is not significantly different. The CONTROL run hailstones produced are slightly larger, as the slightly slower initial fall speed allows them to be lofted higher into a region with more supercooled water.

Fig. 7.

(left) CONTROL and CONSDENSE900 and (right) CONTROL and CONSDENSE500 hailstone trajectories from hailstones in the strong updraft (Fig. 5a). Rows and line colors for the 7.5- and 10-mm embryos are as in Fig. 6, and the orange lines are from a 2-mm embryo inserted at −13°C. Solid lines are from CONTROL and dashed lines are from CONSDENSE900 and CONSDENSE500.

Fig. 7.

(left) CONTROL and CONSDENSE900 and (right) CONTROL and CONSDENSE500 hailstone trajectories from hailstones in the strong updraft (Fig. 5a). Rows and line colors for the 7.5- and 10-mm embryos are as in Fig. 6, and the orange lines are from a 2-mm embryo inserted at −13°C. Solid lines are from CONTROL and dashed lines are from CONSDENSE900 and CONSDENSE500.

Conversely, the 2-mm embryo paths show large differences. If the 2-mm embryo were a frozen raindrop with a bulk density of 900 kg m−3 instead of a graupel particle with an initial bulk density of 500 kg m−3 (Heymsfield 1982), its larger fall speed could allow it to remain within the prime growth layer. The faster fall speed thus ensures that the CONSDENSE900 embryo is able to fall into the supercooled water layer (Fig. 5b) and grow in mass before the updraft multiplier increases. The CONTROL 2-mm embryo remains lofted above the supercooled water layer until the updraft is shut off when the HAILCAST time exceeds , here 2000 s, and then falls to the surface. Allowing hailstone density to vary results in a surface hail diameter difference of 20 mm for this embryo, although it is again noted that such results are likely sensitive to the assumed size, bulk density, and initial altitude of the five embryos in WRF-HAILCAST.

The right column of Fig. 7 corresponds to a sensitivity test with a constant hailstone bulk and layer density of 500 kg m−3 (CONSDENSE500). This test removes the factor of different initial embryo fall velocities due to different initial densities. Here the 7.5- and 10-mm embryos produce 25-mm larger hailstones in CONDSDENSE500 than in CONTROL. The hailstones take very similar trajectories, suggesting their accumulated mass is similar, but in CONSDENSE500 the new growth occurs at a lower density so the same amount of mass translates to a larger volume and diameter. The 2-mm embryos are both lofted above the supercooled water layer, and although the density of the CONSDENSE500 hailstone is still significantly smaller than the CONTROL, the growth rates of both are so small that the density differences are unimportant.

These results indicate that the effects of the variable density in WRF-HAILCAST are different for different initial embryo sizes. Ziegler et al. (1983) found similarly that variable hail density hailstones could be larger or smaller than constant density hailstones. For example, they found that the variable density could either make the hailstone heavier, forcing it to fall out of the updraft faster, or heavy enough to allow it to hover within the prime growth region.

c. Updraft multiplier

The sensitivity test ONOFF-UPDRAFT explores the effect of the updraft multiplier (1). In this test, the vertical wind speed is set to its full, constant value, before being shut off at (2000 s). Figure 8 shows hailstone trajectories for five different embryos in the strong updraft (Fig. 5a). All five embryos are quickly lofted above their insertion points; the 0.9- and 5-mm embryos in particular are lofted above the top of the supercooled water region (Fig. 5a). At such temperatures with a dry hailstone surface the ice collection efficiency is also quite low. Consequently all three hailstones hover in the updraft and grow very little. When the updraft is shut off at 2000 s, the ONOFF hailstones simply fall out of the column, with the 0.9-mm diameter embryo taking over 40 min to reach the surface (Fig. 8a). Such trajectories are not physically representative of hailstone growth (Heymsfield et al. 1980; Ziegler et al. 1983). The 7.5- and 10-mm embryos follow a more physically realistic trajectory as they are able to grow fast enough to fall back through the supercooled water region, reaching a large enough size that their fall speed overcomes the updraft speed and they can reach the surface as hail.

Fig. 8.

CONTROL (solid) and ONOFF-UPDRAFT (dashed) hailstone trajectories from hailstones in the strong updraft (Fig. 5a). Rows and line colors are as in Fig. 6, but one of the 7.5-mm embryos is omitted for clarity.

Fig. 8.

CONTROL (solid) and ONOFF-UPDRAFT (dashed) hailstone trajectories from hailstones in the strong updraft (Fig. 5a). Rows and line colors are as in Fig. 6, but one of the 7.5-mm embryos is omitted for clarity.

d. Ice collection efficiency

The new ice collection efficiency () function (Fig. 3) accounts for the larger size and, therefore, higher collection efficiency of small precipitating ice particles (labeled snow in WRF microphysical parameterizations) at warmer temperatures. The largest differences in hailstone behavior would be expected to be in the −5° to −20°C range where the new is significantly larger than the old , or at very cold temperatures where the old is larger than the new . Figure 9 shows hailstone trajectories in both situations, where the STEP-EI sensitivity test uses the old step function for ice collection efficiency. In the left column of the strong updraft, the 0.9-mm diameter embryos are lofted to the very cold region of the column. The larger in STEP-EI allows the hailstone to continue growing and begin falling out of the updraft due to its large size. Such growth at extremely cold temperatures (−35°C and colder) is physically unlikely due to the lack of a known effective mechanism for the impacting ice particle to stick or interlock with the riming. The CONTROL hailstone does not grow, only falling out of the updraft as the updraft slows and shuts off entirely after 2000 s. In the right column in the weak updraft, the reverse situation occurs. The CONTROL 0.9-mm diameter embryo initially grows more quickly due to higher values in the −15° to −20°C range, allowing the hailstone to hover in the prime growth layer. The larger CONTROL hailstone then falls out of the updraft faster than the more slowly growing STEP-EI hailstone, which grows to a larger size by virtue of remaining aloft for a longer time. The larger 5-, 7.5-, and 10-mm embryos, which spend most of their time in the layer with high concentrations of supercooled water, grow mostly by riming and, hence, see little difference in the CONTROL and STEP-EI runs.

Fig. 9.

CONTROL (solid) and STEP-EI (dashed) hailstone trajectories from hailstones in (left) the strong updraft (Fig. 5a) and (right) the weak updraft (Fig. 5c). Rows and line colors are as in Fig. 8.

Fig. 9.

CONTROL (solid) and STEP-EI (dashed) hailstone trajectories from hailstones in (left) the strong updraft (Fig. 5a) and (right) the weak updraft (Fig. 5c). Rows and line colors are as in Fig. 8.

5. 27 April 2011 case study

a. Background

A devastating tornado outbreak occurred over the southeastern United States on 27 April 2011. A full description of the tornado outbreak and its damage is available in Knupp et al. (2014). Much of the hail was of 1-in. (approximately 2.5 cm) diameter but some was as large as 4 in. (NCDC 2011). The day 1 severe hail outlook from the National Oceanic and Atmospheric Administration (NOAA) Storm Prediction Center along with hail storm reports as recorded in the NOAA/National Climatic Data Center (NOAA/NCDC) Storm Data are presented in Fig. 10. This specific event provides the opportunity to test WRF-HAILCAST’s ability to simulate convection without hail, and convection with multiple sizes of hail in different convective modes, all in a single case study.

Fig. 10.

SPC day-1 hail probability outlook valid from 1300 UTC 27 Apr to 1200 UTC 28 Apr 2011 compared with final validated hail sizes from Storm Data indicated in the key at right (image courtesy of J. Correia, NOAA/NWS Storm Prediction Center). The hail size key indicates the total number of displayed hail reports and includes the following symbols (diameter range): small gray-filled circle (1 in.), green-filled circle (1–1.75 in.), green-filled square (1.75–2.5 in.), and black triangle (2.5 in.).

Fig. 10.

SPC day-1 hail probability outlook valid from 1300 UTC 27 Apr to 1200 UTC 28 Apr 2011 compared with final validated hail sizes from Storm Data indicated in the key at right (image courtesy of J. Correia, NOAA/NWS Storm Prediction Center). The hail size key indicates the total number of displayed hail reports and includes the following symbols (diameter range): small gray-filled circle (1 in.), green-filled circle (1–1.75 in.), green-filled square (1.75–2.5 in.), and black triangle (2.5 in.).

A maximum expected size of hail (MESH) product is produced by the National Severe Storms Laboratory (NSSL) by merging the radar data from the National Weather Service (NWS) Weather Surveillance Radar-1988 Doppler (WSR-88D) network in the continental United States into a three-dimensional Cartesian grid at 1-km grid spacing. The hail detection algorithm described in Witt et al. (1998) and Stumpf et al. (2004) is then used to produce a gridded “observed” maximum hailstone diameter. Verisk Insurance Solutions creates a specific version of this product, called Respond, which is quality controlled unlike the raw MESH product available from NSSL. Hail size values are screened to ensure they correspond to a nonzero reflectivity, and are also manually validated against spotter reports. The Respond product from 0600 UTC 27 April to 0600 UTC 28 April 2011 displays the maximum hail size analyzed at any grid point during the 24-h period (Fig. 11f). The MESH algorithm was not designed to detect hail below sizes of 0.75 in. (Witt et al. 1998). Thus, the Respond product was not used to verify hail diameters below 0.75 in. To directly compare total area covered by certain hail sizes, the Respond product hail analysis was reprojected to the 4-km WRF grid by assigning the maximum hail size in the Respond product within a radius of 2 km of each WRF grid point to that point. It should be noted that the Respond product is a single-radar MESH product; single-radar products have been found to overforecast hail size (Wilson et al. 2009; Cintineo et al. 2012). MESH products nevertheless retain significant advantages over the Storm Data severe weather database, which has a heavy necessary reliance on public reporting and biases inherent therein.

Fig. 11.

WRF-HAILCAST forecast hail size (in) from 0600 UTC 27 Apr to 0600 UTC 28 Apr 2011. Maximum hail diameter over the 24-h period at each grid point displayed. Sensitivity tests: (a) CONTROL, as described in section 2, (b) CONSDENSE900, (c) CONSDENSE500, (d) STEP-EI, and (e) ONOFF-UPDRAFT. (f) Respond product estimated hail diameter (in.) from 0600 UTC 27 Apr to 0600 UTC 28 Apr 2011. Maximum hail diameter over the 24-h period at each grid point is displayed.

Fig. 11.

WRF-HAILCAST forecast hail size (in) from 0600 UTC 27 Apr to 0600 UTC 28 Apr 2011. Maximum hail diameter over the 24-h period at each grid point displayed. Sensitivity tests: (a) CONTROL, as described in section 2, (b) CONSDENSE900, (c) CONSDENSE500, (d) STEP-EI, and (e) ONOFF-UPDRAFT. (f) Respond product estimated hail diameter (in.) from 0600 UTC 27 Apr to 0600 UTC 28 Apr 2011. Maximum hail diameter over the 24-h period at each grid point is displayed.

b. Model setup

The National Centers for Environmental Prediction (NCEP) Final Operational Global Analysis data, available every 6 h at 1° by 1° horizontal grid spacing, are used for initial and boundary conditions. The WRF parent domain is run at 20-km horizontal grid spacing, with an inner nest of 4-km grid spacing (shown in Fig. 11). WRF-HAILCAST is run only on the inner domain, and all results shown are from that domain. The domains have 35 vertical levels with decreased spacing in lower levels. The model is initialized at 0000 UTC 27 April 2011, and run until 0600 UTC 28 April 2011, but only hail forecasts from at least 6 h after initialization are displayed to provide time for convective spinup. WRF version 3.6.1 is used for these tests. WRF-HAILCAST has previously been tested with a variety of WRF microphysical parameterizations with similar results, but results presented here will be limited to the Thompson parameterization (Thompson et al. 2008). Other configuration choices are listed in Table 2.

Table 2.

Sensitivity test model configuration.

Sensitivity test model configuration.
Sensitivity test model configuration.

c. Sensitivity tests

Figures 11a–e show the hail forecast from 0600 UTC 27 April to 0600 UTC 28 April 2011 for the CONTROL WRF-HAILCAST model and four additional sensitivity tests. A forecasted hail diameter is calculated within each grid column and model time step by calculating the maximum hail diameter resulting from the five different embryos. The forecasted hail diameter is then aggregated into a “hail swath” product by taking the maximum hail size forecast at any grid point during the 24-h period, as in Fig. 11f and Kain et al. (2010). The locations of the three updrafts discussed in the previous section are shown as dots labeled strong updraft (Fig. 11a), dry updraft (Fig. 11b), and weak updraft (Fig. 11c). The CONTROL forecast (Fig. 11a) failed to predict some of the convection and hailfall that occurred in north-central Louisiana, and south-central Mississippi and Alabama. This failure highlights a key issue of WRF-HAILCAST; namely, that WRF-HAILCAST will only predict hail if WRF predicts deep convection. The results are favorable when compared to the Respond product in areas of successfully predicted convection.

The CONTROL forecast correctly identifies regions of larger-diameter hail in southern Arkansas, northern Mississippi, and eastern Tennessee (Fig. 11a). Hail diameters within these convective areas in the Respond product are generally between 1.5 and 2.0 in., with smaller embedded pockets of 2.5-in. and larger hail. Figure 12a shows a histogram of the number of grid points that contain certain hail sizes for the Respond product and each of the sensitivity test simulations. Compared to the Respond product, the CONTROL run overforecasts the area of hail of diameters 0.75–2.25 in., owing mainly to overforecasted convection in general. For sizes 2.5 in. and above (Fig. 12b), the CONTROL distribution agrees relatively well with the Respond product distribution. Most importantly, the distributions of hail occurrence frequency versus hail size in both the CONTROL run and the Respond product have an overall inverse exponential shape, whereas the CONDENSE500, STEP-EI, and ONOFF-UPDRAFT tests possess much different hail frequency distributions.

Fig. 12.

(a) A histogram of forecast hail diameters from Figs. 11a–e as well as the Respond product, regridded to the WRF grid as described in section 5. (b) A zoomed-in view for hail diameters 2.0–3.5 in., with the CONSDENSE500 test omitted.

Fig. 12.

(a) A histogram of forecast hail diameters from Figs. 11a–e as well as the Respond product, regridded to the WRF grid as described in section 5. (b) A zoomed-in view for hail diameters 2.0–3.5 in., with the CONSDENSE500 test omitted.

The importance of including variable hail density is further illustrated by comparing tests CONSDENSE900 and CONSDENSE500 (Figs. 11b and 11c). It appears that the higher fixed hailstone density generally resulted in the hailstones falling out of the updraft sooner, allowing less time for growth, a process also suggested by the 7.5- and 10-mm embryos in the sensitivity tests shown in Fig. 7. It is also evident that if the constant ice layer density is set at 500 kg m−3 (CONSDENSE500), the hailstones grow much larger than CONTROL or CONSDENSE900. New layer growth in this situation is prescribed a lower layer density that in turn results in significant overprediction of larger sizes (Figs. 12a,b).

The STEP-EI and ONOFF-UPDRAFT tests (Figs. 11d,e) underforecast all hail sizes (Fig. 12a). The relative reduction in the ice collection efficiency at temperatures near 0°C, particularly near the embryo insertion temperatures, led to smaller hailstone growth in STEP-EI. The stronger updrafts in the ONOFF-UPDRAFT test frequently lofted the embryos above the supercooled water layer, as seen in the 5-mm trajectories in Fig. 8.

6. Forecast verification

WRF-HAILCAST was run by the lead author for two multiweek periods—7–21 May 2014 and 4–22 May 2015—to exemplify hail forecasting during a fairly active hail period. WRF is configured as described in section 5b, except the nested domain has been enlarged to cover the entire continental United States. All runs are initialized at 0000 UTC and run until 0600 UTC the following day. As also described in section 5b, only hail size forecasts from 6 h after initialization are included for verification to allow for convective spinup. The Method for Object-based Diagnostic Evaluation [MODE; available online at http://www.dtcenter.org/met/users; Davis et al. (2006a,b)] is used to compare gridded hail forecasts from the WRF runs to the Respond product. Hail forecasts are generated using the maximum hail diameter from the five embryos as the forecast hail diameter. The hail forecasts are aggregated into a hail swath product by taking the maximum hail size forecast at any grid point between 0600 and 0600 UTC the next day (as in Fig. 11).

The Respond product is reprojected from its native 1-km grid to the 4-km WRF domain, as described above in section 5a, by assigning each WRF grid point the maximum hail diameter in the Respond product data within a 2-km radius. This method ensures that the maximum hail size associated with a hail swath is conserved during the reprojection. It should be noted that this method also results in an artificial broadening of the hail swath as a hail value could be assigned to a grid cell in the reprojection up to 2 km away. To minimize this effect, the original Respond product is also reprojected to the WRF grid using the maximum hail diameter within a 1-km radius of each WRF grid point. Although this method does not conserve the maximum hail value in each hail swath, it minimizes the artificial broadening of the swath. However, the verification results using the 1-km reprojection are not significantly different than those calculated using the 2-km reprojection, so only the 2-km reprojection verification statistics will be presented here. During 2014 and 2015, the Respond product was binned into 0.25-in. intervals, so the model data are binned similarly before calculating the verification statistics.

The goal of the MODE verification is to compare groups of hail swath objects in the forecast and observed (radar derived) hail fields. This MODE configuration was designed to compare clusters of objects approximately on the scale of individual hail swaths produced by either a single mesoscale convective system or a family of supercells. An example of matched object clusters for the observed and forecast data from the 7 May 2014 model run is shown in Figs. 13a and 13b along with the Respond product from the same time (Fig. 13c). MODE initially generates a smoothed forecast or observed field by averaging over a radius of a user-defined number of grid points; a radius of four grid points was used in the present study as recommended by Davis et al. (2006a,b) to ensure all the identified objects are sufficiently resolved by the model. Any contiguous area in the smoothed field that covers at least four grid points and contains values larger than 0.5 in. defines a hail swath object.

Fig. 13.

Object clusters for (a) observed and (b) forecast hail diameter from 0600 UTC 7 May to 0600 UTC 8 May 2014. Clusters that are the same color in (a) and (b) were “matched” by the MODE algorithm. Unmatched clusters are in gray. (c) Respond product from the same time period for comparison.

Fig. 13.

Object clusters for (a) observed and (b) forecast hail diameter from 0600 UTC 7 May to 0600 UTC 8 May 2014. Clusters that are the same color in (a) and (b) were “matched” by the MODE algorithm. Unmatched clusters are in gray. (c) Respond product from the same time period for comparison.

Figure 14 displays the frequency with which the observed and forecast maximum and 90th percentile hail diameters from each cluster fell within 0.25-in. bins. Ideally the colored bins would all be located along the black line, which would indicate observed and forecast cluster diameters are equal. Both the maximum and 90th percentile plots show little bias for under- or overforecasting, with a possible exception of underforecasting the maximum hail sizes in the swath for extremely large (3.5 in.) hail events. For example, by examining Fig. 14a, we can see that MODE identified 12 observed clusters and no forecast clusters with a maximum diameter greater than 3.5 in. during this period. Unfortunately, these severe hail events are relatively rare in this dataset, making further conclusions difficult, as does the potential overanalysis of large hail by the MESH algorithm (Wilson et al. 2009; Cintineo et al. 2012).

Fig. 14.

(a) Maximum and (b) 90th percentile hail size for all observed vs forecast matched clusters for 7–21 May 2014 and 4–22 May 2015. Observed data were binned in 0.25-in. increments, necessitating the binning of the forecast data.

Fig. 14.

(a) Maximum and (b) 90th percentile hail size for all observed vs forecast matched clusters for 7–21 May 2014 and 4–22 May 2015. Observed data were binned in 0.25-in. increments, necessitating the binning of the forecast data.

To better understand the results of these two methods, a traditional objective verification method was also used. Such methods have frequently been used to verify hail forecasts. Brimelow et al. (2006) provided objective verification statistics of HAILCAST hail forecasts by binning the hail forecast into categories. Unfortunately, the hail diameter categories employed by Brimelow et al. (2006) are mostly smaller than 0.75 in., and thus cannot be used here. Instead, the present study employs hail diameter categories for objective verification in the intervals 0.75 d 1.0 in., 1.0 d 1.5 in., and at 0.5-in. intervals for sizes greater than 1.5 in. At each WRF-HAILCAST grid point, all Respond product points within 40 km are searched and the maximum hail diameter within that radius returned (e.g., Hitchens et al. 2013). The two hail diameters are compared if both the WRF-HAILCAST and Respond product diameters are at least 0.75 in. Limiting the diameters evaluated avoids penalizing WRF-HAILCAST when the underlying WRF Model does not forecast convection. Table 3 provides percentages of how often the WRF-HAILCAST hail diameter is within a category, too small, or too large, in relation to the Respond product hail diameter. It is encouraging that WRF-HAILCAST is within one category (essentially 0.5 in.) of the Respond product hail diameter 66% of the time. The mean hail diameter error is −0.39 in., with a mean absolute error of 0.67 in. These errors are comparable to those obtained by Brimelow et al. (2006) when forecasting severe hail using an older version of HAILCAST with proximity model soundings. WRF-HAILCAST fails to predict some very large Respond product hail diameters (3.5 in.) as also seen in Fig. 14a, which contributes to that error.

Table 3.

Summary of neighborhood objective verification statistics. Hail size categories were defined as 0.75–1.0 in., 1.0–1.5 in., and every 0.5-in. interval thereafter. Events were only recorded if both WRF-HAILCAST and Respond product contained hail 0.75 in.

Summary of neighborhood objective verification statistics. Hail size categories were defined as 0.75–1.0 in., 1.0–1.5 in., and every 0.5-in. interval thereafter. Events were only recorded if both WRF-HAILCAST and Respond product contained hail  0.75 in.
Summary of neighborhood objective verification statistics. Hail size categories were defined as 0.75–1.0 in., 1.0–1.5 in., and every 0.5-in. interval thereafter. Events were only recorded if both WRF-HAILCAST and Respond product contained hail  0.75 in.

7. Conclusions

The HAILCAST hail model of Poolman (1992) and Brimelow et al. (2002) has been significantly updated and incorporated into the WRF-ARW model. At each WRF Model grid point containing sufficiently strong convection, the vertical motion, liquid, and ice water profiles are passed to the WRF-HAILCAST hail model. Five embryos, of a range of initial sizes and insertion temperatures based on observations, are released into these WRF grid columns and allowed to grow into hailstones. The embryos are assumed to initiate outside the updraft, and an updraft multiplier is applied to the vertical motion profile to emulate horizontal advection of the hailstone across the updraft as the hailstone simultaneously moves vertically. Other updates to the HAILCAST hail model include incorporating variable hail density that reflects regimes of wet and dry growth, temperature-dependent ice collection efficiency, mass growth by vapor deposition or condensation, and an improved liquid water shedding scheme.

The model test runs reveal a pronounced sensitivity of the model to the initial embryo size, emphasizing the importance of using a range of initial embryo sizes. Use of a variable hail bulk density is also shown to be of first-order importance for simulating physically realistic growth of a hailstone. Imposing an unphysically constant rime layer density results in widely variable forecast hailstone diameters depending on the arbitrarily chosen value of rime layer density. Finally, the use of the time-dependent updraft multiplier facilitates the realistic simulation of growing hailstones as they are advected from the edge across the main updraft rather than being unphysically forced to grow within a strong, one-dimensional updraft. The time-dependent updraft multiplier also facilitates the use of a more realistic range of initial hail embryo sizes, and still produces realistic hail forecasts even at varying horizontal model resolutions.

WRF-HAILCAST outputs the maximum, mean, and standard deviation hail diameters for the hailstones produced by the five embryos. The maximum diameter is used as a hail forecast, and the mean and standard deviation are provided for uncertainty information. Verification of WRF-HAILCAST hail forecasts during multiweek periods in May 2014 and 2015 showed that when WRF successfully forecasts convection, the WRF-HAILCAST hail forecast was within 0.5 in. of the observed hail size over 66% of the time.

A future article is planned to evaluate and compare hail size forecasts from the present WRF-HAILCAST version to the previous versions run as part of the 2014, 2015, and 2016 NOAA Hazardous Weather Testbed (HWT) Spring Forecasting Experiments. The 2014 HWT WRF-HAILCAST version used information from the microphysical parameterization to fix the embryo sizes, and neither the 2014 nor the 2015 HWT WRF-HAILCAST versions included the time-dependent updraft multiplier or many of the other improvements detailed herein. Additional future work will focus on verification efforts to evaluate the potential underforecasting of extremely large hail (3.5 in.) by WRF-HAILCAST, and the evolution of hail forecasts in time throughout convective development. These efforts will be conducted during time periods coinciding with the availability of high-quality, high-resolution hail observation datasets such as the Multi-Radar Multi-Sensor MESH product (Cintineo et al. 2012) or the Severe Hazards Analysis and Verification Experiment (Ortega et al. 2009).

Acknowledgments

This work was performed as part of the Systems Engineering Management and Sustainment contract with the Air Force Life Cycle Management Center, and the Cooperative Research Data Agreement between the 557th Weather Wing and Atmospheric and Environmental Research, Inc. Computing resources were provided by the Navy Department of Defense Supercomputing Resource Center (Navy DSRC), which is sponsored by the DoD High Performance Computing Modernization Program. The authors thank Ryan Jewell for his copy of the HAILCAST code, and David Gagne and Chris Melick for discussions about hail verification. The authors also gratefully acknowledge formal reviewer Julian Brimelow, two additional formal reviewers, and Adam Clark for their insightful reviews of the manuscript.

REFERENCES

REFERENCES
Brimelow
,
J. C.
,
G. W.
Reuter
, and
E. R.
Poolman
,
2002
:
Modeling maximum hail size in Alberta thunderstorms
.
Wea. Forecasting
,
17
,
1048
1062
, doi:.
Brimelow
,
J. C.
,
G. W.
Reuter
,
R.
Goodson
, and
T. W.
Krauss
,
2006
:
Spatial forecasts of maximum hail size using prognostic model soundings and HAILCAST
.
Wea. Forecasting
,
21
,
206
219
, doi:.
Bryan
,
G. H.
,
J. C.
Wyngaard
, and
J. M.
Fritsch
,
2003
:
Resolution requirements for the simulation of deep moist convection
.
Mon. Wea. Rev.
,
131
,
2394
2416
, doi:.
Chalon
,
J.-P.
,
J. C.
Fankhauser
, and
P. J.
Eccles
,
1976
:
Structure of an evolving hailstorm. Part I: General characteristics and cell structure
.
Mon. Wea. Rev.
,
104
,
564
575
, doi:.
Chong
,
S.-L.
, and
C. S.
Chen
,
1974
:
Water shells on ice pellets and hailstones
.
J. Atmos. Sci.
,
31
,
1384
1391
, doi:.
Cintineo
,
J. L.
,
T. M.
Smith
, and
V.
Lakshmanan
,
2012
:
An objective high-resolution hail climatology of the contiguous United States
.
Wea. Forecasting
,
27
,
1235
1248
, doi:.
Clark
,
A.
,
J.
Kain
,
P.
Marsh
,
J.
Correia
Jr.
,
M.
Xue
, and
F.
Kong
,
2012
:
Forecasting tornado pathlengths using a three-dimensional object identification algorithm applied to convection-allowing forecasts
.
Wea. Forecasting
,
27
,
1090
1113
, doi:.
Davis
,
C.
,
B.
Brown
, and
R.
Bullock
,
2006a
:
Object-based verification of precipitation forecasts. Part I: Methods and application to mesoscale rain areas
.
Mon. Wea. Rev.
,
134
,
1772
1784
, doi:.
Davis
,
C.
,
B.
Brown
, and
R.
Bullock
,
2006b
:
Object-based verification of precipitation forecasts. Part II: Application to convective rain systems
.
Mon. Wea. Rev.
,
134
,
1785
1795
, doi:.
Dennis
,
A. S.
,
C. A.
Schock
, and
A.
Koscielski
,
1970
:
Characteristics of hailstorms of western South Dakota
.
J. Appl. Meteor.
,
9
,
127
135
, doi:.
Donavon
,
R. A.
, and
K. A.
Jungbluth
,
2007
:
Evaluation of a technique for radar identification of large hail across the upper Midwest and central plains of the United States
.
Wea. Forecasting
,
22
,
244
254
, doi:.
Edwards
,
R.
, and
R. L.
Thompson
,
1998
:
Nationwide comparisons of hail size with WSR-88D vertically integrated liquid water and derived thermodynamic sounding data
.
Wea. Forecasting
,
13
,
277
285
, doi:.
Ek
,
M. B.
,
K. E.
Mitchell
,
Y.
Lin
,
E.
Rogers
,
P.
Grunmann
,
V.
Koren
,
G.
Gayno
, and
J. D.
Tarpley
,
2003
:
Implementation of Noah land surface model advances in the National Centers for Environmental Prediction operational mesoscale Eta model
.
J. Geophys. Res.
,
108
,
8851
, doi:.
English
,
M.
,
1973
: Growth of large hail in the storm. Alberta Hailstorms, Meteor. Monogr., No. 36, Amer. Meteor. Soc., 37–98.
FAA
,
1975
: Aviation weather. Federal Aviation Administration, 220 pp.
Fankhauser
,
J. C.
,
L. J.
Miller
,
B. E.
Martner
, and
Z.
Levin
,
1982
: The 25 July 1976 case study: Environmental conditions, reflectivity structure, and evolution. Hailstorms of the Central High Plains, P. Squires and C. Knight, Eds., Vol. 2, The National Hail Research Experiment, National Center for Atmospheric Research (in association with Colorado Associated University Press), 197–209.
Foote
,
G. B.
,
1984
:
A study of hail growth utilizing observed storm conditions
.
J. Climate Appl. Meteor.
,
23
,
84
101
, doi:.
Foote
,
G. B.
, and
C. G.
Wade
,
1982
:
Case study of a hailstorm in Colorado. Part I: Radar echo structure and evolution
.
J. Atmos. Sci.
,
39
,
2828
2846
, doi:.
Gallo
,
B. T.
,
A. J.
Clark
, and
S. R.
Dembek
,
2016
:
Forecasting tornadoes using convection-permitting ensembles
.
Wea. Forecasting
,
31
,
273
295
, doi:.
Garcia-Garcia
,
F.
, and
R.
List
,
1992
:
Laboratory measurements and parameterizations of supercooled water skin temperature and bulk properties of gyrating hailstones
.
J. Atmos. Sci.
,
49
,
2058
2073
, doi:.
Gilmore
,
M. S.
,
J. M.
Straka
, and
E. N.
Rasmussen
,
2004
:
Precipitation uncertainty due to variations in precipitation particle parameters within a simple microphysics scheme
.
Mon. Wea. Rev.
,
132
,
2610
2627
, doi:.
Herndon
,
R.
,
2007
: Summary of natural hazard statistics for 2007 in the United States. National Climatic Data Center, accessed 22 October 2015. [Available online at http://www.ncdc.noaa.gov/IPS/sd/sd.html.]
Herndon
,
R.
,
2010
: Summary of natural hazard statistics for 2010 in the United States. National Climatic Data Center, accessed 22 October 2015. [Available online at http://www.ncdc.noaa.gov/IPS/sd/sd.html.]
Heymsfield
,
A. J.
,
1972
:
Ice crystal terminal velocities
.
J. Atmos. Sci.
,
29
,
1348
1357
, doi:.
Heymsfield
,
A. J.
,
1978
:
The characteristics of graupel particles in northeastern Colorado cumulus congestus clouds
.
J. Atmos. Sci.
,
35
,
284
295
, doi:.
Heymsfield
,
A. J.
,
1982
:
A comparative study of the rates of development of potential graupel and hail embryos in high plains storms
.
J. Atmos. Sci.
,
39
,
2867
2897
, doi:.
Heymsfield
,
A. J.
,
1983a
:
A technique for investigating graupel and hail development
.
J. Climate Appl. Meteor.
,
22
,
1143
1160
, doi:.
Heymsfield
,
A. J.
,
1983b
:
Case study of a hailstorm in Colorado. Part IV: Graupel and hail growth mechanisms deduced through particle trajectory calculations
.
J. Atmos. Sci.
,
40
,
1482
1509
, doi:.
Heymsfield
,
A. J.
, and
D. J.
Musil
,
1982
:
Case study of a hailstorm in Colorado. Part II: Particle growth processes at midlevels deduced form in situ measurements
.
J. Atmos. Sci.
,
39
,
2847
2866
, doi:.
Heymsfield
,
A. J.
, and
M. R.
Hjelmfelt
,
1984
:
Processes of hydrometeor development in Oklahoma convective clouds
.
J. Atmos. Sci.
,
41
,
2811
2835
, doi:.
Heymsfield
,
A. J.
, and
J. C.
Pflaum
,
1985
:
A quantitative assessment of the accuracy of techniques for calculating graupel growth
.
J. Atmos. Sci.
,
42
,
2264
2274
, doi:.
Heymsfield
,
A. J.
,
A. R.
Jameson
, and
H. W.
Frank
,
1980
:
Hail growth mechanisms in a Colorado storm. Part II: Hail formation processes
.
J. Atmos. Sci.
,
37
,
1779
1813
, doi:.
Hitchens
,
N. M.
,
H. E.
Brooks
, and
M. P.
Kay
,
2013
:
Objective limited on forecasting skill of rare events
.
Wea. Forecasting
,
28
,
525
534
, doi:.
Hong
,
S.-Y.
,
Y.
Noh
, and
J.
Dudhia
,
2006
:
A new vertical diffusion package with an explicit treatment of entrainment processes
.
Mon. Wea. Rev.
,
134
,
2318
2341
, doi:.
Iacono
,
M. J.
,
J. S.
Delamere
,
E. J.
Mlawer
,
M. W.
Shephard
,
S. A.
Clough
, and
W. D.
Collins
,
2008
:
Radiative forcing by long-lived greenhouse gases: Calculations with the AER radiative transfer models
.
J. Geophys. Res.
,
113
,
D13103
, doi:.
Janjić
,
Z. I.
,
1994
:
The step-mountain eta coordinate model: Further developments of the convection, viscous sublayer, and turbulence closure schemes
.
Mon. Wea. Rev.
,
122
,
927
945
, doi:.
Janjić
,
Z. I.
,
2000
:
Comments on “Development and evaluation of a convective scheme for use in climate models.”
J. Atmos. Sci.
,
57
,
3686
3686
, doi:.
Jewell
,
R.
, and
J.
Brimelow
,
2009
:
Evaluation of Alberta hail growth model using severe hail proximity soundings from the United States
.
Wea. Forecasting
,
24
,
1592
1609
, doi:.
Kain
,
J. S.
,
S. J.
Weiss
,
J. J.
Levit
,
M. E.
Baldwin
, and
D. R.
Bright
,
2006
:
Examination of convection-allowing configurations of the WRF model for the prediction of severe convective weather: The SPC/NSSL Spring Program 2004
.
Wea. Forecasting
,
21
,
167
181
, doi:.
Kain
,
J. S.
,
S. R.
Dembek
,
S. J.
Weiss
,
J. L.
Case
,
J. J.
Levit
, and
R. A.
Sobash
,
2010
:
Extracting unique information from high-resolution forecast models: Monitoring selected fields and phenomena every time step
.
Wea. Forecasting
,
25
,
1536
1542
, doi:.
Kessler
,
E.
,
1969
: On the Distribution and Continuity of Water Substance in Atmospheric Circulation. Meteor. Monogr., No. 10, Amer. Meteor. Soc., 84 pp.
Knight
,
C. A.
, and
N. C.
Knight
,
1970
:
The falling behavior of hailstones
.
J. Atmos. Sci.
,
27
,
672
681
, doi:.
Knight
,
C. A.
,
W. A.
Cooper
,
D. W.
Breed
,
I. R.
Paluch
,
P. L.
Smith
, and
G.
Vali
,
1982
: Microphysics. Hailstorms of the Central High Plains, P. Squires, and C. Knight, Eds., Vol. 1, The National Hail Research Experiment, National Center for Atmospheric Research (in association with Colorado Associated University Press), 151–193.
Knight
,
C. A.
,
P. T.
Schlatter
, and
T. W.
Schlatter
,
2008
:
An unusual hailstorm on 24 June 2006 in Boulder, Colorado. Part II: Low-density growth of hail
.
Mon. Wea. Rev.
,
136
,
2833
2348
, doi:.
Knight
,
N. C.
,
1981
:
The climatology of hailstone embryos
.
J. Appl. Meteor.
,
20
,
750
755
, doi:.
Knight
,
N. C.
, and
A. J.
Heymsfield
,
1983
:
Measurement and interpretation of hailstone density and terminal velocity
.
J. Atmos. Sci.
,
40
,
1510
1516
, doi:.
Knupp
,
K.
, and Coauthors
,
2014
:
Meteorological overview of the devastating 27 April 2011 tornado outbreak
.
Bull. Amer. Meteor. Soc.
,
95
,
1041
1062
, doi:.
Lesins
,
G. B.
, and
R.
List
,
1986
:
Sponginess and drop shedding of gyrating hailstones in a pressure-controlled icing wind tunnel
.
J. Atmos. Sci.
,
43
,
2813
2825
, doi:.
List
,
R.
,
1990
:
Physics of supercooling of thin water skins covering gyrating hailstones
.
J. Atmos. Sci.
,
47
,
1919
1925
, doi:.
List
,
R.
, and
J.-G.
Dussasult
,
1967
:
Quasi steady state icing and melting conditions and heat and mass transfer of spherical and spheroidal hailstones
.
J. Atmos. Sci.
,
24
,
522
529
, doi:.
Ludlam
,
F. H.
,
1958
:
The hail problem
.
Nubila
,
1
,
12
96
.
Macklin
,
W. C.
,
1962
:
The density and structure of ice formed by accretion
.
Quart. J. Roy. Meteor. Soc.
,
88
,
30
50
, doi:.
Macklin
,
W. C.
,
C.
Knight
,
H.
Moore
,
N.
Knight
,
W.
Pollock
,
J.
Carras
, and
S.
Thwaiters
,
1977
:
Isotopic, crystal, and air bubble structures of hailstones
.
J. Atmos. Sci.
,
34
,
961
967
, doi:.
Magono
,
C.
, and
T.
Nakamura
,
1965
:
Aerodynamic studies of falling snowflakes
.
J. Meteor. Soc. Japan
,
43
,
139
147
.
Milbrandt
,
J. A.
, and
H.
Morrison
,
2013
:
Prediction of graupel density in a bulk microphysics scheme
.
J. Atmos. Sci.
,
70
,
410
429
, doi:.
Miller
,
L. J.
,
J. D.
Tuttle
, and
C. A.
Knight
,
1988
:
Airflow and hail growth in a severe northern High Plains supercell
.
J. Atmos. Sci.
,
45
,
736
762
, doi:.
Miller
,
L. J.
,
J. D.
Tuttle
, and
G. B.
Foote
,
1990
:
Precipitation production in a large Montana hailstorm: Airflow and particle growth trajectories
.
J. Atmos. Sci.
,
47
,
1619
1646
, doi:.
Mlawer
,
E. J.
,
S. J.
Taubman
,
T. D.
Brown
,
M. J.
Iacono
, and
S. A.
Clough
,
1997
:
Radiative transfer for inhomogeneous atmospheres: RRTM, a validated correlated-k model for the longwave
.
J. Geophys. Res.
,
102
,
16 663
16 682
, doi:.
Morrison
,
H.
, and
J. A.
Milbrandt
,
2015
:
Parameterization of cloud microphysics based on the prediction of bulk ice particle properties. Part I: Scheme description and idealized tests
.
J. Atmos. Sci.
,
72
,
287
311
, doi:.
NCDC
,
2011
: Storm Data. Vol. 53, No. 4, 1212 pp.
Nelson
,
S. P.
,
1983
:
The influence of storm flow structure on hail growth
.
J. Atmos. Sci.
,
40
,
1965
1983
, doi:.
Nelson
,
S. P.
, and
N. C.
Knight
,
1983
:
The hybrid multicellular-supercellular storm—An efficient hail producer. Part I: An archetypal example
.
J. Atmos. Sci.
,
40
,
1965
1983
, doi:.
NSSL
,
2014
: Severe weather 101: Frequently asked questions about hail. National Severe Storms Laboratory. [Available online at http://www.nssl.noaa.gov/education/svrwx101/hail/faq.]
Ortega
,
K. L.
,
T. M.
Smith
,
K. L.
Manross
,
K. A.
Scharfenberg
,
A.
Witt
,
A. G.
Kolodziej
, and
J. J.
Gourley
,
2009
:
The severe hazards analysis and verification experiment
.
Bull. Amer. Meteor. Soc.
,
90
,
1519
1530
, doi:.
Phillips
,
V. T. J.
,
A.
Khain
,
N.
Benmoshe
, and
E.
Ilotoviz
,
2014
:
Theory of time-dependent freezing. Part I: Description of scheme for wet growth of hail
.
J. Atmos. Sci.
,
71
,
4527
4557
, doi:.
Poolman
,
E. R.
,
1992
: Die voorspelling van haelkorrelgroei in Suid-Afrika (The forecasting of hail growth in South Africa). M.S. thesis, Faculty of Engineering, University of Pretoria, 113 pp.
Pruppacher
,
H. R.
, and
J. D.
Klett
,
1997
: Microphysics of Clouds and Precipitation. 2nd ed. Springer, 954 pp.
Rasmussen
,
R. M.
, and
A. J.
Heymsfield
,
1985
:
A generalized form for impact velocities used to determine graupel accretional densities
.
J. Atmos. Sci.
,
42
,
2275
2279
, doi:.
Rasmussen
,
R. M.
, and
A. J.
Heymsfield
,
1987
:
Melting and shedding of graupel and hail. Part I: Model physics
.
J. Atmos. Sci.
,
44
,
2754
2763
, doi:.
Rosenfeld
,
D.
, and
W. L.
Woodley
,
2000
:
Deep convective cloud with sustained supercooled liquid water down to −37.5°C
.
Nature
,
405
,
440
442
, doi:.
Skamarock
,
W. C.
, and
M. L.
Weisman
,
2009
:
The impact of positive-definite moisture transport on NWP precipitation forecasts
.
Mon. Wea. Rev.
,
137
,
488
494
, doi:.
Skamarock
,
W. C.
, and Coauthors
,
2008
: A description of the Advanced Research WRF version 3. NCAR Tech. Note NCAR/TN-475+STR, 113 pp., doi:.
Sobash
,
R. A.
,
J. S.
Kain
,
D. R.
Bright
,
A. R.
Dean
,
M. C.
Coniglio
, and
S. J.
Weiss
,
2011
:
Probabilistic forecast guidance for severe thunderstorms based on the identification of extreme phenomena in convection-allowing model forecasts
.
Wea. Forecasting
,
26
,
714
728
, doi:.
Stumpf
,
G. J.
,
T. M.
Smith
, and
J.
Hocker
,
2004
: New hail diagnostic parameters derived by integrating multiple radars and multiple sensors. 22nd Conf. on Severe Local Storms, Hyannis, MA, Amer. Meteor. Soc., P7.8. [Available online at http://ams.confex.com/ams/pdfpapers/81451.pdf.]
Tao
,
W.-K.
, and Coauthors
,
2003
:
Microphysics, radiation and surface processes in the Goddard Cumulus Ensemble (GCE) model
.
Meteor. Atmos. Phys.
,
82
,
97
137
, doi:.
Thompson
,
G.
,
R. M.
Rasmussen
, and
K.
Manning
,
2004
:
Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part I: Description and sensitivity analysis
.
Mon. Wea. Rev.
,
132
,
519
542
, doi:.
Thompson
,
G.
,
P. R.
Field
,
R. M.
Rasmussen
, and
W. D.
Hall
,
2008
:
Explicit forecasts of winter precipitation using an improved bulk microphysics scheme. Part II: Implementation of a new snow parameterization
.
Mon. Wea. Rev.
,
136
,
5095
5115
, doi:.
Vali
,
G.
, and
E. J.
Stansbury
,
1965
: Time-dependent characteristics of the heterogeneous nucleation of ice. Science Rep. MW-41, McGill University, Montreal, QC, Canada, 31 pp.
Weisman
,
M. L.
,
W. C.
Skamarock
, and
J. B.
Klemp
,
1997
:
The resolution dependence of explicitly modeled convective systems
.
Mon. Wea. Rev.
,
125
,
527
548
, doi:.
Wilson
,
C. J.
,
K.
Ortega
, and
V.
Lakshmanan
,
2009
: Evaluating multi-radar, multi-sensor hail diagnosis with high resolution hail reports. 25th Conf. on Interactive Information Processing Systems, Phoenix, AZ, Amer. Meteor. Soc., P2.9. [Available online at http://ams.confex.com/ams/pdfpapers/146206.pdf.].
Witt
,
A.
,
M. D.
Eilts
,
G. J.
Stumpf
,
J. T.
Johnson
,
E. D.
Mitchell
, and
K. W.
Thomas
,
1998
:
An enhanced hail detection algorithm for the WSR-88D
.
Wea. Forecasting
,
13
,
286
303
, doi:.
Ziegler
,
C. L.
,
P. S.
Ray
, and
N. C.
Knight
,
1983
:
Hail growth in an Oklahoma multicell storm
.
J. Atmos. Sci.
,
40
,
1768
1791
, doi:.