Graupel and Hail Terminal Velocities: Does a ”Supercritical” Reynolds Number Apply?

ABSTRACT

This study characterizes the terminal velocities of heavily rimed ice crystals and aggregates, graupel, and hail using a combination of recent drag coefficient and particle bulk density observations. Based on a non-dimensional Reynolds number (Re)-Best number (X) approach that applies to atmospheric temperatures and pressures where these particles develop and fall, the authors develop a relationship that spans a wide range of particle sizes. The Re-X relationship can be used to derive the terminal velocities of rimed particles for many applications. Earlier observations suggest that a ''supercritical'' Reynolds number is reached where the drag coefficient for large spherical ice-hail-drops precipitously and the terminal velocities increase rapidly. The authors draw on observations and model simulations for slightly roughened large ice particles that suggest that the critical Reynolds number is dampened and that the rapid increase in the terminal velocity of smooth spherical ice particles rarely occurs for natural hailstones.

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Hail produces considerable monetary damage in the United States and throughout many parts of the world, including Italy, South Africa, and Russia. Damage is mostly to crops and buildings [see Changnon (2009) and articles cited therein]. One primary source of hail in- formation in the United States is the historical records of the crop-hail insurance industry, kept since 1948. The national annual values of insured crop-hail losses have shown ever-increasing amounts (2014 dollars), from $20 million in 1948 to $174 million in 1974, jumping to $358 million in 1980, approaching $550 million in the early 1990s, and with continuing increases since. In 2010, there were 468 000 claims for all types of hail damage filed in the United States. That number increased to 862 000 in 2012. Crop-hail losses for 1949-98, as measured by insurance data, averaged $575 million (2000 dollars) per year (Changnon and Hewings 2001). The intensity of hail at a point or over an area is a function of the fre- quency of hail, the size of the hailstones as a result of their fall speeds, their total cross-sectional area, and, of course, a function of the hail particle size distribution (their number concentration as a function of size) and the horizontal wind speed. A search of the literature reveals that there is a wide spread in estimates of graupel and hail terminal velocities as a function of their size (maximum dimension). The goal of our study is to im- prove these estimates, thereby leading to improved representations for use in climate models and hail damage predictions.

Accurate estimates of the terminal velocities Vt of graupel and hail are also important for improving the representation of precipitation development in weather forecast models. For example, using hail with a bulk density of 0.9 g cm23 rather than graupel with a bulk density of 0.4 g cm23 as the rimed ice species, with ap- propriate adjustments to the Vt, a hail-producing squall- line simulation resulted in a much narrower convective line and slightly smaller stratiform region (Bryan and Morrison 2012). Comparing graupel and hail simula- tions with similar adjustments, Milbrandt and Morrison (2013) and van Weverberg et al. (2012) found that, because of the higher Vt of this hail than graupel, the rate of precipitation fallout was larger than for graupel, and melting extended to the surface rather than not far from the melting layer. However, most schemes do not explicitly consider density but use a single Vt -D relationship.

Terminal velocity depends on the drag coefficient Cd and the particle mass m, which in turn depends on the bulk density of the particles rb (the mass of the particle divided by the volume of a sphere of the same maximum dimension D). For a particle at its equilibrium velocity at a given temperature T and pressure P,

... (1)

where ?f is the density of air, g is the gravitational ac- celeration constant, and A is the cross-sectional area normal to the airflow. Note that rb actually considers two aspects: a reduction in density from that of a sphere due to a nonspherical shape (e.g., spherical versus con- ical graupel), and a reduction in density due to air pockets and voids inside of the particle. Defined as in (1), rb considers both effects.

Because Vt } Cd, accurate estimates of Vt must be based on reliable Cd. For certain sizes of graupel and especially hail, however, there is a sudden drop and flattening of Cd followed by an increase at larger sizes. This anomaly in Cd may be best illustrated by the ear- liest measurements of particles simulating hailstones. To ''at least furnish a rough idea of the order of magnitude of the terminal velocity actually occurring in nature,'' Bilham and Relf (1937, hereafter BR37, p. 150) reported on the form of the relationship between terminal ve- locity and diameter ''deduced from values of the Cd obtained from observations on 4-14-cm spheres towed by airplanes'' (see Table 1). They reported a rapid dropoff in Cd for Reynolds numbers 300 000 , Re , 400 000 and a leveling off of Cd for Re . 4000 (Fig. 1a). The Reynolds number is given by

... (2)

where ? is the kinematic viscosity of air. Note that the actual data in the BR37 study were not reported, only the form of the relationship. The actual data used to derive the curve they show in their article were derived from a pioneering study by Millikan and Klein (1933), although this data source was not acknowledged in their article and took hunting to find.

With points taken from their Fig. 2 (and as shown in their Table 1) converted to appropriate units, Vt can be readily derived assuming the hail particles are solid ice spheres and falling at standard sea level conditions (Fig. 1b). Note that the BR37 definition of the Cd is one- half of the conventional definition, but we have adjusted their values upward by a factor of 2. Rather than showing a monotonic increase in Vt with size from 0 to above 15 cm and a constant Cd of 0.4, which is approxi- mately what is usually assumed (dashed line, Fig. 1b), the resulting Vt show a rather large increase for sizes beginning at 4 cm and below 10 cm, associated with the decrease in Cd. The resulting Vt is due to the trend of Cd with Re and the interplay of Cd and D and suggests that there may be an unexpected change in the Vt-D relationship. BR37 specifically note that ''Over a certain range of diameters, depending on the densities of air and ice and the kinematic viscosity of air, three values of the terminal velocity would therefore appear possible.'' The dip in the Cd-Re relationship above Re ; 2 3 105 has been referred to as a ''supercritical Reynolds number'' regime (Willis et al. 1964).

Following in the footsteps of Laurie (1960), the building industry has widely used the BR37 Cd-Re and/ or the Vt-D relationships to assess and infer hail impact damage. Therefore, we need to ask: does this anomaly apply to real hailstones? Are there other unexpected changes in graupel and hail fall speeds that should be addressed? Can the values of Cd-Re based on smooth hail be applied to natural hailstones?

To answer these questions, this article uses a general- ized, nondimensional Reynolds number-Best number (X) approach often used to derive Vt that depend on characteristic properties of the particles and the local temperature and pressure (see, e.g., Heymsfield and Westbrook 2010). In what follows, we relate Cd to Re and Re to X, so that for a given ice particle and environ- mental conditions, X can be computed directly, Re can be estimated from X, and thus the fall speed can be derived directly.

In section 2, we present many of the observations of rimed crystals or aggregates, graupel, and hail drag co- efficients and associated masses collected to date. Sec- tion 3 attempts to explain the observations and draw on theoretical models, resulting in the development of the nondimensional relationships for deriving the terminal velocities. The results are summarized and conclusions drawn in section 4.

Awordonthe nomenclatureusedinthis article: Drawing on the American Meteorological Society Glos- sary of Meteorology (available online at http://glossary. ametsoc.org/wiki/Main_Page), hail is precipitation in the form of balls or irregular lumps of ice greater than 5 mm; small hail is hail less than 6.4 mm. Graupel are heavily rimed snow particles, often indistinguishable from very small soft hail, except for the size convention that these particles are less than 5 mm. Heavily rimed particles and aggregates are noted where applicable.

2. Observations of graupel and hail

This section presents many of the earlier Cd-Re ob- servations and estimations from numerical simulations of the drag coefficients of rimed crystals and aggregates, graupel, and hail. The primary datasets from which we draw are summarized in Table 1.

According to (1), terminal velocity is proportional to particle mass. Particle mass is proportional to the par- ticle bulk density, which is a highly variable and un- certain parameter. The graupel mass has a major effect on its fall speed, and there have been relatively few measurements of graupel rb. As shown in Fig. 2, graupel masses and the associated rb average about 0.1 g m-3. The rb values from the Knight and Heymsfield (1983) study (Fig. 2) are above the large end of the graupel size designation and are within the small end of the hail size range-small hail-although the rb values are relatively low. They derived the actual densities through volu- metric measurements, thereby considering the ice-free volume inside the particle. Densities so derived were an average of 0.44g cm23 for the actual particles and an average of0.31gcm23for the corresponding values of rb, or about 50% larger. Their rb values are higher than for most of the graupel studies to date and may suggest that the rb values for hail are higher than those for graupel.

When graupel are riming in relatively high-liquid water content regions and at subfreezing temperatures but close to the 08C level, rb values are probably higher than those in Fig. 2. Heymsfield and Pflaum (1985) measured the densities of graupel beginning on frozen drops with diameters of 0.3 and 0.6 mm and growing in a vertical wind tunnel with air temperatures from 2 3 8 to 2158C and liquid water contents from 0.5 to 2.6 g cm23. The relationship that they developed was directly pro- portional to the graupel density and the ''impact veloc- ity'' of the cloud droplets on the graupel (which is approximately equal to the graupel terminal velocity) and inversely proportional to the surface temperature of the graupel. The density of the deposited rime at the warmest temperatures was about 0.6 and 0.25 g cm23 for most temperatures and liquid water contents. Cober and List (1993) have also quantified the density of rime from laboratory measurements. Based on these experiments and the observations of Knight and Heymsfield (1983),it is reasonable to assume that, in convection with rela- tively high liquid water content and warmer cloud-base conditions than found in most of the collections used in Fig. 2 and most of the experiments reported in the Heymsfield and Pflaum (1985) study, graupel densities are closer to 0.3 g cm23.

A curve fit to the graupel mass versus diameter data in Fig. 2, plotted as a line and listed in the figure, indicates that on average, the graupel density increases with size (exponent in the mass power law greater than 3). As- suming for simplicity that the graupel are approximately spherical rather than conical, one can derive an associated power-law density-diameter relationship for graupel:

... (3)

where D is in centimeters and r is in grams per cubic centimeters. Note that diverse datasets are included in Fig. 2 and that the graupel density for one type of scenario (e.g., graupel developed from frozen drops in deep con- vection) may not conform to the results shown in Fig. 2.

The few measurements of rb of hail suggest that they are often close to those of solid ice density. Prodi (1970) used an x-ray absorption technique to measure the in- ternal density (e.g., cavities and hollows) of natural and artificial hailstones, obtaining values between 0.82 and 0.87 g cm23. List et al. (1970) measured rb of naturally occurring 2.54-4.82-cm-diameter hailstones to be ap- proximately 0.9 g cm23. By comparison, Knight and Heymsfield (1983) found rb from 0.31 to 0.61 g cm23 with an average of 0.44 g cm23 for hail from 0.63 to 1.54 cm (see Fig. 2). These particles developed in a springtime Colorado convective storm with surface temperatures of 128C, implying that little melting had occurred and providing an indication of how much in- terior volume is free of ice.

No major sudden decreases have been found in the drag coefficients and the resulting terminal velocities of rimed crystals, rimed aggregates, and graupel particles. Variable parameters for Vt include the graupel di- ameter, shape (conical or spherical), fall orientation, and thus the cross-sectional area, mass, and associated bulk density. Figure 3 shows observations of Cd for particles identified as heavily rimed crystals, rimed aggregates, graupel, and small hail are shown as a func- tion of their derived Reynolds numbers (Locatelli and Hobbs 1974; Kajikawa 1975; Knight and Heymsfield 1983; Takahashi and Fukuta 1988; see Table 1 for identification of the various studies). With increasing Re or size, the Cd decrease by almost an order of magnitude, signifying that all else being equal, Vt would increase by about a factor of 3 rather than remaining constant.

Drawing on fluid dynamic tank studies and observa- tions, Heymsfield and Westbrook (2010) developed an empirical relationship between Cd and Re for graupel as a function of its area ratio Ar , defined as the ratio of the graupel cross-sectional area normal to the airflow rela- tive to that of a circle with the same maximum diameter. This formulation predicts that Cd decrease from about 4 to 0.7 with increasing Re over the typical range of typical graupel Re (Fig. 3). Their formulation yields Cd that are comparable to those derived from a more recent theoretical study (Wang and Kubicek 2013), where in Fig. 3a we plot both formulations assuming Ar 5 1. Both the Heymsfield and Westbrook (2010) and Wang and Kubicek (2013) formulations give a relation- ship between Cd and Re that agrees well with those derived from direct measurements (Fig. 3, from studies reported in Table 1). The main differences between the observations, those from tank experiments, and the theoretical Cd (Re) are differences in Cd at the lower Re, possibly because of the graupel shape (spherical versus conical).

Observations of the Cd for hailstone-size particles indicate that the particle's external structure has a major effect on the resulting relationship between Cd and Re. Observations of the Cd for spherical, smooth ice are plotted in Fig. 4a. The rapid dropoff in Cd with Re noted in the observations of BR37 (Fig. 4a, curve 1) has since been corroborated by Achenbach (1972; curve 2 and others). However, as the roughness of the ice sphere (hailstone) increases, the dip in Cd with Re becomes increasingly flattened [cf. filled and open circles, labeled 11 and 12, from Achenbach (1974, hereafter A74) and curves 7 and 8 from List et al. (1973); Fig. 4b]. As illus- trated schematically in Figs. 4b and 5b, the amount of roughening is quantified by d/D, where d is the diameter or height of the nodules on the surface of the hailstone (visualized as the reverse of the diameter of the dimples on a golf ball). In the A74 experiments, the rough sur- face was obtained by pasting glass spheres of diameter d onto the surface or by abrading the surface with coarse emery paper. Indeed, for the more typical ''spiky'' giant hailstones and from theoretical considerations (curves 7 and 13 of Fig. 4b), the supercritical Reynolds number and the very low values of drag coefficients observed for smooth ice spheres are not observed.

Roughening damps out the supercritical Reynolds number observed for the smooth ice spheres. To show this effect more clearly, Figs. 5a and 5b plot Cd versus Re for smooth spheres and lightly through very rough spheres, respectively. The Re where Cd dips de- creases with increasing roughening and is clearly damped out.

A question remains as to what happens to the Cd of hail as it falls through the melting layer and partially melts. In an ingenious experiment, Willis et al. (1964) lofted ice spheres to subfreezing temperatures on bal- loons, and when the balloons popped, the spheres de- scended through the melting level. A 5.1-cm slightly roughened ice sphere with a roughness parameter (ratio of the diameter of the roughened surface to the diameter of the sphere) less than 0.02 initially had a drag co- efficient of 0.26 and Re of 1.3 3 105. After melting, Cd increased to 0.56 and Re dropped by about 30%. For a 7-cm lightly roughened sphere with an initial Cd of 0.24 and Re of about 1.6 3 105 and just below the critical Reynolds number, the drag coefficient remained con- stant during melting, presumably because the ice sphere diameter changed little during the melting process. Be- cause of the effect of roughening on the fall speeds of the simulated hailstones, they concluded that the rapid in- crease in the terminal velocity observed for smooth ice spheres may only be ''attained rather infrequently in nature'' (Willis et al. 1964, p. 107).

In summary, graupel rb values increase with size or Re, and Cd decreases with increasing Re or size in a systematic fashion that is reasonably well predicted from fluid tank (empirical) data and theoretical con- siderations. Hail of sizes 2 cm and above have densities close to those of solid ice. Experimental data indicate that there is a critical Reynolds number for smooth ice spheres, but this rapid decrease in the Cd becomes more damped and the dip occurs at increasingly lower Re as the ice sphere becomes more roughened. Given this observation and the observations of Browning and Beimers (1967), whose data suggest that most hailstones are likely to be oblate and spiky, it can be concluded that only in occasional instances would a critical Reynolds number be reached by a naturally occurring hailstone.

3. Synthesis of results leading to the calculation of graupel and hail fall speeds

For calculation of terminal velocities, it is convenient to use the nondimensional Davies or Best number,

... (4)

where

... (5)

In other words, X may be calculated from the known properties of the particle and the air temperature and pressure. The objective of this section is to relate X to Re, so that for a given ice particle and environmental condition, X can be computed directly, Re can be esti- mated from X, and thus the fall speed can be derived using (2).

Drawing on the rimed crystal, graupel, and small-hail datasets, and empirical and theoretical relationships identified in Table 1 and used in Fig. 3, Fig. 6 shows the relationship between the Best and Reynolds numbers for rimed through small hail particles. There is a mono- tonic increase in Re with X and no surprising deviations across the range of Re. The theoretical (Wang and Kubicek 2013)andtheHeymsfield and Westbrook (2010) relationships agree quite closely except at the lower Re values, and both provide a reasonably good match with the observations. The Heymsfield and Westbrook (2010) data provide a reasonably good match to the data across the full range of approximately 1 , Re , 105.

The Reynolds number-Best number relationships for smooth and roughened spheres from the observations and for oblate spheroids based on theoretical consider- ations are compared in Fig. 7. The dotted straight lines in the two panels show the corresponding Cd. For smooth spheres, the discontinuity in the Re-X relationship as X increases above 1011 clearly reflects the abrupt decrease in Cd that would be associated with a rapid increase in Vt (Fig. 7a). This effect is appreciably muted or nonexistent for lightly to heavily roughened ice spheres and, where noted, occurs at much lower values of X (Fig. 7b). A similar effect is noted from dimples on a golf ball (Choi et al. 2006), where the dimples cause local flow separa- tion and trigger the shear-layer instability along the separating shear layer, resulting in the generation of large turbulence intensity. In any event, the theoretical Re-X curve in Fig. 7 falls considerably below the data for roughened spheres.

To prevent discontinuities in the fall speeds of graupel and hail for modeling studies and other applications, it is desirable to develop a continuous relationship across all values of the Best number. A power-law Re-X curve, fitted over different ranges of X with coefficient of fits of 0.95 and 0.99, respectively, are

... (6a)

and

... (6b)

The Re-X relationship developed here conforms well to the one developed by Knight and Heymsfield (1983) for small hail and hail of sizes 0.63-1.54 cm (Fig. 8). Because their relationship was developed for the actual particles both as they fell and then after soaking them in oil that fills the particle, it is probably applicable for both small hail as it falls through the melting level and fol- lowing partial melting to fill void cavities. For rimed crystals and graupel, with sizes below those they con- sidered, their relationship would overestimate Re, and for large hail at sizes above those they considered, it would underestimate Re.

We now apply the new Re-X relationship to derive graupel and hail Vt. For simplicity, we assume a pressure level of 1000hPa and temperature of 258C. To illustrate the dependence of Vt on D, two densities are assumed: 0.2 and 0.9 g cm.23 The results indicate that the logarithm of Vt is nearly a linear function of the logarithm of D, indicating that they can be represented by the following power laws:

... (7a)

and

... (7b)

where D is in centimeters and Vt is in centimeters per second. From (6), these relationships change with P by the ratio of (1000/P)0.55.

It is also straightforward to estimate the effect of the particle density on Vt. The density-diameter relation- ship developed for graupel and small hail in Fig. 2 [see (3)] is also applied in Fig. 9.TheVt-D relationship becomes

... (8)

thus, with a steeper slope comes an expected result. Note that this equation would only apply to particles that originate as rimed crystals.

Our Vt-D relationship, for the same particle proper- ties, is quite similar to those of Milbrandt and Morrison (2013) for about the same bulk densities (Fig. 9). Their relationships are developed from a prognostic graupel density scheme that uses the Reynolds number-Best number approach of Mitchell and Heymsfield (2005) assuming that particles are spherical. Also shown is a fit to the terminal velocities derived from the Mitchell and Heymsfield (2005) representation for graupel, assuming solid ice spheres with an area ratio of 1.0. These conform nicely to our new relationship out to 0.1 cm and then diverge from it. Our new Re-X relationship, fitted across the full range of Re-X for rimed particles through hail, extends across a wide range of diameters and can be readily used with variable densities (e.g., see light blue solid curve; Fig. 9). Also shown in Fig. 9 are Vt-D relationships used in weather forecast modeling by Thompson et al. (2008) and Lim and Hong (2010).Al- though not intended to be used at such large diameters, those relationships do produce Vt that are only somewhat higher than those developed here. The range of diameters considered, although too large, is plotted because other modeling studies may apply them in this way.

As suggested by the experiments of Willis et al. (1964), melting will lead to a decrease in the graupel/hail di- ameter, more so for graupel than hail, with the net result being that X, Re, and hence Vt, will decrease. From those experiments with 5.1- and 7-cm hail, there is no evidence that there is an abrupt change in the form of the Re-X relationship. Soaking of graupel or hail with densities below those of solid ice (Knight and Heymsfield 1983) may change the mass but not the diameter, and this in- creased mass could increase the terminal velocity.

4. Summary and conclusions

The goal of this study is to develop accurate expressions for the terminal velocities of rimed crystals, graupel, and hail over size ranges from several tenths of a millimeter to above 10 cm in diameter that can be used in cloud mod- els or engineering-related applications and are general enough to be used for most atmospheric pressures and temperatures. Using observations of the masses and terminal velocities of these particles reported in the liter- ature, we developed a nondimensional Reynolds number- Best number relationship that spans a broad range of Re and corresponding rimed crystal-graupel-hail sizes.

Necessary inputs into the development of this re- lationship are the particle masses or bulk densities and drag coefficients. For heavily rimed crystals or aggregates and graupel, rb values are found to average about 0.1 g cm23. In convection with relatively high liquid water contents and warmer cloud-base conditions than most of the data used in our study, graupel densities are likely to be closer to 0.3 g cm23. The interior volume of graupel undergoing melting will fill with water, leading to densities of solid ice or above. Also, nonspherical graupel will be- come more spherical, with higher masses for a given size. The few observations of the rb of hail are closer to those of solid ice, about 0.5-0.9 g cm23, and possibly higher for low-density hail soaked with liquid water during a melting period. Also, because hail is often oblate and not spheri- cal, most hail will have smaller masses for a given size, leading to lower terminal velocities. Detailed calculations of rb using models with explicit consideration of the density of accreted water (e.g., Mansell et al. 2010; Milbrandt and Morrison 2013) are needed to further quantify particle densities under different scenarios.

Using rimed crystal and graupel mass and terminal velocity data, it is found that for increasing Re, Cd of heavily rimed crystals through graupel particles de- crease by about an order of magnitude, from about 10 to 1. Drag coefficients developed recently based on theo- retical treatments and laboratory tank data for graupel are somewhat lower than the observations. The Cd values of hailstones of about 0.4-0.45 are somewhat lower than the often-assumed value of about 0.5. The so-called hail ''critical Reynolds number'' observed ex- perimentally and theoretically for smooth ice spheres above 10 cm in diameter is damped and occurs at in- creasingly smaller diameters as the particle surface becomes rougher. It can be concluded that only in oc- casional instances would a critical Reynolds number be reached by naturally occurring hailstones.

Acknowledgments. The authors thank Hugh Morrison and Jason Milbrandt for their helpful comments. We especially thank Meg Miller for her editorial assistance. This research was partially supported by NCAR.

REFERENCES

Achenbach, E., 1972: Experiments on the flow past spheres at very high Reynolds numbers. J. Fluid Mech., 54, 565-575, doi:10.1017/S0022112072000874.

<p>_____,1974: The effects of surface roughness andtunnel blockage on the flow past spheres. J. Fluid Mech., 65, 113-125, doi:10.1017/ S0022112074001285.

Auer, A. H., Jr., J. D. Marwitz, G. Vali, and D. L. Veal, 1970: Final report to the National Science Foundation in fulfillment of grants GA-1527 and GA-19105, Department of Atmospheric Resources Rep., University of Wyoming, 92 pp.

Bilham, E. G., and E. F. Relf, 1937: The dynamics of large hail- stones. Quart. J. Roy. Meteor. Soc., 63, 149-162, doi:10.1002/ qj.49706326904.

Browning, K. A., and J. G. D. Beimers, 1967: The oblateness of large hailstones. J. Appl. Meteor., 6, 1075-1081, doi:10.1175/ 1520-0450(1967)006,1075:TOOLH.2.0.CO;2.

Bryan, G. H., and H. Morrison, 2012: Sensitivity of a simulated squall line to horizontal resolution and parameterization of microphysics. Mon. Wea. Rev., 140, 202-225, doi:10.1175/ MWR-D-11-00046.1.

Changnon, S. A., 2009: Increasing major hail losses in the U.S. Climatic Change, 96, 161-166, doi:10.1007/s10584-009-9597-z.

_____,and G. J. Hewings,2001: Losses from weather extremes in the United States. Nat. Hazards Rev., 2, 113-123, doi:10.1061/ (ASCE)1527-6988(2001)2:3(113).

Cheng, K.-Y., and P. K. Wang, 2013: A numerical study of the flow fields around falling hails. Atmos. Res., 132-133, 253-263, doi:10.1016/j.atmosres.2013.05.016.

Cheng, N.-S., 2009: Comparison of formulas for drag coefficient and settling velocity of spherical particles. Powder Technol., 189, 395-398, doi:10.1016/j.powtec.2008.07.006.

Choi, J., W-P Jeon, and H. Choi, 2006: Mechanism of drag re- duction by dimples on a sphere. Phys. Fluids, 18, 041702--, doi:10.1063/1.2191848.

Cober, S. G., and R. List, 1993: Measurements of the heat and mass transfer parameters characterizing conical graupel growth. J. At- mos. Sci., 50, 1591-1609, doi:10.1175/1520-0469(1993)050,1591: MOTHAM.2.0.CO;2.

Heymsfield, A. J., and J. C. Pflaum, 1985: A quantitative as- sessment of the accuracy of techniques for calculating graupel growth. J. Atmos. Sci., 42, 2264-2274, doi:10.1175/ 1520-0469(1985)042,2264:AQAOTA.2.0.CO;2.

_____, and C. D. Westbrook, 2010: Advances in the estimation of ice particle fall speeds using laboratory and field measurements. J. Atmos. Sci., 67, 2469-2482, doi:10.1175/2010JAS3379.1.

Kajikawa, J., 1975: Measurement of falling velocity of individual graupel particles. J. Meteor. Soc. Japan, 53, 476-481.

Knight, N. C., and A. J. Heymsfield, 1983: Measurement and in- terpretation of hailstone density and terminal velocity. J. Atmos. Sci., 40, 1510-1516, doi:10.1175/1520-0469(1983)040,1510: MAIOHD.2.0.CO;2.

Laurie, J. A. P., 1960: Hail and its effects on buildings. National Building Research Institute Council for Scientific and In- dustrial Research Rep. 176, 12 pp.

Lim, K.-S. S., and S.-Y. Hong, 2010: Development of an effective double-moment cloud microphysics scheme with prognostic cloud condensation nuclei (CCN) for weather and climate models. Mon. Wea. Rev., 138, 1587-1612, doi:10.1175/ 2009MWR2968.1.

List, R., P. R. Kry, and U. W. Rentsch, 1970: On the tumbling of spheroidal hailstones. Proc. Conf. on Cloud Physics, Fort Collins, CO, Amer. Meteor. Soc., 67-68.

_____, U. W. Rentsch, A. C. Byram, and E. P. Lozowski, 1973: On the aerodynamics of spheroidal hailstone models. J. Atmos. Sci., 30, 653-661, doi:10.1175/1520-0469(1973)030,0653: OTAOSH.2.0.CO;2.

Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles. J. Geophys. Res., 79, 2185-2197, doi:10.1029/JC079i015p02185.

Mansell, E. R., C. L. Ziegler, and E. C. Bruning, 2010: Simulated electrification of a small thunderstorm with two-moment bulk microphysics. J. Atmos. Sci., 67, 171-194, doi:10.1175/ 2009JAS2965.1.

Milbrandt, J. A., and H. Morrison, 2013: Prediction of graupel density in a bulk microphysics scheme. J. Atmos. Sci., 70, 410- 429, doi:10.1175/JAS-D-12-0204.1.

Millikan, C. B., and A. L. Klein, 1933: The effect of turbulence: An investigation of maximum lift coefficient and turbulence in wind tunnels and in flight. Aircr. Eng. Aerosp. Technol., 5, 169-174, doi:10.1108/eb029703.

Mitchell, D. L., and A. J. Heymsfield, 2005: Refinements in the treatment of ice particle terminal velocities, highlighting ag- gregates. J.Atmos. Sci.,62, 1637-1644,doi:10.1175/JAS3413.1.

Prodi, F., 1970: Measurements of local density in artificial and natural hailstones. J. Appl. Meteor., 9, 903-910, doi:10.1175/ 1520-0450(1970)009,0903:MOLDIA.2.0.CO;2.

Roos, D. S., 1972: A giant hailstone from Kansas in free fall. J. Appl. Meteor., 11, 1008-1011, doi:10.1175/1520-0450(1972)011,1008: AGHFKI.2.0.CO;2.

_____, and A. E. Carte, 1973: The falling behavior of oblate and spiky hailstones. J. Rech. Atmos., 7, 39-52.

Takahashi, T., and N. Fukuta, 1988: Observations of the embryos of graupel. J. Atmos. Sci., 45, 3288-3297, doi:10.1175/ 1520-0469(1988)045,3288:OOTEOG.2.0.CO;2.

Thompson,G.,P.R.Field,R.M.Rasmussen,andW.D. Hall,2008: Implicit 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:10.1175/2008MWR2387.1.

van Weverberg, K., A. M. Vogelmann, H. Morrison, and J. A. Milbrandt, 2012: Sensitivity of idealized squall-line simula- tions to the level of complexity used in two-moment bulk microphysics schemes. Mon. Wea. Rev., 140, 1883-1907, doi:10.1175/MWR-D-11-00120.1.

Wang, P. K., and A. Kubicek, 2013: Flow fields of graupel falling in air. Atmos. Res., 124, 158-169, doi:10.1016/j.atmosres.2013.01.003.

Willis, J. T., K. A. Browning, and D. Atlas, 1964: Radar observations of ice spheres in free fall. J. Atmos. Sci., 21, 103-108, doi:10.1175/ 1520-0469(1964)021,0103:ROOISI.2.0.CO;2.

Young, R. G. E., and K. A. Browning, 1967: Wind tunnel tests of simulated spherical hailstones with variable roughness. J. At- mos. Sci., 24, 58-62, doi:10.1175/1520-0469(1967)024,0058: WTTOSS.2.0.CO;2.

ANDREW HEYMSFIELD

NCAR,* Boulder, Colorado

ROBERT WRIGHT

Robert L. Wright & Associates, Inc., Houston, Texas

(Manuscript received 1 March 2014, in final form 13 April 2014)

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Andrew Heymsfield, NCAR, 3450 Mitchell Lane, Boulder, CO 80301.

E-mail: [email protected]

DOI: 10.1175/JAS-D-14-0034.1