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  • View in gallery

    Map of the continental United States with the collection locations of hailstones over the 6-yr span plotted as blue circle markers. Most of the locations have multiple hailstones at each location.

  • View in gallery

    Plot of the natural hailstone maximum dimension (Dmax) distributions for (a) manually measured hailstones, where n = 3689, and (b) 3D-scanned hailstones, where n = 150.

  • View in gallery

    Examples of 3D-scanned hailstones. In each, the grid boxes represent 1 cm. These were selected from the dataset to highlight hailstone shape diversity and to show some particularly noteworthy examples.

  • View in gallery

    Errors (defined as 3D-scanner measurement minus caliper measurement) for (a) Dmax, (b) Dint, and (c) Dmin, shown as a function of the 3D-scanner measurement of that dimension. The magenta dotted line shows the median error, whereas the orange solid lines encompass plus and minus the RMSE value for each set. A kernel density estimate of the errors is shown to the right of each panel.

  • View in gallery

    Joint PDFs of maximum dimension vs (a),(c) minimum or (b),(d) intermediate aspect ratio for the (top) manually measured hailstones, where (a) n = 3689 (b) n = 1528, and (bottom) 3D-scanned hailstones, where (c),(d) n = 150. Distributions are shown as 2D kernel density estimates, with the densities normalized to the maximum value as indicated using the color bars to the right of each figure. The y axis has an upper limit of 1.1, which allows for some overshooting owing to how the kernel density is calculated.

  • View in gallery

    Plots of Dmax vs minimum aspect ratio for the manually measured hailstones and the Knight (1986) hailstones from Colorado (n = 1790), Oklahoma (n = 2675), and Alberta (n = 1743). Vertical bars represent the 95% confidence interval about the mean within each 5 mm size bin.

  • View in gallery

    Joint PDF of the intermediate vs minimum aspect ratios (shown as KDEs) for (a) the manually measured hailstones (n = 1528) and (b) the scanned hailstones (n = 150). The diagonal line represents prolate spheroids, the right vertical line represents oblate spheroids. Color shading shows density, with yellow values indicating the highest density, and dark blues the lowest. Individual data points shown in (b) as gray circle markers.

  • View in gallery

    Scatterplot of Deq vs Dmax for the manually measured hailstones (n = 3689). The best-fit line is shown in orange, and the cyan shading represents the 95% confidence interval about the best-fit line. The inset box shows the equation for the best-fit line, the 95% confidence interval about the linear correlation coefficient r for the best-fit line, the RMSE, and the 95% confidence interval about the slope of the best-fit line.

  • View in gallery

    As in Fig. 8, but for the 3D-scanned stones (n = 150).

  • View in gallery

    Dmax vs Ψ for the (a) manually measured stones (n = 3689) and (b) 3D-scanned stones (n = 150). The individual data points are shown as gray circles.

  • View in gallery

    Computed Reynolds number NRe vs drag coefficient Cd, based on measured sphericity values. The two green curves represent estimates for spheres from Ganser (1993) (solid) and Cheng (2009) (dashed). Two estimates for velocity are used: Morrison et al. (2005) (stars) and Heymsfield et al. (2018, corrigendum) (squares). The purple color indicates Dmax is used; green indicates Deq is used. Kernel density contours of 10%, 25%, 50%, 75%, and 90% of the data are overlaid in blue.

  • View in gallery

    Hailstone swath from 5 Jun 2014 in eastern Colorado showing (a) the average intermediate aspect ratio, (b) the average minimum ratio, (c) the average maximum dimension, and (d) the overall maximum dimension. n = 82. Two different teams were observing hailstones for this deployment: the top two points were one team and the bottom three were another team.

  • View in gallery

    Schematic conceptual model for how hailstone melting on the ground (brown region) could lead to decreasing φmin. Time increases from left to right. At the first time (t1), the hailstone is freshly fallen and sits with its Dmax (indicated by the green arrow) along the ground, and its Dmin vertical (purple arrow). Owing to contact with the underlying surface, the bottom of the hailstone experiences a larger thermal energy flux (red arrows with sizes suggestive of magnitude). By time t3, the Dmin has undergone a more substantial decrease than Dmax, leading to smaller φmin. Also of note is the diminished protuberance or lobe length, leading to larger Ψ.

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Hailstone Shapes

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  • 1 Department of Meteorology and Atmospheric Science, The Pennsylvania State University, University Park, Pennsylvania
  • | 2 Insurance Institute for Business and Home Safety, Richburg, South Carolina
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Abstract

Hailstone growth results in a variety of hailstone shapes. These shapes hold implications for modeling of hail processes, hailstone fall behaviors including fall speeds, and remote sensing signatures of hail. This study is an in-depth analysis of natural hailstone shapes, using a large dataset of hailstones collected in the field over a 6-yr period. These data come from manual measurements with digital calipers and three-dimensional infrared laser scans. Hailstones tend to have an ellipsoidal geometry with minor-to-major axis ratios ranging from 0.4 to 0.8, and intermediate-to-major axis ratios between 0.8 and 1.0. These suggest hailstones are better represented as triaxial ellipsoids as opposed to spheres or spheroids, which is commonly assumed. The laser scans allow for precise sphericity measurements, for the first time. Hailstones become increasingly nonspherical with increasing maximum dimension, with a typical range of sphericity values of 0.57 to 0.99. These sphericity values were used to estimate the drag coefficient, which was found to have a typical range of 0.5 to over 0.9. Hailstone maximum dimension tends to be 20%–50% larger than the equivalent-volume spherical diameter. As a step toward understanding and quantifying hailstone shapes, this study may aid in better parameterizations of hail in models and remote sensing hail detection and sizing algorithms.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Matthew R. Kumjian, kumjian@psu.edu

Abstract

Hailstone growth results in a variety of hailstone shapes. These shapes hold implications for modeling of hail processes, hailstone fall behaviors including fall speeds, and remote sensing signatures of hail. This study is an in-depth analysis of natural hailstone shapes, using a large dataset of hailstones collected in the field over a 6-yr period. These data come from manual measurements with digital calipers and three-dimensional infrared laser scans. Hailstones tend to have an ellipsoidal geometry with minor-to-major axis ratios ranging from 0.4 to 0.8, and intermediate-to-major axis ratios between 0.8 and 1.0. These suggest hailstones are better represented as triaxial ellipsoids as opposed to spheres or spheroids, which is commonly assumed. The laser scans allow for precise sphericity measurements, for the first time. Hailstones become increasingly nonspherical with increasing maximum dimension, with a typical range of sphericity values of 0.57 to 0.99. These sphericity values were used to estimate the drag coefficient, which was found to have a typical range of 0.5 to over 0.9. Hailstone maximum dimension tends to be 20%–50% larger than the equivalent-volume spherical diameter. As a step toward understanding and quantifying hailstone shapes, this study may aid in better parameterizations of hail in models and remote sensing hail detection and sizing algorithms.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Matthew R. Kumjian, kumjian@psu.edu
Keywords: Hail

1. Introduction

Natural hailstones take on a rich variety of shapes depending on their growth history, trajectory, and fall behavior through their parent storm. In particular, strikingly prominent lobes or protuberances are possible when hailstones undergo vigorous wet growth (e.g., Knight and Knight 1970). Dry growth conditions may render smaller-scale perturbations or cusped lobes (Browning 1966; Knight and Knight 1970) across the hailstone’s surface, as well. Despite this natural variability, hailstones are represented as or assumed to be spheres or spheroids in a wide range of applications, including hailstone growth and melting models (e.g., Ziegler et al. 1983; Nelson 1983; Rasmussen and Heymsfield 1987; Adams-Selin and Ziegler 2016; Dennis and Kumjian 2017; Kumjian and Lombardo 2020), radar-based hail detection and sizing algorithms (e.g., Ryzhkov et al. 2013a,b; Ortega et al. 2016), and microphysics parameterization schemes employed in numerical weather prediction models (e.g., Hong and Lim 2006; Thompson et al. 2004; Morrison and Milbrandt 2015).

Hailstone shape matters for a number of reasons. For example, it affects the thermal energy transfer to and from the environment (e.g., Macklin 1963; Browning 1966; Bailey and Macklin 1968a; Schuepp and List 1969), which governs hailstone growth and melting rates in models (e.g., Nelson 1983; Rasmussen and Heymsfield 1987; Kumjian and Lombardo 2020). A hailstone’s shape—including irregularities and protuberances—also affect its electromagnetic scattering properties (Jiang et al. 2019), which has important implications for radar- or satellite-based hail detection algorithms. An irregular particle’s shape affects its drag coefficient (e.g., Chhabra et al. 1999), and thus fall speed and fall behavior (i.e., tumbling, gyrating, etc.). Hailstone fall speed is an important determinant of its kinetic energy and thus damage potential (Heymsfield et al. 2018), and its falling behavior in turn affects its scattering properties for polarimetric radar (e.g., Ryzhkov et al. 2011; Kumjian 2018). Further, parameterizations of hail fall speed play an important role in simulations of convective storm structure and life cycle (e.g., Morrison and Milbrandt 2011; Bryan and Morrison 2012).

However, despite its importance, only a few studies have quantified the shape of natural hailstones (e.g., Barge and Isaac 1973; Macklin 1977; Knight 1986). This is in large part owing to the difficulty in measuring the often highly irregular shapes of natural hailstones, necessitating simplified models. For example, Barge and Isaac (1973) and Knight (1986) investigated hailstone minor-to-major axis ratios, but characterizing hailstones with a single aspect ratio implicitly assumes a spheroidal shape. More recent field campaigns, such as the Insurance Institute for Business and Home Safety (IBHS) Hail Project (Brown-Giammanco and Giammanco 2018), have begun measuring three orthogonal axes for hailstones, thereby implicitly assuming an ellipsoidal shape.

In this paper, we present an analysis of a more comprehensive dataset of these natural hailstone shapes. The recent advent of infrared laser scanning technology also makes it possible to obtain high-resolution three-dimensional (3D) renderings of hailstone shapes (Giammanco et al. 2017), including precise measurements of natural hailstone volume and surface area, for the first time. Here, we report on two characteristic axis ratios for a large population of hailstones, quantify the relationship between hailstone maximum dimension and equivalent-volume spherical diameter, hailstone sphericity, and discuss implications of how hailstone roughness and irregularities potentially affect thermal energy transfer and drag coefficients.

2. Data and methods

Since 2012, IBHS has conducted an annual field project sampling hailstones in the U.S. Great Plains, and partnered with The Pennsylvania State University (PSU) beginning in 2015. The experiment has sampled more than 3600 hailstones on 42 different days (Table 1) from a large region of the central United States (Fig. 1). Additional samples come from a hail event at the IBHS research center in South Carolina. The hailstones were collected from the ground after the passage of their parent storm, and the major (Dmax) and minor (Dmin) axes were measured manually using digital calipers. Starting in 2014, a third dimension (“intermediate” dimension, Dint) was measured, as well. This intermediate dimension was taken orthogonal to the plane containing the major and minor axes, to the best of the project participants’ abilities. The distribution of hailstone Dmax measurements is shown in Fig. 2a; they range from <0.5 cm to nearly 12 cm, with the vast majority <5 cm.

Table 1.

Numbers of hailstones collected over the 6-yr time span as part of the IBHS field project. There is a discrepancy between the total number of hailstones in the dataset and the total number represented per year. This is because there are 33 stones in the dataset without a specified date or time. Adding those in brings the total number to the 3689 value that is seen elsewhere throughout the study.

Table 1.
Fig. 1.
Fig. 1.

Map of the continental United States with the collection locations of hailstones over the 6-yr span plotted as blue circle markers. Most of the locations have multiple hailstones at each location.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

Fig. 2.
Fig. 2.

Plot of the natural hailstone maximum dimension (Dmax) distributions for (a) manually measured hailstones, where n = 3689, and (b) 3D-scanned hailstones, where n = 150.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

As part of the data collection efforts, IBHS has introduced 3D infrared laser scanning capabilities to provide high-resolution digital models of hailstone shapes (Giammanco et al. 2017). These digital models are used herein to provide more detailed information about hailstone shapes. Such 3D scanning has been used to document and preserve information on giant hailstones (Kumjian et al. 2020) and to compute electromagnetic scattering calculations for natural hailstone shapes (Jiang et al. 2019). Since 2015, 150 hailstones have been scanned, including plaster casts of several record hailstones: Coffeyville, Kansas; Aurora, Nebraska; and Vivian, South Dakota. The 3D-scanned stones ranged in maximum dimension from 1.79 to 16.90 cm (Fig. 2b). In general, these are larger than the manually measured hailstones given the limitations of the 3D-scanner system (the mount used to support the hailstones could not hold stones <1 cm in maximum dimension; see Giammanco et al. 2017) and the time needed to scan a stone, given that smaller hailstones would melt. An example of some of these 3D-scanned hailstones is shown in Fig. 3, with their Dmax shown in Table 2.

Fig. 3.
Fig. 3.

Examples of 3D-scanned hailstones. In each, the grid boxes represent 1 cm. These were selected from the dataset to highlight hailstone shape diversity and to show some particularly noteworthy examples.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

Table 2.

Hailstone shape properties for the sample of 3D-scanned stones shown in Fig. 3. These include equivalent-volume spherical diameter (Deq, in mm), maximum dimension (Dmax, in mm), and sphericity (Ψ).

Table 2.

Many (n = 118) of the 3D-scanned hailstones were also manually measured with digital calipers, affording us the opportunity to estimate the caliper measurement error, considering the 3D-scanned measurement as “truth” for the three hailstone dimensions. The distribution of these errors, defined as the 3D-scanned dimension minus the caliper measurement, are centered on approximately 0 mm for Dmax, Dint, and Dmin (Fig. 4), with median errors of 0.46, −0.08, and 0.87 mm, respectively. The RMSE values for these three dimensions are 2.83, 6.69, and 5.70 mm, respectively. These values suggest the digital caliper measurements are unbiased, and that the Dmax measurements are most accurate; Dmin measurements are the least accurate. These 118 hailstones were included in both datasets, though with independent measurements: calipers versus 3D scans. As such, we should expect relatively good agreement between the statistics from both datasets, aforementioned measurement errors notwithstanding. Though we show both datasets separately in the analysis below, our conclusions are based on the combined data.

Fig. 4.
Fig. 4.

Errors (defined as 3D-scanner measurement minus caliper measurement) for (a) Dmax, (b) Dint, and (c) Dmin, shown as a function of the 3D-scanner measurement of that dimension. The magenta dotted line shows the median error, whereas the orange solid lines encompass plus and minus the RMSE value for each set. A kernel density estimate of the errors is shown to the right of each panel.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

For the manually measured hailstones, both the minimum aspect ratio φmin = Dmin/Dmax and intermediate aspect ratio φint = Dint/Dmax were calculated. From 2012 through part of 2014, only two dimensions were measured in the field, and so for those hailstones (n = 1730), only a minimum aspect ratio could be calculated. For the remainder of the dataset (n = 1959), and for the 3D-scanned stones (n = 150), both aspect ratios were calculated.

We can obtain several additional quantities of interest, including the hailstone equivalent-volume spherical diameter, Deq, defined as the diameter of a sphere with the same volume as the hailstone, the surface area of this equivalent-volume sphere, SAeq, and the actual hailstone surface area, SAmsr. For the manually measured hailstones, we obtained Deq from their estimated volume, V. For hailstones with three measured axes, ellipsoidal geometry was used:
V=π6(Dmax×Dmin×Dint),
whereas for the hailstones with two measured axes, oblate spheroidal geometry was used (e.g., Knight 1986):
V=π6(Dmax×Dmax×Dmin).
These volumes were then used to obtain Deq and SAeq.
To determine SAmsr, we used an expression for the oblate spheroidal surface area SAobl (Beyer 2002) in the case of two-axis measurements:
SAobl=πDmax22+πDmin24εln(1+ε1ε),
where ε=1φmin2 is the eccentricity, or, in the case of three-axis measurements, we used the Thomsen–Cantrell formula for the ellipsoidal surface area SAell:
SAellπ[(DmaxDint)1.6+(DmaxDmin)1.6+(DintDmin)1.63]1/1.6.

For the 3D-scanned hailstones, precise V and SAmsr values were obtained directly from the scanner measurements. The Deq and SAeq directly follow from the measured V.

We also calculated sphericity Ψ (Wadell 1935):
Ψ=SAeqSAmsr
for both datasets. In essence, Ψ is a measure of the hailstone’s shape irregularity. For example, a smooth, spherical hailstone has Ψ = 1, whereas a smooth oblate spheroidal hailstone with φmin = 0.7 (e.g., Knight 1986) has Ψ = 0.977. Note that Ψ−1 can be considered the “surface area enhancement” over the equivalent-volume sphere, so the spheroidal hailstone just described would have a surface area 2.4% greater than a sphere of the same volume. Further, a smooth ellipsoidal hailstone with φmin = 0.7 and φint = 0.85 has Ψ ~ 0.983. Thus, Ψ values substantially less than ~0.98 suggest irregular shapes and/or surfaces compared to smooth spheroidal or ellipsoidal hail of typical axis ratios. The Ψ values for the sample hailstones in Fig. 3 are also provided in Table 2 for reference.

Given the assumed spheroidal or ellipsoidal shape used to characterize the manually measured hailstones, we expect the reported Ψ values to be underestimates, as they do not account for the sometimes highly irregular lobe structures on real hailstones. The 3D-scanned hailstones provide more robust Ψ estimates given the precision of their surface area measurements.

3. Analysis and results

a. Aspect ratios

Figure 5 shows 2D kernel density estimates (KDE) of the distributions of aspect ratios as a function of Dmax for both sets of hailstones. The highest density of data for the manually measured hailstones φmin (Fig. 5a) lies between 0.4 and 0.8, peaking between 0.6 and 0.65. Similarly, the 3D-scanned stones have φmin values peaking between 0.5 and 0.6 (Fig. 5c). These maxima are collocated with some of the smaller hailstone sizes (<4 cm) for the manually measured hailstones and between 2 and 6 cm for the 3D-scanned hailstones, given the distribution of Dmax for these two datasets. For both datasets, some hailstones have a φmin as low as 0.2 or as high as 1. In the case of the φint (Figs. 5b,d), the highest density of data is between 0.8 and 1. Some φint values were as low as 0.4 for both datasets and as low as <0.2 in some isolated instances for the manually measured hailstones. Taken together, there is some evidence of decreasing aspect ratio values with increasing Dmax. The modest positive correlations observed between the Dint errors and Dint (r = 0.69; Fig. 4b), and between the Dmin errors and Dmin (r = 0.53; Fig. 4c) suggest measurement errors would contribute to overestimates of φint and φmin for smaller hailstones, and underestimates for larger hailstones. Such errors could contribute to the observed relationship of decreasing aspect ratios with increasing Dmax; however, these errors are less correlated with Dmax (r = 0.40 and 0.22, respectively). Thus, the digital caliper error characteristics are unlikely to be a dominant contributor to the observed relationship.

Fig. 5.
Fig. 5.

Joint PDFs of maximum dimension vs (a),(c) minimum or (b),(d) intermediate aspect ratio for the (top) manually measured hailstones, where (a) n = 3689 (b) n = 1528, and (bottom) 3D-scanned hailstones, where (c),(d) n = 150. Distributions are shown as 2D kernel density estimates, with the densities normalized to the maximum value as indicated using the color bars to the right of each figure. The y axis has an upper limit of 1.1, which allows for some overshooting owing to how the kernel density is calculated.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

We can further elucidate this relationship by determining the variability of the hailstone φmin with increasing Dmax. Knight (1986) analyzed φmin as a function of Dmax using photographs of hailstone slices from Colorado, Oklahoma, and Alberta, Canada. Figure 6 shows the results from the Knight (1986) study along with the manually measured hailstones used for this study. Similar to the Knight (1986) study, the manually measured hailstones were binned into size ranges of 5 mm, with any hailstone >61 mm in the same bin owing to the small sample size above that Dmax. The mean φmin value for the given range of sizes is plotted along with a bias-corrected and accelerated bootstrapped 95% confidence interval about this mean value using 2000 iterations (Efron and Tibshirani 1993). The Knight (1986) 95% confidence intervals about the mean are also shown.1 In every panel of Fig. 6, the manually measured hailstones φmin were below that of Knight (1986), though the variability in each bin is consistent with that found in Knight (1986). Though the errors in Dmin measurements made with digital calipers were the largest (see section 2), and thus would propagate uncertainty into the φmin estimates, we can rule out a systematic bias in the measurements large enough to explain the discrepancy: recall the median error magnitude was <1 mm. We speculate on a possible reason for this discrepancy in the next section. Both datasets, however, do indicate a decreasing trend of the aspect ratio with increasing Dmax. Knight (1986) found mean φmin values ranging from 0.86 to 0.95 for hailstones with 1 < Dmax < 5 mm, which decreased to a range of 0.58–0.75 for hailstones with 56 <Dmax <60 mm; these values represent the range of means for all three of their sample locations. In contrast, our data show φmin of 0.76 for the 1 <Dmax <5 mm size interval, decreasing to 0.49 for Dmax > 61 mm. In all cases, φmin declines by >0.15 for the full range of Dmax values; averaging Knight’s three datasets with ours results in a decline of 0.25 across the range of Dmax values.

Fig. 6.
Fig. 6.

Plots of Dmax vs minimum aspect ratio for the manually measured hailstones and the Knight (1986) hailstones from Colorado (n = 1790), Oklahoma (n = 2675), and Alberta (n = 1743). Vertical bars represent the 95% confidence interval about the mean within each 5 mm size bin.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

The joint distribution of aspect ratios (φmin, φint) are shown in Fig. 7, along with comparisons to oblate and prolate spheroids. For the manually measured hailstones (Fig. 7a), only the stones with three nonidentical measurements were used, resulting in n = 1528. The highest density of data is located between these two annotated lines, suggesting hailstones typically are triaxial ellipsoids (or scalene oblate spheroids), with φint between 0.8 and 0.9 (modal value of 0.84), and φmin between 0.5 and 0.7 (modal value of 0.63). These measurements agree well with Macklin (1977), who reported the findings of various studies that collectively suggest hailstones tend to be triaxial ellipsoids with only 10%–20% difference in the length of the two major axes (i.e., a φint of 0.8 to 0.9), and φmin down to 0.5.

Fig. 7.
Fig. 7.

Joint PDF of the intermediate vs minimum aspect ratios (shown as KDEs) for (a) the manually measured hailstones (n = 1528) and (b) the scanned hailstones (n = 150). The diagonal line represents prolate spheroids, the right vertical line represents oblate spheroids. Color shading shows density, with yellow values indicating the highest density, and dark blues the lowest. Individual data points shown in (b) as gray circle markers.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

b. Dmax versus Deq

Figure 8 shows Dmax versus Deq for the manually measured hailstones. A strong positive correlation is evident. Again using a bias-corrected and accelerated bootstrapping technique (Efron and Tibshirani 1993) with 2000 iterations to compute the 95% confidence interval about the linear correlation coefficient, we find r = [0.895, 0.940]. A linear fit to the data is also provided in the figure, and has an RMSE of 4.2 mm. The slope of this fitted line (1.175) indicates that Dmax tends to be ~20% larger than Deq. The bootstrapped (n = 2000 iterations) 95% confidence interval about this slope of the best-fit line is [1.1596, 1.1960].

Fig. 8.
Fig. 8.

Scatterplot of Deq vs Dmax for the manually measured hailstones (n = 3689). The best-fit line is shown in orange, and the cyan shading represents the 95% confidence interval about the best-fit line. The inset box shows the equation for the best-fit line, the 95% confidence interval about the linear correlation coefficient r for the best-fit line, the RMSE, and the 95% confidence interval about the slope of the best-fit line.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

Given the imposed assumption of ellipsoidal geometry for the manually measured hailstone dataset, the volume of these stones, and thus Deq, may be overestimated. An overestimation of Deq would lead to an underestimation of the slope of the best-fit line. Giammanco et al. (2015) faced the same issue when estimating hailstone densities: overestimated volume and Deq combined with their mass measurements led to inaccurate estimates of hailstone density.

The relationship between Deq and Dmax for the 3D-scanned stones is shown in Fig. 9. Again, a linear relationship reveals a very good fit as suggested by a high correlation coefficient r = [0.950, 0.978], though with larger RMSE (7.2 mm), compared to the larger dataset of manually measured hailstones. Interestingly, the slope of the fitted line is greater for the 3D-scanned stones, indicating that Dmax is ~40% larger than Deq for these stones, where the bootstrapped (n = 2000 iterations) 95% confidence interval for the slope of the best-fit line is [1.319, 1.486], encompassing larger values than what was obtained from the manually measured dataset.

Fig. 9.
Fig. 9.

As in Fig. 8, but for the 3D-scanned stones (n = 150).

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

Given the limited time and resources during the field campaign, comparatively larger stones were chosen for 3D scanning (cf. Fig. 2). Larger hailstones almost always have substantial wet growth layers (e.g., Knight and Knight 2005), during which more prominent icicle lobes may arise. Because the larger hailstones could be more irregularly shaped, could the 3D-scanner measurements be biased toward greater Dmax? To test this, we recomputed the bootstrapped 95% confidence interval about the slope of the best-fit line to the Dmax as a function of Deq data for the manually measured hailstones, but with all stones with Deq < 15 mm removed (i.e., removed any stone smaller than the smallest one in the 3D-scanned dataset). The removal of these smaller stones did not significantly change the mean slope, and only slightly widened the confidence interval to [1.141, 1.203]. Thus, the ellipsoidal geometry likely contributes more to the discrepancy between the manually measured and 3D-scanned datasets in best-fit line slopes. As such, we suggest that the “true” ratio between Dmax and Deq is somewhere between about 1.2–1.5.

Additionally, for the 3D-scanned stones, as Dmax increases, the difference δD = DmaxDeq tends to increase (r = [0.826, 0.918]; not shown). The slope of the best-fit line (0.33) implies δD tends to be about a third of Dmax. Similarly, the manually measured hailstones feature a moderate correlation between δD and Dmax (r = [0.645, 0.694]; not shown), with a best-fit linear slope of 0.265. Large values of δD can imply greater irregularities or protuberances. This means that hailstones with larger Dmax tend to be more irregular (i.e., nonspherical). Indeed, very large hailstones must have undergone significant periods of wet growth (e.g., Knight and Knight 2005; Kumjian et al. 2020), where prominent lobes or protuberances are thought to form.

c. Ψ

The Ψ distributions as a function of Dmax are shown in Fig. 10 for both sets of hailstones. These are the first known reported Ψ values for natural hailstones.2 For the manually measured hailstone dataset (Fig. 10a), a large majority of the hailstones have Ψ nearing 1, with a mean value of 0.9374. However, the distribution is broad, extending to much smaller values. Additionally, we see that Ψ decreases with increasing Dmax, which again suggests that larger hailstones are more likely to be less spherical in nature compared to their smaller counterparts.

Fig. 10.
Fig. 10.

Dmax vs Ψ for the (a) manually measured stones (n = 3689) and (b) 3D-scanned stones (n = 150). The individual data points are shown as gray circles.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

For the 3D-scanned stones (Fig. 10b), Ψ varies from just below 1.0 to below 0.6, with the bulk of the distribution of points lying between 0.79 and 0.97. In other words, the bulk of the 3D-scanned hailstones exhibit 3%–27% greater surface areas than their equivalent-volume spherical counterparts, although some hailstones have >40%–50% enhancements. These 3D-scanned hailstones also reveal a decrease in Ψ, implying more irregular shapes or surfaces, with increasing Dmax. The one outlier with the largest maximum dimension is the Vivian hailstone: the scan came from a plaster cast, which had a deteriorated shape relative to the actual stone (C. Knight 2014, personal communication). Thus, its Ψ value is likely an overestimate. With this outlier removed, the bootstrapped 95% confidence interval about the linear correlation is r = [−0.799, −0.631] and the best-fit line of Ψ as a function of Dmax has a decrease in Ψ of 0.023 mm−1.

d. Drag coefficient and Reynolds number for 3D-scanned stones

The shapes of particles have an important role in their fall behavior, through the influence of shape on the drag coefficient, Cd. A hailstone’s Cd is inversely proportional to its terminal velocity, υt, so increasing the Cd will decrease the υt and subsequently its kinetic energy. The lobes on a hailstone, which increase the roughness of the stone and can lead to more irregular shapes, have been shown to have an impact on its Cd. For example, a recent wind tunnel study using artificial, symmetrically lobed 3-cm maximum dimension particles (Theis et al. 2020) found that lobes of length 20% of the maximum dimension (i.e., 6 mm) resulted in Cd increases to ~0.57, larger than the value for the smooth 3-cm spheres of the same mass (~0.45).

Often, Cd is parameterized as a function of the particle’s Reynolds number
NRe=υtDpν,
where Dp is the particle’s characteristic linear dimension, and ν is the kinematic viscosity of the fluid within which the particle is embedded. Heymsfield et al. (2014) reviewed studies that found a “supercritical” NRe exists, such that the Cd of the hailstone rapidly decreases at this value, and thus υt rapidly increases; however, this only applied for smooth spheres. Roughening of spherical particles to emulate hailstones tends to damp out this Cd decrease. This suggests that natural hailstones, which, as we have shown above are not smooth spheres, do not have such a supercritical NRe. Further, many of these studies (e.g., Heymsfield et al. 2014, 2018; Theis et al. 2020) employ a method using the Best number, which depends on the hailstone maximum dimension and cross-sectional area. The latter is challenging to estimate, given the uncertainties in hailstone fall behavior and detailed information on its shape. In contrast, the review study of Chhabra et al. (1999) found that the best predictor in NReCd relationships used the particle sphericity Ψ, which is also difficult to obtain in many cases. However, with the advent of 3D scans and the ability to accurately obtain Ψ, we can now explore how Cd might vary with NRe for natural hailstones. As such, for this analysis, only the 3D-scanned hailstones are used.

Determining the hailstone NRe is also challenging owing to the uncertainty associated with υt (e.g., Heymsfield et al. 2018), as well as with Dp (i.e., which characteristic linear dimension to use?). For example, Heymsfield et al. (2018) recommend using Dmax, whereas Chhabra et al. (1999) argue Deq should be used. To account for this uncertainty, we will apply both in the analysis below. Additionally, we apply two hailstone fall speed–dimensional relationships: the latest from Heymsfield et al. (2018, corrigendum), and the one used in the Morrison 2-moment microphysics scheme (e.g., Morrison et al. 2009). In summary, we apply these two fall speed–dimensional relationships, and apply Dmax and Deq as the hailstone characteristic dimensions to both for a total of four combinations.

The distribution of Cd as a function of NRe is shown in Fig. 11. Overlaid for reference are the Cd-NRe relationships for spheres from Ganser (1993) and Cheng (2009). There is no clear relationship between Cd and NRe in the 3D-scanned data (r < 0.25 for all combinations of Dp and υt). The distribution is centered on Cd values between 0.55 and 0.60, with the largest values exceeding 0.8. The Cd values calculated for natural hailstones with irregular shapes are distinctly greater than the expected values for spheres, indicating the influence of natural hailstone shape on Cd. These Ψ-based Cd values are in decent agreement with those found experimentally by Bailey and Macklin (1968b) (<0.66), Knight and Heymsfield (1983) (a bulk of their measurements were between 0.55 and 1.0), and Roos and Carte (1973) (range of 0.36 to 0.9), though somewhat higher than values reported in Heymsfield and Wright (2014) (0.4–0.45 for smaller hail) and lower than those reported in Matson and Huggins (1980) (0.65 to 1.3, with a mean of 0.87).

Fig. 11.
Fig. 11.

Computed Reynolds number NRe vs drag coefficient Cd, based on measured sphericity values. The two green curves represent estimates for spheres from Ganser (1993) (solid) and Cheng (2009) (dashed). Two estimates for velocity are used: Morrison et al. (2005) (stars) and Heymsfield et al. (2018, corrigendum) (squares). The purple color indicates Dmax is used; green indicates Deq is used. Kernel density contours of 10%, 25%, 50%, 75%, and 90% of the data are overlaid in blue.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

4. Discussion

The nonspherical nature of the hailstone shapes described above can give some insight into how such hailstones grow. Spheroidal hailstone shapes may indicate growth while rotating about the minor axis, which presumably is aligned in the horizontal (e.g., Macklin 1977). Asymmetries likely indicate hailstone fall behavior changes during its growth. Such a behavior is at odds with the conceptual model of radar-based studies, which often assume that hailstones are tumbling randomly (e.g., Bringi et al. 1984; Wakimoto and Bringi 1988; Kumjian and Ryzhkov 2008), or wobbling with their major axes oriented in the horizontal on average (e.g., Ryzhkov et al. 2011, 2013a,b). The nonspherical hailstone shapes indicate symmetric tumbling is not occurring.

Further, many studies in the radar community employing electromagnetic scattering calculations for polarization applications (e.g., Ryzhkov et al. 2011, 2013a,b; Kumjian et al. 2019) assume the hailstones to be spherical or spheroidal, in part owing to computational or software limitations. Assuming such rotationally symmetric particles limits the diversity of shapes within the polarization plane, which results in the inability of these studies to accurately simulate observed polarimetric radar signatures. Instead, the use of triaxial ellipsoids in such studies would improve the ability to simulate radar signatures closer to what is observed. Of course, highly irregular or lobed particles further can affect the polarization characteristics of scattered radiation from hailstones and require more sophisticated treatments [see Jiang et al. (2019) for a discussion].

The simple linear relations between Deq and Dmax, as well as between δD and Dmax could be useful for parameterizing Dmax in hailstone growth models (e.g., Adams-Selin and Ziegler 2016; Dennis and Kumjian 2017; Kumjian and Lombardo 2020), which invariably predict spherical shapes and thus Deq. Such a mapping to Dmax affords more useful comparisons of hail growth models to hail report databases, which contain reports of hailstone Dmax.

The precisely measured Ψ values available from the 3D-scanned hailstone dataset reveal surface area enhancements, in some cases of over 50%, compared to equivalent-volume spheres, and that Ψ generally decreases with increasing Dmax. This has important implications for growth and melting rates (e.g., Browning 1977). First, increased surface irregularities, as reflected in smaller Ψ values, imply a larger area of the surface capable of shedding turbulent wake, especially for the larger and thus faster-falling hailstones. Shedding of turbulence enhances the thermal energy transfer to or from the hailstone, and is typically quantified by the ventilation coefficient. Quantifying real hailstone Ψ values could help constrain idealized hailstone models used in wind tunnel studies, electromagnetic scattering calculations, or fluid dynamics computations, which often use rather unrealistic models (e.g., Mirković 2016; Wang and Chueh 2020; Theis et al. 2020). Second, the larger surface areas in general lead to enhanced thermal energy transfer rates because the heating/cooling rates are given by the thermal energy flux integrated over the surface area of the particle. These two factors lead to enhanced energy transfer rates and thus increased growth/melting rates compared to smoother or more spherical hail.

This study also showed the variability of Cd for a given NRe, ranging from 0.5 to 0.9. These values are larger than for spheres, which is unsurprising, given the rich nature of hailstone shapes and irregularities. This suggests that a one-size-fits-all approach (or one-Cd-fits-all, as it were) may not adequately capture the true variability of natural hailstones. This is important, because Kumjian and Lombardo (2020) found a large sensitivity to Cd in their hailstone growth trajectory model, and Cd can strongly impact hailstone damage potential (e.g., Heymsfield et al. 2014). Further, variability in rimed ice fall speeds have enormous impacts on storm-scale numerical simulations (Morrison and Milbrandt 2011; Bryan and Morrison 2012). We suggest that including variability in Cd, especially for larger hailstones and/or those spent in significant periods of wet growth, is warranted for hail growth models, and for assessing hailstone damage potential (e.g., Heymsfield et al. 2014).

The dataset analyzed here provides insights into the natural variability of hailstone shapes. However, no dataset is without limitations. One such limitation of the dataset in our study comes from the potential of the hailstones melting in the field both before and during measurements. We can illustrate the potential impact melting may have on the data using the variability of hailstone properties across a hail swath during an example measurement IOP. Figure 12 shows such an example from a hail swath targeted during the IBHS field campaign from 5 June 2014 in eastern Colorado. The Storm Prediction Center issued a slight risk for severe thunderstorms that day, with a 15% chance for hail with Dmax >1 in. (>2.5 cm). A severe thunderstorm warning was issued for this area at 2250 UTC (1650 MDT), warning for quarter-sized (1-in. or 2.5-cm) hail. The storm passed over the collection area between 2313 and 2346 UTC, with peak lowest-elevation angle radar reflectivity values >60 dBZ, suggestive of hail. Measurements by two IBHS teams were completed within 2.5 h of the storm passage, with the northern team moving south to north, and the southern team moving north to south. The largest hailstone of the 83 measured across this swath was 1.9 cm (or about 0.75 in.), which was at the southernmost location shown in Fig. 12. Because all 3 hailstone dimensions were measured during this IOP, both the intermediate and minimum aspect ratios were calculated. These aspect ratios, along with the Dmax, were averaged over each of the 5 collection sites and plotted in Fig. 12 at their respective locations.

Fig. 12.
Fig. 12.

Hailstone swath from 5 Jun 2014 in eastern Colorado showing (a) the average intermediate aspect ratio, (b) the average minimum ratio, (c) the average maximum dimension, and (d) the overall maximum dimension. n = 82. Two different teams were observing hailstones for this deployment: the top two points were one team and the bottom three were another team.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

Overall, there is a decreasing trend in φmin as the teams progressed: the northern team’s measurements reveal decreasing φmin of 0.0048 min−1 as they moved northward, and the southern team’s measurements reveal decreasing φmin of −0.0052 min−1 as they moved southward. These trends are coupled with a general decrease in both the average Dmax and the overall Dmax, as one moves northward across the hail swath. This is despite the southernmost point being the latest measurement site for the southern team. That the maximum hail size is located on the southern periphery of the hail swath is consistent with expectations of fallout locations based on conceptual models of hail size sorting in right-moving supercell storms in the Northern Hemisphere (e.g., Browning and Foote 1976) and trajectory model calculations (e.g., Kumjian and Lombardo 2020). In contrast, the northward decrease in φmin in this case is more puzzling. We speculate that such a time-dependent signal in the measurements seems to be plausibly an artifact of melting. Though we are unaware of detailed studies of hail melting while on the ground, it seems reasonable to speculate that surface melting could potentially impact the measured hailstone shapes. Specifically, we expect hailstones to “flatten out” or become more oblate as the part of the hailstone in contact with the warm ground3 preferentially melts faster than the sides or top, which are exposed to the air. It is also expected that protuberances or lobes may melt faster than the hailstone body (e.g., Browning 1966), possibly increasing the sphericity and/or resulting in less exaggerated aspect ratios compared to freshly fallen hail. A conceptual model for such preferential melting is depicted schematically in Fig. 13. Additionally, rain may also lead to uneven melting of hailstones on the surface, affecting the hailstone shapes. The preferential melting and decreasing of the φmin with time could explain the discrepancy between our data and Knight (1986) seen in Fig. 6. Recall that the Dint measurement errors were consistent with those of Dmin; interestingly, for φint, the southern collection team observed an increase with location, whereas the northern team observed a decrease similar to the other measured variables. This may be a reflection that φint is less affected by melting while sitting on the surface.

Fig. 13.
Fig. 13.

Schematic conceptual model for how hailstone melting on the ground (brown region) could lead to decreasing φmin. Time increases from left to right. At the first time (t1), the hailstone is freshly fallen and sits with its Dmax (indicated by the green arrow) along the ground, and its Dmin vertical (purple arrow). Owing to contact with the underlying surface, the bottom of the hailstone experiences a larger thermal energy flux (red arrows with sizes suggestive of magnitude). By time t3, the Dmin has undergone a more substantial decrease than Dmax, leading to smaller φmin. Also of note is the diminished protuberance or lobe length, leading to larger Ψ.

Citation: Journal of the Atmospheric Sciences 78, 2; 10.1175/JAS-D-20-0250.1

This case is an illustrative example of trends sometimes observed in the field. Note that these trends were not consistent from swath to swath and in some cases, no clear hailstone shape and size trend was present. Nonetheless, we believe some data limitations are highlighted in this case.

5. Conclusions

Large datasets of manually measured hailstones (n = 3689) and 3D laser-scanned hailstones (n = 150) from the United States were analyzed to characterize and quantify the shapes of natural hailstones. The hailstones range in maximum dimension (Dmax) from about <0.5 to >16 cm. Our analysis yields the following conclusions:

  • Rather than spheres or spheroids, as is often assumed, triaxial ellipsoids or scalene oblate spheroids better represent hailstone shapes. These findings are consistent with previous studies with smaller sample sizes (e.g., Macklin 1977). In particular, the minor-to-major axis ratio tends to be between 0.4 and 0.8, with intermediate-to-major axis ratios between 0.8 and 1.0. The minor-to-major axis ratios are somewhat smaller than other datasets in the literature (Knight 1986), and may reflect prolonged melting on the surface prior to measurement.

  • Hailstones become increasingly nonspherical with increasing size, as the aspect ratios and the sphericity of the hailstones decline with increasing Dmax.

  • Strong, statistically significant positive correlations are found for the relationship between Dmax and spherical volume-equivalent diameter Deq, suggesting that Dmax is typically 20%–50% larger than Deq.

  • For the first time, precise sphericity values are reported for real hailstones, obtained using 3D laser scanning technology. The sphericity values range from 0.57 to 0.99, with the peak in the distribution near 0.85. These sphericity values indicate enhancements in the hailstone surface area over that of the equivalent-volume sphere, which enhances vapor and thermal energy transfer to/from the hailstones, thus increasing the growth/melting rates over those assumed for spheres.

  • These sphericity values are also used to obtain estimates of the drag coefficients of natural hailstones. The drag coefficients range from 0.5 to over 0.9; this range is substantially higher than the values for smooth spheres of similar Reynolds numbers (i.e., sizes), and has important implications for irregular hailstone fall speeds.

Quantification of hailstone shapes is important because of their importance and relevance to microphysical processes, fall behaviors and fall speeds, and radar/satellite remote sensing. The data presented and analyzed here may help inform the parameterization or representation of hail in models and algorithms. We advocate for additional detailed measurements of hailstone shapes, especially with laser scanning. In particular, measurements of freshly fallen hail are needed to overcome the potential bias melting introduces in our dataset.

Acknowledgments

Support for the first and second authors of this research comes from a grant from the Insurance Institute for Business and Home Safety. The second author is also supported by NSF Grants AGS-1661679 and AGS-1855063. We thank the constructive comments from the three anonymous reviewers, which helped to clarify and strengthen this study. This work is part of the first author’s research as an undergraduate student at The Pennsylvania State University.

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1

The Knight (1986) confidence intervals were computed with a t test, which strictly applies to a population characterized by a normal distribution.

2

Note that, although reported as Ψ, the values shown in Fig. 5 of Heymsfield et al. (2018) are of measured volume divided by equivalent spherical volume, which is an alternative and not equivalent definition to the standard from Wadell (1935).

3

This assumes hailstones have their Dmax approximately level with the ground, which seems reasonable.

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