CORRIGENDUM

Andrew Heymsfield National Center for Atmospheric Research, Boulder, Colorado

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Miklós Szakáll Institute for Atmospheric Physics, Johannes Gutenberg University, Mainz, Germany

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Alexander Jost Institute for Atmospheric Physics, Johannes Gutenberg University, Mainz, Germany

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Ian Giammanco Insurance Institute for Business and Home Safety, Richburg, South Carolina

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Robert Wright RLWA and Associates, Houston, Texas

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Julian Brimelow Environment and Climate Change Canada, Edmonton, Alberta, Canada

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Abstract

This corrigendum improves upon the size-dependent representation of graupel and hail terminal velocities, kinetic energies, and mass fluxes that were reported in the Heymsfield et al. (2018) study. In particular, representation of these dependencies on diameter over the full range of particle sizes is improved upon by correcting minor errors and by developing representations that cover different size ranges.

© 2020 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: Andrew Heymsfield, heyms1@ucar.edu

Abstract

This corrigendum improves upon the size-dependent representation of graupel and hail terminal velocities, kinetic energies, and mass fluxes that were reported in the Heymsfield et al. (2018) study. In particular, representation of these dependencies on diameter over the full range of particle sizes is improved upon by correcting minor errors and by developing representations that cover different size ranges.

© 2020 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: Andrew Heymsfield, heyms1@ucar.edu

1. Introduction

Heymsfield et al. (2018, hereafter H18) developed size-dependent relationships for the terminal velocity (Vt), kinetic energy (KE), and mass flux (MF) of natural and 3D printed graupel and hailstones, drawing on a dataset based on collections of graupel and hail at the ground and measurements of printed replicas in a vertical wind tunnel. Because of a minor computational error, the equations [H18, Eqs. (7) and (8)] that were developed to derive the terminal velocity from the particle diameter (D), and KE [H18, Eq. (10)] and MF [H18, Eq. (13)] from D, as well as the fitting parameters for the power laws for different percentiles given in Table 2 in H18, are in need of a correction. The computational error came about because one of the Interactive Data Language (IDL) structures for graupel particle mass and terminal velocity had omitted one of the data sources for particle mass. This problem, although minor, factored into the calculations for not only graupel but when graupel and hail particles were combined. Accordingly, we have carefully reexamined all aspects of our study and have made slight adjustments to equations where deemed desirable.

2. Results

a. Reynolds number–Best number

Figure 1 shows the revised Reynolds number (Re)–Best number (X) relationship derived from observations and theory. In Fig. 1a, Re–X values that are based on the direct measurements (i.e., observations) of Matson and Huggins (1980), the theoretically based curve of Böhm (1989), and those from the widely used Laurie (1960) study are shown. Figure 1b includes graupel observations from Locatelli and Hobbs (1974), Pflaum (1978), Knight and Heymsfield (1983), and Heymsfield and Kajikawa (1987); observations for hail from Matson and Huggins (1980) and Roos and Carte (1973); and the H18 wind tunnel results, corrected for the identified error previously noted. Table 1 lists the curves for graupel from H18 (row 1), and fitted to the observations for graupel, modified to account for the error in H18 (row 2); the observations for hail (row 3); the hail observations combined with the wind tunnel data (row 4); and the analysis of H18 for graupel combined with hail (row 5). The curves for graupel (387 data points), and graupel and hail combined (1214 points), have almost the same coefficients; because the number of graupel data points weighted the size distribution of the combined dataset toward the small diameters; this weighting accounted for most of the error.

Fig. 1.
Fig. 1.

Reynolds number vs Best number (a) for hail and (b) for the revised dataset, with graupel and hail combined.

Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0185.1

Table 1.

Relationships developed between the Reynolds number and Best number, and how well the fitted curves represent the data. The following abbreviations are used: calculated, calc; measured, meas; and observed, obs.

Table 1.

Metrics for the performance of each curve, given by the ratio of Re as predicted by the fitted curve to Re observed (or observed combined with the wind tunnel–derived values), are shown in Table 1. The combined and H18 relationships both provide good fits to the data, but the combined dataset (i.e., observed hail and wind tunnel observations) has a much smaller standard deviation. The graupel and hail relationships for the observed data, shown separately in the table, each provide a good fit.

Figure 2a shows how well the relationships based on the observations for graupel, and the combined observational and wind tunnel data, comply with the data. The results for the combined graupel and hail data differ little from the separate relationships (Fig. 2b). The use of the observations and wind tunnel results combined
Re=0.38X0.56
provides a good fit to the data.
Fig. 2.
Fig. 2.

Evaluation of the quality of the curve fits (a) using the observations for graupel and hail separately and (b) for the combined observations and wind tunnel results.

Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0185.1

b. Terminal velocity

Figure 3a shows the dependence of graupel and hail terminal velocities on their maximum dimension. One set of points is based on the direct physical measurements (observations). A second set uses the Re–X relationship [Eq. (1)], with inputs for X (diameter, mass) from the Insurance Institute for Business and Home Safety (IBHS) measurements, to derive Vt. These are
Vt=7.6D0.89,D<1.5cm
and
Vt=8.4D0.67,D>1.5cm.

The curve from H18 is plotted for comparison. Note that the new curve for hail conforms better to the larger hailstones than the H18 curve. (See H18 for the data sources for the >5-cm hailstones.) If Vt for graupel only was needed, the single curve extrapolated to D > 1.5 cm could readily be used. For both graupel and hail, the two curves, over the respective diameter limits, can be used to estimate Vt.

Fig. 3.
Fig. 3.

(a) Observed terminal velocities, or calculated from the wind tunnel measurements using the Re–X relationship [Eq. (1)], as a function of diameter. Different curve fits identified in the text are described and presented in Table 3. (b) Ratio of the terminal velocity calculated from the fit to the observed together with the calculated terminal velocity divided by the observed or calculated terminal velocity, as a function of diameter. Assessment of the quality of each of the fits is summarized in Table 2.

Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0185.1

Table 2 compares fitted curves for graupel and hail for different scenarios, and indicates how well each of the curves fit the respective dataset: the observations for graupel, using the H18 curve (row 1) and the revised curve (row 2); a fit to the observations for graupel and hail combined (row 3); for hail, observed and from the wind tunnel (row 4); and the H18 curve, compared to the observations (row 5) and wind tunnel measurements (row 6). It is noted that the new curve fitted to the observations for graupel more accurately fits the data than the H18 relationship for graupel. A fit to the combined graupel and hail observations match those data extremely well. Our new, two-relationship fit to the observations together with the wind tunnel results also match the data extremely well. They are a significant improvement over the use of the single H18 relationships.

Table 2.

Relationships developed between the terminal velocity (Vt; m s−1) and diameter (D; cm), and how well the curve fits represent the data.

Table 2.

It is important, in our view, for numerical modeling studies to represent the natural variability of the terminal velocities of graupel or hail at any given size. Two particles of the same diameter but different masses or cross-sectional areas are likely to be transported into different regions of an updraft, with consequently different trajectories and growth. For this reason, in Table 3 we provide size-dependent relationships for Vt at percentiles that enclose the population at given sizes (as in H18).

Table 3.

Power-law fits of terminal velocity (Vt; m s−1) to maximum diameter (D; cm), where Vt = aDb.

Table 3.

c. Kinetic energy

The graupel and hail KE–D relationship of H18 has been reworked to better conform with the revised analysis of the Re–X and VtD relationships. Consistent with the latter analysis, two curves were fitted over different size ranges to provide better agreement with both the observations (mainly 1–2 cm) and the wind tunnel results (many with diameters above 2 cm) (Fig. 4a):
KE=0.0107D4.59,D<2.21cm
and
KE=0.0243D3.5,D>2.21cm.

The H18 relationship is almost identical to the new relationship that was fitted mainly to the wind tunnel measurements for diameters above 2.21 cm, the diameter where the two new relationships cross over. Figure 4b shows how well the new relationships match the observations and wind tunnel results, and Table 4 summarizes the results and compares them with the H18 parameterization. The new curves provide better agreement with the particles at both the smaller and larger diameters. Relative to the new curves, the Laurie relationship produces much larger KE (Fig. 4b).

Fig. 4.
Fig. 4.

(a) Kinetic energy for particles of graupel and hail sizes as measured (observed) and also derived from the IBHS mass and cross-sectional area data and the Re–X relationship. Two least squares curves, and the relationships derived, are shown. Also shown is the curve derived from the Laurie (1960) data. (b) Assessment of the quality of each of the fits, with the results summarized in Table 4.

Citation: Journal of the Atmospheric Sciences 77, 1; 10.1175/JAS-D-19-0185.1

Table 4.

Relationships developed between graupel and hail kinetic energy (KE; J) and diameter (D; cm), and how well the curve fits represent the measurements (observations), and the measurements + wind tunnel measurements.

Table 4.

It is important to represent accurately the KE of hailstones for different diameters when considering the values of KE used in impact testing of materials for hail resistance and for damage modeling. Single power-law fits to the distribution of KE–D for different size percentiles are shown in Table 5. These values are the same as those in H18, except for the 50th percentile.

Table 5.

Power-law fits of kinetic energy (KE; J) to maximum diameter (D; cm) in percentiles, where KE = aDb.

Table 5.

d. Mass flux

Given the potential importance of the hail precipitation rate, Table 6 shows the fractional distribution of the graupel and hail mass flux as a function of diameter. The results are similar to those in H18.

Table 6.

Coefficients in mass flux (MF; g cm s−1)–diameter (D; cm) power-law relationships, where MF = aDb at pressure P = 1000 hPa.

Table 6.

3. Concluding remarks

This article has developed relationships that relate different properties of graupel and hail to their maximum diameter, drawing on a combination of measurements of natural particles and those printed from 3D scans of natural hailstones. This work corrects a minor problem found in the study by H18, and in so doing improves upon the work reported in that study. Careful attention is given to developing relationships that provide better fits to the data over specific size ranges and to characterizing how well the relationships match the data together with how they improve upon other possible solutions. Further testing is needed in order to more accurately represent the orientation of falling graupel and hail as a factor that alters the terminal velocities of individual hailstones during their fall.

REFERENCES

  • Böhm, H. P., 1989: A general equation for terminal fall speed of solid hydrometeors. J. Atmos. Sci., 46, 24192427, https://doi.org/10.1175/1520-0469(1989)046<2419:AGEFTT>2.0.CO;2.

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    • Export Citation
  • Heymsfield, A. J., and M. Kajikawa, 1987: An improved approach to calculating terminal velocities of plate-like crystals and graupel. J. Atmos. Sci., 44, 10881099, https://doi.org/10.1175/1520-0469(1987)044<1088:AIATCT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., M. Szakáll, A. Jost, I. Giammanco, and R. Wright, 2018: A comprehensive observational study of graupel and hail terminal velocity, mass flux, and kinetic energy. J. Atmos. Sci., 75, 38613885, https://doi.org/10.1175/JAS-D-18-0035.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Knight, N. C., and A. J. Heymsfield, 1983: Measurement and interpretation of hailstone density and terminal velocity. J. Atmos. Sci., 40, 15101516, https://doi.org/10.1175/1520-0469(1983)040<1510:MAIOHD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Laurie, J. A. P., 1960: Hail and its effects on buildings. Council for Scientific and Industrial Research Rep. 176, 12 pp.

  • Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles. J. Geophys. Res., 79, 21852197, https://doi.org/10.1029/JC079i015p02185.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Matson, R. J., and A. W. Huggins, 1980: The direct measurement of the sizes, shapes and kinematics of falling hailstones. J. Atmos. Sci., 37, 11071125, https://doi.org/10.1175/1520-0469(1980)037<1107:TDMOTS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pflaum, J. C., 1978: A wind tunnel study on the growth of graupel. Ph.D. dissertation, University of California, Los Angeles, 107 pp.

  • Roos, D. S., and A. E. Carte, 1973: The falling behavior of oblate and spiky hailstones. J. Rech. Atmos., 7, 3952.

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  • Böhm, H. P., 1989: A general equation for terminal fall speed of solid hydrometeors. J. Atmos. Sci., 46, 24192427, https://doi.org/10.1175/1520-0469(1989)046<2419:AGEFTT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., and M. Kajikawa, 1987: An improved approach to calculating terminal velocities of plate-like crystals and graupel. J. Atmos. Sci., 44, 10881099, https://doi.org/10.1175/1520-0469(1987)044<1088:AIATCT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Heymsfield, A. J., M. Szakáll, A. Jost, I. Giammanco, and R. Wright, 2018: A comprehensive observational study of graupel and hail terminal velocity, mass flux, and kinetic energy. J. Atmos. Sci., 75, 38613885, https://doi.org/10.1175/JAS-D-18-0035.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Knight, N. C., and A. J. Heymsfield, 1983: Measurement and interpretation of hailstone density and terminal velocity. J. Atmos. Sci., 40, 15101516, https://doi.org/10.1175/1520-0469(1983)040<1510:MAIOHD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Laurie, J. A. P., 1960: Hail and its effects on buildings. Council for Scientific and Industrial Research Rep. 176, 12 pp.

  • Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles. J. Geophys. Res., 79, 21852197, https://doi.org/10.1029/JC079i015p02185.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Matson, R. J., and A. W. Huggins, 1980: The direct measurement of the sizes, shapes and kinematics of falling hailstones. J. Atmos. Sci., 37, 11071125, https://doi.org/10.1175/1520-0469(1980)037<1107:TDMOTS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pflaum, J. C., 1978: A wind tunnel study on the growth of graupel. Ph.D. dissertation, University of California, Los Angeles, 107 pp.

  • Roos, D. S., and A. E. Carte, 1973: The falling behavior of oblate and spiky hailstones. J. Rech. Atmos., 7, 3952.

  • Fig. 1.

    Reynolds number vs Best number (a) for hail and (b) for the revised dataset, with graupel and hail combined.

  • Fig. 2.

    Evaluation of the quality of the curve fits (a) using the observations for graupel and hail separately and (b) for the combined observations and wind tunnel results.

  • Fig. 3.

    (a) Observed terminal velocities, or calculated from the wind tunnel measurements using the Re–X relationship [Eq. (1)], as a function of diameter. Different curve fits identified in the text are described and presented in Table 3. (b) Ratio of the terminal velocity calculated from the fit to the observed together with the calculated terminal velocity divided by the observed or calculated terminal velocity, as a function of diameter. Assessment of the quality of each of the fits is summarized in Table 2.

  • Fig. 4.

    (a) Kinetic energy for particles of graupel and hail sizes as measured (observed) and also derived from the IBHS mass and cross-sectional area data and the Re–X relationship. Two least squares curves, and the relationships derived, are shown. Also shown is the curve derived from the Laurie (1960) data. (b) Assessment of the quality of each of the fits, with the results summarized in Table 4.

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