• Anderson, M. E., L. D. Carey, W. A. Petersen, and K. R. Knupp, 2011: C-band dual-polarimetric radar signatures of hail. Electron. J. Oper. Meteor., 2011-EJ02. [Available online at http://www.nwas.org/ej/pdf/2011-EJ2.pdf.]

    • Search Google Scholar
    • Export Citation
  • Atlas, D., and F. H. Ludlam, 1961: Multi-wavelength radar reflectivity of hailstorms. Quart. J. Roy. Meteor. Soc., 87, 523534.

  • Aydin, K., and Y. Zhao, 1990: A computational study of polarimetric radar observables in hail. IEEE Trans. Geosci. Remote Sens., 28, 412422.

    • Search Google Scholar
    • Export Citation
  • Aydin, K., and V. Giridhar, 1991: Polarimetric C-band radar observables in melting hail: A computational study. Preprints, 25th Int. Conf. on Radar Meteorology, Paris, France, Amer. Meteor. Soc., 733736.

  • Aydin, K., T. A. Seliga, and V. Balaji, 1986: Remote sensing of hail with a dual linear polarization radar. J. Climate Appl. Meteor., 25, 14751484.

    • Search Google Scholar
    • Export Citation
  • Aydin, K., V. N. Bringi, and L. Liu, 1995: Rain-rate estimation in the presence of hail using S-band specific differential phase and other radar parameters. J. Appl. Meteor., 34, 404410.

    • Search Google Scholar
    • Export Citation
  • Balakrishnan, N., and D. S. Zrnić, 1990a: Estimation of rain and hail rates in mixed-phase precipitation. J. Atmos. Sci., 47, 565583.

    • Search Google Scholar
    • Export Citation
  • Balakrishnan, N., and D. S. Zrnić, 1990b: Use of polarization to characterize precipitation and discriminate large hail. J. Atmos. Sci., 47, 15251540.

    • Search Google Scholar
    • Export Citation
  • Beard, K. V., 1976: Terminal velocity and shape of cloud and precipitation drops aloft. J. Atmos. Sci., 33, 851864.

  • Boodoo, S., D. Hudak, M. Leduc, A. V. Ryzhkov, N. Donaldson, and D. Hassan, 2009: Hail detection with a C-band dual-polarization radar in the Canadian Great Lakes region. Preprints, 34th Conf. on Radar Meteorology, Williamsburg, VA, Amer. Meteor. Soc., 10A.5. [Available online at http://ams.confex.com/ams/pdfpapers/156032.pdf.]

  • Borowska, L., A. Ryzhkov, D. Zrnić, C. Simmer, and R. Palmer, 2011: Attenuation and differential attenuation of the 5-cm-wavelength radiation in melting hail. J. Appl. Meteor. Climatol., 50, 5976.

    • Search Google Scholar
    • Export Citation
  • Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41, 674685.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 636 pp.

  • Bringi, V. N., J. Vivekanandan, and J. D. Tuttle, 1986: Multiparameter radar measurements in Colorado convective storms. Part II: Hail detection studies. J. Atmos. Sci., 43, 25642577.

    • Search Google Scholar
    • Export Citation
  • Carras, J. N., and W. C. Macklin, 1973: The shedding of accreted water during hailstone growth. Quart. J. Roy. Meteor. Soc., 99, 639648.

    • Search Google Scholar
    • Export Citation
  • Cheng, L., and M. English, 1983: A relationship between hailstone concentration and size. J. Atmos. Sci., 40, 204213.

  • Cheng, L., M. English, and R. Wong, 1985: Hailstone size distributions and their relationship to storm thermodynamics. J. Climate Appl. Meteor., 24, 10591067.

    • Search Google Scholar
    • Export Citation
  • Depue, T. K., P. C. Kennedy, and S. A. Rutledge, 2007: Performance of the hail differential reflectivity (HDR) polarimetric radar hail indicator. J. Appl. Meteor. Climatol., 46, 12901301.

    • Search Google Scholar
    • Export Citation
  • Eccles, P. J., and D. Atlas, 1973: A dual-wavelength radar hail detector. J. Appl. Meteor., 12, 847854.

  • Feral, L., H. Sauvageot, and S. Soula, 2003: Hail detection using S- and C-band radar reflectivity difference. J. Atmos. Oceanic Technol., 20, 233248.

    • Search Google Scholar
    • Export Citation
  • Ganson, S., 2012: Investigation of polarimetric radar characteristics of melting hail using advanced T-matrix computations. M.S. thesis, School of Meteorology, University of Oklahoma, 73 pp.

  • Gu, J.-Y., A. Ryzhkov, P. Zhang, P. Neilley, M. Knight, B. Wolf, and D.-I. Lee, 2011: Polarimetric attenuation correction in heavy rain at C band. J. Appl. Meteor., 50, 3958.

    • Search Google Scholar
    • Export Citation
  • Heinselman, P. L., and A. V. Ryzhkov, 2006: Validation of polarimetric hail detection. Wea. Forecasting, 21, 839850.

  • Joe, P. I., and Coauthors, 1976: Loss of accreted water from a growing hailstone. Preprints, Int. Conf. on Cloud Physics, Boulder, CO, Amer. Meteor. Soc., 264269.

  • Jung, Y., G. Zhang, and M. Xue, 2008: Assimilation of simulated polarimetric radar data for a convective storm using the ensemble Kalman filter. Part I: Observation operators for reflectivity and polarimetric variables. Mon. Wea. Rev., 136, 22282245.

    • Search Google Scholar
    • Export Citation
  • Kaltenboeck, R., and A. Ryzhkov, 2013: Comparison of polarimetric signatures of hail at S and C bands for different hail sizes. Atmos. Res., 123, 323336.

    • Search Google Scholar
    • Export Citation
  • Kamra, A. K., R. V. Bhalwankar, and A. B. Sathe, 1991: Spontaneous breakup of charged and uncharged water drops freely suspended in a wind tunnel. J. Geophys. Res., 96, 17 15917 168.

    • Search Google Scholar
    • Export Citation
  • Khain, A., A. Pokrovsky, M. Pinsky, A. Seifert, and V. Phillips, 2004: Simulation of effects of atmospheric aerosols on deep turbulent convective clouds using a special microphysics mixed-phase cumulus cloud model. Part I: Model description and possible applications. J. Atmos. Sci., 61, 29632982.

    • Search Google Scholar
    • Export Citation
  • Khain, A., D. Rosenfeld, A. Pokrovsky, U. Blahak, and A. Ryzhkov, 2011: The role of CCN in precipitation and hail in a mid-latitude storm as seen in simulations using a spectral (bin) microphysics model in a 2D frame. Atmos. Res., 99, 129146.

    • Search Google Scholar
    • Export Citation
  • Kumjian, M. R., J. C. Picca, S. M. Ganson, A. V. Ryzhkov, J. Krause, D. Zrnić, and A. Khain, 2010: Polarimetric characteristics of large hail. Preprints, 25th Conf. on Severe Local Storms, Denver, CO, Amer. Meteor. Soc., 11.2. [Available online at https://ams.confex.com/ams/pdfpapers/176043.pdf.]

  • Lesins, G. B., R. List, and P. I. Joe, 1980: Ice accretions. Part I: Testing of new atmospheric icing concepts. J. Rech. Atmos., 14, 347356.

    • Search Google Scholar
    • Export Citation
  • Lim, S., V. Chandrasekar, and V. N. Bringi, 2005: Hydrometeor classification system using dual polarization radar measurements: Model improvements and in situ verification. IEEE Trans. Geosci. Remote Sens., 43, 792801.

    • Search Google Scholar
    • Export Citation
  • Meischner, P. F., V. N. Bringi, D. Heimann, and H. Holler, 1991: A squall line in southern Germany: Kinematics and precipitation formation as deduced by advanced polarimetric and Doppler radar measurements. Mon. Wea. Rev., 119, 678701.

    • Search Google Scholar
    • Export Citation
  • Milbrandt, J. A., and M. K. Yau, 2005: A multimoment bulk microphysics parameterization. Part II: A proposed three-moment closure and scheme description. J. Atmos. Sci., 62, 30653081.

    • Search Google Scholar
    • Export Citation
  • Ortega, K. L., T. M. Smith, K. L. Manross, A. G. Kolodziej, K. A. Scharfenberg, A. Witt, and J. J. Gourley, 2009: The Severe Hazards Analysis and Verification Experiment. Bull. Amer. Meteor. Soc., 90, 15191530.

    • Search Google Scholar
    • Export Citation
  • Park, H.-S., A. Ryzhkov, D. Zrnić, and K.-E. Kim, 2009: The hydrometeor classification algorithm for polarimetric WSR-88D: Description and application to an MCS. Wea. Forecasting, 24, 730748.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., A. Pokrovsky, and A. Khain, 2007: The influence of time-dependent melting on the dynamics and precipitation production in maritime and continental storm clouds. J. Atmos. Sci., 64, 338359.

    • Search Google Scholar
    • Export Citation
  • Picca, J., and A. Ryzhkov, 2011: Polarimetric radar discrimination between small, large, and giant hail at S band. NOAA/NNSL Rep., 13 pp. [Available online at http://www.nssl.noaa.gov/publications/wsr88d_reports/FINAL2011-Tsk1-Hail.pdf.]

  • Picca, J., and A. Ryzhkov, 2012: A dual-wavelength polarimetric analysis of the 16 May 2010 Oklahoma City extreme hailstorm. Mon. Wea. Rev., 140, 13851403.

    • Search Google Scholar
    • Export Citation
  • Prodi, F., 1970: Measurements of local density in artificial and natural hailstones. J. Appl. Meteor., 9, 903910.

  • Rasmussen, R. M., and A. J. Heymsfield, 1987a: Melting and shedding of graupel and hail. Part I: Model physics. J. Atmos. Sci., 44, 27542763.

    • Search Google Scholar
    • Export Citation
  • Rasmussen, R. M., and A. J. Heymsfield, 1987b: Melting and shedding of graupel and hail. Part II: Sensitivity study. J. Atmos. Sci., 44, 27642782.

    • Search Google Scholar
    • Export Citation
  • Rasmussen, R. M., V. Levizzani, and H. R. Pruppacher, 1984: A wind tunnel and theoretical study on the melting behavior of atmospheric ice particles: III. Experiment and theory for spherical ice particles of radius > 500 μm. J. Atmos. Sci., 41, 381388.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., 2001: Interpretation of polarimetric radar covariance matrix for meteorological scatterers: Theoretical analysis. J. Atmos. Oceanic Technol., 18, 315328.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., S. E. Giangrande, and T. J. Schuur, 2005: Rainfall estimation with a polarimetric prototype of WSR-88D. J. Appl. Meteor., 44, 502515.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., P. Zhang, D. Hudak, J. L. Alford, M. Knight, and J. W. Conway, 2007: Validation of polarimetric methods for attenuation correction at C band. Preprints, 33rd Conf. on Radar Meteorology, Cairns, Australia, Amer. Meteor. Soc., P11B.12. [Available online at https://ams.confex.com/ams/pdfpapers/123122.pdf.]

  • Ryzhkov, A. V., S. Ganson, A. Khain, M. Pinsky, and A. Pokrovsky, 2009: Polarimetric characteristics of melting hail at S and C bands. Preprints, 34th Conf. on Radar Meteorology, Williamsburg, VA, Amer. Meteor. Soc., 4A.6. [Available online at http://ams.confex.com/ams/pdfpapers/155571.pdf.]

  • Ryzhkov, A. V., M. Pinsky, A. Pokrovsky, and A. Khain, 2011: Polarimetric radar observation operator for a cloud model with spectral microphysics. J. Appl. Meteor. Climatol., 50, 873894.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., M. R. Kumjian, S. M. Ganson, and P. Zhang, 2013: Polarimetric radar characteristics of melting hail. Part II: Practical implications. J. Appl. Meteor. Climatol., 52, 28712886.

    • Search Google Scholar
    • Export Citation
  • Smith, P. L., D. J. Musil, S. F. Weber, J. F. Spahn, G. N. Johnson, and W. R. Sand, 1976: Raindrop and hailstone distributions inside hailstorms. Preprints, Int. Conf. on Cloud Physics, Boulder, CO, Amer. Meteor. Soc., 252257.

  • Spahn, J. F., and P. L. Smith Jr., 1976: Some characteristics of hailstone size distributions inside hailstorms. Preprints, 17th Conf. Radar Meteorology, Seattle, WA, Amer. Meteor. Soc., 187191.

  • Srivastava, R. C., 1987: A model of intense downdrafts driven by the melting and evaporation of precipitation. J. Atmos. Sci., 44, 17521773.

    • Search Google Scholar
    • Export Citation
  • Tabary, P., G. Vulpiani, J. J. Gourley, A. J. Illingworth, R. J. Thompson, and O. Bosquet, 2009: Unusually high differential attenuation at C band: Results from a two-year analysis of the French Trappes polarimetric radar data. J. Appl. Meteor. Climatol., 48, 20372053.

    • Search Google Scholar
    • Export Citation
  • Tabary, P., and Coauthors, 2010: Hail detection and quantification with C-band polarimetric radars: Results from a two-year objective comparison against hailpads in the south of France. Proc. Sixth European Conf. on Meteorology and Hydrology: Advances in Radar Technology, Sibiu, Romania, ERAD, 98102. [Available online at http://www.erad2010.org/pdf/oral/tuesday/radpol2/2_ERAD2010_0046.pdf.]

  • Ulbrich, C. W., and D. Atlas, 1982: Hail parameter relations: A comprehensive digest. J. Appl. Meteor., 21, 2243.

  • Vivekanandan, J., V. N. Bringi, and R. Raghavan, 1990: Multiparameter radar modeling and observation of melting ice. J. Atmos. Sci., 47, 549563.

    • Search Google Scholar
    • Export Citation
  • Wakimoto, R. M., and V. N. Bringi, 1988: Dual-polarization observations of microbursts associated with intense convection: The 20 July storm during the MIST project. Mon. Wea. Rev., 116, 15211539.

    • Search Google Scholar
    • Export Citation
  • Wolfson, M. M., R. L. Delanoy, B. E. Forman, R. G. Hallowell, M. L. Pawlak, and P. D. Smith, 1994: Automated microburst wind-shear prediction. Lincoln Lab. J., 7, 399426.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Initial density of ice particles as a function of their size in the cases of variable density (solid line) and fixed density (dashed line).

  • View in gallery

    Dependence of diameters of melting hailstones (solid lines) and effective diameters of their ice cores (dashed lines) on height for different initial sizes of hailstones in the cases of (a) constant initial density of hail (ρ = 0.917 g cm−3) and (b) variable density of hail.

  • View in gallery

    Distribution of mass water fraction across size spectrum at four different heights for (a) high initial density of hail and (b) variable initial density of hail.

  • View in gallery

    Examples of graupel/hail size distribution aloft in the cases of no hail, small hail, moderate hail, and large hail for which simulations were made.

  • View in gallery

    Size distributions of ice particles at H = 4 km (thin solid gray line), raindrops and melting hailstones at H = 0 km (thick solid line), and ice cores at H = 0 km (dashed line) for moderate hail with high density. Note the enhancement of 8-mm drops.

  • View in gallery

    Vertical profiles of mass content associated with ice (thick solid black line), melted water (solid gray line), shed water (dashed black line), and water resulting from breakup of large raindrops (dash–dotted gray line in lower-left corner) in the case of large hail with high density.

  • View in gallery

    Size distributions of melted water (thick line), shed water (dashed line), and water resulting from breakup of large raindrops (thin line) at the ground in the case of large hail with high density.

  • View in gallery

    (a),(b) Normalized radar reflectivity, (c),(d) differential reflectivity, and (e),(f) normalized specific differential phase of (left) dry and (right) melting hail as a function of size at three radar wavelengths: λ = 11.0 cm (S band; black curves), λ = 5.45 cm (C band; dashed dark gray curves), and λ = 3.2 cm (X band; light gray curves). For the melting hailstones, the vertical dotted line at particle size 0.8 cm represents the cutoff between fully melted raindrops and melting hailstones.

  • View in gallery

    As in Fig. 8, but for (a),(b) normalized specific attenuation and (c),(d) normalized specific differential attenuation.

  • View in gallery

    Relative contributions of different parts of the particle size spectrum to S-band and C-band Zh (solid lines) and Zυ (dashed lines) at four height levels for the case of large, high-density hail.

  • View in gallery

    Relative contributions of different parts of particle size spectrum to S-band KDP, Ah, and ADP at four height levels for the case of large, high-density hail.

  • View in gallery

    As in Fig. 11, but for C band.

  • View in gallery

    Simulated vertical profiles of (left) ZH and (right) ZDR at S, C, and X bands for large hail (solid lines) and small hail (dashed lines) of different density. The thickest lines are for S band, and the thinnest lines are for X band. Simulations are made for Ng = 8000 m−3 mm−1 and Λg = 1.6 mm−1.

  • View in gallery

    Simulated vertical profiles of KDP, Ah, and ADP for large (solid lines) and small (dashed lines) high-density hail at C band. Simulations are made for Ng = 8000 m−3 mm−1 and Λg = 1.6 mm−1.

  • View in gallery

    Size dependencies of (left) normalized and (right) differential reflectivity for rain/melting hail computed by three different methods. Green curves correspond to the two-layer version of the T-matrix code, red curves are for uniformly filled particles assuming water as matrix, and orange curves are for uniformly filled particles with ice as matrix.

  • View in gallery

    Simulated vertical profiles of ZDR for the cases of small hail (SH) and moderate hail (MH), along with median observed profiles at C band reported by Anderson et al. (2011) and Kaltenboeck and Ryzhkov (2013). Simulations are made for Ng = 1500 m−3 mm−1 and Λg = 1.1 mm−1.

  • View in gallery

    Frequency distributions of large hail mass (shading) and small hail mass (dashed contours) on the ZHZDR plane that are based on output from the HUCM at S and C bands. The title above each panel indicates the height interval (AGL) from which the distributions are computed. The environmental freezing level is at 2.5 km.

  • View in gallery

    Dependence of vertical profiles of C-band ZDR below the freezing level on downdraft speeds in the case of small hail. Shown are (a) vertical profiles of downdraft velocity and (b) vertical profiles of ZDR corresponding to the downdraft velocities in (a). The black solid line in (b) corresponds to the absence of downdraft.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 805 375 50
PDF Downloads 665 323 36

Polarimetric Radar Characteristics of Melting Hail. Part I: Theoretical Simulations Using Spectral Microphysical Modeling

View More View Less
  • 1 Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, and NOAA/OAR National Severe Storms Laboratory, Norman, Oklahoma
  • | 2 The Hebrew University of Jerusalem, Jerusalem, Israel
Full access

Abstract

Spectral (bin) microphysics models are used to simulate polarimetric radar variables in melting hail. Most computations are performed in a framework of a steady-state, one-dimensional column model. Vertical profiles of radar reflectivity factor Z, differential reflectivity ZDR, specific differential phase KDP, specific attenuation Ah, and specific differential attenuation ADP are modeled at S, C, and X bands for a variety of size distributions of ice particles aloft. The impact of temperature lapse rate, humidity, vertical air velocities, and ice particle density on the vertical profiles of the radar variables is also investigated. Polarimetric radar signatures of melting hail depend on the degree of melting or the height of the radar resolution volume with respect to the freezing level, which determines the relative fractions of partially and completely melted hail (i.e., rain). Simulated vertical profiles of radar variables are very sensitive to radar wavelength and the slope of the size distribution of hail aloft, which is correlated well with maximal hail size. Analysis of relative contributions of different parts of the hail/rain size spectrum to the radar variables allows explanations of a number of experimentally observed features such as large differences in Z of hail at the three radar wavelengths, unusually high values of ZDR at C band, and relative insensitivity of the measurements at C and X bands to the presence of large hail exceeding 2.5 cm in diameter. Modeling results are consistent with S- and C-band polarimetric radar observations and are utilized in Part II for devising practical algorithms for hail detection and determination of hail size as well as attenuation correction and rainfall estimation in the presence of hail.

Current affiliation: Advanced Study Program, National Center for Atmospheric Research,+ Boulder, Colorado.

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

Corresponding author address: Dr. Alexander Ryzhkov, National Weather Center, Suite 4911, 120 David L. Boren Blvd., Norman, OK 73072. E-mail: alexander.ryzhkov@noaa.gov

Abstract

Spectral (bin) microphysics models are used to simulate polarimetric radar variables in melting hail. Most computations are performed in a framework of a steady-state, one-dimensional column model. Vertical profiles of radar reflectivity factor Z, differential reflectivity ZDR, specific differential phase KDP, specific attenuation Ah, and specific differential attenuation ADP are modeled at S, C, and X bands for a variety of size distributions of ice particles aloft. The impact of temperature lapse rate, humidity, vertical air velocities, and ice particle density on the vertical profiles of the radar variables is also investigated. Polarimetric radar signatures of melting hail depend on the degree of melting or the height of the radar resolution volume with respect to the freezing level, which determines the relative fractions of partially and completely melted hail (i.e., rain). Simulated vertical profiles of radar variables are very sensitive to radar wavelength and the slope of the size distribution of hail aloft, which is correlated well with maximal hail size. Analysis of relative contributions of different parts of the hail/rain size spectrum to the radar variables allows explanations of a number of experimentally observed features such as large differences in Z of hail at the three radar wavelengths, unusually high values of ZDR at C band, and relative insensitivity of the measurements at C and X bands to the presence of large hail exceeding 2.5 cm in diameter. Modeling results are consistent with S- and C-band polarimetric radar observations and are utilized in Part II for devising practical algorithms for hail detection and determination of hail size as well as attenuation correction and rainfall estimation in the presence of hail.

Current affiliation: Advanced Study Program, National Center for Atmospheric Research,+ Boulder, Colorado.

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

Corresponding author address: Dr. Alexander Ryzhkov, National Weather Center, Suite 4911, 120 David L. Boren Blvd., Norman, OK 73072. E-mail: alexander.ryzhkov@noaa.gov

1. Introduction

Polarimetric weather radars reveal great potential for detection of hail and determination of its size. The combination of radar reflectivity factor Z and differential reflectivity ZDR proves to be efficient for discrimination between pure rain and dry hail because of the more random orientation of tumbling hailstones and their lower dielectric constant relative to raindrops. Hence, low ZDR associated with high Z is a certain indication of dry hail. Once hail starts melting, it gradually acquires polarimetric attributes of rain, and detection of melting hail mixed with rain becomes less straightforward because ZDR increases as hail progressively melts. Such an increase is especially pronounced at C band because of the strong effects of resonance scattering by large raindrops and smaller-sized partially melted hailstones.

It is hail of sufficiently large size and high density that inflicts substantial damage so that, in addition to hail detection, discrimination between hail of different sizes and densities is required. According to the U.S. National Weather Service standard, hail with size exceeding 2.5 cm (1.0 in.) is considered to be high impact and dangerous (i.e., “severe”), and hail in excess of 5.0 cm (2.0 in.) is considered to be “significantly severe.”

Polarimetric algorithms for hydrometeor classification, which utilize a combination of Z, ZDR, specific differential phase KDP, and the cross-correlation coefficient ρhv, demonstrate good skill for hail detection at S band as limited validation studies have shown (e.g., Heinselman and Ryzhkov 2006; Depue et al. 2007). Polarimetric hail-detection algorithms that were originally developed for S band need significant modification for applications at C band, primarily because ZDR of small- and medium-size melting hail at C band is greater than at S band (Ryzhkov et al. 2007; Tabary et al. 2010; Anderson et al. 2011). Wet hail is typically mixed with rain, and anomalously high ZDR due to resonance scattering associated with large raindrops and small, melting hail at C band offsets the low intrinsic ZDR of moderate-to-large hail. The modeling studies of Ryzhkov et al. (2009, 2011) and Kumjian et al. (2010) support this interpretation and show that ZDR of melting hail is very sensitive to radar wavelength. Direct comparisons of polarimetric hail signatures observed by closely located S- and C-band radars (Borowska et al. 2011; Gu et al. 2011; Picca and Ryzhkov 2012; Kaltenboeck and Ryzhkov 2013) are generally consistent with results of these theoretical simulations. Another important feature that must be considered for hail detection using shorter wavelengths is that Z of hail at these wavelengths may be significantly lower than at S band. The corresponding difference has been termed the “hail signal” in previous studies employing dual-wavelength observations (Atlas and Ludlam 1961; Eccles and Atlas 1973; Bringi et al. 1986; Feral et al. 2003).

Validation studies of polarimetric hail-detection algorithms are rare. Notable exceptions include the works of Heinselman and Ryzhkov (2006) and Depue et al. (2007) at S band and Boodoo et al. (2009) and Tabary et al. (2009, 2010) at C band. To our knowledge, the study of Depue et al. (2007) is probably the only one in which the correlation of the maximal size of ground-truth hail observations with polarimetric signatures has been examined systematically.

Determination of hail size remains challenging. The most recent version of the hydrometeor classification algorithm (HCA) developed at the National Severe Storms Laboratory for polarimetrically upgraded Weather Surveillance Radar-1988 Doppler (WSR-88D) instruments detects “rain mixed with hail” (Park et al. 2009) and does not distinguish between large and small hail. The Colorado State University HCA distinguishes between “graupel/small hail” and “hail” without specifying the borderline size between “small hail” and “hail” (Lim et al. 2005). Depue et al. (2007) recommended using the hail differential reflectivity parameter HDR, which incorporates both Z and ZDR (Aydin et al. 1986). Picca and Ryzhkov (2012) mention that the scheme of Depue et al. (2007) does not account for the hail melting process, which has a very strong impact on the vertical profile of ZDR. Kumjian et al. (2010), Picca and Ryzhkov (2011), and Kaltenboeck and Ryzhkov (2013) also suggest possible ways to identify “giant” hail with size exceeding 5 cm utilizing the pronounced reduction of ρhv and appearance of slightly negative ZDR in the hail generation zone above the freezing level that indicates growth of hail in the “wet regime” for which all accreted water does not freeze on the surface of hailstones because of substantial latent heat release Water coating on the surface of hailstones accentuates the effects of resonance scattering (Balakrishnan and Zrnić 1990b).

The methods for estimating hail size should be substantiated by retrievals from cloud models that explicitly treat the microphysics of melting hail (e.g., Khain et al. 2011; Ryzhkov et al. 2011). Melting of graupel and hail strongly affects the vertical profiles of polarimetric radar variables in convective storms. The impact of graupel/hail melting on the vertical distribution of ZDR and linear depolarization ratio LDR was first studied by Bringi et al. (1986), Vivekanandan et al. (1990), and Meischner et al. (1991) on the basis of the thermodynamic model of Rasmussen and Heymsfield (1987a). In this study, we use a similar thermodynamic model with several modifications regarding drop shedding and breakup, and we account for vertical air motion.

Melting hail causes significant attenuation of the radar signal that is different at orthogonal polarizations. Such attenuation can be significant even at S band. Polarimetric methods for attenuation correction are based on the measurements of differential phase ΦDP and imply knowledge of the ratios α = Ah/KDP and β = ADP/KDP, where Ah is specific attenuation, ADP is specific differential attenuation, and KDP is specific differential phase. As opposed to the case of pure rain, the variability of the ratios α and β in hail is not known, and, as a result, no reliable procedures for attenuation correction in hail exist at the moment.

Another practical challenge is radar rainfall estimation in the presence of hail. It is generally assumed that KDP is less affected by hail than is radar reflectivity Z and that the use of the R(KDP) relation is preferable for the rain/hail mixture (Balakrishnan and Zrnić 1990a; Aydin et al. 1995; Bringi and Chandrasekar 2001; Ryzhkov et al. 2005). This, assumption, however, should be better justified both theoretically and observationally. One of the problems is that KDP may become very noisy in hail-containing areas of the storm where ρhv is reduced because of the presence of large mixed-phase hydrometeors for which the effects of resonance scattering are particularly pronounced.

This three-part series of papers is organized as follows. In the first part, both microphysical and scattering models of melting hail are described. Vertical profiles of the polarimetric radar variables are simulated and examined at three major frequency bands: S, C, and X. The sensitivity of vertical profiles to various physical factors such as 1) temperature lapse rate, 2) humidity profile, 3) strength of descending air motions (downdrafts), 4) size distributions of graupel/hail aloft, 5) density of ice particles at the freezing level, and 6) radar wavelength is investigated. Two spectral microphysical models of melting hail coupled with the polarimetric radar observation operator are utilized in this investigation, and the results of theoretical simulations at S and C bands are compared with observations in hail.

In the second part (Ryzhkov et al. 2013), the strategy for hail detection and determination of its size at different wavelengths is discussed and the algorithm for hail detection and determination of its size at S band is described. We utilize our polarimetric model of melting hail to examine attenuation effects in melting hail and the quality of the rain estimate if rain is mixed with hail. Part III (A. Ryzhkov et al. 2013, unpublished manuscript) contains results of testing and validation of the proposed algorithm using an extensive dataset of polarimetric measurements and ground observations collected in different parts of the United States with a number of polarimetric WSR-88D instruments utilizing the Severe Hazards Analysis and Verification Experiment validation method (Ortega et al. 2009).

2. Thermodynamic model description

Two thermodynamic models of melting hail are utilized in this study. Similar to earlier works of Aydin and Zhao (1990), Aydin and Giridhar (1991), and Vivekanandan et al. (1990), one of the models (model 1) makes use of the Rasmussen and Heymsfield (1987a,b) study of the physics of melting of individual hailstones. It assumes a specified distribution of graupel/hail at the freezing level and follows the change of the size distribution of partially melted ice particles/raindrops and the corresponding polarimetric radar variables as hydrometeors (totally or partially melted) reach the ground. This is a steady-state 1D model that takes into account shedding of excessive water from the surface of melting hailstones and spontaneous breakup of large raindrops but does not allow for interactions/collisions between the particles.

The second model (model 2) is the 2D nonhydrostatic mixed-phase spectral bin Hebrew University of Jerusalem Cloud Model (HUCM; e.g., Khain et al. 2004, 2011). The model contains seven classes of hydrometeors, and each class is represented by size distribution functions in 43 size bins. As opposed to model 1, this model explicitly describes both generation and melting of hail and takes into account all kinds of interaction between hydrometeors.

Model 1 is relatively simple and convenient to examine the sensitivity of the polarimetric signatures to variations of initial hail size distributions, the density of hail, and thermodynamic profiles, whereas model 2 is more sophisticated and computationally expensive. It is expected to better capture the effects of size sorting that have a strong impact on polarimetric variables.

a. Melting of individual hailstones

The 1D explicit bin microphysical model of melting hail is based on the work of Rasmussen et al. (1984) and Rasmussen and Heymsfield (1987a,b, herein RH87a,b). Initial hailstone diameters range from 0 to 40 mm in increments of 0.1 mm (400 total size “bins”). At the top of the model domain (4 km AGL, with a temperature of 0°C), an initial distribution of hailstones is prescribed. These hailstones are allowed to fall through the domain, where the temperature and humidity are prescribed on the basis of an initial sounding interpolated to the model grid (vertical resolution is 10 m) or on the basis of predefined temperature and humidity lapse rates. Any updraft or downdraft profile w may be administered. The equations for terminal velocities of melting hailstones and heat transfer equations determining the rate of melting depending on particle size (or Reynolds number) are summarized in appendixes A and B.

In our study, we perform simulations while assuming that initial density of ice particles aloft is either equal to the density of solid ice (917 kg m−3) or varies across the size spectrum such that the density of graupel-size particles is lower than the density of larger hailstones (Fig. 1). The latter assumption reflects the general perception that graupel is softer than large hail (e.g., Prodi 1970) and that the selected size dependence of density is consistent with the results in RH87b (their Fig. 16). In the case of high-density hail, the melted water accumulates on the surface of the hailstone and sheds after its mass exceeds a certain “shedding threshold.” If the initial density of graupel/hail is lower than 917 kg m−3 then melted water first soaks into the interior of the particle and starts accumulating on the surface only after all air cavities are filled up with melted water. Again, excessive water sheds after reaching a critical mass mwmax (RH87a; Phillips et al. 2007):
e1
where mi is mass of ice and mws is mass of retained soaked water, both expressed in kilograms. We also assume that the ratio of the mass of soaked water and the mass of ice after the particle is fully soaked is constant and equal to the mass when full soaking is attained (i.e., mi and mws decrease proportionally before full melting occurs). This assumption is needed to ensure that the total mass of the resulting water drop does not exceed the mass of an 8-mm raindrop [or 2.68 × 10−4 kg according to Eq. (1)].
Fig. 1.
Fig. 1.

Initial density of ice particles as a function of their size in the cases of variable density (solid line) and fixed density (dashed line).

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

The dependencies of diameters of melting hailstones and their ice cores on height for high-density and variable-density hail estimated with model 1 are illustrated in Fig. 2. The computations have been performed while assuming that the freezing level is at 4 km, temperature lapse rate is 6.5°C km−1, and relative humidity is 100%. If the hailstone is spongy (i.e., originally has lower density), then the effective diameter of the ice core is defined as an equivolume diameter of a spheroid with the mass of unmelted ice and density of solid ice. For a given size of dry hailstone aloft, lower-density hail melts faster and its total diameter and effective diameter of ice core decrease more rapidly toward the ground.

Fig. 2.
Fig. 2.

Dependence of diameters of melting hailstones (solid lines) and effective diameters of their ice cores (dashed lines) on height for different initial sizes of hailstones in the cases of (a) constant initial density of hail (ρ = 0.917 g cm−3) and (b) variable density of hail.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Figure 2 shows that hailstones with initial diameters (i.e., diameters aloft) of less than 14 mm (for high-density hail) and 15 mm (for variable-density hail) melt completely before reaching the ground, whereas ice cores of much bigger hailstones remain large. Shedding of water from the surface of larger melting hailstones causes reduction of their mass and diameter. At each height below the freezing level, mass water fraction fm is equal to 1 for totally melted particles and gradually decreases with size of partially melted graupel and hail (Fig. 3). Once the maximal size of a raindrop resulting from melting hail reaches 8 mm (at about 2.5–3 km below the freezing level), the fm(D) dependence, where D is diameter, practically does not change. This is a consequence of the shedding condition in Eq. (1).

Fig. 3.
Fig. 3.

Distribution of mass water fraction across size spectrum at four different heights for (a) high initial density of hail and (b) variable initial density of hail.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Figure 2 also shows that hailstones with initial sizes between 8 and 14 mm (in the case of high-density hail) or between 9 and 15 mm (in the case of variable-density hail) end up as 8-mm raindrops, which causes an enhancement of the total concentration of 8-mm raindrops near the ground. Such an enhancement of very large drops originated from melting hail is also predicted by the simulations with HUCM (Khain et al. 2011; Ryzhkov et al. 2011). It has to be taken into account, however, that very large raindrops with diameters of 8 mm and larger are unstable and tend to break up spontaneously. In our simulations with model 1, the loss of these large drops due to spontaneous breakup is described probabilistically using an exponential factor
e2
where h0m is the height at which hailstones within the original mth size bin completely melt and become 8-mm raindrops. According to Eq. (2), at the height h = h0mHb 50% of 8-mm raindrops break up. Note that in the spectral (bin) approach we assume conservation of the concentration flux within the given size bin unless breakup occurs; that is, the product of particle concentration N(m, h) and terminal velocity U(m, h) is constant. This condition implies that the particles in separate size bins do not interact with each other. Because we use a Lagrangian approach in our simulations, both the center of a particular bin with index m (mean diameter of melting particle) and the width of the mth bin vary with height. Taking into account the breakup of 8-mm raindrops is implemented by multiplying N(m, h) by Pb(m, h) for h < h0m. Laboratory measurements in Kamra et al. (1991) indicate that average time of survival of an 8-mm raindrop before breakup is about 40 s. Hence, a reasonable choice of the parameter Hb in Eq. (2) is 400 m for terminal velocity of 8-mm drops of about 10 m s−1.

b. Size distribution of melting hailstones and raindrops

The advantage of model 1 is that it allows the study of the impact of the size distribution of graupel/hailstones aloft on the vertical profiles of radar variables in the most direct and straightforward way. In situ measurements of size distributions of ice particles aloft in hailstorms often reveal a biexponential type of particle spectra with different slopes for graupel and hail (Smith et al. 1976; Spahn and Smith 1976). In our simulations with model 1, we prescribe a biexponential size distribution of graupel/hail at the freezing level as
e3
where subscripts g and h stand for graupel and hail, respectively. The parameters Ng = 8000 m−3 mm−1 and Λg = 1.6 mm−1 in Eq. (3) are selected in such a way that the “graupel” part of the size spectrum yields a size distribution of raindrops at the surface that is close to the Marshall–Palmer distribution, and the corresponding values of Z and ZDR at S band are 52.2 dBZ and 2.29 dB, respectively. These are in agreement with typically observed values of Z and ZDR in heavy rain without hail in Oklahoma (Ryzhkov et al. 2005).

The choice of parameters Nh, Λh, and Dmax (maximal hail size at which size distribution is truncated) is dictated by the need to match resulting size distributions of ice cores close to the surface with the observed hail size distributions reported in the literature (Ulbrich and Atlas 1982; Cheng and English 1983; Cheng et al. 1985). In this study, we present results of model simulations for four different size distributions at the freezing level H = 4 km (shown in Fig. 4) with the following parameters characterizing distribution of hail aloft:

  1. no hail aloft and at the surface (Nh = 0),

  2. “small” hail, for which hail is present aloft (at H = 4 km) but is totally melted at the surface (Dmax = 14 mm, Λh = 0.99 mm−1, and m−3 mm−1),

  3. “moderate” hail, with larger hail aloft with Dmax = 24 mm so that maximal hail size at the surface is about 19 mm (Λh = 0.42 mm−1 and m−3 mm−1), and

  4. “large” hail, for which Dmax = 35 mm so that the maximal size of hail at the surface is about 30 mm (Λh = 0.27 mm−1 and m−3 mm−1).

Fig. 4.
Fig. 4.

Examples of graupel/hail size distribution aloft in the cases of no hail, small hail, moderate hail, and large hail for which simulations were made.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Such a parameterization of hail size distributions seems reasonable for the North American high plains (e.g., Colorado and Alberta) where the cited observations of size distributions of hail at the surface have been conducted. Prevalent size distributions can be different in areas with more humid climate where smaller-size hail is usually generated.

The initial biexponential size distribution of hydrometeors at the freezing level is modified in the process of melting. As an example, size distributions of graupel/hail at H = 4 km, rain and partially melted hail at H = 0 km, and ice cores at H = 0 km are compared in Fig. 5. The size distribution of rain and melting hail at H = 0 km (thick solid curve) exhibits a discontinuity around a particle diameter of 8 mm, as a result of shedding. In addition, note the enhanced concentration of surviving 8-mm drops resulting from the processes described above. Parameters Dmax and Λh at the freezing level are selected in such a way that the product of the corresponding values for size distributions of ice cores at the surface is equal to 7.9—its most likely value as reported by Ulbrich and Atlas (1982).

Fig. 5.
Fig. 5.

Size distributions of ice particles at H = 4 km (thin solid gray line), raindrops and melting hailstones at H = 0 km (thick solid line), and ice cores at H = 0 km (dashed line) for moderate hail with high density. Note the enhancement of 8-mm drops.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

All raindrops in the combined rain/hail size spectrum are formed from three sources: (i) direct melting of graupel/smaller-size hail, (ii) shedding of water from the surface of larger hailstones, and (iii) spontaneous breakup of 8-mm raindrops. The size distribution of raindrops from source i is determined by the assumed size distribution of graupel/small hail aloft, whereas size distributions of raindrops coming from sources ii and iii should be prescribed using the total masses of shed drops and fragments of drop breakup. An example of vertical profiles of total mass associated with ice, melted water, shed water, and breakup water is presented in Fig. 6 for the case of large-category hail with high density. Shed water and breakup water start accumulating at heights 2 and 3 km below the freezing level, respectively. According to several observational studies (e.g., Carras and Macklin 1973; Joe et al. 1976; Lesins et al. 1980; Rasmussen et al. 1984), shed drops are typically between 0.5 and 2.0 mm in diameter, with a modal diameter of about 1 mm. We parameterized the shed raindrop size distribution (DSD) as a gamma function,
e4
with spectral shape parameter μ = 2.0 [as in Milbrandt and Yau (2005)] and the slope parameter Λsh = 2.0 mm−1. The intercept parameter N0,sh is determined by using the total mass of shed water at each level. The DSD of the fragments from drop breakup is specified by following the approach of Kamra et al. (1991):
e5
where Λbu = 0.453 mm−1 and the intercept parameter is calculated from the total mass of breakup fragments, the total concentration of which is obtained by summing up N(m, h)[1 − Pb(m, h)] for size bins producing 8-mm raindrops [see Eq. (2)]. Size distributions of melted water, shed water, and water resulted from breakup of large raindrops at the ground in the case of large hail with high density are displayed in Fig. 7.
Fig. 6.
Fig. 6.

Vertical profiles of mass content associated with ice (thick solid black line), melted water (solid gray line), shed water (dashed black line), and water resulting from breakup of large raindrops (dash–dotted gray line in lower-left corner) in the case of large hail with high density.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Fig. 7.
Fig. 7.

Size distributions of melted water (thick line), shed water (dashed line), and water resulting from breakup of large raindrops (thin line) at the ground in the case of large hail with high density.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

3. Scattering model and vertical profiles of polarimetric radar variables in melting hail

a. Description of the scattering model

The forward-scattering operator described in Ryzhkov et al. (2011, R11 hereinafter) is utilized to compute polarimetric radar variables from the outputs of the cloud models 1 and 2. Melting hailstones and raindrops are modeled as spongy, water-coated, or liquid oblate spheroids with aspect ratios defined as in R11. The aspect ratio of a melting particle depends on its initial size and mass water fraction fm. In the case of high-density hail, the melting particle is modeled as a water-coated spheroid with an ice core having a dielectric constant equal to that of solid ice. In the case of variable-density hail, a spongy particle (before all air pockets are filled with water) is considered to be a uniformly filled spheroid with a dielectric constant that depends on volume fractions of ice, water, and air. The Maxwell Garnett mixing formula is used for the dielectric constant, in which water is treated as the matrix and ice/air are treated as inclusions (see details in R11). The melting particle is modeled as a water-coated spheroid once all air pockets are filled up and melted water starts accumulating at the surface, with the inner core treated as a uniformly filled spheroid with water matrix and ice inclusions.

Radar polarimetric variables depend on the orientation of hydrometeors. The distributions of particle orientations are not obtained from the cloud model and should be prescribed. It is assumed that melting hailstones are characterized by a Gaussian distribution of orientations with the mean orientation of their rotation axis along the vertical axis. The width of canting angle distribution σ linearly depends on mass water fraction for particles in a particular size bin and gradually changes from 40° for dry graupel/hailstones to 10° for completely melted hydrometeors. Because fm is a function of size at any given height, σ also varies with the size of melting hailstones.

The scattering amplitudes in two principal planes for backward and forward directions [ and , respectively] are computed using a T-matrix code for two-layer spheroids as in Depue at al. (2007). Then, polarimetric radar variables are estimated as
eq1
eq2
e6
where the subscript a stands for the scattering amplitude if the incident electric vector is parallel to the axis of rotation (smaller axis of an oblate spheroid) and subscript b is for the scattering amplitude for the orthogonal direction, λ is the radar wavelength, Kw = (εw − 1)/(εw + 2) (εw is the dielectric constant of water), and A1A5 are angular moments defined as (Ryzhkov 2001; R11)
e7
where r = exp(−2σ2), with σ in radians. In Eq. (6), the scattering amplitudes are expressed in millimeters, λ is in millimeters, radar reflectivities at horizontal and vertical polarization Zh,υ are in millimeters to the sixth power per meter cubed, KDP is in degrees per kilometer, and Ah and ADP are in decibels per kilometer.

b. Dependencies of different radar variables on size for dry and melting hail

The dependencies of different radar variables on the size of dry and high-density melting hail at S, C, and X bands for monodispersed size distributions are illustrated in Figs. 8 and 9. The values of Zh, KDP, Ah, and ADP normalized by particle concentration N are computed as functions of equivolume diameter D and assuming that hydrometeors of a given size have a variety of orientations that are described by the Gaussian distribution of canting angle, with the parameter σ depending on mass water fraction, as specified above. The size dependencies for melting hail are presented at the surface level where smaller hailstones are completely melted (i.e., all particles with sizes of less than 8 mm are actually liquid raindrops). It is important that the size dependency of mass water fraction practically does not change in the height interval 0–1 km above ground (or 3 km below the freezing level; cf. Fig. 3) and that, therefore, the size dependencies of normalized radar variables change very little in this interval of heights.

Fig. 8.
Fig. 8.

(a),(b) Normalized radar reflectivity, (c),(d) differential reflectivity, and (e),(f) normalized specific differential phase of (left) dry and (right) melting hail as a function of size at three radar wavelengths: λ = 11.0 cm (S band; black curves), λ = 5.45 cm (C band; dashed dark gray curves), and λ = 3.2 cm (X band; light gray curves). For the melting hailstones, the vertical dotted line at particle size 0.8 cm represents the cutoff between fully melted raindrops and melting hailstones.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Fig. 9.
Fig. 9.

As in Fig. 8, but for (a),(b) normalized specific attenuation and (c),(d) normalized specific differential attenuation.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Several important conclusions can be drawn from Figs. 8 and 9. Radar reflectivity is generally a nonmonotonic function of hail size for larger hail diameters, which is the consequence of resonance scattering. Note that Zh(S band) is larger than Zh(C band) and Zh(X band) for dry hail with 1 < D < 5 cm. The differences between Zh(S band), Zh(C band), and Zh(X band) can be as high as 40 dB for certain sizes of melting hail. It should be emphasized that Zh(C band) > Zh(S band) for large raindrops and smaller melting hail with D < 2 cm, whereas the opposite is true for larger hail size (Figs. 8a,b). The positive difference Zh(S band) − Zh(X band), called the “hail signal,” was utilized for hail detection in earlier studies with dual-frequency radars (Eccles and Atlas 1973).

Differential reflectivity of dry hail is small because of its tumbling falling behavior and low dielectric constant. The ZDR can become negative at certain resonance sizes even if the horizontal dimension of the stone is larger than the vertical dimension (Fig. 8c). Water-coated melting hailstones have higher ZDR relative to dry hailstones because of the increase of effective dielectric constant and because the film of water on their surface tends to stabilize their orientation. The difference in ZDR is particularly large for smaller-sized hail, which melts faster and has a higher fraction of water, as shown in Fig. 3. Mass water fraction rapidly decreases with hail size, and the thin film of water on the surface of large melting hailstones is not capable of changing their orientation significantly; thus they are still oriented more chaotically than smaller melting hailstones. As a result, smaller melting hailstones with sizes of less than 10–12 mm have relatively high ZDR, similar to that of large raindrops, whereas larger melting hailstones are characterized by lower ZDR, which is not very different from the one for dry hailstones of similar size. Notable is a strong maximum of C-band ZDR for raindrops with sizes of ~6 mm (Fig. 8d).

Normalized KDP of dry graupel/small hail is about two orders of magnitude less than normalized KDP for raindrops with similar size, and KDP of hail becomes increasingly negative for hail diameters exceeding resonance sizes for each radar wavelength. This is somewhat similar to the behavior of ZDR for larger hailstones (Figs. 8e,f). A vertical dotted line corresponding to D = 8 mm separates pure-raindrop and melting-hail parts of the size spectrum in Fig. 8.

Similar to KDP, attenuation variables Ah and ADP are much lower for dry graupel/hail with sizes of less than 1 cm than for raindrops of similar sizes, but they increase rapidly for larger-size hail (dry and wet) (Fig. 9). Such an increase is particularly significant for specific attenuation Ah, which is comparable at all three radar wavelengths for large hailstones, as opposed to raindrops and small hail for which Ah(S band) < Ah(C band) < Ah(X band). It is known that attenuation (or extinction) of electromagnetic waves is caused by absorption and scattering of microwave radiation by atmospheric particles. The absorption is dominant for hydrometeors that are smaller than radar wavelength, whereas the “scattering loss” due to the wave energy scattered by hydrometeors in directions different from the direction of wave propagation is prevalent for larger particles such as big hailstones. Normalized Ah in hail is much higher than in rain and is mainly caused by the “scattering loss,” which is heavily impacted by the effects of resonance scattering. The latter fact explains why dry hailstones may cause higher attenuation than melting hailstones for certain hail sizes. Note that specific differential attenuation exhibits oscillatory behavior and does not increase dramatically in large hail.

c. Relative contributions of different parts of the hail/rain spectrum to different radar variables

As Figs. 8 and 9 show, radar characteristics of hailstones can be very different from those of raindrops, and the values of radar variables after integration over the whole hail/rain spectrum of sizes strongly depend on relative contributions of raindrops and hailstones of different sizes and their relative concentrations. To clarify the issue, we select the case of large, high-density hail (i.e., Dmax = 35 mm, Λh = 0.27 mm−1, m−3 mm−1, and ρ = 917 kg m−3) and estimate separate contributions of smaller raindrops (0–4 mm), larger raindrops (4–9 mm), and four size categories of hail (9–14, 14–19, 19–25, and >25 mm) to Zh, Zυ, KDP, Ah, and ADP. The results for S and C bands are illustrated in Figs. 1012 for the heights of 0, 1, 2, and 3 km.

Fig. 10.
Fig. 10.

Relative contributions of different parts of the particle size spectrum to S-band and C-band Zh (solid lines) and Zυ (dashed lines) at four height levels for the case of large, high-density hail.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Fig. 11.
Fig. 11.

Relative contributions of different parts of particle size spectrum to S-band KDP, Ah, and ADP at four height levels for the case of large, high-density hail.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Fig. 12.
Fig. 12.

As in Fig. 11, but for C band.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Figure 10 shows that the contribution of melting hail to integral values of Zh and Zυ at S band is dominant at all heights. As a result, ZDR of the mixture of hail and rain remains relatively low because the intrinsic ZDR of hail is lower than that of rain. In contrast, the contribution of hail with sizes larger than 25 mm is insignificant at C band at all altitudes, whereas contributions from large raindrops and smaller-size hailstones are comparable at H = 0 and 1 km. Because ZDR of large drops at C band can be very high as a result of resonance effects, the resulting ZDR of a rain/hail mixture is significantly larger at C band than at S band at lower levels.

Similar plots for KDP, Ah, and ADP are presented in Figs. 11 and 12 for S and C bands, respectively. The KDP is little affected by the presence of melting hail at all four heights (Figs. 11 and 12, left panels) and is almost exclusively determined by rain in the mixture with hail. Larger-size hail makes a tangible contribution to KDP at S band, however. On the contrary, the contribution of hail to Ah is comparable with the contribution of rain at C band (especially at higher levels; see Fig. 12, middle panels) and may overwhelm the one at S band (Fig. 11), which is consistent with results of observations (e.g., Borowska et al. 2011; Picca and Ryzhkov 2012; Kaltenboeck and Ryzhkov 2013). Similar to KDP, the bulk of ADP comes from large raindrops, although hail generally adds more to ADP than KDP, especially at S band (Figs. 11 and 12, right panels).

The relative insensitivity of measurements at C band to large hail with sizes exceeding 25 mm was confirmed in the studies of Borowska et al. (2011) and Picca and Ryzhkov (2012), where simultaneous polarimetric radar measurements at S and C bands in hailstorms are examined. These studies show that the presence of very large hail does not result in significant increases of Zh or Ah at C band. Our simulations show that a similar conclusion holds for X band as well, although in this study we do not perform analysis at X band to the same extent as at S and C bands for two major reasons. First, attenuation in hail at X band is so overwhelming that intrinsic backscattering hail signatures at X band are almost completely masked by the effects of propagation, and reliable detection of hail and determination of its size is not feasible for larger-size hail. Second, when examining polarimetric characteristics of hail at S and C bands, we operate with a substantial amount of well-documented polarimetric data, including the data obtained simultaneously with S-band and C-band radars, that can be used for model validation. We unfortunately do not have similar data collected simultaneously at X-band and longer-wavelength radars.

d. Vertical profiles of polarimetric radar variables in melting hail of different sizes

Vertical profiles of ZH, ZDR, KDP, Ah, and ADP computed from the output of model 1 for the cases of large and small hail are displayed in Figs. 13 and 14. Henceforth, notation ZH is used for the radar reflectivity factor at horizontal polarization expressed in logarithmic scale; that is, ZH = 10 log(Zh). The set of parameters determining large and small hail in our simulations is defined in section 2b. Vertical dependencies of ZH at S and C bands for large hail are characterized by a maximum at the height about 2 km below the freezing level, where shedding of water from larger hailstones starts (Fig. 13, left panels). As melting of hail progresses, the ZH first gradually increases as a result of an increase in dielectric constant and then decreases once shedding starts and the size of larger hailstones diminishes. In the simulations for small hail, ZH at S and C bands is 10–15 dB lower and the ZH maximum mostly disappears because of a smaller impact of shedding. The value of ZH at X band is 5–10 dBZ lower than at S and C bands in the case of large hail. A decrease in hail density at the lower end of the size spectrum as defined in Fig. 1 (variable-hail-density case) causes a reduction of ZH (except for large hail at S band), which is particularly pronounced in the case of small hail (Fig. 13, bottom-left panel). This reduction is due to the overall decrease in effective dielectric constant of spongy melting hailstones and because lower-density dry hailstones melt into smaller raindrops than do high-density hailstones of the same size.

Fig. 13.
Fig. 13.

Simulated vertical profiles of (left) ZH and (right) ZDR at S, C, and X bands for large hail (solid lines) and small hail (dashed lines) of different density. The thickest lines are for S band, and the thinnest lines are for X band. Simulations are made for Ng = 8000 m−3 mm−1 and Λg = 1.6 mm−1.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Fig. 14.
Fig. 14.

Simulated vertical profiles of KDP, Ah, and ADP for large (solid lines) and small (dashed lines) high-density hail at C band. Simulations are made for Ng = 8000 m−3 mm−1 and Λg = 1.6 mm−1.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

The ZDR of melting hail increases with decreasing height within the first 2 km below the freezing level and then stabilizes without much change toward the ground (Fig. 13, right panels). As expected, ZDR near the surface is significantly lower for larger maximal hail size (large hail) than for smaller maximal hail size (small hail). For small hail, ZDR near the surface at C band approaches 4 dB for high-density hail, which is more than 1.0 dB higher than the corresponding value of ZDR at S band. Vertical profiles of ZDR and absolute values of ZDR for high-density and variable-density hail do not differ much.

Vertical dependencies of KDP, Ah, and ADP for large and small hail at C band are shown in Fig. 14. The KDP rapidly increases with decreasing height within the first 1.5–2 km below the freezing level as smaller-size graupel/hail transforms into raindrops. The increase slows down below 2 km. It is interesting that both Ah and ADP reach their maxima at the height about 2 km below the freezing level. This is consistent with observational evidence that the largest attenuation/differential attenuation at C band occurs at a certain height above the ground (e.g., Borowska et al. 2011; Kaltenboeck and Ryzhkov 2013). The difference between vertical profiles of attenuation variables for large and small hail is dramatic (especially for Ah) at all three radar wavelengths, which is consistent with the general perception that larger-size hail contributes significantly to the overall extinction of electromagnetic waves.

Note that a T-matrix code for two-layer spheroids has been used in our simulations, the results of which were presented in Figs. 814. For comparison, we also perform computations by assuming a model of a uniformly filled spheroid in which the effective dielectric constant was estimated using the Maxwell Garnett mixture formulas with water or ice as a matrix (R11) using the same mass water content. Size dependencies of normalized reflectivity factor and differential reflectivity for melting hail at S, C, and X bands computed three different ways are displayed in Fig. 15. It is obvious that the results of computations of radar variables for individual hail sizes using the three methods can differ significantly (as was also mentioned in R11). The difference is especially large between the two-layer model and the model with ice as matrix (green and orange curves in Fig. 15). After integration over the whole spectrum of sizes, such differences generally “wash out” and the computations performed with the T-matrix code for two-layer spheroids and the T-matrix code for uniformly filled spheroids assuming water as matrix [utilized by Jung et al. (2008)] yield close results in terms of integral values of radar variables. For example, the difference in ZH computed by the two methods is within 1–2 dB, and the corresponding difference in ZDR is usually less than 0.2 dB at S band. More detailed results of such a comparison can be found in the thesis of Ganson (2012).

Fig. 15.
Fig. 15.

Size dependencies of (left) normalized and (right) differential reflectivity for rain/melting hail computed by three different methods. Green curves correspond to the two-layer version of the T-matrix code, red curves are for uniformly filled particles assuming water as matrix, and orange curves are for uniformly filled particles with ice as matrix.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

e. The impact of resonance scattering on differential reflectivity in melting hail

As mentioned, ZDR of the mixture of hail and rain is high at C band. Further enhancement of ZDR can be achieved by changing the “graupel” part of the initial size spectrum. Analysis of raindrop size distributions in hail-bearing storms indicates that for ZH approaching 50 dBZ they are more consistent with smaller slopes Λg of the graupel part of the ice spectrum aloft, say 1.1 mm−1 instead of the 1.6 mm−1 used in the previous simulations. The use of combination of Λg = 1.1 mm−1 and Ng = 1500 m−3 mm−1 results in higher (up to 0.5 dB) ZDR near the surface. The corresponding simulated profile of ZDR for small hail at C band matches well the median profiles retrieved from the statistics of observations by Anderson et al. (2011) and Kaltenboeck and Ryzhkov (2013) in melting hail and are consistent with measurements of Tabary et al. (2010) (Fig. 16). In other words, the model 1 of melting hail combined with scattering computations yields realistic profiles of ZDR in hail at C band.

Fig. 16.
Fig. 16.

Simulated vertical profiles of ZDR for the cases of small hail (SH) and moderate hail (MH), along with median observed profiles at C band reported by Anderson et al. (2011) and Kaltenboeck and Ryzhkov (2013). Simulations are made for Ng = 1500 m−3 mm−1 and Λg = 1.1 mm−1.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

Although a simple 1D model explains the radar wavelength dependence of ZDR in melting hail, it falls short of reproducing anomalously high ZDR (up to 8 dB) that is often measured at C band. A more sophisticated 2D cloud model with spectral (bin) microphysics (HUCM) was utilized for simulating the fields of polarimetric variables in severe hailstorm in the study of R11. At each model grid cell, the masses of large hailstones with diameters exceeding 2.5 cm [large hail mass (LHM)] and smaller hailstones with diameters between 1 and 2.5 cm [small hail mass (SHM)] have been calculated for this storm. Using the computed ZH and ZDR and the distributions of LHM and SHM, two-dimensional frequency distributions for ZH and ZDR occurrence in grid cells that contain appreciable LHM and SHM are constructed for different height intervals for S and C bands (Fig. 17). It is obvious that in the first-kilometer layer above ground, the highest ZDR at C band (i.e., values approaching 8 dB) are associated with smaller hail with ZH between 50 and 60 dBZ. The contrast between S and C bands in terms of ZDR is particularly striking. As opposed to the 1D model, the HUCM more adequately reproduces size sorting effects in the proximity of convective updrafts, which cause additional enhancement of ZDR. The changes in the graupel part of the size distribution of hydrometeors aloft mostly affect the resulting rain part of the size spectrum at lower levels. A more flattened graupel size distribution accentuates the contribution from larger raindrops, which causes additional increases in ZDR. A more detailed analysis shows that the HUCM generally produces a lower slope of raindrop size distribution and more pronounced enhancement of raindrop concentration within the size interval between 5 and 9 mm containing “resonance” sizes of raindrops at C band. This can be seen, for example, from the comparison of Fig. 7 in this paper with Fig. 12 in the R11 paper. A secondary maximum of C-band ZDR at 6.5 dB in Fig. 16 is associated with large hail mixed with very big raindrops in the vicinity of the updraft (see also R11).

Fig. 17.
Fig. 17.

Frequency distributions of large hail mass (shading) and small hail mass (dashed contours) on the ZHZDR plane that are based on output from the HUCM at S and C bands. The title above each panel indicates the height interval (AGL) from which the distributions are computed. The environmental freezing level is at 2.5 km.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

4. Sensitivity to temperature, humidity, and vertical motions

In addition to evaluating the dependency of polarimetric signatures of melting hail on its initial size distributions, density, and radar wavelength, we examined the sensitivity of vertical profiles of radar variables to 1) temperature lapse rate, 2) humidity profile, and 3) strength of descending air motions (downdrafts). The sensitivity analysis demonstrates that the factors 1–3 have generally weaker (but not negligible) effect on radar variables when compared with variability in size distribution, density, and radar wavelength.

A more steep temperature lapse rate speeds up the process of melting and the transition from partially melted to completely melted particles occurs at higher levels. This means that the transition from low ZDR at the freezing level to high ZDR would occur in a shallower layer. The reduction of relative humidity toward the ground produces the opposite effect: slowing down the process of melting due to evaporative cooling and increasing the depth of ZDR transition or decreasing vertical gradient of ZDR. Our simulations show that changing temperature lapse rate from 6.5° to 4.5° km−1 would lower the height of the “ZDR step” (i.e., the level at which ZDR sharply slows down its increase with decreasing height) by about 500 m. In a similar way, if relative humidity linearly decreases from 100% at the freezing level down to 60% at the surface (as in the example in RH87b), the ZDR step would descend by about 200 m relative to the case of 100% humidity through the whole depth of the 0–4-km layer.

Melting of hail is a major driving force for downdrafts, which may potentially produce damaging wet microbursts (Srivastava 1987). Downward air motion transports partially melted hailstones to lower height levels and, as a result, the “ZDR hole” [the term coined by Wakimoto and Bringi (1988)] of lower ZDR below the freezing level stretches farther down to the surface. We simulated vertical profiles of C-band ZDR for the case of small hail in the absence of downward motions and for three different profiles of downdraft velocity. Figure 18 shows that the ZDR contour of 1.5 dB descends by about 0.5–1 km for typical profiles of vertical velocity within strong downdrafts.

Fig. 18.
Fig. 18.

Dependence of vertical profiles of C-band ZDR below the freezing level on downdraft speeds in the case of small hail. Shown are (a) vertical profiles of downdraft velocity and (b) vertical profiles of ZDR corresponding to the downdraft velocities in (a). The black solid line in (b) corresponds to the absence of downdraft.

Citation: Journal of Applied Meteorology and Climatology 52, 12; 10.1175/JAMC-D-13-073.1

The ZDR hole may serve as a more reliable microburst predictor than the subsidence of the center of mass (CM) computed from a vertical profile of ZH in a traditional method for microburst prediction (Wolfson et al. 1994). This is because the height of the ZDR depression with respect to surrounding air is a local characteristic that depends only on the downdraft velocity, whereas CM is a function of the vertical distribution of hydrometeor concentration in a whole depth of convective cell. In other words, it is very possible that apparent CM may not descend in the presence of downdraft if additional buildup of mass occurs near the top of the cloud.

5. Conclusions

The one-dimensional thermodynamic model of melting hail by Rasmussen and Heymsfield (1987a,b) is utilized to simulate vertical profiles of polarimetric radar variables at S, C, and X bands using the polarimetric radar observation operator for a cloud model with spectral microphysics described by Ryzhkov et al. (2011) and the T-matrix code for computing the scattering amplitudes of water-coated and spongy spheroidal hydrometeors. The model realistically reproduces vertical profiles of various radar variables and their dependencies on radar wavelength and maximal size of melting hail below the freezing level. It notably explains high values of ZDR that are routinely observed in melting hail (particularly at C band) as opposed to dry hail. A more sophisticated spectral cloud model of The Hebrew University of Jerusalem is capable of generating anomalously high values of ZDR at C band and large differences between C-band and S-band ZDR in even better agreement with dual-wavelength polarimetric radar measurements in hail-bearing storms (Borowska et al. 2011; Kaltenboeck and Ryzhkov 2013; Picca and Ryzhkov 2012). This is attributed to the fact that the HUCM more adequately treats size sorting of raindrops and melting hailstones in the proximity of convective updrafts than does the simplified model 1, which does not explicitly address size sorting.

The 1D model predicts an enhancement in the concentration of very large raindrops originating from hail with initial sizes below 15 mm and a flattening of the raindrop spectrum at its higher end in the presence of hail. This is one of the key factors causing an increase of ZDR at the periphery of hail cells at all three radar wavelengths. Additional enhancement of ZDR (especially at C band) is attributed to intense size sorting of raindrops and smaller-size melting hailstones, processes that are well captured by the HUCM.

It is shown that specific differential phase KDP is least affected by the presence of hail in a mixture with rain when compared with other radar variables. This makes it an attractive parameter for quantifying the rainfall amount in the mixture. Hail contributes significantly to specific attenuation Ah, and this contribution increases with increasing maximal hail size so that it becomes tangible even at S band. Specific differential attenuation ADP is less affected by the presence of hail than is Ah. Its increase is more attributed to large raindrops generated from smaller-sized melting hail.

The sensitivity of the vertical profiles of radar variables to the temperature lapse rate, humidity, size distributions of graupel/hail aloft (e.g., maximal hail size), hail density, and vertical air motions was examined as part of the study. Variability of the size distribution of dry ice aloft is a primary factor affecting polarimetric signatures of the melting hail. Lower intercepts of the exponential distribution of hailstones at the freezing level [which are strongly correlated with the maximal hail size according to Ulbrich and Atlas (1982)] result in higher ZH and Ah as well as lower ZDR beneath. Decreasing the initial density of hail causes a noticeable reduction of ZH, and decreased temperature lapse rate and relative humidity slow down the melting process and shift the ZDR step downward. The dependence of the depth of the ZDR hole on the downdraft velocity shows good promise to utilize this signature to predict microbursts driven by melting of hail.

The polarimetric model of melting hail offers a theoretical basis for development of practical algorithms for hail detection and determination of its size, attenuation correction, and rainfall estimation in the presence of hail at S, C, and X bands, which are discussed in Part II of this paper (Ryzhkov et al. 2013).

Acknowledgments

Funding for the study was provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce, and by the National Weather Service, Federal Aviation Administration, and U.S. Department of Defense program for modernization of NEXRAD. Additional support came from the Binational U.S.–Israel Science Foundation (Grants 2006437 and 2010446) and from the U.S. Department of Energy, Atmospheric System Research.

APPENDIX A

Equations for Particle Fall Speeds

Particle velocities are determined following a simplified version of RH87a, by first computing the Best number:
ea1
where m, g, ρa, and η are the hailstone mass (kg), the gravitational acceleration (m s−2), air density (kg m−3), and dynamic viscosity of air (kg m−1 s−1), respectively. The Best number is then used to compute the Reynolds number NRe of the particle:
ea2
For dry hailstones,
ea3
where ν is the kinematic viscosity of air (=η/ρa; m2 s−1), D is the equivalent spherical diameter (m), and the density correction factor is accounted for in the definition of the Best number [Eq. (A1)]. The equation used to compute the fall speed of the equilibriumA1 melting hailstone ueq is determined by NRe as follows. For NRe < 5000,
ea4a
which is the Brandes et al. (2002) fall speed relation for raindrops, converted so that the mass of the melting hailstone m is used instead of the equivalent diameter of the raindrop. This relation lets small melting hailstones fall at the same velocity as raindrops, using an updated fall speed relation for drops [RH87a use the relation of Beard (1976)]. For larger hailstones (i.e., for 5000 ≤ NRe < 25 000),
ea4b
and for NRe ≥ 25 000 the melting hailstone fall speed is the same as that of a dry hailstone of the same size:
ea4c
Note that this assumes a drag coefficient of 0.6, or that nearly all liquid water is shed. Following RH87a, the instantaneous fall speed of the melting hailstone u varies between the “dry” fall speed ud [Eq. (A3)] and the equilibrium fall speed ueq [Eqs. (A4)] as a linear function of the fraction of equilibrium water mass on or in the particle:
ea5
The equilibrium mass of water on or within a hailstone is interpreted to be the critical mass of water the hailstone can retain before the onset of shedding, given by Eq. (1) in the text.

APPENDIX B

Heat Transfer Equations

The heat transfer equations governing the physics of hail melting are dependent on NRe, following RH87a. Instead of solving time-dependent equations for the particle radius, however, we assume steady-state conditions and solve height-dependent equations for the ice core volume Vi,
eb1
where dq/dh is the rate of change of enthalpy with height for a hailstone, ρice is the density of ice, Lm is the latent enthalpy of melting, and Vi is the hailstone’s volume of ice. The heat transfer equations are as follows. For NRe < 250,
eb2a
for 250 ≤ NRe ≤ 3000,
eb2b
for 3000 < NRe < 6000,
eb2c
for 6000 ≤ NRe ≤ 2 × 104,
eb2d
for NRe > 2 × 104,
eb2e
The heat transfer Eqs. (B2) above are used with Eq. (B1) to determine the reduction in ice volume for each height level and hailstone size bin. This change in ice volume δVi is converted to an increase in liquid water volume δVw via conservation of mass:
eb3
The mass of liquid meltwater is then calculated.
In the heat balance Eqs. (B2), there are several thermodynamic parameters used. Some have functional dependencies on the ambient air temperature T and are given below in mks units for convenience. Parameter ka is the thermal conductivity of air (J m−1 s−1 K−1):
eq6
T0 = 273.15 K is the reference 0°C temperature, is the thermal ventilation coefficient:
eq7
is the vapor ventilation coefficient:
eq8
Lυ is the latent enthalpy of vaporization (J kg−1):
eq9
Lm is the latent enthalpy of melting (J kg−1):
eq10
Dυ is the diffusivity of water vapor in air (m2 s−1):
eq11
p is air pressure and p0 is surface pressure (hPa), ρυ,∞ is the ambient vapor density (kg m−3), ρυ,0 is the vapor density at temperature T0 (kg m−3), NRe = uD/ν is the dimensionless Reynolds number, NPr = ν/Ka is the dimensionless Prandtl number, NSc = ν/Dυ is the dimensionless Schmidt number, Ka is the thermal diffusivity of air (m2 s−1):
eq12
and kw is the thermal conductivity of water (J m−1 s−1 K−1):
eq13

REFERENCES

  • Anderson, M. E., L. D. Carey, W. A. Petersen, and K. R. Knupp, 2011: C-band dual-polarimetric radar signatures of hail. Electron. J. Oper. Meteor., 2011-EJ02. [Available online at http://www.nwas.org/ej/pdf/2011-EJ2.pdf.]

    • Search Google Scholar
    • Export Citation
  • Atlas, D., and F. H. Ludlam, 1961: Multi-wavelength radar reflectivity of hailstorms. Quart. J. Roy. Meteor. Soc., 87, 523534.

  • Aydin, K., and Y. Zhao, 1990: A computational study of polarimetric radar observables in hail. IEEE Trans. Geosci. Remote Sens., 28, 412422.

    • Search Google Scholar
    • Export Citation
  • Aydin, K., and V. Giridhar, 1991: Polarimetric C-band radar observables in melting hail: A computational study. Preprints, 25th Int. Conf. on Radar Meteorology, Paris, France, Amer. Meteor. Soc., 733736.

  • Aydin, K., T. A. Seliga, and V. Balaji, 1986: Remote sensing of hail with a dual linear polarization radar. J. Climate Appl. Meteor., 25, 14751484.

    • Search Google Scholar
    • Export Citation
  • Aydin, K., V. N. Bringi, and L. Liu, 1995: Rain-rate estimation in the presence of hail using S-band specific differential phase and other radar parameters. J. Appl. Meteor., 34, 404410.

    • Search Google Scholar
    • Export Citation
  • Balakrishnan, N., and D. S. Zrnić, 1990a: Estimation of rain and hail rates in mixed-phase precipitation. J. Atmos. Sci., 47, 565583.

    • Search Google Scholar
    • Export Citation
  • Balakrishnan, N., and D. S. Zrnić, 1990b: Use of polarization to characterize precipitation and discriminate large hail. J. Atmos. Sci., 47, 15251540.

    • Search Google Scholar
    • Export Citation
  • Beard, K. V., 1976: Terminal velocity and shape of cloud and precipitation drops aloft. J. Atmos. Sci., 33, 851864.

  • Boodoo, S., D. Hudak, M. Leduc, A. V. Ryzhkov, N. Donaldson, and D. Hassan, 2009: Hail detection with a C-band dual-polarization radar in the Canadian Great Lakes region. Preprints, 34th Conf. on Radar Meteorology, Williamsburg, VA, Amer. Meteor. Soc., 10A.5. [Available online at http://ams.confex.com/ams/pdfpapers/156032.pdf.]

  • Borowska, L., A. Ryzhkov, D. Zrnić, C. Simmer, and R. Palmer, 2011: Attenuation and differential attenuation of the 5-cm-wavelength radiation in melting hail. J. Appl. Meteor. Climatol., 50, 5976.

    • Search Google Scholar
    • Export Citation
  • Brandes, E. A., G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41, 674685.

    • Search Google Scholar
    • Export Citation
  • Bringi, V. N., and V. Chandrasekar, 2001: Polarimetric Doppler Weather Radar: Principles and Applications. Cambridge University Press, 636 pp.

  • Bringi, V. N., J. Vivekanandan, and J. D. Tuttle, 1986: Multiparameter radar measurements in Colorado convective storms. Part II: Hail detection studies. J. Atmos. Sci., 43, 25642577.

    • Search Google Scholar
    • Export Citation
  • Carras, J. N., and W. C. Macklin, 1973: The shedding of accreted water during hailstone growth. Quart. J. Roy. Meteor. Soc., 99, 639648.

    • Search Google Scholar
    • Export Citation
  • Cheng, L., and M. English, 1983: A relationship between hailstone concentration and size. J. Atmos. Sci., 40, 204213.

  • Cheng, L., M. English, and R. Wong, 1985: Hailstone size distributions and their relationship to storm thermodynamics. J. Climate Appl. Meteor., 24, 10591067.

    • Search Google Scholar
    • Export Citation
  • Depue, T. K., P. C. Kennedy, and S. A. Rutledge, 2007: Performance of the hail differential reflectivity (HDR) polarimetric radar hail indicator. J. Appl. Meteor. Climatol., 46, 12901301.

    • Search Google Scholar
    • Export Citation
  • Eccles, P. J., and D. Atlas, 1973: A dual-wavelength radar hail detector. J. Appl. Meteor., 12, 847854.

  • Feral, L., H. Sauvageot, and S. Soula, 2003: Hail detection using S- and C-band radar reflectivity difference. J. Atmos. Oceanic Technol., 20, 233248.

    • Search Google Scholar
    • Export Citation
  • Ganson, S., 2012: Investigation of polarimetric radar characteristics of melting hail using advanced T-matrix computations. M.S. thesis, School of Meteorology, University of Oklahoma, 73 pp.

  • Gu, J.-Y., A. Ryzhkov, P. Zhang, P. Neilley, M. Knight, B. Wolf, and D.-I. Lee, 2011: Polarimetric attenuation correction in heavy rain at C band. J. Appl. Meteor., 50, 3958.

    • Search Google Scholar
    • Export Citation
  • Heinselman, P. L., and A. V. Ryzhkov, 2006: Validation of polarimetric hail detection. Wea. Forecasting, 21, 839850.

  • Joe, P. I., and Coauthors, 1976: Loss of accreted water from a growing hailstone. Preprints, Int. Conf. on Cloud Physics, Boulder, CO, Amer. Meteor. Soc., 264269.

  • Jung, Y., G. Zhang, and M. Xue, 2008: Assimilation of simulated polarimetric radar data for a convective storm using the ensemble Kalman filter. Part I: Observation operators for reflectivity and polarimetric variables. Mon. Wea. Rev., 136, 22282245.

    • Search Google Scholar
    • Export Citation
  • Kaltenboeck, R., and A. Ryzhkov, 2013: Comparison of polarimetric signatures of hail at S and C bands for different hail sizes. Atmos. Res., 123, 323336.

    • Search Google Scholar
    • Export Citation
  • Kamra, A. K., R. V. Bhalwankar, and A. B. Sathe, 1991: Spontaneous breakup of charged and uncharged water drops freely suspended in a wind tunnel. J. Geophys. Res., 96, 17 15917 168.

    • Search Google Scholar
    • Export Citation
  • Khain, A., A. Pokrovsky, M. Pinsky, A. Seifert, and V. Phillips, 2004: Simulation of effects of atmospheric aerosols on deep turbulent convective clouds using a special microphysics mixed-phase cumulus cloud model. Part I: Model description and possible applications. J. Atmos. Sci., 61, 29632982.

    • Search Google Scholar
    • Export Citation
  • Khain, A., D. Rosenfeld, A. Pokrovsky, U. Blahak, and A. Ryzhkov, 2011: The role of CCN in precipitation and hail in a mid-latitude storm as seen in simulations using a spectral (bin) microphysics model in a 2D frame. Atmos. Res., 99, 129146.

    • Search Google Scholar
    • Export Citation
  • Kumjian, M. R., J. C. Picca, S. M. Ganson, A. V. Ryzhkov, J. Krause, D. Zrnić, and A. Khain, 2010: Polarimetric characteristics of large hail. Preprints, 25th Conf. on Severe Local Storms, Denver, CO, Amer. Meteor. Soc., 11.2. [Available online at https://ams.confex.com/ams/pdfpapers/176043.pdf.]

  • Lesins, G. B., R. List, and P. I. Joe, 1980: Ice accretions. Part I: Testing of new atmospheric icing concepts. J. Rech. Atmos., 14, 347356.

    • Search Google Scholar
    • Export Citation
  • Lim, S., V. Chandrasekar, and V. N. Bringi, 2005: Hydrometeor classification system using dual polarization radar measurements: Model improvements and in situ verification. IEEE Trans. Geosci. Remote Sens., 43, 792801.

    • Search Google Scholar
    • Export Citation
  • Meischner, P. F., V. N. Bringi, D. Heimann, and H. Holler, 1991: A squall line in southern Germany: Kinematics and precipitation formation as deduced by advanced polarimetric and Doppler radar measurements. Mon. Wea. Rev., 119, 678701.

    • Search Google Scholar
    • Export Citation
  • Milbrandt, J. A., and M. K. Yau, 2005: A multimoment bulk microphysics parameterization. Part II: A proposed three-moment closure and scheme description. J. Atmos. Sci., 62, 30653081.

    • Search Google Scholar
    • Export Citation
  • Ortega, K. L., T. M. Smith, K. L. Manross, A. G. Kolodziej, K. A. Scharfenberg, A. Witt, and J. J. Gourley, 2009: The Severe Hazards Analysis and Verification Experiment. Bull. Amer. Meteor. Soc., 90, 15191530.

    • Search Google Scholar
    • Export Citation
  • Park, H.-S., A. Ryzhkov, D. Zrnić, and K.-E. Kim, 2009: The hydrometeor classification algorithm for polarimetric WSR-88D: Description and application to an MCS. Wea. Forecasting, 24, 730748.

    • Search Google Scholar
    • Export Citation
  • Phillips, V. T. J., A. Pokrovsky, and A. Khain, 2007: The influence of time-dependent melting on the dynamics and precipitation production in maritime and continental storm clouds. J. Atmos. Sci., 64, 338359.

    • Search Google Scholar
    • Export Citation
  • Picca, J., and A. Ryzhkov, 2011: Polarimetric radar discrimination between small, large, and giant hail at S band. NOAA/NNSL Rep., 13 pp. [Available online at http://www.nssl.noaa.gov/publications/wsr88d_reports/FINAL2011-Tsk1-Hail.pdf.]

  • Picca, J., and A. Ryzhkov, 2012: A dual-wavelength polarimetric analysis of the 16 May 2010 Oklahoma City extreme hailstorm. Mon. Wea. Rev., 140, 13851403.

    • Search Google Scholar
    • Export Citation
  • Prodi, F., 1970: Measurements of local density in artificial and natural hailstones. J. Appl. Meteor., 9, 903910.

  • Rasmussen, R. M., and A. J. Heymsfield, 1987a: Melting and shedding of graupel and hail. Part I: Model physics. J. Atmos. Sci., 44, 27542763.

    • Search Google Scholar
    • Export Citation
  • Rasmussen, R. M., and A. J. Heymsfield, 1987b: Melting and shedding of graupel and hail. Part II: Sensitivity study. J. Atmos. Sci., 44, 27642782.

    • Search Google Scholar
    • Export Citation
  • Rasmussen, R. M., V. Levizzani, and H. R. Pruppacher, 1984: A wind tunnel and theoretical study on the melting behavior of atmospheric ice particles: III. Experiment and theory for spherical ice particles of radius > 500 μm. J. Atmos. Sci., 41, 381388.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., 2001: Interpretation of polarimetric radar covariance matrix for meteorological scatterers: Theoretical analysis. J. Atmos. Oceanic Technol., 18, 315328.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., S. E. Giangrande, and T. J. Schuur, 2005: Rainfall estimation with a polarimetric prototype of WSR-88D. J. Appl. Meteor., 44, 502515.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., P. Zhang, D. Hudak, J. L. Alford, M. Knight, and J. W. Conway, 2007: Validation of polarimetric methods for attenuation correction at C band. Preprints, 33rd Conf. on Radar Meteorology, Cairns, Australia, Amer. Meteor. Soc., P11B.12. [Available online at https://ams.confex.com/ams/pdfpapers/123122.pdf.]

  • Ryzhkov, A. V., S. Ganson, A. Khain, M. Pinsky, and A. Pokrovsky, 2009: Polarimetric characteristics of melting hail at S and C bands. Preprints, 34th Conf. on Radar Meteorology, Williamsburg, VA, Amer. Meteor. Soc., 4A.6. [Available online at http://ams.confex.com/ams/pdfpapers/155571.pdf.]

  • Ryzhkov, A. V., M. Pinsky, A. Pokrovsky, and A. Khain, 2011: Polarimetric radar observation operator for a cloud model with spectral microphysics. J. Appl. Meteor. Climatol., 50, 873894.

    • Search Google Scholar
    • Export Citation
  • Ryzhkov, A. V., M. R. Kumjian, S. M. Ganson, and P. Zhang, 2013: Polarimetric radar characteristics of melting hail. Part II: Practical implications. J. Appl. Meteor. Climatol., 52, 28712886.

    • Search Google Scholar
    • Export Citation
  • Smith, P. L., D. J. Musil, S. F. Weber, J. F. Spahn, G. N. Johnson, and W. R. Sand, 1976: Raindrop and hailstone distributions inside hailstorms. Preprints, Int. Conf. on Cloud Physics, Boulder, CO, Amer. Meteor. Soc., 252257.

  • Spahn, J. F., and P. L. Smith Jr., 1976: Some characteristics of hailstone size distributions inside hailstorms. Preprints, 17th Conf. Radar Meteorology, Seattle, WA, Amer. Meteor. Soc., 187191.

  • Srivastava, R. C., 1987: A model of intense downdrafts driven by the melting and evaporation of precipitation. J. Atmos. Sci., 44, 17521773.

    • Search Google Scholar
    • Export Citation
  • Tabary, P., G. Vulpiani, J. J. Gourley, A. J. Illingworth, R. J. Thompson, and O. Bosquet, 2009: Unusually high differential attenuation at C band: Results from a two-year analysis of the French Trappes polarimetric radar data. J. Appl. Meteor. Climatol., 48, 20372053.

    • Search Google Scholar
    • Export Citation
  • Tabary, P., and Coauthors, 2010: Hail detection and quantification with C-band polarimetric radars: Results from a two-year objective comparison against hailpads in the south of France. Proc. Sixth European Conf. on Meteorology and Hydrology: Advances in Radar Technology, Sibiu, Romania, ERAD, 98102. [Available online at http://www.erad2010.org/pdf/oral/tuesday/radpol2/2_ERAD2010_0046.pdf.]

  • Ulbrich, C. W., and D. Atlas, 1982: Hail parameter relations: A comprehensive digest. J. Appl. Meteor., 21, 2243.

  • Vivekanandan, J., V. N. Bringi, and R. Raghavan, 1990: Multiparameter radar modeling and observation of melting ice. J. Atmos. Sci., 47, 549563.

    • Search Google Scholar
    • Export Citation
  • Wakimoto, R. M., and V. N. Bringi, 1988: Dual-polarization observations of microbursts associated with intense convection: The 20 July storm during the MIST project. Mon. Wea. Rev., 116, 15211539.

    • Search Google Scholar
    • Export Citation
  • Wolfson, M. M., R. L. Delanoy, B. E. Forman, R. G. Hallowell, M. L. Pawlak, and P. D. Smith, 1994: Automated microburst wind-shear prediction. Lincoln Lab. J., 7, 399426.

    • Search Google Scholar
    • Export Citation
A1

Here we adopt the terminology of RH87a, in which “equilibrium” refers to the stage at which an equilibrium mass of water is contained in or on the melting hailstone. In other words, the equilibrium fall speed occurs just after the onset of shedding.

Save