Mechanisms of formation of differential reflectivity columns are investigated in simulations of a midlatitude summertime hailstorm with hailstones up to several centimeters in diameter. Simulations are performed using a new version of the Hebrew University Cloud Model (HUCM) with spectral bin microphysics. A polarimetric radar forward operator is used to calculate radar reflectivity and differential reflectivity ZDR. It is shown that ZDR columns are associated with raindrops and with hail particles growing in a wet growth regime within convective updrafts. The height and volume of ZDR columns increases with an increase in aerosol concentration. Small hail forming under clean conditions grows in updrafts largely in a dry growth regime corresponding to low ZDR. Characteristics of ZDR columns are highly correlated with vertical velocity, hail size, and aerosol concentration.
Hailstorms are a serious atmospheric hazard—the National Weather Service estimated the total property and crop damage caused by hail in the United States during the 2000–08 periods to be ~$10 billion (www.weather.gov/os/hazstats.shtml). There is a great interest in the prediction of severe hailstorms and in hail suppression technology (e.g., Wieringa and Holleman 2006). During the last few decades, much effort has been devoted to understanding the processes favoring hail formation [see Cotton and Anthes (1989) for a comprehensive review]. Most studies point out that, in general, high convective potential instability, high atmospheric humidity, and moderate wind shear favor the development of intense hailstorms (Cotton and Anthes 1989). Foote (1984) stresses that hailstone size is strongly affected by the width and tilt of the main updraft. It was shown in many observational and numerical studies that hail reaches large sizes in the course of recycling, descending around the right flank of the updraft core reentering the updraft (Browning and Foote 1976; Ziegler et al. 1983; Nelson 1983; Tessendorf et al. 2005). The hail grows in updrafts by accretion of cloud droplets.
The crucial role of recycling during hail growth was discussed in detail by Ilotoviz et al. (2016) in simulations of a hailstorm using the Hebrew University Cloud Model (HUCM) with spectral bin microphysics. Strong instability is needed to create updrafts that are able to lift and hold large hailstones (Cotton and Anthes 1989; Khain et al. 2011). The rate of hail growth in cloud updraft depends also on the amount of supercooled liquid available for accretion at subfreezing temperatures. In turn, the amount of supercooled liquid depends on aerosol concentration in the boundary layer.
Effect of aerosols on hail mass and size have been discussed in several studies (Tao et al. 2012, 2016; Loftus and Cotton 2014; Noppel et al. 2010; Khain 2009; Khain et al. 2011, 2015; Ilotoviz et al. 2016). Ilotoviz et al. (2016) found that large hailstones with diameters of several centimeters form in “polluted” (i.e., high aerosol concentration) convective storms containing significant mass of supercooled liquid. In contrast, it was demonstrated that hail particles are substantially smaller in deep convective clouds developing in “clean” air containing low concentration of cloud condensation nuclei (CCN). Despite high practical importance, prediction of large hail, even for short-term forecasts, remains problematic (Marzban and Witt 2001; Brimelow et al. 2006; Manzato 2013).
Remote measurements from polarimetric Doppler radars are an important source of knowledge about microphysical processes in convective storms and, in particular, about hail formation processes. One of the important parameters measured by dual linearly polarized radar is differential reflectivity ZDR, which is equal to the difference between radar reflectivity in the horizontal and vertical planes, thus serving as a measure of particle nonsphericity. Comparatively narrow zones of enhanced ZDR have been identified above the ambient 0°C level; these so-called ZDR columns are often found in strong convective storm updrafts (e.g., Illingworth et al. 1987; Wakimoto and Bringi 1988). The maximal magnitudes of ZDR in such columns can reach 4–6 dB, and the height of such columns may reach a few kilometers above the 0°C level. The height of a ZDR column has been observed to increase with an increase in updraft intensity (e.g., Picca et al. 2010; Kumjian et al. 2012; Snyder et al. 2015). Observations show that the appearance of ZDR column may provide insight into the location and favorability of hailfall (e.g., Picca and Ryzhkov 2012; Carlin et al. 2016).
The values of ZDR depend not only on the shape of the particles but also on the variability of particle orientation (e.g., canting angles; Matrosov et al. 1996, 2001; Reinking et al. 2002). Dry hailstones and graupel are comparatively close to spherical, and these particles tend to tumble during their fall (Knight and Knight 1970). Later theoretical and experimental studies (Kry and List 1974a,b; Stewart and List 1983; Lesins and List 1986; List 1990) have established that hailstones gyrate while freely falling. Because of these reasons, ZDR from dry hail is usually less than 1 dB and is often near or even below 0 dB. In contrast, large raindrops are nonspherical (e.g., Brandes et al. 2002, 2006) and have a mean canting angle close to 0° with relatively little variability (i.e., symmetry axis is vertical; e.g., Beard and Jameson 1983); consequently, ZDR in large raindrops often exceeds 2 dB. These differences in the values of ZDR allow distinguishing zones of rain and dry hail despite the fact that the radar reflectivity of hail may be similar to that of heavy rain (Heinselman and Ryzhkov 2006).
The mechanisms of ZDR column formation have been investigated in several studies [see Kumjian et al. (2013a,b) for a detailed review]. Kumjian et al. (2014) analyzed ZDR columns’ formation and evolution using the HUCM. They found that ZDR columns in updrafts of polluted clouds are caused by large raindrops penetrating the updrafts in the course of recirculation. According to Kumjian et al. (2014), the ZDR values in the lower part of ZDR columns are determined by large raindrops, while in the upper part, the dominant source of ZDR is from freezing drops (FDs) and wet hail. This conclusion coincides with that of Snyder et al. (2015).
In the study by Kumjian et al. (2014), the maximum raindrop size was determined by collisional breakup as described by Seifert et al. (2005). The role of spontaneous and collisional breakup of raindrops is typically analyzed in relation to their impact on surface precipitation and formation of raindrop size distributions by drop–drop collisions (Tzivion et al. 1989; Pruppacher and Klett 1997; Seifert et al. 2006; Straub et al. 2010). The role of collisional breakup is often considered as dominating. This consideration typically follows from calculations of the time evolution of the drop size distribution in an air volume containing a significant number of cloud droplets and small raindrops (which makes drop–drop collisions frequent). The largest raindrops typically form as a result of melting of large snow or hail. In the HUCM version used by Seifert et al. (2005), the maximum hail diameter was about 8 mm, and the spontaneous breakup was not of crucial importance. Accordingly, the maximum raindrop sizes were also limited by these hail sizes. The current HUCM describes hail with diameters up to 6.8 cm. Despite the presence of collisional breakup, melting and shedding of large hail leads to appearance of raindrops with diameters slightly exceeding 10 mm.
At the same time, the maximal possible raindrop diameter typically does not exceed 8.2 mm (Kamra et al. 1991). The observations of raindrops with diameter ≥9 mm are rare and observed near or within hail shafts (Thurai et al. 2014). Supposedly, such huge raindrops contain some embedded ice mass (i.e., they are not 100% liquid water).
The appearance in the simulations of raindrops with diameters ≥10 mm indicates the necessity to take into account spontaneous breakup. Indeed, raindrops with diameters ≤8.2 mm produce substantially lower ZDR than those of 10 mm in diameter, at least at S band, where resonance effects in rain are not pronounced. The reduction in ZDR from raindrops that spontaneously break up requires reconsideration of the relative contribution of raindrops and hail to ZDR. The study is a further development following investigations performed by Kumjian et al. (2014) and Ilotoviz et al. (2016). It differs from that of Kumjian et al. (2014) since we implement a spontaneous drop breakup, as well as a new calculation of the aspect ratio of hail, containing a liquid water fraction. It differs from the study by Ilotoviz et al. (2016) since we consider formation of ZDR columns and aerosol effects on parameters of the columns. As a result, the study shows a relationship between aerosols, hail, and ZDR columns in hailstorms.
Although the present study focuses on analysis of ZDR columns, it is important to mention that others polarimetric parameters such as correlation coefficients are also useful for identifying hail regions (e.g., Kumjian 2013a).
2. Model description
The HUCM is a two-dimensional nonhydrostatic model that uses spectral bin microphysics, main components of which are described by Ilotoviz et al. (2016). Here, we present only a short description of the model, with new aspects of the scheme described in more detail. The model solves kinetic equations for size distributions of eight hydrometeor types: liquid drops (LDs), plates, columnar ice crystals, branch-type ice crystals (dendrites), snow (aggregates), graupel, hail, and freezing drops. The size distributions are defined on the logarithmic equidistant mass grid containing 43 mass-doubling bins for each microphysical species. The mass corresponding to the smallest bin is equal to that of a liquid droplet with a radius 2 μm. The size distribution of CCN contains 43 bins and describes soluble particles with radii ranging from 0.005 to 2 μm.
The initial (t = 0) CCN size distribution is calculated using the empirical dependence NCCN = N0 (Twomey 1959), as described by Khain et al. (2000). The prognostic equation for the size distribution of nonactivated aerosol particles (APs) is solved for t > 0. In the present study, CCN consist of NaCl. As shown in overview by Khain et al. (2000), the main difference between the numbers of droplets nucleated on APs of different chemical compositions is caused not by the difference in the chemical composition itself but by the differences in the shapes of the AP size distributions and by the values of their soluble fraction. In this study, we assume that the soluble fraction is equal to one.
At cloud base, the concentration of droplets is calculated using the method described by Pinsky et al. (2012) and Ilotoviz and Khain (2016), which uses an analytical relationship between the supersaturation maximum near cloud base, droplet concentration, and vertical velocity. Diffusion growth of all particles is simulated by solving analytically the equation system for particle size and supersaturations with respect to water and ice.
Collisions between different particles are calculated by solving a system of stochastic kinetic equations with height-dependent, gravitational collision kernels for drop–drop and drop–graupel interactions (Pinsky et al. 2001; Khain et al. 2001). Collisions between ice crystals are described following studies by Khain and Sednev (1995) and Khain et al. (2004). Calculation of sticking efficiency between ice particles still represents a serious problem in cloud physics. Sticking efficiencies between ice crystals, ice crystals and snow, and snow and snow were calculated as follows:
where T is air temperature (°C), P is the pressure, and Q is the mixing ratio. A little above freezing level (up to −6°C), f (T, P, Q) is a function that depends on temperature, pressure, and mixing ratio (Khain and Sednev 1996). Within this layer, Es ranges from 0.5 to 1. The temperature range −12.5° > T ≥ −17°C is the dendritic growth range, where we assume Es = 1. In the layer of −20° > T > −40°C, Es decreases linearly to zero at T = −40°C. Phillips et al. (2015b) showed that sticking efficiency depends on collision kinetic energy, which makes the sticking efficiency vary with size and type of ice particles. Such dependence will be included in future studies. Note, however, that dependencies and magnitudes of the sticking efficiency calculated according to (1) are on average close to those presented by Mitchell (1988), Connolly et al. (2012), and Phillips et al. (2015b). It was assumed that at T > 0°C, sticking efficiency between all melting ice particles increases to unity. Precipitating ice particles such as graupel, hail, and FDs may contain unfrozen liquid at freezing temperatures. The freezing process consists of two stages. The first nucleation stage is described using the parameterization of immersion drop freezing proposed by Vali (1994) and Bigg (1953). Drops with radii below 80 μm that freeze are assigned to plates, whereas larger drops undergoing freezing are assigned to freezing drops. Time-dependent freezing (second stage of freezing) of the liquid is described by solving heat balance equations at the particle surface and at the ice–liquid interfaces as described by Phillips et al. (2014, 2015a). At air temperature T < 0°C, the water mass within FDs, graupel, and hail is advected, mixed, and settled exactly as the size distributions of corresponding particles. For snow, liquid water fraction is only tracked for T > 0°C, whereas for FDs, it is only tracked for T < 0°C. For graupel and hail, water fraction is tracked within the entire temperature range. The implementation of the mass of water within ice particles allows for the calculation of successive processes of melting and freezing during particle oscillation around the 0°C level.
It was assumed that, if hail grows in the wet growth regime, it can collect graupel and ice crystals. Otherwise, no coalescence between hail (and graupel) with ice particles of any type is allowed. Effects of turbulence on collisions between cloud drops are included following Benmoshe et al. (2012).
The ice nuclei activation depends on supersaturation with respect to ice, as described by the empirical expression of Meyers et al. (1992). Primary nucleation of each ice crystal type occurs within its characteristic temperature range (Takahashi et al. 1991). Secondary ice generation is represented by the Hallett–Mossop mechanism (Hallett and Mossop 1974; Mossop 1976).
It is assumed that at freezing temperatures, supercooled water collected by snow soaks and freezes, forming rimed mass that increases the snow density. To calculate snow density, the mass of rimed (frozen) water within each snow bin is calculated. This mass is advected, mixed, and settled exactly as the size distribution of snow. When the snow density exceeds 0.2 g cm−3, the rimed snow is converted to graupel. At present, graupel in the model is characterized by the typical graupel density of 0.4 g cm−3. Time-dependent melting of snow, graupel, and hail as well as shedding of water from hail is treated following the work of Phillips et al. (2007).
Hail in HUCM forms in three ways: total freezing of liquid within FDs, conversion of graupel that undergoes wet growth, and conversion of graupel particles exceeding 1 cm in diameter. Transformation of particle types is described in detail by Ilotoviz et al. (2016).
The growth regime of FDs, graupel, and hail depends on the air temperature, particle size, and intensity of riming. If particle size or the mass of supercooled water are not large enough, all accreted liquid freezes at the particle surface, and the particle grows in the dry growth regime. If the mass of supercooled water is high enough and riming is intense, latent heat released by freezing heats the particle’s surface above a certain critical value, and a film of liquid water remains on the particle’s surface. This regime is known as wet growth. The latter changes surface roughness and the shape of the particle. The description of wet growth in HUCM is based on the theory of wet growth of hail developed by Phillips et al. (2014).
Following Rasmussen and Heymsfield (1987), drop shedding takes place if both the mass of exterior liquid water and the mass of the hail particle exceed their critical values. No shedding from snow takes place (Phillips et al. 2007). The fall velocity of FDs and hail particles is calculated following Rasmussen and Heymsfield (1987) and Phillips et al. (2015a). The approach is based on the calculation of the drag coefficient Cd using the Cd − Re relationship, where Re is the Reynolds number.
In addition to collisional breakup of raindrops included in the HUCM according to Seifert et al. (2005, 2006) and used by Kumjian et al. (2014), spontaneous breakup has been included. Two schemes based on laboratory experiments by Komabayasi et al. (1964) and Kamra et al. (1991) were tested in supplemental experiments. The size distribution of fragments was introduced following the study by Srivastava (1971). Supplemental simulations showed that, despite the differences in the probabilities of breakup of the largest raindrops reported by Komabayasi et al. (1964) and Kamra et al. (1991), results obtained using the two different methods were practically identical. The similarity can be attributed to the fact that, in both experiments, the probability of breakup of the largest raindrops is so high and the lifetime of the largest drops is so short that both methods rapidly eliminate raindrops with diameters exceeding ~8.2 mm. In the present study, the breakup probability measured by Kamra et al. (1991) was used. In simulations, we used the dependence of raindrop lifetime on drop diameter found by Kamra et al. (1991) in their laboratory experiments. The lifetime of a raindrop with diameter of 8.2 mm is about 14 s. Note that raindrops of larger sizes are allowed in the model. However, their lifetime is substantially shorter than 14 s. The aspect ratio of a raindrop as a function of its equivalent volume diameter D (mm) is defined in this study following Brandes et al. (2002, 2006).
Another improvement in the microphysical scheme was the calculation of aspect ratio of hail containing liquid water fraction. Following Rasmussen and Heymsfield (1987), we assume that in case of wet growth of hail, liquid on the surface of hail forms a horizontally oriented torus, which leads to an oblate-spheroidal shape. Figure 1 shows the dependence of aspect ratio of hail particle on hailstone equivolume diameter and liquid water fraction. The aspect ratio used in simulations by Kumjian et al. (2014) is presented in Fig. 1 for comparison. Overall, the aspect ratio for small, dry hail decreases from 1.0 (at the smallest size) to 0.8 (for a diameter of 10 mm); small hail that has nonzero mass water fraction varies linearly between that of a hailstone with the same mass that is dry and that of a raindrop (i.e., a completely melted hailstone of the same mass). The aspect ratio for larger hail decreases less quickly with increasing size and is, in fact, fixed for dry hailstones with diameters exceeding ~3 cm. As with small, wet hailstones, larger hailstones that are wet have aspect ratios between those of dry hailstones of the same mass and that of an ~8-mm raindrop (for completely melted hailstones). Smaller hailstones generally have larger aspect ratios in the “new” scheme, whereas large hailstones generally have smaller aspect ratios compared to those used by Ryzhkov et al. (2011) and Kumjian et al. (2014).
Note that the size–aspect ratio relationship has generally always been ad hoc in forward operators because of the variability seen in observations of hailstones at the ground; there is no “best” relationship because we do not actually have that many observations of hail shape (at least not in many different parts of the world, in many different environments, etc.), and the shapes that we do have are based on hail found on the ground (which may or may not be shaped differently than hail at higher altitudes). The study by Giammanco et al. (2014) analyzes results from about 2500 hailstone measurements from 33 convective storms. The hailstones reported there generally had a lower aspect ratio (i.e., were more oblate) than those reported by Knight (1986).
It is necessary to take into account that hail aspect ratio is measured at the surface, where hail does not contain a film of water. At the same time, the role of the liquid film is assumed to be of high importance. So the values of aspect ratio shown in Fig. 1 represent an expert evaluation of specialists in polarimetric radars. The application of these relationships gives realistic results and makes it possible to understand and explain the contribution of hail to formation of ZDR columns.
The microphysical output of the HUCM is used to calculate the polarimetric radar quantities by the use of a polarimetric radar forward operator that is generally the same as that described by Ryzhkov et al. (2011). The operator includes the calculation of polarimetric radar quantities at common radar wavelengths. The dependencies of radar variables on particle shapes, composition (liquid water fraction, the thickness of water film, etc.), and masses are used. The effective dielectric constants of dry and wet snow, graupel, and hail were calculated as in Ryzhkov et al. (2011). In the calculations, the two-layer (liquid and ice) structure of hail growing in the wet growth regime was assumed. For the calculation of scattering amplitudes associated with electromagnetically large particles (where the Rayleigh formulas are not appropriate), homogeneous-mixture T-matrix code was used (Mishchenko 2000). Liquid on the surface of hail forms a torus, which leads to an oblate-spheroidal shape and (generally, save for resonance effects) an increase in ZDR. Moreover, wet hail has a larger effective dielectric constant than dry hail, which also increases ZDR. Both factors are taken into account in the polarimetric operator. The reason why the differential reflectivity depends on the dielectric constant is described in the appendix.
3. Design of simulations
All simulations were performed within a computational domain of 153.9 km × 19.2 km (horizontal × vertical) with a grid spacing of 300 m in the horizontal direction and 100 m in the vertical direction. The new microphysical scheme has been tested in simulations of a thunderstorm observed in Villingen-Schwenningen, Germany, on 28 June 2006. Meteorological conditions of this storm were described by Khain et al. (2011). The relative humidity near the ground was high (~85%). The temperature at cloud base was 17°–18°C about 800 m above ground level. The meteorological situation was characterized by strong vertical wind shear, with northerly to easterly winds near the surface and a strong wind from the southwest at levels above 3000 m. The freezing level was located around 3.3 km above ground level. The maximum size of reported hailstones was about 5 cm.
Deep moist convection in the model was initiated by a 20-min temperature decrease (cooling) near the left boundary of the model domain (i.e., from 18 to 43 km in the horizontal direction and 2-km depth). The rate of cooling was 0.0084 K s−1. This method is traditionally used to initiates squall lines (Rotunno and Klemp 1985). Five simulations were performed for different values of N0 ranging from 100 (lowest CCN concentration) to 3000 cm−3 (highest CCN concentration); the slope parameter k = 0.5 was used for all simulations (Fig. 2). The aerosol size distributions calculated using the Twomey formula contain very high concentrations of the smallest CCN, and the concentration of the smallest aerosols was decreased, as shown in Fig. 2. All simulations were performed for 2 h.
a. Model validation
Before discussing the effects of aerosols on ZDR, hail, and the obtained correlation dependencies that can be of practical importance, we present some evidence of the ability of HUCM combined with the forward radar operator to reproduce observed radar observations. The comparison was performed at S band (i.e., 11-cm wavelength).
The scattering diagrams of radar reflectivity versus rainwater content (RWC; ZH vs RWC; Fig. 3a) and differential reflectivity versus radar reflectivity (ZDR vs ZH) obtained in simulations of the thunderstorm observed in Villingen-Schwenningen on 28 June 2006 are shown in Fig. 3. To plot the scattering diagrams, the results of storm simulations performed at different aerosol concentrations were combined. The simulations included those cases when hailstones with diameters up to 4–5 cm were produced. An empirical Z–RWC dependence given by RWC = 1.74 × 10−3Z0.640 (from Carlin et al. 2016) and obtained for convective storms in Oklahoma, which are not always accompanied by hail, is shown by the black curve in Fig. 3a. The red line in Fig. 3a represents the results of a one-dimensional (1D) spectral bin melting hail model (Ryzhkov et al. 2013). Note that the red curve (calculated for hail conditions) is above the black one because melting hail produces higher amount of large raindrops. In Fig. 3b, the mean observed dependencies between ZDR and Z are presented for different types of storms (Schuur et al. 2005). Figure 3 shows that HUCM combined with the forward radar operator is able to reproduce basic microphysics–radar relationships.
To assess the quality of HUCM results, the height dependencies of ZH in the model were compared with the dependencies derived from radar data and from more than 3000 surface hail reports obtained in 2010–14 from the Severe Hazards Analysis and Verification Experiment (SHAVE; Ortega et al. 2016). In the classification presented below, maximum hail size reported in observations was used. Vertical profiles of ZH in SHAVE were plotted for the following surface precipitation types: no hail, small hail (diameter D < 2.5 cm), large hail (2.5 < D < 5 cm), and giant hail (D > 5 cm). The data were segregated into 6 height classes/layers with respect to altitude h at which the wet-bulb temperature TW = 0°C. The height classes are z < h − 3 km (class 1), h − 3 < z < h − 2 km (class 2), h − 2 < z < h − 1 km (class 3), h − 1 < z < h km (class 4), h < z < z(Tw = 25°C) (class 5), and z > z(Tw = 25°C) (class 6). Height classes 1–4 represent areas of the atmosphere below the 0°C wet-bulb temperature height (an approximate height of the start of hail melting); height classes 5 and 6 are located above this altitude at colder temperatures.
Figure 4 shows percentiles of radar reflectivity and differential reflectivity derived from the model (top-left panel) and from observations (top-right panel). The thick vertical lines correspond to median values (i.e., 50th percentile) of radar reflectivity. The left and right boundaries of the colored boxes correspond to the 25th and 75th percentiles. A reasonably good agreement of simulations with the observations is seen. The reflectivity values are highest for giant hail with maximum ZH ~ 65 dBZ in both model and observations. The maximum ZH values are nearly constant with height, indicating the ascent of giant hail to high levels.
A good agreement between model simulations and observations is also found in terms of differential reflectivity. The ZDR values are highest for raindrops (no hail) with maximum of ~3.4 and ~3 dB in the model and observations, respectively. The giant hail cases have the lowest ZDR. The median value of ZDR is 2 dB in the first class both in the model and in observations. A tendency of ZDR to decrease with height is seen in both model and observations. Reduced ZDR values associated with hail is related to the fact that large hailstones fall to the surface with high terminal velocity and have little time to melt. As a result, larger hailstones below the freezing level are manifested by lower ZDR and higher ZH.
The difference in the ranges of ZDR variations in observations and in the model simulation can be partially attributed to the fact that the observations include a larger volume of data from different storms in various locations throughout the United States, while the model simulates only a single storm under specific thermodynamic conditions.
To justify the validity of the polarimetric operator and the differences in polarimetric signatures measured by dual-polarimetric radars of different wavelengths, we present Fig. 5 showing the fields of ZH and ZDR at S band (radar wavelength is 11 cm) and at C band (wavelength is 5.6 cm), as well as the differences between these fields. The values of ZH are higher for S band than for C band. In contrast, ZDR is lower in the S-band simulations compared to the corresponding simulations at C band. These results are in agreement with the ones reported in simultaneous S- and C-band radar observations of hailstorms (Feral et al. 2003; Kaltenboeck and Ryzhkov 2012).
b. The effect of spontaneous breakup of drops on polarimetric radar quantities
From a microphysical point of view, say, for determination of accumulated rain amount, the fact that the maximum raindrop diameter slightly exceeds 8.2 mm when spontaneous breakup is included and 10 mm when only collisional breakup is included might not be of high importance since the number concentrations of such large drops are small. However, since radar variables are highly sensitive to maximal particle sizes, such a difference in maximal drop size can have a large impact on the output of the forward operator. For example, as one can see in Fig. 6, the spontaneous breakup leads to a ~10-dBZ decrease in ZH from raindrops. Spontaneous breakup decreases the maximum of ZDR from unrealistically high values exceeding 6 dB to realistic values of 3–3.5 dB. The analysis of DSDs shows that spontaneous breakup leads to the disappearance of the largest raindrops (i.e., those with diameters near 10 mm) and to the formation of a larger amount of small raindrops. This example shows the importance of using a sufficient number of mass bins for raindrops in bin microphysical models.
c. Formation of ZDR columns and hail in the case of high CCN concentration
The fields of radar reflectivity and differential reflectivity simulated at C band will be illustrated throughout the rest of the paper in order to facilitate comparison with Kumjian et al. (2014), in which all simulations were performed at C band.
Results of the simulation with N0 = 3000 cm−3 will be discussed first. The process of ZDR column formation is illustrated in Fig. 7. First, raindrops form in updrafts at z ~ 6 km (Fig. 7a) and descend along the cloud edges (Figs. 7b,c). These raindrops form the first sign of a ZDR column that grows from above (Figs. 7b,c). Some of the drops penetrate into the updraft, where they grow by collection of cloud droplets (Figs. 7d,e). The maximum ZDR is located at a height of 4–4.5 km, where raindrops lofted above the 0°C level grow to diameters up to 6 mm. Figure 7f shows the DSDs within the cloud updraft at t = 42 min in locations marked by white circles in Fig. 7b. From the analysis of the size distributions, one can see that large raindrops reach high altitudes. These raindrops lead to the appearance of ZDR > 2 dB at these levels. The process of the raindrop formation associated with enhanced ZDR in cloud updraft via raindrop recirculation was first discussed by Kumjian et al. (2014). Herein, we emphasize that such process is efficient only in polluted clouds.
Fields of cloud water content (CWC) and RWC, freezing drop mass content, and hail mass content at the mature stage (78 min) of the storm in the polluted simulation are shown in Fig. 8. Freezing drops form at about 4.5-km altitude. Collection of small cloud droplets by freezing drops is efficient above the freezing level where CWC is high. Freezing drops are comparatively large with diameters of about 1 cm (Fig. 9). Complete freezing of FDs leads to the formation of hailstones ascending in the updraft up to 9 km. Hail particles fall largely to the left of the zone of maximum RWC (see Figs. 8b,d). The zone of hailfall is shown by arrow in Fig. 8b. Hail particles at this time largely melt, producing raindrops at z = 1.8–2 km (i.e., 1 km below the melting level).
Figure 9 shows the ZH and ZDR fields, as well as hydrometeor mass distributions within in the updraft, during the mature stage (78 min) of storm evolution. The maximum ZH exceeds 65 dBZ and is attributed to hail (see Fig. 13 for more detail). Strong updraft transports hail upward, so ZH remains high (e.g., exceeding 50 dBZ) up to ~10-km height. This is a typical feature of natural hailstorms (Noppel et al. 2010; Khain et al. 2011). One can see a pronounced ZDR column with top height exceeding 6 km.
The microphysical composition of hydrometeors comprising the ZH and ZDR fields is shown in the left column (Fig. 9), where the mass distributions of different hydrometeors at several altitudes within the updraft are presented. Mass distributions of liquid drops indicate the presence of raindrops as well as a significant mass of cloud droplets (i.e., high CWC) with a peak at droplet diameter of ~20 μm. The high concentration of small cloud droplets is caused by the high aerosol concentration. One can see that hail particles exist in the updraft at elevations as low as 3 km, where temperatures are greater than 0°C. These hail particles entered the cloud updrafts in the convergence zone in the lower part of the deep convective cloud. Since the onset of freezing results in the conversion of raindrops to FDs, the concentration of rain decreases with height in the updraft. Complete freezing of FDs and accretion of small droplets lead to an increase in hail mass and hail size with height in the updraft. The largest raindrops near the surface produce ZDR of about 3 dB. Maximum ZDR above the freezing level at the considered time (78 min) is about 3.5 dB and is reached at ~1 km above freezing level. Since ZDR from dry hail is less than 1 dB, a higher value of ZDR is caused by hail containing liquid water and growing in the wet growth regime. Indeed, as can be seen from hail size distributions in the right panels of Fig. 9, hail particles of the majority of sizes grow in the wet growth regime (i.e., they are covered by a film of liquid water). Even at an altitude of 6.5 km, hailstones with diameters exceeding 1 cm continue growing in the wet growth regime.
Figure 10 illustrates the evolution of the ZDR field during the 76–82-min time period when hail falls to the surface along cloud edge (in the figure, to the left from cloud updraft). As shown in Fig. 9, high ZDR values in the upper half of the column are formed by hail in the wet growth regime. Above the column, hail grows in the dry growth regime, and ZDR decreases with height. Hail falling to the left of the updraft at x ~ 60 km (green contours in Fig. 10) is dry because of a lack of cloud droplets, and the associated ZDR is less than 1 dB. Below the melting level, smaller hail particles start melting, which increases ZDR in the narrow layer around z = 1.5 km (Figs. 10a,b). When this significant mass of hail reaches the ground, the zone of hailfall is identified by a “ZDR hole” akin to those seen in observations (e.g., Ryzhkov et al. 2011). In areas with large hail, ZH is generally quite high, around ~65 dBZ.
The liquid water fraction (LWF) of FDs and hail, averaged over the particle size spectra during the same time period as in Fig. 9, is presented in Fig. 11. Contours of ZDR are plotted in white. In the polluted case under consideration, hail and FDs have high LWF below heights of 5 km. The high LWF in hail at negative temperatures indicates that hail is growing in wet growth regime; high LWF in FDs and in hail is caused by intense accretion of supercooled droplets that delays complete water freezing in both FDs and hail. The LWF in hail is also high within a narrow layer just below the melting layer, but these large values of LWF do not lead to high ZDR because they are associated with the rapid melting of the smallest hail particles.
Figure 12 presents the ZDR fields produced by each hydrometeor type separately. The ZDR from water is substantial from the ground up to 4.8 km where T ~ −6.5°C (Fig. 12a). The ZDR produced by hail increases in the upper portion on the ZDR column up to ~6.5 km (T ~ −18.5°C). In addition, increased ZDR is found at the edge of cloud, where falling hail collects droplets, as well as in the narrow layer below the melting level. FDs produce moderate ZDR (e.g., 2–3 dB) within a comparatively wide layer from 4 to 6.5 km (T between −2° and up to −18.5°C). This significant thickness of the layer where FDs exist and contribute to ZDR can be attributed to intense accretion of supercooled liquid droplets (Fig. 12c). The “net” ZDR that results from the contributions of all hydrometeor species clearly shows the ZDR column extending to an altitude of 6.5 km where T ~ −18.5°C (Fig. 12d). Hail reaches the surface occasionally for short periods of time. For instance, hail reaches the surface at 82 min (Fig. 10d). The ZDR columns appear about 15–20 min before the hail reaches the surface. The statistical results (Figs. 4, 19, and 20) are based on the entire period of the simulations. Khain et al. (2011, their Fig. 3) and Ilotoviz et al. (2016, their Fig. 25) reported results similar to those obtained in the present study as regards masses of accumulated hail. At CCN of 3000 cm−3, accumulated hail at the surface reaches 0.14 mm.
Hail contribution to the total ZH is dominant in the upper portion of the column (e.g., above ~5 km), as well as the left edge of the column (and total echo) down to about 1.5 km (Fig. 13). Rain dominates the total ZH below the melting level as well as the center of the column up to 1 km above the 0°C level. This is indicative of the existence of large raindrops in the cloudy updraft in this case of high aerosol concentration. The FDs are not a dominant contributor to the total ZH anywhere within the ZDR column except for a very small area between 4.5 and 5 km. Nevertheless, FDs contribute substantially (up to 20%) to the total reflectivity within the 4–6-km layer (Fig. 13, right panel). These results corroborate those reported by Kumjian et al. (2014).
The following sentence is included in the revised paper: “The role of shedding in decrease of hail size is explored in detail by Ilotoviz et al. (2016).” Our experiments (not described in the paper) indicate that the maximum values and the height of ZDR column do not change much when the shedding process is switched off in the model. Shedding takes place only if the mass of water on the surface of the ice core exceeds the critical value (Rasmussen and Heymsfield 1987). Following Phillips et al. (2007), we assumed the shed water mass cannot exceed 50% of the critical water mass. Although the role of shedding on raindrop formation is not analyzed in the study, these limitations on the mass of shed liquid may decrease the role of shedding in production of raindrops as compared to the cases when shedding of the entire liquid mass is assumed as, for instance, in Kumjian et al. (2015).
d. Formation of ZDR columns and hail in the case of low CCN concentration
Next, we discuss results of the simulation in which the CCN activation spectrum is characterized by N0 = 100 cm−3. Fields of CWC, RWC, FD mass content, and hail mass content typical for this low CCN case are shown in Fig. 14. The main difference between the low and high CCN concentration cases is a reduced amount of supercooled CWC in the clean case (Fig. 14a). The CWC maximum in the clean case is at about half the altitude and about half the magnitude of those in the polluted case; CWC exceeding 1 g m−3 reaches 4 km in the clean case as compared to 7 km in the polluted case (Fig. 8a), and maximum CWC is ~1.3 g m−3 in the clean case as compared to ~2.75 g m−3 in the polluted case. A much lower supercooled CWC in the clean case is the primary factor that drives the difference in cloud microphysics and in the structure of the ZDR field as compared to the polluted case. Because of the low CCN concentration, cloud droplets are relatively large, and raindrops form in updrafts below or very near the freezing level; therefore, some drops fall to the ground without participating in ice processes (Fig. 14b). Raindrops ascending above the 0°C level are large enough to begin freezing and convert to FDs. Since supercooled CWC is low, accretion is inefficient, which leads to a rapid complete freezing of FDs and their subsequent conversion to hail. As a result, the mass content of freezing drops in the clean case is lower than in the polluted case by an order of magnitude (Fig. 14c; cf. Fig. 8c). Because accretion is inefficient, hail particles remain relatively small. Note that in some studies, hail is defined as particles larger than 1 cm in diameter. Using this definition, total freezing of FDs leads in the case of low CCN concentration to frozen drops but not to hail. In HUCM, hail and frozen drops are assigned to the hail category because of similar high (0.91 g cm−3) density. Since hail particles and frozen drops are relatively small, they spread over a larger area than in the polluted case (cf. Fig. 8d and Fig. 14d). Therefore, Fig. 14e is plotted for a wide horizontal range (40–90 km). As was shown by Ilotoviz et al. (2016), the total mass of hail above the freezing level in the clean case may be larger than in the polluted case. However, the maximum value of hail mass content in the clean case is more than twice as small as in the polluted case (cf. Fig. 14d and Fig. 8d). Since hail particles are smaller in the clean case, they rapidly melt below the melting level (Fig. 15) and do not reach the surface.
The aforementioned microphysical differences determine the difference in the structures of the ZH and ZDR fields in cases of low and high CCN concentrations. Figure 15 shows the fields of ZH and ZDR, as well as hydrometeor mass distributions within the updraft at the mature stage of the storm in the low CCN concentration case. Note first that the maximum of ZH is about 55 dBZ, which is 10 dBZ lower than in the high CCN case (Fig. 9). The lower reflectivity is the result of the smaller sizes of hydrometeors in the clean case. The values of ZH rapidly decrease above 8 km in the clean case, whereas they extended to ~10-km height in the polluted case. In contrast to the polluted case, the maximum of ZH occurs below the freezing layer. This reflectivity maximum is most associated with raindrops, while the reflectivity maximum is most associated with hail in the polluted case. There is no pronounced ZDR column in the clean case: a zone of enhanced ZDR only slightly extends above the 0°C isotherm.
The mass distributions of different hydrometeor types are plotted (left panels in Fig. 15) at altitudes of 3, 4, and 5 km in the maximum updraft. The mass distributions of liquid drops (black curves) do not contain small (e.g., 20 μm) cloud droplets, which dramatically differs from the polluted case. The concentration of the largest raindrops rapidly decreases with height because of freezing because of the higher probability of larger raindrops to freeze. Owing to the lack of supercooled droplets (CWC is ~1 g m−3 at 4 km in the clean case, while in the polluted case, the CWC is twice as large at the same altitude; Figs. 14 and 8), the accretion rate is comparatively low, so FDs rapidly freeze to produce hail with sizes that do not exceed 1.3 cm in diameter. This hail is substantially smaller than in the polluted case where D reached 4–5 cm. The size of freezing drops is also smaller in this clean case. The mass distributions of FDs and hail at two height levels showing the type of growth regimes are presented in the right column of Fig. 15. Freezing drops and hail grow in the wet growth regime only in vicinity of the 0°C isotherm. At altitude of 5 km, hail particles of all sizes grow in the dry growth regime because of the lack of supercooled water. Since ZDR values produced by dry hail and remaining small raindrops are low, ZDR is also low above 4 km. In agreement with growth regimes shown in Fig. 15, nonzero LWF in FDs is found only within the 3.3–4.5-km layer. Nonzero LWF in hail occurs within a narrow (500-m depth) layer just below the freezing level, where small hail particles melt.
Figure 16 shows ZDR fields produced by hydrometeors of different types. We see again that, at low CCN concentration, a robust ZDR column (above freezing level) does not form. The largest ZDR values produced by raindrops are not located in the updraft but in the zone of hailfall and melting (Fig. 16a). FDs rapidly freeze completely, so there are only a few FDs above 5 km (Fig. 16c). Since hail (Fig. 16b) grows in the wet regime near the freezing level, the ZDR values (Fig. 16d) are substantially lower than in the polluted case.
e. Dependencies of ZDR on aerosol concentration
The height of the ZDR column is typically defined as the distance between the maximal height where ZDR ≥ 1 dB and the environmental 0°C level (e.g., Snyder et al. 2015), although no formal criteria exist. Although the level of the 0°C isotherm is perturbed upward within a narrow updraft zone by several hundred meters relative to the environment, the height of the ZDR columns was determined here with respect to the unperturbed 0°C level. The choice of such reference altitude leads to some increase in the evaluated height of ZDR columns. As a result, the estimated differences between the ZDR column heights in the high and low CCN concentration cases decrease. Nevertheless, the difference in the ZDR column heights remains substantial, as seen in Fig. 17, which shows the time dependence of ZDR column heights determined by the maximum height of (Fig. 17a) 1- and (Fig. 17b) 2-dB ZDR contours for two aerosol concentrations of N0 = 100 cm−3 and N0 = 3000 cm−3, respectively. One can see that the height of the ZDR column is substantially larger in the case of high aerosol concentration. On average, the heights of 1- and 2-dB contours in polluted case are higher than in the clean case by about 1 and 1.5 km, respectively. Note, however, that in the course of time changes, contours ZDR of 1 and 2 dB in the clean case may reach larger heights. Figure 17c shows that these cases are rare when vertical velocities are very high (above 30 m s−1) and the local 0°C isotherm is lifted by a few hundred meters.
A comparison of the maximum vertical velocities between the cases with high and low CCN concentration shows that the maximum of vertical velocities in the polluted case are 5–10 m s−1 higher than in the clean case (Fig. 17c). This higher vertical velocity reflects higher latent heat release by the diffusion growth of droplets and drop freezing during accretion of droplets by FDs and hail in polluted case. This effect is known as convective invigoration and has been widely discussed in the literature (e.g., Rosenfeld et al. 2008; Khain 2009). It is widely assumed, however, that significant convective invigoration takes place only in tropical clouds with warm cloud bases and high freezing levels. The calculations with the current model with detailed description of hail processes show that convective invigoration is significant also for midlatitude storms. The effects of aerosols on microphysics and dynamics of storms were reported also in several studies (Li et al. 2008; Storer et al. 2010; Loftus and Cotton 2014). Loftus and Cotton (2014) reported pronounced convective invigoration of hailstorm in polluted air.
Figure 17d shows time dependencies of ZDR column volumes at aerosol concentrations of 100, 1000, and 3000 cm−3. The ZDR column volume is defined as the product of the height of ZDR contour of 1 dB over the reference 0°C isotherm and the maximum ZDR within this column above this reference level. This value characterizes both the ZDR value and the extent of the column. As can be seen, ZDR column volume is generally larger in the polluted case during the entire period of simulations. The lowest volumes appear in the clean case.
All time dependencies reveal the existence of three main peaks corresponding to the development of new convective cells. Since the peaks form at different times in simulations with different aerosol concentrations, the dependencies of time-averaged characteristics of ZDR columns on aerosol concentrations are plotted in Fig. 18. One can see a clear increase of all ZDR column parameters with increasing aerosol concentration.
f. Relationships between ZDR, vertical velocity, and hail parameters
Figure 19 presents scatterplots showing the relationship between the maximum height of (Fig. 19a) 1- and (Fig. 19b) 2-dB contour of ZDR (above local 0°C level) and the vertical velocity calculated at these heights under different aerosol concentrations. In all cases, correlations are high. It is interesting that slopes for all CCN concentrations turned out to be close, which allowed us to combine all cases into a single scatterplot. The lower CCN cases generally had smaller ZDR columns, but they also generally had weaker updrafts; the higher CCN cases had larger ZDR columns and strong updrafts. The results indicate that ZDR columns can be likely used for evaluating vertical velocity and even vertical velocity profiles in deep convective clouds.
The difference in the mechanisms of hail growth at different aerosol concentrations was discussed in detail by Ilotoviz et al. (2016). The relation between mass and size of hail and ZDR columns was described by Kumjian et al. (2014). The present study with the latest version of the spectral bin microphysics also indicates a close relationship between ZDR columns and the mass and size of hail. To reveal the maximum correlation, the correlation coefficients between different hail parameters and maximum ZDR above the 0°C level were calculated at different time lags. The maximum correlation coefficients were found for time lags of 15 min. The time lag arises because ZDR maxima and maximum ZDR column height take place in cloud updrafts during hail growth. It takes time for this hail to grow to maximum sizes and to fall to the surface. Kumjian et al. (2014) noted that in real storms, such delay is 20–30 min. The underestimation of the time delay in the simulations is related, perhaps, to the fact that the model is two-dimensional. This problem will be investigated in more detail in the future.
Figure 20 shows scatterplots of maximum values of ZDR above the 0°C level versus maximum hail mass and hail mean volume radius near the surface. The correlations were calculated only for polluted cases (1000–3000 cm−3) because of the lack of sufficient statistics in rare cases when hail reaches the surface in clean cases. The correlation coefficients for the two relationships are 0.67 and 0.76, respectively. Regardless, the increase in height and volume of a ZDR column can be a good predictor of a prominent hail shaft. Taking into account that the height and volume of ZDR columns increase with aerosol concentration, one can see clear dependence of the total mass and size of hail on aerosols.
5. Discussion and conclusions
This paper describes the relationships between aerosols, hail, and ZDR in a midlatitude hailstorm. The analysis was performed using a two-dimensional cloud model (HUCM) with spectral bin microphysics. The main improvements to the model in this study as compared to the study by Kumjian et al. (2014) include (i) implementation of spontaneous raindrop breakup, (ii) more accurate calculation of aspect ratios of melted hail as well as hail growing in a wet growth regime, and (iii) the accurate calculation of droplet concentration at cloud base based on analytic determination of local supersaturation maximum.
Taking into account spontaneous breakup of raindrops substantially improved representation of both ZH and ZDR; spontaneous breakup led to a ~10-dBZ decrease in the maximum values of ZH from raindrops and to a 3-dB decrease (from 6 to 3 dB) in the ZDR maximum caused by raindrops. Accordingly, the contribution of hail to the total radar echo (i.e., ZH) and, in particular, to ZDR increased. The values of ZDR produced by hail depend on the amount of liquid water within hail particles, which is strongly affected by the type of hail growth (i.e., wet vs dry). The ZDR produced by hail growing in the dry growth regime tends to be lower (i.e., generally below 1 dB in C band) because dry hail has a larger aspect ratio with more random tumbling behavior; in contrast, ZDR can reach 6 dB at C band in hail undergoing wet growth. The values of ZDR produced by hail growing in wet growth depend on the aspect ratio of hail particles (i.e., on the width of the liquid water torus arising during wet hail growth). In the present study, we have chosen the dependence of aspect ratio on hail size and liquid water fraction that provides better agreement with observations than the dependencies used by Kumjian et al. (2014).
Several examples of model validation using observed data showed that the model combined with the polarimetric radar forward operator is able to describe accurately the observed relationships between ZH and RWC and between ZH and ZDR. Finally, it was shown that the model can describe realistically ZH and ZDR distributions at different heights and for different hail sizes. The satisfactory results of validation allowed us to use HUCM for simulation of the effects of aerosols on hail and ZDR.
The crucial role of aerosols on cloud microphysics is seen in simulations of ZDR. In the case of high aerosol concentration, supercooled cloud water content is high, which promotes wet growth of hail at heights ≥ 6 km. Intense accretion of supercooled cloud liquid delays complete freezing and increases the size of FDs and hail. Intense accretion on large hail supports larger areas of wet growth of hail at altitudes as high as 6.5 km. As a result, high CCN concentration leads to tall ZDR columns and high ZDR values.
In the case of low aerosol concentration, the amount of supercooled liquid is low, and hail growth by riming is not intense. As a result, accretion is weak, and small amount of accreted liquid rapidly freezes at the hail surface. As a result, hail and FDs grow slowly, mainly in a dry growth regime. The low accretion leads to fast freezing of liquid drops and slow hail growth. As a result, ZH is ~10 dBZ lower than in the high CCN concentration case. The ZDR columns are much lower (if any exist) than in the high aerosol concentration case. In the case of low CCN concentration, hail particles are small, and they generally melt before reaching the surface.
The results of this study indicate a close relationship between the structure of ZDR and various cloud microphysical and dynamical parameters, such as vertical velocity, aerosol concentration, hail mass content, and hail size. It is shown that there are high correlations between ZDR-related quantities (e.g., maximum ZDR values aloft and the height and volume of ZDR columns) and vertical velocity, hail mass, and hail size at the surface. These relationships mean that the structure of ZDR fields contains information about aerosol concentration, the mass of supercooled water, and potentially the mass and size of hail. In particular, our results may provide plausible explanation for why deep tropical convective clouds generally do not contain high values of ZDR and deep ZDR columns.
The effects of soluble aerosols (CCN) on cloud structure are discussed in many studies (Rosenfeld et al. 2008; Khain 2009; Tao et al. 2012). It is usually assumed that aerosol effects are maximized in tropical clouds. In this study, it is shown that effects of soluble aerosol on cloud microphysics and dynamics may also be substantial for midlatitude storms. It is usually stressed that strong updrafts are needed to form large hailstones. In this study, we found a synergetic effect between the dynamic mechanism and the microphysical mechanism of aerosols. In the case of high CCN concentration, an increase in the concentration and mass of supercooled droplets in cloud updrafts determines the location of the zone where hail intensively grows by accretion of supercooled droplets. This accretion leads to an increase in latent heat release by freezing and to a corresponding increase in the vertical velocity of the updraft. In turn, the increase in the vertical velocity leads to an increase in the amount of supercooled water. This synergetic mechanism leads to an increase in the ZDR columns’ height. In case of low CCN concentration, supercooled water content at high levels is low, and hail spreads over a large area without growing significantly by accretion of small liquid droplets. As a result, hail remains comparatively small, and latent heat release due to accretion is lower in the clean case. Note that atmospheric instability affects storm updrafts and all radar polarimetric signatures. So aerosol effects are better seen under similar thermodynamic conditions.
In the study, we used the 2D model to investigate the structure of a hailstorm. It is possible that a 2D framework is not able to reproduce the extreme ZDR values that result from size sorting in a 3D shear environment. Multiple comparisons between observations and the results of simulations of isolated clouds, squall lines, and storms (such comparisons are presented in the present study as well) show that the 2D HUCM produces a realistic cloud microphysical structure and realistic values of the vertical velocities, as well as the fields of radar polarimetric signatures. So we believe that our results are qualitatively and, to a large extent, quantitatively correct. In our case, the 2D simulations allowed revealing the main factors affecting ZDR values. To get better quantitative evaluation of 2D effects, the bin microphysics containing the calculation of liquid water fractions within ice hydrometeors during their melting should be implemented into 3D cloud models.
We would like to draw attention to the importance of the proper choice of mass grid to describe size distributions of liquid drops. We saw that appropriate treatment of specific practically important problems, such as calculation of differential reflectivity from liquid drops, requires high resolution of mass grids within specific ranges of drop size. The resolution used is high enough to describe spontaneous breakup. However, the accuracy of the calculations of maximum raindrop sizes decreases because of the lack of mass bins between 8.2 and 10.3 mm. Fortunately, drops exceeding 8.2 mm in diameter are unstable, and spontaneous breakup rapidly leads to the formation of fragments with sizes well described by the logarithmic mass grid.
In this study, we focused on effects of aerosols in the structure of ZDR columns. Note that the structure of ZDR columns and other polarimetric characteristics depend also on environmental conditions (Van Den Broeke 2016).
Partial support of A. K., A. R., and E. I. for this work comes from Grants DE-SC008811 and DE-SC0014295 from the U.S. Department of Energy Atmospheric System Research program. Funding for A. R. and J. S. was also provided by NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA11OAR4320072, U.S. Department of Commerce.
Dependence of the Ratio ZH/ZV on the Dielectric Factors
The dependence of the ratio ZH/ZV in the dielectric factors can be derived from the description of the polarimetric operator presented by Ryzhkov et al. (2011). Below, we present this derivation with corresponding comments. Radar cross sections of a spheroidal particle in the directions along their principal axes a and b are defined as
where sa is the scattering amplitude if the electric field vector of the incident wave is parallel to the symmetry axis of the hydrometeor and sb stands for the scattering amplitude if the electric vector is perpendicular to the symmetry axis.
Differential reflectivity Zdr of the particle (expressed in linear scale) is defined as
If the hydrometeors are relatively small compared to the radar wavelength λ and are modeled as oblate or prolate spheroids, then simple analytical formulas for the scattering amplitudes sa,b can be obtained in the Rayleigh approximation:
where D = (ab2)1/3 is the equivolume diameter of the particle (a is rotation axis of spheroid), ε is the dielectric constant, and La and Lb are the shape parameters defined as
for oblate spheroids (a < b) and
for prolate spheroids (a > b). In the case of spherical particle (a = b), La = Lb = 1/3 and
that is, differential reflectivity depends on the dielectric constant and shape parameters La,b. If the particle is spherical, then La = Lb and ZDR = 1.
For a given shape of the hydrometeor when La and Lb are fixed, the magnitude of ZDR strongly depends on dielectric constant ε. The value of ZDR is close to 1 if |ε − 1| tends to zero as in the case of dry snow so that |ε − 1|La,b ≪ 1. In the opposite case of very high absolute value of ε, the magnitude of ZDR approaches its upper limit (La/Lb)2. Therefore, a raindrop has higher ZDR compared to the snowflake or graupel with the same shape. This fundamental property of ZDR is very beneficial for discrimination between liquid and frozen hydrometeors with dual-polarization radar.