## 1. Introduction

As raindrops descend toward the surface, the evolution of their distribution of sizes is governed by a number of microphysical processes. These processes include sedimentation or size sorting, evaporation, aerodynamic and collisional breakup, and coalescence. Understanding the physics governing the evolution of the drop size distribution (DSD) is important because changes of the DSD affect the mass flux or rainfall rate experienced at the surface. Further, understanding how these processes affect radar measurements is critical for optimizing remote quantitative precipitation estimates as well as studies of precipitation microphysics using remote sensing.

Another major impetus for understanding the evolution of raindrop size spectra is that storm dynamics can be strongly influenced by precipitation microphysics (Srivastava 1985, 1987; Markowski et al. 2002; Gilmore et al. 2004; Dawson et al. 2010; Van Weverberg et al. 2011; Bryan and Morrison 2012; Morrison et al. 2012). This is because processes like precipitation loading, evaporation of raindrops, and melting of hailstones generate negative buoyancy that can affect storm behavior and evolution via downdraft and cold pool production. Thus, providing the best approximation to the physics governing precipitation processes is crucial for storm-scale modeling. Model simulations of convective storms have been found to be quite sensitive to microphysical parameterizations, including warm-rain processes (Ferrier et al. 1995; Morrison et al. 2009; Dawson et al. 2010; Bryan and Morrison 2012). As an example, Morrison et al. (2012) found large sensitivity to the drop breakup parameterization, including its substantial effects on storm characteristics such as cold pool strength, propagation speed, and precipitation accumulation. Though these parameterizations are based on kernels and efficiencies developed from laboratory and bin model studies, there still exists considerable uncertainty (Morrison et al. 2012).

The processes that are the subject of the present study are the collisional processes of coalescence and breakup. There has been much experimental work in the laboratory with colliding drop pairs (McTaggart-Cowan and List 1975; Low and List 1982a, hereafter LL82a; Ochs et al. 1995; Barros et al. 2008; among others) and, more recently, direct numerical simulation of colliding drops (Schlottke et al. 2010). These studies have been instrumental in leading to the determination of collision efficiencies between drop pairs of various sizes and the development of coalescence and breakup kernels used in model parameterizations (Low and List 1982b, hereafter LL82b; Feingold et al. 1988; Beard and Ochs 1995; McFarquhar 2004; Prat et al. 2012). Analytic and numerical models of the evolution of raindrop spectra undergoing these processes have also been widely studied (List and Gillespie 1976; Valdez and Young 1985; Brown 1987, 1988, 1993, 1997; List et al. 1987; Tzivion et al. 1989; List and McFarquhar 1990; Hu and Srivastava 1995; Seifert et al. 2005; Prat and Barros 2007a,b, 2009; Prat et al. 2012), as have numerical methods for efficiently and accurately solving the stochastic collection and breakup equation (Bleck 1970; Bott 1998, 2000, 2001; Prat and Barros 2007b; Jacobson 2011).

In contrast to the extensive modeling and laboratory studies mentioned above, there has been relatively little work in validating these models and kernels in nature. The emergence of dual-polarization radar offers added information regarding the evolution of the shape of the DSD. The variables available from polarimetric radar measurements that will be discussed in this study are radar reflectivity factor at horizontal polarization *Z*_{H}, differential reflectivity *Z*_{DR}, specific differential phase *K*_{DP}, and the copolar correlation coefficient *ρ*_{hυ}. A detailed description of these polarimetric variables may be found in various textbooks (e.g., Doviak and Zrnić 1993; Bringi and Chandrasekar 2001) or review articles (e.g., Herzegh and Jameson 1992; Zrnić and Ryzhkov 1999; Ryzhkov et al. 2005; Kumjian 2013a,b,c). Briefly, *Z*_{H} is strongly dependent on the equivalent volume diameter *D* of particles; it is proportional to *D*^{6} for particles that are electromagnetically small compared to the radar wavelength. In addition, *Z*_{H} depends on the concentration of particles in the radar sampling volume. In rain, *Z*_{H} generally increases for increasingly heavy rainfall rates and varies from near 0 dB*Z* or less in light drizzle to >50 dB*Z* in extremely heavy downpours. The reflectivity-weighted shape of particles in the radar sampling volume is measured by *Z*_{DR}, where values of 0 dB indicate spherical or randomly oriented hydrometeors and increasing magnitudes of positive or negative values indicate increasing particle anisotropy. In rain, values vary from 0 dB in drizzle to 4–5 dB in heavy continental rain at S band (and up to 6 dB or more at C band). Although *Z*_{DR} is independent of particle concentration, it does depend on the shape of the DSD. The accumulated phase lag between the horizontally and vertically polarized waves per unit radial distance is given by *K*_{DP} and thus measures the concentration of anisotropic particles within the sampling volume. It is dependent on particle concentration, shape, size, and composition but is not affected by spherical or randomly oriented particles. Values of *K*_{DP} vary from 0° km^{−1} in drizzle to >4° km^{−1} (at S band) in very heavy rain. It is also inversely proportional to the radar wavelength, so shorter wavelengths like C and X band will produce proportionally larger *K*_{DP} values for the same sampled rainfall. In rain, *ρ*_{hυ} is near unity, except for heavy continental rain at C band, in which values may drop as low as 0.93 owing to resonance scattering effects by large raindrops.

These polarimetric variables are affected by changes in the distribution of sizes and shapes of particles and thus can provide information about the physical processes acting on the precipitation. Such new information may be used to validate microphysical models of rain physics as well as to “fingerprint” the dominant ongoing processes in precipitation. Studies along these lines have investigated the other warm-rain processes of evaporation (Kumjian and Ryzhkov 2010) and size sorting of raindrops (Kumjian and Ryzhkov 2012). Thus, one of the objectives of this study is to quantify the impact of coalescence and breakup of raindrops on the dual-polarization radar variables, thereby determining the microphysical fingerprints of the processes in polarimetric radar data.

To do this, we use the one-dimensional version of an explicit bin-microphysical model (Prat and Barros 2007a,b; Prat et al. 2012) coupled with T-matrix electromagnetic scattering calculations (e.g., Waterman 1969; Mishchenko 2000) and the polarimetric radar operator of Ryzhkov et al. (2011). Though idealized and operating within a simplified (one dimensional) framework, this type of modeling approach allows for efficient exploration of parameter space and isolation of particular processes. Results of the coupled microphysics–scattering model are compared to disdrometer and radar observations.

The next section briefly describes the models used in this study. Section 3 presents the modeling results, and section 4 compares these theoretical results with observations. Section 5 provides a discussion of the results, and a summary of conclusions is presented in section 6.

## 2. Models

### a. Microphysical model

This work uses the one-dimensional version of a bin-spectral model from Prat et al. (2012) that resolves explicitly the evolution of the DSD under the influence of microphysical processes throughout the rain column. The column is 3 km deep with a vertical grid spacing of 10 m. Simulations are allowed to run for 1 h, integrated at a time step of 1 s. The discretization of the stochastic collection–breakup equation is achieved using a number and mass conservative scheme (Prat and Barros 2007b). The raindrop spectrum is divided into 40 bin categories with diameters ranging from 0.1 to 7.0 mm. The discretization grid is irregular: geometric (grid parameter *s* = 2) for the lower diameter range (*d* ≤ 0.8 mm) and regular (Δ*d* = 0.2 mm) in the higher diameter range (*d* > 0.8 mm) to minimize numerical diffusion in the higher diameter range (Prat and Barros 2007b).

The rain rate is given by *N*_{i}(*z*, *t*) is the total number density of drops in the *i*th class size, *V*_{i}(*z*) is the drop fall velocity for a drop of the *i*th bin (Best 1950), and *d*_{i} is the equivolume drop diameter of drops in the *i*th bin. An exploration of the effect of the raindrop fall speed parameterization on the collisional processes will be the subject of a companion study.

The model uses a selection of coalescence–breakup kernels found in the literature. In this work, the coalescence kernel *E*_{coal} is taken from Seifert et al. (2005). It combines the LL82a expression in the large diameter drop range (*d* > 0.06 cm) and the Beard and Ochs (1995) expression in the small diameter drop range (*d* < 0.03 cm). The fragment distribution function *κ*_{ijk} is derived from the parameterization proposed by McFarquhar (2004). The collisional breakup parameterization includes the three types of breakups (filament, sheet, disc) identified by McTaggart-Cowan and List (1975). The occurrence and number of resulting fragments from a drop–drop interaction is provided by LL82b. The spontaneous (aerodynamic) breakup of large drops (*k*th class size) is also accounted for, and the expression for the probability of breakup *P*_{k} and the resulting distribution function *n*_{ik} is given by Srivastava (1971). Note that evaporation is excluded in this study but was treated separately in Kumjian and Ryzhkov (2010).

The interested reader will refer to the following references for the discretization of the stochastic collection and stochastic breakup equations (SCE/SBE), kernel formulations, and the one-dimensional Lagrangian implementation (Prat and Barros 2007a,b).

### b. Electromagnetic scattering model

Output from the microphysical model is converted into profiles of the polarimetric radar variables using electromagnetic scattering calculations. Raindrops are assumed to consist of pure liquid water at a temperature of 20°C. Note that the complex relative permittivity of liquid water is a function of both radar wavelength and temperature; we perform calculations for S-, C-, and X-band radar wavelengths (10.97, 5.4, and 3.2 cm, respectively). Though temperature can also affect the scattering results, we maintain a constant drop temperature to isolate the impact of the microphysical processes. A realistic model of drop temperature and its impact on vertical profiles of polarimetric variables is beyond the scope of the present study and will be the subject of future work.

Raindrops are treated as oblate spheroids, using the corrected diameter–shape relation of Brandes et al. (2004). We assume a mean canting angle of 0° (i.e., on average, the symmetry axis of the drops is aligned with the vertical) and a 2D Gaussian distribution of canting angles with rms width of 10°, following Ryzhkov et al. (2002, 2011). Complex scattering amplitudes for forward and backward scattering at horizontal and vertical polarizations are calculated using a T-matrix code (e.g., Waterman 1969; Mishchenko 2000). The polarimetric radar variables are computed following Ryzhkov (2001) and Ryzhkov et al. (2011), with DSDs explicitly predicted by the microphysical model. Artifacts observed in polarimetric radar data such as attenuation, differential attenuation, or depolarization effects (e.g., Kumjian 2013c) are not considered in this study.

## 3. Results

In the first simulation, a Marshall–Palmer DSD aloft is allowed to run for 1 h (the DSD aloft is steady in time). The evolution of the vertical profiles of the polarimetric radar variables in the 3-km domain is investigated. The simulated profiles of *Z*_{H}, *Z*_{DR}, *K*_{DP}, and rainfall rate are shown in Fig. 1 at six different times. The simulation considers drop settling, coalescence, collisional breakup, and aerodynamic breakup—collectively what we will call “full physics.” The initial transient differential sedimentation (size sorting) is clearly seen in the large vertical gradients (i.e., much larger than the typical variability in vertical profiles) of all variables in the first 5 min of the simulation, including a large increase in *Z*_{DR} and large decreases in *Z*_{H}, *K*_{DP}, and rainfall rate toward the bottom of the descending rain shaft. These results are in agreement with the various size sorting models in Kumjian and Ryzhkov (2012). After 1 h of simulation, and well after steady state has been achieved, the vertical profiles exhibit changes over the domain that are much smaller in magnitude. Note that the profiles achieve steady state after about 28 min of simulation time for the DSD initialized aloft. Heavier (lighter) rainfall rates achieve steady state faster (slower), as described in Prat and Barros (2009). Although the time necessary to achieve steady state highlights the transient nature of most rain events observed (Prat and Barros 2009), a steady-state (or near-steady) situation will allow for assessing the impact of the different microphysical mechanisms on the shape of the DSD and the derived polarimetric variables.

Next, we allow or disallow certain processes, allowing the evolution of the DSD to be governed by individual processes. Doing so enables us to study the impact of each individual process on the resulting DSD and vertical profiles of the radar variables. In this way, the polarimetric fingerprint of each process may be quantified. Figures 2a–d show the vertical profiles for the rain shaft undergoing only aerodynamic breakup. Decreases toward the ground in the *Z*_{H}, *Z*_{DR}, and *K*_{DP} profiles are evident. Such decreases arise from the spontaneous breakup of the largest drops (>5 mm), as shown in Fig. 2e by the decreased number concentration in the tail of the surface DSD after an hour of simulation. Collisional breakup (Figs. 2f–j) reveals a similar fingerprint of *Z*_{H}, *Z*_{DR}, and *K*_{DP} decreasing toward the ground. However, the decreases are much larger in magnitude because more of the drop spectrum is affected by collisions and subsequent breakup. There is a noticeable decrease in number concentration of drops larger than about 3 mm in diameter (Fig. 2j). Also note in Fig. 2j the large shift of mass from larger drop size bins to the smallest drop sizes. This is a result of the breakup of large drops creating numerous smaller drop fragments (LL82a; LL82b; Beard and Ochs 1995).

If instead we consider coalescence only, the opposite fingerprint to that of breakup is revealed (Figs. 2k–o): increases in *Z*_{H}, *Z*_{DR}, and *K*_{DP} toward the ground as mass from smaller drops is shifted to larger sizes. Note that the magnitudes of these increases are small because only the smallest drops (<1 mm) are significantly depleted (Fig. 2o). The full-physics simulation (Figs. 2p–t) demonstrates the fingerprint expected from the interaction of all collisional processes for this initial DSD and rainfall rate, which is a decrease in *Z*_{H} and *Z*_{DR} toward the ground. Note that *K*_{DP} first decreases aloft before starting to increase again toward the ground. Such nonmonotonic behavior arises because large drops aloft first breakup, decreasing *K*_{DP}. Then, drops with sizes from ~0.6 to 1.7 mm are depleted by coalescence, contributing to an increase in concentration of drops with sizes from ~1.7 to 3.1 mm, which increases *K*_{DP}. Note that because mass is strictly conserved in the model, RR is constant in height for the steady-state shaft.

Next, we investigate the sensitivity of the polarimetric fingerprints to the initial DSD aloft (Fig. 3). For each case, a nominal rainfall rate aloft of 20 mm h^{−1} is prescribed, but with varying DSD shape parameters. After an hour of evolution, the resulting steady-state DSD at the ground is quite similar, irrespective of the initial DSD aloft. This is in agreement with Prat and Barros (2009). The most notable differences are with the concentration of larger drops >4 mm (cf. the right side of the final DSDs in Figs. 3a and 3d). However, the resulting polarimetric fingerprint is determined by the evolution of the shape of the DSD, which can be thought of as a combination of three effects: a decrease in the concentration of large drops (which tends to shift the median diameter to smaller values), a decrease in the number of small drops (which tends to shift the median diameter to larger values), and an increase in the number of medium-sized drops, which tends to draw in the median diameter from larger or smaller sizes. For the smallest initial drop sizes (Fig. 3d), a shift in mass of small and large drops to the medium-sized drops leads to the largest increase in *Z*_{DR} and an increase in *K*_{DP} toward the ground.

Note that the variability of changes in *Z*_{DR} is larger than of changes in *K*_{DP}. Because *K*_{DP} is dependent on the concentration of drops, it is more strongly weighted by the medium-sized drops (1–3 mm), which increase by about the same amount in each case. (The very tiniest drops, which are nearly spherical, contribute very little to *K*_{DP}.) In contrast, *Z*_{DR} is independent of number concentration but is reflectivity weighted, making *Z*_{DR} especially sensitive to changes in the large-drop tail of the distribution. Thus, the overall changes in height of *Z*_{DR} depend on (i) the decreasing contribution of large drops to the total *Z*_{H}, which would tend to decrease *Z*_{DR}; (ii) the increasing contribution of medium-sized drops to the total *Z*_{H}, which could have a positive or negative effect on *Z*_{DR} depending on the initial shape of the DSD; and (iii) the decreasing contribution of small drops to the total *Z*_{H}, which would tend to increase *Z*_{DR} values. These considerations imply that for DSDs with initially large concentrations of big drops (i.e., large initial *Z*_{DR}), *Z*_{DR} should decrease when collisional processes are dominant.

To quantify this effect, we next perform simulations for a wide variety of initial DSDs, rainfall rates, and for the three weather radar frequencies (S, C, and X bands). In addition to the Marshall–Palmer DSDs with varying rainfall rates (1, 2, 5, 10, 20, 50, 100 mm h^{−1}), we make use of normalized-gamma DSDs (Testud et al. 2001) for a similar range of rainfall rates and for shape parameters of −1, 0, 2, and 5, with varying normalized intercept parameter and median volume diameter *D*_{0} values. These simulations include full physics, which herein means settling and all collisional processes. The changes in *Z*_{H} and *Z*_{DR} over the 3-km domain for given initial values of *Z*_{H} and *Z*_{DR} aloft are presented in Fig. 4.

In general, increasing values of *Z*_{H} aloft lead to decreasing changes in *Z*_{H} and *Z*_{DR} over the rain shaft, eventually leading to a reversal of the sign of the change (Figs. 4a,b). Similarly, larger initial *Z*_{DR} aloft generally means more negative changes in *Z*_{H} and *Z*_{DR} (Figs. 4c,d), as anticipated by the arguments presented above. The reversal in sign of the changes in *Z*_{H} occurs between about 1.0 and 1.5 dB in *Z*_{DR} (Fig. 4c) whereas the reversal in sign of Δ*Z*_{DR} occurs between 0.7 and 1.0 dB (Fig. 4d). Notice that the initial *Z*_{DR} aloft is better correlated to the resulting change over the rain shaft than the initial *Z*_{H} aloft. This is because *Z*_{DR} is better related to the shape and slope of the DSD than *Z*_{H}. The stratification of points especially evident in Fig. 4a is by rainfall rate. This demonstrates that rainfall rate is not a good predictor of the evolution of the DSD within the rain shaft; rather, the initial shape of the DSD observed by the initial *Z*_{DR} aloft is best (Figs. 4c,d).

There is not much of a distinction among the different radar wavelengths in terms of the initial *Z*_{H} and resulting change in *Z*_{H} (Fig. 4a), though there are slightly larger decreases in *Z*_{H} possible at C and X bands compared to S band. The C-band simulations reveal the largest range of initial *Z*_{DR} and also largest range of changes possible in *Z*_{DR} (Figs. 4b–d), owing to the strong sensitivity of *Z*_{DR} to big drops (5–6 mm in diameter) at C band that is a result of resonance scattering effects. Also note the divergence of the scatter of Δ*Z*_{H} for initial *Z*_{DR} values larger than about 2 dB (Fig. 4c); in particular, the X-band points reveal a much larger decrease in *Z*_{H} compared to C band for a given initial *Z*_{DR}. This is because larger raindrops (>3 mm) are non-Rayleigh scatterers at X band and contribute more to the observed *Z*_{H} than the same raindrops do at S band. Thus, their breakup leads to larger overall reductions in *Z*_{H} at X band than at S band. Despite the changes in the radar variables, rainfall rate does not change over the domain in the steady-state simulations. This is because of two factors: (i) mass is explicitly conserved because of the mass and number conservative discretization scheme used in the microphysical model (Prat and Barros 2007a,b), and (ii) there are compensating changes in the flux of mass as mass is redistributed from the tails of the distribution to the middle (i.e., increases in RR owing to small drops being collected by larger drops is offset by decreases in RR as larger drops break up into smaller drops).

Figure 5 displays the same simulations in terms of their polarimetric fingerprints (changes over the 3-km domain). From Fig. 5a, it is clear that for the range of DSDs and rainfall rates tested, the sign of the change in *Z*_{H} and *Z*_{DR} over the 3-km rain shaft can be either positive or negative (i.e., increases or decreases are possible). Similarly, the sign of *K*_{DP} can be positive or negative (Fig. 5b). When both *Z*_{H} and *Z*_{DR} (*K*_{DP}) decrease toward the ground, the signal from drop breakup is dominant. Likewise, increases in both *Z*_{H} and *Z*_{DR} (*K*_{DP}) toward the ground signify a dominance of the coalescence fingerprint. Because of the aforementioned difference in changeover points for where Δ*Z*_{H} and Δ*Z*_{DR} reverse signs, there is a limited range of conditions at which Δ*Z*_{H} may be positive and Δ*Z*_{DR} (or Δ*K*_{DP}) negative. These could indicate the fingerprint of a “balance” between coalescence and breakup. Intriguingly, the signal for evaporation [an increase in *Z*_{DR} and decrease in *Z*_{H} and *K*_{DP}, documented previously in Kumjian and Ryzhkov (2010)] does not overlap with the signal from coalescence–breakup simulations (Fig. 5a). Similarly, there is no overlap in the collisional processes and evaporation in the Δ*Z*_{DR}–Δ*K*_{DP} phase space (Fig. 5b). Note also that the signal from size sorting is similar to that of evaporation, but larger in magnitude (Kumjian and Ryzhkov 2012), and thus also distinguishable from the fingerprints attributed to the collisional processes.

Comparing the different wavelengths, we see the largest differences for the decreases in *Z*_{H} and *Z*_{DR} (Fig. 5a, bottom left). Though decreases in *Z*_{H} are larger at X band and C band relative to S band, the C-band simulations reveal the largest changes in *Z*_{DR}. This is likely owing to the resonance scattering effect at C band for large drops, which makes this frequency more sensitive to changes in the large-drop end of the DSD. In contrast, there is little distinction between the *Z*_{H}*–Z*_{DR} fingerprints at different wavelengths when both *Z*_{H} and *Z*_{DR} increase (i.e., coalescence-dominated cases). The Δ*Z*_{DR}–Δ*K*_{DP} space (Fig. 5b) reveals larger scatter, but demonstrates that the largest *K*_{DP} increases are expected at X band and the largest *K*_{DP} decreases expected at C band.

Thus, though the polarimetric fingerprints of the collisional processes at a given wavelength may be different from those at other wavelengths, the breakup and coalescence processes have a polarimetric fingerprint distinct from that of evaporation and size sorting. This may allow for accurate diagnoses of the physical process dominating the signal in radar observations, particularly those made with radars operating at different frequencies.

## 4. Comparison to observations

In this section, the results of our idealized simulations are compared to polarimetric radar and 2D video disdrometer observations.

### a. Polarimetric radar observations

Owing to a relative lack of evaporation, warm, humid tropical environments provide a desirable natural laboratory for studying warm-rain collisional processes. We use S-band data from the National Center for Atmospheric Research (NCAR) dual-wavelength polarimetric SPolKa (herein “SPol”) radar that was collected during the Dynamics of the Madden–Julian Oscillation (DYNAMO)/Atmospheric Radiation Measurement Program Madden–Julien Oscillation (MJO) Investigation Experiment (ARM-AMIE), which took place in the central Indian Ocean in late 2011 into early 2012 [for details, see Yoneyama et al. (2013)]. These data have gone through an extensive quality control process at NCAR, including calibration and correction for attenuation and differential attenuation.

A number of stratiform and convective rain cases that were observed during DYNAMO/ARM-AMIE are used in this study (14, 16, 28, and 31 October; 15, 16, and 17 November; 10 and 22 December 2011). Here, a “stratiform” case is loosely defined as one containing a well-defined, horizontally homogeneous melting layer bright band, whereas “convective” cases are those with a cellular, isolated appearance and a lack of a well-defined melting-layer bright band. As will be shown, the partitioning of stratiform and convective does not matter for our purposes. The idea is to explore a range of rainfall rates and initial DSDs for comparison with the simulations rather than to simulate specific events. The initial *Z*_{H} aloft in these cases varies from <20 dB*Z* in the weakest stratiform case to >50 dB*Z* in the stronger convective cases.

The median *Z*_{H} and *Z*_{DR} vertical profiles in the 0.5–3.5-km-AGL layer were taken over 1-km-range windows, with *ρ*_{hυ} ≥ 0.98 used as a threshold to ensure the precipitation was pure rain and that the data did not suffer from any contamination from ground clutter, the melting layer bright band, or nonuniform beamfilling. Changes in *Z*_{H} and *Z*_{DR} over the 3-km layer are plotted in Fig. 6 along with the simulations presented earlier (these include the wide range of initial DSD shapes and rainfall rates—i.e., those shown in Figs. 4 and 5). There is good agreement between the observations and the simulations over a range of initial *Z*_{DR} values (Fig. 6a) in that DSDs characterized by smaller drops initially (i.e., smaller initial *Z*_{DR}) tend to lead to increases in *Z*_{DR} over the rain shaft.

We note two discrepancies between the microphysical fingerprints in the simulations and the observations (Fig. 6b). First, in the breakup-dominant quadrant (negative Δ*Z*_{H} and negative Δ*Z*_{DR}, or bottom left), the observed decreases in *Z*_{DR} generally are smaller for a given decrease in *Z*_{H} than predicted by the model. This may signify that the simulations are too aggressive in their treatment of drop breakup. In the coalescence-dominant quadrant (positive Δ*Z*_{H} and positive Δ*Z*_{DR}, top right), we see substantially larger increases in *Z*_{H} and *Z*_{DR} than predicted by the model. It is not clear how much of this is attributable to more efficient coalescence in the observations than in the simulations, additional drop growth by collection of cloud water droplets, or the possible role of vertical air motions (note that many of these points are from the convective cases). In particular, some of the differences observed could be explained by the relative simplification of the column model configuration. Accurate measurements of the air motion could be incorporated dynamically into the model to account for updrafts and/or downdrafts (e.g., Prat et al. 2008). Also note that overaggressive breakup in the simulations may place a limit on the amount of coalescence possible, thus limiting the maximum increases in *Z*_{H} and *Z*_{DR}.

The SPol observations reveal that both the fingerprint for coalescence and the fingerprint for breakup are possible in either convective or stratiform rainfall; the physics of drop collisions does not care whether the system is characterized as convective or stratiform. Rather, the initial shape of the DSD (observed mainly through the initial *Z*_{DR} aloft) is the primary indicator of the subsequent behavior of the rain shaft. The rainfall rate (or initial *Z*_{H} aloft) is of secondary importance, similar in stratification to the simulations. In the tropical DYNAMO/ARM-AMIE dataset, much of the convective rain data reveal the coalescence fingerprint. This primarily is because of the initially small drops aloft. More continental convection, in which large drops are formed aloft via melting of hail and graupel, are expected to produce more breakup signatures. Likewise, a variety of stratiform cases reveal fingerprints for breakup, coalescence, and (sometimes) evaporation.

The next comparison uses data from the C-band Scanning ARM Precipitation Radar (CSAPR) and features the stratiform region of a mesoscale convective system observed on 20 May 2011 during the Midlatitude Continental Convective Clouds Experiment (MC3E). These data have been corrected for attenuation and differential attenuation. Also, the case was selected because of a lack of significant differential propagation phase Φ_{DP} in the rain region, implying that differential attenuation should be minimized. Again, median vertical profiles are constructed from the RHI scans, though here 5-km windows are used to smooth out noisy fluctuations in the data. The “pure rain” segments were selected based on the vertical profile of *ρ*_{hυ} ≥ 0.98, as before. As such, the vertical depth of the profiles varies as a function of range from the radar (and all are less than 3 km in depth). To make comparisons with the 3-km rain shaft simulations, a least squares linear^{1} fit was applied to the average vertical profiles in each 5-km window. Using the slope of *Z*_{H} and *Z*_{DR}, the change in these variables over a 3-km distance is computed. Note that a linear slope assumption may introduce an overestimation of the magnitudes of the changes over the profiles. However, when the same technique was applied to the SPol data as a sanity check, the changes based on the slope of the linear fit were in good accord with the observed actual 3-km changes.

The resulting comparison between the radar data and simulated C-band results are shown in Fig. 7. Despite having some overlap, there are numerous points in the radar data showing larger increases in *Z*_{DR} (Fig. 7a) than the simulations. Note that many (but not all) of these points are associated with the fingerprint for evaporation or size sorting (Fig. 7b). Evaporation is expected to play a larger role in this continental case compared to the tropical SPol cases, so the results are not surprising. Other points reveal larger increases in *Z*_{H} and *Z*_{DR} than predicted by model simulations (Fig. 7b), which could indicate more efficient coalescence or less efficient breakup in real rainfall situations.

Both SPol and CSAPR data from two very different climatological regions reveal similar comparisons with the simulations, including suggestions that the simulations are too aggressive with drop breakup and/or less efficient at coalescence than what is observed. These data also demonstrate that it is an oversimplification to assert that convective or stratiform rainfall always exhibits one fingerprint or the other. Similarly, such statements based on rainfall rate or initial *Z*_{H} alone also are unwarranted. For example, consider a rainfall rate of 50 mm h^{−1}, which would by nearly any standard be considered convective in nature. If the initial DSD is skewed toward smaller drops (i.e., more “tropical” rain), the expected fingerprint based on simulations and observations is an increase in *Z*_{DR} toward the ground, indicating a dominance of drop coalescence. On the other hand, a more continental convective storm with a rainfall rate of 50 mm h^{−1} and larger initial *Z*_{DR} is expected to produce the breakup fingerprint. Thus, stratiform and convective partitioning does not matter for determining the expected behavior of microphysical processes in the rain shaft; what matters most is the initial shape of the DSD aloft and whether it is skewed toward smaller or larger drops, which is measurable using *Z*_{DR}. This has implications for rainfall estimation algorithms such as vertical profiles of reflectivity (VPR) techniques that treat the rain physics of convective and stratiform precipitation differently.

Prat and Barros (2009) have shown that governance of the evolution of *Z*_{H} in the rain shaft and the resulting DSD at the ground transitions from a DSD-dependent regime to a rainfall-rate-dependent regime for large nominal rainfall rates prescribed aloft. Specifically, for rainfall rates in excess of 30 mm h^{−1}, DSDs would converge to the same *Z*_{H}–rainfall rate space. Whereas *Z*_{H} and *K*_{DP} were found to behave similarly in our study, *Z*_{DR} displays contrasting behavior, indicating that the initial DSD remains an important factor on its subsequent evolution. Thus, combined use of *Z*_{H} and *Z*_{DR} should better characterize the microphysical structure and evolution of the rain shaft.

More detailed analyses over a wider range of precipitation intensities, environmental conditions, and initial values of *Z*_{DR} aloft should be conducted to investigate the possible contributions of (i) measurement errors and/or problems with the representativeness of the linearly extrapolated radar profiles in the CSAPR case, (ii) limitations of the coalescence–breakup parameterizations, (iii) simplifications induced by the idealized column model to the simulation–observation discrepancies, and/or (iv) limitations due to the transient nature of most rain events observed.

### b. 2D video disdrometer observations

For Marshall–Palmer DSDs with different rainfall rates, we compute the *Z*_{H} and *Z*_{DR} at every height level and at every time and plot them in *Z*_{H}–*Z*_{DR} space (Fig. 8). In Figs. 8a and 8b, the full-physics simulations are shown in the black markers, whereas we isolate collisional breakup only (Fig. 8a) and coalescence only (Fig. 8b) as shown in the colored markers. The initial transient effects of differential sedimentation are evident by the high-*Z*_{DR}, low-*Z*_{H} points. As the DSD evolves, the *Z*_{H}–*Z*_{DR} points progress to the right and down in Fig. 8, until the steady-state (equilibrium) DSD is achieved. We can compare the resulting *Z*_{H}–*Z*_{DR} pairs to published observations of many thousands of DSDs measured with a 2D video disdrometer. The best-fit curves for the relation between *Z*_{H} and *Z*_{DR} in rain are shown for two locations: Oklahoma (Cao et al. 2008, solid blue line) and Florida (Zhang et al. 2006, dashed blue line). As expected, the more tropical environment of Florida results in lower *Z*_{DR} for a given *Z*_{H} than the more continental Oklahoma environment, particularly for heavier rain. Long-term observations of this sort exhibit quite a bit of scatter, which is represented by the dotted blue curves depicting the envelope of observations from Florida in Brandes et al. (2004).

Of note is that simulations for rainfall rates higher than about 10 mm h^{−1} have resulting *Z*_{DR} values below both the Florida and Oklahoma relations for a given *Z*_{H}. This demonstrates that the simulations produce DSDs characterized by smaller drops than in the disdrometer-based observations (i.e., a lower *Z*_{DR} for a given *Z*_{H}). Note that for the heaviest rainfall rate of 100 mm h^{−1}, the final *Z*_{H}–*Z*_{DR} points lie outside the envelope of observations from Brandes et al. (2004). Because the breakup of drops is dominant at heavier rainfall rates in the model (Prat and Barros 2009), the results suggest that the breakup of drops may be too aggressive in the simulations. Evaporation becomes decreasingly important with higher rainfall rates (Hu and Srivastava 1995), so it is unlikely that evaporation (which would increase *Z*_{DR}) is playing a significant role in the discrepancy.

The dominance of breakup is especially evident when we compare the full-physics runs to the coalescence-only simulations (Fig. 8b), as there is a large separation between the final *Z*_{H}–*Z*_{DR} points for heavier rainfall rates. In contrast to the full-physics and breakup-only simulations, the coalescence-only runs (Fig. 8b) produce final *Z*_{H}–*Z*_{DR} values in much better agreement with the observed *Z*_{H}–*Z*_{DR} relations.

## 5. Discussion

While the combination of a microphysical and electromagnetic model allowed us to determine unambiguously the impact of selected microphysical processes on the dual-polarization variables, additional insight has been provided on the physical parameterization of these processes. The observations presented herein suggesting that the breakup is overaggressive are consistent with some previous laboratory and numerical work. For example, Hu and Srivastava (1995) found that the modeled equilibrium DSD slope was far steeper than observed equilibrium DSDs, and thus suggested that the Low and List (LL; LL82a; LL82b) breakup parameterization might greatly overestimate the number of fragments produced during breakup. This is echoed in the more recent laboratory work of Barros et al. (2008), who report systematic overestimations of the number of drop fragments produced in breakup events, particularly for the fragment diameter range 0.5–1.0 mm. Such an enhancement of the small-drop end of the DSD steepens the DSD slope, contributing to a decrease in *Z*_{DR}. Using direct numerical simulations of drop collisions, Straub et al. (2010) reported that, under stationary conditions, their equilibrium DSD featured many more large drops than the LL parameterization. Additionally, they found more drops larger than 2.8 mm in diameter than the McFarquhar (2004) modification to the LL parameterization. In this case, the Straub et al. (2010) parameterization would be expected to produce larger *Z*_{DR} for their equilibrium DSD compared to the LL and McFarquhar (2004) parameterizations.

An overestimation of the number concentration of small drops has ramifications for modeling of convective storms. Small drops evaporate more rapidly than larger drops owing to their larger surface-area-to-volume ratio (e.g., Pruppacher and Klett 1997). So, small-drop-dominated DSDs have a large evaporative potential. Overestimating the number of small drops (and/or slope of the DSD) in numerical models would then exaggerate negative buoyancy production. Thus, improving the parameterization of drop breakup has important implications for modeling of cold pool generation and subsequent storm behavior and evolution (e.g., Morrison et al. 2012).

The results of this study may also be useful for quantitative precipitation estimation (QPE). Typically, QPE techniques based on VPR assume that *Z*_{H} increases toward the ground in heavy convective rain. The recent study of Cao et al. (2013) found that this was the case for rainfall rates in excess of 18 mm h^{−1}; for lower rainfall rates and stratiform precipitation, they found that the *Z*_{H} profile decreases toward the ground. The argument is that coalescence and/or accretion growth is dominant in convective precipitation, whereas evaporation and/or breakup dominate in stratiform precipitation. This approach, based on observations and a conceptualized microphysics model, disagrees with rain shaft model simulations of the collisional processes, wherein equilibrium DSDs resulting from heavier rainfall rates are associated with the dominance of breakup and thus decreasing *Z*_{H} toward the ground (Prat and Barros 2009). Additionally, the present study has shown that *Z*_{H} can decrease or increase toward the ground when collisional processes are acting, depending on the DSD shape and rainfall rate. Thus, it is important to consider the vertical profiles of *Z*_{H} and *Z*_{DR} to more effectively diagnose the dominant physical processes governing the evolution of the vertical structure of precipitation. In particular, the initial *Z*_{DR} aloft seems to be the best predictor of the behavior of the vertical profiles of *Z*_{H} and *Z*_{DR}. By considering the profiles of both *Z*_{H} and *Z*_{DR}, cases where evaporation may or may not be occurring can be identified and quantitative precipitation estimates adjusted accordingly. For example, the microphysics model results suggest that the collisional processes themselves do not significantly alter the rainfall rate, whereas evaporation (which depletes rainwater mass into water vapor) does [e.g., see Fig. 11 in Kumjian and Ryzhkov (2010)]. So, using polarimetry to distinguish between cases of evaporation or the collisional processes, both of which can cause decreases in *Z*_{H}, can be quite useful for QPE and hydrometeorological applications.

## 6. Summary

We have explored the impact of the warm-rain collisional processes on the resulting vertical profiles of the polarimetric radar variables. This was accomplished by employing an explicit microphysical model describing the warm-rain processes of drop settling, coalescence, collisional breakup, and aerodynamic breakup in a 1D rain shaft. This microphysics model was coupled to an electromagnetic scattering model for raindrops. The coupled electromagnetic–microphysical model was used to determine the time-varying vertical profiles of *Z*_{H}, *Z*_{DR}, and *K*_{DP} at S, C, and X bands. The polarimetric “fingerprint” of each process (acting in isolation and in combination) was quantified.

The results demonstrate that the initial *Z*_{DR} aloft is a better predictor than *Z*_{H} or rainfall rate for the subsequent changes in *Z*_{H} and *Z*_{DR} over the 3-km domain. For initial *Z*_{DR} values less than about 1.0 dB aloft, the *Z*_{H} and *Z*_{DR} increase toward the ground, indicating the dominance of coalescence. For larger *Z*_{DR} values aloft, the simulations predict decreases in *Z*_{H} and *Z*_{DR} toward the ground as the breakup processes dominate. Changes in *Z*_{DR} are largest in magnitude at C band, suggesting that C-band radars may be best able to observe these fingerprints in nature.

The fingerprints for the collisional processes depend on the rainfall rate and initial DSD aloft and even display some variability between different radar wavelengths. However, they do not overlap with the signatures for evaporation and size sorting (i.e., increases in *Z*_{DR} and decreases in *Z*_{H} and *K*_{DP} toward the ground), suggesting that a distinction between the dominant ongoing processes may be made using polarimetric radar data.

The simulations were compared to disdrometer and polarimetric radar observations. These comparisons suggest that the breakup kernels employed (LL82b; McFarquhar 2004) may be too aggressive, and the coalescence parameterizations used (LL82a; Beard and Ochs 1995) may be too weak, especially for the higher rainfall rates, producing erroneously large concentrations of small raindrops. This is manifested as very low *Z*_{DR} for a given *Z*_{H} value, or rain that has very “tropical” characteristics. Indeed, simulated *Z*_{DR} values for the highest rainfall rates considered lie outside the bounds of published disdrometer observations from Florida. In contrast, the “coalescence only” simulations produce resulting *Z*_{H}–*Z*_{DR} pairs that are in better agreement with the disdrometer data, though data from SPol in a tropical environment indicate that coalescence fingerprints can be substantially larger than predicted by the model. Therefore, because of the unequivocal dependency between selected microphysical processes and the polarimetric radar variables, real-time rainfall field observations could be of great help to improve microphysical parameterizations of drop–drop interactions via inverse problem modeling techniques. Dual-polarization radar observations for different storm types could be used as “real laboratory” experiments to develop and improve parameterizations. Future work will investigate newer breakup kernels that lessen the efficiency of breakup and the number of fragments produced from breakup events as well as the observed variability of microphysical fingerprints in a larger number of different environmental conditions, rainfall intensities, and initial DSD shapes.

SPolKa radar data were collected as part of DYNAMO/ARM-AMIE, which was sponsored by NSF, NOAA, ONR, DOE, NASA, and JAMSTEC. The involvement of the NSF-sponsored NCAR Earth Observing Laboratory (EOL) is acknowledged. Scott Ellis and Bob Rilling (NCAR-EOL) are thanked for providing access to the SPolKa data, which are archived at the DYNAMO Data Archive Center maintained by NCAR EOL. The authors also would like to thank Dr. Scott Collis (Argonne National Laboratory) for providing the CSAPR data. Support for this work for the first author comes from Grant ER#65459 from the Department of Energy Atmospheric System Research program and from the National Center for Atmospheric Research (NCAR) Advanced Study Program. The second author is supported by the NOAA/NCDC Climate Data Records and Science Stewardship Program through the Cooperative Institute for Climate and Satellites–North Carolina under the Agreement NA09NES4400006. We would also like to thank Dr. Hugh Morrison (NCAR MMM), Dr. Angela Rowe (University of Washington), and two additional anonymous reviewers who provided a constructive review of the manuscript.

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^{1}

Other options, such as a parabolic or polynomial fit, are possible as well. However, the linear fit was deemed sufficiently representative for the vast majority of the profiles used.