Abstract

The objective of this study is to evaluate the impact of a new parameterization of drop–drop collision outcomes based on the relationship between Weber number and drop diameter ratios on the dynamical simulation of raindrop size distributions. Results of the simulations with the new parameterization are compared with those of the classical parameterizations. Comparison with previous results indicates on average an increase of 70% in the drop number concentration and a 15% decrease in rain intensity for the equilibrium drop size distribution (DSD). Furthermore, the drop bounce process is parameterized as a function of drop size based on laboratory experiments for the first time in a microphysical model. Numerical results indicate that drop bounce has a strong influence on the equilibrium DSD, in particular for very small drops (<0.5 mm), leading to an increase of up to 150% in the small drop number concentration (left-hand side of the DSD) when compared to previous modeling results without accounting for bounce effects.

1. Introduction

The transient evolution of the drop size distribution (DSD) under the influence of microphysical processes such as coalescence, breakup, evaporation, condensation, raindrop clustering, and mixing effects has been the object of extensive research over the past decades (e.g., Testik and Barros 2007). While drop–drop collisions leading to coalescence or drop breakup are considered to be the main driving mechanism behind the transient evolution of the DSD, there are yet a significant number of unanswered questions. One of the main challenges is the relatively limited number of experimental observations of drop collisions. The pioneering experimental work of McTaggart-Cowan and List (1975), later extended by Low and List (1982a,b; these are parts I and II of an article and are hereafter jointly referred to as LL82), revealed the complex nature of collision-induced breakup of raindrops. Despite a limited number (10) of colliding drop pairs, LL82 proposed a parameterization for drop coalescence and collisional breakup including a formulation of the different types of breakup (filament, sheet, and disk). Later, the parameterization was slightly modified to tackle mass conservation problems (Valdez and Young 1985; Hu and Srivastava 1995; Brown 1997). More recently, McFarquhar (2004, hereafter MF04) proposed a new parameterization derived from the original experimental data from LL82. The use of a modified Monte Carlo technique with bootstrap to randomly select the results of individual collision of arbitrary drops pairs allowed generalizing experimental results of collisional breakup and enforcing mass conservation for each collisional/breakup event (MF04). The incorporation of the original LL82 parameterization and subsequent adjustments into quasi-stochastic numerical models of DSD dynamics showed high sensitivity of equilibrium size distribution to the parameterizations of coalescence and breakup process. In particular the existence of two (MF04; Prat and Barros 2007a,b, hereafter PB07a,b; Straub et al. 2010) or three peaks (Valdez and Young 1985; List et al. 1987; Feingold et al. 1988; List and McFarquhar 1990; Hu and Srivastava 1995; Brown 1997) in the equilibrium DSD remains an open question, although more physically based and general parameterizations tend to yield double-peaked equilibrium DSDs (MF04; PB07a,b; Straub et al. 2010). Beyond the inherent simplifications in using a zero-dimensional (0D), transient in time model to assess parameterization performance, additional difficulties come from the transient nature of rainfall and from the fact that equilibrium DSDs are rarely observed in nature. To overcome the limitations associated with a small number of observations, Beheng and coworkers used direct numerical simulation (DNS) based on the volume of fluid method to predict the resulting fragment size distribution (Beheng et al. 2006) of collisions among 32 drop pairs with diameters ranging from 0.035 to 0.46 cm (Schlottke et al. 2010). A new parameterization was developed from those numerical experiments (Straub et al. 2010), which was found to be in close agreement with other formulations derived from laboratory experiments (LL82; MF04).

In an attempt to further generalize the result of colliding raindrops, Testik (2009, hereafter T09) proposed a delineation of the physical conditions for the occurrence of drop–drop interaction outcomes (bounce, coalescence, and breakup) in the form of a regime diagram in the We–p plane (where We is the Weber number and p is the raindrop diameter ratio). Using the laboratory experiments described by Barros et al. (2008), Testik et al. (2011, hereafter TBB11) evaluated the fitness of this outcome regime diagram to describe drop–drop collisions in realistic rainfall, and further identified the physical conditions for the occurrence of the different collision subregimes—that is, breakup types: filament (also referred to as “neck” in TBB11), sheet, and disk—by including angle of impact considerations. These physical conditions are expressed in terms of relationships between Weber number and the diameter ratio of the two interacting raindrops and provide a new formulation for the drop collision outcome parameterization used in dynamical models of raindrop size distribution. Moreover, the physics-based parameterization delineated a region in the We–p space where bounce occurs that is confirmed by laboratory observations. Bounce was not predicted by the DNS experiments (Schlottke et al. 2010), most probably because only 5 drop pairs out of the 32 selected were located at the boundary of bounce/coalescence/filament-breakup regimes. Using the Straub et al. (2010) parameterization, Jacobson (2011) included bounceoff in a microphysical model by imposing a threshold condition that only small drops (d < 0.005 cm) bounced when colliding with other drops of any size.

Here, the new parameterization (hereafter referred to as We–p kernel; TBB11) was implemented in an existing microphysical numerical model with explicit representation of coalescence and breakup mechanisms (PB07a,b). The model was modified to include the drop bounce process as a function of the drop size. The evaluation of the We–p kernel is focused on the implications of the new parameterization on the shape of the equilibrium DSD and related integral properties such as drop number concentration M0, rain rate R, and radar reflectivity factor Z. The paper is organized as follows. First, a brief description of the zero-dimensional bin-spectral model (PB07a,b) is provided along with the presentation of the physics-based parameterization of drop–drop interaction (TBB11). Next, the new parameterization based on regime delineation outcomes is evaluated against previous formulations and the impact of bounceoff is quantified. An assessment of model sensitivity to regime outcome delineation uncertainties is also performed. Finally, the main findings are summarized.

2. Model description and adjusted parameterization of coalescence/breakup kernels

a. The homogeneous in space/transient in time model (0D bin-spectral box model)

The model used here, a zero-dimensional bin-spectral model that is mass and number conservative, was extensively described elsewhere (PB07a,b). The stochastic collection–breakup equation is solved using an explicit scheme with a time step Δt = 1 s. The numerical scheme is found stable for time steps (Δt > 60 s) compatible with the duration of rainfall events encountered in nature. The diameter discretization consists of 100 class sizes and spans a drop size range from 0.01 to 0.79 cm. The grid is irregular: geometric (geometric grid parameter s = 1.25) in the lower diameter range (d ≤ 0.117 cm) and regular with grid size Δd = 0.01 cm in the higher diameter range (d > 0.117 cm). The discretization grid was optimized to provide a good description of the DSD at small drop sizes and to avoid numerical diffusion in the higher diameter range (PB07a,b). Note that the discretization used here is more refined than previous applications of the model (total of 40 bins for an irregular grid with s = 2 and Δd = 0.02 cm) (PB07b; Prat et al. 2008; Barros et al. 2008; Prat and Barros 2009, 2010a,b) in order to eliminate possible numerical artifacts of coarse bin resolution on the comparison of results using the existing parameterization of drop collision outcomes (LL82) and the We–p kernel. In the  appendix, a simplified expression of the temporal evolution for the drop number concentration under the influence of coalescence and breakup is provided. This expression accounts for theoretical and observed (TBB11) outcomes of bouncing effects of large drops. For detailed derivation of each term of the expression and definition of the kernels, the interested reader is referred to PB07a.

b. Improved formulation of the dynamic kernels

Experimental observations of the interaction of free-falling raindrops are fundamental for accurate microphysical parameterizations of drop–drop interactions used in dynamical modeling of raindrop size distribution. Extending the original laboratory work on drop–drop collision performed by McTaggart-Cowan and List (1975), LL82 developed parameterizations for drop coalescence (Low and List 1982a) and collisional breakup (Low and List 1982b) between two drops to describe drop size evolution under steady-state conditions. Recently, TBB11 evaluated and extended the theoretical model for binary collision outcomes (T09) using data from recent raindrop collision experiments (Barros et al. 2008). The collision diagram proposed by TBB11 combines laboratory observations and theoretical modeling of binary collision outcomes, and thus provides the basis for a more general physics-based parameterization of drop–drop interaction investigated here. In this section, the available parameterizations for drop–drop interactions based on LL82 and TBB11 are intercompared using a modified version of the bin-spectral model described in the previous section.

Figure 1 displays the theoretical domain (gray area) in the (d1, d2) space (where d1 is the diameter of the larger of the colliding drops and d2 is the diameter of the smaller) that corresponds to each regime (bounce, coalescence, and breakup) and breakup subregimes (filament, sheet, and disk/crown breakup), according to the LL82 parameterization (Fig. 1a) and the We–p parameterization (TBB11; Fig. 1b) based on drop collision observations at the National Aeronautics and Space Administration (NASA) Wallops Island Facility (NWIF) rain laboratory (Barros et al. 2008). Substantial differences are observed for the delineation of the occurrence of each type of regime (coalescence and breakup) and types of breakup (filament and disk breakup). The most noticeable difference between the original formulation (Low and List 1982b) and the We–p regime diagram is the prediction of drop bounce occurrences that is not available in the original LL82 parameterization. The We–p parameterization implies that all drop sizes considered (0.01 < d < 0.79 cm) could experience bounce that was observed for six drop pairs ranging from (d1 = 0.131 cm, d2 = 0.079 cm) to (d1 = 0.35 cm, d2 = 0.23 cm). Other important differences are observed for coalescence and breakup (and regardless of the breakup type) that cover a larger portion of the (d1, d2) domain in the original LL82 parameterization (Fig. 1a). The original LL82 parameterization indicates the possibility of occurrence for filament breakup for every drop pair in the (d1, d2) domain, whereas based on theoretical considerations breakup is not predicted when bounce is present. Interestingly, an almost perfect agreement is observed for sheet breakup between LL82 and We–p (TBB11), and a generally good agreement is observed for disk breakup, including the separation of the crown breakup subtype (TBB11).

Fig. 1.

Regime domains (gray area) in (d1, d2) space corresponding to each regime (bounce, coalescence, and breakup) and breakup subregimes (filament, sheet, and disk breakup) derived from (a) the LL82 parameterization and (b) the TBB11 We–p parameterization against drop–drop collision observations at NWIF (Barros et al. 2008). For clarity, drop pairs observed for each regime at NWIF are reported (black squares) in the lower portion of the (d1, d2) space only.

Fig. 1.

Regime domains (gray area) in (d1, d2) space corresponding to each regime (bounce, coalescence, and breakup) and breakup subregimes (filament, sheet, and disk breakup) derived from (a) the LL82 parameterization and (b) the TBB11 We–p parameterization against drop–drop collision observations at NWIF (Barros et al. 2008). For clarity, drop pairs observed for each regime at NWIF are reported (black squares) in the lower portion of the (d1, d2) space only.

Differences between predicted and observed domains (Fig. 1b) are mostly due to inherent uncertainty between the idealized theoretical configuration of the box model [uncertainties in drop fall velocity (Best 1950), kernels for coalescence and breakup, etc.] and the experimental setup. TBB11 reported the concurrence of bounce, coalescence, and different breakup types in the regions of transition from one collision outcome to another in the We–p plane [We = ρ(d2/2)(ΔV)2/σ, p = d2/d1]. A closer look at those realizations (see our Fig. 1 and Fig. 12 in TBB11) suggests there is some handicap in the We–p regime diagram at the interfaces of the different domain delineations—that is, near lines DE1 = (p2We)/[6(1 + p3)] = 1 and DE2 = We/[6(1 + p3)] = 1. For instance, sheet and crown (i.e., disk) breakups are observed in a domain (DE1 < 1 and DE2 > 1 in Fig. 12 of TBB11) where only the presence of filament breakup is predicted. Similarly, bounce is observed in (DE1 < 1 and DE2 > 1) whereas it is only predicted in (DE2 < 1) (TBB11). However, those differences are minimal from a modeling perspective because treating drop–drop interactions in terms of precisely observed occurrences does not capture the inherently stochastic nature of drop–drop interactions. In a “winner takes all” situation, this translates into the delineation of a large region of the (d1, d2) space with a unique observed occurrence for bounce in the region (DE2 > 1) for the drop pair (d1 = 0.22 cm, d2 = 0.08 cm) (Fig. 1b).

Furthermore, despite the fact that original (LL82) and predicted (TBB11) domains are substantially different for the occurrence of different outcome regimes, the drop pairs observed for each regime at NASA Wallops Island Facility rain laboratory (indicated by black squares in Fig. 1) do not allow us to assert whether one parameterization is better than the other (Barros et al. 2008). Indeed, note that the major differences observed between the two formulations (those of LL82 and TBB11) occur in the portion of the (d1, d2) domain where laboratory observations are unavailable. This suggests that those less populated regions of the (d1, d2) domain correspond to physical process of low probability of occurrence. Nevertheless, and regardless of the kernel formulation, coalescence, breakup and breakup subcategories (filament, sheet, disk, and crown breakups), and bounce to a lesser extent, all occur within the expected We–p domain boundaries. To quantify the impacts of the different parameterizations (LL82 and TBB11) on the equilibrium DSD, we use a modified version of the microphysical model (PB07a,b). The uncertainties between theoretical and experimental (TBB11) delineations of the outcome regimes are quantified by performing a sensitivity analysis on the location of domain separation lines DE1 and DE2.

3. Model results and evaluation

To compare the results of the new We–p physical parameterization against LL82, several microphysical configurations were tested via numerical simulations that accounted for different combinations of bounce–coalescence–breakup mechanisms described above (Fig. 1). Table 1 summarizes all the microphysical configurations tested and the collision outcome kernels used for each case. Configuration 1 (Table 1) refers to the original microphysical parameterization (henceforth referred to as the original configuration) based on LL82 and MF04 used in previous studies either with the 0D or 1D model (PB07a,b; Prat et al. 2008; Prat and Barros 2009). An exponential (Marshall–Palmer) drop size distribution (Marshall and Palmer 1948, hereafter MP48) with nominal rain rates ranging from low/moderate (1 mm h−1) to high rain rates (50 mm h−1) was used to specify initial DSD conditions.

Table 1.

Summary of parameterizations tested in this paper. For each configuration, the origin of the coalescence/breakup kernels (collision K, coalescence efficiency Ecoal, ratio breakup Rfila-sheet-disk, fragment number Ffila-sheet-disk, fragment distribution function Pfila-sheet-disk) used as well as microphysical mechanisms involved (coalescence, breakup, and bounce) are listed.

Summary of parameterizations tested in this paper. For each configuration, the origin of the coalescence/breakup kernels (collision K, coalescence efficiency Ecoal, ratio breakup Rfila-sheet-disk, fragment number Ffila-sheet-disk, fragment distribution function Pfila-sheet-disk) used as well as microphysical mechanisms involved (coalescence, breakup, and bounce) are listed.
Summary of parameterizations tested in this paper. For each configuration, the origin of the coalescence/breakup kernels (collision K, coalescence efficiency Ecoal, ratio breakup Rfila-sheet-disk, fragment number Ffila-sheet-disk, fragment distribution function Pfila-sheet-disk) used as well as microphysical mechanisms involved (coalescence, breakup, and bounce) are listed.

a. Equilibrium DSD and modifications when compared to the original configuration

Figure 2 displays the coalescence efficiency Ecoal, breakup efficiency Ebrkp, and ratio for each breakup type (filament Rfila, sheet Rsheet, and disk Rdisk) obtained with the classical LL82 parameterization (case 1: PB07a,b) and using the We–p parameterization without (case 2) and with (case 3) the inclusion of bounce (Table 1). The key differences between the classical parameterization (case 1; PB07a,b) and Ecoal and Ebrkp according to the We–p kernel (cases 2 and 3) are along the d1 = d2 region (i.e., when the two drops have equal diameter). These modifications in coalescence and breakup kernels correspond to the domain where DE2 < 1. The modification of each breakup ratio is closely linked with the limit DE1 < 1 with significant changes in the (d1, d2) domain for Rfila. In addition, breakup ratios are also impacted by the boundary (DE2 < 1) that separates the no-breakup (null breakup ratio) and breakup outcomes. Regardless of the parameterization (cases 1–3), Rsheet remains unchanged while Rdisk is modified in a manner comparable to Rfila. Finally, the occurrence of disk breakup is reduced to a smaller region of (d1, d2). This is due to the fact that the classical parameterization defines the disk breakup ratio as the complement of the two other types of breakup (filament and sheet).

Fig. 2.

Kernels for coalescence efficiency Ecoal, breakup efficiency Ebrkp, and ratio for each breakup type—filament Rfila, sheet Rsheet, and disk Rdisk—for cases (a) 1, (b) 2, and (c) 3 (Table 1). Modifications induced by the delineation of outcome regimes (bounce, coalescence, and breakup) and breakup types (filament, sheet, and disk) are marked by shaded areas. The darker gray and lighter gray areas indicate the impact of the delineation DE1 = 1 and DE2 = 1, respectively.

Fig. 2.

Kernels for coalescence efficiency Ecoal, breakup efficiency Ebrkp, and ratio for each breakup type—filament Rfila, sheet Rsheet, and disk Rdisk—for cases (a) 1, (b) 2, and (c) 3 (Table 1). Modifications induced by the delineation of outcome regimes (bounce, coalescence, and breakup) and breakup types (filament, sheet, and disk) are marked by shaded areas. The darker gray and lighter gray areas indicate the impact of the delineation DE1 = 1 and DE2 = 1, respectively.

A detailed analysis of the influence of the different parameterizations (cases 1–3) and kernel modifications on the equilibrium drop size distribution is presented in Fig. 3. Regardless of the consideration of bounce (cases 2 and 3), the simulated equilibrium drop size distribution (Fig. 3a) using the We–p parameterization is characterized by a lighter tail (i.e., a lower number of large drops on the right-hand side of the DSD) and by a less pronounced midrange peak (“shoulder”) for drop diameters d ≥ 0.19 cm when compared to the classical equilibrium DSD (case 1; PB07a,b; Prat et al. 2008; Prat and Barros 2009). In the lower drop diameter range (left-hand side of the DSD), equilibrium DSDs obtained for all configurations (cases 2–6; Table 1) exhibit a small-range peak around d = 0.024–0.026 cm. Differences with respect to previous studies (PB07a) that reported the location of the first peak at d = 0.026 cm can be explained by differences in the bin grid size resolution (40 bins vs 100 bins here). However, the changes in the left-hand side of the DSD (small drops) are significantly different when compared to the classical equilibrium DSD (case 1) with an increase on the order of 50% when bounce is not included (case 2), and an even more significant increase changes when bounce is included (+150%; case 3). More importantly, we note that none of the DSDs generated using the We–p parameterization (cases 2–6) exhibit the presence of the midrange peak of lesser magnitude located around d = 0.25 cm that is produced by the classical parameterization (MF04; PB07a,b), and for which observational evidence has long been lacking (see discussion in PB07a). Therefore, while there is concurrence regarding the existence of a peak in the lower diameter range of the DSD around d = 0.025 cm, which results from the lack of a source term to describe the evolution of cloud droplets to raindrops in such models, the existence of one midsize peak (MF04; PB07a,b; Straub et al. 2010; Jacobson 2011) or even a second (Valdez and Young 1985; List et al. 1987; List and McFarquhar 1990; Hu and Srivastava 1995) additional peak for the equilibrium DSD in the higher diameter range seems to be linked to the parameterization of collision outcomes, and not to the physics. The equilibrium DSD using the We–p parameterization exhibits a roughly exponential behavior within the intermediate diameter range (i.e., 0.1 < d < 0.3 cm).

Fig. 3.

(a) Equilibrium DSD for cases 1–3 (Table 1). (b) Relative difference for the equilibrium DSD induced by new regime delineation (TBB11; cases 2, 3, 5, and 6), bounce mechanism (cases 3 and 6), and a different coalescence kernel (cases 4–6) when compared to original parameterization (case 1). Results are presented for an initial Marshall–Palmer DSD with a nominal rain rate of 50 mm h−1.

Fig. 3.

(a) Equilibrium DSD for cases 1–3 (Table 1). (b) Relative difference for the equilibrium DSD induced by new regime delineation (TBB11; cases 2, 3, 5, and 6), bounce mechanism (cases 3 and 6), and a different coalescence kernel (cases 4–6) when compared to original parameterization (case 1). Results are presented for an initial Marshall–Palmer DSD with a nominal rain rate of 50 mm h−1.

b. Effect of bounce mechanism on the equilibrium DSD

Previous laboratory experiments of drop–drop interactions at terminal fall velocity (McTaggart-Cowan and List 1975; LL82) did not report drop bounce. This can be explained in part by the differences in experimental configurations and the limited number of drop pairs investigated by McTaggart-Cowan and List (1975) and LL82 [see Barros et al. (2008) for discussion]. More recently, numerical investigations of drop–drop interactions using DNS also did not predict the occurrence of bounce (Beheng et al. 2006; Schlottke et al. 2010; Straub et al. 2010). Despite a limited number of experimental observations (6 out of 322 drop collisions) in the experiments described by Barros et al. 2008), the experimental observations are in line with the theoretical considerations of bounce outcomes by T09 and confirmed in TBB11 (Fig. 1). Moreover, it should be noted that T09 reported 51 bounce occurrences out of 163 collisions, albeit among drops not falling at terminal velocity and at bench scale, which were nevertheless in good agreement with the We–p regime diagram. From a modeling perspective, bounce contributions to the equilibrium DSD are noticeable in the large drop diameter range (d > 0.26 cm) with a slightly heavier tail for the right-hand side of the DSD (see Fig. 3a). However, the most significant impact of including bounce (case 3) is on the left-hand side of the DSD at equilibrium where the proportion of small drops (d ≤ 0.05 cm) increases by more than 150% for smaller drops, when compared to the simulated equilibrium DSD without bounce (case 2) (Fig. 3a). It is important to point out that this drop size range (d ≤ 0.05 cm) corresponds to drop size range for which detection in experimental conditions is tied to the measurement sensitivity of the observing system. Although laser optical disdrometers (Löffler-Mang and Joss 2000) have a nominal diameter detection range from 0.062 to 24.5 mm, the effective detectable minimum drop diameter is d = 0.2 mm because of instrumental noise (Tapiador et al. 2010). The common Joss–Waldvogel disdrometer used for ground-based DSD measurements worldwide has a detection threshold d > 0.03 cm (Prat et al. 2008). In addition, because equilibrium DSDs are rarely observed in nature, it is difficult to evaluate the differences between model configuration predictions using field data. Likewise, numerical models such as the one used here are naturally limited in resolving the DSD for the smaller drops if there is no source term, hence the sharp artificial drop on the left-hand side of the distribution. On the other hand, recent inverse modeling applications to explain observed rain rates in radar retrieval do suggest similar changes in the number concentration of small drops (e.g., Prat and Barros 2010a,b).

Combined effects of coalescence–breakup–bounce mechanisms on the DSD

Figure 3b displays the differences in calculated DSDs for each configuration (cases 2–6) when compared to the DSD calculated with the classical model configuration (case 1). The use of a different coalescence kernel (case 4; Seifert et al. 2005, hereafter SKBB05) generates fewer small drops than the classical configuration (case 1) along with a slightly lighter tail (Fig. 3b). The influence of the coalescence kernel is mainly limited to the small diameter range (d < 0.06 cm), while the DSD remains unchanged (case 4) in the intermediate drop diameter range (0.06 < d < 0.302 cm) vis-à-vis case 1. Furthermore when bounce is considered (cases 3 and 6), the number of small drops created in the small diameter range (d < 0.06 cm) increases when compared to the simulations without bounce (cases 2 and 5, respectively), where the LL82 classical kernel (case 3) tends to yield more drops than the SKBB05 coalescence kernel (case 6). The opposite is observed in the moderate diameter range (0.1 < d < 0.225 cm) with a relative drop depletion when bounce is accounted for (cases 3–6 vs cases 2–5). Furthermore, we note that the equilibrium DSD is insensitive to the coalescence parameterization for d > 0.06 cm as indicated by the similar curves for cases 2–5 and cases 3–6.

c. Effect of the uncertainties on the theoretical delineation of the outcome regimes

As mentioned earlier, a few occurrences of bounce–coalescence–breakup and breakup types were observed outside of their respective theoretical domains (Fig. 12 of TBB11). Here, we attempt to investigate the impact of uncertainty in the delineation of the regime boundaries (DE1 = 1; DE2 = 1) by considering a ±20% variation in the location of coalescence–breakup–bounce regimes consistent with TBB11 (DE1 = 1 ± 0.2 and DE2 = 1 ± 0.2). Figures 4a and 4b display the corresponding equilibrium DSDs (Fig. 4a) and their relative differences (Fig. 4b) when ±20% variations in the regime delineation criteria are introduced. As it can be seen in Fig. 4a, the effect of delineation uncertainty is most prominent toward the right-hand side of the DSD and for large drop sizes (d ≥ 0.3 cm). For the remainder of the drop spectrum (0.01 ≤ d ≤ 0.3 cm) the uncertainty in the delineation of the regions for regime outcomes (coalescence–breakup–bounce) generates an average relative difference of less than 30%. In the larger drop diameter range, this difference is at least 50% and increases with increasing drop diameter (Fig. 4b), which illustrates the governing role of breakup processes in determining the number concentration of large drops as argued by Barros et al. (2010). Furthermore, we note that for a given parameterization [no bounce (case 2) or bounce (case 3)], the differences with respect to the classical configuration (case 1) vanish around d ≈ 0.19 cm.

Fig. 4.

(a) Equilibrium DSD for cases 1 and 2 with a ±20% uncertainty as to the regime’s delineation (coalescence–breakup: DE1 = 1 ± 0.2 and DE2 = 1 ± 0.2). (b) Relative difference for the equilibrium DSD induced by a ±20% uncertainty on the delineation of different regimes (coalescence, breakup, and bounce) when bounce is not (case 2) or is accounted for (case 3). Results are presented for an initial Marshall–Palmer DSD with a nominal rain rate of 50 mm h−1.

Fig. 4.

(a) Equilibrium DSD for cases 1 and 2 with a ±20% uncertainty as to the regime’s delineation (coalescence–breakup: DE1 = 1 ± 0.2 and DE2 = 1 ± 0.2). (b) Relative difference for the equilibrium DSD induced by a ±20% uncertainty on the delineation of different regimes (coalescence, breakup, and bounce) when bounce is not (case 2) or is accounted for (case 3). Results are presented for an initial Marshall–Palmer DSD with a nominal rain rate of 50 mm h−1.

d. Transient DSD dynamics

To evaluate the microphysical contribution of each mechanism (coalescence–breakup–bounce), the bin spectral model was used with a Marshall–Palmer (MP) drop size distribution as boundary condition for different nominal rain rates (RMP48 = 1, 5, 15, and 50 mm h−1); the model was left to run until an equilibrium DSD emerged. The equilibrium DSD was defined as that which is achieved when relative variations of the integral properties of the DSD (i.e., the rain rate, the drop number concentration, and the reflectivity) are less than 0.01% (0.0001) between two consecutive time steps. Recall that only drop size distributions with rain intensity R > 10 mm h−1 closely approach equilibrium within 1 h, and that the time necessary to approach numerical equilibrium increases with decreasing rain intensity (Prat and Barros 2009).

The evolution of the moments of the DSD until reaching equilibrium is reported in Fig. 5 for the radar reflectivity factor (the sixth-order moment with respect to the diameter; Fig. 5a) and for the drop number concentration (the zeroth-order moment with respect to the diameter; Fig. 5b) as a function of rain intensity. Regardless of the configuration (cases 1–6 in Table 1), the equilibrium Z–R and M0R relationships are organized along parallel lines. As reported earlier by Prat and Barros (2009), the evolution of the DSD is associated with increasing reflectivity values at low to moderate rain rates (R < 20 mm h−1) due to the predominance of coalescence, and decreasing reflectivity values at higher rain rates (R > 20 mm h−1) due to the predominance breakup. The opposite is true for the drop number concentration that evolves toward lower values for low rain rates due to coalescence, and toward higher values at higher rain rates due to the production of small drops by breakup (Fig. 5b). The rain rate at which the inversion of the dynamic behavior occurs (i.e. the rain rate below and above which Z increases and decreases respectively during the DSD evolution with the opposite behavior for M0) depends on the microphysical parameterizations considered in the simulations (Table 1). This inversion takes place for lower rain intensity values for M0 (3.5 < R < 26 mm h−1) than for Z (10.8 < R < 26.5 mm h−1) (Figs. 5a,b).

Fig. 5.

Transient evolution for an MP DSD in the (a) rain rate–reflectivity RZ domain and (b) rain rate–drop number concentration R–M0 domain for cases 1–6 (Table 1) as a function of the MP48 nominal rain rate (1–5 and 15–50 mm h−1). The thick black lines indicate the RZ and R–M0 relationships for the initial MP DSD at the beginning of the simulations.

Fig. 5.

Transient evolution for an MP DSD in the (a) rain rate–reflectivity RZ domain and (b) rain rate–drop number concentration R–M0 domain for cases 1–6 (Table 1) as a function of the MP48 nominal rain rate (1–5 and 15–50 mm h−1). The thick black lines indicate the RZ and R–M0 relationships for the initial MP DSD at the beginning of the simulations.

A synthesis of the nonlinear implications of including bounce effects (coalescence, breakup, and bounce), and alternative parameterizations (coalescence kernels and breakup ratio), and the uncertainty (±20%) in the delineation of the We–p kernel outcome regimes (DE1 = 1 ± 0.2, DE2 = 1 ± 0.2) on the equilibrium DSD is presented in Fig. 6. Regardless of the uncertainty (±20%) in the delineation of outcome regimes, the influence of each mechanism (coalescence, breakup, and bounce) and coalescence kernel (LL82 vs SKBB05) and the theoretical delineation of outcome regimes are clearly identifiable. The coalescence kernel has an influence on the value of M0 at which the dynamic inversion occurs , while no significant influence is observed for the dynamic inversion Z value RZ. Depending on the coalescence kernel used [cases 1–3 (LL82) vs cases 4–6: (SKBB05)], and independent of other differences, about 50% more drops are created using the original LL82 coalescence kernel compared to the SKBB05 kernel [a combination of the LL82 kernel for larger drops (d > 0.06 cm) and the Beard and Ochs (1995, hereafter BO95) kernel for smaller drops (d < 0.03 cm), and a composite kernel (LL82 + BO95) in between (0.03 < d < 0.06 cm)].

Fig. 6.

Rain intensity for the inversion of the dynamic behavior with respect to the reflectivity factor RZ and the drop number concentration for cases 1–6 (Table 1). The original parameterization (case 1) is compared against the adjusted parameterization based on the theoretical delineation of outcome regimes (T09; TBB11). The figure summarizes the influences of the coalescence kernel (LL82 vs SKBB05: cases 1–3 vs cases 4–6), the delineation of outcome regimes (LL82 vs TBB11; cases 1 and 4 vs cases 2 and 5), or the influence of no-bounce/bounce mechanism (cases 2 and 5 vs cases 3 and 6). For each case of the adjusted parameterization (cases 2, 3, 5, and 6), the error bars represent the influence of the uncertainty of the regime delineation (±20% on DE1 and DE2).

Fig. 6.

Rain intensity for the inversion of the dynamic behavior with respect to the reflectivity factor RZ and the drop number concentration for cases 1–6 (Table 1). The original parameterization (case 1) is compared against the adjusted parameterization based on the theoretical delineation of outcome regimes (T09; TBB11). The figure summarizes the influences of the coalescence kernel (LL82 vs SKBB05: cases 1–3 vs cases 4–6), the delineation of outcome regimes (LL82 vs TBB11; cases 1 and 4 vs cases 2 and 5), or the influence of no-bounce/bounce mechanism (cases 2 and 5 vs cases 3 and 6). For each case of the adjusted parameterization (cases 2, 3, 5, and 6), the error bars represent the influence of the uncertainty of the regime delineation (±20% on DE1 and DE2).

Simulation results obtained with the new TBB11 We–p parameterization show the dynamic inversion both in the M0R and the Z–R spaces occurring at lower values of the rain intensity. From a microphysics point of view, this indicates that collision–breakup dynamics control the evolution of the raindrop spectra at lower rainfall rates (R ≈ 11 mm h−1) than previously predicted (R ≈ 30 mm h−1) using the classical LL82 parameterization (Prat and Barros 2009). The coalescence parameterization has a significant influence on the drop number concentration dynamics, while only negligible influence is observed on the reflectivity dynamics (more sensitive to size than number). Furthermore, when bounce is accounted for, the relative influence of collision–breakup dynamics is shifted to the right toward slightly higher values of the rain intensity (R ≈ 12.5 mm h−1).

Outcome regimes (coalescence, breakup, and bounce) of drop–drop interactions have a clear influence on the evolution dynamics of raindrop spectra, and this influence is beyond the sensitivity to an uncertainty (±20%) on the regime delineation (Fig. 6). While the original configuration indicated that for most of the rain events encountered in nature (R < 20 mm h−1), coalescence was the dominant microphysical process (Prat and Barros 2009), an improved delineation of outcome regimes shows that the dominance of coalescence over collision–breakup dynamics shifts toward smaller rain intensity values (R ≈ 11–12.5 mm h−1) with or without accounting for bounce.

4. Summary and conclusions

In this work a new parameterization (TBB11) of drop collision outcomes (coalescence, breakup, and bounce) is compared against the classical drop–drop interaction parameterization (LL82) widely used in dynamical raindrop microphysical models. The main conclusions can be summarized as follows:

  • Comparison between the classical parameterization (LL82) and the We–p parameterization (TBB11) of drop collision outcomes. Globally, a good agreement was found among drop–drop interaction observations and predictions from laboratory experiments (LL82; Barros et al. 2008; TBB11), empirical parameterizations (LL82), and theoretical parameterizations (T09; TBB11). While the original parameterization predictions (LL82) agree satisfactorily with laboratory observations (Barros et al. 2008), the We–p parameterization allows a refinement in terms of occurrence of outcome regimes (coalescence, breakup, and types of breakup: filament/neck, sheet, and disk/crown). As a result, disk breakup is defined as an independent outcome and not as the complementary occurrence of filament and sheet breakups (LL82). In addition, the conditions of bounce occurrence are introduced in drop size distribution evolution simulations for the first time. Numerical results obtained using the adjusted parameterization, regardless of the coalescence and breakup kernels, produced drop size distributions with significantly lighter tails (smaller number of large drops) when compared to DSDs obtained using the original LL82 parameterization. From a microphysics perspective, this indicates that collision-induced breakup dynamics controls the transient evolution of the DSD for a lower range of rain intensity (R ≈ 10–15 mm h−1) than previously predicted (R > 20 mm h−1) with the original parameterization (Prat and Barros 2009).

  • Effect of the bounce mechanism. Based on theoretical considerations and laboratory observations, the outcome of bouncing effects was accounted for in microphysical simulations and results indicate that the bounce mechanism has a strong influence on the equilibrium DSD, in particular in the small drop diameter range (d < 0.05 cm) with an increase of at least 150% for the DSD drop number concentration. However because only a limited number of bounce occurrences was observed in laboratory experiments (2%: 6 drop pairs for a total of 322 drop–drop interaction: Barros et al. 2008; TBB11), caution is warranted in the interpretation of numerical results. Nevertheless, these results are consistent with model-based estimates derived from radar observations (Prat and Barros 2010a,b).

Finally, these results further stress the need for better observations, measurements, and understanding of raindrop microphysics in the very small diameter range of the DSD (Barros et al. 2008) and for lower rainfall intensities.

Acknowledgments

This research was supported in part by the Pratt School of Engineering, and NASA Grants NNX07AK40G and NNX10AH66G at Duke University with the second author. The first author was a postdoctoral fellow in the Barros group at Duke University when this research was conducted. The authors express their gratitude to two anonymous reviewers for their valuable and constructive comments.

APPENDIX

Modified Coalescence–Breakup Parameterization

Here we report the general expression of the discretized general stochastic collection–breakup equation (SCE-SBE) for warm rain as a reminder of the terms’ definition and changes introduced by the adjusted parameterization. The discretization technique uses a mass and number conservative scheme and step-by-step derivations can be found in PB07a. The interested reader will also refer to PB07a for a more complete definition of the kernels described below.

The drop number concentration (cm−3) for the ith drop size interval is given by

 
formula

where Cj,k = Kj,kEcoalj,k and Bj,k = Kj,kEbrkpj,k = Kj,k(1 − Ecoalj,k) are respectively the coalescence and breakup kernels, and is the gravitational collision kernel [Vj and Vk are the drop (volume xj and xk) fall velocity] (Pruppacher and Klett 1978, hereafter PK78). Also, δj,k is the Dirac symbol; the terms η and κi,j,k are obtained from the discretization of the general SCE-SBE and insure mass and number conservation after each single coalescence and breakup event, respectively (PB07a,b). The term κi,j,k is a function of the fragment distribution function P(υ, xj, xk) that includes the three types of breakup: filament, sheet, and disk (MF04). The fragment distribution accounts for the weighted contribution of each breakup type (proportion of each type of breakup: Rfila, Rsheet, and Rdisk; LL82) and is a function of the number of fragments resulting from each type of breakup event (Ffila, Fsheet, and Fdisk; LL82). The term Ecollj,k is the collection kernel with Ecollj,k = 1 (PB07a,b), and Ecoalj,k is the coalescence efficiency. For Ecoalj,k we use two different expressions (LL82; SKBB05). The second expression (SKBB05) combines the LL82 formulation for large drops (d > 0.06 cm) with the BO95 formulation for small drops (d < 0.03 cm) and a combination of both for intermediate drop diameter (0.03 < d < 0.06 cm) to insure consistency.

a. Delineation of regime outcomes (TBB11)

The collision Weber number is given by We = ρ(d2/2)(ΔV)2/σ, where p = d2/d1 is the ratio of the smaller (diameter d2) and the larger (diameter d1) drops and ΔV is the difference in drop fall velocity. In the We–p plane, the lines DE1 = (p2We)/[6(1 + p3)] = 1 and DE2 = We/[6(1 + p3)] = 1 delineate three regions for regime (coalescence–breakup–bounce) outcomes. The symbols ρ and σ are the density and the surface tension of water, respectively. Calculations and experimental observations are performed under normal conditions of temperature (20°C) and pressure (1 atm). The modifications of the model (PB07a,b; case 1 in Table 1) accounting for regime outcomes (TBB11) are reported next. To avoid confusion modified kernels are noted as PBT12 (representing the authors and year of the current paper).

b. Coalescence and breakup only (cases 2 and 5; Table 1)

The modified coalescence kernel Ecoalj,k is given by

 
formula

The modified ratio for filament breakup Rfila is given by

 
formula

In the We–p domain where no breakup is observed (DE1 < 1 and DE2 < 1), the situation is equivalent to a filament breakup (Rfila = 1) that would result in an unchanged daughter distribution of two drops (Ffila = 2) that is the same as the two parent drops (xj, xk).

Finally, the modified ratio for disk breakup Rdisk is given by

 
formula
c. Bounce effects included (cases 3 and 6; Table 1)

When bounce is considered, the two parent drops (xj, xk) remain intact after interaction and no drops are gained/loss in any drop class (ith). The bounce mechanism is an interaction (Ecollj,k = 1) between two noncoalescing (Ecoalj,k = 0) drops (xj, xk) and for which the daughter drop distribution remains unchanged after temporal contact.

The modified coalescence kernel Ecoalj,k is given by

 
formula

The breakup kernel is modified accordingly. Bounce outcome (DE2 < 1 in the We–p plane) is equivalent to a filament breakup (Rfila = 1) resulting in two drops (Ffila = 2) similar to the parent drops (xj, xk). Note that Rfila is the same as shown in appendix section b above.

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