Raindrop collision and breakup is a stochastic process that affects the evolution of drop size distributions (DSDs) in precipitating clouds. Low and List have remained the obligatory reference on this matter for almost three decades. Based on a limited number of drop sizes (10), Low and List proposed generalized parameterizations of collisional breakup across the raindrop spectra that are standard building blocks for numerical models of rainfall microphysics. Here, recent laboratory experiments of drop collision at NASA’s Wallops Island Facility (NWIF) using updated high-speed imaging technology with the objective of assessing the generality of Low and List are reported. The experimental fragment size distributions (FSDs) for the collision of selected drop pairs were evaluated against explicit simulations using a dynamical microphysics model (Prat and Barros, with parameterizations based on Low and List updated by McFarquhar). One-to-one comparison of the FSDs shows similar distributions; however, the model was found to underestimate the fragment numbers observed in the smallest diameter range (e.g., D < 0.2 mm), and to overestimate the number of fragments produced when small drops (diameter DS ≥ 1mm) and large drops (diameter DL ≥ 3mm) collide. This effect is particularly large for fragments in the 0.5–1.0-mm range, and more so for filament breakup (the most frequent type of breakup observed in laboratory conditions), reflecting up to 30% uncertainty in the left-hand side of the FSD (i.e., the submillimeter range). For coalescence, the NWIF experiments confirmed the drop collision energy cutoff (ET) estimated by Low and List (i.e., ET > 5.0 μJ). Finally, the digital imagery of the laboratory experiments was analyzed to determine the characteristic time necessary to reach stability in relevant statistical properties. The results indicate that the temporal separation between particle (i.e., single hydrometeor) and population behavior, that is, the characteristic time scale to reach homogeneity in the NWIF raindrop populations, is 160 ms, which provides a lower bound to the governing time scale in population-based microphysical models.
McTaggart-Cowan and List’s (1975a, hereafter ML75) experimental research on raindrop breakup led to finding drop collision as the dominant mechanism of breakup compared to breakup induced by strong electrical forces and aerodynamic instability. ML75 reported that only 712 collisions out of 25 000 attempts resulted in breakup with a relative frequency distribution of 27% of filament (also referred to as neck) breakup, 55% of sheet breakup, and 18% of disc breakup. Visual depictions of the dynamical evolution of each type of breakup captured with a high-speed camera can be found in Testik and Barros (2007) and herein (Figs. 1a–c). The specific realization of filament, sheet, and disc breakup depends on the value of the collision kinetic energy (CKE; a function of the drops’ diameter), the impact location (edge versus center), and the angle between the two colliding drops, the so-called parent drops. Briefly, a filament breakup occurs when the contact between two drops forms a bridge (neck) of water connecting the two drops. The disintegration of the bridge generates smaller fragments (satellite drops), but the identity of the parent drops persists after collision, albeit with smaller size (Testik and Barros 2007, their Fig. 6; Fig. 1a). For sheet breakup, the bulk of the larger parent drop is spread to form a sheet of water after impact. The smaller parent drop usually disappears as a part of the new population of satellite drops (Testik and Barros 2007, their Fig. 7; Fig. 1b). When the impact occurs in the vicinity of the large drop center, a disc of water that includes both drops is generated. The disc spreads out until rupture, generating a large number of satellite drops among which both parent drops are undistinguishable (Testik and Barros 2007, their Fig. 8; Fig. 1c).
Because of the complexity of the physical processes involved, a complete theoretical description of the dynamic evolution of fragment size distributions (FSDs) has thus far eluded investigators, and some have argued that it is not likely to be deduced in the near future (McFarquhar 2004, hereafter MF04). On the basis of the original ML75 data and additional experiments, Low and List (1982a, b, hereafter LL82a, LL82b, respectively) developed parameterizations for collisional breakup between two drops to describe drop size evolution under steady-state conditions. Prat and Barros (2007a, b, hereafter PB07a,PB07b, respectively) developed a spectral model to solve the stochastic equation for the evolution of the drop size distributions (DSDs) in warm rain in the presence of coalescence and breakup. The numerical model was tested using many of the coalescence and breakup kernels available in the literature, including aerodynamical breakup and collisional breakup after LL82a’s, LL82b’s, and MF04’s parameterizations.
Previously, numerical experiments carried out by PB07a and PB07b showed a strong dependence of DSD evolution on the breakup kernels used in the stochastic collection equation (SCE)–stochastic breakup equation (SBE, which are the fit parameters proposed by LL82a and LL82b. It is therefore critical to assess the robustness and generality of the parameterizations. This is the focus of this article. Specifically, we investigate the robustness of the parameterizations for different types of breakup and coalescence using a dataset collected at the National Aeronautics and Space Administration (NASA) Wallops Island Facility (NWIF) and the Prat–Barros (PB07a; PB07b) microphysical model.
The manuscript is organized as follows: Section 2 provides a brief description of the microphysics model and the methodology used to obtain the FSD for selected drop pairs. In section 3, a basic description of the experimental setup and processed data, including scaling analysis, are presented. Finally, the results and conclusions are presented in sections 4 and 5, respectively.
2. Description of the microphysical model
Only a brief description of the spectral bin model developed for the dynamic simulation of the DSD in the presence of coalescence and spontaneous and collisional breakup is provided here. A more complete description can be found in PB07a and PB07b. The DSDs discretization scheme (i.e., bin distribution) is internally consistent with respect to any two selected moments of the DSD, and in particular the drop number concentration and the drop total mass, respectively, the zeroth- and first-order moments (with respect to the drop volume). In addition, the discretization scheme is very flexible and allows for a wide selection of discretization grids (geometric, regular, irregular) and bin refinement. As in PB07a and PB07b, a 0D version of the model is implemented here, that is, the equations are solved for an isolated control volume, and thus an idealized box.
Parameterizations of coalescence efficiency are taken from LL82a, the gravitational collision kernel is from Pruppacher and Klett (1978), and the drop fall velocity is estimated from Best (1950). The overall fragment distribution function includes three types of breakup (filament: FI, sheet: SH, disc: DI) identified by ML75. The relative frequency RXX of breakup type XX and the number of drops created by a collisional event FXX (XX = FI, SH, or DI) follow LL82b. For each type of breakup between a large drop of diameter DL and a small drop of diameter DS, the fragment distribution functions P(D, DL, DS) are derived from the parameterization proposed by MF04, who provides analytical expressions for the parameters (height: H, standard deviation: σ, modal diameter: μ) of the Gaussian and lognormal forms of the fragment size distribution. The overall fragment distribution is obtained from the weighted contribution of each type of breakup. The comparison of the experimental data with the numerical model is conducted for specific pairs of drops (DL–DS) and specific types of breakup. By selecting a ratio RXX = 1 for a selected type of breakup (XX = FI, SH, or DI), individual contributions can be isolated and evaluated independently.
3. Experimental data
Small drops generated by needles of calibrated nominal diameter (NS) were injected at an angle from a height of 5 m into the trajectory of a large drop generated by a needle of calibrated nominal diameter (NL) falling from a height of 14 m. Both heights are consistent with the minimum requirement to attain terminal velocity for the respective drop sizes at the measurement height (Guzel and Barros 2001). Six different combinations of initial needle sizes (NL, NS) were used to study the effect of initial conditions on drop evolution (Table 1). The collisions were captured by a high-speed imaging camera operated at 500 frames per second. The high-speed camera setup for filming raindrop collisions in the laboratory environment is similar to that in Testik et al. (2006). The captured video (300 GB corresponding to 225 000 individual frames) shows the evolution of drop population after the initial collision. A light source covered by a sanded glass shield was used to illuminate the raindrops from behind, and the resulting image projected an opaque drop in a bright background. Each captured frame was analyzed visually for categorizing the breakup and coalescence. Drop sizes (DL–DS) were measured before and after collision using image-processing software.
A frame-by-frame inspection of the video allowed the original detection of 237 filaments, 175 sheets, 39 disc breakups, and 84 coalescence events. Subsequent screenings consisted of verifying whether a minimum sequence of collision stages was available (i.e., was recorded) for each of the following events: (i) one frame where the two parent drops could be unambiguously identified; (ii) one frame capturing the actual collision; (iii) one frame capturing the postcollision drop–drop interaction (i.e., neck formation); and (iv) one frame capturing disintegration with clear visualization of the fragment distribution to permit evaluation of mass conservation requirements. Of the original set of events, only 154 filaments, 56 sheet breakups, and 58 coalescence collisions were ultimately determined to fulfill the minimum collision sequencing requirement.
The drop pairs involved in the collision process deviated from the initial drop sizes consistent with the DSD evolution during the fall. Through image processing of sheet breakup frames, it was determined that the collision process took on average 14 ms within a measurement height of 12 cm. After further quality control preceding automatic computer-based quantitative analysis, a total of 142 filaments and 47 sheet breakups were finally selected. Mass calculations were conducted immediately before collision and immediately after breakup, assuming a spherical shape for the drops. Measurement errors arising as a result of the digital distortion of large drops and the accuracy of the image-processing algorithm produced a 5% excess mass for 44% of the collisions, and a 5% loss in mass for 32% of the collisions. This error is consistent with uncertainty in drop size estimates derived independently from similar digital imagery using glass beads of known size (not shown).
The equivalent diameters of the colliding drop pairs were used for the calculation of the total mass. The fragment numbers are identified by visual inspection of each frame after the collision, and the equivalent diameter of each fragment is determined automatically. The postcollision mass difference (in excess/deficit) was redistributed according to a simple inverse mass-weighted average among all of the detected fragments, and the drop diameters were then adjusted accordingly. Note that, as in LL82a, LL82b, and ML75, the NWIF data also show conspicuous occurrence of filament breakup across the raindrop spectrum. The NWIF data also show evidence of sheet breakup across the entire spectrum.
a. Scaling analysis
Scaling analysis was performed to determine the number of frames over which the statistical distribution of drops in the field of view of the camera remains stable, an expected condition to reach homogeneity. This is equivalent to determining the spatial scale above which the drops exhibit population behavior that can be described by a robust statistical distribution as opposed to discrete behavior that depends on individual hydrometeors. For this purpose, we pursued the idea of trading time for space by collecting frames and we developed a series of Nc composite time series Aλ with spatial resolution λ1/2, where λ is the number of frames in the original time series that were used to generate each new frame of Aλ, and Nc is the total number of frames for each experiment (tens of thousands) divided by λ; that is, frame Aλ,i is obtained by compositing randomly original frames Aiλ through Aiλ+λ.
Subsequently, the statistical moments for each composite series were calculated and their scaling behavior was analyzed. The zeroth- and first-order moments, respectively, the drop number (M0) and drop volume (M1) as a function of scale λ, are shown in Fig. 2 for a selected needle configuration (NL = 4.8 mm, NS = 2.52 mm). A scaling break at λ ∼ 81 is generally observed for all experiments (Figs. 2b,d, others not shown). On the contrary, for M0, the standard deviation does not seem to reach a final stable value. The different scaling behavior between M1 (volume based) and M0 (number based) is due to the fact that variations in number are important in the case of very small drops that result from breakup, whereas this effect is much smaller when we consider the volume and the standard deviation for M1 (proportional to the volume of drops; thus, larger particles play a bigger role).
The utility of scaling in establishing the final fragment size distribution for each drop is illustrated in Fig. 3, which displays the drop counts and volume-weighted drop counts for the configuration (NL = 4.8 mm, NS = 2.52 mm). The histograms evolve from a random DSD (λ = 1 × 1 = 1) toward a more organized pattern (λ = 9 × 9 = 81, λ = 17 × 17 = 289), with peaks located around the values of the needle diameters (NL, NS). Finally, the stable cumulative DSD is used to describe the coalescence–breakup mechanism and obtain the corresponding fragment distribution function that will be compared with expressions proposed by LL82a, LL82b, and MF04. An important finding from this analysis is that the characteristic temporal scale for averaging hydrometeor observations is on the order of 160 ms [(≈ 81 (frame) × 1/500 (s frame−1)], for the characteristic observation length of 12 cm, which is in the range of the outer scales of atmospheric turbulence (Consortini et al. 2002). Thus, this result can be viewed as a lower bound for the time step in numerical integration studies.
4. Experiment–model intercomparison
In contrast to the experimental setup and observations in LL82a and LL82b, the collision experiments in this study result from the interaction between drops that were generated somewhere in the rain shaft after an unknown number of collisions following the first interaction between the two drops produced at the needles, not excluding spontaneous breakup. This is the reason why the results obtained correspond to a wide spectrum of drops (Figs. 4a,b), whereas the data collected by ML75, LL82a, and LL82b are for specific drop pairs that were physically constrained to collide within a limited control volume. The current experimental setup produced a total of 142 individual collisions among 101 different drop pairs (DL–DS) for filament breakup and 47 collisions among 39 different drop pairs (DL–DS) for sheet breakup, which could be used for systematic quantitative analysis. The colliding parent drops ranged between 0.1–2.2 mm for DS and 0.8–5.6 mm for DL. Finally, the maximum number of collisions for any one specific drop pair (DL–DS) was 5 and 3 for filament and sheet breakup, respectively.
It is important to stress the fundamental difference between the experimental data presented in this paper and the previous experiments reported elsewhere (ML75; LL82a; LL82b). In this work, we observe mostly single occurrence breakup events (1–5: filament, 1–3: sheet) for a large number of drop pairs (101: filament; 39: sheet) encompassing most of the (DL–DS) spectrum. Previous studies (ML75; LL82a; LL82b ,Table 2) observed a relatively large number of breakup events (23–80: filament, 11–88: sheet, 15–26: disc) for a limited number of nominal (DL–DS) colliding drop pairs (10: filament, 7: sheet; 5: disc). Whereas the current experimental setup reduces the number of events (samples) over which averaging can be conducted, the wide spectrum of drops more closely resembles natural rainfall conditions. Finally, the level of accuracy in the determination of the drop diameter using the automated image-processing software is 0.1 mm (i.e., one pixel), though depending on lighting conditions smaller drops may be counted; previously, ML75 expressed the FSD in intervals of 0.5 mm, and LL82a and LL82b reported an accuracy of 0.1 mm. Given the averaging of datasets with different accuracy (ML75; LL82a; LL82b), the overall accuracy is controlled by the lowest value, that is, ML75 (0.5 mm).
a. Breakup experiments
Figure 4 shows the spectrum of fragment numbers for colliding drop pairs via filament (Fig. 4a) and sheet (Fig. 4b) breakup. The experimental observations are compared with the LL82a- and LL82b-based model simulation for the respective drop pairs at steady state. Generally, the overall number of fragments produced for a single breakup event (DL–DS) observed during the NWIF experiments was found to be systematically smaller than the average number of fragments obtained with the LL82a and LL82b parameterization, even more so in the case of filament breakup. More specifically, the model overestimates the numbers of fragments for D > 0.5 mm, especially in the 0.5–1.0 mm range for the drop pairs produced by the collision of large drops (DL > 3 mm, DS > 1 mm). This observation is further illustrated in Figs. 4c,d, which displays the distribution of drop pairs for which the model overestimates/underestimates the fragment numbers with regard to a given threshold. The threshold was set as 2 and 3 for filament and sheet breakup, respectively, based on the maximum standard deviation represented in the fragment numbers of colliding drop pairs from LL82a and LL82b. Specifically, in LL82a and LL82b, the average standard deviation in the fragment numbers is ±15% for filament breakup (averaged over 10 drop pairs for which filament breakup occurs, 5 of which were from ML75), ±30% for sheet breakup (averaged over 7 drop pairs, 4 of which were from ML75), and ±50% for disc breakup (averaged over 5 drop pairs, 4 of which were from ML75). A review of LL82a and LL82b indicates that in the specific case of filament breakup the data for collisions of the larger drops was obtained originally by ML75 (Table 2). Indeed, Figs. 11a–e in LL82a show that the number of observed small drops (D ∼ 0.5 mm) in the average observational spectra is overestimated by the FSD diagnosed by the parameterization for the collision of large drops. Refer also to Fig. 5d in LL82a for overall results, including all breakup types that further confirm the larger uncertainty for D < 0.5 mm.
The NWIF experimental FSDs were compared with LL82a and LL82b experimental data for the same drop pairs (DL–DS) whenever possible. Only 3 drop pairs (DL–DS) match LL82a and LL82b in the case of filament breakup, and the observed fragment number is consistent with their experiment (Fig. 5a). However, for the two matching drop pairs for sheet breakup, the FSD for one case is completely outside the range derived from LL82a and LL82b experimental data (Fig. 5b). More generally, Figs. 4c,d show that for cases where the large parent drops are small (DL < 3 mm), 30% and 60% of the breakup events (DL–DS) are within the limits of the predicted fragment numbers determined by LL82a and LL82b for filament and sheet breakup, respectively. On the other hand, for larger DL (DL > 3 mm), the ratio is as low as 5% and 7% for filament and sheet breakup, respectively.
The significance of this result for drops in the small diameter range (D < 0.5 mm) is that uncertainties in the number and distribution of small drops have impact upon the dynamics of microphysical processes, such as accretion, coalescence, and drop growth, and can be critical for the theoretical study of aerosol–cloud–raindrop interactions and interpretation of field data (e.g., Ramanathan et al. 2001). Although this will not affect significantly the estimation of the integral properties of rainfall, such as radar reflectivity factor (Z) or the estimated rain-rate (RNT) values that are proportional to higher-order moments of the DSD, it will certainly affect the interpretation of radar measurements at high frequency (e.g., 95 GHz, W band). Nevertheless, uncertainty in the observed fragments of the FSDs produced in the representation (number, shape) of the population of larger drops (DL > 3 mm) may have dramatic effects on the predicted integral properties of rainfall when using a microphysical model with breakup parameterization (e.g., MF04, PB07a).
Finally, and despite the limited number of independent matching cases for intercomparison, the quantitative differences between NWIF and LL82a and LL82b datasets for filament and sheet breakup bring to the forefront issues related to the experimental setup [14-m gravitational setting tower for NWIF vis-à-vis the short-type acceleration system (McTaggart-Cowan and List 1975b) in ML75 and LL82a and LL82b], as well as measurement philosophy (spectrum based in NWIF vis-à-vis controlled repetition of collisions among specific drop pairs). Another relevant question in this context is whether the collision between a drop pair would produce different fragments in natural rainfall as compared to controlled (quiescent) conditions in the laboratory, especially when the collision kinetic energy is very high. That is, could (can) the transient coalesced drop pair dissipate the extra energy more or less efficiently in the natural environment? For example, raindrop oscillations and lateral drift effects documented by Testik et al. (2006) can also influence drop collision–breakup dynamics, and thus could possibly produce a different fragment number under natural rainfall conditions when compared with the laboratory environment.
b. Breakup parameterization
Figure 6 shows an example of the number of the averaged FSD of model simulations for filament (Fig. 6a) and sheet breakup (Fig. 6b) for the drop pair (DL = 2.5 mm, DS = 0.9 mm) against NWIF observations. The location and the width of the peaks simulated by the model are in close agreement with those obtained from the experiment. This indicates that the fit parameters developed by MF04 from LL82a and LL82b experimental data are able to predict the FSD with considerable accuracy for a given drop pair. Note that this agreement takes place for DS < 1 mm and DL < 3 mm, corresponding to a range measured by LL82a, which underscores the issue raised earlier about the importance of reporting accuracy in the two families of experiments used for the parameterization (i.e., ML75 and LL82a).
The average difference (with respect to all drop pairs) for predicted (model) and observed (NWIF experiment) drop number is displayed for filament (Fig. 6c) and sheet (Fig. 6d) breakup using a regular bin resolution of 0.2 mm in diameter. For both cases, the parameterization using the fit parameters proposed by LL82a, LL82b, and MF04 predicts an excessive number of drops (per 0.2-mm size interval) as compared to the NWIF experiment. As can be seen for the drop pair (DL = 2.5 mm, DS = 0.9 mm), the difference between the model and NWIF experiment decreases when averaged over eight and four drop pairs, respectively for filament (Fig. 6c) and sheet (Fig. 6d) breakup. Recall that the results are consistent with the nature of the fit (overestimation of the number of small drops D < 0.5 mm) of the average histograms of observed FSDs in LL82a and LL82b as discussed before. Ultimately, although it can be argued that the observation of a large number of breakup events for the same drop pairs (DL–DS) might lead to an improved agreement between the LL82a-, LL82b-, and MF04-based model and NWIF experiments, the discrepancy for specific simulations begs the question of translation of the laboratory-based parameterization as a proxy of natural rainfall, especially under conditions favorable to intermittency (short duration showers, mixed light and heavy rainfall intensity, strong winds, high turbulence intensity, etc.).
Further, for a significant number of collisions (DL–DS), it was found as expected from previous discussion that the model underestimates systematically the fragment numbers for drop diameters below 0.2 mm (bin centered at 0.1 mm; see Figs. 6c,d), thus, for diameters below the detection threshold. One overarching implication of these analyses is that systematic measurements of drop dynamics with high-resolution, high-speed imaging capabilities are necessary before a definitive parameterization of raindrop microphysics can be settled in the submillimeter range.
c. Coalescence experiments
LL82a and LL82b suggest that for coalescence to occur, both CKE and ΔSσ (the variation in surface energy, i.e., the difference between the surface energy of incident drops and surface energy of the spherically equivalent coalesced drop) must be adequately dissipated by the coalesced drop. The authors further state that no coalescence was obtained for a total energy ET < 5.0 μJ, where ET = CKE + ΔSσ [see Testik and Barros (2007) for mathematical expressions].
Figure 7 shows that the 58 coalescence events from the NWIF experiment provide similar results. The total energy (ET) is below the limit except for two collision cases (DL = 3.4 mm, DS = 1.5 mm; DL = 3.4 mm, DS = 2.2 mm], where energy exceeded the threshold value of 5.0 μJ with, respectively, 7.6 and 5.4 μJ. A careful survey of the path line (not shown here) of the coalesced drops does not permit to conclude whether the colliding drops were temporarily coalesced, or the newly formed drop was able to dissipate the energy without breakup.
A comparative study of recent laboratory experiments of raindrop breakup against previous results by LL82a, LL82b, and ML75 was presented. The experimental fragment size distributions (FSD) for selected drop pairs (DL–DS) were evaluated against explicit simulations using a dynamical microphysics model (PB07a; PB07b) that uses breakup functions (kernels) based on LL82a, LL82b, and MF04. The main results of the present study are summarized as follows:
A total of 142 individual collisions between 101 drop pairs for filament breakup and 47 collisions between 39 drop pairs for sheet breakup were quantified. The comparison between NWIF observations and modeling results using either the LL82a/MF04 or LL82b/MF04 parameterizations shows good agreement in that it concerns the width and location of the peaks of the FSD. However, the number of fragments observed in the NWIF experiments is systematically lower than the one predicted by the LL82a and LL82b parameterization for D > 0.5 mm; more importantly, in the case of filament breakup, which based on NWIF and LL82a and LL82b, it is the most conspicuous type of breakup simulated in the laboratory.
Using the fit parameters developed by LL82a, LL82b, and MF04 for the lognormal function describing the daughter fragment component of the general FSD, the simulations underestimate the fragment numbers observed in the smallest diameter range (e.g., ∼0–0.2 mm) for both filament and sheet breakup. Whereas it might be argued that an increase in the sample size should increase the agreement between the formulation derived from LL82a, LL82b, and MF04 and the NWIF experiments, such expectation cannot be supported for the small drop sizes because of the measurement constraints in LL82a, LL82b, and ML75. The importance of this disagreement stems from the feedback with regard to cloud drop formation and growth with direct implications the optical and physical properties of clouds independently of rainfall.
The observation of 58 separate cases of drop coalescence has confirmed the upper limit of 5.0 μJ as the total energy (ET) cutoff proposed by LL82a.
A scaling analysis, performed by trading time (individual frames) for space (combined λ frames), suggests that the characteristic time scale to reach homogeneity in raindrop populations, that is, the separation between single hydrometeor and drop population behavior, is 160 ms. This finding should be of interest to design experimental and numerical simulations.
Results from ongoing research using the PB07a and PB07b model suggest that small drops play an important role in determining whether or not heavy rainfall can be produced within the characteristic time scales of persistence in the cloud environment. This issue is even more relevant in the context of aerosol–rainfall interactions and raindrop production and evolution (e.g., Ramanathan et al. 2001). The present study both highlights the contributions of LL82a and LL82b and raises fundamental questions related to laboratory experiments of drop breakup, and, in particular, the role played by the experimental setup and technology. These questions pertain to the observation and measurement of very small drops, and to the representation of the tails of the fragment size distribution, that is, for both very small and very large drop diameters. The NWIF experiments, though innovative in character, did not provide the volume of observations and accuracy necessary to propose an alternative or modified parameterization to LL82a and LL82b. It is therefore appropriate to say that revisiting LL82a and LL82b remains an open proposition that requires a more powerful dataset to quantify the contribution of large drop collisions to the submillimeter range of the DSD.
This research was supported in part by NSF Grant ATM 97-530093 and NASA Grant NNGO04GP02G with the first author and by the Pratt School of Engineering at Duke University, and by NASA through internal funding of the Rain-Sea Interaction Laboratory at the Wallops Flight Facility.
Corresponding author address: Dr. Ana P. Barros, Duke University, Box 90287, 2457 CIEMAS Fitzpatrick Bldg., Durham, NC 27708. Email: firstname.lastname@example.org