1. Introduction
Prediction of observed droplet size distributions (DSDs) growing by condensation in small warm cumuli, critically important for modeling the onset of collision–coalescence and the progression to precipitation within the time frame observed, has proven difficult to achieve. The inability of adiabatic parcel models to reproduce even the order of magnitude of observed DSD widths (before the onset of coalescence) has long been acknowledged (e.g., Howell 1949). Parcel models with simplistic representations of entrainment and mixing (e.g., Mason and Chien 1962; Warner 1973) produced distributions broadened toward smaller sizes. Later treatments artificially forced dynamical sequences of air parcel ascent and descent, broadening the distributions toward larger sizes (e.g., Telford and Chai 1980; Jonas and Mason 1982). Meanwhile, Latham and Reed’s (1977) laboratory results suggested that the time scale for droplet evaporation could be less than that for mixing of the cloudy and entrained air, allowing droplets to experience different degrees of subsaturation. Calculations using this “inhomogeneous” droplet evaporation also broadened distributions toward larger sizes (Baker et al. 1980). Brenguier and Grabowski (1993) used inhomogeneous evaporation to model narrow and broad DSDs within a 2D simulated cloud, but conducted no observational comparison. Su et al. (1998) successfully reproduced some observations from Hawaiian clouds but had to assume specific sizes for entrained parcels of air, lacking representation of the large-scale cloud dynamics.
Compounding the difficulty of predicting DSDs is uncertainty in the observations themselves. The earliest aircraft observations required a microscope to analyze droplets collected on glass slides; such analysis was too manually intensive to obtain representative samples. Later, aircraft-mounted electronic probes such as the forward scattering spectrometer probe (FSSP) counted and sized droplets in real time, but limitations of the components and design incorrectly sized some droplets, artificially broadening the DSDs (e.g., Baumgardner et al. 1990). Some limitations can be overcome by careful calibration, laser beam mapping, and postprocessing during times of high probe activity. However, droplets coincident in the FSSP beam can result in overestimation of droplet sizes and is not straightforward to correct. Postprocessing algorithms may try to correct “coincidence” effects (e.g. Perrin et al. 1998; Lasher-Trapp and Cooper 2000) but are time-consuming to implement and rely upon major statistical assumptions.
Developments in numerical modeling, as well as better observations, present an opportunity to reexamine the prediction of DSDs from condensation. Lasher-Trapp et al. (2005, hereafter LTCB05) presented a modeling framework combining a 3D dynamical cloud model with a Lagrangian microphysical model to investigate variability in droplet saturation histories that result from entrainment and mixing. The microphysical model was run along trajectories derived from a cloud simulation; it performed calculations of droplet condensation and evaporation, using kinematic and thermodynamic variables consistent with the simulated 3D cloud dynamics, lacking in previous studies. LTCB05 successfully replicated general features of observed DSDs—large widths, multiple peaks, and small droplets far above cloud base1—but did not conduct a direct observational comparison. Recently, observations in warm trade wind cumuli were collected during the Rain in Cumulus over the Ocean (RICO) field campaign (Rauber et al. 2007), where low droplet number concentrations limited artificial DSD broadening by droplet coincidence. The goal of this study is to use the LTCB05 modeling framework to predict the breadth of these observed DSDs, constraining the calculations with measurements of cloud condensation nuclei (CCN), general characteristics of the clouds, and their thermodynamic environment. While the observed DSDs may still be somewhat artificially broad, or broadened by some droplet coalescence (which is not modeled here), they are the best available for this modeling comparison. The aim is not to reproduce a particular cloud and its DSDs, but to predict the widths characteristic of DSDs over a field of cumuli on a given day.
2. Observations
The Caribbean trade wind cumuli observed on 10 December 2004 were 1.5 km wide, 1 km deep, with bases near 500 m MSL. The National Center for Atmospheric Research (NCAR) C-130 aircraft penetrated clouds at 150, 480, and 800 m above the bases, yielding data on 52, 43, and 23 developing clouds, respectively, including cloud droplet number concentrations and sizes from an SPP-100,2 liquid water content (LWC) from a Particle Volume Monitor (PVM)-100A (Gerber et al. 1994), and updraft speed w from a vertical gust probe. Data were processed using pre- and postflight calibrations and quality control algorithms. Clouds were selected by requiring droplet number concentrations to exceed 20 cm−3 within regions of positive updraft speed, with raindrop concentrations less than 0.1 L−1 [as detected by a two-dimensinal precipitation (2DP) optical array probe].3 Small LWC (Table 1), substantially less than adiabatic values (Fig. 1a), low updraft speeds (Table 1) with maxima rarely exceeding 7 m s−1, and cloud droplet number concentrations rarely exceeding 120 cm−3 (Fig. 1b) were generally observed. Statistics on the 3545 observed DSDs used in this study are presented in section 4.
Simulated and observed cloud characteristics.
Range of (a) maximum LWC values (black line denotes adiabatic values), and (b) maximum Nd observed at each of three flight levels within clouds on 10 Dec 2004. Box top and bottom bound the 5th and 95th percentiles, dashed lines show 25th and 75th percentiles, and solid center line is median value. Number of clouds sampled at each altitude is noted in parentheses along abscissa.
Citation: Journal of the Atmospheric Sciences 68, 12; 10.1175/JAS-D-11-0153.1
3. Modeling
The modeling framework is that of LTCB05, and specific details not discussed below can be found there, or in Cooper et al. (2011, manuscript submitted to Atmos. Chem. Phys., hereafter CLB).
a. Cloud simulation and trajectories
The Straka Atmospheric Model as modified by Carpenter et al. (1998) produced a 3D simulation representative of the observed clouds using a domain 2.73 × 2.73 × 3.03 km3, with 30-m grid spacing and a 0.3-s time step. The model was modified to force a cloud by convergence of surface winds, set to a maximum of 4.25 m s−1, 690 m from the center of the domain, decreasing to 0 m s−1 at the center and at a height of 30 m. The cloud thermodynamic environment was initialized with the 1501 UTC 10 December dropsonde.
The model produced a reasonable simulation but overestimated the LWC (Table 1). The second of three successive pulses during the 1.5-h simulation is used here. The base, top, and width of the modeled cloud are respectively 600, 1620, and 1800 m, close to the observed values of 500, 1500, and 1500 m. The updrafts and downdrafts are slightly weaker than observed. The LWC is more than double that observed at 1800 m but shows substantial variability (Fig. 2). Attempts to tune constants in the subgrid turbulence parameterization failed to reduce the LWC while still matching the observed updraft speed and cloud-top height. Overestimation of simulated LWC can result from limited spatial resolution [e.g., Carpenter et al. 1998; tests here at coarser resolution support this, as do observations by Gerber et al. (2008) showing structure in cloud water at scales less than 2 m in RICO clouds]. This problem is pervasive among cloud models, as shown by an intercomparison of 12 different large-eddy simulations of RICO clouds (van Zanten et al. 2010). Effects of the overestimated LWC on the model predictions are discussed in section 4.
Cloud water mixing ratio at 14 min after the start of the second thermal in the simulation. Value ranges are <1.3 (blue, green), 1.3–1.8 (yellow, orange), and 1.8–2.6 g kg−1 (red, pink).
Citation: Journal of the Atmospheric Sciences 68, 12; 10.1175/JAS-D-11-0153.1
Trajectories were initiated at “target points” spaced 30 m apart at a given altitude at 14 min into the simulation, and computed backward in time ending beneath cloud base (Fig. 3a). Variability in trajectories departing from a point results from superimposed small velocity fluctuations (scaled to the local subgrid-scale turbulent kinetic energy) at each time and location along the trajectory as in LTCB05. Fall velocities of the droplets are neglected.
(a) Photorealistic rendering of the simulated trade wind cumulus with a few trajectories shown for three target points (green and aqua dots). Color (scale to right) along trajectories represents θq (K) experienced by droplets within the microphysical model run along trajectories. (b)–(d) Predicted DSDs resulting from representations of droplet evaporation as (left) 100% homogeneous or (right) 100% inhomogeneous, for each of the three target points shown in (a) from left to right.
Citation: Journal of the Atmospheric Sciences 68, 12; 10.1175/JAS-D-11-0153.1
b. Predicted DSDs
The microphysical model of Cooper et al. (1997) as adapted by LTCB05 was initialized at each trajectory end point beneath cloud base with a CCN supersaturation spectrum (C = sk; C = 100 cm−3, k = 0.8) consistent with RICO observations (Hudson and Mishra 2007) and a power law (slope = 2.5) for giant aerosol, and run forward to the target point (Fig. 3a).4 When the microphysical model encounters a region along a trajectory where the wet-equivalent potential temperature θq decreases from the previous time step (indicating entrainment occurred in the 3D simulation), environmental air (including CCN similar to cloud-base values except that C = 70 cm−3) is entrained into the microphysical model to match the new θq value. If θq increases or stays constant, no adjustments to the microphysical model are made.5 Droplet evaporation resulting from entrainment can be set as homogeneous, where all droplets experience the same saturation after mixing, or inhomogeneous, where some droplets experience subsaturated conditions and evaporate while others are unaffected. A combination can also be set by weighting each contribution. Recent studies by Andrejczuk et al. (2006) and Lehmann et al. (2009) found that regions of higher (lower) turbulent kinetic energy were characterized by homogeneous (inhomogeneous) droplet evaporation, so the model’s ability to represent a combination of these representations is important, given that both regions exist in a cloud. After mixing, newly entrained CCN, or CCN from previously evaporated droplets (neglected by LTCB05), can be activated if their critical supersaturation is attained. After running the microphysical model along all trajectories, the individual DSDs (up to 800) are averaged to generate one DSD per target point (Figs. 3b–d). Testing showed similar widths are produced by averaging as few as 15 DSDs, but smoother, more realistic DSDs result from averaging 50 or more DSDs from the different trajectories reaching a target point.
The predicted DSD width (Figs. 3b–d) increases with the variability among the trajectories (Fig. 3a). Peaks at smaller sizes are produced by the activation of newly entrained CCN, and peaks at larger sizes consist of droplets activated at cloud base, experiencing varying degrees of entrainment. Inhomogeneous evaporation produces broader DSDs because more entrained CCN are activated, and growth of the surviving original droplets is favored by limited competition.
Entrainment and mixing effects dominate the broadening of the DSDs. Trajectories enter the cloud base at different times (those starting closer to cloud base in Fig. 4a entered the cloud later), and thus droplets can experience different vertical accelerations (Fig. 4a) and growth times, resulting in broadening without entrainment (Figs. 4b,d). When entrainment is considered, however, the averaged DSD (Fig. 4e) is an order of magnitude broader, caused by droplets that traveled through a major eddy (loop in Fig. 4a) and experienced over a 2-K decrease in θq, while others experienced less (Fig. 4c).
(a) Photorealistic rendering of the simulated cloud shown with a subset of 100 trajectories to the target point (aqua dot), colored as indicated by w, and values of θq along each trajectory vs time (b) for adiabatic growth and (c) for growth allowing effects of entrainment. (d),(e) Resulting DSDs averaged from the entire set of trajectories (d) for adiabatic growth and (e) for growth allowing effects of entrainment. For each DSD in (d) and (e), N, mean diameter Dmean, and DSD width σD are noted in legend.
Citation: Journal of the Atmospheric Sciences 68, 12; 10.1175/JAS-D-11-0153.1
4. Results
Prediction of the observed DSDs by representing differences in droplet saturation histories resulting from entrainment is evaluated using droplet number concentration Nd, the mean diameter of the DSD 
Relative frequency distribution (expressed as a percentage of the total number of values) of (a),(c),(e) Nd and (b),(d),(f) σD. Curves denote values calculated from observations (purple, solid) and modeled DSDs when considering droplet evaporation as 100% homogeneous (blue, long dash), 50% homogeneous/inhomogeneous (pink, dotted), and 100% inhomogeneous (green, short dash), at (a),(b) 150, (c),(d) 450, and (e),(f) 800 m above cloud base. Results for single-parcel adiabatic ascent are denoted by the solid vertical orange line.
Citation: Journal of the Atmospheric Sciences 68, 12; 10.1175/JAS-D-11-0153.1
a. Near cloud base
At 150 m above cloud base, the predicted modes of Nd are near the observed but lack the mode at lower Nd (Fig. 5a). The predicted Nd sometimes exceeds the adiabatic value, as a result of activated entrained CCN, producing some DSDs narrower than observed, especially when droplet evaporation is represented as homogeneous. The predicted mode of σD (Fig. 5b) is identical to the observations when inhomogeneous evaporation is included. The predicted mode of 
b. Upper levels
Higher above cloud base, the predicted Nd agrees better with the observations, including occurrences of small Nd, especially when modeling some droplet evaporation as inhomogeneous (Figs. 5c,e). The largest observed values are not predicted (Fig. 5e), as deeper clouds at these heights likely had faster updrafts than that modeled, activating more CCN at cloud base. When droplet evaporation is modeled as partly inhomogeneous, σD (Figs. 5d,f) is predicted better; the pure inhomogeneous case tends to exaggerate the broadest DSDs. Some predicted DSDs are still very narrow at these levels, symptomatic of underestimating entrainment, also driving overestimates of 
5. Summary and future work
Predicting the breadth of observed DSDs was successful when representing entrainment-produced variability in saturation histories along droplet trajectories, consistent with the 3D dynamics of the cloud, as proposed by Cooper (1989). No other broadening mechanism was required to reproduce the observed DSD widths. The best predictions occurred when the model represented droplet evaporation as partly inhomogeneous.
Future work is still required before the evolution of DSDs from condensational growth can be accurately predicted or parameterized using these results. Despite having measurements of CCN and the thermodynamic environment to initialize the models, and observed DSDs less susceptible to artificial broadening, success was limited by the long-standing difficulty of cloud water overestimation in cumulus simulations, especially in predicting the mean droplet diameter. It is unknown if this result would occur for clouds in other environments with higher droplet number concentrations. Additional progress will require advances in numerical modeling of cumulus entrainment, coincident with overcoming deficiencies in its general understanding (e.g., review by Blyth 1993; Heus et al. 2008), as well as improvements in instrumentation for measuring DSDs with more confidence.
Acknowledgments
William Cooper provided his microphysical model and made suggestions for the observational comparison; Jerry Straka and Richard Carpenter provided the 3D cloud model. Observations were collected by the participants of RICO. Alan Blyth hosted JLB’s collaborative visit to the University of Leeds. David Ebert and the Purdue University Rendering & Perceptualization Lab (PURPL) collaborated on visualization techniques. Amanda Sheffield performed the statistical analysis of Fig. 1. This study was funded by ATM-0342421 and IIS-0513464. Computing support was from the National Center for Atmospheric Research’s (sponsored by the National Science Foundation) Scientific Computing Division, and observation analysis software was from their Research Air Facility.
REFERENCES
Andrejczuk, M., W. W. Grabowski, S. P. Malinowski, and P. K. Smolarkiewicz, 2006: Numerical simulation of cloud–clear air interfacial mixing: Effects on cloud microphysics. J. Atmos. Sci., 63, 3204–3225.
Baker, M. B., R. G. Corbin, and J. Latham, 1980: The influence of entrainment on the evolution of cloud droplet spectra: I. A model of inhomogeneous mixing. Quart. J. Roy. Meteor. Soc., 106, 581–598.
Baumgardner, D., W. A. Cooper, and J. E. Dye, 1990: Optical and electronic limitations of the forward-scattering spectrometer probe. Liquid Particle Size Measurements Techniques, Vol. 2, American Society for Testing and Materials, 115–127.
Blyth, A. M., 1993: Entrainment in cumulus clouds. J. Appl. Meteor., 32, 626–641.
Brenguier, J.-L., and W. W. Grabowski, 1993: Cumulus entrainment and cloud droplet spectra: A numerical model within a two-dimensional dynamical framework. J. Atmos. Sci., 50, 120–136.
Carpenter, R. L., Jr., K. K. Droegemeier, and A. M. Blyth, 1998: Entrainment and detrainment in numerically simulated cumulus congestus clouds. Part I: General results. J. Atmos. Sci., 55, 3417–3432.
Cooper, W. A., 1989: Effects of variable droplet growth histories on droplet size distributions. Part I: Theory. J. Atmos. Sci., 46, 1301–1311.
Cooper, W. A., R. T. Bruintjes, and G. K. Mather, 1997: Calculations pertaining to hygroscopic seeding with flares. J. Appl. Meteor., 36, 1449–1469.
Feingold, G., and P. Y. Chuang, 2002: Analysis of the influence of film-forming compounds on droplet growth: Implications for cloud microphysical processes and climate. J. Atmos. Sci., 59, 2006–2018.
Gerber, H. E., B. G. Arends, and A. S. Ackerman, 1994: New microphysics sensor for aircraft use. Atmos. Res., 31, 235–252.
Gerber, H. E., G. M. Frick, J. B. Jensen, and J. G. Hudson, 2008: Entrainment, mixing and microphysics in trade-wind cumulus. J. Meteor. Soc. Japan, 86A, 87–106.
Heus, T., G. van Dijk, H. J. J. Jonker, and J. E. A. Van den Akker, 2008: Mixing in shallow cumulus clouds studied by Lagrangian particle tracking. J. Atmos. Sci., 65, 2581–2597.
Howell, W. E., 1949: The growth of cloud drops in uniformly cooled air. J. Meteor., 6, 134–149.
Hudson, J. G., and S. Mishra, 2007: Relationships between CCN and cloud microphysics variations in clean maritime air. Geophys. Res. Lett., 34, L16804, doi:10.1029/2007GL030044.
Jonas, P. R., and B. J. Mason, 1982: Entrainment and the droplet spectrum in cumulus clouds. Quart. J. Roy. Meteor. Soc., 108, 857–869.
Lasher-Trapp, S. G., and W. A. Cooper, 2000: Comparison of theory and observations of broadening of cloud droplet size distributions in warm cumuli. Proc. 13th Int. Conf. on Clouds and Precipitation, Reno, NV, ICCP, 90–93.
Lasher-Trapp, S. G., W. A. Cooper, and A. M. Blyth, 2005: Broadening of droplet size distributions from entrainment and mixing in a cumulus cloud. Quart. J. Roy. Meteor. Soc., 131, 195–220.
Latham, J., and R. L. Reed, 1977: Laboratory studies of the effects of mixing on the evolution of cloud droplet spectra. Quart. J. Roy. Meteor. Soc., 103, 297–306.
Lehmann, K., H. Siebert, and R. A. Shaw, 2009: Homogeneous and inhomogeneous mixing in cumulus clouds: Dependence on local turbulence structure. J. Atmos. Sci., 66, 3641–3659.
Mason, B. J., and C. W. Chien, 1962: Cloud-droplet growth by condensation in cumulus. Quart. J. Roy. Meteor. Soc., 88, 136–142.
Perrin, T., J.-L. Brenguier, and T. Bourrianne, 1998: Modeling coincidence effects in the Fast-FSSP with a Monte Carlo model. Preprints, Conf. on Cloud Physics, Amer. Meteor. Soc., Everett, WA, 112–115.
Rauber, R. M., and Coauthors, 2007: Rain in shallow cumulus over the ocean: The RICO campaign. Bull. Amer. Meteor. Soc., 88, 1912–1928.
Shaw, R. A., W. C. Reade, L. R. Collins, and J. Verlinde, 1998: Preferential concentration of cloud droplets by turbulence: Effects on the early evolution of cumulus cloud droplet spectra. J. Atmos. Sci., 55, 1965–1976.
Su, C.-W., S. K. Krueger, P. A. McMurtry, and P. H. Austin, 1998: Linear eddy modeling of droplet spectral evolution during entrainment and mixing in cumulus clouds. Atmos. Res., 47–48, 41–58.
Telford, J. W., and S. K. Chai, 1980: A new aspect of condensation theory. Pure Appl. Geophys., 118, 720–742.
vanZanten, M. C., and Coauthors, 2010: Controls on precipitation and cloudiness in simulations of trade-wind cumulus as observed during RICO. J. Adv. Model. Earth Syst., 3, M06001, doi:10.1029/2011MS000056.
Warner, J., 1973: The microstructure of cumulus cloud. Part IV: The effect on the droplet spectrum of mixing between cloud and environment. J. Atmos. Sci., 30, 256–261.
CLB has extended the modeling framework to include coalescence and sedimentation effects, but these features were not available at the time this study was conducted.
An updated version of the FSSP, manufactured by Droplet Measurement Technologies, Boulder, CO.
A value of 0.1 L−1 is a conservative lower threshold because values as high as 120 L−1 were recorded when targeting rain shafts on other RICO days.
More details regarding the CCN initialized in the model can be found in appendix A of CLB.
Tests conducted by CLB with algorithms that limited decreases, or allowed increases and decreases in θq along the trajectories, were found to bracket the solution of the algorithm used here.
Before compiling frequency distributions, modeled DSDs were “re-binned” so the bin sizes were identical to the SPP-100, ranging from 2.35 to 45.75 μm. Droplets outside this range, or having a Nd less than the minimum detectable Nd (10−2 cm−3), were excluded. Additionally, observed DSDs were averaged over a 30-m distance, equivalent to the cloud model resolution.
