1. Introduction
The cloud physics community has grappled with the “growth gap” problem in warm-rain formation for a few decades now. Turbulence has been considered as one of the mechanisms to overcome this bottleneck between cloud droplet growth by diffusion and growth by collision–coalescence (Shaw 2003; Grabowski and Wang 2013). Early studies in the 1960s had proposed already that stochastic condensation itself may be able to bridge this growth gap by means of turbulence-induced fluctuations in supersaturation and diffusive mixing [e.g., see the recent summary by Mazin and Merkulovich (2008)]. This can produce a broad size distribution and depart from conventionally imagined narrow droplet size distributions due to uniform condensational growth. However, Bartlett and Jonas (1972) and others argued that such a process cannot explain broadening of the size distribution since the required supersaturation fluctuations within clouds are closely correlated with updrafts. Hence, a droplet experiencing higher supersaturations will be in a stronger updraft, thereby reducing the amount of time it will get to grow before reaching a level with steady-state supersaturation again. Manton (1979) proposed that the broadening of size distributions may then be primarily due to turbulence induced by entrainment and mixing instead of vertical velocity. Cooper (1989) argued that cloud turbulence due to vertical velocity variations may still induce broadening of droplet size distributions by considering variations in the integral radius (and thus the phase relaxation time of droplets, both of which we shall consider in this paper). Further refinements to the theory of stochastic condensation have been made, and it has been suggested that these approaches are attractive for implementation within a computational framework, for example, for large-eddy simulation (LES; Khvorostyanov and Curry 1999; Jeffery et al. 2007).
Other approaches linked to stochastic condensation have dealt in particular with the source of supersaturation variability. Srivastava (1989) put forward the argument that the local supersaturation around a drop and not the average supersaturation of a cloud parcel should be taken into account while calculating droplet growth rates. This local supersaturation may be significantly different for every drop from the average value because of local variability in the droplet number concentration and vertical air velocity. Since then, some attention has been given to the possible influence of spatial and temporal variability in droplet concentration due to finite droplet inertia (e.g., Shaw et al. 1998; Vaillancourt et al. 2002; Lanotte et al. 2009). Furthermore, Gerber (1991) and Korolev and Isaac (2000) have shown that broadening due to supersaturation variability from isobaric mixing or even mixing between vertically cycling parcels (Korolev et al. 2013) can be important. Whatever the source, careful in situ measurements by Cooper (1989), Politovich (1993), and Brenguier and Chaumat (2001) have found broadening of the droplet size distribution even in cumulus cloud cores.
Recent studies by Sardina et al. (2015) showed that turbulence causes an increase in the variance of the droplet size distribution with time, producing a broad droplet size distribution. This broadening may provide enough large droplets to cross the growth gap and start the collision–coalescence process. They used the stochastic approach suggested by Paoli and Shariff (2009) followed by direct numerical simulation (DNS) and LES to show that the variance grows as t1/2 but did not include any effects of changing aerosol or droplet number concentration. Siewert et al. (2017) performed a study using three-dimensional DNS and compared it with a similar stochastic Lagrangian model. They found different regimes of broadening depending on how the droplet growth time scale and supersaturation field response time scale compared with the turbulent mixing time scale. However, most of these modeling approaches assume a constant phase relaxation time. Using laboratory measurements, Chandrakar et al. (2016) produced a steady-state warm turbulent cloud and showed that the width of the droplet size distribution increases with a decrease in the aerosol concentration because of larger variability in supersaturation fluctuations. However, to characterize cloud microphysics, they used a phase Doppler interferometer, which requires averaging times that are much greater than the large-eddy time scale. Thus, they too obtained a droplet size distribution that is assumed to be spatially constant within the entire chamber, leading to a spatially constant phase relaxation time τc.
Variation in τc, with the resulting fluctuations in the local supersaturation, can occur for a large number of reasons. Baker and Latham (1979) found that inhomogeneous mixing events due to entrainment can greatly reduce the droplet number concentration in some regions. Shaw et al. (1998) suggested that turbulent mixing can cause clustering of particles away from regions of high vorticity, causing broadening of the droplet size distributions due to supersaturation fluctuations. Lasher-Trapp et al. (2005) combined three-dimensional cloud model with a Lagrangian microphysical parcel model that mapped droplet trajectories that ended up at the same point. They found large size widths along with small droplets high in the clouds, because of different entrainment and mixing zones the trajectories encountered. Grabowski and Abade (2017) further investigated this mechanism for size distribution broadening by imposing droplet mixing and different growth histories, resulting in significant increase in broadening compared to adiabatic parcel models.
In this paper, we ask the following questions: How reasonable is the approximation of constant τc? How does the size distribution vary spatially? Can spatial variability in droplet number and size lead to a significant change in τc, and how does this affect the mean and width of the droplet size distribution? We will not directly address the causes of variations in number concentrations but rather study their influence on the droplet size distribution through condensation growth. We will therefore not consider possible effects of variations in number concentration on droplet growth by collision–coalescence. We attempt to answer these questions by creating a turbulent cloud in a laboratory cloud chamber called the Π chamber, similar to Chandrakar et al. (2016). [See Chang et al. (2016) for details about the chamber.] Such a controlled experiment is crucial for evaluating the theory because long-time averages are required, thereby necessitating statistically stationary conditions that are difficult to attain in the atmosphere. The theory validated under such idealized conditions can then be applied to more complex atmospheric conditions. Furthermore, in the laboratory, we can adjust the parameter space to simulate desired cloud conditions optimal for the analysis. For example, since our cloud exists in an enclosed space, we can be assured that observed broadening of the size distribution will not be due to entrainment effects. The estimated cloud droplet collision times for droplet sizes obtained during the experiment are on the order of 1 h, while the droplet residence times are on the order of 1–10 min. Hence, we are also confident that the effects on droplet size are not due to collision–coalescence but primarily due to stochastic condensation. LES calculations have also confirmed this result (Chandrakar et al. 2016). Finally, a crucial aspect of the experimental approach is the ability to measure the droplet size distribution from a spatially localized volume without temporal averaging, using digital in-line holography. This approach allows the questions about microphysical variability, as expressed through the integral radius and the phase relaxation time, to be investigated.
The paper proceeds as follows: First, in section 2, we present the theoretical approach based on stochastic differential equations (SDEs) for supersaturation and droplet growth, with nonconstant phase relaxation time. In section 3, we describe the laboratory experiments, emphasizing the holographic system and how it enables measurement of the quantities that come out of the SDEs. In section 4, we present the results of the experiments, and finally, in section 5, we discuss the findings and implications of those results.
2. Stochastic condensation for variable 
































3. Experimental approach
a. Chamber setup and instruments
The microphysical variability of a steady-state cloud in a turbulent environment is studied by creating a turbulent mixing cloud in the Michigan Technological University Π chamber. A buoyancy-driven convective flow environment (Rayleigh–Bénard convection) is created by applying an unstable temperature difference between the top and bottom surfaces of the chamber. To achieve supersaturated conditions for cloud formation and growth, the top and bottom boundaries are kept saturated with liquid water. The turbulent Rayleigh–Bénard convection mixes air parcels from the top and bottom boundaries; this isobaric mixing of air parcels containing different water vapor concentrations at distinct temperatures creates a supersaturated environment for aerosol activation and subsequently cloud droplet growth. Stated in a slightly different way, in the context of Eq. (4), the top and bottom boundaries are held at fixed temperatures and equilibrium vapor pressures, and the resulting heat and vapor fluxes tend to force a background supersaturation
In this set of experiments, a 19°C temperature difference between the top (Tt = 7°C) and bottom (Tb = 26°C) boundaries is used to drive the turbulent moist convection. Both these boundaries are saturated with water, and the sidewalls are maintained at the mean temperature between these boundaries. The sidewalls are also covered by 3.2-mm-thick polycarbonate sheets to minimize heat flux (relative to the top and bottom boundaries). Once the turbulent and thermodynamic properties, such as temperature gradient between the boundaries, water vapor concentration, and velocity fields, reach a steady state, salt aerosol injection is started at a constant rate (5000 cm−3 at 2 L min−1 of inlet airflow). A fraction of injected aerosol is activated once they experience supersaturation more than the critical value, and then they grow in this fluctuating supersaturated environment. As the droplets grow, gravitational sedimentation becomes a significant loss mechanism, and this limits the lifetime of individual cloud droplets inside the chamber. The chamber is allowed to run in this state for a couple of hours until the rate of activation of new droplets matches the rate of droplet sedimentation, giving us a steady-state droplet number concentration and droplet size distribution as measured by the phase Doppler interferometer. As noted in the previous section, the fact that the cloud is in a dynamic steady state implies that even though collective properties are constant in time, individual droplets experience growth by condensation at an average rate given by Eq. (2).
Thermodynamic properties required for this study are air temperature and water vapor concentration. Resistance thermometers (RTDs; Minco) and a LI-COR hygrometer (7500A) are used to measure these properties. To characterize the turbulent flow properties, a sonic anemometer is employed, which measures the flow velocity in all three directions at a high frequency (20 Hz). Aerosol particles are generated by atomizing an NaCl–water solution using an atomizer (3076 TSI) and subsequently passing it through a diffusion dryer. During this experiment, the measured turbulent kinetic energy (TKE) was approximately 0.004 m2 s−2, dissipation rate ε = 10−3 m2 s−3, a Reynolds number (Re) of 80, and a Rayleigh number for dry conditions (Ra) of 2 × 109. Most of the values are comparable to atmospheric clouds, whereas TKE and Re are smaller because of the chamber dimensions compared to typical large-eddy lengths in the atmosphere (Chang et al. 2016). Details of the microphysical measurement approach are discussed in the subsequent section.
b. Microphysical measurements and holographic setup
The key measurements in this study are of the cloud droplet size distribution and number concentration, in order to assess microphysical variability and its contribution to stochastic condensation. Specifically, from Eqs. (15) and (16), it can be seen that
The holographic setup consists of a collimated Crystal Laser Systems (CryLaS) 532-nm laser that is passed through the top access port of the Π chamber (Fig. 1). This beam passes through the chamber and is received by a K2 Distamax Lens and a Photron Fastcam SA2 camera looking up through the bottom access port of the chamber. The lens provides a magnification of 2.85 times at the focal plane, which is flush with the bottom of the inner chamber. The camera has a 2048 × 2048 pixel detector with a single pixel pitch of 10 μm. The equivalent pixel size thus obtained is 3.5 μm; however, at least two pixels are needed to confidently distinguish a particle from the background image. Hence, we have rejected any possible particle smaller than 7 μm. A burst of seven consecutive images is taken every 30 s for 2.5 h at 500 frames per second (fps). The time interval of 30 s was chosen since it was sufficiently longer than the turbulence decorrelation time scale (≈10τt) measured by the sonic anemometer. The seven images at every instant are used for averaging and background division used for improvement of signal-to-noise ratio. Bench tests showed that the resolution of the holograms does not depreciate much through 20 cm beyond the focal plane. This gives us a total measurement volume of approximately 10 cm3 and an average number concentration of 80 droplets per cubic centimeter. This measurement volume is on the order of the large-eddy length scale and therefore is consistent with the ability to resolve the turbulent fluctuations in microphysical properties. The digital holograms are then reconstructed numerically at every 100-μm depth into the focal plane using the convolution method in Fourier space (Fugal et al. 2004). This procedure allows us to find droplets and their diameters using a light intensity threshold (Lu et al. 2008). Lu et al. (2008) and Henneberger et al. (2013) also showed that the error in diameter calculation using digital holography is approximately the square root of the equivalent pixel size. In our case, this is approximately 1.87 μm.
(top) Optical assembly above the Π chamber. (bottom left) A side-view schematic of the digital in-line holographic setup; not to scale. (bottom right) The imaging assembly below the Π chamber. A steady-state warm mixing cloud exists throughout the volume of the chamber. Cloud parcels passing the measurement volume are imaged and analyzed for the study.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0158.1
The uncertainty in the mean radius r for each hologram can be obtained using a normal distribution as
4. Results
a. Local microphysical variability
Fundamentally, this study is about a quantity very difficult to access in both naturally occurring and laboratory clouds: What does the cloud droplet size distribution look like on the “local” scale at which droplets are interacting through vapor and temperature fields? As discussed already, this information is inaccessible to typical droplet-by-droplet measurements, which require averaging over long times or, in the case of field measurements, over long distances. For example, Chandrakar et al. (2016) needed a minimum of a 10-min average to obtain reasonable estimate of the size distribution but used a 100-min average for better statistics. However, measurements provided by the digital holographic system described in section 3 allow us to observe the instantaneous and spatially local size distribution and how it varies in time, as illustrated in Fig. 2. Later, we will consider the consequences of that microphysical variability. Figure 2 shows eight examples of instantaneous droplet size distributions compared to an average over the eight frames. The time separating each realization is several large-eddy turnover times, and indeed, we observe significant changes in the number concentration and mean size of the droplets.
Variation in droplet size distribution within consecutive holograms taken 30 s apart. Steady-state cloud conditions had already been attained at t = 0 s when the measurement was started. We see considerable variation in the droplet size distribution even for this small sample set of eight measurements with respect to the mean over the same eight holograms.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0158.1
How does this variability affect the value of the local phase relaxation time τc? It is worth noting again that τc has typically been assumed to be constant in theoretical treatments, with relatively few exceptions (Cooper 1989). We can construct a probability density function (pdf) of τc by calculating τc from the droplets in each single hologram, and repeating for a large number of statistically independent realizations, under statistically stationary conditions. We emphasize that the simultaneous achievement of statistically independent samples and statistically stationary conditions is difficult to achieve in the atmosphere (Wyngaard 2010, chapter 2) and can therefore be considered an advantage of the laboratory approach. The pdf thus obtained is displayed in Fig. 3 and has a mean of approximately 17 s and a standard deviation of 10 s. Significantly, the pdf has a positive skewness, with the right tail exceeding 60 s. The parcels with such large τc lie in the extreme “slow microphysics” limit and may therefore be important for broadening of the droplet size distribution (Chandrakar et al. 2016).
Probability density function of τc shows a long tail for higher values of τc. These higher values correspond to holograms with smaller droplet number concentrations and corresponding higher supersaturation fluctuations.
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0158.1
The cause of variability in τc comes from some combination of fluctuations in the droplet number density and the mean droplet radius. Figure 4 shows the local-mean droplet radius 〈r〉 as a function of
(top) Observed values for mean radius
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0158.1
Cubed radius against number concentration with lines of constant LWC (g m−3). Here,
Citation: Journal of the Atmospheric Sciences 75, 1; 10.1175/JAS-D-17-0158.1
Finally, the relative dispersion of cloud droplet radius,
b. Quantities relevant to stochastic condensation













The units for the first two quantities are in square microns, while the rest of the quantities have been nondimensionalized. The last two ratios are evaluated using the stochastic model and are not measured directly. We see very little change due to variability for the mean size but a significant change for the width of the size distribution.
We now continue working toward estimating Eqs. (8) and (16) and how much they change for assumed constant or variable τc. Measurements from the LI-COR give us an initial variance of the supersaturation distribution
The resulting values for Eq. (15), however, show that the growth rate of the width of the droplet size distribution is significantly enhanced when turbulent fluctuations in τc are taken into account (by approximately 60%, as shown in the last line of Table 1). Thus, we may expect a significantly larger width in the droplet size distribution than predicted using constant τc. Qualitatively, this seems straightforward, since fluctuations in τc will affect the local supersaturation response, but quantitatively, this can be considered a rare instance in which stochastic condensation theory has been directly linked to measurements.
c. Comparison between theory and measurements
The logic thus far has been to use the holographic system to measure






5. Discussion
Most previous studies on stochastic condensation (Cooper 1989; Field et al. 2014; Sardina et al. 2015; Grabowski and Abade 2017) introduced supersaturation fluctuations by implementing the Langevin noise term through vertical velocity similar to Mazin and Smirnoff (1969). This approach had the drawback as mentioned by Bartlett and Jonas (1972) that larger vertical velocity fluctuations will lead to larger supersaturation values but the time spent by droplets in these regions will be smaller. This negates the effect of higher supersaturations, and the droplets will not grow as large. Here, we have implemented the noise term directly through supersaturation similar to Paoli and Shariff (2009) and Chandrakar et al. (2016). The fluctuations in droplet number concentration are likely due to a combination of localized droplet activation in regions of high supersaturation and differential droplet sedimentation. The number concentration fluctuations, in turn, influence the supersaturation field independently from vertical velocity.
In this paper, we have described an experiment to investigate the importance of the assumption of constant phase relaxation time τc on the process of stochastic condensation. We produced a steady-state warm turbulent mixing cloud and observed the effect of droplet number and size variations within the cloud on the phase relaxation time, using a digital holographic measurement method. The data are analyzed in the context of stochastic theory with and without constant τc. The results suggest that for our conditions, the assumption of constant τc is a good enough approximation for the growth rate of the mean squared radius but not such a good approximation for calculating the rate of growth of the width of the droplet size distribution. The result is interesting because it suggests that in addition to the importance of supersaturation variability, which was observed by Chandrakar et al. (2016), the microphysical variability can also play an important role.
Physically, we can interpret the finding as follows. Consider a small parcel of cloudy air that suddenly finds itself with lower droplet number concentrations compared to the average because of turbulent fluctuations. Let us assume that most of the droplets in this parcel begin with a size close to the mean. This parcel has different properties from its surroundings and, hence, soon gets mixed with neighboring parcels, bringing the droplet number concentration back to the mean. The few droplets that existed within this parcel were able to grow for a small amount of time during which the parcel was isolated, because of the decreased competition for available water vapor. But the enhanced growth halted as soon as the parcel underwent turbulent mixing. When averaged together with other droplets, including those that may have experienced suppressed growth on account of being contained in a different parcel with a higher-than-mean droplet concentration, a significant change in
With respect to the applicability of these results to clouds, this study motivates a comparison with in situ cloud measurements where the fluctuations in integral radius can be recorded. The conditions created in the laboratory are very close to those expected within nearly steady-state stratocumulus clouds. We expect that large variations in supersaturation will exist in clouds with low droplet number concentrations (Chandrakar et al. 2016). But what level of microphysical variability is observed? This motivates the need for in situ observations of fluctuation terms like
Acknowledgments
We thank D. Ciochetto for assistance in setting up the holographic system, J. Fugal for mentoring with the hologram reconstruction, and J. Lu and F. Yang for helpful discussions on the theory. This work was supported by the National Science Foundation, Grant AGS-1623429.
APPENDIX
Additional Derivation Steps
c. Steps between Eqs. (7) and (9)





e. Steps for Eq. (11)




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