Cloud-droplet growth due to supersaturation fluctuations in stratiform clouds

Condensational growth of cloud droplets due to supersaturation fluctuations is investigated by solving the hydrodynamic and thermodynamic equations using direct numerical simulations with droplets being modeled as Lagrangian particles. The supersaturation field is calculated directly by simulating the temperature and water vapor fields instead of being treated as a passive scalar. Thermodynamic feedbacks to the fields due to condensation are also included for completeness. We find that the width of droplet size distributions increases with time, which is contrary to the classical theory without supersaturation fluctuations, where condensational growth leads to progressively narrower size distributions. Nevertheless, in agreement with earlier Lagrangian stochastic models of the condensational growth, the standard deviation of the surface area of droplets increases as $t^{1/2}$. Also, for the first time, we explicitly demonstrate that the time evolution of the size distribution is sensitive to the Reynolds number, but insensitive to the mean energy dissipation rate. This is shown to be due to the fact that temperature fluctuations and water vapor mixing ratio fluctuations increases with increasing Reynolds number, therefore the resulting supersaturation fluctuations are enhanced with increasing Reynolds number. Our simulations may explain the broadening of the size distribution in stratiform clouds qualitatively, where the mean updraft velocity is almost zero.


Introduction
The growth of cloud droplets is dominated by two processes: condensation and collection. Condensation of water vapor on 15 active cloud condensation nuclei is important in the size range from the activation size of aerosol particles to about a radius of 10 µm (Pruppacher and Klett, 2012;Lamb and Verlinde, 2011). Since the rate of droplet growth by condensation is inversely proportional to the droplet radius, large droplets grow slower than smaller ones. This generates narrower size distributions (Lamb and Verlinde, 2011). To form rain droplets in warm clouds, small droplets must grow to about 50 µm in radius within Therefore, collection, a widely accepted microscopical mechanism, has been proposed to explain the rapid formation of rain droplets (Saffman and Turner, 1956;Berry and Reinhardt, 1974;Shaw, 2003;Grabowski and Wang, 2013). However, collection can only become active when the size distribution reaches a certain width. Hudson and Svensson (1995) observed a broadening of the droplet size distribution in Californian marine stratus, which 5 was contrary to the classical theory of condensational growth (Yau and Rogers, 1996). The increasing width of droplet size distributions were further observed by Pawlowska et al. (2006) and Siebert and Shaw (2017b). The contradiction between the observed broadening width and the theoretical narrowing width in the absence of turbulence has stimulated several studies. The classical treatment of diffusion-limited growth assumes that supersaturation depends only on average temperature and water mixing ratio. Since fluctuations of temperature and the water mixing ratio are affected by turbulence, the supersaturation fluc-10 tuations are inevitably subjected to turbulence. Naturally, condensational growth due to supersaturation fluctuations became the focus (Srivastava, 1989;Korolev, 1995;Sardina et al., 2015;Siewert et al., 2017;Grabowski and Abade, 2017). The supersaturation fluctuations are particularly important for understanding the condensational growth of cloud droplets in stratiform clouds, where the updraft velocity of the parcel is almost zero (Hudson and Svensson, 1995;Korolev, 1995). When the mean updraft velocity is not zero, there could be a competition between mean updraft velocity and supersaturation fluctuations. This 15 may diminish the role of supersaturation fluctuations (Sardina et al., 2018).
Condensational growth due to supersaturation fluctuations was first recognized by Srivastava (1989), who criticized the use of a volume-averaged supersaturation and proposed a randomly distributed supersaturation field. Using direct numerical simulations (DNS), Vaillancourt et al. (2002) found that turbulence has negligible effect on condensational growth and attributed this to the decorrelation between the supersaturation and the droplet size. Paoli and Shariff (2009) considered three-dimensional (3-20 D) turbulence as well as stochastically forced temperature and vapor fields with a focus on statistical modeling for large-eddy simulations. They found that supersaturation fluctuations are responsible for the broadening of the droplet size distribution, which is contrary to the findings by Vaillancourt et al. (2002). Lanotte et al. (2009) conducted 3-D DNS for condensational growth by only solving a passive scalar equation for the supersaturation and concluded that the width of the size distribution increases with increasing Reynolds number. Sardina et al. (2015) extended the DNS of Lanotte et al. (2009) to higher Reynolds 25 number and found that the variance of the size distribution increases in time. In a similar manner as Sardina et al. (2015), Siewert et al. (2017) modelled the supersaturation field as a passive scalar coupled to the Lagrangian particles and found that their results can be reconciled with those of earlier numerical studies by noting that the droplet size distribution broadens with increasing Reynolds number (Paoli and Shariff, 2009;Lanotte et al., 2009;Sardina et al., 2015). Neither Sardina et al. (2015) nor Siewert et al. (2017) solved the thermodynamics that determine the supersaturation field. Both Saito and Gotoh (2017) and 30 Chen et al. (2018) solved the thermodynamics equations governing the supersaturation field. However, since collection was also included in their work, one cannot clearly identify the roles of turbulence on collection or condensational growth, nor can one compare their results with Lagrangian stochastic models (Sardina et al., 2015;Siewert et al., 2017) related to condensational growth. Grabowski and Wang (2013) proposed the eddy-hopping mechanism to explain the broadening and investigated it in Grabowski and Abade (2017).

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Recent laboratory experiments and observations about cloud microphysics also confirm the notion that supersaturation fluctuations may play an important role in broadening the size distribution of cloud droplets. The laboratory study by Chandrakar et al. (2016) suggested that supersaturation fluctuations in the low aerosol number concentration limit are likely of leading importance for precipitation formation. The condensational growth due to supersaturation fluctuations seems to be more sensitive to the integral scale of turbulence (Götzfried et al., 2017). Siebert and Shaw (2017a) measured the variability of tempera-5 ture, water vapor mixing ratio, and supersaturation in warm clouds and support the notion that both aerosol particle activation and droplet growth take place in the presence of a broad distribution of supersaturation (Hudson and Svensson, 1995;Brenguier et al., 1998;Miles et al., 2000;Pawlowska et al., 2006). The challenge is now how to interpret the observed broadening of droplet size distribution in warm clouds. How does turbulence drive fluctuations of the scalar fields (temperature and water vapor mixing ratio) and therefore affect the broadening of droplet size distributions (Siebert and Shaw, 2017a)?

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In an attempt to answer this question, we conduct 3-D DNS experiments of condensational growth of cloud droplets, where turbulence, thermodynamics, feedback from droplets to the fields via the condensation rate and buoyancy force are all included.
The main aim is to investigate how supersaturation fluctuations affect the droplet size distribution. We particularly focus on the time evolution of the size distribution f (r, t) and its dependency on small and large scales of turbulence. We then compare our simulation results with Lagrangian stochastic models (Sardina et al., 2015;Siewert et al., 2017). For the first time, the 15 stochastic model and simulation results from the complete set of equations governing the supersaturation field are compared.

Numerical model
We now discuss the basic equations where we combine the Eulerian description of the density (ρ), turbulent velocity (u), temperature (T ), and water vapor mixing ratio (q v ) with the Lagrangian description of the ensemble of cloud droplets. The water vapor mixing ratio q v is defined as the ratio between the mass density of water vapor and dry air. Droplets are treated 20 as superparticles. A superparticle represents an ensemble of droplets, whose mass, radius, and velocity are the same as those of each individual droplet within it (Shima et al., 2009;Johansen et al., 2012;Li et al., 2017). For condensational growth, the superparticle approach (Li et al., 2017) is the same as the Lagrangian point-particle approach (Kumar et al., 2014) since there is no interactions among droplets. Nevertheless, we still use the superparticle approach so that we can include more processes like collection (Li et al., 2017(Li et al., , 2018 in future. Another reason to adopt superparticle approach is that it can be 25 easily adapted to conduct Large-eddy simulations with appropriate sub-grid scale models (Grabowski and Abade, 2017). To investigate the condensational growth of cloud droplets that experience fluctuating supersaturation, we track each individual superparticle in a Lagrangian manner. The motion of each superparticle is governed by the momentum equation for inertial particles. The supersaturation field in the simulation domain is determined by T (x, t) and q v (x, t) transported by turbulence.
Lagrangian droplets are exposed in different supersaturation fields. Therefore, droplets either grow by condensation or shrink 30 by evaporation depending on the local supersaturation field. This phase transition generates a buoyancy force, which in turn affects the turbulent kinetic energy, T (x, t), and q v (x, t). PENCIL CODE is used to conduct all the simulations.

Equations of motion for Eulerian fields
The background air flow is almost incompressible and thus obeys the Boussinesq approximation. Its density ρ(x, t) is governed by the continuity equation and velocity u(x, t) by Navier-Stokes equation. The temperature T (x, t) of the background air flow is determined by the energy equation with a source term due to the latent heat release. The water vapor mixing ratio q v (x, t) is transported by the background air flow. The Eulerian equations are given by Du 10 where D/Dt = ∂/∂t+u·∇ is the material derivative, f is a random forcing function (Haugen et al., 2004), ν is the kinematic viscosity of air, is the traceless rate-of-strain tensor, p is the gas pressure, ρ is the gas density, c p is the specific heat at constant pressure, L is the latent heat, κ is the thermal diffusivity of air, C d is the condensation 15 rate, B is the buoyancy, e z is the unit vector in the z direction (vertical direction), and D is the diffusivity of water vapor.
To avoid global transpose operations associated with calculating Fourier transforms for solving the nonlocal equation for the pressure in strictly incompressible calculations, we solve here instead the compressible Navier-Stokes equations using highorder finite differences. The sound speed c s obeys c 2 s = γp/ρ, where γ = c p /c v = 7/5 is the ratio between specific heats, c p and c v , at constant pressure and constant volume, respectively. We set the sound speed as 5 m s −1 to simulate the nearly 20 incompressible atmospheric air flow, resulting in a Mach number of 0.06 when u rms = 0.27 m s −1 , where u rms is the rms velocity. Such a configuration, with so small Mach number, is almost equivalent to an incompressible flow. It is worth noting that the temperature determining the compressibility of the flow is constant and independent of the temperature field of the gas flow governed by Equation (3). Also, since the gas flow is almost incompressible and its mass density is much smaller than the one of the droplet, there is no mass exchange between the gas flow and the droplet, i.e., the density of the gas flow 25 ρ(x, t) is not affected by T (x, t). Thus, the source terms S ρ and S u in Equations (1) and (2) are neglected (Krüger et al., 2017).
The buoyancy B(x, t) depends on the temperature T (x, t), water vapor mixing ratio q v (x, t), and the liquid mixing ratio q l (Kumar et al., 2014), where α = M a /M v − 1 ≈ 0.608 when M a and M v are the molar masses of air and water vapor, respectively. The amplitude of 30 the gravitational acceleration is given by g. The liquid water mixing ratio is the ratio between the mass density of liquid water and the dry air and is defined as where ρ l and ρ a are the liquid water density and the reference mass density of dry air. N △ is the total number of droplets in a cubic grid cell with volume (∆x) 3 , where ∆x is the one-dimensional size of the grid box. The temperature fluctuations are given by and the water vapor mixing ratio fluctuations by We adopt the same method as in Kumar et al. (2014), where the mean environmental temperature T env and water vapor mixing ratio q v,env do not change in time. This assumption is plausible in the circumstance that we do not consider the entrainment, 10 i.e., there is only mass and energy transfer between liquid water and water vapor. The condensation rate C d (Vaillancourt et al., 2001) is given by where G is the condensation parameter (in units of m 2 s −1 ), which depends weakly on temperature and pressure and is here assumed to be constant (Lamb and Verlinde, 2011). The supersaturation s is defined as the ratio between the vapor pressure e v 15 and the saturation vapor pressure e s , Using the ideal gas law, Equation (10) can be expressed as, In terms of the water vapor mixing ratio q v = ρ v /ρ a and saturation water vapor mixing ratio q vs = ρ vs /ρ a , Equation (11) can 20 be written as: Here ρ v is the mass density of water vapor and ρ vs the mass density of saturated water vapor, and q vs (T ) is the saturation water vapor mixing ratio at temperature T and can be determined by the ideal gas law, 25 The saturation vapor pressure e s over liquid water is the partial pressure due to the water vapor when an equilibrium state of evaporation and condensation is reached for a given temperature. It can be determined by the Clausius-Clapeyron equation, which determines the change of e s with temperature T . Assuming constant latent heat L, e s is approximated as (Yau and Rogers, 1996;Götzfried et al., 2017) where c 1 and c 2 are constants adopted from page 14 of Yau and Rogers (1996). We refer to Table 1 for all the thermodynamics constants. In the present study, the updraft cooling is omitted. Therefore, the assumption of constant latent heat L is plausible.

Lagrangian model for cloud droplets
In addition to the Eulerian fields described in Section 2.1 we treat cloud droplets as Lagrangian particles. In the PENCIL CODE, they are invoked as non-interacting superparticles.

Kinetics of cloud droplets
Each superparticle is treated as a Lagrangian point-particle, where one solves for the particle position x i , and its velocity V i via 15 in the usual way; see (Li et al., 2017) for details. Here, u is the fluid velocity at the position of the superparticle, τ i is the particle inertial response or stopping time of a droplet i and is given by The correction factor (Schiller and Naumann, 1933;Marchioli et al., 2008), 20 models the effect of non-zero particle Reynolds number Re i = 2r i |u − V i |/ν. This is a widely used approximation, although it does not correctly reproduce the small-Re i correction to Stokes formula (Veysey and Goldenfeld, 2007).

Condensational growth of cloud droplets
The condensational growth of the particle radius r i is governed by (Pruppacher and Klett, 2012;Lamb and Verlinde, 2011) 25 3 Experimental setup

Initial configurations
The initial values of the water vapor mixing ratio q v (x, t = 0) = 0.0157 kg · kg −1 and temperature T (x, t = 0) = 292 K are matched to the ones obtained in the CARRIBA experiments (Katzwinkel et al., 2014), which are the same as those in Götzfried et al. (2017). With this configuration, we obtain s(x, t = 0) = 2%, which means that the water vapor is initially supersaturated. The 5 time step of the simulations presented here is governed by the smallest time scale in the present configuration, which is the particle stopping time defined in Equation (17). The thermodynamic time scale is much larger than the turbulent one. Table 1 shows the list of thermodynamic parameters used in the present study.
Initially, 10 µm-sized droplets with zero velocity are randomly distributed in the simulation domain. The mean number density of droplets, which is constant in time since droplet collections are not considered, is n 0 = 2.5 × 10 8 m −3 . This gives an 10 initial liquid water content,

DNS
We conduct high resolution simulations for different Taylor micro-scale Reynolds number Re λ and mean energy dissipation rateǭ (see Table 2 for details of the simulations). The Taylor micro-scale Reynolds number is defined as Re λ ≡ u 2 rms 5/(3νǭ).
For simulations with different values ofǭ at fixed Re λ , we vary both the domain size L x (L y = L z = L x ) and the amplitude 20 of the forcing f 0 . As for fixedǭ, Re λ is varied by solely changing the domain size, which in turn changes u rms . In all simulations, we use for the Prandtl number Pr = ν/κ = 1 and for the Schmidt number Sc = ν/D = 0.6. For our simulations with N grid = 512 3 meshpoints, the code computes 55,000 time steps in 24 hours wall-clock time using 4096 cores. For N grid = 128 3 meshpoints, the code computes 4.5 million time steps in 24 hours wall-clock time using 512 cores.

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Figure 1(a) shows time-averaged turbulent kinetic-energy spectra for different values ofǭ at fixed Re λ ≈ 130. Since the abscissa in the figures is normalized by k η = 2π/η, the different spectra shown in Figure 1(a) collapse onto a single curve. Here, η is the Kolmogorov length scale. Figure 1(b) shows the time-averaged turbulent kinetic-energy spectra for different values of Re λ at fixedǭ ≈ 0.039 m 2 s −3 . For larger Reynolds numbers the spectra extend to smaller wavenumbers. A flat profile corresponds to Kolmogorov scaling (Pope, 2000) when the energy spectrum is compensated byǭ −2/3 k 5/3 . For the largest Re λ in our 30 simulations (Re λ = 130), the inertial range extends for about a decade in k-space.
Ma (g · mol −1 ) 28.97 Mv (g · mol −1 ) 18.02 Tenv (K) 293 Next we inspect the response of thermodynamics to turbulence. In Figure 2, we show time series of fluctuations of temperature T rms , water vapor mixing ratio q v,rms , buoyancy force B rms , and the supersaturation s rms . All quantities reach a statistically steady state within a few seconds. The steady state values of T rms , q v,rms , and s rms increase with increasing Re λ approximately linearly, and vary hardly at all withǭ. On the other hand. B rms changes only by a few percent as Re λ orǭ vary.  Table 2 for details).
Note, however, that the buoyancy force is only about 0.3% of the fluid acceleration. This is because T rms is small (about 0.1 K in the present study). Therefore, the effect of the buoyancy force should indeed be small.
When changingǭ while keeping Re λ fixed, the Kolmogorov scales of turbulence varies. Therefore, the various fluctuations quoted above are insensitive to the small scales of turbulence. However, when varying Re λ while keepingǭ fixed, their rms values change, which is due to large scales of turbulence. Indeed, temperature fluctuations are driven by the large scales 5 of turbulence, which affects the supersaturated vapor pressure q vs via the Clausius-Clapeyron equation; see Equation (13).
Therefore, supersaturation fluctuations result from both temperature fluctuations and water vapor fluctuations via Equation (12).
Both q v,rms and T rms increase with increasing Re λ , resulting in larger fluctuations of s. Supersaturation fluctuations, in turn, affect T and q v via the condensation rate C d .
Our goal is to investigate the condensational growth of cloud droplets due to supersaturation fluctuations. Figure 3 shows the 10 time evolution of droplet size distributions for different configurations. The conventional understanding is that condensational growth leads to a narrow size distribution (Pruppacher and Klett, 2012;Lamb and Verlinde, 2011). However, supersaturation fluctuations broaden the distribution. More importantly, the width of the size distribution increases with increasing Re λ , but decreases slightly with increasingǭ over the range studied here. This is consistent with the results shown in Figure 2 in that supersaturation fluctuations are sensitive to Re λ but are insensitive toǭ. In atmospheric clouds, Re λ ≈ 10 4 , which may result 15 in an even broader size distribution.
We further quantify the variance of the size distribution by investigating the time evolution of the standard deviation of the droplet surface area σ A for different configurations. In terms of the droplet surface area A i (A i ∝ r 2 i ), Equation (19)  written as It can be seen from Equation (20) that the evolution of the surface area is analogous to Brownian motion, indicating that its standard deviation σ A ∝ √ t. A more detailed stochastic model for σ A is developed by Sardina et al. (2015). Based on Equation (19), σ A is given by Sardina et al. (2015) adopted a Langevin equation to model the supersaturation field and the vertical velocity of droplets, resulting in the scaling law:  Table 2) and (b) Re λ atǭ = 0.039m 2 s −3 (Runs A, B, and C in Table 2). Same simulations as in Figure 1.
where C(τ L , τ s,Re λ ) is a constant for given τ L , τ s , and Re λ . Under the assumptions that τ s ≪ T L and a negligible influence on the macroscopic observables from small-scale turbulent motions, Sardina et al. (2015) obtained an analytical expression for σ A as: where τ s is the phase transition time scale given by and τ L is the turbulence integral time scale. The model proposed that condensational growth of cloud droplets depends only on Re λ and is independent ofǭ. In terms of the size distribution f (r, t), σ A can be given as: where a ζ is the moment of the size distribution, which is defined as: Here, ζ is a positive integer. As shown in Figure 4, the time evolution of σ A agrees with the prediction σ A ∝ t 1/2 . Sardina et al. (2015) and Siewert et al. (2017) solved the passive scalar equation of s without considering fluctuations of T and q v . Feedbacks to flow fields from cloud droplets were also neglected. They found good agreement between the DNS and the stochastic model.

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Comparing with Sardina et al. (2015) and Siewert et al. (2017), our study solve the complete sets of the thermodynamics of supersaturation. It is remarkable that a good agreement between the stochastic model and our DNS is observed. This indicates that the stochastic model is robust. On the other hand, modeling supersaturation fluctuations using the passive scalar equation seems to be sufficient for the Reynolds numbers considered in this study. We recall that τ s in Equation (23) is constant. In the present study, τ s is determined by Equation (24). Therefore, τ s varies with time as shown in the inset of Figure 4(a).

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Nevertheless, since the variation of τ s is small, we still observe σ A ∼ t 1/2 except for the initial phase of the evolution, where Comparing panels (a) and (b) of Figure 4, it is clear that changing Re λ has a much larger effect on σ A than changingǭ.
In fact, asǭ is increased by a factor of about 8, σ A decreases only by a factor of about 1.6, so the ratio of their logarithms is about 1/5, i.e., σ A ∝ǭ −1/5 . By contrast, σ A changes by a factor of about 5 as Re λ is increased by a factor of nearly 3, so λ . This quantifies the high sensitivity of σ A to changes of Re λ compared toǭ. Two comments are here in order. First, we emphasize that we observe here σ A ∝ Re 3/2 λ instead of σ A ∝ Re λ . Therefore, there could be a critical Re λ , beyond which σ A ∝ Re λ and below which σ A ∝ Re 3/2 λ . However, the highest Re λ in our DNS is 130. To verify this proposal, a large parameter range of Re λ is required. Second, we note that σ A ∝ǭ −1/5 . This is because the Damköhler number increases with decreasingǭ (see Table 2), which is defined as the ratio of the fluid time scale to the 20 characteristic thermodynamic time scale associated with the evaporation process Da = τ L /τ s . Vaillancourt et al. (2002) also found that σ A decreases withǭ, even though the mean updraft cooling is included in their study.

Discussion and conclusion
Condensational growth of cloud droplets due to supersaturation fluctuations is investigated using DNS. Cloud droplets are tracked in a Lagrangian framework, where the momentum equation for inertial particles are solved. The thermodynamic equa- 25 tions governing the supersaturation field are solved simultaneously. Feedback from cloud droplets onto u, T , and q v is included through the condensation rate and buoyancy force. We resolve the smallest scale of turbulence in all simulations. Contrary to the classical condensation theory, which leads to a narrow distribution when supersaturation fluctuations are ignored, we find that droplet size distributions broaden due to supersaturation fluctuations. For the first time, we explicitly demonstrate that the size distribution becomes wider with increasing Re λ , which is, however, insensitive toǭ. Supersaturation fluctuations are 30 subjected to both temperature fluctuations and water vapor mixing ratio fluctuations. We observe that σ A ∝ √ t when the complete sets of the thermodynamics equations governing the supersaturation are solved, which are consistent with the findings by Sardina et al. (2015) and Siewert et al. (2017). Even though fluctuations of temperature and water vapor mixing ratio, buoyancy force, and droplets feedbacks to the field quantities are neglected in their studies.
This indicates that the stochastic model of condensational growth developed by Sardina et al. (2015) is robust. For the first time, to our knowledge, the stochastic model (Sardina et al., 2015) and simulation results from the complete set of thermo-5 dynamics equations governing the supersaturation field are compared. The broadening size distribution with increasing Re λ demonstrates that condensational growth due to supersaturation fluctuations is an important mechanism for droplet growth.
The maximum Re λ in the present study is 130, which is about two orders of magnitude smaller than the one in atmospheric clouds (Re λ = 10 4 ). Since the width of the size distribution increases dramatically with increasing Re λ , the supersaturation fluctuation facilitated condensation may easily overcome the bottleneck barrier (Grabowski and Wang, 2013).

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The stochastic model developed by Sardina et al. (2015) assumes that the width of droplet size distributions is independent ofǭ. Our result shows that the width decreases slightly with increasingǭ. However, the largestǭ in warm clouds is about 10 −3 m 2 s −3 (Grabowski and Wang, 2013). Therefore, neglecting the smallest scales in the stochastic model is indeed acceptable. Vaillancourt et al. (2002) also found that the width of droplet size distribution decreases with increasingǭ, which ranges from 1.9 × 10 −4 m 2 s −3 to 1.61 × 10 −2 m 2 s −3 . However, their Re λ varies at the same time asǭ changes from 12 to 34. It 15 is unclear if their shrinking of the size distribution with increasingǭ is related toǭ or Re λ . Nevertheless, theirǭ changes by three orders of magnitude while their largest Re λ is 34. Therefore, the contradiction between Vaillancourt et al. (2002) and the works of others (Paoli and Shariff, 2009;Lanotte et al., 2009;Sardina et al., 2015) could be related to poor scale separation in the simulations of Vaillancourt et al. (2002), who were unable to capture the effect of larger scales on condensational growth.
It could also be due to the mean updraft cooling included in the model of Vaillancourt et al. (2002), which was excluded in the present study and in the work of others. The present study may help resolve this contradiction.
In the present study, the simulation box is stationary, which means that the volume is not exposed to cooling, as no mean updraft is considered. Therefore, the condensational growth is solely driven by supersaturation fluctuations. This is similar to the condensational growth of cloud droplets in stratiform clouds, where the updraft velocity of the parcel is close to zero 5 (Hudson and Svensson, 1995;Korolev, 1995). The observational data shows that the width of the size distribution is wider than the one expected from condensational growth with a mean supersaturation (Hudson and Svensson, 1995;Brenguier et al., 1998;Miles et al., 2000;Pawlowska et al., 2006;Siebert and Shaw, 2017a). Qualitatively consistent with observations, we show that the width of droplet size distributions broadens due to supersaturation fluctuations.
Entrainment of dry air is not considered here. It may lead to rapid changes of the supersaturation fluctuations and result 10 in an even faster broadening of the size distribution (Kumar et al., 2014). Activation of aerosols in a turbulent environment is omitted. This may provide a more physical and realistic initial distribution of cloud droplets. Incorporating all the cloud microphysical processes is computationally demanding, and will have be explored in future studies.
Swedish Research Council grants 2012-5797 and 2013-03992, the University of Colorado through its support of the George Ellery Hale visiting faculty appointment, and the grant "Bottlenecks for particle growth in turbulent aerosols" from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048. The simulations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Royal Institute of Technology in Stockholm and Chalmers Centre for Computational Science and Engineering (C3SE). This work also benefited from computer resources made available through the Norwegian NOTUR program, under award NN9405K.