Cloud-droplet growth due to supersaturation fluctuations in stratiform clouds

Xiang-Yu Li^{1,2,3,4,5}, Gunilla Svensson^{1,3,6}, Axel Brandenburg^{2,4,5,7}, and Nils E. L. Haugen^{8,9}

1Department of Meteorology and Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden 2Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden 3Swedish e-Science Research Centre, Stockholm, Sweden 4Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA 5JILA, Box 440, University of Colorado, Boulder, CO 80303, USA 6Global & Climate Dynamics, National Center for Atmospheric Research, Boulder, CO 80305, USA 7Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden 8SINTEF Energy Research, 7465 Trondheim, Norway 9Department of Energy and Process Engineering, NTNU, 7491 Trondheim, Norway Correspondence: Xiang-Yu Li (xiang.yu.li@su.se), December 18, 2022, Revision: 1.218

Abstract.

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

5 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

10 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.

1 Introduction

15 The growth of cloud droplets is dominated by two processes: condensation and collection. Condensation of water vapor on active cloud condensation nuclei is important in the size range from the activation size of aerosol particles to about a radius of 10μm (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 50μm in radius within 20 15–20 minutes (Pruppacher and Klett, 2012; Devenish et al., 2012; Grabowski and Wang, 2013; Seinfeld and Pandis, 2016).

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 25 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 (2017). 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 fluctu-

30 ations are inevitably subjected to turbulence. Naturally, condensational growth due to supersaturation fluctuations became the focus (Sedunov, 1965; Kabanov and Mazin, 1970; Cooper, 1989; Srivastava, 1989; Korolev, 1995; Khvorostyanov and Curry, 1999; Sardina et al., 2015; 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

35 between mean updraft velocity and supersaturation fluctuations. This 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. Cooper (1989) proposed that droplets moving in clouds are exposed to a varying supersaturation field. This results in broadening of droplets size

40 distribution due to supersaturation fluctuations. Grabowski and Wang (2013) called the mechanism of Cooper (1989) the eddyhopping mechanism, which was then investigated by Grabowski and Abade (2017). Using direct numerical simulations (DNS), Vaillancourt et al. (2002) found that the mean energy dissipation rate of turbulence has a 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-D) turbulence as well as stochastically forced temperature and vapor fields with a focus on statistical

45 modeling for large-eddy simulations. They found that supersaturation fluctuations due to turbulence mixing are responsible for the broadening of the droplet size distribution. 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 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.

50 (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 (2018) and Chen et al. (2018) solved the thermodynamics equations governing the supersaturation field. However, since collection was 55 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.

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 studies of

60 Chandrakar et al. (2016) and Desai et al. (2018) suggested that supersaturation fluctuations in the low aerosol number concentration limit are likely of leading importance for the onset of precipitation. 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 (2017) measured the variability of temperature, 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

65 (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, 2017)?

In an attempt to answer this question, we conduct 3-D DNS experiments of condensational growth of cloud droplets, where 70 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 stochastic model and simulation results from the complete set of equations governing the supersaturation field are compared.

75 2 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 as superparticles. A superparticle represents an ensemble of droplets, whose mass, radius, and velocity are the same as those

80 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, 2018) in future. Another reason to adopt superparticle approach is that it can be easily adapted to conduct Large-eddy simulations with appropriate sub-grid scale models (Grabowski and Abade, 2017). To

85 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 by evaporation depending on the local supersaturation field. This phase transition generates a buoyancy force, which in turn affects

90 the turbulent kinetic energy, T (x,t), and q_{v}(x,t). Pencil Code (Brandenburg, 2018) is used to conduct all the simulations.

2.1 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

95 transported by the background air flow. The Eulerian equations are given by

\frac{∂ρ}{∂t} + ∇ · (ρu) = S_{ρ}, (1)

\frac{Du}{Dt} = f − ρ^{−1}∇p + ρ^{−1}∇ · (2νρS) + Be_{z} + S_{u}, (2)

100 \frac{DT}{Dt} = κ∇^{2}T − \frac{g}{c_{p}} w + \frac{L}{c_{p}} C_{d}, (3)

\frac{Dq_{v}}{Dt} = D∇^{2}q_{v} − C_{d}, (4)

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, S_{ij} = \frac{1}{2}(∂_{j}u_{i} + ∂_{i}u_{j}) − \frac{1}{3}δ_{ij}(∂_{k}u_{k}) is the traceless rate-of-strain tensor, p is the gas pressure, ρ is the gas

105 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 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_{s}^{2} = γp/ρ, where γ = c_{p}/c_{v} = 7/5 is the ratio between specific heats,

110 c_{p} and c_{v}, at constant pressure and constant volume, respectively. We set the sound speed as 5ms^{−1} to simulate the nearly incompressible atmospheric air flow, resulting in a Mach number of 0.06 when u_{rms} = 0.27ms^{−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

115 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 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),

B(x,t) = g(T′/T + αq′_{v} − q_{l}), (5)

120 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 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

q_{l} (x,t) = \frac{4πρ_{l}}{3ρ_{a}(Δx)^{3}} ∑_{j=1}^{N_{△}}r (t)^{3} = \frac{4πρ_{l}}{3ρ_{a}} ∑_{j=1}^{N_{△}}f(r,t)r (t)^{3} δr, (6)

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 125 a cubic grid cell with volume (Δx)^{3}, where Δx is the one-dimensional size of the grid box. The temperature fluctuations are

given by

T′ (x,t) = T (x,t) − T_{env}, (7)

and the water vapor mixing ratio fluctuations by

q′_{v} (x,t) = q_{v} (x,t) − q_{v,env}. (8)

130 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, 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

C_{d} (x,t) = \frac{4πρ_{l}G}{ρ_{a}(Δx)^{3}} ∑_{j=1}^{N_{△}}s(x,t)r (t) = \frac{4πρ_{l}G}{ρ_{a}} ∑_{j=1}^{N_{△}}s(x,t)f(x,t)r (t)δr, (9)

135 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} and the saturation vapor pressure e_{s},

s = \frac{e_{v}}{e_{s}} − 1. (10)

Using the ideal gas law, Equation (10) can be expressed as,

140 s = \frac{ρ_{v}R_{v}T}{ρ_{vs}R_{v}T} − 1 = \frac{ρ_{v}}{ρ_{vs}} − 1. (11)

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 be written as:

s(x,t) = \frac{q_{v} (x,t)}{q_{vs} (T)} − 1. (12)

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 145 water vapor mixing ratio at temperature T and can be determined by the ideal gas law,

q_{vs} (T) = \frac{e_{s} (T)}{R_{v}ρ_{a}T} . (13)

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,

150 1996; Götzfried et al., 2017)

e_{s}(T) = c_{1} exp(−c_{2}/T), (14) 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. 2.2 Lagrangian model for cloud droplets

155 In addition to the Eulerian fields described in Section 2.1we treat cloud droplets as Lagrangian particles. In the Pencil Code, they are invoked as non-interacting superparticles. 2.2.1 Kinetics of cloud droplets Each superparticle is treated as a Lagrangian point-particle, where one solves for the particle position x_{i},

\frac{dx_{i}}{dt} = V_{i}, (15)

160 and its velocity V_{i} via

\frac{dV_{i}}{dt} = \frac{1}{τ_{i}} (u − V_{i}) + ge_{z}, (16)

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 τ_{i} = 2ρ_{l}r_{i}^{2}/[9ρν D(Re_{i})]. (17)

165 The correction factor (Schiller and Naumann, 1933; Marchioli et al., 2008), D(Re_{i}) = 1 + 0.15Re_{i}^{2/3}, (18) 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). 2.2.2 Condensational growth of cloud droplets

170 The condensational growth of the particle radius r_{i} is governed by (Pruppacher and Klett, 2012; Lamb and Verlinde, 2011)

\frac{dr_{i}}{dt} = \frac{Gs(x_{i},t)}{r_{i}}. (19)

3 Experimental setup

3.1 Initial configurations

The initial values of the water vapor mixing ratio q_{v}(x,t = 0) = 0.0157kgkg^{−1} and temperature T (x,t = 0) = 292K are 175 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 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.

180 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 initial liquid water content, ∫_{0}^{∞} f(r,t = 0)r^{3} dr, which is 0.001kgm^{−3}. The simulation domain is a cube of size L_{x} = L_{y} = L_{z}, the values of which are given in Table 2. The number of superparticles N_{s} satisfies N_{s}/N_{grid} ≈ 0.1, where N_{grid} is the number of lattices depending on the spatial resolution of the simulations. Setting N_{s}/N_{grid} ≈ 0.1, on one hand, is still within

185 the convergence range N_{s}/N_{grid} ≈ 0.05 (Li et al., 2018). On the other hand, it can mimic the diluteness of the atmospheric cloud system, where there are about 0.1 droplets per cubic Kolmogorov scale. This configuration results in N_{s,128} = 244140 when N_{grid} = 128^{3}.

3.2 DNS

We conduct high resolution simulations (Li et al., 2019) for different Taylor micro-scale Reynolds number Re_{λ} and mean 190 energy dissipation rate ϵ̄ (see Table 2 for details of the simulations). The Taylor micro-scale Reynolds number is defined as

Re_{λ} ≡ u_{rms}^{2}\sqrt{5/(3νϵ̄)}. For simulations with different values of ϵ̄ at fixed Re_{λ}, we vary both the domain size L_{x} (L_{y} = L_{z} = L_{x}) and the amplitude of the forcing f_{0}. As for fixed ϵ̄, 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.

195 For N_{grid} = 128^{3} meshpoints, the code computes 4.5 million time steps in 24 hours wall-clock time using 512 cores.

4 Results

Figure 1(a) shows time-averaged turbulent kinetic-energy spectra for different values of ϵ̄ 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}

200 at fixed ϵ̄ ≈ 0.039m^{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 by ϵ̄^{−2/3}k^{5/3}. For the largest Re_{λ} in our simulations (Re_{λ} = 130), the inertial range extends for about a decade in k-space.

Table 1. List of constants for the thermodynamics: see text for explanations of symbols.

Quantity  Value
ν (m^{2}s^{−1})  1.5 × 10^{−5}
κ (m^{2}s^{−1})  1.5 × 10^{−5}
D (m^{2}s^{−1})  2.55 × 10^{−5}
G (m^{2}s^{−1})  1.17 × 10^{−10}
c_{1} (Pa)  2.53 × 10^{11}
c_{2} (K)  5420
L (Jkg^{−1})  2.5 × 10^{6}
c_{p} (Jkg^{−1}K^{−1})  1005.0
R_{v} (Jkg^{−1}K^{−1})  461.5
M_{a} (gmol^{−1})  28.97
M_{v} (gmol^{−1})  18.02
ρ_{a} (kgm^{−3})  1
ρ_{l} (kgm^{−3})  1000
α  0.608
Pr = ν/κ  1
Sc = ν/D  0.6
q_{v}(x,t = 0) (kgkg^{−1})  0.0157
q_{v,env} (kgkg^{−1})  0.01
T (x,t = 0)(K)  292
T_{env} (K)  293

Table 2. Summary of the simulations; see text for explanation of symbols.

Run f_{0} L_{x} (m) N_{grid} N_{s} u_{rms} (ms^{−1}) Re_{λ} ϵ̄ (m^{2}s^{−3}) η (10^{−4}m) τ_{η} (s) τ_{L} (s) τ_{s} (s) Da
A 0.02 0.125 128^{3} N_{s,128}  0.16  45 0.039  5.4  0.020 0.25 0.014 0.053
B 0.02 0.25 256^{3} 2^{3}N_{s,128}  0.22  78 0.039  5.4  0.020 0.37 0.014 0.081
C 0.02 0.5 512^{3} 2^{6}N_{s,128}  0.28  130 0.039  5.4  0.020 0.58 0.014 0.125
D 0.014 0.6 512^{3} 2^{6}N_{s,128}  0.24  135 0.019  6.5  0.028 0.81 0.014 0.174
E 0.007 0.8 512^{3} 2^{6}N_{s,128}  0.17  138 0.005  8.9  0.053 1.47 0.014 0.312

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

205 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 with ϵ̄. On the other hand. B_{rms} changes only by a few percent as Re_{λ} or ϵ̄ vary.

Figure 1. Time-averaged kinetic energy spectra of the turbulence gas flow for (a) different ϵ̄ = 0.005m^{2}s^{−3} (blue dash-dotted line), 0.019 (magenta dash-dotted line) and 0.039 (dashed cyan line) at fixed Re_{λ} = 130 (see Runs C, D, and E in Table 2 for details) and for (b) different Re_{λ} = 45 (black solid line), 78 (red dashed line), and 130 (cyan dashed line) at fixed ϵ̄ = 0.039m^{2}s^{−3} (see Runs A, B, and C in 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 0.1K in the present study). Therefore, the effect of the buoyancy force should indeed be small.

When changing ϵ̄ while keeping Re_{λ} fixed, the Kolmogorov scales of turbulence varies. Therefore, the various fluctuations 210 quoted above are insensitive to the small scales of turbulence. However, when varying Re_{λ} while keeping ϵ̄ fixed, their rms

values change, which is due to large scales of turbulence. Indeed, temperature fluctuations are driven by the large scales 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,

215 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 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

220 decreases slightly with increasing ϵ̄ 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 to ϵ̄. In atmospheric clouds, Re_{λ} ≈ 10^{4}, which may result 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_{i}^{2}), Equation (19) can be

Figure 2. Time series of the field quantities: (a) T_{rms}, (b) q_{v},rms, (c) B_{rms}, and (d) s_{rms}. Same simulations as in Figure 1.

225 written as

\frac{dA_{i}}{dt} = 2Gs. (20)

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} ∝ \sqrt{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

230 \frac{dσ_{A}^{2}}{dt} = \frac{d}{dt} ⟨A^{′2}⟩ = \frac{d}{dt} ⟨A^{2} − ⟨A⟩^{2}⟩ = 4G⟨s′A′⟩. (21)

Sardina et al. (2015) adopted a Langevin equation to model the supersaturation field and the vertical velocity of droplets, resulting in the scaling law: σ_{A} ∼ C(τ_{L},τ_{s},Re_{λ})t^{1/2}, (22)

Figure 3. Comparison of the time evolution of droplet size distributions for different (a) ϵ̄ at Re_{λ} = 130 (Runs C, D, and E in Table 2) and (b) Re_{λ} at ϵ̄ = 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 235 on the macroscopic observables from small-scale turbulent motions, Sardina et al. (2015) obtained an analytical expression for

σ_{A} as:

σ_{A} ∼ τ_{s}Re_{λ}t^{1/2}, (23)

where τ_{s} is the phase transition time scale given by

τ_{s}^{−1}(t) = 4πG∫_{0}^{∞}rfdr, (24)

240 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 of ϵ̄. In terms of the size distribution f(r,t), σ_{A} can be given as:

σ_{A} = \sqrt{a_{4} − a_{2}^{2}}, (25) where a_{ζ} is the moment of the size distribution, which is defined as:

a_{ζ} = ∫_{0}^{∞}f r^{ζ} dr/∫_{0}^{∞}f dr. (26)

245 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. 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

250 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). 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 s(t = 0) = 2%.

255 Comparing panels (a) and (b) of Figure 4, it is clear that changing Re_{λ} has a much larger effect on σ_{A} than changing ϵ̄. In fact, as ϵ̄ 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} ∝ ϵ̄^{−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 σ_{A} ∝ Re_{λ}^{3/2}. This quantifies the high sensitivity of σ_{A} to changes of Re_{λ} compared to ϵ̄.

Two comments are here in order. First, we emphasize that we observe here σ_{A} ∝ Re_{λ}^{3/2} instead of σ_{A} ∝ Re_{λ}. Therefore, 260 there could be a critical Re_{λ}, beyond which σ_{A} ∝ Re_{λ} and below which σ_{A} ∝ Re_{λ}^{3/2}. 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} ∝ ϵ̄^{−1/5}. This is because the Damköhler number increases with decreasing ϵ̄ (see Table 2), which is defined as the ratio of the fluid time scale to the characteristic thermodynamic time scale associated with the evaporation process Da = τ_{L}/τ_{s}. Vaillancourt et al. (2002) also found that σ_{A} decreases with ϵ̄, even though the mean updraft cooling is included in their study.

265 5 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 equations 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

270 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 to ϵ̄. Supersaturation fluctuations are subjected to both temperature fluctuations and water vapor mixing ratio fluctuations.

Figure 4. Time evolution of σ_{A} for different (a) ϵ̄ at Re_{λ} = 130 and (b) Re_{λ} at ϵ̄ = 0.039m^{2}s^{−3}. Same simulations as in Figure 1.

We observe that σ_{A} ∝ \sqrt{t} when the complete sets of the thermodynamics equations governing the supersaturation are solved, 275 which are consistent with the findings by Sardina et al. (2015) and Siewert et al. (2017) even though fluctuations of tempera-

ture 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 thermodynamics equations governing the supersaturation field are compared. The broadening size distribution with increasing Re_{λ}

280 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).

The stochastic model developed by Sardina et al. (2015) assumes that the width of droplet size distributions is indepen-285 dent of ϵ̄. Our result shows that the width decreases slightly with increasing ϵ̄. However, the largest ϵ̄ 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 the droplet size distribution decreases with increasing ϵ̄ and attributed this to the decorrelation between supersaturation fluctuations and surface area of droplets. Sardina et al. (2015), however, found stronger correlation between supersaturation fluctuations and surface area of droplets with increasing Re_{λ}. The

290 present study is consistent with both the works of Vaillancourt et al. (2002) and Sardina et al. (2015). Therefore, we emphasize that there is no contradiction between both papers.

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

295 (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, 2017). 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 300 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.

Code and data availability. The source code used for the simulations of this study, the Pencil Code (Brandenburg, 2018), is freely available on https://github.com/pencil-code/. The DOI of the code is http://doi.org/10.5281/zenodo.2315093. The last access to the code is De-

305 cember 16, 2018. The DNS setup and the corresponding data (Li et al., 2019) are freely available at https://doi.org/10.5281/zenodo.2538027.

Author contributions. Xiang-Yu Li developed the idea, coded the module, performed the simulations, and wrote the manuscript. Axel Brandenburg and Nils Haugen contributed to the development of the module and commented on the manuscript. Gunilla Svensson contributed to the development of the idea and commented on the manuscript.

Competing interests. The authors declare that they have no conflict of interest.

310 Acknowledgements. We thank Wojtek Grabowski, Andrew Heymsfield, Gaetano Sardina, Igor Rogachevskii and Dhrubaditya Mitra for stimulating discussions. This work was supported through the FRINATEK grant 231444 under the Research Council of Norway, SeRC, the 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

315 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.

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