Editorial Type:
Article
Article Type:
Research Article

Broadening of the Cloud Droplet Size Distribution due to Thermal Radiative Cooling: Turbulent Parcel Simulations

Mares Barekzai Meteorological Institute, Ludwig-Maximilians University, Munich, Germany

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Bernhard Mayer Meteorological Institute, Ludwig-Maximilians University, Munich, Germany

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Online Publication:
21 May 2020
Print Publication:
01 Jun 2020
Collections:
Waves to Weather (W2W)
DOI:
https://doi.org/10.1175/JAS-D-18-0349.1
Page(s):
1993–2010
Free access

Abstract

Despite impressive advances in rain forecasts over the past decades, our understanding of rain formation on a microphysical scale is still poor. Droplet growth initially occurs through diffusion and, for sufficiently large radii, through the collision of droplets. However, there is no consensus on the mechanism to bridge the condensation coalescence bottleneck. We extend the analysis of prior methods by including radiatively enhanced diffusional growth (RAD) to a Markovian turbulence parameterization. This addition increases the diffusional growth efficiency by allowing for emission and absorption of thermal radiation. Specifically, we quantify an upper estimate for the radiative effect by focusing on droplets close to the cloud boundary. The strength of this simple model is that it determines growth-rate dependencies on a number of parameters, like updraft speed and the radiative effect, in a deterministic way. Realistic calculations with a cloud-resolving model are sensitive to parameter changes, which may cause completely different cloud realizations and thus it requires considerable computational power to obtain statistically significant results. The simulations suggest that the addition of radiative cooling can lead to a doubling of the droplet size standard deviation. However, the magnitude of the increase depends strongly on the broadening established by turbulence, due to an increase in the maximum droplet size, which accelerates the production of drizzle. Furthermore, the broadening caused by the combination of turbulence and thermal radiation is largest for small updrafts and the impact of radiation increases with time until it becomes dominant for slow synoptic updrafts.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Waves to Weather (W2W) Special Collection.

Corresponding author: Mares Barekzai, mares.barekzai@lmu.de

Abstract

Despite impressive advances in rain forecasts over the past decades, our understanding of rain formation on a microphysical scale is still poor. Droplet growth initially occurs through diffusion and, for sufficiently large radii, through the collision of droplets. However, there is no consensus on the mechanism to bridge the condensation coalescence bottleneck. We extend the analysis of prior methods by including radiatively enhanced diffusional growth (RAD) to a Markovian turbulence parameterization. This addition increases the diffusional growth efficiency by allowing for emission and absorption of thermal radiation. Specifically, we quantify an upper estimate for the radiative effect by focusing on droplets close to the cloud boundary. The strength of this simple model is that it determines growth-rate dependencies on a number of parameters, like updraft speed and the radiative effect, in a deterministic way. Realistic calculations with a cloud-resolving model are sensitive to parameter changes, which may cause completely different cloud realizations and thus it requires considerable computational power to obtain statistically significant results. The simulations suggest that the addition of radiative cooling can lead to a doubling of the droplet size standard deviation. However, the magnitude of the increase depends strongly on the broadening established by turbulence, due to an increase in the maximum droplet size, which accelerates the production of drizzle. Furthermore, the broadening caused by the combination of turbulence and thermal radiation is largest for small updrafts and the impact of radiation increases with time until it becomes dominant for slow synoptic updrafts.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Waves to Weather (W2W) Special Collection.

Corresponding author: Mares Barekzai, mares.barekzai@lmu.de
Keywords: Cloud droplets; Cloud microphysics; Drop size distribution; Longwave radiation; Clouds

1. Introduction

Warm rain plays an important role for tropical precipitation (Hou et al. 2014; Lau and Wu 2003; Liu and Zipser 2009), but the precise processes that convert cloud droplets to rain are not well understood. The formation of warm rain begins with the condensation of water on cloud condensation nuclei (CCN) to form haze droplets. If the environment reaches a critical saturation, the haze particle nucleates and grows freely by condensation. The droplets grow until they are sufficiently large to collide; at that point the collision and coalescence process dominates the growth and creates drizzle and eventually raindrops. Although there is agreement on these principal steps the details of drizzle formation remain an open question. Drizzle is defined as hydrometeors with a radius between 20 and 250 μm (Feingold et al. 1999; Hudson and Yum 2001; Rasmussen et al. 2002) and its formation cannot be explained solely by diffusional growth and subsequent collisions (Illingworth 1988; Beard and Ochs 1993; Laird et al. 2000). The classical diffusional growth process is proportional to the inverse of the droplet radius (dr/dt ∝ 1/r), which results in very slow growth speeds for droplets larger than 15 μm. Collision is generally considered efficient for radii larger than 20 μm (Brenguier and Chaumat 2001). Furthermore, diffusional growth decreases the standard deviation of the droplet size distribution, which slows down the onset of collisions by narrowing the fall-speed spectrum. The classical approaches to growth by diffusion and collision are therefore insufficient to explain the formation of drizzle in realistic time scales and also to reproduce the observed broad and multimodal droplet distributions (Warner 1969a,b). The aforementioned problem is called “the condensation coalescence bottleneck” (Brewster 2015; Wang and Grabowski 2009). Previous attempts to pass the bottleneck and simulate more realistic droplet distributions investigated the role of giant cloud condensation nuclei (GCCN) (Feingold et al. 1999; Houghton 1938; Johnson 1982; Yin et al. 2000), turbulence (Fouxon et al. 2002; Grabowski and Wang 2013; Pinsky and Khain 1997) and radiation (Brewster 2015; Guzzi and Rizzi 1980; Klinger et al. 2019; Lebo et al. 2008; Marquis and Harrington 2005; Rasmussen et al. 2002; Roach 1976). We would like to highlight the often underrepresented role of radiation in the diffusional growth process in combination with turbulence. The key idea is that both heat diffusion and thermal radiation allow the droplet to dissipate latent heat released during condensation. Therefore, the addition of radiative cooling can reduce the temperature of the droplet below the temperature of the surrounding air. Consequently, droplets can continue to grow even in slightly subsaturated environments. We will refer to this process as the “radiatively enhanced diffusional” growth (RAD), which is not considered in the classical diffusional growth theory and is not included in the current microphysical parameterizations, despite being already proposed by Roach (1976). (Harrington et al. 2000; Hartman and Harrington 2005a,b) investigated the impact of RAD on the formation of drizzle in arctic stratus clouds. They applied large-eddy simulations (LESs) and a Trajectory model to compare the effects of radiation using a bin microphysics. Radiative fog simulations have also been conducted as proposed in the fundamental paper of Roach (1976) (Brown and Roach 1976; Duynkerke 1991; Roach et al. 1976). Here, we use a parcel model with Lagrangian microphysics to investigate RAD in combination with turbulence induced saturation fluctuations, which can serve as a foundation for the interpretation of more elaborated LESs. We use a Lagrangian microphysics representation, which applies so-called superdroplets, each representing a group of droplets with the same aerosol properties throughout the parcel (Cooper et al. 2013; Shima et al. 2009; Vaillancourt et al. 2002). In contrast to passive tracers, the superdroplets nucleate and subsequently grow and shrink by interacting with the surrounding moisture field. The advantage of this approach is that it allows the explicit treatment of the growth by diffusion and the implementation of a turbulence parameterization, resulting in different growth histories for each superdroplet. The thermal radiative cooling is only relevant near the cloud edges, because at the cloud center the emitted radiation is in balance with the absorbed radiation. The distance from the cloud edges at which cooling is relevant depends on the liquid water path and ranges from 50 to 100 m (Klinger and Mayer 2016). The implementation details of the parcel model are based on Grabowski and Abade (2017). The current paper can be understood as a continuation of that study.

2. Parcel model

The parcel model equations for temperature T, water vapor mixing ratio qυ, and the pressure p are implemented according to (Grabowski and Abade 2017; Grabowski and Wang 2009; Grabowski et al. 2011)
cpdTdt=−gw−Lυdqυdt,
dpdt=−ρ0wg
(see Table 1 for the notations used throughout the equations and text). The temperature of the parcel decreases moist adiabatically as the parcel ascends due to the coupling to the latent heat of condensation [Eq. (1)]. We assume a constant density of air with ρ0 = 1 kg m−3, according to the approximation of small vertical displacement (~1 km). The initial conditions are T = 281.7 K, p = 89 880 Pa, and qυ = 0.0077 kg kg−1, which result in a relative humidity of ϕ = 99%. Accordingly, the ascending parcel will rapidly begin with the development of a cloud.
Table 1.

Symbols and descriptions for variables and constants in this study.

Table 1.

a. Microphysics

We used the superdroplet approach in the context of warm clouds (Andrejczuk et al. 2008; Shima et al. 2009). It is based on Lagrangian particles, which carry information about the aerosol type in the dry radius rdry, chemical composition defaulted here to sodium chloride, droplet size after nucleation and multiplicity factor. The multiplicity factor Ni is the represented number concentration of aerosols, and after nucleation, of droplets by a superdroplet. The multiplicity factors were chosen to be the same for all superdroplets and constant over time with Ni = 103 m−3 and can therefore be seen as a scaling factor for the liquid water mixing ratio. The maximum possible number of nucleated superdroplets is set to Nsd = 105 m−3 with a total CCN number density of nCCN = 100 cm−3. The droplet size starts at the dry radius and diverges after nucleation due to diffusional growth. The superdroplets grow only by condensation and the combined condensation rates of all nucleated superdroplets can be written as
dqυdt=−ddt∑i43πri3Niρwρ0.
The advantage of the superdroplet approach is the combination of a global representation of distributed cloud droplets with the local, explicit treatment of nucleation and diffusional growth processes.

1) Nucleation

The dry radius is sampled from two lognormal distributions with mean radii of 20 and 75 nm and geometric standard deviations of 1.4 and 1.6. The relative concentrations of 60% and 40% correspond to CCN number concentrations of n20 = 60 cm−3 and n75 = 40 cm−3, if nccn = 100 cm−3 (from Grabowski and Abade 2017). The aerosol size information and chemical composition determine the critical supersaturation Scr that must be reached to nucleate the corresponding aerosol. The nucleated superdroplets start to grow freely from the critical radius rcr, which are obtained by calculating the maximum of the Koehler equation (Rogers and Yau 1996):
Seq*=C1r−C2r3.
The critical supersaturation and radius are
rcr=min⁡[⁡(3C2C1)1/2,1 μm],Scr=⁡(4C1327C2)1/2
with the corresponding parameters:
C1=2γMwRυTρw,C2=2rdry3ρNaClMwρwMNaCl.
Critical radii larger than 1 μm are cut off to eliminate the impact of the aerosol distribution tail on the evolution of the droplet population. This condition allows us to isolate the effect of radiation from possible GCCN. The resulting cutoff value is a rough estimate of the average droplet embryo size after one model time step.

2) Diffusional growth

After the nucleation process, the droplets begin to grow by diffusion. Classical diffusional growth considers the diffusion of water molecules and latent heat to and from the droplet. Both processes come together in the diffusion equation:
dridt=1Aρw⁡(S*ri).
The subscript i denotes the ith superdroplet. Equation (7) was extended by Roach (1976) to include the emission and absorption of radiation. It becomes the RAD equation:
dridt=1Aρw⁡(S*ri−DR).
The parameter R is the radiative power per droplet surface area, defined as the difference of absorbed and emitted power. Positive R indicate radiative heating of the droplet and negative values indicate cooling. Equation (8) is obtained with the approximation of [Lυ/(RυT)] ≫ 1 and contains the temperature-dependent parameters A=Lυ2/RυκT2+RυT/Dwes and D = Lυ/RυκT2.
An individual droplet grows with decreasing speed as it increases in size, proportional to 1/r from Eq. (7). This statement still holds for RAD if S*/r>−DR with small droplets and large supersaturations. The differences compared to the classical diffusional growth can be seen in the asymptotic
limr→∞ drdt|classic=0,
limr→∞ drdt|enhanced=−DRAρw.
Large droplet radii under classical diffusional growth show zero growth speed [Eq. (9)], in contrast to RAD, which converges to a term proportional to R [Eq. (10)]. The behavior now strongly depends on the sign of R. If only thermal radiation is considered, R becomes negative, and therefore, large droplets continue to grow (except close to the lower cloud boundary). The additional cooling due to the emission of thermal radiation lowers the temperature of the droplet compared to the environment, effectively reducing the saturation vapor pressure at the droplet surface. For completeness, it should be mentioned that thermal radiation causes considerable cooling at the cloud top and moderate warming at the cloud base. Here, we focus on the cloud top.
It is known that the standard deviation of the droplet size distribution will decrease, because smaller droplets grow faster and catch up with larger drops under classical diffusional growth. The addition of radiative cooling to the diffusional growth is expected to cause more complex behavior: The standard deviation of the distribution will initially decrease, until the parcel becomes subsaturated. Subsequently, the subsaturated parcel will contain large droplets, which continue to grow, and small droplets, which start to evaporate. The radius, which separates growing from shrinking droplets is calculated with dr/dt = 0 from Eq. (8):
rsep=S*DR.
Therefore, the standard deviation of the droplet distribution increases due to thermal radiation (Harrington et al. 2000; Hartman and Harrington 2005a,b). Finally, some droplets will completely evaporate, which again decreases the standard deviation. Figure 1 shows the dependence of the droplet growth speed on the radius, according to Eq. (8) with and without the radiative term. The gray lines show the solution for the droplet growth speed, under constant super or subsaturated conditions. The left panel, without radiation, shows that the growth (I) and evaporation (II) is fastest for small droplets and symmetric with respect to dr/dt = 0. The right panel, with thermal radiation, adds a third growth regime (III) in which the environment is subsaturated, but the droplets keep growing, particularly with large droplets that grow faster than smaller ones. The zero crossings of the isosaturation lines indicate below which radius and subsaturation the droplets evaporate. The diffusional growth equation is evaluated for a model time step of 0.2 s with the Euler forward schema.
Fig. 1.
Fig. 1.

Droplet growth speed evaluated for the RAD equation [Eq. (8)]. The radiative term is set (left) to zero and (right) according to Eq. (14) (with f = 1/6 and T = 282 K from the American standard atmosphere at 1 km). The isosaturation lines are centered around S* = 0 and 10 times incremented and decremented by ΔS* ± 0.02%. The numbers I, II, and III indicate different droplet growth regimes: The small droplets grow faster (regime I), evaporate faster (regime II), grow more slowly (regime III) than the large droplets.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

b. Radiation

The additional term in Eq. (8) contains the net radiative power per droplet surface area (R). This term could comprise solar and thermal contributions. Here, we focus on thermal radiation only, which can be thought of as a nocturnal setting. It was shown in Roach (1976) that R can be directly calculated from the radiative fluxes. In particular, Eq. (11) of Roach (1976) shows that R is directly related to differences of the actinic flux or “average intensity” Fact=∫4πI⁡(λ) dΩ and the blackbody emission:
R=Qa⁡(r)⁡[14 Fact−σsbT4],
where Qa(r) is the absorption efficiency, weighted with the spectral actinic flux and averaged over wavelength. The value of Qa(r) is typically close to 1 for the droplet sizes under consideration. Roach (1976) further showed that the actinic flux can be approximated by the sum of the upward flux Eup and downward flux Edn [or F↓ and -F↑, in the notation of Roach (1976), where the upward component is actually negative]:
Fact≈2⁡(Edn+Eup).
For our parcel model we want to estimate the effect of radiation close to the cloud top. In the following, we show that the heating rate at cloud top can be simply approximated by
R≈−fσsbT4.
We assume Tdrop ≈ T, because Marquis and Harrington (2005) estimated that the difference between droplet and environment temperature is ΔT ≤ 1 K for droplets with r ≤ 200 μm and a maximum of 3 K is reached for r ≥ 1000 μm. In our simulations, the droplet radii are well below 1000 μm and show only small temperature differences. The radiative factor f allows for a first and simple approximation of the surrounding atmosphere and the geometry of the parcel. The value f = 1 would describe a parcel in vacuum. For a parcel at cloud top, f reduces to 1/2, because the downward emitted radiation is balanced by the upward emitted radiation from the droplets below. Finally, only the atmospheric window regions contribute to the cooling, which adds a factor of 1/3 for shallow cumulus cloud tops. In summary, this yields f = 1/6.
To check the validity of f, we calculated the actinic flux for a number of typical atmospheric profiles: pressure, temperature, water vapor, other trace gases (Anderson et al. 1986) and cloud-top heights (Fig. 2). The value of R was calculated following (12). More specifically, rather than using the approximated Qa(r), we integrated over wavelength
R=∫Qa⁡(λ)⁡[14 Fact⁡(λ)−πB⁡(λ,T)]dλ,
where B(λ,T) is the Planck function. The corresponding simulations were done with the radiative transport library libRadtran (Mayer and Kylling 2005; Emde et al. 2016) using the correlated-k distribution by Fu and Liou (1992) for the molecular absorption and assuming an optically thick cloud with an effective droplet radius of 10 μm. Cloud optical properties were calculated by Mie theory (Mie 1908). Figure 2 illustrates that f = 1/6 is a valid approximation for shallow cumulus clouds with top heights of approximately 2 km. Clearly f depends on the amount of water vapor above the cloud: For the midlatitude summer and tropical atmospheres with high temperature and absolute humidity, f is considerably smaller than for the dry subarctic winter and midlatitude winter atmospheres with low temperatures, which is due to the higher atmospheric transmission for less water vapor. For the same reason, f increases with increasing cloud-top height until it reaches values slightly above 0.5. For the following simulations we will use f = 1/6, if not stated otherwise. This approach allows us to investigate the underlying mechanisms in a idealized way, which will serve as a interpretation basis for more elaborated LESs.
Fig. 2.
Fig. 2.

Estimation of the cloud-top height dependence of f [Eq. (14)] for an optically thick cloud and a number of typical atmospheric profiles: U.S. standard atmosphere (afglus), subarctic winter (afglsw), midlatitude winter (afglmw), subarctic summer (afglss), midlatitude summer (afglms), and tropical (afglt). The vertical black line marks the reference value of f = 1/6.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

c. Turbulence

The turbulence parameterization is based on the work of Grabowski and Abade (2017) and references therein, and will only be briefly summarized here. The aforementioned paper describes saturation fluctuations for isotropic homogeneous turbulence, which are implemented for each superdroplet. The evolution for the saturation fluctuations Si*′ is given by
dSi*′dt=a1wi′−Si*′τrelax.
This equation uses the phase relaxation time scale of the droplet distribution (τrelax) (Squires 1952) and the fluctuations of the vertical velocity field (wi′), which are described by the Gaussian stationary process (Ornstein–Uhlenbeck process):
wi′⁡(t+dt)=wi′⁡(t) exp⁡(−dtτ)+1−exp⁡(−2 dtτ)23 E χ.
The solution of the random process depends only on the standard normal distributed random variable χ, the turbulent kinetic energy E(ε, L), and the integrated turbulent time scale τ(ε, L) (Schumann 1991; Lasher-Trapp et al. 2005). The last two quantities can be formulated as functions of the dissipation rate ε = 50 cm2 s−3 and the length scale of the adiabatic parcel L = 50 m. The values for L and ε correspond to an adiabatic core of a cumulus cloud and are taken from the predecessor studies of (Grabowski and Abade 2017; Jonas 1996; Lasher-Trapp et al. 2005). For direct comparison between of the two studies, we will adopt these values. Furthermore, the impact of thermal radiation should be largest at cloud edges; therefore, we included simulations with ε = 10 cm2 s−3, which is a estimate for the turbulent dissipation rate at the stratocumulus cloud top (Moeng et al. 1996).

3. Results

First, we want to emphasize, that this study is a theoretical one. The nature of parcel models allows only for a limited range of scales and processes to be included. However, the parcel model reduces complexity and therefore improves our understanding of the individual processes, which might be concealed in a chaotic LES. A important idealization is the neglection of cloud-edge mixing, which may be included in the statistics of the saturation fluctuations in future studies. A first approach to include the process of mixing is shown in Abade et al. (2018). Furthermore, the sensitivity studies may show a range in parameter space that extends beyond the values found in nature.

a. Distributions and time series

In this section, we compare the temporal development of the droplet distribution for different parcel model setups. The first one, for a cloud edge environment, is w = 1 m s−1, ε = 50 cm2 s−3, L = 50 m, and f = 1/6, which is also used by Grabowski and Abade (2017) and allows us to validate our results and to directly compare the impact of thermal radiation in combination with turbulence. The second one, for a cloud-edge environment, is w = 0.1 m s−1, ε = 10 cm2 s−3, L = 50, and f = 1/6. Here, the turbulent dissipation rate is taken from Moeng et al. (1996) for a stratocumulus cloud top. Furthermore, the updraft is an order of magnitude smaller. If not stated otherwise the figures have the following conventions for labels and colors: The base case applies classical diffusional growth (0, green), the turbulent parcel is identical to the setup of Grabowski and Abade (2017) (T, blue), the simulations with radiation only (RAD, yellow), and the combination of turbulence and radiation (RAD&T, red). The histograms in Figs. 3 and 4 are evaluated 15 min after the onset of condensation. The first two droplet size histograms, which evolved under classical diffusional growth or RAD (green or yellow) show sharply peaked distributions centered around radii that decrease with decreasing updraft, due to less adiabatic cooling. The standard deviation of the droplet size distribution σ is small and the sole addition of radiation introduces just a small amount of spread into the droplet distribution (σ < 0.5 μm). The spreading is caused by the subsaturated environment, due to radiative cooling, which happens earlier for the w = 0.1 m s−1 than for the w = 1 m s−1 case, as can be seen in the S* panel of Fig. 5.

Fig. 3.
Fig. 3.

The (top) droplet size distributions and (bottom) corresponding mean growth speeds, evaluated 15 min after the onset of condensation. The simulations are run with w = 0.1 m s−1, f = 1/6, and ε = 10 cm2 s−3. A bin-size of 1 μm centered around integer values was applied. The shaded area in the bottom panels marks two standard deviations around the mean growth speed. The colors and labels correspond to the simulation: green and 0 for the reference simulation without radiation or turbulence, yellow and RAD for the reference simulation including radiation, blue and T for the reference simulation including turbulence, and red and RAD&T for the combination of radiation and turbulence. Also shown in the top panels is σ, the standard deviation of the droplet distribution.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

Fig. 4.
Fig. 4.

As in Fig. 3, but the simulations are run with w = 1 m s−1, f = 1/6, and ε = 50 cm2 s−3.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

Fig. 5.
Fig. 5.

The time series of several parcel model quantities from the onset of condensation until 15 min. The simulations are run with (left) w = 0.1 m s−1, f = 1/6, and ε = 10 cm2 s−3 and (right) w = 1 m s−1, f = 1/6, and ε = 50 cm2 s−3. Quantities are (top to bottom) the standard deviation σ, the mean radius of the largest droplets rmax (representing a number density of nmax cm−3), the mean radius ⟨r⟩, and the supersaturation S*. The color and label conventions are as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

The droplet growth in a subsaturated environment changes the radius dependence of droplet growth: When classical diffusional growth in a supersaturated environment is considered, smaller droplets grow faster and catch up with larger ones, due to the 1/r dependence of Eq. (7) (regime I in Fig. 1). However, in a subsaturated environment, smaller droplets evaporate faster than larger droplets (regime II in Fig. 1). Finally, the addition of thermal radiation introduces a third growth regime with subsaturated environment, where smaller droplets grow slower than larger droplets, which causes σ to increase (regime III in Fig. 1). The earlier and the longer the parcel is subsaturated, the larger the increase in σ. Therefore, the radiation only simulation (yellow) with w = 0.1 m s−1 produces larger σ values than the w = 1 m s−1 simulations. The difference in σ is small, because the evaporation in regime II has not yet started. The two histograms on the right (blue and red) allow to estimate the impact of radiation added on top of turbulence. The histogram with turbulence (blue) shows the increase of σ due to turbulence alone, which evolves according to the balance between the spread introduced by saturation fluctuations and the narrowing of the droplet spectrum due to classical diffusional growth. The saturation fluctuations show a larger effect for small droplets, which can be seen in the diverging isosaturation lines in the regimes I and II of Fig. 1. Therefore, the simulations with low updraft (w = 0.1 m s−1) show a larger spread in the droplet size distribution, despite of having a lower turbulent dissipation rate (ε = 10 cm2 s−3). The red histograms combine the impact of turbulence with thermal radiation and show that larger vertical motions produce larger supersaturations, which will produce larger droplets and is less sensitive to turbulence or radiation (σ ~ 2 μm). In contrast, radiation with turbulence is most effective in the low updraft environment and substantially broadens the standard deviation. The addition of radiation to the simulations approximately doubles the droplet size standard deviation. Furthermore, the droplet size distribution is symmetric for w = 1 m s−1, in contrast to the w = 0.1 m s−1, where droplets are more likely to deactivate, resulting in a asymmetric droplet size distribution. The broadening of the droplet distribution is critical for the initiation of collision because a narrow droplet size spectrum also has a narrow fall-speed spectrum. Since differences in fall speed are required to initiate the collection process, narrow spectra tend to suppress collisions.

Figure 5 shows the time series for the standard deviation σ, the maximum radius rmax, the mean radius ⟨r⟩, the droplet number concentration n, and the saturation S* of the environment. The maximum radius rmax is defined as the mean radius over the largest droplets
rmax=1nmax∫r0∞rn⁡(r) dr,
where r0 is chosen by the condition ∫r0∞n⁡(r) dr=!nmax with nmax = 0.1 cm−3. This is similar to the approach of Feingold and Chuang (2002) and Feingold et al. (1999), who propose that a drizzle number concentration of ndrizzle = 10−3 cm−3 of droplets with radii larger than 20 μm is required for drizzling and eventually raining parcels. We chose a more cautious value for nmax to increase the number of droplets representing the tail of the droplet distribution, which would otherwise lead to strong noise. The standard deviation initially increases due to the nucleation of droplets and then develops according to the applied parameterizations. The subsequent narrowing due to the classical diffusional growth (green and yellow) is followed by broadening due to a subsaturated environment (yellow only). The resulting σ are small (<1 μm) for all updrafts, because the spread introduced by nucleation, which increases with increasing w, is reduced by the narrowing diffusional growth (regime I in Fig. 1). In this case, thermal radiation is acting only on narrow distributions; therefore, the differential growth is weak. However, the simulations with turbulence (blue and red) show a larger increase in the standard deviation after nucleation. The time series confirms that turbulence introduces spread most effectively at the beginning of the simulations, when droplets are small. The reason is that the growth of small droplets is more sensitive to saturation fluctuations, which is shown by the diverging growth speeds in Fig. 1 for small radii. Consequently, turbulence has a larger impact for small updrafts. Although, after 15 min, the mean droplet size is only about 10 μm for w = 0.1 m s−1 compared to 18 μm for w = 1 m s−1 (a consequence of the adiabatic growth assumption), the standard deviation of the size distribution is a factor of 3 larger with lower updraft. Additionally, thermal radiation (red line) complements the turbulent growth by subsaturating the environment, which further supports the growth of large droplets. For both cases, the standard deviation of the droplet size distribution approximately doubles after 15 min, when radiation is added. Therefore, the largest standard deviations are found for w = 0.1 m s−1 simulations with the combination of turbulence and radiation. The maximum radius is introduced as a measure for the tail of the distribution, because it may harbor the rain droplet candidates, also known as embryonic drizzle from Hobbs and Rangno (1998). For the base and radiation simulations (green and yellow), the maximum radius rmax is similar to the mean radius, because the distribution is sharply peaked and rmax is close to ⟨r⟩. The turbulent simulations (blue and red) reach similar values for rmax after 15 min, for both updraft cases although the mean radius is different by a factor of 2.

The mean radius ⟨r⟩ and the droplet number concentration n depend on the updraft and are hardly affected by radiation or turbulence in the w = 1 m s−1 case. For the low updraft simulations (w = 0.1 m s−1), this is only true for the base and radiation only simulations (green and yellow). After nucleation, turbulence causes the evaporation of droplets due to negative saturation fluctuations and leaves the droplets with positive saturation fluctuations. This bias initially subsaturates the environment. However, after a few minutes the environment saturates again, due to additional sign switches of the saturation fluctuations, which remove the bias from the distribution and S*′ becomes equally distributed. Simultaneously, small droplets nucleate, which lowers ⟨r⟩. The combination of radiation and turbulence (red) is similar in the development, but shows less renucleation of small droplets due to initially negative saturation fluctuations (number concentration in Fig. 5). The environment stays subsaturated, due to thermal radiation, and therefore, the critical saturation needed for renucleation is only rarely reached. However, the mean radius decreases, due to the evaporation of droplets at the benefit of a few large ones, which was also found in Guzzi and Rizzi (1980).

b. Sensitivity to the updraft speed

Updraft speeds are varied between 0.01 and 1.5 m s−1, which correspond to slow (synoptic) and shallow convective vertical motions. The simulations are evaluated 15 min after nucleation and result in clouds with vertical extents varying between 10 and 1250 m. Figure 6 shows the standard deviation, the maximum radius, the mean radius, and the droplet number concentration. The shaded regions (blue and red) connect the range of turbulent dissipation rates from 10 to 50 cm2 s−3. The two black vertical lines at w = 0.1 and w = 1 m s−1 highlight the setups for which the time series and histograms were already shown in the previous section. The simulations with classical diffusion only (green) show a small increase of σ with increasing updraft, due to a increasing spread introduced by nucleation. The simulations with radiation (yellow) approximately double the spread from nucleation by subsaturating the environment and entering regime III (Fig. 1). For low updrafts, the spread increases stronger, due to evaporating droplets, which are below the separation radius [Eq. (11)]. Nonetheless, the standard deviations for simulations without turbulence are small (<0.1 μm). For simulations with turbulence (blue and red) the standard deviations are larger [σ ∈ (1,6) μm] and increase with decreasing updrafts. The impact of turbulence and thermal radiation on σ for lower updrafts increases due to diverging growth rates for small droplets.

Fig. 6.
Fig. 6.

A sensitivity study with respect to the updraft velocity. The simulations are evaluated 15 min after condensation onset. Shown are (top to bottom) the standard deviation σ, the mean radius of the largest droplets rmax (representing a number density of nmax = 0.1 cm−3), the mean radius ⟨r⟩, and the droplet number density n. The color and label conventions are as in Fig. 3. For simulations with turbulence, two dissipation rates are shown: ε = 50 cm2 s−3 (solid) and ε = 20 cm2 s−3 (dotted). Additionally, the blue and red shaded regions mark intermediate values of ε. The vertical black lines at w = 0.1 and 1 m s−1 indicate where distributions and time series are evaluated.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

For the simulations without turbulence (green and yellow), the updraft dependence of the maximum radius is determined by the increase of adiabatic cooling with increasing updraft. Note that rmax is nearly identical to ⟨r⟩, because the distributions are sharply peaked. The simulations with turbulence only (blue) show larger values of rmax with strongest impact for low updrafts, if compared to the base simulations (green). It also reveals an intermediate w range, where w-dependence of rmax shows a minimum. The minimum is caused by an increasing rmax at high updrafts due to increasing adiabatic cooling, which increases ⟨r⟩. However, σ is largest at low updrafts; therefore, rmax shows a minimum at intermediate updrafts. The same can be found for the combination of turbulence and radiation (red), resulting in standard deviations of approximately 22.5 μm. The mean radius is determined by adiabatic cooling for simulations with w > 0.5 m s−1. For w < 0.5 m s−1, slightly smaller ⟨r⟩ appear in turbulent simulations (blue and red) (Δ⟨r⟩ < 2 μm). The reason is, that the droplets start to evaporate right after nucleation followed by a period of secondary nucleation, which lowers the mean by introducing small droplets into the population. The combination of turbulence and thermal radiation (red) is similar to turbulence only (blue) but with less secondary nucleation due to the subsaturated environment. Furthermore, the parcel with radiation continues to evaporate droplets over time, which also introduces smaller droplets into the population that lower the mean. Additionally, the initial droplet number concentration increases with larger updrafts, according to higher reached peak saturations in the nucleation process. The dependence on the turbulent dissipation rate (blue and red shaded regions) shows smaller impact on σ and rmax then the impact of adding radiation (T and RAD&T).

c. Sensitivity to the simulation time

Figure 7 compares clouds with the same vertical extent of 100 m, which develop under different updraft conditions ranging from w = 0.01 to 1.5 m s−1 and therefore in different time intervals (1 to 160 min). For w > 0.5 m s−1, all simulations are approximately parameterization independent and the observed increase in σ and rmax with increasing w for simulations without turbulence is due to the increasing spread introduced by nucleation. Additionally, simulations with turbulence shift σ and rmax to larger values (<1 μm). For w < 0.5 m s−1, the behavior is influenced by turbulence and radiation and the impact increases with simulation time and therefore decreasing w. The base simulations (green) show less nucleated droplets due to a smaller peak supersaturations reached in the process of nucleation. Therefore, ⟨r⟩ and rmax increase, because the condensed water is shared among less droplets. The simulations with radiation only (yellow) are similar to the base simulations except for long simulation times corresponding to w < 0.2 m s−1, where ⟨r⟩, rmax, and σ strongly increase. The reason is that many droplets from the initially sharply peaked droplet distribution fall at the same time below the separation radius [Eq. (11)] and start to evaporate until they eventually denucleate (regime III). The turbulent simulations (blue) increase in σ and rmax with simulation time. The droplet number concentration does not drop as low as for the base simulations, because droplets with positive saturation fluctuations keep nucleating over time, resulting in an approximately constant mean radius. For simulations with the combination of turbulence and radiation (red) the increasing impact of radiation on σ and rmax happens earlier compared to radiation only simulations (yellow), because droplets smaller than the separation radius are introduced early on from saturation fluctuations. The droplet number concentration is smaller than for the base simulations (green) because the droplets that nucleate do not grow in the subsaturated environment. Therefore, thermal radiation acts as an secondary nucleation inhibitor. The shaded regions, which indicate dissipation rates between ε = 10 and 50 cm2 s−3 (blue and red), show only a small impact. Most notably, the standard deviation is decreases, but with a decreasing impact at low updrafts.

Fig. 7.
Fig. 7.

As in Fig. 6, but the simulations were evaluation after the clouds reached a vertical extent of 100 m under different updraft velocities, and simulations with turbulence show two dissipation rates: ε = 50 cm2 s−3 (solid) and ε = 10 cm2 s−3 (dotted).

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

d. Collection initiation

Following a similar approach as Hartman and Harrington (2005a), we calculate the collection initiation time scale τcoll as the time needed to grow droplets with a number concentration of nmax = 0.1 cm−3 that have a radius larger than rdrizzle = 20 μm. The noise of τcoll, due to a small number of droplets in the tail, decreases with increasing nmax, but the time scale itself increases. We chose a trade-off value for nmax balancing the two impacts. Figure 8 shows that turbulence and radiation lower τcoll. Turbulence alone lowers τcoll significantly (blue) while radiation alone has only a small impact (yellow). However, the combination of turbulence and radiation (red) shows significantly shorter time scales for collection initiation than turbulence alone. A interesting feature is the peak of the collection initiation time centered around w = 0.2 m s−1 (depending on ε) for the turbulent simulations (blue) and around w = 0.6 m s−1 for RAD&T simulations (red). For larger updraft speeds, adiabatic cooling due to w dominates the collision initiation time scale and the differences between parameterizations become small. For turbulent simulations (red and blue) and updrafts below the peak, the additional spread in the droplets populations significantly reduces τcoll. The parameters of the time series in Fig. 5 are indicated by black vertical lines in the Fig. 8 and show that both updrafts (0.1 and 1 m s−1) produce drizzle at a similar rate, if turbulence is included (blue and red).

Fig. 8.
Fig. 8.

A sensitivity study of the collision initiation time scale with respect to the updraft velocity: τcoll is defined as the time needed to have nmax = 0.1 cm−3 with size r > rdrizzle. Two dissipation rates are shown: ε = 50 cm2 s−3 (solid) and ε = 10 cm2 s−3 (dotted). Additionally, the blue and red shaded regions mark intermediate values of ε. The vertical black lines at w = 0.1 and w = 1 m s−1 mark where the previously shown distributions and time series are evaluated. The color and label conventions are as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

e. Sensitivity to the radiation factor

In this section, we increase f from 0 to 1 and investigate the impact on the droplet size distribution. Values for f larger than 0.5, that are not reached in the atmosphere are shaded gray. For estimates of possible f values see the Fig. 2. We show the results for simulations with w = 1 m s−1, ε = 50 cm2 s−3 (solid lines) and w = 0.1 m s−1, ε = 10 cm2 s−3 (dotted lines) in Fig. 9. The vertical black lines indicate the value f = 1/6, which is used in the previous sections for the distributions and time series results. The general impact of higher radiation factors is that the subsaturation of the environment increases, and therefore, the separation radius becomes larger, which increases σ and rmax of the droplet population and decreases the droplet number concentration. All simulations show a decreasing droplet number concentration with increasing f. The droplet number concentration decreases stronger for simulations with turbulence (red) and starts at lower values for simulations with lower w. Therefore, all simulations show increasing mean radii with increasing f. Only the simulations with w = 1 m s−1 and without turbulence (solid yellow) show a decreasing mean, because small droplets are introduced by passing rsep, but complete evaporation has not yet started. The impact of passing rsep is strongest for sharply peaked droplet distributions with a large number of droplets at once below rsep. σ and rmax increase strongly with increasing f, as expected.

Fig. 9.
Fig. 9.

A sensitivity study with respect to the radiative factor. The simulations are evaluated after 15 min. The colors and labels represent the combination of radiation and turbulence (red, RAD&T) and the reference simulations including radiation (yellow, RAD). The solid and dotted lines show updraft speeds of 1 and 0.1 m s−1 with ε = 50 or 10 cm2 s−3 if the simulations include turbulence. Additionally, the gray shaded areas indicate f values that are not present in the atmosphere. For more information, see the caption of Fig. 6.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

f. Sensitivity to the aerosol distribution

This section is intended to access the impact of the aerosol distribution. In Fig. 10 we show the time series results for simulations with different CCN number concentrations (left panels) and ratios of n20/nccn (right panel). The ratio n20/nccn = 0 signifies that the CCN are only taken from the larger aerosol mode and n20/nccn = 1 that the CCN are only taken from the smaller aerosol mode, with a constant nccn = 100 cm−3. The standard deviation of the droplet size distribution is influenced by the evolution of the saturation in the nucleation process. A higher and broader peak in saturation gives rise to more diverse saturation growth histories of the droplets and therefore a larger standard deviation. The peak saturation increases with a decreasing droplet number concentration, because less droplets bind less water in short time intervals. The experiments either decrease the droplet number concentration directly by decreasing the CCN number concentration (left panel) or indirect by using smaller CCN sizes, which need higher peak saturations to nucleate. Smaller droplet number concentrations also lead to a larger mean radius, because the condensed water is shared among less droplets. Consequently, the increasing mean radius and standard deviation lead to a increasing maximum radius. Figure 11 confirms that larger mean radii, standard deviations, and maximum radii result from smaller droplet number concentrations in all parameterization combinations.

Fig. 10.
Fig. 10.

A sensitivity study of the relative sedimentation distances evaluated after 15 min. In the Gaussian approximation, 95.45% of the droplets have a relative vertical distance below 4σdz. For simulations with turbulence, two dissipation rates and updrafts are shown: ε = 50 cm2 s−3 with w = 1 m s−1 (solid) and ε = 10 cm2 s−3 with w = 0.1 m s−1 (dotted). The vertical black lines at w = 0.1 and 1 m s−1 indicate where previously shown distributions and time series are evaluated. The color and label conventions are as in the Fig. 3.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

Fig. 11.
Fig. 11.

The time series are evaluated from the beginning of condensation over 15 min with w = 1 m s−1, f = 1/6, and ε = 50 cm2 s−3 and with the combination of radiation and turbulence (RAD&T). The red lines represent the reference simulations with nccn = 100 cm−3 and n20/nccn = 0.6. (left) A sensitivity study with respect to the CCN number concentration nccn. The gray colors represent nccn values below and the blue colors above 100 cm−3. (right) A sensitivity study with respect to the CCN number concentration ratio n20/nccn with nccn = n20 + n75 = 100 cm−3 from the two aerosol modes. The gray colors represent larger ratios and therefore smaller CCN, and the blue colors represent smaller ratios and therefore larger CCN compared to the reference ratio of 0.6.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

4. Conclusions and discussion

Our investigation of the idealized parcel model showed the following results: Cooling by the emission of thermal radiation can cause a doubling of the droplet size standard deviation, with particularly strong effects in combination with turbulence (Fig. 5). The updraft sensitivity shows that turbulence and RAD are more important at small updraft speeds with approximately equal contributions to the standard deviation (Fig. 6). Furthermore, the longer the radiation can operate, the larger the impact. It even becomes the dominant contributor to the droplet size standard deviation, for slowly developing clouds with w = 0.01 m s−1 (Fig. 7). Radiation acts as a nucleation inhibitor, due to the subsaturated environment, which suppresses the nucleation of cloud droplets. Finally, the results for the collision initiation time scale suggest that thermal radiation may play a role in bridging the condensation coalescence bottleneck by increasing the droplet size standard deviation and accelerating the creation of larger droplets.

The simple parcel model allows fast studies, which illustrate the dependencies of a number of parameters. In comparison, LESs are more expensive and introduce complexity, which requires additional statistical analysis. Nevertheless, the simplistic approach comes with limitations and in the following we will highlight some possible issues and give an outlook on subsequent work.

Sedimentation is not included. All droplets are approximated to stay in the parcel if they do not completely evaporate. An estimate of droplet fall speeds using the Stokes approximation shows that droplets with radius of 10 and 20 μm would have fallen 10 and 50 m in 15 min. This is suggesting that sedimentation can be neglected for the time period of 15 min, but longer simulations may become increasingly unrealistic without sedimentation. The sedimentation analysis of the parcel simulations can be found in the appendix. Additionally, the nucleation process depends on RAD and turbulence, which has the most impact for low updraft speeds. The dependence is not separately evaluated here, but details can be found in Marquis and Harrington (2005). Furthermore, the statistics of the turbulence parameterization is kept constant while changing the updraft speed from 0.01 to 1.5 m s−1. The saturation and corresponding updraft fluctuations for ε = 10 and ε = 50 cm2 s−3 are based on stratocumulus cloud-top and shallow cumulus core cases. de Lozar and Mellado (2015) showed that cloud-top regions of stratiform clouds have updraft speed fluctuations similar to those of a shallow cumulus cloud base. There are uncertainties with respect to turbulence, which may be due to unresolved scale interactions or dependence on atmospheric composition. Furthermore, turbulent mixing of moist and dry air is neglected, even though it is a very important process at the cloud edges. The correlation of dry downdrafts and wet updrafts may result in even larger saturation fluctuations. Finally, the parcel position is kept at cloud top, which might be unrealistic, but serves as a maximum impact scenario for thermal radiation. The current simple approach can be investigated without explicitly solving the radiative transfer equation and serves as a baseline for more complicated scenarios. The next step will be to run RAD and the turbulence parameterization in LESs with resolved radiation. In an LES, the saturation fluctuations can be calculated with the prognostic subgrid turbulent kinetic energy. The positions of the droplets inside the scene will be resolved, as well the cloud edges with realistic radiative cooling rates (e.g., Jakub and Mayer 2015). Alternatively, it would be interesting to investigate the impact of radiative cooling on ice clouds, because higher clouds show higher radiative cloud-top cooling.

Acknowledgments

We acknowledge support from the DFG Collaborative Research Center TRR 165, Project B4: Radiative heating and cooling at cloud scale and its impact on dynamics and would like to thank F. Jakub for fruitful discussions, T. Koelling for computational support, P. Polonik for proofreading, and two anonymous referees whose constructive comments improved the manuscript considerably.

APPENDIX

Sedimentation

This section is intended to quantify the vertical droplet dispersion due to sedimentation in the context of a parcel model of 50-m vertical extent. The sedimentation speed of the droplets can be calculated with a piecewise function of the droplet radius as shown in Eq. (A1) and provided by Rogers and Yau (1996):
υ⁡(r)={1.19×108×r2,ifr<40×10−6 m8×103×r,if40×10−6<r<0.6×10−3 m2.01×102×r1/2,else.
A spectrum of droplet radii will lead to a spectrum of sedimentation speeds. The sedimentation distance after 15 min for each droplet can be obtained by integrating the sedimentation speeds over time. Figure A1 shows the standard deviation of the total sedimentation distances obtained over all droplets, more precisely, 4 times the standard deviation, which states that 95.4% of the droplets have a relative distance below 4σdz m, in the approximation of a Gaussian distribution of sedimentation distances. The intermediate updrafts between w = 0.2 and w = 0.4 m s−1 show the largest spread, with a maximum of 30 m, which is below the assumed vertical parcel length of 50 m.
Fig. A1.
Fig. A1.

The simulations are evaluated after 15 min and run with w = 1 m s−1, f = 1/6, and ε = 50 cm2 s−3. (left) A sensitivity with respect to the CCN number concentration nccn. (right) A sensitivity study with respect to the CCN number concentration ratio n20/nccn with nccn = n20 + n75 from the two aerosol modes. The vertical black lines indicate the reference values of n = 100 cm−3 and n20/nccn = 0.6. For more information, see the caption of Fig. 6.

Citation: Journal of the Atmospheric Sciences 77, 6; 10.1175/JAS-D-18-0349.1

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    • Export Citation

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    • Crossref
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    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Droplet growth speed evaluated for the RAD equation [Eq. (8)]. The radiative term is set (left) to zero and (right) according to Eq. (14) (with f = 1/6 and T = 282 K from the American standard atmosphere at 1 km). The isosaturation lines are centered around S* = 0 and 10 times incremented and decremente