Journal of Geophysical Research: Atmospheres

Investigation of Turbulent Entrainment-Mixing ProcessesWith a New Particle-Resolved DirectNumerical Simulation ModelZheng Gao1, Yangang Liu1,2 , Xiaolin Li1, and Chunsong Lu3

1Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY, USA,2Brookhaven National Laboratory, Upton, NY, USA, 3Key Laboratory for Aerosol-Cloud-Precipitation of ChinaMeteorological Administration, Nanjing University of Information Science and Technology, Nanjing, China

Abstract A new particle-resolved three-dimensional direct numerical simulation model is developedthat combines Lagrangian droplet tracking with the Eulerian field representation of turbulence nearthe Kolmogorov microscale. Six numerical experiments are performed to investigate the processes ofentrainment of clear air and subsequent mixing with cloudy air and their interactions with cloudmicrophysics. The experiments are designed to represent different combinations of three configurationsof initial cloudy area and two turbulence modes (decaying and forced turbulence). Five existing measuresof microphysical homogeneous mixing degree are examined, modified, and compared in terms of theirability as a unifying measure to represent the effect of various entrainment-mixing mechanisms on cloudmicrophysics. Also examined and compared are the conventional Damköhler number and transition scalenumber as a dynamical measure of different mixing mechanisms. Relationships between the variousmicrophysical measures and dynamical measures are investigated to search for a unified parameterizationof entrainment-mixing processes. The results show that even with the same cloud water fraction, thethermodynamic and microphysical properties are different, especially for the decaying cases. Furtheranalysis confirms that despite the detailed differences in cloud properties among the six simulationscenarios, the variety of turbulent entrainment-mixing mechanisms can be reasonably representedwith power law relationships between the microphysical homogeneous mixing degrees and thedynamical measures.

Plain Language Summary This paper first presents a new particle-resolved direct numericalsimulation model by combining the Lagrangian tracking of individual droplets with the Eulerian fieldrepresentation of turbulence. The model is then used to address the outstanding problem of entrainment-mixing processes and their interactions with cloud microphysics. Based on the comprehensive analysisof the model results, an expression is recommended to parameterize the effect of entrainment-mixingprocesses on cloud microphysical properties in larger-scale models such as large eddy simulation andclimate models. This study is a rare, valuable addition to both the development of particle-resolveddirect numerical simulation model and study/parameterization of various turbulent entrainment-mixingprocesses.

1. Introduction

Reliable knowledge of cloud droplet size distributions and related microphysical properties (e.g., dropletconcentration, liquid water content, and relative dispersion) is crucial for many cloud-related areas such asprecipitation, weather and climate modeling, and remote sensing. A long-standing problem in cloud physicsis that observed droplet size distributions are generally much broader than those predicted by the classicaluniform model (e.g., Howell, 1949; Hudson & Yum, 1997; Yum & Hudson, 2005). Understanding the issue ofso-called spectral broadening has been a fundamental focus of cloud physics over the last decades, and anumber of ideas have been proposed, including stochastic condensation theory that considers the growth ofdroplet populations as a stochastic process and relates the spectral broadening to turbulence-related fluctu-ations (Hsiu-chi, 1964; Khvorostyanov & Curry, 1999; McGraw & Liu, 2006; Sedunov, 1974), systems theory thatapplies statistical physics ideas to cloud physics (Liu et al., 1995; Liu & Hallett, 1997, 1998; Liu & Daum, 2002;Yano & Moncrieff, 2016), turbulence-induced preferential concentration of droplets (Shaw et al., 1998),

RESEARCH ARTICLE10.1002/2017JD027507

Special Section:Fast Physics in Climate Models:Parameterization, Evaluationand Observation

Key Points:• A new particle-resolved direct

numerical simulation model isdeveloped

• Initial configuration of cloudy areasand turbulence mode exert notableimpacts on cloud properties

• An expression is presented forparameterizing the effect ofentrainment-mixing processes oncloud microphysics

Supporting Information:• Supporting Information S1

Correspondence to:Y. Liu,lyg@bnl.gov

Citation:Gao, Z., Liu, Y., Li, X., & Lu, C.(2018). Investigation of turbulententrainment-mixing processes witha new particle-resolved directnumerical simulation model.Journal of GeophysicalResearch: Atmospheres, 123.https://doi.org/10.1002/2017JD027507

Received 25 JUL 2017

Accepted 26 JAN 2018

Accepted article online 6 FEB 2018

©2018. American Geophysical Union.All Rights Reserved.

LIU ET AL. 1

BNL-203223-2018-JAAM

Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

and turbulent entrainment-mixing processes (Baker et al., 1980; Hicks et al., 1990; Lu, Liu, Niu, & Vogelmann,2013; Telford & Chai, 1980; Su et al., 1998; Warner, 1973). Another outstanding problem is related to the for-mation of warm rain (Liu et al., 2004; McGraw & Liu, 2003). It is observed that precipitation in warm cloudscan be initiated within 30 min after cloud formation (Yau & Rogers, 1996). However, according to adiabaticcondensational growth theory, too much time is required for cloud droplets to grow large enough to initiatethe collision-coalescence process, and moreover cloud droplet size distribution becomes narrowed as clouddroplets grow, hampering realistically fast growth of cloud droplets to raindrops.

Despite the progress (see Devenish et al., 2012; Grabowski & Wang, 2013, for recent reviews), details ofthe processes involved remain poorly understood and elusive. Furthermore, it is commonly accepted thatthe accuracy and reliability of climate models in projecting the climate change caused by climate forc-ing depend heavily on cloud feedbacks and thus on parameterizations of still poorly understood cloudprocesses. The situation worsens when interactions with natural and anthropogenic aerosols are included.Indeed, the latest Intergovernmental Panel on Climate Change continues to assign “very low confidence” toaerosol-cloud-precipitation interactions, with even the sign of the resulting climate forcing remaining uncer-tain. Understanding such complex processes and upscaling them to adequate representation in weather andclimate models present additional challenges to the scientific community, which become more acute forextreme precipitation and weather events and as models progress to ever increasing resolutions. In particular,many key processes that are either not represented at all or represented in a very rudimentary way occur atscales smaller than typical grid sizes of even large eddy simulation (LES) models (e.g., 100 m) or cloud-resolvingmodels (e.g., 1 km) including turbulence fluctuations, turbulent entrainment-mixing, and their interactionswith cloud microphysics.

Despite their differences, virtually all the ideas tie the outstanding problems to turbulence-related processesthat occur on sub-LES scales (e.g.,<100 m) such as turbulent entrainment-mixing processes and turbulence-microphysics interactions. There are significant knowledge gaps on such sub-LES scale processes because ofthe limitations in observations and computer modeling. Fully addressing these vital knowledge gaps at thefundamental level calls for a particle-resolved direct numerical simulation (DNS) model that not only resolvesthe smallest turbulent eddies in clouds but also tracks motion and growth of individual particles as a supple-ment to measurements. Since the pioneering works in the early 2000s (Vaillancourt & Yau, 2000; Vaillancourtet al., 2002), a few studies have contributed to developing and applying DNS to study cloud microphysics. TheDNS presented in Vaillancourt and Yau (2000) and Vaillancourt et al. (2002) solves the forced incompressibleNaiver-Stokes equations in 3-D by using the method of Sullivan et al. (1994) and tracks individual droplets.The authors investigated the influence of turbulence and nonuniformity in the spatial distribution of sizesand positions of cloud droplets on the droplet size distribution under adiabatic conditions. The relationshipamong preferential concentration, sedimentation, and the Stokes number were also discussed. However, theeffect of entrainment-mixing processes on cloud microphysics was not addressed in these studies. In a seriesof publications (Andrejczuk et al., 2004, 2006, 2009), Andrejczuk and his coauthors developed a DNS to studythe cloud-clear air interfacial mixing and effects of mixing processes on cloud microphysics in decaying moistturbulence. They examined the effects of initial turbulence kinetic energy (TKE), cloud fraction, and dropletsizes. The relationship between the mixing mechanisms and the Damköhler number was also explored. Theinitial cloud filaments and velocity field were preset to focus on the details of the decaying turbulence. Bothbulk microphysics and detailed bin microphysics were used in the model. Malinowski et al. (2008) comparedlaboratory measurements with the results of Andrejczuk et al. (2004, 2006). de Lozar and Mellado (2013)added more features such as sedimentation and particle inertia in the bulk formulation. In Lanotte et al. (2009)and Celani et al. (2005), a model combining the Eulerian description of the turbulent velocity and super-saturation fields with a Lagrangian population of cloud droplets was used to study the condensation andevaporation of cloud droplets in turbulent flows. Similar to Vaillancourt et al. (2002) and Lanotte et al. (2009),Kumar, Schumacher, and Shaw (2012) and Kumar, Janetzko, et al. (2012) developed a particle-resolved DNS tostudy turbulent entrainment-mixing processes. In their work, a slab-like vapor field was adopted to mimic thesupersaturated cloudy area and subsaturated environment. The effects of temperature and buoyancy wereignored, while an artificial isotropic volume forcing is introduced to maintain the turbulence. Kumar et al.(2014) extended their previous work to both forced and decaying turbulence and found that the buoyancydue to droplet evaporation played a minor role in the mixing process in their simulations. Kumar et al. (2017)further used their DNS with the decaying turbulence setup constrained by the thermodynamic conditionsobserved in monsoon convective clouds over the Indian subcontinent during the Cloud Aerosol Interaction

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and Precipitation Enhancement EXperiment to investigate cloud-edge mixing and its impact on the clouddroplet size distribution.

Despite the progress, many questions regarding turbulence-microphysics interactions and turbulententrainment-mixing processes remain elusive, still posing challenges for fundamentally understanding andrepresenting clouds in coarse-resolution models. This work expands on these previous studies with three pri-mary objectives. First, as for the entrainment and mixing study, it is interesting to note that the settings inKumar, Janetzko, et al. (2012) and Andrejczuk et al. (2004) are similar, except the initial configuration of thecloudy area. The cloudy area consists of worm-like structures determined by the velocity field of the turbulentflow in Andrejczuk et al. (2004), while a slab-like cloud filament is adopted in Kumar, Janetzko, et al. (2012).Differences in the configuration of cloudy area may cause differences in the results. However, no study hasbeen concentrated on such configuration impacts. Thus, the first objective is to explore the effects of initialconfiguration of cloudy area. Second, different mixing mechanisms can occur and change during cloud evolu-tion between the limiting homogeneous and extreme inhomogeneous mixing types (Andrejczuk et al., 2009;Burnet et al., 1992; Lehmann et al., 2009), and thus it is essential to have a unified parameterization of the vari-ous mixing processes for larger-scale models such as large eddy simulation (LES) models and climate models.However, such studies are rare. Andrejczuk et al. (2009) examined their DNS results to explore this possibil-ity in terms of the relationship between microphysical properties to the Damköhler number. Lu et al. (2011)proposed the transition scale number to measure the occurrence probability of homogeneous or inhomo-geneous entrainment-mixing process. Lu, Liu, Niu, Krueger, and Wagner (2013) and Lu et al. (2014) proposedsome microphysical measures of homogeneous mixing degree and showed a positive relationship betweenthe transition scale number and the microphysical homogeneous mixing degree using both in situ observa-tions and numerical simulations with the Explicit Mixing Parcel Model (EMPM) (Krueger et al., 1997; Su et al.,1998). Thus, the second objective is to systematically investigate the potential of unifying the parameteriza-tion of different entrainment-mixing mechanisms for larger-scale models. Last but not least, to the best ofour knowledge, all the particle-resolved DNS models have been based on pseudospectral methods (Celaniet al., 2005; Kumar, Janetzko, et al., 2012; Orszag, 1972; Rogallo, 1981). However, the standard pseudospectralmethod has some limitations. As pointed out in Kumar, Janetzko, et al. (2012), the spectral method requiressmooth initial conditions to avoid the Gibbs phenomenon (numerical overshoots at sharp interfaces), andthus unable to address sharp or zeroth-order discontinuous interfaces that likely exist in real clouds (Brenguier,1993). Moreover, the spectral method requires a periodic boundary condition in each direction and thus can-not be applied to flows that require a nonperiodic, physical boundary condition. The third objective is todevelop a new particle-resolved DNS using the finite difference method coupled with the WENO (weightedessentially non-oscillatory) scheme (Jiang & Shu, 1996) that is flexible enough to deal with sharp cloud-airinterfaces as well as different boundary conditions and thus enable more general and realistic applications.This paper focuses on applying the model to address the first two objectives; applications to more realisticboundary conditions and effects of interface structure are deferred to future investigation.

The rest of the paper is organized as follows. Section 2 introduces the system of equations and the numericalschemes used to solve these equations. Section 3 describes the design of the numerical experiments, includ-ing configurations of different cases studied and initial and boundary conditions for numerical simulations.The results and discussion are provided in section 4. Concluding remarks are presented in section 5.

2. Description of New Particle-Resolved DNS2.1. Equations of Thermodynamical and Dynamical FieldsSimilar to most previous DNS models (e.g., Andrejczuk et al., 2004), our new DNS is based on the incompress-ible Boussinesq fluid system. Briefly, the dynamical field is given by

𝜕tu + (u ⋅ ∇)u = − 1𝜌0

∇p + 𝜈∇2u + fb + fe (1a)

∇ ⋅ u = 0 (1b)

where u is the velocity field, p is the pressure field, 𝜈 = 1.5 × 10−5 m2 s−1 is the kinematic viscosity, and 𝜌0 isthe density of dry air. Here fb is the buoyancy force given by

fb = −g[

T − T0

T0+ 0.608(qv − qv0) − qc

](2)

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where g is the gravity, and T and qv are temperature and vapor mixing ratio field, respectively, with thesubscript “0” denoting the reference value. The force fe is introduced as an external, “large”-scale forcingto maintain a statistically stationary homogeneous turbulence and is determined by the low-wavenumberforcing in the Fourier space. The Fourier transformation of fe is defined as

fe(k, t) = 𝜖inu(k, t)∑

kf ∈𝜅|u(kf , t)|2

𝛿k,kf(3)

where u(k, t) is the Fourier transformation of velocity function , k is the wavenumber, kf is chosen from thewavenumber space𝜅 containing (2𝜋∕Lx , 2𝜋∕Ly, 4𝜋∕Lz)plus all permutations with respect to components andsign, and 𝜖in is the input energy rate (Ghosal et al., 1995). 𝛿k,kf

is a delta function. fe is obtained by applyinginverse fast Fourier transformation to fe. For decaying turbulence simulations, fe is set to 0.

The temperature T and vapor mixing ratio qv are described by the following equations (Kumar, Schumacher,& Shaw, 2012):

𝜕tT + (u ⋅ ∇)T =Lh

cpCd + 𝜇T∇2T (4)

𝜕tqv + (u ⋅ ∇)qv = −Cd + 𝜇v∇2qv (5)

where Lh is the latent heat of water vapor condensation, cp is the specific heat at constant pressure, and𝜇T = 𝜇v are the molecular diffusivity for temperature and water vapor, respectively, and assumes to be equalto 2.16 × 10−5 m2 s−1. The condensation rate Cd denotes the rate of exchange between liquid and vapor andis described by

Cd(X, t) =d(ml(X, t))

madt=

4𝜋𝜌lA

𝜌0a3Σn

i=1S(Xi, t)Ri(t) (6a)

where A is a function of temperature and pressure given by

A = 1∕[(

Lh

Gv T− 1

)Lh𝜌l

𝜇T T+

𝜌lGvT

𝜇v es(T)

](6b)

where Gv = 461.5 J K−1 kg−1 is the individual gas constant for water vapor and es(T) is the saturation vaporpressure. The supersaturation S(X, t) is calculated directly from the water vapor mixing ratio and temperaturebased on the definition

S(X, t) =qv(X, t)

qv,s(X, t)− 1 (6c)

where qv,s is the corresponding saturation water vapor mixing ratio. The droplets grow or shrink depending onthe sign of supersaturation S. A positive and a negative Cd mean condensation and evaporation, respectively.

The liquid water mixing ratio is given by

qc(X, t) =4𝜋𝜌l

3𝜌0a3

n∑i=1

R3i (t) (7)

where a is the size of a grid cell, n is the number of droplets in the grid cell, 𝜌l and 𝜌0 are the densities of waterand air, and Ri(t) is the radius of the ith droplet.

2.2. Lagrangian Droplet Growth and MotionTo describe the motion and condensation(or evaporation) of cloud droplets, we use

Ri(t)dRi(t)

dt= A ⋅ S(Xi, t) (8)

dXi(t)dt

= Vi(t) (9)

dVi(t)dt

= 1𝜏p[u(Xi, t) − Vi(t)] + g (10)

where Xi(t) and Vi(t) are the position coordinate and velocity of the ith droplet, respectively; g = 9.8 m s−2

is the gravitational acceleration. The particle response time 𝜏p measures the droplet inertial effect and isgiven by

𝜏p =2𝜌lR

2i

9𝜌0𝜈(11)

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Equation (11) is appropriate for the Stokes particles whereby the Reynolds number based on the relativevelocity between the particle and fluid is significantly less than 1 and the drag follows the Stokes law (Eaton& Fessler, 1994). For the Stokes particles, the particle diameter is also smaller than the Kolmogorov microscale𝜂, the smallest length scales of the turbulent flow field. The last term in equation (10) is the sedimentationterm that accounts for the effect of gravity on droplets motion. When 𝜏p is set to be 0, equation (10) becomesVi(t) = u(Xi, t), which implies that the droplets exactly follow the turbulent flows. It is assumed that directinteractions between droplets are negligible during condensation/evaporation considering that the dropletsizes are too small compared to the mean distance between two droplets. The fluid velocity u(Xi, t) is obtainedthrough trilinear interpolation of the Eulerian field at position Xi.

2.3. Numerical ImplementationThe numerical code consists of three main modules to represent the Eulerian fluid, Lagrangian droplet, andcoupling between the two. The dynamic equations (1a) and (1b) are solved following the fraction step algo-rithm (Brown et al., 2001). This method is second-order accurate and uses the projection method to decouplethe pressure and velocity field. The thermodynamical fields equations (4) and (5) are solved with semi-implicitmethod coupling with fifth-order WENO scheme for the discretization of the hyperbolic term. The use ofWENO scheme here is critical since it can well handle the numerical overshoots as well as keep the high orderoverall accuracy. A central difference discretization is used to approximate the derivatives to obtain a consis-tent accuracy. To simplify the implementation, we adopt the external package Portable Extensible Toolkit forScientific Computation (Balay et al., 2016) as the parallel linear solver and Parallel High Performance Precon-ditioners (Falgout & Yang, 2002) as the preconditioner. These two packages are widely used in the communityof computational fluid dynamics and has a good parallel scaling in both Linux clusters and supercomputers.The droplet position equation (9) and motion equation (10) are solved by implicit Euler method in consid-eration of efficiency and stability. The dynamical and thermodynamical fields are represented on a Eulerianrectangular grid, while the particles are explicitly tracked during the simulation by utilizing the informationof the Eulerian field through trilinear interpolation. The Lagrangian particles impact on the Eulerian fieldthrough equation (6a), which acts as a source or sink term in equations (4) and (5). The fluid field is not directlyaffected by the particle ensemble but indirectly by the thermodynamical field through the buoyancy forcedescribed by equation (2). The time step size is adaptive to satisfy the Courant-Friedrichs-Lewy conditionand is 0.003 s on average. The parallel algorithm is designed by decomposing the domain into subdomains.The computation of a subdomain is conducted on an independent processor, and the exchange of bound-ary information between the processors is needed after solving the equation. This algorithm is implementedusing the Message Passing Interface (MPI) library, and a linear scaling has been achieved up to 128 cores ona Linux cluster.

The current numerical domain is set to be 0.512 m3 with triply periodic boundary conditions. The compu-tational grid is 2563, corresponding to grid spacing of 2 mm, close to the typical Kolmogorov length (seesupporting information for a detailed discussion).

3. Description of Numerical Experiments3.1. Initial Velocity FieldThe initial velocity field is constructed in the Fourier space and then transformed to the physical space. Fol-lowing the procedure proposed in Rogallo (1981), one can generate a solenoidal isotropic velocity field withprescribed energy spectrum (Rosales & Meneveau, 2005)

E(k) = 16√𝜋∕2

u20k4

k50

exp

(−2k2

k20

)(12)

where u20 is the initial Root Mean Square (r.m.s.) velocity, and k0 is the wavenumber at which the maximum of

E(k) occurs. The parameters u0 and k0 determine the power spectral shape. The parameters u0 = 0.35 m/s andk0 = |(1, 1, 2)| ≈ 2.4 are used in simulations presented in this paper, which allows one to generate an initialturbulence field with reasonable Reynolds number and narrow energy band in large wavelengths. Differentfrom the commonly used Kolmogorov spectrum, this function enforces the kinetic energy to be concentratedin a relatively narrow band at the initial time, so as to not affect the turbulence behavior in larger wavenumberspace. As turbulence evolves, the spectrum will quickly spread to the inertial range and dissipation rangeaccording to the Navier-Stokes equation.

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3.2. Initial Fields of Vapor Mixing Ratio and TemperatureThree different initial configurations of cloudy area are used to investigate the impact of cloudy area config-uration. Case 1 follows that used in Andrejczuk et al. (2004) whereby water mixing ratio is defined accordingto the sign of the velocity function in physical space such that

Case 1:qv(x, t = 0) ={

qmaxv , u(x)> 0

qv,e, u(x) ≤ 0(13)

where qmaxv = 3.95 g/kg is the maximum amplitude of qv , which exceeds qv,s by 2%, and qv,e = 0.03 g/kg is

the vapor mixing ratio of the clear air. u(x) is the x component of the fluid velocity.

In Kumar, Schumacher, and Shaw (2012), the author investigated a slab-like cloud configuration approximatedwith a smooth function to avoid the Gibbs phenomenon. Similarly, our Case 2 is designed to study the slab-likeconfiguration but approximated with a simple discontinuous function given by

Case 2: qv(x, t = 0) ={

qmaxv , (L − d)∕2 ≤ x < (L + d)∕2

qv,e, elsewhere(14)

where the qmaxv and qv,e are the same as in Case 1. L is the length of computational domain, and d = L∕2 is the

width of the cloud slab.

It is well known that entrainment-mixing processes can also occur near cloud tops, especially for stratiformclouds (Lu et al., 2011; Yum et al., 2015). To mimic the cloud-top entrainment-mixing process, herein we adda new cloud configuration Case 3 by rotating Case 2 by 90∘.

Case 3: qv(z, t = 0) ={

qmaxv , (L − d)∕2 ≤ z < (L + d)∕2

qv,e, elsewhere(15)

The temperature field is initialized by imposing the neutral buoyancy condition such that (Kumar et al., 2014)

T(x, t = 0) = T0 − 0.608T0[qv(x, t = 0) − qv0] (16)

where the reference values are defined by the domain averages T0 = ⟨T(t = 0)⟩V and qv0 = ⟨qv(t = 0)⟩V .This neutral buoyancy condition ensures the initial cloudy area having higher water vapor mixing ratio butlower temperature compared to the environment. Note that this procedure is only performed for the initialtemperature field; later temperature field completely follows equation (4) afterward. Figure 1 compares theinitial fields of supersaturation and temperature for the three cases. Note that all the three initial configura-tions have the same dynamical field and initial cloud fraction of 0.5. The differences in initial water vapor andtemperature fields between the different cases are self-evident, allowing for examination of the impacts ofthe initial configuration of cloudy area on entrainment-mixing processes. By design, the initial fields of tem-perature and supersaturation in Case 1 are closely related to the initial velocity field, whereas those in Case 2and Case 3 are independent of the initial velocity field instead.

3.3. Initial DropletsAt beginning, a total of 107 droplets with the same radius of 15 μm are randomly placed in the cloudy areaaccording to the Poisson point process, giving a droplet number concentration of 153 cm−3. Note that for thefored turbulence mode, the velocity field needs a few time steps (5 s) to relax to a steady state. Therefore, thedroplets are released to move and change their sizes according to the physical laws after this spin-up period.For the decaying turbulence, the droplets are released at time t = 0 s since there is no need to reach a steadystate. The simulation is terminated when all the droplets completely evaporate or the field becomes nearlyuniform (the standard deviation of supersaturation is less than 0.0002).

For convenience, Table 1 summarizes the key quantities and initial conditions.

4. Numerical Results

Six numerical experiments are performed to represent six scenarios (denoted by D1, D2, D3, F1, F2, and F3)according to the combination of the two different turbulence modes (decaying versus forced) and three dif-ferent initial configurations of cloudy area (Case 1, Case2, and Case 3). This section presents the results, with afocus on entrainment-mixing processes. Note that droplets with radius smaller than 1 μm are excluded in themicrophysical analysis.

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Case 1

x [m]

z [m

]

0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

Case 2

x [m]

z [m

]

0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

Case 3

x [m]

z [m

]

0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

[%]

−80

−60

−40

−20

0

x [m]

z [m

]

0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

x [m]

z [m

]

0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

x [m]

z [m

]

0 0.2 0.40

0.1

0.2

0.3

0.4

0.5

[K]

270.8

270.9

271

271.1

271.2

271.3

Figure 1. Cross sections of the initial supersaturation (top row) and temperature (K) field (bottom row) for differentcases. The cloudy part occupies about half of the computational domain.

4.1. Evolution of Dynamical and Thermodynamic FieldsFigure 2 compares between the six simulations the temporal evolution of the domain mean, standard devi-ation, and relative dispersion of turbulent kinetic energy (TKE, first row), temperature (second row), watervapor mixing ratio (third row), and supersaturation (fourth row). Relative dispersion is defined as the ratio ofthe standard deviation to the mean of the corresponding variables. As expected, the mean TKE and its stan-dard deviation for the three forced turbulence simulations (F1, F2, and F3) remain approximately constantdetermined by the large-scale forcing after a short relaxation from the initial time. However, it is interestingto observe a transient turbulence enhancement before gradually decaying to 0 in the decaying cases, espe-cially for D2 and D3. This transient enhancement results likely from the buoyancy effect, which is caused bythe deviation of temperature and vapor mixing ratio to the reference value according to equation (2). The D3simulation exhibits the strongest enhancement, followed by D2. But for D2 the enhancement lasts longer. Themixing in D3 is accelerated by the sedimentation effect, making it a slightly stronger and faster than D2. Notethat D1 can be regarded as the intermediate stage of mixing process in D2 or D3, and therefore the buoyancyeffect quickly disappears and shows little enhancement in the figure. Most of the droplets have a chance toenter the clear air and evaporate at an early stage. Evaporation process absorbs latent heat from the environ-ment, resulting in deviation of the temperature field from the mean value. The transient enhancement can beseen more clearly from the standard deviation of temperature. The transient enhancement is weaker for thethree forced simulations F1, F2, and F3 (but still stronger than that of TKE). It is noteworthy that the behav-ior of transient turbulence enhancement does not appear in the field of water vapor mixing ratio, which is

Table 1Summary of Key Model Parameters and Initial Conditions

Quantity Symbol Value Quantity Symbol Value

Grid points N 256 Droplet radius R0 15 μm

Box length L 0.512 m Environment supersaturation Se −99%

Grid size a 0.002 m Cloud supersaturation Sc 2%

Viscosity 𝜈 1.5 × 10−5 m2 s−1 Number concentration Nc 153 cm−3

Dissipation rate 𝜖 2.0 × 10−3 m2 s−3 Eddy turnover time 𝜏L 4.27 s

Dissipation length 𝜂 10−3 m Evaporation time 𝜏evap 2.09 s

Dissipation time 𝜏𝜂 0.087 s Reaction time 𝜏react 4.52 s

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Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

0 5 10 15 20 25 30 350

0.005

0.01

0.015

0.02

0.025

Time [s]

Mea

n of

trub

ulen

ce k

inet

ic e

nerg

y [m

/s]

D1D2D3F1F2F3

0 5 10 15 20 25 30 350

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Time [s]

Sta

ndar

d de

viat

ion

of tu

rbul

ence

kin

etic

ene

rgy

[m/s

]

D1D2D3F1F2F3

0 5 10 15 20 25 30 350.7

0.75

0.8

0.85

0.9

0.95

Time [s]

Rel

ativ

e di

sper

sion

of t

urbu

lenc

e ki

netic

ene

rgy D1

D2D3F1F2F3

0 5 10 15 20 25 30

269

269.5

270

270.5

271

Time [s]

Mea

n te

mpe

ratu

re [K

]

D1D2D3F1F2F3

0 5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

0.3

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Figure 2. Temporal changes of key properties. The left, middle, and right columns are the mean, standard deviation, and relative dispersion, respectively. Therows are (first row) turbulent kinetic energy, (second row) temperature, (third row) vapor mixing ratio, and (fourth row) supersaturation.

consistent with Kumar et al. (2014). Note that the vapor mixing ratio in the clear air is much lower than in thecloudy air. The droplets entering the clear area quickly evaporate, while the droplets staying in the cloudyarea continue to grow by condensation. This phase transition process reduces the difference of vapor mix-ing ratio between clear air and cloudy air; thus, the transient growth of the deviation can hardly be observed.The behavior of supersaturation reflects the combination of temperature and water vapor mixing ratio, asexpected. The variations manifest themselves in the plots of relative dispersion.

LIU ET AL. 8

Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

0 5 10 15 20 25 30 35 40 450

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Figure 3. Evolution of droplet size distribution for decaying turbulence (left column) and forced turbulence (rightcolumn). The color denotes the droplet concentration normalized by the maximum concentration. The panels from topto bottom are Case 1, Case 2, and Case 3, respectively.

4.2. Evolution of Microphysical PropertiesFigure 3 shows the temporal variation of the cloud droplet size distribution for all the six simulations. Severalpoints are evident. First, the droplets start with a monodisperse distribution droplet radius of 15 μm. As theturbulent mixing between the subsaturated environment and the supersaturated cloudy air proceeds, somedroplets evaporate, and the size distributions gradually shift to small sizes and broaden until all droplets com-pletely evaporate. Due to the initial configuration, the final states of all the cases contain no droplets and arein subsaturated environments. Second, the three cases in the mode of decaying turbulence (D1, D2, and D3)are quite different in their evolution of size distributions. However, the difference between the three forcedturbulence cases (F1, F2, and F3) almost disappear, demonstrating that the buoyancy effect is overwhelmedby the external forcing in the cases of forced turbulence. The differences in the decaying cases are caused

LIU ET AL. 9

Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

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Figure 4. Temporal evolutions of (a) droplet concentration, (b) liquid water content, (c) mean volume radius, (d) meanradius, (e) standard deviation, and (f ) relative dispersion.

by the buoyancy term in equation (2). The role of buoyancy in spectral broadening is also evident from thecomparison of the corresponding simulations between modes of decaying and forced turbulence. It is note-worthy that some droplets grow in the first few seconds for all the cases, with the largest droplet radius ofabout 15.12 μm found in F2. However, this phenomenon is too difficult to see from the figure.

To better illustrate the impacts of different simulation scenarios, Figure 4 shows the temporal evolution ofthe droplet concentration (a), domain mean liquid water content (b), mean volume radius (c), mean radius(d), standard deviation (e), and relative dispersion of the cloud droplet size distribution (f ). It is evident thatin all the simulations, LWC and droplet concentration decrease as turbulent mixing and droplet evaporationproceed. The mean volume radius and mean radius also decrease with time because the decrease of liquidwater content is stronger/faster than that of droplet concentration. It is noteworthy that in all the simulations,standard deviations first increase, peak at some time, and then decrease beyond the peak time. The occur-rence of maximum standard deviation stems primarily from the combined spectral broadening related to

LIU ET AL. 10

Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

entrainment-mixing processes and the shrinking of droplet populations due to evaporation of small droplets.Also noteworthy is that the peak standard deviations occur between 5 s and 10 s for all the six simulationscenarios. The coupled variations of mean radius and standard deviation result in relative dispersion peakingat a much later time compared to standard deviation. Despite the commonalities, the differences among thedifferent scenarios are evident. Since the configuration of Case 1 is close to an already mixed case, its numberconcentration and mean radius decay at a faster rate, and the standard deviation of the droplet size is lowerthan the other cases. Case 2 and Case 3 exhibit no notable difference except for the number concentration.Since the mixing process of Case 3 is accelerated by the buoyancy effects in the vertical direction, the numberconcentration of Case 3 exhibits a stronger decrease than Case 2. Comparison between the forced cases anddecaying cases further shows two contrasting phases. In the first phase, the forced cases contain more liq-uid water content and larger number concentration and mean radius than the corresponding decaying cases,while the opposite is true in the later stage. This implies that the decaying cases initially have faster mixingand evaporation but are later overtaken by the forced turbulence.

5. Turbulent Entrainment-Mixing Processes

The impact of entrainment and subsequent cloudy-clear air mixing on the cloud droplet size distributionremains an important yet still unresolved issue in cloud physics. Although conservation of the total waterand moist static energy is often adequate to determine the temperature, water vapor, and cloud water mix-ing ratio of the homogenized mixture of cloudy and cloud-free unsaturated air, predicting the evolution ofcloud droplet size distribution requires additional constraints because the same amount of cloud water canbe distributed over either a large number of small droplets or a small number of large droplets. The concen-tration and size of cloud droplets depend critically on the specific entrainment-mixing processes. Whethercloud dilution is associated with homogeneous or extreme inhomogeneous mixing has been shown to sig-nificantly affect radiative properties of stratocumulus (Chosson et al., 2007) and shallow convective clouds(Grabowski et al., 2006; Slawinska et al., 2008). However, current models ranging from large eddy simulationsto climate models lack adequate representation of the mixing mechanisms, causing deficiencies in simulatedcloud microphysical properties (Endo et al., 2015). This section aims at developing physical understandingand parameterization of the entrainment-mixing processes using the DNS simulations.

5.1. Mixing Diagram AnalysisTurbulent entrainment of dry environmental air and subsequent turbulent mixing between cloudy air andenvironmental air and associated droplet evaporation are likely one of the primary factors that affect theevolution of droplet size distributions and corresponding microphysical properties. There are two limitingentrainment-mixing mechanisms proposed in the literature. One is that the entrained air and the cloudyair are mixed evenly and all cloud droplets evaporate at the same environment (Warner, 1973). This typeof mixing is referred to as homogeneous mixing. The other is extreme inhomogeneous mixing, where theentrained air mixes with only some portion of cloud parcel and all droplets in this portion completely evapo-rate while the droplets in the rest of the cloud parcel remain intact (Baker et al., 1980). Ambient clouds oftenfall between the two limiting mechanisms. To characterize the effect of turbulent entrainment-mixing pro-cesses on microphysical properties, the so-called R3

v − Nc diagram (Burnet & Brenguier, 2007; Jensen et al.,1985) has been widely used to identify the homogeneous/inhomogeneous entrainment-mixing process inobservational studies. The R3

v − Nc diagram was firstly applied to the analysis of DNS simulations with binmicrophysics in Andrejczuk et al. (2004, 2006, 2009). Kumar et al. (2014) further applied the mixing diagramto analyze their particle-resolved DNS simulations. In addition to their model differences, Andrejczuk et al.(2004, 2006, 2009) used Case 1 initial configuration of cloudy area, whereas Kumar et al. used a configurationsimilar to our Case 2. This section extends these pioneering DNS studies to use the mixing diagram to analyzethe results of all the six scenarios and to examine the potential influence of initial cloud configurations andmodes of turbulence treatments.

In addition to the domain mean as examined by Andrejczuk et al., we also examine smaller averaging boxesto obtain better ideas of statistics by following Kumar et al. (2014) to divide the computational domain into 64equal-sized sample boxes. We keep tracking the volume mean radius and number concentration in each sam-ple box and at each time step. Figure 5 shows the mixing diagrams for the six scenarios. The solid green dotrepresents the value sampled in each sample box at each time step; the red curve denotes the DNS domainaverage, with arrows indicating the direction of temporal evolution. The corresponding homogeneous mix-ing line (black dot) and extreme inhomogeneous mixing line (black solid) are also plotted in the diagram as

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Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

0 0.2 0.4 0.6 0.8 10

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Figure 5. Microphysical mixing diagram for scenarios D1, D2, D3, F1, F2, and F3. The green circles represent the resultsfor different sample boxes, and the red triangles represent the mean over the entire domain, with arrow indicating thedirection of time. The solid and dashed black lines denote the extreme inhomogeneous and homogeneous mixing,respectively. Only the boxes with nonzero droplets at the initial time are considered.

references. Note that in the top row, the mixing diagrams for D1 and F1 do not start from point (1, 1) sincethe initial droplets in a sample box have already been diluted and their number concentration are thus lessthan the adiabatic value. As pointed out by Andrejczuk et al. (2004), the configuration of initial cloudy areaexcludes the initial dilution process and can only be used to simulate the mixing process after dilution. Thedroplet number concentration remains nearly unchanged as the droplet size decreases until some time haselapsed, suggesting an extreme homogeneous mixing. The difference is not obvious between forced turbu-lence and decaying turbulence, except for a wider range of variability in the shape of the mixing trajectoriesfor the decaying turbulence D1, since the forced turbulence will foster the mixing process, resulting in similarstates in different sample boxes.

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Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

Figure 5 (middle row) shows the mixing diagrams for case D2 and F2. These cases have virtually the same con-figurations of the initial cloud areas with Kumar et al. (2014), except for the sharpness of the initial distributionof vapor mixing ratio: Kumar et al. used a smooth distribution to avoid the potential overshooting problem intheir model, whereas a sharper discontinuous distribution is used in this study because our model eliminatesthis problem. As expected, our simulations show overall resemblance with Kumar et al. For example, the phe-nomenon of inhomogeneous offset described by Kumar et al. (2014) can also be observed in the figures: themixing trajectories tend to shift to smaller values of Nc∕Nc,a. This inhomogeneous offset is due to the initialdilution process, in which the droplet number concentration in the sample boxes is diluted while the dropletmean radius in the sample box does not change much. As turbulent mixing proceeds, the turbulent time scalein the decaying case continues to increase, while the time scale for the forced turbulence remains unchanged.Therefore, the inhomogeneous mixing is more likely to occur in D2, leading to a slightly stronger deviationfrom the homogeneous mixing line. A similar conclusion can be obtained in Figure 5 (bottom row) for caseD3 and F3.

The mixing diagrams for the new case D3 and F3 are displayed in Figure 5 (bottom row). The results of Case 3resemble most properties of Case 1 and Case 2, except that its points are more scattered than the other twocases. This phenomenon is likely due to the sedimentation effect, which enhances the droplet motion in thevertical direction. Since the initial cloudy area in Case 3 is vertically sandwiched between two clear air regions,the cloud droplets moving up and down can easily enter a different environment (clear region) in the earlystage, therefore increasing the variability of the scatterplots. It is noteworthy that the difference between themixing diagrams lies more in the cloudy area configurations (especially Case 1 versus Case 2 or Case 3) thanthe mode of turbulence (forced or freely decaying) unlike the temporal evolutions of droplet size distributionshown in Figure 3. The strong similarity in the mixing diagrams between the six different simulations shouldbe highlighted, as it suggests the potential for a unified parameterization of different mixing mechanismsdetailed next.

5.2. Analysis of Microphysical Measures for Mixing MechanismsSince the real entrainment-mixing process can fall anywhere between the two limiting mixing mechanisms, itis desirable to define some measure that can cover all the possible mixing mechanisms. We generically calledsuch a measure as homogeneous mixing degree since a larger homogeneous mixing degree indicates that themixing process is closer to the limiting homogeneous mixing process. Based on the fact that the horizontal linein the R3

v −Nc mixing diagram corresponds to the extremely inhomogeneous mixing whereas the vertical lineimplies extremely homogeneous mixing whereby the number concentration remains constant during mixingand evaporation. Andrejczuk et al. (2009) realized that the homogeneous mixing degree can be quantified bythe instantaneous slope of the trajectories in the mixing diagram and is calculated using backward differencein time:

𝜓1 =R3

v,j∕R3v,a − R3

v,j−1∕R3v,a

Nj∕Na − Nj−1∕Na(17)

where Rv is the mean volume radius, N is the number concentration, and the subscript “a” denotes the adi-abatic value of the droplet population in the initial cloudy region. Note that 𝜓1 is in fact the inverse of theparameter defined by Andrejczuk et al. (2009) such that a larger value of𝜓1 indicates a higher degree of homo-geneous mixing, in line better with intuition and the other microphysical measures discussed below. It can bereadily shown that 𝜓1 equals 0 for the extremely inhomogeneous mixing, but approaches ∞ as the mixingprocess approaches the extreme homogeneous mixing. A shortcoming of this measure lies in the fact thatit is not explicitly constrained by the limiting homogeneous and/or inhomogeneous mixing types. To over-come this deficiency, Lu, Liu, Niu, Krueger, and Wagner (2013), Lu, Liu, Niu, and Vogelmann (2013), and Lu et al.(2014) introduced four more measures of homogeneous mixing degree. These measures are based on themixing of adiabatic cloudy air and clear air and slightly modified here to consider the instantaneous homoge-neous mixing degree between two adjacent temporal states in time tj and tj−1, with tj − tj−1 = 0.5s. Briefly, asschematically illustrated by Figure 6, state 1 represents the state at tj ; state 2 is the cloud state diluted by theenvironmental but before further evaporation. From state 2 to the final state 3, mixing and evaporation occur.The state 3 is located along the contour of the same water dilution ratio LWC/LWCi−1. The superscripts primeand double prime denote the possible final states corresponding to the limiting extreme inhomogeneous andhomogeneous mixing scenarios, respectively. The state made by the superscript triple prime corresponds to

LIU ET AL. 13

Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

Figure 6. Schematic illustration of the sequence of states involved in anentrainment and isobaric mixing event and the definition of thehomogeneous mixing degree. The three black solid lines correspond toextreme inhomogeneous mixing, homogeneous mixing (relative humidityof the dry air is 66%), and the contour of a constant 𝛾 = LWCj∕LWCj−1 = 0.2.The different superscripts above the state 3 denote the possible final statescorresponding to different mixing mechanisms. See text for the meanings ofthe other lines and symbols.

the situation whereby the final droplet concentration is larger than thehomogeneous one NH. The definition of the second measure 𝜓2 is basedon the angle (𝛽) between the linking states 2 and 3 and the line for extremeinhomogeneous mixing such that

𝜓2 = 𝛽

𝜋∕2(18a)

𝛽 = tan−1

(R3

v,j∕R3v,j−1 − 1

Nj∕Nj−1 − NH∕Nj−1

)for Nj ≤ NH, or (18b)

𝛽 = 𝜋 + tan−1

(R3

v,j∕R3v,j−1 − 1

Nj∕Nj−1 − NH∕Nj−1

)for Nj >NH (18c)

𝜓3 = 0.5

(Nj − NI

NH − NI+

R3v,j − R3

v,j−1

R3v,H − R3

v,j−1

)(19)

𝜓4 =ln R3

v,j − ln R3v,j−1

ln R3v,H − ln R3

v,j−1

(20)

𝜓5 =1 − R3

v,j∕R3v,j−1

1 − LWCjNj−1∕(NHLWCj−1)(21)

where all the variables are calculated from a sample box; LWC is the liquid water content; the subscript “j”means the value is calculated from the jth data set at time tj . The subscripts I and H indicate that the valuesare calculated based on the assumption of inhomogeneous and homogeneous mixing, respectively. Briefly,

NH = 𝜒Nj + (1 − 𝜒)Nej (22)

R3v,H =

NjR3v,j

NH(23)

NI =R3

v,j

R3v,j−1

Nj (24)

The mixing fraction 𝜒 of cloudy air is computed according to the mass conservation of total water betweenstate j and j − 1:

𝜒(qj−1vc + qj−1

lc ) + (1 − 𝜒)(qj−1ve + qj−1

le ) = qjlc + qj

vc (25)

where the subscripts c and e stand for the mean value of a sample box and its environmental air, and l andv stand for the liquid water and water vapor. The environmental air is chosen as the mean of the four gridsextended from the original sample box; the subscript j indicates the state of the jth data set collected attime tj . Note that equation (25) is slightly modified to considers the fact that the cloudy air may have beendiluted and the environmental air may contain droplets, unlike Lu, Liu, Niu, Krueger, and Wagner (2013), Lu,Liu, Niu, and Vogelmann (2013), and Lu et al., 2014 (2014) where there is a clear demarcation between thecloud and environmental air. The new equation reduces to the commonly used one if no cloud droplets existin environment air.

Figure 7 compares the five measures of microphysical homogeneous mixing degree, where each dot repre-sents an instantaneous domain mean and the different color denotes the different scenario. Evidently, 𝜓3,𝜓4, and 𝜓5 are virtually equivalent, with most points falling around the perfect one-to-one line. The relation-ship between 𝜓2 and 𝜓3 is tight until both are larger than 1. Despite being still largely positively correlated,the relationships between 𝜓1 and the other measures is much weaker, which is expected because 𝜓1 is notconstrained by the two limiting homogeneous and inhomogeneous mixing mechanisms.

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Figure 7. Comparison between the five measures of microphysical homogeneous mixing degree. Each dot representsan instantaneous domain mean; different color denotes the six different scenarios.

5.3. Analysis of Dynamical Measures for Mixing MechanismsThe entrainment-mixing process has been dynamically characterized with the Damköhler number defined asthe ratio of turbulent mixing time scale to droplet evaporation time scale (Baker et al., 1980; Grabowski, 1993;Krueger et al., 1997):

Da =𝜏mix

𝜏evap(26)

where the turbulence mixing time scale can be estimated as 𝜏mix = (𝜆2∕𝜖)1∕3; the length scale 𝜆 is repre-sented by the mean Taylor microscale for the cloud water, 𝜆 = (𝜆1 + 𝜆2 + 𝜆3)∕3, 𝜆i = ⟨q2

c⟩1∕2∕⟨(𝜕qc∕𝜕xi)2⟩1∕2,and the dissipation rate is estimated with 𝜖 = 2𝜈⟨(∇ × u)2⟩ (Andrejczuk et al., 2009). The evaporation timescale 𝜏evap represents the time that the droplet population needs to complete the evaporation and is givenby (Andrejczuk et al., 2009; Burnet & Brenguier, 2007)

𝜏evap = Rv

(dRv

dt

)−1

=R2

v

−ASe(27)

where Se is the supersaturation of the dry air. In general, Da ≪ 1 and Da ≫ 1 correspond to the homogeneousand extremely inhomogeneous mixing, respectively; ambient clouds often have Da between these two limits.

Recognizing that the turbulent mixing time scale depends on the entrained eddy sizes, Lehmann et al. (2009)introduced the transition length scale defined as the length scale at which Da = 1, such that

l∗ = 𝜖1∕2𝜏3∕2evap (28)

A larger transition scale length suggests a higher degree of homogeneous mixing.

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Journal of Geophysical Research: Atmospheres 10.1002/2017JD027507

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Figure 8. Scatterplots of the five microphysical measures (𝜓1, 𝜓2, 𝜓3, 𝜓4, and 𝜓5) of homogeneous mixing degree as afunction of the inverse of the Damköhler number (Da). Each point denotes the domain mean value for thecorresponding scenario and time; the symbol and color denote the different simulation scenario and the normalizedsimulation time, respectively.

Lu et al. (2011) further introduced the transition scale number by normalizing the transition scale length withthe corresponding Kolmogorov length scale:

NL =l∗

𝜂(29)

where the Kolmogorov length scale is given by

𝜂 =(𝜈3

𝜖

)1∕3

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Besides the evaporation time, reaction time (Lehmann et al., 2009) and phase relaxation time (Kumar,Janetzko, et al., 2012) have also been used as the microphysical time scale in defining the Damköhler numberand characterizing entrainment-mixing processes. Our analysis shows that for the simulations, the reaction

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Figure 9. Scatterplots of the five microphysical measures of homogeneous mixing degree as a function of the transitionscale number (NL). Each point denotes the domain mean value for the corresponding scenario and time; the symbol andcolor denote the different simulation scenario and the normalized simulation time, respectively.

time yields similar results with the evaporation time, whereas the results with phase relaxation time seems atodds with our conventional understanding. For consistency, the following results are hereafter based on theevaporation time; detailed discussion on the results based on reaction time and phase relaxation time andthe reason for choosing evaporation time can be found in the supporting information.

5.4. Parameterization of Entrainment-Mixing ProcessesThe various microphysical measures of homogeneous mixing degree, as given by equations (17), (18a), and(19)–(21), are expected to be related to the dynamical measures (the transition scale number as given byequation (29) and Damköhler number equation (26)) since the two types of measures quantify the homoge-neous mixing degree from different perspectives. Up to now, only few studies have examined some specificrelationships and explored the potential of using such relationships to parameterize mixing mechanisms.Andrejczuk et al. (2009) examined the relationship between the inverse of 𝜓1 and Da according to their DNS

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Table 2Summary of Power Law Fits and the Corresponding Values of R2

Variables Damköhler number Da Transition scale number NL

𝜓1 (98.01,−0.50, 0.26) (63.93, 0.27, 0.25)𝜓2 (81.98,−0.28, 0.84) (64.01, 0.16, 0.83)𝜓3 (94.62,−0.47, 0.43) (62.97, 0.26, 0.43)𝜓4 (94.98,−0.46, 0.37) (64.09, 0.25, 0.37)𝜓5 (95.32,−0.41, 0.34) (66.72, 0.23, 0.34)

Note. The three values in the parentheses represent a and b in the power lawfit y = axb and the determination coefficient R2, respectively.

simulations with bin microphysics, and found a power law relationship. Lu, Liu, Niu, Krueger, and Wagner(2013) and Lu et al. (2014) analyzed the observational data and EMPM simulations for the relationshipsbetween the other microphysical measures of homogeneous mixing degree (𝜓1, 𝜓2, 𝜓3, 𝜓4) and the transi-tion scale number. Here we expand these studies to systematically examine the relationships between thefive microphysical measures of homogeneous mixing degree, Da, and transitional scale number using all thenumerical data produced from the six simulation scenarios.

To examine the usefulness of the conventional Damköhler number, Figure 8 shows all five microphysical mea-sures (𝜓1, 𝜓2, 𝜓3, 𝜓4, 𝜓5) against the inverse of the Damköhler number, with color indicating the simulationtime normalized by the final time. The positive correlations are evident, suggesting that a smaller Damköhlernumber corresponds to a higher homogeneous mixing degree. The results (especially for 𝜓1) are consis-tent with that reported by Andrejczuk et al. (2009). Furthermore, according to the determination coefficient(R2), the 𝜓2-Da relationship has the best power law fit (R2 = 0.84). As expected from Figure 8, the relation-ships for 𝜓3, 𝜓4 and 𝜓5 are similar, with R2 being 0.43, 0.37, and 0.34, respectively. The worst correlation isbetween 𝜓1 and Da, with R2 being 0.26. To examine the potential of the transition scale number, Figure 9shows the five microphysics measures against the transition scale number. Similarly, all the relationshipsare positively correlated and can be fitted with power law relationships. In terms of their corresponding R2

values, the fitness order of the relationships are 𝜓2(R2 = 0.83), 𝜓3(R2 = 0.43), 𝜓4(R2 = 0.37), 𝜓5(R2 = 0.43),and 𝜓1(R2 = 0.25), respectively. These results are consistent with the heuristic argument that a smallerDamköhler number or larger transition scale number corresponds to more homogeneous mixing mechanismbecause turbulent mixing is relatively faster than droplet evaporation. For convenience, Table 2 summarizesthe empirical power law expressions and the corresponding values of R2. The preceding results suggest thatthe two nondimensional numbers, Damköhler number and transition scale number, are virtually equivalentin terms of their usefulness in developing parameterization for the microphysical effects of various differententrainment-mixing processes.

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Figure 10. Comparison between the Taylor length scale and transition length scale (a) and between transition scalenumber and Damköhler number (b). Each point denotes the domain mean value for the corresponding scenario andtime. The symbol and color denote the different simulations scenario and the normalized simulation time, respectively.

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Further inspection of the definitions of the Damköhler number and transition scale number reveals that theiressential difference lies in their use of different turbulent mixing length scales: the Damköhler number isbased on the Taylor length scale, while the transition scale number is based on the transition scale. Figure 10shows the close relationship between the two characteristic length scales (a), which leads to closer relation-ship between the two nondimensional numbers (b). Although the Damköhler number and transition scalenumber are equivalent in principle. The latter is easier to estimate in practice, because the calculation of theTaylor length scale requires information on the spatial gradient of liquid water mixing ratio in addition to dissi-pation rate. Thus, all together, these results suggest that entrainment-mixing processes is best parameterizedusing 𝜓2 as a function of transition scale number, reinforcing the result of Lu, Liu, Niu, Krueger, and Wagner(2013) based on the observational analysis and EMPM simulations. The similarity between our DNS results andthe observational results is noteworthy, which somewhat echoes the results for the mixing diagram reportedin Kumar et al. (2017).

6. Conclusions

A new particle-resolved three-dimensional DNS model is developed that combines the Lagrangian droplettracking with the Eulerian field representation of turbulence near the Kolmogorov microscale. The newparticle-resolved DNS uses the finite difference method coupled with WENO scheme instead of the pseu-dospectral method on which most DNS models have been based on, thus providing more flexibility to dealwith sharp cloud-air interfaces and nonperiodic boundary conditions in future.

Six numerical experiments are performed to investigate the processes of entrainment of clear air and subse-quent mixing with cloudy air and their interactions with cloud microphysics. The experiments are designedto represent different combinations of three initial cloudy areas (Case 1, Case 2, and Case 3), and two turbu-lence modes (decaying and forced turbulence). Case 1 corresponds to Andrejczuk et al. (2004), which aims tostudy the final stage of the mixing process. Case 2 tries to mimic the idealized cloud slab in Kumar, Janetzko,et al. (2012). However, due to the Gibb’s phenomena of the pseudospectral method, there is an inconsistencybetween the initial fields of cloud droplets and vapor mixing ratio in Kumar, Janetzko, et al. (2012), in which anartificial continuous function was used to connect the area of cloudy air and clear air, while the cloud dropletswere treated as a simple slab. This inconsistency is not desirable and is overcome here by taking advantage ofthe high-resolution finite difference WENO scheme, which is designed for problems with piecewise smoothsolutions containing discontinuities. Case 3 is created by rotating Case 2 with 90∘ clockwise to better mimicthe cloud-top entrainment-mixing process and show the sedimentation effect on the entrainment-mixingprocesses and cloud droplet size distributions. All the simulation have been tested in modes of both decay-ing turbulence and forced turbulence. The thermodynamics and cloud microphysics are compared betweendifferent cases. The transient growth of turbulence kinetic energy due to buoyancy effects in the decayingcases agrees with the observation in Kumar et al. (2014). Analysis of the temporal changes of the dropletsize distribution and key microphysical properties reveals that the initial configuration of cloudy areas effec-tively influence the mixing and microphysical processes in decaying turbulence. However, the effect of initialconfiguration seems to be much smaller in the forced turbulence.

Five existing measures of microphysical homogeneous mixing degree are systematically examined, mod-ified, and compared in terms of their ability as a unifying metric to represent the effect of variousentrainment-mixing mechanisms on cloud microphysics. Also examined and compared are the conventionalDamköhler number and transition scale number as a dynamical measure of different mixing mechanisms.Further analysis of the relationship between the microphysical homogeneous mixing degree and the dynam-ical measure confirms the few previous studies (Andrejczuk et al., 2009; Lu, Liu, Niu, Krueger, & Wagner, 2013;Lu et al., 2014) that despite the detailed differences among the six simulation scenarios in cloud proper-ties, the variety of turbulent entrainment-mixing mechanisms can be reasonably represented with a powerlaw relationship between one of the microphysical homogeneous mixing degrees and one of the dynamicalmeasures. The relationship between 𝜓2 and transition scale number is recommended for practical use.

Several points are noteworthy. First, this study is focused on exploring the potential of developing a unifiedparameterization for effect of various turbulent entrainment-mixing processes on cloud microphysics usingthe new particle-resolved DNS model. The new model has other great potentials as well (e.g., studying theeffect of sharp interfaces and different boundary conditions), which will be a focus of our future research.Second, like most previous DNS studies, this work investigates the turbulent entrainment-mixing processes

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and their effects on cloud microphysics with a uniform level of fluid turbulence inside and outside the cloud.The more realistic configuration of stronger turbulence intensity inside the cloud slab, and a weaker turbu-lence intensity in the clear air region is worth pursuing (Götzfried et al., 2017; Tordella & Iovieno, 2011). Anotherimportant topic to be examined is the interaction between turbulence and cloud microphysics and the rela-tive importance of turbulence and entrainment-mixing processes in shaping droplet size distributions. Third,the derived parameterization is directly applicable to improving microphysical representation in large eddysimulation models whose model resolution is similar to the size of the DNS model domain. Application tolarger-scale models such as Numerical Weather Prediction (NWP) and climate models requires further inves-tigation in view of the potential of homogeneous mixing degree on averaging scales (Lu et al., 2014). Finally,similar to other DNS models, the domain DNS size used in this study is still too small to cover the full range ofturbulent eddy sizes, which will require the model to run on high-performance supercomputing platform.

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AcknowledgmentsThe research was supported bythe U.S. Department of Energy’sAtmospheric System Research(ASR),Earth System Modeling (ESM), andAdvanced Scientific ComputingResearch (ASCR) programs viaDE-SC00112704, and by the U.S.Army Research Office underARO-DURIP grant W911NF-15-1-0403.Lu is supported by the NationalNatural Science Foundation ofChina and Jiangsu (91537108and BK20160041). The programfor the simulation is available athttps://github.com/antdvid/climate,and the simulation data can beobtained through https://doi.org/10.6084/m9.figshare.c.3986244.v1.

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