Eulerian and Lagrangian approaches to multi-dimensional1

condensation and collection2

Xiang-Yu Li 1,2,3,4, A. Brandenburg2,4,5,6, N. E. L. Haugen7,8, and G. Svensson1,3,93

1Department of Meteorology, and Bolin Centre for Climate Research, Stockholm University, Stockholm, Sweden42Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden5

3Swedish e-Science Research Centre, www.e-science.se, Stockholm, Sweden64Laboratory for Atmospheric and Space Physics, University ofColorado, Boulder, CO 80303, USA7

5JILA and Department of Astrophysical and Planetary SciencesUniversity of Colorado, Boulder, CO 80303, USA86Department of Astronomy, Stockholm University, SE-10691 Stockholm, Sweden9

7SINTEF Energy Research, 7465 Trondheim, Norway108Department of Energy and Process Engineering, NTNU, 7491 Trondheim, Norway11

9Global & Climate Dynamics, National Center for Atmospheric Research, Boulder, CO 80305, USA12

Key Points:13

• Eulerian Smoluchowski and Lagrangian superdroplet/superparticle approaches to cloud14

droplet growth through condensation and collection are compared using DNS tech-15

niques16

• Size spectra agree well for both approaches, especially in case of turbulence17

• The Lagrangian scheme with symmetric collection is found tobe optimal and compu-18

tationally most efficient19

Corresponding author: Xiang-Yu Li,xiang.yu.li@su.se, Revision: 1.765

–1–

Abstract20

Turbulence is argued to play a crucial role in cloud droplet growth. The combined problem21

of turbulence and cloud droplet growth is numerically challenging. Here, an Eulerian scheme22

based on the Smoluchowski equation is compared with two Lagrangian superparticle (or su-23

perdroplet) schemes in the presence of condensation and collection. The growth processes are24

studied either separately or in combination using either two-dimensional turbulence, a steady25

flow, or just gravitational acceleration without gas flow. Good agreement between the differ-26

ent schemes for the time evolution of the size spectra is observed in the presence of gravity or27

turbulence. The Lagrangian superparticle schemes are found to be superior over the Eulerian28

one in terms of computational performance. However, it is shown that the use of interpolation29

schemes such as the cloud-in-cell algorithm is detrimentalin connection with superparticle30

or superdroplet approaches. Furthermore, the use of symmetric over asymmetric collection31

schemes is shown to reduce the amount of scatter in the results. For the Eulerian scheme,32

gravitational collection is rather sensitive to the mass bin resolution, but not so in the case33

with turbulence.34

1 Introduction35

In the context of raindrop formation, it is generally accepted that turbulence plays a36

crucial role in bridging the size gap between efficient condensational growth of small parti-37

cles (radii below 10µm) and efficient collectional growth due to gravity of larger ones (radii38

around 100µm and above) [Shaw, 2003;Grabowski and Wang, 2013;Khain et al., 2007]. Im-39

proving the understanding of this important problem in meteorology [Berry and Reinhardt,40

1974;Pinsky and Khain, 1997;Falkovich et al., 2002;Naumann and Seifert, 2016] might also41

shed light on how to bridge the even more severe size gap in theastrophysical context of42

planetesimal formation [Johansen et al., 2007, 2012]. To address these questions numeri-43

cally, one has to combine direct numerical simulations (DNS) of turbulent gas motions with44

those of particles. The particles are cloud droplets in the meteorological context and dust45

grains in astrophysics. A possible approach to treat collection is to solve the Smoluchowski46

equation (also known as the stochastic collection equationin the meteorological context)47

[Ogura and Takahash, 1973;Svensson and Seinfeld, 2002;Bec et al., 2016], which couples48

the spatio-temporal evolution equations of the particle distribution function for different par-49

ticle sizes. The particle motion can be treated using a fluid description for each particle size,50

which we refer to as the particle fluid. Thus, not only does onehave to solve the Smolu-51

chowski equation at each meshpoint, but, because heavier particles have finite momenta and52

speeds that are different from those of the gas, one has to solve corresponding momentum53

equations for each mass species. In the meteorological context, it is also referred to as a54

binned spectral method, although in that case the momentum equations for the particle bins55

are normally ignored [Xue et al., 2008]. An Eulerian approach is technically more straightfor-56

ward than a Lagrangian one, but it becomes computationally demanding as the size range of57

cloud droplets is large.58

The Eulerian approach also has conceptual difficulties if the collection probability de-59

pends on the mutual velocity difference. This is due to the fact that particles of the same60

size are described by the same momentum equation and have therefore the same velocity at61

a given position in space, so the velocity difference vanishes. This means that particles of62

the same size are not allowed to collide. This is not a problemfor freely falling particles of63

the same size, which would have the same terminal velocity and are not expected to collide.64

This would however be an unrealistic restriction when particles are subject to acceleration by65

turbulence. More importantly, as was emphasized in the recent review ofKhain et al.[2015],66

the Smoluchowski equation is a mean-field equation and cannot capture the random properties67

of the collections if the collision kernel is prescribed a priori. Nevertheless, most numerical68

cloud microphysical approaches are based on the Smoluchowski equation, which therefore69

raises questions regarding the accuracy of the basic equations [Khain et al., 2015]. Thus, new70

–2–

approaches based on inherently different equations are required to model the cloud microphys-71

ical processes.72

An alternative approach is the Lagrangian one, where one solves for the motion of in-73

dividual particles and treats collections explicitly. In atmospheric clouds, the number density74

of micrometer-sized cloud droplets is of the order of 108 m−3, so in a volume of 1 m3, one has75

100 million particles, which is the typical size that can be treated on modern supercomputers.76

A domain of this size is also about the largest that is possible in direct numerical simulations77

(DNS) of atmospheric turbulence; the Reynolds number basedon the length scaleℓ = 1 m78

and the corresponding velocity scaleuℓ ≈ 0.2 m/ s isuℓ ℓ/ν ≈ 20,000, whereν ≈ 10−5 m2 s−179

is the viscosity of the gas flow. Such a large Reynolds number is just within reach on cur-80

rent supercomputers, but larger domains would remain out ofreach for a long time. Sev-81

eral earlier works investigated condensational growth of cloud droplets using Lagrangian82

tracking in DNS [Paoli and Shariff , 2009;Sardina et al., 2015;de Lozar and Muessle, 2016;83

Lanotte et al., 2009], but those neglected the collectional growth and only proposed to study84

the collectional growth in future work. An intermediate approach involves the use of La-85

grangian “superparticles” [Johansen et al., 2012;Pruppacher et al., 1998;Shima et al., 2009;86

Zsom and Dullemond, 2008], which represent a “swarm” of particles of certain size and num-87

ber density. Depending on the values of particle size and number density, there is a certain88

probability that an encounter between two superparticles leads to collectional growth of some89

of the particles in each swarm (or superparticle). This superparticle approach has been applied90

in a recent LES model to represent the cloud microphysical condensation [Andrejczuk et al.,91

2008] and collection [Andrejczuk et al., 2010;Riechelmann et al., 2012;Naumann and Seifert,92

2015] processes.93

The purpose of the present paper is to compare the Eulerian approach involving the94

Smoluchowski equation with the Lagrangian superparticle approach with the aim of identify-95

ing a promising DNS scheme for tackling the bottleneck problem of cloud droplets growth.96

This has been done in the astrophysical context [Ohtsuki et al., 1990;Drazkowska et al., 2014],97

where the principal problem with the Eulerian approach was emphasized in that it requires98

high mass bin resolution (MBR) to avoid artificial speedup ofthe growth rate. Here we also99

compare with the superdroplet approach ofShima et al.[2009]. The original work on this100

approach was restricted to the case of vanishing particle inertia, but this restriction is not a101

principal limitation of this scheme, which is in fact well applicable to the case of finite particle102

inertia.103

2 Lagrangian and Eulerian approaches104

In the following, we refer to the superparticle or superdroplet approaches as theswarm105

model, where each superparticle represents a swarm of physical particles. By contrast, the106

Eulerian approach is also referred to as theSmoluchowski model. Here we compare the two107

approaches in the meteorological context of water dropletsusing, however, simplifying as-108

sumptions such as constant supersaturation and ideal collection efficiency. In this paper,109

we generally refer to particles and superparticles, which are thus used interchangeably with110

droplets and superdroplets, respectively. We begin with a discussion of the gas flows that are111

being used in some of the models.112

2.1 Evolution equations for the gas flow in both approaches113

In all the experiments reported below, where a nonvanishinggas flow is used, we restrict114

ourselves to two-dimensional (2-D) flows. However, we also perform several experiments115

with no gas flow (u = 0). In those cases the system is spatially uniform and therefore zero-116

dimensional (0-D), which is discussed for the Eulerian case. In our implementation of the117

swarm model, however, we assume that each swarm occupies onegrid cell, so we choose to118

use at least one dimension (1-D). In the following, we use higher-dimensional swarm models,119

which are computational cheaper because we can take advantage of better parallelization.120

–3–

2.1.1 Momentum equation of the gas flow121

To obtainu at each meshpoint, we solve the usual Navier-Stokes equation

∂u∂t+ u · ∇u = f − ρ−1

∇p+ F(u), (1)

where f is a forcing term,p is the gas pressure,ρ is the gas density, which in turn obeys thecontinuity equation,

∂ρ

∂t+ ∇ · (ρu) = 0, (2)

the viscous forceF(u) is given by

F(u) = ν(∇2u + 13∇∇ · u + 2S · ∇ ln ρ), (3)

whereSi j =12(∂ jui + ∂iu j)− 1

3δi j∇ · u is the traceless rate-of-strain tensor. We assume that the122

gas is isothermal and has constant sound speedcs so that the pressurep = c2sρ is proportional to123

the gas densityρ. Note that gravity has been neglected in equation (1), but this is not a principal124

restriction and can be relaxed once suitable non-periodic boundary conditions are adopted. For125

the relatively small domains that can be handled by DNS, gravity will nevertheless have only126

minor effects on the fluid flow for atmospheric conditions.127

2.1.2 Straining flow128

To obtain a non-vanishing flow, we apply volume forcing via the term f . In the case ofa time-independent 2-D divergence-free straining flow,

ustr = u0 (sinkxcoskz,0,− coskxsinkz), (4)

we takef = νk2ustr, whereu0 determines the amplitude andk the wavenumber of the flow.129

2.1.3 Turbulence130

In the case of a turbulent flow,f is delta-correlated in time and consists of random wavesin space [Haugen et al., 2004]. The flow is characterized by a typical forcing wavenumberkf

(√

2k for the straining flow or the average wavenumber from a narrowband of wavevectors)and the root-mean-square (rms) velocityurms. As a relevant timescale characterizing such aflow, we define

τcor = (urmskf )−1 , (5)

which is an estimate of the correlation time. This definitionis also used for the straining flow,131

which is a special case in that it is time-independent and thereforeτcor would no longer char-132

acterize the correlation time of the flow, but it would still be proportional to the turnover time.133

A simulation without spatial extent can be adopted to investigate the statistical convergence134

properties of the Eulerian model regarding its computational efficiency.135

2.2 Condensational growth136

The growth of the particle radiusr i by condensation is governed by [Lamb and Verlinde,2011]

dr i

dt=

Gsr i, (6)

wheres is the supersaturation andG is the condensation parameter. Boths andG are in prin-137

ciple dependent on the flow and the environmental temperature and pressure [see Chapter 8138

of Lamb and Verlinde, 2011], but these dependencies are here neglected, becauseit would139

complicate the comparison of different numerical schemes even further. Therefore, the con-140

densational growth is driven by constant water vapor flux without latent heat release in the141

present study. We adopt the valueG = 5 × 10−11 m2 s−1 [Lanotte et al., 2009]. The assumed142

constancy ofsalso implies that the total liquid water content is not conserved.143

–4–

2.3 The swarm model144

The swarm model is a Monte Carlo type approach that handles particle collections in aswarm of particles in a statistical manner [Zsom and Dullemond, 2008]. Each swarmi has aparticle number densityni , and occupies a volumeδxD, which equals the volume of a fluid gridcell of sizeδx in D dimensions. All particles in a given swarm have the same mass, radius, andvelocity. Following the description ofJohansen et al.[2012], the swarm is transported alongwith its “shepherd particle”, which is also referred to as the corresponding superparticle. Theswarm is treated as a Lagrangian point-particle, where one solves for the particle positionxi

viadxi

dt= V i (7)

and the velocity viadV i

dt=

1τi

(u − V i) + g (8)

in the usual way. Here,g is the gravitational acceleration,τi is the particle inertial response orstopping time of a particle in swarmi and is given by

τi =2ρwr2

i

9ρνeffi, (9)

wherer i is the radius of particles in swarmi, ρw is the mass density of the water in the droplet,ρ is the density of the gas and the effective viscosity is given bySullivan et al.[1994]

νeffi = ν (1+ 0.15 Re0.687i ), (10)

whereν is the ordinary (microphysical) fluid viscosity, and Rei = 2r i |u − V i |/ν is the particle145

Reynolds number, which provides a correction factor to the particle stopping time.146

A given swarm may only interact with every other swarm withinthe same grid cell. The147

computational cost associated with such collections scales asN2pg, whereNpg is the number of148

swarms within a grid cell, but this is computationally not prohibitive as long asNpg is not too149

large.150

We now consider two swarmsi and j residing within the same grid cell. Consider firstcollections of particles within swarmj with a particle of swarmi. The inverse mean free pathof i in j is given by

λ−1i j = σi j n j Ei j , (11)

whereσi j is the collectional cross section with

σi j = π(r i + r j)2, (12)

andEi j is the collision efficiency, but in the following we assumeEi j = 1 in all cases. Theparticle number density in swarmj is n j andr i and r j represent the radii of the particles inthe two swarms. From this, one can find the typical rate of collections between a particle ofswarmi and particles of swarmj as

τ−1i j = λ

−1i j

∣

∣

∣V i − V j

∣

∣

∣ = σi j n j

∣

∣

∣V i − V j

∣

∣

∣Ei j , (13)

whereV i andV j are the velocities of swarmsi and j. The probability of a collection betweenthe swarmi and any of the particles of swarmj within the current time step∆t is then givenby

pi j = τ−1i j ∆t. (14)

This effectively puts a restriction on the time step, since the probability cannot be larger than151

unity. For each swarm pair in a grid cell, one now picks a random number,ηi j , and compares152

it with pi j . A collection event occurs in the case whenηi j < pi j . It is worth noting that153

in Shima et al.[2009], the mean free path is defined by invoking the swarm with the larger154

number density of physical particles; see section 2.3.2 fordetails.155

–5–

2.3.1 Collection scheme I156

For the swarm model, several collection schemes have been proposed in the astrophys-157

ical and meteorological contexts. Most recently,Unterstrasser et al.[2016] modified and ex-158

tended the collection scheme to allow for weighting factorsthat depend on the number of159

particles within each swarm. They do this by introducing a socalled weak threshold, where160

even swarms with very few particles can occur with a certain probability. In this way, they161

obtain higher accuracy with less particles. In the present paper, however, we will focus on the162

comparison of two basic superparticle approaches. We begindiscussing the former (scheme I),163

which is similar to that described byJohansen et al.[2012] in that it maintains a constant mass164

of the individual swarms. In the context of mathematical probability, this approach is also165

known as mass flow algorithm [Eibeck and Wagner, 2001;Patterson et al., 2011]. Scheme II166

is discussed in section 2.3.2.167

If ηi j < pi j , one assumes thatall the particles in swarmi have collided with a particle inswarm j. In this collection scheme, all swarms are treated individually. This means that eventhough the particles in swarmi have collided with the particles in swarmj, swarm j is keptunchanged at this stage. Instead, swarmj is treated individually at a different stage. Hence,all collections are asymmetric, i.e.,pi j , p ji . The new mass of the particles in swarmi nowbecomes

mi = mi +mj , (15)

wheremi is the mass before the collection and the tilde represents the new value after col-lection. In order to ensure mass conservation, thetotal mass of swarmi is kept unchanged,i.e.,

nimi = nimi , (16)

which implies that the new particle number density, ˜ni , is given byni = nimi/mi ; see equa-tion (17) ofPatterson et al.[2011] for the corresponding treatment in the mass flow algorithm.By invoking momentum conservation,

V imi = V imi + V jmj , (17)

the new velocity of any particle in swarmi is given byV i = (V imi + V jmj)/mi .168

2.3.2 Collection scheme II169

In the meteorological context, the following collection scheme has been proposed [Shima et al.,170

2009]. Assume two swarmsi and j, and consider (without loss of generality) the casen j > ni .171

The collection probability of particles in swarmi with swarm j is, again, given by equa-172

tion (14). If the two swarms are found to collide, the new masses of the particles in the two173

swarms are given by174

mi = mi +mj ,

mj = mj , (18)

but now their new particle number densities are175

ni = ni ,

n j = n j − ni . (19)

In other words, the number of particles in the smaller swarm remains unchanged (and their176

masses are increased), while that in the larger one is reduced by the amount of particles that177

have collided with all the particles of the smaller swarm (and their masses remain unchanged).178

This implies that in equation (11), the mean free path is defined with respect to the swarm with179

the larger number density of physical particles, as explained in Shima et al.[2009]. Finally,180

the new momenta of the particles in the two swarms are given by181

V imi = V imi + V jmj ,

V jmj = V jmj . (20)

–6–

In contrast to scheme I, these collections are symmetric, i.e. pi j = p ji . Consequently, both182

swarms are changed during a collection. However, the asymmetric collection property of183

scheme I ofJohansen et al.[2012] may not have been previously recognized, nor has its ac-184

curacy been compared with other models, which we will further discuss below.185

2.3.3 Initial particle distribution186

We recall that particles within a swarm may interact with particles of another swarm onlyif both swarms occupy the same grid cell. The effective volume of each swarm is thereforeequal toδxD, whereD is the spatial dimension introduced in section 2.3. The total numberof particles in our computational domain is thereforeδxD times the sum ofni over all Np

swarms. This must also be equal tonLD, wheren is the total number density represented bythe simulation andL is the size of the computational domain. Thus, we have

nLD = δxDNp∑

i=1

ni . (21)

Initially ( t = 0), the particle number densities of all swarms are the same and since (L/δx)D =

Ngrid is the total number of grid points, we havenNgrid = niNp. Thus, the initial number densityof particles within one swarm must be

ni = nNgrid/Np (at t = 0). (22)

In the following, we choose the initial particle size distribution of total physical particles inthe domain to be lognormal, i.e.,

f (r i ,0) =(

n0

/

(√

2πσpr))

exp{

−[ln(r i/r ini)]2/2σ2

p

}

, (23)

wherer ini andσp are the center and width of the size distribution, respectively; n0 = n(t = 0)187

is the initial total number density of physical particles. For each particle, the logarithm of its188

radius is drawn from a normal distribution. These particlesare distributed uniformly in space189

within the computational domain. This means that particlesin each swarm are of the same190

size, but different from swarm to swarm.191

2.4 Eulerian approach192

To model the combined growth of particles through condensation and collection in amulti-dimensional flow in the Eulerian description, we describe the evolution of particles ofdifferent radiir (or, equivalently, of different logarithmic particle mass lnm) at different po-sitions x and timet. We employ the particle distribution functionf (x, r, t), or, alternativelyin terms of logarithmic particle mass lnm, f (x, ln m, t), such that the total number density ofparticles is given by

n(x, t) =∫ ∞

0f (x, r, t) dr, (24)

or, correspondingly forf , we haven(x, t) =∫ ∞−∞ f (x, ln m, t) d lnm. Sincem = 4πr3ρw/3, we

have f = f dr/d lnm= f r/3. Note thatn(x, t) obeys the usual continuity equation,

∂n∂t+ ∇ · (nv) = Dp∇2n, (25)

wherev is the mean particle velocity (i.e., an average over all particle sizes) andDp is a Brow-nian diffusion term, which is enhanced for numerical stability and will be chosen dependingon the mesh resolution. The evolution of the particle distribution function is governed by asimilar equation, but with additional coupling terms due tocondensation and collection, i.e.

∂ f∂t+ ∇ · ( f v) + ∇r ( fC) = Tcoll + Dp∇2 f , (26)

–7–

where∇r = ∂/∂r is the derivative with respect tor, C ≡ dr/dt = Gs/r, as given in equation (6),193

andTcoll describes the change of the number density of particles for smaller and larger radii,194

as will be defined below. Furthermore,v(x, r, t) is the particle velocity within the resolved grid195

cell, which is discussed below. It also determines the mean particle velocityv =∫

f vdr/n in196

equation (25).197

The modeling of condensation and collection implies coupling of the evolution equa-tions of f (x, r, t) for different values ofr. The advantage of usingf (x, ln m, t) is that it al-lows us to cover a large range inm, because we will use then an exponentially stretched gridin m such that lnm is uniformly spaced [Pruppacher et al., 1998;Suttner and Yorke, 2001;Johansen, 2004]. The total number density within a finite mass interval δ ln m is then givenby f (x, ln m, t) δ ln m. Thus, the total number density of particles of all sizes at positionx andtime t is given by

n(x, t) =kmax∑

k=1

fk δ ln m=kmax∑

k=1

fk, (27)

where fk = f (ln mk) δ ln m is the variable used in the simulations andkmax is the number of198

logarithmic mass bins. To compare with the Lagrangian model, we choose the lognormal199

distribution of equation (23) as the initial distribution of particles.200

2.4.1 Condensation201

Let us first consider the process of condensation, which is described in equation (26) bythe term∇r ( fC), where fC is the flux of particle from one size bin to the next. Evidently,the total number density is only conserved if the particle flux fC vanishes forr = rmin andr = rmax, which is the case if the range ofr is sufficiently large. In particular, (fC)min → 0,becausen→ 0 for m→ 0. In practice, however, we consider finite lower cutoff values ofmand therefore expect some degree of mass loss at the smallestmass bins. The same is also truefor the largest mass bin once the size distribution has grownto sufficiently large values. In allcases with pure condensation, it is convenient to display solutions in non-dimensional form bymeasuring time in units of

τcond= r2ini/2Gs (28)

andr in units of r ini . We refer to Appendix A for more details on the condensation equation202

for the Eulerian approach.203

2.4.2 Collection204

Next, we consider collection, which leads to a decrease ofn, but does not change the205

mean mass density of liquid water. The evolution off (x, ln m, t) due to collection is governed206

by the Smoluchowski equation207

Tcoll =12

∫ m

0K(m−m′,m′) f (m−m′) f (m′) dm′

−∫ ∞

0K(m,m′) f (m) f (m′) dm′. (29)

Here,K is a kernel, which is proportional to the collision efficiencyE(m,m′) and a geometriccontribution. As mentioned above, we assumeE = 1 and soK is given by

K(m,m′) = π(r + r ′)2|v− v′|, (30)

wherer andr ′ are the radii of the corresponding mass variables,m andm′, while v andv′ are208

their respective velocities, whose governing equation is given below.209

In the following, we define the mass and radius bins such that

mk = m1δk−1, rk = r1δ

(k−1)/3. (31)

–8–

Unfortunately,δ = 2 is in many cases far too coarse, so we take

δ = 21/β, (32)

whereβ is a parameter that we chose to be a power of two. For a fixed massbin range, thenumber of mass binskmax increases with increasingβ. In terms of fk, equation (29) reads

T collk = 1

2

∑

i + j ∈ k

Ki jmi +mj

mkfi f j − fk

kmax∑

i=1

Kik fi , (33)

where we have adopted the nomenclature ofJohansen[2004], wherei + j ∈ k denotes allvalues ofi and j for which

mk−1/2 ≤ mi +mj < mk+1/2 (34)

is fulfilled. The term (mi + mj)/mk in equation (33) comes from the fact that collections210

between cloud droplets from two mass bins may not necessarily result in a cloud droplet mass211

being exactly in the middle of the nearest mass bin.Johansen[2004] therefore included this212

factor so that mass is strictly conserved. The discrete kernel is thenKi j = π(r i + r j)2|vi − v j |.213

The corresponding momentum equations for particle fluidk, i.e., for the velocitiesvk, isgiven byvk(x, t) = v(x, ln mk, t) for each logarithmic mass value lnmk is

∂vk

∂t+ vk · ∇vk = g− 1

τk(vk − u) + Fk(vk) +Mk, 1 ≤ k ≤ kmax. (35)

Here,u is the gas velocity,τk (for k = i) is defined by equation (9), and

Fk(vk) = νp∇2vk (36)

is a viscous force of the particle fluid, which is due to the interaction between the individual214

particles. This viscous force should be very small for dilute particle suspensions, but is nev-215

ertheless retained in equation (35) for the sake of numerical stability of the code. It is not216

to be confused with the drag force,−τ−1k (vk − u) between particles and gas. In principle, the217

expression forFk(vk) should be based on the divergence of the traceless rate-of-strain tensor218

of vk, similarly to the corresponding expression for the viscousforce of the gas discussed in219

equation (3). However, since the termFk(vk) is unphysical anyway, we just use the simpler220

expression proportional to∇2vk instead.221

The linear momentum of all particles is given by∑〈 fkmkvk〉, where angle brackets de-

note volume averages. In order that this quantity is conserved by each collection, the targethas to receive a corresponding kick, which leads to the last term in equation (35), but it leavesthe velocities of the collection partners unchanged. It is therefore only related to the first termon the right-hand side of equation (33) and not the second, soit is given by (see Appendix B)

Mk =1

2 fkmk

∑

i + j ∈ k

Ki j fi f j

[

mivi +mjv j − (mi +mj)vk

]

. (37)

To our knowledge, this momentum-conserving term has not been included in any of the very222

few earlier works that include a momentum equation for each particle species [cf.Suttner and Yorke,223

2001;Elperin et al., 2015]. The reason why this has apparently not previously been discussed224

in the literature is that in meteorological applications one usually works with the averaged ker-225

nel and neglects the evolution of the velocities for the different mass bins [Grabowski and Wang,226

2013]. This correction term is evidently zero when the momentum of the two collection con-227

stituents (= mivi + mjv j) is equal to that of the resulting constituent [= (mi + mj)vk]. Never-228

theless, as is shown in Appendix B, the momentum conserving correction changes the time229

evolution of the droplet spectrum in an unexpected way when the MBR is high, but the results230

are similar forβ = 2. Furthermore, for turbulent flows, as is discussed below, these correction231

terms become insignificant.232

–9–

2.5 Boundary conditions and diagnostics233

In the present work, we use periodic boundary conditions forall variables in all direc-234

tions. Therefore, no particles and no gas are lost through the boundaries of the domain. This235

approximation is reasonable as long as we are interested in modeling a small domain well236

within a cloud where also heavier particles can be assumed toenter from above. The use of237

periodic boundary conditions requires us to neglect gravity in equation (1), which could be238

relaxed if non-periodic boundary conditions were adopted.239

To characterize the size distribution, we consider the evolution of different normalizedmoments of the size spectra,

aζ =

kmax∑

k=1

⟨

fk rζk⟩

/ kmax∑

k=1

⟨

fk⟩

1/ζ

, (38)

whereζ is a positive integer. The mean radiusr is given bya1, the maximum radius is max(r) =240

a∞, and the droplet mass is proportional to the third power ofa3.241

In the case of collection, the condensation timescaleτcond, defined in equation (28), isno longer relevant, but it is instead a collection timescalethat can be defined in the Eulerianmodel as

τ−1coll =

kmax∑

k=1

⟨

T collk

⟩

/ kmax∑

k=1

⟨

fk⟩

, (39)

which is, in this definition, a time-dependent quantity. In the Lagrangian model, this quantitycan be defined by the collection frequency. Unlike the case ofpure condensation, whereτcond is the appropriate time unit,τcoll can only be used a posteriori as a diagnostic quantity.However, given that the speed of pure collection is proportional to the mean particle densityn, it is often convenient to perform simulations at increasedvalues ofn and then rescale timeto a fixed reference densitynref and use

t = t n0/nref. (40)

In the following we usenref = 108 m−3, which is the typical value ofn in atmospheric clouds.242

Analogously we also define ˜τcoll = τcoll n0/nref. Finally, the number of particles in the total243

simulation domain isN(t) =∫

n(x, t) dDx.244

2.6 Computational implementation245

We use the P C, which is a public domain code where the relevant equations246

have been implemented [Johansen, 2004;Johansen et al., 2004;Babkovskaia et al., 2015].247

We refer to Appendix A for a description of an important modification applied to the imple-248

mentation of equation (6). The implementation of equation (33) has been discussed in detail249

by Johansen[2004], and follows an approach described earlier bySuttner and Yorke[2001].250

However, momentum conservation during collections was previously ignored in the Eulerian251

model. The current revision number is 73563 when checking out the code via the svn bridge252

on the public github repository.253

When traditional Lagrangian point particle particle tracking is employed, it is usually254

beneficial to employ higher order interpolation between theneighboring grid cells to find the255

value of a given fluid variable at the exact position of the particle. By default, the cloud-in-cell256

(CIC) algorithm is used, which involves first order interpolation for the particle properties on257

the mesh. In the swarm approach, however, the particles in each swarm fill the volume of a258

grid cell in which the shepherd particle is located. The distribution of the swarm throughout259

the grid cell is homogeneous and isotropic, and as such the swarm has no particular position260

within the grid cell. It is true that there is a particular position associated with the swarm,261

namely the position of the shepherd particle, but this position has no purpose other than to262

determine in which grid cell the swarm resides. The positionis also needed in the integration263

of equation (7). Below we shall show that it isnot better to use any kind of interpolation264

–10–

Table 1. Summary of the simulations.271

Run Scheme Dim Np Ngrid IM Processes β n0 (m−3) Case Dp (m2/s) νp (m2/s)

1A SwI 3-D 104 163 CIC Con – 1010 – – –1B Eu 0-D – – – Con 128 1011 – – –2A Eu 0-D – – – Col 128 1011 grav – –2B SwI 3-D 32Ngrid 323 CIC Col – 1010 grav – –2C SwII 3-D 32Ngrid 323 CIC Col – 1010 grav – –3A SwII 2-D 3× 105 642 CIC Col – 1010 strain – –3B SwII 2-D 3× 105 1282 CIC Col – 1010 strain – –3C SwII 2-D 3× 105 2562 CIC Col – 1010 strain – –3D Eu 2-D – 1282 – Col 2 1010 strain 0.05 0.013E SwII 2-D 3× 105 802 NGP Col – 1010 strain – –3F SwII 2-D 3× 105 1602 NGP Col – 1010 strain – –4A SwII 2-D 5× 104 1282 CIC Both – 108 strain – –4B SwII 2-D 5× 104 1282 NGP Both – 108 strain – –4C Eu 2-D – 1282 – Both 2 108 strain 0.02 0.104D Eu 2-D – 1282 – Both 2 108 strain 0.01 0.054E Eu 2-D – 2562 – Both 2 108 strain 0.005 0.055A Eu 2-D – 5122 – Col 2 1010 turb 0.001 0.0015B SwII 2-D 1.2× 106 5122 NGP Col – 1010 turb – –

“IM” denotes the interpolation method, “Col” refers to collection, “Con” refers to condensation,“Eu” refers to Eulerian model, “SwI” refers to collection scheme I of swarm model, “SwII” refersto collection scheme II of swarm model, “Both” refers to condensation and collection, “grav”refers to gravity (u=0), “strain” refers to straining flow, “turb” refers to turbulence, and “Dim”refers to the dimension. “Case” refers to the mechanisms driving the collection or condensation.Simulations with gravity and turbulence are performed in a box of sizeL=0.5 m, while the simula-tions with straining flow are performed in a box with sizeL = 2πm.

in determining the value of the fluid variables at the position of the swarm, but rather to use265

the values of the grid cell in which the swarm resides. This method is technically referred to266

as nearest grid point mapping (NGP). Details concerning each experiment are summarized in267

Table 1.268

3 Results269

3.1 Condensation experiments270

We compare the Eulerian and Lagrangian models for the pure condensation processwithout motion, i.e., zero gas velocity. In the case of homogeneous condensation, we cancompare the numerical solution with the analytic solution of Seinfeld and Pandis[2006]; seetheir Fig. 13.25. To this end, we make use of the fact that solutions of the condensationequation (6) obey

f (r, t) = (r/r) f (r ,0), (41)

where ˜r is a shifted coordinate with ˜r2 = r2 − 2Gst. With the lognormal initial distributiongiven by equation (23), this yields

f (r, t) =n0√2πσp

rr2

exp

− (ln r − ln r ini)2

2σ2p

, (42)

wherer ini denotes the position of the peak of the distribution andσp = lnσSPdenotes its width,272

whereσSP is the symbol introduced bySeinfeld and Pandis[2006]. What is remarkable here273

–11–

Figure 1. Comparison of the numerically obtained size spectra with the analytic solution for condensation

with a lognormal initial condition given byr ini = 5µm, andσp = 0.2. Simulations of pure condensation

(no turbulence nor gravity;s=0.02) with the Eulerian model (a) usingβ=128 andkmax=1281 mass bins in

the range 2–20µm and the Lagrangian swarm model (b) withNp = 10000 andNgrid = 163. The solid lines

correspond to the analytic solution given by equation (42) while the black dots represent the numerical results.

See run 1A and 1B of Table 1 for simulation details.

289

290

291

292

293

294

is the fact thatf (r, t) vanishes forr < r∗ ≡√

2Gst. This is because in this model, no new274

particles are created and even particles of zero initial radius will have grown to a radiusr∗275

after timet. Furthermore, the small particles withr = r∗ grow faster than any of the larger276

ones, which leads to a sharp rise in the distribution function at r = r∗. Thus,∂ f /∂r has a277

discontinuity atr = r∗. This poses a challenge for the Eulerian scheme in which the derivative278

∂/∂r is discretized; see equation (26). In Figure 1, we compare solutions obtained using both279

Eulerian and Lagrangian approaches. It is evident that ther-dependence obtained from the280

Eulerian solution is too smooth compared with the analytic one, even though we have used281

1281 mass bins withβ=128 to representr on our logarithmically spaced mesh over the range282

2µm ≤ r ≤ 20µm, which corresponds toδ ≈ 1.0054; see equation (32). Better accuracy could283

be obtained by using a uniformly spaced grid inr, but this would not be useful later when the284

purpose is to consider collection spanning a range of several orders of magnitude in radius.285

By comparison, the Lagrangian solution shown in the right-hand panel of Figure 1 (here with286

n0 = 1010 m−3) has no difficulty in reproducing the discontinuity in∂ f /∂r at r = r∗. Moreover,287

the Lagrangian solution agrees perfectly with the analytical solution.288

3.2 Purely gravitational collection experiments295

We now consider uniform collection with no spatial variation of the velocity and density296

fields for both the gas and the particles. For the purely geometrical kernel, no analytic solution297

exists. However, we can compare the convergence propertiesof our two quite different numer-298

ical approaches and thereby get some sense of their validityin cases when the two agree. We299

consider pure collection experiments, starting again witha lognormal distribution. The results300

are presented in terms of normalized time; see equation (40).301

3.2.1 Comparison between swarm collection schemes I and II305

In Figure 2, we compare schemes I and II of the swarm model together with the Eulerian306

model. The simulations have been performed withNgrid = 323 grid points andNp = 32Ngrid307

swarms (the statistics is converged forNp/Ngrid ≥ 4, as discussed in Appendix C). The left-308

hand panel of Figure 2 shows that for ¯r the results of the swarm simulations with scheme I309

agree with those of scheme II at early times, but depart at late times. However, fora3, the310

agreement is excellent, as shown in the right-hand panel of Figure 2. The evolution of ¯r with311

–12–

Figure 2. 3-D simulations with the swarm model and 323 grid points using schemes II (red) and I (orange),

compared with the Eulerian model withβ = 128 (solid blue line) for (a) ¯r and (b)a3. The collection is driven

by gravity. See Runs 2A, 2B, and 2C of Table 1 for simulation details.

302

303

304

scheme I shows considerable scatter at late times. We recallthat the main difference between312

schemes I and II is the geometry of collections. The collections simulated with scheme I313

are asymmetric, while those with scheme II are symmetric. Thus, in scheme II both swarms314

change either their total mass or their total particle number, while in scheme I the total mass of315

a swarm is kept constant by adjusting the particle number correspondingly. This property of316

scheme I may be responsible for creating stronger fluctuations in the mean radius. Therefore,317

to keep the amount of scatter comparable, scheme II is effectively less demanding. In the318

following, we will mainly adopt scheme II to save computational time.319

3.2.2 Comparison between collection scheme II and the Eulerian model320

As we have seen above, the swarm simulations follow the Eulerian results rather well321

for a3 (see the right-hand panel of Figure 2), but are somewhat different for ¯r. At early times,322

on the other hand, the evolution ofr obtained with the swarm model with collection scheme I323

follows more closely that of the Eulerian model. At later times, however, the evolution ofr324

obtained with the swarm model departs from the one simulatedwith the Eulerian model. This325

is surprising and might hint at a false convergence behavior.326

We show in Appendix D that, in the case of purely gravity-driven collections, ¯r con-327

verges only for very large MBR. Thus, the MBR dependency of the numerical solution using328

the Eulerian scheme appears to be a serious obstacle in studying particle growth not only by329

condensation, but also by collection. This is a strong argument in favor of the Lagrangian330

scheme. The evolution ofa3, on the other hand, agrees rather well between the swarm and331

Eulerian models.332

The mean radiusr is not well suited for characterizing the collectional growth toward335

large particles. As is shown in the following sections, the mean particle radius often increases336

by not much more than a factor of three (see also the left-handpanel of Figure 2), while the337

size distribution can become rather broad and its tail can reach the size of raindrops within338

a relatively short time. In addition to the mean radius, we now also consider size spectra to339

address the collectional growth to larger particles.340

–13–

Figure 3. Same simulations as in Figure 2, but here we only compare scheme II with the Eulerian model.

Size spectra are given fort=0 s, 1000 s, 2000 s and 3000 s.

333

334

The evolution of size spectra simulated with the Eulerian scheme with 3457 mass bins341

(β = 128) is shown as blue lines in Figure 3, while the corresponding size spectra obtained342

with the swarm model (collection scheme II) with 32 particles per grid point are shown as343

red curves. The agreement between the Eulerian and Lagrangian schemes is good at early344

times (t ≤ 2000 s), but at late times (t = 3000 s) the size spectra from the Eulerian approach is345

broader for the largest sizes (rmax = 1000µm). Shima et al.[2009] found that the results of the346

super-droplet method (collection scheme II) agree fairly well with the numerical solution of a347

binned spectral method. We also found that the size spectra simulated with the swarm model348

(scheme II) converge to those obtained with the Eulerian model with increasingNp/Ngrid. This349

can simply be explained by the fact that more swarms contribute as potential collectional part-350

ners and thus ensure more reliable statistics, which was also shown in the work ofShima et al.351

[2009].352

3.3 Inhomogeneous collection in a straining flow353

Spatial variation in the flow leads to local concentrations and thus to large peak values354

of f (x, r, t) that shorten the collection timeτcoll [Saffman and Turner, 1956]. Before studying355

the turbulent case, we consider first collectional growth ina steady two-dimensional (2-D)356

divergence-free straining flow. The straining flow is numerically inexpensive and easy to357

control and analyze compared with turbulence.358

3.3.1 Pure collection365

We consider first the case of pure collection. In Figure 4 we show the time evolution of366

r for the swarm model with collection scheme II at different grid resolutions ranging from 642367

to 2562 meshpoints. Surprisingly,r grows more slowly as we increase the mesh resolution of368

the swarm model. Given that the swarm models seem to convergetoward the Eulerian model,369

we are confronted with the question of what causes the growthof r in the swarm model to slow370

down at higher mesh resolution. In this connection, we must emphasize that by default we use371

–14–

Figure 4. Comparison of the evolution of (a) the mean particle size and (b)a3 in a straining flow for

simulations with the swarm approach at different grid resolutions. Here, pure collection with CIC particle

interpolation algorithm has been used. The total number of swarms isNp = 300, 000 whileDp = 0.05 m2/s

andνp = 0.01 m2/s are adopted in the Eulerian model. The inset shows the case with NGP mapping instead

of the CIC first order interpolation for particle properties. See Runs 3A,3B, 3C, 3D, 3E, and 3F of Table 1 for

simulation details.

359

360

361

362

363

364

the CIC algorithm to evaluate the gas properties at the position of each Lagrangian particle.372

As explained in section 2.6, the position of the shepherd particle has no purpose other than to373

determine in which grid cell the swarm resides. It is therefore not better to use any kind of374

interpolation in determining the value of the fluid variables at the position of the swarm, but375

rather to use NGP mapping. This will play an important role, as will be discussed now. For the376

sake of solving equations (7) and (8), the use of the CIC algorithm is perfectly valid, but this377

would only be relevant for a direct Lagrangian tracking algorithm. This can be understood378

by realizing that in the special case of particles with vanishingly small inertia, the particles379

will follow their local fluid cell, and hence, two particles will in the real world never collide.380

However, if the CIC scheme is used for equations (7) and (8), two swarms residing at different381

positions within thesamegrid cell may have different velocities, and hence, equation (13) may382

yield a collection.383

Since the swarms are filling the entire volume of the grid cell, this means that the two384

swarms will have different velocities and exist in the same volume, and hence, theswarms may385

collide. The larger grid cells yield potentially larger velocity differences between the particles,386

which explains why the collectional growth is larger for thecoarser resolutions. When NGP387

mapping is adopted, the artificial speedup disappears, as shown in the insets of Figure 4.388

However, the discrepancy between Lagrangian and Eulerian particle descriptions is still389

strong for collectional growth in the straining flow as shownin the insets of Figure 4. This390

is because that in a steady flow, the particles will end up nearthe vertices of converging flow391

vectors and will therefore be much more concentrated in the swarm model than what is possi-392

ble to represent in the Eulerian model. This is evident by looking at the spatial distribution of393

superparticles belonging to a certain radius (here 128µm); see Figure 5, where we also show394

the corresponding number density in the Eulerian model.395

–15–

Figure 5. Visualization of flow and particle field (t=1000 s) in a straining flow for simulations with the

swarm approach (left panel; red dotted curve in Figure. 4) and Eulerian approach (right-hand panel; blue

curve in Figure. 4). Here the radius of the particles isr=128µm. The swarms are represented by the red dots

in the left-hand panel. The contour map shows the spatial distribution of the number density in the right panel.

The black and white arrows represent the velocity vectors of the strainingflow.

396

397

398

399

400

3.3.2 Combined condensation and collection401

When both condensation and collection play a role, it is no longer possible to define402

a unique timescale, and the solution depends on bothτcond andτcoll. We consider here the403

straining flow usingr ini = 12µm,G = 5×10−11 m2/ s ands= 0.01, which yieldsτcond= 144 s.404

We investigate the role that particle viscosity and Brownian diffusion play in simulations using405

the Eulerian model. The Brownian motion of the particles is usually small, so the particle406

diffusion coefficientDp in equation (26) should be finite, but small. Since it is assumed that the407

particle flows are relatively dilute, there should be very little interaction between the different408

particle fluids, except for the occasional collections. This implies that the particle viscosity409

νp in equation (36) should be close to zero. For the Smoluchowski approach, bothνp and410

Dp have to be made large in order to stabilize the simulations inspatially extended cases. It411

turns out that the values of these diffusion coefficients have a surprisingly strong effect on the412

solutions, which is shown in Figure 6. This could be due to thefact that the viscosity between413

the particle fluids diffuses the momentum of the particles and thereby modifies the collection414

rate.415

Comparing now with the swarm approach, which avoids artificial viscosity and en-416

hanced Brownian diffusion altogether, we see from Figure 6 that Eulerian and Lagrangian417

approaches agree with each other at early times (t < 1000 s). After 1000 s, both swarm and418

the Eulerian models follow the same trend in the sense that the evolution of ¯r shows a bump;419

see the dashed and dotted lines att ≈ 8× 104 s. The bump occurs earlier for the swarm model420

than the Eulerian model. In the extreme case that the artificial viscosity in the Eulerian model421

were zero, the evolution of ¯r, as obtained from the swarm model, may come closer that of422

the Eulerian model. However, owing to the absence of a pressure term for particles, discon-423

tinuities would develop in the Eulerian model that destabilize the code if the viscosity and424

Brownian diffusion are too small. Again, this may be a strong argument in favor of using the425

swarm model for studying the collectional growth of cloud droplets.426

To relate the speed of evolution in Figure 6 to ˜τcoll, we plot in the inset of panel (a) the427

inverse of its unscaled value,τcoll, as a function of time. On average, we haveτcoll ≈ 100.428

–16–

Figure 6. Evolution of (a)r and (b)a3 in the straining flow with combined condensation and collection.

The different blue lines correspond to different amounts of artificial viscosity and enhanced Brownian dif-

fusivity. The inset of (a) shows the evolution of the inverse collection timescaleτ−1coll. The inset of (b) shows

the evolution of the mass ratio. The monotonic growth of the mass ratio demonstrates that particles have not

yet populated at the largest mass bin. The initial mean radius, supersaturation, and condensation parameter is

given byr ini=12µm, s=0.01, andG = 5 × 10−11 m2/ s, respectively, andkmax=53 withβ=2. See Runs 4A, 4B,

4C, 4D, and 4E of Table 1 for simulation details.

448

449

450

451

452

453

454

It is comparable toτcond = 144 s and both are long compared withτcor ≈ 1.4. The relevant429

quantity is the scaled value, ˜τcoll, which is much larger≈ 10,000. This may suggest that the430

speed of growth is not governed by the spatially averaged kernel, but by its value weighted431

toward regions where the concentration is high.432

We recall that growth of cloud droplets driven by pure collections in the straining flow433

depends on the models (Eulerian and Lagrangian models; see detailed comparisons in sec-434

tion 3.3.1). This suggests that condensation has a “regularizing” effect in that it makes the435

overall evolution ofr much less dependent on the initial conditions and other model details.436

This is due to the fact that the condensation process with constant positive supersaturation437

value leads to narrow size spectra of cloud droplets.438

Another interesting aspect is the bump in the evolution of the mean radius. At first439

glance it seems counterintuitive thatr can actually decrease during some time interval. In440

Appendix E we consider an example of four particles, two large ones and two small ones. If441

two small ones collide, we still have the two large ones, but only 3 particles in total after the442

collection, so the average radius increases from 1/2 to 2/3. On the other hand, if two large ones443

collide, we are still left with the two small ones and one particle whose radius has only grown444

by a factor of 21/3 ≈ 1.26 (the radius scales with the mass to the 1/3 power). The average445

radius is therefore 21/3/3 ≈ 0.42, which is less than the original mean radius, which is halfthe446

radius of the large ones.447

3.4 Growth of droplets in 2-D turbulence455

Turbulence is generally believed to help bridging the size gaps in both cloud droplet and456

planetesimal formation. In this section, pure turbulence-generated collections are simulated457

using both the Eulerian and Lagrangian models. We consider a2-D squared domain of side458

lengthL = 0.5 m at a resolution of 5122 meshpoints, with viscosityν = 5×10−4 m2 s−1 (which459

is about 50 times the physical value for air), average forcing wavenumberkf ≈ 40 m−1, i.e.,460

kf L/2π ≈ 3, and a root-mean-square velocityurms = 0.8 m s−1, resulting in a Reynolds number461

–17–

Figure 7. Comparison of size spectra for Lagrangian (red lines) and Eulerian (blue lines) approaches at

different times in the presence of 2-D turbulence and no gravity nor condensation. The largest departure be-

tween both approaches occurs fort = 2000 s and is plotted separately in the right-hand panel. See Runs 5A

and 5B of Table 1 for simulation details.

468

469

470

471

of Re = urms/νkf ≈ 40. Our choice ofkf L/2π ≈ 3 corresponds to forcing at large scales462

that are not yet too large to be affected by constraints resulting from the Cartesian geometry.463

The rate of energy dissipation per unit volume isǫ = 2ν〈S2〉 ≈ 0.1 m2 s−3 and the turnover464

time is τto = (urmskf )−1 ≈ 0.03 s. For the Lagrangian model, we use NGP mapping while465

for the Eulerian model we adopt artificial viscosity and enhanced Brownian diffusivity for the466

particles (νp = Dp = 10−3 m2 s−1).467

Figure 7 shows the comparison of size spectra for the swarm and Eulerian models in 2-D472

turbulence. In general, the agreement is good at small radii, but less good at large radii. The473

swarm model predicts larger particles than the Eulerian model at t = 2000 s. A similar trend474

is already seen att = 1000 s. On the other hand, fort = 3000 s, the two agree reasonably well475

at all radii. By contrast, in the case with pure gravity (Figure 3), we found that the Eulerian476

model predicted larger particles att = 3000 s. The reason for this is unclear, but it is possible477

that there are opposing trends that cancel each other and thus lead to reasonable agreement478

between the two models at late times.479

3.4.1 Other numerical aspects480

It is worth noting that the MBR convergence of the Smoluchowski equation depends on481

the flow pattern. While gravitational collection is rather sensitive to MBR (see Appendix D),482

it is much less sensitive for the straining flow and convergesatkmax ≈ 55 in turbulence.483

We emphasize that for the swarm model, the interpolation scheme of the tracked swarms484

does affect the results, but this does not seem to be the case for turbulence. Turbulence con-485

tinues to mix particles all the time while the straining flow tends to sweep up particles into486

predetermined locations that do not change. We may therefore conclude that the restriction on487

the interpolation scheme depends on the spatio-temporal properties of the flow. Nevertheless,488

a high-order interpolation is not strictly applicable to the swarm model.489

It is worth noting that in the case with pure gravity, the Eulerian model is rather sensitive490

to the presence or absence of theMk term. This is neither the case for turbulence nor for the491

straining flow as will be discussed in Appendix B.492

–18–

3.4.2 Comparison of computational cost493

Comparing our Lagrangian and Eulerian models in Figure 7, itis worth noting that the494

Lagrangian one is clearly superior to the Eulerian one in terms of CPU time for simulating495

the collectional process in 2-D turbulence. A similar conclusion was drawn byShima et al.496

[2009], who found the Lagrangian model to be computationally faster than the Eulerian one.497

We compare the computational cost between Eulerian and Lagrangian models using the 2-D498

turbulence runs 5A and 5B (runs in Figure 7), which have comparable accuracy; see Table 1499

for details of these runs. The Lagrangian model with 1.2× 106 superparticles covers 217 s in500

physical time within 24 hours of wall-clock time on 512 CPUs,while the Eulerian model with501

53 mass bins covers only 48 s in physical time within 24 hours wall-clock time on 1024 CPUs.502

This example demonstrates that our Lagrangian model is roughly ten times more efficient than503

a comparable Eulerian one. This will not be generally true for Lagrangian models that are not504

based on a superdroplet approach.505

3.4.3 Combined condensation and collection506

The combined condensational and collectional growth in turbulence is investigated as507

well. Again, the results are similar to the case with pure collectional growth due to the fact that508

the condensation process in the present study with constantsupersaturation is homogeneous.509

In future studies, the supersaturation should be calculated self-consistently and the effects510

of turbulence on the condensational growth should be considered, similar to what was done511

previously [Kumar et al., 2014;Sardina et al., 2015].512

4 Conclusion513

The combined collectional and condensational growth of cloud droplets is studied in514

numerical simulations where the gas phase is solved on a mesh, while the particle phase is515

approximated by a point particle approach and is treated either by an Eulerian or a Lagrangian516

formalism. In the absence of any flows, the Lagrangian approach is found to agree well with517

the analytic solution of condensational growth. By contrast, the Eulerian approach requires518

high resolution in the number of mass bins to avoid artificialspeedup of the growth rate, which519

agrees with previous findings [Ohtsuki et al., 1990;Drazkowska et al., 2014]. It is worth not-520

ing that the MBR dependency is closely related to the temporal and spatial properties of the521

flow. The dependency is strongest for gravity [u = 0 in equation (1)], less strongly for the522

straining flow, and weak for turbulence.523

A detailed comparison of the collectional size spectra between the Lagrangian and Eu-524

lerian models demonstrates consistency between the two, especially when both condensation525

and collection are included. This suggests that condensation has a regularizing effect and526

makes the overall evolution of the mean radius less dependent on details such as the width of527

our lognormal initial distribution or discretization errors that might affect the early evolution.528

However, the evolution of the mean radius, i.e., the ratio ofthe two lowest (first and zeroth)529

moments of the size distribution function, is a rather insensitive measure of particle growth.530

This is also seen in the fact that the mean particle radius often increases by not much more531

than a factor of three, while the size distribution can become rather broad and even millimeter-532

sized particles can be produced within a relatively short time. The mean particle radius is also533

not the most relevant diagnostics in that it does not characterize properly the growth of the534

largest particles. In fact, as we have shown in Appendix E, the mean radius actuallydecreases535

when two large particles collide. This is somewhat counterintuitive, but actually quite natural.536

When two very small particles collide, the sum of all radii does basically not change, but the537

number of particles decreased by one, so the average increases. By contrast, when two large538

particles collide, the particle number again decreases by one, but the sum of the radii decreases539

from 2 to 21/3 ≈ 1.26, so the average also decreases.540

–19–

When studying pure collection, the Eulerian approach yieldssatisfactory results only541

when the mass bins are sufficiently fine. Furthermore, for collections in the case of a straining542

flow, it is found that the Eulerian approach requires artificially large viscosity and Brownian543

diffusivity for keeping the resulting shocks in the particle fluid resolved. Because of this,544

it seems that for future studies of the effect of turbulence on condensational and collectional545

growth of particles, the Lagrangian swarm approach would bemost suitable. However, several546

precautions have to be taken. First, the symmetric collection scheme II [Shima et al., 2009] is547

to be preferred because it shows less scatter in the mean radius than the asymmetric scheme I.548

This is because in scheme I the particle number is adjusted tokeep the total mass in the549

swarm constant. Second, when interpolation of the gas properties at the position of each550

Lagrangian particle is invoked (for example the CIC algorithm or the triangular shaped cloud551

scheme), both collection schemes yield artificially increased collection rates. This is because552

two swarms within the same grid cell may always collide sincethe interpolation of the fluid553

velocity results in a velocity difference between the two swarms. This causes a speedup of554

the collection rate already at early times. At higher grid resolution, the interpolated velocity555

differences are smaller, which reduces the collectional growth. Therefore, it is best to map the556

gas properties to just the nearest grid point, which is foundto yield converged results even at557

low resolution.558

The discrepancy between Lagrangian and Eulerian particle descriptions is particularly559

strong in the time-independent straining flow. This is because particles tend to be swept into560

extremely narrow lanes, which leads to high concentrationsthat can never be achieved with561

the Eulerian approach, in which sharp gradients must be smeared out by invoking artificial562

viscosity and large Brownian diffusivity. On the other hand, we are here primarily interested563

in turbulent flows that are always time-dependent, which limits the amount of particle concen-564

tration that can be achieved in a given time. In that case, thediscrepancies between Eulerian565

and Lagrangian approaches are smaller at early times, but there are still differences in the evo-566

lution of the mean radius at late times. This can easily be caused by changes in the relative567

importance of collections of large and small particles. This is confirmed by the fact that the568

size distribution spectra in the turbulent case are more similar for Lagrangian and Eulerian569

approaches than in the straining flow.570

Our present work neglects local and temporal changes in the supersaturation. In fu-571

ture studies, we will take into account that the supersaturation increases (decreases) as a fluid572

parcel rises (falls) and that droplet condensation (evaporation) act as sinks (sources) of super-573

saturation. We would then be able to account for the fact thatthe total water content should re-574

main constant and that the supersaturation would become progressively more limited as water575

droplets grow by condensation. Another important shortcoming is our assumption of perfect576

collection efficiency, which resulted in artificially rapid cloud dropletsgrowth. Alleviating577

these restrictions will be important tasks for future work.Furthermore, we have here only578

considered 2-D turbulence. Extending our work to 3-D is straightforward, but our conclusions579

regarding the comparison of different schemes should carry over to 3-D.580

Acknowledgments581

We thank Nathan Kleeorin, Dhrubaditya Mitra, and Igor Rogachevskii for useful dis-582

cussions. We also thank the anonymous referees for constructive comments and suggestions583

that lead to substantial improvements in the manuscript. This work was supported through584

the FRINATEK grant 231444 under the Research Council of Norway, the Swedish Research585

Council grant 2012-5797, and the grant “Bottlenecks for particle growth in turbulent aerosols”586

from the Knut and Alice Wallenberg Foundation, Dnr. KAW 2014.0048. The simulations587

were performed using resources provided by the Swedish National Infrastructure for Com-588

puting (SNIC) at the Royal Institute of Technology in Stockholm. This work utilized the589

Janus supercomputer, which is supported by the National Science Foundation (award num-590

ber CNS-0821794), the University of Colorado Boulder, the University of Colorado Den-591

ver, and the National Center for Atmospheric Research. The Janus supercomputer is op-592

–20–

erated by the University of Colorado Boulder. G. Svensson also thanks the Wenner-Gren593

Foundation for their support. The source code used for the simulations of this study, the594

P C, is freely available onhttps://github.com/pencil-code/. The input files595

as well as some of the output files of the simulations listed inTable 1 are available under596

http://www.nordita.org/˜brandenb/projects/SwarmSmolu_numerics/.597

Appendices598

A Upwinding scheme for a nonuniform mesh599

In the presence of condensation alone, the evolution equation for f (r, t) as a function ofradiusr and timet is given by

∂ f∂t= − ∂∂r

( fC), (A.1)

whereC ≡ dr/dt and is given by equation (6). Thus, we have

∂ f∂t= −A

∂

∂r

(

fr

)

(A.2)

where A = Gs is assumed independent ofr; see equation (13.14) ofSeinfeld and Pandis[2006]. It can be seen from the form of the analytic solution that there will be a discontinuityat r2 = 2At, which is numerically difficult to handle. In particular, it is difficult to ensure thepositivity of f . For these reasons, a low-order upwind scheme is advantageous. Furthermore,expanding the RHS of equation (A.2) using the quotient rule,

∂ f∂t=

Ar2

f − Ar∂ f∂r, (A.3)

it is obvious that the first term in isolation would lead to exponential growth off proportional600

to exp(At/r2), which must be partially canceled by the second term. If thecancellation is601

numerically imperfect,f (r, t) will indeed grow exponentially, which tends to occur in regions602

wherer2 < 2At, i.e., wheref should vanish. For nonuniform mesh spacing,rk with k = 1, 2,603

...,kmax, the first-order upwind scheme can be written as604

∂ fk∂t= c+k

fk+1

rk+1+ c0

k

fkrk+ c−k

fk−1

rk−1(A.4)

withc±k = ± 1

2

|A| ∓ Ark±1 − rk

, c0k = −c+k − c−k . (A.5)

On the boundaries of the radius bins atk = 1 andkmax, equation (A.4) cannot be used unlesswe make an assumption about the nonexisting points outside the interval 1≤ k ≤ kmax. Forexample, fork = kmax, the coefficient c+k would multiply fk+1/rk+1, which is not defined.Therefore, a simple assumption is to setc+k = 0. However,c+k also enters in the expressionfor c0

k, which is the factor in front offk/rk. The coefficientc+k can only be nonvanishing whenA < 0. If we were to omitc+k in the expression forc0

k, then, forA < 0, the value offk would notevolve atk = kmax and would be frozen. Thus, the non-existing points lead to anunphysicalsituation. It would be natural to assume that atk = kmax, fk should decay with time at a rate−(|A| − A)/rk. Therefore, assume

c+k = 0, c0k = −(|A| − A)/rk − c−k (for k = kmax) (A.6)

andc−k unchanged, and analogously

c−k = 0, c0k = −(|A| + A)/rk − c+k (for k = 0) (A.7)

andc+k unchanged.605

–21–

Table B.1. Total particle momentum in kgm−2s−1 after three different times using the Eulerian model.607

case T M g t = 0.0 s t = 0.1 s t = 1 s t = 10 s

A 0 0 0 0.8042 0.8042 0.8042 0.8042B , 0 0 0 – 0.3386 0.0012 0.00C , 0 , 0 0 – 0.8035 0.8032 0.75D 0 0 , 0 – 0.3070 −4.1679 −48.92E , 0 0 , 0 – −0.1586 −4.9709 −49.72F , 0 , 0 , 0 – 0.3063 −4.1673 −45.51

The initial parameters are:v1=1 m s−1 andv2=2 m s−1 atradius binsr1=100µm andr2=112µm (×21/6 larger) withn0=108 m−3 distributed evenly over the first two mass bins.

B Momentum conservation solution of the Eulerian model606

The purpose of this appendix is to derive the momentum-conserving velocity kickMk inequation (35) and to demonstrate how it works. Each collection event involves three partners,which we denote by subscriptsi, j, andk, wherek is the result of the collection betweeniand j. Mass conservation implies thatfimi + f jmj + fkmk is constant, i.e., its time derivativevanishes. Likewise, momentum conservation implies that

∂

∂t

(

fimivi + f jmjv j + fkmkvk

)

= 0. (B.1)

The time derivatives off caused by collections isT , while that ofv isM. However, only theresulting particlek will suffer a kick, whilei and j do not, so we have

Timivi + T jmjv j + Tkmkvk + fkmkMk = 0. (B.2)

As seen from equation (33), for the collection ofi and j, the increase infk is given by

Tk = Ki j fi f jmi +mj

mk, (B.3)

while the corresponding decreases in bothfi and f j are

Ti = T j = −Ki j fi f j , (B.4)

which evidently obeys mass conservation, i.e.,Timi + T jmj + Tkmk = 0. Inserting equa-tions (B.3) and (B.4) into equation (B.2) and solving forMk yields

Mk =1

fkmkKi j fi f j

[

mivi +mjv j − (mi +mj)vk

]

. (B.5)

We give in Table B.1 the values of the total momentum of all particles in the Eulerian608

model,∑

fimivi , at three different times for a model without spatial extent (0-D). Initially, we609

have two mass bins with velocities 1 m s−1 and 2 m s−1, which leads to collectional growth if610

T , 0. Drag with the gas is here neglected. In the absence of gravity, the total momentum611

is the same for all three times when there is no collection (T = 0, case A). ForT , 0, there612

is a dramatic change of momentum if theM term is neglected (case B). With theM term613

included, momentum is reasonably well conserved (compare case C with case A). In the pres-614

ence of gravity, the momentum changes just because of gravitational acceleration (cases D–F).615

However, we would not expect the total momentum to change dramatically when we allow for616

collection (T , 0). We see that without theM term the total momentum departs signifi-617

cantly from the case without collection (case E), while withtheM term included, the values618

–22–

Figure B.1. Same as Figure 3, but with the Eulerian model with momentum conservation (blue dashed

lines, denoted by “EulerMC”) included. Here we only compare EulerMC and Euler. Thick lines:β = 8; thin

lines:β = 2. See additional Runs AppE1, AppB4, and AppB5 in Table B.2 for simulation details.

635

636

637

Figure B.2. The effect of the momentum conserving term for a turbulent flow (dashed lines,denoted by

“EulerMC”) compared with the case without it (denoted by “Euler”), same as in Figure 7. Thick lines:β = 8;

thin lines:β = 2. See additional Runs AppB1, AppB2, and AppB3 in Table B.2 for simulation details.

638

639

640

of total momentum are similar to those without collection (compare case F with case D). This619

validates the implementation of the momentum conserving term.620

Let us now discuss the effect of the momentum conserving correction in the context of621

gravitational collection. This is shown in Figure B.1, where we compare size spectra forβ = 2622

and 8 with and without theM term. It turns out that without theM term, the growth of large623

droplets is increased when the MBR is large (β = 8). This is not the case, however, when the624

M term is included, which leads to a much slower growth of the largest droplets. On the other625

hand, as demonstrated above, theM term leads to a decrease of the momentum of the large626

droplets, which explains the absence of particles above 1 mmat t = 3000 s and the increase at627

smaller radii.628

Remarkably, in turbulence, the evolution of the size spectra are almost the same with629

or without momentum correction term. This is shown in FigureB.2. It is still unclear why630

the effect of the momentum correction term depends so strongly on the flow pattern. Further631

investigation is required to understand this in the future work. However, one might specu-632

late that the momentum conservation correction accumulates numerical errors with increasing633

number of mass bins, so it is unclear that this procedure leads to more accurate results.634

C Statistical convergence of the swarm model642

The purpose of this appendix is to investigate the statistical convergence with respect to643

the number of grid cells and swarms. First we inspect the convergence property ofNp/Ngrid.644

The simulations have been performed with 323 grid points and different average numbers of645

swarm particles per grid point (Np/Ngrid = 2–8). It can be seen from the upper panels of Fig-646

–23–

Table B.2. Summary of additional simulations presented in the appendix.641

Run Scheme Dim Np Ngrid IM Processes β n0 (m−3) Case Dp (m2/s) νp (m2/s)

AppB1 EuMC 2-D – 5122 – Col 2 1010 turb 0.001 0.001AppB2 EuMC 2-D – 5122 – Col 4 1010 turb 0.001 0.001AppB3 Eu 2-D – 5122 – Col 4 1010 turb 0.001 0.001AppB4 EuMC 0-D – – – Col 2 1010 grav – –AppB5 EuMC 0-D – – – Col 128 1011 grav – –AppC1 SwII 3-D 2× 103 Np/4 CIC Col – 1010 grav – –AppC2 SwII 3-D 16× 103 Np/4 CIC Col – 1010 grav – –AppC3 SwII 3-D 442× 103 Np/4 CIC Col – 1010 grav – –AppC4 SwII 3-D 1024× 103 Np/4 CIC Col – 1010 grav – –AppC5 SwI 3-D 2Ngrid 323 CIC Col – 1010 grav – –AppC6 SwI 3-D 8Ngrid 323 CIC Col – 1010 grav – –AppC7 SwII 3-D 2Ngrid 323 CIC Col – 1010 grav – –AppC8 SwII 3-D 4Ngrid 323 CIC Col – 1010 grav – –AppC9 SwII 3-D 8Ngrid 323 CIC Col – 1010 grav – –AppC10 SwI 3-D 4Ngrid 323 CIC Col – 1010 grav – –AppE1 Eu 0-D – – – Col 2 108 grav – –AppE2 Eu 0-D – – – Col 32 108 grav – –AppE3 Eu 2-D – 802 – Col 2 1010 strain 0.01 0.05AppE4 Eu 2-D – 802 – Col 4 1010 strain 0.01 0.05AppE5 Eu 2-D – 1282 – Both 4 108 strain 0.01 0.05

Here, the abbreviations are the same as the ones in Table 1 butwith additional abbreviations listedbelow. “EuMC” refers to the Eulerian model with momentum conservation invoked.

ure C.1 that the swarm simulations with collection scheme IIalmost converge forNp/Ngrid = 4.647

The details of these additional runs are summarized in TableB.2648

From the lower panels of Figure C.1 it can be seen that for simulations withNp/Ngrid =658

4, the results are more or less converged when the total number of swarms reaches 128× 103.659

Since all fluid variables are spatially uniform in these simulations, the number of grid points660

has no effect on the fluid. The number of swarms can therefore be changedby increasing the661

total number of grid points while maintainingNp/Ngrid = 4 (the value ofni is approximately662

the same in all cases;ni ≈ 109). However, as reported byArabas and ichiro Shima[2013],663

when the swarm model is used in an LES simulation, certain macrophysical features of their664

simulated could field does not show convergence regarding grid resolution.665

D MBR dependency for collection666

In Figure D.1, we compare the evolutions of ¯r using different MBR and thus different667

values ofβ for the pure collection experiment with different flow patterns. The MBR conver-668

gence strongly depends on the flow pattern. The evolution of the mean radius ¯r does depend669

on MBR strongly in the case with gravity, but only weakly in case with the straining flow, and670

almost not at all in the case of a turbulent flow. It is worth noting that the case with com-671

bined condensation and collection depends on MBR only weakly. We also tested the MBR672

dependency using a constant kernel. In that case, it turns out that the results converge only for673

kmax ≥ 50.674

–24–

Figure C.1. Same as Figure 2, but here we only study the statistical convergence properties of swarm

model. Upper panels: orange (red) lines represent the swarm modelwith collection scheme I (II). The line

types indicate the mean number of swarms per grid point (Np/Ngrid); the total number density of physical

particles is kept the same for all simulations by changing the number densityof particles in each swarm and

the number of swarms; see Runs AppC5, AppC6, AppC7, AppC8, AppC9 and AppC10 of Table B.2 for

simulation details. Lower panels: similar to the upper panels, but for scheme II with Np/Ngrid=4 and different

total numbers of swarms, as indicated by the line types; the corresponding Ngrid is 83 (solid curve), 163 (dotted

curve), 323 (dashed curve), 483 (dash-dotted curve) and 643 (dash-triple-dotted curve); see Runs AppC1,

AppC2, AppC3, and AppC4 of Table B.2 for simulation details.

649

650

651

652

653

654

655

656

657

E The “bump” in the evolution of the mean particle radius684

For the following discussion, it is convenient to introducethe unscaled moments

Mζ =∑

f (r) rζ . (E.1)

so thataζ = (Mζ/M0)1/ζ and r = a1, as before. Let us now assume a situation with pure685

collection such that the total volume of water in the droplets is conserved. This implies that686

M3 is constant, whileM0 andM1 will always decrease with time. However, the relative rates687

at which M0 and M1 decrease can change. Indeed, a bump inr is observed ifM1 switches688

from decreasing more slowly with time thanM0 to decreasing faster thanM0. An example of689

such a situation will be presented in the following.690

For a flow with two small and two large particles, with radiirS andrL , respectively, the694

size distribution is given byf (r) = 2δr rS+2δr rL , whereδi j denotes the Kronecker delta (δi j = 1695

for i = j and 0 otherwise). From equation E.1 it can then be found that the initial number of696

particles and sum of particle radii is given byM0(0) = 4 andM1(0) = 2rS + 2rL , respectively.697

This yields a mean initial particle radius ofr(0) = M1(0)/M0(0). In the following, we assume698

thatrS≪ rL , so thatr(0) ≈ 2rL/4 = 0.5rL .699

When two particles of radiusr0 collide, their combined mass is unchanged, so 2r30 = r3,

i.e., the target radius becomesr = 21/3r0 [Lamb and Verlinde, 2011]. Let us now consider two

–25–

Figure D.1. MBR dependency for simulations with different flow pattern. Upper left panel: collection

driven by gravity usingkmax=3457 andβ=128 (solid),kmax=865 andβ=32 (dotted), as well askmax=55 and

β=2; see Runs AppE1 and AppE2 of Table B.2 and 2A of Table 1 for simulation details. Upper right panel:

collection driven by straining flow usingkmax=109 andβ=4 (dashed line),kmax=55 andβ=2 (solid line); see

Runs AppE3 and AppE4 of Table B.2 for simulation details. Low left panel:collection driven by turbulence

usingkmax=109 andβ=4 (dashed line),kmax=55 andβ=2 (solid line); see Runs AppB3 of Table B.2 and 5A

of Table 1 for simulation details. Low right panel: collection driven by straining flow with condensation us-

ing kmax=109 andβ=4 (dashed line),kmax=55 andβ=2 (solid line); see Runs AppE5 of Table B.2 and 9D of

Table 1 for simulation details.

675

676

677

678

679

680

681

682

683

Figure E.1. Sketch illustrating the growth ofr when two small particles collide (A) and the decrease ofr

when two large particles collide (B). Filled black symbols denote actual particle sizes and open red symbols

and red text refer tor.

691

692

693

different collection scenarios; cf. Figure E.1. In scenarioA, two smaller particles collide suchthatM0(A) = 3 andM1(A) = 21/3rS+2rL , while in scenarioB two larger particles collide suchthatM0(B) = 3 andM1(B) = 2rS + 21/3rL . SincerL ≫ rS, we find forr in both scenarios

r(A) = (21/3rS + 2rL)/3 ≈ 2rL/3 ≈ 0.67rL > r(0), (E.2)

r(B) = (2rS + 21/3rL)/3 ≈ 21/3rL/3 ≈ 0.42rL < r(0). (E.3)

–26–

This means that for scenarioA the mean particle radius is increasing, while for scenarioB it is700

decreasing. After the time when the bump appears in the time evolution of the mean particle701

radius (see Figure 6), it is primarily the heavier particlesthat continue collecting.702

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