Introduction
The evolution of the size distribution of coalescing particles has often
been described by the kinetic collection (hereafter KCE) or Smoluchowski
coagulation equation, known under a number of names (“stochastic
collection”, “coalescence”). The discrete form of this equation has the
form (Pruppacher and Klett, 1997)
∂N(i,t)∂t=12∑j=1i-1K(i-j,j)N(i-j)N(j)-N(i)∑j=1∞K(i,j)N(j),
where N(i,t) is the average number of droplets with mass xi, and K(i,j) is the
collection kernel related to the probability of coalescence of two droplets
of masses xi and xj. In Eq. (1), the time rate of change of the
average number of droplets with mass xi is determined as the difference
between two terms: the first term describes the average rate of production
of droplets of mass xi due to coalescence between pairs of drops whose
masses add up to mass xi, and the second term describes the average rate
of depletion of droplets with mass xi due to their collisions and
coalescence with other droplets.
Within the kinetic approach (Eq. 1), it is assumed that fluctuations are
negligibly small. This assumption can only be correct if the volume and the
number of particles are infinitely large. An alternative approach considers
the coalescence process in a system of finite number of particles, with
fluctuations that are no longer negligible. This finite-volume description
is intrinsically stochastic and has been pioneered by Marcus (1968) and Bayewitz
et al. (1974) and studied in detailed by Lushnikov (1978, 2004) and Tanaka
and Nakazawa (1993).
Within the finite volume description a system of particles whose total mass
is MT is considered. The mass distribution of the particles is
described by giving the number ni of particles with mass i, i.e., n1,n2,n3, …, nN. Then, the state of the mass distribution of the
particle system is described by the N-dimensional state
vector n¯=(n1,n2,…,nN). The time evolution of the
joint probability P(n1,n2,…,nN;t) that the system is in
state n¯=(n1,n2,…,nN) at time t is calculated
according to the equation (Tanaka and Nakazawa, 1993)
∂P(n¯)∂t=∑i=1N∑j=i+1NK(i,j)(ni+1)(nj+1)×P(…,ni+1,…,nj+1,…,ni+j-1,…;t)+∑i=1N12K(i,i)(ni+2)(ni+1)×P(…,ni+2,…,n2i-1,…;t)-∑i=1N∑j=i+1NK(i,j)ninjP(n¯;t)-∑i=1N12K(i,i)ni(ni-1)P(n¯;t).
The master Eq. (2) is a gain–loss equation for the probability of each
state n¯=(n1,n2,…,nN). The sum of the first two
terms is the gain due to transition from other states, and the sum of the
last two terms is the loss due to transitions into other states. The gain
terms show that the system may be reached from any state with an i-mer and a
j-mer more, and one (i+j)-mer less. In Eq. (2) K(i,j) is the collection
kernel and the transition rates are K(i,j)(ni+1)(nj+1) if i≠j and K(i,i)(ni+1)(ni+2) if i=j. From conservation of the total
probability, P(n¯;t) must satisfy the relation
∑n‾P(n¯;t)=1,
where the sum is taken over all states. Moreover, the total mass MT of
the system must be conserved, and the particle number ni should be
non-negative for any mass xi:
∑i=1Nxini=MT,ni≥0,i=1,…,N.
Exact solutions of Eq. (2) are only known for a limited number of cases
(constant, sum and product kernels) and for monodisperse initial conditions.
For these special cases the master equation has been solved by Lushnikov
(1978, 2004) and Tanaka and Nakazawa (1993) in terms of the generating
function of P(n¯;t). For general, multidisperse initial conditions,
the solution of Eq. (2) is not known.
Additionally, for stochastic coagulation, approximate solutions were
calculated by using Van Kampen's system size expansion or Ω expansion (Van Dongen and Ernst, 1987; Van Dongen, 1987) which permits
finding solutions of Eq. (2) valid in the limit of a large system. However, the
system size expansion gives less reliable results when applied to systems
with a low number of particles or small volumes.
Then, in order to obtain solutions for more realistic kernels (Brownian
motion, differential sedimentation, etc.), a small number of particles and
general multidisperse initial conditions, it has to be solved numerically.
In this paper, we present an algorithm that can be applied to obtain the
solution of Eq. (2) for any type of kernel and initial conditions. By
applying this method, numerical solutions of the master equation were
obtained for realistic kernels relevant to cloud physics, along with
calculation of the correlations for the number of droplets for different
sizes.
It is worth mentioning that the stochastic simulation algorithm (SSA)
developed by Gillespie (1975) also accurately reproduces the master
equation. In Gillespie's method, the master equation is not solved directly,
but a statistically correct trajectory (possible solution) of the master
equation is generated. At any time, expected values at each droplet size can
be obtained by averaging over many runs. However, a large number of
realizations are necessary in order to obtain the desired accuracy at the
large end of the droplet size distribution. A detailed comparison between
the two methods will be made in Sect. 3.
The problem of calculating correlation coefficients was also addressed by
Wang et al. (2006), who derived what they called the “true stochastic
collection equation” (TSCE), which is a mean field equation at the first
order and contains correlations among instantaneous droplets of different
sizes. The problem with this equation and similar ones is that the rate of change of
moments of order n depends on moments of order (n+1), as was remarked by
Marcus (1968).
In our work, we overcome this drawback by calculating the true stochastic
averages directly from the solution of the master equation. The main idea is
to reduce the dimensionality by restricting the state space only to those
states which have a finite probability of being accessed. It turns out that
this provides a considerable improvement in numerical efficiency.
The paper is organized as follows: in Sect. 2, the numerical algorithm is
explained in detail. Numerical solutions for the sum and constant kernels
with a comparison with analytical solutions and with the method of Gillespie (1975) are presented
in Sect. 3. The numerical results for mass-dependent kernels along with
calculation of correlations for different droplet sizes are presented in
Sect. 4. Finally, in Sect. 5 we briefly discuss the results and the
possible applications of the numerical algorithm.
The numerical algorithm
To solve Eq. (2) by brute force, the joint probability P(n1,n2,…,nN;t) must be discretized into a multi-dimensional array. The
main drawback of this approach is its susceptibility to the curse of
dimensionality (Bellman, 1961), i.e., the exponential growth in memory and
computational requirements in the number of problem dimensions.
For example, for a system with a monodisperse initial
condition P(50,0,0,…,0;0)=1, even considering the restriction (4), we
would be in need to define a 50-dimensional array with about 1.34×1016
elements, which is computationally prohibitive.
Calculation of all possible states
Instead of the brute force discretization of the multi-dimensional joint
probability distribution, the solution for this problem lies on the
generation of all possible states from an initial configuration, and the
posterior calculation of the time evolution of the probability
P(n¯;t) for each generated configuration by using the master
equation. From an arbitrary initial condition P(n01,n02,…,n0N;0)=1 all possible states can be generated numerically.
This can be performed by taking into account that the only transitions allowed
are of the form n¯1(+)→n¯1 if
i≠j and n¯2(+)→n¯2 if i=j,
where n¯1(+), n¯1 and n¯2(+),
n¯2 are the state vectors:
n¯1(+)=(n1,…,ni+1,…,nj+1,…,ni+j-1,…,nN),n¯1=(n1,…,ni,…,nj,…,ni+j,…,nN),n¯2(+)=(n1,…,ni+2,…,n2i-1,…,nN),n¯2=(n1,…,ni,…,n2i,…,nN).
For a system consisting of N monomers at t=0, R(N) states (or
N-dimensional vectors) can be realized, where R(N) is the number of
solutions in integers n¯ of the Eq. (4) for conservation of
mass. The number of possible configurations can be approximated from the
equation (Hall, 1967)
R(N)∼14N3expπ2N/31/2.
Note that, although R(N) increases very quickly with N (for example,
R(50)=217 590 and R(100)=190 569 232), a number of states that is manageable
with an average computer is obtained (compare with the 50-dimensional array
with 1.34×1016 elements required for N=50). Although Eq. (6)
slightly overestimates the number of states, it gives estimates that can be
used in order to check the performance of the algorithm. For N=6, 10, 20,
and 30 we obtained 11, 42, 627, and 5604 with the numerical algorithm, and 13,
48, 692, and 6078 by using Eq. (6). As an example, the 11 possible
configurations generated from the initial state (6,0,0,0,0,0) are displayed
in Fig. 1.
State space obtained from the initial condition
P(6,0,0,0,0,0;0)=1 with the constraint ∑i=16ini=6.
Time evolution of the probabilities P(n‾;t) for each state
At t0=0 for the initial state P(n01,n02,n03,n04,…,;t0)=1, and the probabilities for the rest of the states
are set equal to 0. The probabilities of all generated configurations are
updated according to the first-order finite difference scheme:
Pn¯;t0+Δt=Pn¯;t0+Δt∑i=1N∑j=i+1NK(i,j)(ni+1)(nj+1)×P(…,ni+1,…,nj+1,…,ni+j-1,…;t0)+Δt∑i=1N12K(i,i)(ni+2)(ni+1)×P(…,ni+2,…,n2i-1,…;t0)-Δt∑i,j=1NK(i,j)ninjP(n¯;t0)-Δt∑i=1N12K(i,i)ni(ni-1)P(n¯;t0).
It is clear from Eq. (7) that the state probabilities Pn¯;t0+Δt at t=t0+Δt will increase
if the states from which transitions are allowed have a non-zero
probability at t=t0 (second and third terms in the right-hand side of
Eq. 6) and will decrease due to collisions of particles from the same
state at t=t0 (fourth and fifth terms in the right-hand side of
Eq. 7) if P(n¯;t0) is positive. The finite difference
equation for P(1,0,0,0,1,0) was written to illustrate the method. As can be
seen from the generation scheme displayed in Fig. 1, the only allowed
transitions to (1,0,0,0,1,0) are from the states (1,1,1,0,0,0)
and (2,0,0,1,0,0). Consequently, at t=t0+Δt,
P(0,1,0,1,0,0;t0+Δt) will increase if P(1,1,1,0,0,0;t0)
and P(2,0,0,1,0,0;t0) are positive at t=t0. On the other hand,
P(1,0,0,0,1,0;t0+Δt) will decrease due to collisions from
particles within the same state at t=t0 if P(1,0,0,0,1,0;t0) is
positive. Then, P(1,0,0,0,1,0;t0+Δt) is calculated from the
equation
P(1,0,0,0,0,1,0;t0+Δt)=P1,0,0,0,0,1,0;t0+ΔtK(2,3)(n2+1)(n3+1)×P(1,1,1,0,0,0;t0)+ΔtK(1,4)(n1+1)(n4+1)×P(2,0,0,1,0,0;t0)-ΔtK(1,5)(n1)(n5)×P(1,0,0,0,1,0;t0).
In the second term on the right-hand side of Eq. (8), n2+1 and
n3+1 are set equal to 1, as they are the number of particles in the
second and third bins in the configuration (1,1,1,0,0,0) at t=t0. In the third term, n1+1=2 and n4+1=1 as they are
defined from the state (2,0,0,1,0,0) and, finally, n1=n5=1 in
the fourth and last term. As an exercise, the time evolution of each state
probability was calculated for the coalescence kernel
K(i,j)=(i1/2+j1/2)/40 from Marcus (1968). The results for 5 of the
11 possible configurations are displayed in Fig. 2.
Probability distribution P(n,1;t) of finding n particles of size
m=1 at time t, for a system with the initial condition P(6,0,0,0,0,0;0)=1.
Probability distribution P(n,1;t)
n=0
P(0,1;t)=P(0,1,0,1,0,0,t)+P(0,0,2,0,0,0)+P(0,0,0,0,0,1)
n=1
P(1,1;t)=P(1,1,1,0,0,0,t)+P(1,0,0,0,1,0)
n=2
P(2,1;t)=P(2,2,0,0,0,0;t)+P(2,0,0,0,1,0;t)
n=3
P(3,1;t)=P(3,0,1,0,0,0;t)
n=4
P(4,1;t)=P(4,1,0,0,0,0;t)
n=5
P(5,1;t)=0
n=6
P(6,1;t)=P(6,0,0,0,0,0;t)
Time evolution of the probability for 5 of the 11 the states for the
initial condition P(6,0,0,0,0,0;0)=1 and the collection
kernel K(i,j)=(i1/2+j1/2)/40.
Calculation of the expected values of the number of particles for each particle mass
The number of particles for a given mass n1, n2, …, nN
are discrete random variables whose probability distributions can be
obtained from
P(n,m;t)=∑ExceptnmPn1,n2,…,nm=n,…nN;t.
Usually, the numerical implementation of Eq. (9) would involve calculating
the sum of all elements of a multi-dimensional array, which is
computationally very expensive. Our approach is simpler: once the
probabilities of all possible states are determined for all times, P(n,m;t)
can be calculated just by summing over all states that have
nm=n:
P(n,m;t)=∑allstateswithnm=n,Pn1,n2,…,nm=n,…nN;t.
The expected values nm for the number of
particles of mass m are then calculated from the equation
nm=∑nnP(n,m;t).
As an example, for the system from Fig. 1, the probability distribution
P(n,1;t) of having n particles with mass m=1 is displayed in Table 1.
Comparison with analytical solutions and the stochastic simulation algorithm (SSA) of Gillespie
Comparison with analytical solutions
The expected values for each particle mass calculated with the numerical
algorithm were tested against the analytical solutions of the master
equation reported in Tanaka and Nakazawa (1993) for the constant (Eq. 12)
and sum (Eq. 13) kernels (K(i,j)=A, K(i,j)=B(xi+xj)) obtained
for the monodisperse initial condition P(N0,0,0,…,0;0)=1. They are
〈nm〉=CmN0m!∑l=1N0-m+1∑k=lN0(-1)k-1(2k-1)Cl-1N0-1Cl-1N0-mCN0-kN0-l(k+l-1)Ck+l-1N0+k-1×(l-1)/∏i=1mN0-ie-k(k-1)2τ,〈nm〉=CmN0iN0m-11-mN01-eTN0-m-1×(1-e-T)m-1e-T.
In Eqs. (12) and (13), N0 is the initial number of particles,
CmN0 is the binomial coefficient and nm values are the true stochastic averages for each particle mass m at
time t. In Eq. (12) τ=AN0t, where A=1.2×10-4 cm3s-1 is the constant collection kernel. Finally, in Eq. (13), T=BN0v0t, where v0 is the initial volume of droplets and
B=8.82×102 cm3g-1s-1. Turning to a
concrete numerical example, the evolution of a cloud system with an initial
monodisperse droplet size distribution of N0=10 droplets of 10 µm
in radius (droplet mass 4.189×10-9 g) at t0, and a
volume of 1 cm3 was calculated with the numerical algorithm. The time
step was set equal to Δt=0.1 s. Comparisons between the
numerical and analytical results for both the sum and constant kernels at
t=1200 s are shown in Figs. 3 and 4 with an excellent agreement between the
two approaches.
For the sum kernel, size distribution obtained from the analytical
solution of the master equation (triangles) and the numerical algorithm
(squares) at t=1200 s. Calculations were performed with the initial
condition P(10,0,0,…,0;0)=1 and the sum kernel K(i,j)=B(xi+xj),
with B=8.82×102 cm3g-1s-1.
Same as Fig. 3 but for the constant kernel K(i,j)=1.2×10-4 cm-3s-1.
At t=1200 s, comparison between the droplet size distributions
obtained from the analytical solution of the master equation (line) and
(a) the SSA of Gillespie for 103 realizations (circles) and (b) the numerical
algorithm (circles). Calculations were performed with the initial condition
P(30,0,0,0,…,0;0)=1 for the sum kernel K(i,j)=B(xi+xj), with B=8.82×102 cm3g-1s-1.
Comparison with the SSA of Gillespie
As was mentioned in the introduction, the algorithm of Gillespie generates a
statistically correct trajectory of the stochastic master equation. It was
presented in Gillespie (1975), and popularized in Gillespie (1977) were it
was used to simulate chemical systems. As we know, in Gillespie's SSA, the
ensemble mean for the number of droplets at each droplet mass is calculated
from the expression (Gillespie, 1975)
N(m;t)=1Nr∑i=1NrNi(m;t),
where Nr is the number of realizations of the stochastic algorithm,
Ni(m;t) is the number of droplets of mass m in the i realization at
time t, and N(m;t) is the ensemble mean. From expression (14) it
is clear that in order to obtain the correct expected values (N(m;t)) at
the large end of the droplet size distribution, we will need a large number
of realizations of the SSA.
To further investigate this question, the evolution of a cloud system with
an initial monodisperse droplet size distribution of N0=30 droplets
of 14 µm in radius (droplet mass 1.1494×10-8 g) at
t0, and a volume of 1 cm3 was calculated with both the numerical
algorithm and Gillespie's SSA for the sum kernel (K(i,j)=B(xi+xj), with B=8.82×102 cm3g-1s-1). The results
obtained by the two methods were then compared with the analytical solution
of the master equation (Eq. 13) obtained by Tanaka and Nakazawa (1993) for
the same conditions.
The averages calculated from Gillespie's method for Nr=103
realizations and the analytical solution at t=1200 are displayed in Fig. 5.
As can be observed, both the Monte Carlo averages and the analytical
solution are closely coincident for the small end of the droplet size
distribution. However, due to the small number of realizations, the SSA
fails to reproduce the distribution for the expected values at the large end
(see Table 1).
For a more detailed analysis, the expected number of particles for each
droplet size calculated from the analytical solution, the numerical
algorithm and the SSA of Gillespie (for 1000 and 10 000 realizations) are
displayed in Table 1. As can be seen in the table, the size distributions
are almost identical for the small end. However, they differ substantially
at the large end since the SSA produces no particles larger than
12 v0 and 16 v0 for 1000 and 10 000 realizations,
respectively (v0=1.1494×10-8 g, mass of a 14 µm
droplet).
For 1000 realizations, the Monte Carlo averages differ from the analytical
solution for bin numbers larger than 8. For 10 000 realizations we have the
same situation for bin numbers larger than 13.
As expected, for 1000 and 10 000 realizations, no states with droplets 30
times larger than monomer-sized ones were realized. The numerical algorithm
described in this paper performed very well at the large end, with expected
values that are very close to the analytical solution (see Fig. 5 and Table 3).
It can be concluded that our method will be suitable if we need to
accurately calculate the large end of the droplet spectrum for small systems
(with <50 monomer droplets in the initial state). As the SSA
requires a large number of realizations, it will be computationally very
expensive. Then, for a small number of particles, our algorithm will be a
good alternative, as it provides the desired accuracy to detect the possible
small differences between different numerical approaches. It can also work
as a benchmark for different Monte Carlo methods for the collision–coalescence process.
Time evolution of the correlation coefficients ρ1,2 and
ρ2,3 for the constant, sum and product kernels (in panels a,
b and c, respectively) for two systems with a volume of 1 cm-3 and
containing 10 and 40 droplets of 14 µm.
Kinetic vs. stochastic approach: calculation of correlation coefficients and numerical results for mass-dependent collection kernels
Numerical calculation of correlation coefficients
The evolution equation for the expected values of the random variables can
be obtained by multiplying Eq. (7) by nk and summing over all states
(see Bayewitz et al., 1974):
∂∂tnk=12∑i+j=kK(i,j)ninj-niδi,j-∑jK(j,n)njnk-njδj,k.
The KCE is obtained from Eq. (15) by assuming that ninj=ninj, i.e., that the correlation between the random variables is
zero. A form of Eq. (15) was deduced in Tanaka and Nakazawa (1993) and in
Wang et al. (2006) for a general type of kernel. Bayewitz et al. (1974)
have quantified the deviation of the size distributions calculated with the
KCE from the exact distribution obtained from the master equation for a
constant kernel. From Eq. (15) it can be concluded that as long as the
correlations remain appreciable, the results of the KCE will not match the
true stochastic averages. The correlation (or correlation coefficient)
between two random variables ni and nj denoted as ρi,j is
ρi,j=cov(ni,nj)Var(ni)Var(nj)=σninjσniσnj.
In Eq. (16), the covariance (cov(ni,nj)) is calculated
according to
cov(ni,nj)=Eni-ninj-nj=Eninj-ninj.
Where Eninj is the expected value of the
product ninj which, for the bivariate case, is
Eninj=∑ni∑njninjf(ni,nj).
In Eq. (18), f(ni,nj) is the two-dimensional joint probability
mass function (pmf) which was calculated similarly to how it was done in the
univariate case (see Eq. 10):
f(n,l)=P(n,i;l,j;t)=∑allstateswithni=nandnj=l,Pn1,n2,…,ni=n,…,nj=l,…,nN;t.
In the former equation, P(n,i;l,j;t) is the probability of having
n droplets of mass i and l droplets of mass j.
Comparison of the size distributions obtained from the stochastic
master equation (solid line) with that of the KCE (dashed line) at t=1800 s
for a 1 cm3 system containing initially 40 droplets of 14 µm.
The expectation values are shown for the constant, sum and product kernels
(in panels a, b and c, respectively). For the small end the size
distributions are closely coincident, for the large end the two equations
give different values.
For the turbulent hydrodynamic kernel, comparison of the size
distributions obtained from the stochastic master equation (solid line) with
that of the KCE (dashed line) at t= 1200 and 1800 s for a 1 cm3 system
containing initially 20 droplets of 14 µm and 10 droplets of 17 µm. For the small end the size distributions are closely coincident, for the
large end the two equations give different values.
Time evolution of the correlation coefficients ρ1,3 and
ρ1,5 for a 1 cm3 system modeled with the turbulent
hydrodynamic kernel and containing initially 20 droplets of 14 µm and
10 droplets of 17 µm.
For the turbulent hydrodynamic kernel, comparison of the expected
values nk obtained from the stochastic
master equation (solid line) with that of the KCE (dashed line), for a
1 cm3 system containing initially 20 droplets of 14 µm and 10
droplets of 17 µm. The time evolution of the expected values are shown
for k=1, 5, 15 and 20 (panels a, b, c and d, respectively). For the
small masses k=1 and 5, both solutions are closely coincident up to 1800 s.
For the larger masses k=15 and 20, the results are different at all
times.
Numerical results for the constant, sum and product kernels
Correlation coefficients (ρ1,2 and ρ2,3) were obtained
by Wang et al. (2006) using the analytical solution obtained by Bayewitz et
al. (1974) for a constant collection kernel. They found that, even for this
case, the magnitude of correlations could be quite large. We will extend
their analysis by calculating the time evolution of the correlation
coefficients ρ1,2 and ρ2,3 for the constant, sum and
product kernels (see Fig. 6). For each case, the simulations were conducted
for two systems containing 10 and 40 droplets of 14 µm in radius,
respectively, and a volume of 1 cm-3. As can be observed in the figure,
in all the cases we have non-zero correlations. From the evolution of ρ1,2 for all the kernels, we can infer that the random variables
n1 and n2 are, at the beginning of the simulation, strongly
anticorrelated. This is due to the fact that in the initial stage of
evolution of the system we have mainly collisions between size 1 droplets to
form size 2 droplets. On the other hand, the random variables n2 and
n3 are also anticorrelated, because a decrease of n2 due to
collisions with size 1 droplets will increase the number of size 3 droplets
(Wang et al., 2006).
Analytical size distributions of the kinetic collection equation
(KCE) calculated with monodisperse initial conditions.
K(xi,xj)
N(i,t)
B(xi+xj)
N0(1-ϕ)(iϕ)i-1Γ(i+1)exp(-iϕ)
ϕ=1-exp(-BN0ν0t)
C(xi×xj)
N0(iT)i-1iΓ(i+1)exp(-iT)
T=CN0ν02t
A
4N0(T)i-1(T+2)i+1
T=AN0t
Note: parameters β, B and C are constants, x and y are the masses of the
colliding drops. N0 is the initial concentration and v0 is the
initial volume of droplets. The index i represents the bin size.
Expected values for each droplet mass obtained at t=1200 s for
the analytical solution, the numerical algorithm proposed in this work, and
Gillespie's SSA (for Nr=1000, 10 000 realizations). Calculations
were performed with the initial condition P(30,0,0,0,…,0;0)=1, and the
sum kernel K(i,j)=B(xi+xj) with B=8.82×102 cm3g-1s-1.
Expected values for each droplet size: <ni>, t=1200 s
Bin number
Analytical solution
Numerical algorithm
SSA
SSA
(Nr=1000)
(Nr=10 000)
1.000
1.5633E+01
1.5622E+01
1.5612E+01
1.5619E+01
2.000
3.5302E+00
3.5303E+00
3.5250E+00
3.5425E+00
3.000
1.1754E+00
1.1762E+00
1.1870E+00
1.1712E+00
4.000
4.5543E-01
4.5609E-01
4.4800E-01
4.5050E-01
5.000
1.9017E-01
1.9057E-01
2.2300E-01
1.9600E-01
6.000
8.2592E-02
8.2824E-02
7.2000E-02
8.2000E-02
7.000
3.6583E-02
3.6709E-02
3.6000E-02
3.6800E-02
8.000
1.6320E-02
1.6387E-02
1.6000E-02
1.6100E-02
9.000
7.2696E-03
7.3034E-03
3.0000E-03
6.5000E-03
10.000
3.2117E-03
3.2284E-03
2.0000E-03
3.5000E-03
11.000
1.3997E-03
1.4077E-03
1.0000E-03
1.2000E-03
12.000
5.9891E-04
6.0263E-04
0.0000E+00
4.0000E-04
13.000
2.5049E-04
2.5216E-04
0.0000E+00
4.0000E-04
14.000
1.0197E-04
1.0269E-04
0.0000E+00
3.0000E-04
15.000
4.0229E-05
4.0529E-05
0.0000E+00
0.0000E+00
16.000
1.5312E-05
1.5431E-05
0.0000E+00
1.0000E-04
17.000
5.5954E-06
5.6404E-06
0.0000E+00
0.0000E+00
18.000
1.9526E-06
1.9687E-06
0.0000E+00
0.0000E+00
19.000
6.4672E-07
6.5217E-07
0.0000E+00
0.0000E+00
20.000
2.0189E-07
2.0361E-07
0.0000E+00
0.0000E+00
21.000
5.8917E-08
5.9419E-08
0.0000E+00
0.0000E+00
22.000
1.5913E-08
1.6048E-08
0.0000E+00
0.0000E+00
23.000
3.9295E-09
3.9622E-09
0.0000E+00
0.0000E+00
24.000
8.7349E-10
8.6634E-10
0.0000E+00
0.0000E+00
25.000
1.7127E-10
1.7176E-10
0.0000E+00
0.0000E+00
26.000
2.8809E-11
2.8765E-11
0.0000E+00
0.0000E+00
27.000
3.9922E-12
3.9906E-12
0.0000E+00
0.0000E+00
28.000
4.2746E-13
4.2803E-13
0.0000E+00
0.0000E+00
29.000
3.1450E-14
3.1525E-14
0.0000E+00
0.0000E+00
30.000
1.1930E-15
1.1962E-15
0.0000E+00
0.0000E+00
At t=1800 s, the true stochastic averages (see Eq. 11) obtained
numerically from the master equation are displayed in Fig. 7, together with
the mean values for each droplet mass calculated from the analytical
solutions of the KCE (see Table 2). For the three cases, at the large end
of the spectrum, results differ substantially. This is in agreement with the
analytical study of Tanaka and Nakazawa (1994), who demonstrated that the
true stochastic averages coincide well with those obtained from the kinetic
collection Eq. (1) if the bin mass k satisfies the inequality k2≪M0, where M0 is the total mass of the system.
Numerical results for the turbulent hydrodynamic collection kernel
Collisions between droplets under pure gravity conditions are simulated with
a collection kernel of the form
Kg(xi,xj)=π(ri+rj)2V(xi)-V(xj)E(ri,rj).
The hydrodynamic kernel (Eq. 19) does not take into account the turbulence
effects and considers that droplets with different masses (xi and
xj and corresponding radii ri and rj) have different settling
velocities. In Eq. (20), E(xi,xj) are the collection efficiencies
calculated according to Hall (1980). In turbulent air, the hydrodynamic
kernel should be enhanced due to an increase in relative velocity between
droplets (transport effect) and an increase in the collision efficiency (the
drop hydrodynamic interaction). These effects were taken into account by
implementing the turbulence-induced collision enhancement factor PTurb(xi,xj) calculated in Pinsky et al. (2008) for a cumulonimbus cloud
with dissipation rate, ε=0.1 m2s-3, and Reynolds
number, Reλ=2×104 for cloud droplets with radii
≤21 µm. Then, the turbulent collection kernel has the form
KTurb(xi,xj)=PTurb(xi,xj)Kg(xi,xj).
In the simulation for turbulent air, a system corresponding to a cloud
volume of 1 cm3 and a bidisperse droplet distribution was considered: 20
droplets of 14 µm in radius and another 10 droplets of 17.64 µm in
radius, corresponding to a liquid water content (LWC) of 0.436 gm-3.
For the turbulent collection kernel the true stochastic averages at
t=1200 and 1800 s are displayed in Fig. 8, and compared with the mean values
for each droplet mass calculated numerically from the KCE with kernel (20). Also, for this case, at the large end of
the spectrum, results obtained from the KCE differ substantially from the
stochastic means. The time evolution of the correlation coefficients ρ1,5 and ρ1,3 displayed in Fig. 9 confirms the fact that
correlations cannot be neglected.
Finally, the time variations of n1, n5, n15 and n20 were calculated
and compared with the time evolution of the averages calculated from the KCE
with the same initial conditions and coalescence rate. We can see from Fig. 10
that for the small masses k=1 and 5, both solutions are closely
coincident up to 1800 s, and that for the larger masses k=15 and 20, the
results are different at all times.
Discussion and conclusions
The full stochastic description of the growth of cloud droplets in a
coalescing system is a challenging problem. For finite volume systems or in
systems of small populations, statistical fluctuations become important and
the mathematical description relies on the master equation which has
analytical solutions for a limited number of cases. In an effort to solve
this problem, we have introduced a new approach to numerically calculate the
solution of the coalescence multivariate master equation that works for any
type of kernel and initial conditions.
For the constant, sum and product kernels, the true stochastic averages
calculated numerically were compared with analytical solutions of the master
equation, with an excellent agreement between the two approaches.
A numerical procedure to calculate the correlation coefficients was
implemented, which were calculated for mass-dependent kernels (sum, product,
and kernels modified by turbulent processes). Also numerical solutions of
the master equation for bivariate initial conditions and collection kernels
modified by turbulent processes were obtained and compared
with size distributions obtained from the numerical integration of the KCE.
The two equations give different values at the large end of the droplet size
distribution. It was also shown that, for small k, the true stochastic
averages nk and the solution of the KCE
are closely coincident up to 1800 s. For larger masses, the results are
different at all times.
A topic of discussion can be the limits of applicability of the finite
volume approach to problems of precipitation formation, since such small
volumes would not remain undisturbed for a long time in a real cloud.
However, in defense of the finite system approach, it might be argued that
in the early stages of cloud development, due to small terminal velocities
of the droplets, the coalescence process is a fairly localized process; i.e., two droplets in widely separated parts of the cloud are not going to
be coalescing with each other. This was the approach followed by Bayewitz et
al. (1974) (and endorsed in Gillespie, 1975). In their paper, for comparing
the stochastic and kinetic approaches, they partitioned the cloud into many
sub-volumes, with no collisions being permitted for two droplets of
different sub-volumes. However, interactions between sub-volumes through
sedimentation, diffusion or other physical processes were not considered.
For a constant collection kernel, a more complex model that uses the master
equation formalism and introduces the interactions between the sub-volumes
was developed by Merkulovich and Stepanov (1990, 1991). This model is based
on a scheme proposed by Nicolis and Prigogine (1977) for chemical reactions.
Within this theory, the whole system is subdivided into sub-volumes
(coalescence cells) that can be considered spatially homogeneous.
Coalescence events are permitted only between droplets from the same
sub-volume, and interactions between neighbors occur through the diffusion
process. That leads to a set of master equations for each sub-volume.
Although very complex, it could be a starting point in order to consider the
interactions between small coalescence volumes through sedimentation or
other physical mechanisms.
However, fluctuations will be also very important, if the collection kernel
K(i,j) increases sufficiently rapidly with i and j and a giant droplet with
mass comparable to the total mass of the system is formed. In that case, the
total mass predicted by the KCE starts to decrease. This is usually
interpreted to mean that the system exhibits a phase transition (also called
gelation). After this moment, the true averages calculated from the master
equation will differ from the averages obtained from Eq. (1) and there is a
transition from a system with a continuous droplet distribution to one with
a continuous distribution plus a giant cluster (Alfonso et al., 2013). After
the sol–gel transition the KCE breaks down: the second moment of the size
distribution diverges at the gel point and, as was remarked, the first
moment decays, i.e., mass is not conserved.
The limitation of the KCE equation arises from the fact that it is a
deterministic equation with no fluctuations or correlations included. Then
it describes an inherently stochastic process with a single metric, the mean
cluster distribution (Matsoukas, 2015). Then, in order to model properly the
system behavior after the giant cluster is formed, the role of fluctuations
should be considered.
By using the finite volume approach, the expected values at the large end of
the droplet size distribution can be obtained in the post-gel region
(Lushnikov, 2004; Matsoukas, 2015) and be compared with the expected values
obtained from the kinetic approach. As a result, it is expected to obtain
broader droplet mass distributions by using the stochastic approach. A
follow-up paper will be devoted to a more detailed analysis of all these
problems.