Recently, several Lagrangian microphysical models have been developed which use a large number of (computational) particles to represent a cloud. In particular, the collision process leading to coalescence of cloud droplets or aggregation of ice crystals is implemented differently in various models. Three existing implementations are reviewed and extended, and their performance is evaluated by a comparison with well-established analytical and bin model solutions. In this first step of rigorous evaluation, box model simulations, with collection/aggregation being the only process considered, have been performed for the three well-known kernels of Golovin, Long and Hall.

Besides numerical parameters, like the time step and the number of simulation particles (SIPs) used, the details of how the initial SIP ensemble is created from a prescribed analytically defined size distribution is crucial for the performance of the algorithms. Using a constant weight technique, as done in previous studies, greatly underestimates the quality of the algorithms. Using better initialisation techniques considerably reduces the number of required SIPs to obtain realistic results. From the box model results, recommendations for the collection/aggregation implementation in higher dimensional model setups are derived. Suitable algorithms are equally relevant to treating the warm rain process and aggregation in cirrus.

The collection of cloud droplets and the aggregation of ice crystals are important processes in liquid and ice clouds. By changing the size, number and, in the case of ice, the shape of hydrometeors, collection and aggregation affect the microphysical behaviour of clouds and thereby their role in the climate system.

The warm rain process (i.e. the production of precipitation in clouds in the
absence of ice) depends essentially on the collision and subsequent
coalescence of cloud droplets. At its initial stage, however, condensational
growth governs the activation of aerosols and the following growth of cloud
droplets, which might initiate the collection process if they become
sufficiently large. Then, collection produces drizzle or raindrops, which are
able to precipitate from the cloud, affecting lifetime and organisation of
clouds

In ice clouds, sedimentation, deposition growth and in particular radiative
properties depend on the ice crystals' habits

The temporal change of an infinite system of droplets by collision and
subsequent coalescence (or any other particles) is described by the
stochastic collection equation (SCE), also known as the kinetic collection
equation, coagulation equation, Smoluchowski or population balance equation

For several kernel functions (mostly of polynomial form), analytic solutions
exist for specific initial distributions

Solving (1) demands simplifications in the representation of the droplet
spectrum for which several numerical models have been developed. Spectral-bin
models

Due to their specific construction, LCMs offer a variety of advantages in
comparison to spectral-bin and bulk cloud models. Their representation of
aerosol activation and subsequent diffusional growth closely follows
fundamental equations and therefore avoids the possible perils of
parametrisations

To our knowledge, five fully coupled LCMs for warm clouds exist, which are
described in

So far, no consistent terminology has been used in the latter publications.
Various names have been used for the same things by various authors. We point
out that super droplet, computational droplet and simulation particle (SIP)
all have the same meaning and refer to several identical real cloud
droplets (or ice crystals) represented by one Lagrangian particle. The number
of real droplets represented in a SIP is denoted as the weighting factor or
multiplicity. Moreover, Lagrangian approaches in cloud physics have been
named the Lagrangian cloud model (LCM), super droplet method (SDM) or particle-based
method. In this paper, we use the terms SIP, weighting factor

Usually, only the liquid water or the ice of a cloud are described with a
Lagrangian representation, whereas all other physical quantities (like
velocity, temperature and water vapour concentration) are described in
Eulerian space

Lists of used symbols and abbreviation are given in Tables

List of symbols.

List of abbreviations.

We use the terminology of

The moments of order

The initialisation is successful for a given parameter set if the moments of
the SIP ensemble

Throughout this study, the initial parameters of the droplet size
distribution (DSD) are DNC

In our test cases, all microphysical processes except collection are neglected and an exponential DSD is initialised. In the results section, we will demonstrate that the outcome of the various collection algorithms critically depends on how this initial, analytically defined, continuous DSD is translated into a discrete ensemble of SIPs. Hence, the SIP initialisation is described in some detail.

First, the mass distribution is discretized on a logarithmic scale. The
boundaries of bin

For each bin, the droplet number is approximated by

Following

So far, we introduced initialisation techniques with a strict lower threshold

Using the probabilistic version and a weak lower threshold is particularly
important if different realisations of SIP ensembles of the same analytic DSD
should be created. The number of SIPs

Moreover, the singleSIP-init is used in a hybrid version, where different

Table

Number of SIPs for the probabilistic singleSIP-init method (and
variants like the multiSIP-init) as a function of

The accumulated PDF

For the case of an exponential distribution, the following holds for the SIPs

The third approach allows specifying the spectrum of weighting factors that
should be covered by the SIP ensemble. Similar to the

SIPs with weighting factors

For the case of an exponential distribution, the following holds for the interval boundaries and the SIPs

Experimenting with the SIP-init procedure, several optimisations have been
incorporated. First, the

Going through the list of SIPs, the droplet masses increase, and hence the
individual SIPs contain gradually increasing fractions of the total grid box
mass. This can lead to a rather coarse representation of the right tail of
the DSD. Two options to improve this have been implemented. In the

Characteristics of the various SIP initialisation methods (as given
on top of each panel): weighting factors

Figure

The

The second row shows average

To display DSDs represented by a SIP ensemble, a SIP ensemble must be
converted back into a bin representation. For this, we establish a grid with
resolution

First, we present a hypothetical algorithm for the treatment of
collection/aggregation in an LCM, which would probably yield excellent
results. However, it is prohibitively expensive in terms of computing power
and memory, as

Whereas condensation/deposition and sedimentation may be computed using
interpolated quantities which implicitly assume that all droplets of a SIP
are located at the same point, the numerical treatment of collection usually
assumes that the droplets of a SIP are spatially uniformly distributed, i.e.
well-mixed within the grid box. An approach, where the vertical SIP position
is retained in the collection algorithm and the process of larger droplets overtaking
smaller droplets is explicitly modelled, is described in

Following

For SIPs

In the hypothetical algorithm, the weighting factor of SIP

The droplet mass

For each

The rate equations for the weighting factors can be numerically solved by a simple Euler forward step.
The weighting factor of existing SIPs is reduced by

For new SIPs

In each time step,

In the following subsections, algorithms are presented that include various approaches to keep the number of SIPs in an acceptable range.

In the following, the various algorithms are described and pseudo-code of the implementations is given. For the sake of readability, the pseudo-code examples show easy-to-understand implementations. The actual codes of the algorithms are, however, optimised in terms of computational efficiency. The style conventions for the pseudo-code examples are as follows: commands of the algorithms are written in upright font with keywords in boldface. Comments appear in italic font (explanations are enclosed by {} and headings of code blocks are in boldface).

Treatment of a collection between two SIPs in the remapping algorithm (RMA), average impact algorithm (AIM) and all-or-nothing algorithm (AON).

First, the remapping algorithm is described, as its concept follows closely
the hypothetical algorithm introduced in the latter section. RMA
is based on ideas of

Instead of creating a new SIP

Figure

At the end of each time step, after treating all possible

Optionally, a lower threshold

We pick up the concept of a weak threshold introduced earlier and adjust it
such that on average the total mass is conserved (instead of total number as
before). We introduce the threshold

Time steps typically used in previous collection/aggregation tests are around

Hence, several extensions to RMA allowing larger time steps are proposed in the following.

The default version uses the algorithm as outlined in
Algorithm (1)
(i.e. nothing is changed). Negative

Clipping simply ignores bins with negative

In adaptive time stepping, instead of reducing the general time step, only the
treatment of SIPs with

By using the reduction limiter (abbreviated as RedLim), the effect of an
adaptively reduced time step can be reached with simpler and
cheaper means. We introduce a threshold parameter

Update on the fly (abbreviated as OTF) is another option that
effectively eliminates negative

Top: (

The average impact algorithm by

Moreover, same-size collisions are considered in each SIP. These decrease the weighting factor of each SIP but not its total mass. Accordingly, the radius of the SIP increases.

Both processes are represented in the following two equations which are
solved for all colliding SIPs (assuming that

Using a Euler forward method for time integration, the above equations read as

Figure

The ratio

Negative values of

The same as Fig.

The all-or-nothing algorithm (AON) is based on the ideas of

The treatment of the special case

Another special case appears if both SIPs have the same weighting factor
which regularly occurs when the

Moreover, self-collections can be considered for kernels with

So far, we explained how a single

For most

As for AIM in Fig.

For the generation of the random numbers, the well-proven

The current implementation differs slightly from the version in

Moreover, in Shima's formulation, the weighting factors are considered to be
integer numbers. In contrast, we use real numbers

The study of

In RMA and AIM, SIPs with negative weights may be generated depending, e.g.
on the condition

In this section, box model simulations of the three algorithms introduced in
the latter section are presented, starting with the results of RMA, then those
of AIM and finally AON. The results of each algorithm are tested for
three different collection kernels (Golovin, Long and Hall). As default,
probabilistic SIP initialisation methods are used. For each parameter
setting, simulations are performed for 50 different realisations. Simulations
with the Golovin kernel are compared against the analytical solution given by

In all simulations, collision/coalescence is the only process considered in
order to enable a rigorous evaluation of the algorithms. The evaluation is
based on the comparison of mass density distributions and the temporal
development of the zeroth, second and third moments of the droplet distributions. The first
moment is not shown since the mass is conserved in all algorithms per
construction. The Supplement contains
a large collection of figures that systematically report all sensitivity
tests that have been performed. The behaviour of the second and third moments
is similar, and the

Mass density distributions obtained by RMA for the
Golovin kernel from

SIP number and moments

Figure

Next, RMA simulations with the Long kernel are discussed. As already
mentioned, the default RMA version would require tiny time steps which would
rule out RMA from any practical application. Both approaches introduced
before, update on the fly (OTF) and reduction limiter (RedLim), succeed
in eliminating negative

We tested the algorithm for many parameter settings varying all of the
aforementioned parameters:

Mass density distributions obtained by RMA for the
Long kernel from

Hence, our RMA implementation is not capable of producing reasonable results for the Long kernel. It is not clear whether the oscillations are inherent to the original RMA or caused by the introduction of the Reduction Limiter. The latter might introduce discontinuities which could trigger instabilities.

At least, the Golovin RMA simulations with the reduction limiter do not show any
signs of instability and agree well with the reference. However, this is not
surprising. Clearly, the RedLim correction is only performed for SIPs, where
negative weights are predicted. In Golovin simulations, this happens less
frequently than in Long simulations. Only, in the very end, the abundance of
the largest droplets is underestimated (see top right panel in
Fig.

Another RMA variant uses update on the fly, which also effectively eliminates
negative weights. Such Golovin RMA simulations can be close to the reference;
however, the results depend on the order in which the SIP combinations are
processed. If collections between the smallest SIPs are treated first within
each time iteration (OTF

SIP number and moments

Long kernel simulations with OTF

Note that the Golovin simulations used

We conclude that, for time steps feasible in operational terms, none of the
tested RMA implementations are capable of producing reasonable results with
the Long kernel.

RMA simulations with the Hall kernel are similarly corrupted by oscillations and do not produce useful simulations either (not shown).

Mass density distributions obtained by AIM for the
Golovin kernel from

Moments

Figure

The algorithm performs, in general, better for the Long and Hall kernels, as is
detailed in the following. Figure

Mass density distributions obtained by AIM for the
Long kernel from

Now we draw the attention to the importance of the SIP-init method. The right
panel of Fig.

The

All quantities shown in Figs.

Moments

Next, simulations with the Hall kernel are shortly discussed (figures are
only shown in the Supplement). Compared to the Long simulations, the
reference solution reveals that small droplets are much more abundant, as the
collection of small droplets proceeds at a lower rate. This makes the
simulation less challenging from a numerical point of view and AIM DSDs come
closer to the reference than in the Long simulations. Consequently, the AIM
moments agree very well with the reference. For

Mass density distributions obtained by AON for the
Golovin kernel from

Moments

Figure

Mass density distributions obtained by AON for the
Long kernel from

Moments

Figure

There are two sources that are potentially responsible for the large ensemble
spread: the probabilistic SIP initialisation and the probabilistic AON
approach. In a sensitivity test, 50 realisations are computed, all using the
same SIP initialisation obtained by a deterministic singleSIP-init.
Figure

Figure

Figure

Mass density distributions obtained by AON for the
Long kernel from

Moments

In the conventional version, SIPs are initialised down to a radius of

Further tests with the singleSIP-init include a variation of the threshold
parameter

Finally, the AON performance for other SIP initialisations is discussed
(right column of Fig.

Droplet number as a function of time obtained by AON
for the Long kernel. The black symbols show the moments of the reference
solution. In each panel, the dotted curves depict the results with the
regular singleSIP-init as already shown in column 2 of
Fig.

Let us consider the possible weighting factors the SIPs can attain in the
course of a simulation. In the beginning, all SIPs have

To complete the analysis for the Long kernel, the right column of
Fig.

As already noted in the AIM section, Hall simulations are not as challenging as Long simulations from a numerical point of view. As the collection of small droplets proceeds at a lower rate for the Hall kernel, disenabling multiple collections in the AON simulations does not deteriorate the results as much as in the Long simulations (see the Supplement). Besides this, simulations with the Hall kernel led to similar conclusions as for the Long simulations and are therefore not discussed in more detail.

The presented box model simulations can be regarded as a first evaluation
step of collection/aggregation algorithms in LCMs. The final goal is the
evaluation in (multidimensional) applications of LCMs with full
microphysics. In order to isolate the effect of collection, other
microphysical processes like droplet formation and diffusional droplet growth
have been switched off, and all box model simulations started with a
prescribed SIP ensemble following a specific exponential distribution. In
Sect.

The initialisation techniques for the SIP population generation are mostly
probabilistic and, by default, each simulation was performed for 50 different
realisations. For RMA and AIM, we found the ensemble spread to be small; hence, a
single realisation is as good as the ensemble mean. AON is
inherently probabilistic, and we highlighted the substantial ensemble spread.
Reasonable results are only obtained only by averaging over many
realisations. One may argue that this precludes the usage of AON in
real-world applications, as it is not feasible to run 50 realisations in each
grid box of a 2-D/3-D model simulation. However, we are not that pessimistic.
In such simulations, many grid boxes have similar atmospheric conditions and
averaging will occur across such grid boxes. We made a similar experience in
simulations of contrail cirrus, where we tested the

RMA simulations for the Long kernel require around a factor of 1000 smaller time
steps than the respective AON and AIM simulations (

If the initial SIP ensemble is created with the singleSIP-init, 50 to 100
SIPs are needed for convergence in any of the three algorithms. This value is
similar to the number of bins used in traditional algorithms for spectral-bin
models

For a given

The time complexity of all presented algorithms is

All in all, the time step

In this section, we provide further insight and discuss the implications from the box model tests. Since our results have been gained with typical assumptions for warm clouds, we discuss their representativeness for ice clouds.

The evaluation of different initialisation methods showed that the
performance of the collection/aggregation approaches depends essentially on
the way the SIPs are initialised, a problem which is inherently absent in
spectral-bin models. Their initialisation resembles the singleSIP technique
used here; i.e. the number concentration (the weighting factor) within a bin
(for a certain mass range represented by one SIP) is directly prescribed.
However, LCMs exhibit a larger variety of how an initial droplet spectrum can
be translated into the SIP space. The study showed that the singleSIP is
advantageous for the correct representation of the collisional growth, since
they initialise large SIPs with small weighting factors, which are
responsible for the strongest radius growth. On the other hand, the

In this idealised study, we were able to control (to a certain extent) the
representation of droplet spectra by various initialisation methods. In
more-dimensional simulations with full microphysics, however, this is not
straightforward nor has it been intended. So far, convergence tests in
real-world LCM applications simply included variations of the SIP number
and have not focused on more detailed characteristics of the SIP ensemble
(i.e. the properties that have been discussed in Fig.

Generally, the performance of the algorithms is better when the SIP ensemble
features a broad range of weighting factors. One viable option to achieve
this is the introduction of a SIP splitting technique

Normalised SIP mass

Mass fractions represented by individual SIPs,

Moments

In all simulations so far, the mean radius of the initial DSD was

In multidimensional models, collection/aggregation might be further
influenced by the movement of SIPs due to sedimentation or flow dynamics. For
instance, sedimentation removes the largest SIPs with the potentially
smallest weighting factors, while turbulent mixing may add SIPs with their
initial weighting factor into matured grid boxes, where collection has
already decreased the weighting factors of the older SIPs. Indeed, the
additional variability in more-dimensional simulations might compensate for
the missing variability in the weighting factors usually present in
simulations using the

It is not clear which findings of our evaluation efforts are the most relevant aspects that control the performance of collection/aggregation algorithms in more complex LCM simulations. Nevertheless, the idealised box simulations are an essential prerequisite towards more comprehensive evaluations, as they disclosed the potential importance of the SIP initialisation (an aspect that is inherently absent in spectral-bin models). All in all, we can state that the behaviour of Lagrangian collection algorithms in more complex simulations demands further investigation. Nevertheless, we have already learned a lot from the box model simulations. A summary will be given in the concluding section.

Besides the academic Golovin kernel, our simulations used the hydrodynamic
kernel with collection efficiencies that are usually employed for warm clouds
(Long and Hall). We found that Hall simulations are not as challenging as
Long simulations from a numerical point of view. For ice clouds, usually a
constant aggregation efficiency

In the recent past, Lagrangian cloud models (LCMs), which use a large number
of simulation particles (SIPs, also called super droplets in the literature)
to represent a cloud, have been developed and become more and more popular.
Each SIP represents a certain number of real droplets; this number is termed
the weighting factor (or multiplicity) of a SIP. In particular, the collision
process leading to coalescence of cloud droplets or aggregation of ice
crystals is implemented differently in the various models described in the
literature. The present study evaluates the performance of three different
collection algorithms in a box model framework. All microphysical processes
except collection/aggregation are neglected, and an exponential droplet mass
distribution is used for initialisation. The box model simulation results are
compared to analytical solutions (in the case of the Golovin kernel) and to a
reference solution obtained from a spectral-bin model approach by

LCMs exhibit a large variety of how an initial droplet spectrum can be translated into the SIP space and various initialisation methods are thoroughly explained. The performance of the algorithms depends crucially on details of the SIP initialisation and various characteristics of the initialised SIP ensemble (an issue that is inherently absent in spectral-bin models and has not been paid much attention in previous LCM studies).

The remapping algorithm

The average impact algorithm

The probabilistic all-or-nothing algorithm

Many current (multidimensional) applications of LCMs use such SIP ensembles
with a narrow spectrum of weighting factors causing a poor performance of the
collection/aggregation algorithms. This should be clearly avoided in order to
have collection/aggregation algorithms to work properly and/or efficiently.
The time step and the bin resolution

The programming language IDL was used to perform the simulations and produce the plots. The source code can be obtained from the first author. Pseudo-code of the algorithms is given in the text.

The authors declare that they have no conflict of interest.

The DFG (German Science Foundation) partly funded the first author (contract number UN286/1-2) and the second author (RA617/27-1). We thank A. Bott for providing us with his Fortran code; L.-P. Wang for simulation data; M. Andrejczuk, S. Shima, I. Sölch and P. L'Ecuyer for discussions; and the reviewers for their constructive reviews.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: S. Remy Reviewed by: A. Jaruga and one anonymous referee