Lagrangian cloud models (LCMs) are increasingly used in the cloud physics community. They not only enable a very detailed representation of cloud microphysics but also lack numerical errors typical for most other models. However, insufficient statistics, caused by an inadequate number of Lagrangian particles to represent cloud microphysical processes, can limit the applicability and validity of this approach. This study presents the first use of a splitting and merging algorithm designed to improve the warm cloud precipitation process by deliberately increasing or decreasing the number of Lagrangian particles under appropriate conditions. This new approach and the details of how splitting is executed are evaluated in box and single-cloud simulations, as well as a shallow cumulus test case. The results indicate that splitting is essential for a proper representation of the precipitation process. Moreover, the details of the splitting method (i.e., identifying the appropriate conditions) become insignificant for larger model domains as long as a sufficiently large number of Lagrangian particles is produced by the algorithm. The accompanying merging algorithm is essential to constrict the number of Lagrangian particles in order to maintain the computational performance of the model. Overall, splitting and merging do not affect the life cycle and domain-averaged macroscopic properties of the simulated clouds. This new approach is a useful addition to all LCMs since it is able to significantly increase the number of Lagrangian particles in appropriate regions of the clouds, while maintaining a computationally feasible total number of Lagrangian particles in the entire model domain.

Lagrangian cloud models (LCMs) are a recently developed approach to simulate
cloud microphysics

The present study introduces and verifies a splitting algorithm designed to
improve the precipitation process. Additionally, an accompanying merging
algorithm is proposed that is able to unite superdroplets that are not
required for an adequate representation of the precipitation process. Thus,
the merging algorithm is essential to improve the computational performance
of the LCM. Both algorithms are tested in zero-dimensional box simulations, a
three-dimensional simulation of a single cumulus cloud, and an established
shallow cumulus test case. This paper is structured as follows. The next
section briefly summarizes the collection algorithm and the basic
framework of the applied LCM. Section

This section gives a short overview of the LCM basic equations. The
applied LCM was initially developed by

In the following, the applied collection algorithm will be summarized to show
how collection affects a superdroplet's weighting factor, and to understand
how collection and splitting interact. The reader is referred to

If a collection takes place, each droplet of superdroplet

If

Transport of each superdroplet is described by

The following subsections will introduce techniques for the interactive
modification of the number of superdroplets by splitting and merging. This is
different from most previous LCM approaches, in which the number of
superdroplets is set at the beginning of the simulation and remains constant
thereafter (unless precipitation scavenges superdroplets). Note that the
splitting and merging algorithms will be tested for the all-or-nothing
collection approach, but they are similarly applicable to the average-impact
approach introduced by

Splitting takes place if a superdroplet fulfills certain criteria. First, the
radius of the superdroplet needs to be greater than or equal to a threshold

The numerical implementation of the splitting can be understood as cloning of
the superdroplet that has been determined to be split. In addition to the
already existing superdroplet,

In a first straightforward approach, the thresholds

In a more advanced method (abbreviated

The assumed drop size distribution (DSD) is calculated from

It is assumed that the weighting factor of a superdroplet should be smaller
than or equal to the approximated number of droplets in the corresponding bin
of the discretized gamma distribution. Thus, the weighting factor threshold
is determined by

No matter which splitting mode is chosen, the splitting operations are
executed at each time step of the LCM. Due to limited computational
resources, the generation of new superdroplets must be restricted to a
feasible amount. Hence, two limitations are introduced. The first restriction
is the maximum splitting factor

As a consequence of the potentially massive generation of new superdroplets due to splitting, the total number of superdroplets may increase sharply, which makes simulations computationally very expensive. For this reason, a merging algorithm was developed to decrease the number of superdroplets in order to reduce the required computational resources.

To avoid an impact of merging on micro- or macrophysical properties of the
cloud, the algorithm is only executed in non-cloudy grid boxes (liquid water
is lower than

The algorithm is designed as follows. Based on the thresholds

The use of the merging algorithm inside certain regions of the cloud where collection plays only a subordinate role is also conceivable. However, the (probably sophisticated) determination of necessary thresholds is not within the scope of this study. Furthermore, it must be mentioned that the merging algorithm does not conserve size and chemical composition of the aerosol. Therefore, for studies that explicitly consider aerosols, the merging algorithm needs to be adapted.

In the following box simulations, the sensitivity of the LCM collection process to the number of simulated superdroplets, different splitting approaches, and the approaches' specific parameters is investigated. Therefore, the box model simulation considers collection as the only microphysical process.

Although zero-dimensional simulations do not have a spatial extent,
allocating a certain weighting factor requires a reference volume to
represent a defined droplet concentration. Therefore, the volume of a grid
box is

Besides the traditional single-box approach, a new multi-box approach is
introduced. In contrast to the calculation of independent grid boxes, the
multi-box approach allows superdroplets to move from one grid box to the next
by prescribing a stochastic velocity (but no mean motion) in
Eq. (

This multi-box approach has one distinct advantage over the ensemble mean of
the same amount of individual box model simulations (single-box model), which
results from the difficulties to initialize a DSD with superdroplets of a
constant weighting factor, as it is done in many applications of LCMs in the
literature

The impact of different numbers of superdroplets per grid box and the use of
splitting for the traditional single-box approach will be discussed first;
then, the new introduced multi-box approach will be presented for both
splitting and non-splitting cases. Box model simulations will be compared to
the results of

As reference, the “singleSIP” initialization of

In addition to analyzing the DSD directly, the temporal development of the
zeroth and second moment of the mass density distributions is examined. Due
to mass conservation in all applied approaches, the first moment is constant
in time and will not be shown. The moments of the mass distribution

For a given superdroplet ensemble the moments for each grid box are
calculated with

First, the sensitivity of the collision algorithm to the number of superdroplets is examined using the LCM as a single-box model. Second, the improvements by the splitting method on collisional growth is evaluated. Subsequently, those investigations are repeated for the multi-box approach.

Figures

Mass density distribution for the single-box approach after

Moments of the mass density distribution as a function of time
obtained from the single-box simulations for the “singleSIP” initialization.
The black solid line denotes the solution of

Mass density distribution for the single-box approach after
3600

Moments of the mass density distribution as a function of time
obtained from the single-box simulations. The black solid line denotes the
solution of

Now, Figs.

Mass density distribution for the single-box approach after

Moments of the mass density distribution as a function of time
obtained from single-box simulations. The black solid line denotes the
solution of

In Figs.

The black dashed line (Const.) shows the reference LCM case in which no
splitting is applied. Comparing the non-splitting case to splitting cases, the
results are significantly improved with respect to the reference solution.
More precisely, the fluctuations that occur for large droplet radii are
successfully removed by splitting. Furthermore, a better representation of
the second maximum is also achieved by splitting. Independent of the
splitting mode, simulations with the same splitting radius provide similar
results. The only exception is between the simulations

Similar conclusions are possible from Fig.

These results exhibit how strongly collisional growth suffers from the
initialization with a constant weighting factor, consistent with

Same as Fig.

Same as Fig.

Same as Fig.

Figure

In Fig.

To maintain a reasonable amount of superdroplets, these box simulations will be repeated now, using the splitting approach. Here all parameters (initializing all simulations with 87 superdroplets per grid box) and splitting thresholds are identical as for the single-box approach described above but the superdroplets are now allowed to move between grid boxes.

Figure

All in all, it is shown that collisional growth is better represented by
using the splitting method in both the single- and multi-box simulations.
Furthermore, the choice of the splitting mode is secondary, but the splitting
radius is identified as the most crucial parameter. The multi-box simulations
exhibit a distinct advantage over the single-box simulations. Due to the
presence or absence of sufficiently large droplets that might initiate
collision and coalescence, as a result of the initialization, collisional
growth can be overestimated in certain grid boxes while it is
underestimated in others. Splitting and the subsequent stochastic exchange
are able to distribute these so-called precipitation embryos among the entire
ensemble where they are able to initiate collision and coalescence as
sketched in Fig.

Same as Fig.

Schematic representation on how splitting affects the spatial
distribution of large superdroplets. The squares outline the different grid
boxes with superdroplets of the size of cloud droplets (blue) and
superdroplets representing rain drops (dark red). Without splitting

The limiting parameters of the splitting algorithm are now examined in
sensitivity studies using the multi-box approach. For this purpose, the
parameters of the maximum possible number of superdroplets per grid box

Figure

The sensitivity studies for the maximum splitting factor show that this has
no influence on the results (Fig.

As shown before, the development of the spectrum is highly sensitive to the
choice of the splitting radius. Figure

In this case, we are simulating an idealized shallow cumulus cloud in the form of
a rising warm air bubble as in

Mass density distribution for the box simulation after
3600

The initial profiles for temperature and specific humidity are based on the
shallow cumulus case by

At the surface, superdroplets are absorbed if their radius is larger than
1.0

Summary of the main parameters for the single-cloud simulations.

Mass density distribution after 1800

Figure

This behavior can be ascribed to different requirements on the superdroplet number for the convergence of different growth processes. The left part of the spectrum is dominated by diffusional growth which can be sufficiently represented by just a couple of superdroplets per grid box. By contrast, collisional growth is highly sensitive to the superdroplet number and the correct representation of large droplets. An improved representation of these droplets is ensured by the splitting algorithm, no matter what splitting mode is used.

The improved statistics of large superdroplets are also shown in
Fig.

In Fig.

Total number of superdroplets per logarithmic radius bin after

Figure

Figure

Figure

All in all, the splitting of large droplets, which results in an improved representation of the collision process and thus the DSD, also partly influences the macroscopic properties of the cloud. In particular, rain water content, radar reflectivity, and precipitation rate are represented in a more realistic manner. Due to the improved statistics, the temporal and spatial variance of these parameters is significantly reduced. However, the whole cloud life cycle, which is driven by the general dynamics and thermodynamics, is largely unaffected by splitting. Additionally, the merging shows no influence on the physical outcomes.

To estimate the increase in computing time due to splitting, we conducted
three simulations (

Time series of different variables for the idealized single-cloud
simulation for different initial numbers of superdroplets and splitting
configurations. In

Vertical cross sections of the precipitation rate for the reference
case

The setup for simulating a shallow cumulus field is based on the LES
intercomparison study by

Time series of

Probability density function of precipitation rates for different initial numbers of superdroplets and splitting configurations.

Three different simulations will be presented. In the cases

Based on the previously presented results, the maximum number of particles
per grid box is set to

The analysis is focused on the influence of splitting on the macroscopic
properties of the shallow cumulus field. Figure

Considering the temporal variability in the precipitation rate and total
precipitation (Fig.

The main objective of this paper was the development and verification of a
splitting algorithm to improve collisional growth in Lagrangian cloud models
(LCMs). These models are able to represent collision and coalescence well

The splitting and merging algorithms have been validated using
box simulations, a simulation of a single cumulus cloud, and an established
shallow cumulus test case. The box simulations confirmed that the capability
of an LCM to represent the temporal evolution of a DSD due to collision and
coalescence depends crucially on the number of simulated superdroplets

In the idealized single-cloud simulation, splitting improved the representation of collisional growth with up to 70 % larger maximum radii. Moreover, splitting improves the spatial and temporal representation of precipitation by distributing the precipitable water on more superdroplets with an accordingly smaller weighting factor. It is important to note, however, that the life cycle and domain-averaged macroscopic properties are almost not affected by the splitting process. If applied, the merging algorithm has been shown to reduce the computing time by 18 % and the storage demand by at least 7 % in comparison to simulations with splitting alone. Since merging is restricted to cloud-free regions, its application did not alter the simulated physics. Similar findings on the effect of splitting on the production of rain have been made for the shallow cumulus test case.

In light of the fact that LCMs become increasingly important in the field
of modeling cloud microphysics, it is necessary to minimize the (typically)
large demand of memory and computing time required for their application.
Thus, a fixed number of superdroplets needs to be replaced by a dynamic
number, which adapts interactively to the given physical and numerical
requirements. In this regard, the presented methods follow the approaches by

The LES model used in this study (revision 2263) is
publicly available at

JS carried out the analysis. JS, FH, and SR developed the basic ideas, discussed the results, and wrote the manuscript.

The authors declare that they have no conflict of interest.

All simulations have been carried out on the Cray XC40 systems of the
North-German Supercomputing Alliance (HLRN,