Lagrangian cloud models (LCMs) are considered the future of cloud microphysical modelling. Compared to bulk models, however, LCMs are computationally expensive due to the typically high number of simulation particles (SIPs) necessary to represent microphysical processes such as collisional growth of hydrometeors successfully. In this study, the representation of collisional growth is explored in one-dimensional column simulations, allowing for the explicit consideration of sedimentation, complementing the authors' previous study on zero-dimensional collection in a single grid box. Two variants of the Lagrangian probabilistic all-or-nothing (AON) collection algorithm are tested that mainly differ in the assumed spatial distribution of the droplet ensemble: the first variant assumes the droplet ensemble to be well-mixed in a predefined three-dimensional grid box (WM3D), while the second variant considers the (sub-grid) vertical position of the SIPs, reducing the well-mixed assumption to a two-dimensional, horizontal plane (WM2D). Since the number of calculations in AON depends quadratically on the number of SIPs, an established approach is tested that reduces the number of calculations to a linear dependence (so-called linear sampling). All variants are compared to established Eulerian bin model solutions. Generally, all methods approach the same solutions and agree well if the methods are applied with sufficiently high resolution (foremost is the number of SIPs, and to a lesser extent time step and vertical grid spacing). Converging results were found for fairly large time steps, larger than those typically used in the numerical solution of diffusional growth. The dependence on the vertical grid spacing can be reduced if AON-WM2D is applied. The study also shows that AON-WM3D simulations with linear sampling, a common speed-up measure, converge only slightly slower compared to simulations with a quadratic SIP sampling. Hence, AON with linear sampling is the preferred choice when computation time is a limiting factor.

Most importantly, the study highlights that results generally require a smaller number of SIPs per grid box for convergence than previous one-dimensional box simulations indicated. The reason is the ability of sedimenting SIPs to interact with a larger ensemble of particles when they are not restricted to a single grid box. Since sedimentation is considered in most commonly applied three-dimensional models, the results indicate smaller computational requirements for successful simulations, encouraging a wider use of LCMs in the future.

Clouds are a fundamental part of the global hydrological cycle, responsible for the transport and formation of precipitation. While we expect a global increase in precipitation due to climate change, our knowledge on its spatial distribution, even including decreasing rainfall in some regions of the globe, is still uncertain

The representation of these microphysical processes in climate models is impelled by the available computational resources, requiring necessary idealisations. Primarily, this is the case for computationally efficient Eulerian bulk models that predict only a small number of statistical moments for each hydrometeor class

In the last decade, Lagrangian cloud models (LCMs) emerged as a valid alternative to bin models

However, many aspects of this relatively young modelling approach in cloud physics have not been tested thoroughly. One important message of our previous box simulations in U2017 was that the representation of collisional growth exhibits considerably more freedom in setting up a simulation than in bin models. Accordingly, in this study, we are going to extend the box simulations of U2017 by analysing collisional growth in a vertical column, including sedimentation, as it has been done in previous studies for Eulerian bulk and bin models

The paper is structured as follows. First, Sect.

List of frequently used abbreviations.

Two column models, which consider collection and sedimentation, have been implemented; the first one represents a traditional Eulerian bin scheme and the second model uses a particle-based approach. Before we describe both models in some detail, we will write down basic relations, which will help disentangle the effects of particular parameter variations later.

We use a column with

The droplets are assumed to be spherical with a density of

Following

The hydrodynamic collection kernel, driven by differences in the droplet vertical velocity, is given by

The average number of collisions from

An exponential DSD is used to prescribe the cloud droplets in the beginning

The function

For an exponential DSD, the moments can be expressed analytically as

Schematic plot of how a droplet size distribution is discretised in a bin model and represented by a SIP (simulation particle) ensemble in a Lagrangian cloud model (LCM). The red and green stars shows two different realisations of a SIP ensemble.

Using the terminology of

The DSD is usually discretised using exponentially increasing bin sizes. In analogy to U2017, the bin boundaries are defined by the masses

In an LCM, real droplets are represented by simulation particles (SIPs, also called super-droplets).
Each SIP has a discrete position (vertical coordinate

The moments

How does one represent an ensemble of droplets in an Eulerian or Lagrangian cloud model? Their size distribution can be uniquely described in a bin model by simply accounting for each real droplet in its respective bin, where its boundaries are given by the bin model (see illustration in Fig.

Various techniques to generate a SIP ensemble in an LCM for a given (analytically prescribed) DSD exist (see Sect. 2.1 in U2017). In this study, we use a SIP initialisation technique (termed “singleSIP-init” in U2017), for which Lagrangian collection algorithms, and in particular AON, achieved the best results in box model tests. In the singleSIP-init, the DSD, more specifically

Eulerian column models have been widely employed in cloud physics, and the present bin implementation is conceptually similar to previous ones

The variable

In a second step, the mass concentrations are advected vertically according to the classical advection equation

For its numerical solution, two different positive definite advection algorithms have been used. The first option is the classical first-order upwind scheme (known for its inherent numerical diffusivity).
For

Evaluating

Irrespective of the chosen advection solver, the prediction of the “new”

If the prescribed

After one call of Bott's algorithm,

The moments are computed by

In a Lagrangian model, the inclusion of sedimentation (obeying the transport equation

For the collection process, it assumed that each SIP belongs to a certain GB

In the version with explicit overtakes (WM2D; see Sect.

In the remainder of this paper, the classical approach is referred to as AON-regular and the new approach as AON-WM2D.
Figure

Grid box with a SIP pair in the LCM world

AON is probabilistic and an individual realisation does not usually reproduce the mean state as predicted by deterministic methods like Eulerian approaches. The extent of deviations from the mean state is exemplified in Fig. 15 of U2017 for a box model application of AON.
Hence, the AON results discussed in the present study are usually ensemble averages over

Pseudo-code of both algorithm 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 brace brackets {}, and headings of code blocks are in boldface).

Here we basically repeat the AON description of U2017 (their Sect. 2.5).

Pseudo-code of the regular all-or-nothing (AON) algorithm; style conventions are explained right before Sect.

“Figure

The treatment of the special case

Treatment of a collection between two SIPs in the all-or-nothing (AON) algorithm, partially adopted from Fig. 2 of

The AON treatment of collection of droplets within one SIP, as well as the collection of two SIPs with equal weighting factors, is described in U2017. In the simulations presented here these aspects are not relevant and thus omitted.

The current implementation differs in several aspects from the version in

Pseudo-code of the AON-WM2D algorithm; style conventions are explained right before Sect.

We now introduce the AON version based on an idea by

Unlike the classical case where 3D well-mixedness has to be assumed, droplets of a SIP are now assumed to be well-mixed on the

The AON algorithm is split into two steps:

Based on the evaluation of the vertical positions

In case of such an overtake: compute the average number of droplet collections by

Similarly to the WM3D version, it happens that

Specifically to WM2D, it is also possible that a SIP interacts with other SIPs located in not only one but several GBs. Accordingly, it is not only necessary to check overtakes of other SIPs in the original GB (more specifically, SIPs that lie in the same GB at time

In a Lagrangian model, the time step choice is not numerically restricted by the CFL criterion and in particular the largest collecting drops may fall through several GBs during the time period

In a naive implementation, this would dramatically increase the computational costs. In the regular (WM3D) version,

For the smallest SIPs, which often travel only a small distance inside a GB, the list of SIPs that may be overtaken is commensurately small and overtakes have to be checked for a fraction of SIPs of the GB only (that means the actual computational work is smaller than in the regular version). On the other hand, imagine the largest SIPs travel through three GBs – then overtakes have to be tested for roughly 3 times more SIPs than in the regular version. Moreover, testing for overtakes (step 1) is computationally less demanding than calculating the potential collections (step 2). In WM3D we always have the workload of step 2 for all tested combinations, whereas in WM2D only the cheaper step 1 is executed in case of no overtake.

Besides the weaker assumption of 2D well-mixedness, the present approach is actually more intuitive (even though it may first be regarded counter-intuitive by those who are familiar with traditional Eulerian grid-based approaches). Moreover, this approach complies better with the Lagrangian paradigm of a grid-free description (the present approach is independent of

For more sophisticated kernels, including for example turbulence enhancement, the present approach may not be adopted easily as the driving mechanism for collisions to occur in the current model is differential sedimentation. Related to this are studies on cylindrical vs. spherical formulations of kernels in

Finally, we briefly summarise the differences between the WM2D and WM3D approach. The standard kernel

From a technical point of view, it might be challenging to implement the WM2D version in full 2D/3D cloud models, as one has to keep track of all SIPs in a grid box column. If domain decomposition is used in a vertical direction, collision candidates had to be searched across multiple processors.

The regular AON version can be sped up by introducing a linear sampling technique (LinSamp) as done in

If

The Supplement demonstrates that how the limiter is implemented is critical. We thank reviewer Shin-ichiro Shima for pointing us to a better limiter implementation, which has been already described in

Employing a limiter is recommended for all AON versions (even though we never encountered a limiter event in QuadSamp simulations), but it is particularly significant in the LinSamp version due to the upscaling of

In addition to the favourable linear computational complexity, LinSamp can be easily parallelised, in particular on shared-memory multi-processor architectures as used by

At the lower boundary, droplets leave the domain according to their fall speed. Using the LCM, the moment outflow

In both models, a non-zero influx at the model top can also be prescribed.
One option is to use periodic boundary conditions. In the Lagrangian approach this is done by increasing the altitude

Summary of AON versions.

Before we start discussing the results, we outline the terminology of the various model versions.
On a first level, we differentiate between Eulerian (“BIN”) and Lagrangian approaches (“LCM”), which can be both applied in a box (“0D”) or column model (“1D”) framework.
By default, BIN uses the MPDATA advection algorithm (clearly only in 1D) and Bott's collection algorithm. Alternatively, MPDATA can be replaced by the first-order upstream scheme (“US1”) and Bott's collection algorithm by Wang's algorithm (“Wang”).
The Lagrangian model versions differ only in the way AON is employed. The various model versions are summarised in Table

By switching off sedimentation in the column model source code (as partially done in Sect.

If the space in figure legends is limited, abbreviations “LS” and “nS” are used for “LinSamp” and “noSedi”, respectively.

Before we start comparing collisional growth in column model applications, we should first demonstrate that the differences introduced by the different numerical treatment of the sedimentation process are small to negligible. This exercises is deferred to the Appendix.

We find the discrepancies introduced by the different sedimentation treatments small enough as long as the MPDATA advection algorithm is employed in BIN. Hence, all following BIN simulations rely on MPDATA and we can attribute the differences that we may see in the following validation exercises to the different numerical treatment of collisional growth.

In this section, we choose a column model set-up that is supposed to produce results that are similar to box model results. For this, we initialise the default DSD in all GBs of the column and use periodic boundary conditions. In LCM1D, different SIP ensemble realisations of this DSD are initialised in each GB.

The deterministic BIN1D model predicts identical DSDs in all GBs, as in each GB the divergence of the sedimentation flux is zero. Hence, for this specific set-up, the BIN1D results attained are identical to those of a corresponding BIN0D model or the data of

In LCM1D, the combination of homogeneous initial conditions and periodic BCs results in statistically identical results across all GBs. However, the averaged results may not be the same as in LCM0D, as lucky droplets or SIPs

Within the LCM1D model, pure box model results can be obtained by switching off sedimentation (“noSedi”). Without sedimentation, the GBs of the column are not interconnected and the collisional growth process proceeds independently.

All figures related to the box model emulation set-up start their caption with the label “BoxModelEmul set-up”.

By default, we use

Moreover, we use the Long kernel

This subsection presents results obtained with the regular AON, i.e. with quadratic sampling of SIP combinations (“QuadSamp”) and 3D well-mixed assumption (WM3D). Sedimentation is switched on unless noted (for better discrimination from the “noSedi” cases, these simulations will be referred to as “full”).

BoxModelEmul set-up: temporal evolution of column-averaged moments

BoxModelEmul set-up: temporal evolution of column-averaged moment

Figure

Next, we discuss the sensitivity to further physical and numerical parameters.
Generally, we find faster convergence for higher moments than for

In a first simple step, we vary

In the noSedi simulations (panel a), the moment evolution is not affected by varying (

The second row shows a variation of

Contrarily, the full simulations (panel d) give nearly identical results independent of

How strongly SIPs are interconnected across GBs in LCM1D should also depend on geometrical properties of the column. In the next set-up, we investigate the

In an even more academic experiment, sedimentation is turned off, but SIPs are randomly redistributed inside the column after each time step (panel f) similar to

In bin models, the Smoluchowski equation, which is strictly valid only for an infinite volume and hence an infinite number of well-mixed droplets, is solved. Accordingly, only concentrations are prescribed in bin model algorithms. Neither

For our SIP-initialisation procedure,

In the present simulations, where SIPs with weights

The noSedi

In the discussion of the subsequent sensitivity studies, we refrain from showing time series of

BoxModelEmul set-up: this figure summarises results of many sensitivity studies for various AON versions and BIN simulations by displaying DNC after 1 h as a function of resolution

This subsection discusses the AON version with linear sampling. Both full simulations and noSedi simulations have been carried out.
The first row of Fig.

In the full simulations (solid lines), simulations converge for any

The

Next, we will discuss the results of the AON-WM2D version with explicit overtakes. Results are presented in Fig.

The dotted, green curve in panel (d) shows results for the version where only intra-GB overtakes are considered. Results are far off the benchmark curve; only for the smallest time step of

Overall, we can conclude that the feasibility and correct implementation of the WM2D version was demonstrated, with the caveat that overtakes have to be considered in the full column.
Checking for overtakes outside of the “own” GB can cause some computational overhead in implementing the WM2D version in higher-dimensional cloud models, which are typically parallelised. If the chosen time step for collection obeys the CFL criterion

As noted in Sect.

So far, all simulations were initialised with the same initial DSD and the same collection kernel, and the results have always been compared to the same BIN reference simulation.

Accordingly, in this section, we perform simulations with modified LWC

BoxModelEmul set-up:
sensitivities to the initial size distribution parameters LWC

In a first experiment, we increase LWC

In a next step, the characteristics of the initial DSD are more systematically varied for fixed

A more detailed presentation of simulation results with time series of the mean diameter,

As a last AON sensitivity study, the default Long kernel is replaced by the Hall kernel. Figure

So far, all reference BIN results were obtained with Wang's algorithm, using a time step

We find that Bott's algorithm converges for

Wang's algorithm, on the other hand, requires

Comparing the various collisional growth algorithms, we find that Bott's algorithm has the least requirements in terms of bin resolution and time step as we have converged results for

BoxModelEmul set-up: time series of a number of events in the various AON versions. Shown are the number of tested SIP combinations, of overtakes, of no collection, of a single collection and of a multiple collection in every time step. Additionally, the number of limiter cases, where

BoxModelEmul set-up: a number of events for various AON versions for the parameter set-up given in the text.

Now, we turn the attention to an algorithm profiling of the various AON versions.

Figure

Moreover, the workload per time step is constant in both WM3D versions and determined solely by

In the table, the ratios

Finally, we focus on the WM2D version (block no. 2). Here, the sum of

Note that the relative occurrence frequency of

Figure S14 demonstrates that all five AON simulations converge and show a basically identical time evolution of

Several of the above findings may hold only for the specific set-up used here.
To put the findings into a broader context, we next derive scaling relations for basic numerical quantities and, in particular, discuss their sensitivity to the time step and the number of SIPs.
For a simplified presentation, we limit ourselves to the regular and LinSamp version and assumed converged simulation results and no limiter events. Moreover, we assume that an increase in

For the following basic quantities we have

Accordingly,

The box model emulation simulations presented in Sect.

HalfDomLinDec set-up: temporal evolution of

We initialise droplets in the upper half of a

Figure

HalfDomLinDec set-up: vertical profiles of moments

Figure

In the upper half,

HalfDomLinDec set-up: size distribution

Figure

For a cleaner presentation, AON-LinSamp results were not shown in Figs.

Overall, the agreement between the four model versions is remarkable given the completely different numerics of the Eulerian and Lagrangian approach.

Next, the vertical resolution

It follows that the results of the AON-WM2D version should be independent of

Given a constant column height

EmptyDom set-up: vertical profiles of moments

In this section, the

Over time the column fills with droplets, a distinct size sorting is established and DSDs at a specific altitude are expected to be rather narrow.
Hence, choosing a vertical resolution that is too coarse may result in overestimating collections as the droplets are not supposed to be well-mixed within such deep GBs.
In such a case, the AON-WM2D version has a conceptional advantage as it does not assume well-mixedness in the vertical direction. The chosen set-up specifically aims to demonstrate the possible improvement by this.
Again, the further parameter settings are

Figure

EmptyDom set-up: temporal evolution of

Figure

EmptyDom set-up: temporal evolution of column-averaged moments

Fig.

The

This undesired

Collection, i.e. the coalescence, accretion and aggregation of hydrometeors, is an important process for the development of precipitation in liquid-, mixed- and ice-phase clouds, respectively. The correct representation of these processes in cloud microphysical models is, therefore, of utmost importance. In this study, we investigated and validated the representation of collection in LCMs, a relatively new approach that uses simulation particles, so-called SIPs or super-droplets, to represent cloud microphysics.

This study is a continuation of U2017, in which we analysed various representations of collisional growth algorithms in LCMs using zero-dimensional box model simulations. Here, this analysis is extended to one-dimensional column simulations that allow the effects of sedimentation to be explicitly considered. This study focuses on the AON algorithm

Furthermore, two variants of AON-WM3D are tested that differ in the number of SIP combinations that need to be tested during collection. In its simplest form, AON-WM3D depends quadratically on the number of SIPs since every SIP may interact with any other SIP inside a GB (QuadSamp). Additionally,

All results are compared to established Eulerian bin model results

As a first step and link to U2017 simulations, box model simulations are emulated in the column model. This is done by initialising each GB of the column with the same droplet size distribution and applying cyclic boundary conditions at the surface and the top. By using this framework, we were able to show that sedimentation increases the model convergence rate significantly compared to box model simulations without sedimentation; i.e. fewer SIPs per GB are required in the column model. The reason for this behaviour is that the largest and hence fastest-falling droplets are no longer confined to the same GB and to the same potential collection partners, which increases the ensemble of potential collection partners. A similar observation has been made by

In general, a remarkably good agreement of the LCM results with the bin reference has been found for all AON versions (regular AON, AON-WM2D and AON-LinSamp). AON-LinSamp results are only slightly worse compared to regular AON simulation of the same time step and SIP number. However, these stronger restrictions on the time step do not at all outweigh the computational benefit gained by the favourable linear computational complexity, making the LinSamp version the preferred choice if computation time is a critical factor. In an operational setting, the QuadSamp approach is a valuable alternative to LinSamp as long as the number of SIPs is not prohibitively high.

We further compared the computational requirements for the WM2D and WM3D implementations of AON. We found that WM2D requires checking for overtakes in the entire column, not only in the GB in which the SIP is located, as is the case for WM3D. However, this seeming disadvantage is turned into an advantage, since only a minority of SIPs overtakes other SIPs. Accordingly, the overall number of calculations necessary for the application of WM2D is reduced compared to WM3D. The physical reason for this effect is the typical bimodal structure of droplet spectra, which consist of only a few large droplets that sediment and collect other droplets efficiently, while the remaining droplets are usually too small to sediment and collect other droplets.

Finally, we applied the various AON versions to two more realistic column cases. While both cases use a prescribed inflow of droplets from the top, the first case is initialised with a linearly increasing liquid water content, and the second case is completely devoid of any initial droplets. Overall, the agreement of AON-regular, AON-WM2D, AON-LinSamp and the bin references is remarkable. Only in the second case, which is designed to be heavily prone to size-sorting, is a dependence on the vertical grid spacing detectable for WM3D and the bin reference, which both assume droplets to be well-mixed within a GB, while the WM2D results are found to be completely independent of the vertical grid spacing.

In all AON variants, simulation results converge for fairly large time steps

All in all, this study has shown that the representation of collisional growth in LCMs using AON successfully reproduces established Eulerian bin results. This ability, of course, depends foremost on the number of SIPs and the applied time step as already indicated in previous zero-dimensional box model studies. Compared to these zero-dimensional studies, the application of an LCM in a column decreases the required number of SIPs significantly. The consequently lower computational costs raise hopes to use LCMs more frequently in large-scale, multidimensional models in the future.

This Appendix presents pure sedimentation test cases that are suited to demonstrating that minor differences are introduced by the different numerical treatment of the sedimentation process.
Two simple set-ups with an influx of an exponential DSD with

Pure sedimentation test case: comparison of BIN and LCM (solid) advection. BIN uses either MPDATA (dashed) or first-order upstream scheme (dotted). EmptyDom

Pure sedimentation test case: comparison of BIN and LCM advection. EmptyDom set-up with an exponential distribution with

Moreover, we tested the sensitivity to

The source code of the Lagrangian column
model is hosted on GitHub (

The supplement related to this article is available online at:

SU designed the study, programmed the Lagrangian column model, carried out the simulations and wrote most parts of the paper. FH discussed the results with the first author and wrote the introduction and conclusions. A first code version and preliminary results were obtained during the master's thesis of ML.

The authors declare that they have no conflict of interest.

We thank Lian-Ping Wang and Andreas Bott for providing box model versions of their collection algorithms. The first author thanks Jan Bohrer (Tropos Leipzig) for carefully examining the AON code and spotting the bug mentioned in Sect.

The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Samuel Remy and reviewed by Sylwester Arabas, Shin-ichiro Shima, and one anonymous referee.