Numerical convergence of the collision–coalescence algorithm used in Lagrangian particle-based microphysics is studied in 2D simulations of an isolated cumulus congestus (CC) and in box and multi-box simulations of collision–coalescence. Parameters studied are the time step for coalescence and the number of super-droplets (SDs) per cell. A time step of

Particle microphysics (also known as Lagrangian particle-based microphysics, Lagrangian cloud models, or super-droplet microphysics) is a class of Lagrangian methods for numerical modeling of cloud microphysics that has been developed in the last decade

A recent study has found discrepancies in rain production between different particle models that use AON, although the models agree well in modeling condensational growth

We begin with a study of numerical convergence of AON in simulations of collision–coalescence in a box model. Although a similar study was done by

Next, we study convergence of AON in multi-box simulations. This simulation type, introduced by

Finally, we study numerical convergence of AON in 2D simulations of isolated cumulus congestus. It is a much more realistic simulation than was used before for convergence tests of AON. The same processes are included as in an LES, with the only difference being smaller dimensionality. The reason why we use 2D instead of 3D is that this decreases the required computational power and memory size, allowing us to study a broader range of parameters. AON is a stochastic algorithm, so it gives different realizations of collision–coalescence in independent simulation runs. LES runs often also differ due to random differences in initial conditions. These differences in initial conditions include random perturbations of thermodynamic variables (e.g., temperature and humidity) and random initialization of SD attributes. Stochasticity in AON, as well as in initial conditions, leads to differences in flow fields, which can strongly impact results. To isolate the effect of stochasticity of AON from stochasticity of initial conditions, we use the same initial conditions in ensembles of simulations. Moreover, to facilitate studying convergence of AON, we use the same flow field for different simulations. This way, flow field is not affected by different realizations of AON. We also perform reference “dynamic” simulations with differences in initial conditions and without a prescribed flow field. This allows us to assess the importance of stochasticity of AON relative to other sources of stochasticity in LESs.

We start with a presentation of the particle microphysics scheme, with emphasis on AON and on the SD initialization procedure (Sect.

In Lagrangian particle microphysics methods, particles in the air (aerosols, haze particles, cloud droplets, raindrops, ice particles) are represented by computational objects called super-droplets. In most cases, each SD represents numerous identical real particles. The number of real particles an SD represents is called its multiplicity

Multiplicities and radii of SDs are initialized from a prescribed initial size distribution. In this section, we describe common methods for doing this. The prescribed radius can either be wet or dry radius. We denote the initial number of SDs per grid cell with

In one initialization method, all SDs have the same multiplicities and their initial radii are drawn from the distribution using inverse sampling

Another method of initialization is to divide the initial distribution into bins of equal sizes on a logarithmic scale. We denote the number of bins with

Instead of using the algorithm for finding bin edges, one can simply prescribe them. We call this method “constant SD fixed-init”. In this method, no SDs are added to represent the part of the distribution to the right of the largest bin.

The AON algorithm, developed by

In some implementations of AON, the number of super-droplet pairs tested for coalescence per time step is equal to

We model collision–coalescence of droplets in a well-mixed box. For simplicity, we use

Three types of collision–coalescence models are compared: the AON algorithm, one-to-one simulations, and the stochastic coalescence equation (SCE). AON is discussed in Sect.

One-to-one and AON simulations are stochastic. We run an ensemble of one-to-one simulations and ensembles of AON simulations for different values of

First, we check how numerical time step

Next we check how results are affected by the number of SDs for

Mean and standard deviation of the mass density function

Differences in

Differences between

Initialization of droplet radii in Lagrangian particle microphysics is often stochastic

As in Fig.

Box simulations of collision–coalescence with AON show convergence of

Box model tests of the mean DSD in the AON algorithm were previously done by

Using the singleSIP initialization,

The AON implementation from the

The simulation setup is the same as in box simulations, but the domain is divided into

Mean mass density function at the end of multi-box simulations from

We run ensembles of one-to-one and AON simulations for differing numbers of cells. The ensemble size is

In line with the conclusions of

In this section, we analyze AON in a two-dimensional simulation of an isolated cumulus congestus cloud. Conclusions about convergence of AON in box and multi-box simulations that were presented in the previous sections do not necessarily apply to higher-dimensional simulations or simulations that include more processes affecting the DSD (e.g., condensational growth). For example,

The CC simulations are done with the University of Warsaw Lagrangian Cloud Model (UWLCM). UWLCM is an LES tool that allows 2D and 3D simulations with Lagrangian particle (or Eulerian bulk) microphysics. Thermodynamic variables (potential temperature, water vapor mixing ratio, velocity) are modeled in an Eulerian manner. The Lipps–Hemler anelastic approximation

We use an isolated cumulus congestus modeling setup that was one of the cases studied at the International Cloud Modeling Workshop 2020. It is an adaptation of the setup developed by

We use a 2D instead of 3D LES because it allows us to study much larger values of

Typically, in LESs there is a random perturbation of initial conditions, e.g., of temperature and humidity. In LESs with particle microphysics, initial conditions may also differ in SD attributes because they are often randomly initialized. This randomness in initial conditions leads to differences in results between simulation runs, independently of AON. To understand the role of AON, we isolate its effect by comparing dynamic and kinematic simulations. In dynamic simulations, the pressure equation is solved, meaning that different realizations of microphysics lead to different flow fields. In kinematic simulations, the flow field is prescribed. Our strategy is to run an ensemble of dynamic simulations, denoted with D, with random differences in initial conditions. We consider this ensemble to be a control group because this is the way an LES is usually done. From dynamic simulations, we select three realizations: one with little rain, one with medium rain, and one with a high amount of rain (LR, MR, and HR, respectively). Flow fields from these simulations are used to run ensembles of kinematic simulations. In kinematic simulations, initial conditions do not change within an ensemble. Therefore, any variability within a kinematic ensemble is solely caused by AON. Our goal is to study convergence of precipitation, which is a variable sensitive to modeling collision–coalescence. We also study convergence in simulations without collision–coalescence to make sure that convergence in simulations with collision–coalescence is only related to AON. The number of simulations of each type is given in Table

To generate velocity fields for kinematic simulations, we run numerous dynamic simulations. Our goal is to find three velocity fields that would give significantly different amounts of rain. In a single dynamic simulation, the amount of precipitation depends not only on the realized flow field, but also on the realization of the AON algorithm. This means that rain from a single dynamic simulation is not representative of the expected amount of rain from a series of simulations with the same velocity field. To be sure that we selected velocity fields that will give different amounts of rain, first we chose a candidate velocity fields based on the amount of rain in the single dynamic run, and then we ran 20 kinematic simulations and used the average from these 20 simulations as the expected amount of rain for a given velocity field. Note that these 20 simulations were just a preliminary ensemble to estimate the expected amount of precipitation and that the final number of simulations was much larger (it is given in Table

Frequency histogram of accumulated surface precipitation (

Besides using different flow fields, we study sensitivity to the model of SGS advection of SDs to the SD initialization method, and we run simulations without collision–coalescence. A list of all simulation types is given in Table

Configuration of CC simulations. Columns, from left to right, show the configuration name, type of flow field (dynamic or kinematic), collision–coalescence on/off flag, the method for SD initialization, and the model of SGS advection and aerosol relaxation method (discussed in Sect.

In this section we discuss time series of general cloud properties in the D, LR, MR, and HR scenarios (with collision–coalescence). This is done to give readers an idea about how the modeled cloud develops. Time series of cloud-top height (CTH), cloud cover (cc), cloud water path (CWP), rainwater path (RWP), and surface precipitation are plotted in Fig.

Time series of CTH, cc, and CWP are smoother in dynamic than in kinematic simulations. In dynamic simulations, there are differences between simulation runs at the moment when the cloud starts to develop. When averaged over simulation runs, the results are smooth. In kinematic simulations, cloud develops in a very similar way in all simulations within an ensemble. Therefore, the ensemble average resembles a single dynamic simulation in that it changes significantly at short timescales. This illustrates that, unsurprisingly, CTH, cc, and CWP are more sensitive to the airflow than to the realization of collision–coalescence.

In all scenarios, cloud starts to develop at around

Simulation ensemble sizes for all simulation types (defined in Table

Time series of ensemble averages of cloud-top height, cloud cover, cloud water path, rainwater path, and surface precipitation for D, LR, MR, and HR scenarios with

To reliably study convergence of the collision–coalescence algorithm, we first need to make sure that simulations without the collision–coalescence process have converged. The time step for condensation used in all simulations,

Note that the relative dispersion around

Profiles of cloud droplet concentration (top row), cloud droplet mean radius (center row), and relative dispersion of cloud droplet radius (bottom row) from simulations without collision–coalescence. Profiles are averaged over cloudy cells, over the time interval between

From now on, only simulations with collision–coalescence will be discussed. Profiles of precipitation flux for different numbers of SDs are shown in Fig.

Profiles of precipitation flux in simulations with collision–coalescence. Profiles are averaged over all cells, over the time interval between

Figure

Ensemble mean and standard deviation of accumulated precipitation at the end of a simulation against time step for coalescence in four scenarios of a CC simulation with

Mean surface precipitation for differing numbers of SDs is shown in Fig.

Ensemble mean of precipitation against number of super-droplets for four scenarios: D, LR, MR, and HR. In

Changes in

The increase in

Probability density function of rainwater mass in cloudy cells at four moments in time, averaged from the HR simulation ensemble.

Ensemble standard deviation

The standard deviation of precipitation for differing numbers of SDs is shown in Fig.

In kinematic simulations (panels b–d) the standard deviation of precipitation is more sensitive to

Collision–coalescence in particle microphysics is sensitive to the way SD attributes are initialized. Therefore, the way precipitation changes with the number of SDs could depend on SD initialization. To check this, we test convergence for three types of SD initializations that were introduced in Sect.

The

Mean

In multi-box simulations, mixing of droplets between cells helps achieve convergence of collision–coalescence modeling. In CC simulations discussed so far, intercell mixing was caused by the resolved-scale motion and by sedimentation, but there was no SGS motion of SDs. Here, we consider simulations in which SGS velocity of SDs is modeled using the OU method, which should make mixing more efficient. In the OU model, we assume a constant and uniform TKE dissipation rate of

Mean and relative standard deviation of accumulated precipitation against the number of SDs for simulations with (red) and without (black) the OU model of SGS motion of SDs.

Our study shows that using particle microphysics it is more difficult to reach numerical convergence of precipitation in cloud simulations than it is to reach convergence of mean DSD in an ensemble of box or multi-box simulations of collision–coalescence. In general, convergence requirements are less strict in strongly precipitating clouds than in lightly precipitating clouds.

It is relatively easy to have convergence with

It is more difficult to reach convergence with the number of SDs per cell. In box simulations, mean DSD converges for

It is not clear why convergence is slower in CC than in box simulations. In CC, convergence of mean precipitation coincides with convergence of the spatial distribution of rain. This may suggest that mean precipitation is dependent on the spatial distribution of droplet sizes, probably because of interaction between cells. However, in multi-box simulations we observe that intercell mixing helps reach convergence. Increasing the rate of intercell mixing in CC by using an SGS model does not help with convergence. This does not necessarily indicate that intercell mixing is not important for convergence in CC. It is possible that the increase in intercell mixing caused by the SGS model is small in relation to intercell mixing caused by resolved eddies and by sedimentation. Precipitation is sensitive to the super-droplet initialization procedure. In this study initial radii were almost evenly distributed on a logarithmic scale. If droplet radii are randomly drawn from the initial distribution, it is more difficult to reach convergence with

Variance of precipitation in an ensemble of cloud simulations decreases with

In multi-box simulations, incompressible isotropic turbulence is modeled as a sum of random Fourier modes. The model is similar to that used in

Validity of the periodic model is tested by comparing pair separation statistics with results from the non-periodic model of

Pair separation statistics from the periodic and non-periodic synthetic turbulence models for different values of the TKE dissipation rate. Solid lines show the ensemble mean, and shading shows 1 standard deviation interval.

Cloud simulations were done using UWLCM, and box and multi-box simulations were done with

The supplement related to this article is available online at:

PZ and PD conceived the idea of the study. Simulations, analyses, and data visualization were done by PZ for CC simulations and by PD for box and multi-box simulations. All authors contributed to the paper. Funding was secured by PD and HP.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We thank Gustavo Abade for help with the periodic synthetic turbulence model and its description. We gratefully acknowledge Poland's high-performance infrastructure PLGrid (HPC centers: ACK Cyfronet AGH, PCSS, CI TASK, WCSS) for providing computer facilities and support under computational grant no. PLG/2022/015886. The calculations were made with the support of the Interdisciplinary Center for Mathematical and Computational Modeling of the University of Warsaw (ICM UW) under computational grant no. GR84-48.

This research has been supported by the Polish National Science Center (grant nos. 2018/31/D/ST10/01577 and 2016/23/B/ST10/00690).

This paper was edited by Simon Unterstrasser and reviewed by Yign Noh and one anonymous referee.