A key constraint of particle-based methods for modeling cloud microphysics is the conservation of total particle number, which is required for computational tractability. The process of collisional breakup poses a particular challenge to this framework, as breakup events often produce many droplet fragments of varying sizes, which would require creating new particles in the system. This work introduces a representation of collisional breakup in the so-called “superdroplet” method which conserves the total number of superdroplets in the system. This representation extends an existing stochastic collisional-coalescence scheme and samples from a fragment size distribution in an additional Monte Carlo step. This method is demonstrated in a set of idealized box model and single-column warm-rain simulations. We further discuss the effects of the breakup dynamic and fragment size distribution on the particle size distribution, hydrometeor population, and microphysical process rates. Box model experiments serve to characterize the impacts of properties such as coalescence efficiency and fragmentation function on the relative roles of collisional breakup and coalescence. The results demonstrate that this representation of collisional breakup can produce a stationary particle size distribution, in which breakup and coalescence rates are approximately equal, and that it recovers expected behavior such as a reduction in precipitate-sized particles in the column model. The breakup algorithm presented here contributes to an open-source pythonic implementation of the superdroplet method, PySDM, which will facilitate future research using particle-based microphysics.

The superdroplet method (SDM) for cloud microphysics is a high-fidelity particle-based (Lagrangian) representation of aerosols and hydrometeors that offers notable advantages over traditional bulk and bin microphysics schemes. Particle-based methods were initially used in atmospheric simulations to represent ice nucleation

Many implementations of the SDM do not include the process of collisional breakup of droplets. Not only is collisional breakup a highly uncertain process in existing bin and bulk parameterizations

This superdroplet-conserving breakup implementation draws inspiration from an analogous “superparticle” representation of phytoplankton

The contents of this paper proceed as follows: Sect.

Two colliding liquid hydrometeors in the atmosphere can break up via several physical pathways, including filament, sheet, and disk breakup

Conceptual view of a real filament breakup event

The superdroplet-conserving method of collisional breakup is illustrated in Fig.

Diagram of the Monte Carlo decision pathway during a collision–coalescence–breakup event in the proposed algorithm.

The proposed breakup algorithm unifies the representation of collisional coalescence and breakup and builds on the original coalescence Monte Carlo steps in

All superdroplets within a cell are randomly ordered in a list of non-overlapping pairs

The probability of collision between droplets

In the original SDM, particles coalesce as long as

Compute the dynamic that occurs: coalescence, breakup, or bounce (nothing). A second uniform random number

Sample a fragment size

Finally, updating of multiplicities and attributes proceeds based on the selected dynamic, number of collisions, and sampled fragment size (if applicable).

This method of breakup allows for the splitting of a coalesced transition state into a non-integer number of fragments (

The presence of a breakup deficit in the case where

In order to validate the proposed Monte Carlo algorithm which encompasses both coalescence and breakup, we compare it against an analytical solution to the generalized stochastic collection equation (SCE, or Smoluchowski equation). The approach is similar to that of

The deterministic solution to the SCE relates the evolution in time of the ratio

Two caveats are involved in comparing SDM results against SCE solutions in this test setup. First, the constant collision kernel admits collisions of same-sized particles, whereas collisions with a single superdroplet are not included in the present SDM implementation. This discrepancy should diminish with increasing numbers of superparticles (and hence decreasing values of multiplicities) down to zero for a one-to-one simulation with multiplicities of unity. The second caveat involves the assumption of breakup resulting in a only a single fragment size,

We analyze three comparison cases in a zero-dimensional box setting: breakup only (

Coalescence rate (

Mean (solid line) and standard deviation (shading) for the time evolution of the breakup-only dynamics, including the analytical solution of

Mean (solid line) and standard deviation (shading) for the time evolution of the coalescence-only dynamics, including the analytical solution of

Mean (solid line) and standard deviation (shading) for the time evolution of the coalescence–breakup combined dynamics, including the analytical solution of

Figures

In Fig.

To demonstrate the behavior of a particle population under the proposed breakup algorithm, we focus here on sensitivity studies in a zero-dimensional box setting and in a one-dimensional rain-shaft setup. The simulations presented in this section use a geometric collision kernel, where the rate of collisions

The coalescence efficiency is specified to be either a constant value (for sensitivity studies) or the empirical coalescence efficiency of

The zero-dimensional box simulations include collisional-coalescence and collisional-breakup dynamics only. The droplet size distribution is initialized to an exponential distribution in mass

Particle size distributions are displayed as the number distribution or as the marginal mass distribution

First we investigate the sensitivity of the PSD evolution to the coalescence efficiency, using four values of a constant-valued efficiency

Particle size distribution with varying coalescence efficiencies under a geometric collision kernel after 100 s

By contrast, the PSD for the Straub 2010 parameterization of

Sensitivity to fragmentation function of PSDs following collisions with a geometric kernel and fixed coalescence efficiency of

Next we consider the PSD evolution when the coalescence efficiency is held fixed at a constant value and the fragmentation function is varied. In Fig.

When the fragment size is sampled from an exponential distribution (Fig.

Finally, we consider an empirically derived coalescence efficiency and corresponding fragmentation function whose parameters depend on the colliding droplet properties

Initial PSD (dashed black) and PSDs following collisions with a geometric kernel, the Straub 2010 collection efficiency, and the Straub 2010 fragmentation function

Initial PSD

This empirical parameterization provides an additional opportunity for validation of the breakup algorithm on top of the analytical results presented in Sect.

Next we consider the impact of collisional breakup in a one-dimensional warm-rain setting that includes condensation/evaporation (including aerosol activation/deactivation), collisions, and transport of particles within the column through advection and sedimentation/precipitation.
These 1D simulations are based on the kinematic framework of

The test cases demonstrated here include a no-breakup case, a property-independent breakup case where the coalescence efficiency is fixed and fragment sizes are sampled from a fixed distribution, and the particle-property-dependent empirical coalescence efficiency and fragmentation parameterizations from

The mixing ratio of cloud droplets (activated droplets of no more than

Hydrometeor concentrations without breakup

Vertically averaged particle size spectra (as a marginal mass distribution) at four times selected from the 1D rain-shaft simulations for each set of dynamics (colors).

When property-independent breakup is included, a higher concentration of cloud-sized droplets persists at cloud base, and the surface precipitation is delayed and spread out relative to the case with no breakup. This behavior indicates that rain droplets favorably break up within the cloud and especially near cloud base, fragmenting into smaller cloud droplets (the mean of the fragment size distribution is

The empirical property-dependent breakup case using the

The relatively short updraft time and simple one-dimensional representation of this setup produce a short-lived cloud that is precipitating for only a few minutes. The likelihood of breakup in the Straub parameterization is strongly correlated with the size of the colliding droplets, with

Collisional dynamic rates for the one-dimensional case with

Figure

Aerosol processing rates for (left to right) no breakup, property-independent breakup, and

All three cases display similar rates of collision and coalescence, with the highest of these rates occurring in the cloud among rain droplets and below cloud base among precipitating rain droplets. In the no-breakup case, every feasible collision results in a coalescence, and the breakup rate is zero. When property-independent breakup is included, the time–space distribution of the breakup rate is nearly identical to that of the collision rate. This trend is consistent with the use of a uniform coalescence efficiency

As noted in the discussion of Fig.

The property-independent and no-breakup cases have nearly identical behavior in aerosol processing, consistent with the correspondence between their hydrometeor concentrations and collision process rates. In all three cases, a few superdroplets at cloud top encounter humidity close to their critical supersaturation, which results in the “ripening” processes of fluctuation between an activated and deactivated state due to competition when the supersaturation is insufficient to activate all aerosols (e.g.,

This work presents a superdroplet algorithm for collisional breakup that is scalable in avoiding creation of new superdroplets and physical in its ability to produce results in a box and one-dimensional setting that are consistent with the expected reduction and delay in rain formation. Furthermore, the algorithm produces hydrometeor populations and process rates that differ between a property-independent approach (with a fixed coalescence efficiency and fixed fragment size distribution) and a property-dependent approach using empirical parameterizations. These differences indicate the importance of random, stochastic events in warm-rain microphysics, a trait which has also been documented in other microphysical phenomena such as giant cloud
condensation nuclei

This work provides the basis for a more complete representation of microphysical processes in particle-based simulations. For instance, a superdroplet representation that considers ice-phase hydrometeors and properties could probe processes of secondary ice production such as fragmentation and freezing of supercooled water droplets upon collision with ice

In implementing collisional breakup for superdroplets, we suggest imposing a few limiters to enforce physical constraints and maintain stability of the code. If the user-selected time step for the SDM implementation is too large, collisional breakup may quickly become a runaway process with superdroplet multiplicities increasingly rapidly and unphysically, leading to numerical overflow. As an example, suppose a droplet of multiplicity

In addition, the process of collisional breakup is physically constrained such that the resulting superdroplet (the fragment) volume should not exceed the volume of the coalesced transition state, nor should it drop below a realistic size for a liquid water droplet (molecule scale, for instance). These physical constraints can be imposed by setting a minimum and maximum allowable fragment size resulting from breakup.

The first of these constraints can then be imposed during computation of

Sampling a fragment size

Suppose the unnormalized fragmentation function

Next, to sample a single fragment size

Next, the fragment size is chosen by sampling at random from the CDF of the mode

We note that there are several methods of sampling a fragment size from distributions composed of several modes. The presented implementation is used in generating results in this work and is included as one such example.

Implementation of this breakup algorithm in the SDM is available at

EdJ led the code development, generation of results, interpretation, and writing. BM contributed to code development and the underlying methodology. OB contributed to the algorithm and code review, to validation against the analytic solution, and to porting the code for the GPU backend of PySDM. AJ contributed to the underlying methodology and interpretation of results. SA contributed to the code development and interpretation and leads the maintenance of the PySDM codebase.

At least one of the (co-)authors is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Tapio Schneider and Shin-ichiro Shima for feedback, insights, and discussion. The study benefited from peer-review comments provided by Axel Seifert, Christoph Siewert, and two anonymous reviewers. Additional thanks go to Piotr Bartman for ongoing support of PySDM.

Emily de Jong was supported by a Department of Energy Computational Science Graduate Fellowship. Sylwester Arabas and Oleksii Bulenok were supported by the Polish National Science Centre (grant no. 2020/39/D/ST10/01220). This research was additionally supported by Eric and Wendy Schmidt (by recommendation of Schmidt Futures) and the Heising-Simons Foundation.

This paper was edited by Po-Lun Ma and reviewed by two anonymous referees.