Substantial differences exist between conventional Eulerian models for cloud microphysics. Excessive degrees of freedom within one- and two-moment bulk schemes, the least expensive of existing microphysics models, allow for varying definitions of condensate categories, as well as the parametrisations for how they interact with one another, and their assumed size distributions . Increasing in sophistication, even the most conventional spectral bin schemes still vary in how they define condensate categories and their parametrisations. They also introduce differences in numerical implementations, for example to tackle numerical diffusion , such that even if two bin schemes solve the same equations they could calculate different results. These differences matter for more than just microphysics.

The differences between conventional Eulerian, bulk and bin, models have a wide-range of macrophysical ramifications. In global storm resolving models e.g., only changing fall velocity parameters in one- and two- moment bulk schemes can substantially alter the relationship between moisture and convection, which has consequences for the Madden–Julian oscillation and the radiation balance of the entire atmosphere . Varying definitions of condensate size distributions in bulk schemes are also problematic because they alter shallow-cloud fraction and albedo, even in non-precipitating clouds, and their underlying assumptions, regarding the functional form of each distribution and distinguishing between “rain” and “cloud” condensates, are known to be inconsistent with observations and with bin scheme models of cloud microphysical processes e.g.. When Large-Eddy Simulations (LES) and observations are compared, differing precipitation patterns are frequently attributed to differences in microphysics schemes e.g., and bin schemes have been shown to be just as disparate as bulk ones . Indeed there is extensive literature showing the many ways in which microphysics can appreciably affect larger-scale dynamics e.g., and thus as long as there are differences between microphysics schemes we can expect there to be differences in macrophysical outcomes. Differences inevitably arise from gaps in our knowledge of cloud microphysics, however, the differences highlighted above are not caused by knowledge gaps, but rather by “model uncertainty”, uncertainty inherent to the way that conventional Eulerian cloud microphysics models represent what we already understand .

The Super-Droplet Method SDM; is a Lagrangian particle method for modelling cloud microphysics where the condensate population is represented by “superdroplets”. Superdroplets have the same properties as real condensates as well as an extra one, the multiplicity, which states how many real condensate particles that superdroplet stands for. In a somewhat analogous way to bulk schemes where the accuracy of the condensate size distributions is bounded by the number of moments, and to bin schemes where it is bounded by the number of bins, it is the number of superdroplets which limit the accuracy of SDM. The major limitation of SDM is its computational cost, both in terms of its memory consumption and its number of computations, which presumably is the reason that LES using SDM have so far been limited to domains O(10km) or smaller (to the best of our knowledge, the largest LES with SDM to-date are and ). Historically, Lagrangian particle-based microphysics schemes have struggled computationally e.g. whilst bin and bulk schemes have advanced not only cloud process understanding but also numerical weather and climate prediction in ways that would not otherwise have been possible, by working within the computational constraints of the time . Nevertheless, advances in high-performance computing combined with SDM's embarrassingly parallel algorithms, make SDM a promising model for cloud microphysics on current and upcoming computer architectures , potentially changing the perception that Lagrangian particle-based cloud microphysics is too computationally expensive.

Part of what makes SDM appealing is that SDM eradicates the biggest sources of conventional model uncertainty. Whilst SDM is not free of modelling assumptions and hence still suffers from its own sources of model uncertainty , SDM makes no assumptions about condensate categories or size distributions unlike bulk schemes, and it does not suffer from same intrinsic problems as bin schemes, such as numerical diffusion and the “curse of dimensionality” . Furthermore, SDM has an important convergence property. As the number of superdroplets increases, a simulation tends towards what we expect from explicit individual particle simulations , and the mean of many simulations of its collision-coalescence algorithm approaches the solution to the Smoluchowski equation . This is in stark contrast to bulk schemes, where increasing refinement does not converge to a generic solution, and also to bin schemes which model collisions by solving the Smoluchowski equation directly. Bulk schemes therefore suffer from the limitations of that equation; namely its mean-field approximation assumes that the locations of differently sized droplets are uncorrelated , even though this is only variably true , and prevents modelling “lucky drops” which may be crucial for precipitation onset . In SDM, the stochastic collision-coalescence algorithm does not make these assumptions and can represent lucky drops . Finally, the link between our fundamental (particle-based) understanding of microphysical processes and their representation in SDM is clearer and more direct than in Eulerian microphysics models, meaning that variations between SDMs hold a stronger relation to differences in our understanding of the underlying physics.

We have created a novel computational implementation of SDM, Cleo. Our goal is to make a simplistic, ordinary representation of warm-cloud microphysics according to SDM which is well-suited to high-performance computers. Consequently we hope to make SDM feasible for large-enough-domain LES to resolve the interactions between warm-cloud microphysics and shallow mesoscale cloud organisation O(100km). In the companion to this paper we described the fundamental computational design of our SDM and its computational performance . In this paper we document Cleo's numerical methods for warm-cloud microphysics and demonstrate their behaviour in known test-cases. Since the link between the numerical methods and fundamental theory is so clear in SDM, by carefully describing our numerical methods we assist the physical interpretation of studies using Cleo and comparisons with other microphysics models.

Cleo models all the major microphysical processes of warm-clouds: condensation/evaporation, collisions between droplets and droplet motion. Most of the methods are already found in the literature, but in addition to documenting which of those Cleo adopts, we highlight particular novelties of our approaches and explain how the flexibility we incorporated reflects the uncertainties in our current understanding of warm-cloud microphysics. More specifically, we clarify the methods of and for modelling condensation/evaporation and extend them to include ventilation effects. For collision between droplets, Cleo can use the original collision-coalescence algorithm according to , however Cleo also facilitates various definitions for the collision kernel and the outcome of collisions between droplets. The breakup of raindrops as a consequence of collisions is included in and , but is neglected by the majority of SDMs (and some bulk and bin microphysics schemes), despite the fact it could be an important influence on the rate of evaporation in downdraughts . Cleo therefore offers an extended version of the collision-coalescence algorithm to permit the study of rebound and breakup.

In this paper the ith superdroplet is defined by its spatial coordinates, xi, its multiplicity, ξi, and its attributes, ai, wherein Ri and Ms,i are its radius and its mass of solute (aerosol), respectively. The total mass of each superdroplet is given by 1MT,i=Ms,i1-ρlρs,i+43πρlRi3, to account for the solute's volume whilst neglecting its change due to dissolution, where ρl and ρs,i are the densities of liquid water and the solute, respectively. a′i symbolises xi as well as all the superdroplet's intensive attributes, for example ρs,i.

Cleo's numerical methods for microphysics and droplet motion are found in Sects.  to ; Sect.  details how Cleo couples to a host dynamical driver; and the verifications of Cleo's numerical methods are found in Sect. . Within Sects.  to , Sect.  describes how Cleo models condensation/evaporation, including ventilation effects and an adaptive sub-time-stepping algorithm; Sect.  explains how we have extended the collision-coalescence algorithm of to include a framework for collisional breakup and rebound; and Sect.  explains how Cleo determines (super-)droplet motion.

We use Köhler theory to describe droplet growth due to evaporation/condensation, but also account for the enhancement of evaporation of moving droplets by including a ventilation factor. The change of the ith superdroplet's radius, Ri, is therefore 2RidRidt=fv(S-1)-aRi+bRi3Fk+Fd, where fv is the ventilation factor; S is the ambient saturation ratio of the grid-box; Fk and Fd are, respectively, the heat conductivity and vapour diffusion factors; aRi represents the effect of curvature on the saturation at a droplet's surface; and bRi3 accounts for the reduction in water vapour pressure due to the presence of solute. We use the same formulations for Fk, Fd, a and b as in . Supersaturation fluctuations caused by sub-grid-scale turbulence are not taken into account.

We obtain the ventilation factor, fv, by fitting the curve 3fv(R)=1+1α1Rβ1+1α2Rβ2-1 to the curve from , such that α1=6.954×107 m-β1, α2=1.069×103 m-β2, β1=1.963, β2=0.702. This gives similar values as given the terminal velocity parametrisation from but makes the size dependence more explicit whilst neglecting the small dependence of fv on pressure variations. Additionally we constrain fv≤20.0 to reflect the fact that droplets with a radius larger than approximately 3 mm have the same terminal velocity (see also Fig. ). As shown in Fig. , fv is negligible for droplets with radii less than about 0.1 mm, but can increase the evaporation rate of large droplets by an order of magnitude.

Careful consideration of the numerical methods for integrating Eq. () is required because this ordinary differential equation is stiff. We use an implicit Euler method with a Newton-Raphson root-finding algorithm similar to to ensure stability when integrating the modified version of the ODE in Eq. (). Our method saves computational cost by permitting a comparatively large default time-step (Δtcond∼1s) and very often keeping the number of Newton-Raphson iterations below three. Explicitly, given the time-step for condensation/evaporation, Δtcond, and the notation Ri(tn)=Rin, we make the approximation fv(R)≈fv(Rin) and let fv(Rin)=fv, the ODE is discretised as 40=Z-(Rin)2Δtcond-2fv[(S-1)-aZ-1/2+bZ-3/2]Fk+Fd; where Z=(Rin+1)2. The Newton-Raphson method is then used to find the real positive root of the polynomial, 5g(Z)=Z-(Rin)2Δtcond-2fv[(S-1)-aZ-1/2+bZ-3/2]Fk+Fd; such that the radius of each superdroplet at the subsequent time-step is given by Rin+1=+Z when g(Z)=0. We use g(Z) rather than the higher order polynomial in terms of +Z because it converges more rapidly – in fact in the limit a=b=0, the Newton-Raphson method can find the true root of Eq. () with one iteration. A reasonable initial guess close to the true solution for g(Z)=0 can make convergence fast and so we choose our first guess for Z to be (Rin)2 unless S>1+4a327b, in which case the superdroplet is already activated and we make the initial guess very large: at least 10-6m2.

There are up to three real roots of g(Z) and so we must ensure that the genuine solution for Rin+1 is converged upon. If the droplet was previously and is currently un-activated, and in an environment with a supersaturation less than its activation supersaturation, i.e. if 6S≤1+4a327b,andRin<3ba, then the uniqueness of the solution to g(Z)=0 is guaranteed. Likewise, if the time-step is small enough, 7Δtcond≤25b(Fk+Fd)2a2fv5ba, uniqueness is guaranteed (as in modulo the ventilation factor). In either of these cases we attempt two iterations of the Newton-Raphson method and then perform a standard local error test for convergence, whereby the method has converged if g(Z)<α+βgRin)2 for some absolute and relative tolerances, α and β, respectively

By default we choose α=0.01 and β=0.0, and the maximum number of iterations as 50.

Modelling collisions between droplets in Cleo can be done with or without collisional breakup and rebound. Collisional breakup during rainfall has been observed in situ and has been suggested as an important control on the timing and intensity of rain in LES . Since breakup converts large droplets into significantly smaller ones, its effect on the higher moments of the droplet size distribution can be appreciable e.g.. This motivates two alternative treatments for collisions in Cleo, schematised in Fig. , which are: a.

the original algorithm of which models only collision-coalescence,

In both treatments, the steps to determine whether or not a collision occurs is the same as in . All the superdroplets in a grid-box are first randomly paired with one-another. Then for each pair the probability that they collide is compared to a random number, ϕα, to determine whether or not a collision is enacted. The probability that two superdroplets collide, pα, is the probability that two real droplets collide, Pjk, scaled by a factor dependent on the superdroplets' multiplicities. Usually Pjk can be calculated given a collision kernel, Kjk, as 8Pjk=KjkΔtcollΔV, for a pair of droplets j and k, where Δtcoll and ΔV are the collision time interval and volume, respectively . The probability two superdroplets collide is then 9pα=max⁡(ξj,ξk)ns(ns-1)/2⌊ns/2⌋Pjk, where ns is the number of superdroplets in the collision volume, ΔV.

Cleo defines the collision probability flexibly to make it easy to switch between different formulations of Pjk. There are large uncertainties in determining the collision probability of a pair of superdroplets' because the collision kernel is so poorly constrained. As such, several formulations for Kjk exist, which differ for example in their treatment of turbulence e.g.. The function to calculate Pjk in Cleo can therefore be changed before compilation as long as it still obeys a set of technical constraints on the function's signature defined by a C++20 concept ; see also. There are two Pjk calculations implemented at the time of writing: one for Golovin's kernel , and the other for the hydrodynamic kernel with the formulation for the terminal velocity from and collision efficiencies from , as also done by . There is no default calculation for the collision probability in Cleo, the user must always specify it during compilation.

If a collision occurs, treatments (a) and (b) differ in their outcome. In the original algorithm, coalescence is assumed and the superdroplets are updated as in Sect. . In contrast, if a collision occurs in the extended algorithm, the last step of the original collision-coalescence algorithm is modified as shown in Fig. b. Rather than assuming a successful collision of two superdroplets results in coalescence, the attributes of the original superdroplets are used to determine if the collision causes rebound, coalescence, or breakup. We then modify the superdroplets to reflect the appropriate outcome: as in Sect.  for coalescence and as in Sect.  for breakup. In the event of rebound, the superdroplets remain unchanged.

Cleo has several options for how the attributes of colliding superdroplets determine whether the outcome of the extended algorithm is rebound, coalescence, or breakup. This is physically motivated since properties of droplets, such as their velocities, shapes, and masses, determine whether a collision has sufficient energy to result in coalescence or breakup instead of rebound, but exactly how such properties determine the outcome of a collision is disputed. Our algorithm can therefore easily interchange various formulations from the literature, at the time of writing from , combined with , and .

As an example, Fig.  demonstrates how the outcome of a collision is determined based on combined with . Let the pair of droplets j and k be re-labelled small, S, and large, L, based on their radii, RS and RL respectively. Then, according to , there are three possible regimes depending on the magnitude of the collision kinetic energy, TE, relative to the surface tension energy of the smaller and larger droplet, SS and SL respectively. Using the definitions: 10RS=min(Rj,Rk),11RL=max(Rj,Rk),12TE=23πρlRS3RL3RS3+RL3|vS,∞-vL,∞|2,13Si=4πσlRi2 for i∈{S,L}, where ρl is the density and σl is the surface tension of liquid water, these three regimes are:

when TE<SS either coalescence or rebound occurs,

We use a simple predictor-corrector method to model droplet motion. One of the most computationally expensive parts to Lagrangian microphysics is the motion of particles throughout the domain . However, since a superdroplet's location is not the true location of the droplets it represents, there is no imperative to precisely resolve a superdroplet's position. Indeed looking at the conceptual picture of SDM, perhaps the most suitable choice for superdroplet motion would be probabilistic, similar to . However, we save expense by choosing Heun's method, a second-order predictor-corrector method, with a simple linear interpolation of the wind velocity. This method by construction preserves the divergence of the flow field, and gives physically consistent results at a 1s time-step .

In summary, we integrate the equation for the coordinates xi of the ith superdroplet, 33dxi(t)dt=ui(xi,t), where the superdroplet's velocity, ui(xi,t), is 34ui(xi,t)=w(xi,t)+vi,∞(xi), and vi,∞(xi)=-vi,∞(xi)z^ is the superdroplet's terminal velocity, and w(xi,t) is the wind velocity, obtained by simple linear interpolation of the wind velocity at a grid-box's faces. In Cleo the terminal velocity is defined, analogously to the collision probability (Sect. ), by a set of technical constraints on the signature of the function which calculates it using a C++20 concept ; see also, meaning that different formulations can be easily interchanged. Also analogously to the collision probability, there is also no default calculation for the terminal velocity in Cleo, the user must always specify which one to use during compilation. The options currently available are depicted in Fig. . Using the notation xi(tn)=xin, and the time-step for superdroplet motion, Δtm, the predicted coordinates at the subsequent time-step are 35x̃in+1=xin+uin(xin)Δtm, which are corrected to 36xin+1=xin+uin(x̃in+1)+uin(xin)Δtm2.

This formulation for droplet motion could easily be extended to account for the effects of particle inertia and sub-grid scale turbulence, for example following , which would increase the fidelity of the particle trajectories, especially for large droplets whose inertial relaxation timescale is larger than Kolmogorov time scale.

Cleo is designed to run SDM microphysics concurrently to any host dynamical driver which it couples to for the computational implementation see. For a certain simulation, there are therefore two distinct grids composing the domain at play: Cleo's Eulerian grid and the Eulerian grid of the driver, and these are completely independent except during coupling when information is sent between the two. Cleo's grid can be any kind of Arakawa C-grid and is composed of grid-boxes which define the volumes of the domain and the necessary Eulerian thermodynamic and wind fields so that the Lagrangian superdroplets can enact microphysics and move around. These are: the temperature, pressure, water vapour mass mixing ratio, and liquid water mass mixing ratio defined at the centre of each of Cleo's grid-boxes, as well as the wind velocity defined on the faces of each grid-box. (The meridional wind component is defined at the centre of the faces perpendicular to north-south, and analogously for the zonal and vertical wind components.)

Cleo can be one-way or two-way coupled to a host dynamical driver. In both cases, at the start of each coupling time-step the host dynamical driver, via the coupler, communicates each of the necessary thermodynamic and wind fields to Cleo's grid-boxes. Cleo then enacts SDM microphysics and droplet motion, which potentially alters the temperature, water vapour and liquid water in each grid-box but not the pressure nor winds (e.g. condensation increases temperature and converts water vapour to liquid). In a two-way coupling Cleo then communicates the thermodynamic fields back to host dynamical driver, whereas in a one-way coupling it does not. As long as it can provide the required fields for Cleo, the host dynamical driver can be anything, from analytical formulae, to reading data from files, all the way to a fully-fledged numerical weather or climate model. Any potential advection of thermodynamic fields, momentum feedbacks, radiation or turbulence schemes are the responsibility of the particular host dynamical driver which Cleo is coupled to. For further details of Cleo's time-stepping, coupling or domain composition, readers should refer to .

This section presents three test-cases which demonstrate the behaviour of our numerical methods for condensation/evaporation, collisions between droplets, and droplet motion individually. Thereafter we present a final integrated test-case applying all three methods at once using the 1-D Kinematic Driver (KiD) modelling framework . Additionally, the Supplement to this paper contains two further validations, of droplet collisions and of droplet motion.

Cleo is a concise and fully-functioning novel implementation of SDM for warm-cloud microphysics. Our primary motivation for creating Cleo is to provide a computationally efficient, “standard” SDM, which we could use in large-enough-domain LES to decipher the interactions between warm-cloud microphysics and shallow mesoscale cloud organisation. Cleo models collisions between droplets and condensation/evaporation, including (de-/re-)activation, similarly to existing SDM implementations albeit with some differences as summarised below. Cleo models Lagrangian particle motion following the SDM of but with the ability to switch between different terminal velocity calculations. In this paper we detailed Cleo's numerical methods for SDM and verified their behaviour against known test-cases for condensation/evaporation, collisions between droplets, and droplet motion, and using an integrated test-case including all three processes in the 1-D KiD framework.

Our numerical methods for condensation/evaporation and collisions between droplets have subtle but noteworthy differences to existing SDM implementations. For condensation/evaporation we solve the Köhler theory ODE explicitly with a number of cost-saving measures including adaptive sub-time-stepping. Our method for condensation/evaporation is thus purposefully very similar to and , except that we additionally incorporate ventilation effects into the ODE we solve. For collisions between droplets, Cleo can use the original collision-coalescence algorithm according to , however Cleo also has various other options, which reflect the uncertainty in the current knowledge about collision probabilities and their outcomes. Therefore Cleo can easily switch between different formulations of the collision kernel and, motivated by a need to investigate the possible influence of collisional breakup on rain evaporation and downdraughts, Cleo includes a framework to model collisional breakup and rebound. We do this in a way that still respects the key features of the original collision-coalescence algorithm by conserving superdroplet number and decreasing the variance in the collisions with decreasing superdroplet multiplicity.

Cleo's numerical methods have some limitations in comparison to some existing SDMs. Regarding condensation/evaporation, perhaps the most significant is that Cleo's method for condensation/evaporation currently does not account for multiple aerosol types. Regarding collisions, we have not incorporated any sub-time-stepping algorithm. Whilst Cleo's simple predictor-corrector method for droplet motion is less expensive than higher-order methods and still preserves the flow-field, is non-diffusive, and has increasing accuracy with increasing grid resolution, it does not yet incorporate turbulent effects and is less precise than a higher order method. All Cleo's limitations and assumptions make it poorly applicable to highly detailed microphysics studies requiring more complex processes, but make it well-designed, as intended, as a simplistic, ordinary representation of warm-cloud microphysics according to SDM.

Not withstanding these limitations Cleo is useful to a broad group of researchers that require a model for warm-cloud microphysics. Simulations can be up to 3-D and can include any combination of the primary microphysical processes behind warm-rain, each with their own independent time-step and with adaptive time-stepping between different processes, as explained in. Cleo can be stand-alone, perform “piggybacking” of LES , or be used as a fully fledged microphysics scheme. Like other SDMs, Cleo provides a precise representation of the droplet size distribution and an intuitive depiction of microphysical processes. We have also made sensitivity studies easy to conduct through Cleo's high-degree of flexibility, whereby microphysics and data output can be easily customised in order to assess the impact of individual microphysical processes , potentially for probing microphysical uncertainties and especially those related to droplet collisions. Finally, since Cleo is also well-suited to high-performance computers, it has the potential to be used in more demanding simulations which probe how cloud microphysics and its uncertainties propagate to larger spatial and temporal scales.