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

“Figure  illustrates how a collection between two SIPs is treated. SIP i is assumed to represent fewer droplets than SIP j, i.e. νi<νj. Each real droplet in SIP i collects one real droplet from SIP j. Hence, SIP i contains νi=4 droplets, now with mass μi+μj=15. SIP j now contains νj-νi=8-4=4 droplets with mass μj=9. Following Eq. (), only νcoll=2 pairs of droplets would, however, merge in reality. The idea behind this probabilistic AON is that such a collection event is realised only under certain circumstances in the model, namely such that the expectation values of collection events in the model and in the real world are the same. This is achieved if a collection event occurs with probability 21pcrit=νcoll/νi in the model. Then, the average number of collections in the model, 22ν‾coll=pcritνi=(νcoll/νi)νi, is equal to νcoll as in the real world. A collection event between two SIPs occurs if pcrit > rand(). The function rand() provides uniformly distributed random numbers ∈[0,1]. Noticeably, no operation on a specific SIP pair is performed if pcrit<rand().

The treatment of the special case νcoll/νi>1 needs some clarification. This case is regularly encountered when SIPs with large droplets and small νi collect small droplets from a SIP with large νj. The large difference in droplet masses μ led to large kernel values and high νcoll with νi<νcoll<νj. […] If pcrit>1, we allow multiple collections, as each droplet in SIP i is allowed to collect more than one droplet from SIP j. In total, SIP i collects νcoll droplets from SIP j and distributes them on νi droplets. A total mass of νcollμj is transferred from SIP j to SIP i and the droplet mass in SIPs i becomes μinew=(νiμi+νcollμj)/νi. The number of droplets in SIP j is reduced by νcoll and νjnew=νj-νcoll. Keeping with the example in Fig.  and assuming νcoll=5, each of the νi=4 droplets would collect νcoll/νi=1.25 droplets. The properties of SIP i and SIP j are then νi=4, μi=17.25, νj=3 and μj=9. […] So far, we explained how a single i–j combination is treated in AON. In every time step, the full algorithm simply checks each i-j combination for a possible collection event. To avoid double counting, only combinations with i<j. Pseudo-code of the algorithm is given in Algorithm (1). The SIP properties are updated on the fly. If a certain SIP is involved in a collection event in the model and changes its properties, all subsequent combinations with this SIP take into account the updated SIP properties. […] For the generation of the random numbers, the well-proven Mersenne Twister algorithm by is used.”

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 . First, they use a linear sampling approach (which will be described in Sect. ). Second, the weighting factors are considered to be integer numbers, whereas we use real numbers ν. Integer values are appropriate in discrete test cases of small sample volumes such as the validation test case in Sect. 3 of . For comparing AON with bin model references, usually continuous DSDs are prescribed. Then a SIP ensemble with real-value weighting factors is more appropriate in our opinion. Third, multiple collections (MCs) are differently treated. For pcrit=(νcoll/νi)>1, either ⌊pcrit⌋νi or ⌈pcrit⌉νi droplets of SIP j merge with νi droplets of SIP i depending on the probability pcrit-⌊pcrit⌋. This maintains the integer property of the SIP weights. As the latter feature is not required in our approach, we deterministically merge pcritνi=νcoll droplets from SIP j with νi droplets of SIP i. This is computationally more efficient than the integer-preserving implementation. Test simulations showed that both MC treatments produce similar results.

We now introduce the AON version based on an idea by where the vertical position zp of the SIPs is explicitly considered. The approach and its implications will be detailed next. Pseudo-code of this AON version (“WM2D”) is given in Algorithm .

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 x–y plane at z=zp within the GB (horizontally well-mixed instead of the traditional well-mixed assumption for the entire three-dimensional GB) and represent a “concentration” of n2D=ν/δA (units L-2, where L is a length scale). We introduce an adapted kernel definition where the relative velocity term |wsed,i-wsed,j| is dropped from Eq. (): 23KijWM2D:=Ec(ri,rj)π(ri+rj)2.

The AON algorithm is split into two steps:

Based on the evaluation of the vertical positions zi and zj at times t and t+Δt, a check is made to see if SIP i overtakes SIP j within a time step Δt. Given zi(t)≥zj(t) (otherwise swap i and j) an overtake takes place in the time interval Δt if zi(t+Δt)<zj(t+Δt).

Similarly to the WM3D version, it happens that νcoll is larger than νi, and multiple collections are also considered in AON-WM2D.

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 t), but also the SIPs that are located underneath, depending on the prescribed time step.

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 Δt. Hence, their collections are underrated unless potential overtakes are checked among all NSIP,tot SIPs of the entire column. Even if the CFL criterion is obeyed, SIPs close to the lower GB boundary will mostly collect SIPs from the GB underneath. Hence, seeking collision candidates only in the present GB is never a good choice.

In a naive implementation, this would dramatically increase the computational costs. In the regular (WM3D) version, nz calls of AON with O(NSIP,GB2) (for simplicity lets assume NSIP,GB is the same in all GBs) give a total cost of nz×O(NSIP,GB2). Contrarily, AON-WM2D is called once for all SIPs of the column. Hence the cost is 1×O(NSIP,tot2)=nz2×O(NSIP,GB2) and a factor nz higher than the regular AON version. However, the WM2D version can be sped up by first sorting all SIPs by their position (if sorting is done independently in each GB, the complexity is nz×O(NSIP,GBlog⁡(NSIP,GB))), and second by taking into account that the final position zi(t+Δt) of the potentially overtaking SIP i must be below the initial position zj(t) of SIP j. Finding possible candidates for SIP i within the sorted SIP list can be stopped once a SIP j with zj(t)<zi(t+Δt) is encountered (see condition in line 10 of Algorithm ).

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 nz and Δz, and yet some horizontal “mixing area” ΔA has to defined, over which the droplets of a SIP are assumed to be dispersed). In the regular AON, the aspect ratios of the grid box do not matter, and only the grid box volume ΔV enters the computations. In WM2D, on the other hand, the value of ΔV is insignificant, and ΔA enters the computations. In a column model with sedimentation, results also depend on Δz as it determines the travel time through a grid box. Note that a variation of Δz can also implicitly change ΔV or ΔA.

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 and . A possible route to consider the effects of sub-grid motions on collision in LCMs has recently been presented by . Their one-dimensional approach is able to represent droplet clustering and turbulence-induced relative droplet velocities in a realistic manner, and its implementation in already applied LCM sub-grid-scale models e.g. is deemed straightforward. However, further research is required on how the limited number of SIPs in current LCM applications may corrupt the correct representation of such processes.

Finally, we briefly summarise the differences between the WM2D and WM3D approach. The standard kernel KWM3D as given by Eq. () has units L3/T (where L and T are a length scale and timescale, resp.). Multiplying it by concentrations ni and nj (units L-3), one obtains the rate of a concentration increase in merged droplets (L-3/T) which is finally multiplied by δt (unit T) to obtain ncoll (see Eq. ). Since SIPs represent droplet concentrations of ni=νi/δV and nj=νj/δV, Eq. () follows. In the WM2D approach, the kernel KWM2D as given by Eq. () has units L2. Multiplying it by “2D” concentrations n2D,i and n2D,j (units L-2) one obtains the collected 2D concentration n2D,coll (units L-2). Since SIPs represent “2D” droplet concentrations of n2D,i=νi/δA and nj=ν2D,j/δA, Eq. () follows. A collection can only occur if a larger droplet (or SIP) i overtakes a smaller droplet (or SIP) j. First, zi>zj and wsed,i>wsed,j must hold and second the overtake time ΔtOT:=(zi-zj)×(wsed,i-wsed,j)-1 must fulfil ΔtOT≤δt. One can define the overtake probability pOT being 0 for ΔtOT>δt and 1 for ΔtOT≤δt, and the “2D” collection probability pijWM2D=KijWM2DδA-1. Simulations in the Supplement demonstrate that the WM2D and WM3D formulations are statistically equivalent; i.e. pOT×pWM2D equals pWM3D, under certain conditions (see Fig. S9).

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 or . ⌊NSIP/2⌋ combinations of pairs i-j are randomly picked, where each SIP appears in exactly one pair (if NSIP is odd, one SIP is ignored). As only a subset of all possible combinations is numerically evaluated, the extent of collisions is underestimated. To compensate for this, the probability pcrit (or equivalently νcoll) is upscaled by a scaling factor 25γcorr=NSIP(NSIP-1)/(2⌊NSIP/2⌋) to guarantee an expectation value as desired. Clearly, this reduces the computational complexity of the algorithm from O(NSIP2) to O(NSIP). Multiple collections are more likely than in the regular AON version. The LinSamp version becomes the preferred choice if NSIP is large.

If νcoll is larger than both νi and νj, all AON versions as introduced so far would produce negative weights. In order to prevent this, νcoll is artificially reduced to νj in such a case (let us assume that νi<νj). The standard procedure would then produce a SIP j with zero weight, which allows the updated SIP i with weight νi (the weight νi remains unchanged during the update) to be split into two SIPs. We choose a 60 %–40 % partitioning, and the operations are as follows. 26aμj:=(νiμi+νjμj)/νi26bμi:=μj26cνj:=0.6νi26dνi:=0.4νi

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 . There, a 50 %–50 % partitioning was implemented. We avoid this equal splitting as it produces two identical SIPs. In our implementation with floating point weights, SIPs with identical weights are extremely rare and no special care is taken for this. Hence, including an operation that produces identical weights is unfavourable. The dependence of the AON-LinSamp performance on the limiter definition is showcased in the Supplement (Figs. S3–S7, S15 and S16 and Table S1).

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 pcrit. Moreover, note that LinSamp can be reasonably used only in conjunction with AON-WM3D, not AON-WM2D.

In addition to the favourable linear computational complexity, LinSamp can be easily parallelised, in particular on shared-memory multi-processor architectures as used by or . Once the SIP pairs are determined in the beginning of each time step, each processor treats a subset of SIP pairs. After a collection event, SIP properties are updated on the fly. Note that the need to do updates on the fly precludes simple parallelisation strategies in the quadratic sampling version, where all SIPs are interconnected.

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”).

Figure  shows the temporal evolution of column-averaged LCM1D moments λl (l=0 and 2) over 1 h for various time steps Δt. The box model data serve as orientation in this and following Figs. –. We find that in terms of λ0 and λ2 LCM1D results converge for Δt≤10 s. The noSedi simulations show a similar time step dependence (not shown). Hence, AON works well even for large time steps, a fact that was already shown with the AON box model (see Fig. 18 of U2017).

Next, we discuss the sensitivity to further physical and numerical parameters. Generally, we find faster convergence for higher moments than for λ0 (not shown). Hence in the following, we confine our analysis to the most “critical” quantity, and Fig.  displays the λ0 evolution for various sensitivity experiments. Even though we analyse the results in some detail, we want to mention that the observed differences are in principle not substantial. In fact, results differ much more due to a different collection kernel or slightly varied initial DSDs (see Sect. ). Nevertheless, the analysis will help to understand more deeply how collisional growth works in an LCM with AON. This pronounced effort is justified, as precipitation initiation is still not fully understood and a well-validated Lagrangian approach may lead to new insights .

In a first simple step, we vary nz (see Fig. a and b), which changes two aspects of the numerical set-up: the number of GBs over which interactions can occur and the height of the column. This implicitly changes the time it takes for SIPs to fall through the total column and hence changes the ”recycling” timescale Lz/wsed. Together with nz, nrinst is varied such that nz×nrinst is always 1000. Accordingly, all simulation results are averaged over the same number of GBs, and we avoid cases of simulations with smaller nz producing noisier data.

In the noSedi simulations (panel a), the moment evolution is not affected by varying (nz, nrinst). This is trivial, as in any case the average is taken over 1000 independent GBs. At least, these results demonstrate that averaging over that many GBs suffices by far to produce robust averages. In the full simulations (panel b), the λ0 decrease is more pronounced, and the various set-ups produce nearly identical results (except for the case with nz=2, which is in between the other full simulations and the noSedi simulations). From this finding alone one may argue that the collisional growth process is more efficient in LCM1D than in LCM0D.

The second row shows a variation of κ which reveals qualitatively different convergence properties of the noSedi simulations (panel c) and the full simulations (panel d). In the noSedi simulations, an increase in κ (and NSIP; see extra legend for corresponding NSIP values) leads to a faster decrease in λ0. Large differences between κ=5 and κ=40 simulations are apparent; above κ=40, an increase in κ leads only to marginal improvements. Also, for the highest κ, the λ0 values remain above the BIN0D reference. For the smallest κ value, only 24 SIPs are created according to Eq. (), and interactions among those few computational particles overemphasise the impact of correlations. It is well-known that for small ensembles of real droplets correlations become important . Analogously, we introduced correlations in our numerical approach by using too few computational particles. We speculate that this hinders the formation of lucky droplets and fewer droplets get collected (hence λ0 is larger for smaller κ). Another more technical explanation is that the νp distribution of the SIP ensemble is such that the formation of lucky SIPs is not supported. Ideally, there is a reservoir of SIPs with small ν values that can become lucky SIPs. There might be too few SIPs with small ν for small κ.

Contrarily, the full simulations (panel d) give nearly identical results independent of κ. We obtain converged results with as few as 24 SIPs in each GB. Compared to κ=200 with 1000 SIPs, the simulations are a factor of 402 faster. The reason for the much faster convergence in terms of NSIP,GB is that the GBs are interconnected, which effectively raises the number of potential collision partners. Drops with radii of 100 and 500 µm have fall speeds of around 0.7 and 4 m s-1, respectively. Thus it takes them around 14 and 2.5 s to fall through a Δz=10 m GB and they enter a new GB every time step or every few time steps given Δt=10 s.

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 κ sensitivity in a column with nz=10 and Δz=100 m instead of nz=50 and Δz=10 m (panel e). Then, SIP interactions can occur only across 10 GBs, and overall 5 times fewer SIPs are present in the column than for the default case with nz=50. Moreover, the domain is stretched by increasing Δz to 100 m, which increases the residence time of a SIP in a GB by a factor of 10, additionally slowing down SIP interactions across GBs. Those two changes introduce a weak κ dependence, and yet it is much weaker than in the corresponding noSedi simulations (panel c).

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 . Again, we find converged results for small κ values as small as 5 (panel f). This elucidates that convergence is improved once some process exchanges SIPs between GBs, be it for physical reasons like sedimentation or by an artificial operation such as the randomised SIP re-location. We speculate that in full 2D/3D LCM simulations turbulent motions and sedimentation increase the SIP exchange across GBs and hence may additionally increase the performance of AON. The two last simulation series are promising, as they suggest that in a column model (and probably also 2D or 3D models) convergence is potentially reached with fewer SIPs per GB than in a box model. Nevertheless the tests also highlight that convergence with κ depends on many circumstances and convergence tests are a prerequisite to any LCM simulation with AON.

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 ΔV nor the absolute number of droplets is considered in this approach. At least in the limit of all SIPs having weighting factor ν=1, the AON algorithm solves the master equation , which takes into account ΔV, and results may depend on the actual number of involved droplets. Clearly, correlations (which are accounted for in the master equation) are larger in smaller volumes .

For our SIP-initialisation procedure, NSIP,GB depends solely on the chosen κ values and is independent of ΔV. By construction, a ΔV variation does not affect at all the simulation results, as all SIP weights are simply rescaled. Indeed, we obtain nearly bit-identical results for a ΔV variation. To explore the ΔV sensitivity in our LCM1D, the SIP-init procedure has to be adapted. In the adapted version the SIP number increases proportionally with ΔV as it would in reality. As computational requirements increase quadratically with NSIP,GB, the variation of ΔV and NSIP,GB can be performed only for a small range of ΔV values. ΔV is increased by a factor of 5 or 10. As a base case, we use the simulations with κ=20 and κ=100 and define ΔV:=1 m3. The fourth row shows results for the noSedi (panel g) and the full simulations (panel h). Apparently, the noSedi simulations with larger ΔV converge to the solution we obtained before by using a sufficiently large κ. In full simulations, a ΔV variation has basically no effect. The κ=100, ΔV=10 m3 simulation considered on average collisions between 5000 SIPs in each GB. Yet, the results are basically identical to the case κ=5, ΔV=1 m3 with 24 SIPs in each GB (which runs nearly 40 000 times faster).

In the present simulations, where SIPs with weights ν>1 are used, variations of the numerical parameter κ and the physical parameter ΔV are interconnected and their effects cannot be disentangled. Hence, the AON algorithm can only answer whether correlations matter in systems with a certain number of SIPs. These correlations are not necessarily the correlations one would see in a real system with millions to billions of real droplets. Nevertheless, the last sensitivity series implies that at least in our model system the importance of correlations is likely the same as in a system with NSIP,GB=24 and with NSIP,GB≈5000. Assuming that the importance of correlations in a real system with billions of droplets is similar to that of a system with 5000 SIPs, the latter finding demonstrates that LCMs can capture the collisional growth process with astonishingly few SIPs.

The noSedi κ sensitivity series as shown in panel (c) was already presented in Fig. 18 of U2017. There we found that for high enough κ the LCM0D results lie below the BIN0D reference contradictory to the present noSedi simulations. The reason for this inconsistency is a programming bug in the LCM0D-AON version used in U2017. The Hall and Long kernel values are stored in look-up tables and were wrongly accessed (overestimating the actual mass of the involved droplets by 2 %). Hence, the collisional growth process proceeded more rapidly in U2017. Despite this flaw, the main findings of U2017 remain valid. Yet, the more rapid collisional growth of LCM0D-AON in U2017 should clearly not be attributed to conceptual differences of AON and BIN algorithms.

In the discussion of the subsequent sensitivity studies, we refrain from showing time series of λ0 as done in Fig. . Instead we only evaluate λ0 at t=1 h as this is a suitable metric for the algorithm performance in the BoxModelEmul set-up. Figure  comprises Δt and κ sensitivity series of all subsequent BoxModelEmul simulations. The black dotted (horizontal) line depicts the reference BIN result obtained with Wang's algorithm with s=16 and Δt=1 s and was already added in Fig.  for orientation.

This subsection discusses the AON version with linear sampling. Both full simulations and noSedi simulations have been carried out. The first row of Fig.  shows sensitivity of λ0(t=1h) to κ (a) and Δt (b), respectively. The grey curves repeat the regular AON results (i.e. with quadratic sampling); they show the endpoints of curves shown in Fig.  (top) and Fig. c and d. We find that the qualitative behaviour does not differ between LinSamp and regular AON.

In the full simulations (solid lines), simulations converge for any κ, whereas for the noSedi simulations (dotted, “nS” in the legend) convergence is reached only for the largest κ values. Using the default time step Δt=10 s, the LinSamp results (orange curves) are slightly further away from the BIN reference (black dots) than the regular results. A second LinSamp series with Δt=1 s (blue) produces better results than the regular AON version with Δt=10 s.

The Δt sensitivity series shown in the right panels of Fig.  demonstrates that LinSamp results are slightly worse than the regular results for the default resolution κ=40. Using LinSamp with a finer resolution of κ=100 produces better results than the regular AON with κ=40. In LinSamp simulations with large time steps, limiter cases occur quite often and one may expect that the artificial reduction of collection events strongly deteriorates the model outcome. However, we see that the performance in the high-Δt range drops similarly in the LinSamp and regular AON version.

Next, we will discuss the results of the AON-WM2D version with explicit overtakes. Results are presented in Fig. c and d. For the chosen set-up with homogeneous initial conditions and periodic boundary conditions, 3D well-mixedness of the SIPs is expected to be maintained over the course of the simulation. Hence, the AON-WM3D and AON-WM2D version are supposed to produce similar outcomes.

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 Δt=0.5 s do they become close to the reference. The solid, green curve shows a Δt variation (down to Δt=2 s) for the version where overtakes are considered across the full column. In the present example, it was also necessary to check for overtakes across the periodic boundary. Then, convergence is reached for Δt≤10 s, very similar to the regular (WM3D) version (see grey curve for comparison). Panel (c) shows a slight dependence on κ, and yet the performance of AON-WM2D is almost comparable to that of the regular AON results.

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 argued in, SIPs can at most travel from one GB to the one right below. Then, potential collision partners can only appear in two different GBs.

As noted in Sect. , the WM2D version can only be used in conjunction with kernels where the differential sedimentation term |wsed,i-wsed,j| is explicitly included and can be dropped. Typically, this is not fulfilled for kernels accounting for turbulence enhancement, in which motions in all spatial directions need to be accounted for. Turbulence in cirrus clouds is often weak. Moreover, cirrus clouds often show a strong layering by ice crystal size, possibly making the 3D well-mixed assumption overly simplistic. Hence, the WM2D version appears to be a reasonable alternative to the regular (WM3D) version. Furthermore, the mixed-phase LCM of used for the simulation of a cumulonimbus employs a hydrodynamic kernel. Hence, the WM2D version would be applicable in this context as well.

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 LWCinit, rinit and DNCinit. Moreover, we highlight the effect of the employed kernel on the AON performance. And finally, we also present BIN sensitivities (namely, we switch from Bott's algorithm to Wang's algorithm and vary the bin resolution and the time step).

In a first experiment, we increase LWCinit by a factor of 1.5 and repeat the κ sensitivity test; see Fig. e. We keep DNCinit fixed and hence the mean radius is rinit=9.3µm×1.5(1/3)=10.7 µm. Compared to the base case with LWCinit=1 g m-3, λ0 starts to decrease after 20 min (instead of 40 min; see Fig. S10). Eventually, λ0 decreases below 104 cm-3 (instead of 106 cm-3). In the full simulations (all solid curves), we again find results nearly independent of κ for all tested AON versions (regular, LinSamp and WM2D). In the noSedi simulations (grey, dotted curve), fewer SIPs are necessary to obtain reasonable results compared to the base case in panel (a).

In a next step, the characteristics of the initial DSD are more systematically varied for fixed κ=40. For such a κ value the noSedi simulation of the base case was considerably off the reference. LWCinit=λ1(t0) is varied, for either fixed droplet number or fixed mean radius. The default value is scaled by a factor of 1.5,2.0 or 2.5. Similarly, DNCinit is varied by a factor of 0.5,0.7 or 1.5 keeping LWCinit constant.

A more detailed presentation of simulation results with time series of the mean diameter, λ0 and λ2 over 100 min is deferred to the Supplement (see Fig. S11). Here, we focus on a single metric again. We define Tcross as the time, when λ0 drops below 107 m-3. The smaller Tcross, the faster precipitation sets in. Figure  shows Tcross for all three sensitivity series (see lower left legend for the various line styles). Simulations with the BIN are contrasted with the regular AON, AON-WM2D, AON-noSedi and AON-LinSamp (see the top-left of the Fig.  for the various colours). Tcross and with it precipitation onset changes strongly with LWCinit and DNCinit. Generally, we find a similar behaviour across all tested models. The AON-noSedi version features the largest Tcross values. This is consistent with previous noSedi results in Fig.  where the decrease in λ0 lags behind. All other AON versions match well and are close to the BIN results. Only for the largest DNCinit value does some spread in Tcross exist. Figure S11 shows that BIN predicts in all cases slightly lower droplet numbers similar to what we already observed for the default microphysical initialisation in Fig. . Nevertheless, we can confirm the very good agreement of BIN and all full AON simulations.

As a last AON sensitivity study, the default Long kernel is replaced by the Hall kernel. Figure f shows the corresponding results. The decrease in λ0 occurs at a slower rate (the y scale now uses a linear scale). For the full simulations (solid curves), we obtain perfect agreement for any chosen κ value and for all three model versions. Moreover, convergence with κ in the noSedi simulations (dotted curve) is less critical than in the base case (compare with panel a again) and results converge for κ≥40. Time series of λ0 of all Hall kernel simulations are shown in Fig. S12.

So far, all reference BIN results were obtained with Wang's algorithm, using a time step Δt=1 s and resolution s=16. We conclude the box model emulation section by showing the sensitivities of two BIN versions. For this, we vary the bin resolution s and the time step for the base case with LWCinit=1 g m-3 and Long kernel and apply either Bott's or Wang's algorithm. The default time step is Δt=10 s as in the AON simulations and the bin resolution is s=4. Figure g–h show results obtained with Bott's and Wang's algorithm, respectively. Again, λ0 time series of these BIN simulations are shown in Fig. S13.

We find that Bott's algorithm converges for s≥2 (Fig. g). Wang's algorithm, on the other hand, does not produce stable results for higher resolutions and Δt=10 s. Thus, the time step had to be reduced (see inserted legend, for the combination of s and Δt). For s≥8 results have converged to the reference. Figure h shows the time step dependency for a medium resolution of s=4. While Bott yields stable results for Δt≤100 s, the results only converge for Δt≤20 s. We can even see a slight dependence of λ0(t=1h) on Δt. As a side note, this is a clear indication that the BIN reference values used for orientation so far should not be interpreted as absolute reference, and it would be premature to discredit AON results being slightly above the BIN reference.

Wang's algorithm, on the other hand, requires Δt≤10 s for stable results, and convergence is reached for Δt≤5 s. Overall, we can conclude that both algorithms converge to basically the same values, given a sufficiently high s and small Δt is chosen. As Bott's algorithm appears to be more robust than Wang's algorithm, all following BIN simulations are carried out with this algorithm.

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 t up to 100 s and s as low as 2. AON simulations may converge for κ=5 (corresponds roughly to s=2) and Δt=10 s if GBs of the column are sufficiently interconnected and averaging over several realisations is done. Wang's algorithm produces correct solutions for s=4 and Δt=5 s, and yet increasing the bin resolution has to be done hand in hand with a reduction of the time step.

We initialise droplets in the upper half of a 4 km column. In each GB the mean radius of the DSD is fixed at the default value rinit=9.3 µm. LWCinit (and with it DNCinit) decreases linearly from 3 g m-3 at the model top to zero at z=2 km. At the model top, a constant influx of a DSD with LWCinit=3 g m-3 is prescribed, which guarantees a smooth profile over time. Otherwise, a discontinuity would occur at the top-most GB which may cause problems in the BIN model. The further settings are nz=400, Δz=10 m, Δt=10 s, nrinst=20, κ=40. All figures related to this set-up start their caption with the label “HalfDomLinDec set-up”.

Figure  shows the temporal evolution of the mean diameter and the moments λ0, λ1 and λ2. Due to the influx condition, the total mass increases during the first 10 min, barely visible in the third panel. During this period, however, collisional growth is already efficiently reducing the droplet number. This is accompanied by an increase in the mean diameter and radar reflectivity. Soon after, the first droplets reach the surface, the mass declines rapidly, and the whole column is more or less washed out after 30 min. We find an excellent agreement among the four model versions BIN, AON-regular, AON-WM2D and AON-LinSamp.

Figure  shows vertical profiles of DNC, LWC, Z and NSIP,GB for times t=0, 10, 20, 30 and 60 min. In the upper half, droplet number is roughly homogeneously distributed and decreases over time. In the lower half, droplet number concentrations are several orders of magnitude smaller than in the upper half and increase over time. The profile of the radar reflectivity shows the highest values after 10 min with a pronounced peak in the middle of the domain. Soon after, the Z profiles become smooth and increase monotonically towards the surface. The sedimentation flux also increases towards the surface, and hence λ2 values decrease over time.