We thank the three reviewers for their detailed assessment. In the
following we explain in detail the changes made to the paper. All the
changes in response to the comments are marked in blue.

----------------------------------------------------------------------------
> Reviewer #1
> Review of "Collision fluctuations of lucky droplets with superdroplets"
> by Xiang-Yu Li et al. This manuscript could be considered for publication
> after a major revision. This is an interesting study assessing the
> collision fluctuation of the Monte Carlo algorithm of super-droplet method
> through a unique approach. Their results suggest that super-droplet
> method can faithfully capture the behavior of lucky droplets if
> "jumps" (artificially enhanced coalescence between lucky droplets)
> do not occur. The condition tested in this study is based on the so-called
> lucky droplet model and very idealized. Some more future works have to be
> done to clarify the consequence and relevance of their findings to more
> realistic simulations in cloud scales, but the results of this study is
> very insightful and encouraging to the cloud modeling community. However,
> very unfortunately, the manuscript is not at all well organized and not
> clearly written. A lot more elaboration is required to make it into its
> final form. For example, the notations are not consistent and confusing.
> The numerical setup is not thoroughly explained. I have to say I had a
> hard time reading this manuscript. It is like an early draft not ready
> for a review. Nevertheless, recognizing that this is a cutting-edge
> study, I look forward to reading the revision of this manuscript.

We thank the referee for a detailed assessment and the many concrete
suggestions. We recognize the shortcomings identified by the referee
and have now responded to all items identified below. We hope that the
present version addresses the concerns regarding confusing notation and
the lack of explanations noted above.

> Major Comments
> 1) [request] Table 1 The definition of Np/s is not clear and
> confusing. The number of droplets in a superdroplet can differ in each
> superdroplet i, and it varies in time. The definition of Np is also
> confusing. The total number of droplets varies in time, and Np=Np/s Ns
> does not hold all the time. Please use appropriate notations and symbols
> throughout the manuscript.

Shima et al used "M_i" to denote the "mass of the solute contained in
the droplet." and "xi" for multiplicity. Dziekan and Pawlowska used
"xi" for multiplicity as well, but used "m" as a symbol for the mass in
general to describe the mass density distribution instead of the specific
"mass of superdroplet". We now follow the notation of Shima et al to
describe the algorithm and use consistent notation throughout. Thus,
we have now changed N_{p/s} to "xi". Also, we have now changed them
to N_d/s^i(t), Nd(t), and Ns(t) all throughout the paper. xi_i(t=0),
Nd(t=0), and Ns(t=0) represent the initial numbers. The new symbols are
now in blue in Table 1, where we also now use the word multiplicity.

> 2) [request] P. 8, ll. 133--134
> To reduce the computational cost, Shima et al. (2009) introduced two
> techniques; multiple coalescence trick and sample reduction trick.
> Dziekan and Pawlowska (2017) and Unterstrasser et al. (2020) confirmed
> that these techniques work efficiently. Please explicitly mention that
> these are not adopted in this study. Note also that when comparing
> the results with Dziekan and Pawlowska (2017), you have to take this
> difference into account.

According to the scheme, we are only considering collisions within a mesh
point, which is perhaps what the referee refers to as multiple coalescence
trick. We have now acknowledged that the superdroplet algorithm, as
implemented in the Pencil Code, does not use the random permutation
technique. This is now said in the last paragraph of section 2.a.

> 3) [request] P. 8, l. 135, "which pairs of droplets collide." This
> should be "which pairs of superdroplets collide and coalesce."
> It should be also mentioned that all pairs in have the possibility to
> collide and coalesce.

We have now changed this to superdroplets. We have also now stated
"All pairs of superdroplets within one mesh point may collide."; see
the paragraph just before Eq.(3).

> 4) [request] P. 8, ll. 138--139, "To avoid a probability ..."
> Is a fixed constant? Or, do you adjust it adaptively? Please clarify
> this point.

Yes, so we have now added the sentence "This is one of several other
time step constraints that are applied adaptively during the simulation."
This is in the paragraph just before Eq.(4).

> 5) [request] Pp. 8--9, ll. 139--141, Eq. (4).
> It is explained that "N_{p/s} is the largest initial number of droplet per
> superdroplet or (Table 1)". However, this has to be "the largest
> number of droplet per superdroplet or". Further, the above
> definition of conflicts with that of Table 1. Please resolve this issue.
> Please also clarify how "\delta x^3" is assigned in this study.  Around here or
> elsewhere, what about explaining explicitly that background droplets do
> not coalesce each other? It must be informative to the readers.

We agree with the referee and have now removed "initial" and have
introduced the superscript max to clarify that the larger one of
superdroplets i and j is to be used, which also resolves the conflict
with Table 1. We have also now added "being the volume of a grid cell"
after "\delta x^3". We have also now added the statement "Note that Eq.(4)
implies that our background droplets, which all have the same radius,
can never collide among themselves." This is in the paragraph just
below Eq.(4).

> 6) [request] P . 9, Eq. (7).
> Please clarify if N_p/s^i is an integer or a real number in this study. In Shima
> et al. (2009), it is defined as an integer, and they use Eq. (16)
> in their paper when splitting a superdroplet to guarantee that they
> remain integers.

We have now removed our statement about non-integer multiplicities.

> 7) [request] P. 10, Sec. 2b. "Numerical setup".
> Please explain the numerical setup in more detail. How big is the
> domain? How many grids do you have in the domain? What is the boundary
> condition? How the superdroplets are initialized? How do you solve the
> equation of motion (1)? How big is the time step? What is the difference
> between 1-D and 3-D superdroplet simulations?

Superdroplets are initialized randomly. This was said at the first
paragraph of section 2.b. Domain size, boundary conditions, and time
steps are now given in the last paragraph of section 2. Yes, the
smaller one is removed once it collides with a bigger one as presented
just below Eq.7. For a 12.6um-sized (radius) droplet to grow to 50um,
125 collisions is needed. We therefore used 256 droplets, so that we don't
run out of droplets for the 1-D simulation and 128 for the 3-D simulation.
This is said just below Eq.(7).

We've also added more detailed information of the numerical setup
in captions of figures showing the results of superdroplet simulations.

> 8) [request] P. 11, ll. 196--197, "In 3-D, however, the number
> density ..." Please elaborate. I do not understand why there is no
> fluctuation of number density in 1-D.

In 3-D, we have independent vertical columns. They are independent because
the droplets fall vertically. The mean density in each column can be
different. This was already mentioned in the penultimate paragraph of
Section 2.b and the last paragraph of Section 3.b. It is also the
topic of Section 4.d, where we quantify this variation and are able
to reproduce the 3-D superdroplet model with a correspondingly adapted
version of approach II; see also appendix A.2.

> 9) [request] P. 12, ll. 215--218, "The rate \lambda_k ..."
> The explanation here is incorrect and misleading. Please revise
> it. If I understand correctly, \lambda_k is the coalescence rate
> that the lucky droplet coalesce with any one background droplet. And
> the definition of \lambda_k1 is very unclear; in Eq. (4) i and j
> are used for superdroplet indices, but k here represents the k-th
> coalescence of the lucky droplet, and the second subscript 1 seems
> to be representing the background droplet. Therefore, by any means,
> the statement "lambda_k=lambda_k1" is wrong. Perhaps this is what
> you mean: Let N' be the number of droplets in "\delta x^3", then
> \lambda_{k}&=N'\pi(r_k+r_1)^2\lvert \vec v_k - \vec v_1\rvert
> E(r_k,r_1)/\delta x^3\\\\ &\approx \pi(r_k+r_1)^2\lvert \vec v_k -
> \vec v_1\rvert E(r_k,r_1) n.

Our original intention was not to repeat an equation that looks very
similar to Eq.(4). We have now done this, as suggested by the referee,
and have added the additional clarifications. This is now shown as
Eq.(10) in Section 3.a.

> 10) [suggestion] P. 13, ll. 226--227, "The actual time until ..."
> It should be informative to point out that the variance is 1/lamk^2.

We have now started the sentence like so "Given that the variance of
lambda_k^{-1} is large for small k, the actual time until the first
collision can be very long, but it can also be very short, depending
on fluctuations."

> 11) [request and question] P. 14, Fig.4. Could you explain how you
> calculated P(T) of LDM? Is it possible to derive the analytic form?
> Or, did you plot it numerically? Is <T> equal to T_125^MFT?

Yes, we have done this numerically, and have now added some explanations
in the paragraph after Eq.(15), where we also refer to the new Appendix
A.1, where we describe a convenient method to compute the sums efficiently
for 10^10 realizations using the Pencil Code.

> 12) [request] P. 17, Eq. (16).
> Again, the meaning of the subscript is different in Eq. (16) and in
> Eq. (2). Please clarify.

We have now addressed this just above Eq.16 as "Here we use the
subscript k to represent the stopping time of the k-th collision,
which is equivalent to the i-th droplet."

> 13) [question and request] Approaches I--IV
> Let me confirm: approach I = LDM; approach II = explicit collision model;
> approach III = Monte-Carlo model described in Sec. 3e; approach IV =
> superdroplet method.  In approaches I (LDM) and III (Monte-Carlo 3e),
> background droplets are not considered explicitly.  In approaches
> I, II, and III, superdroplets are not used, i.e., all .  Are these
> correct? Please explain these points more clearly in the manuscript.
> It seems a tall box domain is used for approach II. Please specify the
> size. Please also clarify the boundary condition. Was this domain also
> used for approach IV (superdroplet method)? Or, was some different
> geometry used for approach IV?

We agree with the statements of the referee, although we regard LDM as the
model examined with all four approaches. We have therefore now rephrased
the sentence to say "By contrast, approach I is different from either of
the two,..."  We have also now added details regarding approach II and
write "For our solution using approach~II, we use a non-periodic domain
of size $10^{-4}\times10^{-4}\times700\m^3$, containing thus on average
700 droplets. This was tall enough for the lucky droplet to reach $50\um$
for all the $10^7$ realizations in this experiment."

> 14) [question] P. 17, l. 318, "LDM" Do you mean "approach III"?

Yes, so we have now written "include this effect in solutions of the
LDM using approach III and compare with ...". We have now reviewed and
adapted the usage of LDM throughout the paper.

> 15) [request] P. 18, l.327, "N_{p/s}=1". Please clarify. I suppose
> you set the initial multiplicity of all the background superdroplets
> and the lucky superdroplet equal to 1, i.e., for all i, N_{p/s}^i=1.

Yes, this applies to all droplets. This is now said in the last paragraph
of section 2.b.

> 16) [request] Caption of Figure 6
> The configuration of the superdroplet simulation is partly explained for
> the first time in the caption, but not in the main text. Please describe
> all the detailed information necessary to reproduce the result in the
> main text, such as the domain size, boundary conditions, and time steps.
> In the caption, it is explained that the number of superdroplets used
> for this simulation is N_s = 256. I assume that 1 superdroplet is for lucky
> superdroplet. In the next sentence, it is explained that the mean number
> density of droplets is n_0=2.28E9/m^3. This must be the INITIAL mean number density. Is
> lucky superdroplet included in n_0? I have estimated the size of the domain
> by \delta x^3 = (255 or 256)/n_0~1.1E-7/m^3. Is this correct?
> If my interpretation above is correct, and also
> because the lucky superdroplet has to coalesce 124 times to reach the size 50um,
> the number density of droplets n will be almost half of n_0
> at the end of the simulation. I think this is not the situation that
> you want to simulate. Please clarify all these points in the main text.

Domain size, boundary conditions, and time steps are now given in
the last paragraph of section 2. Yes, the smaller one is removed
once it collides with a bigger one as presented just below Eq.(7).
For a 12.6um-sized (radius) droplet to grow to 50um, 125 collisions
is needed. We therefore used 256 droplets, so that we don't run out
of droplets for the 1-D simulation and 128 for the 3-D simulation.
This is said just below Eq.(7).

> 17) [request] Appendix A1 and Fig. 15.
> The numerical setup tested here is very unclear. Suddenly, Ngrid and Nd
> (=?Np) were introduced without any explanation. Please provide all the
> details so that the readers can reproduce the results.

We have now explained why we study these statistical convergences
and full details are given in Appendix A1. 

> Please add "(a)" and "(b)" to Fig. 15. Replace "Figure
> 7(a)" at the end of the caption of Fig. 15 by "Figure 7".

We have now added "(a)" and "(b)" to what is now Fig.A1 (Fig.15
in the original format) and replaced "Figure 7(a)" at the end of
the caption of Fig.15 by "Figure 7".

> 18) [comment] P. 19 and the rest of the manuscript
> Because the sufficient detail of the simulations conducted are not
> provided, it is difficult to understand and evaluate the rest of the
> manuscript accurately.

Details are now provided; see, in particular, the end of Section 2.b.

> 19) [request] P. 19, l. 349, "Figure 8 where N_{p/s}^luck=N_{p/s}^back=2..."
> First of all, you have to say that the INITIAL CONDITION OF MULTIPLICITY is
> N_{p/s}^luck=N_{p/s}^back=2. You may consider it almost obvious, but
> such a small lack of explanation is piled up high in this manuscript. And,
> again, the numerical setup is unclear. What is the number of superdroplets
> used for this test? The same domain size as before? What are the time
> steps?

Again, all these details are now provided at the end of Sec.2.b.
We have also elaborated on the setup for Figure 8 in the first
paragraph of Sec.4.b.

> 20) [question] P. 19, l. 359, "N_p^luck=3 superdroplets"
> Do you mean "droplets"? If I understand correctly, approach III does
> not use superdroplets.

That is not correct; approach III is probabilistic and applies to
what happens when two superdroplets collide. In our (extreme) model
(approach III), we have just 2 superdroplets, one (or a few, when we
study different values of epsilon) for the lucky droplets and one for
the background droplets.

> 21) [suggestion] P. 19, Eq. (17). It is better to give lambda_ij^luck
> simply by lambda_ij=pi*(ri+rj)^2|vi-vj|/dx^3. The newly introduced
> variable n_luck satisfies n_luck=epsilon*n/N_p^luck=1/dx^3.
> Further, more importantly, your definition of n_luck is confusing,
> because it does not correspond to the number density of lucky
> droplets, N_p^luck/dx^3.

This is now explained in more detail in the text just after Eq.(19).

> 22) [request] P. 19, Eq. (18)
> The definition of epsilon is also confusing. It seems to me that
> epsilon is defined by the initial ratio of lucky droplets and
> background droplets, Np^luck(t=0)/Np^back(t=0). But, if so,
> we cannot apply this epsilon to Eq. (17).
> If I understand correctly, in approach III (Monte-Carlo 3e),
> background droplets are not considered explicitly, hence the
> number density of background droplets is a fixed constant.
> Further, superdroplet is not used for the lucky droplets in
> approach III. Then, it is confusing to use N_s in Eq. (18).
> Please clarify.

Approach III is purely probabilistic and describes the collision
between superdroplets. Their number stays constant and therefore
also the number of superdroplets containing luck droplets stays fixed.
Therefore, we do not say anything about an initial number.
Equation (17) contains n_luck, i.e., the number density of heavier
droplets relevant for their mutual collisions. This is proportional
to epsilon, and applies therefore to Eq.(17). The number density of
background droplets does not have to be fixed, but including this
would make a negligible difference. Furthermore, regarding the use
of N_s, the relevant number is the actual number of lucky droplets.
It becomes relevant when estimating epsilon for the superdroplet
approach, which we do on the next page, where we discuss a case
with a multiplicity of 2.

> 23) [request] P. 19, ll. 364-366, "we used N_s=256..."
> This information must be explained much earlier.

This information is now provided at the end of Section 2.b.

> 24) [question] P. 20, ll. 367--373
> In approach III, will you reduce the number of lucky droplets when they
> coalesce each other?  It is explained that Fig.9 was produced by the
> approach II. Is this correct? I do not understand how multiple lucky
> droplets were introduced to the approach II.  I cannot find any results
> of approach III with multiple lucky droplets. Where is it?

As explained above, approach III models superdroplets, and therefore
their number does not change after a collision. We did write incorrectly
approach~II, but did actually mean approach~III, and have now corrected
this. In the following sentence, we now write "We see that for small
values of epsilon, this model has similar cumulative distribution
functions, so the effect of jumps is very small."

> 25) [request] P. 20, ll. 374--385
> It seems you suddenly switched the target and started talking about the
> superdroplet model. Please declare more explicitly which one of the four
> models you are currently talking about.

Yes, so we have now added the sentence "Let us now compare with the
jumps found using the full superdroplet approach (approach~IV)."

> 26) [request] P. 24, l. 470, "LDM (approaches I, II, and III)"
> If I understand correctly, approach I = LDM; approach II = explicit
> collision model; approach III = Monte-Carlo model described in
> Sec. 3e. Please use the same definitions throughout the manuscript.

We have now added a table to summarize more concisely the four
different approaches. We have also checked that we are using the name
LDM consistently and have explained it in the beginning of the
section "The effects of various approximations". We also clarify that
lateral density variations can be addressed with the LDM using all four
approaches, as is now said just before the section "Relation to the
superdroplet algorithm". We have therefore also corrected the
relevant sentence in the beginning of that section.

> Minor Comments:
> 27) [suggestion] P. 5, ll. 64--69
> Perhaps you can also cite Jaruga and Pawlowska (2018), Sato et al. (2018),
> Seifert et al. (2019), Shima et al. (2020), and Unterstrasser et
> al. (2020).

We have now cited the recommended references; see paragraph 2 of the
introduction and at the end of Section 2.a, where the Unterstrasser et
al. (2020) paper is mentioned another time.

> 28) [typo] P. 8, l. 137, "p_ij < eta" -> "eta < p_ij"

We have now corrected this; see the blue piece after Eq.(3).

> 29) [question] P. 15, ll. 259--261, "P(T) can be approximated by a
> lognormal ..." How good is the approximation?

The departure from a lognormal distribution can be judged qualitatively
by just inspective the shape and comparing with that of an inverted
parabola, as was explained in the text. In the sentence starting with
"To quantify the shape of $P(T)$", we have now also added "... and its
departure from a lognormal distribution,...". This is where we refer
to the table and, for clarity, we have now added "We recall that, for
a perfectly lognormal distribution, skew X = kurt X = 0."

> 30) [question] P. 15, l. 271 and Table 2, "T_K^mf " Do you mean
> T_125^MFT?

Yes, this is what we meant and we have now corrected this;
which is also marked in blue.

> 31) [suggestion] Figure 7
> Perhaps you had better label the vertical axis as P(T/<T>).

We have now updated the vertical label to as P(T/<T>).

> References
> Jaruga, A. and Pawlowska, H.: libcloudph++ 2.0: aqueous-phase chemistry
> extension of the particle-based cloud microphysics scheme, Geosci. Model
> Dev., 11, 3623–3645, https://doi.org/10.5194/gmd-11-3623-2018, 2018.

This is now quoted in paragraph 2 of the introduction.

> Sato, Y., Shima, S., & Tomita, H. (2018). Numerical convergence of shallow
> convection cloud field simulations: Comparison between double-moment
> Eulerian and particle-based Lagrangian microphysics coupled to the
> same dynamical core. Journal of Advances in Modeling Earth Systems, 10,
> 1495-1512. https://doi.org/10.1029/2018MS001285

This is now quoted in paragraph 2 of the introduction.

> Seifert, A., Leinonen, J., Siewert, C., and Kneifel, S.: The Geometry of
> Rimed Aggregate Snowflakes: A Modeling Study, J. Adv. Model. Earth Sy.,
> 11, 712-731, https://doi.org/10.1029/2018MS001519, 2019.

This is now quoted in paragraph 2 of the introduction.

> Shima, S., Sato, Y., Hashimoto, A., and Misumi, R.: Predicting
> the morphology of ice particles in deep convection using the
> super-droplet method: development and evaluation of SCALE-SDM
> 0.2.5-2.2.0, -2.2.1, and -2.2.2, Geosci. Model Dev., 13, 4107–4157,
> https://doi.org/10.5194/gmd-13-4107-2020, 2020.

This is now quoted in paragraph 2 of the introduction.

> Unterstrasser, S., Hoffmann, F., and Lerch, M.: Collisional growth in
> a particle-based cloud microphysical model: insights from column model
> simulations using LCM1D (v1.0), Geosci. Model Dev., 13, 5119–5145,
> https://doi.org/10.5194/gmd-13-5119-2020, 2020.

This is now quoted in paragraph 2 of the introduction and at the end
of Section 2.a.

----------------------------------------------------------------------------
> Reviewer #2: General comments: This manuscript returns to the problem
> of "lucky droplets" as discussed in Telford (1955) and Kostinsky and
> Shaw (2005) by considering growth of an ensemble of lucky droplets
> and applying a superdroplet method (SDM). As the cloud simulation
> community expands the use of Lagrangian microphysics to study
> cloud and precipitation processes, understanding its limitations is
> important. This paper contributes to such an effort. Moreover, it is
> also important to recognize that the authors represent two distant
> communities that apply Lagrangian method in collision/coalescence
> simulations and making these two communities aware of the progress is
> important. I thus recommend publication after my general and specific
> comments are addressed.

We thank the reviewer for their positive assessment.

> General comments:
> 1. I feel some aspects of the paper discussion (e.g., the four approaches,
> I to IV) seem to detract from the main thrust of the paper. In my view,
> the narrative can be substantially tightened up, focusing on the new
> aspects, like the extension of previous studies by allowing many initially
> larger droplets that lead to jumps. The abstract does not give justice
> to the discussion in the main text. This is perhaps my personal taste,
> but if other reviewers echo my assessment, then the authors should
> seriously consider significant rewriting focusing on the key new findings.

We do not agree that the jumps are the main new contribution of this
paper. Our main point is to say that the superdroplet algorithm models
the lucky droplet problem correctly. We do this by decomposing this
algorithm into approaches II and III, and demonstrate that both of them
give, on their own, an accurate solution, which is also the same when
combined into approach four. The aspects of jumps occurs as an extra
complication that we now can explain. Likewise, the extension from 1-D
to 3-D is another such complication that we are also able to explain,
which is necessary to compare with the right reference solution.

To explain the significance of approaches II and III early on, we have
now added the following to the abstract, "The superdroplet algorithm
incorporates fluctuations in two distinct ways: through the random
distribution of superdroplets and through the random choice of whether
or not a collision occurs when two superdroplets reside within the volume
of one mesh point. Through specifically designed numerical experiments,
we show that both sources of fluctuations on their own give an accurate
representation of fluctuations.

> 2. The notation used in the paper should be improved. For instance,
> having "N" with various sub- and superscripts makes reading and
> understanding difficult. This is why those various "N" symbols are
> listed in Table 1. One can use N and M for number of superdroplets and
> multiplicity, respectively, and use "m" for mass (not M as it is now).
> Or maybe use the same notation as in Shima et al.? I think this is
> what Dziekan and Pawlowska used, correct? The word "multiplicity" never
> appears in the manuscript and I think it should as this is the best way
> in my view to represent the essence of superdroplets.

Shima et al used "M_i" to denote "mass of the solute contained in the
droplet." and "xi" for multiplicity. Dziekan and Pawlowska used "xi" for
multiplicity as well but used "m" as a symbol for the mass in general to
describe the mass density distribution instead of the specific "mass of
superdroplet". Thus, we follow the notation of Shima et al to describe the
algorithm for consistency. We have now changed N_{p/s} to "xi". Also,
we have now changed them to xi_i(t), Nd(t), and Ns(t) all across the
paper. xi_i(t=0), Nd(t=0), and Ns(t=0) represent the initial numbers.

> 3. The key idea of the SDM is somehow lost in the paper. In real world,
> two colliding droplets create a new droplet of a larger size. The same
> applies to superdroplets. In a numerical implementation, one then expects
> the number of superdroplets to increase in time, up to the point when the
> problem becomes computationally intractable. The beauty of the Shima et
> al. stochastic algorithm is that the number of superdroplets stays the
> same because one waits until the droplets with smaller number are replaced
> by the product of their collisions with larger-number superdroplets. The
> other aspect, stochastic selection of superdroplet pairs involved in
> collisions, is also important, although probably irrelevant for the
> problem considered in the manuscript under review. Maybe this comment
> reflects the fact that it is not clear to me where the superdroplet
> approach enters the study presented in the paper. In other words, most
> of the paper does not require reference to superdroplets at all, correct?

This is not correct; the model of Shima et al is what is implemented
and what is tested, typically with up to 1000 independent realizations.
This is barely enough to see the inaccuracies in the model and already
computationally expensive. This is what is referred to as approach IV.
However, we explain that the superdroplet algorithm invokes two separate
sources of randomness, and we show that both of them on their own
(approaches II and III) describe the LDM correctly. It is therefore
not correct that we do not make references to superdroplets. To clarify
this further, we have now added Table 3 to summarize these aspects.

> 4. Jumps. I think this is where the paper moves forward when compared to
> Telford and Kostinsky/Shaw. This is because Li et al. study considers
> a growth of an ensemble of initially-larger droplets (rather than a
> single larger droplet as in those other studies), and thus collisions
> between those larger droplets are possible. This where the jumps come
> from, correct? If so, this needs to be appropriately stressed. That
> said, I am surprised by the difference in the time axes in figure 6 and
> 8. I would expect jumps accelerate the growth, but this is not the case
> comparing the two figures (the mean reaches 50 microns in about 100 s
> in Fig. 6 and about 700 s in Fig. 8; why?). I think there are other
> differences than just inclusion of more-than-one initial larger droplet.
> Please explain. Also, rather than presenting details of each simulation
> in the figure caption, the main text should provide those. Figure 6 is
> a good example, but the same applies to several other figures.

Yes, this explanation of jumps is correct. The reason for the change in
the typical time scale of growth was caused by a change in n_0. We have
now rerun this case and updated Figures 6 and 7 correspondingly.

> 5. I think the fact that the study applies completely unrealistic
> collision efficiencies (equal to 1) needs to be stressed
> throughout. Collision efficiencies for such small droplets are just a few
> percent and they rapidly increase to about 70% for collisions between 10
> and 50 micron droplets (see Table 1 in Hall JAS 1980). The authors argue
> that they use collision efficiency of 1 to compare to previous studies,
> but this aspect (the comparison) is not really discussed in detail
> in the manuscript (maybe I missed it?). I feel both including large
> number of initially-larger droplets (per 3 above) and realistic collision
> efficiencies are important to bring this study closer to reality.

We did discuss efficiencies different from unity in Equation (15) and
presented corresponding results in Figure 5. To stress this, we have now
added the sentence "To assess the effects of this assumption, we also
compare with result where the efficiency increases with droplet radius
(Lamb & Verlinde_2011). In particular, we adopt a simple power law
prescription that was previously considered by Kosinski & Shaw (2005)
and Wilkinson (2016)." at the end of the paragraph after Eq.(4).

> Specific comments:
> 1. L. 89/90, "When the number...". This sentence is unclear. What
> correlations? Also, this is a good place to introduce multiplicity as
> suggested above.

We have now removed this sentence, which was essentially a quote from
the paper by Dziekan & Pawlowska (2017). We have now changed this and
introduced the multiplicity, as suggested by the referee, so: "This number
is what is called the multiplicity, which we denote by xi. When this
number is larger than 9, they found that a residual error remains."

> 2. L. 142. As stated above, this is an unrealistic assumption and I do
> not see direct comparison with previous models. In fact, adding realistic
> collision efficiencies, together with many initially-larger droplets
> would benefit the presentation.

We refer here to the model of Kosinski & Shaw (2005) as well as that of
Wilkinson (2016). This is first mentioned at the end of the paragraph
after Eq.(4). We return to this when explaining Eq.(16). The results
are described in the following paragraph; see also Figure 5 for the
quantitative results.

> 3. L. 152. A discussion of Fig. 1 would be appropriate here.

This is the content of the two paragraphs involving Eqs.(5)-(7).
We have now incorporated the discussion of Fig.1 in the text
around Eq.(5).

> 4. L. 156. I do not understand how one can get "fractional number of
> droplets". Please explain.

You are right, in this work we do not consider fractional droplet numbers.
We have now removed this statement, which was previously just below Eq.(7).

> 5. L. 158. When and why do you remove a droplet from the calculations
> (per 4 above)? How often does that happen? Is the loss of the total
> mass significant?

We have now revised this sentence to the following, "It is then assumed
that, when two superdroplets with less than one physical droplet collide,
the superdroplet containing the smaller physical droplet is collected by
the larger one and thus is removed from the computational domain after
the collision."

> 6. L. 190. Where the viscosity given in this line enters the picture?

We have now moved the viscosity of the airflow right after Eq.(2).

> 7. L. 192. Where do the fall velocity of the initial larger droplet (3.5
> cm/s) comes from? 10 micron droplet falls with about 1.2 cm/s at typical
> surface conditions (air temperature, density, etc). 12.6 micron droplet
> would then fall around 1.9 cm/s following the Stokes law. Please explain.

The fall velocity is given by g*tau, where g=9.81 m/s^2 and,
according to our Eq.(2), tau=2*1000*r1^2/(9*rho*nu), so
tau=2*1000*12.6e-6^2/(9*1*1e-5)=0.00352s, and v=9.81*0.00352=0.035m/s.

> 8. L. 219-221. But rk >> r1 not a valid approximation for the case
> considered here! Neither is E=1.

We have now added "While the LDM is well suited for addressing
theoretical questions regarding the significance of rare events, it
should be emphasized that it is at the same time highly idealized.
Collisions with $r_i\gg r_1$ are not very realistic. Furthermore,
it is well known that..."

> 9. Eq. 10-12. The Stokes flow regime that is assumed here is not
> valid beyond droplet radius of about 30 microns. This should be
> pointed out here.

At the end of Section 3.a, we have now added "Note that for droplets
with r >= 30 um, the linear Stokes drag is not valid." below Eq.(10).

> 10. Fig. 15, right panel. Green and blue symbols are difficult to
> distinguish. I suggest using red for either blue or green.

We have now changed the green color to red.

----------------------------------------------------------------------------
> Review #3
> Dear Editor, Dear Authors,
> In what follows, I provide my comments to the manuscript entitled
> "Collision fluctuations of 1 context of cloud droplets. The motivation
> revolves around understanding the role of fluctuations in the process, and
> in assessing the performance of the numerical coagulation schemes used in
> particle- based cloud microphysics modelling. The topic clearly matches
> the scope of the journal. In several aspects, the submitted version of
> the paper presents the findings in juxtaposition to those reported in
> Dziekan and Pawlowska 2017 (the discussion section is solely devoted to
> it). The study uses a simple quiescent setup with mostly monodisperse
> particles settling and colliding under gravity but not affecting the flow
> (i.e., there is no flow). In what follows, I suggest several points to
> address when revising the paper before publication (citations that are
> not listed in the reviewed manuscript are given below).

We thank the reviewer for the detailed assessment and have now improved
the paper in the ways explained below in detail.

> 1. The "superdroplets" mentioned already in the title are presented
> in a misleading way in my opinion. The authors interchangeably use
> "superdroplet method", "superdroplet algorithm", "superdroplet
> simulation", "the superdroplet model" to refer to either general
> or specific aspects of probabilistic particle-based simulations of
> flow-coupled phenomena and/or coagulation. While the study does not
> feature flow coupling, it is still of great value to clearly distinguish:
> (a) super-particle approach to simulations; (b) inclusion of coagulation
> process in such simulations, (c) particular algorithm used to numerically
> represent coagulation (e.g., SDM); and (d) its implementation (here, the
> super-particle coagulation module of Pencil-Code). It is worth noting that
> the idea "to combine physical aerosol droplets into superdroplets"
> (p5/l61) and its application to atmospheric simulations clearly predates
> the cited works of Zsom and Dullemond, 2008 and Shima et al., 2009
> (see e.g., [Lan78; Zan84; CO97; PHP04]). While I agree that the aughts
> brought breakthroughs with the introduction of scalable Monte-Carlo
> schemes in the referenced works (but also owing to other developments
> surveyed e.g. in [DRW11]), the background information presented in
> the study and the nomenclature used seems misleading. The particular
> algorithm, SDM, introduced in Shima et al. 2009 is a precisely-defined
> numerical representation of Monte-Carlo coagulation. SDM has the unique
> feature of having computational complexity linear with the number of
> super-particles used (among other aspects due to candidate pair selection
> method). It seems to me to be essential to discuss such "details"
> as the favorable scaling of SDM comes at a trade-off (see e.g.,
> discussion in [UHL20] where SDM is referred to as "AON with linear
> sampling"). All the more, given that the authors state in the abstract
> that they "quantify the accuracy of the super-droplet method".

We have now used the word "superdroplet algorithm" throughout, except when
we talk about "superdroplet simulation", which refers to a simulation
performed with the superdroplet algorithm.  We thank the referee for
the other references that we have studied with interest. Quoting them in
the context of the present lucky droplet work seemed to us a bit remote,
so have not mentioned them here.

The work of Zannetti 1984  and Paoli et al 2004 indeed
introduced the superpartcile concept but they didn't tackle
the coagulation problem.
We have said this now in the paper.
Since we only address the application of Monte-Carlo coagulation
scheme in this manuscript, those work are beyond the scope of
this study. As far as we understand, Lange 78
(https://doi.org/10.1175/1520-0450(1978)017%3C0320: ATDPIC%3E2.0.CO;2.)
does not discuss the Monte-Carlo collison scheme. It is a Lagrangian
tracking method for particles advected in a Eulerian turbulent flow.
CO97 (https://rmets.onlinelibrary.wiley.com/doi/epdf/10.1017/S1350482797000455)
does not discuss the superdroplet method either.

> 2. There is no discussion on the discrepancy between the presented
> simulations involving "several" (as already acknowledged in
> the abstract) real drops per super-droplet, and the common values of
> billions of particles per super-particle which is found in many of the
> referenced works.

We say that it doesn't matter much; see Figs.10 and A.1.
 
> 3. The notion of Direct Numerical Simulation (DNS) – the very first
> words of the paper – seem to be used in a somewhat different way then
> it is often assumed in the domain (which is not wrong, but calls for
> discussion). In the introduction, the super-particle method is introduced
> 1The paper seems to be an updated version of an e-print published on
> arXiv in 2018 [Li+18] where it has a "Q. J. R. Meteorol. Soc" header,
> and which constituted one of five papers contained in the PhD thesis of
> the first author [Li18] where it is labelled as "Phys.Rev. E., to be
> submitted". 1 lucky droplets with superdroplets" submitted to JAS .
> Presented study deals with the process of collisional growth of particles
> and is discussed in the by contrasting it with DNS: "Compared with DNS,
> the superdroplet method is distinctly more efficient". However, it
> seems common to associate the DNS qualifier with the type of continuous-
> phase representation, regardless of the way the dispersed phase is treated
> (in particular also when using super-particle/weighting-factor approach
> as in [RPW20], but also when using bulk or bin representation for cloud
> water as in [And+04]).

We have now added an explanation of what DNS means; see the second
sentence of the introduction, where we have now added the sentence "Here,
DNS refers to the realistic representation of all relevant processes
including the use of a realistic viscosity along with proper operators
for the viscous flux and a realistic modeling of all droplets." This
connected naturally with the already existing explanation of Lagrangian
tracking. Regarding the history of the present paper, the referee's
observation is correct in that it was originally submitted t0 QJRMS,
but received unfavorable comments. In the following 3 years, we have
substantially reworked the paper. The paper was never sent to Phys Rev
E, but the template of that journal was used in the thesis.  To refer
to RPW20 and And+04.

> 4. The choice and usage of references is puzzling. On the one hand,
> there are just four papers cited on the very topic of "lucky
> droplets" announced in the title. On the other hand, a dozen of
> papers on astrophysical aggregation mechanisms are among the referenced
> literature. At times, references are misleading and off target, e.g.:
> (*) Jokulsdottir and Archer 2016 is listed among "meteorological
> literature" while the study deals with the biological pump mechanism
> in the oceans;
> (*) Patterson & Wagner 2012 seem also not to involve the meteorological
> context either, while listing 20 other "meteorological" works in one
> parenthesis (p5/l64-69) without a hint of explanation why these and not
> others were picked is confusing – one may come up with a different
> set of 20 works applying super-particle approach in "meteorological
> literature" (starting with the above-mentioned works predating Zsom
> & Dullemond and Shima et al. papers);
> (*) there is a mention of "the stochastic coagulation equation
> of Gillespie (1972)" despite that Gillespie’s model is the very
> alternative to SCE;
> (*) works not even dealing with Monte-Carlo coalescence are listed
> as using "the superdroplet algorithm": Andrejczuk et al. 2008 does
> not feature coalescence at all (it’s a condensation- only study),
> Andrejczuk et al. 2010 does not feature Monte-Carlo coalescence (it uses
> SCE-like approach) – these works use particle-based cloud microphysics,
> super-particle approach, but not the Super-Droplet Method algorithm
> (please do differentiate the terms).
> Overall, the first 20 lines of the text contain 40 references, while
> the whole bibliography totals 46 items, of which a sheer majority is
> cited once on the first page. Not that the actual numbers matter here,
> but it would be worth to bring some more balance. Intriguingly, unlike
> in the case of the submitted text, the initial version of the manuscript
> [Li+18] and the PhD thesis it was a part of [Li18] do provide somewhat
> clearer background sections pointing to several review papers on the
> topic to which references are not present in the present version (e.g.,
> [Sha03; Da12; GW13; PW16]). Technical issues with references: some
> reference entries include DOIs, some not; several include doubled URLs;
> numerous entries include "n/a-n/a" page ranges; acronyms and proper
> names have bogus spelling (Mcsnow, lagrangian, kuiper, slams, neptunian,
> ...); capitalisation is not consistent.

We are aware of only 4 papers addressing the lucky droplet model.
We mention astrophysical papers to address the wide application of
the superdroplet approach beyond meteorology. We have now removed the
references to Jokulsdottir and Archer 2016 and Patterson & Wagner 2012,
as well as Andrejczuk et al. 2008 and 2010. We have now listed other
meteorological references and hope that this addresses the referee's
concern regarding the lack of suitable papers.

We mentioned Gillespie et al just to describe the work of Dziekan et al 2017.
We have now corrected those entries with "n/a-n/a" page ranges.

> 5. Effectively, the paper seem to lack proper definition of the simulation
> mesh. It is just said that simulations are performed in 1D and in 3D.

We have now added the mesh points in Sec.2.b.

> 6. Reproducibility. Information on the software used to perform the
> simulations (multi-purpose CFD solver, not the actual setups or analysis
> scripts) can only be found in Acknowledgements, without version number or
> permanent archive. Please follow the AMS guidelines regarding archiving
> of core research output including software. Github does not qualify
> as permanent archive, please use e.g., zenodo.org. To make the study
> reproducible, the code archive should include all scripts needed to set
> up, run and analyse the simulations to obtain the presented figures.
> Hope that helps.

We have now assembled all the run scripts on the web page mentioned at
the end of the paper. We have also now put a copy with this information
on Zenodo.