16 Jun 2021
16 Jun 2021
A strong statistical link between aerosol indirect effects and the selfsimilarity of rainfall distributions
 Met Ofﬁce, Exeter, UK
 Met Ofﬁce, Exeter, UK
Abstract. We use convectivescale simulations of monsoonal clouds to reveal a selfsimilar probability density function that underpins surface rainfall statistics. This density is independent of clouddroplet number concentration and is unchanged by aerosol perturbations. It therefore represents an invariant property of our model with respect to cloudaerosol interactions. For a given aerosol concentration, if the dependence of at least one moment of the rainfall distribution on clouddroplet number is a known input parameter, then the selfsimilar density can be used to reconstruct the entire rainfall distribution to a useful degree of accuracy. In particular, we present both singlemoment and doublemoment reconstructions that are able to predict the responses of the rainfall distributions to changes in aerosol concentration. In doing so we show that the seemingly highdimensional space of possible aerosolinduced rainfalldistribution transformations can be parametrized by a surprisingly small (at most three) independent “degrees of freedom”: the selfsimilar density, and auxiliary information about two moments of the rainfall distribution. This suggests that, although aerosolindirect effects on any specific hydrometeorological system may be multifarious in terms of rainfall changes and physical mechanisms, there may, nevertheless, be a universal constraint on the number of independent degrees of freedom needed to represent the dependencies of rainfall on aerosols.
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Kalli Furtado and Paul Field
Status: final response (author comments only)

RC1: 'Comment on acp2021443', Anonymous Referee #1, 22 Jul 2021
The manuscript describes the derivation of an invariant distribution of surface rainfall rate as a function of the precipitation flux. The derivation is based on a set of numerical model simulations where the aerosol number concentrations are varied. The resulting invariant distribution appears to be independent of the amount of aerosol and has the ability to predict the response of the rainfall statistics to a perturbation of the aerosol. The manuscript describes an interesting work to construct such an invariant and fits into the scope of ACP. I recommend the manuscript for publication after the authors have addressed the following comments and revised the manuscript accordingly.
General comments: In the manuscript, you state at multiple locations that you construct a universal distribution for the rainfall statistics. However, the whole analysis is based on three simulations of one case. At the very end (starting at line 455) you mention some of these serious assumptions that might limit the conclusion to only this single simulated case. At the moment, your results show that for your model and for this simulated case there is a universal distribution. To get an idea about the true universality of the result, one should
 simulate more cases with your model
 in particular also at different locations in the world (maybe the universal distribution is different in the tropics or in the arctic?)
 ideally using also different models.
I know that this is maybe far too much work, however I think at least a second case should be simulated and the predictions made with your present universal distribution be compared to the actual rainfall statistics in the second scenario. But even then, the rainfall statistics might be much more dependent on the implemented microphysics scheme.  Reading through your data analysis, I wondered how robust the statistics are, i.e. are there enough clouds to sample from in each of the regimes you mentioned?
Specific comments:
 Line 25 to 28: There are also studies that indicate that, on average, the amount of precipitation is not influenced due to the buffering effect of clouds. The idea of buffering is described in Stevens and Feingold (2009) and a study indicating that there might, on average, be no influence is Seifert et al (2012) (although the latter study is based on an operational NWP model, i.e. the coupling to aerosol is limited). Hence at this point it is important to state the context more clearly: Do you ask the question for a specific cloud? Do you ask the question for the amount of precipitation averaged over an area? See also the paragraph starting at line 90.
 Line 30 to 32: I do have problems understanding this sentence. Not every cloud produces rain and a cloud is an example of a system with unbalanced sources and sinks (otherwise the cloud would not have formed)? Please clarify.
 Line 120 to 121: You refer to the domain of your simulation by pointing the reader to a plot of radiative fluxes. You should either only state the domain of your simulation by indicating the geographical coordinates or adding a geographical map. I prefer the latter.
 Line 127 to 128: To which degree do your results degree on the choice of the lateral boundary conditions for the aerosols?
 Line 296: You refer the reader to a figure in the supplementary material; please include the figure in the main text.
 Line 317: The predictions do not always reproduce the simulated values, e.g. the black circle in panel b is off; also in panel d a more stagnant behaviour is predicted instead of the decrease that is visible in the solid line.
 Caption of Figure 4, fourth line: The sentence within the brackets appears exaggerated to me. You can only assess the sensitivity to your experiments, which is different from a "universal" sensitivity.
 Figure 6: Is there a motivation for the thresholds used? Why not use "simpler" values, e.g. 0.5 instead of 0.4 or 29 instead of 28.7?
Technical corrections:
 Line 36: "it is" should read "is"
 Line 86: interpreted
 Line 118: "were" instead of "where"
 Line 175: based on
 Line 182: becomes
 Line 221: "colored lines in figure 4"
 Lines 248 and 288: The section numbering should read 5.1 and 5.2 instead of 5.0.1 and 5.0.2
 Line 267: There is a period missing after the equation.
 Line 292: In particular
 Line 310: It should read M_0, ..., M_3
 Line 323: fractions
 Line 331: Delete one of the "because"
 Line 343/344: It should read "...precipitating (and highly cloudy) regime, ..."
 Line 356: that the universal
 Line 363: It should read \gamma_0  \gamma_3 instead of \gamma_{03}
 Line 366: Index k should only range between 0 and 3.
 Line 387: "...parameter space is a..."
 Line 392: Delete "of"
 Line 396: "if the distribution"
 Line 460: next step
 Figures 1, S1, S2, : There are missing labels for the axes.
 Caption of Figure 2, second line: columnaveraged
 Figure 2b: Units are missing in the legend. I suggest to add the units in the caption.
 Caption of Figure 4, fourth line: "sensitivity" instead of "sensitive"
 Figure 9: I suggest to also indicate the regime in the axes label instead of only the values for L, M, H.
 Caption of Figure 9, third line: distribution
 Caption of Figure S4, first line: M_1 should read M_0?References:
 Seifert, A., Köhler, C., and Beheng, K. D.: Aerosolcloudprecipitation effects over Germany as simulated by a convectivescale numerical weather prediction model, Atmos. Chem. Phys., 12, 709–725, https://doi.org/10.5194/acp127092012, 2012.
 Stevens, B., Feingold, G. Untangling aerosol effects on clouds and precipitation in a buffered system. Nature 461, 607–613 (2009). https://doi.org/10.1038/nature08281

AC1: 'Reply on RC1', Kalli Furtado, 06 Oct 2021
We agree with you that to get an idea about the true universality of the result, we should ideally simulate more cases, at different locations, and with different models.
We have added a new section (Section 7.3) that:
1) constructs the universal distribution for 20 case studies of midlatitude weather, using a convectionpermitting (1.5km resolution) regional model over the UK;
2) constructs universal pdfs for three large (17 x 12 degree) regions, covering tropical Pacific, northeastern Pacific, and Southern Ocean, from 20years of daily mean precipitation from AMIP simulations with a global climate model.The inclusion of the UK cases samples a variety of different meteorological regimes. The global simulation samples a different microphysics scheme (a singlemoment scheme), and convectionscheme rainfall, in three different climatological backgrounds (tropical deep convection, subtropical stratocumulus, and midlatitude storm tracks). The results show that the ‘universal’ distribution is approximately independent of the factors sampled. Please see our revised manuscript for the full details.
We also agree with your suggestion to try and strengthen the statistical robustness of the results. The new analysis of other model configurations, different regions, and more cases, supports the conclusion that the existence of a widely applicable rescaling is statistically robust.
To physically motivate this statistical property, we have also added a new section (Section 7.4) which gives a theoretical argument for the existence of a universal scaling, based on a stochastic rainfall generator. The generator treats rainfall at each point as a sum of independent precipitation ‘events’ with durations and intensities drawn at random from underlying probability distributions. We show that this stochastic model also has datacollapse to a universal distribution. Please see the revised manuscript for full details.You made several, very helpful specific points which are addressed inline below (and in the manuscript):
Specific comments:
RC: Line 25 to 28: There are also studies that indicate that, on average, the amount of precipitation is not influenced due to the buffering effect of clouds. The idea of buffering is described in Stevens and Feingold (2009) and a study indicating that there might, on average, be no influence is Seifert et al (2012) (although the latter study is based on an operational NWP model, i.e. the coupling to aerosol is limited).
Response: we've added the following “Furthermore, Stevens and Feingold (2009) suggested that in some situations the answer to this question [of whether ACIs increase/decrease precipitation] is ‘neither’, because systems of clouds adjust to counteract the aerosolinduced changes in precipitation. This implies that although an individual cloud may have a large response to aerosols, changes in the amount of precipitation average over an area may be much smaller. This was illustrated by Seifert et al (2012) who showed that aerosols had negligible effects on precipitation over a large range of region numerical weather predictions.”
RC: Hence at this point it is important to state the context more clearly: Do you ask the question for a specific cloud? Do you ask the question for the amount of precipitation averaged over an area?
Response: We now address this explicitly after the bulletpoints 'aims' in the Introduction:
“We will primarily consider these questions in relation to the area and timeaveraged statistics of rainfall over a large domain for a case of typical case of monsoon rainfall over East Asia.”
RC: See also the paragraph starting at line 90.
Response: we've added a citation to Siefert et al (2012) in this paragraph.RC: Line 30 to 32: I do have problems understanding this sentence. Not every cloud produces rain and a cloud is an example of a system with unbalanced sources and sinks (otherwise the cloud would not have formed)? Please clarify.
Response: I think there is an implicit steady state assumption in Khain’s 2008 framework which implies balanced (i.e., steadystate) systems where sources and sinks are approximately equal over a period of time; it therefore probably doesn’t apply well to the formation/dissipation of individual clouds. I’ve tried to clarify this as follows:
“If the precipitation rate adjusts over time to a slowly varying state in which sources and sinks of condensate approximately cancel out, then an aerosol change which increases the sources more than it increases the sinks will necessarily lead to an increase in the amount of precipitation.
Therefore, if we consider a two systems both precipitating at rate, P, and subject one to an aerosol perturbation, the perturbed system will evolve to a new state with a (possibly) different precipitation rate, P+dP, where dP is due to a change in the net source of condensate relative to the
unperturbed system.”
RC: Line 120 to 121: You refer to the domain of your simulation by pointing the reader to a plot of radiative fluxes. You should either only state the domain of your simulation by indicating the geographical coordinates or adding a geographical map. I prefer the latter.
Response: I hope it's okay that I’ve opted to specify the coordinates in the text (17 ◦ –35 ◦ N,97 ◦ –126 ◦ E) because there are already quite a few figures.RC: Line 127 to 128: To which degree do your results degree on the choice of the lateral boundary conditions for the aerosols?
Response: The new results for a regional NWP over the UK (Section 7.3) which uses a different aerosol model, suggest that scaling of the distributions is independent of LBC details.
RC: Line 296: You refer the reader to a figure in the supplementary material; please include the figure in the main text.
Response: If possible, I’d like to keep this in the supplementary (there are so many figures already)
RC: Line 317: The predictions do not always reproduce the simulated values, e.g. the black circle in panel b is off; also in panel d a more stagnant behaviour is predicted instead of the decrease that is visible in the solid line.
Response: Agreed! The original was too strong – I’ve revised it to:
“In most cases, the predictions are able to reproduce the simulated values of the moments reasonably well. The agreement is slightly less good for some values of the singlemoment reconstructions and for the highestorder moment tested.”RC: Caption of Figure 4, fourth line: The sentence within the brackets appears exaggerated to me. You can only assess the sensitivity to your experiments, which is different from a "universal" sensitivity.
Response: Changed “universal” to “dimensionless”.
RC: Figure 6: Is there a motivation for the thresholds used? Why not use "simpler" values, e.g. 0.5 instead of 0.4 or 29 instead of 28.7?
Response: We deliberately chose cloudregime boundaries that isolate the three rainfall regimes shown in S3 as well as possible (See discussion on lines 357–360)RC technical corrections:
Response: Thank you very much for finding all these! All corrected (with the exception of Fig. 9 axislabels for which I’d rather keep the full labels, if possible, because these are more precise descriptions of the regimes).
 In the manuscript, you state at multiple locations that you construct a universal distribution for the rainfall statistics. However, the whole analysis is based on three simulations of one case. At the very end (starting at line 455) you mention some of these serious assumptions that might limit the conclusion to only this single simulated case. At the moment, your results show that for your model and for this simulated case there is a universal distribution. To get an idea about the true universality of the result, one should

RC2: 'Comment on acp2021443', Anonymous Referee #2, 27 Jul 2021
Summary
This study describes what the authors term a “universal” selfsimilar probability density function of rescaled rainfall rate/intensity and its invariance under aerosol perturbations. It is an intriguing idea that seems to be wellsupported by the analysis of a single largedomain simulation. While I agree with the authors that it is beyond the scope of this work to simulate a multitude of cases with a variety of models, etc., I find the assertion of “universal” scaling from one simulation to be quite a stretch. Either the language needs to be changed, or some other evidence given that the “universal function” will hold for different storm types/climatological contexts. Another obvious perturbation that would increase confidence in the universality assertion would be to perturb sensitive parameters of the microphysics scheme (akin to what the authors did in Furtado et al. 2018). As it is, I believe the title puts it best: there is evidence for a strong statistical link between AIE and selfsimilar distributions, but I am not convinced that this is the last word on the characteristics of the underlying distribution. I recommend the study for publication pending the authors’ response to the above critique and several minor and typographical comments below.
Minor/typographical comments
 L118: “were performed” instead of “where performed”
 L182: “suppression becomes stronger”
 L199: “referred to as rainfall intensity”
 L200: “where rain is falling”
 L201: “CDNCconditioned mean rainfall rate”
 L208: “up to four orders”
 L252: “the sum of CDNCconditioned”
 L270: “assumption corresponds to the simplification”
 L292: “fewer than two moments”
 L301: I am confused – the parameters in Table 1 have different symbols. Please correct.
 L301: Do you mean Figure 6b? Hard to tell because it looks like the axis labels are wrong.
 L305: There is no factor of M_{2}(n) in Eq. 7. Do you mean M_{1}(n)?
 L310: I think you mean “M_{0},…,M_{3}”
 L319321: What is your metric for “capturing the trends” in Fig. 7? It looks to me like you get a great fit for M_{0} and then the fit degrades with increasing moment order. Even M_{1} is pretty far off for single moment N_{a}=1.
 L351353: Scale breaks are common in systems like these due to both statistical noise and violations of scaling laws. Can you rule out the latter?
 L391: “the probability distribution of”…of what?
 L396: “We do not if the distribution is…”
 L404: Should there be an “and” between the definitions of r_{1} and r_{2}?
 L414415: This sentence is confusing. I suggest you break it into two and reword.
 L416: “a family of powerlaw relationships”
 L424: “a detailed understanding of how aerosols…”
 L427: “rather than seeking a physical reason for why aerosols…”
 L434: “choose these moments”
 L460: “next” instead of “nest”
 Fig. 4 caption: “and hence the sensitivity of…”
 Fig. 6a: should axes read “M1,fit” and “M1” instead of referencing M2?
 Fig. 9: I am confused about which regime is which in the figure. Could you descriptively label the xticks instead of the visuallydistracting cloud fraction bounds?

AC2: 'Reply on RC2', Kalli Furtado, 06 Oct 2021
We agree with your suggestion that our assertion of “universal” scaling from one simulation is perhaps quite a stretch. Rather than completely revise the language, we give some further evidence that the “universal function” holds for different storm types/climatological contexts and different microphysics schemes/parameters.
To strengthen the claim of ‘universality’, we added a new section (Section 7.3) which includes the following:
1) the universal distribution for 20 case studies of midlatitude weather, using a convectionpermitting (1.5km resolution) regional model over the UK;
2) universal pdfs for three large (17 x 12 degree) regions, covering tropical Pacific, northeastern Pacific, and Southern Ocean, from 20years of daily mean precipitation from AMIP simulations with a global climate model.The inclusion of the UK cases samples a variety of different meteorological regimes. The global simulation samples a different microphysics scheme (a singlemoment scheme), and convectionscheme rainfall, in three different climatological backgrounds (tropical deep convection, subtropical stratocumulus, and midlatitude storm tracks). The results show that the ‘universal’ distribution is approximately independent of the factors sampled.
The comparison to the UK NWP cases encompasses a different aerosol microphysics scheme (but the same cloud microphysics).
The global AMIP simulations use a different cloud microphysics (the single moment scheme used in Furtado 2018) and a convection parametrization. So, the similarity of the nondimensional pdf across these simulations strengthens the claim that some form of ‘universality’ is at work (at least for the Unified Model). Please see the revised manuscipt for the full details.RC: As it is, I believe the title puts it best: there is evidence for a strong statistical link between AIE and selfsimilar distributions, but I am not convinced that this is the last word on the characteristics of the underlying distribution.
Response: we agree with this; I’ve added sentences to the last paragraph of conclusions to this effect.You made several very helpful Minor/typographical comments; these are addressed in bold below (and in the manuscript):
L270: “assumption corresponds to the simplification” Corrected
L292: “fewer than two moments” Corrected
L301: I am confused – the parameters in Table 1 have different symbols. Please correct.
Thanks for noticing this! These are parameters needed to get the prefactor x and exponent y parametrically from the aerosol number, Nsol; we’ve corrected the description in the text (Eq. 11) and symbols used in Table 1.
L301: Do you mean Figure 6b? Hard to tell because it looks like the axis labels are wrong.
I’ve corrected the axis labels in 6a.
L305: There is no factor of M2(n) in Eq. 7. Do you mean M1(n)?
Yes, M1(n); corrected
L310: I think you mean “M0,…,M3” Corrected
L319321: What is your metric for “capturing the trends” in Fig. 7? It looks to me like you get a great fit for M0 and then the fit degrades with increasing moment order. Even M1 is pretty far off for single moment Na=1.
Agreed! We’ve revised the discussion of this Figure:
“In most cases, the predictions are able to reproduce the simulated values of the moments reasonably well. The agreement is slightly less good for some values of the singlemoment reconstructions and for the highestorder moment tested.”L351353: Scale breaks are common in systems like these due to both statistical noise and violations of scaling laws. Can you rule out the latter?
We can’t rule this out – it’s definitely possible; I think this possibility is covered by our concluding remarks about “understanding the extent to this [universality] holds”L391: “the probability distribution of”…of what? corrected
L396: “We do not if the distribution is…”
L404: Should there be an “and” between the definitions of r1 and r2? Yes! corrected
L414415: This sentence is confusing. I suggest you break it into two and reword.
Simplified to: “The choice of M0 and M1 is arbitrary: as shown by Field and Shutts (2009), any pair of moments could be used for the reconstructions.”
L416: “a family of powerlaw relationships” corrected
L424: “a detailed understanding of how aerosols…” corrected
L427: “rather than seeking a physical reason for why aerosols…” corrected
L434: “choose these moments” corrected
L460: “next” instead of “nest” corrected!
Fig. 4 caption: “and hence the sensitivity of…” corrected
Fig. 6a: should axes read “M1,fit” and “M1” instead of referencing M2?
corrected
Fig. 9: I am confused about which regime is which in the figure. Could you descriptively label the xticks instead of the visuallydistracting cloud fraction bounds?
I hope it’s ok, I’ve left the quantitively descriptive labels; I know they are a bit cumbersome, but I think they help with reproducibility of the results.
Kalli Furtado and Paul Field
Kalli Furtado and Paul Field
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