Research article 14 Mar 2022
Research article  14 Mar 2022
A strong statistical link between aerosol indirect effects and the selfsimilarity of rainfall distributions
 Met Office, Exeter, UK
 Met Office, Exeter, UK
Correspondence: Kalli Furtado (kalli.furtado@metoffice.gov.uk)
Hide author detailsCorrespondence: Kalli Furtado (kalli.furtado@metoffice.gov.uk)
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 cloud–aerosol 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 parameterised by surprisingly few (at most 3) independent “degrees of freedom”: the selfsimilar density and auxiliary information about two moments of the rainfall distribution. Comparisons to convectionpermitting forecasts of midlatitude weather and atmosphereonly global simulations show that the selfsimilar density is also independent of model physics and background meteorology. A theoretical explanation for this invariance is given, based on numerical results from a stochastic rainfall simulator. This suggests that, although aerosol indirect 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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The indirect effects of aerosols on precipitation influence the Earth's hydrological, energy and geochemical cycles on a range of timescales. Moreover, because of diversity in model representations of aerosol–cloud interactions, they are a large source of uncertainty in weather and climate predictions. These uncertainties have their proximal origins in what is essentially an engineering problem: different communities of developers have adopted discrepant approaches for encoding aerosol indirect effects into their models. However, these discrepancies are in part rooted in a lack of scientific consensus as to the mechanisms by which aerosols affect clouds and precipitation. These discrepancies were starkly highlighted by Khain et al. (2008), who reviewed a large number of earlier studies and classified them according to whether increasing the concentration of aerosols increased or decreased the surface precipitation. They concluded that the precipitation response depends on such a large range of factors, including the macroscale cloud regime, cloud microphysics and the thermal and dynamic conditions of the ambient atmosphere, and that a general, systemindependent answer to the following question is not possible: do aerosols increase or decrease precipitation? Furthermore, Stevens and Feingold (2009) suggested that the answer to this question may often be “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 range of regional numerical weather predictions.
In the absence of a general theory, much research has focused on elucidating the physical mechanisms that determine the precipitation response (or lack thereof) on a casebycase basis. Khain et al. (2008) suggested that this could be done by considering how, for each specific system, the various sources and sinks of condensed water respond to aerosol perturbations. 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 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+δP, where δP is due to a change in the net source of condensate relative to the unperturbed system. If we denote the response of the rate of production of condensate to an aerosol perturbation by δG and the response of the loss rate by δL, then response of the mean precipitation rate, P, can be written heuristically as
The mean precipitation, P, therefore increases or decreases according to whether δG−δL is positive or negative. In general, δG is due to changes in condensation and atmospheric dynamics (i.e. convergence), and δL is due to changes in evaporation and dynamics. In general the responses of these factors are strongly coupled and are highly systemdependent, from which the diversity in aerosol indirect effects originates. As an example of this diversity, we may follow Khain et al. (2008) in contrasting subtropical stratocumulus with tropical deep convection over oceans. Stratocumulus clouds, when capped by drier air in the free troposphere, exist in an evaporationdominated regime, where an increase in aerosol concentration increases drying of the boundary layer due to enhanced cloudtop mixing (Ackerman et al., 2004; Xue and Feingold, 2006). By contrast, because maritime deep convective clouds exist in humid environments, their response is dominated by increased generation of condensate as cloud droplets ascend. Khain's source–sink approach focuses on understanding mean precipitation changes. In some situations, the sensitivities of other hydrometeorological quantities are of equal importance, for example, the sensitivity of the rainfall frequency, occurrences of extremely heavy rainfall or other characteristics of rainfall variability. In the most general case, we are therefore interested in the response of the frequency distribution of surface rainfall rates, f(p), as the concentration of aerosols changes.
Given sufficiently detailed observations or models of a cloud or system of clouds, the source–sink framework of Khain et al. (2008) provides a detailed understanding of why a particular precipitation response was measured or simulated. To the best of our knowledge, it cannot at present provide prior predictions of why a particular combination of G and L changes was the necessary response to an aerosol perturbation. In other words, except for cases of simplified theoretical models (Albrecht, 1989; Twomey, 1966), we can only explain why a particular response was observed, not why it had to occur in preference to any other candidate for the response. Tao et al. (2012) give several examples which highlight this difficulty and show how it arises from the multiplicity of different cloud–aerosol interaction mechanisms, or “pathways”, which can operate in the same system and compete to determine the overall precipitation response. For example, in deep convective clouds, because increasing aerosols reduces the size of the cloud droplets but increases the liquid water path, warm rain generation is suppressed, but cold rain generation may be enhanced. Therefore, a theory for the precipitation response of deep convection requires explanations for why a particular balance was struck between these competing effects in a given situation. This balance involves adjustments in many microphysical processes, so the theory should explain why these processes responded as they did (and not in some other way). A detailed theory for an aerosol indirect effect should therefore be able to predict the following: for a given cloud system, which processes will respond to an aerosol perturbation? How large will these responses be, relative to each other? How and why do these processlevel responses determine the overall adjustment in sources and sinks of condensate? In purely formal terms, such a theory would predict the δG and δL as functions of an arbitrary aerosol perturbation, δN_{a}. To the best of our knowledge, there is currently no theory of cloud–aerosol interactions capable of doing this. Hence we need to resort to numerical simulations to determine the relationship between δG, δL and δN_{a} empirically. Given the wide variety of clouds, the multiplicity of cloud–aerosol interaction pathways available to these systems and the potential sensitivity of the pathways to cloud regime and ambient environment, it is not surprising that the reviews by Khain et al. (2008) and Tao et al. (2012) found that many such empirical theories exist in the literature.
In this paper, rather than investigating the mechanisms by which aerosols affect precipitation, we will address two simpler questions the answers to which provide insights into the structure that an eventual mechanistic theory should have. These questions are as follows:

What are the fundamental ingredients that a mechanistic theory needs to predict?

Are there any constraints on how these ingredients can be transformed due to an aerosol perturbation?
We will deliberately avoid the question of why rainfall changes in given way, for a given cloud regime. Rather we will take a set of simulated sensitivities of rainfall distributions to aerosol perturbations as given and ask how much information is actually needed to describe (“parameterise”) these sensitivities. As such, we are not concerned with predicting by how much rainfall increases or decreases in response to aerosol changes (this fundamental question is outside the scope of the framework that we will propose) but rather with identifying a minimal set of information that is required to make such predictions. We will primarily consider these questions in relation to the area and timeaveraged statistics of simulated rainfall over a large domain for a case of typical case of summer monsoon rainfall over East Asia.
The first of the above questions is related to the concept of “degrees of freedom”: how many independent variables are needed to describe the effect of an aerosol perturbation? For this to be nontrivial, we obviously need to seek as minimal a set of fundamental parameters as possible. For example, the physical mechanisms governing the dynamics of stratus clouds are different from those governing deep convective clouds; hence it may not seem surprising that they can respond differently to aerosols. But which properties of these systems are essential to their differing responses? What is the minimal set of properties that a mechanistic theory of cloud–aerosol interactions needs to predict in order to specify the aerosol response of a cloud system? We will show that, even if an “aerosol effect” is interpreted in the general sense of “any aerosolinduced change in the rainfall rate distribution”, a relatively small number of parameters is sufficient. The parameters that we identify are related to the dependencies of the first and second moments of the rainfall rate distribution on clouddroplet number concentration (CDNC) and aerosol number concentration (AC) – information that is readily available from simulations and satellite retrievals.
The second question is related to the concept of invariants: are there any properties which remain unchanged when a system is perturbed into a higher or lower aerosol state? If we restrict attention to large enough volumes, and short enough timescales, then the total mass of water (including accumulated surface precipitation) and the total energy are obvious atmospheric quantities that are approximately conserved when the aerosol amount changes. These quantities are however of limited use for aerosol–cloud interactions because we are interested in the partitioning of mass between the condensed and gaseous phases and the effect that this partitioning has on the precipitation rate. A slightly more useful invariant would be a statistical property of rainfall that did not change. A candidate would be the total rainfall accumulated over a large enough area and a long enough time. In situations where this variable is conserved, it does furnish a useful constraint: that the frequency with which rainfall occurs must be inversely proportional to the mean rainfall rate, as aerosol varies (e.g. if the intensity of rainfall increases, the frequency of occurrence must decrease to conserve the amount of rainfall). However, there are many situations, not least when considering individual clouds or shorttimescale responses, when accumulated rainfall is not conserved. In this paper we will show that the rainfall rate distributions simulated by our cloud–aerosol interacting mesoscale model have a common, underlying “shape” that is independent of CDNC and independent of aerosol concentration. This underlying distribution can be extracted by rescaling simulated rainfall rate distributions into a dimensionless form. Selfsimilarity of this kind is common in systems where dynamical processes occurring on different scales are stretched (or spedup) copies of each other. Examples from atmospheric science include aggregation of ice particles (Field et al., 2005) and boundary layer turbulence. Moreover, Field and Shutts (2009) and Lovejoy et al. (2008) showed that observed rainfall also has a selfsimilar rate distribution.
The existence of an invariant rainfall distribution in our simulations implies that the cloud–aerosol interactions act by squashing or stretching this distribution. We will show that these stretches amount to transforming how frequently rainfall occurs and how much rainfall occurs for a given number of cloud droplets. It is therefore because an invariant distribution exists that a succinct specification of cloud–aerosol interactions in terms of the CDNC dependencies of only two moments of the rainfall rate distribution is possible. In addition we will show that a single moment, e.g. rainfall frequency, is sufficient if it is supplemented by an empirically determined frequency–amount relationship which can be used to predict the other required moment. We will show that these frequency–amount relationships can be parameterised as power laws with aerosoldependent parameters. This implies that systems with the same powerlaw exponents and the same invariant distribution will have the same sensitivities to aerosols.
The simulations analysed were performed with a convectionpermitting (0.03^{∘} horizontal resolution) configuration of the Met Office Unified Model. A description of the model setup can be found in Furtado et al. (2018), together with a detailed description of the nonaerosol components of the microphysics scheme (see also Grosvenor et al., 2017). The simulated domain is situated over central and southern China (17–35^{∘} N, 97–126^{∘} E). The simulated period is from 12:00 UTC on 17 May to 12:00 UTC on 22 May 2016, during which time a cyclonic vortex formed in the lee of the Tibetan Plateau (105^{∘} E, 30^{∘} N) and propagated southeast across China over a 48 h period (see Fig. 1; Furtado et al., 2018, 2020). The simulations use a doublemoment version of the Clouds and AeroSols Interaction Microphysics scheme (CASIM), in which five species of hydrometeor (cloud, rain, ice, snow and graupel) have prognostic masses and number concentrations. The aerosol concentrations are initialised with horizontally homogeneous but vertically varying values that are allowed to evolve via advection, turbulent mixing and twoway coupling between cloud microphysics and aerosols. The initial vertical profiles are retained along the lateral boundaries to maintain a source of aerosol that is constant in time. Lateral boundary conditions for nonaerosol prognostic fields are derived hourly from a global model forecast with the Met Office Unified Model. The primary meteorological fields (including winds, temperature and moisture) are reinitialised on a 36 h cycle from Met Office operational global atmospheric analyses. Only hours 12 to 36 of each forecast are used in this study to avoid a spinup period at the beginning of each cycle. This reinitialisation does not apply to the aerosol fields, which are freerunning for the entire period. Hence, for each forecast after the 12:00 UTC 17 May cycle, the initial aerosol field is taken directly from the proceeding cycle.
The method for coupling clouds and aerosols is described in Miltenberger et al. (2018): interstitial aerosol particles are removed by activation of cloud droplets; following activation, a prognostic variable for dissolved aerosol mass is coadvected with the hydrometeors so that it is transported conservatively through clouds. During evaporation of hydrometeors, the soluble mass is redeposited into the air with a number concentration equal to the number of evaporated particles. Therefore, redeposited aerosols usually have a mean size exceeding that of the previously activated aerosols (because collision–coalescence gives rain drops that are fewer in number than the cloud droplets from which they develop). Hence, aerosols that were activated from the “accumulation” mode can be converted to coarser particles during evaporation. Icenucleating aerosol particles are not simulated by prognostics, but CDNCs can modulate ice hydrometeor concentrations via a temperaturedependent parameterisation of immersion freezing.
We will compare the results from three simulations with different initial and boundary aerosol concentrations. These will be called the high, intermediate and lowaerosolconcentration experiments. Particularly in the figures, it will be convenient to label these as aero+, aero∘ and aero, respectively. In the atmospheric boundary layer, the orderofmagnitude mass and number concentrations in the highaerosol experiment are 10^{−8} kg kg^{−1} and 10^{10} kg^{−1}, respectively. The typical values of the sphericalequivalent mean radius are 0.1 and 1 µm for accumulation and coarsemode aerosols, respectively. The vertical profile for the intermediateconcentration regime was identified by taking average aerosol concentrations, over the target domain, from a global model simulation with multispecies aerosols and atmospheric composition scheme. The initial profiles for the highconcentration experiment are obtained from the intermediateaerosol profiles by increasing mass and number of accumulation and coarsemode aerosols at each level by a factor of 10 (thereby leaving the aerosol particle size unchanged). A factor of 10 reduction was applied to generate the initial profiles for the low aerosol experiment. Vertical profiles of initial aerosol in the experiments can be found in the Supplement.
We focus firstly on the deep convective regime, which we identify as columns with relatively large fractions of low, mid and highlevel cloud (defined as area fractions of low, mid and highlevel clouds greater than 0.1, 0.6 and 0.8, respectively). These criteria pick out features, such as cloud bands around the cyclonic vortex, and convective clouds over high terrain, which have unbroken layers of condensate distributed throughout the depth of the troposphere. Such features are the main producers of heavy rainfall in the simulation domain (further information on their characteristics can be found in the Supplement).
For this regime, the effects of aerosols on precipitation can be seen in Fig. 2a, which shows the rainfall rate frequency distribution functions for the three aerosolconcentration experiments. The frequency decreases with increasing aerosol concentration for most rainfall rates. In particular, light and heavy rain are more frequent when there is more aerosol. The differences between the simulated distributions reflect the totality of aerosol effects on precipitation in our model, in the selected cloud regime. Statistically, these differences are a result of either changes in the frequency of occurrence of cloudy columns in the selected cloudfraction regime or changes in the rainfall rates produced in these columns.
The aerosol indirect effects that are included in our model are mediated via changes in clouddroplet number concentration (CDNC). Hence, a natural first step for investigating the relationship between aerosol and rainfall rate is to decompose the rainfall rate distributions into components from model columns with different CDNCs. We will use the vertically averaged CDNC in each column to represent the droplet numbers. Mathematically, we may write
where n ranges over a set of prescribed CDNC intervals with widths Δn, and f_{n}(p) is the frequency density of rainfall rate per unit precipitation flux, per unit CDNC. We will call the component distributions, f_{n}, the “conditionally sampled rainfall distributions” (or just CDNCconditioned distributions, for brevity) to indicate that they are constructed by subsampling the total rainfall rate distribution, based on CDNC.
The different coloured lines in Fig. 2b show the CDNCconditioned distributions which contribute to the rainfall distributions shown in Fig. 2a. The set of five logarithmically spaced CDNC intervals shown in the figure's legend has been used for the decomposition. For each colour (CDNC interval), the three lines (solid, dashed and dotted–dashed) correspond to the three aerosolconcentration experiments. We see that increasing aerosol concentration increases the frequencies of rainfall rates occurring at relatively high CDNCs (see, for example, the purple and red lines). This is consistent with an increase in the prevalence of high CDNCs when the aerosol number concentration increases. For smaller CDNCs (see, for example, the orange and blue lines), heavy rainfall is suppressed by increasing the aerosol concentration. The suppression becomes stronger and extends to smaller rainfall rates as the CDNC is lowered. The differences in the statistical properties of rainfall occurring at different CDNC and for different aerosol concentrations can also be seen in the moments of the CDNCconditioned distributions (Fig. 3). The kth moment of the nth conditional distribution is defined by
(where, numerically, the integral can be approximated for any set of precipitation flux intervals). The zeroth moment, M_{0}(n), for each CDNC interval, is related to the frequency of rainfall, ℳ_{0}, via
(Note that the occurrence of the CDNC interval width, Δn, is due to the definition of the f_{n}s as frequency densities, i.e. frequencies per unit CDNC, in Eq. 2.) In general, the kth moment of the rainfall rate distribution is ${\mathcal{M}}_{k}\left(n\right)={\sum}_{n}{M}_{k}\left(n\right)\mathrm{\Delta}n$. In particular, ℳ_{1} is the domainaveraged rainfall amount, which is closely related to the rate of convergence of moisture into the domain. It is convenient for the following to note that Δℳ_{k}(n)=M_{k}(n)Δn is the contribution to the total moment ℳ_{k} that comes from the subsample of rainfall rates that occur with a CDNC of n.
Because they contribute to the rainfall frequency and rainfall amount, the CDNCconditioned rainfall frequencies, Δℳ_{0}(n), and rainfall amounts, Δℳ_{1}(n) (and hence also their equivalent densities, M_{0}(n) and M_{1}(n)), are of particular interest. Figure 3a and b show how M_{0} and M_{1} vary with CDNC for each simulation. For high CDNCs, increasing the aerosol concentration increases the frequency and amount of rainfall. When the CDNC is low, aerosol increases have the opposite effect and suppress rainfall. Another defining characteristic of rainfall is the mean rainfall rate (sometimes referred to as rainfall “intensity”), which measures the average flux of rain at surface points where rain is falling, neglecting points with no or negligible rainfall. For each value, n, of the CDNC, we can define a CDNCconditioned mean rainfall rate, λ_{n}, by the ratio
This is the mean rainfall rate for the subsample of columns where rainfall reaches the surface and the columnaveraged CDNC is n kg^{−1}. The CDNCconditioned mean rain rates are plotted in Fig. 3c, which shows that rainfall intensity increases with CDNC for large aerosol concentrations but decreases with CDNC in cleaner conditions.
A noticeable feature of CDNCconditioned distributions shown in Fig. 2 is the very large (6 orders of magnitude) spread in rainfall frequency density as the CDNC varies. The intersimulation spread for a fixed CDNC is also large: up to 4 orders of magnitude, for some rainfall rates. However, there is some visual indication that the “shapes” of the conditioned distributions are quite similar to each other. This suggests that a suitable simultaneous scaling of precipitation flux and frequency density might reveal that these distributions are rescaled instances of a single, underlying frequency distribution. On dimensional grounds, we expect such a scaling to map rainfall rate to a dimensionless precipitation flux and map the frequency densities, f_{n}, to dimensionless distribution functions.
An obvious candidate for rescaling the precipitation flux for each CDNC is the mean rainfall rate, λ_{n}, because it is the only combination of M_{1}(n) and M_{0}(n) that has the dimensions of a mass flux. Similarly, the quantity
(plotted in Fig. 3d) is the unique combination of the zeroth and first moments that has the same dimensions as the frequency densities. For each aerosol concentration and CDNC, we will use these parameters to define a dimensionless rainfall rate, ${r}_{n}=p/{\mathit{\lambda}}_{n}$, and a dimensionless frequency density, Φ, as follows:
The coloured lines in Fig. 4 show the dimensionless distributions, derived for each CDNC, n, and each aerosol concentration, as functions of their corresponding dimensionless rainfall rates, r_{n}. It is clear that this scaling results in a significant amount of data collapse: the disparate distributions shown in Fig. 2b give rise to very similar distributions in the rescaled variables. The degree of similarity between the scaled distributions is sufficient for us to regard the ensemble mean of the scaled distributions as defining a single (“universal”) dimensionless density function that is independent of both aerosol concentration and CDNC. The histogram of this universal distribution is shown by the black line in Fig. 4. We have tried unsuccessfully to fit a functional form to the universal histogram. This is not to say that no such form exists, but in the absence of one, a pragmatic approach is to use the histogram itself to define the universal distribution. We therefore view the distribution as the probability density of a random variable defined by randomly sampling the empirical histogram shown in Fig. 4. The data specifying the histogram are available in the Supplement.
If the proposed universal distribution is independent of CDNC and aerosol concentration, then we expect to be able to use the scale transformations in Eq. (7) to approximately reconstruct the rainfall rate distributions. In this section we assess the accuracy of these reconstructions. The simplest case occurs if both M_{0}(n) and M_{1}(n) are known functions of the CDNC, n. In this case, Eq. (7) can be used directly to estimate f_{n} for each n, and the total rainfall rate distribution, f, can be estimated as the sum of the f_{n}s using Eq. (2). We will call this case a doublemoment closure because two moments of the CDNCconditioned rainfall distribution are required. If only one moment of the first two moments is a known function of n, then additional information is required that parameterises the other moment (or more generally the aerosolconditioned mean rain rate, λ_{n}, and normalisation, ν_{n}) in terms of the known moment. Here we will consider the case where the CDNCconditioned frequency M_{0} is the known moment and derive an empirical closure relation that specifies M_{1} in terms of M_{0}. This case will be called a singlemoment closure because only one moment of the CDNCconditioned distributions needs to be specified. The doublemoment closure is essentially just a further test of the validity of the data collapse affected by the rescaling the rainfall rate distributions. However, we show in Sect. 5.1 that it permits an insight into aerosol indirect effects because it separates the contributions of changes in the rainfall intensities, λ_{n}, from changes in the frequencies of occurrence rainfall (as represented by ν_{n}). The singlemoment closure (Sect. 5.2) is a more stringent test of the universality of the invariant distribution; moreover, it allows us to identify the minimal information that is needed to parameterise the effects of aerosols on rainfall rate distributions.
5.1 Doublemoment closure
If the first two moments of the CDNCconditioned frequency distributions, f_{n}, are known, then the total rainfall distribution for each simulation can be reconstructed using Eqs. (2) and (7). The black circles in Fig. 5a show this reconstruction for the highconcentration simulation. The solid black line shows the rainfall rate distribution obtained from the simulation. The total frequency distribution is the sum of CDNCconditioned contributions, f_{n}Δn, from each of the specified CDNC intervals. These contributions are shown by the coloured lines in Fig. 5a. The estimated CDNCconditioned densities are shown by the coloured circles. In general, the doublemoment reconstructions can reproduce the conditioned distributions and the rainfall rate frequency distribution. The accuracies of the reconstructions for the other two aerosolconcentration experiments are similar.
Overall, the reconstructions are accurate enough to predict the effects of aerosols on the rainfall distributions. These effects can be seen in Fig. 5b and c, which show the fractional changes in the rainfall rate distributions, compared to the highconcentration simulation. The dashed black lines and symbols show the fractional changes in the simulated and predicted total rainfall rate frequency distribution. The dashed coloured lines and symbols show the fractional changes in each CDNCconditioned distribution. The doublemoment reconstructions are able to capture the changes in rainfall rate frequencies due to the aerosol perturbations. Moreover, the relative contributions to the overall changes coming from each CDNC interval are also predicted well. For example, the reductions in the frequencies of heavy rainfall are predicted as aerosol concentration decreases, and the relative importance of high CDNCs for driving these reductions (red lines and symbols) is also captured.
The doublemoment reconstructions allow the contributions of the CDNCconditioned mean rainfall rate changes to be separated from changes in the CDNCconditioned rainfall frequencies. These contributions can be inferred as follows. Firstly, we denote the highaerosolconcentration experiment as
For one of the other aerosol experiments, we may estimate the rainfall rate distribution after a reduction in aerosol concentration, based on the assumption that the conditional intensities, λ_{n}, do not change from their reference values, λ_{ref}(n). This assumption corresponds to the simplification that aerosol perturbations can alter the number of cloud droplets but that this does not affect the intensity of rainfall for a given CDNC. Making use of the universal distribution, Φ, this estimate is given by
Note that only the normalised frequencies, ν(n), have changed from their reference values, whilst the intensities, λ(n), remain the same as in the highaerosol experiment. Eq. (9) says that if an aerosol perturbation does not affect the intensity of rainfall for a fixed CDNC, then the aerosol effect amounts to a rainfallrateindependent reweighting of the relative contributions from each CDNCconditioned distribution. For a given value of n, the reweighting factor, ${\mathit{\alpha}}_{n}={\mathit{\nu}}_{\text{exp}}\left(n\right)/{\mathit{\nu}}_{\text{ref}}\left(n\right)$, will enhance (α_{n}>1) or suppress (α_{n}<1) rainfall frequency, for that number of CDNC, uniformly across the rainfall rate spectra. The estimated fractional changes, assuming no changes in the CDNCconditioned rainfall intensities, are shown by the solid black lines in Fig. 5b and c. By comparing these to the dashed black lines, it can be seen that they significantly overestimate the suppression of rainfall frequency with decreasing aerosol concentration and do not capture the dependence of the fractional changes on the rainfall rate, p. For each CDNC, n, the short coloured horizontal lines show frequency suppression factors, α_{n}. For CDNCs such that α_{n}<1, the effects of aerosol on normalised rainfall frequency are such that rainfall is suppressed. Where the fractional change in a CDNCconditioned distribution (the dashed coloured lines) is greater than the corresponding value of α_{n}, the mean rainfall rate, λ_{n}, for this CDNC has increased (i.e. the rainfall produced for this CDNC is becoming more intense in response to decreasing aerosol). The inability of the constantintensity estimates to predict the simulation results implies that the suppression of rainfall frequency by decreasing aerosol is therefore partly offset by the simultaneous intensification of rainfall intensities for some value of CDNC.
5.2 Singlemoment closure
The doublemoment closure described in Sect. 5.1 provides a test of validity of the universal distribution. It also allows us to separate aerosolinduced changes in rainfall intensity (at a fixed CDNC) from changes in the relative frequencies of rainfall occurring at different CDNCs. We will now investigate whether the effects of aerosols can be predicted using information about fewer than two moments of the CDNCconditioned distributions. In particular, we will show that the CDNCconditioned rainfall frequency, M_{0}, is sufficient, if it is supplemented by an empirical closure relation for M_{1}.
We have found that M_{0} and M_{1} can be related to each other by
where the parameters x and y vary between the experiments (see Fig. S4a in the Supplement). Furthermore, we found that the parameters x and y are not independent and are related by $\mathrm{log}\left(x\right)={n}_{\mathrm{0}}+ay$, where n_{0} and a are constants that are independent of the aerosol concentration, implying that a single, aerosoldependent parameter (either x or y) is sufficient to specify the relationship between M_{0} and M_{1} (Fig. S4b). Equivalently, it is convenient to express both y and x parametrically as functions of the ratio, ${N}_{\mathrm{a}}/{N}_{\text{ref}}$, of the initial aerosol concentration in each experiment to the concentration, N_{a}, in the intermediateaerosolconcentration experiment:
The parameters δ, γ, ϵ and η are given in Table 1. Figure 6a shows that there is good agreement between simulated first moments, M_{1}(n), and the prediction obtained from the empirical fits.
The empirical relationship between moments can be used to replace M_{1}(n) in Eq. (7) by a function of M_{0}. This gives the rainfall frequencies distributions as functions of M_{0} only. The utility of the singlemoment closure is assessed in Fig. 6b, which compares the simulated rainfall amounts, ${\mathcal{M}}_{\mathrm{1}}^{>{p}_{\mathrm{0}}}={\int}_{{p}_{\mathrm{0}}}f\left(p\right)\phantom{\rule{0.125em}{0ex}}\mathrm{d}p$, above each of a range of rainfall rate thresholds, p_{0}, to the predictions obtained from the singlemoment closure. The good agreement obtained indicates that the simulated aerosol indirect effects can be fully parameterised by the CDNC dependence of the rainfall frequency.
We will summarise the overall indirect effects of aerosols on precipitation by the first four moments, ${\mathcal{M}}_{\mathrm{0}},\mathrm{\dots},{\mathcal{M}}_{\mathrm{3}}$, of the total rainfall rate frequency distributions in each of the simulations. The black lines in Fig. 7 show these moments as functions of the initial aerosol concentrations (where the latter are expressed relative to the intermediateaerosol experiment). The variations of these moments express different aspects of the hydrological sensitivity of the system to perturbing the aerosols: the changes in the zeroth and first moments correspond to the changes in the frequency of occurrence and amount of rainfall, respectively; the second and third moments, ℳ_{2} and ℳ_{3}, express changes in the width of the rainfall distributions, particularly the relative frequencies of occurrence of large rainfall rate. The symbols in Fig. 7 show the values of these moments predicted by the doublemoment (blue) and singlemoment (black) closure relations. In most cases, the predictions are able to reproduce the simulated values of the moments reasonably well. Moreover, the predictions capture the increasing trends in the moments as the aerosol concentration increases. The agreement is slightly less good for some values of the singlemoment reconstructions and for the highest order moment tested. This indicates that the predictions are accurate enough to reproduce the sensitivity for the simulated rainfall to aerosol perturbations. The doublemoment closure is typically more accurate than the singlemoment closure, as expected, because it contains more information about the CDNC dependence of rainfall statistics.
So far we have considered a deep convective regime, where there are relatively large cloudarea fractions at low, middle and high levels. Specifically we selected only model columns where the low, mid and highcloud area fractions exceeded 10 %, 60 % and 80 %, respectively. This is a computationally simple way of selecting deep convective columns, such as those associated with the eastwardpropagating vortex (see Figs. S1 and S2 in the Supplement). There are, however, other regimes of clouds and precipitation occurring within the domain during simulations which may respond to aerosols differently from the deep convective regime.
Regimes for which we may expect aerosol–cloud interactions to differ from those in deep convection are stratus clouds and shallow convection. In general, we may expect different responses to aerosols in regimes where mid and highlevel clouds are present because this may be related to the role of icephase processes. For example, Figs. 1, S1 and S2 show that there is a region of precipitating lowlevel clouds in the wake of the cyclonic vortex. These clouds are readily identified as emitting higher fluxes of longwave radiation than the deep convective cloud regime (Figs. 1c and S2), suggesting that they exist in a regime dominated by warmcloud microphysical processes. In this section, we will extend the proceeding analysis to a range of cloudfraction regimes, chosen to span the cloud types present in the simulations. We will assess the extent to which aerosols affect these regimes differently and the extent to which they exhibit the selfsimilarity in rainfall statistics identified above for the deep convective regime.
We will denote the low, mid and highlevel cloud fractions by L, M and H; these are defined from the model's subgrid cloudarea fractions using the ISCCP cloudheight pressure classification. We divided the model output into three mutually exclusive categories: the deep convective cloud regime, with high fractional cloudiness, described above for which L>0.1, M>0.6 and H>0.8; a regime dominated by low clouds, where L>0.9, M<0.2 and H<0.2; and a transitional/“marginal” regime with intermediate values of midlevel and high clouds (L>0.4, $\mathrm{0.4}<M<\mathrm{0.6}$ and $\mathrm{0.2}<H<\mathrm{0.8}$). In the Supplement, we show that these categories divide the rainfall frequency distribution into a heavily precipitating (and highly cloudy) regime, a moderately precipitating regime with intermediate cloudiness and less cloudy and more lightly precipitating regime (Fig. S3 in the Supplement). We also consider the totality of all cloudy columns, for which at least one L, M or H is nonzero.
Figure 8a shows rainfall rate distributions for each of the selected regimes. We see that the deep convective regime (black) accounts for the majority of the rainfall occurring in the simulations. The other two regimes produce progressively less precipitation as the amount of high and midlevel cloud decreases. For each regime, the CDNCconditioned rainfall rate distributions are calculated then rescaled to their dimensionless forms using their corresponding mean rainfall rates and normalised frequencies. The rescaled distributions for each regime are plotted in Fig. 8b, from which it can be seen that the universal distribution is highly consistent across the regimes (except for the largest values of the dimensionless rainfall flux, where the universal histograms become regimedependent, perhaps because these values are relatively undersampled for the moderate and lightly precipitating regimes). However the empirical relationships between the CDNCconditioned zeroth and second moments are regime dependent (Fig. S4 in the Supplement). Hence the parameters in the singlemoment closure vary across the cloud regimes (Table 1). This is to be expected because the relationship between rainfall frequency and rainfall amount depends on the specific cloud microphysical and macrophysical processes leading to rainfall, whereas we claim that universal distribution does not.
For each cloud regime, we can define the overall sensitivity of rainfall to aerosols by the differences in the moments of the rainfall frequency distribution between the high and lowaerosolconcentration experiments; e.g. for the kth moment we have
where a Δ denotes the difference in its antecedent quantity between the two experiments. For moments ${\mathcal{M}}_{\mathrm{0}},\mathrm{\dots},{\mathcal{M}}_{\mathrm{3}}$, these quantities describe the sensitivities for rainfall frequency, rainfall amount and rainfall variability to aerosol perturbations. The sensitivities, ${\mathit{\gamma}}_{\mathrm{0}},\mathrm{\dots},{\mathit{\gamma}}_{\mathrm{3}}$, for each regime are plotted in Fig. 9 for the simulations (lines) and single and doublemoment closures (symbols). It can be seen that the predictions are in quantitative agreement with the simulation results in the three regimes and for the totality of precipitating, cloudy columns. We see that the deep convective cloud regime has a positive aerosol indirect effect for all four moments (γ_{k}>0, $k=\mathrm{0},\mathrm{\dots},\mathrm{3}$). This is because increasing the aerosol concentration increases the rainfall frequency, amount (Fig. 7a and b) and the occurrence of heavy rainfall (Figs. 7c and d and 2a) in this regime. By contrast, the lowclouddominated and intermediate regimes have negative sensitivity, particularly for rainfall frequency and amount, because increasing aerosols reduces rainfall in these regimes (Fig. 8a). Interestingly, when the domain is considered as a whole, the overall sensitivity of the entire system to aerosol perturbations is small. This is because of the opposing signs of the aerosol effects in different parts of the domain.
Cloud–aerosol interacting systems show a range of responses to aerosol perturbations (Khain et al., 2008; Tao et al., 2012), from precipitation suppression (in, for example, stratocumulus, Xue and Feingold, 2006, Ackerman et al., 2004; shallow cumulus and some deep continental clouds, Khain et al., 2008) to precipitation enhancement in deep convection over oceans and, for some cases, deep convection over land. Ultimately, the change in the frequency distribution of rainfall rates induced by a change aerosol is a function of how all the hydrometeorological processes occurring within the system respond to the aerosols. This includes modifications of the rates of condensation, C, and evaporation, E, and adjustments in the dynamical state, D, of the system. The aerosolinduced change, δf, in the rainfall rate distribution is given by a generalisation of Khain's source–sink framework that also includes dynamical factors:
Unfortunately, since δC, δE and δD are complicated combinations of many variables (Khain et al., 2008), and therefore $\mathit{\delta}(CED)$ is not a simple function of p, Eq. (13) cannot be used directly, and we typically resort to numerical experiments to determine the factors influencing δf in each particular case. The aim of such analyses is often to investigate how individual process rates have responded to an aerosol perturbation and to understand the effects that these have on precipitation. However, to the best of our knowledge, several more basic questions are overlooked by this procedure. Firstly, how many parameters are actually needed to specify δf? (The variety of possible responses suggest that this parameter space is a highdimensional one.) Secondly, can δf be an arbitrarily complex perturbation, or are there any constraints on how cloud processes can adjust to redistribute rainfall over a range of intensities? In particular, are there any properties of rainfall rate distributions that we can expect to be unchanged by aerosol perturbations?
7.1 Is there a statistical property of rainfall that is invariant under aerosol perturbations?
We showed in Sect. 4 that, for any CDNC, n, and aerosol concentration, N_{a}, the probability distribution defined by
where ${\mathit{\nu}}_{n}={M}_{\mathrm{0}}^{\mathrm{2}}\left(n\right)/{M}_{\mathrm{1}}\left(n\right)$ and ${\mathit{\lambda}}_{n}={M}_{\mathrm{1}}\left(n\right)/{M}_{\mathrm{0}}\left(n\right)$, is independent of n and N_{a}. In Sect. 6 we showed that this distribution is also quasiindependent of the cloud regimes simulated. For example, it is approximately the same in lowclouddominated regions and in regions of deep convection (except for large and small r). We do not know if the distribution is independent of the modelling system used, but this will be interesting to investigate in future work. Similarly, our model does not include prognostic icenucleating aerosols; it is possible that these might alter the universal shape. The existence of this distribution implies that rainfall events occurring with different CDNCs are statistically similar, in the mathematical sense that their frequency distributions can be transformed to each other by a change of scales. Equivalently, suppose we have a pair of equally long time series of precipitation values from two rainfall events, one with CDNC, n_{1}, another with CDNC, n_{2}. We can view both these time series as realisations of two different random variables, ${p}_{{n}_{\mathrm{1}}}$ and ${p}_{{n}_{\mathrm{2}}}$. The invariant distribution implies that the random variables ${r}_{\mathrm{1}}={p}_{{n}_{\mathrm{1}}}/\mathit{\lambda}\left({n}_{\mathrm{1}}\right)$ and ${r}_{\mathrm{2}}={p}_{{n}_{\mathrm{2}}}/\mathit{\lambda}\left({n}_{\mathrm{2}}\right)$ are identically distributed (with distribution Φ) and independent of the value of n.
7.2 How many degrees of freedom are needed to describe aerosol indirect effects on precipitation?
The singlemoment and doublemoment reconstructions in Sect. 5 show that precipitation state of a cloud–aerosolinteracting system is specified by

the universal distribution, Φ, for the system, which is independent of the CDNC and aerosol concentrations in the system, and

either
 a.
the zeroth moment, M_{0}(n), and first moment, M_{1}(n) of the CDNCconditioned rainfall rate distributions, f(p,n), as functions of the CDNC, n, or
 b.
a single moment (e.g. M_{0}(n)) and an aerosoldependent, frequency–amount relationship, e.g. ${M}_{\mathrm{1}}\left(n\right)={I}_{{N}_{\mathrm{a}}}\left[{M}_{\mathrm{0}}\right(n\left)\right]$, for the system, which diagnoses the remaining moment (e.g. M_{1}(n)) in terms of the known moment.
 a.
From this information, the rainfall rate distributions of the system can be reconstructed to the degrees of accuracy demonstrated in Sect. 5. The choice of M_{0} and M_{1} is arbitrary: as shown by Field and Shutts (2009), any pair of moments could be used for the reconstructions.
For our simulations, the moment relation, ${I}_{{N}_{\mathrm{a}}}$, can be parameterised as a family of powerlaw relationships of the form
between the CDNCconditioned moments, where only one of the parameters, x or y, needs to be specified directly in terms of the aerosol concentration. The remaining parameter (in this case, x) is a function of the other. We found that a further pair of power laws, x∼y^{a} and $y\sim {N}_{\mathrm{a}}^{\mathit{\gamma}}$, were sufficient to specify the parameters in the moment relations. Hence a total of four constants (see Table 1) is needed to specify the mapping from the universal distribution to the dimensional rainfall distributions.
7.3 How universal is the invariant distribution?
Based on a single set of simulations, it is not possible to evaluate the “universality” of Φ distribution. We have shown that the distribution is approximately independent of CDNC and cloud regime, but dependencies on the modelling system, parameterisation schemes and background meteorology have not been explored. In this section we present evidence which suggests that the distribution Φ is a statistically robust feature of global and regional simulations with the Met Office Unified Model. Firstly, we show that rainfall rate frequency distributions from 23 case studies of midlatitude weather systems forecast with a regional model configuration over the United Kingdom (UK) have the same Φ distribution as the (subtropical) May 2016 case study. Secondly we show that the frequency distributions of daily mean rainfall over three $\mathrm{17}{}^{\circ}\times \mathrm{12}{}^{\circ}$ regions from a 20year global climate simulation also give the same Φ as the regional models. In addition we show that in the global simulation the rainfall rates from the model's microphysics and convection schemes, both rescale to the same dimensionless distribution.
The regional model used for the UK case studies has the same physical parameterisations as the configuration described in Sect. 2, except for the representation of aerosols. In the UK forecasts, aerosols are modelled with a single mass prognostic (representing the combined mass of aerosols) which is produced from surface sources and advected. An aerosol number concentration is diagnosed as described by Wilkinson et al. (2012) and then passed to the microphysics scheme, which calculates activation increments to the CDNCs. The same cloud microphysics scheme (CASIM) is used for both the UK and China cases.
The global model configuration is a version of the Met Office Global Atmosphere (GA; Walters et al., 2019). This configuration uses a singlemoment microphysics scheme (Wilson and Ballard, 1999), coupled to the UK Chemistry and Aerosols (UKCA) model, which provides diagnosed CDNCs to the microphysics schemes. Hence the cloud and aerosol microphysics in the global model is structurally different from that used in the regional models. Moreover, the global configuration includes parameterised, subgridscale convection, which is an additional source of diversity between models.
For the UK case studies, we partition the rainfall into CDNC intervals, as described in Sect. 4, and cloudfraction regimes. We choose the cloudfraction partition to reflect the most common local precipitation regimes, which are (very broadly) frontal systems (with large amounts of high clouds), stratiform cloud decks or shallow convection (with lowlevel cloudiness but little high cloud) and an “intermediate” regime, encompassing other combinations. The results are not sensitive to this classification. Figure 10a shows that the rainfall rate distributions are less sensitive to CDNC than in the China case. The sensitivity of rainfall to CDNC is muted because the simplified aerosol physics in the UK configuration produces less variable aerosol concentrations. However, the combined variability across all CDNC intervals and cloud regimes is still many orders of magnitude. Figure 10b shows that nondimensionalisation collapses the frequency distributions to single distribution. Moreover, this distribution is the same function, Φ, of nondimensional rainfall, that was found in the China case study (the latter is reproduced in Fig. 10b as the red histogram).
The UK and China forecasts both use the CASIM microphysics scheme. To assess the effects of differences in cloud microphysics, we calculate the nondimensional distributions of daily mean rainfall rates from a 20year global, atmosphereonly simulation using the Atmospheric Model Intercomparison Project (AMIP) protocol. Rainfall rates during June, July and August are selected for three geographical regions: the northeastern Pacific (20–37^{∘} N, 220–245^{∘} E), the western tropical Pacific (2–19^{∘} N, 132–157^{∘} E) and the Southern Ocean (52–70^{∘} S, 206–231^{∘} E). This samples subtropical stratocumulus, tropical deep convection and midlatitude cyclones; thus each region tests for occurrence of universality in a different background climate. Because a daily CDNC diagnostic is not available in the output from the simulation used, we partition the rainfall distributions in each region into intervals of the cloud albedo calculated by the radiation scheme. Although not equivalent to the CDNC partitioning used for the regional simulations, this provides another way of classifying the rainfall with a property of clouds that is sensitive to aerosol–cloud interactions. Figure 11a and b shows the frequency distributions of rainfall from largescale clouds (i.e. the microphysics scheme) and convective clouds (the subgrid convection scheme), in each albedo interval (shading), for each region (colour). As expected, relatively heavier precipitation is associated with higher albedo in all regions, and the balance between largescale and convective precipitation varies between the regions. The large diversity of rainfall distributions is removed by nondimensionalisation, which approximately collapses all the frequency distributions onto the same dimensionless distribution. In particular, distributions from different albedo intervals and in different regions become very similar after rescaling. For convective and largescale rainfall, we define a universal distribution by the means of the corresponding dimensionless distributions. These are shown by the dashed and dotted–dashed histograms in Fig. 11c. We see that, despite structurally different treatments of precipitation processes in largescale microphysics and subgridscale convection, the nondimensional histograms for the two schemes are approximately the same. Moreover, these distributions are also very similar to the universal distribution obtained from the convectionpermitting forecasts over China (the solid black histogram). The degree of data collapse (across regions, models and cloud properties) suggests that the distribution Φ may be considered a universal property of the Met Office model.
7.4 An explanation for the simulated rainfall universality
The existence of a widely applicable nondimensional rainfall distribution suggests that it is due to underlying physical characteristics of rainfall producing processes that are independent of cloud properties, cloud type, meteorological conditions and model parameterisations. These factors determine the dimensional rainfall frequency distribution at a particular location and time, but their effects can be described by a twoparameter rescaling of the universal, underlying Φ distribution. In this section we will support this claim by showing that a simplified, “toy” model, based on modelling rainfall as a multiscale stochastic process, can produce synthetic rainfall statistics which also have a twoparameter family of distributions. The simplified model is based on the stochastic rainfall generator analysed by RodriguezIturbe et al. (1984). It simulates a discrete rainfall time series with integer length T as a sum of independent, temporally overlapping rainfall “events”. Each event, e, is described by a duration, d_{e}, and intensity, λ_{e}. At each time step, $k=\mathrm{1},\mathrm{\dots},T$, a new rainfall event is initialised with a fixed probability q_{i}. Hence the total rainfall, $\widehat{p}\left(k\right)$ is the sum of λ_{e} for all events whose durations span across the time k. This generates rainfall time series that are sums of rectangular pulses with random lengths and heights. Models of this type are known to exhibit a rich statistical behaviour and have been used to generate synthetic rainfall rates for hydrological applications (e.g. Burton et al., 2008).
We make two further modelling assumptions: the event durations are numbers of time steps chosen independently from a powerlaw distribution (hence, $\text{Pr}[{d}_{e}=t]\sim {t}^{\mathit{\alpha}}$, where α<0); the event intensities are uniformly distributed in an interval [1,l_{1}]. The parameters in the model are therefore the mean event intensity ($\stackrel{\mathrm{\u203e}}{{\mathit{\lambda}}_{e}}=({l}_{\mathrm{1}}\mathrm{1})/\mathrm{2}$), the powerlaw exponent for the events durations (α) and the initiation probability (q_{i}). These parameters can be varied to imitate external factors (e.g. aerosol perturbations) that affect rainfall event characteristics. For example, increasing the mean intensity (or, equivalently, the maximum rainfall intensity parameter, l_{1}) corresponds to a factor or process that increases the probability of heavier precipitation events and reduces the probability of lighter rainfall events. This is similar to increasing CDNC in our regional model simulations.
In Fig. 12 we show that the stochastic rainfall process can generate rainfall distributions with a selfsimilarity property that is reminiscent of Unified Model simulations. We do not attempt a detailed parametric description of the model's behaviour because it suffices to show that parameter regimes with universal distributions exist. Figure 12a shows examples of rainfall time series for a selection of values of $\stackrel{\mathrm{\u203e}}{{\mathit{\lambda}}_{e}}$ and α (with q_{i}=0.05). As the mean intensity increases, the peak values of the precipitation time series increase; as the duration exponent increases, the rainfall rates become more correlated in time. Figure 12b shows the frequency distributions for several parameter combinations. Increases in heavy rainfall result from more intense rainfall events or from longer duration events. The differences between the distributions are qualitatively similar to the effects of aerosols on the Unified Model simulations (e.g. Fig. 2). After renormalisation, these distributions become statistically nearidentical (Fig. 12c), suggesting the existence of an exact selfsimilarity of rainfall statistics in this part of the parameter space. This similarity can be presumed to be a geometric property of the time series generated by the model.
The stochastic model encodes some basic properties of precipitation physics, namely, that rainfall at a location is a superposition of independent precipitation events with random durations and intensities. External factors, such as aerosols, alter the statistical distributions of the properties of rainfall events (e.g. increasing aerosols may make rainfall events heavier, on average). Hence, by analogy with the stochastic model, we propose that the universality of rainfall statistics in our regional and global simulations is consistent with the following ingredients:

Rainfall statistics are due to the accumulation of quasiindependent rainfall “events” with random durations and intensities.

External factors (e.g. aerosols) and cloud characteristics (e.g. stratiform or convective) affect rainfall event properties, but as these properties vary, the rainfall distributions remain within a family of selfsimilar distributions.
We defer further investigation of these claims to future works.
There is no general theory for how aerosols affect precipitation. Instead, analyses of different cloud regimes have revealed a range of behaviours. Hence, at a theoretical level, a detailed understanding of how aerosols affect a particular cloud regime, or type of cloud system, is probably the most that can be achieved. Such theories usually rely on numerical models to understand the physical mechanisms, or pathways, by which aerosols modify precipitation rates, in a given situation (or class of situations).
In this paper, rather than seeking a physical reason as to why aerosols affect precipitation in a particular way, we have instead investigated the structure that any mechanistic theory of an aerosol–cloud–precipitation interaction needs to have in order to describe an arbitrary change in surface rainfall statistics. We have shown that, despite the diversity in the possible precipitation responses, there is a fairly minimal set of statistical quantities which can describe any response. (At least, we hypothesise that this is the case, based on results from simulations with an aerosol–cloud microphysics scheme.) This set of quantities can be two moments of the rainfall rate distribution as functions of CDNC (a doublemoment closure) or a singlemoment and an intermoment powerlaw relationship that predicts a second moment (a singlemoment closure). It is convenient to choose these moments to be the frequency of occurrence of rainfall (the zeroth moment) and the rainfall amount (the first moment). Given these two quantities (or only rainfall frequency, in the singlemoment case), the entire rainfall rate distribution can be reconstructed with an accuracy that is sufficient to resolve changes in the distribution due to aerosolconcentration changes. This approach, using a small number of moments, is possible because there is an invariant, i.e. universal, frequency density function for a normalised (nondimensional) rainfall rate that is independent of CDNC and is unaffected by aerosol perturbations. The existence of this distribution, particularly its independence from background aerosol concentrations, significantly restricts the number of independent degrees of freedom that an aerosolinduced modification of rainfall can have. In particular, we have shown that if rainfall is partitioned into CDNC intervals, then an aerosol perturbation can affect the number of precipitating points and the mean rainfall rate in each interval, but it cannot alter the probability distribution of rainfall fluctuations relative to the mean rainfall rate. This is because the fluctuations are apparently governed by a universal distribution. Hence, two variables for each CDNC interval are sufficient to specify the rainfall distribution and its response to aerosol perturbations.
This analysis cannot predict whether or not precipitation increases or decreases in response to aerosol. For example, for our simulations we have not tried to explain why precipitation frequency increases with aerosol concentration for the deep convective cloud regime. Rather, we have attempted to understand the relationship between changes in rainfall frequency and changes in other moments of the rainfall distribution. This relationship is fixed by the four aerosoldependent constants needed to specify the frequency–intensity relationship of the regime and by the universal distribution, Φ. Any two systems in which these four constants were the same would respond to aerosol perturbations in structurally the same way; i.e. the dependencies of their rainfall distributions on rainfall frequency would be the same. Moreover, a theory that predicted these four constants, and predicted the response of rainfall frequency to aerosol, as a function of CDNC, would also predict the rainfall distribution change because this could be determined via the universal distribution.
Understanding how universal the Φ distribution actually is therefore constitutes a valuable question for future work. For the simulations used here, the distribution is shown to be approximately independent of cloud regime; e.g. it is the same for lowcloud only and highclouddominated regimes. We have also shown that convectionpermitting forecasts of subtropical and midlatitude weather systems and global climate simulations all share the same nondimensional rainfall distribution. The global climate simulations also show that the universal distribution is independent of whether rainfall is from the microphysics scheme or from the convection parameterisation. A highly simplified model of rainfall as a multiscale, stochastic process provides the tentative theoretical explanation that the universality is a result of geometric properties of rainfall time series. Perturbing an external factor (such as aerosols) modulates the properties of individual rainfall events, but the overall geometry of the rainfall time series changes in a statistically selfsimilar way. Understanding the extent to which this holds over a range of regimes, climate backgrounds, processes and timescales can be investigated further using observations and simulations with other models. We may expect that the invariant distribution is not independent of modelling system and that it will break down for extreme parameter settings (e.g. no aerosols, or fixed CDNCs), so a multimodel analysis and evaluation against observations would be a useful next step.
The Python code and postprocessed model data used are available for download from https://doi.org/10.5281/zenodo.6347484 (Furtado, 2022).
The supplement related to this article is available online at: https://doi.org/10.5194/acp2233912022supplement.
KF and PF jointly conceived the work and wrote the manuscript.
The contact author has declared that neither they nor their coauthor has any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research has been supported by the UK–China Research and Innovation Partnership Fund through the Met Office Climate Science for Service Partnership (CSSP) China as part of the Newton Fund.
This paper was edited by Franziska Glassmeier and reviewed by two anonymous referees.
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 Abstract
 Copyright statement
 Introduction
 Model configuration and simulation setups
 The effects of aerosol number concentration on the simulated rainfall distributions
 Selfsimilarity of simulated rainfall statistics
 Reconstructing the rainfall distributions
 Regime dependencies
 Discussion
 Conclusions
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Financial support
 Review statement
 References
 Supplement
 Abstract
 Copyright statement
 Introduction
 Model configuration and simulation setups
 The effects of aerosol number concentration on the simulated rainfall distributions
 Selfsimilarity of simulated rainfall statistics
 Reconstructing the rainfall distributions
 Regime dependencies
 Discussion
 Conclusions
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Financial support
 Review statement
 References
 Supplement