Aerosol particles affect Earth’s energy balance through exerting effective radiative forcings (ERFaer) via both aerosol–radiation interactions and aerosol–cloud interactions1. Specifically, aerosols can alter radiative balance directly by scattering and absorbing sunlight (ERFari) and also indirectly by acting as cloud condensation nuclei (CCN), in turn modifying cloud properties and precipitation (ERFaci;2,3,4). ERFaci has been known as the most uncertain component in anthropogenic forcing, in which the sensitivity of cloud droplet number concentration (Nd) to aerosol plays a fundamental role5,6.

The latest report by the Intergovernmental Panel on Climate Change provided a best estimate (5 to 95% confidence interval) of the total ERFaer at −1.1 (−1.7 to –0.4) W m−2 (ref. 1). The resulting cooling due to aerosols in turn counteracts about half of the ERF due to CO2 from 1750 to 20197. To mitigate risks of air pollution, most industrial regions have implemented air quality regulations, for example, emissions reductions in Europe (EU) and North America (NA) since the mid-1980s8,9 and in China since 2006–201010,11. This could bring additional ‘unmasked’ warming due to mitigated cooling from aerosols, leading to what has been called a climate ‘penalty’12. Modelling studies showed that environmental policies induced warming of 0.25 ± 0.12 °C globally from 2015 to 205513, 0.35 °C over the eastern United States from 1980 to 201014, 0.45 ± 0.11 °C over EU from 1970 to 201015 and 0.12 ± 0.02 °C over East Asia (only attributed to emissions reductions in China) from 2006 to 201716. Interestingly, more recent evidences from multi-satellite observations together with models also show a clear increase in negative ERFaer by 0.1 to 0.3 W m−2 from 2001 to 2019, confirming that the climate penalty has now clearly started to happen17.

To confidently predict the future climate penalty, a better observational constraint on ERFaci is vital because ERFaci contributes most to the uncertainty of ERFaer18,19. In this regard, long-term trends in aerosol loading and cloud variables observed by satellites may be particularly helpful. There have been some attempts to link long-term trends in cloud variables with emissions change. Along with the emissions reduction in EU and the United States, Nd declines were observed correspondingly20,21,22,23, while trends in liquid water path and cloud fraction (CF) were not always consistent with emissions change22,23 due to their large natural variability compared to the variability driven by Nd. Consistent trends in Nd and aerosols were also reported over India (IN)21,23 and the East China (EC) Sea22,24,25 but were not evident over China21,23. Those findings support theoretical hypotheses2,3 but only in terms of qualitative consistency in signs across regions17.

Here our study provides quantitative estimates of the Nd sensitivity from decadal variations via filtering out high-frequency signals. Together with the analyses based on daily variations of aerosol and Nd, this study comes up with a robust depiction of the Nd sensitivity that the aerosol–Nd relation is nonlinear with a saturated Nd under polluted conditions even in log–log scale. This, in turn, suggests that the commonly used ordinary least-squares (OLS) linear fit on ln aerosol–ln Nd (OLS method) might be problematic to accurately predict the Nd change on the basis of aerosol changes. To tackle this, a sigmoidal fit is proposed as a better alternative and then used to describe the past and predict the timing of cloud-mediated climate penalty with future emissions reduction. Furthermore, we show that current climate models have a limited ability to reproduce the past Nd variation, mainly owing to the insufficient representation of the saturation effect. Our findings highlight the importance of improving the estimation of aerosol–cloud interactions from both observational and modelling sides, especially in terms of the nonlinearity of Nd sensitivity.

Observed decadal trends

Unlike qualitatively comparing the consistency between trends in aerosol and cloud variables17, here we compute the Nd sensitivity from decadal variations in a quantitative sense. To isolate the signal that is solely relevant to decadal emissions change, a de-seasonalization and a locally weighted scatterplot smoothing were used (Methods).

Decadal variations of normalized anthropogenic aerosol emissions, aerosol index (AI) and Nd from 2001 to 2020 over major industrial regions (Supplementary Fig. 1) are shown in Fig. 1. Continuous declines in sulfur (SO2) emissions are seen with changes, obtained as linear trends, of –5% per year over EU and –9% per year over NA. In contrast, the SO2 emissions over IN increase throughout the whole period by 4% per year; in EC it has been increasing until around 2008 (11% per year) and decreasing thereafter (–9% per year). Black carbon (BC) and organic carbon (OC) emissions have similar trends as SO2 but are less pronounced.

Fig. 1: Decadal variations of emissions, aerosol index and cloud droplet number concentration during 2001–2020 over major industrial regions and their adjacent oceans.
figure 1

ad, Anthropogenic SO2, BC and OC emissions from the Community Emissions Data System (CEDSv_2021_04_21;44) over EU (a), NA (b), IN (c) and EC (d). eh, AI and Nd from MODIS, Nd from CMIP6 hist/ssp245-GHG and hist/ssp245-NAT experiments over EU (e), NA (f), IN (g) and EC (h). il, Same as eh but for adjacent oceans: EUO (i), NAO (j), INO (k) and ECO (l). Here the variables are normalized against the 20-year mean value in each region, meaning the percent deviation from the mean value (in units of %). The time series are smoothed using robust locally weighted regression algorithm (LOWESS) with a seven-year time window. The numbers in the top right corner of each plot show relative trends (% per year). The corresponding multi-year means and absolute linear trends are displayed in Supplementary Table 1. Also shown is the Nd sensitivity (SAI) calculated from decadal variations of AI and Nd excluding any high-frequency noises.

Due to the short lifetime of aerosols, the variations of AI—a proxy of fine-mode aerosol concentration—are tightly following the emissions changes (Fig. 1e–l). However, Nd changes are not always consistent with AI, which is mostly aerosol-environment dependent. For relatively clean regions (AI < 0.25), the Nd variations in general well fit aerosol changes; over the rather clean ocean of NAO, a 1:1 Nd-to-AI sensitivity (\({S}_\mathrm{AI}=\frac{{{{\rm{d}}}}\ln {N}_{{{{\rm{d}}}}}}{{{{\rm{d}}}}\ln\mathrm{AI}}\), unitless) is even observed; NA and EUO also have high SAI of 0.49 and 0.57, respectively. An exception is EU, where an increasing AI is seen until 2010, presumably due to the increasing anthropogenic OC emissions (Fig. 1a) and continuous wildfire events26. In this case, the AI variation is linked more to increasing non-CCN aerosols (BC and OC) rather than the expected declining CCN (sulfate), hence hampering the SAI estimate. In contrast, for polluted regions, Nd is less responsive to aerosol changes; over IN, a 53% increase in AI from 2001 to 2020 results only in 13% enhancement in Nd. This is particularly severe over EC, where Nd does not change at all with such a strong switch in aerosol trends (135% increase before 2008 and then −95% decline afterwards). Similar behaviours also appear in their adjacent oceans (Fig. 1k–l). The corresponding SAI for those polluted regions are lower than 0.15, much smaller than the lower bound (0.3) given by ref. 6.

In addition to aerosols, clouds also respond to global warming and internal climate variability on interannual and decadal timescales1, which were reported to largely affect observed trends in CF and liquid water content27. To confirm if the Nd changes are solely a manifestation of aerosol impacts so that the decadal variations can be confidently used to calculate SAI, we examine Coupled Model Intercomparison Project Phase 6 (CMIP6) multi-model simulations with external radiative forcing only from greenhouse gases (CMIP6-GHG) and only from natural solar variations and volcanic aerosol (CMIP6-NAT; Supplementary Table 2 includes model information). The negligible trends (less than 0.05% per year) generated by the two experiments support the idea that aerosol emissions governed Nd changes on a decadal scale.

Implications for N d sensitivity

It is important to note that SAI is calculated based on decadal variations of regional means, in turn limiting aerosol variability, which ties SAI to a certain aerosol condition. There is a strong negative dependence of SAI on regional mean AI (Fig. 2a). In very polluted regions, which is generally updraft limited, SAI is close to 0, while for relatively clean oceans, which is more likely aerosol limited, it can be almost 1, that is, the maximal value we would expect. This reflects a transition from aerosol-limited to updraft-limited regime with increasing aerosol28.

Fig. 2: Observed relationships between aerosol index and cloud droplet number concentration and implied Nd sensitivity.
figure 2

a, Dependence of SAI on AI, where SAI is inferred from decadal variations and the mean ± standard deviation (error bar) of AI is calculated based on 20-year annual means (sample sizes n = 20) for each region. b, The upper panel depicts the joint AI–Nd histogram constructed from 5 km × 5 km daily observations of non-precipitating liquid clouds over global oceans, where each column is normalized so that it sums to 1. The blue dot indicates the median Nd in each AI bin. Also shown are OLS linear and sigmoidal fits on blue dots (n = 30), with the shaded area representing the 95% credible interval (according to a Student’s t test) that represents the fitting uncertainty. Corresponding fitting functions are displayed as well where x = ln AI and y = ln Nd. The lower panel in b shows the derivative of the sigmoidal fit function that is equivalent to SAI, along with the dependence of SAI on AI shown in a. c, Schematic explaining the advantage of sigmoid method over OLS method in capturing Nd change with AI. The vertical black dashed line is at an AI of 0.05, with unreliable aerosol retrievals lying to the left of the line31. The dotted cyan and grey curves in b and c indicate the plausible predictions considering the issue of large retrieval error under clean conditions. Data used to generate a are 20-year observations (2001–2020) of AI and Nd from MODIS Level 3 1° × 1° products. Data used to construct b and c are one-year (2008) MODIS 5 × 5 km2 daily Nd and AI (interpolated from 1° × 1°) observations over global oceans between 60° S and 60° N, in which the 2B-CLDCLASS product45 from CloudSat radar is utilized to exclude precipitation pixels to reduce uncertainty.

However, what we have been doing is to compute the Nd sensitivity as a fixed value with aerosol varying for a specific regime. Here the regimes are mainly used to constrain the co-variability of aerosol type and meteorological background29. The problem is, no matter how accurately those regimes can be constrained, as long as S is kept as a constant with the aerosol regime, the aerosol-updraft transition is naturally fixed. But according to the finding here, it is apparently not true. By continuously increasing aerosol, we will eventually go to the updraft-limited regime, for example, the insensitive Nd seen in EC and IN. A reduced S with aerosol increasing or a saturated Nd at high aerosol loading even in log–log scale is thus theoretically expected.

By considering the linear aerosol–Nd relation in logarithmic space, the OLS method can capture the tendency towards saturation as reflected by a power-law relation in regular space, but it is insufficient to capture the absolute saturated Nd under high aerosol loading unless the sensitivity is set to zero. Although this nonlinear ln aerosol–ln Nd relation has been observed in previous studies that used different CCN proxies30,31,32, the OLS method was still used to fit the relation. As shown in Fig. 2b (the upper panel), the AI–Nd joint histogram constructed from non-precipitating clouds over global oceans is well characterized by an S shape. Note that the flat Nd under clean conditions is not physically meaningful and instead is caused by the problematic aerosol retrieval33. The conventional OLS method is unable to capture the ‘shallow’ relation on both low and high ends (Fig. 2b). Besides this, the OLS method is sensitive to the probability distribution of aerosol as it puts more weight to commonly occurring aerosol values. This explains why the low sensitivity was often observed over the pristine Southern Ocean31,34, where a strong Nd response is expected instead.

To overcome the issues raised by the OLS method, we propose using a sigmoidal function to fit the ln AI–ln Nd relation. Here the median Nd in each AI bin is taken as the input of regression, ensuring that the equal weight is put to each AI. It is exciting to see that the sigmoid method perfectly captures the Nd variation. The peak SAI of sigmoid method is 1 at AI = 0.16, which is a demarcation between the aerosol-limited and transition regime, assuming first-order unchanged updraft with aerosol. In the aerosol-limited regime, the reduced SAI was reported as an artefact of the aerosol-retrieval issue33, thus the plausible value should be 1, given the nature of the aerosol-limited regime28. Then, the plausible Nd prediction is inferred accordingly (dotted cyan/grey line in Fig. 2b). This corrected sigmoid curve will be used in the rest of the paper. Surprisingly, in the aerosol-updraft transition regime (AI > 0.16), the SAI from daily variations are overall well consistent with ones derived from decadal trends—decreasing sensitivity as aerosol increases, except for EU, where the AI variation cannot reflect the CCN decadal trend. The two lines of evidence confirm the nonlinearity of ln AI–ln Nd relation.

Capturing this nonlinearity is crucially important when predicting Nd evolution and/or estimating radiative forcing due to aerosol–cloud interactions (RFaci). According to the observations, Nd starts to be largely saturated around AI = 0.35, but the OLS method gives a continuously increasing Nd (Fig. 2c). This means that the OLS method tends to overestimate Nd change from pre-industrial (PI) to present day (PD) over very polluted regions, while underestimating it over moderately polluted regions, which would bias the spatial patterns of Nd change30 and hence RFaci34,35,36.

Past and future trends in N d and RFaci

The OLS method has been often employed to estimate the PD–PI Nd change (ΔNd) and RFaci, however, it is difficult to evaluate its accuracy due to the impossibility of observing the PI Nd. In this regard, satellite-observed Nd trends over past decades provide a great opportunity to evaluate the predictability of OLS and sigmoid methods. Moreover, the future trends are also of particular interest. As recently documented by ref. 17, on a global scale, the reduction in aerosol emissions since 2000 brought additional warming radiative effects through both direct and indirect effects. Nevertheless, at a regional scale over China, we do not actually see the climate penalty mediated by the indirect effects, as Nd nearly did not change because of the saturation effect at high aerosol. It is expected that with the future emissions reduction, the climate penalty will eventually manifest at increasing rates. Thus, what is interesting now is when this will happen.

The agreement between the AI predicted by the historical SO2 emissions and observed by MODerate Resolution Imaging Spectroradiometer (MODIS) lends credibility to future AI predictions (Fig. 3). From these, in turn, the transient changes in Nd can be inferred. The OLS-diagnosed Nd strongly scales with aerosol changes (Fig. 3b,e) and thereby is unable to reproduce the saturated (slightly increasing) Nd over EC (IN) in the past. Also, for the SSP2–4.5 and SSP3–7.0 pathways in IN, where the AI remains higher than the critical AI (Fig. 3d and Extended Data Fig. 1), the updraft-limited behaviours are expected, however, the OLS-diagnosed Nd is still highly sensitive to aerosol.

Fig. 3: Observed and predicted aerosol index, cloud droplet number concentration and radiative forcing during 2000–2100.
figure 3

af, Past and future mean regional changes in AI (a,d), Nd (b,e) and RFaci (c,f) over EC (top) and IN (bottom) diagnosed from past (grey) and future aerosol emissions under strong-mitigation (SSP1–2.6; green) and high-emissions (SSP3–7.0; orange) scenarios. To enhance the conciseness and readability, the results for mid-emissions (SSP2–4.5) are not displayed here and instead are shown in Extended Data Fig. 1. Historical observations of AI and Nd from MODIS Level 3 1° × 1° products (2001–2020) are shown in cyan. Solid and dashed lines indicate the predictions using OLS and sigmoid methods, respectively, for Nd and RFaci, with the shaded areas representing the 95% confidence interval (according to a Student’s t test) that represents the regression uncertainty. The horizontal dashed lines in a and d denote the critical AI (0.35) from the sigmoidal fit in Fig. 2b, below which the updraft-limited regime is transiting towards the aerosol-limited regime, that is, the aerosol effect starts to appear. Here the critical AI is defined as the AI value when SAI equals to 0.2, in which case the aerosol effect on Nd is assumed to be negligible.

It is interesting to see that the sigmoid method performs very well in reproducing the past trends over each region—not only the saturated Nd in EC and IN (Fig. 3b,e) but also strong declines in NA and EU (Extended Data Fig. 2). Looking at the near-term future, the sigmoid-diagnosed Nd stay almost unchanged under the scenarios SSP3–7.0 for EC and the SSP2–4.5 and SSP3–7.0 for IN. When going to the cleanest scenarios (SSP1–2.6), the decline in Nd would occur around 2025 in EC and 2050 in IN. The regression uncertainty can alter the predicted year by ten years for EC (2020–2030) and seven years for IN (2045–2052) (Fig. 3a,d).

The Nd changes are consequently reflected in RFaci evolutions. The strong changes in aerosol emissions in EC over the past two decades did not exert any appreciable RFaci (Fig. 3c). Similarly in IN, the strong sulfur emissions resulted in a regional mean RFaci of merely –0.45 W m−2 by 2020 relative to 2001 (Fig. 3f). However, for the regions that already transited out of the updraft-limited regime due to air quality policy implementions (including NA and EU in Extended Data Fig. 2), aerosol reductions efficiently led to stronger warming with RFaci of 1.37 W m−2 and 0.91 W m−2 by 2020, respectively.

According to the sigmoid-based predictions, if the strongest air quality policy (SSP1–2.6) will be applied, the rapid warming via RFaci are estimated to occur around 2025 in EC and 2050 in IN (the same time as Nd falling), with ~17 and ~30 years delayed compared to the conventional OLS predictions. Note that we considered aerosol-limited SAI = 1 in the sigmoidal prediction. Sensitivity experiments further show that the diverse treatments of aerosol-limited SAI lead to spreads in the RFaci of ~1–2 W m−2 at 2100 (Supplementary Fig. 2), but this does not affect the prediction of timing when the climate penalty will occur. Similarly, the warming in EC will be postponed by ~22 years under SSP2–4.5; but under SSP3–7.0, no additional forcing will be imposed to EC and IN owing to the saturated Nd. Beyond this delayed warming in polluted regions, the sigmoidal predictions reveal an accelerated warming in clean regions (NA and EU), with nearly double increase rates in RFaci relative to the OLS results (Extended Data Fig. 2). As cloud adjustments approximately scale with RFaci6,37, the predicted evolutions are relevant for ERFaci, too.

Performance of CMIP6 models in reproducing N d trends

Modelling studies demonstrated enhanced future warming and precipitation from aerosol reductions14,16. However, the lack of a proper representation of the observed nonlinearity of the Nd response to aerosols (in log–log space) in climate models may imply they overestimate the warming from aerosol reductions in polluted regions that are still in the updraft-limited regime. For example, the climate models suggested a regional ERFaer of ~3 W m−2 induced by emissions reductions during 2006–2017 over EC, along with strong enhancements of surface air temperature (~0.3 °C) and precipitation (~70 mm per year) (ref. 16). Such strong climate effects are unlikely to occur in the absence of aerosol indirect forcing, as it contributes the largest part of ERFaer1,6.

Therefore, it is essential to evaluate the performance of current climate models in reproducing observed Nd trends. We see that none of the CMIP6 models can reproduce the observed unchanged Nd trends in EC (Fig. 4). Though some models perform better for the regions where Nd is not completely saturated (IN, NA and EU), the majority of models still tend to systematically overestimate the relative changes compared to observations and sigmoid-based predictions. Under the future polluted scenario (SSP3–7.0), the saturated Nd is expected over EC and IN (Supplementary Fig. 3), according to our analysis, but the CMIP6 models instead show fairly strong Nd changes.

Fig. 4: Linear trends in cloud droplet number concentrations from MODIS, observation-based predictions and CMIP6 models.
figure 4

Trends in Nd retrieved from MODIS Level 3 1° × 1° products and predicted by OLS and sigmoid regression methods (group I) and simulated by 17 CMIP6 models with empirical parameterizations (group II) and detailed activation schemes (group III), along with 95% confidence intervals (according to a Student’s t test) depicted as error bars. ad, The whole period of 2001–2020 (n = 20) is chosen for calculating the trends in IN (b), NA (c) and EU (d), but 2001–2008 (n = 8) for EC (a), because from 2008 onwards, there was a trend reversal and also the SO2 emissions used in CMIP6 models were incorrect46. Blue dashed lines highlight the 95% confidence intervals of trends from MODIS. Green shaded areas mark GISS-E2-1-G/H models that have both the empirical parameterization (OMA) and the advanced activation scheme (MATRIX).

When focusing on models that have the same dynamical core, turbulent diffusion and cloud microphysics but only differ in activation treatments (green shaded areas), it is clear that the detailed activation scheme consistently performs better than the OLS empirical parameterization. The detailed activation scheme accounts for the depletion of the maximum supersaturation under high aerosol loading38, thus allowing the simulation of the reduced Nd sensitivity with more aerosol abundance39,40 to some extent. This, however, is still insufficient to fully capture the nonlinearity as observed by satellites.


Our results based on both decadal and daily variations demonstrate that the aerosol–Nd relation even in log–log space is not linear as commonly assumed. Specifically, the Nd sensitivity is shown to be negatively correlated with aerosol loading in a region, indicating a transition from aerosol-limited to updraft-limited regime with aerosol increasing. This finding suggests that the widely used OLS method and corresponding RFaci estimate30,34,35,36 are problematic because they assume a constant Nd sensitivity despite the mean aerosol varying.

As a step forward, a sigmoidal function is proposed and shown to fit the S-shaped ln AI–ln Nd relation seen by satellite statistics. The optimized sigmoid method overcomes two major issues: (1) the nonlinearity of the ln aerosol–ln Nd relation and (2) the poor retrieval capability under clean conditions, which drives part of the uncertainty in aerosol–cloud radiative forcing39,41. As a result, the sigmoid method performs very well in reproducing the past Nd trends—not only the saturated Nd in EC and IN but also strong declines in NA and EU, lending credibility to the near-term Nd and RFaci predictions.

Although the cloud-mediated climate penalty of air quality improvements was not observed in EC over past decades because of the saturation effect, it is expected to manifest eventually with future mitigation of air pollution (scenario SSP1–2.6). According to the sigmoid-based predictions, the rapid warming via RFaci is estimated to start occurring from around 2025 in EC and from 2050 in IN. In relatively clean regions (NA and EU), the sigmoid-based predictions show much larger increasing rates in RFaci compared to the conventional OLS method. This highlights the urgency of mitigating greenhouse gas emissions to avoid strong temperature increase rates.

ERFaci can be faithfully projected only if the historical Nd evolution is faithfully simulated. However, we show that none of the CMIP6 models can reproduce the saturated Nd over EC satisfactorily. Generally, the majority of models tend to overestimate the relative changes of Nd. Although the detailed activation scheme is better than the OLS empirical parameterization, it is still insufficient. Failure to simulate the saturation effect might explain why climate models suggest stronger cooling from aerosols in the 1970s42. The findings here emphasize that further improvements to the current activation scheme are needed to achieve better ERFaci projections.

A few potential caveats to the sigmoid predictions should be noted. Considering the large uncertainties in retrievals over land, the sigmoidal curve was inferred from well-filtered single-layer liquid clouds over the global ocean. The cost of retrieval quality is to discard a portion of clouds, which is acceptable compared to the large confounding effects caused by retrieval errors32,43. Additionally, the differing backgrounds of meteorological conditions and aerosol types among regions might lead to a difference in the threshold of AI for the updraft-limited regime that cannot be reflected in the current sigmoidal fit. However, this effect appears small (Supplementary Fig. 4). The reproduction of historical Nd trends over land largely supports the utility of this sigmoidal relationship over land. However, to thoroughly fix the above issues, using improved CCN and cloud retrievals over land31 and constraining updraft regimes32 would be useful ways forward. Additionally, to eliminate the confounding effect of precipitation on the aerosol–Nd relation, only non-precipitating clouds were analysed, so the sigmoid-based Nd and RFaci projections did not account for the impact of changing precipitation under global warming. Although those caveats might add uncertainties or alter the predicted year when the climate penalty will occur, it does not hamper the robustness of the optimized sigmoidal fit as a promising alternative for the conventional OLS method that has been shown to be problematic.


Aerosol and N d observations

Level 3 1° × 1° products of the MODIS Collection 6.1 (MOD08 and MYD08) provide aerosol optical depth (AOD) and Ångström exponent (AE) retrieved at several wavelengths globally47,48. In this study, the AOD is from Dark Target and Deep Blue combined product. Compared to AOD, the AI is considered a better proxy for CCN because it is more weighted towards smaller particles49,50. The AI is derived from AOD and Ångström exponent (AI = AOD × AE), in which AE is calculated from AOD at wavelengths of 460 and 660 nm. To minimize the influence of heterogeneous aerosol distribution, we discarded aerosol retrievals where the relative standard deviation is larger than unity51.

Cloud droplet effective radius (CER) and cloud optical depth from the MODIS Level 2 products (MOD06 and MYD06;52) are used to calculate Nd based on the adiabatic approximation53. The retrievals at 3.7 μm are used as expected to produce more accurate CER retrievals in inhomogeneous conditions54. The following sampling strategy is applied to obtain confident retrievals, that is, only including single-layer liquid clouds with (1) cloud-top temperature higher than 268 K, (2) CER > 4 μm and cloud optical depth > 4 to reduce the uncertainty in weak optical signal conditions53, (3) CF at 5 km resolution > 0.9 and sub-pixel inhomogeneity index (cloud_mask_SPI) < 30 to reduce the retrieval errors in broken cloud inhomogeneous conditions54 and (4) solar zenith angle < 65° and sensor zenith angle < 41.4° to minimize the uncertainties raised by cloud 3D effects and multiple scattering55.

The 2B-CLDCLASS product45 from CloudSat radar for the year 2008 is employed to identify precipitation. The precipitation data at a 1.4 × 2.5 km2 resolution are matched to the nearest MYD06 5 × 5 km2 pixels to obtain co-located precipitation flag and Nd. To exclude the possible confounding effect of aerosol–precipitation interactions on Nd sensitivity estimate32, only non-precipitating clouds (with the flag of ‘no precipitation’) are analysed to generate the AI–Nd joint histogram, which is then used for linear and sigmoidal fits between AI and Nd.

For the analysis on long-term trends, aerosol and cloud observations in the morning (MOD08 and MOD06 from Terra for 2001–2020) and afternoon (MYD08 and MYD06 from Aqua for 2002–2020) are averaged to represent daily-mean conditions. However, for the AI–Nd joint histogram (Fig. 2b) that is used for the predictions, the simultaneous precipitation observation from CloudSat is necessary for excluding precipitating clouds, thus only observations from the A-Train constellation of satellites in the afternoon (2B-CLDCLASS, MYD08 and MYD06 from CloudSat and Aqua for 2008) are used. We show that one-year data are sufficient to capture the sigmoidal relationship with only a slight change in the statistics compared to the analysis on a longer period (Supplementary Fig. 5).

Emissions data

For the historical period (2001–2019), anthropogenic emissions of sulfur dioxide (SO2), OC and BC are obtained from the newest version of the Community Emissions Data System (CEDSv_2021_04_21;44). An important update compared to the old version of CEDS used in CMIP656 is a correction on aerosol emissions trends in China. Comparing to somewhat unchanged SO2 in the old CEDS from 2006 onwards, the newest version is able to capture strong declines in reality44.

For the future period (2020–2100), anthropogenic aerosol emissions from different shared socio-economic pathway (SSP) scenarios are used57, including the low-emission SSP1–2.6 (‘sustainability’), intermediate SSP2–4.5 (‘middle of the road’) and high-emissions SSP3–7.0 (‘regional rivalry’) scenarios. Notably, the officially released SSP emissions are harmonized to old CEDS emissions (CEDS-v2017-05-18) in the year 201558, which causes a discontinuity in the emissions between the newest CEDS data and the future scenarios. To perform a consistent analysis, we thus utilize the software Aneris59 to harmonize regional-integrated SSP emissions trajectories to the newest CEDS; the methodology of Aneris are described by ref. 60.

Climate model data

The Nd simulated by climate models are taken from CMIP6 ‘historical’, ‘hist-GHG’ and ‘hist-NAT’ experiments for the period 2001–201461,62 and then extended by the Scenario Model Intercomparison Project (ScenarioMIP) ‘sspxxx’, ‘ssp245-GHG’ and ‘ssp245-NAT’ experiments until 210063. Here ‘sspxxx’ means different SSP scenarios (SSP1–2.6, SSP2–4.5 and SSP3–7.0). The ‘historical’ and ‘sspxxx’ experiments are driven by all forcings, whereas ‘hist-GHG’ and ‘ssp245-GHG’ experiments only by greenhouse gas forcing and ‘hist-NAT’ and ‘ssp245-NAT’ experiments only by natural forcing. Due to the difficulty in determining the top height of liquid clouds from CMIP6 outputs, alternatively, we make use of the maximal Nd in a vertical atmospheric column to compare with the cloud-top Nd from satellite64. To investigate the impact of aerosol activation treatments on the simulated trend in Nd, the models are further divided into two subsets that using: (1) empirical approach that directly links Nd to aerosol concentration and (2) detailed activation schemes, respectively. The details of CMIP6 models analysed in this study are given in Supplementary Table 2.

Isolating decadal signal from high-frequency noise

The systematic change in anthropogenic aerosol emissions occurred over a quasi-decadal timescale. To identify the aerosol–cloud signals solely related to the decadal emissions change, possible noises (that is, high-frequency variability) need to be excluded, specifically (1) the seasonality caused by co-varied meteorological fields and emissions and (2) the occasional fluctuation due to natural emissions, for example, volcano eruptions, wild fires and dust storms. To eliminate (1), we first aggregate the filtered daily AI and Nd into monthly means at each grid point and then remove the monthly cycle from the monthly series by subtracting the difference value between monthly climatology and full-period mean. After regional averaging, we obtain a de-seasonalized monthly series with sample size of 240 (12 months × 20 years) for each grid point. Furthermore, the locally weighted scatterplot smoothing (LOWESS) method is applied to the de-seasonalized series to exclude (2). At each data point, LOWESS conducts a weighted regression within a prescribed span width that determines the number of nearby data used in each local fit65. The choice of span width depends on the timescales of interest; here a seven-year time window is chosen to smooth out high-frequency noise. It is illustrated in Supplementary Fig. 6 that the derived Nd sensitivities only change slightly if a different time window is chosen (for example, 4 and 14 years).

Regression models for capturing aerosol–N d relation

Two regression models are employed to fit aerosol–Nd relations in this study. The first model is the widely used OLS linear regression

$$\ln {N}_\mathrm{d}=\beta \ln \alpha +c$$

where β is the sensitivity of Nd to CCN proxy, α. The OLS method describes a power-law relationship between aerosol and Nd, which has been broadly observed by in situ aircraft measurements, ground- and satellite-based remote sensing, however, mostly over clean to moderately polluted regions66. The second model is a sigmoidal regression

$$\ln {N}_\mathrm{d}=\frac{{a}_{1}}{1+\mathrm{e}^{{a}_{2}\ln \alpha +{a}_{3}}}+{a}_{4}$$

which has a characteristic S-shaped curve, thereby is able to capture the markedly saturated Nd at high aerosol as seen in satellite observations30,31,32,41.

N d prediction

The key idea of predicting Nd from an observational perspective is to bridge two connections: (1) anthropogenic aerosol emissions to aerosol and (2) aerosol to Nd, by which future aerosol and Nd can be inferred from different SSP scenarios.

It has been documented that sulfate (thus CCN) concentrations are less sensitive to SO2 precursor emissions under polluted conditions because of the limited availability of oxidants in the atmosphere and the intensive condensation sink of nucleating H2SO4 vapour onto existing aerosols67,68. Therefore, we use a power-law relationship to describe this nonlinearity. Because non-sulfate aerosols have been found to contribute only minimally to the long-term trends in AOD69,70 and therefore to AI, only SO2 emissions are used to predict AI (treated as CCN proxy). In this case, a linear fit on a logarithmic scale is applied:

$$\ln {{{\rm{AI}}}}={a}_{1}\ln {{{{\rm{SO}}}}}_{2}+c$$

For each individual region, the regression models are trained by annual-mean CEDS emissions and MODIS aerosol (2001–2019) and then applied on SSP emissions to predict AI in different future scenarios (2020–2100). Using the CMIP6 model output as a direct source for obtaining aerosol in the near-term future is also an alternative approach; however, considering the incorrect emissions used in CMIP6 models (‘Emissions data’ section), we choose not to do so here.

Regarding the aerosol-to-Nd relations, both sigmoidal and OLS methods (as detailed earlier) are utilized but with the preference being put on the former. Data used to construct the regressions are one-year (2008) MODIS 5 × 5 km2 daily Nd and AI (projected from 1° × 1° without altering the original values) observations over global oceans between 60° S and 60° N, which are carefully filtered for retrieval errors and precipitation to reduce uncertainty. For the OLS method, the data are grouped into 20 AI bins, with each bin having an equal number of samples. The median values of Nd and AI in these bins are then used in OLS regression as in refs. 31 and 32. As for the sigmoid fit, an AI–Nd joint histogram is constructed first with fixed intervals in a logarithmic scale; then the medians of Nd in each AI interval together with central AI are considered as inputs for the regression.

Given the AI–Nd relationship over land is quite unreliable due to the retrieval issues; for example, even the ‘anti-Twomey effect’ can be seen from satellites in most continental regions43,71,72, here we apply the aforementioned AI–Nd relationship over global oceans to different continental regions. Note that using a single global relation would cause differences between observed and predicted Nd for individual regions. This is probably attributable to the varying (1) cloud adiabaticity, (2) surface reflectance and (3) meteorological background across regions, in which (1) and (2) appear to impact Nd retrieval, while (3) reflects the capability of droplet activation. It is also interesting to note that the difference between observed and predicted Nd was found to be close to a certain constant for each region and independent of long-term temporal evolution30. Building on this idea, we add a correction constant—that is, defined as the difference in multi-year averaged Nd between observations and predictions—to predicted Nd for each individual region to overcome this issue.

Radiative forcing

From predicted Nd, the change in ln Nd relative to 2001 (Δ ln Nd) is computed. On the basis of Δ ln Nd, the change in cloud albedo is derived using the Twomey formula73; then radiative forcing due aerosol–cloud interactions (the Twomey effect; RFaci) relative to 2001 can be thereby computed:

$$\mathrm{R{F}}_{{{{\rm{aci}}}}}=-{F}^{\downarrow }{f}_{{{{\rm{liq}}}}}\frac{{\alpha }_{{{{\rm{cld}}}}}(1-{\alpha }_{{{{\rm{cld}}}}})}{3}\Delta \ln {N}_{{{{\rm{d}}}}}$$

where F and αcld are mean incoming solar radiation flux and the cloud albedo from the Clouds and the Earth’s Radiant Energy System and fliq is the CF of liquid clouds from the MODIS Level 3 product.