Abstract
A large fraction of the uncertainty around future global warming is due to the cooling effect of aerosolliquid cloud interactions, and in particular to the elusive sign of liquid water path (LWP) adjustments to aerosol perturbations. To quantify this adjustment, we propose a causal approach that combines physical knowledge in the form of a causal graph with geostationary satellite observations of stratocumulus clouds. This allows us to remove confounding influences from largescale meteorology and to disentangle counteracting physical processes (cloudtop entrainment enhancement and precipitation suppression due to aerosol perturbations) on different timescales. This results in weak LWP adjustments that are timedependent (first positive then negative) and meteorological regimedependent. More importantly, the causal approach reveals that failing to account for covariations of cloud droplet sizes and cloud depth, which are, respectively, a mediator and a confounder of entrainment and precipitation influences, leads to an overly negative aerosolinduced LWP response. This would result in an underestimation of the cooling influence of aerosolcloud interactions.
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Introduction
Aerosols are airborne particles that modify the planetary radiative budget, either directly, by absorbing or scattering radiation, or indirectly, by acting as cloud condensation nuclei (CCN) for the formation of cloud droplets and subsequently modifying the albedo of clouds^{1}. Between preindustrial times and 2019, anthropogenic aerosol emissions caused a negative effective radiative forcing (ERF) of −1.1[−1.7; −0.4] W m^{−2}^{2}, thereby offsetting part of the global warming induced by greenhouse gases. The exact magnitude of the aerosolinduced cooling is hard to quantify, with large uncertainties originating from the understanding of aerosolcloud interactions (ACI). Liquid stratocumulus clouds particularly contribute to this uncertainty because of their moderately high albedo (≈40%) and extensive coverage in oceanic regions with high insolation^{3,4}. In liquid clouds, \({{{{\rm{ERF}}}}}_{{{{\rm{ACI}}}}}\) is a combination of the instantaneous Twomey effect and rapid adjustments of cloud macrophysical properties, namely liquid water path (LWP) and cloud fraction (CF)^{1}. The Twomey effect describes how, at an initially constant LWP, an increase in the cloud droplet number concentration (N_{d}, used as a proxy for aerosol concentrations) will cause a decrease in the effective cloud droplet radius (r_{eff}^{5}) but an increase in the total cloud droplet surface area and therefore an increase in cloud albedo^{6}. This initial change in N_{d} can later trigger LWP adjustments (\(\frac{\partial {{{\rm{ln}}\,{\rm{LWP}}}}}{\partial \,{{{{\ln }}}}\,{N}_{{{{\rm{d}}}}}}\)), which are hard to constrain because they result from the superposition of two counteracting physical processes. On the one hand, a decrease in r_{eff} leads to precipitation suppression and subsequent increases in LWP and cloud albedo^{7}. On the other hand, increased droplet concentrations and smaller radii will lead to suppressed droplet sedimentation and enhanced radiative cooling at the top of the cloud, thereby driving turbulence and entrainment of warm (and dry) air from the free troposphere into the cloud. This leads to evaporation of the smaller droplets, enhanced evaporative cooling and even stronger entrainment, resulting in decreases in LWP and cloud albedo^{8,9,10,11,12}. Precipitation suppression and cloudtop entrainment enhancement are difficult to disentangle as these two processes occur simultaneously and they both involve a feedback loop between cloud microphysical properties and dynamical processes. This is illustrated in Fig. 1a.
The sign and magnitude of LWP adjustments depend on temporal and spatial scales, cloud regimes and environmental conditions, making it hard to interpret mere correlations as causal effects of aerosols on clouds. The main difficulty consists in removing confounding, i.e. the effect that a variable Z (e.g. an environmental factor) has on both a causevariable X (e.g. aerosol properties) and an effectvariable Y (e.g. cloud properties), thereby inducing a spurious correlation between X and Y. Ideally, causality is inferred from randomized controlled trial experiments. Unfortunately, this is rarely feasible in the field of Earth sciences. A good alternative are the socalled opportunistic experiments (e.g. ship tracks or volcano eruptions)^{13,14,15}, or the use of climate models to simulate the response of the climate system to a given forcing while keeping all other forcings constant (i.e. evaluating counterfactual climate scenarios)^{16}. However, opportunistic experiments and model simulations have drawbacks, especially when it comes to representativity, or to computational cost for simulations. For this reason, many studies use nonopportunistic observational data, and in particular, satellite observations that have representative coverage in both time and space.
Previous ACI studies have tried to explicitly identify and remove sources of confounding in observational data, such as the confounding effect of relative humidity (RH) on CF adjustments^{17}, or the confounding effect of rainfall on convective cloud invigoration^{18}. Methods from causal inference^{19,20,21,22} have also been applied to satellite studies of LWP adjustments^{23,24,25,26,27,28,29}. These studies use multivariable regressions of the effectvariable on the causevariable as well as the environmental covariates, or data binning as a function of a given covariate in order to remove spurious confounding effects. Most of these satellite studies find a negative N_{d}LWP sensitivity, i.e. a decrease in LWP following an increase in N_{d}, causing a warming effect that can compensate part of the cooling due to the Twomey effect. The suggested reason is the prevalence of cloudtop entrainment enhancement over precipitation suppression. However, the strength of these two physical processes is modulated by environmental conditions. The above studies (ibid.) have shown that LWP adjustments can become less negative (or even reverse to positive) when conditioning the analyses on given environmental variables: large lower tropospheric stability (LTS) and high free tropospheric RH (RH_{FT}) conditions, precipitating cloud regimes (characterized by low droplet number concentrations, large droplet sizes and deep clouds), i.e. conditions where entrainment enhancement becomes weaker and/or precipitation suppression becomes stronger. This illustrates how accounting for environmental conditions can change the magnitude and sign of a correlation and yield causal effects that can be different from the original correlation. However, environmental variables are often treated equally as variables that can impact the causal effect of N_{d} on LWP, without necessarily specifying whether they act as confounders, mediators, colliders, etc. (Supplementary Fig. 3). A causal graph (Fig. 1b) can help to explicitly describe the relationships between cloud and environmental variables. This allows us to know which environmental factors should be conditioned on, and which factors should not. In fact, Simpson’s paradox describes how, depending on the causal structure underlying the data, one can sometimes draw false conclusions when conditioning on the wrong variable^{20} (Supplementary Fig. 3).
Recent studies have also investigated the temporal development of LWP adjustments in order to use the precedence of cause with respect to effect^{30,31,32,33}. Geostationary satellite data are a promising resource to resolve causality for ACI as their temporal resolution (Δt ≈ 10–15 min) matches the process timescale of macrophysical cloud adjustments to aerosol perturbations. For a stratocumulus with a typical geometrical height H of 300 m and a typical updraft speed of 0.5 m s^{−1}^{34}, the expected circulation time of an air parcel through the cloud height is \(\frac{300\,{{{\rm{m}}}}}{0.5\,{{{\rm{m}}}}\,{{{{\rm{s}}}}}^{1}}=600\,{{{\rm{s}}}}\approx 10\,\min\). At this resolution, it becomes possible to resolve feedback loops^{35} (Supplementary Fig. 4) involved in LWP adjustments. It should be noted that the choice of spatial scale can also introduce confounding. In particular, spatial aggregation could lead to spurious correlations resulting from Simpson’s paradox if performed over an area encompassing different cloud types (Supplementary Fig. 5). The impact of spatial aggregation on ACI has already been addressed in other studies^{36,37,38}. However, Bender et al.^{39} observed that stratocumulus albedo variability is more related to temporal rather than spatial variability (using monthly satellite data on a 1^{∘} × 1^{∘} grid). For this reason, and because less attention has been paid to temporal developments, we chose to focus on temporal developments of domainaveraged cloud properties in this study.
In this study, we apply a transparent causal methodology to investigate LWP adjustments in stratocumulus clouds. We propose the causal graph in Fig. 1b, which encodes physical knowledge about cloud processes (Fig. 1a) and which we apply to geostationary satellite data of the Namibian stratocumulus deck. We then showcase a method to derive causal effects, i.e. causalitygrounded sensitivities that go beyond simple correlations (Fig. 1c) and can shed some light on the conflicting estimates found in the literature by focusing on physical processes rather than state variables. Instead of focusing on precipitation and entrainmentdominated regimes (low vs. high N_{d}) separately, we disentangle LWP adjustments that are simultaneously mediated by rain rates (RR) and entrainment rates (approximated by the entrainment velocity w_{e}). We do not explicitly include further environmental covariates (e.g. LTS, RH_{FT}) as variables of the causal graph to keep it (relatively) interpretable, but instead we investigate the LTS/RH_{FT}specific effects, i.e. how LTS/RH_{FT} modulate the influences of RR and w_{e} on the causal effect of N_{d} on LWP, denoted \({\beta }_{{N}_{{{{\rm{d}}}}},\,{{{\rm{LWP}}}}}\,=\,\frac{\partial {{{\rm{ln}}\,{\rm{LWP}}}}}{\partial \,{{{{\ln }}}}\,{N}_{{{{\rm{d}}}}}}\).
Results
Physical description of the causal graph
Figure 2 shows linear direct causal effects \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\), computed using Wright’s approach^{40,41} applied to timeseries of the Namibian stratocumulus deck (c.f. methods). \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\) represents the direct effect of a variable X_{i} on another variable X_{j} on the l_{ij}lagged arrow linking X_{i} and X_{j} in the graph. The \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\) are similar to linear regression slopes between X and Y except that the graph is used to detect and remove any source of confounding prior to the regression. The results confirm the physical plausibility of the proposed causal graph, as the (statistically significant) signs of the direct causal effects agree well with the physical processes expected to underlie each arrow (marked as “correctly detected” in Table 1). A complete result table is provided in the Supplementary material (Supplementary Table 1). It should be noted that, because the data are adjusted for seasonal and diurnal cycles and standardized (see methods), the absolute magnitude of causal effects derived here cannot be directly compared to other studies that use nonstandardized data. However, one can still comment on the physical relevance of the sign and relative magnitude of causal effects.
Lag0 positive arrows from N_{d}, r_{eff} and H to LWP (arrow A in Fig. 2) simply correspond to the definition of LWP as a vertical integral of the liquid water content^{5}. The negative arrow from N_{d} to r_{eff} (arrow B) describes how an increase in N_{d} causes a decrease in r_{eff} at a constant LWP^{6}. The arrow from H to r_{eff} (arrow C) is positive, in line with the continuous condensational growth of cloud droplets as they are carried upwards in an adiabatic cloud^{42}. Although, in reality, arrows (B) and (C) might be lagged, these effects are considered to be contemporaneous here, as the three variables are derived simultaneously from the same cloudtop satellite measurements.
The cloudtop entrainment enhancement feedback is well described by the direct causal effects, with a negative and significant effect of r_{eff} on w_{e} (arrow D). This describes the fact that larger cloud droplets tend to sediment, thereby moving cloud water away from the inversion level and preventing turbulence induced by cloudtop radiative and evaporative cooling to enhance entrainment^{8,9,10,11}. In turn, w_{e} has a negative effect on r_{eff} and N_{d} (arrow E), indicating the evaporation of entire cloud droplets due to mixing with warm (and dry) free tropospheric air at cloud top. This suggests a mixture of homogeneous and extreme inhomogeneous entrainment regimes^{43}, as was also observed by ref. ^{44} in direct numerical simulations of stratocumulus clouds. The effect of w_{e} on H is also negative and significant, meaning that entrainment of dry and warm free tropospheric air reduces cloud depth. It can be noted that, although the causal effects of w_{e} on r_{eff}, N_{d} and H (Supplementary Table 1) are significant, they are quite weak, potentially due to the largescale approximation used for the computation of w_{e} (see methods). Even though we did not explicitly include environmental variables (RH_{FT}, LTS) in the causal graph, causal effects can be evaluated for data binned by environmental factors. This reveals a regimedependence of entrainment: entrainment mixing becomes more homogeneous (r_{eff} is reduced but N_{d} remains constant)^{43} under moist free tropospheric conditions or polluted conditions (lines 3 and 8 in Supplementary Table 1), which agrees with ref. ^{44} and ref. ^{45}. Under such conditions, the evaporative timescale becomes longer than the mixing timescale, thus evaporating all cloud droplets homogeneously. On the contrary, under dry free tropospheric conditions, entrainment mixing becomes more inhomogeneous (N_{d} is reduced but r_{eff} remains constant) as evaporation becomes faster. Our results also seem to indicate that entrainment becomes more inhomogeneous in unstable boundary layers, although one would expect the opposite, as the mixing timescale should become shorter. This might be due to the fact that, in an unstable boundary layer, where lateral entrainment becomes dominant, the largescale estimate of w_{e} used here is not a good proxy for mixing. A lag 1 was chosen for arrows (D) and (E) to indicate that entrainment does not happen instantaneously at the entrainment interfacial layer.
The precipitation suppression feedback is also detected by the direct causal effects. The positive lag1 arrow from r_{eff} to RR (arrow F) indicates that larger droplets are likely to initiate precipitation at the next timestep. The negative lag0 effect of RR on N_{d} (arrow G) indicates that rain onset at cloud base immediately removes droplets from the cloud. Although, in theory, there could also be a lag0 arrow from RR to r_{eff}, we make the approximation that collection efficiency is roughly independent of size. In fact, for stratocumulus drizzle drops in the range 50–100 μm^{34}, the collection efficiency of cloud droplets >10 μm only varies between 60 and 70% ^{46}. The lag1 arrow from RR to N_{d} (arrow H) describes processes that occur below the cloud and only impact N_{d} at the next timestep. Arrow H is found to be weakly positive and insignificant, although one would expect wet scavenging to make this arrow negative. It is possible that dynamical effects (e.g. updraft enhancement around cold pool edges^{47,48}) somehow counterbalance N_{d} losses from wet scavenging, although this remains speculative and would need to be investigated further.
All variables (except LWP) were chosen to have causal autodependencies (arrow from X(t − 1) to X(t)) to illustrate the inertia of the physical variables that they represent. LWP was chosen to have null autodependency as LWP is fully determined by N_{d}, r_{eff} and H at each timestep. Note that a null causal autodependency does not prevent LWP from having nonzero statistical autocorrelation.
Temporal developments of causal effects
Direct causal effects \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\) confirm the validity of the proposed causal graph but do not provide a full picture of causal effects. Total causal effects β_{X,Y,l} (i.e. between two variables X(t − l) and Y(t) that are not directly linked by a single arrow in the causal graph) can be derived from the direct causal effects using Wright’s path approach, and these total effects can inform us about temporal LWP changes following aerosol perturbations. Figure 3 shows the temporal evolution of the β_{X,Y,l} (see methods) for a selection of (X, Y) pairs, where a positive (negative) β_{X,Y,l} means that an increase in X causes a increase (decrease) in Y after a lag l.
The temporal evolution of precipitation suppression (Fig. 3a) shows a peaking negative \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{RR}}}},l}\) observed 4–6 h after the initial N_{d} perturbation. The N_{d}RR sensitivity then decays back to 0 within 24 h, describing the return of the cloud system to an equilibrium state as other microphysical and dynamical processes take over. Cloud top entrainment enhancement is also well detected (Fig. 3b), with the strongest positive \({\beta }_{{N}_{{{{\rm{d}}}}},{w}_{{{{\rm{e}}}}},l}\) observed about 12 h after the initial N_{d} perturbation and continuing entrainment enhancement well beyond 24 h. The timescale of precipitation suppression is faster than that of cloud top entrainment enhancement, which agrees with theoretical calculations^{49} and observations of ship tracks^{31}.
Also shown on Fig. 3a–b is the regime dependence of \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{RR}}}},l}\) and \({\beta }_{{N}_{{{{\rm{d}}}}},{w}_{{{{\rm{e}}}}},l}\). The boundary layer stability does not seem to impact \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{RR}}}},l}\) so much. Dryer free tropospheric conditions seem to be associated with a more negative \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{RR}}}},l}\), even though there is no obvious physical explanation. Entrainment enhancement is stronger in unstable and drier free tropospheric conditions. Under such conditions, entrainment is favored, and the entrained air causes more evaporation, thereby enhancing evaporative cooling and downdrafts, and in turn further entrainment. In cases of a moist free troposphere, \({\beta }_{{N}_{{{{\rm{d}}}}},{w}_{{{{\rm{e}}}}},l}\) is slightly negative (entrainment suppression), which could be a result of additional moisture being transported into the cloud via entrainment, leading to cloud growth and sedimentation of the cloud top further away from the inversion level. We also evaluated the clean vs. polluted conditions (using a threshold N_{d}), but we do not include them in Fig. 3a, b because conditioning on N_{d} results in a subcase of Simpson’s paradox (the biased lines are provided in Supplementary Fig. 6).
Figure 3c shows that the total causal effect of N_{d} on LWP is initially positive, but quickly becomes negative and remains negative for up to 24 h. This temporal development offers a more complete picture than the negative N_{d}LWP correlation from Fig. 1c. Figure 3d shows the fractions of the total causal effect of N_{d} on LWP that are mediated by RR and w_{e}. Although the RRmediated effect on the LWP is initially slightly positive, it becomes slightly negative after a few hours, which is inconsistent with LWP buildup from precipitation suppression. The slightly negative effect could be due to the lag0 RR → N_{d} arrow being too weakly negative compared to the other arrows in Fig. 2. There could be datarelated or causal modelrelated reasons why the RR variable does not fully behave as expected. For instance, it is possible that RR retrievals are too noisy due to the difficulty in measuring precipitation from lightlydrizzling stratocumulus clouds using satellites^{50,51}, leading to incomplete confounding removal (Supplementary Fig. 7). In particular, RR is an estimate of surface precipitation, while, ideally, RR would measure the cloudbase precipitation to better evaluate its impact on the cloud water budget. Additionally, there could be biases associated with the use of the adiabatic assumption, which is not valid for strongly precipitating clouds^{52}. Finally, there could be unknown sources of confounding that causal graph A does not capture. The w_{e}mediated effect on LWP is negative, which is consistent with water loss by evaporation of cloud droplets due to enhanced entrainment of warm (and dry) free tropospheric air at cloud top. The w_{e}mediated effect is longlasting, with a peaking negative sensitivity after ~20 h. This agrees with the timescales of negative LWP adjustments that were calculated in^{30}.
A comparison of the yaxis scales of Fig. 3c and d shows that the longlasting negative N_{d}LWP sensitivity is mainly driven by entrainment enhancement, confirming the conclusions of multiple studies^{8,9,12}. However, although previous studies have demonstrated the importance of precipitation controls on the mesoscale structure and water budget of stratocumulus clouds^{7,53,54,55}, this study only detects weakly positive (then negative) influences of precipitation suppression on LWP. As explained above, this could be caused by datarelated or causal modelrelated issues.
The positive lag0 \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{LWP}}}},l = 0}={\alpha }_{{N}_{{{{\rm{d}}}}},{{{\rm{LWP}}}},0}+{\alpha }_{{N}_{{{{\rm{d}}}}},{r}_{{{{\rm{eff}}}}},0}\times {\alpha }_{{r}_{{{{\rm{eff}}}}},{{{\rm{LWP}}}},0}\) (path rule) could be explained by increased condensation rates of cloud droplets due to an increased cloud droplet surface area (more numerous and smaller)^{56}. Alternatively, it is possible that the use of a noisy RR variable led to an incomplete removal of confounding from RR, leading to a positively confounded contemporaneous effect of N_{d} on LWP \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{LWP}}}},0}\). It is also possible that the temporal resolution of these data, although high, is still not high enough, and that precipitationmediated aerosol influences are already noticeable at lags l < 15 min, although the graph says they should not, because of the lag1 arrow (F) from r_{eff} to RR. In ref. ^{35}, Runge describes how temporal resolutions coarser than process timescales can lead to confounded causal estimates.
With this potentially incomplete removal of confounding from RR, it is essential to take a critical look at the temporal development of the causal effect of N_{d} on LWP from Fig. 3c, as the strong positive sensitivity of LWP to N_{d} at lag 0 might be an artifact due to an incorrect diagnosis of RRmediated influences on LWP. Instead, we might imagine a \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{LWP}}}},l}\) that is closer to 0 at lag 0, quickly becomes positive due to fast precipitation suppression, then slowly reverses to negative due to longlasting entrainment enhancement.
Causal graph sensitivity study
To test the sensitivity of the results to the initial graph assumption, three other plausible causal graphs (graphs B, C and D) are evaluated (Fig. 4). Graph B does not include cloud depth H, graph C includes neither H nor r_{eff}, while graph D includes neither H, r_{eff} nor RR and w_{e}.
The first row (Fig. 4a–c) shows that direct causal effects for graphs B, C and D are similar to the ones found for graph A (Fig. 2), with one notable difference: the direct effect of N_{d} on LWP is negative in graphs C and D, which is not physical, as one would expect the negative effect of N_{d} on LWP to be exclusively mediated by entrainment w_{e}. This simple comparison suggests that graphs A and B (which contain r_{eff}) are more physical than graphs C and D.
The second and third rows (Fig. 4d–g) show that precipitation suppression and entrainment enhancement are correctly detected in graphs B and C, as \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{RR}}}},l}\) and \({\beta }_{{N}_{{{{\rm{d}}}}},{w}_{{{{\rm{e}}}}},l}\) look very similar to the ones derived from graph A (Fig. 3a, b). This highlights the robustness of the results concerning precipitation suppression and entrainment enhancement, i.e. the microphysics to dynamics branch of the feedback loops.
However, the physical description of the dynamics to microphysics branch is not accurate with graphs B and C, as shown in the fourth row (Fig. 4h–i). The w_{e}mediated effect on LWP is positive in graph C, which disagrees with the evaporation of liquid water due to entrainment enhancement. In graph B, the w_{e}mediated effect is initially correctly detected as negative, similarly to graph A, but the effect is weak and eventually becomes positive. The RRmediated effect on LWP is even more negative in graphs B and C than in graph A, which disagrees with the buildup of LWP due to precipitation suppression. This suggests that the effects of precipitation and entrainment on LWP following aerosol perturbations can be best captured when considering changes in both r_{eff} and H (i.e. with causal graph A). In particular, the pivotal role of r_{eff} for an accurate representation of how w_{e} influences the LWP is probably due to the mixed homogeneous/extreme inhomogeneous entrainment regime^{43}, as discussed previously (Fig. 5a). As H is a confounder of the causal link from r_{eff} to LWP, it is essential to add H in the causal graph alongside r_{eff}. In fact, including H avoids confounding due to aggregation over different precipitation regimes (Fig. 5b).
The last row (Fig. 4j–l) shows the temporal development of the total N_{d}LWP effect. This development differs greatly between the different graphs, which is a direct consequence of the differences observed in Fig. 4a–c and h–i. For graphs C and D, the N_{d} to LWP causal effect is always negative, with strong negative sensitivities reached at lag0, which slowly decay back to zero over the course of 24 h. Consistently negative developments have also been observed in ship track studies^{31}. Including r_{eff} in the graph (graphs A and B) permits to remove the confounding influence of w_{e} on the initial sensitivity and changes the initial sign of the causal effect of N_{d} on LWP from negative to positive. For graph B, the N_{d}LWP temporal development is similar to graph A, although with a much stronger magnitude. Compared to graphs B, C and D, graph A yields weaker LWP adjustments (note the different yaxis scales in Fig. 3c and Fig. 4j–l). In particular, when ignoring all sources of confounding (graph D), LWP adjustments are predicted to be strongly negative, implying a strong compensation of the cooling Twomey effect by LWP adjustments. When covariations in r_{eff} and H are taken into account with graph A, LWP adjustments are weaker. By integrating \({\beta }_{{N}_{{{{\rm{d}}}}},{{{\rm{LWP}}}},l}\) over time, we find that LWP decreases predicted from graph A only represent about 6% of LWP decreases predicted from graph D (after 24 h, in response to a 1standard deviation increase in N_{d}). This means that the cooling effect of aerosolliquid cloud interactions (including Twomey and cloud adjustments) could be much stronger than previously thought.
Discussion
This study proposes a physicsinformed causal graph to quantify the causal effect of N_{d} on LWP in marine stratocumulus clouds. We evaluated the causal graph on daytime geostationary satellite data colocated with reanalysis data, at a temporal resolution that is expected to match the process timescale at which macrophysical changes are propagated through stratocumulus clouds. Contrary to other studies that looked directly at the temporal evolution of the N_{d}LWP sensitivity^{30,31,32}, we divided this sensitivity into its physical components, by separately investigating the entrainment and precipitationmediated responses. This physical processoriented approach (as opposed to a state variableoriented approach) allows us to remove environmental confounding that targets these physical processes, and hence to calculate causal effects (as opposed to correlations), while checking the physical plausibility of the results. We were able to disentangle LWP adjustments due to precipitation suppression and entrainment enhancement on different timescales (fast vs. slow), leading to LWP adjustments that are both regime and timedependent. We confirmed cloudtop entrainment enhancement as a key control for LWP adjustments, and noticed issues associated with precipitation that deserve to be addressed in future research. The methodology adopted in this study showcases how to conduct a thorough causal effect estimation analysis: from discussing physical assumptions behind the causal graph to a systematic investigation of lagged causal effects, mediation and regimedependence with a focus on the sensitivity of the results on the assumed graph.
Of course, all the results in this study are contingent upon a set of assumptions being met: (1) validity of the causal graph, (2) linearity of causal effects, (3) absence of hidden confounders, (4) stationarity of the causal effects, (5) trustworthiness of data.
We partially tested the implications of assumption (1) by comparing the results for four different plausible causal graphs.
Although the short time lags (0 or 15 min) might justify the use of a linear assumption (2) as a first order approximation, it might be worthwhile to investigate nonlinearities with the adjustment approach (see methods) in future research.
Concerning assumption (3), if there were any unobserved confounders, other than the ones already included in the graph, the method presented here would have to be adapted to deal with hidden confounding^{57}. For instance, this study ignored the confounding that can arise from the use of the adiabatic assumption to derive cloud properties. This retrieval assumption implies deterministic relationships that causally differ from the physical relationships between the variables (Supplementary Fig. 8), and a causal framing of this issue is still lacking in the literature. Moreover, it has been demonstrated that the use of this assumption can introduce correlated noises in satellite retrievals. This can introduce spurious correlations in the N_{d}LWP relationship (Supplementary Fig. 9)^{35}. For instance^{58}, showed how an initially positive N_{d}LWP can be falsely interpreted as negative because of such retrieval noises. Further confounders might need to be included in graph A. For example, including cloudbase updraft speeds, or another proxy for the influence of cold pools^{47} in the causal graph could help to solve some of the issues encountered with RR in this study.
Although the physical processes in the causal graph are expected to be stationary (4), the passive satellite instruments only provide daytime cloud property retrievals, and, for the sake of this analysis, we assumed that the magnitude of the causal effects remained unchanged through the night.
Finally, there might be additional datarelated issues (5). The physical processbased causal approach used here allows us to diagnose where some datarelated issues might arise. For instance, the analyses suggested potential issues with precipitation retrievals (a difficult task for lightlyprecipitating stratocumulus clouds^{50,51}), leading to incorrect detection of precipitationmediated LWP adjustments. Moreover, the coarse spatial averaging of the data is a limitation of this study, and the impact of spatial aggregation on causal effects will be evaluated in future work. Other issues might include: the lack of vertical information in passive satellite retrievals; the failure of the adiabatic assumption in cases of strong entrainment or precipitation^{52}; the sampling strategy chosen for the calculation of N_{d}^{59}; or an imperfect colocation of the different satellite and reanalysis products over the stratocumulus region.
Future causal studies of ACI could focus on evaluating the effect of datarelated issues (3–5) by applying the methodology presented here to other sources of data, e.g. model data or insitu data. In that sense, a causal approach can be used as a physicsinformed diagnosis tool to identify the sources of discrepancies in ACI estimates between different data sources, similarly to what is done for model evaluation in^{60}. A complete study of how errors related to retrieval assumptions propagate from the data to the causal effects would also be of interest.
Despite these limitations, the causal inference method presented here provides a helpful framework to address confounding. In particular, the graph sensitivity analysis allows to identify which variables need to be included to obtain physically plausible causal effects. It revealed that the aerosolinduced LWP response is overly negative if environmental confounding is not properly removed with the use of timeseries and with the appropriate consideration of covariations in cloud droplet sizes and cloud depths. This implies that the cooling effect of ACI could be underestimated when failing to account for the effects of meteorological covariations on LWP adjustments in marine stratocumulus regions. This agrees with other studies that used causal approaches (e.g. opportunistic experiments, such as refs. ^{14,15,61}). These results put into perspective the vastly different N_{d}LWP sensitivities found in the literature and highlight the importance of considering confounding as well as longterm developments to accurately calculate cloud sensitivities.
Methods
Data
The data used in this work are 2year timeseries (2016–2017) of satellite cloud retrievals colocated with reanalysis data over the Namibian stratocumulus deck (10^{∘}–20^{∘} S, 0^{∘}–10^{∘} E, as defined by ref. ^{3}). The level 2 cloud properties (r_{eff}, LWP) were obtained from the Cloud Physical Properties (CPP) product of the CLoud property dAtAset using SEVIRI (CLAAS 2.1)^{62}, where SEVIRI is the Spinning Enhanced Visible and InfraRed Imager aboard the eleventh Meteosat Second Generation geostationary satellite. Low liquid clouds were filtered by using the cloud type product. N_{d} and H were calculated using the adiabatic assumption^{63,64}:
where N_{d} is assumed to be constant along the cloud depth, τ is the cloud optical depth, k = 0.72 ^{65} is a factor accounting for the width of the droplet size distribution, Q_{ext} ≈ 2 is the scattering coefficient, ρ_{w} is the density of water and C_{w} is the adiabatic condensation rate. C_{w} is a function of temperature and pressure^{66}, which we calculate using the cloud top temperature and pressure from the CTX product of SEVIRI. For the calculation of N_{d}, pixels where r_{eff} < 4 μm or τ < 4 are filtered out because of high retrieval uncertainties associated with these lower values^{59}. Although geostationary data are still underused in ACI studies, it should be noted that the SEVIRI cloud products have been extensively validated with other more commonly used polarorbiting satellite data products^{67}. Because few studies have specifically validated cloud droplet number concentrations from SEVIRI (e.g. ref. ^{68}), we compared N_{d} from SEVIRI and from the level 3 MODIS Terra satellite^{69} over our study region and period and found a very good agreement of the two derived products (Supplementary Fig. 10). The slight positive bias in N_{d} from SEVIRI should not be an issue since the data are standardized in this study.
Precipitation rates (RR) were obtained from the Global Precipitation Measurement (GPM) Integrated MultisatellitE Retrievals Version 6 (iMERG V06)^{70}. Meteorological variables were downloaded from ERA5^{71,72}. LTS was calculated as the difference in potential temperatures between 700 and 1000 hPa^{3}. The entrainment velocity of free tropospheric air into the boundary layer (w_{e}) was calculated using a largescale boundary layer continuity equation, following^{73}:
where w_{subs} is the rate of largescale subsidence, taken at 700 hPa. This yields w_{e} values on the order of a few millimeters per second (Supplementary Table 2), which is comparable to other studies that employ the same equation^{8,74,75}.
The different data products were first colocated on the 0.25° × 0.25° ERA5 grid and linearly interpolated to the temporal resolution of the SEVIRI data, i.e. Δt = 15 min. Then, because the emphasis of this study is on the temporal resolution, and in order to simplify the causal analysis, the data are spatially averaged over the larger 10° × 10° stratocumulus region. It should be noted that the averaging of cloud properties is performed on incloud properties, not on allsky properties, i.e. we exclude clearsky zero values before averaging.
The diurnal and seasonal cycles might be a source of confounding, i.e. they can introduce correlations in meteorological properties and cloud properties that may not be causal in nature (Supplementary Fig. 11 for an illustration). For this reason, the data were adjusted for the diurnal and seasonal cycle by computing a seasonal instantaneous mean value and standard deviation for all variables (i.e. we take the mean and standard deviations of all data points for a given timestamp, e.g. 10:15 am), and using these means and standard deviations to standardize all data points with the same timestamp. Because of the standardization, the units of causal effects are not on a natural physical scale, but rather in units of deviation of the variable from its seasonal instantaneous mean per unit of the seasonal instantaneous standard deviation. As a consequence, the interpretations focus on the sign of the causal effects and their relative strength, but not on their absolute magnitude. The cloud properties were logtransformed prior to the standardization, for consistency with previous studies in which sensitivities are expressed as the derivatives of the logarithms of the variables. Zero precipitation values were removed from the dataset prior to the application of the logtransform and precipitation values smaller than the first percentile were removed from the dataset. The whole precipitation timeseries were shifted backwards by 1 timestep in order to approximate cloudbase precipitation rates instead of the surfacelevel precipitation rates.
The original timeseries, as well as the adjusted timeseries, are shown in Supplementary Fig. 12 and average values for all variables are shown in Supplementary Table 2.
Causal effect estimation
Causal effect estimation^{20,21,22,57,76} requires three ingredients:

1.
A causal graph describing all direct causal links between the variables. Causal graphs can be obtained using causal discovery methods, or can be drawn from domain knowledge. In this study, we preferred the second option as causal discovery is subject to large uncertainties given the finite nature of datasets and the potential existence of hidden confounders^{22}. The proposed causal graph (Fig. 1b) is a stationary directed acyclic graph, meaning that: (1) it is resolved in time, i.e. it contains lagged causal links and autodependency links; (2) considered to be stationary in time, i.e. the graph structure and the associated causal effects do not vary with time; (3) the direction of all causal links is known, with no hidden confounding variable; and (4) the graph does not present any feedback loop within the same timestep. In this work, all causal links are hypothesized to have lags l = 0 or 1 timestep (i.e. 0 or 15 min). A structural causal model (SCM) is implicitly associated with the causal graph. The SCM is a set of equations that describes the causal relationships between the n variables.
$$\begin{array}{c}{X}_{j}(t):= {f}_{j}(pa({X}_{j}(t)),{U}_{j}(t))\\ {{{{\rm{if}}}}\,{{{\rm{linear}}}}}\,{=}\mathop{\sum}\limits_{\begin{array}{c}{X}_{i}(t{l}_{ij})\atop \in pa({X}_{j}(t))\end{array}}{\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\times {X}_{i}(t{l}_{ij})+{U}_{j}(t)\,{{{\rm{for}\, {\rm{j}}}}}\,\in [1:n]\end{array}$$where f_{j} is a linear or nonlinear function that determines the value of the effectvariable X_{j}(t) based on the values of its direct causal parents \(pa({X}_{j}(t))={({X}_{i}(t{l}_{ij}))}_{i}\), i.e. those variables with arrows (lag l_{ij}) pointing directly towards X_{j}(t) in the causal graph. U_{j}’s are jointly independently distributed noise variables. In the linear case, \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\) is the coefficient of the SCM for the direct causal effect of the parentvariable X_{i} on variable X_{j} at lag l_{ij}. In this study, the SCM is assumed to be linear as a first order approximation, and the \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\) coefficients of the SCM, also called path coefficients or direct causal effects, correspond to the weights on each single arrow of the causal graph. They are the target metric of the causal effect estimation performed in this study.

2.
Observational data for all the variables in the causal graph. Timeseries are ideal as the precedence of cause on effect can be exploited. As explained in the introduction, the 15 min temporal resolution of geostationary data is reasonable for stratocumulus clouds as it is close to the expected process timescale at which microphysical changes occur and are propagated throughout the cloud.

3.
An estimation method for causal effect quantification given a causal graph and its associated data. There are two methods: the adjustment approach^{20,57} and Wright’s path approach^{40,41}. The adjustment method is a nonparametric approach that allows to treat graphs with hidden variables. In the linear adjustment method, the total causal effect of X on Y is:
$$\begin{array}{c}{\beta }_{X,Y}=\frac{\partial {\mathbb{E}}(Y do(X=x))}{\partial x},\\ {{{\rm{where}}}}\,{\mathbb{E}}(Y do(X=x))={{\mathbb{E}}}_{Z}\left[{{\mathbb{E}}}_{Y X,Z}[(Y X=x,Z=z)]\right]\end{array}$$The formulation of β_{X,Y} can be extended to nonlinear cases, by using a nonlinear estimator (e.g. a neural network). The dooperator signals that we are calculating causal effects (not correlations) from the observational distribution. This is done by using a set of adjustment variables Z. Z is determined from the causal graph and contains the variables that block all noncausal paths from X to Y, thereby removing any confounding. Importantly, Z does not contain any descendants of Y, or any mediators, thereby avoiding introducing collider bias. In the linear case, β_{X,Y} simply corresponds to the partial linear regression slope for X in the regression of Y with respect to both X and Z. The Wright method only applies to the linear case and generally cannot handle hidden confounding. It is primarily concerned with the estimation of the direct causal effects \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\), i.e. the arrow coefficients in the causal graph. They are calculated as the partial regression slopes in the multiple linear regression of X_{j} on its causal parents pa(X_{j}(t)), thereby removing any source of confounding that is implied by the causal graph. The Wright method differs from the adjustment method in its computation of total effects, as the total effect of X on Y (llagged) is derived from precomputed direct effects using the path tracing formula:
$$\beta_{X,Y,l} = \mathop{\sum}\limits_{\begin{array}{c}\scriptstyle{\mathrm{causal}}\,{{\mathrm{paths}}}\\ {{{\mathrm{from}}}\,X(tl)}\atop{{\mathrm{to}}\,Y(t)}\end{array}}\left(\mathop{\prod}\limits_{{X_i} {\mathop\rightarrow\limits^{l_{ij}}} X_j{\rm{arrow}}\atop{\mathrm{in}\,{{\mathrm{path}}}}} \alpha_{X_i,X_j,l_{ij}}\right)$$(2)See Supplementary Fig. 4b for an illustration of the path tracing formula. The path tracing formula can be applied to derive contemporaneous total causal effects or lagged total causal effects, i.e. the temporal development of causal effects. It should be noted that the temporal developments of causal effects are not recalculated from the data at each timestep. Instead, the data are used once in combination with the graph to calculate the direct causal effects (including autodependency coefficients), and then, these direct effects are propagated in time through the graph using the path tracing formula. At the timescales at which direct causal effects are computed (lags l = 0 or 1, i.e. 0 or 15 min), there is not much advection of the cloud fields, so we can consider the cloud system to be stationary. Assuming stationarity of the causal effects, we can extend temporal development calculations past 12 h even though the satellite only provides daytime data. 24h developments of cloud processes should therefore be understood as hypothetical cloud developments should clouds persist for so long in the study region, and they should not be understood as a direct measurement of cloud lifetime. All results of this study were derived using the Wright method, as it allows for an easier decomposition of direct and mediated effects, and the variance of this method is asymptotically smaller than the adjustment method^{77}.
Causal effect quantification analyses were all carried out with the Tigramite package in Python (https://github.com/jakobrunge/tigramite).
Confidence intervals, masking
The first step of the analysis is to compute the direct causal effects between the variables using Wright’s estimator, i.e. we calculate the \({\alpha }_{{X}_{i},{X}_{j},{l}_{ij}}\) coefficients of the linear SCM. Confidence intervals for direct causal effect estimates were calculated using a bootstrapping method with 500 members. Direct causal effects are considered significant when the bootstrap confidence interval does not include 0. Once direct causal effects have been estimated, total causal effects β_{X,Y,l} between two variables X(t − l) and Y(t) can be estimated using Wright’s path tracing formula (Eq. (2)). The confidence intervals for the temporal development plots were calculated by bootstrapping with n = 100 (the number of bootstrap ensemble members was scaled down due to the number of calculations).
We evaluate the regimedependence of causal effects using binary masks, i.e. by binning the data points by low/high value of the masking variable (by comparison to the median value). Using the lower and upper fiftieth percentiles (instead of quartiles for instance) allows to have enough consecutive timesteps to carry out the causal effect calculations. Specifically, LTS, RH_{FT}, N_{d} are used as masking variables for the boundary layer stability, free tropospheric humidity and aerosol background regimes.
Data availability
The timeseries used for the analyses were generated from colocated SEVIRI (Copyright (c) (2020) EUMETSAT), GPM and ERA5 data (generated using Copernicus Climate Change Service information [2022]). MODIS Level 3 data (used for comparison purposes) were downloaded from https://ladsweb.modaps.eosdis.nasa.gov/archive/allData/61/MOD08_D3. SEVIRI data are freely available from https://wui.cmsaf.eu/safira, GPM data from https://disc.gsfc.nasa.gov/datasetsand ERA5 data from https://cds.climate.copernicus.eu. The timeseries and analyses outputs are provided on Zenodo (https://doi.org/10.5281/zenodo.7692695).
Code availability
Code for the data processing and analysis is provided on Zenodo (https://doi.org/10.5281/zenodo.7692232).
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Acknowledgements
This work was supported by the European Union’s Horizon 2020 research and innovation program under Marie SklodowskaCurie grant agreement No. 860100 (iMIRACLI). J.R. has received funding from the European Research Council (ERC) Starting Grant CausalEarth under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 948112).
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E.F. developed the concept of the study together with U.L. and D.N. J.R. wrote the core python package for the causal analyses and provided guidance on its usage. E.F. wrote the code for the data processing, for the causal workflow (using J.R.’s package) and for the data postprocessing. E.F. and U.L. worked on to the interpretations of the results. E.F. drafted the manuscript with contributions from all other coauthors.
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Fons, E., Runge, J., Neubauer, D. et al. Stratocumulus adjustments to aerosol perturbations disentangled with a causal approach. npj Clim Atmos Sci 6, 130 (2023). https://doi.org/10.1038/s4161202300452w
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DOI: https://doi.org/10.1038/s4161202300452w