the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 13 Feb 2026
A potential emergent constraint on cloud liquid water path adjustments to aerosol–cloud interactions
Abstract. Emergent constraints are relationships between an observable in the present-day climate (such as cloud state variables) and an unobservable response (such as adjustments to radiative forcing) of the climate system to a perturbation. Here we present a candidate emergent constraint arising from the relationship across members of a perturbed parameter ensemble (PPE) between the observable present-day cloud droplet number–liquid water path (𝑁𝑑–𝓛) correlation and the unobservable liquid water path adjustment to anthropogenic aerosol–cloud forcing (RA𝓛). Emergent constraint candidates require scrutiny to distinguish them from, for example, spurious correlations. The candidate presented here meets several criteria delineated by Klein and Hall (2015): high correlation coefficient, plausible underlying physical mechanism, and emergence from a PPE that perturbs the physical parameters relevant to both the observable and the climate response. Constraining the observable 𝑁𝑑–𝓛 regression slope to present-day satellite estimates yields a constrained estimate for the ratio of present-day to preindustrial cloud liquid water path 𝓛PD/𝓛PI = 0.976 ± 0.009 (PPE regression slope uncertainty only) with a regression coefficient of 0.92. The constrained 𝓛PD/𝓛PI implies a robustly positive RA𝓛; this disagrees in sign with all other general circulation model (GCM) estimates, but agrees with non-GCM lines of evidence. However, the constrained estimate requires extrapolating the emergent-constraint relationship past the minimum 𝓛PD/𝓛PI produced by any of the PPE members.
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Status: closed (peer review stopped)
- RC1: 'Comment on egusphere-2026-748', Anonymous Referee #1, 09 Mar 2026
-
RC2: 'Comment on egusphere-2026-748', Anonymous Referee #2, 20 Mar 2026
Summary
In this paper, Mülmenstädt and Ma propose a new emergent constraint for GCMs: the present day to preindustrial cloud liquid water path ratio (LWPpd/LWPpi) vs. the slope of the downward branch in the PD droplet number (Nd) vs. LWP (“inverted v”) relationship. This typically negative slope is thought to be determined by the sedimentation-entrainment feedback. The authors utilize a perturbed parameter ensemble (PPE) in the DOE E3SM model where each ensemble member has been perturbed in their parametrizations of liquid phase microphysics, mixed phase microphysics, activation, and/or turbulence. The strength of the proposed emergent constraint relationship is heavily driven by the liquid phase microphysics perturbed members which have widely varying ratio and slope values. However, this is one of the key requirements from the Klein and Hall emergent constraint checklist: support of the underlying physical mechanism through perturbations of related physics. The relationship between the LWPpd/LWPpi ratio and the positive branch of the inverted v (typically thought of as controlled by precipitation-suppression) is surprisingly non-existent, possibly supporting the idea that precipitation alone is not driving this part of the relationship. Using the satellite-based estimation for the negative slope produces a constrained LWPpd/LWPpi ratio (if extrapolating beyond the PPE members) that implies a significant, positive value for LWP adjustment since the PI. GCMs generally disagree with this significant positive value but these results are consistent with other lines of evidence from the literature.
Overall, the paper presents a clear and elegant case for a new emergent constraint, carefully following the Klein and Hall criteria. The authors have created an exceptional paper, integrating multiple, sometimes competing ideas from the literature into an insightful investigation leading to several thought-provoking results that resolve some of the tension in the field. Using these results to address and reframe the two key questions that have been discussed around the inverted v (see conclusions) is particularly interesting. These results and new emergent constraint have the potential to significantly advance the field and modify the approach to GCM evaluations and parameterizations. I recommend prompt publication after some minor clarifications about the mechanism and specifics of the methodology.
Minor and Typographical Comments:
Line 42-43: Surprising, does this imply that the underlying model physics is unable to produce this positive response in keeping with Bellouin? But then how can it achieve this in aggregate across the members?
Line 140-146: Thank you for explaining the logic of how you handle the base state and formulate the constraint. How much of an issue do you expect the lower order contributions to be here? Would comparing with other PPEs that have different base states help? No need to include further work, just curious about the implications.
Line 147-148: Is there an uncertainty on the slope? If so, please include as that potentially changes how much overlap the PPE members have with reality (i.e. the satellite estimate). It might also help to encapsulate some of the other factors potentially contributing to this satellite value (e.g. Goren et al 2024 as discussed).
Line 147-149: Apologies if I have missed this, but would it be possible to spell out the physical mechanism (i.e., lines 137-139) you expect to connect the LWP ratio and the slope here? I understand that they are both being driven by aci processes (sedimentation-entrainment feedback specifically?) and changes in aerosol from PI to PD but if there is a clearer linkage that you have in mind that would be very helpful to state and/or diagram (here or at the end). This seems particularly important to be precise about since there was a stronger relationship with mh than ml, despite the latter being more expected based on the physical mechanism prior.
Line 155: Is it feasible to quantify this agreement with a goodness of fit statistic for all the members? I would be curious if that influences the ratio vs mh result (e.g., do outliers have worse fits to the inverted v?).
Line 165: In general, thank you for a detailed discussion of implications from the PPE member responses, very interesting.
Line 175-178: Please expand on how the Nd outliers are potentially impacting this ml regression while also having a much lower peak location, I don’t quite follow.
Line 185: It looks like there is a negative relationship between these, not zero relationship, so is there some precipitation contribution? I think I’m missing your point, apologies.
Line 250, broader paragraph: Framing the slope as having value as a continuous variable really turns this question on its head and appears, throughout this paper, to be helping us progress beyond this stalemate. I do still wonder what the mechanism is exactly, see earlier point.
Line 215-217, 251-253, 261-263: Very thought-provoking conclusion. Because extrapolation to the satellite estimate of mh is needed, this implies that no members satisfy this constraint, right? It will be interesting to see if there are any GCMs or PPE members that do overlap the observed range, presumably a topic of future work based on the discussion.
Line 218: Typo: “candidates for emergent constraints”
Line 223: Typo: “constraint”
Citation: https://doi.org/10.5194/egusphere-2026-748-RC2 -
RC3: 'Comment on egusphere-2026-748', Anonymous Referee #3, 08 Apr 2026
The manuscript proposes an emergent constraint between the observable correlation of the liquid water path LWP and the cloud droplet number N_d and the unobservable liquid water path adjustment to anthropogenic aerosol–cloud forcing. The authors use a perturbed parameter ensemble for the E3SM model to investigate this dependency and describe the reasoning behind these parameter perturbations in detail.
A key point in the analysis is that the proposed emergent constraint is mostly reliant on the perturbations in the liquid microphysics. The result implies a positive radiative adjustment regarding LWP. This is in contrast to the negative adjustment usually simulated by most GCMs, but is in line with evidence from observations and process modelling. The strong correlation is identified as one important factor in the reasoning for an emergent constraint as described in Klein and Hall (2015). The remaining conceptual difficulties associated with establishing new emergent constraints are explained clearly.
Major points
- In Figure 2 the variable on the y-axis (L_PD / L_PI) is relative, which probably reduces the influence of systematic biases in L. However, for m_h the systematic bias of the aerosol activation is not accounted for, which might be the reason that the runs with a perturbation in the aerosol activation (E12, 13, 18) are distant from the other runs. Could a suitable scaling for m_h, e.g., d ln(N) / d ln(A), resolve this? Moreover, Figure 2 shows the “activation” category forming its own cluster. It seems that with more models here, this could form its own regression line parallel but shifted compared to the red line. Have you investigated this further?
- With the scaling mentioned in the former point, one would have L_PD/L_PI as a function of d ln(N) / d ln(A) and d ln(L) / d ln(N). Could it be that this relationship already follows from the framework established in Bellouin et al (https://doi.org/10.1029/2019RG000660)?
- You mention that you nudge winds towards MERRA (lines 107-109). Does this constrain the results you have obtained?
Minor points
- You refer to Mülmenstädt 2024a,b multiple times. If the concepts of these publications are important here, it might help to at least shortly explain them in the current manuscript.
- Could the results of Hoffmann et al 2025 (https://acp.copernicus.org/articles/25/8657/2025) help to explain the large influence of the liquid microphysics?
- line 127: You mention “three independent shape variables”. To me it is not clear what the three are. m_l, m_h, and what else? The apex point as a single variable? Can you clarify this?
- Line 130: the “to an anthropogenic perturbation.” seems wrong at the end of the sentence, since you already say “linking an X to a Y” in that sentence.
- Line 136: the “a” before “statistical techniques” seems wrong.
- Lines 142: I do not understand the term “logarithmic difference” completely, but also do not understand why it is necessary/helpful. The only reference to the concept seems to be line 160. Out of curiosity, what does it help to understand?
- Lines 172 and 199: The interpretation of what the legs of the inverted-v represent would be helpful when the inverted-v equation is first explained.
Citation: https://doi.org/10.5194/egusphere-2026-748-RC3
Status: closed (peer review stopped)
-
RC1: 'Comment on egusphere-2026-748', Anonymous Referee #1, 09 Mar 2026
This manuscript proposes and examines an emergent constraint between the radiative adjustment associated with the liquid water path (LWP) adjustment and the empirical LWP-drop number concentration (LWP-Nd) relationship. The latter is measurable in the present day while the former is the unknown target variable of great relevance to climate change. LWP-Nd relationships have been studied using multiple data sources including satellite remote sensing, climate model output and large eddy simulation (LES). An obvious concern is whether LWP-Nd relationships represent causality or simply correlation.
The manuscript is appropriately brief and well-written. There is a lot of information packed into the list of simulations that were performed and it is hard to figure out what they all mean. On the other hand, perhaps it doesn’t matter much, for reasons I will discuss below:
Major concerns:
- The methodology is so briefly mentioned. The set-up seems to be a modern era simulation that does not allow for significant circulation changes (fixed SST, sea ice, nudging to MERRA-2 winds) so this isn’t a PD vs PI study. I appreciate the need to constrain the system so that clouds are not changing as they might in a true PD vs PI study. But this then begs the question as to why the authors stick with such coarse resolution/such poorly resolved clouds and cloud processes? How long were these simulations run for?
- The one-off PPE set up is troubling because we know that it is very difficult to assess the influence of processes when a limited number of settings are changed. The interactive influence of different parameter settings can provide unexpected responses. It means that drawing conclusions about the importance of a given process from the difference between the experiments in Table 1 is highly uncertain. Also, the number of parameters in this model is huge and I suspect that different combinations of parameter settings could give similar results, leading (possibly) to different conclusions.
- Following from 1), have the authors looked at an equivalent set of E3SM simulations at finer resolution/presumably better resolved cloud processes? For that matter, why not take a large set of LES simulations where processes are (mostly) resolved https://rmets.onlinelibrary.wiley.com/doi/10.1002/gdj3.70049. These were developed in-house and should be easily accessible. If results point in similar directions, one would have more confidence in the current results. Absent this, the current simulations are extremely difficult to parse.
- m_l is dismissed based on Fig. S1. Why, other than it helps to limit the analysis to m_h? The two branches of LWP-Nd supposedly represent physical processes and if the model physics doesn’t show a clear emergent constraint, shouldn’t we question the representation of the warm phase physical processes that you have shown to be dominant in the emergent constraint with respect to m_h? The autoconversion and accretion processes are, after all, central to m_l.
- Following from 1)-4), my concern is that the current manuscript presents another set of climate modeling results of limited use to the community because they are not sufficiently well-rooted in real-world clouds and aerosol-cloud interactions. My advice would be to pursue the ‘structural’ rather than the ‘parameteric’.
Other:
- If my understanding of the model set-up is correct–i.e., this is not an historical PD vs PI set of simulations, doesn’t some of the text need to be more carefully written? E.g., the CMIP6 models are historical PD vs PI runs. In that case there are other reasons why a model might exhibit a negative LWP adjustment in PD while at the same time LWP in PD is higher than that in PI.
- Coloring the points by class will be more effective than the change in symbol.
- Long sentence starting on line 232 is hard to follow.
Citation: https://doi.org/10.5194/egusphere-2026-748-RC1 -
RC2: 'Comment on egusphere-2026-748', Anonymous Referee #2, 20 Mar 2026
Summary
In this paper, Mülmenstädt and Ma propose a new emergent constraint for GCMs: the present day to preindustrial cloud liquid water path ratio (LWPpd/LWPpi) vs. the slope of the downward branch in the PD droplet number (Nd) vs. LWP (“inverted v”) relationship. This typically negative slope is thought to be determined by the sedimentation-entrainment feedback. The authors utilize a perturbed parameter ensemble (PPE) in the DOE E3SM model where each ensemble member has been perturbed in their parametrizations of liquid phase microphysics, mixed phase microphysics, activation, and/or turbulence. The strength of the proposed emergent constraint relationship is heavily driven by the liquid phase microphysics perturbed members which have widely varying ratio and slope values. However, this is one of the key requirements from the Klein and Hall emergent constraint checklist: support of the underlying physical mechanism through perturbations of related physics. The relationship between the LWPpd/LWPpi ratio and the positive branch of the inverted v (typically thought of as controlled by precipitation-suppression) is surprisingly non-existent, possibly supporting the idea that precipitation alone is not driving this part of the relationship. Using the satellite-based estimation for the negative slope produces a constrained LWPpd/LWPpi ratio (if extrapolating beyond the PPE members) that implies a significant, positive value for LWP adjustment since the PI. GCMs generally disagree with this significant positive value but these results are consistent with other lines of evidence from the literature.
Overall, the paper presents a clear and elegant case for a new emergent constraint, carefully following the Klein and Hall criteria. The authors have created an exceptional paper, integrating multiple, sometimes competing ideas from the literature into an insightful investigation leading to several thought-provoking results that resolve some of the tension in the field. Using these results to address and reframe the two key questions that have been discussed around the inverted v (see conclusions) is particularly interesting. These results and new emergent constraint have the potential to significantly advance the field and modify the approach to GCM evaluations and parameterizations. I recommend prompt publication after some minor clarifications about the mechanism and specifics of the methodology.
Minor and Typographical Comments:
Line 42-43: Surprising, does this imply that the underlying model physics is unable to produce this positive response in keeping with Bellouin? But then how can it achieve this in aggregate across the members?
Line 140-146: Thank you for explaining the logic of how you handle the base state and formulate the constraint. How much of an issue do you expect the lower order contributions to be here? Would comparing with other PPEs that have different base states help? No need to include further work, just curious about the implications.
Line 147-148: Is there an uncertainty on the slope? If so, please include as that potentially changes how much overlap the PPE members have with reality (i.e. the satellite estimate). It might also help to encapsulate some of the other factors potentially contributing to this satellite value (e.g. Goren et al 2024 as discussed).
Line 147-149: Apologies if I have missed this, but would it be possible to spell out the physical mechanism (i.e., lines 137-139) you expect to connect the LWP ratio and the slope here? I understand that they are both being driven by aci processes (sedimentation-entrainment feedback specifically?) and changes in aerosol from PI to PD but if there is a clearer linkage that you have in mind that would be very helpful to state and/or diagram (here or at the end). This seems particularly important to be precise about since there was a stronger relationship with mh than ml, despite the latter being more expected based on the physical mechanism prior.
Line 155: Is it feasible to quantify this agreement with a goodness of fit statistic for all the members? I would be curious if that influences the ratio vs mh result (e.g., do outliers have worse fits to the inverted v?).
Line 165: In general, thank you for a detailed discussion of implications from the PPE member responses, very interesting.
Line 175-178: Please expand on how the Nd outliers are potentially impacting this ml regression while also having a much lower peak location, I don’t quite follow.
Line 185: It looks like there is a negative relationship between these, not zero relationship, so is there some precipitation contribution? I think I’m missing your point, apologies.
Line 250, broader paragraph: Framing the slope as having value as a continuous variable really turns this question on its head and appears, throughout this paper, to be helping us progress beyond this stalemate. I do still wonder what the mechanism is exactly, see earlier point.
Line 215-217, 251-253, 261-263: Very thought-provoking conclusion. Because extrapolation to the satellite estimate of mh is needed, this implies that no members satisfy this constraint, right? It will be interesting to see if there are any GCMs or PPE members that do overlap the observed range, presumably a topic of future work based on the discussion.
Line 218: Typo: “candidates for emergent constraints”
Line 223: Typo: “constraint”
Citation: https://doi.org/10.5194/egusphere-2026-748-RC2 -
RC3: 'Comment on egusphere-2026-748', Anonymous Referee #3, 08 Apr 2026
The manuscript proposes an emergent constraint between the observable correlation of the liquid water path LWP and the cloud droplet number N_d and the unobservable liquid water path adjustment to anthropogenic aerosol–cloud forcing. The authors use a perturbed parameter ensemble for the E3SM model to investigate this dependency and describe the reasoning behind these parameter perturbations in detail.
A key point in the analysis is that the proposed emergent constraint is mostly reliant on the perturbations in the liquid microphysics. The result implies a positive radiative adjustment regarding LWP. This is in contrast to the negative adjustment usually simulated by most GCMs, but is in line with evidence from observations and process modelling. The strong correlation is identified as one important factor in the reasoning for an emergent constraint as described in Klein and Hall (2015). The remaining conceptual difficulties associated with establishing new emergent constraints are explained clearly.
Major points
- In Figure 2 the variable on the y-axis (L_PD / L_PI) is relative, which probably reduces the influence of systematic biases in L. However, for m_h the systematic bias of the aerosol activation is not accounted for, which might be the reason that the runs with a perturbation in the aerosol activation (E12, 13, 18) are distant from the other runs. Could a suitable scaling for m_h, e.g., d ln(N) / d ln(A), resolve this? Moreover, Figure 2 shows the “activation” category forming its own cluster. It seems that with more models here, this could form its own regression line parallel but shifted compared to the red line. Have you investigated this further?
- With the scaling mentioned in the former point, one would have L_PD/L_PI as a function of d ln(N) / d ln(A) and d ln(L) / d ln(N). Could it be that this relationship already follows from the framework established in Bellouin et al (https://doi.org/10.1029/2019RG000660)?
- You mention that you nudge winds towards MERRA (lines 107-109). Does this constrain the results you have obtained?
Minor points
- You refer to Mülmenstädt 2024a,b multiple times. If the concepts of these publications are important here, it might help to at least shortly explain them in the current manuscript.
- Could the results of Hoffmann et al 2025 (https://acp.copernicus.org/articles/25/8657/2025) help to explain the large influence of the liquid microphysics?
- line 127: You mention “three independent shape variables”. To me it is not clear what the three are. m_l, m_h, and what else? The apex point as a single variable? Can you clarify this?
- Line 130: the “to an anthropogenic perturbation.” seems wrong at the end of the sentence, since you already say “linking an X to a Y” in that sentence.
- Line 136: the “a” before “statistical techniques” seems wrong.
- Lines 142: I do not understand the term “logarithmic difference” completely, but also do not understand why it is necessary/helpful. The only reference to the concept seems to be line 160. Out of curiosity, what does it help to understand?
- Lines 172 and 199: The interpretation of what the legs of the inverted-v represent would be helpful when the inverted-v equation is first explained.
Citation: https://doi.org/10.5194/egusphere-2026-748-RC3
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- 1
This manuscript proposes and examines an emergent constraint between the radiative adjustment associated with the liquid water path (LWP) adjustment and the empirical LWP-drop number concentration (LWP-Nd) relationship. The latter is measurable in the present day while the former is the unknown target variable of great relevance to climate change. LWP-Nd relationships have been studied using multiple data sources including satellite remote sensing, climate model output and large eddy simulation (LES). An obvious concern is whether LWP-Nd relationships represent causality or simply correlation.
The manuscript is appropriately brief and well-written. There is a lot of information packed into the list of simulations that were performed and it is hard to figure out what they all mean. On the other hand, perhaps it doesn’t matter much, for reasons I will discuss below:
Major concerns:
Other: