Probabilistic energy-balance attribution of observed warming: the anthropogenic–natural partition under unresolved aerosol forcing, efficacy, and time-varying feedback uncertainties

The planetary energy budget ties radiative forcing, heat storage, and temperature response into a single conservation constraint, so observed changes in Earth's energy balance can in principle attribute global warming to its drivers without relying on spatial warming patterns. Huber and Knutti (2012) implemented this idea as a Bayesian, observation-constrained ensemble attribution of warming since the mid-twentieth century, and themselves identified what limits its precision: the magnitude and efficacy of aerosol forcing (especially indirect aerosol–cloud effects), sparse depth-limited ocean heat observations, the assumptions of time-constant linear feedbacks and equal forcing efficacy, structural and prior sensitivity of the inference, and the model-sampled magnitude of multi-decadal internal variability. The subsequent literature bears on each limitation: the qualitative conclusion (anthropogenic dominance of observed warming) is now settled at assessment level, and ocean heat observations have been greatly improved, but the quantitative agent-level partition — above all aerosol effective radiative forcing and its efficacy, and time-varying (pattern-dependent) climate feedbacks — remains open, leaving the uncertainty-bounded partition of historical warming an active problem.

Background

A forcing of the climate system is a perturbation of Earth's top-of-atmosphere energy budget, such as added greenhouse gases, aerosols, solar changes, or volcanic eruptions; the effective radiative forcing (ERF) is the resulting net radiative imbalance after rapid atmospheric adjustments, before the surface temperature responds. Different forcings can produce different warming per watt per square metre, a property called forcing efficacy. The warming response itself is shaped by climate feedbacks — changes in water vapour, clouds, sea ice, and lapse rate that amplify or damp the direct forcing — conventionally summarized in the linear feedback parameter $\lambda$ (watts per square metre per degree Celsius), so that global energy conservation gives the relation $F = Q + \lambda T$, where $F$ is net radiative forcing, $Q$ is the rate of heat uptake by the climate system (dominated by the ocean; ocean heat content, OHC, is its integral), and $T$ is global-mean surface temperature change. The Earth energy imbalance (EEI) is the observed net top-of-atmosphere flux; the equilibrium climate sensitivity (ECS) is the equilibrium warming for doubled CO$_2$, equal to the CO$_2$ forcing divided by $\lambda$. Classical detection and attribution used optimal fingerprinting — regressing observed spatio-temporal temperature patterns onto model-simulated response patterns — which depends on the realism of simulated spatial patterns. Huber and Knutti (2012) proposed an alternative and largely independent attribution method built on energy conservation alone: a massive ensemble of simulations with the Bern2.5D Earth system model of intermediate complexity (EMIC — a zonally averaged coupled model, cheap enough for thousands of runs), emulated by neural networks and constrained by observations of global temperature and ocean heat uptake via Markov chain Monte Carlo (MCMC) Bayesian inference, with the likelihood including measurement error, internal variability estimated from unforced control runs of the Coupled Model Intercomparison Project Phase 3 (CMIP3) models, and emulator error. They found that since the mid-twentieth century greenhouse gases contributed 0.85 °C of warming (5–95%: 0.6–1.1 °C), about half offset by aerosol cooling, against an observed change of about 0.56 °C, and that internal variability alone explains the observed trends with probability below 5% even if models underestimated variability threefold. The authors quantified what limits the precision of this partition: (1) aerosol forcing magnitude and efficacy, particularly the indirect aerosol–cloud effect, which dominates the uncertainty range of aerosol cooling and of the cumulative forcing partition; (2) ocean heat uptake observations that were sparse and limited to the upper 700 m, leaving residual uncertainty in the total planetary heat uptake $Q$; (3) the idealization of a time-constant linear feedback parameter $\lambda$ and equal efficacy across forcing agents; (4) dependence of the inference on the Bern2.5D model structure and on prior assumptions (they showed a more informative prior for climate sensitivity shifts conclusions little); and (5) reliance on CMIP3 control runs to gauge multi-decadal internal variability. Later work transformed parts of this landscape: forcing efficacy and its ocean counterpart were formalized (Hansen et al. 2005; Winton et al. 2010); forcing and feedback were shown to depend on the evolving pattern of surface warming (Andrews et al. 2015; Armour 2017 — the pattern effect, i.e. time-varying feedbacks that bias historical energy-budget inferences low); ocean heat storage was re-quantified to full depth (Cheng et al. 2017; von Schuckmann et al. 2020) and the EEI was tracked from space (Loeb et al. 2024); aerosol ERF was re-bounded but not narrowed decisively (Bellouin et al. 2020); and attribution itself was updated with dedicated model intercomparisons (Gillett et al. 2021) and annual indicator assessments (Forster et al. 2024). The unresolved question is therefore no longer whether human influence dominates observed warming, but whether the agent-level quantitative partition — above all the aerosol contribution and its interaction with feedbacks assumed constant — can be bounded tightly and robustly within the energy-balance framework.

Problem Statement

Determine, with quantified uncertainty, the partition of observed global warming (and of the energy accumulated in the climate system) since the mid-twentieth century among greenhouse gases, aerosols, other anthropogenic forcings, and natural forcings, together with the residual role of internal variability, using observational constraints on the planetary energy balance $F = Q + \lambda T$ — where $F$ is net radiative forcing, $Q$ the rate of heat uptake by the climate system, $\lambda$ the linear climate feedback parameter, and $T$ the global-mean surface temperature change — rather than spatial warming patterns. The open elements are: (1) the magnitude and efficacy of aerosol forcing, in particular indirect aerosol–cloud effects, which dominate the uncertainty of the partition; (2) the treatment of ocean heat uptake, now observed to full depth but uncertain in pre-instrument and deep-ocean reconstructions; (3) the assumption of a time-constant linear feedback parameter $\lambda$ and equal efficacy across forcing agents, given demonstrated pattern-dependent and time-varying feedbacks; (4) robustness of the probabilistic inference to model structure and prior choices; and (5) the magnitude of multi-decadal internal variability. The partition is meaningful only as posterior probability distributions — the agent-level contributions and the anthropogenic versus natural/internal-variability share — with intervals that propagate the errors of the observational inputs, the radiative-forcing inputs, internal variability, and model inadequacy, and with these five uncertainty sources treated and quantified explicitly, aerosol forcing magnitude and efficacy dominating the uncertainty of the partition, rather than assumed away.

References

1. Huber, Markus, and Reto Knutti (2012). Anthropogenic and natural warming inferred from changes in Earth's energy balance. Nature Geoscience 5, 31–36. doi:10.1038/ngeo1327. https://doi.org/10.1038/ngeo1327
2. Hansen, J., M. Sato, R. Ruedy, et al. (2005). Efficacy of climate forcings. Journal of Geophysical Research: Atmospheres 110, D18104. doi:10.1029/2005JD005776. https://doi.org/10.1029/2005JD005776
3. Winton, Michael, Ken Takahashi, and Isaac M. Held (2010). Importance of ocean heat uptake efficacy to transient climate change. Journal of Climate 23, 2333–2344. doi:10.1175/2009JCLI3139.1. https://doi.org/10.1175/2009JCLI3139.1
4. Andrews, Timothy, Jonathan M. Gregory, and Mark J. Webb (2015). The dependence of radiative forcing and feedback on evolving patterns of surface temperature change in climate models. Journal of Climate 28, 1630–1648. doi:10.1175/JCLI-D-14-00545.1. https://doi.org/10.1175/JCLI-D-14-00545.1
5. Armour, Kyle C. (2017). Energy budget constraints on climate sensitivity in light of inconstant climate feedbacks. Nature Climate Change 7, 331–335. doi:10.1038/nclimate3278. https://doi.org/10.1038/nclimate3278
6. Bellouin, N., J. Quaas, E. Gryspeerdt, et al. (2020). Bounding global aerosol radiative forcing of climate change. Reviews of Geophysics 58, e2019RG000660. doi:10.1029/2019RG000660. https://doi.org/10.1029/2019RG000660
7. von Schuckmann, Karina, Lijing Cheng, Matthew D. Palmer, et al. (2020). Heat stored in the Earth system: where does the energy go? Earth System Science Data 12, 2013–2041. doi:10.5194/essd-12-2013-2020. https://doi.org/10.5194/essd-12-2013-2020
8. Cheng, Lijing, Kevin E. Trenberth, John Fasullo, Tim Boyer, John Abraham, and Jiang Zhu (2017). Improved estimates of ocean heat content from 1960 to 2015. Science Advances 3, e1601545. doi:10.1126/sciadv.1601545. https://doi.org/10.1126/sciadv.1601545
9. Skeie, R. B., T. Berntsen, M. Aldrin, M. Holden, and G. Myhre (2014). A lower and more constrained estimate of climate sensitivity using updated observations and detailed radiative forcing time series. Earth System Dynamics 5, 139–175. doi:10.5194/esd-5-139-2014. https://doi.org/10.5194/esd-5-139-2014
10. Gillett, Nathan P., Megan Kirchmeier-Young, Aurélien Ribes, Hideo Shiogama, Gabriele C. Hegerl, Reto Knutti, Guillaume Gastineau, Jasmin G. John, Lijuan Li, Larissa Nazarenko, Nan Rosenbloom, Øyvind Seland, Tongwen Wu, Seiji Yukimoto, and Tilo Ziehn (2021). Constraining human contributions to observed warming since the pre-industrial period. Nature Climate Change 11, 207–212. doi:10.1038/s41558-020-00965-9. https://doi.org/10.1038/s41558-020-00965-9
11. Andrews, Timothy, and Piers M. Forster (2020). Energy budget constraints on historical radiative forcing. Nature Climate Change 10, 313–316. doi:10.1038/s41558-020-0696-1. https://doi.org/10.1038/s41558-020-0696-1
12. IPCC (2021). Human Influence on the Climate System. Chapter 3 of Climate Change 2021: The Physical Science Basis. Working Group I Contribution to the Sixth Assessment Report. Cambridge University Press. doi:10.1017/9781009157896.005. https://doi.org/10.1017/9781009157896.005
13. Loeb, Norman G., Seung-Hee Ham, Richard P. Allan, Tyler J. Thorsen, Benoit Meyssignac, Seiji Kato, Gregory C. Johnson, and John M. Lyman (2024). Observational Assessment of Changes in Earth's Energy Imbalance Since 2000. Surveys in Geophysics 45, 1757–1783. doi:10.1007/s10712-024-09838-8. https://doi.org/10.1007/s10712-024-09838-8
14. Forster, P. M., C. J. Smith, T. Walsh, et al. (2024). Indicators of Global Climate Change 2023: annual update of key indicators of the state of the climate system and human influence. Earth System Science Data 16, 2625–2658. doi:10.5194/essd-16-2625-2024. https://doi.org/10.5194/essd-16-2625-2024