The origins of this scheme can be traced back to the 1970s at least. A study by tabulated the refractive indices of dust-like, water-soluble, soot, oceanic, sulfate, and mineral aerosol components at wavelengths from 0.300–40 µm, derived using Mie theory. The refractive indices of soot, which is frequently used interchangeably with BC, were drawn from the tabulation of which itself compiled data from a number of pre-existing measurements. Following its publication in , this scheme was included in the Optical Properties of Aerosols and Clouds OPAC; database and entered widespread usage in the climate modeling community, where it is most frequently attributed to one of these two publications. It has since been demonstrated that the dA1991 scheme is inconsistent with observations ; nevertheless, it remains in use in many models (Table ), and it is the default scheme in CanAM5.1-PAM.

In this work, the spectrally varying refractive index is drawn directly from the tabulation. However, we modify the scheme by using a density of 1.6 g cm⁻³. The original scheme assumed a density of 1.0 g cm⁻³ to account for the fact that the particles they measured contained a great deal of air . This density is far lower than the accepted 1.8±0.1 g cm⁻³ and is an unreasonable value of use in an Earth system model. A density of 1.6 g cm⁻³ is selected as a compromise.

compiled and reviewed laboratory measurements of the optical properties of BC. From these data, they used the RDG approximation and the accepted density of black carbon, 1.8 g cm⁻³, to propose a range of BCRI values lying along a “void fraction line” which describes BC with varying degrees of air included within its structure. We select m550nm=1.75–0.63i and m550nm=1.95–0.79i, the lowest and highest values proposed by , for this analysis.

Unlike , only provide estimates of the BCRI at 550 nm. obtained full spectral information for m550nm=1.95–0.79i using the expressions derived by . We use this dataset for our BB2006high scheme and apply an equivalent scaling to the equations of to obtain the BB2006low spectrum given m550nm=1.75–0.63i. For both schemes we assume a density of 1.8 g cm⁻³ as used by .

The BB2006high scheme is used by a number of aerosol models. To our knowledge, no models currently use the BB2006low scheme, although some use intermediate values from , most commonly m550nm=1.85–0.71i (Table ). Nevertheless we select m550nm=1.75–0.63i as our intermediate BCRI in order to span the range of estimates from .

acknowledged in their review that their recommended BCRI could not reproduce the observed mass absorption cross section of black carbon when used in combination with the accepted density of 1.8 g cm⁻³. investigated their hypothesis that the discrepancy was related to shortcomings of the RDG model used in their calculations and demonstrated that the choice of optical model was insufficient to explain the underestimation. More recently, reviewed estimates of the refractive index published since in the context of current estimates of the mass absorption cross section and the absorption function E(m) (Sect. ). Based on this assessment, they recommended refractive indices with E(m)>0.32 in the visible and near-infrared, which would rule out the three BCRI schemes described above.

One scheme recommended by was the estimate of m532nm=1.48–0.84i, derived from measurements of ethylene flame using a bulk density of 1.74 g cm⁻³ and a modified version of the RDG approximation which accounts for some internal scattering effects . This scheme is substantially more absorbing than the previous three at all wavelengths, and to our knowledge it has not been used in any Earth system models. The Besc2016 scheme may not be representative of the BC being simulated by Earth system models since most atmospheric BC comes from more complex sources such as coal, propane, or biomass burning and because BC undergoes rapid morphological transitions after its emission which alter its optical properties. However, its inclusion in this analysis provides a useful upper bound for the likely impacts of varying the BCRI in Earth system models. Other recent BCRI estimates are discussed in Sect. .

report the refractive index at a subset of wavelengths between 266 and 1064 nm. Estimates of the refractive index at other wavelengths, which were derived through application of the Kramers–Kronig relation, were obtained though personal communication with the authors and are presented in .

The four schemes span a range of absorption, which can be quantified by the wavelength-dependent absorption function E(m) : 1E(m)=Imm2-1m2+2. Higher values of E(m) indicate an increased tendency for absorption, and the mass absorption cross section of an aerosol is a linear function of E(m) although the details of this relationship depend on aerosol morphology . At all wavelengths, the dA1991 scheme has the lowest absorption and the Besc2016 the highest. E(m) in the first three schemes increases to both the ultraviolet and infrared, while the Besc2016 scheme decreases slightly to the infrared. All four schemes are fairly constant through the visible and near-infrared .

Although our experiments vary the BCRI at all wavelengths, our analysis predominantly focuses on the characteristics of these schemes at 550 nm. This is the wavelength for which Earth system models typically publish aerosol optical data and for which many satellite retrievals are available. In a complementary analysis, assess the optical properties of BC in the dA1991, BB2006high, and Besc2016 schemes, including the dependence on wavelength, particle size, and mixing state. The analysis relies on theoretical calculations and a one-dimensional radiative transfer model, while the work presented here explores the impacts of the BCRI on Earth system model simulations.

The Canadian Atmospheric Model version 5 CanAM5; is the atmospheric component of the Canadian Earth System Model CanESM5. Here we use CanAM5.1, which contains a number of technical and process representation updates as described in . Most importantly for aerosol modeling, these updates eliminate the occasional formation of spurious tropospheric dust storms seen in CanESM5.

CanAM5.1 can be run with either of two aerosol schemes: a bulk scheme, which simulates aerosol mass budgets and is used in most applications of the model, or the PLA (piecewise lognormal approximation) Aerosol Model (PAM), which we use here. PAM uses the PLA method to simulate aerosol size distributions using a series of truncated, non-overlapping lognormal modes within specified aerosol size sections. Each truncated mode has a specified geometric standard deviation; the magnitudes and mode radii are calculated from the predicted mean masses and number concentrations in each mode at each time step.

Black carbon and organic carbon are emitted as externally mixed, hydrophobic aerosol, represented by one mode each. Upon ageing (time constant τ=1 h during the day and 24 h during the night) these tracers are merged with pre-existing internally mixed aerosol, which is represented by three modes. Sulfate aerosol can form from the ternary homogeneous nucleation of water vapour, gaseous sulfuric acid (H₂SO₄), and ammonia, after which point it grows by Brownian coagulation , or by condensation of water vapour, H₂SO₄, and secondary organic aerosol precursor gases . All sulfate aerosol is contained within the three internally mixed modes and assumed to be fully neutralized by ammonium. Dust and sea salt are externally mixed and are represented by two modes each .

Aerosol activation and cloud droplet growth are determined using pre-calculated solutions to the cloud droplet growth equation for an adiabatically rising air parcel near the cloud base . These solutions are stored in lookup tables and referenced using a numerically efficient iterative approach. The simulated cloud droplet number concentration is used to compute the effective radius of the cloud droplets according to the first indirect effect . In the model configuration used here, the second indirect effect is included via the autoconversion of cloud to rain droplets .

Aerosol sinks in PAM include dry deposition, which depends on ground-surface properties and the near-surface aerosol concentration ; wet deposition by below-cloud scavenging ; and in-cloud scavenging in convective and layer clouds . Both below-cloud and in-cloud wet deposition rates are proportional to the precipitation formation rate.

Aerosol optical properties in PAM are determined from pre-computed lookup tables. The tables are generated using Mie theory to determine optical properties as a function of relative humidity, wavelength, and particle size. Although the assumption of spherical particles may be inappropriate for freshly emitted BC, the majority of the BC simulated by an Earth system model is hours to days old and will be relatively compact and/or internally mixed, making Mie theory a reasonable approximation. For internally mixed aerosol an effective refractive index is computed using the Maxwell-Garnett approximation , which has been demonstrated to describe the optical properties of coated BC better than volume-weighted or core-shell approximations .

Other treatments of aerosol morphology and mixing state would likely yield different absolute values of simulated AAOD and ERFari. However, the focus of this work is on the difference in these quantities between simulations conducted with different BCRI schemes with other parameterizations held constant. As it is, CanAM5.1-PAM's simulated AAOD is in good agreement with other Earth system models and with observations to the extent that the latter can be determined given the associated uncertainties (Fig. ). The report found that CanAM5-PAM reproduced the observed vertical profile of BC in both the Arctic and northern midlatitudes particularly well, relative to other models.

In the simulations conducted for this analysis, transient historical sea surface temperatures and sea ice concentrations were specified using the PCMDI observational dataset , and historical sea ice thicknesses for the Northern Hemisphere and Southern Hemisphere were taken from PIOMAS and ORAP5 reanalyses respectively.

We simulate four sets of “core ensembles”: one control ensemble and one perturbed ensemble for each of the BCRIs described above. Each ensemble consists of nine short simulations (2014–2019) and one long simulation (1949–2019). The first year of each is discarded as spinup. In the control ensemble, all emissions of aerosols and greenhouse gases are transient; in the perturbed ensemble, BC emissions are fixed at 1850 levels, while other emissions evolve as in the control scenario. ERF is then calculated from the difference in top-of-atmosphere flux between pairs of control and perturbed runs, following the “ERF_trans” method of . This method of calculating ERF is chosen over the alternative approach in which the control scenario uses preindustrial emissions and the perturbed scenario adds transient emissions of the forcer of interest because the latter method does not account for interactions between species. Finally, the total BC ERF is decomposed into contributions from aerosol–radiation (ERFari), aerosol–cloud (ERFaci), and albedo (ERFalb) interactions following . In this work we exclusively consider shortwave ERF. Longwave BC ERF is small in CanAM5.1-PAM, consistent with previous findings for models that do not parameterize aerosol impacts on ice- and mixed-phase clouds . Although PAM includes representations of the albedo effects on BC deposited on snow and ice and absorption of solar radiation by BC-containing cloud droplets , in this work we only vary the refractive index of atmospheric BC. ERFaci and ERFalb are thus expected to be similar between the three core ensembles.

The influence of the BCRI on AAOD, ERFari, and tropospheric temperature is quantified by comparing these fields between the four core ensembles. We then compare the changes in ensemble-median AAOD and ERFari that arise from the choice of BCRI to the variation in comparison datasets due to other factors. This comparison is not intended as a comprehensive analysis of BC uncertainties but rather presents an illustrative sample of relevant uncertainties other than the BCRI.

To contextualize the sensitivity of AAOD and ERFari to variations in the BCRI, we consider three different comparison datasets characterizing aspects of the uncertainties in these quantities.

Our first comparison investigates the impact that recent updates to aerosol emission inventories have on simulated AAOD. In the core ensembles, anthropogenic aerosol emissions are taken from the Community Emissions Data System (CEDS) 21 April 2021 release . CEDSv2021 emissions not only extend the historical emissions used in CMIP6 to more recent years but also include several back corrections, most notably reducing the emissions of BC, organic carbon, and sulfur dioxide over China . Biomass burning emissions in the core ensembles are taken from the CMIP6 historical inventory for 1950–2014 and the Global Fire Emissions Database GFED v4.1s; for 2015–2019. To investigate the impact of these selections, we run a single simulation forced with the more commonly used CMIP6 historical and SSP2-4.5 anthropogenic and biomass burning emissions, otherwise identical to the low-absorption core ensemble. We compare the AAOD from this simulation against each realization of the low-absorption ensemble in turn, yielding a nine-member ensemble of differences. We emphasize that this comparison does not represent the total uncertainty in either anthropogenic or biomass burning aerosol emissions, as we are comparing two versions of the same anthropogenic emissions inventory and extending the biomass burning emissions with the same observational product as was used in the creation of the CMIP6 historical and SSP inventories. Instead this comparison shows the sensitivity of AAOD to recent improvements in both sets of emissions.

Our second comparison considers observational uncertainty. We compare estimates of AAOD from the Multi-angle Imaging SpectroRadiometer MISR; and the Polarization and Directionality of the Earth's Reflectances instrument with the Generalized Retrieval of Atmosphere and Surface Properties algorithm POLDER-GRASP;, selected for their availability in level 3 gridded format. MISR and POLDER-GRASP bracket the range of AAOD simulated by current Earth system models (Fig. ). We consider satellite observations rather than ground-based or in situ measurements due to the challenge in comparing point-source data against gridded model results . POLDER-GRASP data are not available for the 2015–2019 study period, so we instead compare these satellites over the 5-year period of 2007–2011. The difference between MISR results for 2015–2019 and 2007–2011 is substantially smaller than the difference between MISR and POLDER-GRASP data for 2007–2011, indicating that the use of these alternate years is unlikely to affect our conclusions. This comparison is not intended to be a detailed evaluation of the uncertainty in remotely sensed AAOD; such assessments can be found in, for example, . Instead it provides an estimate of the range of AAOD that can be obtained from different instruments. As such it does not account for differences in sampling between the two satellites or between the observed and simulated AAOD fields.

Our final comparison considers the range of AAOD and ERFari reported in recent multimodel assessments from the literature. This comparison folds in many sources of model uncertainty, including differences in the treatment of mixing state which has been shown to have a substantial impact on simulated BC absorption , as well as differences in the parameterization of aerosol transport and deposition. The individual assessments are described in Sect. .

Modifying the BCRI directly modifies BC absorption, and we thus begin by assessing its impact on AAOD (Fig. ). Increasing BC absorption from the dA1991 scheme to the BB2006low scheme increases global-mean 2015–2019 AAOD by 27 %, increasing from dA1991 to BB2006high increases AAOD by 42 %, and increasing from dA1991 to Besc2016 increases AAOD by 59 %. Absolute increases are, unsurprisingly, largest over major source regions, but the medium- and high-absorption ensembles are statistically significantly different from the low-absorption ensemble everywhere (Fig. b, c, d). The four ensembles of global-mean AAOD are clearly separated, with no overlap between annual-mean AAOD (Fig. e) and little overlap between monthly-mean AAOD (Fig. f). As absorption increases, so too does the magnitude of the trend in AAOD over 1950–2019 (Fig. e).

Figure g compares the regional increases in AAOD from varying the BCRI to the increases obtained by varying the aerosol emissions and to the differences in observed AAOD from the MISR and POLDER-GRASP satellites. Two BCRI-induced changes are shown: solid bars give the AAOD difference between dA1991 and BB2006high ensembles and hashed bars the difference between dA1991 and Besc2016. Region definitions are provided in Appendix . The “near-global” and “Northern Hemisphere” domains exclude latitudes poleward of 60° where the satellite retrievals are poorly sampled. Most of the regional variation in the increase in AAOD caused by varying the BCRI comes from differences in the local BC burden (Fig. ); the relative increase is fairly consistent, varying from 39 %–47 % when comparing BB2006high to dA1991.

The choice of aerosol emission inventory has the greatest impact in regions where the two inventories are the most different and where the baseline emissions are high. The largest emissions-based AAOD increase, both in absolute terms and relative to the BCRI-induced increase, occurs in East Asia where both conditions are satisfied; the smallest occurs in South Asia where the two inventories are nearly identical. These regional increases are a factor of 1.5 larger than, and a factor of 16 smaller than, the AAOD differences between dA1991 and BB2006high ensembles respectively. The 60° S–60° N average change resulting from the updated emissions is approximately 60 % of the BCRI increase. Note that these values do not indicate the overall uncertainty in global or local emissions, only the spatial distribution of updates to the inventories being considered. For example, report a factor of 7 difference in Southeast Asian biomass burning emissions from different inventories (a more accurate representation of emission uncertainties in this region), whereas we see almost no change between the inventory versions considered in this assessment.

In all regions the range of AAOD between different satellites is substantially larger than the range of simulated AAOD in CanAM5.1-PAM under different configurations. This is expected given the challenges in constraining remotely sensed AAOD . Even so, in regions where the BC burden is high the impact of the BCRI on AAOD can be as much as two-thirds as large as the difference between satellites. Averaged over the near-global domain, the inter-satellite discrepancy is a factor of 5 larger than the AAOD difference between dA1991 and Besc2016. The satellite differences are expected to be largest in regions where characteristics of the geography increase retrieval uncertainty (e.g. regions with more reflective surfaces) and in regions where the satellites differ in their sensitivity to the dominant aerosol type.

Finally, for the near-global domain only we compare with the spread in global-mean AAOD from AeroCom Phase III models . AeroCom Phase III estimates of the global-mean AAOD in simulations forced with 2010 emissions range from 2.04×10-3 to 9.78×10-3 for an overall range of 7.75×10-3 or approximately 6 times that obtained by varying the BCRI alone. This is similar to the range of AAOD between satellites, and as shown in Fig. , the distribution of zonal-mean AeroCom Phase III AAOD is approximately bracketed by the zonal-mean AAOD from MISR and POLDER-GRASP. The 15 AeroCom models have BCRI values ranging from our dA1991 to BB2006high schemes, but the BCRI alone does not explain the intermodel spread, as discussed further in Sect. .

Overall, the sensitivity of simulated AAOD to the choice of a BCRI is comparable to its sensitivity to recent updates to the aerosol emission inventory within the same model. Between models, or between satellites, the global-mean AAOD spread is a factor of 5 to 7 larger.

Despite the fact that BC generally makes up a small fraction of total aerosol extinction , we do see regionally significant increases in total aerosol optical depth (AOD) when BC absorption is increased (Appendix ). These changes are particularly evident in central and southern Africa, South America, and the northern latitudes. These are all regions with high emissions from biomass burning, which are associated with a higher ratio of BC emissions. Heavily polluted regions, such as South and East Asia, do not show an AOD dependence on a BCRI, likely because AOD in these regions is dominated by sulfate. There is not a statistically significant difference between globally averaged AOD in the four core ensembles.

The BCRI directly affects the interaction of BC with incoming solar radiation and therefore the aerosol–radiation forcing ERFari. Figure  summarizes the variation in shortwave BC ERFari across our four ensembles.

Increasing BC absorption from the dA1991 scheme to the BB2006low scheme increases global-mean 2015–2019 BC ERFari by 32 %, increasing from dA1991 to BB2006high increases BC ERFari by 47 %, and increasing from dA1991 to Besc2016 increases BC ERFari by 100 %. These increases are largely confined to the Northern Hemisphere, and for the BB2006high and Besc2016 ensembles, they are statistically significant in most regions where the dA1991 BC ERFari is significantly nonzero (Fig. c, d). Global-mean BC ERFari values in the BB2006low, BB2006high, and Besc2016 ensembles are significantly different from the dA1991 ensemble at the 5 % level. There is more interannual variability in the time series of global-mean BC ERFari than there is in AAOD, likely due to the dependence of ERFari on cloud fields (Fig. e, f). Despite this variability, ERFari trends over 1950–2019 increase with the BCRI as the AAOD trends do.

The uncertainty in BC ERFari attributable to the choice of the BCRI can be compared to results from two recent literature assessments. We first compare with the multimodel range of BC ERFari assessed by and then with the statistical uncertainty in BC ERFari reported by the Arctic Monitoring and Assessment Program (AMAP) 2021 report. We emphasize that these are fundamentally different quantities with different interpretations. Furthermore, and estimate ERF for different time periods, assuming different emission inventories, and using different methodology in their calculations, so the studies' best estimates are not directly comparable to our results or to each other. The comparisons are nevertheless useful in contextualizing how much of an impact uncertainty in the BCRI has on that of BC ERFari.

calculated ERF for numerous aerosol and greenhouse gas species for 1850–2014 based on results from AerChemMIP . Eight models provided estimates of total BC ERF, but only four decomposed this ERF into contributions from radiation, cloud, and surface albedo forcings. These four models (CNRM-ESM2, MRI-ESM2, NorESM2, and UKESM1) simulated BC ERFari of 0.37, 0.13, 0.35, and 0.26 W m⁻² respectively for an overall range of 0.24 W m⁻². This is a factor of 1.3 (2.6) larger than the difference in ERFari simulated by our dA1991 and Besc2016 (BB2006high) ensembles. The four models use a narrow range of the high-absorption BCRI: m550nm=1.95–0.79i (MRI-ESM2 and NorESM2), m550nm=1.85–0.71i (UKESM), and m550nm=1.83–0.74i (CNRM-ESM2). The assessed range thus did not stem primarily from differences in the BCRI, although differences in the treatment of mixing states could still result in different levels of BC absorption. The results of provide the basis for the assessed BC ERFari in the latest Intergovernmental Panel on Climate Change (IPCC) assessment report, AR6 ; as AR6 did not itself quote an uncertainty range for BC ERFari, we do not otherwise include it in this comparison.

assessed radiative forcing for 1850–2015 using a different aerosol emission inventory than . Five models contributed data for BC ERF, but only two provided the decomposition into ERFari: MRI-ESM2 and CanAM5-PAM, with BCRI values of m550nm=1.95–0.79i and m550nm=1.75–0.44i respectively (equivalent to the BB2006high and dA1991 schemes assessed here). The reported ERFari derived from these two models was 0.27±0.04 W m⁻². Unlike the and AeroCom results discussed above, this ±0.04 W m⁻² is an estimate of statistical uncertainty rather than a multimodel range. It represents the precision to which ERFari can be determined when derived from 100-year integrations of two Earth system models. The overall uncertainty range of 0.08 W m⁻² is ∼ 80 % of the spread we obtain by varying the BCRI across the same range within one model, suggesting that a different choice of BCRI in either model could have materially impacted the assessed ERFari.

Varying the BCRI, and thus the absorption, of atmospheric BC is found not to have a statistically significant impact on the aerosol–cloud forcing in this experiment. We did not vary the BCRI within cloud droplets, so the only impact on clouds would be via the impact of changes in atmospheric temperature profiles, discussed below. These changes are found to be small relative to the variability in simulated cloud fields. Similarly, the radiative forcing from albedo changes was not found to vary with BCRI scheme because we did not vary the refractive index of BC deposited on snow and ice. The only change in the total BC ERF was thus from the aerosol–radiation component. For the BB2006low and BB2006high schemes, this change was too small to result in a statistically significant increase in total ERF. However, the Besc2016 scheme led to a statistically significant increase in global-mean BC ERF relative to the dA1991 scheme, from -0.02 to +0.24 W m⁻².

Black carbon influences global and regional temperature , mean and extreme precipitation , and the Asian monsoon . The sensitivity of these fields to the BCRI cannot be assessed in this study as the four ensembles used the same set of prescribed sea surface temperatures. We can, however, examine temperature changes from the radiative heating of the atmosphere by BC in the mid- and upper troposphere and draw on other works to estimate potential precipitation changes.

Figure  illustrates the vertical distribution of temperature changes obtained by varying the BCRI. Although near-surface air temperatures are constrained by the prescribed sea surface temperatures, statistically significant zonal-mean warming is evident over the northern midlatitudes starting at about 850 hPa. At 600 hPa and above, this warming is mostly confined to regions above and downwind of East and South Asia (not shown); at lower altitudes, significant warming is also apparent over or downwind of Africa and South America. Large temperature responses are apparent at the poles in Fig. a and b, but note that these changes are not statistically significant and that they correspond to small geographic areas. The mid-tropospheric warming does not appear sufficient to impact local cloudiness in our model: there are not statistically significant differences between the low- and high-absorption cloud fractions at these levels.

To estimate the potential impact of the BCRI on precipitation, we draw on the results of , who derived an expression for global-mean precipitation suppression as a function of changes in AAOD: 2ΔP=dPdAbs×dAbsdAAOD,ΔAAOD=-4646±1600mmyr-1(unitAAOD)-1, where P denotes the precipitation rate (in mm yr⁻¹) and “Abs” denotes atmospheric absorption, defined as the difference between top-of-atmosphere and surface radiative forcings (in units of W m⁻²). A numerical estimate of the first factor in Eq. () was derived from historical simulations in AeroCom Phase II and CMIP6 models, and the second factor was drawn from the results of and . Applying Eq. () to the global-mean ensemble-median increase in AAOD obtained by changing from the dA1991 to BB2006high scheme yields an estimated precipitation suppression of 5.9±2.0 mm yr⁻¹. It is extremely unlikely that this signal would be detectable in the global mean in our simulations. It is possible, however, that regional changes could be considerably larger.