The experiments were conducted with the NorESM1-Happi model , which is an upgraded version of the NorESM1-M model used for CMIP5 . In the current experiments, sea surface temperatures and sea ice fraction were prescribed based on observations for the years 2012–2017, and sea ice thickness was set to 2 m (1 m) for the Northern (Southern) Hemisphere. The following model components were included:

The atmospheric component is the Oslo version of the Community Atmosphere Model (CAM4-Oslo), which differs from the original CAM4 through modified chemistry–aerosol–cloud–radiation interaction schemes . The finite-volume dynamic core is employed. The horizontal resolution is 1.9∘ latitude and 2.5∘ longitude, with 26 levels in the vertical and the model top at 2.19 hPa. An error in the aerosol life cycle scheme was corrected for NorESM1-Happi . This acts to reduce the atmospheric residence time of BC and upper tropospheric BC concentration.

In the present experiments, the aerosol life cycle scheme in NorESM1-Happi is run in an offline mode . The physical and chemical processes influencing aerosol concentrations are simulated interactively, but the simulated aerosol concentrations do not influence the simulated climate. Instead, in the atmosphere model, CCSM4's prescribed standard aerosols are used for integrating the model forward in time. In this study, we extended the offline treatment of aerosols to BC in snow. The BC concentration in snow is simulated interactively, based on BC deposition fields provided by NorESM's aerosol scheme. Snow albedo is computed both with and without BC; the latter is used for integrating the model forward in time and the former for diagnostics. The offline approach is crucial for the present study, since it allows for an essentially noise-free evaluation of RFs associated with even minor changes in emissions. If, instead, the emissions were allowed to influence the simulation, the RFs could easily be swamped by radiative flux changes related to internal climate variability.

A limitation of the offline approach is that it can only be used for evaluating BC instantaneous RF. Since BC does not influence the simulated temperature, the effect of rapid adjustments (including, but not limited to, the semi-direct effect of BC on clouds; ) cannot be evaluated. The RFs are evaluated based on extra diagnostic calls to the model shortwave radiation scheme. 1dirRF=Fair+snow-Fsnow2snowRF=Fair+snow-Fair-Fair+snow,0-Fair,03indirRF=Faie-Faie,0. Here, F denotes net (down - up) solar radiative fluxes, and the subindices air + snow, air, and snow refer to calculations in which BC is included in both air and snow, only in air, and only in snow, respectively, and aie refers to a calculation targeted at evaluating the aerosol indirect effect on liquid-phase clouds. Note that, for snowRF, the additional pair of terms on the right-hand side Fair+snow,0-Fair,0 is the BC snowRF in the absence of BC emissions. It is nonzero (≈0.004 W m-2 in the global mean) because, in the model initial conditions, there is some BC stored in snow in regions with permanent snow, mainly in Greenland. Likewise, Faie for the experiment with no BC emissions has to be subtracted to define BC indirRF. Further details on how Fair+snow, Fair, Fsnow, and Faie are calculated are provided in Appendix A. The model also includes a calculation of longwave indirRF for liquid water clouds. This effect is, however, very small (nearly 2 orders of magnitude smaller than the shortwave indirRF for BC), and for simplicity, it has been ignored throughout this paper.

While Eqs. ()–() can be applied to define the RFs at any level in the atmosphere at any spatiotemporal scale, we will only consider here annual mean RFs at the top of the atmosphere (TOA). Furthermore, we found that indirRF depends very nonlinearly on BC emissions and pre-existing aerosol concentrations, which would make the interpretation of the experiments discussed in Sect.  difficult. Therefore, the focus of this study will be on BC dirRF and snowRF.

The chemical components in the NorESM1 aerosol scheme include BC, sulfate, organic matter (OM), sea salt, and mineral dust. The BC emissions are discussed in Sect.  below. For the other aerosol components, a realistic spatiotemporal distribution of emissions was aimed at. The emissions follow , with modifications for sulfur dioxide SO2 (a precursor of sulfate) and OM. The SO2, OM, and BC emission sources are divided in NorESM1 into fossil fuel and biomass burning emissions, with a somewhat different treatment of mixing and other physicochemical processes in the aerosol scheme (see ; Fig. 1). For the biomass burning SO2 and OM emissions, RCP8.5 emissions for the year 2015 are assumed in this study. Correspondingly, the fossil fuel SO2 and OM emissions are taken from the ECLIPSE V6b current legislation (CLE) data (; see also ) for the year 2015, with the agricultural waste burning (AWB) sector eliminated due to overlap with biomass burning emissions. Sensitivity tests indicated that the impacts of SO2 and OM emissions on the BC burden simulated by NorESM1-Happi are very small. Likewise, BC emissions have very little impact on the burdens of SO4 and OM.

We first briefly evaluate the experiment REAL in which a realistic global spatiotemporal distribution of emissions is assumed. Figures a and a show the annual mean dirRF and snowRF at the TOA for REAL. The global mean dirRF is 0.651 Wm-2, which is close to the observationally constrained best estimate of 0.61 Wm-2 (90 % confidence interval 0.16 to 1.40 Wm-2) derived by . The global mean snowRF in REAL is 0.0285 Wm-2. This is closer to the lower end of the range of instantaneous snowRF quoted in the Intergovernmental Panel on Climate Change (IPCC) AR5 0.02–0.09 Wm-2;, while falling between the post-AR5 model-based estimates of 0.013 Wm-2 by and 0.045 Wm-2 by . (The IPCC AR5 and estimates include only anthropogenic BC, while natural BC is also included in our RF values. However, this does not bias the comparison substantially because, in the REAL experiment, ca. 90 % of snowRF is associated with fossil fuel BC emissions.) The global mean BC burden in REAL is 2.78×10-7 kgm-2 (equivalent to 0.142 Tg), while the global mean BC lifetime is 6.2 d. These values are slightly above the multimodel median (BC burden 0.131±0.047 Tg; BC lifetime 5.5±1.9 d; median ± standard deviation) of the ensemble of 14 AeroCom phase III model simulations for the year 2010 analyzed by .

Overall, the global mean BC dirRF, snowRF, burden, and lifetime in REAL are well within the range of previous studies. Even so, the results may be subject to model-specific biases. One known issue is that NorESM1-Happi likely overestimates upper-tropospheric BC concentrations, although less severely than NorESM1-M (Fig. S1 in ; see also ). This may be caused by overly efficient convective transport of BC to the upper troposphere . Since upper-tropospheric BC contributes disproportionately to dirRF , it appears likely that NorESM1-Happi somewhat overestimates dirRF, especially for BC emissions in convective regions.

In Sect. , the dependence of BC RF on the emission region and season is evaluated based on idealized experiments in which BC emissions are limited to a single region/season with a fixed emission rate ϵideal=10-12 kgm-2s-1. This approach involves the assumptions that the BC RF is linear with regard to the emission rate and additive with regard to emissions from different regions/seasons. To evaluate to which extent these assumptions are violated in practice, we apply the results from the idealized experiments for reconstructing the BC RF in cases in which the emissions occur all over the globe and throughout the year. The reconstructed RFs (RFreconst) are calculated from the idealized experiments as follows: 4RFreconst=∑i=112∑j=116∑s=14ϵijsϵidealRFideal,ijs, where i, j, and s are indices for longitude, latitude, and season, RFideal,ijs is the RF in the idealized experiment with the constant BC emission rate ϵideal applied in the lat–long box (i,j) for the season s, and ϵijs is the average BC emission rate in the same lat–long box and season for the experiment whose RF is being reconstructed. Note that, besides the RFs, this formula can also be applied to other BC-related fields, such as emission and deposition rates and column BC burden.

The patterns of dirRF and snowRF reconstructed for the REAL experiment are shown in Figs. b and b. They each correlate strongly with the dirRF and snowRF simulated in REAL, with spatial correlation coefficients of 0.952 and 0.919, respectively. Nevertheless the reconstruction errors for dirRF (Fig. d, g) and for snowRF (Fig. d, g) feature both substantial small-scale variations and larger-scale biases. The relative global mean bias is small for dirRF (-2.5 %) but rather large and positive for snowRF (+44.9 %). The reconstruction errors in Figs. d, g and d, g arise from two distinct sources which we denote as the emission error and the additivity error.

The emission error arises because the reconstruction cannot represent accurately the distribution of emissions in REAL. First, the emissions are represented at 1.9∘×2.5∘ spatial resolution and monthly temporal resolution in REAL, while, in the idealized experiments used for reconstruction, the spatial resolution is ca. 11.4∘×30∘, and the temporal resolution is 3-monthly. Second, the idealized experiments treat all BC as fossil fuel BC (see ; Fig. 1) and place the emissions in the lowermost model layer, while for REAL, ca. 30 % of the emissions are released as biomass burning BC, with a different treatment in the aerosol scheme and larger emission heights. Since the reconstructed emissions for REAL are equal to both the actual and reconstructed emissions for COARSE, the emission error can be evaluated as the difference between COARSE and REAL (Figs. e, h and e, h). It explains most of the locally large positive/negative reconstruction errors seen in Figs. d, g and d, g. It also influences the global mean values, especially for snowRF, for which it explains roughly one half of the reconstruction bias (0.0062 out of 0.0128 Wm-2). This bias mainly originates from the Tibetan region and is associated with overestimated BC deposition (Figs. S2d, e, g, h and S3d, e, g, h in the Supplement) due to the misrepresentation of the spatial emission patterns from India and China by the reconstruction (Fig. S1d, e, g, h). Furthermore, the emission error results in underestimated dirRF in regions where BC mostly originates from biomass burning in REAL, especially over Angola and off its western coast and over the western Amazonas and downwind of it in the eastern Pacific (Fig. d, e, g, h).

The additivity error is more fundamental than the emission error in the sense that it directly addresses the validity of the basic assumption of the reconstruction approach, that is, how large is the impact of nonadditivity and nonlinearity when combining the effect of BC emissions from different regions and seasons? In other words, how large are the reconstruction errors if the emission errors are eliminated? This question can be answered by considering the reconstruction errors for the COARSE experiment, shown in Figs. f, i and f, i. Note that, by the design of the experiment, the reconstructed fields for COARSE are identical to those for REAL (Figs. b and b) and are not shown separately.

For the COARSE experiment, the reconstruction reproduces the spatial patterns of dirRF and snowRF extremely well, with correlation coefficients of 0.9996 and 0.996, respectively. The reconstruction errors are relatively smooth and positive almost throughout (Figs. f, i and f, i). For dirRF, the global mean reconstruction bias is 8.5 %, in contrast to the slightly negative bias of -2.5 % for the reconstruction estimate of REAL. For snowRF, the global mean reconstruction bias (19.0 %) is significantly smaller than that for the REAL experiment (44.9 %). The positive errors suggest that dirRF and, in particular, snowRF, increase slightly more weakly than linearly with BC emissions. This is demonstrated explicitly in Fig. . We argue in Sect.  that this occurs primarily due to partial saturation of BC absorption. For dirRF, a slight overestimate of the atmospheric BC burden in the reconstruction (on average 2.6 %; see Fig. S4i) also contributes to the positive reconstruction errors.

The above reconstruction tests, especially for the COARSE experiment, indicate that the separate consideration of emissions from different regions and seasons tends to result in overestimated RF and more so for snowRF than dirRF. Nevertheless, spatial patterns of RF are reproduced very well, and the magnitude of the bias (on average below 10 % for dirRF and below 20 % for snowRF) appears reasonably small. It is concluded that the separate consideration of emissions from different regions/seasons in the idealized experiments provides acceptable results for dirRF and snowRF. In contrast, for the BC indirRF diagnosed by NorESM1-Happi, the dependence on emissions is strongly nonlinear. The reconstruction overestimates the global mean values severely by 410 % for REAL and by 228 % for COARSE (Fig. S5). Accordingly, indirRF is omitted from the main analysis addressing the impact of emission location and season. However, some more discussion on the BC RF nonlinearity, including also indirRF, is provided in Sect. .

Figure  displays the spatial distribution of the BC burden, BC deposition rate, dirRF, and snowRF for the experiment in which a seasonally uniform BC emission rate of 10-12 kgm-2s-1 was assumed for the lat–long box 56.84–68.21∘ N, 1.25–31.25∘ E. This area covers most of the Fennoscandian region and some of its surroundings. Not surprisingly, the simulated BC burden (Fig. a) peaks in the emission region, with a local maximum of ca. 88 mgm-2. Some BC is, however, transported further away (especially towards the east/northeast due to the dominant wind directions), and the BC burden exceeds 1 mgm-2 (i.e., roughly 1 % of the maximum value) in most of the Arctic, in parts of northern midlatitudes, and in northern Africa. The pattern of BC deposition is broadly similar to that of BC burden, although slightly more confined to the vicinity of the emission region (Fig. b). The dirRF follows largely the same pattern as the BC burden, with some modulation due to, for example, differences in surface albedo, how much of the BC is above clouds, and the availability of solar radiation (Fig. c; see also Fig. S6). The snowRF is constrained both by the distribution of BC deposition and the availability of snow (Fig. d).

Due to the offline treatment of aerosols, the simulated patterns of RF are quite smooth even though the RFs themselves are very small. The global mean values are 1.90 mWm-2 for dirRF and 0.86 mWm-2 for snowRF. Of course, these RF values depend on the area A of the lat–long boxes used for emissions and on the emission rate ϵ employed. For this reason, we will consider global mean specific forcings (SFs), defined as the global annual mean RFs (RFglob) normalized by the global annual mean emission rate ϵglob resulting from emissions in a given region as follows: 5dirSF=dirRFglob/ϵglob6snowSF=snowRFglob/ϵglob7ϵglob=ϵftimeA/Aglob, where ftime is the time fraction with emissions (1 for annual emissions and ca. 0.25 for seasonal emissions), and Aglob is the global surface area. This is similar to the concept of the specific forcing pulse (SFP) introduced in , but we leave out the word “pulse” because the specific forcings refer here to continuous emissions. The SFs will be expressed in units of terajoules per kilogram (TJkg-1, i.e., the amount of energy added to the climate system in terajoules per 1 kg of BC emitted). BC burdens will likewise be normalized by the emissions, yielding the BC lifetime expressed in days. In the case of Fig. , the normalized global mean values are as follows: 0.50 TJkg-1 for direct specific forcing (dirSF), 0.23 TJkg-1 for snow specific forcing (snowSF), and 3.15 d for the BC lifetime.

Figure  summarizes the dependence of BC SF on the emission location, based on the experiments in which BC emissions are uniform over the year. Each lat–long box is colored according to the global mean SF associated with BC emissions within that lat–long box, and Fig. a shows BC dirSF and Fig. b BC snowSF. (A corresponding figure for BC indirect specific forcing (indirSF) is provided in the Supplement (Fig. S7), but it is not discussed here due to the strong nonlinearity noted above). The impact of the emission location is very large, both for dirSF and snowSF. Considering first dirSF, the values vary by more than a factor of 10, depending on the emission location, from 0.18 TJkg-1 (for emissions at 45.47–56.84∘ S, 28.75∘ W–1.25∘ E) up to 2.33 TJkg-1 (for emissions at 0–11.37∘ N, 151.25–181.25∘ E). The dirSF tends to be larger for emissions at low latitudes than at high latitudes; however, relatively large values also occur for emissions over Antarctica and Greenland. There are also substantial variations as a function of longitude, with dirSF being typically larger for emissions over land than over the ocean. Nevertheless, the very largest values occur for emissions located east of Indonesia over the Tropical Warm Pool. This, together with the high values over low-latitude land areas, points to the important role of convective transport of BC.

The patterns of dirSF bear a strong resemblance to the corresponding patterns of BC lifetime, as shown in Fig. a. The area-weighted spatial correlation between them is as high as 0.94. BC lifetime varies from less than 2 d for emissions in parts of the Southern Ocean and the Arctic Ocean up to 11 d in the Tropical Warm Pool. Figure b further shows the ratio of dirSF to BC lifetime. This ratio varies by a factor of 4, from 0.10 to 0.40 TJkg-1d-1, indicating that factors other than BC lifetime also play a substantial role for dirSF. BC dirSF is enhanced when BC is located above a high-albedo surface (or, alternatively, above reflective clouds). The role of high surface albedo is seen in the locally large values of the dirSF to lifetime ratio for emissions over Greenland, the Sahara, and especially Antarctica. Another factor that influences the dirSF to lifetime ratio is the distribution of incoming solar radiation. Much of the emitted BC remains relatively close to the latitude of emission, and therefore, BC emitted at low latitudes is, on average, exposed to higher levels of solar radiation than BC emitted at high latitudes. This effect is, however, partially offset by the solar zenith angles being larger at high latitudes, which makes the slantwise aerosol optical depth larger for direct solar radiation.

For BC snowSF (Fig. b), the impact of the emission location is even larger than that for dirSF. For low-latitude emissions between 34.11∘ S and 22.74∘ N, snowSF is minimal (generally below 0.01 TJkg-1). Although some of the emitted BC is transported to the upper troposphere at high latitudes, its contribution to BC deposition on snow is negligible. The BC emissions from the northern India area (22.74–34.11∘ N, 61.25-91.25∘ E) are associated with a slightly larger BC snowSF (0.20 TJkg-1), due to deposition in Tibet. In contrast, at high northern and southern latitudes, snowSF reaches very high values, with the maxima over the permanent snow-/ice-covered regions of Antarctica and the Arctic Ocean exceeding 7 TJkg-1. These maxima greatly exceed the maximum of BC dirSF (2.33 TJkg-1; Fig. a). This suggests that, for a given mass of BC emitted, BC total RF and, very likely, the climate effects are largest for high-latitude emissions. This will be discussed further in Sect. .

It is also worth noting that BC snowSF shows a substantial dependence on the longitude of the emissions, for example, in the subarctic region (56.84–68.21∘ N). There, snowSF varies from values close to 0.2 TJkg-1 for emissions in the Atlantic and Fennoscandian regions to 1–1.2 TJkg-1 for Siberian emissions and even up to 1.9 TJkg-1 for emissions from northeastern Canada. A key factor for these differences is how long snow persists into spring, since large snowSF requires a combination of extensive snow cover and abundant solar radiation in the regions where BC is deposited. A further factor is the shielding of snow by vegetation. The main BC deposition regions for Fennoscandian emissions (Fig. b) are characterized by both relatively early snowmelt and extensive forest cover, which both act to decrease the BC snowSF, as compared with Siberian and North American emissions originating from the same latitudes.

In this section, the impact of the emission season on the global mean BC dirSF, BC snowSF, and BC lifetime is presented. Figure  summarizes the general features of the seasonal dependence. Land-weighted and sea-weighted zonal mean values were defined as averages over the 12 emission regions in each latitude band weighted by the land fraction and sea fraction of the emission regions, respectively. The land-weighted and sea-weighted averages help to illustrate some differences between the impact of seasonal emissions over land and sea. This separation is, however, not complete because many of the 192 emission regions contain both land and sea areas. Detailed lat–long maps showing the dependence on the emission season are presented in the Supplement (Figs. S8–S10).

It is seen from Fig. a, b that BC dirSF depends substantially on the emission season. The dependence is particularly large at high and middle latitudes, owing to the strong seasonal cycle of insolation. BC dirSF is generally largest for emissions in the local summer and smallest for emissions in winter. As an exception, for emissions in the Arctic and in the southern parts of Southern Ocean, dirSF peaks already for emissions in spring, presumably because the snow and ice cover is larger in spring than in summer, and therefore, the surface albedo is higher. Furthermore, the dependence of dirSF on the emission season is generally larger for emissions over land than over sea. Over land, the impact of increased insolation is reinforced by increased BC lifetime, especially in the subtropics and at midlatitudes (Fig. e). This applies to emissions from (for example) the USA, southern/central Canada, northern India, China, and southern/central Siberia in JJA (Fig. S10c) and emissions from Australia and southern Africa in DJF and SON (Fig. S10a, d). The increased lifetime is associated with lesser atmospheric stability and stronger convection in summer.

For BC snowSF (Fig. c, d), the impact of emission season is rather different from that for dirSF. The fundamental difference is that, while the BC atmospheric lifetime is short (mostly 1 week or less), BC deposited on snow remains in the snowpack until snow melts. Therefore, BC deposited on snow in the dark season can absorb solar radiation in spring as long as there is snow left. This explains the relatively large annual mean BC snowSF associated with Arctic BC emissions in DJF (and, for the high Arctic, also SON). The largest snowSF by Northern Hemisphere high-latitude emissions is, however, caused by emissions in spring, as they maximize the concentration of BC in the uppermost snow in spring. The snowSF for Northern Hemisphere emissions is smallest for emissions in summer (in fact, near zero, except for Arctic emissions) due to the lack of snow. In contrast, for emissions in Antarctica, where snow cover is permanent, snowSF is largest for emissions in summer (DJF), followed by spring (SON).

The NorESM1-Happi experiments indicate that the global mean SF associated with BC emissions depends greatly on both the location and season of the emissions. The findings regarding emission season (dirSF is larger for BC emissions in summer than in winter, while snowSF is larger for BC emissions in winter than in summer, except for permanently snow-covered regions) are largely consistent with previous work . However, even when considering seasonally uniform emissions, dirSF ranges by more than a factor of 10, depending on the emission location. This is a much larger range than found in previous studies, where the reported interregional differences have been less than a factor of 2 . The origin of this difference probably mostly lies in the size of the emission regions considered. For continental/subcontinental regions, compensation between locations at which BC emissions give rise to low or high normalized dirSF is likely to occur. The large differences in how the emission regions (and the distribution of emissions within them) are defined makes a quantitative comparison with the aforementioned studies challenging. However, qualitatively, our Fig. a is consistent with the findings of and in that the normalized dirRF is relatively large for African emissions and relatively small for European and Russian emissions and with the notion that enhanced dirRF is associated with BC emissions in convective regions . The details sometimes differ. For example, while Fig. a suggests that BC dirSF is larger for East Asian than European emissions, Fig. 6 in shows broadly similar specific RF for airborne BC for these two emission regions. This comparison is slightly ambiguous because did not report the contributions from the aerosol–radiation and aerosol–cloud interaction separately. However, the difference could also be related to the experimental setup. considered 20 % reductions from near-present baseline emissions (which makes the baseline substantially more polluted for East Asia), while we added equally strong BC emissions in each region on top of a zero-BC baseline case. Furthermore, in the results of , snowSF is more than twice as large for European emissions as for Asian emissions, while in our results (Fig. b), this difference is less obvious.

provide estimates of BC dirRF efficiency at an even higher resolution than the present study. To make their results quantitatively comparable to ours, the values in their Fig. 1a should be normalized by the grid cell area in GEOS-Chem, but qualitatively, substantial differences are apparent. In particular, in contrast to our results (and also those of , , and ), the BC dirRF efficiencies reported by are generally small for low-latitude emissions, especially over oceans but also over continents, which suggests a negligible role for the convective transport of BC. The root cause of this discrepancy remains unclear.

For the total SF (defined here as the sum of dirSF and snowSF), the impact of the emission location is even larger than for dirSF. The values shown in Fig. a vary by a factor of more than 30 (from 0.27 to 8.8 TJkg-1). While most real-world BC emissions take place at low- to midlatitudes, the total SF (i.e., RF normalized by emitted mass) is largest for high-latitude emissions. This occurs because, for high-latitude emissions, a substantial fraction of the emitted BC is deposited on snow and sea ice. Indeed, as shown in Fig. b, the large total SF resulting from high-latitude BC emissions is strongly dominated by snowSF, while for tropical and subtropical BC emissions, the contribution from snowSF is very small.

The longitude of the emissions also matters, which should be considered in climate policy. For example, for the subarctic region, snowSF (Fig. b) and the total SF (Fig. a) are substantially larger for BC emissions in Siberia and northern North America than for equally large emissions in Fennoscandia, due to the longer snow season and lesser vegetation masking in the former regions. In this respect, it is worth noting the role of BC emissions from the Russian oil industry in western Siberia. Not only are these emissions the largest high-latitude BC emission source in the ECLIPSE V6b CLE dataset (Fig. S1a), they are also located in a region where snow cover prevails relatively long in spring, which gives rise to a large RF per emitted mass of BC. This highlights the potential climate benefits of curbing the BC emissions from the Russian oil industry (see ). The role of Arctic BC emissions was also pointed out by , who estimated the Arctic temperature response to emissions of SLCFs from various regions and emission sectors. While they found that, in absolute terms, the largest Arctic warming source stems from Asian BC emissions, the largest warming per unit emissions was associated with BC emissions from flaring and other activities within the Arctic nations.

This study has focused on the instantaneous RF of BC only, including dirRF and snowRF. The problem with this is that, for BC, the instantaneous RF is, in fact, not a very good predictor of the ensuing climate response. Previous studies have shown that the climate response to BC dirRF may depend strongly on the spatial distribution of BC and, especially, its vertical profile . In particular, demonstrated that, while dirRF is largest when BC is located at high altitudes, the temperature response is largest for BC close to the surface. The disconnection between the temperature response and dirRF is related to rapid adjustments in atmospheric stability, humidity, and cloudiness. In general, for a realistic global distribution of BC, these adjustments act to make the BC effective radiative forcing (ERF) smaller than the instantaneous RF, thereby leading to a reduced global mean temperature response . This translates to a low efficacy (the ratio of global mean temperature responses to BC and CO2 for a given RF) when defined with regard to the instantaneous or stratospherically adjusted RF . In contrast, high efficacy has been reported for BC snowRF due to albedo feedbacks associated with accelerated snowmelt and due to warming concentrated near the surface.

Unfortunately, a direct evaluation of the climate response arising from BC emissions in the 192 lat–long boxes and for the five seasonal distributions of emissions considered in this work would be unfeasible computationally. However, in order to obtain a rough estimate of the global mean temperature response ΔTglob, we resorted to the use of regional climate sensitivity (RCS) coefficients (also known as regional climate response coefficients) employed in previous research . The RCSs describe the temperature response in four wide latitude bands to the dirRF and snowRF averaged separately over each of these bands (90–28∘ S, 28∘ S–28∘ N, 28–60∘ N, and 60–90∘ N), and they implicitly include the effect of rapid adjustments. The details of this analysis are presented in Appendix B.

Figure c shows ΔTglob normalized by BC emissions (ΔTglob,norm; in units of mK(kgs-1)-1) for all 192 emission regions, and the fractional contribution to ΔTglob,norm by snowRF is displayed in Fig. d. These results involve substantial uncertainties, not least due to the very coarse spatial resolution at which the RCSs were defined. Nevertheless, three qualitative conclusions appear to be robust. First, the estimated ΔTglob,norm depends very strongly on the emission location (Fig. c). Second, the largest ΔTglob,norm occurs for BC emissions in polar regions. Third, the large ΔTglob,norm resulting from high-latitude BC emissions is strongly dominated by BC snowRF (Fig. d), while for tropical and subtropical BC emissions, the contribution from snowRF is very small. Thus, the same general features apply to ΔTglob,norm as to the total SF and RF (Figs. a, b). In fact, ΔTglob,norm varies even more widely than the total RF, by a factor of over 160 (from 0.20 to 32.5 mK(kgs-1)-1). This mainly occurs because the impact of Arctic BC emissions is further exacerbated by the high efficacy of BC snowRF.

One uncertainty factor in our quantitative results is the assumption that the efficacy for BC snowRF is 3 times as large as that for BC dirRF (except for the Arctic temperature response to local BC RF in the Arctic, which was modeled explicitly by ). Previous studies have indicated that the efficacy of BC snowRF defined with regard to the stratospheric-adjusted RF (which is almost the same as the instantaneous RF) is well above 1. give an efficacy of roughly 2, 1.71, and 2.1–4.5, depending on the experiment. Furthermore, as noted above, due to rapid adjustments, the efficacy of BC dirRF defined with regard to the stratospheric adjusted RF is very likely below 1. In particular, report values of 0.58–0.78 for GISS ModelE-R, which employed to derive the RCS coefficients used for BC dirRF here. Therefore, the value of 3 for the snowRF vs. dirRF efficacy ratio seems well justified. Yet it might be an overestimate for regions with permanent snow (such as Antarctica and the interior of Greenland), where BC-induced snow cover changes are absent and, therefore, positive feedbacks due to albedo and lapse-rate changes are probably greatly reduced. If the snowRF vs. dirRF efficacy ratio were set to 1, which is almost certainly too low, the normalized temperature response in Fig. c would be reduced by ∼ 30 % for emissions in the Arctic and by 60 %–65 % for emissions in Antarctica, yet even then the qualitative features seen in Fig. c, d would remain largely the same.

Recently, and interactively computed the temperature response to regional BC emissions using the Community Earth System Model (CESM) and NorESM1, respectively, although for much larger emission regions and strongly inflated emissions to improve the signal-to-noise ratio. When converted to the units of Fig. c, the values of ΔTglob,norm in range from 0.7–0.9 mK(kgs-1)-1 for European emissions to ca. 0.5 mK(kgs-1)-1 for South Asian emissions, while obtained values of 1.2 mK(kgs-1)-1 for Arctic (60–90∘ N) emissions and 0.35 mK(kgs-1)-1 for Northern Hemisphere midlatitude (28–60∘ N) emissions. While these studies show a tendency for higher-latitude BC emissions to cause a larger ΔTglob,norm, broadly consistent with our results, especially the values obtained by are lower than those in Fig. c. This could be partially related to the use of much larger than realistic BC emissions, which may lead to underestimated ΔTglob,norm and underestimated SF (Fig. ). However, much uncertainty also remains in the model-specific responses related especially to clouds.

The normalized temperature response in Fig. c can also be qualitatively compared with , who computed the global temperature potential (GTP) and absolute regional temperature potential metrics for BC emissions in Europe and East Asia, based on the RF data from . Overall, their results suggest that the global mean temperature is more sensitive to changes in European than East Asian BC emissions, especially if the enhanced efficacy of BC snowRF is taken into account . In contrast, our results rather show a slightly larger ΔTglob,norm for BC emissions in East Asia than in Europe. These differences are consistent with the respective RF differences between our study and discussed above.

As a final check of our approach, we employed it for computing the BC contribution to the global mean temperature (for details, see Appendix B). We obtained estimates of ΔTglob=0.30 K for the year 2015 emissions (totaling 8.4 Tgyr-1) employed in REAL and ΔTglob=0.10 K when using year 1850 emissions (totaling 3.1 Tgyr-1; ). This yields an estimate of 0.20 K for the BC-induced warming between 1850 and 2015. This value is within the uncertainty range given for the BC contribution to warming in the IPCC AR6 WG1 report (from ca. -0.07 to 0.26 K for the temperature change from 1850–1900 to 2010–2019; Fig. SPM.2 in ) but closer to its upper bound.

It was noted in Sect.  that summing the RFs computed for individual regions/seasons without considering BC emissions from elsewhere tends to slightly overestimate dirRF and especially snowRF. To provide some more insight on this finding, we consider, in Fig. , experiments in which the spatiotemporal pattern of BC emissions was fixed but their magnitude varied. One set of experiments was based on REAL, but the emissions were scaled by a factor between 0.1 and 10 so that the global mean emission rate varied between 5.2×10-14 and 5.2×10-12 kgm-2s-1. In another set of experiments, uniform (UNIF) BC emission rates ranging from 5.0×10-14 to 5.0×10-12 kgm-2s-1 were considered. It is noteworthy that, for similar global mean BC emissions, all quantities considered in Fig.  differ substantially between the REAL and UNIF experiments. This is consistent with the main result of our study that the BC RF depends strongly on the spatiotemporal distribution of the emissions. Here, we focus on the issue of nonlinearity.

The global mean BC lifetime is almost independent of the magnitude of emissions (Fig. a), which means that the BC burden increases linearly with emissions. This result should be taken with caution due to the offline treatment of aerosols in our experiments. When BC is allowed to influence the simulated climate, the increase of BC burden with emissions is not necessarily linear .

BC dirSF shows a slight decrease with an increasing emission rate (Fig. b), and for snowSF, the decrease is stronger (Fig. c). For example, for emissions cut to 1/10 of the realistic case REAL, dirSF is 4 % larger, and snowRF is 20 % larger, while for 10-fold emissions, dirSF is smaller by 23 %, and snowSF is smaller by 47 %. Since both the atmospheric BC burden (Fig. a) and the BC burden in snow (not shown) increase quasi-linearly with emissions, this implies that both dirRF and especially snowRF increase more weakly than linearly with the increasing BC burden. Indeed, considering Beer's law (exponential decay of transmission with optical depth), a linear increase in absorption with increasing BC mass can only be expected at the limit of weak BC absorption, that is, when the BC absorption optical depth is small at all relevant wavelengths. With increasing BC burden, the absorption becomes partly saturated. This provides a plausible explanation for the tendency of the reconstruction to overestimate snowRF (Fig. f, i) and, to a lesser extent, dirRF (Fig. f, i). These results are consistent with the snow albedo calculations by and with the climate model simulations of dirRF by .

Figure d confirms that the dependence of BC indirRF on the emission rate is strongly nonlinear. A number of factors could potentially contribute to this nonlinearity, including at least (1) the sublinearity of the Twomey effect (cloud optical depth increases more weakly than linearly with increasing cloud droplet number concentration and cloud reflectance more weakly than linearly with optical depth; ) and (2) the competition effect for water vapor (increasing availability of condensation nuclei acts to decrease the maximum supersaturation reached in the cloud; ). CAM5.1 simulations by see their Table 1 also indicate some sublinearity of BC indirRF with regard to the magnitude of emissions but not nearly as much as that seen for NorESM1-Happi.

Furthermore, for NorESM1-Happi, indirRF is positive (e.g., 0.168 Wm-2 for the REAL experiment), thereby reinforcing the effects of dirRF and snowRF. BC indirRF is known to strongly vary between different models and be negative in most, but not all, models . It also depends heavily on the co-emitted aerosol precursors and aerosol species which can affect the aerosol hygroscopic properties and mixing state . A possible explanation for the positive indirRF in NorESM1-Happi is that fossil fuel BC can actually reduce the cloud droplet number concentration by acting as a condensation sink for sulfuric acid and water vapor while not being large enough to activate into cloud droplets . In NorESM1-Happi, 90 % of the fossil fuel BC mass is emitted to the nucleation mode, with a diameter 23.6 nm, and 10 % to the fractal agglomerate mode, with a diameter 200 nm, while in other models, fossil fuel BC is emitted at sizes between 30 and 100 nm . When numerous small fossil fuel BC particles become coated with SO4, they reduce the amount of condensable sulfuric acid available for the growth of aerosols into activation sizes, and they also become hygroscopic and act as a sink for water vapor. Thus, they can both inhibit the formation of large activation-sized aerosol particles and slightly reduce the maximum supersaturation, which then further reduces the number of aerosol particles activated into cloud droplets.

The nonlinearity of BC RF with regard to the magnitude of emissions has implications concerning studies where BC climate effects are evaluated. In such studies, higher than realistic BC emissions are commonly used in order to obtain a robust signal for ERF or the climate response in the presence of noise related to internal climate variability. For example, scaled present-day global BC emissions by a factor of 2, by a factor of 10, and by a factor of 25, while considered regional scaling factors of 3.5–14 for midlatitude emissions and 75–150 for Arctic emissions. The decrease in SF with BC emissions in Fig.  suggests that the simplest approach to estimating the climate response to present-day BC emissions from very high-emission experiments (i.e., scaling down the climate response by the multiplication factor applied to BC emissions) would lead to an underestimate.