Introduction

As a ubiquitous aerosol component, black carbon (BC) is a major contributor to global warming owing to its strong light absorption1,2,3,4,5,6,7, which strongly depends on its mixing state2,8,9,10,11,12,13,14,15. Undergoing multiple atmospheric processes, freshly emitted BC can be internally mixed with other aerosol components (i.e., BC coating) and its light absorption is enhanced by the “lensing effect”1,16,17,18,19. In the real atmosphere, the mixing state of BC-containing particles is a complex property related to several processes, including condensation, coagulation, and deposition. Many studies have characterized mixing state properties from different aspects based on field observations8,9,20,21,22,23,24,25,26. However, the overall effect of these dynamic processes on BC mixing state is not well understood. Moreover, the complexity and diversity of BC mixing states in the real atmosphere cannot be represented in global climate models, and therefore these models generally use simplified schemes, assuming either an internal or external mixture of aerosols2,27, leading to a wide range of estimated BC mass absorption cross-sections (MAC, a typical indicator of BC light absorption ability) from 3.1 to 18.0 m2/g (at 550 nm) on global average27,28. Therefore, a precise description of BC mixing state becomes the determinant factor of model performance when estimating BC optical properties and radiative forcing.

In this study, we built a theoretical framework linking dynamic processes to BC coating thickness distribution and discovered the self-similarity of BC mixing states, which was verified in eight different observation sites globally. The size distribution of BC-containing particles is found to follow a universal law. This self-similarity allows us to greatly simplify the characterization of BC mixing states in both model simulations and field observations. A new mixing state scheme was established for model simulation, which can precisely represent the BC mixing state. Model estimated BC absorption and radiative forcing is substantially reduced, which fits well with available field observation results. Our study links observations with model simulations in both mixing state and light absorption.

Results

A universal law of BC mixing state

To characterize the BC mixing state, which is controlled by complex atmospheric processes, we performed a theoretical derivation considering the main physical processes affecting BC in the atmosphere. We discovered that the size distribution of BC-containing particles follows a universal law and is independent of BC core size. Figure 1 provides a conceptional scheme describing the main physical processes and evolution of aerosols in the atmosphere. We assume a monodisperse aerosol population (consisting of BC cores only) emitted into the atmosphere at time zero with diameter Dc and number concentration n(Dc). After being emitted into the atmosphere, the particles experience both growth and deposition progresses continuously, which form a steady state29, that is, the size distribution of BC-containing particles is approximately steady (although the overall mass concentration may change). The growth of BC-containing particles via condensation and coagulation results in an enlarged particle size, whose change as a function of time is represented by the growth rate (GR).

$$\frac{{{{{{\rm{d}}}}}}({D}_{{{{{{\rm{p}}}}}}})}{{{{{{\rm{d}}}}}}t}={{{{{\rm{GR}}}}}}$$
(1)

The time evolution of the diameter of BC-containing particles (Dp) can be integrated to give

$${D}_{{{{{{\rm{p}}}}}}}={{{{{\rm{GR}}}}}}\cdot t+{D}_{{{{{{\rm{c}}}}}}}$$
(2)

At the same time, BC particles are removed by deposition process with the rate of Dep. The number concentration of particles at Dp, i.e., n(Dp), decays due to deposition process is

$$dn({D}_{{{\rm{p}}}})=-{{\rm{Dep}}}\cdot n({D}_{p})dt$$
(3)

Then, the time evolution of n(Dp) can be integrated as

$$n({D}_{{{{{{\rm{p}}}}}}})=n({D}_{{{{{{\rm{c}}}}}}})\cdot {e}^{-{{{{{{\rm{Dep}}}}}}}\cdot t}$$
(4)

Based on the steady-state approximation, Eqs. 2 and 4 can be combined and time term t can be eliminated. Taking the logarithm on both sides, we obtain the following equation:

$${{{{{\rm{ln}}}}}}(n({D}_{{{{{{\rm{p}}}}}}}))={{{{{\rm{ln}}}}}}(n({D}_{{{{{{\rm{c}}}}}}}))-\frac{{{{{{\rm{Dep}}}}}}}{{{{{{\rm{GR}}}}}}}({D}_{{{{{{\rm{p}}}}}}}-{D}_{{{{{{\rm{c}}}}}}})$$
(5)

The slope \(k=\frac{{{{{{{\rm{Dep}}}}}}}}{{{{{{{\rm{GR}}}}}}}}\) is determined by the deposition rate and the growth rate, and the intercept is determined by n(Dc). Note that we adopted a simplified derivation in the above theoretical analysis for better understanding. A more rigorous theoretical derivation as well as the interpretation of the dependency of GR and Dep on Dp and time can be found in the Supplementary Information (SI).

Fig. 1: Similarity of black carbon (BC) size distribution controlled by combined effects of growth and sink.
figure 1

The yellow shaded area and arrow show the emission source of BC where BC core size (Dc) is approximately equal to the diameter of BC-containing particles (Dp). The blue shaded area and line represent the size distribution of Dp controlled by growth (purple arrow) and sink (gray arrow) processes. The slope of the blue line equals to \(-\frac{{{{{{{\rm{Dep}}}}}}}}{{{{{{{\rm{GR}}}}}}}}\) in a ln(n(Dp))−Dp coordinate system.

Full size image

Equation 5 demonstrates that for different particle sizes, ln(n(Dp)) and Dp − Dc (ΔDp, defined as coating thickness) are in a linear relationship (Fig. 1). The average coating thickness can be derived as 1/k (the detailed formula is shown as Eq. 7 in the “Methods” section). Furthermore, we find that the slope k is independent of Dc, indicating the self-similarity of BC size distributions, that is, BC-containing aerosol with different core sizes should have similar distributions of coating thickness. Such self-similarity allows us to greatly simplify the description of BC mixing states since BC coating thickness can be fully described with known slope k and the BC core number-size distribution.

We verified our theoretical model by field observations of BC size using the single particle soot photometer (SP2, Droplet Measurement Technologies, USA), which is a well-recognized instrument to measure BC mixing state) from eight sites covering different environments globally24,30,31. As presented in Fig. 2, the BC size distribution follows an exponential law at all sites despite their different regions and properties (e.g., urban, regional background), which agrees well with our theoretical model. The slope k of linear regression, ranging from 0.008 to 0.020, provides a useful parameter to quantify BC size distribution and its absorption enhancement.

Fig. 2: Exponent distribution of black carbon (BC) size distributions from field measurements using single particle soot photometers (SP2) at different sites.
figure 2

The red dots and black lines represent the normalized size distribution of the diameter of BC-containing particles (Dp) and linear regression, respectively. Observations in Nanjing were conducted in different seasons. Data in Lulang and Maqu were observed over the Tibetan Plateau in spring 2021. Observations in Shaoguan, Beijing31, Tokyo30, Sacramento24, and Amazon Tall Tower Observatory (ATTO) were performed in December 2020, November 2014, August 2012, June 2010, and October 2019, respectively. The map shows simulated BC radiative forcing at top of atmosphere (TOA).

Full size image

The Dp distributions with different BC core sizes from SP2 measurements at four sites are presented in Fig. 3. We selected four Dc bins (110–120 nm, 120–130 nm, 130–140 nm, and 140–150 nm) and calculated the average Dp distribution in each bin. The result shows that the BC size distributions with different Dc have approximately the same slope of ln(n(Dp))~Dp. This phenomenon can be observed at all four sites, validating our theoretical model presented as Eq. 5, i.e., the shape of the BC size distribution is independent of Dc. The presence of the same pattern in Nanjing (suburban), Maqu (remote background), Tokyo (urban), and at the Amazon Tall Tower Observatory site (affected by biomass burning) further indicates that the self-similarity of BC size distribution is ubiquitous in the real atmosphere.

Fig. 3: Black carbon (BC) size distributions with different Dc ranges at four sites.
figure 3

Single particle soot photometers (SP2) observational data from a Nanjing (April), b Maqu, c Tokyo, and d Amazon Tall Tower Observatory (ATTO) sites is used. Circles are the diameter of BC-containing particles (Dp) distributions with four selected BC core size (Dc) ranges. Dashed lines represent the linear regression of each distribution. n(Dp) with each Dc range normalized.

Full size image

Improved estimation of BC absorption and radiative effect

When evaluating BC absorption and radiative effect, simply assuming either internal or external mixture of aerosols may induce large discrepancies and uncertainty. Also, it is difficult to provide an accurate description of mixing state due to its great complexity. Based on our theory, the description of BC mixing state can be greatly simplified, which is applicable to the light absorption calculation in both climate models and chemical transport models. The absorption enhancement factor, Eabs, is the ratio between the aerosol absorption coefficients before and after removal of coating, which is a widely used parameter to represent BC light absorption amplification. As shown in Fig. S3, the relationship between Eabs and ΔDp is approximately linear when ΔDp is small. Therefore, the BC coating thickness distribution can be replaced by a monodisperse coating thickness, 1/k, when calculating the black carbon absorption (a detailed demonstration can be found in the “Methods” section). Light absorption coefficients over all Dp and with mean Dp show good agreement (Fig. S4), which further validates this simplification. Hence, this approximation can be applied directly in global and regional models for optical estimation.

Based on the above findings, we established a new mixing state module and applied it in a global climate model (CESM-CAM6) and a chemical transport model (WRF-chem) as examples. Model simulations of Eabs and BC direct radiative forcing (DRFBC) were performed using these two models alternatively with our module and the conventional assumption of mixing state (Figs. 4 and 5). Comparing the simulated Eabs with observational data shown in Fig. 4, the simulated Eabs using the conventional assumption of mixing state in both CESM-CAM6 and WRF-Chem (blue squares) are significantly higher than observations (1.0 to 1.7, shown as black squares). CESM-CAM6 uses the volume mixing assumption for Eabs calculation and the simulated result is higher than 2.5. WRF-Chem has two types of mixing state assumption, which are volume mixing and core-shell mixing. However, the simulated Eabs using both types of assumptions ranges from 2.0 to 2.5, which is also nearly twice that of observational results. Using the new module, the calculated Eabs in CESM-CAM6 and WRF-Chem are 1.4 (1.3–1.6) and 1.4 (1.3–1.7), respectively, which agrees well with observational data, demonstrating the good performance of our mixing state description in the quantification of Eabs.

Fig. 4: Model simulated black carbon (BC) absorption enhancement using the new scheme in this study (red squares) and the conventional scheme (blue squares), compared with observations.
figure 4

WRF-Chemv and WRF-Chemcs stand for WRF-Chem simulations with volume mixing and core-shell mode, respectively. The black squares with error bars (standard deviation) are Eabs observed using the thermodenuder (TD) method at different sites obtained from previous studies9,13,48,49,50,51,52. There are two exceptions, which are Knox et al.48 and Ueda et al.50. The error bars in Knox et al.48 represent the Eabs of aerosol with different age categories and the error bars in Ueda et al.50 cover the 25th−75th percentiles. The Eabs reported in Cappa et al.51 were observed in two cities (Fresno and Fontana, California, respectively).

Full size image
Fig. 5: Model simulated black carbon (BC) direct radiative forcing (DRFBC) using the new scheme and the conventional scheme in different regions.
figure 5

The map shows the global distribution of DRFBC using the new scheme. The blue and red bars represent the average simulated DRFBC within each region (gray squares in dash line) using the conventional scheme and the new scheme, respectively.

Full size image

Figure 5 presents simulated DRFBC using the new scheme and the conventional scheme in CESM-CAM6. We selected four typical regions (Europe, North America, South America, and Asia) to perform statistical analyses. Calculated DRFBC using our scheme is greatly reduced in all four regions (40–50%) compared with adopting the conventional scheme. This result is in agreement with the recently-found overestimation of radiative warming by BC in global climate models, largely due to the treatment of aerosol mixing state32. Figure 5 demonstrates that the new scheme based on our theoretical framework can efficiently resolve the existing overestimation of model-simulated radiative warming by BC.

Discussion

We built a unified theoretical framework based on a steady-state assumption to describe the complex mixing state of BC and find its self-similarity confirmed across a wide range of field observation sites. This universal law is the result of a balance of multiple atmospheric processes. Our findings link the representation of particle diameter (from field observations) and dynamic parameters (generally used in models), making observational data applicable in model simulations. Moreover, this unified theoretical framework reduces the dimension of mixing state descriptions. The mixing state module established in this study can be embedded in various types of atmospheric models and efficiently improves the accuracy of aerosol climate effect estimations without increasing computational complexity. We find that BC absorption enhancement and warming effect are much lower than current estimates.

Methods

Light absorption enhancement

We preformed the optical calculation using the core-shell Mie method. We used a lognormal distribution of Dc. The geometric standard deviation was set to 1.8. The mean diameter of Dc was 70 nm and the wavelength was 550 nm. The refractive indices (RI) of the BC and scattering components were set to 1.85 + 0.7i according to Bond et al.1 and 1.53 + 0i according to Pitchford et al.2. To derive the response of MAC to ΔDp (Fig. S3), ΔDp from 10 nm to 200 nm with 10 nm intervals was adopted to calculate the mass absorption coefficient.

For the optical calculations in Fig. S4, we used the integral method (ΔDp varied from 1 nm to 1000 nm with 1 nm interval) and the k-value method to calculate absorption coefficients. The Dp size distribution represented as Eq. 5 was adopted and k was set to 0.014. The setting of RI, wavelength, and Dc size distribution were same as in Fig. S3.

As shown in Fig. S3, the relationship between Eabs and ΔDp is approximately linear when ΔDp is small. Therefore, Eabs could be represented as Eq. 6.

$${E}_{{{{{{\rm{abs}}}}}}}({D}_{{{{{{\rm{c}}}}}}},\Delta {D}_{{{{{{\rm{p}}}}}}})=\alpha ({D}_{{{{{{\rm{c}}}}}}})\Delta {D}_{{{{{{\rm{p}}}}}}}$$
(6)

where α(Dc) is the linear coefficient of Eabs and ΔDp. With the known calculation formula of \(\overline{{D}_{{{{{{\rm{p}}}}}}}}\), we can derive that

$$\overline{\Delta {D}_{{{{{{\rm{p}}}}}}}}=\frac{{\int }_{{D}_{{{{{{\rm{p}}}}}}}={D}_{{{{{{\rm{c}}}}}}}}^{\infty }{D}_{{{{{{\rm{p}}}}}}}\cdot n({D}_{{{{{{\rm{p}}}}}}})\cdot {{{{{\rm{d}}}}}}({D}_{{{{{{\rm{p}}}}}}})}{{\int }_{{D}_{{{{{{\rm{p}}}}}}}={D}_{{{{{{\rm{c}}}}}}}}^{\infty }n({D}_{{{{{{\rm{p}}}}}}})\cdot {{{{{\rm{d}}}}}}({D}_{{{{{{\rm{p}}}}}}})}=\frac{1}{k}$$
(7)

If Eq. 6 is integrated over all Dp, the average Eabs with given Dc is found to be

$${E}_{{{{{{\rm{abs}}}}}}}({D}_{{{{{{\rm{c}}}}}}}) =\frac{{\int }_{{D}_{{{{{{\rm{p}}}}}}}={D}_{{{{{{\rm{c}}}}}}}}^{\infty }{c}_{{{{{{\rm{abs}}}}}}\mbox{-}{{{{{\rm{external}}}}}}}({D}_{{{{{{\rm{c}}}}}}})\cdot {E}_{{{{{{\rm{abs}}}}}}}({D}_{{{{{{\rm{c}}}}}}},\Delta {D}_{{{{{{\rm{p}}}}}}})\cdot n({D}_{{{{{{\rm{c}}}}}}},{D}_{{{{{{\rm{p}}}}}}})\cdot {{{{{\rm{d}}}}}}({D}_{{{{{{\rm{p}}}}}}})}{{\int }_{{D}_{{{{{{\rm{p}}}}}}}={D}_{{{{{{\rm{c}}}}}}}}^{\infty }{c}_{{{{{{\rm{abs}}}}}}{\mbox{-}}{{{{{\rm{external}}}}}}}({D}_{{{{{{\rm{c}}}}}}})\cdot n({D}_{{{{{{\rm{c}}}}}}},{D}_{{{{{{\rm{p}}}}}}})\cdot {{{{{\rm{d}}}}}}({D}_{{{{{{\rm{p}}}}}}})}\\ =\alpha ({D}_{{{{{{\rm{c}}}}}}})\cdot \overline{\Delta {D}_{{{{{{\rm{p}}}}}}}}\\ =\alpha ({D}_{{{{{{\rm{c}}}}}}})\cdot \frac{1}{k}\\ ={E}_{{{{{{\rm{abs}}}}}}}({D}_{{{{{{\rm{c}}}}}}},\overline{\Delta {D}_{{{{{{\rm{p}}}}}}}})$$
(8)

where cabs-external represents the light absorption coefficients of BC core. Equations 7 and 8 show that 1/k plays a similar role with coating thickness.

Field observations and site descriptions

Observational data of BC mixing states was collected from different sites, including Nanjing (a regional background site in the Yangtze River Delta region in China), Beijing (an urban site in the capital of China), Shaoguan (a regional background site in the Pearl River Delta region in China), the Tibetan Plateau (including three sites with different features), Japan (Tokyo), and the United States (Sacramento, influenced by biomass burning).

Field measurements in Nanjing were performed at the Station for Observing Regional Processes of the Earth system (SORPES, 118°57′10″E, 32°07′14′′ N; 40 m a.s.l.), a regional background station in the western YRD region. A detailed description of SORPES can be found in previous studies33,34. Due to its geographical position, this observation platform is ideal to capture the transport from megacities in the YRD region and North China Plain. Observational data from February 2020, April 2020, and December 2021 was used in this study.

Observations in Lulang and Maqu were made over the Tibetan Plateau (TP) from April to July 2021. The Lulang site is located on the southeastern part of the TP with few traffic emissions nearby. The measurement period in Lulang was from 1 April to 25 May 2021. Maqu can be considered as a background site over the TP. The measurement period was from 26 June to 8 July 2021. The observation locations and periods at other sites are summarized in Table S1.

Besides field observations conducted in this study, observational data from several sites covering different kinds of environment globally was also collected to support our findings. The measurement periods in Shaoguan, Beijing31, Tokyo30, Sacramento9, and the Amazon Tall Tower Observatory (ATTO) were December 2020, November 2014, August 2012, June 2010, and October 2019, respectively. Shaoguan is a regional background site in southeastern China and the Beijing site represents a heavily polluted region in the North China Plain. The Tokyo and Sacramento sites are located in urban areas in Japan and the United States, respectively. The selected observation period at the ATTO site covers a biomass burning episode including some relatively clean days. Detailed information on these observations is summarized in Table S1.

The physical properties of refractory BC particles were measured using single particle soot photometers (SP2, Droplet Measurement Technologies, USA). The operation principle of the SP2 has been well described in previous literature20,35. Briefly, sampled particles pass through a 1064 nm Nd:YAG laser beam. BC-containing particles heat up to their boiling point and incandesce. The BC mass can be computed based on its proportional relationship with the peak intensity of the incandescence signal and the BC mass equivalent diameter can be calculated with the known density of BC (normally assumed to be 1.8 g cm−3 36). The scattering calibration was performed using polystyrene latex spheres (PSL). The rBC mass was calibrated using fullerene soot with known diameter, which was selected by a differential mobility analyzer (DMA, TSI Inc., USA) and its mass was calculated using effective density values presented by Gysel et al.37. The leading-edge-only (LEO) fit method developed by Gao et al.20 was adopted to calculate the scattering cross-sections of BC-containing particles and saturated scattering particles. Therefore, the optical diameters of BC-containing particles can be further determined based on core-shell Mie theory.

Model simulations

There are two typical methods that are extensively applied for optical calculations in global climate models and regional transport models, i.e., the volume-mixing and core-shell Mie methods. The volume-mixing Mie algorithm assumes that all components are mixed in all individual particles and the mean refractive index is calculated as the volume-weighted average of the refractive indices of each species. The core-shell Mie method assumes that BC is in the center, and other components are coated on the BC core. The shell refractive indices are assumed to be the volume-weighted average of the refractive indices of dissolved components. The volume-mixing Mie algorithm is included in CESM-CAM6. Both volume-mixing and core-shell Mie methods can be used to estimate aerosol optical properties in WRF-Chem. The refractive indices for shortwave radiation and densities of aerosol species in CESM-CAM6 and WRF-Chem model are summarized in Table S3. Eabs is the ratio of MACinternal/MACexternal, where MAC stands for the mass absorption cross section of BC. Since there is no external mixing module in CESM-CAM6 and WRF-Chem, the estimation of MACexternal is based on off-line calculation. The default refractive indices and densities in each model are adopted. The BC diameter is assumed to follow a lognormal distribution with a count median diameter of 70 nm2.

The new mixing state and optical scheme based on our theory frame is established and applied in both CESM-CAM6 and WRF-Chem, which cover different model types. CESM-CAM6 and WRF-Chem are examples of global climate models and regional transport models, respectively. Moreover, CESM-CAM6 uses a modal aerosol module and WRF-Chem uses a sectional aerosol module, which are the two most widely implemented modules in atmospheric models. In the new scheme, a monodisperse coating thickness of 70 nm derived from k = 0.014 is adopted. In CESM-CAM6 and WRF-Chem, the BC core diameter is assumed to follow a lognormal distribution with a count median diameter of 70 nm.

We used the Community Atmosphere Model version 2.1.3 of the Community Earth System Model version 6 (CESM2.1.3-CAM6)38 in the simulation of light absorption by BC and the global DRF with Modal Aerosol Module 4 (MAM4)39. MAM4 includes six aerosol components (BC, sea salt, sulfate, POA, SOA, and dust), which are divided into four modes (primary carbon mode, Aitken mode, accumulation mode and coarse mode), and simulates the mass mixing ratios of six components within each mode. The spatial resolution in the global simulation is 1.9° × 2.5° for a latitude and longitude grid with 70 vertical layers (from 50 m to ~140 km). The simulation is performed for four years (2012–2015) with the spin-up in the first three years and analysis in the last year. The radiative transfer module in the shortwave is calculated by the radiation code RRTMG. The diagnostic calculation of CESM-CAM6 is conducted for the radiative properties of one specific component, namely by double running cases with and without that component. The aerosols in the accumulation mode in this study are resolved with a sectional representation (30 size bins) in the optical calculations.

The BC-induced direct radiative forcing (DRFBC) in the conventional models is simulated using the default setting. The DRFBC determined with our new module by using k is performed assuming DRFBC is linear with MAC2,40. Thus, the DRFBC_k can be estimated from the change of MACk and MAC in CESM.

WRF-Chem version 3.7 (Weather Research and Forecasting model coupled with Chemistry) was employed in this study, which is an online-coupled meteorology and chemistry model considering multiple physical and chemical processes, including emission and deposition of pollutants, advection and diffusion, gaseous41 and aqueous chemical transformations, aerosol chemistry, and dynamics42. The model has been incorporated in several studies concerning the estimation of aerosol optical properties and its radiative forcing43,44. The model domain is centered at 35.0°N, 110.0°E with a grid resolution of 20 km that covers eastern China and the surrounding regions. A total of 30 vertical layers extending from the surface to 50 hPa are utilized in the model. The simulation is conducted for the first two weeks of April 2020, each run covers 36 h, and the last 24 h of output were kept for further analysis. The initial and boundary conditions of meteorological fields were updated from the 6-h NCEP (National Centers for Environment Prediction) global final analysis (FNL) data with 1°× 1° spatial resolution. The Rapid Radiative Transfer Model shortwave and longwave radiation scheme (RRTMG) represents the radiation transfer within the atmosphere45. Anthropogenic emissions from power plants, residential combustion, industrial processes, on-road mobile sources, and agricultural activities were derived from the MIX Asian emission inventory database46. Emissions of major pollutants such as carbon monoxide, sulfur dioxide, nitrogen oxides, ammonia, and speciated VOCs are all included. The MEGAN (Model of Emissions of Gases and Aerosols from Nature, version2) model embedded in WRF-Chem is used to calculate biogenic emissions online. Soil derived dust emissions are characterized by the GOCART emission schemes. For numerical representation of atmospheric chemistry, we used the CBMZ (Carbon-Bond Mechanism version Z) photochemical mechanism combined with the MOSAIC (Model for Simulating Aerosol Interactions and Chemistry) aerosol model47. Major aerosol components include BC, organic mass, sulfate, nitrate, ammonium, and other inorganic species. All aerosols were assumed to be spherical particles. The size distribution was divided into four discrete size bins defined by their lower and upper dry particle diameters (0.039–0.156, 0.156–0.625, 0.625–2.5, 2.5–10.0 μm). In each bin, aerosols were assumed to be internally mixed.