Black carbon (BC), a ubiquitous aerosol component, significantly contributes to global warming due to its strong light absorption. This absorption is heavily influenced by its complex mixing state within the atmosphere. Freshly emitted BC becomes internally mixed with other aerosol components, forming a coating that enhances light absorption through a "lensing effect." The mixing state of BC-containing particles is a dynamic property shaped by processes like condensation, coagulation, and deposition. While numerous studies have characterized BC mixing states from various perspectives using field observations, a comprehensive understanding of the combined effect of these dynamic processes remains elusive. This complexity and diversity in real-world atmospheric BC mixing states pose a significant challenge for global climate models, which often employ simplified schemes assuming either internal or external mixtures. These simplifications lead to a wide range of estimated BC mass absorption cross-sections (MAC), resulting in substantial uncertainty in estimates of BC's radiative forcing. The inability to accurately represent BC mixing state in models directly impacts the precision of estimations regarding BC optical properties and its consequent radiative forcing. This research aims to bridge this gap by developing a theoretical framework that links dynamic atmospheric processes to the observed BC mixing state, thereby improving the accuracy of climate effect estimations.

Previous research has highlighted the significant impact of BC's mixing state on its radiative properties and subsequent climate effects. Studies have shown that the "lensing effect" caused by the coating of BC particles enhances their light absorption. However, the complexity of the BC mixing state in the real atmosphere, influenced by various dynamic processes, has made accurate modeling challenging. Existing global climate models often rely on simplified assumptions, such as internal or external mixing, which introduce significant uncertainties in the estimation of BC's mass absorption cross-section (MAC) and its radiative forcing. The range of estimated MAC values from various models highlights the need for a more robust and accurate representation of BC mixing states. The current literature demonstrates a significant need for a unified theoretical framework that incorporates the dynamic processes affecting BC mixing and allows for a more precise representation of its radiative effects in climate models.

This study developed a theoretical framework that connects dynamic atmospheric processes to the distribution of BC coating thickness. The framework begins by considering the main physical processes impacting BC in the atmosphere, specifically growth (via condensation and coagulation) and deposition. A simplified theoretical derivation, assuming a monodisperse aerosol population emitted at time zero, is presented. This derivation shows that the size distribution of BC-containing particles follows a universal law independent of BC core size. The time evolution of the diameter of BC-containing particles (Dp) and their number concentration n(Dp) are modeled, considering growth rate (GR) and deposition rate (Dep). The steady-state approximation allows for the elimination of the time variable, resulting in an equation showing a linear relationship between ln(n(Dp)) and Dp – Dc (ΔDp, defined as coating thickness). The slope of this linear relationship (k = Dep/GR) is key to quantifying BC size distribution and absorption enhancement. A more rigorous derivation and interpretation of GR and Dep dependency are included in the Supplementary Information. The framework’s self-similarity, meaning BC-containing aerosols with different core sizes show similar coating thickness distributions, simplifies the characterization of BC mixing states. This model was validated using field observations from eight sites globally with varying environmental conditions, utilizing single particle soot photometers (SP2). These observations confirmed the exponential law governing BC size distribution across diverse locations. The absorption enhancement factor (Eabs), the ratio of aerosol absorption coefficients before and after coating removal, is another key parameter. The relationship between Eabs and ΔDp is approximately linear when ΔDp is small, allowing for the replacement of the BC coating thickness distribution with a monodisperse coating thickness (1/k) in absorption calculations. This simplification was validated by comparing light absorption coefficients calculated using the integral method and the k-value method. A new mixing state module, incorporating this simplified representation, was implemented into a global climate model (CESM-CAM6) and a chemical transport model (WRF-chem). Model simulations of Eabs and BC direct radiative forcing (DRFBC) were compared with observational data and simulations using conventional mixing state assumptions. Optical calculations utilized the core-shell Mie method with a lognormal distribution for Dp, specific refractive indices for BC and scattering components, and various ΔDc values. In CESM-CAM6, the modal aerosol module with six aerosol components and four modes was used, with a spatial resolution of 1.9° × 2.5°. In WRF-Chem, a sectional aerosol module with four size bins was employed, covering eastern China with a 20 km grid resolution. Both models used the RRTMG radiation scheme for radiative transfer calculations. The DRFBC was calculated using the default settings in conventional models and with the new module, assuming linearity between DRFBC and MAC.

The core finding is the development and validation of a unified theoretical framework describing BC mixing states. This framework reveals a universal law governing the size distribution of BC-containing particles, independent of the BC core size. This universality, confirmed across eight diverse global sites using single particle soot photometer (SP2) measurements, simplifies BC mixing state characterization. The size distribution at all sites followed an exponential law, despite differences in location and properties. The slope (k) of the linear regression of ln(n(Dp)) against Dp – Dc varied from 0.008 to 0.020, providing a quantifiable parameter for BC size distribution and absorption enhancement. Furthermore, the analysis demonstrates self-similarity in BC size distributions, irrespective of the BC core size. This self-similarity is robust, observed across various locations like Nanjing (suburban), Maqu (remote background), Tokyo (urban), and the Amazon Tall Tower Observatory (affected by biomass burning). The study shows that the absorption enhancement factor (Eabs) is approximately linearly related to ΔDp when ΔDp is small. This enabled the simplification of representing the BC coating thickness distribution by a monodisperse coating thickness (1/k) in light absorption calculations. This approximation, validated through comparison with the integral method, greatly facilitates the incorporation of realistic BC mixing states into atmospheric models. The incorporation of the new mixing state module into both CESM-CAM6 and WRF-Chem models significantly improved the accuracy of BC absorption and radiative forcing estimations. The conventional mixing state assumptions in both models overestimated Eabs (nearly twice the observed values), while the new module yielded Eabs values (1.4 in both models) consistent with observations. Similarly, the new scheme considerably reduced (by 40–50%) the simulated BC direct radiative forcing (DRFBC) in four typical regions (Europe, North America, South America, and Asia), correcting for the overestimation of radiative warming found in previous model studies.

The unified theoretical framework presented in this study successfully addresses the long-standing challenge of accurately representing BC mixing state in climate models. The discovery of a universal law governing BC size distribution and its self-similarity across diverse environments provides a significant advancement in our understanding of BC's atmospheric behavior. The simplification of BC mixing state description through the use of a single parameter (k) significantly improves model efficiency without compromising accuracy. The improved agreement between simulated and observed values of Eabs and DRFBC demonstrates the effectiveness of the new mixing state module in reducing uncertainties in the estimation of BC's climate effects. The findings of substantially lower BC absorption enhancement and warming effects compared to previous estimates necessitate a reassessment of BC's overall impact on the climate system. This framework is easily adaptable and applicable to various atmospheric models, offering a significant contribution to climate modeling and prediction.

This research establishes a novel unified theoretical framework for describing black carbon (BC) mixing states, demonstrating self-similarity across diverse global environments. This framework simplifies BC mixing state characterization using a single parameter (k), allowing for more accurate and efficient integration into atmospheric models. The new mixing state module significantly improves estimations of BC absorption and radiative forcing, reducing previous overestimations and enhancing the accuracy of climate effect predictions. Future research could explore the temporal variability of k and further refine the model by incorporating additional factors influencing BC mixing.

The study's theoretical framework relies on a steady-state assumption, which may not perfectly capture the highly dynamic nature of atmospheric processes in all conditions. The simplified theoretical derivation, while providing valuable insights, could be further expanded to incorporate more complex interactions and variables. The validation of the model used limited observational datasets, although these datasets were collected from a diverse range of locations and environmental conditions. While the model provides a significant improvement, further validation with more extensive datasets could enhance its robustness.