Unified theoretical framework for black carbon mixing state allows greater accuracy of climate effect estimation

Black carbon (BC), a ubiquitous aerosol component, is a major contributor to global warming due to its strong light absorption, which depends strongly on particle mixing state. In the atmosphere, freshly emitted BC undergoes condensation, coagulation, and deposition, developing coatings that enhance light absorption via the lensing effect. The mixing state in the real atmosphere is complex and has been characterized from various perspectives in field observations, yet the combined influence of these dynamic processes on BC mixing state remains poorly understood. Global climate models cannot represent the full complexity and diversity of mixing states and typically assume either internal or external mixing, leading to a wide range of estimated global-average BC mass absorption cross-sections (MAC at 550 nm) from 3.1 to 18.0 m2/g. Thus, precisely describing BC mixing state is crucial to improve model estimates of BC optical properties and radiative forcing. This study develops a theoretical framework that links dynamic processes to the BC coating thickness distribution, reveals self-similarity of BC mixing states across diverse sites, and enables simplified, accurate characterization of mixing state in both simulations and observations.

Prior work has highlighted the sensitivity of BC radiative effects to mixing state and particle-to-particle heterogeneity, with studies reporting significant absorption enhancements and uncertainties when assuming simplified internal or external mixing. Observational and modeling efforts have characterized BC mixing state and its optical impacts across environments (urban, regional background, biomass burning), yet global models still commonly rely on simplified mixing assumptions, contributing to wide MAC estimates. Reviews have synthesized measurements and modeling of aerosol mixing state, and field campaigns (e.g., CARES) have provided datasets on BC properties. These studies motivate a framework capable of linking dynamic atmospheric processes to observable size and coating distributions to reduce uncertainty in radiative forcing estimates.

The study derives a theoretical framework under a steady-state approximation that connects atmospheric growth and deposition processes to the coating thickness distribution of BC-containing particles. Starting from a monodisperse BC core population emitted at time zero with diameter Dc and number concentration n(Dc), particle growth via condensation and coagulation is represented by dDp/dt = GR, giving Dp = GR·t + Dc. Deposition removes particles at rate Dep, yielding n(Dp) = n(Dc)·exp(−Dep·t). Eliminating time under steady state leads to ln(n(Dp)) = ln(n(Dc)) − (Dep/GR)·(Dp − Dc), i.e., a linear relation between ln(n(Dp)) and coating thickness ΔDp = Dp − Dc with slope k = Dep/GR. The slope k determines an average coating thickness of approximately 1/k and is theoretically independent of Dc, implying self-similarity across BC core sizes. A more rigorous derivation and the dependencies of GR and Dep on Dp and time are provided in the Supplementary Information. Field verification: The framework was tested using single particle soot photometer (SP2) measurements across eight sites with diverse environments (e.g., Nanjing, Beijing, Shaoguan, Tibetan Plateau sites including Lulang and Maqu, Tokyo, Sacramento, and Amazon Tall Tower Observatory). SP2 provides refractory BC core mass and derives BC-containing particle optical diameters via scattering calibration and core-shell Mie theory. Size distributions of Dp were analyzed in Dc bins (110–120, 120–130, 130–140, 140–150 nm) to test slope invariance with Dc. Optical calculations and approximation: Core-shell Mie calculations were used to quantify absorption enhancement Eabs. A lognormal Dp distribution (geometric standard deviation 1.8, mean Dp 70 nm) at wavelength 550 nm was assumed. Refractive indices used: BC 1.85 + 0.7i and scattering shell 1.53 + 0i. Calculations of MAC response to ΔDc (10–200 nm, 10 nm steps) showed an approximately linear relation between Eabs and ΔDp for small ΔDp, enabling replacement of the coating distribution with a monodisperse coating thickness equal to 1/k when computing absorption. Integral computations (ΔDc 1–1000 nm, 1 nm steps) and a k-based method with k=0.014 showed good agreement in absorption coefficients. Model implementation: A new mixing state module based on the universal Dp distribution (Eq. 5) and the monodisperse effective coating thickness 1/k was implemented in CESM2.1.3-CAM6 (with MAM4) and in WRF-Chem v3.7 (with MOSAIC). In both, BC core diameters followed a lognormal distribution with count median diameter 70 nm. CESM-CAM6: global simulations at 1.9°×2.5° with 70 vertical layers, four-year run (2012–2015; last year analyzed), shortwave radiation via RRTMG, aerosol optics for accumulation mode resolved with 30 size bins; DRFBC estimated assuming linearity with MAC. WRF-Chem: domain centered at 35°N, 110°E, 20 km resolution, 30 vertical layers, two-week April 2020 period (36 h cycles, last 24 h analyzed), meteorology from NCEP FNL (1°×1°), RRTMG radiation, anthropogenic emissions from MIX inventory, biogenic emissions via MEGAN v2, dust via GOCART, CBMZ chemistry with MOSAIC aerosol (four size bins; spherical particles; internal mixing within bins). For comparison, conventional mixing assumptions (volume mixing in CAM6; volume and core-shell mixing in WRF-Chem) were also used. Since models lack explicit external mixing modules, MACexternal was computed offline using default refractive indices and densities.

• The size distribution of BC-containing particles follows a universal exponential law and is independent of BC core size, consistent with ln(n(Dp)) linearly decreasing with coating thickness ΔDp. Field SP2 measurements across eight globally diverse sites exhibit this behavior. • The slope k = Dep/GR characterizes the distribution and is approximately invariant with Dc. Observed k from linear regressions ranged from 0.008 to 0.020 across sites. • Dp distributions for multiple Dc bins (110–120, 120–130, 130–140, 140–150 nm) share nearly identical slopes, confirming self-similarity of BC size distributions across core sizes in suburban (Nanjing), remote (Maqu), urban (Tokyo), and biomass-burning influenced (ATTO) environments. • Conventional model assumptions of mixing state substantially overestimate absorption enhancement: CESM-CAM6 (volume mixing) simulated Eabs > 2.5; WRF-Chem (volume and core-shell mixing) simulated Eabs ≈ 2.0–2.5, versus observed Eabs ≈ 1.0–1.7. • Using the new mixing-state module, simulated Eabs matched observations: approximately 1.4 (1.3–1.6) in CESM-CAM6 and 1.4 (1.3–1.7) in WRF-Chem. • The new scheme reduced model-simulated BC direct radiative forcing (DRFBC) by about 40–50% across Europe, North America, South America, and Asia compared with conventional schemes, aligning with recent findings that models overestimate BC warming due to mixing-state treatment. • The monodisperse effective coating thickness approximation (1/k) reproduces absorption coefficients integrated over size, enabling efficient application in global and regional models.

The unified framework demonstrates that BC mixing state in the atmosphere can be succinctly described by a universal size distribution characterized by a single slope parameter k, reflecting the balance of growth and deposition processes. This self-similarity, validated across diverse environments and independent of core size, bridges observable particle diameters from field measurements with dynamic parameters used in models. By reducing the dimensionality of mixing state representation and enabling a monodisperse effective coating thickness approximation, the framework provides a practical and accurate means to quantify BC light absorption in models without additional computational burden. Implementations in CESM-CAM6 and WRF-Chem show that the approach corrects the overestimation of absorption enhancement and substantially lowers simulated DRFBC, bringing model results into agreement with observations and improving the fidelity of aerosol climate effect estimates.

This study establishes a unified theoretical framework for BC mixing state that links atmospheric dynamics to coating thickness distributions and reveals a universal, self-similar law independent of core size. A new mixing-state module derived from this framework, using an effective coating thickness of 1/k, accurately represents BC absorption enhancement in both global and regional models and substantially reduces overestimated radiative forcing. The approach connects observations and model simulations in a consistent description of mixing state and absorption, improving the accuracy of aerosol climate effect estimations without increasing computational complexity.

The framework relies on a steady-state approximation and a simplified derivation (with more rigorous treatment in the Supplementary Information). The monodisperse effective coating thickness (1/k) approximation is justified by the near-linearity of Eabs versus ΔDp for small ΔDp and may be less accurate for large coatings. Optical calculations adopt specific refractive indices (BC: 1.85+0.7i; shell: 1.53+0i), spherical particle assumptions, and a lognormal BC size distribution with median 70 nm. Model implementations used internal mixing within size bins (WRF-Chem) and required offline estimation of MACexternal due to lack of explicit external mixing modules. These assumptions may affect generalizability to conditions that deviate from these parameterizations.