the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Aug 2026
Fractal Geometry of Aerosol Particles and Its Impact on Atmospheric Optical Properties: Development and Validation of the Fractal Aerosol Cluster Model in a Heavy Haze Event
Abstract. Aerosol particles, as crucial atmospheric components, significantly influence optical properties and play an important role in climate change. Based on Lorenz-Mie's theory, the optical equivalent radius of aerosol particle affects the atmosphere visibility. Traditional aerosol models, which assume spherical or other geometrically regular particle shapes, significantly deviate from observations in visibility simulations. In this study, the mechanism of random diffusion and aggregation of monomeric particles into fractal geometrical clusters was reproduced. The Fractal Aerosol Cluster Model (FACM) was developed to parameterize the optical and aerodynamic sizes of the fractal geometry of aerosol particles. Sensitivity experiments were conducted to simulate a severe haze event in northern China in November 2018 by coupling FACM into WRF-Chem as the experimental case (EXP) while the control case (CTR) by the original WRF-Chem. The simulated near-ground PM2.5 concentrations in both EXP and CTR are similar to the observations (OBS). However, EXP simulated the larger extinction coefficients and lower atmospheric visibility (AV), which is more closely to OBS. The average normalized mean error of AV by EXP to OBS is 163.39 %, compared to 421.62 % by CTR. Thus, considering the fractal geometry of aerosol particles significantly improves simulated AV. Furthermore, a reduction of approximately 60 W·m⁻² in land surface shortwave radiation in EXP than those by CTR was also confirmed by observations. This study of the optical properties of the fractal aerosol cluster will contribute to future research of atmospheric environment and climate change forcing.
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CC1: 'Comment on egusphere-2026-4511', Jing Li, 02 Sep 2026
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AC1: 'Reply on CC1', Zhenxin Liu, 29 Sep 2026
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Our Response to CC1:
Reply to Comment 1
We greatly appreciate the reviewer’s constructive comment. We fully agree that a quantitative assessment of the sensitivity to the adopted value of the fractal dimension would significantly strengthen the robustness of the conclusions of this paper. To this end, we have added a set of sensitivity experiments for , which are presented in detail in the Supplementary Material.
- Design of the sensitivity experiments
Based on the North China case simulation in the main text (24–27 November 2018), we designed four experiments for three selected times (25 November 00:00 and 12:00, and 26 November 00:00), with (CTR, spherical assumption), (EXP, the experimental setup in this paper), , and , respectively. The AOD and near-surface horizontal visibility (VIS) were calculated for the four experiments, and the results are shown in the Supplementary Material (Figures S1–S2).
- Main findings
The sensitivity experiments reveal a key conclusion: even a small change in has an extremely significant impact on AOD and VIS. When is further reduced from 2.7 to 2.4 and 2.1, the AOD increases significantly, the VIS decreases sharply, and the magnitude of these changes far exceeds the realistic range that observations can support. This result is scientifically significant in two respects:
First, it strongly demonstrates that the fractal morphological characteristics of aerosol clusters have a high sensitivity to macroscopic atmospheric optical properties and are a core factor worthy of in-depth study. This is fully consistent with the core argument of this paper, that is, is a key parameter controlling the extinction capability of aerosols.
Second, it indicates that , taken as an extreme assumption of purely free coagulation growth, is theoretically valid but cannot represent the average aerosol state after aging processes in the real atmosphere.
- Physical meaning and limitations of
As described in Section 2 of the main text, the fractal dimension of the clusters simulated by the DLA algorithm is approximately 2.1, which corresponds to the structural characteristics of freshly generated soot aggregates that have not undergone any external compaction and are formed through free coagulation. Numerous electron microscopy observations have also confirmed that the of fresh soot collected at tunnel stations is about 1.8–2.0, while the of aged soot at background stations can rise above 2.0 (Wang et al., 2017) . However, aerosols in the real atmosphere undergo a series of rapid aging processes after emission:
(1) Condensation aging: Sulfuric acid, nitric acid, and organic vapors condense on the soot surface and fill the internal voids of the aggregates. In the Beijing area, the aging timescale dominated by condensation in the daytime surface layer can be as short as less than 2 hours (Peng et al., 2016).
(2) Coagulation aging: Soot coagulates with soluble aerosols such as sulfates and nitrates, further altering the aggregate structure. Although the timescale of coagulation aging at night is longer than that of condensation, it can still be completed within several hours to one day(Chen et al., 2017; Moteki et al., 2007).
(3) Hygroscopic compaction: After the coating absorbs water, the surface tension of liquid water forces the aggregates to restructure toward a denser and more nearly spherical morphology. Experiments show that after sulfuric acid coating and exposure to 80% relative humidity, the scattering of soot can be enhanced by about a factor of 10 and the absorption by nearly a factor of 2 (Zhang et al., 2008). The timescale for hygroscopic restructuring and compaction is estimated to be less than 0.7 days (Mikhailov, Vlasenko, Podgorny, Ramanathan, & Corrigan, 2006).
The combined effect of these aging processes is to shift from about 2.1 in the fresh state toward 3. Therefore, the average fractal dimension of aerosols in the real atmosphere should logically lie between 2.1 and 3. The specific value exhibits pronounced spatiotemporal variability depending on emission sources, chemical conditions, meteorological factors (especially humidity), and other factors.
- Basis for choosing
In the simulation of the winter haze event over the North China Plain, we adopt based on the following considerations: this value lies between that of fresh soot ( ) and that of a fully compact sphere ( ), representing a mixed aerosol population that has undergone partial aging and hygroscopic compaction. Although the mass fraction of fresh black carbon in PM₂.₅ is typically only 5–10%, after several hours to days of aging, the mass fraction of black carbon-containing aerosols in polluted urban atmospheres can reach 32–52%, and most of them are in a thickly coated state (with a coating thickness parameter of about 1.6–2.2), indicating that aging processes are highly efficient in polluted atmospheres(Wu et al., 2019). Therefore, is a relatively conservative engineering approximation for the overall aging degree of aerosols during winter haze events over the North China Plain—if a smaller were adopted, the fractal effect would only be stronger and the improvement from FACM would be even more pronounced, without changing the direction of the conclusions.
- Summary
The sensitivity experiments show that: (1) is a highly sensitive parameter controlling aerosol optical properties; (2) represents freshly generated soot formed by free growth, whereas aging processes in the real atmosphere drive it toward 3; and (3) is a reasonable value based on observational constraints and multiple tests, representing the aged mixed aerosol state during winter haze events over the North China Plain. The cases with and 2.4 can be understood as extreme hypothetical scenarios—namely, assuming that the aging mechanism of fractal aerosols suddenly disappears—and their results demonstrate how exaggerated the changes in atmospheric optical properties would be under such an assumption, thereby further highlighting the importance of accounting for realistic aging processes. We thank the reviewer for prompting us to add this important sensitivity analysis.
Reply to Comment 2
We sincerely thank the reviewer for carefully pointing out this error. You are entirely correct: in the description near Eq. (15), we reversed the equivalent radii that should be used for optical calculations and for sedimentation processes. The correct statement should be: optical calculations must use the optical equivalent radius , while dynamical processes such as sedimentation must use the aerodynamic equivalent radius . The incorrect statement in the original manuscript was a typographical error. We have corrected it in the revised manuscript and have carefully checked the entire text paragraph by paragraph to ensure that all statements involving and are consistent with their definitions. In addition, we have checked all similar statements throughout the manuscript to ensure that no such reversal remains.
Revision in the main text:
Location: Section 2.2, near Eq. (15).
Revision: The erroneous sentence:
"optical calculations must use , while processes such as sedimentation must use "
has been changed to:
"optical calculations must use , while processes such as sedimentation must use ".
Reply to Comment 3: Transferability of the FACM module and its prospects for application in GCMs
We greatly appreciate the reviewer’s forward-looking suggestion. We fully agree that the transferability of FACM and its potential for improving aerosol radiative forcing estimates in global climate models (GCMs) is an important direction for extending this study.
We would like to make the following points:
First, FACM has good portability. The scheme relies on relatively few parameters, mainly including the fractal dimension , the monomer radius , the scaling relationship between the radius of gyration and the number of monomers , and the resulting optical equivalent radius and aerodynamic equivalent radius . These parameters are available or can be diagnosed in most aerosol modules, so FACM can be readily embedded into the aerosol optical calculation component of GCMs.
Second, we have already conducted preliminary sensitivity experiments with the WRF model over the East Asian region to test the applicability of FACM at a larger spatial scale. The results show that after accounting for aerosol fractal morphology, the model produces non-negligible effects on aerosol optical depth, surface shortwave radiation, and surface temperature over East Asia, indicating that FACM has clear potential for improving aerosol radiative forcing estimates. However, due to the scope and workload of this paper, these results are not formally presented in the main text but are instead included in the Supplementary Material S2 (Figures S3–S7).
We greatly appreciate the reviewer’s professional and forward-looking suggestion, and we hope that in the future we or other researchers can, based on the model tool provided in this paper, further systematically evaluate the applicability of FACM in GCMs, including different parameterization schemes, coupling with aerosol mixing-state schemes, and impacts on global radiative forcing estimates.
Corresponding revision:
In the last paragraph of the Conclusions, add a brief discussion on the transferability of FACM and its prospects for application in GCMs, with a reference to Supplementary Material S2:
"Furthermore, FACM has good transferability. The scheme relies on few parameters and is physically well-defined, making it readily embeddable into the aerosol optical modules of other regional models or global climate models. We have conducted preliminary sensitivity experiments over East Asia, and the results show that fractal aerosols have non-negligible impacts on aerosol optical depth, surface shortwave radiation, and surface temperature, demonstrating the potential of FACM for improving aerosol radiative forcing estimates. Due to the scope and workload of this paper, these results are not presented in the main text; see Supplementary Material S2 for details. In future work, we will systematically evaluate the applicability of FACM in GCMs, including different fractal dimension parameterization schemes and their coupling with aerosol mixing-state schemes."
References in this response to comments:
Chen, X., Wang, Z., Yu, F., Pan, X., Li, J., Ge, B., . . . Chen, H. (2017). Estimation of atmospheric aging time of black carbon particles in the polluted atmosphere over central-eastern China using microphysical process analysis in regional chemical transport model. Atmospheric Environment, 163, 44-56. doi:https://doi.org/10.1016/j.atmosenv.2017.05.016
Mikhailov, E. F., Vlasenko, S. S., Podgorny, I. A., Ramanathan, V., & Corrigan, C. E. (2006). Optical properties of soot–water drop agglomerates: An experimental study. Journal of Geophysical Research: Atmospheres, 111(D7). doi:10.1029/2005jd006389
Moteki, N., Kondo, Y., Miyazaki, Y., Takegawa, N., Komazaki, Y., Kurata, G., . . . Koike, M. (2007). Evolution of mixing state of black carbon particles: Aircraft measurements over the western Pacific in March 2004. Geophysical Research Letters, 34(11). doi:10.1029/2006gl028943
Peng, J., Hu, M., Guo, S., Du, Z., Zheng, J., Shang, D., . . . Zhang, R. (2016). Markedly enhanced absorption and direct radiative forcing of black carbon under polluted urban environments. Proceedings of the National Academy of Sciences, 113(16), 4266-4271. doi:10.1073/pnas.1602310113
Wang, Y., Liu, F., He, C., Bi, L., Cheng, T., Wang, Z., . . . Li, W. (2017). Fractal Dimensions and Mixing Structures of Soot Particles during Atmospheric Processing. Environmental Science & Technology Letters, 4(11), 487-493. doi:10.1021/acs.estlett.7b00418
Wu, Y., Liu, D., Wang, J., Shen, F., Chen, Y., Cui, S., . . . Ge, X. (2019). Characterization of Size-Resolved Hygroscopicity of Black Carbon-Containing Particle in Urban Environment. Environmental Science & Technology, 53(24), 14212-14221. doi:10.1021/acs.est.9b05546
Zhang, R., Khalizov, A. F., Pagels, J., Zhang, D., Xue, H., & McMurry, P. H. (2008). Variability in morphology, hygroscopicity, and optical properties of soot aerosols during atmospheric processing. Proceedings of the National Academy of Sciences, 105(30), 10291-10296. doi:10.1073/pnas.0804860105
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AC2: 'AC1 content (PDF version)', Zhenxin Liu, 29 Sep 2026
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Please find attached the AC1 content uploaded again in PDF format, because some characters and equations could not be displayed correctly on the web page.
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AC1: 'Reply on CC1', Zhenxin Liu, 29 Sep 2026
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This is an interesting and potentially useful study. The consideration of aerosol fractal morphology provides a physically meaningful extension to the conventional spherical-particle assumption, and the implementation in WRF-Chem makes the work relevant to both air-quality and aerosol–climate modeling. The manuscript is generally complete, and the main conclusions are reasonably supported by the numerical experiments and observations. I only have a few minor comments that may help improve the clarity and robustness of the paper.