We incorporate the isotopic chemistry module into the Eulerian atmospheric chemistry transport models NAQPMS. The NAQPMS is a Nested Air Quality Prediction Model System developed by the Institute of Atmospheric Physics (IAP), Chinese Academy of Sciences (CAS) (Wei et al., 2019; Chen et al., 2021; Wang et al., 2001), including physical processes (advection, diffusion, and dry/wet deposition), chemistry module (gas and aqueous chemistry, aerosol chemistry and thermodynamics, and mercury chemistry) (Chen et al., 2015), and online emission of dimethyl sulfide, sea salt, and dust. The Advanced Particle Microphysics module (APM) and 1.5-D Volatility Basis Set (VBS) aerosol schemes are also coupled to simulate the aerosol microphysical processes and organic aerosol formation, respectively (Chen et al., 2019; Yang et al., 2019; Chen et al., 2014).

The simulation is performed with a two-way nested configuration. The external domain D1 covers the mainland of China at a horizontal resolution of 45 km, while the innermost domain D2 is centered over eastern China with a finer resolution of 15 km, as illustrated in Fig. 1. The model runs on 20 atmospheric vertical layers, utilizing terrain-following hybrid sigma coordinates from the surface to an altitude of 20 km. The sulfur chemistry from SO₂ to S(VI) involves gas, heterogeneous and aqueous-phase oxidation. The gas-phase chemistry incorporates the Carbon Bond Mechanism Z (CBM-Z) (Zaveri and Peters, 1999). For aqueous chemistry, the Regional Acid Deposition Model (RADM2) is applied, resolving cloud-phase sulfur oxidation through dissoluble S(IV) with O₃, H₂O₂, proxy acetic acid (CH₃OOH), and transition metal-catalyzed O₂ (Stockwell et al., 1997). The oxidation of S(IV) species by dissolved NO₂ is also incorporated into the sulfur aqueous chemistry, referring to the kinetic parameters proposed by Cheng et al. (2016). Heterogeneous reactions of SO₂ on various surfaces, including dust, sea salt, soot and deliquesced aerosols, have also been incorporated using the reactive uptake coefficients parameterization (Li et al., 2018). The gas-particle partitioning of inorganic aerosols and aerosol water content are simulated using the improved ISORROPIA II (Fountoukis and Nenes, 2007; Song et al., 2018). The simulation began in March 2014. The initial 3 months served as a spin-up time. For additional details on the model and its detailed configuration, refer to Sect. S1 and Table S2 in the Supplement.

The sulfur element is widely present throughout all natural environments and possesses four stable isotopes, namely ³²S, ³³S, ³⁴S, and ³⁶S, with relative abundances of 95.02 %, 0.75 %, 4.21 %, and 0.02 %, respectively (De Laeter et al., 2003). In general, the sulfur isotopic composition is denoted as δxS value in per mil (‰), where ^xS represents one of the less abundant isotopes, e.g., ³³S, ³⁴S, or ³⁶S. This notation is defined as the difference between the isotopic ratio of a given sample and the reference standard V-CDT (Vienna Canyon Diablo Troilite): 1δxSSample(‰)=R(xS/32S)SampleR(xS/32S)V-CDT-1×1000 where R represents the isotopic ratio, defined as the atomic abundance ratio of heavier to lighter stable isotope, expressed as R=N(hX)/N(lX), where N(hX) and N(lX) represent the atomic abundances of the heavier stable isotope (hX) and lighter stable isotope (lX), respectively. V-CDT is the international sulfur isotope standard, with isotopic ratios of R(³⁴S/³²S)V-CDT=0.044163 and R(³³S/³²S)V-CDT=0.007877 (Ding et al., 2001).

Isotope effects are observable in physical or chemical processes, especially during equilibrium (or thermodynamic) and kinetic processes, leading to a change in isotope distribution between two substances or different phases of the same substance with distinct isotope ratios. Isotopic fractionation is often a result of the isotope effect. The isotopic fractionation factor α hXP/S is defined as the ratio of isotopic ratio of the instantaneously formed product RPi=(N(hX)/N(lX))Pi and substrate RS=(N(hX)/N(lX))S (Mariotti et al., 1981), 2αhXP/S=RPi/RS

For thermodynamic (equilibrium) or unidirectional kinetic process with a single-step reaction following a first-order rate law, the αhXP/S can be interpreted as the ratio of rate constant k or equilibrium constant K: 3αhXP/S=hk/lk=hK/lK

An isotope enrichment factor, ϵ (in per mil ‰), can be defined as: 4ϵ=(αhXP/S-1)*1000

The αhXP/S determines how much faster or slower the reaction with the heavier isotopologues proceeds relative to the reaction with the lighter isotopologues.

Generally, isotopic fractionation determined in laboratory experiments represents overall isotope effects, resulting from physical, equilibrium, and kinetic fractionation. The isotopic composition between reservoir and product, as a function of reactant extent and fractionation factor, is described by the Rayleigh fractionation equation.

5RS,t/RS,t0=frem(αhXP/S-1) where frem is the fraction of residual reactant, RS,t0 and RS,t are the isotope ratio of substrate at initial t0 and t moment, respectively. The instantaneous isotope ratio of the product RP,t at t moment is given by: 6RP,t/RS,t0=αhXP/S×frem(αhXP/S-1)

The average isotope ratio of accumulated product (RP‾) during a finite reaction interval follows the standard Rayleigh-fractionation expression for accumulated products (Hoefs, 2018): 7RP‾/RS,t0=1-frem(αhXP/S-1)1-frem

The multi-resolution Emission Inventory for China (MEIC, http://meicmodel.org.cn/?p=1579&lang=en, last access: 11 August 2026) provides a comprehensive SO₂ emission inventory, including anthropogenic emissions from power plants, industry, residential, and transportation, respectively. Primary combustion sulfate, which forms as the oxidation product in the chimney and plumes (Sun et al., 2023; Ding et al., 2021), is assumed to be 5 wt % of anthropogenic SO₂ (Berglen et al., 2004). Anthropogenic sulfur-containing sources exhibit a highly variable and overlapping isotopic composition, ranging from -30 ‰ to 30 ‰ (Hoefs and Harmon, 2022). We adopt a constant signature of δ34S =0 ‰ V-CDT for anthropogenic SO₂ emission sources. This simplification isolates the atmospheric chemical isotope effect from poorly constrained source signatures.

The sulfur isotope difference between SO₂ and sulfate (Δδ34S_SO42- / SO2=δ34S_SO42-–δ34S_SO₂) approximates the apparent isotopic fractionation multiplied by 1000 ‰ and is hereafter referred to as the isotope effect. Δδ34S_SO42- / SO₂ is mathematically independent of the absolute emission δ34S and can be therefore validate the oxidation module of isotope-enabled CTM. By contrast, the δ34S_SO₂ and δ34S_SO42- depend on the prescribed emission signature. Direct comparisons between simulated and observed δ34S_SO₂ and δ34S_SO42- are subject to the uncertainties of this assumption and should be interpreted with caution. Jointly examining the simulated and observed δ34S_SO₂ and δ34S_SO42- together with Δδ34S_SO42- / SO₂ provides a diagnostic of whether a given model–observation bias arises primarily from the atmospheric oxidation processes or from the assumed emission-source signatures.

The tracer mass of ³²SO₂, ³⁴SO₂, ³²SO42-, ³⁴SO42- isotopologues can be determined from the SO₂ mass in the emission inventory based on the Eq. (1), with the initial values of RV-CDT at 0.044163.

834S=32S×(δ34S1000+1)×RV-CDT

In previous studies, the initial isotopic composition of reactant (δs,0) was calculated based on the isotopic composition of the product (δp) and the residual fraction of reactants frem using the Rayleigh equation, assuming no addition or mixing. This resulted in isotopic “reservoir effects” as the reaction progressed. The isotopic fractionation factor determines the direction of fractionation: when α>1, heavy isotopes are preferentially enriched in the product and depleted in the residual reservoir; conversely, when α<1, heavy isotopes are preferentially depleted in the product and enriched in the residual reservoir.

To illustrate the magnitude of the reservoir effect under realistic atmospheric conditions, we consider the competing S(IV) oxidation pathways: gas-phase OH radicals, aqueous-phase H₂O₂, O₃, and TMI-catalysis, and heterogeneous reaction. Using model results from Shao et al. (2019), who employed the GEOS-Chem model to quantify the contributions of these competing pathways to sulfate formation during a Beijing pollution episode, their reported contributions are 42.6 %, 2.2 %, 1.3 %, 27.2 %, and 26.7 %, respectively. Based on these pathway contributions, the calculated overall/net sulfur fractionation factor α34S_S(IV) is 1.0063 at 0 °C. Therefore, the sulfate product favors the heavier isotopologues ³⁴SO42- over ³²SO42-, leading to an increase in δ34S_SO42- as the reaction progresses. This relationship depends on the fractionation fraction and the reaction extent. As shown in Fig. 3, with the initial δ34S of SO₂ set to 0 ‰, when frem drops to 70 % and 20 %, the δ34S of the residual SO₂ reaches approximately -2.2 ‰ and -10 ‰, while the δ34S of the accumulated sulfate product reaches approximately 5.2 ‰ and 2.5 ‰, respectively. The resulting Δδ34S_SO42- / SO₂ is 7.4 ‰ and 12.5 ‰. As the reservoir approaches depletion (frem≈0), the apparent isotope fractionation becomes unrealistically large. This challenges the direct application of the Rayleigh equation to systems where mixing and reaction occur simultaneously.

Using the integration method that describes the isotopic evolution of the reservoir in an open system (Sect. S3), Fig. S1 illustrates that as the β value (the ratio of the instantaneous amount added to the removal) increases, the δ value of the reservoir progressively gets higher, with the external sources consistently contributing with a constant isotopic composition (δs) of 10 ‰. This emphasizes the importance of considering the mixing process when calculating the isotope effect. The overall apparent isotope fractionations are indeed the result of combined effects of the Rayleigh-like distillation process and the diffusion-driven isotope distributions (Guan and Liu, 2023). The combined processes with the mixing of fresh material and isotopic fractionation can lead to a smaller variation in Δδ34S_SO42- / SO₂ than estimated by Rayleigh equation. These results indicate that the optimal sub-timesteps effectively reduce the time-step impact and minimize discrepancies between the Rayleigh-based calculation and the integration method.

To mitigate the numerical bias that can arise when the Rayleigh equation is applied directly over a finite CTM time step, we implement an iterative time-splitting method. Firstly, the comparison between this iterative method and integration method in the open system helps determine the optimal sub-timesteps, effectively minimizing the biases in isotopic calculation when employing the Rayleigh equation. Figure S2 illustrates that the largest differences are observed for small reaction fraction and large fraction of fresh mixture relative to the initial substrate. A larger difference is expected when the differences between the isotopic composition of substrate and mixture increase and the isotope effect becomes stronger. For 50 sub-timesteps, good agreements are found in the calculated isotopic value of the reservoir between the iterative time-splitting method and the integrating method, assuming 0.3 ‰ is approximate to the average analytical precision of isotope. This indicates that the combination of the Rayleigh equation with the iterative method is capable of effectively simulating the progressive isotopic evolution of reservoirs with simultaneous mixing and isotopic fractionation.

We also conduct sensitivity tests to check the performance of the simulated isotopic composition of product with the optimized sub-timesteps. We choose 0.3 ‰ as the tolerance of this iterative time-splitting method. Figure S3 illustrates the deviation of δ-values for 1, 10, 50, and 100 sub-steps relative to the reference simulation with 1000 sub-steps. Notably, the largest deviation occurs for a large fraction of reaction f and mixture, attributed to greater depletion in the reservoirs (α>1) and influence of mixing process. For 50 sub-steps, the maximum deviation is less than 0.1 ‰. This comparison confirms that the sub-timestep limiting reaction fraction f to approximately 2 % is acceptable for reducing simulation bias, thus the value is adopted to improve the simulation.

The numerical comparison further shows that the largest differences between the direct Rayleigh calculation and the iterative method occur mainly over remote oceanic regions, where SO₂ emissions are low, reaction extents are high, and the SO₂ reservoir effect is strongest; in these regions, the iterative method reduces the simulated Δδ34S_SO42- / SO₂ by approximately 1.5 ‰–3.0 ‰ relative to the direct Rayleigh calculation applied over the full model time step. In regions with intensive anthropogenic activity, the influence is smaller but still non-negligible, with changes of approximately 0.3 ‰–1.0 ‰ in some polluted areas, demonstrating that the iterative time-splitting method improves the numerical accuracy of simulated sulfur isotope fractionation.

The overestimation of Δδ34S_SO42- / SO₂ at Nanjing (NMB = 35 %–55 %) is attributed to the insufficient representation of the TMI-catalyzed oxidation pathway – the only known oxidation pathway that produces sulfate depleted in ³⁴S relative to SO₂ (α=0.9949,ϵ≈-5.1 ‰). Previous studies have indicated that the aqueous-phase TMI-catalyzed oxidation pathway is underestimated in the current atmospheric chemical transport models, contributing 9 %–18 % to sulfate production globally (Alexander et al., 2009; Itahashi et al., 2022), while heterogeneous reactions via TMI-catalyzed oxidation on deliquesced aerosol particles can contribute 14 %–92 % to sulfate production during the heavy pollution period in northern China (Shao et al., 2019; Wang et al., 2021). Furthermore, model parameterizations of Fe and Mn concentrations are biased low in East Asia (Itahashi et al., 2022), which directly suppresses the simulated TMI-catalyzed sulfate production. Our model does not currently include TMI-catalyzed oxidation in aerosol liquid water. Instead, we employ the obtained sulfur isotope fractionation during heterogeneous oxidation of SO₂ on mineral dust (α>1) (Harris et al., 2012b) to represent the overall isotope effect of all heterogeneous reactions at the interface of deliquesced aerosol particles, which likely introduces additional biases. Currently, laboratory studies also show enhanced multiphase oxidation of SO₂ by NO₂ in deliquesced aerosol particles (Wang et al., 2016; Liu and Abbatt, 2021), but isotopic fractionation factors for this pathway are currently lacking and require further measurement.

The site- and season-dependent δ34S_SO42- biases can be grouped into two categories: Category 1 – Oxidation-driven biases. At Nanjing, Hangzhou, and Tianjin winter, the δ34S_SO42- biases are broadly consistent in direction with the known overestimation of Δδ34S_SO42- / SO₂ by the oxidation module. This confirms that the δ34S_SO2=0 ‰ assumption is a reasonable approximation for the well-mixed regional SO₂ emissions at these locations. The remaining bias is therefore predominantly oxidation-driven, reflecting the underrepresented TMI-catalyzed pathway and simplified heterogeneous chemistry noted above.

Category 2 – Source-driven biases. By contrast, at Beijing winter, the model (3.99 ‰ ± 1.86 ‰) underestimates observed δ34S_SO42- (7.94 ‰ ± 1.41 ‰) by ∼50 % (NMB =-50.5 %). This contrasts with the oxidation module's upward bias in Δδ34S, which would drive δ34S_SO42- above observation. This reversed bias direction provides a clear diagnostic that the dominant error originates from the source assumption rather than from the oxidation module.

As shown in Fig. 1 and Table S1, documented coal δ34S in northern China – including Beijing, Hebei, and Henan – are substantially higher than 0 ‰, with means of 7.4 ‰ ±16.1 ‰, 3.5 ‰ ±6.6 ‰, and 6.4 ‰ ±1.0 ‰, respectively. Enhanced residential heating in northern China contributes about 46 % of the monthly averaged PM_2.5 concentration (Zhang et al., 2017), providing a localized source of SO₂ with elevated δ34S. If Beijing winter SO₂ carries δ34S ≈7 ‰ (the mean coal value) instead of 0 ‰, the modeled δ34S_SO42- would increase from ∼4.0 ‰ to ∼ 6–7 ‰ after accounting for oxidation-driven enrichment, approaching the observed 7.94 ‰. Consequently, the distinct seasonal shift in energy type (coal, biomass, clean energy, etc.) and activity (cooking, heating, etc.) can substantially affect aerosol δ34S_SO42- (Wei et al., 2018), complicating direct model–observation comparisons where strong localized combustion sources are present.

The preceding discussion reflects what is directly resolved by the isotope-enabled model. We now extend the discussion to broader implications inferred by combining our model results with literature evidence on emission-source isotopic signatures, combustion fractionation, and FGD.

Assuming a fixed δ34S_SO₂ of 0 ‰ for all anthropogenic emissions of SO₂, the model's ability to reproduce observed δ34S_SO42- (Sect. 3.2.2) suggests that well-mixed anthropogenic SO₂ emissions are characterized by δ34S values close to 0 ‰. However, compiled anthropogenic δ34S values of SO₂ emission sources in eastern China, including coal, crude oil, and biomass, range from 1 ‰ to 10 ‰ (Hong et al., 1993; Guo et al., 2016; Chen et al., 2017). This discrepancy points to a systematic ³⁴S depletion process in emitted SO₂.

As shown in Fig. 7, residential and industrial combustion experiments show that the sulfate in ash particles enriched ³⁴S, while emitted SO₂ is depleted in ³⁴S (-10 ‰ ∼-5 ‰) relative to the fuel (Hong et al., 1993; Chen et al., 2017). FGD further introduces isotopic fractionation, with the outlet SO₂ isotopically lighter than the inlet SO₂ (calculated apparent sulfur isotope enrichment factors ϵ*=-5.5 ‰) (Derda et al., 2007). These documented combustion and FGD fractionation patterns are temporally consistent with the abrupt decline in δ34S_SO42- with continuous long-term aerosol observation (shifting from 5 ‰ ∼10 ‰ before 1999 to 0 ‰ ∼5 ‰ at Tsuruoka, Japan) (Oduro et al., 2012) and wet precipitation records (from 6.6 ‰ in 2010 to -0.1 ‰ in 2017 at Jiaozuo city, China) (Zheng et al., 2024). This observed δ34S decline coincides with the widespread adoption of FGD technology in coal-fired industries after 2000 (Liu, 2019), although other concurrent factors – such as changes in the relative contribution of coal, oil, and biomass combustion sources, shifts in the geographic distribution of SO₂ emissions, and variations in atmospheric oxidation pathways – may also contribute to the observed trends. Establishing a direct causal linkage between FGD deployment and the recorded δ34S decline would require a dedicated study combining long-term field observations with systematic source sampling and experimental study of isotopic fractionation during combustion and FGD processes, which is beyond the scope of the present analysis. This temporal coherence, nevertheless, suggests that FGD-induced isotopic fractionation is a plausible contributing factor worth further investigation.

Additionally, the sulfur oxidation pathway produces enriched sulfate with Δδ34S_SO42- / SO₂ at 6 ‰ ∼ 8 ‰. The emission-side ³⁴S depletion and the oxidation-side ³⁴S enrichment thus partly counterbalance each other, leading to only a modest net difference between fuel δ34S and airborne sulfate δ34S (typically 1 ‰–8 ‰) (Fig. 7). This emphasizes a complex isotope effect of sulfur chemistry during combustion and FGD. Using fuel δ34S directly as the SO₂ source end-member – without correcting for the ³⁴S depletion introduced by combustion and FGD – artificially elevates the source signature, causing systematic underestimation of the contribution of sulfur-containing fuels with significant combustion fractionation and overestimation of sources with δ34S near 0 ‰. An independent study by Feng et al. (2023) quantified this effect: using high-time-resolution δ34S observations during haze episodes in northern China, they found that neglecting coal-combustion ³⁴S fractionation caused the coal contribution to be underestimated by 17 %–38 % of secondary sulfate. These quantitative findings corroborate the deductive evidence from our model and underscore the necessity of incorporating both emission-side and oxidation-side isotopic fractionation into the end-member framework for reliable source apportionment.