Chemistry-climate simulations in this assessment are performed with the EMAC model. It is a flexible, global model that simulates a plethora of atmospheric processes with interactions between land, ocean and anthropogenic activity via a coupling interface called MESSy (Jöckel et al., 2010) that can connect more than 100 different submodels to ECHAM5, EMAC's base general circulation model (Roeckner et al., 2006). Apart from AIRTRAC, which estimates the contribution of aviation-related pollutants along air parcel trajectories and MADE3, which calculates aerosol dynamics and microphysical processes, there are other submodels that are also relevant in our modeling setup. The full list of applied submodels is included in Table A1 in Appendix A.

Simulations are performed with version 5.3.02 of the base model ECHAM5 and with version 2.55.2 of the MESSy framework. The EMAC model resolution is T42L41DLR, which corresponds to a quadratic Gaussian grid of size ∼ 2.8° × 2.8° (with 128 longitude and 64 latitude grid cells) with 41 discrete, vertical hybrid sigma-pressure levels ranging from the surface to the uppermost layer of the atmosphere centered at 5 hPa. Across the midlatitudes and Tropics, most T42L41 model layers lie in the troposphere. At the Equator, for example, ∼ 80 % of the layers are below the climatological tropopause during July to September. Defining the UTLS here as 100–400 hPa, approximately 16 model layers fall within this pressure range (Fig. S13 of the Supplement). For comparability purposes with Righi et al. (2023), our model output is set to a temporal resolution of 11 h with a model calculation time-step length of 15 min. The meteorology (the temperature, the wind divergence, the vorticity and the logarithm of the sea-level pressure) in our simulations has been nudged by Newtonian relaxation towards ERA-Interim reanalysis data (Dee et al., 2011) for the simulated year. Sea surface temperature and sea ice concentration have been prescribed from the ERA-Interim reanalysis data as well. Two simulations (starting on 1 January and on 1 July 2015) were performed to accompany the transport of aviation SO₂ and H₂SO₄ and their chemical conversion into SO₄ for three months across 28 emission points (Fig. 2) at an altitude of 240 hPa (Fig. 3). To ensure background meteorological conditions are in quasi-equilibrium, each simulation was preceded by a four-month spin-up period. Each emission point releases variable amounts of SO₂ and H₂SO₄ in the form of 15 min pulse emissions. These are initially advected by 50 air parcels originating from the grid box of each emission point, consistent with the recommendation of Grewe et al. (2014a). These 50 air parcels are pseudo-randomly initialized around the coordinates of an emission point according to a uniform distribution between 0 and 1. A simplified background chemistry mechanism is applied with the MECCA submodel (Sander et al., 2019) for the troposphere that involves the most relevant gaseous species like NO_x, HO_x, CH₄ and O₃. Aqueous-phase chemistry is handled by the SCAV submodel (Tost et al., 2006). These chemistry mechanisms are solved automatically by the Kinetic Pre-Processor (KPP) software using Fortran 90 code (Sander et al., 2005). The overall Lagrangian setup is similar to the one used by Maruhashi et al. (2022, 2024), while the applied aerosol chemistry and scavenging mechanisms are the same as those applied by Righi et al. (2023).

The 28 emission locations at which SO₂ and H₂SO₄ pulses are introduced (Fig. 2) are selected to reflect a realistic modern-day spatial distribution of aviation emissions. This distribution was determined according to the 2015 CMIP6 (Feng et al., 2020) aviation emissions inventory that includes the correction for the latitudinal bias found by Thor et al. (2023). The exact coordinates of the 28 points in Fig. 2 are included in Table B1 from Appendix B. Each coordinate has three dimensions – latitude, longitude and altitude – which are found by identifying the locations at which the aviation SO₂ mass flux (kg m⁻³ s⁻¹) is maximum. The emission altitude is determined by establishing the pressure level at which the zonally averaged aviation SO₂ mass flux, which is a function of both latitude and altitude, is the largest (Fig. 3): 238.2≈240 hPa.

We have developed an accompanying Python tool (see EP_selector in Appendix B and the data repository of Maruhashi et al. (2025a) for the complete code) that approximates continuous emissions as a set of distributed pulse emissions by automatically identifying the top 28 points whose grid cells have the largest SO₂ mass flux contributions (defined as a grid cell's area-weighted mean mass flux, see Eq. B1) within a user-defined mesh. The total mass of emitted SO₂ and H₂SO₄ across these points was scaled to yield the approximate global aviation SO₂ produced in a day from aviation in 2015 (Righi et al., 2023). The emitted amount of H₂SO₄ at any given emission point is determined by assuming that it constitutes 2 % of the total SO₂ mass emitted by aviation, which is consistent with measurements of aircraft exhaust plumes at cruise altitudes (Jurkat et al., 2011). Based on this assumption, the amount of H₂SO₄ emitted per simulation is found according to Eq. (B3) in Appendix B.

Aerosol microphysical processes are simulated using the MADE3 (Kaiser et al., 2014, 2019) submodel, a successor of the MADE (Ackermann et al., 1998; Lauer et al., 2005) and MADE-in (Aquila et al., 2011) submodels. The performance of MADE3 has been extensively evaluated. Its ground-level aerosol mass concentrations have been compared to data from a network of measurement stations and satellite observations. Its simulated vertical profiles of the mass mixing ratios and particle number distributions have been evaluated against aircraft campaign data. Overall, the model has demonstrated satisfactory alignment with observational data, although MADE3 tends to produce larger average sulfate concentrations near the surface, with biases ranging from 13 % to 92 % when compared to observations from measurement stations (Kaiser et al., 2019).

The MADE3 submodel considers nine types of aerosols: SO₄, BC, sea spray (Na), H₂O, chloride (Cl), mineral dust (DU), NH₄, NO₃ and particulate organic matter (POM). Each aerosol type is further classified into nine modes, which result from the combination of three size categories (Aitken (subscript “k”), accumulation (subscript “a”) and coarse (subscript “c”)) and three mixing states (soluble (subscript “s”), insoluble (subscript “i”) and mixed (subscript “m”)). MADE3 therefore adopts a modal approach, where the total particle number distribution nln⁡D of an aerosol is obtained by the superposition of its nine lognormal distributions (one for each mode M), according to Eq. (1a) (Aquila et al., 2011; Kaiser et al., 2014; Boucher, 2015): 1anln⁡D=∑M=19NM2πln⁡σg,Mexp-ln⁡D-ln⁡Dg,M22ln2σg,M, where NM is the total particle number concentration for mode M, Dg,M is the median diameter and σg,M is the geometric standard deviation. Assuming spherical particles, the mass distribution mln⁡D is obtained by multiplying Eq. (1a) by the particle density ρM and cubic diameter D3 according to Eq. (1b): 1bmln⁡D=∑M=19π6ρMD3NM2πln⁡σg,Mexp-ln⁡D-ln⁡Dg,M22ln2σg,M. The Aitken mode is the smallest size range resolved by MADE3 and consists of particles around 10 nm, the accumulation mode includes aerosols roughly 100 nm in diameter, and the coarse mode is the largest, with particle sizes of ∼ 1 µm (Kaiser et al., 2014). Figure 4 describes the differences across the three mixing states for SO₄. The soluble state (Fig. 4a) consists entirely of soluble species like SO₄, NO₃ or others. Externally mixed or insoluble (Fig. 4b) aerosols contain a non-volatile core (e.g. black carbon or mineral dust) with a total soluble mass fraction θ below 10 % while internally mixed or simply “mixed” (Fig. 4c) aerosols have the same structure, but θ is larger than 10 %.

The MADE3 submodel calculates tracer tendencies for both mass and particle number mixing ratios across these nine modes and nine species. These tendencies encompass several microphysical processes: gas-particle partitioning (subscript “gtp”), condensation (subscript “cond”), nucleation (subscript “nucl”), coagulation (subscript “coag”), particle growth (subscript “gr”) and aging (subscript “ag”). The general form of the governing differential equations for the mass mixing ratios CI,M of an aerosol species I in mode M is therefore given by Eq. (2) (Aquila et al., 2011): 2∂CI,M∂t=RCI,M+∂CI,M∂tgtp+∂CI,M∂tcond+∂CI,M∂tnucl+∂CI,M∂tcoag+∂CI,M∂tgr+∂CI,M∂tag.

The term RCI,M indicates the variation of the aerosol mass mixing ratio because of removal (e.g. dry and wet deposition) and transport phenomena like convection, advection or turbulent mixing processes that are all handled outside of MADE3 by other submodels. These include sedimentation (handled by the SEDI submodel), dry deposition (handled by the DDEP submodel), wet deposition (handled by the SCAV submodel) as well as mixing from atmospheric turbulence (handled in Eulerian representation by the submodel E5VDIFF (Supplement of Emmerichs et al., 2021) and in Lagrangian representation by the submodel LGTMIX (Brinkop and Jöckel, 2019)).

The focus of this study is on aerosols containing SO₄ (I=SO4) and their nine modes, for which the following simplifications may be applied to Eq. (2):

∂CSO4,M∂tgtp=0. The gas-particle partitioning term is neglected for SO₄, as it is assumed that H₂SO₄ has an equilibrium vapor pressure that is low enough so that it is fully transferred from the gas to the aerosol phase at each time step. The reverse conversion is also not possible, i.e., H₂SO₄ cannot re-evaporate (Kaiser et al., 2014). In MADE3, the condensation of H₂SO₄ is fully accounted for by the “cond” term.

The removal processes included in the term RCSO4,M of Eq. (3) are dry deposition, sedimentation and scavenging of aerosols. Dry deposition refers to the removal of atmospheric aerosols through various interactions with surfaces, such as land or sea, and includes mechanisms such as impaction, interception and diffusion (Farmer et al., 2021). Although sedimentation is a subset of dry deposition, there are two notable differences between them in the coding of each process.

Firstly, sedimentation is handled by the SEDI submodel (Kerkweg et al., 2006a) and applies to the entirety of the simulation's vertical domain, whereas dry deposition, handled by the DDEP submodel (Kerkweg et al., 2006a), only applies to the lowermost layer of the model. Secondly, sedimentation solely affects the removal of aerosols and not of gaseous species due to mass considerations.

The dry deposition flux for aerosols is proportional to the dry deposition velocity vd, which can vary according to the surface type: vegetation (subscript “veg”), soil and snow (subscript “slsn”) and water (subscript “wat”). The dry deposition tendency for the SO₄ aerosol is represented by Eq. (7), where parameters β1, β2 and β3 depend on the surface type: 7∂CSO4,M∂tDDEP∝β1⋅vd,veg(SO4)+β2⋅vd,slsn(SO4)+β3⋅vd,wat(SO4)︸vd. The sedimentation tendency (Eq. 8) is in turn proportional to the terminal sedimentation velocity vt, which depends on the Stokes velocity vStokes, the Cunningham slip flow correction factor (fcsf) and the Slinn factor (fs). The latter is used to correct for the larger average sedimentation velocity of a lognormal population of aerosols when compared to the sedimentation velocity of a particle with an average particle diameter: 8∂CSO4,M∂tSEDI∝vStokes(M)⋅fcsf(M)⋅fs(M)︸vt. The expressions for the terms needed to compute both dry deposition and sedimentation velocities in Eqs. (7) and (8), are documented by Kerkweg et al. (2006a).

Wet deposition of SO₄ refers to scavenging because of the precipitation of either ice or liquid water. Aerosol lifetimes are therefore strongly influenced by the interaction between wet and dry deposition processes. Global studies have shown that wet deposition is more important for removing aerosols, but dry deposition is naturally dominant in cloud-free regions where precipitation is unlikely (Farmer et al., 2021). Wet deposition is calculated by the SCAV submodel (Tost et al., 2006) and considers two mechanisms: nucleation scavenging (rainout) and impaction scavenging (washout). The former involves the removal of chemical species from the atmosphere by means of nucleation and subsequent growth of cloud droplets that dissolve them and are then rained out. The latter process refers to the removal of aerosols and gases via their direct collision with raindrops. In a cloud-free region, nucleation scavenging is disregarded and only impaction scavenging is considered. Both mechanisms are contemplated via parameterizations that depend on, e.g., the Brownian motion of aerosols. These are described in further detail by Tost et al. (2006).

The transport of SO₄ in a Lagrangian framework is made possible primarily by three submodels: ATTILA, LGTMIX and TREXP. The Atmospheric Tracer Transport in a LAgrangian (ATTILA; Reithmeier and Sausen, 2002; Brinkop and Jöckel, 2019) submodel is the transport scheme for the Lagrangian air parcels and it resolves their advection according to the EMAC wind field and their convective motion. The LaGrangian Tracer MIXing (LGTMIX; Brinkop and Jöckel, 2019) submodel estimates the mass exchange of different chemical species from the isotropic mixing that occurs from atmospheric turbulence. Lastly, although the Tracer Release Experiments from Point sources (TREXP; Jöckel et al., 2010) submodel is not directly involved in the transport of tracers, it assists ATTILA and LGTMIX by defining the initial emission conditions (i.e. position, time and amount emitted) for point sources.

AIRTRAC v2.0 leverages the same technical infrastructure of its predecessor, introducing new Lagrangian tracers dedicated to SO₄, SO₂ and H₂SO₄ chemistry. Some of these include tracers specific to individual aerosol microphysical processes such as coagulation, enabling the estimation of each process's influence on an aerosol mode. A new subroutine called “airtrac_aerosol_integrate” was created to handle the calculation of aerosol chemistry along the Lagrangian air parcel trajectories. The AIRTRAC control (CTRL) and coupling (CPL) namelists have also been adapted to now allow the user to select the value of “airtrac_mode”, which should be set to “1” when studying SO₄ aerosols and to “2” for the gas-phase analysis mode.

As SO₂ and H₂SO₄ are both precursors of SO₄ and are directly emitted by aircraft, their evolution throughout the course of a simulation must be tracked to properly account for aviation's impact on sulfate. The key gas-phase, tropospheric reaction involving SO₂ that leads to the production of H₂SO₄ in the chemistry mechanism adopted in this study is represented by the net Reaction (R4): R4SO2+OH→H2SO4+HO2. The evolution of aviation SO₂ is simplified and described in AIRTRAC as a pure loss process, where it is oxidized to form H₂SO₄ according to Reaction (R4). While processes such as scavenging contribute to SO₂ removal, these are excluded from the analysis due to the computational constraints arising from the difficulty of storing all the liquid-phase tracers associated with the production of SO₄ from SO₂ (Tost et al., 2006). These removal processes primarily affect liquid-phase species anyway and thus have a smaller impact on the simulated gas-phase SO₂. Consequently, the mixing ratio of aviation-attributable SO₂ (CSO2avi), is based on the initial amount emitted into the atmosphere (CSO2t=0) and the production rate of H₂SO₄ from SO₂ that is provided by the MECCA submodel (PH2SO4), as is illustrated by Eq. (11). Unlike in Eq. (3), the term RCSO2 only contemplates the effects of isotropic turbulent mixing: 11CSO2avi=CSO2(t=0)︸Emission-CSO2|aviCSO2|avi+CSO2|rem︸Tagging ratio×MSO2MH2SO4︸Molar masses×PH2SO4︸Production rate×Δt+R(CSO2)︸Turbulence. In theory, the tagging ratio that scales the aviation-attributable SO₂ in Eq. (11) should consider the amount of OH emanating from both, aviation and other sources, and therefore be expressed as 12×CSO2aviCSO2avi+CSO2rem+COHaviCOHavi+COHrem, corresponding to the case of bimolecular reactions by Grewe et al. (2010) and Grewe (2013). However, this contribution is omitted in Eq. (11), as this study focuses exclusively on the influence of aviation SO₂ and H₂SO₄ emissions on SO₄. Since OH is not directly emitted by aircraft, fully tracking it would at least require other gas-phase aircraft emissions like NO_x and CO and their chemical cycling in the atmosphere with other compounds like HO₂ and O₃ to be completely followed, which is beyond the scope of this assessment. Lastly, as the production rate PH2SO4 is the amount of H₂SO₄ produced from the oxidation of SO₂, it must be converted into the equivalent amount of SO₂ loss by multiplying it by the ratio of molar masses (MSO2MH2SO4).

To track the evolution of gas-phase, aviation-attributable H₂SO₄, the amount produced from the conversion of SO₂ described in Eq. (11) must be considered along with two primary sinks: the binary nucleation of H₂SO₄ with H₂O and the condensation of H₂SO₄ onto pre-existing particles, as is represented in Eq. (12). As a limitation to the approach represented by Eqs. (11) and (12), only the gas-phase formation of H₂SO₄ (produced from the hydrolysis of SO₃ via Reactions R1–R3) is considered in our analysis. The liquid-phase production that involves the oxidation by O₃ and hydrogen peroxide (H₂O₂) of H₂SO₃, which in turn is formed by the hydrolysis of SO₂ (Sheng et al., 2018; Shostak et al., 2019), is excluded. While this pathway is the dominant source of sulfuric acid (Textor et al., 2006), incorporating it is challenging, due to the cloud evaporation assumption described by Tost et al. (2006), which considers that clouds and aqueous-phase species are fully evaporated at the end of each time step. This assumption is used to avoid the high computation and memory costs that would otherwise be required to advect both gas- and aqueous-phase tracer species. The term RCH2SO4 therefore again only represents isotropic turbulent mixing. 12CH2SO4avi=CH2SO4(t=0)︸Emission+CSO2|aviCSO2|avi+CSO2|rem×PH2SO4×Δt︸OxidationofSO2+……-CH2SO4|aviCH2SO4|avi+CH2SO4|rem×∂CH2SO4∂t|nuclavi×Δt︸NucleationofH2SO4+……-CH2SO4|aviCH2SO4|avi+CH2SO4|rem×∂CH2SO4∂t|condavi×Δt︸CondensationofH2SO4+R(CH2SO4)︸Turbulence

The contribution from aviation to the formation of the soluble Aitken mode of SO₄ via the binary nucleation of H₂SO₄-H₂O is estimated by applying a tagging ratio to the MADE3 nucleation tendency (Eq. 4). This tagging ratio in Eq. (13) naturally involves H₂SO₄, given that sulfuric acid from aviation (CH2SO4avi) drives the nucleation process. 13∂CSO4,ks∂tnuclavi=CH2SO4aviCH2SO4avi+CH2SO4rem×∂CSO4,ks∂tnucl Aviation's contribution to SO₄ via the condensation of H₂SO₄ is estimated in a fashion similar to Eq. (13), as the tagging ratio is the same (Eq. 14) due to the central role of H₂SO₄ also in this process. The main difference is that the condensation process may contribute to increasing any one of the nine SO₄ aerosol modes. 14∂CSO4,M∂tcondavi=CH2SO4aviCH2SO4avi+CH2SO4rem×∂CSO4,M∂tcond

We first derive an expression for aviation's contribution to the soluble Aitken mode (M=ks) of SO₄ via coagulation by applying the tagging formulation in Eq. (9). For clarity and conciseness in representing the full coagulation equation (Eq. 6), vector notation is introduced for the state variables x according to Eq. (15) for all nine aerosol modes, where x1, for instance, is the mass concentration of mode “ks”, i.e., x1≡CSO4,ks. Each state variable xi represents the sum of contributions from all sources, in this study the sources are j=avi,rem.

15x=x1x2x3x4x5x6x7x8x9=x1avi+x1remx2avi+x2remx3avi+x3remx4avi+x4remx5avi+x5remx6avi+x6remx7avi+x7remx8avi+x8remx9avi+x9rem=CSO4,ksCSO4,kmCSO4,kiCSO4,asCSO4,amCSO4,aiCSO4,csCSO4,cmCSO4,ci The coagulation kernels are also more compactly rewritten (Eqs. 16a and 16b), where p and q are the modes of the colliding particles. Additionally, the summation in the denominator of Eq. (6) for an aerosol in mode p with a diameter D1 will be reformulated more succinctly as ∑s=1A=9Cs,p=x1+Cp, where Cp=CNH4,p+CNO3,p+CNa,p+CCl,p+CPOM,p+CBC,p+CDU,p+CH2O,p. We also note that fp,q=fq,p′.

16afp,q=∫0∞∫0∞D13βD1,D2npD1nqD2dD1dD216bfp,q′=∫0∞∫0∞D23βD1,D2npD1nqD2dD1dD2 When evaluating the double summations of Eq. (6) for the case M=ks, it is worth noting that for intermodal coagulation, i.e., p≠q, the value of τpq≠ks. In other words, when two particles from different modes collide, the destination mode will never be the soluble Aitken mode (Table 2 from Kaiser et al., 2014). It follows that since τpq≠ks, the Kronecker deltas depending on τpq will be zero: δks,τpq=0. In contrast, the Kronecker deltas δks,p and δks,q evaluate to “1” whenever either mode p or q is also ks. We note that if p=q, the coagulation tendency is zero, as intramodal collisions do not affect mass concentrations. It is also essential to recall that the destination mode of the coagulated particle depends on the soluble mass fraction θ of the final particle. This dependency implies that the coagulation tendency for each of the nine modes will be a piecewise function (F1) conditioned by θ. Combining Eqs. (6) and (16) leads to a more compact formulation of the coagulation tendency for mode ks (Eq. 17). 17∂x1∂tcoag=F1x=-Ax1x1+C1;θ=1Soluble-A′x1x1+C1;0.1≤θ<1Mixed-A′′x1x1+C1;0≤θ<0.1Insoluble The coefficients A, A′ and A′′ are defined as follows, where the subscripts refer to the colliding modes (see Eq. 15 for details): A=ρ1π6f1,4+f1,7A′=ρ1π6f1,3+f1,5+f1,6+f1,8+f2,1′A′′=ρ1π6f1,3+f1,6+f1,9. The negative sign present across all cases of θ in Eq. (17) reflects that any intermodal coagulation event involving a particle in the ks mode will always result in its conversion to other aerosol modes. Applying Eq. (9) to Eq. (17) results in the following expression for the contribution of aviation to mode ks via coagulation: ∂x1avi∂tcoag=F1xxjT∇F1xxT∇F1x. Evaluating the sensitivity fraction leads to the final expression for ∂x1avi∂tcoag, where K=A,A′,A′′: 18∂x1avi∂tcoag=F1xx1avi,x2avi,x3avi,…,x9avi-KC1x1+C12,0,0,…,0Tx1,x2,x3,…,x9-KC1x1+C12,0,0,…,0T=F1xx1avix1. Based on the result from Eq. (18), the tagging ratio x1avix1 for the coagulation tendency of SO₄ in mode ks according to Grewe (2013) is the ratio of soluble Aitken mode sulfate, CSO4,ksavi, to the total amount of this aerosol across all other sources, i.e., x1avix1=CSO4,ksaviCSO4,ksavi+CSO4,ksrem. Typically, the form of this ratio varies with θ, but for mode ks, the same structure is obtained (only K varies). This result is verified analytically as Eq. (10) is upheld. The derivation for the remaining eight aerosol modes is included in Appendix C.

We highlight that this derivation considers the following simplifying assumptions:

Coagulation kernels β and terms Cp are independent of the state variables xi.

The removal of aviation-produced SO4 through the three processes described in Sect. 2.4 (dry deposition, sedimentation, and scavenging) is scaled by the aviation-attributable SO4 tagging ratio. This reflects the fact that removal processes are directly proportional to the pollutant to be removed and is consistent with the method of IW17. In other words, the removal rate should be linearly proportional to the amount of the aerosol present in the atmosphere as the removal tendency should be approximately zero once the aviation-induced aerosols are nearly depleted. The dry deposition (DDEP), sedimentation (SEDI) and scavenging (SCAV) tendencies for aviation aerosols are therefore defined by Eq. (21): 21∂CSO4,M∂tiavi=CSO4aviCSO4avi+CSO4rem×∂CSO4,M∂ti;i=DDEP,SEDI,SCAV.

By combining the tagging formulations for all aerosol microphysical and removal processes, we provide a consolidated overview of the non-linear tagging differential equations as a function of process tendencies to track the transport of aviation SO₄ across all nine modes. Equation (22) applies only to mode ks, while Eq. (23) to the remaining modes. The isotropic turbulent mixing parameterization from LGTMIX is represented by RCSO4,M. 22∂CSO4,ks∂tavi=CH2SO4aviCH2SO4avi+CH2SO4rem×∂CSO4,ks∂t|nucl︸Nucleation+∂CSO4,ks∂t|cond︸Condensation+∂CSO4,ks∂t|coag︸Coagulation+……+∂CSO4,ks∂t|rename︸Growth & Aging-CSO4aviCSO4|avi+CSO4|rem×∂CSO4,ks∂t|DDEP︸Dry Deposition+∂CSO4,ks∂t|SEDI︸Sedimentation+∂CSO4,ks∂t|SCAV︸Scavenging+R(CSO4,ks)︸Turbulence.

Lastly, for the remaining modes M=km,ki,as,am,ai,cs,cm,ci: 23∂CSO4,M∂tavi=CH2SO4aviCH2SO4avi+CH2SO4rem×∂CSO4,M∂t|cond︸Condensation+∂CSO4,M∂t|coag︸Coagulation+∂CSO4,M∂t|rename︸Growth & Aging+……-CSO4|aviCSO4|avi+CSO4|rem×∂CSO4,M∂t|DDEP︸Dry Deposition+∂CSO4,M∂t|SEDI︸Sedimentation+∂CSO4,M∂t|SCAV︸Scavenging+RCSO4,M︸Turbulence.

As it would be much more computationally demanding to develop a complete chemistry mechanism within every air parcel defined in a simulation, AIRTRAC leverages the background chemistry from the EMAC model by linearly scaling these non-linear background responses with the emission amount. Although this approach introduces simplifications like discarding feedback effects between the emission and background, the computational requirements are decreased by at least one to two orders of magnitude (Maruhashi et al., 2024). To linearize Eqs. (22) and (23), the tagging ratios CH2SO4aviCH2SO4avi+CH2SO4rem and CSO4aviCSO4avi+CSO4rem themselves must be linearized. As they are comparable to the rational function fx=xx+k, where k is a real number, the tagging ratios are approximated with a first-order MacLaurin polynomial: xx+k≈xk⟹CH2SO4aviCH2SO4avi+CH2SO4rem≈CH2SO4aviCH2SO4rem and CSO4aviCSO4avi+CSO4rem≈CSO4aviCSO4rem. We note that this linearization is also applied to Eqs. (11) and (12) regarding the aviation-attributable SO₂ and H₂SO₄, i.e., the tagging ratio involving SO₂ is also approximated: CSO2aviCSO2avi+CSO2rem≈CSO2aviCSO2rem. The quality of this approximation rests with ensuring that the ratios |xk|≡|CjaviCjrem|j=SO2,H2SO4,SO4<1. In Figs. S7–S9 of the Supplement, we observe that this condition is upheld for the January–March simulation period for the complete EMAC grid and all 28 emission points, while Figs. S10–S12 also demonstrate that |xk|<1 holds for the July–September period.

The linearized tagging equation implemented in AIRTRAC v2.0 for mode M=ks is therefore represented by Eq. (24): 24∂CSO4,ks∂tavi=CH2SO4aviCH2SO4rem×∂CSO4,ks∂t|nucl︸Nucleation+∂CSO4,ks∂t|cond︸Condensation+∂CSO4,ks∂t|coag︸Coagulation+∂CSO4,ks∂t|rename︸Growth & Aging+……-CSO4aviCSO4rem×∂CSO4,ks∂t|DDEP︸Dry Deposition+∂CSO4,ks∂t|SEDI︸Sedimentation+∂CSO4,ks∂t|SCAV︸Scavenging+R(CSO4,ks)︸Turbulence.

For the remaining modes M=km,ki,as,am,ai,cs,cm,ci, the tagging equations are given by Eq. (25): 25∂CSO4,M∂tavi=CH2SO4aviCH2SO4rem×∂CSO4,M∂t|cond︸Condensation+∂CSO4,M∂t|coag︸Coagulation+∂CSO4,M∂t|rename︸Growth & Aging+……-CSO4aviCSO4rem×∂CSO4,M∂t|DDEP︸Dry Deposition+∂CSO4,M∂t|SEDI︸Sedimentation+∂CSO4,M∂t|SCAV︸Scavenging+R(CSO4,M)︸Turbulence.

Figure 5 illustrates the differing transport pathways of SO₂ emissions originating from emission points 8 and 10, situated in the North and South Atlantic regions, respectively (see Fig. 2). Analogously, Fig. 6 presents the same analysis for H₂SO₄ for the same emission points. Both figures pertain to a 90 d period encompassing the northern summer (July–September 2015). We have selected these emission points because they exhibit distinct vertical transport behavior: SO₂ emitted at emission point 10 remains entirely within the troposphere, whereas SO₂ emitted at emission point 8 is partially injected above the tropopause and into the lower stratosphere.

The amount of sulfate attributable to aviation emissions of SO₂ and H₂SO₄ is calculated according to Eqs. (24) and (25). Figure 7 displays the spatio-temporal evolution of total sulfate, which is defined as the sum of SO₄ across the nine aerosol modes mentioned in Sect. 2.3: SO4=∑i=19SO4,i. On average, SO₄ production at emission point 10 (Fig. 7b) is nearly three times larger than at emission point 8 (Fig. 7a). This large discrepancy is owed to the significantly larger amount of H₂SO₄ that results from point 10, which is converted to SO₄ via processes like nucleation and condensation. Both Fig. 7a and b further exemplify this, as SO₄ production is initiated when H₂SO₄ production is maximal, which occurs approximately at the 30 d mark for point 8 (Fig. 6a) and at the 20 d mark for point 10 (Fig. 6b). As there is an approximate 20 d delay from the emission of SO₂ and H₂SO₄ until the point during which SO₄ production occurs, sulfate aerosols can form in regions far from the initial emission points, as was seen with H₂SO₄ in Fig. 6. The sulfate lifetimes approximate to 67 and 64 d for points 8 and 10, respectively. Unlike precursor species such as SO₂, secondary sulfate exhibits significantly longer atmospheric lifetimes – ranging from several days to weeks in the troposphere (Textor et al., 2006; Boucher, 2015; Toohey et al., 2025), and extending to several months or even years in the stratosphere (Myhre et al., 2013; Sun et al., 2024; Toohey et al., 2025). The lifetimes estimated in this study are comparable to the upper end of the tropospheric range, likely due to the consideration of pulse emissions in the UTLS and fall well within the expected stratospheric range. A more detailed discussion of these calculations is provided in Sect. 4.3.

Regarding the spatial distribution of SO₄, flying at point 8 on that specific day is likely to lead to the strongest sulfate production in the Northern Hemisphere, between the latitudes of 30 and 60° N (Fig. 7c). The vertical SO₄ profiles for all emission points and both simulation periods may be consulted in Figs. S16 and S19 in the Supplement. According to the corresponding horizontal distribution panel (Fig. 7e), several regions beyond the local North Atlantic emission area are affected, including parts of Europe and Asia. For point 10, a wider latitudinal band is impacted, with the largest impacts noted between 0 and 60° S (Fig. 7d). The Pacific region close to the Equator is likely to experience the largest SO₄ production (Fig. 7f). The Brewer-Dobson Circulation (BDC) patterns might also impact the vertical transport of sulfate. As has been noted by Sun et al. (2023), particles injected in the Tropics at around the upper troposphere and lower stratosphere closer to the Equator will generally experience upwelling and may be transported higher into the stratosphere via the deep branch of the BDC. This phenomenon is likely observed in Fig. 7c and d, where air parcels rise above the emission point (green triangle) and reach the highest altitudes near 0° N. The highest point reached for an emission starting at point 8 is 103 hPa (∼ 16.1 km) and for point 10 is 94 hPa (∼ 16.7 km). Additionally, as emission point 10 is closer to the Equator, more sulfate aerosols are lofted into the stratosphere, which enhances sulfate VMRs at higher altitudes and may even prolong the overall lifetime of aerosols if they remain in the stratosphere.

The production of sulfate aerosols from aviation exhibits a strong seasonal dependence driven by the variability in background chemistry and solar radiation. Seasonal changes in the background water vapor and HO_x levels may influence the oxidation pathways that convert SO₂ into SO₄. It is also worth noting that years 2015 and 2016 were characterized by an extreme El Niño event (Paek et al., 2017). Our analysis is based on a single simulation year, so the differences across both simulation periods shown in Fig. 8 may reflect both seasonal variability and interannual anomalies, rather than the climatological seasonal cycle alone. We therefore interpret the January to March and July to September periods as contrasting background states in different seasons to assess model sensitivity, not to quantify the climatological seasonal cycle.

Among the relevant background variables, OH is particularly important. The SO₂ lifetime depends on the OH radical, which drives its gas-phase oxidation. OH is prevalent in warm and humid locations as its formation relies on the photodissociation of O₃ and subsequent reaction with H₂O (Riedel and Lassey, 2008). Figure 8a depicts the range of SO₂ e-folding times for both periods. The median e-folding time of 22 d in January (winter) is nearly 60 % larger than the median of 14 d in July (summer), which is expected, given the increased solar radiation during summer that contributes to increased OH and faster SO₂ depletion rates. Consequently, this enhanced oxidation of SO₂ results in larger SO₄ production, as shown in Fig. 8b, where its median production efficiency is 144 % larger in July compared to January. The summertime SO₂ median lifetime of 14 d is within the upper limit from past studies (Von Glasow et al., 2009), while our wintertime estimate of 22 d is slightly overestimated. This may be explained by our simplification of not accounting for the rapid aqueous-phase oxidation in the troposphere and scavenging processes, which act as significant SO₂ sinks. However, our wintertime range of SO₂ e-folding estimates agrees with results of the modeling study by Zhu et al. (2022) and with the observations from the 2022 Hunga Eruption from the Infrared Atmospheric Sounding Interferometer (IASI) satellite instrument that also estimated a similar upper limit lifetime of 21.4 d (Sellitto et al., 2024). Due to the skewed nature of the distributions shown in Fig. 8, a non-parametric statistical approach is more appropriate than parametric alternatives that assume normality. Therefore, the Mann-Whitney U test (Mann and Whitney, 1947) was applied to assess the statistical significance of differences across the two seasons. Based on this test, both the SO₂ lifetime (p-value =3.57×10-4) and the SO₄ mean production efficiencies (p-value =6.83×10-3) are found to be statistically significantly larger at the 95 % confidence level during the winter and summer seasons, respectively.

The lifetime of sulfate (Fig. 8c) has a median close to 2 months, with a value of around 62 d in January and 65 d in July. Unlike SO₂, the median SO₄ lifetimes are not statistically significantly different across summer and winter according to the same test statistic and confidence interval mentioned earlier (p-value = 0.55). It is worth noting, however, that the maximum SO₄ lifetime during winter of 123.5 d is almost 30 % larger than the maximum in summer (96.7 d). This causes the mean SO₄ lifetime to be larger in January (69 d) than in July (67 d). The sulfate time series plots for all 28 emissions points for the periods January–March and July–September 2015 from which the e-folding times in Fig. 8 were calculated are shown in Figs. S5 and S6 in the Supplement (Maruhashi et al., 2025b).

Our range of calculated SO₄ lifetimes is consistent with output from another Lagrangian passive tracer pulse experiment from Toohey et al. (2025) that employs the FLEXPART model. In their analysis, stratospheric sulfate aerosol transport and lifetimes were analyzed in the context of volcanic SO₂ eruptions in the Northern Hemisphere. The altitudinal range of tracer injections in December and June that they considered varies from 13–25 km, where SO₄ aerosol lifetimes increased sharply with altitude. Since we emit SO₂ pulses at the UTLS that lead to sulfate maxima reaching around 100 hPa (∼ 16 km) according to Fig. 7c and d, our results will also be comparable to theirs. According to their Fig. 4, sulfate aerosols at an altitude of 13 km between 40 and 60° N exhibit lifetimes ranging from 1.9 to 3.9 months during June (summer) and from 3.4 to 4 months during December (winter). Our Fig. 8c displays a median value of 62 d or ∼ 2.1 months for January and ∼ 2.2 months for July. Our estimates are expectedly lower than their estimated range given our lower emission altitude. Furthermore, seeing as there is a portion of sulfate that reaches the stratosphere at around 16 km in altitude (e.g. Fig. 7), upper bound values in Fig. 8c such as 96.7 d (∼ 3.2 months) in July and 123.5 d (∼ 4.1 months) in January are also justified bearing in mind that, Sun et al. (2023), for instance, for an injection height of 16 km, found particle lifetimes throughout the year that ranged from 2.4 to 9.6 months, which are similar to both of our seasonal estimates from Fig. 8c. Sun et al. (2024) inferred a shorter lower-stratospheric aerosol lifetime of 4.8 months at 65 hPa (∼ 18.5 km). By contrast, particle lifetimes according to Toohey et al. (2025) for an injection height of 15 km were longer, ranging from 6.1–7.5 months. Differences between estimates from AIRTRAC and other models like LAGRANTO are, however, naturally expected as the latter does not consider aerosol microphysical processes like particle growth along trajectories.

To demonstrate the capability of AIRTRAC v2.0 in predicting when aviation-attributable sulfate aerosols will likely interact with low-level liquid clouds, we incorporate satellite data from ESA's Climate Change Initiative (CCI) to identify the approximate locations of these clouds. More specifically, version 3 of the Advanced Very High Resolution Radiometer ante meridiem dataset (AVHRR-AMv3; Stengel et al., 2019) at monthly means is used, which covers the period from 1982 to 2016. This comprehensive dataset comprises 174 variables with a latitude-longitude grid resolution of 0.5° × 0.5°. To pinpoint the location of liquid clouds during the simulated periods in summer and winter, the liquid cloud fraction (LCF) and cloud top pressure (CTP) variables were analyzed.

Based on Fig. D1 from Appendix D, liquid clouds are predominantly found at CTPs below 800 hPa and are primarily concentrated in the Northern Atlantic, Northern Pacific and Southern Tropics, with a notably larger mean LCF observed during the summer months (July–September 2015). During this period, liquid clouds frequently form in the Southern Tropical Belt, between 0–30° S across all longitudes. These horizontal liquid cloud distributions are consistent with those reported by Stengel et al. (2020) for June 2014 and with those from Muhlbauer et al. (2014), who used CloudSat/CALIPSO data from 2006 to 2011. Similarly, Lauer et al. (2007), using annual mean satellite data from the International Satellite Cloud Climatology Project (ISCCP) for the period 1983–2004, confirm a significant presence of low-level liquid clouds in the North Atlantic, North Pacific and South Atlantic regions.

Based on observations from the AVHRR-AMv3 cloud satellite dataset, one of the main regions with predominant low-level liquid cloud formation between July–September 2015 is marked with a blue rectangle in Fig. 9. When emissions occur near the Northern Mid-latitudes during summer (Fig. 9a), SO₄ may have an extended residence time near the cruise altitude, resulting in a slower downward transport to lower altitudes. The dark blue median trajectory shows that air parcels following this path remain close to the emission pressure altitude of approximately 240 hPa for around 8 d before descending towards the surface, eventually reaching latitudes between 30–60° N. Although it is unlikely that SO₄ from emission point 8 will reach the Southern Tropical Belt with liquid clouds indicated by the blue rectangle, it could still interact with some liquid clouds present in the North Atlantic and in the North Pacific (see Fig. D1a and b). In contrast, emissions at point 10 in the Southern Tropics (Fig. 9b) present an increased likelihood for interactions between aviation sulfate and liquid clouds in the Southern Tropical Belt.

AIRTRAC v2.0 is the first Lagrangian sulfate tagging scheme within EMAC, providing the unique capability to track the long-range atmospheric evolution of aviation-emitted sulfur compounds. Starting from the emission of SO₂ and H₂SO₄ at subsonic cruise altitudes, the scheme follows their chemical transformation into SO₄ followed by the subsequent descent of these aerosols into the lower troposphere. By considering aerosol microphysical processes like nucleation, condensation, coagulation, aging and particle growth, AIRTRAC v2.0 can calculate the VMRs across nine sulfate aerosol modes. It is therefore a useful computational tool that may help predict which flight regions are most likely to lead to aerosol-cloud interactions. The quantification of these interactions and how they impact the Earth's radiative budget (i.e. in terms of radiative forcing) is a key next step.

In terms of limitations, the linearization of the production and loss tendencies from MADE3 in AIRTRAC discards climate feedback effects that could be introduced by the SO₂ and H₂SO₄ emissions. This means that locally, for instance, the atmosphere's oxidative capacity will not vary with the magnitude of the pulse emissions, meaning that background OH levels will remain unaffected by the amount of SO₂ introduced by AIRTRAC. The implications of this linearization have been analyzed in greater detail in the context of aviation NO_x-O₃ interactions by Maruhashi et al. (2024). As has been introduced in Sect. 5, the suitability of air parcel mixing parameters that were originally formulated mostly for gas-phase species still needs to be better understood. As smaller aerosols in the Aitken mode are likely to behave more closely to gaseous species when mixing and larger particles in the coarse mode will not mix as rapidly given their much larger momenta, the introduction of size-dependent mixing parameters should be considered.

Another limitation of AIRTRAC v2.0 is that it cannot track sulfate particles that have been scavenged and later re-evaporated back into the atmosphere, which may lead to underestimations of aerosol mixing ratios. Another factor potentially contributing again to the underestimation of sulfate is the exclusion of its aqueous-phase production pathway via SO₂ oxidation by O₃ and H₂O₂, due to the computational limitation discussed by Tost et al. (2006). This not only has implications for sulfate produced, but will also tend to overestimate SO₂ estimated lifetimes (Brodowsky et al., 2021). Longer SO₂ lifetimes may then also lead to longer sulfate lifetimes. Excluding plume-scale processes can lead to an overestimation of around 15 % of the aviation-induced sulfate particle number concentration, but has a negligible impact on sulfate mass when compared to an instant dispersion approach, which is adopted in AIRTRAC v2.0 (Sharma et al., 2025).

A few final considerations to further extend the capabilities of AIRTRAC v2.0 would be to firstly include the indirect impact of aviation NO_x emissions on sulfate via the production of OH by connecting its gas-phase capabilities (version 1.0) to its new aerosol capabilities. AIRTRAC could then provide a more comprehensive assessment of climate effects from aviation NO_x, which will remain relevant even for drop-in sustainable aviation fuels and emerging hydrogen-powered aircraft (Tiwari et al., 2024). This could be contemplated by updating the tagging ratio of Eq. (11) based on the bimolecular reaction formulation of the tagging equation, as was described earlier in Sect. 3.2: 26CSO2avi=CSO2(t=0)︸Emission-12×CSO2|aviCSO2|avi+CSO2|rem+COH|aviCOH|avi+COH|rem︸Revised tagging ratio×MSO2MH2SO4︸Molar masses×PH2SO4︸Production rate×Δt+R(CSO2)︸Turbulence. Secondly, to model the influence of aviation-induced sulfate on the microphysical properties of liquid clouds (e.g. by estimating the aerosol activation into cloud condensation nuclei), it would be necessary to extend the tagging from AIRTRAC v2.0 to also include the aerosol particle number concentration, as this is a key quantity driving aerosol-cloud interactions.

Lastly, AIRTRAC has been previously used as the computational foundation for the climate change functions (CCFs), which associate a change in temperature resulting from a local emission (Grewe et al., 2014a). Thus far, only CCFs for the net NO_x effect, H₂O and contrails have been formulated for a limited set of emission points in the Northern Trans-Atlantic (Frömming et al., 2021). By running AIRTRAC v2.0, we have generated a dataset to analyze the transport of aviation-induced SO₂, H₂SO₄ and nine aerosol modes containing SO₄. This dataset may later serve as the basis for the development of the first aerosol-cloud interaction CCFs, thereby furthering the possibility of applying them to climate-optimal routing (Sausen et al., 1994; Grewe et al., 2014a).

This paper presents the new functionalities and technical implementation of the AIRTRAC v2.0 submodel within the EMAC modeling framework. It applies a Lagrangian tagging approach to estimate the contributions of aviation-induced SO₂ and H₂SO₄ emissions to nine aerosol modes of atmospheric sulfate. To showcase AIRTRAC's capabilities in characterizing the transport patterns of aviation sulfur compounds, two three-month simulation scenarios (across winter and summer) were performed based on the CMIP6 emissions inventory. Furthermore, ESA satellite data were incorporated to demonstrate how AIRTRAC can help predict where aviation-induced sulfate will reach lower-level liquid clouds.

The transport patterns of SO₂, H₂SO₄ and SO₄ are primarily driven by the atmospheric circulation, where trade winds and downdrafts from, e.g., subsidence events play a particularly important role. Emission latitude also influences the vertical transport pathway. In the tropics, emissions typically occur below the climatological tropopause and thus within the troposphere, where pollutants are more rapidly removed and are preferentially transported downward. At higher latitudes, emissions are more likely to be released above the tropopause, entering the stratosphere, where they are more stable and persist for longer. It was found, for instance, that SO₂ that is quickly transported to lower altitudes has a shorter atmospheric lifetime due to locally larger background concentrations of OH, which are responsible for oxidizing SO₂ and other chemical species. Furthermore, SO₂ that is transported to lower altitudes early, may lead to the largest sulfate production efficiency given the increased oxidative potential at lower tropospheric altitudes. This is consistent with past studies that have focused on NO_x (Rosanka et al., 2020). Additionally, the BDC is also influential in determining the amount of sulfate that is lofted into the stratosphere, where its lifetime is longer. For example, species emitted from points closer to the Equator may be transported higher through the upwelling of the BDC in the tropical region over time, which according to our results can reach pressure altitudes above 100 hPa. This phenomenon was also observed in another Lagrangian modeling study (Sun et al., 2023).

The SO₂ lifetime and production efficiency of SO₄ also varied across seasons in a statistically significant manner. The median lifetime of SO₂ during winter (January–March 2015) was approximately 22 d compared to 14 d during the summer (July–September 2015). The production efficiencies of SO₄ also varied intra-annually, due in part, to the dependence of OH concentrations on solar radiation: summer brings increased sunlight that enhances OH production, thereby boosting SO₄ formation. Overall, there was a 40 % reduction of SO₂ lifetimes in summer relative to winter and the SO₄ production efficiency in summer was 144 % larger, these differences were found to be statistically significant at a 95 % confidence level. Although median sulfate lifetimes were found to be larger in summer, by almost 4 %, the maximum SO₄ lifetime during winter was approximately 30 % larger than in summer. The differences in SO₄ lifetimes were not found to be statistically significant.

We also present a case study illustrating how AIRTRAC v2.0 can be used to assess whether SO₂ emitted at cruise altitude is transported downward to levels more conducive to liquid-cloud formation, as indicated by ESA satellite cloud data. A clear difference was found between two emission scenarios: emitting along the North Atlantic Flight Corridor is unlikely to lead to aerosol interactions with the predominant liquid cloud cover in the Southern Tropical Belt, whereas flying at typical subsonic cruise pressure altitudes (∼ 240 hPa) in the Tropics can lead to a greater likelihood of interactions with these clouds.

AIRTRAC v2.0 relies on the aerosol tendencies from the MADE3 submodel, which have been evaluated extensively against satellite observations, in-situ aircraft measurements and ground-based station data. We have evaluated AIRTRAC v2.0 output itself, however, with a combination of other Lagrangian modeling studies and with our results from a perturbation-based modeling approach. For example, lifetime estimates of SO₂ and SO₄ have been compared with both modeling and observational studies of sulfate formation from volcanic eruptions, which can likewise be represented as a pulse emission of SO₂ at an above-surface injection altitude, followed by the production of SO4 and its transport into the stratosphere. Given that some of our sulfate emissions also reach the stratosphere at similar injection heights tested by these studies, we have used them as reference. Our median SO₂ lifetimes agree mostly well with these past estimates ranging between a couple of days to three weeks. Our median SO₄ lifetimes of 2.1 and 2.2 months in January and July, respectively, likewise agree. To assess the spatial distribution patterns of SO₄ predicted by AIRTRAC v2.0, we performed additional perturbation simulations with a comparable setup. Results showed that both approaches are consistent in identifying a similar latitudinal band in which sulfate VMR maxima are found in the lower troposphere. The near-surface estimates differ the most, where such discrepancies are likely owed to the limitations of a Lagrangian approach in having fewer air parcels near the surface (∼ 0.25–0.5 per grid box) compared to higher parts of the atmosphere, where around 12–24 times as many air parcels can exist per grid box.

Overall, AIRTRAC v2.0 is a promising tool that estimates lifetimes of sulfur-based species that drive aerosol-cloud interactions. Additionally, it can map the spatial distribution patterns to predict the emission points that are more likely to lead to impacts on lower-level liquid clouds. It is also highly efficient, capable of calculating up to around 28 emission scenarios in a single simulation, depending on the characteristics of the high-performance cluster.