The mixing state index χ quantifies where an aerosol population lies on the continuum from external to internal mixing – that is, how spread out the chemical species are over an aerosol population. We will focus here on the mixing state of submicron aerosols (PM1.0) due to their relevance for light scattering and absorption and their contribution to CCN formation .

To summarize, the mixing state index χ is given by the affine ratio of the average particle species diversity, Dα, and bulk population species diversity, Dγ, as 1χ=Dα-1Dγ-1.

The diversities Dα and Dγ are calculated as follows. First, the per-particle mixing entropies Hi are determined for each particle by 2Hi=∑a=1A-pialn⁡pia. Here, A is the number of distinct aerosol species and pia is the mass fraction of species a in particle i. These values are then averaged (mass-weighted) over the entire population to obtain the average particle species diversity Dα by 3Hα=∑i=1NppiHi,4Dα=eHα, where Np is the total number of particles in the population and pi is the mass fraction of particle i in the population. Finally, the bulk diversity Dγ is calculated as 5Hγ=∑a=1A-paln⁡pa,6Dγ=eHγ, where pa is the bulk mass fraction of species a in the population.

Note that the definition of “species” for calculating χ is based on application needs. It can be based on operationally defined chemical species , elemental composition , or species groups such as volatile and nonvolatile species or hygroscopic and non-hygroscopic species . Other possibilities include the propensity for aerosols to undergo heterogeneous reactions, quantified by the heterogeneous reaction rate coefficient for a specific reaction. In this paper we consider three different definitions of χ, which we explain in more detail in Sect. .

The framework laid out in Sect.  can be easily generalized to a modal modeling framework (see Fig. ). The bulk mixing entropy, Hγ, and the bulk diversity, Dγ, can be calculated using the bulk mass fractions, pa, of species a from the MAM4 simulation and Eqs. () and (). To calculate the average particle mixing entropy, Hα, and the average particle species diversity, Dα, we use 7Hm=∑a=1A-pmaln⁡pma,8Hα=∑m=1MpmHm,9Dα=eHα, where pma is the mass fraction of species a in mode m, pm is the mass fraction of mode m in the population, and Hm represents the per-mode mixing entropies. Finally, the mixing state index, χ, can be calculated using Eq. (). Note that Eqs. () and () are analogous to Eqs. () and (). A detailed derivation of these equations is provided in the Appendix .

In this study, we consider the mixing states of submicron aerosols including the Aitken, accumulation, and primary carbon modes, and we do not include the coarse mode because the coarse particles are above 1 µm. Since the mixing entropies are mass-weighted (rather than number-weighted), the mixing state index is more representative of the modes with the larger particles, i.e., the accumulation and primary carbon modes.

Here we compare and contrast the aerosol mixing state indices defined in three different ways, namely based on the mixing of optically absorbing and non-absorbing species (χo), based on the mixing of primary carbonaceous and non-primary carbonaceous species (χc), and based on the mixing of hygroscopic and non-hygroscopic species (χh). Table  shows the definitions of these aerosol mixing state indices.

For χo, we considered two surrogate species: black carbon (strongly absorbing, assigned a mass absorption coefficient in CESM2 at 533 nm and 0 % RH of 8.144 m2 g-1) and the five other aerosol species grouped together (less absorbing or non-absorbing, with mass absorption coefficients in CESM2 at 533 nm and 0 % RH of 0.1442, 9.975×10-2, 4.703×10-2, 2×10-6, and 5×10-7 m2 g-1 for POM, SOA, dust, sea salt, and sulfate, respectively). Thus, a lower value in χo refers to the case where the strongly absorbing species black carbon and the sum of the other species (termed “non-absorbing” here for convenience) are more externally mixed.

The index χc is motivated by the primary carbon treatment of MAM4, where the primary particulate organic matter and black carbon are assigned to a separate primary carbon mode . A lower value in χc refers to the situation where the primary carbonaceous species and all other species exist separately in different particles.

Similarly, χh was also calculated from two surrogate species. We combined black carbon, primary organic matter, and dust as one surrogate species, given their comparatively lower hygroscopicities (kappa values of ∼0, ∼0, and 0.068, respectively). Accordingly, NaCl (1.16), SOA (0.14), and sulfate (0.507) were grouped as the other surrogate species. Here, a lower value in χh represents the case where hygroscopic and non-hygroscopic species tend to be present in separate particles.

Let χS,tML and χS,tMAM4 denote the mixing state indices computed by the ML model and by the MAM4 model for each grid cell at timestamp t, respectively. The corresponding time-averaged values for a certain time interval and for each grid cell are χSML‾ and χSMAM4‾. Here we consider the full year as the time-averaging interval. An analysis of the seasonal variation of mixing state indices can be found in .

To compare the annual mean values, we calculated the mean difference (ΔχS‾) and the mean absolute difference (|ΔχS|‾) for each grid cell of the layer closest to the surface: 11ΔχS‾=1T∑t=1T(χS,tMAM4-χS,tML)=χSMAM4‾-χSML‾,12|ΔχS|‾=1T∑t=1T(|χS,tMAM4-χS,tML|), where the subscript S refers to the mixing state index (o, c, or h), and the total number of timestamps is T=2920. Since it only makes sense to quantify mixing state when at least two species are present in a given location, areas where the mass fraction of any one surrogate species was higher than 99 % for χo (due to the low mass fraction of black carbon) and 97.5 % for χc and χh were ignored for the calculation and appear as hatched areas in Fig. . We will first discuss the overall probability density functions of these quantities (Fig. ) and then their spatial distributions (Fig. ).

Figure  shows the probability density functions of the annual averaged mixing state indices computed by the ML model (χSML‾), by MAM4 (χSMAM4‾), their average difference (ΔχS‾), and their average absolute difference (|ΔχS|‾) for each surface-layer grid cell. The results show large discrepancies in mixing state indices between the ML model and the MAM4 model, without a clear relationship between them (see Fig. d–f).

The annual average of the mixing state index χo estimated by the ML model, χoML‾, ranged between 55 % and 96 %, with a mean of 73 %. Calculated by the MAM4 model, χoMAM4‾ varied spatially from 46 % to 99.76 %, with a higher mean of 86 %. The similar mean values of Δχo‾ (14 %) and |Δχo|‾ (18 %) were caused by higher values in χoMAM4 compared to χoML, which is confirmed below with Fig. . The averaged mixing state index χcML‾ ranged between 31 % and 84 % with a mean of 54 %, while χcMAM4‾ had a wider range (from 9 % to 99.81 %) with a mean (of 58 %). Similarly, χhML‾ ranged from 21 % to 81 % with a mean of 58 %, while χhMAM4‾ varied between 10 % and 99.85 % with a mean of 63 %. The large discrepancy between the mean difference (4.8 % for Δχc‾ and 4.7 % for Δχh‾) and mean absolute difference (30 % for |Δχc|‾ and 38 % for |Δχh|‾) indicates that the errors in χc and χh were symmetric (positive and negative) but large. The maximal errors in |Δχc|‾ and |Δχh|‾ between the two methods were up to 59 and 76 percentage points, respectively.

The implications of these discrepancies are more easily discussed with Fig. , which illustrates the global spatial distribution of annually averaged mixing state indices predicted by the ML model (first column), MAM4 (second column), their mean difference (third column), and their mean absolute difference (fourth column). The differences in mixing state indices between the ML model and MAM4 varied strongly across the globe.

High values of χoML‾ occurred in the continental regions (77 %) compared to oceans (69 %). Specifically, the ML model predicted high values for χo in central Africa (20∘ S–15∘ N, 12–30∘ E), the Arctic (66.5–90∘ N), and southern Asia (5–38∘ N, 60–90∘ E). These are also the regions with relatively larger mass fractions of black carbon (∼5 %; see Fig. ). The mixing state index χoMAM4 showed a higher degree of internal mixing over the globe (with a median of 90 %) compared to the ML model. The only exceptions were oceans in the Northern Hemisphere at the mid-latitudes (45–60∘ N, dominated by sea salt, sulfate, and secondary organic aerosol in the accumulation mode) and Antarctica (66.5–90∘ S, dominated by sea salt and sulfate in the accumulation mode as well as sulfate in Aitken mode), where χoMAM4‾ was 75 %. Qualitatively, the MAM4 model captured the trend that areas with high black carbon concentration (defined here as concentrations above the 95 % percentile) tended to have higher χo values.

The ML model estimate χcML‾ suggested a rather homogeneous spatial distribution of the annually averaged mixing state, with values of approximately 50 %. Compared to χcML‾, χcMAM4‾ values were lower (primary carbonaceous aerosol more externally mixed) at high latitudes and higher at low and mid-latitudes (primary carbonaceous aerosol more internally mixed). Note that, while χcMAM4‾ values were similar in the Arctic and Antarctic, the abundance of primary carbonaceous species was predicted to be higher in the Arctic compared to the Antarctic (see Fig. ).

The spatial distributions of χhMAM4‾ were similar to χcMAM4‾. That is, the MAM4 model predicted that the hygroscopic species and non-hygroscopic species were more externally mixed at high latitudes and more internally mixed at low latitudes. In contrast, the spatial distribution of χhML‾ shows qualitative differences compared to χcML‾ in two aspects. First, χhML‾ was higher than χcML‾ at high latitudes, meaning that hygroscopic species and non-hygroscopic appeared more internally mixed than primary carbonaceous and non-carbonaceous species in this region. Second, areas over the North Atlantic Ocean (0–20∘ N, 20–45∘ W), southern Africa (5–32∘ S, 5–20∘ E), and Australia (10–30∘ S, 100–140∘ E) appeared rather externally mixed. These are areas where mineral dust is the dominant aerosol species (see Fig. ).

These two facts lead to the overall finding that χh exhibits the largest differences between the two methods. This applies especially to regions where mineral dust was the dominant aerosol species, which points to an important structural issue of the four-mode setup used in MAM4. While the ML model predicted a more external mixture in these regions (dust externally mixed from sea salt and other species), the MAM4 model could not represent this because the accumulation mode included all six aerosol species in an internal mixture. Figure  illustrates the relationship of the mean absolute difference of χh and the mass fraction of dust for all model grid points. It confirms that grid points with large dust mass fractions were associated with larger mean absolute differences in χh. These results confirm the tradeoff discussed in : MAM3 (and MAM4 in ) intentionally combines dust and sea salt in the same mode to reduce the computational burden; however, this simplification does not always realistically reflect the aerosol mixing state in the ambient atmosphere.

It is interesting to note that the areas where sea salt is present, but not dust, are not associated with large errors, even though sea salt – just like mineral dust – is a primary aerosol type. The reason for this lies in our surrogate species definitions (Table ) for computing the mixing state index. Based on our mixing state definitions, sea salt, secondary organic aerosol, and sulfate are always grouped together. Therefore, none of the mixing state indices as defined here tell us how externally mixed sea salt is when it is considered as a single aerosol type.

Figure  further demonstrates the zonal mean annual aerosol mixing state indices, highlighting that differences between χc and χh tended to be zonally structured, where the MAM4 model overestimated at low latitudes, while it underestimated at high latitudes relative to the ML model. In contrast, the MAM4 model overestimated χo at all latitudes north of 60∘ S.

From Sect. , the following picture emerges: MAM4 overestimates the mixing state index χo except in regions at high latitudes in the Southern Hemisphere. At the same time, χc and χh are overestimated at low latitudes to mid-latitudes and underestimated at high latitudes. These findings point towards too rapid a transfer from the carbonaceous mode to the accumulation mode at low latitudes to mid-latitudes and too slow a transfer at high latitudes.

To conceptually illustrate these relationships, here we use χo and χc as examples and contrast the conditions for high and low latitudes. Figure a–f show conditions representative of high latitudes. A grid cell sampled from the CESM2/MAM4 simulation (73∘ N, 151∘ W) contains 15 % BC and 37 % POM, distributed over the accumulation and primary carbon mode as shown in Fig. a and d. The corresponding value for χo is 80 %. Figure b depicts particle population that was sampled from the MAM4 population in Fig. a. All particles, except for the smallest ones (corresponding to Aitken mode particles), contain BC, which results in the relatively high mixing state index value for χo. Note that in MAM4 BC is not included in the Aitken mode by definition. Considering the same particle population, but now evaluating the mixing state metric χc, which quantifies the degree of mixing of primary carbon and other species, yields the following observation. The entire primary carbon mode, by definition, consists of POM and BC, which results in an appreciable number of particles that contain only primary carbon (BC + POM), giving a mixing state index χc of only 27 %.

We now compare the MAM4-sampled particle populations above to particle populations that were sampled from our PartMC scenario library. We searched for populations with similar mass fractions of BC and POM as in the MAM4 populations and that were simulated at a similar latitude as the grid point location of the CESM2/MAM4 model output. Figure c shows that the PartMC results have comparatively more BC-free particles, and Fig. f shows that comparatively more particles are mixtures of primary carbon and other species. Overall, this means that in MAM4 BC appears too internally mixed (because irrespective of whether BC is placed in the primary carbon or accumulation mode, it is by design mixed with other species) and that at high latitudes the primary carbon mode is not transferring mass to the accumulation mode as quickly as is the case in PartMC simulations.

The reason why MAM4 behaves in this way can be explained by the aging process treatment in MAM4. Aging in MAM4 is formulated using a threshold criteria. That is, BC and POA mass is transferred from the primary carbon mode to the accumulation mode when a certain threshold of sulfate and SOA has condensed. In MAM4 this threshold is set to a relatively large value. This is done to prevent BC from being removed too quickly by wet deposition – because the primary carbon mode has a lower hygroscopicity than the accumulation mode and thus a lower wet scavenging efficiency – thereby counteracting a low bias in BC concentrations in the Arctic regions. From we already know that using such a high threshold may not be appropriate. However, the global model also has biases in other processes that contribute to the low BC bias in the Arctic, and setting the threshold to a high value compensates for these errors. Our results are a reflection of this fact. While adjusting the threshold criteria in MAM4 to a lower value may improve the agreement with the ML simulations in some regions, it may deteriorate the overall results in other areas. This is a good example how structural uncertainty manifests itself, namely by the fact that adjusting a parameter does not fundamentally fix the issue.

Figure g–l show conditions representative of low latitudes. A grid cell sampled from the CESM2/MAM4 simulation (20∘ N, 120∘ E) contains 11 % BC and 24 % POM, distributed over the accumulation and primary carbon mode as shown in Fig. g and j, with most of the mass in the accumulation mode. The corresponding value for χo is therefore 99 %, an almost complete internal mixture. For the same reason, χc is also very high. Similarly to the high-latitude case, Fig. i and l show that the comparable PartMC population has comparatively more BC-free particles and more particles that contain very low amounts of primary carbonaceous material, leading to lower values of both χo and χc compared to the MAM4 results.

The question that arises from Sect.  and  is of course the following: which spatial distribution of aerosol mixing state reflects reality more closely? The validation of simulated mixing state indices with observational data is still challenging since per-particle mass fractions of species are required for calculating the mixing state indices (see Sect. ). These are in principle obtainable from in situ deployments of single-particle mass spectrometers or by using electron microscopy techniques, but their quantitative derivation comes with challenges and is not routinely done, so that only very few data sets exist that allow for a meaningful comparison . Keeping these limitations in mind, reported a qualitative comparison of available measurements of mixing state metrics in locations in developed countries (Paris, France; Pittsburgh, USA; various locations in Japan) with seasonally averaged results from the ML model based on particle-resolved simulations. This showed that the ML model was able to capture the range of values that is consistent with the observations.

We further compared the ML model estimates using recent observations from China. Specifically, we compared χoML and χoMAM4 with χ values from Taizhou and Beijing derived from Single Particle Soot Photometer (SP2) measurements. For both locations, χoMAM4 overestimated the observed χ values, while χoML was in the range of the observations. Specifically, the χ measured at a suburban site Taizhou from 26 May to 18 June 2017 ranged from 62 % to 82 %. During the same time period (but in the year 2011), the values of χoML were between 63 % and 84 %, while χoMAM4 was between 84 % and 96 %. The χ values at the urban site of Beijing ranged between 55 % and 70 % in winter (from 10 November to 10 December 2016) and varied between 60 % and 75 % in summer (from 18 May to 25 June 2017). Using our simulations of the year 2011, χoML varied from 60 % to 88 % in winter and from 59 % to 83 % in summer. As a comparison, χoMAM4 ranged from 92 % to 97 % in winter and from 87 % to 95 % in summer. A caveat when comparing χoML and χoMAM4, respectively, with the observations reported in and is that the definition of χoML and χoMAM4 included BC-free particles, while the χ values in the measurements by and were calculated only considering the subpopulation of BC-containing particles. This might introduce a bias in the mixing state index between the χo index used in this paper and the observations (depending on the fraction of the BC-free particles present at any given location).

We can also relate our χo index qualitatively to the SP2 measurements in the Finnish Arctic during winter 2011–2012 . Although this study did not provide quantitative mixing state index calculations, it is an important finding that BC-containing particles (with various amounts of coatings) co-existed with BC-free particles. As we saw in Sect. , this condition can easily be represented with a particle-resolved approach. However, the modal model with modes configured as in MAM4 puts black carbon in all accumulation-sized particles (Fig. ), which is not consistent with the observations.