The process of small aerosol particles formation and growth (speaking mainly about 1.7–2.3 nm particles) has different time scales, influenced by atmospheric chemistry as well as meteorological conditions (wind, temperature, atmospheric stability). In a typical boreal environment, formation and growth of small particles involve three components (in addition to water): sulfuric acid, a stabilizing base (predominantly ammonia) and biogenic vapours .

Sulfuric acid (H2SO4) is formed on site from sulfur dioxide (SO2) or dimethylsulfide, both having atmospheric lifetimes between about 0.5 and 2 d and thus being transported over relatively long distances, e.g. tens to hundreds of km . SO2 is predominantly an anthropogenic pollutant, while dimethylsulfide is emitted by algae. In the absence of strong nearby sources, especially during the warmer season, the precursors of H2SO4 are spatially relatively evenly distributed. The major pathway to form H2SO4 is through oxidation of precursors by the OH radical, and therefore H2SO4 concentrations are highest during the daytime (sunlight promotes OH formation). In Finland, H2SO4 formed by oxidation of oceanic dimethylsulfide was found to contribute to the formation of aerosol particles over a boreal forest .

The sources of ammonia include animal waste, fertilizers, anthropogenic activities and ammonification of humus, and are commonly associated with agriculture (, and references therein). Recent research suggests that the lifetime of ammonia in the atmosphere is about a day with the global mean 22 h . Similarly to SO2, the ammonia concentration gradient is expected to be modest outside the vicinity of strong local sources, e.g. agricultural fields.

Oxidized organic vapour formation involves volatile organic compounds (VOC) and oxidants (i.e. ozone (O3), hydroxyl radicals (OH), and nitrate radicals (NO3)). In contrast to SO2, dimethylsulfide and ammonia contributing to small particle formation, VOC are relatively short-lived. The lifetimes of the VOC abundantly released in a boreal environment varies from minutes for sesquiterpenes to hours for monoterpenes . Therefore, the distances influenced by local sources are much shorter for the main biogenic VOC compared with either H2SO4 precursors or ammonia, ranging a few km in case of longer-lived monoterpenes. Coniferous forests are important sources of monoterpenes and sesquiterpenes e.g.. Agricultural fields also emit VOC, e.g. isoprene, but the role of isoprene in the formation and growth of small particles considered here is not significant. Some studies suggest that agricultural fields are also sources of sesquiterpenes .

A typical boreal landscape, e.g. in Finland, is a mosaic of agricultural fields, wetlands and especially forests representing patches with scales roughly on the order of 1–10 km, skewed to have a larger fraction of the patches at the lower end of this range. All the three components essential to new particle formation (H2SO4, ammonia, organic vapors) are expected to be present all over the boreal landscape, even though their relative abundances and contributions likely differ. One could expect ammonia to be the highest over agricultural fields, while most condensable organics are likely to have the highest concentration in forests. The roles of these two components in aerosol formation are different. Ammonia stabilizes sulfuric acid molecules within clusters e.g., but does not not directly influence the growth of such a cluster, which is determined by the sulfuric acid concentration. Arguably, the early growth of clusters and aerosol particles via H2SO4-ammonia (amines) clustering mechanism is more typical for polluted sites where both of these components are abundant , and probably also for agricultural sites. and observed a higher ammonia concentration during new particle formation compared to the times with no particle formation. While some studies in polluted environments such as Beijing report that H2SO4 and bases alone would be enough to explain particle growth at 1–3 nm , chamber experiments modelling boreal environment indicate a different story. Indeed, organics can give a totally separate contribution to the particle growth rate, as their thermodynamics (essentially volatility) is not tied to the presence of H2SO4. Heavy organic molecules are able to quickly increase the size of small clusters and particles, but they are not as low-volatile as sulfuric acid and probably contribute to the particle growth only at a bit larger sizes, after 1.8 nm .

The process of aerosol formation and growth outlined above is sketched in Fig.  for agricultural and forest ecosystems. As highly oxygenated molecules (HOM) and ammonia are significant components of aerosol formation and growth directly linked to the ecosystems, we choose to focus on them and not on sulfur compounds or sulfuric acid that can be assumed to be rather similarly available across ecosystems. In the case of forest, this focus makes it possible to consider the activation step from 1.7 to 2.0 nm with the growth times on the order of minutes. The open question is the time scale of conversion of VOC to HOM. This question can also be formulated as how fast can VOC produce condensable organics with concentrations sufficient to contribute to aerosol growth?

In this study, we chose to proceed with species or conditions when the production of HOM from VOC is fast. For example, sesquiterpenes are VOC, emitted by forests, wetlands and other ecosystems, that have a short lifetime and directly satisfy this requirement. However, we argue that longer-lived VOC may also create HOM quickly in non-equilibrium conditions. VOC emission is a non-stationary process influenced by light, temperature and different stress factors (e.g. light stress in spring, drought, or herbivory). In the presence of oxidants, HOM start to form immediately when VOC are released, and their source strength depends on the VOC concentration (see Appendix ). Therefore, strong emissions of VOC, for example, as a result of stress factors, can create concentrations of VOC near the source high enough to quickly produce a sufficient amount of HOM to initiate aerosol growth.

With this in mind, the concept developed in the current study is applicable to situations where VOC quickly form HOM (sesquiterpenes; some fraction of monoterpenes during high emission events), or to emission of ammonia from agricultural fields (ammonia does not need any conversion and condenses as it is). Strong and quick increases in HOM or ammonia concentrations and 2–2.3 nm particles' concentrations could be related to relatively local emissions from the surface. Our data-based estimates show that the total number of the peaks in the 2–2.3 nm concentration time series is larger in forests compared to agricultural fields, and that the number of fast growing peaks comprises about one fourth of all the peaks detected in the forest, and about one third of all the peaks detected in the agricultural fields. Increases in HOM concentrations (at some points, concentration doubles within minutes) were observed at the sites Melpitz and Hyytiälä forest , and increase in HOM and ions concentrations – at Siikaneva .

Before proceeding with the concept of the source contribution function, we would like to emphasize that other approaches to quantify the scales of local aerosol formation are also possible. One such setup is a well-mixed before-noon boundary layer with a near steady-state VOC (e.g. monoterpenes) concentration. In these conditions, solar radiation provides energy to produce more oxidants, which boosts the production of both sulfuric acid and HOM, contributing to the hour-scale formation of small aerosol particles around noon throughout the boundary layer. In this case, the process occurs in the column of the atmosphere within the boundaries set by wind advection and (turbulent) diffusion from the source. Denoting the lifetime of the prevailing VOC species τ, we can estimate that the horizontal scale of the VOC concentration field Lh≈Lf+uτ, where Lf is the forest horizontal scale and the second term is due to atmospheric advection of VOC with wind speed u. The vertical scale of the VOC concentration field can be estimated as Lv≈κτ, representing turbulent diffusion from the vegetated source surface with diffusivity coefficient κ. Assuming that sulfur compounds and ammonia are available, these scales could be used to estimate the area around the forest, over which VOC from the given forest are abundant and aerosol particles can be formed. Note that in the evening or night when decoupling settles or stable stratification is formed, turbulent mixing is limited, so that after the time on the order of the VOC lifetime, the vertical extent of the layer where aerosol can be formed significantly decreases. With this in mind, we do not claim that aerosol formation at small scales always occurs near the surface but rather outline specific situations, when this can be the case.

Based on the above overview, the “footprint-based” approach is likely to be suitable when VOC emissions and increased concentration govern the formation of HOM needed to initiate aerosol formation and growth, whereas the “boundary-layer” approach is suitable when HOM formation is determined by the increased oxidants concentration. The difference is that oxidants are formed in the column whereas VOCs are released from the surface.

Within Lagrangian approach to air flow, the observer follows the trajectories of air parcels. These parcels are defined, in accordance with the continuity of the medium, to be small enough to consider different dynamic processes in the atmosphere (e.g. turbulent vortices at small scales), and at the same time large enough to contain many molecules, i.e. large compared to the mean free path of air molecules. We assume that transport of scalar due to diffusion is low compared to that due to advection, so air parcels contain all the same scalar compounds when moving along their trajectories. This is a typical assumption when Lagrangian theory is applied to quantify scalar characteristics of the flow e.g..

In stochastic Lagrangian transport models, concentration of a scalar is defined by averaging contributions of all the air parcels arriving at the measurement point during the measurement time. When footprint for the uniform surface sources is calculated, it is assumed that parcels originated from the surface have the same amount of scalar within them, whereas the air parcels that were not in contact with the surface do not contain this scalar. The concentration footprint function can be calculated as a sum of the contributions from the parcels originating closer or further from the measurement point, which is analogous to them leaving the surface at different time moments. Essentially, it quantifies the area (or typically the upwind distance) from which the scalar-containing parcels arriving to the measurement point were released e.g.. Contribution from sources at different distances depends on horizontal and vertical turbulent transport from the surface sources to the observation level (Fig. ). Emissions very close to the measurement mast fly off the measurement point as they do not ascend upright. Trajectories originated from far locations at the surface have a small probability to end up at the measurement point; hence a smaller contribution associated with larger traveling times and longer distances. We emphasize here that the time associated with the travel of different parcels from the surface to the measurement point is implicitly included in the footprint concept, even though the footprint function itself is not time dependent.

The stochastic Lagrangian transport models have also been used to study the transport and footprint functions of non-inert trace gases by implementing a simple first order decay to the tracer concentration in each air parcel . Our concept of the source contribution function of local aerosol production is related to the concentration footprint for non-inert trace gases as we consider also the time scales of aerosol dynamics. As aerosol formation and growth at small scales is currently better studied for the forest sites, we take concept as a basis (see green area in Fig. ). Below we list our main premises:

organic vapours are essential for the growth of small particles from about 1.6–1.7 nm diameter to 2 nm. The concentration of sub-2 nm cluster ions to be activated by organics is stable within each air parcel;

To summarize, vegetation surfaces are sources of organic vapours that are lost to formation and growth of small aerosol particles within separate air parcels when travelling in the turbulent atmosphere from the source surfaces to the measurement point. The source contribution area of local aerosol production is therefore based on the formulation for the concentration footprint of gases with one important difference: we are interested not in all air parcels that are arriving at the measurements site from the surface but only in those that could contain aerosol particles grown to the size 2.0–2.3 nm during the transport time. This means an additional time-related constraint that can be introduced using sub-2 nm aerosol distribution and the growth rates of the particles. That said, we do not consider separate organic vapours but assume that they provide a certain average constant growth for aerosol particles. This approach is justified for estimates because the link between growth rates and concentrations of highly oxygenated molecules is still a subject of research. Otherwise, we would have to deal with lots of different components with different lifetimes and also have to know at which step each of these components contributes to aerosol growth.

We combine the contributions to the concentration footprint function associated with parcels travelling from distance x within time t to the observation point, f(x,t) (see examples in Figs.  and a), and sub-2 nm aerosol distributions dN(d)/dlog⁡(d) (see example in Fig. b) to calculate source contribution of the local aerosol production. Since we are only interested in the shape of the aerosol distribution as it provides the weights for the footprint contributions, we approximate number-size ion distributions with a Gaussian function N(d,σ), where d is the mean diameter, and σ is the standard deviation. The source contribution function for the production of 2–2.3 nm particles can then be calculated as: 1fw(x)=∫0Tf(x,t)N(D-GRt,σ)dt∫0TN(D-GRt,σ)dt. where N(d,σ) is the initial distribution of aerosol particles that is assumed to be Gaussian, D=2 nm is the threshold diameter of 2–2.3 nm interval. Aerosol particles in sub-2 nm range are assumed to grow with the same constant rate, GR. This formula can be interpreted as follows: only a certain portion of aerosol distribution can reach 2 nm size during the travel time of an air parcel to the measurement point, therefore, contributions to the footprint function associated with different travel times should be weighted with these portions.

The portion of distribution that can grow to 2 nm during a certain time depends on the initial distribution of sub-2 nm aerosol particles and their growth rates. Previous studies suggest that in the range of diameters between 1.5–2.2 nm the growth rates can vary between 1–4 nmh-1 (Kulmala et al. 2013; same estimates can be obtained from Garmash et al., 2024, Ehn et al., 2014), with 1.5–2 nmh-1 being a more typical value and 4 nmh-1 representing a strong new particle formation event. Since our framework suggests elevated VOC emissions and hence higher concentrations of highly oxygenated molecules, we will make estimates for the range of the larger growth rates, 2–4 nmh-1.

Finally, regarding initial sub-2 nm aerosol distributions used for the calculations: even though Fig. b suggests that the aerosol distribution is a Gaussian distribution with the mean value close to 1.15 nm in summer months, not all these particles are able to grow with organic vapours. It is likely that majority of the particles are not growing but correspond to charged molecules and clusters that are always present in the atmosphere as background. The starting diameter when organic vapours are important for growth is about 1.7 nm according to Kulmala et al. (2013).

The calculations of contribution area were performed for two ecosystems, a closed-canopy forest and an agricultural field. Stratification was assumed neutral, and we prescribed a moderate wind speed (friction velocity u*=0.4 ms-1). Calculations were performed for several measurement heights.

The forest height was taken as 20 m, and it was assumed that 25 % of sources were on forest floor and 75 % of sources were proportional to leaf area density (close to the numbers obtained by ). Observation heights included both subcanopy and above canopy levels: 0.5, 1, 2, 4, 21, and 30 m.

For agricultural field calculations, we assumed a surface with roughness z0=0.05 m (for the typical crops grown in Viikki fields, the vegetation height is between 0–0.7 m, being highest during mid/end of June and again end of July). The source height was assumed to be 1.5×z0. Observation heights were 0.5, 1, 2, and 4 m.

The Lagrangian dispersion model described in Appendix  was used to calculate the contributions to the inert scalar concentration footprint corresponding to different travel times of the air parcels to the measurement point within 60 min numerical experiments. Note that for the forest calculations, our footprint function is not a footprint function in its traditional sense (i.e. function relating surface sources to the measured concentration), but an aggregate footprint function that weighs the contributions from different levels with the source strength.

In accordance with the premises of the source contribution concept formulated in the previous section, the growing part of sub-2 nm aerosol distribution was assumed to be just a portion of the distributions, considered in Fig. b, in their higher end. For simplicity, we assumed it to be Gaussian, with d=1.75 and σ=0.07 nm. Other options and their effect on the resulting contributing area are discussed in Sect. 3.3. The final source contribution function of the local aerosol production was calculated from the convolution of the spatio-temporal contribution function with the aerosol number-size distribution function and normalized with the number-size distribution function, see Eq (1).

Atmospheric ion size distributions (Fig. b) were measured using Neutral cluster and Air Ion Spectrometer NAIS; at the SMEAR II station in Hyytiälä, Finland and at the SMEAR-Agri site in Viikki, Helsinki, Finland. The NAIS is capable of measuring the size distribution of ions with mobility diameters between 0.8–40 nm. The data from Hyytiälä was from 2 February 2010 until 31 December 2024, while the data from Viikki was from 17 June 2022 until 31 December 2024.

The inert scalar concentration footprint for the forest ecosystem is shown in Fig. . The functions corresponding to the below-canopy measurement heights display a high peak near the measurement point. This peak likely originates from the surface emissions, and it is more pronounced compared to above-forest functions because of the dampened subcanopy turbulence. The closer the measurements are to the surface, the higher is the peak, supporting its origin from the surface emissions rather than emissions from the forest canopy.

Oppositely, for the measurement heights above the canopy, even at 1 m above canopy top, the peak is much less pronounced due to developed turbulence above the vegetation in neutral stratification: most of the air parcels leaving the surface quickly move up and miss the measurement point but about equally small fraction of them can end up there, resulting in a long tail. At 10 m above the vegetation (i.e. at 30 m measurement height), the peak is very wide and cannot be easily distinguished.

Furthermore, for all the measurement heights, the contributions from close to the measurement point (this peak discussed above) are not significant compared to the tail contributions (see cumulative footprints in Fig. b), as the cumulative footprint functions continue to grow at 500 m and after that. This means that the concentration footprint function is delocalized, i.e. the parcels which come to the measurement point could originate from far locations. The cumulative concentration footprint of inert gases, for example, CO2, does not saturate with distance.

Figure  shows the source contribution functions of local aerosol formation for different measurement heights and growth rates of aerosol particles. Note that the peak is now moved further from the very close vicinity of the measurement point towards 50–200 m; the larger the growth rate, the more pronounced the peak is and the closer it is to the measurement point. At the same time, the difference between the functions corresponding to the different measurement heights below the canopy is small. This behaviour of source contribution functions is dependent on the initial distribution of sub-2 nm aerosol particles and footprint contributions for different air parcels travel times. The peak is now defined not only by the largest contribution of f(x,t) to the footprint as in Fig. , which of course occurs for short travel times featuring the area in the vicinity of the measurement point, but also by the weighing function. If there are too few aerosol particles close to 2 nm size in the sub-2 nm distribution, it would mean that in the air parcels originating near the measurement point, they do not have time to grow to the 2 nm size, and thus, large contribution from vapours emitted nearby the measurement mast does not appear in the source contribution function of local aerosol production. Instead, the peak corresponds to the parcels originated further from the measurement point, associated with longer travel times. Note that the biggest difference between the vapor concentration footprint functions (peaks in Fig. a) under forest canopy resulted from these close contributions. When the aerosol distribution-weighing dampens it, there is in practice almost no difference between the different measurement heights (Fig a and c, heights 0.5–4 m). Moreover that only part of the aerosol particles can grow to larger sizes (meaning contribution of initially small particles is also dampened) makes the source contribution functions of the local aerosol production more localized compared to the normal vapor concentration footprint functions, and the cumulative functions saturate with distance (see example for 4 m height in Fig. ). Thus, we can now normalize the cumulative concentration footprint function with its saturated value to obtain fractional contributions to the footprint, for example, the 80 % cumulative footprint for aerosol production of 2–2.3 nm diameter particles, meaning the distance where the cumulative function reaches 80 % of its saturation value (Table ).

From Table , under the forest canopy, the area contributing 80 % to concentration of 2–2.3 nm aerosol particles is about 100 m for the fastest growth rates of 3 and 4 nmh-1, about 200 m for the growth rate of 2 nmh-1 and 0.5 km for the slowest growth of 1 nmh-1. For the measurements above the forest (1 m above the canopy), all the distances approximately triple.

In this subsection, we consider source contribution areas of the local aerosol production calculated for the agricultural fields using the same initial aerosol distribution and growth rates as in the previous section.

We again start with the standard inert scalar concentration source contribution functions (Fig. ). It shows a stronger contribution of the nearby area if the measurement point is located closer to the surface (blue curve at 0.5 m above the surface), whereas the peak is much wider at 4 m height. Also for this ecosystem, the cumulative functions exhibit growth without tendency to saturation still at the furthest points shown in the figure, similar to the functions in Fig. .

The source contribution function of the local aerosol production for the agricultural field exhibit behaviour qualitatively similar to that of the functions calculated for the forest: peaks become localized further from the measurement site and tails become dampened. However, it is located much further from the measurement point compared to the forest subcanopy simulations, at 0.5–4 km compared to ca 200–500 m (Fig. ). The cumulative functions reach 80 % of the saturated values within 1–5 km depending on the growth rate (Table , two first rows), about tripling the values obtained for the measurement height 1 m above the forest canopy. At 4 m height, the functions are similar to 0.5 m, also quantitatively (Table ).

In this subsection, we consider the effect of different initial aerosol distributions on the source contribution functions. We perform analysis for agricultural fields, in which case the source contribution functions show higher sensitivities to changes in different model variables (the peak position can be much further from the measurement point, see Figs.  and ).

In Sects. 3.1 and 3.2, we accounted for the fact that only a portion of the sub-2 nm particles is able to grow due to organic vapours, and modelled it using Gaussian function with diameter 1.75 and σ=0.07 nm (Fig. 8). Here we assess the impact of initial sub-2 nm distribution on the final source contribution area of local aerosol production, changing parameters of the Gaussian function: Case 1) wider initial distribution, so that there are already some particles in the tail at 2 nm size (d=1.75 and σ=0.1 nm); Case 2) the whole initial sub-2 nm distribution can grow due to organic vapours (a wide distribution with σ=0.25 and d=1.15 nm far from the 2 nm threshold). Case 1 is close to the points obtained from the measured distributions (Fig. ), whereas Case 2 is an unlikely situation and serves to illustrate the effect of d and σ on the source contribution function. In this section, we consider only one measurement height of 0.5 m, as the functions at different heights are similar to each other.

First, we consider probably the most relevant situation, where a small portion of aerosol particles in the initial distribution is already at 2 nm (see the points obtained from normalized median distributions for the forest and field in Fig. ). The results are shown in Fig. : the main difference to Fig.  is a bigger contribution from the nearby area of 100 m. About one third of the cumulative contribution to aerosol concentration is now from this nearby location of about 100 m whereas the 80 % cumulative source contribution is within 0.9–2.0 km depending on the growth rate (excluding the smallest growth rate GR=1 nmh-1).

When we use a full median initial distribution which is also located closer to 2 nm threshold, meaning that fraction of the particles can grow relatively quickly, we expect both a high peak in the vicinity of the measurement and longer saturation distances because of the width of the function. The results for the Case 2 are shown in Fig. . There is indeed a narrow peak near the measurement site, but also wider peaks at a few km distance, and the saturation occurs much further compared to all the previous cases. The 80 % cumulative functions are at the distances of 6–10 km from the measurement points at GR=3 and 4 nmh-1 so the contributing areas are very large. At GR=1 nmh-1, the cumulative source contribution just starts to grow.

In general, the conclusions from the use of different aerosol distributions are as follows. The functions under integral in Eq (1) both have maxima: there is a maximum in aerosol particle distributions and in the functions f(x,t) contributing to the concentration footprint. The resulting source contribution area of the local aerosol formation has a noticeable maximum in the near vicinity of the measurement point if there are particles in aerosol distribution that can grow quickly. If the share of these particles is small as in the narrow distribution (blue curve in Fig. ), then the near peak is not visible, and there is a dominating peak further from the measurement point due to weighing the tails of f(x,t) by aerosol distribution also having a maximum (Fig. ). In other words, the source contribution function of the local aerosol production has two peaks with their relative contribution depending on the growing aerosol initial distribution and growth rates.