Data sets used in this study are partly the same as in the study of Nieminen et al. (2018) and are supplemented by newer data from the Aerosol, Clouds and Trace Gases Research Infrastructure (ACTRIS) sites (https://www.actris.eu/, last access: 9 October 2019) and SmartSmear (https://smear.avaa.csc.fi/, last access: 31 July 2019). From ACTRIS sites, we have also included new sites that were not included in Nieminen et al. (2018) (Annaberg-Buchholz, Birkenes II, Leipzig, Neuglobsow, Puijo, Schauinsland, and Waldhof) and expanded the data length by including recent years that were missing in Nieminen et al. (2018). In addition, data from Villum Research Station at Station Nord (Villum) and some recent years' data from Puijo and San Pietro Capofiume were received directly from the research groups operating the sites.

In this study, we have used only long-term observations (minimum of 6 years of measurement data) of particle number size distributions. The length of the data sets (6–22 years) and corresponding data coverage (59.6 %–98.4 % of the days of the measurement period) varies between the sites (see Fig. S1 in the Supplement). The measurement sites used in this study are listed in Table 1. For model comparison, in turn, we have included only those sites that have at least 7 years of a common time period with the model simulations (2001–2014) and sufficient data coverage (i.e. coverage >50 % of days). In Table 1, the sites are presented in two separate lists: the first list shows the sites that are used in both trend analysis and comparisons of observational and model trends, and the second list shows sites that were used only in trend analysis.

In this study we use commonly used site classes (polar, high-altitude, remote, rural, and urban) following Nieminen et al. (2018). Site environment classification of each site is adapted from Nieminen et al. (2018) for those sites that were included in their study. For other sites, we have used classifications from the literature (Sun et al., 2020, for German sites; Yttri et al., 2021, for Birkenes II; Leskinen et al., 2012, for Puijo; Schmale et al., 2018, for Vavihill; and Nguyen et al., 2016, for Villum) for environment classification and adjusted their classification according to Nieminen et al. (2018). A detailed description of each site, including the facility and environment descriptions, can be found in the literature (see Table 1).

It should be noted that there is a significant variation in the detected size ranges of the measurement instruments between the sites and within one site over the analysed time period (see Table 1). For those sites where the size range has varied over the investigated time period, we have limited the analysis only to the size range that has been measured over the whole analysis period. This size range is site-specific to maximize the number of data at each site. We have interpolated the data to site-specific, common size resolution; i.e. the size bins of size distribution data were the same for the whole time period. Measurement data size bins were interpolated because otherwise, the size bins can vary during time series, and hence, for example, the calculated modal and sectional representations (see definitions from Sect. 2.1.4) would be calculated from the different size bins.

When the in situ observations and large-scale models are compared, it is important to consider how representative the stations are for the larger areas surrounding them. The polar and remote sites (Zeppelin, Pallas, and Värriö) as well as rural site Hyytiälä can be considered to be representative of a larger regional fingerprint (Hari and Kulmala, 2005; Kyrö et al., 2014; Lohila et al., 2015; Tunved et al., 2013), and no large cities are located close to these sites. It should be noted that the Värriö site can be impacted by pollution transported from the Kola Peninsula mining and industrial areas (200–300 km north-east from the station) at times (Kyrö et al., 2014). Mace Head represents marine environment excellently when the air masses arrive from the Atlantic but on the other hand can be affected by the continental outflow as well (O'Connor et al., 2008). The urban sites Helsinki and Puijo (as urban sites in general) are affected by strong, local sources such as traffic or local industrial activity, and the diurnal variation in the representativeness to the larger areas might be significant (Hussein et al., 2008; Leskinen et al., 2012). The rural (Hohenpeißenberg, K-Puszta, Melpitz, San Pietro Capofiume, Vavihill) sites represent European background well, but their representativeness for the model grid box depends on the placement of the grid box and on how large the fraction of the grid box covered by large cities is. It should be noted that Hohenpeißenberg is located at high altitude (988 m) and is classified as a mountain site in some of the earlier studies (e.g. Rose et al., 2021), while Nieminen et al. (2018) classified it as a rural site.

Multimodal log-normal size distributions were fitted to the measured data, and the trend analysis was performed on the mode parameters. We fitted three log-normal modes (nucleation, Aitken, and accumulation) to the measured data. Before fitting the modes, we first performed a visual examination of the size distribution time series to detect clear errors in the data that could affect the results of the fitting process, e.g. the absence of some modes in the fit due to problems in the data. For example, if a substantial fraction (over 20 % of the size bins) of the number size distribution was not measured during a specific size distribution measurement, the whole distribution was removed. In addition, measurement sites have performed the quality checks routinely on the data before transferring data to the database or server.

Modes were fitted for each particle size distribution using an automatic mode-fitting algorithm (Hussein et al., 2005). Briefly, the algorithm fits a combination of one to three log-normal distributions to the particle number size distribution data separately for each time step at each location. The algorithm assumes three log-normal modes as a starting point and automatically reduces the number of modes if any of the overlapping conditions for modes is true (for more details, see Hussein et al., 2005). For each mode, the algorithm returns three parameters: geometric mean diameter, Dp; geometric variance, σp2; and mode number concentration, N.

For each fit, a quality check was performed. Firstly, we checked that the number concentrations of the fitted modes were reasonable. We used measured size bin diameters as a limit and omitted those cases where the geometric mean diameter of the mode was smaller than the smallest size bin or larger than the largest size bin from the analysis. To avoid possible overestimation of the number concentration of the modes, we assigned the number concentration of the missing or removed modes to be zero, with missing geometric diameter and geometric standard deviation.

We noticed that in cases where the smallest size bin of the measured size distribution had a high number concentration, the mode-fitting algorithm did not perform well and, instead, fitted a nucleation mode that had an unreasonably high number concentration and often also a geometric mean diameter outside of the measured size range. The reason for this was that the geometric mean diameter of the nucleation mode was smaller than the smallest detected size of the instrument, especially in cases where the smallest detected size was relatively large. For the nucleation mode, this limitation removed a median of 17.8 % of the fitted nucleation modes amongst all sites, ranging from 0 % to 41.1 % (Mace Head) between sites. For the accumulation mode, a similar phenomenon was observed, resulting in high number concentrations for large diameters near the largest detected size, although this was less likely (<0.1 % of the fitted accumulation modes).

The fitted modes were sorted into three categories – nucleation, Aitken, and accumulation mode – based on their geometric mean diameter. In the case of three fitted modes, the modes were arranged based on geometric mean diameters, with one mode always being assigned to each category. In cases with one or two fitted modes, the assignment was primarily based on the mean diameter of the mode. Here a cut-off of 20 nm was used for the fitted geometric mean diameter to distinguish between nucleation and Aitken modes, and a cut-off of 100 nm was used to distinguish between Aitken and accumulation modes. Sometimes two fitted modes both fell within the same category. In such cases, the mode was assigned to categories based on the diameter. If both modes had diameters between 20 and 100 nm (1.7 % of the cases), the mode with a diameter farther from those cut-off points was assigned to be Aitken mode, and the other mode, depending on its diameter, was assigned to be nucleation or accumulation mode. If both modes had diameters larger than 100 nm (0.4 % of the cases), the mode with the larger diameter was assigned to accumulation mode and the mode with the smaller diameter to Aitken mode. There were no cases where both modes had diameters below 20 nm.

The time resolution of the measured size distributions, and consequently the fitted modes, varied between sites and ranged from 3 to 60 min. For further analysis, we calculated daily means for each fitted mode parameter (i.e. N, Dp, and σ). For the mean to be calculated, there had to be at least 50 % of measurements available for a day (i.e. 12 h of data).

We further studied when a fraction of the different modes was missing at each site. The absence of a fitted mode at certain time points was dependent on the mode (nucleation, Aitken, or accumulation) and site. The absence was most probably caused by low concentrations of particles within the mode size range. The Aitken mode was most often present, and the nucleation mode was most often missing. Daily percentages of mode occurrence, i.e. in which fraction of measurements a certain mode was fitted for each day, for each measurement site are presented in Table 2 and Figs. S2 and S3. For Aitken and accumulation modes, the mode occurrence was more than 80 % for most of the days at all sites and was close to 100 % (i.e. mode was fitted for every observation) at most of the sites. For the nucleation mode, the mean mode occurrence was around 80 %; however, there are sites where the occurrence was much lower. This can be due to limitations of size distribution measurements for nucleation mode particles (size range starting from >10 nm) or lack of nucleation mode particles, e.g. due to meteorological or emission-related reasons. The latter is suggested by observations of nucleation occurrence in Fig. S2: urban sites had a reasonably high coverage also in the nucleation mode, whereas remote sites had days during which the nucleation mode was fitted for only a few or even zero measurement points per day. More detailed information about coverage as a function of month and hour of day is presented in Fig. S3. There were differences in nucleation mode coverage during a day and during a year, nucleation mode most often being fitted after midday. However, the patterns were not uniform for all the sites, and especially for Mace Head, the lower limit of the detected particle size most probably affected the results.

To conclude, the absence of modes did not drastically affect the daily mean of observed modes in Aitken and accumulation modes. As the fraction of fitted nucleation modes is smaller than for Aitken and accumulation modes, results for nucleation mode number concentrations are more uncertain compared to results for the other modes, which should be kept in mind when interpreting the results.

For comparison between climate models and observations, we also computed monthly means (for trend analysis) and seasonal medians (for SeasC – ratio of maximum and minimum of seasonal median values – calculation; see Supplement) of the fitted log-normal modes to the observational data described above. As global model results were monthly means, the same time resolution was also applied for the mode data. Monthly means of the measured data were calculated using the daily averaged data, with the limitation that at least five daily mean values per month were required. This limitation removed only 2 months from the entire data set, in addition to the months that were completely missing from the observational data. Seasonal means and seasonal medians were computed using monthly means with at least two monthly means per season being required.

As shown later in the results section, the mean diameters of the fitted modes are larger than the corresponding diameters or bins used in climate models. This might affect the model–observation comparison results, especially for the nucleation mode, where the relative difference between the diameters of fitted modes and model modes is largest. Therefore, we calculated separate representations of the measurement data, which are more directly comparable to the model results: for the modal and sectional aerosol schemes, the measurement data were re-binned using the model limits. For comparison, with the Sectional Aerosol module for Large Scale Applications (SALSA), the measured size bins with a mean geometrical diameter of 3 to 7.7 nm were assigned to the nucleation mode. This size range corresponds to the limits of the smallest size bin in a SALSA (Kokkola et al., 2018). Measured size bins from 7.7 to 50 nm (corresponding to the second- and third-smallest size bins in SALSA) were assigned to the Aitken mode and from 50 to 700 nm (fourth- to sixth-smallest size bins in SALSA) to the accumulation mode. In the modal representation for comparison with the modal models, the corresponding size limits were 3 to 10 nm for nucleation, 10 to 100 nm for Aitken, and 100 to 1000 nm for accumulation mode. As can be seen from Table 1, the corresponding diameter range of each mode category from the models is not fully captured by the measurements at every site. If measurements were covering only a part of the model's diameter range, that part has been used as a representative mode from measurements if there are at least three size bins of measurement data available. This limitation was used because the number concentrations from one or two bins have a large variance, resulting in very uncertain trends. If there were fewer bins or no measurement data available, the corresponding nucleation mode is not represented in the results section. For Aitken and accumulation modes, there were always enough data to calculate representative modes, even though the accumulation mode is not always measured up to the diameter of 1000 nm.

The atmospheric component of the global climate model EC-Earth3-AerChem (van Noije et al., 2021) consists of a modified version of the general circulation model used in the Integrated Forecasting System (IFS) cycle 36r4 from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the aerosol and chemistry model TM5. The IFS model version applied in EC-Earth3-AerChem has a horizontal resolution of TL255 (i.e. a spectral truncation at wavenumber 255 with a linear N128 reduced Gaussian grid, corresponding to a spacing of about 80 km) and uses 91 hybrid sigma-pressure levels in the vertical direction with a model top at 0.01 hPa. TM5 uses an atmospheric grid with a reduced resolution of 2∘×3∘ (latitude × longitude) and 34 vertical layers extending to ∼0.1 hPa. The data exchange between the two model components is governed by the OASIS coupler.

The aerosol scheme of TM5 is based on the modal aerosol microphysical scheme M7 from Vignati et al. (2004), which includes sulfate, black carbon, organic aerosols, sea salt, and mineral dust. In TM5, the formation of secondary organic aerosols is described as in Bergman et al. (2022). The concentrations of ammonium, nitrate, and the aerosol water associated with (ammonium) nitrate are calculated assuming equilibrium gas–particle partitioning. In the current model version, this equilibrium is calculated from the Equilibrium Simplified Aerosol Model (EQSAM; Metzger et al., 2002). The chemistry scheme of TM5 accounts for gas-phase, aqueous-phase, and heterogeneous chemistry (van Noije et al., 2021). The sources of mineral dust and sea salt, the oceanic source of DMS, and the production of nitrogen oxides by lightning are calculated online. Emissions from anthropogenic activities and open biomass burning are prescribed using data sets provided by CMIP6. All other emissions are prescribed as documented in van Noije et al. (2021).

ECHAM-HAMMOZ (echam6.3-hammoz2.3-moz1.0) is a global aerosol–chemistry–climate model which consists of the atmospheric circulation model ECHAM (Stevens et al., 2013), the aerosol model HAM (Kokkola et al., 2018; Tegen et al., 2019), and the chemistry model MOZ (Schultz et al., 2018) not used in this study. The model solves atmospheric circulation in three dimensions with spectral truncation of T63, which corresponds to approximately 1.9∘×1.9∘ horizontal resolution and uses 47 vertical layers extending to 0.01 hPa. The model includes the sectional aerosol model SALSA, which describes size distributions using 10 size bins between 3 nm and 10 µm in diameter, with externally mixed parallel size bins between 50 nm and 10 µm for treatment of particles consisting of insoluble material when they are emitted. The ECHAM-HAMMOZ also includes an option of using the modal aerosol model M7, which describes the aerosol size distribution with a superposition of seven log-normal modes. Details of how aerosol processes are calculated in SALSA are described by Kokkola et al. (2018). The same details for M7 are described by Tegen et al. (2019).

Both model configurations (i.e. SALSA and M7) were set up according to the AeroCom (Aerosol Comparisons between Observations and Models) initiative phase III experiment set-up. Anthropogenic aerosol emissions were according to the Community Emissions Data System (CEDS; Hoesly et al., 2018); for biomass burning, we used Biomass Burning Emissions for CMIP6 (BB4CMIP; van Marle et al., 2017). Dust, sea salt, and maritime DMS emissions are calculated online as a function of 10 m wind speed (see Tegen et al., 2019, and references therein). Atmospheric circulation (vorticity, divergence, and surface pressure) was nudged towards ERA-Interim reanalysis data (Berrisford et al., 2011), but temperature was allowed to evolve freely.

NorESM1.2 (Kirkevåg et al., 2018) is an earth system model which consists of the atmospheric model CAM5.3-Oslo, the sea ice model CICE4, the land model CLM4.5, and an updated version of the MICOM ocean model used in NorESM1 (Bentsen et al., 2013). CAM5.3-Oslo is based on CAM5.3 (Liu et al., 2016; Neale et al., 2012) but contains a different aerosol scheme (OsloAero5.3), along with other small modifications. In this study, the model is run with a horizontal resolution of 0.9∘×1.25∘ and 30 layers in the vertical (model top at around 3 hPa).

The aerosol scheme in NorESM1.2 describes aerosols using 12 separate modes, which can consist of sulfate, BC, OM (including SOA), sea salt, or dust (see Kirkevåg et al., 2018, for a detailed description), and their interaction with radiation and clouds. Emission strength of natural aerosol precursors and aerosols such as dust, sea salt, primary marine organic matter, marine DMS, isoprene, and monoterpenes is calculated interactively (Kirkevåg et al., 2018). The nucleation scheme for new particle formation used in NorESM1.2 is described in Makkonen et al. (2014). We have used the anthropogenic emissions from Hoesly et al. (2018) and biomass burning emissions from van Marle et al. (2017). We prescribed sea-surface temperatures and sea ice concentrations based on observations, and in the atmosphere, the horizontal wind (the zonal wind speed U and the meridional wind speed V) and surface pressures were nudged to 6-hourly ERA-Interim reanalysis data.

The United Kingdom Earth System Model (UKESM1) is described in detail by Sellar et al. (2019) and is built around the Global Coupled 3.1 (GC3.1) configuration of the HadGEM3 (Hadley Centre Global Environment Model) physical climate model (Kuhlbrodt et al., 2018; Williams et al., 2018). UKESM1 additionally includes ocean and land biogeochemical processes and a stratospheric–tropospheric chemistry scheme (Archibald et al., 2020) implemented as part of the United Kingdom Chemistry and Aerosol (UKCA) model. In the simulations performed for the CRESCENDO project, UKESM1 was set to operate at a horizontal resolution of 1.25∘×1.88∘ (latitude × longitude), with 85 vertical levels.

The representation of aerosols within UKESM1 is described and evaluated by Mulcahy et al. (2020); UKESM1 employs the modal version of the Global Model of Aerosol Processes (GLOMAP) two-moment aerosol microphysics scheme (Mann et al., 2010). The aerosol number size distribution is represented by soluble nucleation, Aitken, accumulation, and coarse (diameter >1000 nm) modes and an additional insoluble Aitken mode. The above modes are used to carry information about sulfate, black carbon, particulate organic matter, and sea salt, whilst mineral dust is treated using the separate sectional scheme of Woodward (2001). In UKESM1, there is no parameterized new particle formation scheme applied in the boundary layer.

Anthropogenic emissions of aerosols are prescribed from the CMIP6 inventories: SO2 and anthropogenic BC and OC are taken from the Community Emissions Data System (CEDS; Hoesly et al., 2018), and biomass burning emissions are from van Marle et al. (2017). UKESM1 interactively simulates emissions of marine DMS, biogenic volatile organic compounds (BVOCs), and primary marine organic aerosol (Sellar et al., 2019).

We used the dynamic linear model (DLM) for determining the short-term variation in trends, i.e. transient changes in the (long-term) trend in timescales of some months to some years, of different measured mode parameters in the daily data set (Durbin and Koopman, 2012; Laine, 2020; Petris et al., 2009). The main advantage of DLM compared to many other non-parametric and parametric trend estimation methods is that DLM can also detect a non-monotonic trend, and the seasonality of the time series can be estimated simultaneously with the trend.

DLM explains the measured variability in the time series yt of the mode parameter (N, Dp, or σ) with three components: firstly, the level component μt that is locally linear, but the trend αt can change during the measured period; secondly, a seasonality component γt that captures the seasonal pattern of the time series; thirdly, a residual component ηt that uses an autoregressive model (AR(1), ρ) and accounts for autoregression of the time series, i.e. dependence of the daily measurement on that from its previous day; and finally normally distributed random noise components εt, εlevel,t, εtrend,t,εseas,t, and εAR,t, which are related to uncertainties in each component. For each observation yt at time t the DLM model used in this study is given by yt=μt+γt+ηt+εt,t=1,…,T, μt=μt-1+αt+εlevel,t, αt=αt-1+εtrend,t, ∑i=011γt-i=εseas,t, ηt=ρηt-1+εAR,t, where εt∼N(0,σt), εlevel,t∼N(0,σlevel2), εtrend∼N(0,σtrend2), εseas∼N(0,σseas2), and εAR2∼N(0,σAR2). We have used ρ=0.4 as a value for AR(1) coefficient in all model fittings. The initial value of the level has been set to be the yearly mean of the first year. Calculation of the DLM model has been done in the MATLAB environment (MATLAB, 2019) using the DLM MATLAB Toolbox (Laine et al., 2014).

As the applied DLM formulation assumes normally distributed data, we used log10 transformation for mode number concentrations. If number concentration was zero (i.e. no fitted modes were available for that day), we used a value of one as a number concentration for that day to avoid problems with log10 transformation. For mode diameter and geometric standard deviation, no transformations were applied. We investigated the residuals εt after the model fitting, and in most cases, the assumptions of the model are sufficiently fulfilled, with the distribution of the residuals being close to a normal distribution. Before interpreting the level and the trend of the number concentration of each mode, we have transformed the level μt and trend αt back to the original scale by using the exponential back-transformation.

Long-term trends of measured mode parameters in the data set were estimated using the Sen–Theil estimator (Sen, 1968; Theil, 1950). The Sen–Theil estimator is a non-parametric method to estimate a linear trend. The advantages of the Sen–Theil estimator compared to more common linear regression methods are that it does not assume normality of the data, and it is more robust to outliers. Compared to the more complex DLM model, the Sen–Theil estimator also works with a lower number of data points, which is one reason we used it in the model comparison.

Trend estimation was performed using the TheilSen function from openair package in the R environment (Carslaw and Ropkins, 2012; R Core Team, 2021). The calculation of 95 % confidence intervals is based on the bootstrap method (Kunsch, 1989). Trend estimation was done for whole-year data (monthly averages) and seasonal data (monthly averages of a specific season). Before trend estimation for the whole-year data set, the time series was de-seasonalized with seasonal trend decomposition using loess, and autocorrelation for consecutive months was taken into account when calculating the uncertainty in the trend estimates. Seasons have been defined to be 3 months each, winter consisting of December–February, spring March–May, summer June–August, and autumn September–November. In the trend estimation for observational data sets (Sect. 3.1), we have used all months available from each site. In all comparisons of observations and models (Sect. 3.2), we used only those months that were available from the measurement sites.

We have used relative change (% yr-1) as the main parameter for comparing results. Relative change has been calculated for the Sen–Theil estimator and confidence intervals by using the option slope-percent. The function uses the fitted value of a first observation as a reference for calculating relative change (Carslaw and Ropkins, 2012).

The seasonality of particle number concentration and its magnitude is highly varying between different measurement sites, depending on, for example, latitude and environment type of site (Asmi et al., 2013; Rose et al., 2021) and the mode studied. Similarly, parameters such as cloud condensation nuclei (CCN) number concentrations and new particle formation (NPF) frequency have a seasonal cycle (Asmi et al., 2011; Nieminen et al., 2018). Seasonality of the optical properties in models has been studied (Gliß et al., 2021), but for particle number concentrations we are not aware of studies that compare measurements and models based on long-term data sets.

We compared the seasonality of number concentrations in models and measurements by studying modes separately. We used two variables, the normalized interquartile range (NIQR) and SeasC (Rose et al., 2021), to compare seasonality between models and measurements. When calculating these seasonal parameters from measurements and model results, we included only those months for which the measurement and model data were available. We calculated NIQR and SeasC separately for each year to also assess the distribution of values in the studied period.

NIQR, defined as NIQR =3rd quartile-1st quartileMedian, describes the interquartile range of observations for 1 year. NIQR was calculated using monthly averages of concentrations, with at least 10 monthly averages needed to be available. The calculation of NIQR is slightly different from Rose et al. (2021), who used daily values calculating NIQR. As we had only monthly averages from model data, daily values could not be used. Based on the measurement data, we checked whether the time resolution would change the NIQR values, by comparing NIQR values calculated from daily and monthly averages. We found that the NIQR values calculated from daily averages were usually higher, sometimes as much as twice the one calculated from monthly averages. Therefore, NIQR values presented in this study are not comparable to values presented in Rose et al. (2021) but only between the different data sets in this study or others calculated from monthly averages.

SeasC is the ratio of maximum and minimum of seasonal median values, calculated separately for each year and mode in each data set. It was calculated by first taking the seasonal averages for each season. For calculating the seasonal median, at least two monthly means from the season were required. Then, if we were able to calculate all the seasonal medians for the year, SeasC was calculated as the ratio of the maximum and minimum of those seasonal medians.

In general, both SeasC and NIQR describe the distribution of number concentrations within 1 year. SeasC focusses more on utmost values, minimum and maximum of seasonal medians, whereas NIQR focusses on values closer to the yearly median. Neither SeasC nor NIQR considers when the maximum and minimum in number concentrations are achieved. Though the seasonal cycle of the measured and modelled number concentrations might be opposite to each other, the difference in SeasC or NIQR values can be small when comparing measurements and model data.

To assess whether the seasonal maximums and minimums have similarities between measurements and models, we have calculated the seasonal averages, selected the seasons that have most often had seasonal maximum and minimum during the measured time period, and evaluated how modelled results correspond to the measurements.

Figure 6 shows the trends in nucleation mode N; exact numbers for trends are shown in Fig. S13. Unfortunately, at many measurement sites, the minimum detected particle diameter was too large to compute meaningful results for nucleation-mode-sized particles that were comparable to the models. Hence only five of the measurement sites (Hyytiälä, Helsinki, Vavihill, Melpitz, San Pietro Capofiume) could be compared to all models, and three additional sites (K-Puszta, Pallas, Värriö) could be compared to models with modal aerosol representation. Of these sites, Hyytiälä, Helsinki, Vavihill, and San Pietro Capofiume showed comparable trends for all three representations of the measurement data, which were all decreasing and statistically significant. At all four of these measurement sites, the models showed decreasing trends as well, but in many cases, the negative trends were weaker, and sometimes no significant trend was found. Observations at Pallas showed a strong increasing trend for both fitted mode and modal representation of the data, while all models showed slightly decreasing trends, of which one result was statistically significant.

When inter-comparing model results, we found that for most sites all models showed slight to medium decreasing trends (about 0 % to -5 % yr-1) for nucleation mode N (Figs. 6 and S13). This was also expected, as all models used the same anthropogenic emission inventory, which exhibits a steadily decreasing trend in sulfur dioxide emissions over Europe for the modelled period (Hoesly et al., 2018). This directly affects nucleation rates and condensation rates of sulfuric acid in the models. There were only two measurement sites that deviate from this general model trend. At K-Puszta, EC-Earth3 and ECHAM-SALSA showed increasing trends for the nucleation mode concentration. The other exception was a very strong decreasing trend in nucleation mode particle concentration for K-Puszta and Hohenpeißenberg in NorESM1.2. For both sites, however, the accumulation mode showed a positive trend in NorESM1.2, which was not present for the other models. A growing number of accumulation mode particles probably led to a larger condensation sink and therefore to suppression of new particle formation in the model.

Figure 7 shows the yearly trends in Aitken mode N; exact numbers for trends are shown in Fig. S13. When the three representations of observations were investigated it can be concluded that the three different representations of the measurement data qualitatively agreed at most sites. The only exceptions were Pallas, where trends varied between -0.7 % (fitted mode) and +3.1 % yr-1 (sectional representation), and for Zeppelin, where the positive trend was stronger in the sectional representation compared to the other two representations (Fig. S13). Furthermore, except for Zeppelin, Pallas, Mace Head, and Melpitz, all observational trends for all three representations were statistically significant. Of all statistically significant trends, only Hohenpeißenberg showed a positive trend in Aitken mode N for all three observational representations. Mace Head and Zeppelin were quite different, as here the calculated trends for measurements were quite large and positive, but still not statistically significant. This is very likely explained by both sites' close vicinity to the ocean (O'Connor et al., 2008; Tunved et al., 2013).

Most model trends for Aitken mode at sites in northern Europe were not statistically significant, while for the rest of the European sites, most trends were significant (Figs. 7 and S13). Interestingly, the sectional model ECHAM-SALSA showed a significantly decreasing trend at most of the northern sites. This might be due to the different size limits used in the modal and sectional models. At most sites where both measurement and model trends were significant, the models agreed quite well with the measurements in both strength and direction of the trend. However, Hohenpeißenberg was an exception where measurements showed a strong increasing trend, while the modelled trends were negative. The reasons for these differences are not clear.

Figure 8 shows the yearly trends in accumulation mode N; exact numbers for trends are shown in Fig. S13. Again, for most measurement sites, the different representations of the measurement data showed statistically significant trends of equal direction and similar strength. Exceptions were Melpitz and Hohenpeißenberg, which showed fairly weak, insignificant trends altogether; Zeppelin, which showed strong, opposite but, due to high variance, not statistically significant trends; and Puijo, which showed strong positive (but only partly significant) trends for all representations.

Concerning the model data, we did not find trends at any of the measurement sites that were statistically significant in the models. A general but weak tendency was that occurrence of statistical significance increased with decreasing latitude of the site. However, this tendency was not systematic in terms of which model produced significance at which site. Additionally, accumulation mode N depends on wildfire, sea salt, and mineral dust emissions (and atmospheric processes such as cloud processing) and hence on the means of how these emissions are calculated and inserted into the model atmosphere. Considering these factors in combination with the relatively short period analysed here, a strong model internal and inter-model variability is to be expected.

There were only two sites, Helsinki and Vavihill, where all models and measurement representations agreed on the direction of the trend (negative in both cases) in accumulation mode N (Figs. 8 and S13). Some sites stood out because the different models found strong trends in opposite directions there. Hohenpeißenberg and K-Puszta stood out, as here the model trends were mainly negative except for NorESM1.2, which showed positive (albeit not significant) trends for both sites, as was also already discussed in connection with the nucleation mode trends.

In general, the agreement between models and observations in trends of N for all modes varied a lot within the site classes, and no specific factor explaining the variation was found (see Figs. S14–S16).

Figures 9 and 10 show the seasonal trends for Aitken and accumulation mode N, respectively, at all measurement sites analysed in Sect. 3.2.1. Results for nucleation mode are shown in Fig. S17. Seasonal trends of N included more uncertainty than yearly average trends due to fewer data points. Particularly the modelling results rarely showed statistically significant trends, even though the actual magnitudes of the calculated trends were often quite large. In general, the trends derived for the measurement data did not depend strongly on the representation used. The few exceptions to this were Aitken mode trends at Zeppelin, Pallas, and Melpitz and accumulation mode trends at Zeppelin and Hohenpeißenberg. Seasonal model trends varied quite a lot between models, depending on the season, mode, and measurement site. We found that the differences between the models and observations and between models were largest for the sites where the observations show a strong positive trend (Zeppelin, Mace Head, and Hohenpeißenberg). For such stations, models exhibited either negative trends or lower trends than what was observed.

Apart from a few exceptions, the measurements showed decreasing seasonal trends of the Aitken mode N, which were also significant for some sites (Fig. 9). The exceptions were Zeppelin, Hohenpeißenberg, and Mace Head. Additionally, the measurements at K-Puszta showed increasing trends in the autumn. In general, most of the significant model trends were negative and were found during spring and summer. Neither observed nor simulated data showed significant trends in opposite directions for any of the two seasons; i.e. the significant seasonal trends were either decreasing or increasing for the one site and one measurement or model. Insignificant trends for the same site and measurement or model were sometimes decreasing for some seasons and increasing for some other seasons. The clearest difference between trends in modelled and measured data could be seen for the sites located in Finland, especially during winter and autumn, where the measurements showed a decreasing trend, while the models mostly showed an increasing trend. Those differences observed during winter and autumn could affect the differences in yearly trends observed in Fig. 7.

There was no general agreement between different models concerning accumulation mode N trends (Fig. 10). The trends in the measurements for accumulation mode were mostly fairly similar to the Aitken mode trends. For many sites, these trends from measurements were significant only during spring. Aitken mode trends from models were mostly insignificant. As can be expected from the yearly trends, the models reproduced measurement trends rather poorly, with no model performing much better or worse than any other model.

In this section, we describe the seasonality and its pattern for nucleation (Fig. S24), Aitken (Fig. 11), and accumulation (Fig. 12) modes. More quantitative investigation based on SeasC and NIQR described in Sect. 2.3.3 can be found in Sect. S1 in the Supplement.

For pattern of seasonality in modelled data, two models, NorESM1.2 and EC-Earth3, had relatively consistent patterns for all sites, whereas for the other three models the seasonal cycle changed between north and south (Fig. 11 for Aitken mode and Fig. 12 for accumulation mode). NorESM1.2 and EC-Earth3 had relatively constant patterns of seasonality throughout Europe, even though the seasonal maximum variation between the sites varied. For NorESM1.2, nucleation mode had its maximum N in winter (see Fig. S24), whereas Aitken and accumulation mode had their maximum N in summer. EC-Earth3 had also consistent modes among all sites: nucleation mode had its maximum in summer; Aitken and accumulation mode had their maximum in winter or early spring.

The other three models – ECHAM-M7, ECHAM-SALSA, and UKESM1 – showed more clear changes in the patterns of seasonality between sites, typically showing stronger seasonality at northern sites. For Aitken mode (Fig. 11), ECHAM-SALSA showed two maxima in the seasonality in Aitken mode; however the seasonality is weaker at southern sites. ECHAM-SALSA also showed two maxima for nucleation mode (Fig. S24). ECHAM-M7 showed the summer maximum for northern sites (Fig. 11), whereas for southern sites the seasonal curve was constant throughout the year or has the maximum in winter. Looking at the measurement-based representations (modal and sectional representation), the differences in seasonal patterns between the two ECHAM models were not only due to differences in Aitken mode diameter ranges. One likely contributor to the differences between M7 and SALSA was that they use different nucleation parameterizations. M7 uses the parameterization by Kazil et al. (2010), and SALSA uses the activation nucleation parameterizations by Sihto et al. (2006). In addition, it has been shown that solving simultaneously occurring nucleation and condensation within microphysical models will have implications for simulated new particle formation and growth of particles (Kokkola et al., 2009; Wan et al., 2013). Thus, the differences between M7 and SALSA are also related to differences in their numerical methods used for solving nucleation and condensation (see Kokkola et al., 2008, 2009). For the accumulation mode (Fig. 12), these three models show a summer maximum at northern sites. For southern sites, ECHAM-SALSA shows a summer maximum with a weaker seasonal effect, and UKESM1 and ECHAM-M7 show consistent seasonal curves or winter N maxima with weak seasonal effects. For nucleation mode, ECHAM-SALSA and ECHAM-M7 have two maxima in spring and autumn, whereas UKESM1 has typically only one maximum in winter or early spring (Fig. S24).

Additionally, modelled N's for different sites and the ratio between the highest- and lowest-concentration sites varied significantly between the models. Differences in Aitken mode N's between models can be due to differences in model microphysics (see Table 3), and especially in accumulation mode these differences can be due to varying deposition rates that affect the efficiency of long-range transportation of particles or the way emissions are divided into different size ranges. Differences were large, especially in Aitken mode, when we compared how N's were distributed between the sites in models and measurements. Furthermore, there were large variations in measured concentrations between the sites for all three investigated modes. The ratio for Aitken mode yearly median concentrations between the highest- and lowest-concentration sites was between 65 and 90 for different measurement-based representations (fitted modes, modal and sectional representation) and between 4 and 180 for models (see also Fig. 11). For Aitken mode, ECHAM models had the least variation in concentrations between sites, followed by EC-Earth3, UKESM1, and NorESM1.2. For accumulation mode, ratios were smaller, between 34 and 40 for measurement-based representations and between 11 and 111 for models. For accumulation mode, the ratios were between 11 and 15 for UKESM1, EC-Earth3, and ECHAM-M7; 58 for ECHAM-SALSA; and 111 for NorESM1.2. A large difference between ECHAM models might be due to differences in accumulation mode diameters and low concentration of accumulation mode particles at the Zeppelin site in ECHAM-SALSA. The concentrations in sectional model representation (particle diameter 50–700 nm) were higher than for modal representation (100–1000 nm) for both ECHAM models and measurement-based representations.