Wenhao Xue

Jing Zhang

Jing Wei

Yulu Tian

Suzanne Crumeyrolle
email: suzanne.crumeyrolle@univ-lille.fr

Andrey Khlystov

Harry Ten Brink

On the Trend in Below-Cloud Solar Irradiance in The Netherlands versus That in Aerosol Sulfate Concentration

Keywords: aerosol-cloud interaction, cloud chamber, stratocumulus

The below-cloud irradiance in The Netherlands increased by over 10 Wm -2 in the past half century. It was hypothesized that this could be due to a decrease in the aerosol serving as cloud condensation nuclei, on which the cloud droplets form, in the following way. With unchanged macrophysics, clouds with a lower number of droplets are less reflective, in other words, they transmit more solar radiation. This hypothesis cannot be substantiated with data because of a generic absence of long-term information on cloud droplet number concentrations (CDNCs) worldwide. To assess the historic trend in CDNC, we used the Boucher and Lohmann (B&L) empirical relationship between CDNC and the mass concentration of the water-attracting hygroscopic aerosol components. The B&L parameterization was tested and validated with observations from the CHIEF cloud chamber, in which the formation of marine stratocumulus, the most frequent cloud type in The Netherlands, is simulated. This study will focus on summer periods because the irradiance governs the yearly average at this latitude. The summer trend of sulfate, the most dominant hygroscopic component of observed aerosol mass concentration, was analyzed with EBAS data from 1972 onwards. The average summer CDNCs were then calculated via the B&L parameterization, showing an upper limit of 380 cm -3 in the 1970s and decreasing to around 200 cm -3 in the 2010s. The associated increase in transmission for thin marine stratocumulus without overlying clouds would be, at most, 3.5 W m -2 . Unobstructed stratocumuli occur only part of the time, and the change in irradiance based on the reduction in cloud droplet number is certainly small in comparison to the empirically derived trend of 10 W m -2 .

Introduction

Aerosols have a significant impact on the Earth's radiation budget by scattering and absorbing solar radiation (direct effect), leading to negative (cooling) and positive (warming) radiative forcing. Moreover, interactions between aerosols and clouds (indirect effect) have a large influence on the role of aerosols on climate change. The atmospheric aerosols affect cloud formation through their role as cloud condensation nuclei (CCN). Sulfate aerosols are nowadays considered as the most dominant hygroscopic component of aerosols [1,[START_REF] Kim | Optimization of the sulfate aerosol hygroscopicity parameter in WRF[END_REF]. Sulfate is an important secondary aerosol formed from photochemical reactions of sulfur dioxide (SO 2 ) emissions in the atmosphere. Summertime peaks of sulfate concentrations are common for most areas, including The Netherlands [START_REF] Yoon | Seasonal characteristics of the physicochemical properties of North Atlantic marine atmospheric aerosols[END_REF], due to available solar insolation, chemical reactions facilitated in high relative humidity environments, and stagnation events e.g., [START_REF] Hidy | Spatial and temporal distributions of airborne sulfate in parts of the United States[END_REF][START_REF] Tai | Correlations between fine particulate matter (PM2.5) and meteorological variables in the United States: Implications for the sensitivity of PM2.5 to climate change[END_REF]. To reduce SO 2 emissions over the world, regulatory and legislative mandates were established and total annual SO 2 emissions over Europe have decreased by 72% from 1980 to 2004 [START_REF] Engardt | Deposition of sulphur and nitrogen in Europe 1900-2050. Model calculations and comparison to historical observations[END_REF][START_REF] Vestreng | Twenty-five years of continuous sulphur dioxide emission reduction in Europe[END_REF], and by 60% from 1990 to 2010 over the United States [START_REF] Schmalensee | The SO 2 Allowance Trading System: The Ironic History of a Grand Policy Experiment[END_REF].

Over the past 50 years, the solar irradiance over The Netherlands significantly increased, both during clear-sky and cloudy situations [START_REF] Boers | Impact of aerosols and clouds on decadal trends in all-sky solar radiation over The Netherlands (1966-2015[END_REF]. The solar irradiance increase during clear sky conditions could be attributed to a decrease in reflective aerosols within the atmosphere, such as sulfate [START_REF] Hand | Interagency Monitoring of Protected Visual Environments): Spatial and Seasonal Patterns and Temporal Variability of Haze and Its Constituents in the United States; Report V, CIRA Report[END_REF][START_REF] Malm | Characteristics and origins of haze in the continental United States[END_REF]. Whereas, the solar irradiance decrease under cloudy conditions may have been caused by a decrease in CDNC, leading to a reduction in cloud reflectivity [START_REF] Twomey | Atmospheric Aerosols[END_REF][START_REF] Lohmann | Aerosol Effects on Clouds and Climate[END_REF][START_REF] Wood | Stratocumulus Clouds[END_REF][START_REF] Berner | Large eddy simulation of ship tracks in the collapsed marine boundary layer: A case study from the Monterey area ship track experiment[END_REF]. Assuming a constant liquid water path (LWP), a decrease in CDNC would be due to a lower CCN concentration. In this case, CDNC and below-cloud irradiance trends should be similar.

Over the North Sea region, stratocumulus (SC) is the most common [START_REF] Duynkerke | Turbulent Structure of a Shear-Driven Stratus-Topped Atmospheric Boundary Layer: A Comparison of Model Results with Observations[END_REF] and the most frequent cloud type. Together with stratus, it covers the North Sea nearly 50% of the time [START_REF] Duynkerke | Turbulent Structure of a Shear-Driven Stratus-Topped Atmospheric Boundary Layer: A Comparison of Model Results with Observations[END_REF]. SC generally forms at the top of the planetary boundary layer, and is therefore connected to aerosol observed at the surface. Moreover, previous studies, based on remote sensing, parcel modeling, and large eddy simulation (LES), have highlighted that SC clouds exhibit substantial albedo susceptibility to variations in CDNC [START_REF] Sanchez | Meteorological and aerosol effects on marine cloud microphysical properties[END_REF][START_REF] Platnick | Determining the susceptibility of cloud albedo to changes in droplet concentration with the advanced very high-resolution radiometer[END_REF].

Trend analyses are needed to track the evolution of solar radiation and CDNC. In a trend analysis, the annual average of the solar irradiance measured values cannot be used because of its variation, which depends on the solar angle and cloud coverage. For these reasons, cloud cover values were scaled to the yearly average of cloud cover and occurrence of clouds per solar angle [START_REF] Boers | Impact of aerosols and clouds on decadal trends in all-sky solar radiation over The Netherlands (1966-2015[END_REF]. The parameter obtained is defined as a "proxy" for below-cloud irradiance per year. At The Netherlands latitudes, the annual solar irradiance is determined by that in summer because of the long sunshine hours and high irradiance. This leads to a difference of a factor of ten in irradiance in summer and winter [START_REF] Velds | Solar Irradiance in The Netherlands[END_REF].

Contrary to solar irradiation, the CDNC trend cannot be evaluated from measurements because of an absence of monitoring data. This is not a local issue but a worldwide phenomenon. To assess the historic CDNC trend, de Martino et al. [START_REF] De Martino | An Aerosol-Cloud Module for Inclusion in the KNMI Regional Climate Model RACMO2[END_REF] detailed several approaches, including using the mass concentration of the hygroscopic aerosol components as a proxy. In a recent evaluation of the largest dataset in marine clouds in South California [START_REF] Macdonald | On the relationship between cloud water composition and cloud droplet number concentration[END_REF], the parameterization of Boucher and Lohmann (B&L) [START_REF] Boucher | The sulfate-CCN-cloud albedo effect[END_REF], based on the mass concentration of sulfate to retrieve CDNC, was found to provide the best results in comparison to the data. This was also the case in a shorter study off the Canadian coast of the North Atlantic [START_REF] Leaitch | Physical and chemical observations in marine stratus during the 1993 North Atlantic Regional Experiment: Factors controlling cloud droplet number concentrations[END_REF].

There is very limited information on the CDNC over the North Sea region, let alone in combination with sulfate concentrations. Only some data exist on the CDNC in the 1980s [START_REF] Martin | The Measurement and Parameterization of Effective Radius of Droplets in Warm Stratocumulus Clouds[END_REF], mostly collected in continental air masses non-representative of the background concentrations. In view of this, we derived an evaluation of the relationship between CDNC and sulfate concentration in a large cloud chamber (CHIEF) at the northwestern seacoast of The Netherlands, in which the conditions of marine stratocumulus were simulated [START_REF] Crumeyrolle | On the importance of nitrate for the droplet concentration in stratocumulus in the North-Sea region[END_REF].

In the present investigation, we first tested whether the B&L parameterization was appropriate for the relevant marine clouds over The Netherlands via an investigation in a large cloud chamber (CHIEF). Then, the summer trend of the sulfate concentration was reconstructed using EBAS data to infer the CDNC trend using the B&L parameterization. Finally, the CDNC trend was used to estimate the modification in solar irradiance only due to indirect effect, and will be compared to below-cloud irradiance trend. This will provide information on the magnitude of the aerosol-cloud interaction (ACI) effect on climate [START_REF] Yoon | Seasonal characteristics of the physicochemical properties of North Atlantic marine atmospheric aerosols[END_REF]. Before we proceed, it is important to mention that in addition to sulfate, other chemical compounds in aerosols, such as nitrate [START_REF] De Martino | An Aerosol-Cloud Module for Inclusion in the KNMI Regional Climate Model RACMO2[END_REF], might be of importance as components of the cloud nuclei. However, we showed in a recent comprehensive study [START_REF] Crumeyrolle | On the importance of nitrate for the droplet concentration in stratocumulus in the North-Sea region[END_REF] that, in The Netherlands, ammonium nitrate is not relevant during the summertime, and neither are organic compounds or sea salt; even though they significantly contribute to the total aerosol mass, they are not present in the size range of particles that serve as cloud nuclei.

Materials and Methods

Cloud Chamber

The large cloud chamber (CHIEF) used in the tests is situated at the northwestern seacoast of The Netherlands. A full description of the set-up is provided in an earlier publication [START_REF] Crumeyrolle | On the importance of nitrate for the droplet concentration in stratocumulus in the North-Sea region[END_REF]. Here we provide a summary of the relevant aspects. It is a 21 m 3 flow-through chamber in which the low supersaturation conditions of marine stratocumulus are simulated [START_REF] Crumeyrolle | On the importance of nitrate for the droplet concentration in stratocumulus in the North-Sea region[END_REF]. The proper setting of the chamber was derived from simultaneous CDNC measurements within the chamber and aboard a small aircraft flying in stratocumulus over the site. These results are detailed for the first time in Appendix A. The CDNC was measured using a forward scattering spectrometer probe (FSSP-100 from DMT) for droplets within the diameter range of 3 to 47 µm. Size and number concentration were calibrated according to the guidelines in the manual. The sulfate concentration was observed with a MARGA-sizer that works as follows: particles are grown to large droplets by steam condensation, and these are collected in a cyclone [START_REF] Brink | An instrument for semi-continuous monitoring of the size-distribution of nitrate, ammonium, sulphate and chloride in aerosol[END_REF][START_REF] Brink | Visibility in The Netherlands during New Year's fireworks: The role of soot and salty aerosol products[END_REF]. In the drain water, the hygroscopic species dissolve and are then measured online as ions. The accuracy of the MARGA system has been checked against 24-h filter measurements, e.g., [START_REF] Rumsey | An assessment of the performance of the Monitor for AeRosols and GAses in ambient air (MARGA): A semi-continuous method for soluble compounds[END_REF][START_REF] Weijers | Measurements of Secondary Inorganic Aerosols in The Netherlands[END_REF], with an average deviation of less than 3%.

Historic Sulfate Measurements

The sulfate mass concentration has been measured since July 1972 in The Netherlands, with many of the data included in the EMEP/EBAS database [START_REF] Tsyro | PM2.5 and Individual Aerosol Components[END_REF]. A previous study [START_REF] Twomey | Pollution and the planetary albedo[END_REF] already conducted an analysis comparing the trend of observed sulfate mass concentration to the reflection of radiation during summer over the years 1978-1994. Since the publication of this study, the database has evolved to homogenize the mass concentration values as a function of the method used to measure it. In the early years, sulfate concentrations were measured through X-ray fluorescence (XRF), detecting sulfur, while in the period 1984-1987, a colorization reaction of the leached filter was applied [START_REF] Twomey | Pollution and the planetary albedo[END_REF]. Thereafter, ion chromatography was used, though data for the first 4 years of application of the latter method are not in the EBAS database. In 2009, PM 10 samplers were introduced within the national network. An issue with this is that these collect a larger fraction of sea salt than the original samplers. To correct this bias, the PM 10 samples were also analyzed for sodium, which allows the estimation of the sea salt sulfate fraction.

Results and Discussion

Cloud Chamber Study

Figure 1 shows the total sulfate concentrations and CDNC formed within the cloud chamber CHIEF during an intensive observation period in summer 2007. One can see that sulfate mass concentration ranges from 0.5 to 6 µg m -3 , while the corresponding CDNC (only) spans from 180 to 400 cm -3 . The large variability of the sulfate mass concentration is clearly not observed within the CDNC time series. Indeed, the CDNC did not significantly increase for sulfate concentrations exceeding 3 µg m -3 . However, one can see that the tendency is similar in both figures, with two large peaks observed on the 1st and on 12 August 2007 for both parameters. On the 5 August 2007, the CDNC peak reached up to 366 cm -3 , while the sulfate was unfortunately not measured due to technical issues. However, the sulfate concentration before and after this date is rather large (>3 µg cm -3 ), probably suggesting a peak on the 5 August.

As for the functional relationship of the CDNC and the mass concentration, we refer to Figure 2 (blue cross), in which the relationship of Leaitch et al. [START_REF] Leaitch | Physical and chemical observations in marine stratus during the 1993 North Atlantic Regional Experiment: Factors controlling cloud droplet number concentrations[END_REF] is added. These data derive from the concentration of sulfate in the cloud water translated to the corresponding air concentration. For this translation, the liquid water content (LWC) is used, for which the average uncertainty is 20%, as we deduce from the information given in the publication. This constitutes by far the largest measurement uncertainty. The Boucher and Lohmann (B&L) parameterization (dashed lines) for this relationship is provided, which thus holds for polluted coastal aerosol at the northwestern North Atlantic coast. It should be mentioned that the asymptotic value of the CDNC at the highest concentration agrees with that measured by Martin and Johnson [START_REF] Martin | The Measurement and Parameterization of Effective Radius of Droplets in Warm Stratocumulus Clouds[END_REF] in the North Sea area (CDNC ~350-400 cm -3 ), substantiating the choice for the chosen parameterization. It seems that the observed CDNC in the cloud chamber is larger than the B&L parameterization. This can be explained by the fact that CHIEF, with its residence time of 2 min, simulates the situation at the cloud base, where the CDNC is higher up in the cloud, as found in our cloud flight presented in Appendix A and by Leaitch et al. [START_REF] Leaitch | Physical and chemical observations in marine stratus during the 1993 North Atlantic Regional Experiment: Factors controlling cloud droplet number concentrations[END_REF]. Hence, CDNCs measured within the chamber are indeed expected to be higher than B&L retrievals. Incidentally, the presence of a higher concentration at the cloud base is of little relevance for the reflectivity, because it concerns very small droplets that evaporate upon further upward transport. As for the functional relationship of the CDNC and the mass concentration, we refer to Figure 2 (blue cross), in which the relationship of Leaitch et al. [START_REF] Leaitch | Physical and chemical observations in marine stratus during the 1993 North Atlantic Regional Experiment: Factors controlling cloud droplet number concentrations[END_REF] is added. These data derive from the concentration of sulfate in the cloud water translated to the corresponding air concentration. For this translation, the liquid water content (LWC) is used, for which the average uncertainty is 20%, as we deduce from the information given in the publication. This constitutes by far the largest measurement uncertainty. The Boucher and Lohmann (B&L) parameterization (dashed lines) for this relationship is provided, which thus holds for polluted coastal aerosol at the northwestern North Atlantic coast. It should be mentioned that the asymptotic value of the CDNC at the highest concentration agrees with that measured by Martin and Johnson [START_REF] Martin | The Measurement and Parameterization of Effective Radius of Droplets in Warm Stratocumulus Clouds[END_REF] in the North Sea area (CDNC ~ 350-400 cm -3 ), substantiating the choice for the chosen parameterization. It seems that the observed CDNC in the cloud chamber is larger than the B&L parameterization. This can be explained by the fact that CHIEF, with its residence time of 2 min, simulates the situation at the cloud base, where the CDNC is higher up in the cloud, as found in our cloud flight presented in Appendix A and by Leaitch et al. [START_REF] Leaitch | Physical and chemical observations in marine stratus during the 1993 North Atlantic Regional Experiment: Factors controlling cloud droplet number concentrations[END_REF]. Hence, CDNCs measured within the chamber are indeed expected to be higher than B&L retrievals. Incidentally, the presence of a higher concentration at the cloud base is of little relevance for the reflectivity, because it concerns very small droplets that evaporate upon further upward transport. As the CDNC follows the B&L parameterization rather well, the B&L parameterization could be used for the CDNC trend over the Netherlands. Therefore, in the next sections, the B&L parameterization will be used to estimate the trend in CDNC based on As the CDNC follows the B&L parameterization rather well, the B&L parameterization could be used for the CDNC trend over The Netherlands. Therefore, in the next sections, the B&L parameterization will be used to estimate the trend in CDNC based on measured sulfate mass concentrations over the past ~50 years. It is also interesting to note, in Figure 2, that the maximum CDNC values observed within the CHIEF chamber are close to 350 cm -3 for sulfate concentrations larger than 1 µg m -3 . This upper limit can also be seen in Leaitch data, with a maximum observed at 380 cm -3 .

Sulfate Mass Concentration and CDNC Trends

All in all, non-sea-salt (nss) sulfate mass concentrations are available for eleven sites operational at different periods, with relatively often only one location active (see supplementary material for more information). There is a good correlation of data simultaneously collected at the different locations (Figure S2), which is consistent with the previous analysis [START_REF] Twomey | Pollution and the planetary albedo[END_REF] for The Netherlands in the 1980s and for stations, as far apart as 200 km, in western Europe in general [START_REF] Arends | Trends in summer sulphate in Europe[END_REF]. Median nss-sulfate mass concentration per summer from only three sites (Witteveen, Appelscha, Vredepeel) were used to reconstruct the sulfate trend presented in Figure 3a. One can clearly see that the nss-sulfate trend is clearly decreasing with time. The median summer concentration in 1972 was around 10.2 µg m -3 , and decreased down to 1.9 µg m -3 in 2019. This decrease is essentially due to stakeholder decisions at the European scale, especially the gasoline restrictions [START_REF] Hand | Interagency Monitoring of Protected Visual Environments): Spatial and Seasonal Patterns and Temporal Variability of Haze and Its Constituents in the United States; Report V, CIRA Report[END_REF]. Moreover, one can clearly see that the daily average variability (grey dots) is also decreasing with time due to the decrease in daily PM exceedances also due to stakeholder decisions. A similar trend is observed over the southern part of the UK. For the years 1966 to 1971, the average summer concentration of nss-sulfate in the southern UK based on data at Chilton is close to 7 µg m -3 [START_REF] Schaap | Secondary inorganic aerosol simulations for Europe with special attention to nitrate[END_REF]. Currently (2016-2021), the concentration of nss-sulfate, observed in nearby Chilbolton, is just over 2 µg m -3 .

Atmosphere 2022, 13, x FOR PEER REVIEW 6 of 11 To estimate the summer CDNC trend, the relationship (B&L (A) and (D), shown in Figure 2) between CDNC and nss-sulfate mass concentration is used (Figure 3b). With a current-day nss-sulfate concentration of around 1.9 μg m -3 over the Netherlands, the CDNC should be in the order of 200 cm -3 . Meanwhile, in the 1970s, the nss-sulfate concentration was over 10 μg m -3 and the CDNC was retrieved at or even over the asymptotic maximum value of 380 cm -3 . A clear CDNC decrease, by a ratio up to 2, is observed during the last 40 years.

The trend in the measured average annual irradiance [START_REF] Boers | Impact of aerosols and clouds on decadal trends in all-sky solar radiation over The Netherlands (1966-2015[END_REF] is reproduced in Figure 4a. From a plateau at 76 +/-3 W m -2 during the period 1966-1996, it increased to a level of approximately 87 +/-2.5 W m -2 during the period 2000-2015. This difference (increase by 16%) is significant and far above the uncertainties. Therefore, the below-cloud solar radiation is clearly increasing over the Netherlands. As the CDNC is decreasing during the same period, the hypothesis of lower reflectivity clouds due to lower CDNC could explain this tendency. The cloud transmission would be larger, and then the total radiation below To estimate the summer CDNC trend, the relationship (B&L (A) and (D), shown in Figure 2) between CDNC and nss-sulfate mass concentration is used (Figure 3b). With a current-day nss-sulfate concentration of around 1.9 µg m -3 over The Netherlands, the CDNC should be in the order of 200 cm -3 . Meanwhile, in the 1970s, the nss-sulfate concentration was over 10 µg m -3 and the CDNC was retrieved at or even over the asymptotic maximum value of 380 cm -3 . A clear CDNC decrease, by a ratio up to 2, is observed during the last 40 years.

The trend in the measured average annual irradiance [START_REF] Boers | Impact of aerosols and clouds on decadal trends in all-sky solar radiation over The Netherlands (1966-2015[END_REF] is reproduced in Figure 4a. From a plateau at 76 ± 3 W m -2 during the period 1966-1996, it increased to a level of approximately 87 ± 2.5 W m -2 during the period 2000-2015. This difference (increase by 16%) is significant and far above the uncertainties. Therefore, the below-cloud solar radiation is clearly increasing over The Netherlands. As the CDNC is decreasing during the same period, the hypothesis of lower reflectivity clouds due to lower CDNC could explain this tendency. The cloud transmission would be larger, and then the total radiation below the cloud would increase as well.

CDNC should be in the order of 200 cm -3 . Meanwhile, in the 1970s, the nss-sulfate concentration was over 10 μg m -3 and the CDNC was retrieved at or even over the asymptotic maximum value of 380 cm -3 . A clear CDNC decrease, by a ratio up to 2, is observed during the last 40 years. ±

The trend in the measured average annual irradiance [START_REF] Boers | Impact of aerosols and clouds on decadal trends in all-sky solar radiation over The Netherlands (1966-2015[END_REF] is reproduced in Figure 4a. From a plateau at 76 +/-3 W m -2 during the period 1966-1996, it increased to a level of approximately 87 +/-2.5 W m -2 during the period 2000-2015. This difference (increase by 16%) is significant and far above the uncertainties. Therefore, the below-cloud solar radiation is clearly increasing over the Netherlands. As the CDNC is decreasing during the same period, the hypothesis of lower reflectivity clouds due to lower CDNC could explain this tendency. The cloud transmission would be larger, and then the total radiation below the cloud would increase as well. In order to estimate this effect, we used the relationship provided by Schwartz and Slingo [START_REF] Salmon | Retrospective trend analysis of the content of U.K. air particulate material 1957-1974[END_REF]. They established (as demonstrated in Equation ( 1)), based on older studies [START_REF] Twomey | Pollution and the planetary albedo[END_REF][START_REF] Schwartz | Enhanced shortwave cloud radiative forcing due to anthropogenic aerosols[END_REF], that the decrease in reflectivity (ΔR) is associated with CDNC decrease In order to estimate this effect, we used the relationship provided by Schwartz and Slingo [START_REF] Salmon | Retrospective trend analysis of the content of U.K. air particulate material 1957-1974[END_REF]. They established (as demonstrated in Equation ( 1)), based on older studies [START_REF] Twomey | Pollution and the planetary albedo[END_REF][START_REF] Schwartz | Enhanced shortwave cloud radiative forcing due to anthropogenic aerosols[END_REF], that the decrease in reflectivity (∆R) is associated with CDNC decrease considering a constant liquid water path (LWP) and cloud thickness (Zc). This equation is accurate within 10% for the range of cloud-top albedo, 0.28 < R < 0.72, characteristic of the prevalent and climatically important marine stratus clouds. It should also be noted that this relationship applies in the absence of overlying higher clouds. As the cloud profiles are not observed, we assume that there are only unobstructed stratocumulus clouds over The Netherlands, which means that the calculated reflectivity evolution is an upper limit.

(∆R) LWP,Zc = 0.075 * ∆ln(CDNC)

Assuming a constant LWP and cloud thickness throughout the last 50 summers over The Netherlands, the estimated SC reflectivity decreases by 3.5 Wm -2 in 2019 due to the decrease in CDNC. To add some perspective regarding the indirect forcing retrieved here, the total greenhouse gas forcing over the industrial period was about 2.5 W m -2 , with CO 2 contributing 1.5 W m -2 [START_REF] Twomey | The Influence of Pollution on the Shortwave Albedo of Clouds[END_REF]. The indirect forcing on cloud reflectivity is therefore not negligible. As previously stated, this value is an upper estimate, given the fact that SC clouds observed over The Netherlands are not permanently unobstructed. Moreover, this estimated value is far below the observed increase in the below-cloud irradiance (around 10 Wm -2 ). Therefore, the increase in the below-cloud irradiance can only be partly explained by the aerosol and cloud interaction in the lower layers. The rest of the increase could be due to a decrease in LWP and/or a decrease of cloud depth. Another possibility might be a systematic change in cloudiness values as a result of switching from a visual appraisal to objective probing by ceilometers [START_REF] Boers | Impact of aerosols and clouds on decadal trends in all-sky solar radiation over The Netherlands (1966-2015[END_REF]. This latter hypothesis could be checked via data for neighboring countries in which a similar trend in the mass concentration of sulfate occurred, but where cloudiness was very likely appreciated in a different manner over the years as compared to in The Netherlands.

Conclusions

In this study, Boucher and Lohmann parameterizations [START_REF] Boucher | The sulfate-CCN-cloud albedo effect[END_REF] were tested using cloud chamber (CHIEF) measurements of non-sea-salt (nss) sulfate mass concentration and cloud droplet number concentration (CDNC). As expected, the CDNC measured in CHIEF, mimicking the cloud formation at its base, is overestimated in comparison to the B&L retrievals. Despite this overestimation, the results highlight that the B&L retrievals are reasonably comparable to our observations. The CDNC trend is then estimated from the B&L parameterizations, and the nss-sulfate mass concentration observed over The Netherlands for the past 50 years. Both the nss-sulfate and the CDNC trends are decreasing over the years. The CDNC was around 380 cm -3 in the 1970s, and decreased down to 200 cm -3 , while the nss-sulfate mass concentration has decreased by a factor of five over the last 50 years (from 10 to 2 µg m -3 ).

The CDNC evolution was then used to estimate the trend in stratocumulus cloud transmissivity. The below-cloud irradiance estimation increases by 3.5 W m -2 , at most, due to the presence of less CDNC. This increase can only explain part (35%) of the observed increase in below-cloud solar irradiance of 10 W m -2 . Therefore, there must be other parameters, such as cloud density or the appreciation of cloudiness over the years, that would explain this large modification of the solar irradiance received at the surface during cloudy days.

Supplementary Materials:

The following supporting information can be downloaded at: https: //www.mdpi.com/article/10.3390/atmos13122037/s1, Figure S1: Sulfate monitoring sites listed in EBAS for The Netherlands (Table S1). The grey dots correspond to the sites used to build the sulfate long term trend. The diamond corresponds to Cabauw., Table S1: Netherland sites listed in EBAS data platform and the associated instrumentation used to measure sulfate. Table S2: Sulfate concentration (µg m -3 ) measured at EBAS-NL sites (Table S1) since 1972. Figure S2: Comparison of sulfate concentration (µg m -3 ) measured at different EBAS-NL sites (Table S1). The dashed line corresponds to the 1:1 line. The aircraft encountered a closed cloud deck that had entered The Netherlands from the north. Surface winds were from the NNE for the entire period of the measurements (6:00 to 9:00 UTC). The back trajectory indicated that the air masses were mostly coming from Arctic marine regions. The cloud layer was capped by an inversion at about 1100 m according to sounding at a nearby airport. There was no observable precipitation.

Several horizontal tracks were constructed within the cloud in which the droplet number and size distribution were determined. From these measurements, the LWC and effective diameter (D eff ) were derived. The standard deviation of both parameters within one track was rather small (see Table A1). The droplet spectrum shifts towards larger sizes at higher altitudes due to the continuing growth of the cloud droplets. The vertical profile of the LWC based on the droplet spectrum measured by the FSSP in the various horizontal tracks in the layer is shown in Figure S3. The experimental vertical profile of the LWC agrees reasonably well with the adiabatic profile of an almost ideal cloud layer.

One can see that the droplet number concentration (N) is relatively constant over the cloud height, apart from that measured at cloud base. Indeed, the CDNC value at cloud base (210 cm -3 ) is higher than that higher up in the cloud layer. This observation is similar to that of Martin and Johnson [START_REF] Martin | The Measurement and Parameterization of Effective Radius of Droplets in Warm Stratocumulus Clouds[END_REF] within SC over the same region. The cloud base measurements (862 m) are then comparable to the onset of the cloud formation as simulated in the CHIEF chamber.

Operating the CHIEF chamber in scanning mode [START_REF] Shine | Radiative Forcing of Climate[END_REF], the supersaturation is ramped up and down in a cycle of 40 min. The aerosol spectrum was monitored with an SMPS, while the droplet spectrum was assessed with the same type of FSSP as on board the measuring aircraft flying above.

At the time of the overhead flight, the aerosol number concentration was 2800 cm -3 . The aerosol had a bimodal size distribution with an Aitken mode (Dp = 50 nm) and an accumulation mode (Dp = 150 nm). The best agreement of CDNC measured at cloud base (210 cm -3 ) and within the cloud chamber was around a supersaturation of 0.1% (Figure A1). For further details, we refer to the detailed report by Khlystov et al. [START_REF] Khlystov | Micro-Physical Properties of Stratocumulus Clouds during the CLARA Campaign[END_REF]. Operating the CHIEF chamber in scanning mode [START_REF] Shine | Radiative Forcing of Climate[END_REF], the supersaturation is ramped up and down in a cycle of 40 min. The aerosol spectrum was monitored with an SMPS, while the droplet spectrum was assessed with the same type of FSSP as on board the measuring aircraft flying above.

At the time of the overhead flight, the aerosol number concentration was 2800 cm -3 . The aerosol had a bimodal size distribution with an Aitken mode (Dp = 50 nm) and an accumulation mode (Dp = 150 nm). The best agreement of CDNC measured at cloud base (210 cm -3 ) and within the cloud chamber was around a supersaturation of 0.1% (Figure A1). For further details, we refer to the detailed report by Khlystov et al. [40].

11 Figure 1 .

 111 Figure 1. Sulfate mass concentration and cloud droplet number concentration (CDNC) measured within the cloud chamber (CHIEF) during the indicated intensive operational period in summer 2007.

Figure 1 . 11 Figure 2 .

 1112 Figure 1. Sulfate mass concentration and cloud droplet number concentration (CDNC) measured within the cloud chamber (CHIEF) during the indicated intensive operational period in summer 2007. Atmosphere 2022, 13, x FOR PEER REVIEW 5 of 11

Figure 2 .

 2 Figure 2. CDNC as a function of sulfate concentration measured in the CHIEF cloud chamber (blue cross) and observed during the stratocumulus flights of Leaitch et al. [23] (red squares and stars). The dashed line represents the Boucher and Lohmann [22] parameterization.

Figure 3 .

 3 Figure 3. (a) Time series of non-sea-salt (nss) sulfate mass concentration observed over the Netherlands during the summer season (June-August) from three sites (Witteveen, triangles; Appelscha, dots; Vredepeel, squares; see supplementary material for more details). The black signs correspond to the median over the summer season and the grey signs to the daily averages. (b) Time series of CDNC derived from the Boucher and Lohmann relationship [22] applied to the nss-sulfate concentration shown in Figure 3a.

Figure 3 .

 3 Figure 3. (a) Time series of non-sea-salt (nss) sulfate mass concentration observed over The Netherlands during the summer season (June-August) from three sites (Witteveen, triangles; Appelscha, dots; Vredepeel, squares; see supplementary material for more details). The black signs correspond to the median over the summer season and the grey signs to the daily averages. (b) Time series of CDNC derived from the Boucher and Lohmann relationship [22] applied to the nss-sulfate concentration shown in (a) .

Figure 4 .

 4 Figure 4. (a) Absolute cloud base proxy radiation as described by Boers et al. [9] over the indicated years averaged for five measuring sites. (b) Change in solar irradiance as deduced from the decrease in CDNC (Figure 3b).

Figure 4 .

 4 Figure 4. (a) Absolute cloud base proxy radiation as described by Boers et al. [9] over the indicated years averaged for five measuring sites. (b) Change in solar irradiance as deduced from the decrease in CDNC (Figure 3b).

  Figure S3: Vertical profiles of LWC in the vicinity of Petten. Line indicates the ideal adiabatic relation between LWC and altitude. Author Contributions: Conceptualization, H.T.B., A.K. and S.C.; methodology, H.T.B. and A.K.; formal analysis, H.T.B. and A.K.; investigation, H.T.B. and S.C.; writing-original draft preparation, H.T.B. and S.C.; writing-review and editing, H.T.B. and S.C. All authors have read and agreed to the published version of the manuscript. Funding: The first author has been financially supported by the CaPPA project (Chemical and Physical Properties of the Atmosphere), which is funded by the French National Research Agency (ANR) through the PIA (Programme d'Investissement d'Avenir) under contract "ANR-11-LABX-0005-01". Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

Figure A1 .

 A1 Figure A1. CDNC measured within the CHIEF cloud chamber as a function of supersaturation.

Table A1 .

 A1 Average and standard deviation (STD) of droplet number concentration (N), with LWC and effective diameter (D eff ) derived from the indicated horizontal tracks through a SC field in the vicinity of the CHIEF cloud chamber. Grey background is used for data measured at cloud base with higher variability in D eff .

	Altitude	N (# cm -3 )	LWC (mg m -3 )	D eff (µm)
	Average	STD	Average	STD	Average	STD	Average	STD
	862	11	210	60	170	60	6.6	1.5
	895	3	110	35	250	50	18.8	1.3
	895	3	90	20	250	50	19.4	1.0
	942	26	145	70	420	165	19.9	0.6
	970	5	120	30	530	60	22.1	0.9
	1020	20	115	30	730	110	24.4	0.7
	1025	5	120	40	650	145	23.5	0.9

Acknowledgments: This work is a contribution to the CaPPA project (Chemical and Physical Properties of the Atmosphere), funded by the ANR through the PIA (Programme d'Investissement d'Avenir) under contract ANR-11-LABX-0005-01.

Data Availability Statement: Sulfate measurements over The Netherlands are available through the EBAS database (https://ebas.nilu.no, accessed on 9 November 2021).

Conflicts of Interest:

The authors declare no conflict of interest.

Appendix A. Cloud Base and Cloud Chamber CDNC Measurements

We present here the set-up and results of an aircraft study in low clouds over the site of Petten, where the CHIEF cloud chamber is located. The aim was to assess at which setting of the chamber a similar CDNC is observed as in the cloud layer above. The campaign took place in September 1996 with specific evaluation of the flight on September 4th with a closed cloud deck. The research aircraft was a PA 31 Navajo Chieftain two engine turbo-prop. Several horizontal transects were constructed in which the altitude was kept constant. The droplet number concentration was measured with a FSSP-100 (PMS Inc., Las Cruces, NM, USA), working at a frequency of 2 Hz, similar to that installed in the cloud chamber.


Romane Boutillier
email: romane.boutillier@enpc.fr

Laurent Hauswirth

Olivier Baverel

Topology generation of architectural meshes adapted to the support conditions

Keywords: Topology finding, Gridshell, Form Finding

The performances of gridshells depend on the topology of the grid but also on the geometry of the underlying surface. Topology and geometry are linked: the shape of a hanging chain model depends on the pattern of the suspended fabric. In this paper, a method is proposed that generates topologies by varying continuous parameters. The obtained grids are adapted to the support conditions of the considered surface: elements converge towards the supports and arrive perpendicular to the edges otherwise. Grids are then lifted: the spatial mesh is the result of an optimisation to have a funicular surface and flat panels. The spatial mesh is then mechanically evaluated and its performance according to different criteria is represented in a Pareto front. A non-intuitive result shows that a large number of different topologies are situated on the Pareto front. These results show the strength of the method to explore different topologies.

Introduction

Necessity of topology generation

Meshes are everywhere in free form architecture, especially when it comes to constructing surfaces. On so-called discrete surfaces, or gridshells, made of beams and panels, the mesh is clearly visible: the beams are the edges of the mesh, the panels are the faces. The topology of the grid, i.e. the connectivity between the different elements, is thus apparent.

The construction of the surfaces must meet several requirements, of a geometrical nature and relative to manufacturing (flat panels are less expensive and easier to implement, for example), of a mechanical nature (the engineer seeks to reduce deformations, to direct the forces towards the supports) or even aesthetic (one prefers an equal distribution of the elements and panels of regular shape)

The geometry of the grid, its shape, is one of the ways to meet these objectives. But before the geometry, it is necessary to choose the topology of the grid.

The topology of the grid can be derived from the geometry to be discretised. In the case of the Milan Trade Fair (Figure 1), the abrupt change in curvature makes it necessary to introduce higher valence nodes (i.e. nodes that connect more elements, coloured on the image) in order to limit the distortion of the panels [START_REF] Schlaich | New trade fair in milan -grid topology and structural behaviour of a free-formed glass-covered surface[END_REF]. Isler's hanging model [START_REF] Chilton | Heinz Isler: the engineer's contribution to contemporary architecture[END_REF] But the topology of the grid can also control the resulting shape, in a process of form finding as illustrated by the image of a Heinz Isler model (Figure 2). This shape was obtained by suspending a net, whose topology (the textile orientation) influences the deformation under gravity loading. The influence of the topology is also illustrated in [START_REF] Chilton | Rooted in nature: Aesthetics, geometry and structure in the shells of heinz isler[END_REF] where different fabric patterns are lifted, and in Figure 6 where a quad mesh with isolated singularities is submitted to gravity with a constraint on panel planarity.

In general, most form finding methods depend on the definition of an input grid topology, be it thrust network analysis [START_REF] Block | Thrust network analysis: A new methodology for threedimensional equilibrium[END_REF], the Marionette method [START_REF] Mesnil | Marionette meshes: Modelling freeform architecture with planar facets[END_REF] or other historical methods developed by Frei Otto at the Institute for Lightweight Structures [START_REF] Otto | IL 10 Gitterschalen / Grid Shells[END_REF].

Existing methods of topology generation

Topological generation has not been widely studied for architectural structures, but is a subject of research in the computer graphics community (see surveys [START_REF] Bommes | Quad-mesh generation and processing: A survey[END_REF][START_REF] Campen | Partitioning surfaces into quadrilateral patches: A survey[END_REF] and [START_REF] Dong | Harmonic functions for quadrilateral remeshing of arbitrary manifolds[END_REF][START_REF] Jezdimirovic | Quad layouts with high valence singularities for flexible quad meshing[END_REF] which are closer to the method described hereafter).

Topological generation is not to be confused with topological optimisation, which optimises the distribution of material in an element. R. Oval [START_REF] Oval | Rule-based topology finding of patterns towards multi-objective structural design[END_REF][START_REF] Oval | Featurebased topology finding of patterns for shell structures[END_REF] developed a method of topology generation based on the boundary of the surface to be meshed, as well as features such as points or curves chosen by the designer. The exploration of the different grids is then controlled by a grammar of elementary operations on strips.

The method presented here is a continuation of this approach called "topology finding". It proposes a new paradigm where the exploration of the grids is controlled by continuous parameters defined at the boundary. It is therefore intuitive, and starts from the support conditions which are a crucial element in the behaviour of the structure but also its integration in an architectural context.

Importance of boundary conditions

The support conditions are the starting point for the method presented in this article. Beyond aesthetics, the main issue of gridshell design is to carry the load towards the supports where forces are transmitted to the ground. On both surfaces in Figures 3 and4, the grid is formed of regular quadrilaterals. In the case of the Smithsonian Institute, the support conditions are punctual, and the topology of the grid is not changed at the columns. However, the thickness of the beams increases at this point where the forces are concentrated. In the case of Cabot Circus, the regular grid is supported by large edge beams, which interrupt the grid: at the boundary, the beams are of various lengths, and the connections may be unique.

The method, presented in the next section, generates topologies and grids based on parameters defined along the edge. The resulting grids have beams that converge at the supports, and beams that arrive perpendicular to the edge elsewhere. They are therefore optimal from a mechanical point of view.

2 Topology and geometry generation method

Topology generation

The topology finding process is detailed in [START_REF] Boutillier | Topology generation of structural grids for architectural surfaces[END_REF]. It relies on the computation of minimal surfaces that are only an intermediate tool in the topology finding process, but are not connected to the architectural shape. However, the boundary of the minimal surface is chosen in relation to the supports of the real architectural surface, in order to find a topology and a grid that are adapted to the boundary conditions. One example is shown Figure 5, the minimal surface boundary is plotted in red, its vertical components are aligned with the columns shown underneath. The grid is the discretisation of streamlines of a field, namely the gradient field and its conjugate set the level set, on minimal surfaces which boundary is made of horizontal curves and vertical lines. This choice provides thus a proper alignment of the grid to the boundary, and smooth streamlines which intersect at 90 degrees. Moreover, level lines converge at the location of vertical boundary lines, which we choose to locate at the supports. By doing so forces are channelled to the supports. This phenomenon is illustrated Figure 5, where grey lines converge at six points on the boundary, where columns are, and arrive perpendicularly to the edge otherwise.

The boundary of the minimal surface can be modified: it is subdivided into slots whose heights can vary while keeping the same planar projection, and thus the resulting grid changes as well. This relation between the boundary and the grid is at the core of the generation process shown hereafter.

Generation of grid skeleton

In this study, grids with quad panels are generated. Some triangular panels appear at the boundary, and the boundary point they connect is called a pole. Other irregular points, located on the boundary or in the shape, are called singularities. At these points on the minimal surface, the gradient is null: singularities are saddle points. The topology finding process falls out into few parts: the determination of singularities (shown in red in Figure 5) and the computation of a skeleton. The skeleton, represented with blue and orange curves Figure 5, is made of streamlines that connect singularities between them and to the boundary. It defines patches: quadrilateral or triangular zones that can be discretised regularly.

Determination of the geometry

Direct method for determining the strip subdivisions The number of patch subdivisions is the same for all patches of a strip (a strip is a series of patches that are connected along their opposite edges) and is computed to get a target average edge length (of about 1.2 meters in our example).

The resulting grid is made of quads panels (except at poles) that are not distorted because they come from the discretisation of lines that meet at right angle. There are overall distributed nicely in the mesh. However, concentration of lines at poles, and large faces at singularities can be noted. This could easily be remeshed locally, and the former grids are kept for the evaluation, despite this constructive remark.

Direct method for computing an adapted shape In this study, in order to compare topologies, only one shape is computed for each topology. This shape depends on the topology, but is not the only one that is adapted to this topology. It is the result an optimisation under two major constraints: flat panels and a funicular shape. This optimisation is run with Kangaroo [START_REF] Piker | Kangaroo: Form finding with computational physics[END_REF]. The funicular load is proportional to the tributary area in plane, therefore all grids, that have the same planar projection, are submitted to the same loading. Other goals are applied for achieving a smooth shape: mesh edge lengths are controlled, and a stiffness is provided at the ridge beams to smooth curvature. The optimised shape is rescaled to get a maximum target height that is the same for all grids (and corresponds to a ratio height/span of 0.2). Grid heights could also be normalised to get a constant enclosed volume. This lifting process is illustrated on two grids with different topologies in Figure 6: because the shape depends on the topology, the resulting geometries vary.

As mentioned in the introduction, this method for generating topologies can also be used as a form finding method, as the resulting geometry is directly related to the topology of the grid. Therefore, each geometry is also evaluated as an isotropic shell, in order to compare their performance.

Case study: generation of grids on a hexagonal boundary

The method is illustrated on a regular shape, that can be used to cover an interior space or as a building (like the restaurant of Felix Candela in Xochimilco [11]). We consider a point supported structure, whose boundary projection is an hexagon, where supports are located at the discontinuities of the boundary. The crenellated contour is thus made of six slots, and five parameter heights are considered (when removing the rigid body motion). Some topologies that are found are shown Figure 7.

Topologies that can be reached with this method

Even if this contour is very symmetrical, many topologies can appear. The types of topologies that can occur are controlled by the Poincaré Hopf theorem: the sum of the indices of the vertices must be equal to the Euler characteristic of the surface (equal to 1 in this case).

Fig. 7. Selection of topologies that can be achieved with the presented method. Various combinations of vertex indices can be created.

Mechanical model

Structural behaviour Three structural types are tested: a shell (isotropic), a braced gridshell (anisotropic but with a membrane stiffness) and a gridshell without bracing (anisotropic). The gridshell without bracing serves as a reference for equivalent mechanical properties. The bracing section for the diagonals of the braced gridshell is taken as a tenth of the regular beams section. The shell properties are chosen in order to get a similar rigidity than the braced gridshell.

Applied loads and boundary conditions A symmetric gravity load is applied to mesh points and proportional to the tributary area. Hinged supports are set at the six corners.

Initialisation of the model Some parameters are chosen at the first iteration and kept constant for the rest of the generation process (same structure mass and section inertia in all grids). The only variables of the generation are those changing the topology, i.e the slot heights of the boundary of the minimal surface.

Evaluation of the grids The grid geometry is thus given, and its properties can be evaluated. Among all the criteria that can be chosen, dimensionless ratios are built: the thrust divided by the vertical reaction, which can be a design criteria for structures on top of existing building for example, the ratio between axial and bending strain energies, which characterises deformations and forces in the structure, and deflection, which is a design factor for relatively low structures (deflection is kept as it is, since the height and span of the different structures are the same). Since several criteria are involved, the Pareto front of the generated grids is evaluated.

Results

In total, three structural topologies are tested. More than a hundred of grids are generated, but one cannot prove that it constitutes an exhaustive scan of the space of grids achievable with this method. Nevertheless, this sample allows to see the diversity of possible designs, and to compare grids between them (see Figure 8). Pareto Front The Pareto front is calculated for the gridshell without bracing.

Over the 138 tested grid, 24 of them belong to the Pareto front, and are represented Figure 8. These grids can be grouped into topological families. Indeed, the generation method controls both the topology of the grid and the position of the elements in space. Some topologies are also equivalent (with one rotation) and are grouped together. Finally, eleven different topologies are present on the Pareto front. These grids are plotted in the objective space, a three dimensional space whose axis are the supports reactions, the ratio of strain energies and deflection. The grids belonging to the Pareto front are represented in red among all other generated grids represented in blue in Figure 9. The results are represented as a projection in the plane of two criteria (in this case the deflection on the ordinate and the ratio of horizontal to vertical reactions on the abscissa). A representative of each family is also displayed. This visualisation tool shows how the grids are positioned in relation to each other. It also allows the designer to make choices according to the criteria to be met.

Comparison of the structure typologies Three structural typologies are tested and compared in Table 1. Values are computed for two topologies, the symmetrical one with one singularity in the center, and another topology with a singularity in the surface, two in corners, and no symmetry in the grid (these topologies are shown in Figure 11 and the resulting 3D shapes are shown Figure 10).

As expected, adding bracing improves drastically the behaviour of the grid, in particular with regard to strain energies. The relative performance depends on the type of structure considered: with bracing, the symmetrical topology works better than the asymmetrical topology. Deflection is much larger for the shell models than for the grids. The shell model serves as an evaluation of the geometry obtained with the grid topology. Stresslines can be computed on this shell, and compared to the grid that was used to create the geometry. Two examples are shown Figure 11. The stress lines of the two funicular forms are completely different. However, the shape of the lines is close to the underlying grid topology. The deviations may be due to the added planarity objective, or to the method of determining these lines on imperfect meshes. This topic deserves to be explored in more detail.

Conclusion and perspectives

The presented method allows the generation of a variety of grid topologies, all adapted to the support conditions, by varying parameters defined along the edge. This variety is reflected in the Pareto front of the case study: a real choice is offered to the designer according to the chosen criteria. The topology generation tool can also be used upstream of many formfinding methods, for which the topology is a given. Local remeshing techniques and local optimisations could be added to improve grid performance. The link between topology and isotropic shells deserves further investigation.

Fig. 1 .

 1 Fig. 1. Milan Trade Fair (Fuksas/SBP) Fig. 2.Isler's hanging model[START_REF] Chilton | Heinz Isler: the engineer's contribution to contemporary architecture[END_REF]

Fig. 3 .

 3 Fig. 3. Smithsonian Institute, Washington, DC (Foster and Partners / Buro Happold -2001) [8].

Fig. 4 .

 4 Fig. 4. Cabot Circus, Bristol (Chapman Taylor with Nayan Kulkarni / SKM -2008)[START_REF] Schober | Cabot circus, bristol ebene vierecknetze für freigeformte glasdächer[END_REF]

Fig. 5 .

 5 Fig. 5. Topology and grid generation process. A crenelated boundary (in red) is built over the real supports of the surface. This artificial boundary is controlled by height parameters. A minimal surface (in blue) is computed on this boundary. A topological skeleton is then retrieved (top right). It is made of singularities (red dots) of the height gradient field. Streamlines lines of the gradient are shown in blue, the level set is shown in orange. The skeleton defines patches that can be discretised regularly. The field is finally integrated to get patch subdivisions (bottom right).

Fig. 6 .

 6 Fig. 6. Two grid examples that are lifted with a gravity load and constrained to have planar panels. The two lifted shapes differ, because grid topologies are different.

Fig. 8 .

 8 Fig.8. Grids that belong to the Pareto front when evaluated for a symmetric loading. Different topologies appear, but also geometric variations of the same topology, which are framed with colours.

Fig. 9 .

 9 Fig. 9. Projection of the grids without bracing, in the plane of the support reactions and deflection criteria. The grids belonging to the Pareto front are shown in red, one representative of each topology family of the Pareto front is displayed on the side.

Fig. 11 .

 11 Fig.11. Stress lines under gravity loading computed on the symmetric grid and on a grid with no symmetries and singularities located in the surface and on the boundary, with their associated topology.

Table 1 .

 1 Comparison of the performance of two grids, for three structural types (R. for reactions, E. for strain energy and d for deflection). Perspective views of the grid whose performances are compared for the different structural typologies: a symmetric topology with a central singularity (left) and an asymmetric topology with one singularity in the surface and two at corners (right).

		3-symmetrical topology	asymmetrical topology
		R.	E.	d (mm)	R.	E.	d (mm)
	Gridshell	1.84	4.70	0.62	2.13	5.16	1.0
	Braced gridshell	1.79	1.44	0.49	2.11	1.29	0.75
	Shell	1.83	0.47	5.7	2.19	0.59	18
	Fig. 10.


Laurent Michel
email: laurent.michel@astro.unistra.fr

Julien Abid

François Bonnarel

Mark Cresitello-Dittmar

Somia Floret

Gille Landais

Mireille Louys

Gregory Mantelet

François-Xavier Pineau

Jesus Salgado

Mapping VOTable Data on Data Models: Implementation Status and Prospect

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

MIVOT (?) is an XML schema that defines a standard way to connect VOTable (?) data with VODML (?) compliant data models. The MIVOT syntax reproduces the structure of the mapped classes such as sky positions from Meas/Coords models (?, ?) or fluxes from the Mango draft model 1 . It uses @ref XML attributes to establish connections between model leaves and columns of the data tables. Producing instances of the classes outlined in the MIVOT block is fairly straightforward: simply replicate the class description (a compound XML element) and resolve the pending references with the data cells of the current row. Annotating query responses with MIVOT offers the following advantages:

1. The model views generated that way do not depend on the data server since the models are shared by all stakeholders.

2. The MIVOT syntax allows to connect data to each other e.g., associate measures with flags or errors.

3. MIVOT can provide VOTables with possibly missing metadata (coordinate system details, curation tracking, filter descriptions, flags, link semantics) defined in mapped models.

Using MIVOT in the context of a TAP service

The goal here is to enable a TAP (?) server to annotate query responses on the fly. As it was outlined after the ADASS BoF 2021 (?), the requirements hold the following:

• The client application must be able to discover services able to annotate responses. This can be achieved by either a registry capability or a new TAP parameter specific for that purpose.

• The client application must be able to ask the server to generate MIVOT annotations for a given model.

• The query engine must be able to analyse the ADQL query to determine whether the selected columns can be mapped on the requested data model.

• The annotation engine must be able to retrieve MIVOT serializations of the model components that can be mapped.

• The annotation engine must understand the binding between model leaves and table columns to complete the annotation process.

Client Side

We have defined different API levels that can apply to any language (Python or RUST in our case).

• Level 0: extract the MIVOT block from the VOTable: No reference is resolved. The MIVOT block is made available as a parser subset. Level 0 has been implemented in Astropy 6.0.

• Level 1: provide access to a parser tree whose structure matches the model view of the current row. The internal references (e.g. association with coordinate systems) have been resolved. The attribute values have been set with the actual table data. This parser subset can be used for building whatever objects.

• Level 2: entail a few methods to ease the browsing of the level 1 output. Level 2 API allows to retrieve MIVOT elements by either class roles or data types. At this level, the MIVOT block still needs to be handled as parsed XML elements.

• Level 3: transform the layer 1 output into a dynamic object (in Python). Different components can be accessed through object paths. The field names of this object are derived from the roles played by the attributes in the model context.

• Level 4 (Python): build Astropy objects (e.g., SkyCoord) from the level 1 output.

Both of our implementations have a similar architecture. Level 0 is embedded in the VOTable parser and all further levels are implemented in a VO aware package.

Status and Prospect

The presented work is still in progress at the time of writing. The PyVO implementation is working up to layer 3. Current developments are guided by IVOA's request for a high-performance demonstrator processing celestial positions with epoch propagation (position, proper motion, parallax and radial velocity including correlated errors). On server side, our implementation is based on a former prototype (?) that led to an online demonstrator xtapdb2 operated at Strasbourg by the Survey Science Consortium of XMM-Newton (SSC).

Figure 1 .

 1 Figure 1. This figure shows how MIVOT binds model leaves with table columns. This mechanism allows client code to easily obtain a model view on the active data row.

Figure 2 .

 2 Figure 2. On the fly annotation by a TAP server: The annotator module knows the binding of the searched table FIELDS with the model leaves. It searches the MIVOT elements matching the classes to be mapped, sets the appropriate references in the MIVOT ATTRIBUTE, and packs all of these snippets on top of the VOTable data table.

https://github.com/ivoa-std/MANGO

https://xcatdb.unistra.fr/xtapdb

Acknowledgments. We would like to thank M. Marchand (CDS), B. Sipöcz, P.L. Lim and T. Donaldson (STSCI) for introducing us as AstroPy/PyVO contributors and for their careful review of our PRs. We would also like to thank the CDS, the CNES and the SSC for funding and supporting the internships that made this possible.


Kouhei Ichiyanagi
email: ichiyanagi@knlab.msl.titech.ac.jp

Johan Hebert

Loic Toupet

Hérve Cailleau

Eric Collet
email: beric.collet@univ-rennes1.fr

Philippe Guionneau

Jean-François Létard

Structural investigation of the photoinduced spin transition in the [Fe(PM-BiA) 2 (NCS) 2 ] compound

Keywords: phase transition, photoinduced, spin crossover

We investigated the thermo-and photo-induced phase transitions between low spin (LS) and high spin (HS) states of the molecular crystal of [Fe(PM-BiA) 2 (NCS) 2 ] in the orthorhombic form, by using X-ray diffraction. The structure of the photoinduced HS state, generated from the LS state at low temperature, is compared to the structures of the HS and LS phases at thermal equilibrium and to the thermally trapped HS state. The preliminary results presented here show that the structural reorganization is similar in the different HS states.

Introduction

The switching effects driven by light irradiation is of special interest in physics around molecular materials where electronic and structural degrees of freedom are so strongly coupled that photoinduced phase transition can be generated [1,2]. Spin transition materials represent a very nice example of highly cooperative phenomenon around photo excitation [START_REF] Gutlich | [END_REF][START_REF]Spin Crossover in Transition Metal Compounds[END_REF]. Accordingly, a photo-switching in solids can be driven with laser excitation, and the resulting change of the physical properties of the material may be applied in optical devices for digital information storage for instance.

The [Fe (PM-BiA)2(NCS)2] compound investigated here exhibits a phase transition between a diamagnetic low spin state (LS, S=0) and a paramagnetic high spin state (HS, S=2) [START_REF] Létard | [END_REF]. X-ray diffraction investigations evidenced two phases with orthorhombic (phase I) and monoclinic (phase II) structures [START_REF] Létard | [END_REF]6]. The phase I shows an extremely abrupt spin transition with a thermal hysteresis around 170 K (Figure 1) [6]. The phase II shows a gradual spin conversion centered at 205 K [START_REF] Létard | [END_REF]. It was demonstrated that in this family of compound it is possible to photo-switch from the LS state (stable at low temperature) to a metastable HS state, by using the appropriate light irradiation conditions, of continuous or pulsed nature [START_REF] Létard | [END_REF]7]. This photo-induced effect is refereed as LIESST effect (light-induced excited spin state trapping) and was discovered in 1994 [START_REF] Gutlich | [END_REF][START_REF]Spin Crossover in Transition Metal Compounds[END_REF]. The LIESST effect in the compound [Fe(PM-BiA)2(NCS)2] has been studied by magnetic techniques [START_REF] Létard | [END_REF]6]. The present paper deals with the investigation by X-ray diffraction of the photoinduced spin transition in the orthorhombic phase I, and the comparison with the structural changes driven by the thermal effects.

The irradiation with a continuous laser excitation at 830 nm (5mW/cm 2 ) of the LS state at low temperature (10 K) makes it possible to switch from the LS state to the photoinduced HS state. It was also evidenced that the HS state stable at high temperature can be trapped at low temperature by a flash cooling [8]. The lifetime of the metastable photoinduced or trapped HS state in this compound can reach several days at low temperature [6], and is decreasing as the temperature is increased. The T (LIESST) temperature, where the relaxation to the LS state becomes very fast without laser excitation was estimated to be 78 K for

[Fe(PM-BiA) 2 (NCS) 2 ] [6]
. This photoinduced HS state can also be kept while increasing the temperature under laser excitation up to about 80 K. The competition between the thermal relaxation process which depends non-linearly on the fraction of HS molecule and the transformation of HS molecules generated by laser excitation, gives rise to another interesting phenomenon known as LITH (Light-Induced Thermal Hysteresis). As shown in Figure 1, the fraction of HS molecules depends on the thermal history of the material under laser excitation [START_REF] Létard | [END_REF].

Experiment

The photoinduced phase transition was investigated by using a four-circle diffractometer Xcalibur3 with a 2D sapphire 3 CCD (Oxford-Diffraction) using the Mo Kα radiation. The system was coupled to a continuous laser to generate the photo-induced phase (808 nm, 50 mW/cm 2 ), as shown in Figure 2. The typical sample size of the crystals of [Fe(PM-BiA) 2 (NCS) 2 ] compound used for the X-ray diffraction experiments was 50×50 ×100 µm 3 . The single crystal were mounted with a Oxford Diffraction Helijet Helium flow cryostat, allowing to reach 20 K on the crystal, or in an Oxford nitrogen flow cryostat allowing a better control of the temperature above 78 K. The HS and LS structures of [Fe(PM-BiA) 2 (NCS) 2 ] correspond to the same crystallographic space group (Pccn, with Z = 4). The molecular structures were solved and refined with the Sir97 program and SHELXL97 program [START_REF] Sheldrick | The cell parameters and the data reduction were obtained with CrysAlis from Oxford Diffraction and the structure where refined with SHELXL97[END_REF] at 300 K (HS), 150 K (LS), as well as for the thermally trapped and photoinduced HS states at 20 K, in order to compare both to the already known structures of LS and HS states [6,8,[START_REF] Marchivie | [END_REF]. Up to now, such structural investigations of the photoinduced spin transition were performed in a limited number of materials [11][12][13][14].

Results and discussion

An important signature associated with the change of spin state is the variation of the lattice parameters, as shown in Table 1a. The thermal contraction of the lattice parameter b is quite important, but the huge jumps of a and c sign the thermal phase transition between the HS and LS phases around 170 K. It is associated with the pronounced first order aspect of the phase transition. When a sample in the HS phase at room temperature is directly put in a helium flow (20 K), the HS state is trapped by the flash cooling as the molecular structural relaxation is frozen. This thermally trapped HS structure is very similar to the HS state stable at high temperature [START_REF] Marchivie | [END_REF]. The photoinduced HS state was also easily generated at 20 K from the LS state by using a continuous laser excitation (a few minutes with 800 mW/cm 2 at 808 nm). The lattice parameters and the unit-cell volume of the photoinduced HS state and the trapped HS state are almost similar to the HS one (Table 1a). It is well known that the Fe-N 6 octahedral conformation is strongly coupled to the spin states. Therefore the main structural modifications during the spin conversion occur around the Fe atom [START_REF]Spin Crossover in Transition Metal Compounds[END_REF], and are characterized by the change of the parameters <Fe-N> (the average bond length) and Σ (sum of the deviation from 90{ of the N-Fe-N angles). The molecular structure analysis as a function of temperature is shown in Table .1b and Figure 3. The results we obtained on the thermally trapped HS state at low temperature are in good agreement with previous X-ray diffraction measurements [START_REF] Marchivie | [END_REF]13]. The photoinduced HS state is here again very similar to the other HS states. We also present here the inter-molecular hydrogen contacts between the H and S atoms involving the NCS branch. The molecular structures and inter-molecular interactions of the photoinduced HS, thermally trapped HS and thermal HS states are here again almost similar, as shown on Figure 3 with the S•••H-C contact (Table . 1(b)). This inter-molecular hydrogen contact is well depending on the changes of the lattice parameters. It was also reported that this molecular hydrogen bond influences the stability feature of the photoinduced state [START_REF] Marchivie | [END_REF]. All the results presented here indicate a high similarity between the structural changes associated with the thermal phase transition and the ones associated with the photoinduced transition.

To conclude, we have investigated the phase transition in the [Fe(PM-BiA) 2 (NCS) 2 ] compound by using X-ray diffraction on a four-circle X-ray diffractometer. We concluded that the photoinduced and thermally trapped HS states are consistent with the HS state stable at high temperature. The results presented here also stress the coupling between the intra-molecular reorganization and the inter-molecular contacts, known to be playing a key role to stabilize the photoinduced or thermally trapped states. <Fe-N> is the average bond length of Fe-N6 octhahedron. is defined as the sum of the absolute values of the deviation from 90{ of the 12 cis angles in the coordinate sphere [START_REF] Marchivie | [END_REF].

Figure. 1

 1 Figure. 1 Temperature dependence of the magnetic susceptibility measurements in the cooling and warming mode. The arrow line represents the switching from LS state to photoinduced HS state under laser irradiation (830 nm, 5mW/cm 2 )

Figure. 3

 3 Figure. 3View of the molecular structures and intermolecular contacts of LS state at 150 K, photoinduced HS state at 20 K, thermal HS state at 300 K and thermally trapped HS state along the b axis.

Acknowledgement: this work was performed in the frame of the College Doctoral Franco-Japonais, and the MAGMANet network. Authors thank the ANR and Region Bretagne for financial support.