Isabelle Fourel
email: isabelle.fourel@vetagro-sup.fr

François-Xavier Couzi

Virginie Lattard

generation anticoagulant rodenticides reveals a different bioaccumulation of diastereoisomers in the food chain of the Réunion harrier (Circus maillardi)

Keywords:

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

Monitoring the hepatic residues of cis-and trans-diastereoisomers of second generation anticoagulant rodenticides reveals a different bioaccumulation of diastereoisomers in the food chain of the Réunion harrier (Circus maillardi)

Isabelle Fourel, François-Xavier Couzi, Virginie Lattard

To cite this version:

Isabelle Fourel, François-Xavier Couzi, Virginie Lattard. Monitoring the hepatic residues of cis-and trans-diastereoisomers of second generation anticoagulant rodenticides reveals a different bioaccumulation of diastereoisomers in the food chain of the Réunion harrier (Circus maillardi). Science of the Total Environment, 2021Environment, , 779, pp.146287. 10.1016Environment, /j.scitotenv.2021.146287.146287. hal-03299570

Introduction

Rodent populations remain one of the primary causes of economic losses in the agricultural sector. Sugar cane plantations are the main agricultural activity on the Réunion Island, an overseas territory of France in the Indian Ocean, and annual losses due to rodent damages regularly reaches 10 to 15% of the harvest [START_REF] Grollier | Dossier spécial : Lutter contre les ravageurs[END_REF]. Public health authorities also target rodents because they may transmit zoonotic diseases such as leptospirosis (Buckle and Smith, 2015). The risk of rodent-to-human transmission exists on the island because of the abundance of rodents, their close association with humans, and their carrying of pathogenic leptospires without any evident health effects [START_REF] Guernier | Human Leptospirosis on Réunion Island, Indian Ocean: Are rodents the (only) ones to blame?[END_REF]. Moreover, as is often in an insular context, rodents are also a threat to biodiversity conservation (Buckle and Smith, 2015).

The use of anticoagulant rodenticides (ARs) on a worldwide basis constitutes the main method of rodent population control. ARs block blood clotting and lead to the death through induced haemorrhages. The massive increase in rodent genetic resistance to first generation ARs (FGARs), as observed in Western European countries, has led to the development of second generation ARs (SGARs).

Nevertheless, the recognized ecotoxicity is a major adverse effect and induces a limitation of the use of these SGARs. Exposure or poisoning of non-target wildlife are usually due to the ingestion of poisoned rodents by their predators or scavengers [START_REF] Rattner | Adverse outcome pathway and risks of anticoagulant rodenticides to predatory wildlife[END_REF][START_REF] Shore | Monitoring rodenticides residues in wildlife[END_REF], as reported in many studies on raptors throughout the world [START_REF] Hughes | Monitoring agricultural rodenticide use and secondary exposure of raptors in Scotland[END_REF]Ruiz-Suarez et al., 2014;[START_REF] Langford | The occurrence of second generation anticoagulant rodenticides in non-target raptor species in Norway[END_REF][START_REF] Christensen | High exposure rates of anticoagulant rodenticides in predatory bird species in intensively managed landscapes in Denmark[END_REF][START_REF] Coeurdassier | Unintentional wildlife poisoning and proposals for sustainable management of rodents[END_REF][START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF][START_REF] Huang | Increased rodenticide exposure rate and risk of toxicosis in barn owls (Tyto alba) from southwestern Canada and linkage with demographic but not genetic factors[END_REF]Lopez-Perea and Mateo, 2018). The long tissue persistence of SGARs in rodents and wildlife appears to be the main cause of this secondary exposure and, consequently, the main determinant of the ecotoxicity of these molecules. However, such long persistence (for some SGARs, half-lives in liver would be higher than 100 days) is not needed to kill rodents and in fact dramatically increases the secondary exposure.

This ecotoxicological problem has led European authorities to identify these molecules as "candidates for substitution" in rodent management.

In the Réunion Island, rodent eradication campaigns (Norway rats (Rattus norvegicus), black rats (Rattus rattus), and house mice (Mus musculus)) started to be organised in the 1970s. Since then, different ARs (chlorophacinone and different SGARs such as bromadiolone, difenacoum and brodifacoum) have been used, either to limit leptospirosis transmission, or to protect crops and urban settings or even biodiversity (rats being predators of endemic birds species) [START_REF] Grollier | Dossier spécial : Lutter contre les ravageurs[END_REF]. However, [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF] have recently reported the potential poisoning of the Réunion harrier (Circus maillardi) by ARs. This species, endemic to the Réunion Island, is the single breeding raptor of the Island and is classified as endangered (IUCN 2020). Like many raptors, the diet of the Réunion harrier is based on rodents [START_REF] Grondin | Plan de conservation du Busard de Maillard (Circus maillardi)[END_REF]. It has been shown that the Réunion harrier was particularly at risk and multi-exposure to ARs was demonstrated. From 1999 to 2016, 93% of the collected carcasses (n=58) contained ARs hepatic residues, 95% of which were SGARs. This exposure was related to the area of sugar cane plantations and to several active ingredients used in crops, mainly brodifacoum, bromadiolone and difenacoum. Most of the birds (over 60%) had summed liver ARs residues above 100 ng/g, and 34% above 200 ng/g. To determine whether AR exposure could pose a threat, the measured concentrations were compared to toxicity threshold concentrations (> 100-200 ng/g). The use of these limits was based on several experimental observations or probabilistic studies with collections of numerous individuals [START_REF] Newton | Empirical evidence of sideeffects of rodenticides on some predatory birds and mammals[END_REF][START_REF] Thomas | Second generation anticoagulant rodenticides in predatory birds: probabilistic characterisation of toxic liver concentrations and implications for predatory bird populations in Canada[END_REF][START_REF] Huang | Increased rodenticide exposure rate and risk of toxicosis in barn owls (Tyto alba) from southwestern Canada and linkage with demographic but not genetic factors[END_REF]. Hepatic residues of SGARs in carnivorous birds were correlated with internal haemorrhage without associated trauma, and a probability of the lethal risk was statistically characterized as a function of the SGARs residues. As a result, it was defined that ∑SGARs=100 ng/g was representing a lethal risk of 11% for the barn owl (Tyto alba)

which rose to 22% when ∑SGARs=200 ng/g [START_REF] Thomas | Second generation anticoagulant rodenticides in predatory birds: probabilistic characterisation of toxic liver concentrations and implications for predatory bird populations in Canada[END_REF]. This studies were concerning a restricted number of bird species (n=4) and mainly the barn owl.

However, it was extrapolated to other species even though there was evidence of inter-species variations [START_REF] Thomas | Second generation anticoagulant rodenticides in predatory birds: probabilistic characterisation of toxic liver concentrations and implications for predatory bird populations in Canada[END_REF] or inter-individual variations depending on class age and/or sex [START_REF] Thomas | Second generation anticoagulant rodenticides in predatory birds: probabilistic characterisation of toxic liver concentrations and implications for predatory bird populations in Canada[END_REF][START_REF] Huang | Increased rodenticide exposure rate and risk of toxicosis in barn owls (Tyto alba) from southwestern Canada and linkage with demographic but not genetic factors[END_REF]Ruiz-Suarez et al., 2016), or even different molecular toxicity between ARs [START_REF] Eason | Vertebrate pesticide toxicology manual (poisons)[END_REF][START_REF] Erickson | Potential Risks of Nine Rodenticides to Birds and Nontarget Mammals: A Comparative Approach[END_REF].

A major pathway for improving these molecules has been recently proposed based on their stereoisomerism. SGARs have two asymmetric carbons in their chemical structure and thus exist as four stereoisomers (two diastereoisomers, each of them being a pair of enantiomers). In Europe, the proportions of cis-and transdiastereoisomers in baits follows the European directives concerning the placing of biocidal products on the market (98/8/EC Product Type 14). This legislation is valid on French ultramarine territories. For rats, pharmacokinetics were investigated and the half-lives of cis-diastereoisomers and trans-diastereoisomers were shown to be different for all SGARs, although the rodenticide efficiency was similar [START_REF] Damin-Pernik | Management of rodent populations by anticoagulant rodenticides: toward third generation anticoagulant rodenticides[END_REF]2017). Moreover, hepatic residues of SGARs diastereoisomers have been quantified in exposed or intoxicated wildlife feeding on rodents, i.e. red kites, and red foxes (Fourel et al, 2017a;2018)

Material and methods

Sample collection

Sample collection was described in [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. From 1999 until February 2016, 58 Réunion harrier livers were collected. The carcasses were stored at -18 °C from discovery or death of injured animals at the Société d'Etudes

Ornithologique de la Réunion (SEOR). After the collection, the livers were stored at -18°C until analysis.

ARs analysis

The analysis of ARs was carried out following the method recently described by Reaction Monitoring mode for each analyte. Calibration curves were generated for each of the FGARs as well as for each of the SGARs diastereoisomer. The limits of quantification varied between 1 and 2 ng/g wet weight (ww) and the recovery rates were above 70%. The limits of detection varied between 0.5 and 1 ng/g ww as detailed in Fourel et al. (2017b). The proportions of the cis-and trans-isomers in hepatic residues were calculated except when the value for one of them was between the limit of detection and the limit of quantification.

Statistical analysis

The Mann Whitney Wilcoxon test was used to explore the differences between two groups of brodifacoum concentrations when the proportions of trans-brodifacoum were zero or not. The proportions of major-isomer were also compared between positive livers and baits with the Mann Whitney Wilcoxon test. The significance level of the statistical test was set at a value of p < 0.05.

Diastereoisomers proportions in baits

A local industrial group on the Réunion Island was created in1996 and manufactured baits used during rat control campaigns throughout the study period from 1999 to 2016. The baits were produced on the Réunion Island from concentrated active ingredients imported from European chemical groups (Grolliet and Soufflet, 2011).

Consequently, the diastereoisomers proportions in the baits used on the Réunion Island during our study are the ones defined in the European Union. We analysed in However, the information is missing to confirm that maybe some baits might be imported as well from other non-European countries (for biocide use for example), and that their diastereoisomers proportions might be different from the ones of European countries. This has still to be elucidated with local authorities, but it should not represent most of the used baits.

Results

The results of total hepatic residues for each AR were already reported in

Bromadiolone diastereoisomers

Bromadiolone was detected in 41 (70%) Réunion harrier livers [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. The present results show that bromadiolone hepatic residues consisted only of trans-bromadiolone (100% trans-bromadiolone and 0% cis-bromadiolone) in 40 of the Réunion harrier livers. Cis-bromadiolone was detected in the liver of just one Réunion harrier with 99.6% being trans-bromadiolone and 0.4% cis-bromadiolone.

The mean bromadiolone diastereoisomer proportions in the positive samples were then 100 ±0.1% of trans-bromadiolone (97,9 ± 134 ng/g ww) and 0 ±0.1% of cisbromadiolone (table 1, figure 1).

Difenacoum diastereoisomers

Difenacoum has been quantified in 42 (72%) Réunion harrier livers [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. The present results show that difenacoum hepatic residues consisted only of cis-difenacoum (20,6 ± 19,5 ng/g ww) (100 ±0% cis-difenacoum and 0 ±0% transdifenacoum) for the 42 Réunion harriers (table 1, figure 1).

Brodifacoum diastereoisomers

Brodifacoum has been quantified in 29 (50%) Réunion harrier livers [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. Brodifacoum hepatic residues consisted only of trans-brodifacoum for nine Réunion harriers, and that cis-brodifacoum (the proportion was at least 62%) and trans-brodifacoum (with proportion up to 38%) were both present in the liver of 20

Réunion harriers. For seven of them, trans-brodifacoum was detected but concentrations were under the LOQ, and so the diastereoisomers proportions could not be reported. The mean brodifacoum diastereoisomer proportions in the positive samples were 89.5 ±11.8% of cis-brodifacoum (125 ± 132 ng/g ww) and 10.5 ±11.8%

of trans-brodifacoum (11.4 ± 16,4 ng/g ww) (table 1, figure 1).

Difethialone diastereoisomers

Difethialone has been quantified in 10 (17%) Réunion harrier livers [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. The results show that difethialone hepatic residues is only cis-difethialone (100 ±0% cis-difethialone (52,3 ± 63,5 ng/g ww) and 0 ±0% trans-difethialone) for the 10 Réunion harriers (table 1, figure 1).

Considering the results for the whole SGARs, 96.3% of the hepatic residues from the 58 Réunion harriers are major-diastereoisomers and only 3.7% are minordiastereoisomers. 100% of minor-diastereoisomers are trans-brodifacoum (table 1, figure 1). The concentrations (ng/g ww) of the sum of major or minor diastereoisomers of SGARs from the Réunion harrier liver samples are represented in figure 2. A black dotted line represents a toxic hepatic threshold of 100 n/g considered to represent a lethal risk for birds and mammals according to the work from [START_REF] Newton | Empirical evidence of sideeffects of rodenticides on some predatory birds and mammals[END_REF] and [START_REF] Thomas | Second generation anticoagulant rodenticides in predatory birds: probabilistic characterisation of toxic liver concentrations and implications for predatory bird populations in Canada[END_REF].

A significant difference was evidenced between the proportion values of cisbrodifacoum, cis-difenacoum, trans-bromadiolone or cis-difethialone in positive livers and the ones in baits with respective results of Mann Whitney Wilcoxon test, p < 0.0001, p < 0.0001, p < 0.0001, p < 0.0003 (see figure S1). The differences between the proportions of all SGARs diastereoisomers in Réunion harriers' liver and the ones in baits (%liver-%bait) were calculated and represented in figure 3, using the mean proportions in baits as presented in the material and method section. The positive results (%) stands for the diastereoisomers with a gain of proportion in the liver compared to the baits, which is the case for all major diastereoisomers. The negative results (%) stands for the diastereoisomers with a loss of proportion in the liver compared to the baits, which is the case for all minor diastereoisomers. The resulting percentages, either positive or negative, are all at the same level for bromadiolone, difenacoum and difethialone. In all instances, the dispersion of the ratios is null for the positive samples with any of those three active ingredients (figure 3). The gain or the loss of difethialone is very small because baits are composed of 98% of cisdifethialone and only 2% of trans-difethialone. Consequently, the proportion of cisand trans-difethialone is weakly modified.

For brodifacoum, the resulting percentage are dispersed for the samples with cisand trans-brodifacoum (n=12) and at the same level for the ones with cisbrodifacoum only (n=8). No significant difference was evidenced between the concentrations of those two kind of samples (Mann Whitney Wilcoxon test, p = 0.6032) (figure S2).

Discussion

Monitoring of the hepatic residues of cis-and trans-diastereoisomers of SGARs

as an indicator of their differential bioaccumulation in the food chain of the Réunion harrier

In the context of leptospirosis in the Réunion island, several active ingredients (ie., bromadiolone, difenacoum, brodifacoum) have been used in crops to reduce rat population, and prevent the transmission of leptospires from rodents to humans (Coeurdassier et al., 2019, Grolliet and[START_REF] Grollier | Dossier spécial : Lutter contre les ravageurs[END_REF]. This probably enhanced multi-exposure of the Réunion harrier to SGARs, which showed to be important.

Moreover, difethialone residues were also detected in bird livers, probably because of a biocide usage in urban settings. Consequently, the Réunion harrier is a relevant bio-indicator to assess the effects of SGARs on predator populations and especially, in this study, to monitor the changes of the proportions of SGAR diastereomers from bait to predator via the food chain. This modification does not provide any specific information on the role of either the rodent or the predator because the rodent may have eliminated all minor-diastereoisomers (except trans-brodifacoum) before being consumed by the predator. Although no data is available to know to what extend non target species like raptors change proportions, there is evidence that rodents at least do according to [START_REF] Damin-Pernik | Management of rodent populations by anticoagulant rodenticides: toward third generation anticoagulant rodenticides[END_REF]2017. However, this study only gives a global information on the food chain, and the proportions of the two diastereoisomers of SGARs in the liver of the Réunion harriers are different from those that are occurring in the baits (figure S1). Moreover, this is observed for all the SGARs evolved in the exposure i.e. bromadiolone, brodifacoum, difenacoum, and even difethialone. The proportions of the minor diastereoisomers have always been weakened in the liver of the Réunion harriers compared to the baits (figures 1 and 3).

Due to the differentiated pharmacokinetics of the cis-and trans-forms, rodents have be shown to change the diastereoisomers proportions in their body with an observed bioaccumulation of major diastereoisomers in rats [START_REF] Damin-Pernik | Management of rodent populations by anticoagulant rodenticides: toward third generation anticoagulant rodenticides[END_REF]2017).

This was confirmed in this study because the persistence of minor diastereoisomers in the food chain of the Réunion harriers was obviously lower than the persistence of major diastereoisomers. Cis-bromadiolone, trans-difenacoum and trans-difethialone

were not detected and therefore do not appear to be persistent in the food chain. As they were not present even in the most exposed Réunion harriers there was strong evidence that cis-bromadiolone, trans-difenacoum and trans-difethialone were unlikely to play a role in the poisoning.

Trans-brodifacoum residues were found in some of the Réunion harrier livers, but in weaker proportions than in the baits. Therefore, trans-brodifacoum has been shown to be less persistent in the food chain than cis-brodifacoum but more persistent than cis-bromadiolone and trans-difenacoum (figures 1 and 3).

The proportion of diastereoisomers is very homogeneous for all livers with bromadiolone and difenacoum, whereas it is partially dispersed for those with brodifacoum (figure 3). It might be the consequence of the total elimination of cisbromadiolone and trans-difenacoum by rodents (first step of the food chain) and/or the Réunion harrier (second step of the food chain). It is evident that not all harriers ate the same amount of bromadiolone or difenacoum, nor did they all die at the same time after ingesting it. Therefore, for the samples of our study, the total elimination of cis-bromadiolone and trans-difenacoum appears to be independent from those two variables. Rats may have greater capacity to metabolize xenobiotics [START_REF] Saengtienchai | Comparison of xenobiotic metabolism in phase I oxidation and phase II conjugation between rats and bird species[END_REF]. Probably because of their more rapid and perhaps total elimination by rodents, it is conceivable that Réunion Harriers have never been exposed to cisbromadiolone and trans-difenacoum. For brodifacoum, the slower elimination of both diastereoisomers of brodifacoum by the rodents and/or the predator could be responsible for the dispersion. The latter could reflect the duration since last ingestion and/or the ingested quantity for both rodent and predator. However, no significant difference was evidenced between the concentrations of livers with or without transbrodifacoum (p = 0.6032) (figure S2). The presence of two variables (rodent and predator) of both time and quantity makes the establishment of a correlation complex.

Exposure to cis-brodifacoum and trans-bromadiolone above a toxic threshold

As explained above in the introduction section, secondary ARs poisonings have been commonly diagnosed by residue levels above a toxic threshold of 100-200 ng/g [START_REF] Newton | Empirical evidence of sideeffects of rodenticides on some predatory birds and mammals[END_REF][START_REF] Thomas | Second generation anticoagulant rodenticides in predatory birds: probabilistic characterisation of toxic liver concentrations and implications for predatory bird populations in Canada[END_REF], and it was admitted as such for the Réunion harrier [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. Considering that a limit of 100 ng/g could represent a minimum toxic threshold for the Réunion harrier species, it was obvious that the summed major-isomers residues (until 652 ng/g) were above this limit for 60% of the individuals when the minor-isomers (only trans-brodifacoum) (less than 46 ng/g) were always below this limit as shown in figure 2. Brodifacoum and bromadiolone were detected at levels of concern to cause poisoning as shown in [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. According to the results in figure 1, only trans-bromadiolone and cis-brodifacoum have hepatic concentrations above this toxic threshold, and never cis-bromadiolone or trans-brodifacoum. Brodifacoum has been used in crops from 2013 onwards, which is consistent with the evidence of hepatic toxic levels of brodifacoum. However, present work has shown that cis-brodifacoum was more likely to be responsible for the poisoning than trans-brodifacoum as concentrations of trans-brodifacoum were always below a toxic threshold.

Bromadiolone was also suspected of poisining harriers. This work has demonstrated that bromadiolone liver residues were only trans-bromadiolone, and that cisbromadiolone was probably not involved in the poisoning of the Réunion harriers.

Considering the cumulative effects, multi-exposure might probably involve as well difethialone and difenacoum with consequent exposure above a toxic threshold. In this case, the present study showed that trans-difenacoum and trans-difethialone were not involved in the exposure of the Réunion harriers above a toxic threshold, but more likely cis-difenacoum and cis-difethialone. The gain or loss of cis-or transdifethialone relative to the baits is very small because baits are composed of 98% of cis-difethialone and only 2% of trans-difethialone (figure 3). Consequently, the proportion of cis-and trans-difethialone is only slighty modified (figure 1). Further efforts should be made in the future to better document the comparative persistence of cis-and trans-difethialone.

This work, combined with similar results previously obtained on other non-target rodent-predatory wildlife (Fourel et al., 2017a[START_REF] Fourel | Liver and fecal samples suggest differential exposure of red fox (Vulpes vulpes) to trans-and cis-bromadiolone in areas from France treated with plant protection products[END_REF] and on rodents [START_REF] Damin-Pernik | Management of rodent populations by anticoagulant rodenticides: toward third generation anticoagulant rodenticides[END_REF], 2017;Fourel et al., 2017b), suggest that a change in the proportions of diastereoisomers in actual baits could reduce the secondary toxicity of SGARs below a toxic threshold. Consequently, it could decrease the exposure of non-target rodent-predatory wildlife.

Since both diastereoisomeric forms of SGARs are both effective in killing rodents [START_REF] Damin-Pernik | Management of rodent populations by anticoagulant rodenticides: toward third generation anticoagulant rodenticides[END_REF]2017), it then seems clear that the proportions of diastereoisomers in baits should be reconsidered in favour of minor diastereoisomers. It could help in the conservation of predators but retain primary toxicity for the long-term management of rodent populations. It might be an important step towards safer use of anticoagulant rodenticides and should be added to the mitigation measures already recommended in [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF] for the conservation of the Réunion harrier.

Conclusion

The current results on the Réunion harrier have confirmed the previous ones on the red kite and red fox. The concept, which has been observed for the red kite and red fox, both predators of water voles in metropolitan French areas (Fourel et al., 2017a[START_REF] Fourel | Liver and fecal samples suggest differential exposure of red fox (Vulpes vulpes) to trans-and cis-bromadiolone in areas from France treated with plant protection products[END_REF], may be extended to the Réunion harrier. The latter is the predator of other rodent species on the Réunion island, which are Norway rats (Rattus norvegicus), black rats (Rattus rattus), and house mice (Mus musculus). All those results have permitted to draw conclusions on the fact that major SGARs diastereoisomers (trans-bromadiolone, cis-brodifacoum, cis-difenacoum, and cisdifethialone) were more persistent and more likely responsible for secondary exposure or poisoning of rodent predators, although more species have yet to be investigated.

Aknowlegments

We warmly thank the members of the Concentration in livers (ng/g ww)

Legends of figures

Fourel

  et al. (2017b). A LC-MS/MS for the multi-residual quantification of ARs in Réunion harrier livers was used and complete validation was achieved according to the Guideline on Bioanalytical Method Validation of the European Medicine Agency(EMEA, 2011). The ARs analysed in this study were three FGARs (i.e., warfarin, coumatetralyl, chlorophacinone), and the two diastereoisomers (major or minor proportions for the cis-or trans-isomers) of the five SGARs potentially used in European countries: trans-bromadiolone (major) and cis-bromadiolone (minor), cisdifenacoum (major) and trans-difenacoum (minor), cis-brodifacoum (major) and trans-brodifacoum (minor), cis-flocoumafen (major) and trans-flocoumafen (minor), and cis-difethialone (major) and trans-difethialone (minor). The chromatographic separation of the two diastereoisomers of each SGAR was achieved with a semiporous Poroshell 120 StableBond C18 column (2.1*100mm, 2.7µm). Mass spectrometry measurements were performed with a 6410B Triple Quadrupole from Agilent Technologies (Palo Alto, CA, USA) equipped with ElectroSpray Ionization source in negative mode. Two fragment ions were recorded in dynamic Multiple

Fourel

  et al. (2017b) some baits produced by different commercial brands (n=7) from 2009 to 2013 to determine their proportions of major and minor diastereoisomers.These results were implemented with the analysis of two bromadiolone baits manufactured on the Réunion Island in 2016. Bromadiolone baits contained 75.1 ±2.3% of trans-bromadiolone, brodifacoum baits contained 55.4 ±3.9% of cisbrodifacoum, difenacoum baits contained 59.4 ±5.8% of cis-brodifacoum, difethialone baits contained 98.0 ±1.1% of cis-difethialone.

Figure 1 :

 1 Figure 1: Concentrations and proportions of cis-and trans-diastereoisomers for bromadiolone, difenacoum, brodifacoum and difethialone in the liver of the Réunion harriers, and in European commercial baits dosed in Fourel et al., 2017b. Transbrodifacoum is the only minor diastereoisomer present in the liver of the Réunion harriers. Black dotted line: toxic hepatic threshold of 100 n/g considered to represent a lethal risk for birds and mammals.

Figure 2 :

 2 Figure 2: Concentrations (ng/g ww) of major or minor diastereoisomers of SGARs from the Réunion harrier liver samples. Black dotted line: toxic hepatic threshold of 100 n/g considered to represent a lethal risk for birds and mammals

Figure 3 :

 3 Figure 3: Differences between proportions of all SGARs diastereoisomers inRéunion harriers' liver and in baits (%Harrier liver -%bait). Above the dotted line at zero% stands the diastereoisomers with a gain of proportion in the liver compared to the baits. Underneath the dotted line at zero% stands the diastereoisomers with a loss of proportion in the liver compared to the baits.

  Figure 1

  [START_REF] Coeurdassier | Pesticides threaten an endemic raptor in an overseas French territory[END_REF]. A rate of exposure of 93% was characterized and correlated to the intensive usage of ARs on the Island, and particularly of SGARs (all except flocoumafen) because 95% of ARs liver residues were SGARs. In this work,

complementary results on the diastereoisomers composition of each SGARs and their respective detailed concentrations were determined. They are described below and are reported in table

1 and figure 1.

  SEOR Wildlife care centre, Julie Tourmetz, Samantha Renault, Emilie Robert, and all the field workers, volunteers and Réunion citizens of the wildlife rescue network set up by SEOR. We acknowledge Dr Matthieu Le Corre of the University of La Réunion, Dr Mickaël Sage from CD Eau Environnement, Pierrick Ferret, Colombe Valette and Valérie Grondin from the SEOR for the dissection and liver sampling, and Dr Steve Augiron in charge of coordinating the Réunion Harrier conservation program. This study was co-funded by VetAgro Sup (INRAe) and by the DEAL Réunion (EAMES Project).

Table 1 :

 1 Diastereoisomers proportions of SGARs in Réunion harrier livers (n=58), and ARs concentrations (ng/g wet weight) in positive samples. Anticoagulant rodenticides, SGARs: second generation ARs, < LOD: below limits od detection, NA: non applicable.

	Figure 3


Marianne Clausel

Marc Duruflé

Patrick Joly

Sébastien Tordeux
email: se-bastien.tordeux@inria.fr

A mathematical analysis of the resonance of the finite thin slots

Keywords: thin slot, slit, resonance, Helmholtz, wave equation, 1991 MSC: 35J05, 65N15, 78M30, 78M35

In this article we are interested in the asymptotic behaviour of the solution of a wave propagation problem in a domain including a thin slot. Some rate of convergences are obtained and illustrated by numerical results.

Introduction and position of the problem

The analysis of the propagation of waves in media including thin slots poses, as for any PDE problem involving a small scale (here the width of the slot), some difficulties from the numerical point of view and raises exciting related mathematical questions. Moreover, this kind of problem appears in a lot of applications in acoustics and electromagnetism and has already generated an abundant literature in the engineering community.

In [START_REF] Joly | Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots[END_REF], we studied the case of the 2D time harmonic wave equation with Neumann boundary conditions (which is physically relevant in acoustics and electromagnetism) and investigated the question of building an approximate model, in view of its numerical discretization, consisting of coupling a 1D model for the slot to a 2D model for the rest of the domain of propagation. We proposed a coupling strategy that we analyzed in detail in the case of a semi-infinite straight slot. The basic ingredient of the analysis is of course the study of the asymptotic behaviour of the solution when the width ε of the slot tends to 0. The analysis of the same question for a finite slot, that we wish to investigate in this paper, appears to be more delicate. One has in particular to face the question of resonant frequencies (or equivalently critical slot lengthes), a phenomenon which is already known by engineers [START_REF] Harrington | Electromagnetic transmission through narrow slots in thick conducting screens[END_REF] and that we propose to analyze from both analytical and numerical points of view.

Let us first present our model problem and begin by the description of the geometry of the problem. We assume that we are in 2D (x = (x, y) will denote the current point of R 2 ) and that the domain of propagation Ω ε is given by (with L > 0 and 0 < ε < H):

Ω ε = Ω ∪ O ε , Ω = R 2 \ B, (1)
B = [0, L] × [-H, H], (a rectangle) (2)
O ε =]0, L[×]0, ε[, (a straight finite slot)

We shall call "end points" of the slot the two points A -= (a -, b -) ≡ (0, 0) and A + = (a + , b + ) ≡ (L, 0) and denote A the middle of [A -, A + ].

We assume that a time harmonic wave of pulsation ω > 0 (c = 1) is emitted by a source term f with compact support F included inside the open set Ω:

               Find u ε ∈ H 1 (Ω ε ) outgoing such that: -∆u ε -ω 2 u ε = f, in Ω ε , ∂u ε ∂n = 0, on ∂Ω ε . (4)
It is classical and useful to reduce the problem to a bounded domain thanks to a transparent boundary condition on the circle Γ R = {x ∈ Ω / |x -A| = R},

where the radius R > 0 is large enough so that the obstacle and the support F of f are contained in Ω R := {x ∈ Ω / |x -A| < R}. A Dirichlet to Neumann transparent boundary condition can be put on Γ R as:

∂u ε ∂n + T R u ε = 0, on Γ R , (5)
where the operator T R from H

1 2 (Γ R ) to H -1 2 (Γ R
) is given by:

                           T R u = +∞ n=-∞ µ R n (ω) u R n ψ R n , u R n = 2π 0 u(R, θ) ψ R n (θ) R dθ, ψ R n (θ) = 1/(2πR) exp inθ, µ R n (ω) = -ω (H (1)
|n| ) ′ (ωR) H

(1)

|n| (ωR) , Re µ R n (ω) 0 and Im µ R n (ω) 0. (6)
In this paper, we are interested in the behaviour of the solution u ε when the width ε tends to 0. Intuition suggests that u ε should converge to the solution u 0 of the "slotless" problem:

               Find u 0 ∈ H 1 (Ω) outgoing such that: -∆u 0 -ω 2 u 0 = f, in Ω, ∂u 0 ∂n = 0, on ∂Ω. (7)
This is what we proved in [START_REF] Joly | Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots[END_REF] in the case of the semi-infinite slot (i. e. L = +∞), where we proved moreover that

u ε -u 0 = O(ε), in L 2 loc . (8)
In the case of the finite slot, we shall see that such an estimate is correct except for a set of exceptional frequencies ω, called resonant frequencies, corresponding to:

ωL = ℓπ, ℓ ∈ N * , (9)
for which the convergence is much slower. More precisely:

u ε -u 0 = O( 1 |Logε| ), in L 2 loc . ( 10
)
Inside the slot, the solution remains bounded for non resonant frequencies but blows up for resonant frequencies.

The outline of the rest of the paper is as follows. In section 2, we investigate the case of non resonant frequencies and prove, in this case, the convergence of u ε to u 0 . In section 3, we investigate the case of resonant frequencies through numerical results. Our results suggest that the solution u ε slowly converges outside the slot to the limit u 0 -see equation (10)-and blows up inside the slot. Finally, in section 4, we give a formal analysis of these observations, using the technique of matched asymptotic expansions.

2 The case of non resonant frequencies: mathematical analysis

The aim of this section is to prove the following theorem, which formulates in a precise way one of the assertions in the introduction. Note that estimates (11) can be shown to be generically optimal as in [START_REF] Joly | Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots[END_REF]. We conjecture that the constant C blows up as 1/d(ω) where d(ω) denotes the distance to the set of resonant frequencies.

Theorem 1 For all ω, L, such that (9) does not hold and for all R in R * ,+ and for any F closed subset of Ω there exists C ∈ R and ε 0 > 0 such that for all 0 < ε < ε 0 and f ∈ L 2 (Ω) with support included in F , we have:

       u ε -u 0 H 1 (Ω R ) C(ω, L, R, F ) ε | log ε| f L 2 (Ω) , u ε -u 0 L 2 (Ω R ) C(ω, L, R, F ) ε f L 2 (Ω) . (11)
and

u ε L 2 (Oε) C(ω, L, F ) √ ε f L 2 (Ω) . (12)
2.1 Reformulation of the problem (4) in the domain Ω R .

Preliminary material. We first recall precise trace theorems on:

Σ ε = Σ - ε ∪ Σ + ε , Σ - ε = {0}×]0; ε[, Σ + ε = {0}×]0; ε[. (13)
In what follows, we shall often identify Σ - ε and Σ + ε to the segment ]0; ε[. We introduce ((w ε n )) n∈N , the orthogonal basis of H s (]0; ε[) for s ∈ [0; 3 2 [:

w ε 0 (y) = 1/ε and w ε n (y) = 2/ε cos nπy ε , ∀n ∈ N * . ( 14
)
For all u in L 2 (Σ ε ), we denote by u ε n ∈ C 2 the vector of the coefficients (in this basis) of the respective restrictions of u to Σ - ε and Σ + ε :

u ε n = Σ - ε u w ε n dσ, Σ + ε u w ε n dσ t (15)
that allows us to characterize the spaces H s (Σ ε ) for 0 s < 3 2 :

u ∈ H s (Σ ε ) ⇐⇒ u 2 H s (Σε) = +∞ n=0 1 + π 2 n 2 /ε 2 s |u ε n | 2 < +∞ (16)
The dual space (H s (Σ ε )) * can be defined analogously if one replaces the integrals by duality brackets in (15) . We shall also use the following semi-norm in H s (Σ ε ):

u 2 H s * (Σε) = +∞ n=1 1 + π 2 n 2 /ε 2 s |u ε n | 2 . ( 17
)
From now on, we shall often denote by u the trace on Σ ε of a function u ∈ H 1 loc (Ω) and by u ε n the vectors of its generalized Fourier coefficients (see (15)). One can easily adapt the proof of [START_REF] Joly | Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots[END_REF] to obtain the following results.

Lemma 2 For any neighborhood V of A -and A + , for s ∈ R + , there exists C(s) ∈ R and ε 0 > 0 such that for all u ∈ H 1+s (Ω ∩ V ) and ε ∈]0; ε 0 ]:

                 u H 1/2 * (Σε) C(s) ε s u H 1+s (Ω∩V ) , if 0 s < 1, u H 1/2 * (Σε) C(s) ε | log ε| u H 1+s (Ω∩V ) , if s = 1, u H 1/2 * (Σε) C(s) ε u H 1+s (Ω∩V ) , if s > 1. ( 18
)
Moreover, for all u ∈ H 1 2 +s (Ω ∩ V ) and ε ∈]0; ε 0 ]:

                 |u ε 0 | C(s) ε s u H 1 2 +s (Ω∩V ) , if 0 < s < 1 2 , |u ε 0 | C(s) ε | log ε| u H 1 2 +s (Ω∩V ) , if s = 1 2 , |u ε 0 | C(s) √ ε u H 1 2 +s (Ω∩V ) , if s > 1 2 . ( 19
)
Reduction to Ω R . Our objective in this section is now to characterize the restriction of u ε to Ω R (a domain independent of ε) thanks to DtN maps on Σ - ε and Σ + ε . To be able to define these operators, we need to solve the interior problem inside O ε :

         -∆u ε -ω 2 u ε = 0, in O ε , ∂u ε ∂y = 0, if y = 0 or ε, (20)
completed by given Dirichlet data on Σ - ε and Σ + ε . Such a problem is well posed if and only if ω 2 is not in the spectrum of the operator -∆ with domain:

D(-∆) = u ∈ H 1 (O ε ) ∆u ∈ L 2 (O ε ), u = 0 on Σ ε , ∂u ∂n = 0 if y = 0 or ε ,
that is to say:

ω 2 = n 2 π 2 /ε 2 + p 2 π 2 /L 2 n ∈ N and p ∈ N * . ( 21
)
If we assume that the slot is not resonant, i.e. that ωL is not a multiple of π -see equation ( 9) -, it is clear that the condition (21) is satisfied for ε small enough. Then one can define the Dirichlet to Neumann-map (the normal n is exterior with respect to Ω)

T ε (ω, L) : u| Σε → - ∂u ε ∂n | Σε .
This operator is continuous from H 1 2 (Σ ε ) to H -1 2 (Σ ε ) and can be computed explicitly:

T ε (ω, L) : u -→ g ⇐⇒ g ε n = M ε n (ω, L) u ε n , (22)
where the matrices M ε n (ω, L) are defined by (note that sin ωL = 0):

M ε 0 (ω, L) = ω sin ωL     cos ωL -1 -1 cos ωL     , (23)
and, for all strictly positive n, setting

ξ ε n (ω) = π 2 n 2 /ε 2 -ω 2 > 0: M ε n (ω, L) = ξ ε n (ω) sinh ξ ε n (ω)L     cosh ξ ε n (ω)L -1 -1 cosh ξ ε n (ω)L     , ∀n ∈ N * . ( 24
)
It is easy to see that these matrices have the following properties (u • v and |u| denote respectively the usual inner product and associated norm in C 2 ):

         (M ε 0 (ω, L)u) • u ∈ R, (M ε 0 (ω, L)u) • v 2ω sin ωL |u| |v|, (M ε n (ω, L)u) • u ∈ R + , (M ε n (ω, L)u) • v 2 ξ ε n (ω) tanh ξ ε n (ω)L |u| |v|. (25)
By construction, the restriction of u ε to Ω is characterized by the transparent condition on Σ ε :

∂u ε ∂n Σε + T ε (ω, L) u ε | Σε = 0. ( 26
)
Equivalently, the restriction of u ε to Ω R is characterized as the unique solution of the problem (

ϕ f ∈ H 1 (Ω R ) is defined by (ϕ f , v) H 1 (Ω R ) = (f, v) 2 L (Ω R ) for any v ∈ H 1 (Ω R )) Find u ε ∈ H 1 (Ω R ) such that : A R u ε + B ε 0 u ε + R ε u ε = ϕ f , ( 27
)
where the operators

A R , B ε 0 and R ε , H 1 (Ω R ) → H 1 (Ω R )
, are defined, using the Riesz theorem, by :

             (A R u; v) H 1 (Ω R ) = Ω R (∇u ∇v -ω 2 u v) + +∞ n=-∞ µ R n (ω) u R n v R N , (B ε 0 u; v) H 1 (Ω R ) = M ε 0 u ε 0 v ε 0 , (R ε u; v) H 1 (Ω R ) = +∞ n=-∞ M ε n u ε n v ε n . (28)
In the same way,

u 0 ∈ H 1 (Ω R ) is the unique solution of : A R u 0 = ϕ f . (29) 2.2 Convergence of u ε to u 0 .
We only prove the H 1 -error estimate of (11). The L 2 -error estimate is shown by a duality argument as in [START_REF] Joly | Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots[END_REF]. From ( 27) and (29), we deduce:

(A R + B ε 0 + R ε )(u ε -u 0 ) = -B ε 0 + R ε u 0 . ( 30
)
The conclusion follows from the two following lemmas and from (the operator norms below are norms in L(H 1 (Ω R ))):

u ε -u 0 (A R + B ε 0 + R ε ) -1 B ε 0 u 0 + R ε u 0 . ( 31
)
Lemma 3 If the condition ( 9) is not satisfied, there exists ε 0 (ω, R, L) strictly positive and C(ω, R, L) real such that for all 0 < ε < ε 0 :

(A R + B ε 0 + R ε ) -1 L(H 1 (Ω R )) C(ω, R, L). ( 32
)
Lemma 4 If the condition ( 9) is not satisfied, there exists ε 0 (ω, R, L, F ) strictly positive and C(ω, R, L, F ) real such that for all 0 < ε < ε 0 :

         B ε 0 u 0 H 1 (Ω R ) C(ω, R, L, F ) ε | log ε| f L 2 (Ω) , R ε u 0 H 1 (Ω R ) C(ω, R, L, F ) ε f L 2 (Ω) . (33)
Proof of lemma 3. We make a proof by contradiction. If (32) were false, there would exist a sequence

v ε ∈ H 1 (Ω R ), ε → 0, such that: v ε H 1 (Ω R ) = 1 and ϕ ε = (A R + B ε 0 + R ε )v ε → 0, in H 1 (Ω R ). ( 34
)
By compactness, let us extract from v ε a subsequence (still denoted v ε ) with ε → 0 such that:

v ε → v 0 weakly in H 1 (Ω R ) and strongly in L 2 (Ω R ). (35)
For any v ∈ H 1 (Ω R ), one has:

(ϕ ε ; v) H 1 (Ω R ) = (A R v ε ; v) + big(B ε 0 v ε ; v) + (R ε v ε ; v). ( 36
)
Let us introduce the subspace

H 1 00 (Ω R ) of H 1 (Ω R ) made of functions vanishing in a neighborhood V of A -and A + (v = 0 in V ∩ Ω). For any v ∈ H 1 00 (Ω R ), as soon as ε is small enough, (B ε 0 ϕ ε ; v) = 0 and (R ε ϕ ε ; v) = 0. Thus (ϕ ε ; v) H 1 (Ω R ) = (A R v ε ; v). ( 37
)
Taking the limit as ε → 0, we get

(A R v 0 ; v) H 1 (Ω R ) = 0, ∀ v ∈ H 1 00 (Ω R ) which implies, by density of H 1 00 (Ω R ) in H 1 (Ω R ) that A R v 0 =
0 that is to say, since A R is an isomorphism (this is nothing but the well-posedness of the slotless problem), v 0 = 0.

We choose now v = v ε in (36), and consider the real part of the resulting equality. We obtain, by definition of A R , B ε 0 and R ε :

             Re (ϕ ε ; v ε ) H 1 (Ω R ) = Ω R |∇v ε | 2 -ω 2 Ω R |v ε | 2 + Re +∞ n=0 (µ R n (ω)) |u R n | 2 + M ε 0 (ω, L) v ε 0 • v ε 0 + +∞ n=1 M ε n (ω, L) v ε n • v ε n .
Due to properties ( 6) and (25), we thus have:

Ω R |∇v ε | 2 (ϕ ε ; v ε ) H 1 (Ω R ) + ω 2 Ω R |v ε | 2 + 2 ω sin ωL |v ε 0 | 2 . ( 38
)
Besides, from lemma 2 (with s = 1 2 ), we get:

|v ε 0 | 2 C(R) ε | log ε| v ε 2 H 1 (Ω R ) = C(R) ε | log ε|. (39)
Hence, from (35), (38), and (39), we deduce that ∇v ε converges to 0 in L 2 (Ω R ). Therefore v ε converges strongly to 0 in H 1 (Ω R ), which contradicts (34) and concludes the proof.

Proof of lemma 4. As u 0 is a solution of a Helmholtz problem in Ω with a source term compactly supported in F ⊂ Ω, by elliptic regularity there exists a neighborhood V (simply choose

V ∩ F = ∅) of A -and A + such that u 0 ∈ H m (V ∩ Ω)
, for all m > 0 and:

u 0 H m (V ∩Ω) C(m, F ) f L 2 (Ω) , ∀m > 0. ( 40
)
(i) By definition of R ε and property (25), we have for all v ∈ H 1 (Ω R ):

(R ε u 0 ; v) H 1 (Ω R ) +∞ n=1 2 ξ ε n (ω) tanh ξ ε n (ω)L |(u 0 ) ε n | |v ε n |. (41)
For ε < π/2ω, we have the inequalities for all n ∈ N * :

2ξ ε n (ω) tanh(ξ ε n (ω)L) 2 tanh (ωL/2) 1 + π 2 n 2 ε 2 (42)
The Cauchy-Schwartz inequality leads to:

(R ε u 0 ; v) H 1 (Ω R ) 2 tanh (ωL/2) u 0 H 1 2 * (Σε) v H 1 2 * (Σε) . ( 43
)
Thanks to (40) with m = 3, we can use (18) with s = 2 for u 0 , s = 0 for v. We get:

(R ε u 0 ; v) H 1 (Ω R ) C(ω, R, L, F ) ε f L 2 (Ω) v H 1 (Ω R ) . (44)
The result follows from

R ε u 0 H 1 (Ω R ) = sup v∈H 1 (Ω R ) (R ε u 0 ; v) H 1 (Ω R ) v H 1 (Ω R ) .
(ii) By definition of B ε 0 , and due to (25) we have:

(B ε 0 u 0 ; v) H 1 (Ω R ) 2 ω sin (ωL) |(u 0 ) ε 0 | |v ε 0 |. ( 45
)
Thanks to (40) with m = 3 2 , we can use (19) with s = 1 for u 0 , s = 1 2 for v. We get:

(B ε 0 u 0 ; v) H 1 (Ω R ) C(ω, R, L, F ) ε | log ε| f L 2 (Ω) v H 1 (Ω R ) . (46)
One concludes as for point (i).

3 The case of resonant frequencies: some numerical computations Presentation of the experiment. The experiment we simulate is explained in Figure 3. We choose ω = 2π, which corresponds to a unit wave length. We shall make our computations with two values of the length of the slot : L=0.8 (the non resonant case) and L=1 (the resonant case). The height 2H of the rectangle B is 2H = 8.0. The distance from the point source to B is D = 0.5 in the non resonant case and D = 0.4 in the resonant case. The numerical results are obtained by a Q 2 finite element code working with quadrilateral elements [START_REF] Cohen | Mixed spectral elements for the helmholtz equation[END_REF]. The computational domain (the grey region in figure 3) is delimited by a rectangle of respective dimensions 5 in the x direction and 10 in the y direction. The computational mesh is refined in the neighborhood of the slot as represented in figure 4 (each of the triangle appearing in the picture is divided into three quadrilaterals). In this way, one has about 160 points per wavelength close to the slot and 16 points per wavelength far away. A PML of width 1, surrounding Ω c , is used to simulate an infinite domain. Comments of the results. In figures 5 and 6, we represent the (real part of) solution u ε (called total field) in the two cases, for ε = 10 -3 . In this picture, it already appears that much more energy has been transmitted through the slot in the resonant case. This appears much more clearly when we look at the diffracted fields, namely the difference u εu 0 between the solution and the solution in the absence of the slot. In figure 7, we represent these fields for the three values of ε = 10 -1 , 10 -2 , 10 -3 . The diffracted field decays much more strongly with ε in the non resonant case than in the resonant case, for which it is even not so clear that the diffracted field converges to 0 ! In figure 8, we are interested in the solution inside the slot. We represent the variation of the solution along the symmetry line of the slot. Note that the two scales for the two pictures are completely different. Our results show that, while the slot field stabilizes when ε goes to 0, it blows up in the resonant case.

We now vary ε continuously and represent the variations of various norms of the diffracted field and slot fields as a function of ε.

We first look at the results associated to the non resonant case. In figure 9, we represent the L 2 norm in Ω c of the diffracted field. The curves confirm the theory: the L 2 norm of the diffracted field tends to 0 as O(ε). In figure 10, we represent the variations of the H 1 -norm of the diffracted field as a function of ε | log ε|, the linear behaviour in ε | log ε| close to 0 seems to confirm the theory, see theorem 1. In figure 11, we represent the variations of the norm of the solution u ε in L 2 (O ε ). The shape of the curve suggests that this norm decays to 0 sublinearly. When we represent in figure 12 the log of this quantity as a function of log ε, we obtain a straight line of slope ∼ 0.5 which confirms a behaviour in O( √ ε).

In the resonant case, figure 13 shows that the L 2 norm of the diffracted field Resonant, ε = 10 -1

Resonant, ε = 10 -2

Resonant, ε = 10 -3

Fig. 8. The solution inside the slot for ε = 10 -1 , 10 -2 , 10 -3 , left : non resonant case, right: resonant case. converges to 0 sublinearly. The result of figure 17 (we represent the inverse of the L 2 norm of the diffracted field as a function of log ε) suggests in fact that if behaves as 1/(A log ε + B + o(1)). In figures 15 and 16, we are interested into the H 1 norm of the diffracted fields. Our results suggest that this norm decays as 1/ (A log ε + B + o(1)). Finally, figure 17 shows that the L 2 norm of the slot field blows up when ε goes to 0 and figure 18 suggests more precisely that it behaves as 1/( √ ε| log ε|).

ε -u 0 H 1 (Ω R ) = f (ε |
Finally, in figure 19, we fix ε = 10 -3 and make ω vary such that the frequency ω/2π goes from 0 to 2.5. One observes that the two curves depicting respectively the L 2 norm of the diffracted field (normalized by the one of u 0 ) and the L 2 norm of the slot field present very sharp peaks corresponding to the resonant frequencies (ω/2π is an integer multiple of 1 2 ). One also observes the the height of these peaks decays with ω/2π for sufficiently large ω/2π.

Fig. 13. u ε -u 0 L 2 (Ωc) = f (ε) 0 0.
u ε -u 0 L 2 (Ω) / u 0 L 2 (Ω) = f (ω/2π) 0 0.5 1 1.5 2
u ε -u 0 L 2 (Oε) / u 0 L 2 (Oε) = f (ω/2π) 4
The case of resonant frequencies: a formal asymptotic analysis

We now provide a (formal) analysis of the results observed in the previous section thanks to the help of matched asympotic expansions. Of course, it is not our purpose to present in detail this (rather complicated) technique and we refer the reader to the references [START_REF] Crighton | Modern Methods in Analytical Acoustics[END_REF], [START_REF] Il ′ In | Matching of asymptotic expansions of solutions of boundary value problems[END_REF] for more information and to [START_REF] Mciver | Two-dimensional wave-scattering problems involving parallel-walled ducts[END_REF] for an application to the semi-infinite slot problem. We shall restrict ourselves to present the main ideas of the method. In particular, we shall introduce our Ansatz without particular justification, despite the fact it can be done (rather tediously) as in [START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF].

Throughout this section, we assume that:

ωL = ℓπ, ℓ ∈ N * . ( 47
)

The Ansatz of matched asymptotic expansions

The principle is to decompose the domain Ω ε into four (overlapping) subdomains which are:

• a "far field" zone (or exterior zone) Ω F (ε) in which we shall describe the behaviour of the solution outside the slot. • a "slot" zone Ω S (ε) to describe the behaviour of the solution in the slot.

• two "near field" zones (or junction zones) Ω - N (ε) and Ω + N (ε) in which we describe the behaviour of the solution at the neighborhood of A -and A + .

The precise definition of these domains uses four functions of ε:

0 < η - F (ε) < η + F (ε), 0 < η - S (ε) < η + S (ε) (48)
that are supposed to belong to the class of continuous functions η(ε) satisfying:

lim ε→0 η(ε) = 0, lim ε→0 η(ε)/ε = +∞. ( 49
)                      Ω F (ε) = {x ∈ Ω / |x -A -| > η - F (ε) and |x -A + | > η - F (ε)} Ω S (ε) = {x ∈ O ε / x > η - S (ε) and L -x > η - S (ε)} Ω ± N (ε) = {x ∈ Ω/|x -A ± | < η + F (ε)} ∪ {x ∈ O ε /|x -a±| < η + S (ε)} (50)
Note in particular that, when ε tends to 0, Ω - N (ε) and Ω + N (ε) collapse to one point, Ω S (ε) to the segment [A -, A + ] while Ω F (ε) converges to Ω. The next step consists in working with normalized coordinates in each zone and postulating some asymptotic expansion of the solution in these coordinates. For 2D problems, it is well known that powers of ε are not sufficient and one must use the set of gauge functions:

f i,k (ε) = ε i log k ε, where (i, k) are integers. ( 51
)
One of the difficulties is to choose the good set of indices (i, k) in each zone but we shall be guided by our numerical results. In the paragraph below, we present our Ansatz and point out the leading term in each expansion (the one we are interested in in this paper, the other terms being lower order correctors).

Notation. From now on, for any function v ε defined as a formal series:

v ε = i k ε i log k ε v k i (52) we shall set v ε i = k log k ε v k i .
This allows us to distinguish the dependence of v ε with respect to ε and log ε, which is justified by the fact that:

∀k ∈ Z, ε = o(| log ε| k ). ( 53
)
Solution in the far field zone. We keep here the physical coordinates (x, y) and look for u ε of the form:

             u ε (x, y) = +∞ i=0 i k=-∞ ε i log k ε u k i (x, y) + o(ε ∞ ). = +∞ i=0 ε i u ε i (x, y) + o(ε ∞ ) = u ε 0 + o(1) (54)
The leading term corresponds to (i, k) = (0, 0) which means that u ε converges to a limit in Ω. In the expansion below, the functions u k i and u ε i are defined in the canonical (i.e. independent of ε) slotless domain Ω. Inserting (54) into (4), it is easy to see that:

∆u ε 0 + ω 2 u ε 0 = -f, in Ω, ∂u ε 0 ∂n = 0, on ∂Ω \ {A -, A + }. (55)
The fact that u ε 0 satisfies the Neumann condition except at points A -and A + allows u ε 0 to be singular at A -and A + . Therefore, one needs additional informations about the behaviour of u ε 0 in the neighborhood of these two points.

Solution in the slot zone. We use the coordinates (x, y/ε) that maps O ε into O 1 , a slot of unit width, and look for u ε of the form:

             u ε (x, y) = U ε (x, y ε ) = +∞ i=-1 i k=-∞ ε i log k ε U k i (x, y ε ) + o(ε ∞ ) = +∞ i=-1 ε i U ε i (x, y ε ) = 1 ε U ε -1 (x, y/ε) + U ε 0 (x, y/ε) + o(1). (56)
The leading term corresponds this time to (i, k) = (-1, -1) which means that

u ε blows up in O ε as O(1/(ε log ε)).
The first term U ε -1 and the next term U ε 0 will play a role in the calculations. Substituting (56) into (4), one sees that U ε i depends only from x and that:

d 2 U ε i dx 2 + ω 2 U ε i = 0, in O 1 . (57)
Note that, to determine U ε i , one needs additional informations at x = 0 or L. Solution in the near field zones. We present only the case of the zone Ω - N (ε), the case of the zone Ω + N (ε) is obviously similar. We use the coordinates (x/ε, y/ε), a scaling that maps the region Ω - N (ε) in to a domain that converges to an infinite "T-shape" domain Ω - N defined as:

Ω - N = {(X, Y ) ∈ R 2 /X 0} ∪ {(X, Y ) ∈ R 2 /X 0 et 0 Y 1}. ( 58
)
We look for u ε of the form (we point out in the second line the terms that will play a role in the computations):

             u ε (x, y) = U -,ε ( x ε , y ε ) = +∞ i=0 i k=-∞ ε i log k ε U -,k i ( x ε , y ε ) + o(ε ∞ ) = +∞ i=0 ε i U -,ε i ( x ε , y ε ) + o(ε ∞ ) = U -,ε 0 ( x ε , y ε ) + o(1). (59)
(ν ε -) p are complex functions of log ε) :

U -,ε 0 (X, Y ) = µ ε -X + ν ε -+ +∞ p=1 (ν ε -) p exp -pπX cos pπY , X > 0. ( 63
)
Since the slot field satisfies the 1D Helmholtz equation, there exists complex functions (A ε i , B ε i , i = -1, 0) of log ε such that:

U ε i (x) = A ε i cos ωx + B ε i sin ωx, i = -1, 0. (64)
Hence, when x → 0 and x/ε → +∞, one has:

1 ε U ε -1 (x, y ε ) + U ε 0 (x, y ε ) = 1 ε A ε -1 + ω B ε -1 x ε + A ε 0 + o(1), (65)
U -,ε 0 ( x ε , y ε ) = µ ε - x ε + ν ε -+ o(1). (66)
Therefore, (62) leads to:

A ε -1 = 0, ωB ε -1 = µ ε -, A ε 0 = ν ε -. (67)
By symmetry, we have also:

U +,ε 0 (X, Y ) = µ ε + X + ν ε + + o(1), when X → -∞. (68)
with the relations:

A ε -1 = 0, (-1) ℓ ωB ε -1 = µ ε + , (-1) ℓ A ε 0 = ν ε + . (69)
Matching between the far field and the near field On the domain Ω F (ε) ∩ Ω - N (ε), u ε (x, y) and U -,ε (x/ε, y/ε) must coincide. In particular, we need to match the truncated expansions ( in o(1) ) of ( 54) and (59). Let us use the polar coordinates (r -, θ -) associated to the point A -(and such that θ -= 0 or π coincides with the line x = 0). Noticing that, in the overlapping zone, r -→ 0 and r -/ε → +∞, this can be reduced to:

lim r -→0, r - ε →+∞ u ε 0 (r -, θ -) -U -,ε 0 ( r - ε , θ -) = 0. ( 70
)
As f has a compact support include in Ω , there exists 0 < R < inf(H, L), such that, for r -< R, by separation of variables, u ε 0 can be written as a linear combination of:

         J p (ωr -) cos (pθ -), p ∈ N ( smooth functions of r -)
Y p (ωr -) cos (pθ -), p ∈ N ( which are singular close to r -= 0) (71) It can be shown (but the proof is omitted here) that the only function Y p (ωr -) that may appear in the expansion corresponds to p = 0. Therefore, we have an expression of the form (for r -< R):

u ε 0 (r -, θ -) = +∞ p=1 (b ε -) p J p (ωr -) cos (pθ -) + a ε -Y 0 (ωr -) + b ε -J 0 (ωr -). (72)
In the same way, if we use (ρ -, θ -) the polar coordinates associated to the variables (X, Y ) in Ω - N , since U ε 0 satisfies the homogeneous Laplace equation with homogeneous Neumann condition, by separation of variables, in the domain ρ -> 1), U -,ε 0 can be written as a linear combination of:

{(ρ -) p cos (pθ -), (ρ -) -p cos (pθ -), p ≥ 1}, log ρ -, 1.
One can show (once again, the proof is omitted here) that U -,ε 0 cannot grow more rapidly than log ρ -when ρ -goes to +∞. Therefore, we have an expression of the form (ρ -> 1):

U -,ε 0 (ρ -, θ -) = +∞ p=1 (β ε -) p (ρ -) -p cos (pθ -) + α ε -log ρ -+ β ε -. (73)
Recalling that (γ denotes the Euler number [START_REF]Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]):

lim z→0 J 0 (z) = 1, lim z→0 J p (z) = 0 for p ≥ 1, Y 0 (z) = 2 π log( z 2 ) + γ + o(1),
we deduce that:

           u ε 0 (r -, θ -) = 2 π a ε -log ωr - 2 + γ + b ε -+ o(1), (r -→ 0), U -,ε 0 (r -/ε, θ -) = α ε -log r - ε + β ε -+ o(1) (r -/ε → +∞). (74)
Therefore, (70) leads to :

2 π a ε -= α ε -, 2 π a ε -log ω 2 + γ + b ε -= -α ε -log ε + β ε -. (75)
In the same way (with obvious notation for polar coordinates), we have:

           u ε 0 (r + , θ + ) = 2 π a ε + log ωr + 2 + γ + b ε + + o(1), (r + → 0), U +,ε 0 ( r + ε , θ + ) = α ε + log r + ε + β ε + + o(1) (r + /ε → +∞). ( 76
)
with the relations:

2 π a ε + = α ε + , 2 π a ε + log ω 2 + γ + b ε + = -α ε + log ε + β ε + . (77)
We have now all the necessary information to determine the fields u ε 0 , U ε -1 , U -,ε 0 and U +,ε 0 .

4.2 Determination of the leading terms of the expansion.

The analysis of the far field u ε 0 will rely on the following lemma. The proof, omitted here, can be adapted from those in [START_REF] Tordeux | Méthodes asymptotiques pour la propagation des ondes dans les milieux comportant des fentes[END_REF].

Lemma 5 There exists a unique outgoing solution u * of the homogeneous Helmholtz equation in Ω satisfying the homogeneous Neumann boundary condition on ∂Ω \ {A -, A + } such that there exists b * ∈ C with:

           u * (r -, θ -) - 2 π log ωr - 2 + γ → b * , when r -→ 0, u * (r + , θ + ) + (-1) ℓ 2 π log ωr + 2 + γ → (-1) ℓ+1 b * , when r + → 0. (78)
By linearity, we deduce the following

           v(r -, θ -) - 2 π a log ωr - 2
+ γ has a finite limit when r -→ 0, v(r + , θ + ) + (-1) ℓ 2 π a log ωr + 2 + γ has a finite limit when r + → 0, then, one has: v = a u * .

We first notice that, if we integrate the equation (60) over the domain:

Ω - N,1 = {x ∈ Ω - N / 0 < ρ + < 1, 0 < θ + < π}. (79)
One easily obtain the relation:

α ε -= - 1 π µ ε -. (80)
In the same way, we show that:

α ε + = 1 π µ ε + . (81)
Therefore, using (75), ( 67) and (80) (respectively (77), ( 69) and ( 81)), one can obtain the relations:

           2 π a ε -= α ε -= - 1 π µ ε -= - ω B ε -1 π , 2 π a ε + = α ε + = 1 π µ ε + = (-1) ℓ ω B ε -1 π , (82)
which implies in particular:

a ε -= (-1) ℓ+1 a ε + (:= a ε ). (83)
Applying corollary 6 to v = u εu 0 , we obtain:

u ε 0 = u 0 + a ε u * . ( 84
)
Therefore u ε 0 has the following behaviour at the neighborhood of A + and A -:

         u ε 0 (r -, θ -) = u 0 (A -) + a ε 2 π log ωr - 2 + γ + b * + o(1), u ε 0 (r + , θ + ) = u 0 (A + ) + (-1) ℓ+1 a ε 2 π log ωr + 2 + γ + b * + o(1), (85)
Comparing (74,76) and (85), we have:

b ε -= a ε b * + u 0 (A -), b ε + = (-1) ℓ+1 a ε b * + u 0 (A + ). ( 86
)
For the near field we shall use the following result (adapting the proof of [START_REF] Tordeux | Un problème de Laplace non standard en milieu non borné[END_REF]):

Lemma 7 There exists a unique function U * ± , harmonic in Ω ± N , which satisfies, when ρ ± → +∞ and ∓X → +∞: then:

V ± = α ± U * ± + β ± .
Applying corollary 8 to U -,ε 0 and U +,ε 0 , we obtain using equations (74,76), (75,77) and (86):

As can be seen in the formulas (84), ( 90), (91), we know u ε 0 , U ±,ε 0 and U ε -1 as soon as we know a ε . To obtain a ε , we have to ensure:

From equations (67) and (69) we deduce:

ν ε -= (-1) ℓ ν ε + (92)
Equations ( 90) and (88) lead to: (-1) ℓ u 0 (A + )u 0 (A -) log ε , we conjecture the estimate :

                       U -,ε 0 (X, Y ) = a ε -2X +
         u ε -u 0 L 2 (Ω R ) C f L 2 (Ω) / | log ε|, u ε -u 0 L 2 (Oε) C f L 2 (Ω) / (ε 1/2 | log ε|), (96)
Moreover, due to the two logarithmic singularities at the point A -and A + , u * is not in H 1 loc (Ω). Therefore we can not have the same inequality for the H 1 (Ω R )norm as for the L 2 (Ω R )norm. One can expect the (generically optimal) estimate:

u ε -u 0 H 1 (Ω R ) C(ω, L, R, F ) f L 2 (Ω) / | log ε|. ( 97
)
All this results are coherent with those obtained numerically in section 3. One can refer to figure 13, 14, 15, 16, 17, 18. The proof of ( 96) and (97) should follow the lines of the proof of theorem but the stability argument, that used the DtN map, has to be adequately modified. This paper is delayed to a forthcoming work.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Geometry of the propagation domain with the slot

Fig. 3 .2HFig. 4 .

 34 Fig. 3. Presentation of the experiment L

Fig. 5 .Fig. 6 .Fig. 7 .

 567 Fig. 5. The total field, non resonant

Fig. 9 .

 9 Fig. 9. u εu 0 L 2 (Ωc) = f (ε)

Fig. 19 .

 19 Fig. 19. Dependance of the diffracted field with respect to the frequency

Corollary 6

 6 If v is an outgoing solution of the homogeneous Helmholtz equation in Ω with Neumann boundary condition in ∂Ω \ {A -, A + } and satisfies the local behaviours:

Corollary 8 V

 8 U * ± (ρ ± , θ ± ) = log ρ ± + o(1), U * ± (X, Y ) = O(|X|). (87)as well as the homogeneous Neumann boundary condition on ∂ Ω ± N . Moreover, there exists ν * ∈ C such that:U * ± (X, Y ) = ±πX + ν * + o(1), when ∓ X → +∞ (88) Let V ± asolution of the homogeneous Laplace equation in Ω ± N satisfying the homogeneous Neumann boundary condition on ∂ Ω ± N and: ±(ρ ± , θ ± ) = α ± log ρ ± + β ± + o(1), ρ ± → +∞ V ± (X, Y ) = O(|X|),

2 + 2 +

 22 γ + b * + u 0 (A -), γ + b * + u 0 (A + ). (90)Moreover, U ε -1 is given via (64) and (83): U ε -1 (x) = 2 ω a ε sin ωx.

2 π log ωε 2 +π log ωε 2 +

 222 γ + ν * + b * +u 0 (A -) + o(1), U +,ε 0 (X, Y ) = (-1) ℓ+1 a ε 2X + 2 γ + ν * + b * +u 0 (A + ) + o(1).

= a ε 2 π log ωε 2 + 1 ) ℓ+1 a ε 2 π log ωε 2 + 4 . 3

 21243 γ + ν * + b * + u 0 (A -), ν ε + = (-γ + ν * + b * + u 0 (A + ). (94)It follows from (92) and (94): a ε = (-1) ℓ u 0 (A + )u 0 (A -Towards a rigorous proof Since a ε ∼ π 4

  log 10 ε|)

						0.25
						0.2
						0.15
						0.1
						0.05
						0	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
	0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16	0	Fig. 11. u ε 0.01 0.02 0.03 0.04	L 2 (Oε) = f (ε) 0.05 0.06 0.07 0.08 0.09	0.1	Fig. 12. log u ε L -3 -2.8 -2.6 -2.4 -2.2 -2 -4.5	-1.8	-1.6	-1.4	-1.2	-1

2 (Oε) = f (log 10 ε)

where the U -,k i and U -,ε i are defined in Ω - N . Note that this time, it is not clear from the numerical computations how to determine the good leading term of the expansion. Substituting (59) into (4), one easily obtains:

To determine U -,ε 0 , one needs to prescribe its behaviour at infinity in Ω - N . In the same way, to describe the near field in the second junction region using the scaled coordinates (X = (x -L)/ε, Y = y/ε), we introduce an harmonic function U +,ε 0 that is defined in the domain:

and satisfies the homogeneous Neumann condition on ∂ Ω + N . Summary of the first step of the method. Our objective will be to determine the four functions u ε 0 , U ε -1 , U -,ε 0 and U +,ε 0 . We know the PDE's satisfied by these functions in their respective domains of definition, as well as the boundary conditions they satisfy. We still need:

• to know the asymptotic behaviour of u ε 0 at the neighborhood of A -and A + , • to know the boundary conditions for U ε -1 at x = 0 and x = L, • to know the asymptotic behaviour at infinity of U -,ε 0 and U +,ε 0 .

This missing information will be provided by the application of the matching principle which will express, in some approximate way, that the various expansions (54), ( 56), (59) represent the same function in the overlapping zones.

Remark. Although we do not need to determine U ε 0 , this function will naturally appear in the application of the matching principle.

Matching between the near field and the slot field. On the domain Ω S (ε) ∩ Ω - N (ε), U ε (x, y/ε) and U ε (x/ε, y/ε) must coincide. In particular, we need to match the truncated expansions ( in o(1) ) in ( 56) and (59). Noticing that, in the overlapping zone, x → 0 and x/ε → +∞, this can be reduced to:

In the domain [0; +∞[ X ×[0; 1] Y , since the near field U -,ε 0 satisfies the Laplace equation with Neumann boundary condition on the two horizontal boundaries, by separation of variable, if we exclude exponentially growing solutions inside the slot (which can be proved), we know that U -,ε 0 is of the form (µ ε -, ν ε -and