Charlène Heiniger

Sébastien Barot

Jean-François Ponge
email: ponge@mnhn.fr

Sandrine Salmon

David Carmignac

Margot Suillerot

Florence Dubs

Jacques Meriguet

Collembolan preferences for soil and microclimate in forest and pasture communities

Keywords: collembolan communities, habitat preference, forest and pasture soil, microclimate effect, field experiment

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

Introduction

The search for unifying principles in community ecology led to the identification of three processes that interact to shape species assemblages: 1) habitat selection, 2) dispersal and 3) biotic interactions [START_REF] Weiher | Assembly Rules: Perspectives, Advances, Retreats[END_REF][START_REF] Wardle | The influence of biotic interactions on soil biodiversity[END_REF][START_REF] Mayfield | Traits, habitats, and clades: identifying traits of potential importance to environmental filtering[END_REF]. Understanding the factors that determine the preference of a species for a given habitat is thus essential to predict species distribution and local community composition. In most habitats, many different factors (biotic and abiotic) interact, creating environmental conditions that allow or impede species persistence and reproduction [START_REF] Bull | Metapopulation extinction risk is increased by environmental stochasticity and assemblage complexity[END_REF]. Furthermore, different species show different levels of specialization for a given habitat, from specialists which are only found in a restricted array of environmental conditions to generalists which are found in a wide array of environmental conditions [START_REF] Egas | Evolution restricts the coexistence of specialists and generalists: the role of trade-off structure[END_REF][START_REF] Julliard | Spatial segregation of specialists and generalists in bird communities[END_REF]. The extent to which a species is specialist of a given habitat probably depends on how much it is adapted to the different habitat features and the level of specialization is likely to differ between habitat features.

For invertebrate species inhabiting soil and litter layers, habitat is at least twofold. First, the nature of the soil and the humus form are very influential: (1) they determine the availability and quality of resources such as organic matter, which in turn determines the composition and activity of microbial communities, one of the main food sources of soil invertebrates [START_REF] Ponge | Food resources and diets of soil animals in a small area of scots pine litter[END_REF][START_REF] Murray | Dissipation of bacterially derived C and N through the meso-and macrofauna of a grassland soil[END_REF][START_REF] Sabais | Plant species richness drives the density and diversity of Collembola in temperate grassland[END_REF]; (2) soil and humus through several physicochemical properties, such as pH, moisture, structure, carbon content, etc., are critical parameters for collembolan survival [START_REF] Ponge | Biocenoses of Collembola in atlantic temperate grass-woodland ecosystems[END_REF][START_REF] Berg | Dynamics and stratification of functional groups of micro-and mesoarthropods in the organic layer of a Scots pine forest[END_REF][START_REF] Loranger | Does soil acidity explain altitudinal sequences in collembolan communities?[END_REF]. Second, the type of vegetation is also influential: (1) it influences the quality and quantity of organic matter inputs; (2) it influences the local microclimate and interacts with soil and humus to determine temperature and moisture levels which prevail within the soil [START_REF] Chen | Impacts of grassland afforestation with coniferous trees on soil phosphorus dynamics and associated microbial processes: a review[END_REF][START_REF] Ponge | Plant-soil feedbacks by humus forms: a review[END_REF]. For example tree canopy cover in forests prevents most UV radiation from reaching the ground surface and creates lower soil temperatures in forests compared to pastures [START_REF] Scott | Processes influencing soil carbon storage following afforestation of pasture with Pinus radiata at different stocking densities in New Zealand[END_REF].

Collembolan communities have been shown to vary according to vegetation types, e.g. open vs closed vegetation [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF][START_REF] Vanbergen | Scale-specific correlations between habitat heterogeneity and soil fauna diversity along a landscape structure gradient[END_REF]. Forests (closed vegetation) benefit from high inputs of litter which create thick organic (and organic-mineral) layers. High soil carbon content induces both low pH and high soil moisture and creates conditions favouring overall collembolan abundance and diversity [START_REF] Hopkin | Biology of the Springtails (Insecta: Collembola)[END_REF]. In addition, high organic inputs in forests provide abundant trophic resources. In contrast, open vegetation (e.g. any habitat without trees such as pastures or meadows) is characterized by intense export through mowing, grazing, or harvesting, and more active decomposition, which induces lower organic contents and reduced or absent organic layers [START_REF] Compton | Long-term impacts of agriculture on soil carbon and nitrogen in New England forests[END_REF]. Additionally, the absence of tree cover induces higher temperatures in summer and lower soil moisture than in forests [START_REF] Batlle-Aguilar | Modelling soil carbon and nitrogen cycles during land use change. A review[END_REF]. Thus, in collembolan communities, specialists of a given habitat should be intolerant to at least one feature of non-preferred habitats (microclimate, resource quality and/or availability, physicochemical factors): for example, forest specialists should be intolerant either to soil properties or microclimate of open habitats. In contrast, generalist species should be generalist for both soil and microclimate.

In their experiment, [START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF] determined habitat preference and dispersal ability of a large set of collembolan species. Using a soil transplant experiment between a forest and a meadow, they showed that several forest-preferring and forest-strict species actually colonized more efficiently meadow soil transferred to forest than non-transferred forest soil. They suggested that certain forest species, more abundant in the transplanted meadow soil, could not survive in the meadow because of its microclimate. However, in their study the effect of species ability to colonize both soil types through dispersal was difficult to distinguish from the effects of actual preferences for a given habitat. Moreover, [START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF] only transplanted soil cores from one type of habitat to another but did not submit collembolan communities to a different microclimate. This did not allow a full disentanglement of the effects of soil and humus nature from the effects of microclimate determined by plant cover.

The present experiment thus aimed at addressing the two following questions. Are forest or pasture species excluded from (or less abundant in) pastures and forests, respectively, because they do not withstand differences in temperature and related soil moisture (microclimate) in these habitats, or because they do not find appropriate trophic resources and suitable physicochemical conditions (soil nature)? Are generalist species tolerant to both soil and microclimate? We hypothesize that forest and pasture species are not primarily influenced by the same habitat features. Forest species would be absent (or less abundant) in pastures because of physiological requirements for forest microclimate (i.e. higher humidity and lower temperature) whereas pasture species would be absent (or less abundant) in forests because they do not find appropriate trophic resources in them.

Given our choice of a transfer experiment in which animals cannot freely move to find suitable conditions for their growth and reproduction, preferences will be only inferred from their ability to survive and multiply better under certain conditions than others. This is also the sense given to the word "affinity" in similar experiments [START_REF] Huhta | Community of Mesostigmata (Acari) in experimental habitat patches of forest floor[END_REF] but we here refer to the definition given by [START_REF] Pey | Current use of and future needs for soil invertebrate functional traits in community ecology[END_REF] of "ecological preference" as "the optimum and/or the breadth of distribution of a trait on an environmental gradient", considering "ecological preference" as the result of multiple interacting ecophysiological traits each species display and "habitat preference" as a subset of "ecological preference.

Material and methods

Study site

The study was set up in a forest and an adjacent pasture in the Morvan Regional Natural Park at the same location as the experiment reported in [START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF]. The Morvan Natural Park is located in the centre of France (Burgundy) and has a submontane-atlantic climate with continental influence (mean annual rainfall 1000 mm and mean temperature 9 C). The bedrock is granite and soils are moderately to strongly acidic (pH < 5). The forest canopy is comprised of deciduous trees (Fagus sylvatica and Quercus petraea) and has been in place over at least a century, according to stand structure. The forest soil is an Acrisol and the humus form is a dysmoder sensu [START_REF] Brêthes | Classification of forest humus forms: a French proposal[END_REF].

The nearby pasture used to be mowed every year in spring and then grazed by cattle in summer and autumn, but mowing had been abandoned for several years because of poor forage production due to several consecutive drought years. The pasture soil is a Cambisol and the humus form is an eumull.

The transition between forest and pasture is sharp.

Experimental design and soil core manipulation

We designed a soil core transplantation experiment between forest and pasture (closed vs. open vegetation, respectively) coupled with a manipulation of invertebrate communities. Eight treatments (five replicates each) corresponded to all possible combinations of three factors: community origin, COM (forest vs. pasture), soil origin, S (forest vs. pasture) and microclimate, CLIM (forest vs. pasture) (Fig. 1, see also Fig. 2 for a global view of manipulation steps). The setup took place between March and June 2011 (fauna removal, inoculation and transplantation) and the experiment ended in the beginning of November 2011.

Fauna removal and re-inoculation

In order to control the communities present in both soils (forest and pasture), we first removed the fauna and re-inoculated it with a new community extracted from a fresh soil core. This allowed us to have a forest community in the pasture soil and conversely a pasture community in the forest soil.

Thirty soil cores (20 cm diameter x 10 cm depth) were taken in both forest and pasture (60 soil cores in total, i. e. the soil, including the soil biota, was sampled by taking of soil samples) and brought back to the laboratory. Soil fauna was then eliminated by repeatedly freezing soil cores. Each soil core was dipped in liquid nitrogen for 45 min. This was repeated after a week interval, in order to eliminate possible resistant eggs that could have been stimulated to hatch by the first freezing. In between, soil cores were stored in a cold chamber at 15 °C.

We then inoculated each soil core with a new community. To do so, 48 soil cores (24 for each soil) of the same volume (20 cm diameter x 10 cm depth) were taken at the same site. These cores were split into four equal parts in the field, packed into semi waterproof bags (plastic bags with holes allowing gas exchanges) and brought back to the lab within two days. They were immediately stored in a cold chamber at 15 °C before being used as a new community source for re-inoculation. Fourteen defaunated pasture soil cores were inoculated with a community originating from the pasture (4 of which were used as controls, see following section) and 10 pasture soil cores were inoculated with a community originating from the forest. Likewise, 14 defaunated forest soil cores were inoculated with a community originating from the forest (4 of which were used as controls, see following section) and 10 forest soil cores were inoculated with a community originating from the pasture. To re-inoculate communities, we used a Berlese dry-funnel extractor. We placed the fresh soil on the extractor sieve and the soil core which had been previously defaunated under it. This procedure allowed transferring the new community from the fresh to the defaunated soil core. Each quarter of the fresh cores was left one week on the extractor sieve. Re-inoculation thus lasted 4 weeks. Each week, one quarter of the soil cores used for re-inoculation was placed on the extractor sieve after the previous quarter was removed.

Soil cores were watered every week with 100 mL distilled water. After fauna removal and before reinoculation, we watered all soil cores with a soil suspension (10 g of soil sampled the same day per litre distilled water) sieved to 20 µm. Pasture and forest soil cores were watered with a soil suspension prepared with pasture and forest soils, respectively. This procedure was performed in order to reestablish the microbial community in soil cores after fauna removal (freezing).

Soil core enclosure and transplantation to the field

In order to prevent as much as possible exchanges of soil animals between treated soils and the surrounding environment, soil cores were enclosed in PVC pipes covered with a 350 µm mesh at their top and a 20 µm mesh at their bottom. We finally brought the 46 manipulated soil cores back to the field. Each soil-community treatment was transplanted both in the forest and in the pasture and was left in the field from June 15 to November 2, 2011 (four and a half months).

The experimental design thus comprised every combination of three factors (community origin, soil and microclimate) for a total of 8 treatments with 5 replicates each (Fig 1). Additionally, it included 3 types of manipulation controls and 2 types of natural references (3 to 5 replicates depending on the type of control, see next section).

Experimental controls and natural references

At each stage of the experimental setup, controls were implemented. This allowed us to assess the efficiency of: 1) fauna removal, 2) community re-inoculation, 3) exclosure, and allowed us to determine the composition of forest and pasture communities in a non-manipulated situation.

To check for the efficiency of fauna removal, we randomly selected 3 soil cores of each soil directly after fauna removal and we performed fauna extraction (fauna removal controls).

To check for the efficiency of community re-inoculation, 8 soil cores (4 forest and 4 pasture cores inoculated with their own community) were randomly selected directly after re-inoculation and placed in a Berlese dry-funnel extractor (inoculation controls).

To check for the efficiency of exclosure, 6 soil cores (3 for each soil) were randomly selected and directly enclosed after fauna removal (i. e. without inoculation with a fresh community) and brought back to the field for transplantation (exclosure controls).

In order to determine the composition of both communities in the undisturbed (i.e. nonmanipulated) situation, 3 samples (5 cm diameter x 10 cm depth) were taken at the same time in each habitat (forest and pasture) when sampling for the soil material used to re-inoculate experimental soil cores (natural control t 0 ). They were brought back to the laboratory on the same day for fauna extraction. Likewise, 5 samples (5 cm diameter x 10 cm depth) were taken in each habitat (forest and pasture) at the end of the experiment and brought back to the laboratory within three days for fauna extraction (natural controls t end ).

All fauna extractions were performed using a Berlese dry-funnel apparatus and lasted 12 days.

Soil sample treatments

At the end of the experiment, we sampled each core according to three methods. First, a sample 6.3 x 6.3 x 10 (depth) cm was taken at the centre of each core for fauna extraction (fauna samples). Second, a 300-g sample was taken in each core, air dried and sieved (2 mm) for soil analysis (soil pH water , total carbon, and total nitrogen content by gas chromatography). And third, another 300-g sample was taken in each core and immediately packed in waterproof bags for soil moisture measurements.

Fauna samples were brought back to the lab within three days and placed in a Berlese dryfunnel extractor for 12 days. Animals were collected and stored in 70 % ethyl alcohol until identification. Collembola were mounted, cleared in chloral-lactophenol and identified to species level under a light microscope (magnification x 400), according to [START_REF] Hopkin | A Key to Collembola (Springtails) of Britain and Ireland[END_REF], [START_REF] Potapov | Synopses of Palearctic Collembola. III. Isotomidae[END_REF][START_REF] Thibaud | Synopses on Palearctic Collembola. IV. Hypogastruridae[END_REF] and [START_REF] Bretfeld | Synopses on Palaearctic Collembola. II. Symphypleona[END_REF]. Due to the very large number of individuals belonging to this species group, we pooled the two species Folsomia quadrioculata and F. manolachei together.

Calculation of species overall habitat preference

The two ecological traits describing the habitat preference (IndF and IndA, see below) of each species were calculated using the IndVal index [START_REF] Dufrêne | Species assemblages and indicator species: the need for a flexible asymmetrical approach[END_REF] adapted to the measurement of preference for a given habitat type by [START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF]. For this calculation, we used the data set produced in [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF], who worked in exactly the same region. One species present in our study (Detriturus jubilarius) was absent from the study by [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF]. The habitat preference of this species was assessed according to expert knowledge (Salmon, unpublished data).

The IndVal index combines the specificity of a species for a habitat type (maximized when the species is found only in a given habitat) and its fidelity to this habitat (maximized when the species is found in all samples of a given habitat):

Ind ij = A ij * B ij * 100, where
A ij = average abundance of species i in samples of habitat j divided by the average abundance of species i in all samples.

B ij = number of samples of habitat j where the species is present divided by the total number of samples of habitat j.

Ind ij ranges from 0, when species i is absent from habitat j, to 100 (its maximum value), when species i is present in all samples of habitat j and absent in all other habitat samples. We thus obtained two IndVal values for each species, one for forest (IndF) and one for agricultural land (IndA). Classes of habitat preference were then determined using the IndVal values IndF and IndA for each species.

Species present in both habitat types and having a ratio IndF/IndA (or the reverse) higher or equal to 0.25 were classified as "generalists". Species having a ratio IndA/IndF lower than 0.25 were classified as "forest-preferring" and species having a ratio IndA/IndF = 0 were classified as "strict forest" species. Species having a ratio IndF/IndA lower than 0.25 were classified as "agricultural-preferring" and species having a ratio IndF/IndA = 0 were classified as "strict agricultural" species (sensu [START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF].

Data analyses 2.5.1. Assessing the effect of experimental manipulation

In order to detect possible effects of soil manipulation, inoculation, and exclosure on species abundance, we implemented linear models testing the effect of control type (natural controls t 0 and t end , inoculation control, exclosure control, and experimental control, i.e. treated soil cores transplanted in their own microclimate with their own community), habitat type (forest vs. pasture) and the interaction between these factors, on total abundance (type III sum of squares used for unbalanced design). As the soil volumes sampled for natural controls (t 0 and t end ) and experimental controls were different, we transformed the total abundance into areal density (number of individuals per m 2 ). To fulfil linear model assumptions, areal density was log-transformed. In order to compare community structure and composition of all types of controls (natural controls t 0 and t end , inoculation control, exclosure control, and experimental control), we performed a principal component analysis using abundances of common species (i.e. present in at least 10 % of the experimental cores).

In order to detect the effects of experimental treatments on soil properties (total carbon and nitrogen content, soil pH and moisture) we implemented linear and generalized linear models (Gamma link function) testing the effect of soil nature (forest vs. pasture) and microclimate (forest vs. pasture) on soil properties. Data for total carbon and nitrogen content and for soil moisture were logtransformed to fulfil linear model assumptions.

Effect of experimental treatments on collembolan diversity and abundance

In order to detect the effects of experimental treatments on collembolan diversity and abundance, we tested the effect of the three experimental factors (origin of the community, soil nature and microclimate) and the interaction between these factors on species richness, Shannon diversity index, and total abundance using linear models. Abundances were log-transformed to fulfil linear model assumptions. Models were tested after a procedure of automatic model selection based on AIC criterion (stepwise procedure). Combinations of experimental treatments were compared using least square means and associated multiple comparisons of means (Tukey).

Effect of experimental treatments on collembolan community structure and species abundance

In order to detect the effect of experimental treatments on community structure, we implemented a between-group multivariate analysis [START_REF] Baty | Analysis with respect to instrumental variables for the exploration of microarray data structures[END_REF] on abundances of common species in each treatment. Between-group analysis is a particular case of instrumental variables methods where a single qualitative variable is accounted for [START_REF] Baty | Analysis with respect to instrumental variables for the exploration of microarray data structures[END_REF], providing the best linear combination of variables maximizing between-group variance. Between-group analysis was performed using a combination of the three experimental factors (origin of the community COM, soil nature S and microclimate CLIM, 8 combinations) as the explanatory variable. The significance of the composite factor COM/S/CLIM was tested using a Monte-Carlo permutation test (999 permutations).

The effects of experimental factors (COM, S, CLIM and all possible interactions) on the abundance of each common species (i.e. species present in at least 10 % of the experimental cores) were tested using generalized linear models (poisson link function) after a procedure of automatic model selection based on AIC criterion (stepwise procedure). Combinations of experimental treatments were compared using least square means and associated multiple comparisons of means (Tukey). Based on the results of these models, we classified species according to their response to experimental factors. Species being significantly more abundant in a given soil and/or microclimate were considered as preferring this soil and/or microclimate. Species showing similar preferences for soil nature and microclimate were grouped together.

All statistical analyses were performed using vegan, ade4, car, and lsmeans packages of R software (R Development Core Team, 2010).

Results

Experimental controls

In total, 28 species were found (controls included), of which 22 species were present in the experimental treatments (controls excluded). Among these 22 species, 6 were present in less than 10 % of the experimental soil cores (< 4 cores) and were thus excluded from the analysis for improving robustness of the conclusions. Among the 16 species kept for the analysis, 9 were also present in exclosure controls. Among these 9 species, four were present in both pasture and forest exclosure controls (Lepidocyrtus lanuginosus, Mesaphorura macrochaeta, Parisotoma notabilis and Sphaeridia pumilis), four were present in pasture exclosure controls only (Brachystomella parvula, Isotoma viridis, Protaphorura armata and Sminthurides schoetti) and one species was present in forest exclosure controls only (Xenylla tullbergi) (Table 1). Thirteen species were successfully inoculated in the experimental soil cores, among them four species were successfully inoculated in both forest and pasture soils, seven were inoculated in forest soil only and two were successfully inoculated in pasture soil only (Table 1). No Collembola were found in the fauna removal control either in pasture or forest soil.

The linear model testing the effect of treatments on collembolan density showed that the type of control (natural controls t 0 and t end , inoculation control, exclosure control, and experimental control) and the interaction between control type and soil nature exerted an influence on collembolan density (p<0.001 and p<0.01, respectively). Collembolan density was significantly higher in inoculation controls and in experimental controls than in natural controls taken at the end of the experiment (t end ) (Fig. 3). Additionally, post-hoc tests (Tukey) showed that the natural control taken at the end of the experiment (t end ) in the pasture showed a lower collembolan density than both forest and pasture experimental controls. It also showed a lower density than the natural controls taken at the end of the experiment in the forest and than exclosure and inoculation controls in the pasture (Fig 3). The first two axes of principal component analysis (PCA) implemented on species abundances of controls (Fig. 4) extracted 34.5 % of the total variance (29.4 % and 15.1 %, respectively). PCA showed that communities were distinguished according to community origin on axis 1, pasture communities standing on the positive side and forest communities standing on the negative side of axis 1. However, the exclosure control in the forest (TexF) displayed communities closer to the pasture on axis 1. The community in the forest experimental control (FFF) lay close to the community of forest natural reference both at the beginning and at the end of the experiment. In contrast, the community of the pasture experimental control (PPP) lay close to the community of the pasture natural reference at the end of the experiment but far from the one present at beginning of the experiment.

Effects of experimental treatments on soil physicochemical properties

Linear and generalized linear models (Table 2) testing the effect of soil nature (forest vs. pasture) and microclimate (forest vs. pasture) on soil properties (total carbon and nitrogen content, soil pH and moisture) showed that the total carbon content was higher in forest than in pasture soil. In contrast, the total nitrogen content did not differ with soil nature or microclimate. Soil pH was higher in pasture than in forest soil (p<0.001) and soil pH in pasture soil was higher under forest than under pasture microclimate (p<0.001, Fig. 5a). Soil moisture was significantly affected both by soil nature and microclimate (p<0.001, and p<0.01 respectively). Soil moisture was higher in forest than in pasture soil and soil moisture in pasture soil was higher under forest than under pasture microclimate (Fig. 5b).

Effects of experimental treatments on collembolan diversity and abundance

Linear models testing the effect of the three experimental factors (origin of the community, soil nature and microclimate) on species richness, Shannon index, and total abundance (Table 3) showed that the three factors (community origin, soil nature and microclimate) had an effect on total abundance (p<0.01, p<0.05, and p<0.001, respectively). Collembola were more abundant in the pasture than in the forest community (community origin), they were also more abundant in the pasture than in the forest soil, but they were more abundant under forest than under pasture microclimate (Fig. 6a). Only the origin of the community exerted an effect on species richness (p<0.001). The community originating from the forest displayed higher species richness than the community originating from the pasture, whatever the microclimate or the soil in which they were inoculated (Fig. 6b). Finally, these models showed that community origin, soil nature, and the interaction between community origin and microclimate had a significant effect on the Shannon index (p<0.001, p<0.01, and p<0.01 respectively). The Shannon index was higher in forest than in pasture community and was also higher in forest than in pasture soil. Post-hoc tests (Tukey) showed that the interaction between community origin and microclimate was due to the fact that the Shannon index was higher under forest than under pasture microclimate, but only for the community originating from the pasture (Fig. 6c).

Effects of experimental treatments on collembolan community structure and species abundance

The individual response to experimental treatments of the 16 most common species is shown in the Appendix. Between-group analysis (Fig. 7) performed on species abundances taking a combination of community origin, soil nature, and microclimate as the explanatory variable extracted 56 % of the total variance. Axes 1 and 2 accounted for 44 % and 23 % of the variance extracted, respectively. Nine species contributed to the formation of axis 1, four on the positive side (Protaphorura armata, Parisotoma notabilis, Mesaphorura macrochaeta and Pseudosinella alba) and five on the negative side (Folsomia spp., Isotomiella minor, Detriturus jubilarius, Megalothorax minimus and Friesea truncata). Axis 1 discriminated communities according to their origin, pasture on the positive side and forest on the negative side. Only three species mostly contributed to the formation of axis 2, all of them negatively (Sminthurides schoetti, Isotoma viridis and Xenylla tullbergi). Axis 2 discriminated forest communities according to the microclimate in which they were transplanted, forest microclimate on positive side and pasture microclimate on negative side. For the pasture community, treatments were much less discriminative on axis 2 than for the forest community.

A Monte-Carlo permutation test showed that the composite factor COM/S/CLIM significantly affected the community (p=0.001).

Species classification based on the effect of experimental treatments on species abundance

Using generalized linear models testing the effect of the three factors COM, S, and CLIM on species abundance, we classified species into six groups according to their response to the factors (Table 1). Group A was comprised of three species (Isotomiella minor, Megalothorax minimus, and Pseudosinella mauli) that were more abundant in forest community, soil and microclimate. They were labeled "true forest species" (Fig 8a). Group B was comprised of three species (Folsomia spp., Friesea truncata and Detriturus jubilarius), that were more abundant in both forest community and microclimate, but were more abundant in pasture soil (Fig. 8b). They were labeled "forest species preferring pasture soil". Species of groups A and B were classified as forest species except Folsomia spp. that were classified as generalists using the IndVal index calculated with the data set produced in [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF] (Table 1). Group C was comprised of two species (Protaphorura armata and Pseudosinella alba) that were more abundant in the pasture community but more abundant in the forest soil. Additionally, while Protaphorura armata was also more abundant in the forest microclimate (Fig. 8c), microclimate did not exert an effect on the abundance of Pseudosinella alba.

Group C was labeled "pasture species preferring forest soil". Group D was comprised of two species (Mesaphorura macrochaeta and Parisotoma notabilis) which were more abundant both in pasture community and soil but were more abundant in forest microclimate (Fig 8d). They were labeled "pasture species preferring forest microclimate". Group E was comprised of three species (Brachystomella parvula, Lepidocyrtus lanuginosus, and Sphaeridia pumilis) that were more abundant in pasture microclimate. However, in this group, all three species showed a preference for a different component of the forest habitat, either for soil (Lepidocyrtus lanuginosus) or forest community, i.e.

were more abundant in cores inoculated with a forest community (Sphaeridia pumilis), or both components (Brachystomella parvula). Group E was labeled "species preferring pasture microclimate". And finally, group F was comprised of three species (Xenylla tullbergi, Isotoma viridis and Sminthurides schoetti), that were more abundant in both pasture soil and microclimate. Isotoma viridis (Fig. 8e) and Sminthurides schoetti were as abundant in cores inoculated with a pasture community as in cores inoculated with a forest community whereas Xenylla tullbergi (Fig. 8f) was more abundant in cores inoculated with a forest community. This group was labeled "pasture species".

Most species of groups C, D, E and F were classified as agricultural and generalist species except for Xenylla tullbergi that was classified as a forest species according to the IndVal index calculated with the data set produced in [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF] (Table 1).

Discussion

Effect of soil nature and microclimate on collembolan total abundance and community structure

Our results show that collembolan abundance was higher in forest than in pasture microclimate for both forest and pasture communities. Transplantation decreased the moisture content of forest cores when transplanted in the pasture and it increased the soil moisture of pasture cores transplanted in the forest (Fig. 5b, Table 2). Collembola are known to be sensitive to drought [START_REF] Vannier | The porosphere as an ecological medium emphasized in Ghilarov's work on soil animal adaptations[END_REF]. We thus attribute to this physiological trait the overall abundance increase in forest microclimate and decrease in pasture microclimate. It means that forest species are likely to be absent (or less abundant) in pasture mainly because they survive poorly in pasture climate conditions. This may concern only some stages of collembolan life, such as the moisture-sensitive first stadium, stemming in the incapacity of some species to endure moisture and temperature fluctuations which characterize open environments as opposed to closed environments [START_REF] Betsch | Caractérisation des deux phases juvéniles d'Allacma fusca (Collembola, Symphypleona) par leur morphologie et leur écophysiologie[END_REF].

Additionally, we showed that forest communities were different under pasture and forest microclimate whatever the nature of the soil. This means that microclimate conditions are the first driver shaping collembolan communities in the forest. We thus suggest that forest species display physiological traits (namely poor resistance to drought) that prevent them from surviving or growing larger populations in open habitats. The pasture community did not show such a trend, suggesting that microclimate change (pasture to forest) did not affect its species composition. Thus, microclimate conditions are not likely to be the most important constraint shaping the pasture community.

Species preferences for soil and microclimate

Our experimental design enabled us to unravel species responses to soil nature and microclimate. We showed that some species, classified as forest species according to field occurrence data, are more abundant in forest soil and microclimate and that these species are also more abundant in the communities originating from the forest (Group A). These species can thus be regarded as "true forest species" because they need both forest microclimate (moisture, temperature) and soil (food resources and physicochemical properties) to fully develop, at least in the studied region. This is supported by previous experiments showing that I. minor and M. minimus, two out of the three "true forest species" (Group A), are particularly sensitive to drought [START_REF] Makkonen | Traits explain the responses of a sub-arctic Collembola community to climate manipulation[END_REF]. However, some other species, also classified as forest species using the large data set from [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF], are shown to prefer the pasture soil when transplanted to the forest microclimate (Group B). Hence, for these species, preferences for soil and microclimate are not tuned. It means that although they prefer the forest microclimate (temperature and moisture) they prefer trophic resources or physicochemical properties of the pasture soil. Their confinement to forest habitats is thus the result of climate requirements overwhelming soil quality requirements, i.e. least-worst strategy [START_REF] Berger | Ecological constraints on female fitness in a phytophagous insect[END_REF]. Some authors have already underlined the strong influence of microclimate on Collembola [START_REF] Lindberg | Population responses of oribatid mites and collembolans after drought[END_REF][START_REF] Makkonen | Traits explain the responses of a sub-arctic Collembola community to climate manipulation[END_REF][START_REF] Petersen | Collembolan communities in shrublands aklong a climatic gradient in Europe and the effect of experimental warming and drought on population density, biomass and diversity[END_REF]. In their experiment, [START_REF] Krab | Turning northern peatlands upside down: disentangling microclimate and substrate quality effects on vertical distribution of Collembola[END_REF] showed that most species found in a subarctic community tended to select microclimate over substrate quality. Here, we go further and show that some forest species survive better in pasture soil (of mull type) if they can find forest climate conditions. Such conditions (forest microclimate and pasture soil quality) are fulfilled in not or poorly acidic forest soils, as already shown on census basis (but not experimentally demonstrated) by [START_REF] Ponge | Biocenoses of Collembola in atlantic temperate grass-woodland ecosystems[END_REF].

Likewise, we showed that some pasture and generalist species benefit from the forest microclimate (Group D plus Protaphorura armata) but that the abundance of some species decreases when they are transferred to the forest soil (Group D). These species are thus also favoured by higher soil moisture and lower temperature but probably do not find in forest habitat appropriate resources and/or physicochemical features, or are too poorly competitive to maintain populations as large as in the pasture soil. However, pasture species of Group C are more abundant in the forest soil indicating that this soil fulfils their trophic and/or physicochemical requirements. We can thus genuinely ask why these species are more abundant in pasture, given that they seem to be favoured by forest microclimate and soil. Since we eliminated environmental filters (microclimate, soil quality) and dispersal limitation (eliminated in our experimental design) the answer probably relies in species interactions. Although suspected to explain cases of species richness deficit or species turnover at the local scale [START_REF] Hågvar | Reactions to soil acidification in microarthropods. Is competition a key factor?[END_REF][START_REF] Winkler | Local and regional species richness in communities of surfacedwelling grassland Collembola: indication of species saturation[END_REF], competition within soil communities is still a too scarcely investigated topic [START_REF] Bardgett | Causes and consequences of biological diversity in soil[END_REF][START_REF] Decaëns | Macroecological patterns in soil communities[END_REF]. Despite being primarily carried out in laboratory conditions and with a reduced number of species, the few studies trying to shed light on the importance of competition in structuring soil communities suggest that competition occurs and is an important mechanism [START_REF] Christiansen | Competition between collembolan species in culture jars[END_REF][START_REF] Christiansen | Interspecific interactions between collembolan populations in culture[END_REF][START_REF] Theenhaus | Contramensal interactions between two collembolan species: effects on population development and on soil processes[END_REF][START_REF] Postma-Blaauw | Within-trophic group interactions of bacterivorous nematode species and their effects on the bacterial community and nitrogen mineralization[END_REF]. Our experiment does not allow directly testing this hypothesis, but our results

show that some pasture and generalist species would perfectly withstand and even benefit from forest climate conditions and/or soil quality. This suggests that they are prevented from developing larger populations in forests by forest collembolan species that might be more efficient in exploiting forest resources. To these effects of other members of the collembolan community must be added those of members of a much wider community, the complete trophic network in which Collembola are included, still imperfectly known up to present [START_REF] Brose | 1153Into darkness: unravelling the structure of soil food webs[END_REF]. Biotic interactions in which Collembola are dynamically involved include negative interactions such as predation [START_REF] Lawrence | Sprider predation on forest-floor Collembola and evidence for indirect effects on decomposition[END_REF], but also positive interactions such as earthworm attraction [START_REF] Salmon | Earthworm excreta (mucus and urine) affect the distribution of springtails in forest soils[END_REF].

Finally, several species are more abundant under pasture microclimate (Groups E and F). They were all classified as agricultural or generalist species with a single exception: Xenylla tullbergi is the only forest species (present in the original forest community only) that is more abundant in pasture soil and under pasture microclimate. This result may be explained by the fact that Xenylla tullbergi is mostly found in corticolous habitats [START_REF] Ponge | Biocenoses of Collembola in atlantic temperate grass-woodland ecosystems[END_REF]. It is thus drought tolerant but found more abundantly in trees (absent from agricultural plots such as pasture). Its absence in the pasture community may also result from competition with pasture species. All other species of Groups E and F are agricultural or generalist species that prefer pasture microclimate (Group E) or pasture microclimate and soil (Group F). In Group E, two species were more abundant in the forest soil, which suggests that they either prefer resources found in forest soil or are favoured by higher soil moisture (or other physicochemical properties linked to forest soil) as we showed that under pasture microclimate, soil moisture was higher in the forest than in the pasture soil. We only found two species classified as agricultural species that were actually more abundant in pasture soil and under pasture microclimate as supported by previous observations in agricultural habitats [START_REF] Fratello | Effetcs of atrazine on soil microarthropods in experimental maize fields[END_REF][START_REF] Dittmer | Longterm effects of soil compaction and tillage on Collembola and straw decomposition in arable soil[END_REF][START_REF] Frampton | Diel activity patterns in an arable collembolan community[END_REF]. These two "pasture species" are thus likely to be primarily influenced by microclimate, resources, and soil physicochemical properties rather than by interspecific competition. However, conclusions about Sminthurides schoetti (one of the two abovementioned species) must be drawn with caution. Indeed, this species is the only one that did not succeed in re-inoculated samples but was present in the exclosure controls in the pasture. It is thus present in the experimental soil cores as a pure "invader". Therefore, the preference of this species for pasture or forest microclimate could not be ascertained. However we can be fairly certain that this species preferred the pasture soil as it was more abundant than in the forest soil independently of the community that was present in the soil beforehand.

Our results show that all forest species are better represented under forest microclimate, but that some of them prefer the pasture soil. It means that the most important factor constraining forest species is actually the microclimate. This is probably explained by physiological intolerance of forest species to summer drought. Thus, for some forest species (Group B) habitat preference seems to be the result of a trade-off between physiological requirements and requirements for resources and/or the physicochemical environment.

Methodological limitations

We were not able to fully prevent exchanges between experimental cores and their surroundings. More than half of the species present in more than four experimental soil cores (i.e. common species) invaded the mesocosms. All these species but one were agricultural or generalist species. This means that agricultural and generalist species have a greater mobility than forest species, as they had to climb or jump over the mesocosms in order to penetrate them. This is also partly why forest communities transplanted to pasture microclimate largely differed from forest communities transplanted to forest microclimate. The latter communities were not influenced by species invading from the surrounding pasture. Additionally, soil moisture in pasture soil cores was higher under forest than under pasture microclimate and we showed that total collembolan abundance was also higher in pasture experimental control than in natural controls. Thus, microclimate conditions created in the mesocosms seem to overall favour species abundance of the pasture community. Hence some of our results must be interpreted cautiously. First we showed that species richness was not affected by any experimental treatment. Species richness was only lower in the original pasture than in the original forest community. However, the invasion of the forest community by pasture species in microcosms transplanted to the pasture artificially increased species richness. Second, we cannot totally refute that the decrease in the abundance of some forest species under pasture microclimate was due to competitive exclusion from species invading from the surrounding pasture. Besides, we do not know what effects experimental manipulations had on microbial communities. Nevertheless, we were able to successfully re-inoculate most common species despite a long-lasting experimental procedure and to provide responses about soil and microclimate preferences of several collembolan species. This is very encouraging for future experiments dealing with Collembola as more studies are still needed to fully understand mechanisms responsible for patterns of species distribution.

Changes in species composition are known to occur over the year in collembolan communities [START_REF] Chagnon | Community structures of Collembola in sugar maple forests : relations to humus type and seasonal trends[END_REF]. Thus in our transfer experiment starting in spring and ending in autumn temporal variability accompanied the effects of microclimate and soil change, and thus could have blurred these effects. This cannot avoided, because expected effects take necessarily some time to appear at community level, through the combination of growth, reproduction, dispersal and species interactions, adding their effects to immediate mortality. However, natural controls, sampled at the beginning and at the end of our transfer experiment, allowed discerning changes in species composition in the meadow while no discernible change occurred in the forest (Fig. 4). Data collected on the same sites in the abovementioned experiment by [START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF] can be used to support this assessment. A sign-test done on the 16 more common species (unpublished data) showed that over the six months of this experiment (from December to June) the species composition did not change in the forest (exact P value = 0.454) while it significantly changed in the meadow (exact P value = 0.021). Thus temporal changes of collembolan populations are probably included in the observed effects of transfer from forest to pasture but not in the reverse case, to the possible exception of species with genetically coded cycles of egg diapause [START_REF] Leinaas | Egg diapause and demographic strategy in Lepidocyrtus lignorum Fabricius (Collembola; Entomobryidae)[END_REF].

Conclusion

We showed that habitat preference depends on responses to microclimate and soil quality and that environmental constraints have a different importance depending on the overall habitat preference of species. We conclude that an anthropogenic-induced stress, such as habitat conversion (deforestation or afforestation), modifies collembolan communities to a large extent, and that species show different levels of resistance to perturbations and respond to different constraints (e.g. microclimate, soil, interspecific competition). Generally, forest species seem to be primarily influenced by microclimate, whereas pasture species seem more influenced by trophic resources and competition. This suggests that trade-offs between several habitat constraints are at play and structure collembolan communities in open vs. closed vegetation. More insights into the importance of competition and predation in structuring collembolan communities are still needed at community level. the first letter refers to the origin of the community ("F" for forest and "P" for pasture); the second letter refers to the origin of the soil ("F" for forest and "P" for pasture) and the third letter refers to the habitat (microclimate) in which the core has been transplanted ("F" for forest and "P" for pasture). For species codes see Table 1. Labels on the gravity centers correspond to each treatment. TF0: natural reference in forest at beginning of the experiment, TFend: natural reference in forest at end of the experiment, FFF: control experiment for forest community, TeF: inoculation control for forest community in forest soil, TexF: fauna removal control for forest community, TP0: natural reference in pasture at beginning of the experiment, TPend: natural reference in pasture at end of the experiment, PPP: control experiment for pasture community, TeP: inoculation control for pasture community in pasture soil, TexP: fauna removal control for pasture community. Right: Correlation circle plot with species vectors (vector labels correspond to species codes in Table 1). 1). Right: Correlation circle plot with species vectors (vector labels correspond to species codes in Table 1). Eigen values 0.44, 0.24, 0.16 for axes 1 to 3, respectively; Randtest: simulated p-value: 0.001; Explained variance: 0.56. Table 1. Presence/absence in inoculation and exclosure controls of the 16 species which were common in experimental soil cores (X = not present in controls; F = present in forest control only; P = present in pasture control only; FP = present in both forest and pasture controls). Results of generalized linear models testing the effect of three factors (community origin, soil nature and microclimate) on each species abundance (* = p < 0.05; ** = p < 0.01; *** = p < 0.001; / = not significant; F = species more abundant in forest community, soil or microclimate; P = species more abundant in pasture community, soil or microclimate). Habitat preference was calculated using IndVal index with the data set produced in [START_REF] Ponge | Collembolan communities as bioindicators of land use intensification[END_REF]: g = generalist species; f = strict-forest species; fp = forest-preferring species; a = strict agricultural species; ap = agricultural-preferring species.

Response groups correspond to the six groups formed using species responses to the three factors community origin, soil, and microclimate. Group A = true forest species; Group B = forest species preferring pasture soil; Group C = pasture species preferring forest soil; Group D = pasture species preferring forest microclimate; Group E = species preferring pasture microclimate; Group F = pasture species.

Figure captions Figure 1 .

 captions1 Figure captions

Figure 2 .

 2 Figure 2. Summary of manipulation steps.

Figure 3 .

 3 Figure 3. Mean collembolan density in 5 types of controls (experimental, inoculation, natural at the

Figure 4 .

 4 Figure 4. Principal component analysis using abundances of common collembolan species (i.e.

Figure 5 .

 5 Figure 5. Mean soil pH (a) and mean soil moisture (b) in experimental forest (left) and pasture (right)

Figure 6 :

 6 Figure 6: Mean collembolan abundance (a), species richness (b) and Shannon index (c) in

Figure 7 :

 7 Figure 7: Between-group analysis on the abundance of common species, with the composite factor

Figure 8 :

 8 Figure 8: Abundance of six species in the experimental treatments. (a) Isotomiella minor (Group A);

Fig. 1

 1 Fig. 1

Fig. 2

 2 Fig. 2

  Fig. 3

Fig. 7 Fig. 8

 78 Fig. 7

Table 2 .

 2 Effect of soil nature and microclimate (and interaction between both factors) on total carbon (Ctot) and nitrogen (Ntot) content, soil pH and soil moisture of experimental soil cores. Results of linear and generalized linear models (F value/Chi square and degrees of freedom df). Significance levels: ** = p < 0.01; *** = p < 0.001; NS = not significant.

		Species	Exclosure Inoculation Community Soil	Microclimate Habitat	Response
		codes						preference	group
	Isotomiella minor	Iso.min	X	F	F ***	F *** F ***	fp	A
	Megalothorax minimus	Meg.min X	F	F ***	F *** F ***	fp	A
	Pseudosinella mauli	Pse.mau	X	F	F ***	F *** /	f	A
	Folsomia quadrioculata/manolachei Fol	X	FP	F ***	P *** F ***	g	B
	Friesea truncata	Fri.tru	X	F	F ***	P ** F **	fp	B
	Detriturus jubilarius	Det.jub	X	F	F ***	P *** F ***	f	B
	Protaphorura armata	Pro.arm	P	P	P ***	F *	F ***	ap	C
	Pseudosinella alba	Pse.alb	X	X	P ***	F *** /	a	C
	Mesaphorura macrochaeta	Mes.mac FP	FP	P ***	P *** F ***	g	D
	Parisotoma notabilis	Par.not	FP	FP	P ***	P *** F ***	ap	D
	Brachystomella parvula	Bra.par	P	X	F ***	F ** P ***	ap	E
	Lepidocyrtus lanuginosus	Lep.lan	FP	FP	/	F *	P *	g	E
	Sphaeridia pumilis	Sph.Pum FP	F	F *	/	P *	ap	E
	Xenylla tullbergi	Xen.tul	F	F	F ***	P *** P ***	f	F
	Isotoma viridis	Iso.vir	P	P	/	P *** P ***	ap	F
	Sminthurides schoetti	Smi.sch	P	X	/	P *** P **	ap	F

Table 3 .

 3 Effect of community origin, soil nature and microclimate on total abundance, species richness and Shannon index. Results of linear models (F values and degrees of freedom df) tested after a procedure of automatic selection based on AIC criterion. NT = not tested. * = p < 0.05; ** = p < 0.01; *** = p < 0.001; NS = not significant.

			F values
		df Abundance	df Species richness	df Shannon index
	Community (Co) 1	10.98 **	1 15.8 ***	1	33.85 ***
	Soil (S)	1	5.29 *	1 1.9	NS	1	9.33 **
	Climate (Cli)	1	50.02 ***	1 0.06 NS	1	0.88 NS
	Co : S	NT		1 0.8	NS	NT
	Co : Cli	NT		1 0.007 NS	1	11.04 **
	S : Cli	NT		1 0.007 NS	NT
	Co : S : Cli	NT		1 0.16 NS	NT
	model type		Normal	Normal	Normal
	transformation		log 10	none		none

Acknowledgments

This study was sponsored by the R2DS program of the Conseil Régional d'Île-de-France. It also received funds from the French Ministry of Ecology, Energy and Sustainable Development (MEEDDM) through the GESSOL 3 2009 call for projects (TRACES project). We would like to thank the private owner of sampling sites for allowing the study to be carried out on his land. We would like to thank the IRD (Centre France Nord) and the MNHN (Brunoy) for access to their facilities and material; Louis-Cyrille Guillard for his help with performing CHN gas chromatography; Stéphane Sabbe and Marion Meconte for soil pH measurements and their help in sorting fauna material. We would also like to thank Fatima Boucha for her great help in identifying collembolan species and Jean-Christophe Lata for his help with the field work as well as Joshua Lobe for English corrections.

Appendix. Abundances of the 16 common species in the eight experimental treatments together with the correlation circle of between-group analysis performed on the abundances of the 16 common species using a combination of community origin, soil, and microclimate as explanatory variable.

Letters on bars indicate significant differences. Labels under bars correspond to each treatment (codes according to Figure 1).


Olivier Doar
email: olivier.doare@ensta-paristech.fr

Chistophe Eloy
email: eloy@irphe.univ-mrs.fr

THE INFLUENCE OF CHANNEL WALLS ON FLAG FLUTTER

The effect of confinement of a rectangular channel on the flag flutter instability is investigated, both in terms of waves propagating in the infinite medium (local approach) and in terms of modes in the finite length system (global approach). Using a 1D Euler-Bernoulli model for the plate and a 3D potential flow model for the flow, the problem is solved numerically for various sets of the geometrical parameters to compute the critical velocity for instability as function of the mass ratio. It is found that the proximity of walls parallel and perpendicular the plate affects significantly the plate's dynamics and instability. One main conclusion of this work is that models considering a 2D flow can predict correctly the critical velocity only in specific cases such as large mass ratio, large aspect ratio or very confined plates.

INTRODUCTION

The equilibrium state of a cantilevered plate in an axial flow can become unstable once a critical value of the flow velocity is reached, resulting in oscillations of large amplitude. This phenomenon, referred to as flag flutter has been the focus of a large amount of researches, as reported in some recent books and reviews [START_REF] Païdoussis | Fluid-structure Interactions[END_REF][START_REF] Shelley | Flapping and Bending Bodies Interacting with Fluid Flows[END_REF]. Among the various linear models that atempted to give a numerical prediction of the critical flow velocity for apparition of flutter instability, one can distinguish three main families. In the first one, a 2D problem is considered, i.e. an infinite span plate in a 2D potential flow [START_REF] Kornecki | On the aeroelastic instability of two-dimensional panels in uniform incompressible flow[END_REF][START_REF] Huang | Flutter of cantilevered plates in axial flow[END_REF][START_REF] Aurégan | Snoring: Linear stability and in-vitro experiments[END_REF]. The approximation done in these models is referred to as large span approximation. In the second familiy, the typical wavelengths of deformation are considered large enough when compared to the typical size of the cross section of the plate, so that the perturbation of the flow at a given cross section depends only on the plate dynamics at this cross section [START_REF] Lemaitre | Instability of a long ribbon hanging in axial air flow[END_REF]. These models are referred to as slender body models. The third family concerns models that do not consider any of the above approximations, and solve the 3D flow around the plate. This kind of model has shown that there is a strong influence on the aspect ratio of the plate on the critical velocity [START_REF] Eloy | Flutter of a rectangular plate[END_REF]. Works using similar method followed to study the presence of walls perpendicular to the plate (spanwise confinement) and showed that the critical velocity tends slowly to that predicted by a 2D model when the walls are approched [START_REF] Doaré | Flutter of an elastic plate in a channel flow: Confinement and finite-size effects[END_REF]. In the present work, two additional walls, parallel to the plate are considered so that the latter is confined in a rectangular channel flow. The method used in the present paper is the same as developped in previous works [START_REF] Eloy | Flutter of a rectangular plate[END_REF][START_REF] Doaré | Flutter of an elastic plate in a channel flow: Confinement and finite-size effects[END_REF].

Experimentally, flag flutter has also been intensively investigated. The first experimental studies of the phenomenon are reported in references [START_REF] Kornecki | On the aeroelastic instability of two-dimensional panels in uniform incompressible flow[END_REF][START_REF] Taneda | Waving motions of flags[END_REF][START_REF] Datta | Instability of an elastic strip hanging in an airstream[END_REF], among others. A consistent review of all experimental studies before 2004 can be found in reference [START_REF] Païdoussis | Fluid-structure Interactions[END_REF]. These studies have shown that although the plate's dynamics is always two-dimensional, the values of the critical velocity predicted by 2D models are always lower than that found experimentally. Comparison with 3D models taking into account the finite aspect ratio of the plate significantly improved the prediction of the critical velocity [START_REF] Eloy | Flutter of a rectangular plate[END_REF] and a nonlinear model taking into account the initial spanwise curvature [START_REF] Eloy | The origin of hysteresis in the flag instability[END_REF] explained the origin of the hysteresis observed experimentally. Finally, the effect of walls perpendicular to the plate has been investigated experimentally in reference [START_REF] Doaré | Effect of spanwise confinement on flag flutter: Experimental measurements[END_REF] and results are in good agreement with the model developed in [START_REF] Doaré | Flutter of an elastic plate in a channel flow: Confinement and finite-size effects[END_REF].

The paper is organized as follows. In the next section, the problem of a flat plate in a rectangular channel potential flow is formulated. The stability properties of the infinite length problem are then analysed. Next, a finite length system is considered.

FORMULATION OF THE PROBLEM 1.Coupled fluid-structure equations

The problem considered in this paper is sketched on Fig. 1. The solid is modeled as a plate of negligible thickness under the Euler-Bernoulli approximation. It is assumed that the deformation does not depend on the spanwise coordinate, so that the linearized equilibrium equation of the coupled fluid-solid system has the following expression:

B ∂ 4 W ∂ X 4 + µ ∂ 2 W ∂ T 2 = [P] , (1)
where B is the flexural rigidity per unit length in the spanwise direction of the plate and µ its surface density, [P] stands for the pressure jump between each side of the plate and the angles . denote the mean value along the span. The surrounding fluid flows at a constant velocity U 0 in the direction X and is modeled using the potential flow approximation. The velocity potential is noted Ψ.

Using the following dimensionless variables,

x = 2Z H , z = 2Z H , t = 4T B µH 2 , w = 2W H , (2)
p = P ρU 2 , ψ = Ψ µ/B, (3)
and parameters,

m = ρ f H 2µ , u = U H 2 µ B , c = 2C H , d = 2D H , (4)
equation ( 1) becomes,

∂ 4 w ∂ x 4 + ∂ 2 w ∂t 2 = m [p] . (5)
In the fluid domain, the non-dimensional velocity potential satisfies the Laplace equation,

∆ψ = 0, (6)
with the following boundary conditions:

∂ ψ ∂ z = 0 for |y| = d, (7)
∂ ψ ∂ y = 0 for |z| = 1 + c, (8)
∂ ψ ∂ z = ∂ w ∂t + u ∂ w ∂ z on the plate. ( 9
)
The pressure is then linked to the potential through the linearized unsteady Bernoulli equation,

p = - ∂ ψ ∂t -u ∂ ψ ∂ x . ( 10
)

Problem in the Fourier space

Introducing the Fourier transform in space and time ĥ of a function h(x,t),

ĥ(k, ω) = 1 2π 2 +∞ -∞ +∞ -∞ h(x,t)e iωt e ikx dtdx, (11)
equation ( 5) takes the following form in the Fourier space:

k 4 -ω 2 ŵ = -m [ p] . (12)
A normalized potential φ = ψ/i(ω -U R k) ŵ is then introduced. Using the unsteady Bernoulli equation [START_REF] Datta | Instability of an elastic strip hanging in an airstream[END_REF], the pressure jump may be linked to the displacement and the normalised potential φ, The Laplace equation ( 6) with boundary conditions (7-9) can now be expressed for the normalized velocity potential φ. This problem is defined in the plane (y, z) of a cross-section of the plate, as sketched on Fig. 2a. It can be shown that this problem is symmetric with respect to the y-axis and skew-symmetric with respect to the zaxis. Hence it can be rewritten in only one quarter of the domain as sketched on Fig. 2b,

[ p] = (ω -uk) 2 [ φ] ŵ. (13)
∂ 2 φ ∂ y 2 + ∂ 2 φ ∂ z 2 = k 2 φ, for z ∈ [0, 1 + c], y ∈ [0, d](14) ∂ φ ∂ z = 1 for z < 1, y = 0, (15)
∂ φ ∂ z = 0 for y = d, (16)
∂ φ ∂ y = 0 for z = 1 + c and z = 0. ( 17
)
This last problem depends only on k and the geometrical parameters c and d. Combining equations ( 12) and ( 13), the dispersion relation takes the following form,

k 4 -ω 2 = m 2g(k, c, d) |k| (ω -uk) 2 , (18)
where g(k, c, d) reads,

g(k, c, d) = k φ(y = 0 + ) . (19)
In the Fourier space, the work hence consists in obtaining the dispersion relation [START_REF] De Langre | Absolutely unstable waves in inviscid hydroelastic systems[END_REF] for a given set of geometrical parameters. To do so, it is necessary to compute the function g(k, c, d) by solving an Helmholtz problem (equation 14). The end of this section is devoted to recall the analytical form of g in various limit cases found in previous studies and present some computational results in the general case.

Limit cases for the function g found in the litterature

The different limit cases found in the litterature for the function g are presented here. The 2D limit of a plate in an infinite fluid domain is [START_REF] Crighton | Fluid loading with mean flow. I. Response of an elastic plate to localized excitation[END_REF],

g = 1. ( 20
)
The 2D limit of a plate confined by two parallel rigid walls is [START_REF] Guo | Stability of Rectangular Plates With Free Side-Edges in Two-Dimensional Inviscid Channel Flow[END_REF],

g = 1 tanh(kd) . (21)
The slender-body limit is [START_REF] Lighthill | Note on the swimming of slender fish[END_REF],

g = kπ 4 . ( 22
)
Using asymptotic expansions of the 2D and slenderbody limits, a composite function as been introduced to take into account the influence of a 3D flow around the plate [START_REF] Eloy | Flutter of a rectangular plate[END_REF],

g = 1 - 1 2k + exp[(π/4 -2)k] . (23)
In refence [START_REF] Doaré | Flutter of an elastic plate in a channel flow: Confinement and finite-size effects[END_REF], a similar approach has been used to develop asymptotic expansions of the 2D and slender-body limits that take into account the presence of walls perpendicular to the plate (C effect). A composite expansion of these two limits has then been developed, that extends the kernel of equation ( 23) to take into account spanwise confinement.

General case of a plate in a rectangular flow: numerical solution

In the general case of the present paper, the Helmholtz equation is solved using the open source finite element software Freefem [16] for given values of c, d and k. The resulting function g is plotted on Fig. 3a

A ≃ 1/3d, d ≪ 1. ( 24
)
When d is increased, A tends to a consant value which dependence on c has been characterized in a previous work [START_REF] Doaré | Flutter of an elastic plate in a channel flow: Confinement and finite-size effects[END_REF],

A = kπ 4 1 + 0.805 ln c + 0.189 c . ( 25
)
Now that we have the function g(k), and consequently the dispersion relation, in numerical form for any combination of the numerical parameters, it will be used to compute stability boundaries of the system in two different cases: the infinite case (local instability), for which only the dispersion relation is necessary to characterize the medium, and the finite case (global instability), for which subsequent work is needed, as boundary conditions and the length have to be taken into account. In the following, numerical data will be directly used to compute the different stability limits except at low values of k, where a third order polynomial fit is used.

LOCAL STABILITY ANALYSIS

The instability properties of bending waves propagating in an infinite length medium in such channel flow are now studied. In its non-dimensional form, the dispersion relation depends on four parameters, the mass ratio m, the non-dimensional velocity u, the non-dimensional spanwise gap c and the non-dimensional channel height d. The dispersion relation of equation ( 18) is considered, where the g function is obtained by solving the Helmoltz problem (14-17), as explained in the previous section.

The temporal approach consists in solving the dispersion relation, a second order polynomial for the variable ω, associated to real values of k. If for a least one real value of k, the imaginary part of one root ω is positive, the amplitude of the associated wave exp[i(kx -ωt)] grows exponentially with time and the medium is said locally unstable. It can be easily shown that the system bears unstable waves at any value of u = 0. However, one may distinguish two cases of wave instability, depending on the long time impulse response of the plate. If the exponentially growing wave packet is advected by the flow, the instability is said convective. Conversely, if it grows in place, so that the entire space is ultimately dominated by the instability, the latter is said absolute. The concepts of convective and absolute instabilities, and the underlying theory, have been initially introduced in the domain of plasma physics [START_REF] Briggs | Electron-Stream Interaction with Plasmas[END_REF] and successfully applied in the particular domain of fluidstructure interaction [START_REF] Crighton | Fluid loading with mean flow. I. Response of an elastic plate to localized excitation[END_REF]. In the present paper, a numerical method presented in reference [START_REF] De Langre | Absolutely unstable waves in inviscid hydroelastic systems[END_REF] is used to compute the critical curves for transistion between convective and absolute instabilities. The results are plotted on Fig. 5 in the (m, u/m) plane. On Fig. 5a, d = 30 and c varies from 3 × 10 -4 to 3. On Fig. 5b, c = 3 and d varies from 3 × 10 -3 to 30. All curves share a common limit at large values of m. This limit (u/m = 0.148) is that given by a 2D approach, where g = 1. It is actually twice the value found by Crighton & Oswell [START_REF] Crighton | Fluid loading with mean flow. I. Response of an elastic plate to localized excitation[END_REF] in the case of a compliant panel with a flow on one side only, whereas the flow is on both sides in the present study. It has to be noted that in this context of 3D flow, the 2D limit corresponds to large values of the wavenumber. It is consistent with the fact that when m → ∞, the wavenumber at which convective to absolute instability transition arises also tends to infinity.

The infinite derivative of all these curves at low values of m corresponds to the opposite limit where the wavenumber at transition is small. This is the slender body limit. In the case of a fluid-conveying pipe, which is properly described by a similar dispersion relation in the slender-body limit, Kulikoskii [START_REF] Kulikovskii | On the bending oscillations of a long tube filled with moving fluid[END_REF] showed that the transition does not depend on the flow velocity, but only on the mass-ratio. Again, this is consistent with the vertical limit observed on Fig. 5: an added mass effect of the channel confinement modifies the mass ratio in the slenderbody limit and the asymptotic value of m varies. The more the plate is confined (small c and d), the larger is the added mass. Hence the smaller is the critical value of m. Two additionnal asymptotic cases are ploted on Fig. 5: the curve obtained using an analytical solution of the Helmotlz problem in a 2D case with walls (blue dotted) and the curve obtained considering the 3D flow without walls (thick red). The numerical data tends to these two limits in a consistent manner.

GLOBAL STABILITY ANALYSIS

When studying a finite length system during timescales such that boundaries have an influence on the dynamics, boundary conditions must be taken into account. A plate clamped at X = 0 and free at X = L is now considered. The problem is now rescaled using L as a characteristic length, so that the following non-dimensional variables and parameters are introduced,

x = X L , y = Y L , z = Z L , w = W L ,t = U T L , p = P ρU 2 . (26) M * = ρL M ,U * = M B LU, H * = H L ,C * = C L , D * = D L .
(27) In this finite length approach, the dimensionless equation of motion is now

∂ 2 w ∂t 2 + 1 U * 2 ∂ 4 w ∂ x 4 = M * [p] , (28)
and the clamped-free boundary conditions are

w(x = 0) = ∂ w ∂ x x=0 = ∂ 2 w ∂ x 2 x=1 = ∂ 2 w ∂ x 2 x=1 = 0 (29)
Stability analysis is done through a modal analysis of the system. By looking for solutions of the problem in the form

w(x,t) = φ (x)e iωt , (30)
and assuming that pressure forces exerted on the plate can be expressed as linear functions of the displacement w, one end up with a Sturm-Liouville eigenvalue problem that does not have an analytical solution in the general case. If one eigenvalue has a negative imaginary part, the system is unstable, as the temporal evolution of displacement is such that it diverges when time goes to infinity. The corresponding real part gives then the circular frequency. In practice the system has to be put in a form suitable for numerical integration. Equation ( 13) may be rewritten in the form,

k p g = (ω -k) 2 ŵ. ( 31
)
Taking the inverse transform of this expression leads to a solution in the form of a convolution integral for the pressure,

1 2π 1 0 p ′ (v)G(x -v)dv = ∂ ∂t + ∂ ∂ x 2 w, ( 32
)
where prime denotes for derivation with respect to the axial coordinate and G(x) is the inverse Fourier transform of 1/g(k),

G(x) = ∞ 0 sin(kx) g(k) dk. ( 33
)
It has to be noted that as G diverges when x tends to 0, the Cauchy's principal value of the integral in equation (32) has to be taken. As done in previous papers [START_REF] Eloy | Flutter of a rectangular plate[END_REF][START_REF] Doaré | Flutter of an elastic plate in a channel flow: Confinement and finite-size effects[END_REF], the displacement is decomposed on the modes of the plate in vacuum φ i (x),

w(x,t) = e iωt ∑ i a i φ i (x). (34)
Next, the pressure associated with the mode i is expanded on Chebyshev polynomials of the first kind and assumed to have an inverse square root singularity,

p ′ i (x) = ∑ j A i j T j (2x -1) x(1 -x) . ( 35
)
This expansion is then inserted into the integral expression for the pressure (32). A scalar product with Chebyshev polynomials of the second kind U k (2x -1) is applied. This leads to a linear problem for A i j , which is solved numerically for a given set of the geometrical parameters (C * , D * , H * ). This method necessitates the knowledge of the function G(x,C * , D * , H * ). In practice, it is calculated numerically using,

G(x) = 1 x + G 1 (x), (36)
with,

G 1 (x) = ∞ 0 sin(kx) 1 g(k) -1 dk. ( 37
)
As the integrand tends to zero when k tends to infinity, this integral is easily computed numerically. Numerical results are now presented. It has to be noted that they involve a significant amount of numerical steps: Helmotlz equation using finite elements, numerical inverse Fourier transform, various projections (Chebyshev modes and Galerkin modes) and finally an eigenvalue problem. A first set of results is presented on Fig. 6. It consists in comparing the critical curves for apparition of instability in the (M * ,U * ) plane when the channel is large, (c, d) = (3, 30), with that of previous studies in an unbounded flow (C * and D * = ∞). A good agreement is found between the two methods.

Another comparison is done with results of Guo and Païdoussis [START_REF] Guo | Stability of Rectangular Plates With Free Side-Edges in Two-Dimensional Inviscid Channel Flow[END_REF] for D * = 0.2 on Fig. 7a,b. Here again, the critical curves of the present 3D model tend to that predicted by a 2D model when H * is increased. It appears that the smaller is C * , the faster is the convergence. The main result of this comparison is that for low values of M * , the 2D limit of reference [START_REF] Guo | Stability of Rectangular Plates With Free Side-Edges in Two-Dimensional Inviscid Channel Flow[END_REF], can be reached only for very large H * or very low C * . This is in practice difficult to achieve experimentally.

DISCUSSION

In this article, the influence of the walls of a rectangular channel on the critical velocity for flutter instability has been characterized. Local and global stability analyses have been performed. In the local approach the system is always unstable, provided the fluid is not still. Critical values of the parameters for transition between convective and absolute instabilities have been computed. In a general manner, confinement is found to significantly affect the local stability properties of the medium. Next, global stability of the finite length system has been investigated. The crucial importance of the 3D modeling at low values of the mass ratio has been emphasized. Indeed, it is found that 2D models predict the correct critical velocity only for high values of the aspect ratio or low values the spanwise and lateral confinements. When confinement is important, one may wonder if the viscosity has an effect on the instability thresholds. In the case spanwise confinement (small values of C * ), the effect of the existence of a boundary layer has been discussed in reference [START_REF] Doaré | Effect of spanwise confinement on flag flutter: Experimental measurements[END_REF]. In this case, only the edges of the plate are affected by the presence of a boundary layer and the experimental results are found to be in good agreement with the theoretical predictions. Conversely, in the case of significant confinement by walls parallel to the plate (small values of D * ), viscosity may play a more important role. Here the whole surface may be affected by viscous drag and it may be necessary to consider a boundary layer of a Poiseuille-type flow.

Further work is now mainly focused on experiments.

FIGURE 1 :

 1 FIGURE 1: Skematic view of a plate in a rectangular channel.

FIGURE 2 :

 2 FIGURE 2: (a), complete geometry of the problem in Fourier space; (b), numerical domain and boundary conditions after taking into account symetries of the problem.

FIGURE 3 :

 3 FIGURE 3: Function g(k) computed numerically for various sets of the parameters c and d (solid thin lines) and compared with different limit cases; thick dashed line, equation (20); thin dashed lines, equation (21); thick solid line, equation (23).

FIGURE 4 :

 4 FIGURE 4: Linear coefficient of the slender-body limit (i.e. small wavenumbers) as function of d for different values of c. Bold line indicate the limit for low values of d, equation (24), thin line indicates the limit of unconfined flow, A = π/4 and dotted lines indicate the empirical model developed in reference [8], equation (25).

FIGURE 5 :

 5 FIGURE 5: Critical curves for transition between convective and absolute instabilities (thin plain line) compared with that given by using g defined in equation (20), thick dashed line, equation (21), thin dashed lines and equation (23), thick solid line.

2 H * = 1 H * = 5 HFIGURE 6 :

 2156 FIGURE 6: Critical curves of marginal stability at c = 3 and d = 30 obtained by the present numerical method (solid lines), compared to critical curves obtained considering an unbounded flow [7] (dashed lines).

4 FIGURE 7 :

 47 FIGURE 7: Critical curves of marginal stability at D * = 0.2 for different values of H * , compared with the 2D results of reference [14]; (a) c = 3, (b) c = 0.1, (c) c = 0.001. The small irregularities of the curves are due to imprecisions of the numerical method.

  for a large value of d = 30, and values of c in the range [3 × 10 -4 , 3]. As already reported in a previous work [8], the function g is linear at low values of k and tends to unity at large values

of k. When c is large, it is correctly approximated by the infinite fluid domain 3D model

[START_REF] Eloy | Flutter of a rectangular plate[END_REF]

, equation (23). Typical results when c is large are plotted on Fig.

3b

. Here, g is well approximated by equation (

21

) for large values of k while it tends to a linear slender-body type behavior at low values of k. Indeed, the function g is well fitted by a function of the form g = Ak when k ≪ 1. The value of A is plotted as function of d on Fig.

4

for different values of c. When d is small, the value of A does not depend on c and the data is well fitted by,