Charline Pinna
email: charline.pinna1@edu.mnhn.fr

Maëlle Vilbert

Stephan Borensztajn

Willy Daney De Marcillac

Florence Piron-Prunier

Aaron F Pomerantz

Nipam Patel

Serge Berthier

Christine Andraud

Doris Gomez

Stephan Borenztajn

Marianne Elias

Convergence in light transmission properties of transparent wing areas in clearwing mimetic butterflies

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

Introduction

Lepidoptera (butterflies and moths) are characterized by large wings typically covered by scales, as testified by the name of the order (after the ancient greek lepís -scale and pterón -wing). Scales can contain pigments or generate interferential colours, thereby producing colour patterns across the entire wing. Wing colour patterns are involved in thermoregulation [START_REF] Dufour | Divergent Melanism Strategies in Andean Butterfly Communities Structure Diversity Patterns and Climate Responses[END_REF][START_REF] Heidrich | The Dark Side of Lepidoptera: Colour Lightness of Geometrid Moths Decreases with Increasing Latitude[END_REF], sexual selection [START_REF] Kemp | Female Butterflies Prefer Males Bearing Bright Iridescent Ornamentation[END_REF] and anti-predator defences, such as camouflage [START_REF] Arias | Transparency Reduces Predator Detection in Mimetic Clearwing Butterflies[END_REF](Arias et al. , 2020;;[START_REF] Endler | Progressive Background Matching in Moths, and a Quantitative Measure of Crypsis[END_REF], masquerade [START_REF] Skelhorn | Masquerade: Camouflage without Crypsis[END_REF][START_REF] Stoddard | 58 Current Zoology Mimicry and Masquerade from the Avian Visual Perspec-Tive[END_REF], disruptive coloration, and deflection of predator attacks [START_REF] Stevens | The Anti-Predator Function of 'eyespots' on Camouflaged and Conspicuous Prey[END_REF]. Another type of anti-predator defence in Lepidoptera involving wing colour pattern is aposematism, where the presence of secondary defences is advertised by the means of bright and contrasted colour patterns. Because of the positive frequency-dependent selection incurred on aposematic signals [START_REF] Greenwood | Frequencydependent Selection on Aposematic Prey: Some Experiments[END_REF], Chouteau et al. 2016), aposematic species often engage in Müllerian mimetic interactions, whereby species exposed to the same suite of predators converge on the same colour pattern and form mimicry 'rings' [START_REF] Müller | Ituna and Thyridia : A Remarkable Case of Mimicry in Butterflies[END_REF]. Co-mimetic species (species that share a common aposematic colour pattern) are often distantly related, implying that the same colour pattern has evolved independently multiple times.

Among such co-mimetic Lepidopteran species, several studies using visual modelling have shown that analogous colour patches (i. e., those occupying a similar position in the wing and harbouring similar colour) cannot be discriminated by birds, believed to be the main predators [START_REF] Llaurens | Cryptic Differences in Colour among Müllerian Mimics: How Can the Visual Capacities of Predators and Prey Shape the Evolution of Wing Colours?[END_REF][START_REF] Su | Prey from the Eyes of Predators: Color Discriminability of Aposematic and Mimetic Butterflies from an Avian Visual Perspective[END_REF][START_REF] Thurman | A Bird's Eye View of Two Mimetic Tropical Butterflies: Coloration Matches Predator's Sensitivity[END_REF]. Therefore, mimicry selects for convergent colourations, as perceived by predators.

Surprisingly, although most aposematic Lepidoptera species harbour brightly coloured patterns, some unpalatable, aposematic species exhibit transparent wing areas [START_REF] Mcclure | Why Has Transparency Evolved in Aposematic Butterflies? Insights from the Largest Radiation of Aposematic Butterflies, the Ithomiini[END_REF]. In those species, wing colour pattern typically consists in a mosaic of brightly coloured and transparent patches. Notably, in tropical America, many mimicry rings are comprised of such transparent species [START_REF] Beccaloni | Ecology , Natural History and Behaviour of Ithomiine Butterflies and Their Mimics in Ecuador ( Lepidoptera : Nymphalidae : Ithomiinae )[END_REF][START_REF] Elias | Mutualistic Interactions Drive Ecological Niche Convergence in a Diverse Butterfly Community[END_REF][START_REF] Willmott | Maintaining Mimicry Diversity: Optimal Warning Colour Patterns Differ among Microhabitats in Amazonian Clearwing Butterflies[END_REF]. Mimicry among species harbouring transparent patches raises the question of selection for convergence in optical properties, as perceived by predators, in those transparent patches.

A related question is whether transparency in co-mimetic species is achieved by the means of similar structural changes in wings and scales. Previous studies on a handful of species (most of which are not aposematic) have revealed several, non-mutually exclusive means to achieve transparency, through scale modification or scale shedding, with the effect of reducing the total coverage of the chitin membrane by scales. Scales can fall upon adult emergence [START_REF] Yoshida | Nanoprotuberance Array in the Transparent Wing of a Hawkmoth, Cephonodes Hylas[END_REF]; they can be reduced [START_REF] Dushkina | Coloration and Structure of the Wings of Chorinea Sylphina Bates[END_REF][START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF]) and even resemble bristle or hair [START_REF] Binetti | The Natural Transparency and Piezoelectric Response of the Greta Oto Butterfly Wing[END_REF][START_REF] Hernández-Chavarría | The 'Windows', Scales, and Bristles of the Tropical Moth Rothschildia Lebeau (Lepidoptera: Saturniidae)[END_REF][START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF][START_REF] Siddique | The Role of Random Nanostructures for the Omnidirectional Anti-Reflection Properties of the Glasswing Butterfly[END_REF]; they can be either flat on the membrane [START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF] or erected [START_REF] Dushkina | Coloration and Structure of the Wings of Chorinea Sylphina Bates[END_REF][START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF], which also reduces effective membrane coverage by scales.

Reducing scale density could also make wings transparent to some extent [START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF]. Although this has not been reported in transparent lepidoptera, transparent scales, such as those covering coloured scales in the opaque butterfly Graphium sarpedon [START_REF] Stavenga | Butterfly Wing Colors: Glass Scales of Graphium Sarpedon Cause Polarized Iridescence and Enhance Blue/Green Pigment Coloration of the Wing Membrane[END_REF], could also be a means to achieve transparency. In addition to scale modifications, the presence of nanostructures on the surface of the wing membrane may enhance transparency through the reduction of light reflection, by generating a gradient of refractive index between the chitin-made membrane and the air [START_REF] Binetti | The Natural Transparency and Piezoelectric Response of the Greta Oto Butterfly Wing[END_REF][START_REF] Siddique | The Role of Random Nanostructures for the Omnidirectional Anti-Reflection Properties of the Glasswing Butterfly[END_REF]Yoshida et al. 1997). Yet, so far, no study has compared the microstructures (scales) and nanostructures present in transparent patches across co-mimetic species. Furthermore, the diversity of structures described above may lead to a range of transparency efficacy.

Exploring the link between structural features and optical properties can shed light on whether and how different structures might achieve similar degrees of transparency in the context of mimicry.

Here, we investigate the convergence of transmission properties of transparent areas among co-mimetic butterflies and moths and the structural bases of transparency on 62 Neotropical transparent species, which belong to seven different Lepidoptera families and represent 10 distinct mimicry rings. We characterize wing micro-and nanostructures with digital microscopy and SEM imaging and measure transmission properties of transparent patches using spectrophotometry. We implement comparative analyses that account for phylogenetic relatedness, to (1) examine the putative convergence of transparent patches in visual appearance as seen by bird predators, (2) identify which structures are involved in transparency in the different co-mimetic species and finally (3) explore the links between structural features and transmission properties of transparent patches.

Results & Discussion

Convergence among co-mimics in visual appearance of transparent areas as seen by bird predators

To assess whether transparent areas of co-mimetic species were under selection for convergence due to mimicry, we tested whether these transparent areas, as perceived by predators, were more similar among co-mimetic species than expected at random and given the phylogeny. We used spectrophotometry to measure specular transmittance of the transparent areas, which is a quantitative measurement of transparency. We applied bird vision modelling on the resulting spectra and performed a PERMANOVA and a phylogenetic MANOVA, respectively, on the coordinates in tetrahedral colour space and luminance (which is perceived brightness), setting mimicry ring as factor. The results of the most ecologically relevant vision models (Lepidoptera viewed against a leaf in forest shade) are presented in Table 1 but the other vision models yield similar results (Supplementary table 1). We show that whether we control for phylogenetic relationship or not, predators see transparent patches among co-mimetic species as more similar than among species that belong to different mimicry rings, mirroring what has been shown for coloured patches in opaque species [START_REF] Llaurens | Cryptic Differences in Colour among Müllerian Mimics: How Can the Visual Capacities of Predators and Prey Shape the Evolution of Wing Colours?[END_REF][START_REF] Su | Prey from the Eyes of Predators: Color Discriminability of Aposematic and Mimetic Butterflies from an Avian Visual Perspective[END_REF][START_REF] Thurman | A Bird's Eye View of Two Mimetic Tropical Butterflies: Coloration Matches Predator's Sensitivity[END_REF]. The fact that the test remains significant with the phylogenetic correction indicates that such similarity in transparent patches is due to convergent evolution. When investigating separately the effects of achromatic aspects (luminance) and chromatic aspects (tetrahedral coordinates x, y and z), the former appear more significant than the latter (Table 1), suggesting that selection may act more on mean transmittance (degree of transparency) than on colour. Overall, our results suggest that, for mimetic Lepidoptera, transparent areas, especially the degree of transparency, are under selection for convergence and are therefore part of the aposematic signal.

Diversity of structures involved in transparency

Convergence in transmission among co-mimetic species raises the question of the nature and similarity of clearwing micro-and nanostructures among co-mimetic species. We therefore explored the diversity of structures present in the transparent patches in the 62 species. We used digital photonic microscopy and SEM imaging to characterize the structures present in the transparent zones (type, insertion, colour, length, width, and density of scales; type and density of nanostructures; wing membrane thickness).

We find a diversity of microstructural features in transparent areas (Figure 1A). Scales can be coloured (76% of species) or transparent (24%); they can be flat on the membrane (16%) or erected (84%). Scales can be 'standard' (i. e., a two-dimension structure, 55% of species), or hair-like (45%). Forked hair-like scales have been reported in the highly transparent nymphalid species Greta oto [START_REF] Binetti | The Natural Transparency and Piezoelectric Response of the Greta Oto Butterfly Wing[END_REF][START_REF] Siddique | The Role of Random Nanostructures for the Omnidirectional Anti-Reflection Properties of the Glasswing Butterfly[END_REF], which belongs to the mimetic butterfly Ithomiini tribe. In our dataset, hair-like scales appear to be almost exclusively found in the Ithomiini tribe, although one erebid species also harbours bristle-like scales. Erected scales (i. e., with a non-flat angle between the scale basis and the wing membrane) have been previously reported in the riodinid Chorinea sylphina [START_REF] Dushkina | Coloration and Structure of the Wings of Chorinea Sylphina Bates[END_REF] and in the nymphalid Parantica sita [START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF]). Here we describe, as Gomez et al. (2020, in review) did for the first time, some species with coloured erected scales that are completely perpendicular to the wing membrane, such as in the ithomiine Methona curvifascia. Transparent scales have already been reported in the opaque papilionid Graphium sarpedon [START_REF] Stavenga | Butterfly Wing Colors: Glass Scales of Graphium Sarpedon Cause Polarized Iridescence and Enhance Blue/Green Pigment Coloration of the Wing Membrane[END_REF] and as Gomez et al (2020, in review) we are describing them for the first time in transparent Lepidoptera. Other means of achieving transparency reported in the literature are not observed among our species (e. g., wing membrane devoid of scales, [START_REF] Yoshida | Nanoprotuberance Array in the Transparent Wing of a Hawkmoth, Cephonodes Hylas[END_REF]. However, our study is restricted to mimetic transparent butterflies and, as such, spans a relatively small number of families. An exhaustive study on all families comprising species with partially or totally transparent wings is needed to investigate thoroughly the different structures that might be involved in transparency.

We also reveal an unexpected diversity of nanostructures covering wing membrane (Figure 1B). In our sample, we find five types of nanostructures: absent (10% of species), maze (3%), nipple arrays (55%), pillars (21%), and moss (11%). While nipple arrays and pillars have previously been described on the wing of the sphingid Cephonodes hylas (Yoshida et al. 1997) and in the nymphalid G. oto [START_REF] Binetti | The Natural Transparency and Piezoelectric Response of the Greta Oto Butterfly Wing[END_REF][START_REF] Siddique | The Role of Random Nanostructures for the Omnidirectional Anti-Reflection Properties of the Glasswing Butterfly[END_REF], respectively, maze-like nanostructures have only been reported on the corneal surface of insect eyes [START_REF] Blagodatski | Diverse Set of Turing Nanopatterns Coat Corneae across Insect Lineages[END_REF]. Similarly, the moss type of nanostructures is reported here for the first time. Those nanostructures can be related to the classification proposed by [START_REF] Blagodatski | Diverse Set of Turing Nanopatterns Coat Corneae across Insect Lineages[END_REF]: pillars are a subcategory of nipple arrays, with higher and more densely packed nipples with an enlarged basis; moss-like nanostructures are similar to dimples (holes embedded in a matrix), although with much bigger and more profound holes. Nipples, mazes and dimples have been found to be produced by Turing's reaction-diffusion models, a solid framework that explains pattern formation in biology [START_REF] Turing | The Chemical Basis of Morphogenesis[END_REF]. While the principle of formation can be elegantly modelled, developmental studies are needed to understand the process by which nanostructures are laid on butterfly wing membrane.

Phylogenetic signal tests show that both micro-and nanostructure features are highly conserved in the phylogeny (Figure 2, Supplementary Table 2), suggesting the existence of constraints in the developmental pathways underlying micro-and nanostructures.

However, in the nymphalid tribe Ithomiini, which is highly represented is our dataset, microstructures seem to be more conserved (all species but the basal species M. curvifascia only have hair-like scales in transparent areas) than nanostructures (all five types of nanostructures, mixed in the Ithomiini clade).

There is a significant association between mimicry rings and structural features (Fischer exact test on scale type and insertion and mimicry rings: p-value < 0.001 and Fischer exact test on nanostructure type and mimicry ring: p-value < 0.001), meaning that in some mimicry rings one type of micro-or nanostructures is more represented than in other mimicry rings (Supplementary figure 1). For example, in the 'panthyale' and 'theudelinda' mimicry rings, species have only hair-like scales and mostly pillar nanostructures. In the 'confusa' mimicry ring, species often harbour erected scales (five out of seven species) and do not have any nanostructure covering their wing membrane (six out of seven species). These associations may be either the result of common ancestry or the result of convergence. For the 'panthyale' and 'theudelinda' mimicry rings, as species tend to be closely related, their structures are likely inherited from a common ancestor. For the 'confusa' mimicry ring, however, species are distantly related, suggesting evolutionary convergence, even though we cannot rule out that the absence of nanostructures may be ancestral and inherited from a common ancestor. Although different structures can be involved in transparency for each mimicry ring, the structures in transparent areas of co-mimetic species tend to be similar, and likely convergent in some cases.

Mimicry rings are characterized by transmission properties of transparent areas, mainly the degree of transparency, which is under selection for convergence. Are some structures independently selected in different species belonging to some mimicry rings because they confer a peculiar visual aspect, typical of the mimicry ring, and therefore participating in the aposematic signal? This raises the question of the link between transmission properties and structures in the transparent areas.

Link between structural features and transmission properties

To investigate whether transmission properties depend on the structures present in the transparent zones we measured the specular transmittance of transparent areas of each species with a goniospectrophotometer and we calculated the mean transmittance over 300-700nm, hereafter called mean transmittance, for each spectrum. We first confirmed that the physical property 'mean transmittance' (a proxy for the degree of transparency), is correlated to what is perceived by predators (the x, y and z coordinates and luminance from visual models) performing a mixed linear model and a Phylogenetic Generalized Least Square (PGLS) analysis (see Supplementary result 1 for details). Across the 62 species, mean transmittance ranges from 0.0284% in Eresia nauplius to 71.7% in Godyris panthyale (mean: 29.2%, median: 31.6%, supplementary table 3). We performed Phylogenetic Generalized Least Squares (PGLS) to assess the relationship between mean transmittance and micro-and nanostructural features (type, insertion, colour, length, width and density of scales; type and density of nanostructures; wing membrane thickness; including some interactions), while accounting for phylogeny. We retained as best models all models within 2 AICc units of the minimal AICc value. Following this procedure, eight models were retained.

Mean transmittance depends mainly on scale type, scale density and nanostructure density, and to a lesser extent on membrane thickness and scale colour (Table 2, Supplementary table 4). The effect of scale type and insertion is retained in all eight models and is significant in all of them. Wings covered with hair-like scales transmit more light than those covered with standard scales (Figure 3A). Among wings covered with standard scales, those with erected scales transmit more light than those with flat scales.

The effect of scale density is retained in the eight best models and is significant in five of those (Supplementary table 4): mean transmittance decreases as scale density increases.

Mean transmittance thus increases when membrane coverage decreases (due to reduced scale surface and/or scale density), because there is less material interacting (reflecting, diffusing, or absorbing) with light. The effect of nanostructure density is retained in six models and is significant in four of those: mean transmittance increases when nanostructure density increases (Figure 3B). Light transmission is negatively correlated to light reflection and nanostructures are known to have anti-reflective properties, as demonstrated in the sphingid Cephonodes hylas (Yoshida et al. 1997) and in the nymphalid G. oto [START_REF] Siddique | The Role of Random Nanostructures for the Omnidirectional Anti-Reflection Properties of the Glasswing Butterfly[END_REF]. Reflection increases as the difference in refractive index between air and organic materials increases. Nanostructures create a gradient of refractive index between air and wing tissue, and gradient effectiveness in reducing reflection increases with gradient progressiveness. For instance, pillars with conical basis are more effective at cancelling reflection than cylinders because cones produce a smoother air:chitin gradient from air to wing than cylinders [START_REF] Siddique | The Role of Random Nanostructures for the Omnidirectional Anti-Reflection Properties of the Glasswing Butterfly[END_REF]. Nanostructure shape is thus important in creating a smooth gradient.

In our case, nanostructure density is highly correlated to nanostructure type, which we have defined according to their shape (phylogenetic ANOVA on nanostructure density with nanostructure type as factor: F = 26.26, p-value = 0.001, see supplementary result 2 for details): the nanostructures whose shape likely creates the smoother gradient (pillar and moss) are also the denser ones. When nanostructure density increases, light reflection thus decreases. Light can either be transmitted, reflected or absorbed, and assuming that the chitin wing membrane only absorbs a small amount of light between 300 and 700 nm [START_REF] Stavenga | The Colouration Toolkit of the Pipevine Swallowtail Butterfly, Battus Philenor: Thin Films, Papiliochromes, and Melanin[END_REF], when light reflection decreases, due to the presence of nanostructures, light transmission necessarily increases, which explains the positive effect of nanostructure density on mean transmittance.

The interaction between scale density and nanostructure density is retained in three out of eight models and it is marginally significantly different from zero in two of those three models (supplementary table 4). The coefficient is always negative, meaning that the increase in light transmission due to the increase in nanostructure density is not as strong when scale density is high than when scale density is low. This suggests that the contribution of nanostructures to transparency is stronger when scale density is low.

Selection can act on microstructures (scales) and nanostructures. As nanostructures seem more labile than microstructures and allow fine-tuning of transparency, they could therefore evolve more readily in response to selection on the degree of transparency. The interplay between scales and nanostructures can thus modulate the degree of transparency.

The effect of membrane thickness is retained in three out of eight models and is significantly different from zero in one of them: light transmission decreases when membrane thickness increases. Wing membrane is mainly made of chitin and given that chitin absorbs a little amount of light [START_REF] Stavenga | The Colouration Toolkit of the Pipevine Swallowtail Butterfly, Battus Philenor: Thin Films, Papiliochromes, and Melanin[END_REF], thicker membranes, which contain more chitin, absorb more light than thinner ones, thereby reducing light transmission.

Transparent scales, which do not contain pigments, transmit more light than coloured ones, which contain pigments; a relationship which is retained in three out of eight models and is marginally significantly different from zero in one model (supplementary table 4). Pigments, such as melanin or ommochrome commonly found in butterfly scales, absorb some part of the light spectrum, thereby reducing light transmission.

Other variables that were included in the model (scale length and width, nanostructure type, the interaction between scale type and scale density and the triple interaction between scale length, width and density) are not retained in any models (Supplementary table 4). These results suggest that those variables do not have any effect on transparency.

While we show a high structural diversity, future studies should thoroughly quantify the relative contributions of micro and nanostructures on the produced optical effects, notably on reflection in transparent areas.

Conclusion

Our comparative analysis on transparent mimetic Lepidoptera showed that the transmission properties of transparent wing areas are convergent between co-mimetic species, suggesting that visual features of transparent areas are part of the aposematic signal. Despite an unexpected diversity of micro-and nanostructures underlying transparency, these structures are more similar than expected at random among comimetic species, perhaps because they confer typical visual aspects that are characteristic of the aposematic signal of the different mimicry rings. The diversity of nanostructures described may encourage bio-inspired applications for transparent materials.

This study challenges our vision on transparency, which might have evolved under multiple selective pressures in aposematic butterflies. Transparency has been shown to be involved in camouflage and to decrease detectability by predators (Arias et al. 2020), even in aposematic species [START_REF] Arias | Transparency Reduces Predator Detection in Mimetic Clearwing Butterflies[END_REF]). Nevertheless, our results suggest that transparent patches participate in the aposematic signal and that selection acts on the transmission properties of these patches, mainly on the degree of transparency but also on chromatic aspects. Therefore, transparent aposematic Lepidoptera benefit from a double protection from predation, which can act at different distances [START_REF] Barnett | Distance-Dependent Defensive Coloration in the Poison Frog Dendrobates Tinctorius, Dendrobatidae[END_REF][START_REF] Cuthill | Camouflage[END_REF][START_REF] Tullberg | Aposematism and Crypsis Combined as a Result of Distance Dependence: Functional Versatility of the Colour Pattern in the Swallowtail Butterfly Larva[END_REF]. Transparent aposematic species are less detectable than opaque species, but when detected they may be recognized as an unpalatable prey due to their aposematic signal. However, transparency entails strong structural modifications of scales that might impair other functions such as thermoregulation [START_REF] Berthier | Thermoregulation and Spectral Selectivity of the Tropical Butterfly Prepona Meander: A Remarkable Example of Temperature Auto-Regulation[END_REF] or hydrophobicity [START_REF] Perez Goodwyn | Waterproof and Translucent Wings at the Same Time: Problems and Solutions in Butterflies[END_REF].

Transparency may therefore come at a cost in those large-winged insects, which may explain why it is not pervasive among Lepidoptera.

Materials & Methods

Material

In this study, we focus on 62 different species represented by 1 or 2 specimens collected with hand nets in understory forests in Peru and Ecuador by ourselves and private collectors (Supplementary table 5). They belong to 7 different families (Nymphalidae, Riodinidae, Pieridae, Papilionidae, Erebidae, Notodontidae, Geometridae) and represent 10 different mimicry rings, following the classification used in Ithomiini: 'agnosia', 'aureliana', 'banjana-m', 'confusa', 'eurimedia', 'hewitsoni', 'lerida', 'panthyale', 'theudelinda' [START_REF] Chazot | Mutualistic Mimicry and Filtering by Altitude Shape the Structure of Andean Butterfly Communities[END_REF][START_REF] Willmott | Maintaining Mimicry Diversity: Optimal Warning Colour Patterns Differ among Microhabitats in Amazonian Clearwing Butterflies[END_REF][START_REF] Willmott | Correlations between Adult Mimicry and Larval Host Plants in Ithomiine Butterflies[END_REF]. In addition, we call 'blue' a mimicry ring that does not include Ithomiini species.

Phylogeny

We used both published and de novo sequences from one mitochondrial gene and seven nuclear genes, representing a total length of 7433 bp to infer a molecular phylogeny. To improve the phylogeny topology, we added 35 species representing 8 additional families to the dataset (see Supplementary table 5 and SI). We performed a Bayesian inference of the phylogeny using BEAST 1.8.3. We forced the monophyly of some groups and we added eleven secondary calibration points (see Supplementary table 6) according to [START_REF] Kawahara | Phylogenomics Reveals the Evolutionary Timing and Pattern of Butterflies and Moths[END_REF].

Spectrophotometry

Specular transmittance was measured over 300-700 nm, a range to which both birds and butterflies are sensitive [START_REF] Briscoe | The Evolution of Color Vision in Insects[END_REF][START_REF] Hart | The Visual Ecology of Avian Photoreceptors[END_REF]) using a home-built goniospectrophotometer (see SI for details). For each species, we measured five different spots in the transparent areas on the ventral side of the forewing. We computed mean transmittance over 300-700 nm from smoothen spectra using Pavo2 [START_REF] Maia | Pavo 2: New Tools for the Spectral and Spatial Analysis of Colour in R[END_REF], as a proxy for transparency: wing transparency increases as mean transmittance increases. We ensured that all measurements were repeatable (see SI), and we considered that any individual is representative of its species. One specimen per species was therefore used in most subsequent analyses.

High-resolution imaging and structure characterization

We observed structures with a digital photonic microscope (Keyence VHX-5000) to determine scale type (standard scale vs. hair-like scale), scale colour (coloured vs. transparent) and scale insertion (flat vs. erected) on ventral side. Wings were imaged in SEM (Zeiss Auriga 40) to measure scale density, scale length and width, membrane thickness, and nanostructure density (see SI for more details). As scale structural features were shown to be repeatable (see SI) within species we used one specimen per species.

Vision models

We used bird vision modelling on the smoothened transmission spectra to test whether transparent areas of comimetic species are perceived as similar by birds. Birds differ in their sensitivity to UV wavelength: some are more sensitive to UV (UVS vision) than others (VS vision). As predators of neotropical butterflies can belong to either category [START_REF] Dell'aglio | The Appearance of Mimetic Heliconius Butterflies to Predators and Conspecifics[END_REF], we used Puffinus pacificus as a model for VS vision and Cyanistes caeruleus as model for UVS vision. We considered different light environments differing in their intensity and spectral distribution: forest shade, woodland shade and large gap as defined by [START_REF] Endler | The Color of Light in Forests and Its Implications[END_REF]. We also considered different viewing conditions: the transparent patch of the butterfly could be seen against the sky (light is just transmitted through the wing) or it could be seen against a leaf (light is transmitted through the wing, then reflected on the leaf and transmitted again through the wing). In total we calculated 12 different vision models, using R package Pavo 2 [START_REF] Maia | Pavo 2: New Tools for the Spectral and Spatial Analysis of Colour in R[END_REF], depending on the different combinations of bird visual system, light environment and viewing conditions.

We extracted the x, y and z coordinates in the tetrahedral colour space corresponding to each smoothened spectrum in each vision model and the luminance (which corresponds to perceived brightness).

Statistical analyses

All statistical analyses were performed with the software R version 3.6.2. (R Core Team

2019).

Convergence on optical properties

To assess whether transparent areas, as perceived by predators, were more similar than expected at random and convergent among co-mimetic species, we performed a nonparametric PERMANOVA (with the R package 'vegan' [START_REF] Oksanen | Vegan: Community Ecology Packagevegan: Community Ecology Package[END_REF])) and a phylogenetic MANOVA (which accounts for species relatedness with the R package 'geiger' [START_REF] Pennell | Geiger v2.0: An Expanded Suite of Methods for Fitting Macroevolutionary Models to Phylogenetic Trees[END_REF])) on the mean of each coordinate in the tetrahedral colour space and the luminance (i.e. the perceived brightness) across the five spots measured (except for species belonging to the mimicry ring 'agnosia' where we excluded the spot in white band), with mimicry ring as the explanatory variable. To disentangle the effect of chromatic aspects (x, y and z coordinates) and achromatic aspects (luminance), we also performed a Kruskal-Wallis test and a phylogenetic ANOVA implemented in the R package 'phytools' [START_REF] Revell | Phytools: An R Package for Phylogenetic Comparative Biology (and Other Things)[END_REF]) on luminance and a PERMANOVA and a phylogenetic MANOVA on the coordinates x, y and z with mimicry ring as factor, as described above.

Phylogenetic signal

To assess whether transmission properties and structural features were conserved in the phylogeny, we estimated the phylogenetic signal of each variable. For quantitative variable (mean transmittance, scale density, scale length, scale width, nanostructure density and membrane thickness), we calculated both Pagel's λ [START_REF] Pagel | Inferring the Historical Patterns of Biological Evolution[END_REF]) and Blomberg's K [START_REF] Blomberg | Testing for Phylogenetic Signal in Comparative Data: Behavioral Traits Are More Labile[END_REF] implemented in the R package 'phytools' [START_REF] Revell | Phytools: An R Package for Phylogenetic Comparative Biology (and Other Things)[END_REF]. For multicategorical variables (scale type and nanostructure type), we used the delta statistic [START_REF] Borges | Measuring Phylogenetic Signal between Categorical Traits and Phylogenies[END_REF]) and we compared it to the distribution of values of delta when the trait is randomised along the phylogeny to estimate whether the trait is randomly distributed along the phylogeny. Finally, for binary variables (scale colour), we used Fritz and Purvis' D (Fritz and Purvis 2010) implemented in the R package 'caper' [START_REF] Orme | Caper: Comparative Analyses of Phylogenetics and Evolution in R[END_REF].

Association between structures and mimicry ring

We tested whether there was an association between some structure type (scale type and nanostructure type) and mimicry ring by performing Fisher's exact.

Link between transparency (mean transmittance) and structures

To assess the link between structural features and the degree of transparency we only used the spectrophotometric data of the points that correspond to the location of the SEM images (between 1 and 3 points per species) and we calculated the average of mean transmittance over 300-700 nm for each specimen. We tested the link between this average mean transmittance and all the structural features we measured (scale type, scale colour, scale density, scale length, scale width, nanostructure type, nanostructure density, membrane thickness and the following interactions: interaction between scale type and scale density, interaction between scale density and nanostructure density and the triple interaction between scale density, scale length and scale width) while controlling for phylogenetic relationships by performing pgls implemented in the R package 'caper' [START_REF] Orme | Caper: Comparative Analyses of Phylogenetics and Evolution in R[END_REF]. We compared all possible models with all the structural variables, but we prevented some variables from being in the same model because they were highly correlated, using the R package 'MuMIn' [START_REF] Barton | MuMIn: Multi-Model Inference[END_REF]). Among the 308 models, we selected the best models (difference in AICc inferior to 2). Eight such models were retained.

Table 1. Tests of convergence of transparent areas, as perceived by predators, among co-mimetic species

Results of the PERMANOVA, phylogenetic MANOVA, Kruskal Wallis test and phylogenetic ANOVA to test the effect of mimicry rings on predator perception of wing transparent areas. Variables x, y and z are the mean coordinates in the tetrahedral colour space of transparent areas for each species and L is the mean luminance. For phylogenetic analyses, p-value is calculated based on simulations. Results shown are those obtained under the most ecologically relevant vision models (forest shade illuminant and Lepidoptera viewed against a leaf).

Figure 1 .

 1 Figure 1. Diversity of micro-and nanostructures involved in transparency. A. Diversity of microstructures. a. transparent scales of Hypocrita strigifera, b. erected scales of Methona curvifascia and c. hair-like scales of Hypomenitis ortygia. Scale bars represent 100 µm. B. Diversity of nanostructures. a, b, c, d and e represent topviews and a', b', c', d' and e' represent section of wing membrane. Scale bars represent 1 µm. a, a'. absence of nanostructure in Methona curvifascia; b, b'. maze nanostructures of Megoleria orestilla; c, c'. nipple nanostructures of Ithomiola floralis; d, d'. moss nanostructures of Oleria onega; e, e'. pillar nanostructures of Hypomenitis enigma. Each coloured frame corresponds to a scale type or nanostructure type, as defined in figure 2.

Figure 2 .

 2 Figure 2. Phylogeny and distribution of traits along the phylogeny. Mimicry rings are represented by a symbol and a specimen is given as example for each mimicry ring.

Table 2 . Link between structural features and mean transmittance over 300- 700nm.

 2

	Structural features	Estimate	Std.Error t value	p-value
	Mean nanostructure density	1.078	0.383	2.815 0.006712 **
	Mean scale density	-0.006339 0.008515 -0.7444 0.4597
	Scale type and insertion Hair vs.
	Scale	6.898	1.513	4.560	< 0.001 ***
	Scale type and insertion Flat scale vs.
	Erected scale	6.67	1.987	3.357 0.001423 **
	Interaction between nanostructure
	density and scale density	-0.001357 0.0007831 -1.732 0.08875	.

Acknowledgments/funding

We thank Jonathan Pairraire and Céline Houssin for the help with photonic imaging and Josquin Gerber and Edgar Attivissimo for spectroscopic measurements. We thank Benoit Vincent for the identification of some Arctiini specimens. We are thankful to the Institut de Physique du Globe de Paris (IPGP) for giving us access to SEM and to the Peruvian and Ecuadorian governmental authorities for collection permits (021C⁄C-2005-INRENA-IANP, 002-2015-SERFOR-DGGSPFFS, 373-2017-SERFOR-DGGSPFFS, 005-IC-FAU-DNBAPVS/MA, 019-IC-FAU-DNBAPVS/MA). This work was funded by Clearwing ANR project (ANR-16-CE02-0012), HFSP project on transparency (RGP0014/2016) and a France-Berkeley fund grant (FBF #2015--58).

A. Link between mean transmittance and scale type. B. Link between mean transmittance and nanostructure density and scale type.

3

NB. We considered the spot corresponding to the location of the SEM images for mean transmittance. 4


M Hamed

M Muraglia

Y Camenen

X Garbet

O Agullo

Impact of electric potential and magnetic drift on microtearing modes stability

The stability of a microtearing mode (MTM) as a function of collisonality is investigated by means of linear reduced model and numerical simulations using the gyrokinetic code GKW. The study is focused on the role of the electric potential and the magnetic drift, which are potential candidates to explain the destabilization of MTM observed at low collisionality in some recent gyrokinetic simulations. In the simulations, the magnetic drift and electric potential are found destabilizing in the presence of a finite collisionality. This destabilizating role is captured in the analytical calculation, which further highlights the requirement for a finite collisionality.

INTRODUCTION

Improvement of tokamak performance is possible thanks to the spontaneous onset of a high confinement mode (H-mode). The H-mode 1,2 is characterised by the formation of a narrow and insulating region inside the last closed flux surface, called pedestal. In this region, the radial transport is reduced and a steep pressure gradient is observed. The height and width of the pedestal strongly impacts the energy confinement time. Presently the "EPED" model 3,4 , based on the stability of large and small scales MagnetoHydroDynamic (MHD) modes, is most commonly used to characterize the pedestal region. The EPED model has been extensively used and several numerical simulations using experimental data are in fair agreement with the observations. However, recent gyrokinetic simulations using the JET-ILW experimental equilibrium profils suggests that another class of instabilities, called microtearing modes (MTM), may also influence the pedestal height and width by generating a large electron heat transport level 5 . MTM physics is not presently included in the EPED model. This microinstability is a possible candidate to explain the anomalous electron heat transport in tokamaks [6][7][8] . Microtearing modes (MTMs) are small scale tearing instabilities with a toroidal (n) and poloidal (m) mode numbers larger than conventional tearing modes. MTM modifies the magnetic field line topology at the ion Larmor radius scale and leads to the formation of magnetic islands. The discovery of collisional MTM driven by electron temperature gradient is attributed to Hazeltine et al. 9 , in 1975 and has been responsible for several developements of the linear stability theory. Past analytical work predicted a peaked growth rate of MTM at a finite value of collisionality and decrease down to a negative value in the collisionless regime. Hazeltine et al., proposed a kinetic description of a slab current sheet destabilized by an electron temperature gradient leading to a tearing instability in the collisional regime only. In 1977, Drake and Lee 10 , using a cylindrical geometry, studied in details the stability of MTMs driven by an electron temperature gradient in three regimes of collisionality, "collisionless", "semi-collisional" and "collisional". By means of analytical calculations they predicted unstable MTMs in the presence of collisions and stable MTMs in the collisionless regime. In 1980, Gladd et al. 11 established a new analytical calculation in slab geometry including the effect of the electric potential. The electric potential was found destabilizing in the presence of collisions. The role played by the trapped particles in MTMs destabilization has been investigated by Catto et al., in 1981 12 and by Connor et al. 13 in 1990. A derivation of a dispersion relation in realistic tokamak geometry was formulated showing that trapped electrons are destabilizing in the presence of collisions. In 1989, Smolyakov 14 showed that drift effects can give rise to small-scale magnetic islands called "drift magnetic islands". Then, in 1990, Garbet et al. 15,16 suggested analytically that MTM is unstable nonlinearly and generates electron heat transport. In summary, past analytical work predicted MTM stable in the collisionless regime and driven by the electron temperature gradient in a slab magnetised plasma. The analytical calculation was then improved in 2015 by Zocco et al. 17 . A novel hybrid fluid-kinetic model was developped, showing that the electric potential destabilizes MTMs in presence of collisions. The destabilization of MTM in the collisionless regime by the interchange effect was recently investigated by means of a fluid theory neglecting the electric potential 18 . Clearly, the physical mechanisms at play in the MTMs destabilization in both the collisional and collisionless regime are not all well understood owing to the number of parameters (electric potential, interchange, trapped particles, magnetic drift,...) potentially involved. Futhermore, tokamaks operate at high temperature and therefore low collisionality at which MTM are not expected to be unstable. This claim caused a lessened interest in these micro instabilities as a relevant drive of turbulence. However, recent investigations using differents gyrokinetic codes predicted MTMs to be unstable in tokamak experimental conditions. MTM was found unstable in standard tokamak JET 5,20 , ASDEX 21,22 and DIII-D 23 , in spherical tokamaks [6][7][8][24][25][26][27][28] and in Reverse-field pinch 30,31 . The existence of MTMs in tokamaks highlights the need to determine the role played by this micro-instability in differents collisionality regimes. Numerically, MTMs dominate at the top of the pedestal and can generate an electron heat transport. To better understand the mechanisms which can destabilize MTM, a reduced kinetic model must be extended to include the electric potential, the magnetic drift, the trapped particles, the interchange effect and the collisions.

In previous studies, a linear dispersion relation of a slab microtearing mode using a kinetic approach was established and compared with linear gyrokinetic simulations 33 . The linear stability of the collisionless MTMs predicted by the theory [9][10][11][12][13][14]16 is found consistent with numerical simulations using the gyrokinetic code GKW 32 . MTMs are stable in collisionless regime. The purpose of this paper is to establish a kinetic reduced MHD model noted KRMHD model, including electric potential, magnetic drift, interchange effect and collisions to better understand their role in MTM destabilization. The effect of the trapped electrons is neglected in this reduced model. Then, the KRMHD model is compared with numerical simulations using GKW.

The paper is organized as follows. In Section I, a dispersion relation of a current sheet model is derived including electric potential, magnetic drift and collisions in a kinetic framework. This KRMHD model is a derivation of a current sheet model by solving the Fokker-Planck and Maxwell equations. The Fokker-Planck equation, required to evaluate the current inside the resistive layer, is solved in ballooning representation. The full expression of the current inside the resistive layer is rather complex. This complex form can be simplified by using an effective magnetic drift. In section I A a dispersion relation including only the magnetic drift is found. It appears that the magnetic drift destabilizes MTMs only in conjunction with a finite collisionality 34 . In section I B, the full electromagnetic problem is investigated. With both electric potential and magnetic drift, the evaluation of the current inside the resistive layer is obtained from a system of two equations linking the vector potential and the electric potential. This system of equations have been solved numerically using an eigenvalue code, called "Solve_AP". In section II , a description of the gyrokinetic equations used in the GKW code is given. Stressing which terms need to be retained/neglected to solve equations similar to the KRMHD model. Physical mechanisms are added in GKW step by step to identify their effect on MTM stability.

I. KINETIC MODEL OF MICROTEARING MODES

The fundamental aspect of the physics behind the microtearing instability is the creation of a parallel current at a resonant flux surface (q = m/n, where q is the safety factor). Current sheets are strongly localized around the resonant surface. Linear gyrokinetic simulations 33 show that the mode structure is peaked at the low field side mid-plane and rotates in the electron diamagnetic direction. The width of the current layer, typically a few ρ e , and the sensitivity of magnetic reconnection to dissipation make the numerical calculations of MTMs challenging: it requires a high numerical resolution and a very weak numerical dissipation, especially at low collisionality. The improvement of the analytical model is crucial, first to better understand the role played by the different physical parameters, but also because it is free from the sensitivity to numerical resolution and dissipation.

We consider a simple geometry of circular concentric magnetic surfaces where r is the minor radius, ϕ the toroidal angle and θ the straight field line poloidal angle. The ballooning representation 19 appears as the appropriate tool to study MTM including the magnetic drift and the electric potential. The KRMHD model is derived in the "ballooning" space using the field aligned coordinates (r, α, η). Here, α = ϕq(r)θ is a transverse coordinate and η = θ is the coordinate along the unperturbed field lines. Any perturbed field, for instance the vector potential A , at given toroidal wave number n and complex frequency ω, is written:

A (r, α, η, t) = ∞ ∑ p=-∞ Â (η + 2pπ) exp {inα + inq(r) (η -η k -2pπ) -iωt} + c.c. ( 1
)
Where q is the safety factor (near the rational surface q = m/n where m and n are, respectively, the poloidal and the toroidal mode numbers) and η k is the ballooning angle. The evaluation of the perturbed current along the magnetic field line J = ∑ species e d 3 v f nω v is obtained from the distribution function of each charged species whose evolution is given by the Fokker-Planck equation for each charged species, including the magnetic drift, electric potential fluctuations and collisions. The distribution function f nω for each species at given nω is split in an unperturbed part F M taken as an unshifted Maxwellian of temperature T eq , and a perturbed part fnω η, v , µ , where v is the parallel velocity and µ the magnetic moment. The perturbed distribution function fnω is itself written as the sum of an adiabatic part and a resonant part fnω = -F M ĥnω T eq + ḡnω , where ĥnω = e φv  is the perturbed Hamiltonian, φ(η) and  (η) are the n, ω Fourier components of the electric and vector potentials and e is the algebraic charge.  and φ determine the perturbed magnetic and electric fields:

B = ∇ × ( Â b) and Ê = -∇ φ - ∂ Â ∂t .
The resonant part ḡnω , is solution of the following kinetic equation:

ω -k v -ω d ḡnω = F M T eq (ω -ω * ) J ĥnω + iC( ḡnω ) (2)
The complex frequency ω = ωr + i γ contains the mode frequency ωr and the mode growth rate γ. J is the gyro-average operator, C is a linearized Fokker-Planck electron-ion pitch-angle scattering collision operator and

ω * = ω * T ( 1 η e + ζ 2 -3 2 ) is the kinetic diamagnetic frequency with ζ = v/v The and ω * T = k θ ρ i 2 m e m i v The R R L Te
where v The = √ 2T e /m e is the electron thermal velocity. The electron and ion mass and temperature are noted m e , m i , T e and T i , respectively. In ballooning representation, k must be understood as an operator which is a derivative along the magnetic field line -i 1

q 0 R 0 ∂ ∂η .
The magnetic drift frequency is defined as:

ω d = nq 0 r 0 mv 2 + µB 0 eB 0 R 0 (cos η + s 0 (η -η k ) sin η) (3)
where η k is the ballooning angle and s 0 = rdq qdr r=r 0 the magnetic shear at the reference resonant surface. Note that in ballooning representation |∇ ⊥ | 2 should be re- placed by

k 2 ⊥ = k 2 θ 1 + s 2 0 (η -η k ) 2 ,
where k θ = -nq 0 r 0 is the reference poloidal wave number. Finally it is convenient to replace the distribution function ḡnω by ĝnω = ḡnω -F M T eq e ω-ω * ω J φ. The linear dispersion equation for MTM including both magnetic drift and electric potential can be obtained by solving the Ampère law and Poisson equation written in a variational form:

L = - 1 µ 0 d 3 x ∇ ⊥ A 2 + d 3 x(J A * -ρφ * ) (4)
where ρ is the charge density and the complex conjugate of the electric potential and the vector potential are noted respectively φ * and A * . The extremalisation of the functional with any variation of φ * and A * is equivalent to the linearized Poisson and Ampère equations. One important property of L is that it vanishes when φ and A match the solution of Maxwell equations. It turns out that the Hamiltonian formulation is not the most suitable one to get close to the traditional resistive MHD formulation. Indeed, one would like to involve the parallel electric field rather than a perturbed Hamiltonian ĥnω . This is done via the following transformation:

ĥnω e = ω -k v -ω d ω φ + ω d ω φ -v Ê (5)
Where Ê = Âk ω φ is proportional to the parallel electric field. Replacing the expression of the perturbed current in the functional we obtain the following expression:

L = - 1 µ 0 d 3 x ∇ ⊥ Â 2 + dτ F M e 2 T i,eq ω -ω * i ω 1 -J 2 φ 2 + ∑ species dτ F M e 2 T eq ω * ω d ω 2 J φ 2 + ∑ species L res (6)
where

L res = -e dτ ĝnω J ω d ω φ -v Ê * (7)
where dτ = d 3 xd 3 p is the volume element in the ballooning phase space. The first term represents the magnetic energy, and gives rise to the field part of the Ampère equation. The second term is the polarization term (including diamagnetic corrections): it results from both ion and electron contributions. The third term is the interchange drive responsible for the ballooning instability. The last term is associated with the resonant response of ions and electrons. The latter is of particular interest. Ions are considered collisionless and the ion resonant contribution is neglected. An electron-ion pitch-angle scattering collision operator is used in the Eq.( 2):

C( ĝnω ) = 1 2 ν ei (v) ∂ ∂ξ 1 -ξ 2 ∂ ĝnω ∂ξ . ( 8
)
Where ν ei (v) is the electron-ion collisional frequency and ξ = v v is the pitch-angle variable (which varies between -1 and 1 and v the velocity modulus). It is then convenient to expand the perturbed distribution function over a basis of Legendre polynomials restricted to the first two components. Then, the distribution function becomes a solution of a differential equation that is solved using a WKB approximation [START_REF] Heading | An introduction to Phase Integral Methods[END_REF] . More details about this calculation are given in 34 . Eq.( 6) is the exact solution of the problem including electric potential, magnetic drift, interchange effect and collisions. This solution is not tractable. To simplify it, an effective curvature magnetic drift is employed in the following.

A. Effect of Magnetic drift

Asymptotic matching in ballooning space

In this part, we focus on the magnetic drift and neglect the electric potential. Outside the resistive layer (currentless region), A decays as A (0)e -|k θ x| , where x is the radial distance to the resonant surface of the mode. Note that the slab current sheet is very thin and localized around the resonant surface:

δ J = Ls L Te ρ e ν ei ω * << 1 k θ (9)
Where L s is magnetic shear length L s = qR/ ŝ, L T e is the electron temperature gradient length scale and ρ e is the electron Larmor radius. Inside the layer where a current develops, A is assumed to vary slowly consistently with a "constant A " approximation (A is constant in the current layer) 37 . The matching of the internal and external solutions in physical space is presented in 34,37 . Here the same procedure is translated in the ballooning space. The external mode away from the resistive layer in physical space becomes a localized solution, near η = 0 in ballooning space.

 (η) = -µ 0 Ĵ (0) [1 + s 2 0 η 2 ]k 2 θ =  (0) 1 1 + s 2 0 η 2 (10)
Where

∇ 2 ⊥ Â (η) is replaced by k 2 ⊥ Â (η) = k 2 θ [1 + s 2 0 (η -η k ) 2 ]
in the Ampère equation. Eq.( 10) yields a localization of A around η = 0 (low field side midplane) comparable to that obtained in the simulations, see Fig. (4). Then, the internal solution  (x) which is constant in the physical space becomes a function  (η) in the ballooning space. Note that, for circular flux surfaces and small aspect ration = r/R << 1, with r and R the minor and major radius of the surface, respectively the coordinate along the magnetic field lines used in GKW becomes s = η/2π [START_REF] Peeters | GKW how and why[END_REF] . This asymptotic matching gives the new following expression of the functional:

L = L mag + L res (11)
where

L mag = - 1 µ 0 +∞ -∞ dη 2π k 2 θ 1 + s 2 0 η 2 Â (η) 2 ( 12
)
and L res is given by Eq.( 7) with Ê = Â , when the electric potential is neglected. We restrict the analysis to a zero ballooning angle η k = 0, localized mode. Plugging Eq.( 10) into the functional L mag provides the following form of L mag

L mag = - 2 |k θ | µ 0 |d| A (0) 2 ( 13
)
Where d is the distance between resonant surfaces. All calculations done, the resonant term becomes:

L res = i π √ 3 qR 0 A (0) 2 × +∞ 0 dv4πv 3 F M e 2 T eq ω -ω * 1 + i ν ei (v) ω-ω d (v,0) 1/2 (14)
Hence the total functional L = L mag + L res becomes

L = - 2 |k θ | µ 0 |d| A (0) 2 + i π √ 3 qR 0 A (0) 2 × +∞ 0 dv4πv 3 F M e 2 T eq ω -ω * 1 + i ν ei (v) ω-ω d (v,0) 1/2 (15)
Where , where ν ei,th is the thermal collision frequency. The condition L = 0 then reads

ω d (v, η) = ω d (v, 0).
1 β = 8i π 3 +∞ 0 dξξ 9 2 e -ξ 2 Ω -Ω * (ξ) ξ 3 + i νei Ω-Ω d ξ 2 1/2 (16)
where β = 2µ 0 N eq T e,eq B 2

q 0 R 0 s 0 L Te 1 k θ ρ e , ρ e = m e v Te eB 0 , Ω * (ξ) = 1 η e + ξ 2 -3 2 , η e = L ne L Te and Ω d = L Te R 0 4
3 . In the absence of magnetic drift Ω d = 0, the previous result obtained for a slab collisional tearing mode is qualitatively recovered 33 . The magnetic drift is found to be destabilising, but only in conjunction with a finite collisionality 34 . The use of an effective magnetic drift is somewhat justified by the localization of the mode around the low-field-side midplane, Fig. 4. A comparison between the analytical calculation Eq.( 16) and numerical simulations will be presented in section II. The logical next step to understand the MTMs destabilization mechanisms is to add the electric potential in the calculation.

B. Effect of the electric potential in MTM destabilization

We now go back to the full electromagnetic problem. In the reduced model, the Fokker-Planck equation is averaged over the cyclotronic motion (gyroaverage). J is the gyroaverage operator which is defined as the Fourier transform of the Bessel function of zero-th order. For k ⊥ ρ i << 1, the Bessel function is approximated and replaced by 1 -J 2 = 1 2 k 2 ⊥ ρ 2 i for ions and by 1 for electrons, ρ e ρ i = m e m i << 1. In the resistive layer, k ⊥ k r 1/δ e , k ⊥ and k r are the perpendicular and radial wave number and δ e ρ i = ρ e ρ i L s

L Te is the collisionless layer width. The KRMHD model is valid for the following condition, δ e >> ρ i . In the opposite limit (k ⊥ ρ i → +∞), this approximation is not valid. Let us start from Eq.( 6). The ion polarization functional then becomes:

dτ F M e 2 T i,eq Ω -Ω * i Ω 1 -J 2 φ 2 = d 3 x N eq m i B 2 0 Ω -Ω * pi Ω ∇ ⊥ φ 2 (17)
where

Ω * pi = ω * pi ω * Te with ω * pi = - k θ T i,eq e p B 0 L pi
, L pi is the ion pressure gradient length. The final form of the functional then becomes

L = - 1 µ 0 d 3 x ∇ ⊥ Â 2 + d 3 x N eq m i B 2 0 Ω -Ω * pi Ω ∇ ⊥ φ 2 + ∑ species d 3 x N eq m i B 2 0 Ω * Ω d Ω 2 J φ 2 + L res (18)
The complex form of the functional Eq.( 18) can be simplified by assuming as in section I A an effective magnetic drift.

L res = 1 µ 0 d 2 e ∞ -∞ dx |d| σ(x) Â - k ω φ 2 (19)
Where d 2 e = c ω pe is the electron skin depth (ω 2 pe = N eq e 2 /m e 0 , ω pe is the electron plasma frequency). The dimensionless "conductivity" σ is defined as

σ(ρ) = - 2 3 +∞ 0 dζF M ζ 2 Ω -Ω * (ζ) Ω -Ω d ζ 2 + i ν ei ζ 3 -1 3 ρ 2 ζ 2 Ω-Ω d ζ 2 (20)
where Ω d is an effective magnetic frequency calculated at η = 0 and ρ = x/δ e is the radial coordinate. The extremalization of the functional in the physical space provides a set of two equations

∇ 2 ⊥ ψ + β * σ(ρ, ζ) ψ - ρ Ω φ = 0 (21) ∇ 2 ⊥ φ + µ e ρ Ω σ(ρ, ζ) ψ - ρ Ω φ -C int φ = 0 (22)
Where ψ = v the  and

µ e (Ω) = m e m i L 2 s L 2 Te 1 1 Ω ( 1 η e + 1) + 1 and β * = L s L Te 2 β e (23)
The quantities (β * , µ e ) are defined to have compact equations. C int is a coefficient associated with the interchange term:

C int = (1 + 1 η e ) 1 + 1 Ω 1 + 1 η e 1 Ω 2 L s L T 2 m e m i 2L T R (24)
This system is nothing else than the equations of reduced MHD where the Spitzer [START_REF] Spitzer | Physics of Fully Ionized Gases[END_REF] conductivity is replaced by the kinetic conductivity σ(ρ, ζ) 30 . We call it kinetic reduced MHD model (KRMHD). To solve Eqs. (21,22) this system of equations, is written in a matrix form. Using finite differences  is assuming to decay at large |x| as  ∼ exp(-|k θ ||x|). The matching with the external solution is represented by the ∆ parameter, which defined for a microtearing mode by ∆ = -2|k θ |. The asymptotic matching allows fixing the boundary conditions. When x = 0, the boundary conditions are: (21,22) are solved far away from the current layer. These equations are then implemented in an eigenvalue code, called "Solve_AP". The results obtained from "Solve_AP" are directly compared with gyrokinetic simulations using a version of GKW close to the theoretical model. A reasonable agreement is observed in the long wavelength limit, δ e >> ρ i . Fixing revelant physical parameters (Table III) to be in this limit, we seek the growth rate and the frequency of the mode solution of this system of equations. This comparison is presented in section II.

II. NUMERICAL INVESTIGATION OF THE STABILITY OF MTMS

All the results presented in this part use the local version of GKW. The gyrokinetic Fokker-Planck equation of the perturbed distribution function fgy is solved in the electromagnetic case with a perturbed electric potential φ and a pertubed vector potential  in the flux-tube approximation. To be closer to the assumption of the analytical calculation of section I, a linearized collision operator with pitch-angle scattering only is used. Non-linearities, particle trapping, neoclassical transport and magnetic field compression are switched off. The gyrokinetic equation solved in this section can be written in the form:

∂g ∂t = I + I I + V + V I I + V I I I (25)
where I = -v b.∇ fgy (25.1)

I I = -v d .∇ fgy (25.2) V = -v χ .∇F M (25.5) V I I = Ze T v .∇J φF M (25.7) V I I I = - Ze T v d .∇J φF M (25.8)
The notations of these terms follow the notations used in the GKW user guide [START_REF] Peeters | GKW how and why[END_REF] . I is the free streaming motion along the field line, II is the magnetic drift in the perturbed distribution function, V is the perturbed drift into the background distribution, VII is the parallel Landau damping term, VIII is the perpendicular Landau damping. In the terms written above v b is the parallel motion along the unperturbed field, v d the drift motion due to the inhomogeneous field and v χ is a combinaison of the perturbed E × B velocity (v E = b × ∇J φ/B) and the parallel motion along the perturbed field line ratio of plasma kinetic to magnetic pressure) and to the electron temperature gradient R/L Te 9,10,33 . Their values are chosen to have a sufficiently high MTM growth rate at high collisionality. As explained in the introduction the numerical study of MTM is challenging. MTMs are localized around the resonant surface. The current sheet is very thin, which implies having a high numerical resolution and a weak numerical dissipation, in particular at low collisionality. At low collision frequency the artificial numerical dissipation implemented in the code to ensure the stability of the numerical schemes can play the role of collisions leading to a destabilization of MTM in this regime. The numerical resolution used for the numerical simulations is presented in Table II. The number of points per poloidal turns N s and the number of poloidal turns N pol (an equivalently connected radial modes) have been adjusted to enable an accurate description of the mode. Note that for ITG modes, N s = 24 and N pol = 5 is usually sufficient. N µ is the number of magnetic moment grid points, N v is the number of parallel velocity grid points.

(v δB = -b × ∇v J A /B) then v χ = v E + v δB .

III. NUMERICAL INVESTIGATION OF THE ROLE PLAYED BY THE ELECTRIC POTENTIAL AND MAGNETIC DRIFT

A. MTMs unstable in presence of collisions

The role played by the electric potential and the magnetic drift is investigated numerically. The electric potential and magnetic drift are switched on step by step in the GKW to better understand their role in the MTM destablisation. Fig. 1 illustrates the role played by the magnetic drift and the electric potential. Without electric potential and without magnetic drift (black square), it appears that MTMs are stable at both low and high collisionality. For intermediate collisionality, typical of value in the pedestal region, MTMs are unstable, destabilized by the electron temperature gradient. A fair agreement between analytical calculation and numerical simulations was found, details can be found in 33 . The magnetic drift is then added in the code (green plus sign curve). Without the electric potential, the magnetic drift destabilizes MTMs in the intermediate collisionality. The role played by the electric potential is now investigated. The electric potential has been added in GKW and the magnetic drift has been switched on or off (the blue triangle and the red cross curve in the Fig. 1 respectively. It appears, without magnetic drift and with electric potential, another branch with opposite parity appears at high collisionality (purple triangle). These modes are giant electrons tails [START_REF] Hallatschek | [END_REF] . The growth rate definitely vanishes at low collisionality, as shown in Fig. 1 whereas the growth rate of MTMs increases then rolls over and decreases in the strong collisional regime. Finally, with magnetic drift and the electric potential (red cross) MTMs are more unstable but still stabilized at very low electronion collisions. For this set of parameters (Table I), the electric potential and magnetic drift destabilize MTM Figure 1: Growth rate of the microtearing mode as function of the electron-ion collision frequency. Different cases have been studied to better understand the role played by the magnetic drift and the electric potential. First case, without electric potential and without magnetic drift (black square), second case without electric potential and with magnetic drift (green plus sign), third case with electric potential and without magnetic drift (blue triangle) and the last case with both electric potential and magnetic drift (red cross) in the intermediate collisionality. In 28 and [START_REF] Swamy | [END_REF] , MTMs are found unstbale is collisionless regime. It appears that the trapped particle in this regime play an important role in the MTMs destabilization. For the set of parameters presented in the Table I, we add in GKW code the effect of trapped particles which amounts to adding the following term in the Eq.25:

IV = + b m . µ∇B + ∇ξ Ω ∂ f ∂v ( 26
)
The term IV 26 calculates the mirror trapping terms.

In the Fig. 2, when 10 -3 < ν GKW ei < 10 -2 , the trapped particles destabilize MTMs and when ν GKW ei > 10 -2 MTMs are stabilized. At low collisionality, ν GKW ei < 10 -3 , the trapped electrons modes (TEM) are the dominant modes, blue square curve in the Fig. 2. Starting with the Fig. 2, is not possible to conclude about the stabilization of MTMs by the trapped particles in collisionless regime. Before adding an additional difficulty, which is the effect of the trapped particles, in the understanding of the MTMs destabilization, let's look at the first time the role played by the magnetic drift. In the next part, a comparison between the analytical model, considering only the effect of magnetic drift (Eq. ( 16)), with GKW simulation is presented. The red cross curve is the same curve presented in the Fig. 1, the electric potential and magnetic are considered whereas the trapped particles are neglected. The effect of trapped particles is added in the GKW code, blue and green square curves.

B. MTMs destabilized by the Magnetic drift

Using the set of parameters in Table I and the numerical resolution in Table II a comparison between Eq.( 16) and numerical simulations has been done. Fig. 3 shows the numerical and analytical growth rate of MTMs as a function of the electron-ion collisionality frequency, ν GKW ei . To compare the mode frequency and growth rate of MTMs with GKW simulations, the theoretical collision frequency, νei , has been normalized using GKW normalization:

ν GKW ei = νei k θ ρ i 2 m e m i R
L Te . In Fig. 3, it appears that MTMs are stable at both low and high collision frequency. At intermediate collisionality ν GKW ei R/v thi ∼ 10 -2 -10 -1 , typical the pedestal region, MTMs are destabilized. Analytical theory and simulations are found to agree. Quantitatively, the agreement is satisfactory in the intermediate whereas differences start to arise at low and high collisionalities. At low collisionality, the expansion of the perturbed distribution function over a basis of Legendre polynomials restricted to the two first components to express the collision operator could explain the difference observed between the analytical calculation and numerical simulations. At high collisionality, the current sheet width becomes wider, and the "constant ψ" approximation is debatable 11 . The main result here is that the magnetic drift plays an important destabilizing role but only in conjecture with a finite collisionality. The fair agreement between the linear theory and GKW simulations and the structure of the mode presented in the Fig. 4 legitimates a posteriori the use I.

of an effective magnetic drift in the electromagnetic model.

C. Structure of microtearing modes

In the GKW 32 code, MTMs are clearly identified by their eigenfunctions.

Numerical simulations show that the perturbed electric potential φ is odd and extended along the magnetic field lines whereas the vector potential  is even and more localized around the low field side midplane. This agrees with the classical theory of tearing/microtearing mode. Fig. 4 shows the real (blue curve) and imaginary (red curve) parts of the eigenfunctions for linear simulations. The black curve represents the real part of the mode found analytically using the set of parameters given in TableI. In the Fig. 4. (a), the blue and black curve represent the real part of the vector potential  (s) along the magnetic field line.  (s) is even and localized around the resonant surface. The radial structure expected by the theory (Eq.

(η) = Â (0) 1 1+s 2 0 η 2 .
The localization of the vector potential near η = 0 a posteriori justifies the use of an effective magnetic drift in the analytical calculation. Moreover, in the radial direction, away from the current layer A decays as A (0)e -|k θ x| (Fig. 4). (b)).

D. Electric potential and magnetic drifts destabilize MTMs in conjunction with collisions

Now, the full electromagnetic calculation is directly compared with GKW simulations. Fig. 5 shows the real and the imaginary part of the electric potential along the magnetic field and in the radial direction.

φ is odd and extended along the magnetic field lines and localized around the resonant surface. In Fig. 6, the linear growth rate of MTMs obtained from the eigenvalue code that solves KRMHD equations is Table III: Revelant input parameters used for the KRMHD model and the linear simulations with GKW. q s β(%) R/L Te R/L Ti R/L ne k θ ρ i 2.2 0.3 0.9 10 0 0.26 0.22 0.3 compared with the numerical simulations using GKW.

A reasonable agreement between "Solve_AP" and the GKW simulations is found. The main result is, MTMs are only destabilized in the presence of collisions. The magnetic drift and the electric potential play and important role in the MTM destabilization that in the local limit and neglecting trapped particles. There is a fair agreement between "Solve_AP" and numerical simulations despite the approximations done, i.e ψ- III.

constant in the current layer and the use of an effective curvature in the KRMHD model

IV. CONCLUSION

In conclusion, a kinetic dispersion relation of microtearing modes has been developed to model the current sheet including the electric potential, magnetic drift and collisions. It has been compared direclty with numerical simulations using the gyrokinetic code GKW. Electric potential and magnetic drift play an important role in MTM destabilization. The localisation of the vector potential in the ballooning space allows using an effective curvature. Step by step the magnetic drift and electric potential are included in kinetic reduced MHD model and compared with GKW simulations. An analytical dispersion relation including only the magnetic drift is formulated. A fair agreement between the analytical model and GKW simulations is observed. It appears that the magnetic drift can destablize MTM only in conjunction with a finite collisionality. Then, the electric potential is added and a reasonable agreement between the numerical simulations and a reduced kinetic MHD model is found. The main result is that MTMs are destabilized by the electric potential and magnetic drift onlyin conjecture with a finite collisionality. Then, we have compared the radial structure of the vector potential assumed by the theory in real and ballooning space with that obtained from numerical simulations. We have found a good qualitative agreement between the two structures. It is very important to stress that the study of the stability of microtearing especially at low collisionality requires a very high resolution to describe correctly the current sheet. The purpose of the future work is to improve the kinetic reduced MHD model by taking into account the physic of the trapped particles to understand their role in the MTMs destabilization.

2 .

 2 Effect of magnetic drift on microtearing modes stability Starting from Eq. 15), a dispersion relation including the magnetic drift and collisions is formulated. It is convenient to normalize all frequencies to the electron temperature diamagnetic frequency ω * Te = k θ T e,eq eB 0 L Te and all velocities are normalized to the electron thermal velocity v Te = 2T e,eq m e . Hence, Ω = ω ω * Te , Ω d (v, η) = ω d (v,η) ω * Te , Ω * = ω * ω * Te and νei = ν ei,th ω * Te

  d  dx | x=0 = 0 and φ(0) = 0. When |x| → +∞, the conditions are: | φ| → 0 and |  | → exp(-|k θ ||x|). Eqs.

Figure 2 :

 2 Figure 2: Growth rate of the microtearing mode as function of the electron-ion collision frequency. Different cases have been studied to better understand the role played by the trapped particles.The red cross curve is the same curve presented in the Fig.1, the electric potential and magnetic are considered whereas the trapped particles are neglected. The effect of trapped particles is added in the GKW code, blue and green square curves.

Figure 3 :

 3 Figure 3: Left: analytical and numerical linear growth rates. Right: the analytical and numerical real frequency of microtearing modes as a function of electron-ion collision frequency ν GKW ei for the set of parameters presented in TableI.

Figure 4 :

 4 Figure 4: Real and imaginary parts of the numerical eigenfuntion of a MTM at ν GKW ei (v thi /R re f ) = 0.016 using the set of parameters given in Table I. (a): vector potential  (s) along the magnetic field line in the ballooning space. (b): radial structure of the vector potential A (x) in real space. Dashed lines indicate resonant surfaces

Figure 5 :

 5 Figure 5: Real and imaginary parts of the numerical electric potential eigenfunction of a MTM at ν GKW ei (v thi /R re f ) = 0.016 using the set of parameters given in Table I. (a): electric potential φ along the magnetic field line in the ballooning space. (b): radial structure of the electric potential φ(x) in the real space. Dashed lines indicate resonant surfaces.

Figure 6 :

 6 Figure 6: Comparison between linear analytical growth rate (a) and frequency (b) of MTMs with the GKW growth rate of these modes as the function of the electron-ions frequency using the set of parameters presented in TableIII.

Table I :

 I F M is a Maxwellian and ĝgy = fgy + Ze T v .∇J A F M . The revelant physical parameters used in the simulations are presented in TableI. MTMs are sensitive to β (the Revelant input parameters used for the theoretical calculations and the linear simulations with GKW.

	.	q 1.34 1.08 1.55 s β(%) R/L Te R/L Ti R/L ne 8 0 0.26 0.22 0.3 k θ ρ i

Table II :

 II Numerical resolution using in gyrokinetic simulations.

	N µ Nv	N s N pol
	16	60	79 99

V. ACKNOWLEDGMENTS

The authors thank I. Predebon, C. M. Roach, S. Sareelma, J. C. Hillesheim, L. Frassinetti for fruitful discussions. This work was granted access to the HPC resources of Aix-Marseille University financed by the project Equip@Meso (ANR-10-EQPX-29-01) of the program Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. Part of this work was performed by using computational resources provided by the MARCONI-FUSION HPC.