Chantal Amade-Escot

Epistemic Gender Positioning: an Analytical Concept to (Re)Consider Classroom Practices within the French Didactique Research Tradition

The chapter presents a theoretical discussion of the concept of 'epistemic gender positioning' as used in French didactique research. It begins with an overview on the late emergence of a gender focus in the francophone didactic research. Then, drawing on the theoretical idea that knowledge construction and meaning making are a by-product of teacher and students' joint action, it provides examples of the use of the concept of 'epistemic gender positioning' in different school subjects, which sees gender as a fluid, multiple and relational category.

Through a vignette and two examples of research data the chapter outlines the need for detailed and in-depth analyses of classroom practices and participants' interactions. The discussion brings to the fore the idea that gender positioning is always linked with each participant's practical epistemology, in the sense that teacher and students, who are embedded and act within an implicit and differential didactic contract, value or privilege different facets of knowledge depending on context, meanings and interactions. The conclusion summarises

Introduction

This chapter discusses the theoretical concept of 'epistemic gender positioning' as used in French didactique research and its relevance for providing new perspectives on gender in the teaching and learning of specific subjects in preschool, primary and secondary schools. It argues that the strength of a gender-positioning lens lies in its ability to produce in-depth analysis that takes the teacher, the students and the situatedness of knowledge into account simultaneously as interrelated instances. The chapter begins by providing some insights into the French context that leads feminist researchers to precisely characterize, in the everyday life of the class, how boys and girls construct their knowledge differently through actions and discourse. Theoretically, the chapter draws on the idea that the knowledge taught and learned is a by-product of the teacher's and students' didactical joint action. This is illustrated by a vignette which introduces the core concepts used and posits gender as a relational category beyond the reductionist masculinity/femininity binary. The next section outlines the conceptual framework and methodology. The discussion continues through two additional research excerpts, concerning physical education and science education, which further illustrate the potential of gender positioning in accounting for the differential dynamics of the didactic contract. The discussion here indicates that gender differences and inequalities among students go through tiny, almost imperceptible processes in relation to the gendered facets of each particular piece of content. The conclusion summarises the distinctive contribution of the French didactique research tradition to the understanding of how gender order is done/undone in the class and advocates the need not only deconstruct but challenge gender norms in teaching and learning.

The emergence of a gender focus in French didactique research

Research on gender in teaching and learning first appeared in the late 1980s in France, with the development of the new sociology of education. Most of this research was conducted in the areas of sociology and psychology (see [START_REF] Duru-Bellat | Filles et garçons à l'école, approches sociologiques et psychosociales: La construction scolaire des différences entre les sexes[END_REF] for a useful review). Drawing on second-wave feminism, this initial body of work highlighted how schooling reproduces and maintains masculine domination and the gendered aspects of the cultural heritage of French society, notably the contribution of schools to the reproduction of stereotypes related to social sex roles. Based on the notion of 'la construction scolaire des différences entre les sexes' [sex differences as a school construction], three prominent factors were identified: i) the generally male-oriented curriculum and the sexist contents of textbooks; ii) the number and types of interactions between teacher and students; iii) the underestimation of gender essentialism unconsciously attributed by teachers to girls and boys [START_REF] Duru-Bellat | Filles et garçons à l'école, approches sociologiques et psychosociales: La construction scolaire des différences entre les sexes[END_REF]. As in other western countries, feminist researchers in France documented in depth the fact that female and male students are not given equal opportunities to participate at school and are treated differently in relation to gender-based expectations. It has already been pointed out that if the social and cultural ideologies are not challenged and if explicit antisexist pedagogy is not introduced, equity in schools will remain no more than an aspiration. Notwithstanding this, in the French context to date little research has focused on classroom practices in relation to the knowledge taught and little attention has been paid to the issue of how the contents of lessons impact on student gendered learning. This is where the distinctive contribution of a didactical approach when studying gender in teaching and learning comes into play. Subject didactics research in France emerged in the early 1970s in a period of important shifts in curriculum reforms subsequent to the massive spread of secondary education. Calling attention to the building of French didactics, Caillot (2007: 127) argues that the greatest specificity of 'any didactique is that a specific subject is always involved'. In the Frenchspeaking world of educational research didactics is strongly related to: i) the study of the subject-content and its function in the teaching and learning; ii) the way it is embedded in learning environments and brought into play within teacher-student interactions; and iii) the extent to which a student's relationship to knowledge impacts on her/his learning and developmental process. However, until the 2000s, didactic research in France did not pay much attention to gender and could be seen as gender-blind research. Recent years have been marked by an effort at gendering research through various perspectives: curriculum and textbook analysis (see Lebeaume in this book), teacher practices (see Collet in this book), students' relations to knowledge, and teacher education. Yet, despite this, a focus on gender still remains marginal in French didactics.

In this context, studying classroom practices with the purpose of critically describing gendered knowledge construction is an innovative trend. For nearly two decades, a broad research programme called 'Genre & Didactique' has been in progress at the University of Toulouse, aiming to identify, in the everyday life of the class, how boys and girls construct their knowledge differently through actions and discourses. Studies were first conducted in physical education (Amade-Escot, 2017) and indicated that this school subject was not innocent but was, rather, 'one of the most sex-differentiated subjects on the school curriculum that contributes to the social construction of homogeneous gendered categories' (Flintoff and Scraton, 2006: 768). Important for this chapter is that the specific forms of embodiments, the values of strength, power and spatial movement, and cultural experiences [START_REF] Taylor | Objects, bodies and space: gender and embodied practices of mattering in the classroom[END_REF] at the core of student learning in physical education generate empirical conditions for delving further into the issue of how a didactical approach (that is, the specificity of the knowledge intended to be taught) can deepen understanding of how gender order is enacted in the classroom. Nowadays, the 'Genre & Didactique' research programme has been extended to other school subjects and has produced a series of observational case studies of a variety of subjects and settings within the French educational system. These case studies include theoretical and empirical works, doctoral studies and participatory action researches, all of which develop in-depth analysis of didactical interactions. Like the research underpinning this chapter, these studies share the same starting point: that boys and girls are physically and discursively produced to be differently literate according to the various school subjects. This process takes place through: i) gendered differences in interactions between teachers and students; ii) social relations between peers; and, iii) differential academic expectations. The interactions explored in this chapter are aligned with the aims of the research programme which relies on the claim articulated by Mosconi, a feminist educational philosopher who wrote:

To understand how teachers act with girls and boys in the classroom, fine-grained analysis of everyday classroom life and didactical interactions is needed. The differences and even the inequalities of treatment between students according to gender (but also according to their social origin, as well as their standing of excellence in the class) involve tiny, very subtle elements that cannot be seen without a very detailed analysis, including the didactical viewpoint (Mosconi, 2003: 38, our translation).

The French didactique approach and the study of gender order in the classroom

This section begins with a vignette as an empirical example which frames the subsequent discussion of 'epistemic gender positioning' and its importance in looking at didactical practices.

Louise's Story

Louise (not her real name -all names are pseudonyms) was a four and a half year old girl, and this episode occurred during her early preschool learning. Louise belongs to a class called 'moyenne section' (age 4-5 years) in the French preschool system. The learning context is related to individual work monitored by the teacher. After having read a carnival tale and set up an informal conversation about disguises with the whole class, the teacher hands out a worksheet to each child. This worksheet presents nine different characters with animal and human faces (see Figure 1). The teacher then asks pupils 'to circle each animal in green, each feminine character in pink, and each masculine character in blue'.

INSERT CHAPTER 3 FIGURE 1 HERE

Beyond the binary sex-stereotyping assignment provided by the teacher (pink for girls, blue for boys) it is worth noting that the knowledge content embedded in this learning exercise is complex and has various interwoven aspects: i) cognitive, in categorizing and grouping items;

ii) observational, in making distinctions between human and non-human life forms; iii) social, related to the traditional coloured representation of the gender binary; iv) cultural, in relation with the carnival tale read earlier. Moreover, to some extent, two characters of the worksheet look 'enigmatic' (Figure 1, characters 4 and 6).

During the first minutes Louise, strongly engaged in the task, circles characters 3, 5, and7 in green, characters 1 and8 in pink, andcharacters 2, 4, 6, and9 Three ideas emerge from this vignette of Louise's actions that help to introduce the core concepts used to conduct a didactical inquiry on how gendered contents are enacted through classroom practices. The first idea concerns the feminine/masculine dichotomy embedded in the learning environment set up by this preschool teacher: a taken-for-granted feature of Western societies pervading the display of school contents. This is not new. Research has consistently shown that curriculum and classroom practices convey gendered bias, even at preschool [START_REF] Davies | Frogs and snails and feminists tales. Preschool children and gender[END_REF]. The second idea is that Louise has a kind of independence with respect to the didactic contract (a core concept presented below). She feels free to use the purple crayon, thus introducing a breach with the teacher's assignment. It worth noting that Louise, a high achieving pupil, understands pretty well what the teacher's expectations are but plays with them. The last idea is that Louise activates remarkable gender positioning. In some way she jeopardizes the binary dichotomy of feminine and masculine characters introduced by her teacher, thus producing resistance and repositioning. Altogether, these three ideas encourage us to go further into the theoretical tenets underlying the approach.

Using the concept of epistemic gender positioning to account for individual agency within didactical practices

The term 'gender positioning' is used to account for the different ways individuals act in their everyday lives according (or not) to traditional sex-roles. In educational research, it has often been used in this sense, and is at odds with the research related to post-structuralist discourse analysis of power and resistance. Research on positioning emerged in the 1980s in the area of feminist studies (Davies 1989, Harré and[START_REF] Harré | Positioning theory: moral contexts of intentional action[END_REF]. Based on social constructionist psychology, these authors assume that human behaviour is goal-directed by group norms, and that human agency is a product of the history of each individual's interactions with other people. Drawing on the social, symbolic and interactional dimensions of human action, and the importance of context and language, positioning theory considers that positions are not fixed but fluid and can change from one moment to the next, depending on the context, participants, meanings and interactions. [START_REF] Harré | Positioning theory: moral contexts of intentional action[END_REF] criticise the concept of 'social role' in American social psychology, and contend that the theory of 'positioning' goes further and offers a dynamic alternative to it. Broadening the scope of positioning we have argued elsewhere that 'gender positioning' is a fruitful concept in attempts to account for the transactional dynamics of knowledge construction in day-to-day classroom life (Amade-Escot, Elandoulsi and [START_REF] Amade-Escot | Gender positioning as an analytical tool for the studying of learning in physical education didactics[END_REF][START_REF] Amade-Escot | Physical Education in Tunisia: Teachers' Practical Epistemology, Students' Positioning and Gender Issues[END_REF]. Furthermore, the epistemological stance of positioning theory, in paying attention to the fluidity and the shifting forms of gendered actions, resonates with [START_REF] Butler | Gender trouble. Feminism and the politics of subversion[END_REF] idea of gender performativity. However, to be clear, gender positioning in the didactic approach is understood as a knowledge-specific concept (as illustrated by Louise's vignette), which challenges the sex-role theory still strong in educational research.

How gender is understood in the French didactique approach to classroom practices

When investigating gender in classroom practices the didactique approach examines the ways girls and boys construct their knowledge differently through academic expectations and within interactions with the teacher and/or among peers. This approach sees gender as a relational social construct that is fluid, multiple and shifting, and so goes beyond the traditional male and female binary [START_REF] Butler | Gender trouble. Feminism and the politics of subversion[END_REF][START_REF] Chabaud-Rychter | Relectures critiques de Max Weber à Bruno Latour[END_REF][START_REF] Francis | The 'nature of Gender[END_REF]. In questioning the validity of traditional binary categories and the implicit and underlying vestiges of essentialism underpinning many school practices, our research tries to find a path between 'modernist reductionism and post-structuralist relativism' as discussed by [START_REF] Francis | Modernist Reductionism or Post-structuralist Relativism: Can we move on? An Evaluation of the Arguments in Relation to Feminist Educational Research[END_REF]. It does this by drawing on feminist research which accounts for the structural and institutional role of schooling practices in cultural reproduction and, at the same time, it acknowledges that gender must be conceptualized and investigated in terms of the subject's fluid, shifting, and sometimes fragmented experiences that regulate, rather than determine, the enactment of unequal learning trajectories. I next turn to the didactics framework in relation to which the empirical studies are conducted.

Studying gender order in classroom practices through the joint action in didactics framework

The didactical approach to classroom practices sheds light on how gendered learning is coconstructed through a fine-grained analysis of transactions related to the several facets of each particular piece of knowledge taught. It focuses on the differences and even the inequalities of treatment between students according to gender, their social origin, as well as their standing in the hierarchy of achievement in the class (see Verscheure and Debars, in this book). To capture the details of the curriculum in motion in classroom settings, the approach draws on the idea that student learning occurs within the unavoidable tension between student and teacher agencies and a learning environment, called the 'didactic milieu', which is always cultural, institutional, material and social. It aims to account for the complexity of the functioning of the didactical system conceptualized as an interwoven threefold relationship between the teacher, the students and the particular situatedness of knowledge content.

Assuming that the knowledge taught and learned and all the associated meaning-makings are a by-product of teacher and student practices in culturally bounded environments, these practices are, in theoretical terms, envisaged as 'didactical joint actions'. This does not mean that the participants have the same agendas and share the same goals; in fact, much research indicates the uniqueness of participants' actions and their consequences in terms of gendered learning.

The joint action in didactics (JAD) conceptual framework is a descriptive framework within a pragmatist view of classroom practices. It enables teachers' and students' discourses, actions and interactions in a given situation to be studied in order to depict student learning experiences and the meanings constructed through these experiences [START_REF] Ligozat | The Development of Comparative Didactics & Joint Action Theory in the Context of the French-speaking subject didactiques[END_REF][START_REF] Sensevy | Des catégories pour décrire et comprendre l'action didactique[END_REF]. Its purpose is to access the logic of didactical practices, and draws on [START_REF] Bourdieu | Le sens pratique[END_REF] sense of practice as to how the practical concerns of everyday life (here in the classroom) condition the transmission and functioning of social or cultural habits. In the analysis which follows, I focus only on the interrelated aspects of the JAD framework which are relevant for a gendered analysis of didactical transactions: i) 'epistemic gender positioning', which helps our understanding of how individuals (students and teacher) engage themselves in the practice of learning and teaching, ii) 'differential didactical contract', which accounts for students' various learning trajectories, and iii) 'practical epistemology', which allows us to capture the gendered facets of knowledge privileged by participants. I have already delineated the former concept above, and I briefly sketch out the last two immediately below.

Differential didactic contract as a result of teacher and student epistemic gender positioning

In the French didactique tradition, the 'didactic contract' broadly refers to the teacher's and students' implicit system of mutual expectations, joint habits and attribution of intentions.

Focusing on teacher and student joint action related to the knowledge at stake during classroom transactions, this concept links the Didactique perspective with pragmatist and social-interactionist approaches of classroom practices [START_REF] Ligozat | The Development of Comparative Didactics & Joint Action Theory in the Context of the French-speaking subject didactiques[END_REF]. Nevertheless, unavoidable breaches in the didactic contract appear, particularly in relation to transactions that express discrepancies between the lines of actions of teacher and students. As noted by [START_REF] Schubauer-Leoni | Etude du contrat didactique pour des élèves en difficulté en mathématiques. Problématique didactique et/ou psychosociale[END_REF], the didactic contract is inherently differential because it is not implicitly established with all the students of the classroom but with groups of students (and individuals), who have diverse standings of excellence in the class and diverse social backgrounds. The concept of differential didactic contract is often used to conduct relational analyses at the micro-level of didactical transactions. Relevant to the purposes of this chapter, Verscheure shows how various forms of gender positioning and repositioning that teacher and students enact during didactical transactions play a major role in the differential evolution of the didactic contract [START_REF] Verscheure | The gendered construction of physical education content as the result of the differentiated didactic contract[END_REF]. These dynamics are constrained by participants' agency in terms of what each of them brings to the transactional situation. This leads us to envision the third key concept.

Participants' practical epistemologies and their intricate interplay with gender positioning

The notion of practical epistemology in the JAD framework was first elaborated to account for the teacher's action. It underlined that teachers' views on the subject knowledge they teach are part of their actions [START_REF] Sensevy | Des catégories pour décrire et comprendre l'action didactique[END_REF]. Within this pragmatist definition, the concept was extended to students' action. In the JAD framework, it should be understood less as a knowledge base than as a pragmatist propensity to act in certain direction. In this sense, the teacher's actions and discourse reveal what counts as valid knowledge and appropriate ways of performing in a specific social practice. Conversely, students' actions following their own agenda (which, as I have already mentioned, may or may not be aligned with the teacher's one) reveal the sense they give to the knowledge they encounter. Teaching and learning are discursive acts, where participants' attention is directed towards certain events, questions, and relationships, while others are undervalued or ignored. The research posits that participants' practical epistemologies can be fruitfully described through a gendered lens, namely through positioning. To sum up, the concept of practical epistemology helps to understand what facets of knowledge are privileged by teachers and students and the extent to which the knowledge taught and learned is marked by gendered patterns of expectation and perception of the subject.

Analysing epistemic gender positioning: two empirical illustrations

In order to obtain a fine-grained description of the teacher's and students' actions and discourses, a qualitative research design which gathered data from video and audio recordings of lessons and interviews with participants is used. For ethical purposes, information on the study is shared with school teachers and authorities. In addition, parents' consent and assent forms are collected and the use of video data is restricted to research purposes. The data analysis is inspired by [START_REF] Héritier | Dissoudre la hiérarchie[END_REF] anthropological idea that any social practice (and thus any school practice) encompasses gendered interpretation and is marked by what she coined as the 'valence differentielle des sexes' [differential valence of the sexes]. This term acknowledges that gender dichotomy and masculine domination are a taken-for-granted (in various forms) in all societies; yet, at the same time, the term dialectically attests to 'the existence, not of a radical division between the sexes, but of a relatively complex and unstable continuum, on the organic, psychological, emotional, and social planes' (Héritier, 2010: 31, our translation).

The analysis consists in identifying in each subject-content transaction the several facets of each particular piece of knowledge taught and the gendered forms, practices and discourses that underpin them. This enables us to see how subject knowledge is informed by traditional gender binary norms (or not), and at the same time how the teacher and students might challenge these norms during classroom events. Through close analysis we can, then on one hand, identify sex-stereotyped reductionist social constructions, which, most often, reproduce gender binary norms and behaviours and marginalise students who are not clearly identified as acting according to these traditional demands and, on the other, we can begin to see the multiple modes of gender performativity that open up opportunities for resistance through the positioning and repositioning that each participant initiates in relation to knowledge during on-going transactions (as for example, does Louise when she breaches the 'pink and blue didactic contract' using a purple pen!). Assuming that gender positioning evolves during didactical transactions with regard to the knowledge at stake, the crucial point when conducting data analysis is to record all forms of positioning. Following Héritier's assertion about the 'relative complex and unstable continuum' of gender performativity and seeking equity and genuineness, we try to report on all participants' contrasted ways of performing a task and to pinpoint in their actions and discourses which forms of individual gender positioning are activated: those reinforcing traditional gender order and also those undoing gender. The differential learning trajectories are described in the two empirical analyses below -one from physical education the other from science education.

How a teacher's practical epistemology impacts on gendered learning

This video excerpt concerns a physical education unit in a Tunisian high school class (15-16 years) and focuses on the handstand forward roll in gymnastics (Amade-Escot, Elandoulsi and [START_REF] Amade-Escot | Physical Education in Tunisia: Teachers' Practical Epistemology, Students' Positioning and Gender Issues[END_REF]. This subject content encompasses various combinations of aesthetic and acrobatic movements. Reductionist forms of practices associated with dominant gender norms envision aesthetic movements as feminine whereas a masculine stereotype values the acrobatic dimension. This is a gender stereotypical vision because: i) aesthetics and acrobatics are not dichotomous categories and should be combined in a proper handstand; ii) female students may choose to perform acrobatically, and conversely male students may pay attention to the correctness of their body movement; and, iii) learning to perform a handstand forward roll implies controlling both dimensions together in action. These considerations drive the analysis.

In action and the associated discourse, a teacher may value the aesthetic or the acrobatic dimension, or a combination of both when teaching. Students may (or may not) elicit one of these dimensions or both, thus indicating the meanings they are construing. In the selected excerpt, Mohamed, the teacher, gives students instructions related to the acrobatic facet of the handstand forward roll movement: 'you should throw the leg and then roll'. To Salim, a male student who never reaches the handstand position, he says: 'go, go, push hard on your leg'; to the class: 'hey guys, make an explosive push'. Mohamed asks Ouissal, a highly skilled female student, to demonstrate how to reach the handstand position. He comments on her performance to the class: 'you should be as strong as she is'. From his discourse, always related to strength or force, which, according to [START_REF] Whitson | The embodiment of gender: discipline, domination and empowerment[END_REF], illustrates how masculinity is publicly celebrated, a pre-eminent masculine gender positioning stands out. Ouissal's actions repeatedly favour the aesthetic dimension of the gymnastic skills, with body alignment, and feet and toes pointed, thus performing perfect, tight handstands. However, she never engages herself in the roll, thus introducing a breach of the didactic contract (i.e. she does not choose to follow the teacher's instructions). Interestingly, Mohamed never interacts with her on this theme. Ouissal does not find any support or opportunity to experience the acrobatic dimension of a gymnastic movement. An implicit and gendered differential didactic contract emerges through transactions as if the handstand forward roll is not an objective for a female student.

Mohamed's actions indicate what is valuable gymnastic knowledge for her in comparison

with that implicitly negotiated with male students, who are called upon to perform more acrobatic gymnastics. Mohamed's positioning illustrates a teacher's practical epistemology paying tribute to masculinity, and at the same time being benevolent toward female studentsreinforcing gendered dichotomies of body movement as pointed out by [START_REF] Larson | Queering physical education. Between benevolence towards girls and a tribute to masculinity[END_REF].

Gender positioning and students' practical epistemologies during group work

This video excerpt comes from research by [START_REF] Pautal | Interaction verbales et non-verbales : outils de compréhension de la co-construction des savoirs et du genre entre élèves[END_REF], whose work I have permission to re-analyse for the purpose of this chapter. This data concerns the teaching of biology in a French primary school (age 9-10 years), particularly the study of how the human blood circulation functions, within an enquiry-based pedagogy. During the two previous lessons, students identified the different blood vessels and the function of the blood, they saw a short video on heart anatomy and participated in scientific workshops related to these contents. In the following excerpt, the knowledge to be transmitted concerns a scientific debate in groups on human blood flow and its related scientific model. The teacher distributes a paper representing a blank human body to each group. She asks the students to exchange ideas with the purpose of collectively deciding 'how many blood circuits are needed to make the blood function efficient' and to 'draw how it works on the poster'.

The excerpt involves two girls (Lucie and Asmae) and two boys (Marius and John). Marius and Lucie are high achieving students. Asmae is seated between Lucie and Marius. John is on the opposite side of the table. Lucie and Marius place the poster facing them. They take the floor; the other students do not talk much.

1. Lucie: 'There is one [circuit] in the head … and then there's a circuit that comes back ... I think' (Lucie draws two circuits from the heart on the poster) 2. Marius: 'NO ... the circuit goes all over the body ... I believe there's only one tube' 3. Lucie: 'but the dirty blood ... it goes, it goes …' (She sketches the lungs) Lucie provides the first statement and draws two circuits on the poster. Interrupting her, Marius resists Lucie's statement. Lucie objects to Marius's proposal with a rational argument related to 'the dirty blood' and sketches the lungs (3). It can be said that, at this stage, Lucie's blood flow model is compatible with the knowledge that is intended to be established. Marius interrupts her again, adding his own rational claim for a single circuit (4). At the end of these four turns of talk, two models related to the blood flow have been put forward: Lucie believes that there are two circuits, while Marius believes that only one is needed. Nevertheless, their different practical epistemologies are appropriate to the didactic contract of a scientific debate as both rationally argue their own vision. Such differences are entirely appropriate in such a social-constructivist didactic milieu. However, Marius does not draw anything, leaving the girl to do the written work. An additional understanding of this dialogue (at 2 & 4) is that Marius's epistemic gender positioning falls into 'manterrupting' (he interrupts Lucie with two sturdy 'NO's). This analysis is sustained by the next turns: when Asmae intervenes with a rational counter-argument to the mixing of the clean and dirty blood (5), Marius stops contributing any argument to the scientific debate. Unsuccessful in finding support from John (6), he asks for a vote (8). John and Asmae agree with him. Discouraged, Lucie follows the vote, erasing her drawing (11).

In asking for a vote, Marius stepped out of the didactic contract of a scientific debate. By this power act, he blocked the discussion and thus the collective knowledge construction. Marius shifted toward a (traditional) masculine gender positioning while Asmae and Lucie gave up, abandoning their relevant model of two blood circuits and adopting behaviour that may be interpreted as silenced traditional feminine gender repositioning. Consequently, the intended knowledge faded away. This excerpt confirms the interplays of practical epistemology, knowledge and power as in science and technology classrooms (see [START_REF] Danielsson | Knowledge and power in the technology classroom: a framework for studying teachers and students in action[END_REF].

Conclusion and implications for teaching and learning

This chapter explains the distinctive contribution the French didactique tradition makes to feminist dialogues on teaching and learning. In doing so, it adds an important dimension to the debates on gender explored in this book. It outlines the concept of epistemic gender positioning and demonstrates its analytical potential in investigating gender in different school subjects, in relation to a vignette and two examples of research data. The theoretical discussion brings to the fore the idea that gender positioning is always linked with each participant's practical epistemology, in the sense that teacher and students, who are embedded and act within an implicit and differential didactic contract, value or privilege different facets of knowledge depending on context, meanings and interactions. The findings in this chapter illuminate how gender differences in knowledge construction vary depending on the very specific subject content and how they are brought into play during classroom events. At the same time they exhibit similar patterns of interactions in different subjects that align this research with the scope of comparative didactics and 'the relationships between content-specific and generic features of teaching and learning practices' (Ligozat, Amade-Escot and Östman, 2015: 314). Moreover, such studies indicate that gendered learning trajectories among students go through tiny and almost imperceptible processes that need to be analysed.

In mobilising the interrelated concepts of epistemic gender positioning, differential didactic contract and practical epistemology, the French didactique tradition sheds new light upon how gender inequalities are perpetuated in classroom settings. Because teaching and learning specific subjects are situated actions, they also demonstrate the incredible diversity and uncertainty of student learning and how classroom practices offer resources and opportunities but also constraints that limit gender educational benefits.

The events analysed demonstrate the subtle processes entailed in the production of the gender order in the classroom and how this, more often than not, privileges masculinity.

Underscoring the relational process involved in the production of gendered content and the risk of marginalisation of girls and boys who are not clearly identified as acting according to their assigned sex, the approach suggests that greater attention should be paid to teacher and students' joint action related to knowledge during classroom events. A better understanding of the didactical phenomena that are at the core of doing/undoing gender in the class may open the way to new emancipatory and feminist educational projects in which gender norms are not only deconstructed but challenged [START_REF] Francis | Modernist Reductionism or Post-structuralist Relativism: Can we move on? An Evaluation of the Arguments in Relation to Feminist Educational Research[END_REF].

Paired Dialogue

Subjects of learning and pedagogical encounters

Susanne Gannon

The Didactique research tradition draws attention to bodies, relations and knowledge as coconstructed amongst students and teachers in the live space of the classroom. The micropolitical intricacies of classroom life are observed through video and audio recordings and interviews. The examples presented in this chapter provide fine-grained analysis of what children and teachers do in tiny moments in classrooms, and how these are infused by gendered positionings. Bodies feature in the secondary gymnastics lesson, where strength and grace (or acrobatic and aesthetic dimensions) are differentially valued by the teacher for male and female students. Artefacts feature in the early childhood literacy lesson where the worksheet requiring classification as 'male', 'female' or 'animal,' confuses the child -not because of the degendering of animals but because of the gender ambiguity of a human clown. Young people resist and modify the demands that are made of them in both lessons, demonstrating that the 'didactic contract' requires negotiation and adjustment, within particular subject specific conventions.

My own work, in the Anglophone tradition, has drawn on the concept of 'pedagogical encounters' to explore how many orders of things -human and nonhuman, animate and inanimate -come together unpredictably in spaces of learning (Davies & Gannon, 2009). The example below is from narrative fieldnotes from an elective Year 12 English Extension class, during which students develop an externally evaluated creative writing project. The class comprises three 17 year old male students and an experienced female teacher. Bodies are carefully and casually arranged: 'Although the classroom is big, the three students sit at the front of the classroom in a rather cramped row.' Ms M. 'rests against a desk slightly to the side of the students' row, creating an intimate and casual space.' She has brought in 'outrageous openings' from 'philosophical novels' ranging from Adam's Hitchhiker's Guide

  in blue. She stops for a few seconds looking at the worksheet, then grabs another coloured pencil and circles character 6 again, in purple. How might we interpret Louise's actions? Answering a teacher's question about 'why purple?' later, Louise says: 'Umm ... It is … it is a boy … disguised as a girl!'Matching her words with action, Louise takes the purple pencil again and re-circles character 6: '… it should be circled in purple'.

  4. Marius: (raising one arm) 'NO … what I am trying to say is… at the beginning all blood is clean, and it goes around the body. It is cleaned up in the heart, and then ... it leaves again' 5. Asmae: 'but it's not possible ... the clean blood and the dirty blood will mix as a slop' 6. Marius to John: 'and you, you should talk, you should explain ...'

	7. John (silent)
	8. Marius: (to the group) 'who agrees with me?'
	9. John: (raising a hand) 'me'
	10. Asmae, following: 'me'
	11. Lucie, after a second: 'me too' (She erases her previous drawing with two circuits)

to the Galaxy to Dostoevsky's Notes from the Underground. The synergies between the two traditions becomes clear: gymnastics focuses on correct formation of bodies; English focuses on correct formation of minds and literary tastes.

In my fieldnotes, discussion is highlighted as the mode of engagement with knowledge: we see the teacher's questions lead towards the close study of language that characterises literary study. This excerpt, for example, is from discussion of the first sentences of Notes from the Underground: 'I am a sick man. I am a spiteful man. I am an unattractive man.'

Ms M asks about the character of the narrator. Billy (all names are pseudonyms) says that the opening 'gently gives some idea of character'. Charles suggests that 'the first person gives an idea of character.' Ms M asks 'how does the way it is written add to that?' She asks about the effects of 'short, blunt sentences' and highlights the ways in which the speech patterns give the text a self-critical tone, so the reader can access ideas about the character of the narrator.

Miss M returns to her initial question about enticing the reader, asking, 'Do you want to read more about this man?'

Where are the hints of gender in this lesson? In bodies 'cramped' into small plastic chairs? In 'philosophical' texts selected by the teacher? In the invitation extended to young men to feel, express choices and preferences, and become analytical and sensitive readers? Gender intersects with age, experience and power differentials. In this lesson as well artefacts (extracts of first lines from novels), bodies (students and teachers) and the interactions between them form what Amade-Escot terms the 'didactic contract' and what I see as the conditions for 'pedagogical encounters.'


Caroline Cohen

Michael Berhanu

Julien Derr

Sylvain Courrech
email: sylvain.courrech@univ-paris-diderot.fr

Du Pont

Buoyancy driven dissolution of inclined blocks: Erosion rate and pattern formation

The dissolution of a body into quiescent water leads to density stratifications at the interfaces that drive buoyant flows. Where the stratification is unstable, the flow destabilizes into convective solute plumes. By analogy with the Rayleigh-Bénard instability where concentration replaces temperature, this phenomenon is known as the solutal Rayleigh-Bénard instability. Here we report experiments of the dissolution of inclined rectangular blocks made of salt, caramel or plaster in aqueous solutions of various concentrations. The solute flows along the block while forming plumes before they detach and sink. This flow along the block organizes the emission of plumes within longitudinal parallel stripes with a well-defined millimeter scale wavelength. The instability of the flow reflects on the concentration field in the boundary layer, which engraves longitudinal grooves onto the block. These grooves interact with the flow and turn into a paving of 3 dimensional cup like patterns that grow in size and propagate upstream. These bedforms are reminiscent of the scallop bedforms observed on the walls of cave or icebergs. Whereas the block interface is highly dynamical and evolves through time, it remains flat on the global scale and recedes at a stationary rate. We derive scaling laws for the receding velocity and the pattern genesis at the inclined interface that are based on a concentration boundary layer of constant thickness, which is controlled by the flow instability but where neither the patterns nor the flow along the block do not play any role. We apply these results to the formation of sublimation patterns.

I. INTRODUCTION

A water flow, and its interaction with a topography, modifies the rate of phase transition, or of dissolution or precipitation of a body. In nature, the coupling between the geometry and the mass transfer drives the formation of recognizable patterns at different scales. These can be erosion patterns like river meanders, cyclic steps, rillenkarren, lapiaz, ice ripples, dissolution flutes and scallops, regmaglypts on meteorites or deposition shapes like travertines, stalactites, icicles or brinicles. These patterns are not geological curiosities only but are markers of the hydrodynamic processes, which often control the global erosion process. The identification of erosion and deposition patterns is therefore a key to infer flow and thermodynamic conditions. It is especially valuable because these structures are quite resilient and give information on the long term not only for Earth but also for other planetary bodies that are difficult to instrument.

Here we study the dissolution of rectangular blocks immersed in still aqueous solutions in the case where the solute stratification it engenders is buoyancy unstable [START_REF] Cohen | Erosion patterns on dissolving and melting bodies[END_REF]. The concentration boundary layer destabilizes to form plumes that drive a convective flow of solute. This heterogeneous flow influences the concentration field at the interface, which patterns the dissolving body.

Several studies address the growth or dissolution of solids into aqueous solutions and the subsequent turbulent solutal convective flow [START_REF] Schürr | Sur la vitesse de dissolution des sels dans leurs solutions aqueuses (dissolution rate of salts in aqueous solutions)[END_REF][START_REF] Garner | Mass transfer from single solid spheres by free convection[END_REF][START_REF] Thomas | Concentration-gradient-driven convection: experiments[END_REF][START_REF] Grijseels | Hydrodynamic approach to dissolution rate[END_REF][START_REF] Hurle | Effect of solutal convection on the morphological stability of a binary alloy[END_REF][START_REF] Tait | Compositional convection in viscous melts[END_REF][START_REF] Kerr | Melting driven by vigourous compositional convection[END_REF][START_REF] Sullivan | Turbulent solutal convection and surface patterning in solid dissolution[END_REF][START_REF] Davies Wykes | Self-sculpting of a dissolvable body due to gravitational convection[END_REF]. Most of them do the parallel with the Rayleigh-Bénard instability where the buoyancy is not due to the thermal expansion of the liquid but to the gradients of solute concentration. These studies are either motivated by industrial applications from a chemical engineering perspective [START_REF] Grijseels | Hydrodynamic approach to dissolution rate[END_REF][START_REF] Hurle | Effect of solutal convection on the morphological stability of a binary alloy[END_REF] or by geophysical questions such as the sequestration of CO 2 in porous rocks [START_REF] Neufeld | Convective dissolution of carbon dioxide in saline aquifers[END_REF][START_REF] Huppert | The fluid mechanics of carbon dioxide sequestration[END_REF][START_REF] Loodts | Control of convective dissolution by chemical reactions: General classification and application to co 2 dissolution in reactive aqueous solutions[END_REF][START_REF] Slim | Solutal-convection regimes in a two-dimensional porous medium[END_REF] or the formation of patterns like brinicles or corrugations on the ice cap of brackish lakes or in magma chambers [START_REF] Tait | Compositional convection in viscous melts[END_REF][START_REF] Kerr | Melting driven by vigourous compositional convection[END_REF][START_REF] Tait | Compositional convection in a reactive crystalline mush and melt differentiation[END_REF][START_REF] Meakin | Geological pattern formation by growth and dissolution in aqueous systems[END_REF][START_REF] Solari | Morphodynamic modeling of the basal boundary of ice cover on brackish lakes[END_REF].

Our experimental study stands out from these previous studies by addressing the dissolution of inclined blocks. The inclination of the block gives a direction to the dissolution flow. As a result, the instability of the flow and the primary patterns are longitudinal with a well-defined wavelength. The latter interplay between the dissolution flow and the shaped topography gives rise to dynamical three dimensional structures that propagate upstream the dissolution flow. In the next section (sec. II) we detail our experiments with caramel, salt or plaster blocks, which are fast dissolving materials. The caramel being transparent, we can follow the pattern formation in real time. In section III we derive scaling laws for the dissolution rate and for the characteristic time and wavelength of pattern formation in the case of inclined bodies. We compare the scaling laws to the experimental data in section IV. We show in particular that the solute flow along the block and the dynamics of patterns at the block interface have little if any influence on the global dissolution rate. Finally we put our work in perspective in a geomorphological context. In section V we make the connection between the dissolution and sublimation patterns such as penitentes. In conclusion we link our work to the dissolution cavities observed in nature and compare the free dissolution patterns we observe in our experiments with their counterparts shaped by an externally imposed stream.

II. EXPERIMENTS

We investigate the dissolution of rectangular blocks of either caramel, salt (NaCl) or plaster (gypsum) immersed in quiescent aqueous solutions. We systematically vary the inclination of the block and the solute concentration of the solution.

Salt blocks come from Himalayan mines. They are typically 10 cm long, 5 cm wide and 2.5 cm thick and have a density of ρ s = 2348 ± 5 kg.m -3 . We stick to blocks aluminum lugs with silicon sealant to hold them into the saline solution. We mix tap water and table salt to prepare solutions of various concentrations. The measured density of the saturated brine is ρ sat = 1197.3 ± 1 kg.m -3 at 22 • C. Caramel is made with saccharose, tap water and a few drops of lemon juice (acid) to favor the caramelization reaction. We cast hot caramel into silicon molds of 10 × 10 × 3 cm or 5.5 × 2.5 × 2 cm to obtain rectangular blocks. An allen wrench is embedded in the block to hold it in the solution. The color of caramel is controlled during cooking to ensure that blocks are of similar properties and constant density (ρ s = 1540 ± 10 kg.m -3 ). We either mix tap water and caramel or tap water and saccharose to prepare the syrups. The measured density of the saturated caramel solution is ρ sat = 1450 ± 10 kg.m -3 . We made the plaster block by pouring a 5.5 × 2.5 × 2 cm silicon mold with a mix of fine powder of plaster of Paris and water. The density of the block is ρ s = 1859 ± 5 kg.m -3 and we measured a density difference of 1.8 ± 0.6 kg.m -3 between the saturated solution and water.

An experiment consists in immersing a block at room temperature (between 19 and 24 • C) at the center of a glass tank, which is 40 cm long, 20 cm wide and is filled with the solution to a depth of 20 cm. We capture the receding of the bottom and the top interfaces with a camera (see fig. 1). Caramel blocks are transparent so that we can also record the formation and the evolution of patterns on the bottom interface. To do so, we light the block from behind and either take pictures of the bottom interface thanks to an inclined mirror put into the tank or take frontal pictures with the camera positioned underneath and the tank tilted of the same angle as the immersed block (fig. 2). We cannot observe the patterns on the salt and plaster blocks when immersed. They are opaque and interfaces appear blurry because of the gradient of the solute concentration and consequently of the refractive index of the solution in the vicinity of interfaces. We take salt and plaster blocks out of solutions to observe the patterns. water. The block dissolves, interfaces recede and we observe different flows and block dynamics depending on the orientation of the block walls. Threads of solute-laden fluid escape from the block interfaces pointing downwards and sink while the solute flows all along the block walls pointing upwards (fig. 3). The inclination of the block enforces a downward direction to the flow on either interface. On the bottom interface, the solute flows downwards before detaching (see fig. 3). Using Particle Image Velocimetry, we measured a typical along-block downward velocity of 1.5 cm/s at a distance of 10 mm beneath the bottom interface of a caramel block dissolving in water and inclined of 60 • and of 2 cm/s at a distance of 8 mm beneath a salt block dissolving in water and inclined of 40 • . The differences in solute transport depending on the orientation of the block interface goes along with a different erosion velocity and shaping of the block walls. The block dissolves faster at interfaces facing downwards than at interfaces facing upwards. Moreover, while the walls facing upwards remain smooth, patterns form on the walls facing downwards.

For the experiment shown in figure 2, longitudinal parallel stripes with an initial wavelength of 0.4 mm are observed a few second after immersion. These stripes extend on the whole length of the block after a given "entry" distance from the top. Then, the wavelength increases with time, longitudinal stripes evolve into chevrons then into cup-like patterns named scallops by geologists, that propagate upwards, i.e. upstream. After an hour, scallops are 7.6 mm wide, 6.3 mm long and propagate at a velocity of about 3.10 -3 mm/s. They have the typical shape of rounded troughs and acute peaks with an asymmetry between the upstream and downstream sides. Although patterns grow, evolve and propagate on the bottom interface, its overall inclination does not change over time (but in the vicinity of the top corner) and the global receding velocity is found remarkably constant through time as shown in figure 4. A similar result is obtained in simulations of Rayleigh-Bénard convection with a melting boundary, which show a constant receding rate of the mean position of the boundary while patterns are growing [START_REF] Favier | Rayleigh-bénard convection with a melting boundary[END_REF].

Figure 5 shows the evolution of patterns at the bottom interface of a salt block when immersed into water. The dissolution scenario we described applies for all the dissolution experiments we performed. However, the characteristic time and length scales depend on material, initial bath concentration or inclination of the block. We measure the initial wavelength of longitudinal stripes λ s , the characteristic growth time T s for their formation and the global erosion velocity ḣ of the block wall pointing downwards for blocks of caramel or salt for different inclinations and concentrations of solutions and for one block of plaster into water. Figure 6 shows early longitudinal stripes observed on the bottom wall of caramel blocks immersed in syrups of different concentration. The initial wavelength as the characteristic time for their formation increase with the bath concentration. The interface recedes faster when the with the horizontal. One sees the very different solute flows on the interfaces facing upwards and downwards. On the bottom interface, the solute-laden fluid also flows along the block before detaching because the block is inclined. This flow has some inertia as observed at the bottom corner of the block. One also sees that the solute-laden fluid poorly mixes with the bath solution but sink and accumulate at the bottom of the tank. Right: Schematic view of the dissolution-driven flow. concentration of the bath solution is smaller. We also observe that the initial wavelength and growth time increase and the interface recedes slower when the inclination angle of the block increases.

III. SCALING LAWS: A SOLUTAL RAYLEIGH-B ÉNARD INSTABILITY

The rate of dissolution (or precipitation) of a solid block depends on the difference between the concentration of the solution at the block interface and the saturation concentration. The simplest yet reasonable dependency assumes a linear relationship with a constant characteristic dissolution (or precipitation) velocity, which contains the chemical kinetics [START_REF] Lasaga | Kinetic theory in the earth sciences[END_REF][START_REF] Crank | The mathematics of diffusion[END_REF]. The equation of conservation of the mass flux per unit of area reads:

ρ s ∂h ∂t n = α (C i -C sat ) n (1)
where ρ s is the density of the solid and α the characteristic dissolution velocity. C i and C sat are the mass concentrations of the solution at the interface and of the saturated solution, respectively. n is a unit vector, normal to the solid interface. For a flat and horizontal interface, if the dissolution process generates a density stratification that is buoyancy stable, only the diffusion drives the mass transport into the solution. Then, when putting a block in the solution, the concentration at the interface increases towards the saturation value on a characteristic time D/α 2 , where D is the coefficient of diffusion of the solute. The receding velocity of the dissolving block decreases with time and for large times, the interface position withdraws as the square root of time [START_REF] Crank | The mathematics of diffusion[END_REF].

If the dissolution process generates a density stratification that is buoyancy unstable, a completely different scenario occurs. When immersing the block, it starts dissolving and a front of dissolved medium moves away from the solid interface. On small lengths, close to the solid interface, the diffusion drives the transport. But beyond a characteristic length, the gravity takes over the transport. As a result, the concentration front destabilizes into plumes [START_REF] Philippi | Solutal convection induced by dissolution[END_REF] (see fig. 7). This instability is a solutal Rayleigh-Bénard instability where concentration replaces temperature and the same scalings laws apply as previously observed by [START_REF] Schürr | Sur la vitesse de dissolution des sels dans leurs solutions aqueuses (dissolution rate of salts in aqueous solutions)[END_REF][START_REF] Tait | Compositional convection in viscous melts[END_REF][START_REF] Sullivan | Turbulent solutal convection and surface patterning in solid dissolution[END_REF][START_REF] Tait | Compositional convection in a reactive crystalline mush and melt differentiation[END_REF][START_REF] Kerr | Dissolving driven by vigourous compositional convection[END_REF][START_REF] Haudin | Experimental study of a buoyancy-driven instability of a miscible horizontal displacement in a hele-shaw cell[END_REF]. Identically, it can be seen as a Rayleigh- FIG. 7. Sketch of the first steps of the dissolution process in a transverse plane. Time increases from top to bottom. The instability selects the thickness of the concentration layer and the wavelength of solute threads. It produces a modulation of the solute concentration at the interface, which shapes the dissolving block. The concentration at the interface of the block is larger above a thread (+) than between threads (-). The block dissolves faster where the concentration at the interface is smaller (eq. 1).

Taylor instability of a thin film whose thickness and concentration are controlled by diffusion [START_REF] Kerr | Dissolving driven by vigourous compositional convection[END_REF][START_REF] Haudin | Experimental study of a buoyancy-driven instability of a miscible horizontal displacement in a hele-shaw cell[END_REF]. In the classical Rayleigh-Taylor instability, the interface between two fluids of different density destabilizes when the heavier fluid is above the lighter one. The growth rate of perturbations is larger for small wavelengths. However, for non-miscible fluids, surface tension prevents the shorter wavelengths to grow and the most unstable wavelength is given by the capillary length. For a thin film when the surface tension goes to zero, the characteristic wavelength is given by the thickness of the film. H. R. Brown did the linear stability analysis in 2D for a horizontal thin film of a viscous liquid above an inviscid fluid and in contact with a solid wall at its upper limit [START_REF] Brown | Rayleigh-taylor instability in a finite thickness layer of a viscous fluid[END_REF]. The most unstable wavelength λ f and its associated growth rate σ f are [START_REF] Brown | Rayleigh-taylor instability in a finite thickness layer of a viscous fluid[END_REF][START_REF] Limat | Instability of a liquid hanging below a solid ceiling: influence of layer thickness[END_REF]:

λ f 3 δ & σ f 0.32 δ∆ρ g/η f , (2)
where δ is the thickness of the destabilizing film, ∆ρ the apparent density, η f the dynamic viscosity of the fluid in the film and g the gravity acceleration. In Brown calculation there is no diffusion nor tangential stress at the interface because the outer fluid is inviscid. This last simplification holds for a viscous outer fluid when its dynamic viscosity is small compared to that of the fluid in the thin film and/or when the wavelength is large compared to the thickness of the film. Lister and Kerr extended the calculation for two fluids of different finite viscosities [START_REF] Lister | The effect of geometry on the gravitational instability of a buoyant region of viscous fluid[END_REF]. For two fluids with equaled dynamic viscosities, they found λ f 3.8 δ [START_REF] Lister | The effect of geometry on the gravitational instability of a buoyant region of viscous fluid[END_REF]. Hereafter, we denote R λ the ratio between λ f and δ. Naturally, σ f scales as the inverse of the characteristic settling time of a particle of diameter δ in terminal Stokes velocity over a length equal to its diameter.

The front between the layer of dissolved medium and the solvent is unstable when the gravity overcomes the diffusion to transport the dissolved mass and concentration. Thus, the thickness δ of the destabilizing film is such that the growth time scale of the instability 1/σ f equals the diffusion time over δ: δ 2 /D, which gives:

δ ∼ 1.5 D η f ∆ρ g 1/3 . (3)
In a system where the outer bath is much deeper than δ, the solute transport that the dissolution generates is always unstable and the onset of the instability corresponds to a constant value of the solutal Rayleigh number Ra c [START_REF] Tait | Compositional convection in viscous melts[END_REF][START_REF] Sullivan | Turbulent solutal convection and surface patterning in solid dissolution[END_REF][START_REF] Philippi | Solutal convection induced by dissolution[END_REF]:

Ra c = ∆ρ gδ 3 Dη f . (4)
Solute diffusion and viscosity stabilize the interface and tend to increase the film thickness δ while gravity drives the instability and tends to decrease δ. The mean viscosity η f and density ρ f inside the film depend on the concentration of the solute-laden fluid.

We observe in experiments that the dissolution rate reaches a stationary state, i.e. the mass fluxes balance. In one dimension, the conservation of mass in a stationary regime reads:

-ρ s ḣ = -D ∂C ∂x (x) + C(x) v(x) -ḣ , ( 5
)
where x is the normal distance from the solid interface and v the flow velocity in the laboratory frame (like ḣ).

The diffusion coefficient D is taken constant. In the solution, the flow and the diffusion of the solute concentration balance the dissolution of the solid, which is made of one species only. At the solid interface, the solute transport is diffusion-dominated while it is advection-dominated beyond the unstable solute layer, which gives three different equations for the dissolution flux:

-ρ s ḣ = α (C sat -C i ) (6) ∼ D (C i -C b ) δ -β ḣ C i (7)
∼ C f - ρ f -C f ρ b -C b C b 4(ρ f -ρ b )g cos θδ 4 3η f λ f 2 . ( 8
)
The first balance of solute flux per unit of surface is given by the kinetics of dissolution (eq. 1).

The second line is the diffusion-dominated flux of mass at the solid interface. At the interface, the flow does not penetrate the solid but the normal velocity is not necessarily zero because the density of the solid and of the liquid may be different: v(0) = ḣ(1 -β), where β is the ratio between the density of the solid and the density of the liquid at the interface (β = ρ s /ρ i ) [START_REF] Quintard | Convection, dispersion, and interfacial transport of contaminants: Homogeneous porous media[END_REF][START_REF] Wells | Melting and dissolving of a vertical solid surface with laminar compositional convection[END_REF][START_REF] Gagliardi | Thin film modeling of crystal growth and dissolution in confinement[END_REF]. The gravity takes over the transport a distance δ from the solid interface within flowing threads with a wavelength λ f (see fig. 7). Thus, between threads and outside the mixture film, the concentration may be reduced to the one of the outer bath. The unstable solute layer assimilates to a boundary layer. Inside the boundary layer, the diffusion overcomes the advection so that the concentration decreases monotonically from the concentration C i at the interface to the concentration C b of the outer bath, almost linearly if D does not depend on C [START_REF] Philippi | Solutal convection induced by dissolution[END_REF]. The concentration gradient at the interface scales like (C b -C i )/δ.

The third line is the advection flux of mass at the boundary between the solute layer of thickness δ and the outer bath. It is buoyancy driven. The dissolution increases the concentration within the flowing film from the concentration of the outer bath C b to the (mean) one of the film C f . The scaling of the flow velocity corresponds to a two dimensional lubrication flow of a fluid of density ρ f and viscosity η f inside a thin film of thickness δ confined between a rigid wall (the dissolving block) and an inviscid fluid of density ρ b (the outer bath). The flow is driven by a characteristic pressure gradient 2δ(ρ b -ρ i )g /λ f and fed half a thread (see fig. 7). Threads have a wavelength λ f , which scales like δ (eq. 2). g is the effective gravity for the growth of the instability. It should take into account the possible inclination of the dissolving interface. When the dissolving block makes an angle θ with the horizontal: g = g cos θ.

It is the only way we take into account the inclination of the block, we neglect the longitudinal flow. Between threads, a flow of the bath solution at concentration C b renews the water inside the solute layer. The factor in front of C b comes from the conservation of water inside the solute layer. This ratio is always close to one. To obtain the scaling eq. 8, we further assume that the thickness of threads is negligible compared to the distance λ f between threads.

The equilibrium between the dissolution kinetics and the diffusive flux sets the concentration at the solid interface (eqs. 6 and 7). The concentration C i has two asymptotic expressions depending on the ratio between the dissolution velocity α and the characteristic diffusion velocity over δ: D/δ. When the transport at the interface is limited by the diffusion, i.e. D/(αδ)

1, the concentration at the interface is close to the saturation concentration. On the other hand, when the transport is limited by the dissolution kinetics, i.e. D/(αδ)

1, the concentration at the interface is close to the concentration of the outer bath. For the dissolving materials we consider, the dissolution is limited by diffusion and we assume C i = C sat . The interested reader will find the scaling laws in the other limit in [START_REF] Philippi | Solutal convection induced by dissolution[END_REF]. We assume a linear concentration profile in the mixture film, so that the average concentration is

C f = (C sat + C b )/2.
Finally, we assume a linear relationship between density and concentration, i.e. ρ = ρ 0 + C(ρ sat -ρ 0 )/C sat , where ρ 0 is the density of the solution when C = 0. Then, the density contrast reads ρ f -ρ b = (ρ sat -ρ b )/2 and β = ρ s /ρ sat . Combining equations 7 and 8 gives:

δ ∼ 3R λ 2 1/3 ρ s ρ s -βC sat D η f (ρ sat -ρ b ) g cos θ 1/3 C sat -C b C sat -C b ρsat-Csat ρ b -C b 1/3 and (9) ḣ ∼ - 1 3R λ 2 1/3 D 2 (ρ sat -ρ b ) 4 g cos θ η f (ρ s -βC sat ) 2 ρ s 1/3 C sat ρ sat -ρ 0 C sat -C b ρsat-Csat ρ b -C b C sat -C b 1/3 , (10)
where R λ is the ratio between the wavelength λ f of the buoyant instability and the thickness δ of the solute layer as given by the analysis of the Rayleigh-Taylor instability (for non diffusing, homogeneous fluids) [START_REF] Brown | Rayleigh-taylor instability in a finite thickness layer of a viscous fluid[END_REF][START_REF] Lister | The effect of geometry on the gravitational instability of a buoyant region of viscous fluid[END_REF]. The instability of the flow naturally reflects on the field of concentration at the solid interface. Above the flowing thread, the concentration should be maximum and the solid dissolution rate should be minimum (see fig. 7). The instability of the flow prints onto the solid and we expect that the first observed wavelength of patterns at the solid interface corresponds to the wavelength of the flowing threads:

λ s ∼ 3R λ 5 1/3 ρ s ρ s -βC sat D η f (ρ sat -ρ b ) g cos θ 1/3 C sat -C b C sat -C b ρsat-Csat ρ b -C b 1/3 . (11)
Patterns grow because of a differential rate of dissolution imposed by a modulation of the solute concentration at the interface. The differential dissolution velocity should scale like the global erosion velocity ḣ [START_REF] Philippi | Solutal convection induced by dissolution[END_REF] so that the characteristic growth time T s for the formation of patterns, i.e. to observe a given or detectable aspect ratio, should scale like λ s / ḣ:

T s ∼ 9R λ 7 1/3 η f 2 (ρ s -βC sat ) ρ s 2 D (ρ sat -ρ b ) 5 (g cos θ) 2 1/3 ρ sat -ρ 0 C sat C sat -C b C sat -C b ρsat-Csat ρ b -C b 2/3 . ( 12
)
These scaling laws are somewhat different than the previous scalings [START_REF] Sullivan | Turbulent solutal convection and surface patterning in solid dissolution[END_REF][START_REF] Philippi | Solutal convection induced by dissolution[END_REF][START_REF] Kerr | Dissolving driven by vigourous compositional convection[END_REF]. Here, we derive the scaling laws from the conservation of mass assuming a stationary dissolution flux while previous scalings are based on a constant critical Rayleigh number (eq. 4), i.e. derived assuming that the thickness δ of the boundary layer is such that the system is at the instability threshold. The two differences are the terms (ρ s -βC sat ) and

[C sat -C b (ρ sat -C sat )/(ρ b - C b )]/(C sat -C b )
≡ Ω in ours scalings instead of ρ s and 1 respectively in previous scaling laws. The value of Ω increases with the bath concentration. It varies between 1 and 1.13 for salt and between 1 and 2 for caramel. This term does not change the quantitive values of the scaling laws very much. The term with the factor β depends on the dissolving material. For usual natural materials, which have a very low saturation concentration (i.e. C sat ρ sat ), the difference is incidental. It is not negligible for salt and caramel for which β, C sat , ρ sat and ρ s are 1.96, 317 kg.m 3 , 1197 kg.m 3 and 2348 kg.m 3 and 1.06, 968 kg.m 3 , 1450 kg.m 3 and 1540 kg.m 3 , respectively.

IV. RESULTS AND DISCUSSION

To calculate these scaling laws one has to chose values for the coefficient of diffusion D and for the viscosity η f . We take the value of η f that corresponds to the assumed mean concentration in the film, i.e. for (C sat + C b )/2. We measured the viscosity as a function of the solute concentration for solutions of caramel, saccharose and salt. Because the saturation concentration of plaster in water is very weak, we take the viscosity of water for the plaster experiment. For D, we take the value that corresponds to the saturated concentration, i.e. the expected one at the block interface. It is determined from handbooks for salt and plaster and from the tabulated value at zero concentration for saccharose, which we extrapolate to the value at saturation using the Stokes-Einstein relation. These choices are not critical in the case of plaster, whose saturation concentration is very weak and in the case of salt, for which D barely depends on concentration and the viscosity changes by a factor of two between the pure water and the saturated brine [START_REF] Vitagliano | Diffusion coefficients for aqueous solutions of sodium chloride and barium chloride[END_REF]. However, these choices are not insignificant in the case of caramel ; the viscosity value of the saturated syrup is 4 × 10 4 times larger than the value of water. Most previous studies do not address the values of these coefficients, which is justified for materials with a small value of C sat without impact on these parameters.

Figure 8 compares the experimental results to the scaling laws without pre-factors, which include R λ . The various dependencies of the erosion velocity and the initial wavelength of patterns with the density contrast, the block inclination and, for caramel, the viscosity through the solute concentration, are verified over several decades. We find a pre-factor of about 0.18 for the dissolving velocity of salt blocks, which is very close to the expected order of magnitude (∼ 0.29 with R λ = 3.8, eq. 10). For the dissolving velocity of caramel blocks the pre-factor is larger and equals 2.7 when the expected one is about 0.34 (with R λ = 3). The choices we do for the values of D and η f may explain the different pre-factors for experiments with salt and caramel. Note that the scaling laws compare well to the measurements even if derived neglecting the solute flow along the dissolving block and the dynamics of patterns at the interface. Patterns are ignored and the inclination of the block is taken into account only through the projection of the gravity, which gives good agreement with experimental data (fig. 8). This is a noteworthy result for predictions in natural environments.

A horizontal interface dissolves faster than an inclined one. This result suggests that a polygonal or rounded object should not keep a constant shape while dissolving. This is different from recent observations of Davies Wykes et al., who report that the gravity-driven dissolving velocity of a block does not depend on its inclination [START_REF] Davies Wykes | Self-sculpting of a dissolvable body due to gravitational convection[END_REF]. This may be ascribed to the limited range of their experiments and the weak dependency at small angles (-ḣ ∝ cos 1/3 θ, eq. 10) and/or to the relative small size of objects they consider. The instability fully develops after an entry length (see fig. 2) and the differences in macroscopic receding velocity should affect the overall shape of the object if the latter is way larger than the size of patterns.

Our experiments where blocks are inclined clearly show a characteristic wavelength for the initial stripes that longitudinally extend all along the block. The initial wavelength of patterns we measure agree with the scaling law of the most unstable wavelength of the flow instability (fig. 8, eq. 11). Like for the erosion velocity, the observed pre-factor is closer for experiments with salt (∼ 40 instead of ∼ 13) than for experiments with caramel (∼ 75 instead of ∼ 8.8). Note that measurements may deviate from the scaling law at small values. However, this could be ascribed to experimental issues like an initial roughness of the solid surface or to the difficulty of measuring the very first wavelength when the patterns evolve rapidly.

Figure 8 shows that the characteristic growth time of patterns on caramel blocks properly scales like λ s / ḣ (eq. 12). This raises the question of the pattern growth rate. Most of instabilities grow exponentially at first in the linear regime. An exponential growth demands a coupling between the cause and the effect. Such a coupling is not obvious here where the patterning of the block comes from the buoyant instability of the dissolution flow. However, the flow instability can imprint onto the dissolving solid only if the positions of flowing threads are stabilized. A possible interaction between the patterns and the flow could rely on the spatial stabilization of dissolution plumes. The position of plumes may be all the more stable (stationary) that a dissolution peak is developed overhead. It is worth noting that the wavelength of initial patterns does not seem as well-defined as in our experiments with inclined blocks when interfaces are horizontal [START_REF] Sullivan | Turbulent solutal convection and surface patterning in solid dissolution[END_REF]. In our experiments, the dissolution flow along the block could stabilize the transverse position of early plumes and limit their interactions. Indeed, the longitudinal extension of early stripes is striking (fig. 6). Note that an outer flow, transverse to the growth direction of the instability, reduces the flow pattern to two dimensions (longitudinal rolls) in Rayleigh-Bénard experiments with small gaps [START_REF] Kelly | The onset and development of thermal convection in fully developed shear flows[END_REF][START_REF] Pabiou | Wavy secondary instability of longitudinal rolls in rayleigh-bénard-poiseuille flows[END_REF]. In our experiments the transverse-to-growth-direction flow is the dissolution flow along the inclined block itself. It is the opposite case of the Rayleigh-Bénard vortices in granular flows where it is the flow that "heats" the granular medium [START_REF] Forterre | Longitudinal vortices in granular flows[END_REF]. Anyway, without coupling between the amplitude of stripes and their growth velocity, the amplitude of early stripes would grow linearly in time. Numerical simulations could address this question effectively.

We observe experimentally that the initial longitudinal stripes do not last and turn into scallop-like patterns that deepen, enlarge, elongate, and propagate upstream. This is definitely a true interaction between the patterns and the dissolution flow. Figure 1 shows that solute plumes preferentially detach the boundary layer from peaks. This explains the upstream propagation of patterns, the reorientation of stripes and the final concave shape of scallops with acute peaks and rounded troughs as shown in figure 5. The buoyancy driven dissolution tends to form and preserves singularities. Following the solute flow along the solid interface, the concentration is larger, closer to the saturation value, and the concentration gradient is smaller because of a thicker boundary layer, upstream from the peak, before the plume detaches, than downstream. Thus, nearby a peak where a plume detaches, the dissolution flux is larger downstream than upstream from the peak. This differential dissolution makes that the peak (and patterns) propagate upstream. Crests channelizing the solute flow, the initial longitudinal stripes are unstable because of an inhomogeneous propagation velocity, transverse or longitudinal. The propagation velocity is maximum where plumes detach because the spatial variation of the solute flux is maximum there. If we consider for example that a stripe is not perfectly longitudinal but makes a small angle with the downstream flow, it will break and reorientate where plumes detach. The image sequence shown in fig. 9 illustrates the propensity of patterns to orientate transversally.

V. ANALOGY WITH THE FORMATION OF PENITENTES AND SUBLIMATION PATTERNS

Penitentes are ice or snow spikes that form on high altitudes glaciers in sublimation conditions, and have recently been suspected on other planetary bodies [START_REF] Moores | Penitentes as the origin of the bladed terrain of tartarus dorsa on pluto[END_REF][START_REF] Hobley | Formation of metre-scale bladed roughness on europa's surface by ablation of ice[END_REF]. They point in the direction of the sun, which suggests that the sublimation is driven by the heat flux resulting from the light absorption by the ice. Previous studies proposed a coupling between the ice topography and the sublimation rate to explain the pattern formation [START_REF] Lliboitry | The origin of penitents[END_REF][START_REF] Betterton | Theory of structure formation in snowfields motivated by penitentes, suncups, and dirt cones[END_REF][START_REF] Bartels-Raush | Ice structures, patterns, and processes: A view across the icefields[END_REF][START_REF] Cathles | Intra-surface radiative transfer limits the geographic extent of snow penitents on horizontal snowfields[END_REF][START_REF] Claudin | Physical processes causing the formation of penitentes[END_REF]. Because of a curvature effect, a trough receives more light than a peak and a perturbation of the topography grows. The corresponding linear stability analysis, where heat and mass diffuse but without advection, does not predict the selection of an initial wavelength [START_REF] Claudin | Physical processes causing the formation of penitentes[END_REF]. Claudin et al. propose that the wavelength may be selected by the boundary layer of an external air flow above the ice surface [START_REF] Claudin | Physical processes causing the formation of penitentes[END_REF]. However, Bergeron et al. report that an air flow above the snow surface prevents the formation of penitentes in their experiment [START_REF] Bergeron | Controlled irradiative formation of penitentes[END_REF].

We propose that penitentes firstly form by differential sublimation because of a buoyancy instability in the flow of moist air above the ice surface, like for the dissolution patterns we report. The initial ice peaks would then not form because of the coupling between the topography and the radiative heat flux but would simply be a print of the air flow instability. The buoyancy instability would be driven by moisture because moist air is lighter than dry air.

The ice sublimates because the partial pressure of water vapor in the air is lower than the equilibrium vapor pressure. At the ice interface, the air is almost saturated in water vapor with a concentration C sat , which depends on the temperature. The incident light brings energy to sublimate the ice but also to heat the (moist) air at the ice interface. The warming of air at the ice interface enhances the instability because the equilibrium vapor pressure of water in the air increases with temperature and so does the buoyancy. Thereby, the energy balance must be resolved concurrently with the equations of mass conservation. The incident light brings a characteristic heat flux, which is absorbed by the ice. In a steady state of sublimation, the absorbed heat at the interface diffuses in the ice, diffuses and convects in the air and provides the energy for the sublimation. The balance between the sublimation fluxes derived from the conservation of mass and derived from the conservation of energy would allow the sublimation rate, the thickness of the buoyancy unstable layer of moist air (and the wavelength between threads) together with the temperature of the air at the ice / air interface to be calculated.

A full analysis of this problem is beyond the scope of our article. However, we can estimate the first wavelength of buoyancy driven penitentes for given temperatures of the moist air layer and of the dry ambient air, which set the parameters value as the saturation concentration of the water vapor. For a dry atmosphere at -20 • C, if the ice interface is heated up to -5 • C, we find λ s 32 mm according to eq. 11 with the pre-factor we measure for salt dissolution (see fig. 8) 1 . It compares to the typical centimeter scale observed in experiments [START_REF] Bergeron | Controlled irradiative formation of penitentes[END_REF]. Although it might be annihilated in windy conditions, the buoyancy driven sublimation and evaporation might make non negligible contributions to water budgets [START_REF] Shahidzadeh-Bonn | Evaporating droplets[END_REF][START_REF] Dehaeck | Vapor-based interferometric measurement of local evaporation rate and interfacial temperature of evaporating droplets[END_REF][START_REF] Carrier | Evaporation of water: evaporation rate and collective effects[END_REF][START_REF] Dollet | Natural convection above circular disks of evaporating liquids[END_REF].

VI. CONCLUSION, FREE CONVECTION AND PATTERNS IN NATURAL ENVIRONMENT

Our experimental observation of dissolution of various solids in aqueous solutions evidences the succession of two pattern formation mechanisms. First, a longitudinal instability develops with the appearance of stripes carved on the surface and spaced out by a well-defined wavelength. We source it to the buoyancy hydrodynamical instability due to the density stratification that the dissolution generates. It ends up to the emission of solute plumes spatially separated. We show that the inclination of the dissolving blocks stabilizes the location of the plumes in order to print coherently the block surface. We confirm that this observed primary pattern follows scaling laws based on the hydrodynamics only. At later times the stripes eventually turn into scallops. This second mechanism involves strong interactions between the flow and the topography. The final shape displays very stereotyped scallops, similar to some observed in nature, advocating for the importance of the role played by buoyancy in natural systems.

Underground cavities shaped by free convection in nature are reported in several environments including salt, gypsum and limestone rocks. They occur as dissolutional chambers or vertical breakdown structures and are confined systems [START_REF] Klimchouk | Hypogene speleogenesis: hydrogeological and morphogenetic perspective[END_REF]. The dissolutional chambers devellop upwards and the buoyancy-driven flow shapes the cavity into an upsidedown triangular prism or cone with a flat horizontal ceiling [START_REF] Gechter | Genesis and shape of natural solution cavities within salt deposits[END_REF]. This particular shape is another indication that free convection increases the dissolution rate. Onto the top walls, centimeter cups with sharp edges named dissolution pits are observed [START_REF] Kempe | Cave genesis in gypsum with particular reference to underwater conditions[END_REF][START_REF] Kempe | Gypsum karst of germany[END_REF]. In our experiments with inclined blocks, the morphology of the dissolution pits is reminiscent of the scallop like patterns observed in unconfined systems such as caves or icebergs. However, these lasts are likely produced by the turbulent flow of a ground river or an ocean current. An externally imposed flow should prevail on the buoyancy to drive the dissolution when the turbulence of the flow decreases the thickness of the concentration boundary layer below the one imposed by the solutal Rayleigh Bénard instability (eqs. [START_REF] Garner | Mass transfer from single solid spheres by free convection[END_REF][START_REF] Sullivan | Turbulent solutal convection and surface patterning in solid dissolution[END_REF]. Regarding heat transfer, numbers of studies address these questions [START_REF] Bergman | Introduction to heat transfer[END_REF][START_REF] Pirozzoli | Mixed convection in turbulent channels with unstable stratification[END_REF].

Scallops shaped by an external flow have been successfully reproduced in laboratory flumes with a turbulent flow over beds of plaster [START_REF] Allen | Bed forms due to mass transfer in turbulent flows: a kaleidoscope of phenomena[END_REF][START_REF] Blumberg | Experimental and theoretical studies of dissolution roughness[END_REF][START_REF] Villien | Surface dissolution and the development of scallops[END_REF] or ice [START_REF] Ashton | Ripples on underside of river ice covers[END_REF][START_REF] Gilpin | Wave formation and heat transfer at an ice-water interface in the presence of a turbulent flow[END_REF][START_REF] Bushuk | Ice scallops: a laboratory investigation of the ice-water interface[END_REF]. These studies report that mature patterns in experiments have constant amplitude and longitudinal wavelength (periodicity in the stream direction), which scales as one over the flow velocity. In the field, scallops could thus be used as proxies of the flow [START_REF] Curl | Scallops and flutes[END_REF][START_REF] Curl | Deducing flow velocity in cave conduits from scallops[END_REF]. Theoretically, Hanratty and Claudin et al. studied the pattern formation on a bed sheared by a turbulent flow in the limit of deep flows [START_REF] Hanratty | Stability of surfaces that are dissolving or being formed by convective diffusion[END_REF][START_REF] Claudin | Dissolution instability and roughening transition[END_REF]. Patterns arise from the coupling between the dissolving topography and the turbulent viscosity at the solid interface, which increases the solute transport [START_REF] Claudin | Dissolution instability and roughening transition[END_REF]. Interestingly, the positive feedback is restricted to small values of the hydrodynamic roughness of the flow, and to a narrow range of wavelengths, which scale as one over the characteristic velocity of the flow [START_REF] Hanratty | Stability of surfaces that are dissolving or being formed by convective diffusion[END_REF][START_REF] Claudin | Dissolution instability and roughening transition[END_REF]. Considering the relative small aspect ratios of scallops observed in experiments and in the field, this result obtained in the linear regime could explain the longitudinal wavelengths and amplitudes of mature scallops. The 3-D shape of these scallops remains an open question. In our experiments, the convective solute flow driven by the dissolution of inclined blocks shapes bed forms through differential dissolution. Initial bedforms are longitudinal ridges, which turn into 3-D scallop-like patterns. Unlike the scallops in turbulent flows, the amplitude, width and length of the free dissolution scallops are not constant nor saturate in our experiments, but increase with time. Moreover, the free dissolution scallops propagate upstream while scallops in turbulent flow propagate downstream [52-54, 57, 58]. This difference in propagation direction reflects a different shift between the topography and the erosion rate. The erosion must be maximum on the lee side of free dissolution scallops, and on the stoss side of scallops in a turbulent flow. If these last ones propagate downstream but do not grow in size, the erosion is maximum at mid slope. Despite the different space shifts between the erosion rate and the topography, these two patterns look similar. They exhibit sharp peaks and asymmetric profiles with a larger slope on the lee side than on the stoss side. The peaks may however be more acute in our experiments with a rather concave shape upstream while the stoss side of the peaks appear more rounded in flows experiments [START_REF] Allen | Bed forms due to mass transfer in turbulent flows: a kaleidoscope of phenomena[END_REF][START_REF] Blumberg | Experimental and theoretical studies of dissolution roughness[END_REF][START_REF] Curl | Scallops and flutes[END_REF]. The overall resemblance between in flow and free dissolution scallops incites one to link some aspects of their morphogenesis. In both cases, the transverse topography could be due to an instability in the dissolution flow only, while a true feedback between the topography and the solute flow should be responsible for the longitudinal topography. The first observed patterns could be longitudinal ridges in both cases and they could trigger the transverse instability.

FIG. 1

 1 FIG. 1. a) Side view of a caramel block dissolving into water. The initial block is 6 cm-long and makes an angle of 62 • with the horizontal. Except in the vicinity of the top corner, the overall inclination of the receding bottom interface does not change over time. On the contrary, the top interface dissolves faster at the top (upstream) than at the bottom (downstream). b) Detail showing solute threads detaching from tips at the bottom interface.

Figure 1 FIG. 2 .

 12 Figure 1 and 2 show side views and bottom views of the evolution of an inclined caramel block immersed in

FIG. 3 .

 3 FIG.3. Left: Schlieren picture of a salt block dissolving into water. The block is 9 cm long and makes an angle of 45 • with the horizontal. One sees the very different solute flows on the interfaces facing upwards and downwards. On the bottom interface, the solute-laden fluid also flows along the block before detaching because the block is inclined. This flow has some inertia as observed at the bottom corner of the block. One also sees that the solute-laden fluid poorly mixes with the bath solution but sink and accumulate at the bottom of the tank. Right: Schematic view of the dissolution-driven flow.

FIG. 4 .

 4 FIG.[START_REF] Thomas | Concentration-gradient-driven convection: experiments[END_REF]. Position h taken in the middle of the bottom interface (plain points) and of the top interface (empty points) as a function of time t for a caramel block inclined of 62 • dissolving into water. The black line shows a linear fit. We measure positions along lines normal to the initial interfaces. We average the receding velocities from the measurements along five different lines along the block. The mean receding velocity of the bottom interface is ḣ = -2.78 × 10 -3 ± 3 × 10 -5 mm/s.

FIG. 5 .

 5 FIG. 5. Pattern formation. Bottom view of a salt block after 5 min. (a), 10 min. (b), 20 min. (c) and 1 hour (d-e) of immersion into pure water with an inclination of 60 • with the horizontal. Image (d) is lit from the back. Image (e) shows the asymmetry and pointy tips of dissolution patterns. Graphic (f) shows a line profile along this block. The slope on the stoss side seems parallel to the gravity in the vicinity of the tip. After an hour of immersion, the standard deviation of the topography is 0.62 mm.

FIG. 6 .

 6 FIG. 6. Bottom view of caramel blocks (55 × 25 mm) dissolving into aqueous solutions of different caramel concentration. From left to right, the densities of solutions are ρ b = 1006, 1108, 1229, 1303 and 1403 kg.m -3 . The inclination of blocks ranges between 70 • and 75 • .

1 FIG. 8 .

 18 FIG. 8. Measurements of erosion velocity ḣ, first observed wavelength of patterns λs and growth characteristic time Ts as functions of scaling laws (without pre-factor) for experiments with salt and plaster blocks (a) and with caramel blocks (b). Vertical and horizontal error bars show the resolution or dispersion of measurements and the scaling confidence interval due to parameters uncertainties, respectively. Colors stand for the varied parameter. a) Green: pure water with θ ranging between 6 and 87 • , orange: saline solutions of density ρ b ranging between 1000 and 1189 kg.m -3 and θ 30 • , blue: saline solutions of density ρ b ranging between 997 and 1197 kg.m -3 and various inclinations θ. The yellow point corresponds to gypsum into pure water with θ 52 • . The line in the first graphic shows a power law with an exponent 1/3. The lines in the two other graphics show relations of proportionality with constants equal to 0.18 for ḣ and 40 for λs. b -ḣ graphic) Green: pure water with θ ranging between 0 and 88 • , orange: sucrose solution with density ρ b ranging between 1000 and 1320 kg.m -3 and θ 73 -75 • , blue: caramel solution with density ρ b = 1403 kg.m -3 and θ = 75 • . b -λs & Ts graphics) Green: sucrose solution of density ρ b = 1150 kg.m -3 with θ ranging between 45 and 85 • , orange: sucrose solution with density ρ b ranging between 1000 and 1320 kg.m -3 and θ 73 -77 • , blue: caramel solution with density ρ b ranging between 1000 and 1430 kg.m -3 and θ 70 -75 • . The coefficient of proportionality of black lines are 2.7 for ḣ, 75 for λs and 1.5 for Ts.

2mmFIG. 9 .

 9 FIG. 9. Pattern dynamics. Bottom view of a caramel block dissolving into water. A V-shape pointing downstream vanishes. The block makes an angle of 60 • with the horizontal. Gravity points towards the bottom of images. There is a 6 minutes time delay between snapshots. First image is taken 80 minutes after immersion.

This value corresponds to D =

2.14 × 10 -5 m 2 .s -1 , η f = 1.65 × 10 -5 Pa.s, ρsat = 1.314kg.m -3 , ρ b = 1.394kg.m -3 .

ACKNOWLEDGMENTS

We thank Cyril Ozouf who did some experiments during an internship in our laboratory and Marc Durand for fruitful discussions. We thank Jérôme Jovet for his help to do Schlieren imaging and Imane Boucenna for sharing her rheometer. This research was funded by the french Agence Nationale de la Recherche (grants ANR-12-BS05-0001/EXO-DUNES and ANR-16-CE30-0005/ERODISS).