Chris Paola

Seb, Sophie, Damian Marianne Lilian

Adélaïde Lambert

Laura

Florent Sarita Audrey

Keywords: cône alluvial, rivière, seuil de mise en mouvement, archive sédimentaire alluvial fan, river, threshold of motion, sedimentary record

Je tiens, tout d'abord, à remercier François et Laurie qui m'ont fait conance et m'ont donné l'opportunité de réaliser cette thèse. Merci à Laurie qui m'a permis de faire le lien entre les cônes de laboratoire et les cônes naturels et à François dont l'enthousiasme quotidien est une grande source de motivation. J'ai eu la chance de proter du savoir-faire d'Olivier, grâce à qui j'ai énormément appris sur l'art du travail bien fait (j'admets ne pas être sure d'avoir retenu toutes les règles...).

Je tiens aussi à remercier toutes les personnes qui ont contribué à ce travail, notamment les Éric(s): Éric Lajeunesse, toujours disponible pour une discussion autour du café du matin et Éric Gayer le roi du traitement de MNT ! Sans oublier Clément pour ses remarques toujours constructives bien que parfois désagréables (ou l'inverse...). Ainsi que Anaïs, de qui j'ai beaucoup appris au cours de nos nombreuses discussions que ce soit sur la biblio, les manips ou autres..

La dénudation des reliefs, par des processus d'altération et d'érosion, permet la production de sédiment. Ces sédiments sont ensuite transportés depuis les reliefs jusqu'aux zones de dépôts par diérents processus : transport uviatile, glissement de terrains, coulée de boue. Au niveau des surfaces continentales, la majorité de la matière érodée est transportée par les rivières vers des bassins sédimentaires intracontinentaux ou océaniques [START_REF] Allen | Earth surface processes[END_REF]. Cependant, le long de leur parcours les rivières peuvent rencontrer des zones de stockage, dans lesquelles les sédiments peuvent s'accumuler pendant des périodes de temps plus ou moins longues [START_REF] Romans | Environmental signal propagation in sedimentary systems across timescales[END_REF]. Pour quantier les transferts de masse ayant lieu à la surface terrestre, il est nécessaire de comprendre les mécanismes à l'origine de l'érosion, du transport et du dépôt des sédiments.

La compréhension de ces processus passe par l'étude d'objets sédimentaires ayant préservés des marqueurs du transfert de sédiments. À la sortie des bassins versants, on observe souvent la formation d'un dépôt sédimentaire de forme circulaire : le cône alluvial. Les cônes alluviaux permettent l'accumulation et la préservation d'une grande partie de la charge sédimentaire (celle transportée par charriage). Leur position à la sortie des reliefs pourrait donc permettre de reconstituer les ux émanant de ceux-ci.

Dans le cadre de cette thèse, nous nous sommes donc concentrés sur l'étude des cônes alluviaux. En particulier, nous avons essayé de mettre en évidence les relations entre les forçages (débit d'eau, débit de sédiment et taille des sédiments) et la morphologie des cônes construits en aval. L'idée est donc de mettre en évidence les informations, enregistrées dans les dépôts des cônes alluviaux, qui pourraient permettre de reconstruire les paramètres ayant conduits à leur formation. Pour ce faire, nous avons choisi d'utiliser l'approche expérimentale qui permet de considérablement réduire les échelles de temps et d'espace, ainsi que de contrôler les paramètres d'entrée.

Les résultats obtenus par cette étude sont présentés dans ce manuscrit sous la forme de six chapitres. Le premier chapitre fait une synthèse des connaissances. Les trois chapitres suivants sont chacun consacrés à l'étude d'un paramètre en particulier dans des conditions simpliées (paramètres d'entrée maintenus constants). Les deux suivants proposent d'utiliser les résultats des premiers chapitres, pour étudier des cas plus proches de la réalité.

Dans le premier chapitre, nous proposons un état des connaissances sur les cônes alluviaux.

Nous dénissons le cône alluvial et présentons les relations connues entre la morphologie des cônes alluviaux et les paramètres d'entrées (débit d'eau, débit de sédiment et taille des sédiments). Il existe déjà de nombreuses études visant à quantier l'inuence de ces paramètres sur 12 Introduction la morphologie et la dynamique des cônes alluviaux, cependant celles-ci reposent souvent sur des lois empiriques déterminées à partir de cas particuliers. Elles ne sont donc pas généralisables. Ce constat nous permet de présenter les objectifs de ce travail (Chapitre 1).

Le second chapitre est focalisé sur l'étude de l'inuence du débit d'eau et de la taille des sédiments. En utilisant une approche expérimentale, nous montrons que lorsque le débit de sédiment alimentant le cône alluvial est très faible, la pente du cône atteint une valeur seuil.

Cette pente seuil est déterminée par le débit d'eau et la taille des sédiments alimentant le cône (Chapitre 2).

Dans le troisième chapitre, toujours en s'appuyant sur une démarche expérimentale, nous mettons en évidence l'inuence de la composition des sédiments alimentant le cône sur sa morphologie. Nous nous intéressons au cas d'un cône construit par deux types de sédiments, et montrons une corrélation entre la géométrie du dépôt sédimentaire et la nature des sédiments.

Cette description géométrique permet de contraindre la composition initiale du mélange sédimentaire (Chapitre 3).

Le quatrième chapitre reprend le dispositif expérimental utilisé dans le second chapitre, an d'étudier l'inuence du débit sédimentaire sur la pente du cône. Nous montrons que lorsque ce débit augmente, la pente du cône s'éloigne de sa valeur seuil. Cet écart au seuil génère la formation d'un prol concave (Chapitre 4).

Dans le cinquième chapitre, nous étudions numériquement les modications enregistrées dans la stratigraphie et la géométrie du cône alluvial lorsque les paramètres d'entrée, débit d'eau et composition du mélange sédimentaire, varient (Chapitre 5).

Pour nir, le sixième chapitre présente des pistes pour appliquer nos résultats aux objets naturels. Pour ce faire, nous étudions des cônes naturels, déjà bien documentés dans la littérature, an de mettre en évidence les données qu'il est nécessaire d'acquérir sur le terrain pour reconstruire les paramètres d'entrée (Chapitre 6).

Les démarches expérimentales ainsi que les techniques d'acquisitions des données sont présentées en annexe.

Chapitre 1

État de l'art

Les cônes alluviaux

Les cônes alluviaux sont des dépôts sédimentaires qui se mettent en place à la sortie de bassins versants en érosion (Fig. 1.1). Lorsqu'un écoulement uvial ou gravitaire, chargé en sédiment, atteint une zone plane, le changement de pente associé provoque le dépôt des sédiments. L'écoulement, n'étant plus conné dans une vallée, peut s'écouler radialement dans toutes les directions à partir de l'apex (point amont à partir duquel la rivière émerge du relief ). Le dépôt, formé en aval, acquiert une forme caractéristique de portion de cône, d'angle d'ouverture Θ et de rayon R [START_REF] Drew | Alluvial and lacustrine deposits and glacial records of the upper-indus basin[END_REF][START_REF] Bull | The alluvial-fan environment[END_REF][START_REF] Rachocki | Alluvial fans: a eld approach[END_REF]Blair andMcPherson, 1994, 2009, Fig. 1.1b). Lignes pointillées : contour du cône. Les dépôts les plus clairs sont les plus récents.

Prograda�on

On trouve des cônes alluviaux sur toute la surface du globe quel que soit le contexte tectonique et climatique [START_REF] Weissmann | Fluvial form in modern continental sedimentary basins: Distributive uvial systems[END_REF]. Ils sont regroupés en trois catégories dénies principalement par les processus de transport et de dépôt des sédiments [START_REF] Stanistreet | The okavango fan and the classication of subaerial fan systems[END_REF][START_REF] Galloway | Terrigenous clastic depositional systems. applications to fossil fuel and groundwater resources[END_REF]. En suivant la classication de Stanistreet and McCarthy (1993), on distinguera trois types de cônes alluviaux (Fig. 1.2) Les cônes de débris (Fig. 1.2a). Ces cônes sont les plus petits avec une étendue radiale comprise entre 1 et 20 km. Ils sont également les plus pentus avec une pente pouvant atteindre 15°. Ils sont principalement constitués de sédiments grossiers et leur surface est très peu végétalisée. Leur mode de formation est lié à des processus gravitaires (coulées de boue et de débris) ne faisant intervenir qu'occasionnellement le transport de sédiments par les rivières. Ce type de cônes ne sera pas considéré dans cette étude.

Les cônes chenalisés en tresse (Fig. 1.2b). Ces cônes sont alimentés par une rivière en tresse formant des barres longitudinales et transversales. Ils ont généralement une étendue radiale allant de la dizaine à la centaine de kilomètres et des pentes moyennes pouvant varier de 0.1°à 2°. On observe souvent de la végétation sur les berges des chenaux. On distingue deux cas, tout d'abord les cônes avec des écoulements chenalisés en tresse peu profonds. Ces écoulements forment des cônes petits avec une pente forte et transportent principalement des graviers. On les trouve majoritairement dans les zones semi-arides.

Les cônes du Tian Shan en Chine font, par exemple, partie de cette catégorie [START_REF] Guérit | Caractéristiques morpho-sédimentaires des cônes alluviaux et reconstitution de leurs paléo-ux d'eau et de sédiments: Exemples naturels du piedmont nord du Tian Shan (Xinjiang, Chine) et modélisation expérimentale[END_REF]. À l'autre extrémité de cette catégorie on trouve les cônes avec des écoulements chenalisés en tresse profonds qui ont des pentes plus faibles et une grande étendue radiale.

Ces cônes transportent principalement des sédiments ns et se mettent en place dans des zones humides [START_REF] Saito | Slope of alluvial fans in humid regions of japan, taiwan and the philippines[END_REF]. Un exemple type est le cône de la Kosi, Inde.

Les cônes chenalisés à écoulement méandriformes aussi appelés Losimean (Low sinuosity meandering) (Fig. 1.2c). Ces cônes représentent une extrémité du spectre des cônes alluviaux. Ce n'est qu'après l'étude de [START_REF] Mccarthy | The great inland deltas of africa[END_REF] sur le cône de l'Okavango qu'ils ont été classiés en tant que cônes alluviaux. Ils sont généralement très étendus radialement (R 150 km) avec une pente très faible (S 0.1°). Leur surface est végétalisée et présente des chenaux bien identiés. On les trouve préférentiellement dans des zones humides et en extension.

Les cônes alluviaux se mettent en place à l'intersection entre les zones en érosion et les zones de 1.1. Les cônes alluviaux 15 dépôts. Ces espaces peuvent être localisés à la conuence entre deux rivières ou dans des bassins sédimentaires. Dans le premier cas, le dépôt sera de taille limitée et ne sera pas préservé sur de longues échelles de temps. Au contraire, les bassins sédimentaires permettent la formation de plus grands cônes qui sont préservés sur des échelles de temps beaucoup plus longues. Dans ce cas-ci l'espace disponible pour l'accumulation des sédiments est donc contrôlé par l'action des failles qui engendrent la surélévation d'un bloc par rapport à un autre [START_REF] Ritter | Reconciling the roles of tectonism and climate in quaternary alluvial fan evolution[END_REF]. Dans le cas où le cône se jette dans une étendue d'eau (lac ou océan), l'espace disponible pour la sédimentation est contrôlé par les variations du niveau d'eau. Les cônes alluviaux sont construit par l'apport d'un ux sédimentaire, Q s , transporté depuis un relief par une rivière, de débit Q w . Ces trois facteurs (espace disponible pour la sédimentation, débits d'eau et de sédiment) représentent les conditions aux limites qui contrôlent la formation des cônes alluviaux [START_REF] Rachocki | Alluvial fans: a eld approach[END_REF]. Géométriquement, un cône alluvial est caractérisé (1) par son rayon R, qui représente la distance depuis l'apex au pied du cône, (2) un angle d'ouverture Θ qui en fonction de l'espace disponible peut s'étendre latéralement jusqu'à 180°(Fig. [START_REF] Rachocki | Alluvial fans: a eld approach[END_REF][START_REF] Blair | Processes and forms of alluvial fans[END_REF]).

An de diérencier les cônes alluviaux des autres systèmes uviaux, tels que les rivières alluviales ou les deltas, [START_REF] Blair | Alluvial fans and their natural distinction from rivers based on morphology, hydraulic processes, sedimentary processes, and facies assemblages[END_REF] Données SRTM (source : Earth Explorer). D'après [START_REF] Gaurav | Morphology of the Kosi megafan[END_REF]. peut toutefois varier d'un cône à l'autre en fonction des conditions dans lesquelles il se forme. En Chapitre 1. État de l'art eet, [START_REF] Bull | Geomorphology of segmented alluvial fans in western Fresno County, California[END_REF] et Blair (1987a) mettent en évidence trois types de prol radial : (a) une pente relativement constante, (b) une pente qui diminue progressivement le long du cône, et (c) une pente composée de plusieurs segments linéaires dont la pente diminue vers l'aval. Les cônes ayant une pente constante sont associés à un processus de formation simple dans lequel les paramètres d'entrée (ux d'eau et de sédiments et tailles des sédiments) restent constants. Le prol concave des cônes est souvent expliqué soit par une diminution progressive de la taille des grains le long du cône, soit par une diminution du ux de sédiment en aval du cône [START_REF] Stock | Controls on alluvial fan long-proles[END_REF]. Les cônes segmentés, qui ont une pente composée de plusieurs segments, ont un prol également concave similaire à celui engendré par une diminution progressive de la taille des grains, excepté que leur pente présente des ruptures nettes qui sont généralement associées à un changement brutal de la taille des sédiments transportés [START_REF] Bull | Geomorphology of segmented alluvial fans in western Fresno County, California[END_REF]. Ces prols sont caractéristiques des cônes composés de mélanges de graviers et de sables qui présentent une transition nette entre des dépôts grossiers en amont et des dépôts plus n en aval [START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF].

Les cônes alluviaux sont des objets dynamiques qui évoluent dans le temps et dans l'espace.

À grande échelle de temps, l'échelle de la formation du cône, le rayon R des cônes croît (progradation) et leur surface h s'élève (aggradation), sous l'eet du dépôt de sédiment [START_REF] Rachocki | Alluvial fans: a eld approach[END_REF]Harvey et al., 2005, Fig. 1.1). À plus courte échelle de temps, celle de la mobilité de la rivière, les sédiments remplissent le chenal en commençant par la partie aval du cône. Ce front de remplissage remonte ensuite progressivement vers l'amont. Le dépôt des sédiments dans le chenal provoque ainsi une surélévation du lit. En conséquence, la rivière change d'orientation pour s'écouler en direction de la pente la plus forte, on parle alors d'avulsion [START_REF] Field | Channel avulsion on alluvial fans in southern arizona[END_REF][START_REF] Slingerland | River avulsions and their deposits[END_REF][START_REF] Sinha | The great avulsion of kosi on 18 august 2008[END_REF]. Un nouveau chenal se met ensuite en place et le cycle se répète. Ces cycles sont autogéniques : ils se mettent en place à la surface des cônes spontanément sans variation des conditions aux limites. La succession de ces cycles maintient l'équilibre du cône [START_REF] Fraser | Geomorphic controls on sediment accumulation at margins of foreland basins[END_REF]. Entre deux phases chenalisées, on peut observer une période où l'écoulement n'est pas localisé, et forme une nappe qui recouvre entièrement ou partiellement la surface du cône [START_REF] Decelles | Controls on synorogenic alluvial-fan architecture, beartooth conglomerate (palaeocene), wyoming and montana[END_REF]. La mobilité de l'écoulement se caractérise par la présence simultanée de zones actives et inactives à la surface du cône (les zones claires représentent la zone active, Fig. 1.1). L'axis-symétrie du cône est maintenue par les phénomènes d'avulsion qui permettent à des zones inactives de devenir actives et inversement [START_REF] Gómez-Villar | Surface sediment characteristics and present dynamics in alluvial fans of the central spanish pyrenees[END_REF].

Reconstitution des paléo-ux

Les cônes alluviaux sont situés à l'interface entre les zones en érosion (reliefs), et les zones en sédimentation (vallées et plaines alluviales). Cette position en fait les premières archives sédimentaires à la sortie des reliefs terrestres [START_REF] Hinderer | From gullies to mountain belts: a review of sediment budgets at various scales[END_REF][START_REF] Latrubesse | Large rivers, megafans and other quaternary avulsive uvial systems: A potential who's who in the geological record[END_REF]. On peut donc tenter, en les étudiant, de reconstruire les paléo-ux ayant permis leur mise en place. La quantication des ux sédimentaires, émanant des reliefs terrestres, est essentielle pour comprendre les transferts de masse sur les continents et à une échelle plus petite, l'histoire des topographies dont ils sont issus [START_REF] Métivier | Mass transfer between eastern tien shan and adjacent basins (central asia): constraints on regional tectonics and topography[END_REF][START_REF] Kuhlemann | Post-collisional sediment budget history of the alps: tectonic versus climatic control[END_REF]. La technique la plus utilisée pour reconstituer les paléo-ux sédimentaires est, à l'heure actuelle, la reconstruction volumique [START_REF] Kiefer | Gravity-based mass balance of an alluvial fan giant: the arcas fan, pampa del tamarugal, northern chile[END_REF][START_REF] Jayko | Late quaternary denudation, death and panamint valleys, eastern california[END_REF][START_REF] Jolivet | Unbalanced sediment budgets in the catchmentalluvial fan system of the kuitun river (northern tian shan, china): Implications for mass-balance estimates, denudation and sedimentation rates in orogenic systems[END_REF][START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF]. L'utilisation de cette Les cônes alluviaux présentent l'avantage d'être bien délimités spatialement. L'estimation du volume des dépôts peut donc être eectuée en utilisant des modèles numériques de terrain (DEM). Cependant, en fonction du contexte tectonique dans lequel un cône est construit, sa reconstruction 3D à partir de sa surface peut s'avérer inexacte. En eet, l'extrapolation directe du volume du cône à partir de sa surface suppose que celui-ci est construit sur une surface plane, ce qui n'est souvent pas le cas. Pour estimer le volume du dépôt le plus précisément possible, il est nécessaire d'avoir accès à la topographie de la base sur laquelle il repose [START_REF] Jayko | Late quaternary denudation, death and panamint valleys, eastern california[END_REF][START_REF] Jolivet | Unbalanced sediment budgets in the catchmentalluvial fan system of the kuitun river (northern tian shan, china): Implications for mass-balance estimates, denudation and sedimentation rates in orogenic systems[END_REF]Guerit et al., 2016, Fig. 1.4). Une fois le volume de sédiment connu, il est alors nécessaire de connaître l'âge du dépôt pour avoir accès au ux. La période d'activité des cônes alluviaux peut, par exemple, être estimée par des méthodes radiochronologiques, ou encore en Chapitre 1. État de l'art utilisant les isotopes cosmogéniques, tels que le 10 Be [START_REF] France-Lanord | Erosion rates determined from the accumulation of in situ-produced 10be in uvial sediments of himalayan rivers[END_REF].

En ayant accès à toutes ces informations, il est possible d'estimer les paléo-ux sédimentaires en utilisant la relation suivante [START_REF] Guérit | Caractéristiques morpho-sédimentaires des cônes alluviaux et reconstitution de leurs paléo-ux d'eau et de sédiments: Exemples naturels du piedmont nord du Tian Shan (Xinjiang, Chine) et modélisation expérimentale[END_REF],

Q s = V (1 -λ) f P , (1.1)
où Q s est le ux de sédiment émanant du relief, V le volume du cône alluvial, λ la porosité des sédiments, f la fraction de la charge sédimentaire transportée par la rivière qui est déposée dans le cône, et P la période d'activité du cône. Dans l'équation (1.1), la fraction de sédiment piégée dans le cône, f , est encore mal contrainte et rarement évaluée [START_REF] Kiefer | Gravity-based mass balance of an alluvial fan giant: the arcas fan, pampa del tamarugal, northern chile[END_REF][START_REF] Oguchi | Late quaternary sediment budget in alluvial-fansource-basin systems in japan[END_REF][START_REF] Jolivet | Unbalanced sediment budgets in the catchmentalluvial fan system of the kuitun river (northern tian shan, china): Implications for mass-balance estimates, denudation and sedimentation rates in orogenic systems[END_REF]. Pour simplier, on considère généralement que f , la fraction sédimentaire déposée dans le cône, correspond à la fraction du ux sédimentaire transportée en charge de fond [START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF]. Malgré ces incertitudes la reconstruction volumétrique permet d'estimer l'ordre de grandeur du taux de dénudation des bassins versants alimentant les cônes étudiés, si l'aire de ces derniers est connue.

Pour reconstruire l'histoire climatique d'une région, il est également nécessaire de quantier les paléo-ux d'eau [START_REF] Mcdonald | Regional response of alluvial fans to the pleistocene-holocene climatic transition, mojave desert, california. Special papers-geological society of America[END_REF][START_REF] Leier | Mountains, monsoons, and megafans[END_REF][START_REF] Davidson | Towards a quantitative method for estimating paleohydrology from clast size and comparison with modern rivers[END_REF][START_REF] Waters | Climatic controls on late pleistocene alluvial fans, cyprus[END_REF]. Trois méthodes sont principalement utilisées pour estimer les caractéristiques de l'écoulement à partir des dépôts formés : (a) l'analyse de la géométrie de l'écoulement (largeur, profondeur), (b) l'étude des formes sédimentaires présentes dans les chenaux (dunes, rides) et (c) la détermination du seuil de mise en mouvement des sédiments (Métivier and Barrier, 2010, et références incluses). Ces trois méthodes dépendent souvent de relations empiriques déterminées à partir de cas particuliers et ne sont donc pas généralisables. [START_REF] Le Hooke | Geometry of alluvial fans: Eect of discharge and sediment size[END_REF]Parker et al., 1998a;[START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF]. Ces trois paramètres sont dépendants, à plus grande échelle par la tectonique et le climat [START_REF] Lustig | Clastic sedimentation in deep springs valley, California[END_REF][START_REF] Blair | Tectonic and hydrologic controls on cyclic alluvial fan, uvial, and lacustrine rift-basin sedimentation, jurassic-lowermost cretaceous todos santos formation, chiapas, mexico[END_REF]. Dans la section suivante, nous passerons brièvement en revue les relations existantes entre la morphologie des cônes et ces diérents paramètres an de mieux comprendre comment reconstituer les paléo-ux à partir de la morphologie actuelle des cônes alluviaux. [START_REF] Schumm | Experimental uvial geomorphology[END_REF][START_REF] Paola | The unreasonable eectiveness of stratigraphic and geomorphic experiments[END_REF][START_REF] Clarke | Experimental alluvial fans: Advances in understanding of fan dynamics and processes[END_REF].

En laboratoire, les modèles expérimentaux sont conçus en rapport avec les paramètres que l'on veut étudier. C'est, par exemple, dans l'optique d'étudier l'inuence de la tectonique et des variations eustatiques sur la mise en place de bassin sédimentaire que l' Experimental Earthscape Facility (XES) Basin a été conçu au Saint Anthony Falls Laboratory (Fig. 1.5a). Ce dispositif, unique en son genre, permet de créer des zones de subsidence à l'aide de pistons qui déforment la surface sur laquelle le dépôt se forme. Un drain disposé à l'extrémité du bassin permet également d'ajuster le niveau de base de la rivière [START_REF] Paola | Experimental stratigraphy[END_REF][START_REF] Heller | Geomorphology and sequence stratigraphy due to slow and rapid base-level changes in an experimental subsiding basin (xes 96-1)[END_REF]. Pour étudier l'inuence de la quantité de pluie, et donc du climat, sur la formation des cônes alluviaux, le laboratoire Géosciences Rennes a, quant à lui, développé un système couplant bassin versant et cône alluvial (Rohais, 2007, Fig. 1.5b [START_REF] Blair | Tectonic and hydrologic controls on cyclic alluvial fan, uvial, and lacustrine rift-basin sedimentation, jurassic-lowermost cretaceous todos santos formation, chiapas, mexico[END_REF][START_REF] Whipple | Tectonic control of fan size: the importance of spatially variable subsidence rates[END_REF][START_REF] Harvey | The role of base-level change in the dissection of alluvial fans: case studies from southeast spain and nevada[END_REF][START_REF] Pang | Climatic and tectonic controls on late triassic to middle jurassic sedimentation in northeastern guangdong province, south china[END_REF]. Lorsque l'espace disponible pour la sédimentation varie, les frontières du cône évoluent simultanément. Ainsi, les variations tectonique et climatique peuvent être enregistrées dans la stratigraphie du cône par la progradation et la rétrogradation des dépôts du cône [START_REF] Horton | Tectonic control on coarse-grained foreland-basin sequences: An example from the cordilleran foreland basin, utah[END_REF][START_REF] Hartley | Climatic controls on alluvial-fan activity, coastal cordillera, northern chile[END_REF][START_REF] Harvey | Dierential eects of base-level, tectonic setting and climatic change on quaternary alluvial fans in the northern great basin, nevada, usa[END_REF][START_REF] Whittaker | Decoding downstream trends in stratigraphic grain size as a function of tectonic subsidence and sediment supply[END_REF]. Une diminution du niveau de base peut aussi provoquer l'incision du cône [START_REF] Waters | Climatic controls on late pleistocene alluvial fans, cyprus[END_REF][START_REF] Pope | A chronology of alluvial fan response to late quaternary sea level and climate change, crete[END_REF].

Des modélisations numériques et expérimentales corroborent ces observations de terrain [START_REF] Muto | Contributions to sequence stratigraphy from analogue and numerical experiments[END_REF]. En utilisant les équations de conservation de la masse, [START_REF] Whipple | Tectonic control of fan size: the importance of spatially variable subsidence rates[END_REF] montrent que la taille des cônes alluviaux est contrôlée principalement par les événements tectoniques, les variations climatiques n'ayant qu'une inuence secondaire. L'incision et la formation de terrasses peuvent être reproduites en laboratoire en imposant un abaissement continue du niveau de base [START_REF] Muto | Autogenic response of uvial deltas to steady sea-level fall: Implications from ume-tank experiments[END_REF]. À l'aide de modélisations numériques combinant le bassin en érosion et la zone de dépôt, plusieurs études mettent en évidence la signature stratigraphique (progradation et rétrogradation du front de sédiments grossiers) des événements climatiques et tectoniques [START_REF] Coulthard | A cellular model of holocene upland river basin and alluvial fan evolution[END_REF][START_REF] Clevis | Numerical modelling of drainage basin evolution and three-dimensional alluvial fan stratigraphy[END_REF][START_REF] Armitage | Transformation of tectonic and climatic signals from source to sedimentary archive[END_REF].

À l'échelle du bassin versant, la tectonique et le climat agissent conjointement sur les ux d'eau et de sédiment qui atteignent l'apex du cône [START_REF] Bonnet | Landscape response to climate change: Insights from experimental modeling and implications for tectonic versus climatic uplift of topography[END_REF]. En eet, la surrection du relief favorisera une érosion plus forte en mettant plus de matériel à disposition, tandis que lors d'un période glaciaire les ux d'eau seront plus faibles, limitant l'érosion et donc le ux de sédiment [START_REF] Frostick | Climatic versus tectonic controls of fan sequences: lessons from the dead sea, israel[END_REF]. Les études de terrain montrent que les alternances entre érosion (incision du cône) et dépôt (aggradation du cône) sont corrélées aux cycles glaciaires [START_REF] Weissmann | Factors controlling sequence development on quaternary uvial fans, san joaquin basin, california, usa[END_REF][START_REF] Guerit | Reconstruction of the sediment trans-port conditions in the urumqi alluvial fan (northeast tian shan, china)[END_REF][START_REF] Cesta | Timing and nature of alluvial fan development along the chajnantor plateau, northern chile[END_REF]. Enn, les variations climatiques aectent la végétation et ainsi modie la capacité du sol à produire des sédiments [START_REF] Lustig | Clastic sedimentation in deep springs valley, California[END_REF][START_REF] Macklin | Correlation of uvial sequences in the mediterranean basin over the last 200ka and their relationship to climate change[END_REF][START_REF] Candy | Interpreting the response of a dryland river system to late quaternary climate change[END_REF][START_REF] Harvey | Dierential eects of base-level, tectonic setting and climatic change on quaternary alluvial fans in the northern great basin, nevada, usa[END_REF]. [START_REF] Allen | Sediment ux from an uplifting fault block[END_REF][START_REF] Marr | A two-diusion model of uvial stratigraphy in closed depositional basins[END_REF]. Au contraire, le système réagit plus rapidement au variations climatiques, un signal haute fréquence dans la stratigraphie est donc interprété comme le marqueur de variations climatiques [START_REF] Allen | Sediment ux from an uplifting fault block[END_REF].

Enn, à l'échelle du cône alluvial et de son basin versant, la tectonique et le climat contrôlent principalement les paramètres d'entrée (taille des sédiments, d s , ux d'eau, Q w , et de sédiment, Q s ). Il est donc indispensable de comprendre comment ces paramètres inuencent la morphologie et la stratigraphie des cônes pour pouvoir a posteriori reconstruire les paléo-ux.

Dans les sections suivantes, nous présentons brièvement l'état des connaissances concernant l'inuence de ces trois paramètres sur la dynamique de croissance et la morphologie des cônes alluviaux, an d'introduire les objectifs de cette thèse.

La taille des sédiments

Les études de terrains ont permis de mettre en évidence une forte corrélation entre la taille des sédiments composant les cônes alluviaux et la pente de ceux-ci. Ces études montrent que plus les sédiments sont grossiers, plus la pente du cône est forte [START_REF] Blissenbach | Relation of surface angle distribution to particle size distribution on alluvial fans[END_REF][START_REF] Bluck | Sedimentation of an alluvial fan in southern nevada[END_REF][START_REF] Denny | Alluvial fans in the Death Valley region, California and Nevada[END_REF]Rice, 1999, Fig. 1.6). Cependant, sur le terrain, plusieurs facteurs peuvent agir simultanément sur la pente [START_REF] Stock | Controls on alluvial fan long-proles[END_REF]. Il est donc dicile d'isoler l'inuence de la taille des grains. Les modèles expérimentaux ont toutefois permis de lever cette ambiguïté et conrment les observations de terrain [START_REF] Hooke | Model geology: prototype and laboratory streams: discussion[END_REF][START_REF] Le Hooke | Geometry of alluvial fans: Eect of discharge and sediment size[END_REF][START_REF] Ikeda | Experimental study of heterogeneous sediment transport[END_REF]. Pour comprendre cette dépendance, il est nécessaire de s'intéresser au seuil de mise en mouvement des grains. Ce seuil de mise en mouvement correspond à un équilibre entre la Chapitre 1. État de l'art force de friction exercée par le uide sur le grain (qui dépend de la pente) et la force de gravité [START_REF] Shields | Anwendung der ahnlichkeits mechanik und der turbulenz-forschung auf die geschiebebewegung[END_REF]. À conditions identiques, pour qu'un gros grain soit mis en mouvement, la force de friction nécessaire doit être plus importante que pour un petit grain. On comprend donc, intuitivement, que la pente associée à la mise en mouvement d'un gros grain est plus importante que celle d'une rivière transportant des petits grains [START_REF] Ikeda | Experimental study of heterogeneous sediment transport[END_REF]. Dans la plupart des rivières naturelles, la taille des sédiments diminue vers l'aval, ce phénomène, appelé downstream ning, résulte soit de l'abrasion mécanique des grains au cours de leurs transport, soit d'un dépôt diérentiel [START_REF] Paola | Downstream ning by selective deposition in a laboratory ume[END_REF][START_REF] Smith | The gravel-sand transition: ume study of channel response to reduced slope[END_REF][START_REF] Fedele | Similarity solutions for uvial sediment ning by selective deposition[END_REF][START_REF] Singer | Downstream patterns of bed material grain size in a large, lowland alluvial river subject to low sediment supply[END_REF][START_REF] Blom | The graded alluvial river: Prole concavity and downstream ning[END_REF]. La pente d'un cône étant corrélée à la taille des grains transportés par la rivière, une conséquence direct du downstream ning est la formation d'un prol radial concave. En utilisant les données de Stock et al. (2008), on peut montrer que sur les cônes alluviaux, il existe une forte corrélation entre la proportion de sédiment ns qui composent le lit de la rivière et la pente locale. La pente des cônes diminue vers l'aval quand la proportion de grains grossiers diminue (Fig. 1.7).

La présence de plusieurs tailles de grains dans les sédiments naturels soulève le problème du transport de sédiments polydisperses. Les mécanismes de transport de sédiment s'appliquant à un mélange de sédiment de plusieurs tailles sont encore mal compris [START_REF] Meyer-Peter | Formulas for bed-load transport[END_REF][START_REF] Wilcock | Surface-based transport model for mixed-size sediment[END_REF][START_REF] Houssais | Bedload transport of a bimodal sediment bed[END_REF]. Par conséquent, les lois de transport utilisées à l'heure actuelle sont souvent simpliées : on ne prend en compte, par exemple, qu'une taille de sédiment (Parker, 1978a;Parker et al., 1998a;[START_REF] Nicholas | Modeling alluvial landform change in the absence of external environmental forcing[END_REF].

Lorsque les cônes sont composés de deux types de sédiment, de mobilité distincte (gravier et sable par exemple), il a été observé expérimentalement et sur le terrain que les sédiments sont ségrégés en fonction de leur mobilité.

1.3. Relation entre morphologie et forçages 23

Les ux

Inuence du ux de sédiment sur la dynamique des cônes alluviaux Les cônes alluviaux résultent du dépôt de sédiments transportés par une rivière, la quantité de sédiment qui émerge à leur apex contrôle leur vitesse de croissance [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF][START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF]. Au premier ordre, si l'on considère un cône alluvial comme un demi-cône de pente constante, son volume est

V = πSR 3 6 , (1.2)
pour un cône d'angle d'ouverture π, de pente S et de rayon R (Fig. [START_REF] Sinha | On the controls of uvial hazards in the north bihar plains, eastern india[END_REF][START_REF] Assine | River avulsions on the taquari megafan, pantanal wetland, brazil[END_REF]. Plus le ux de sédiment est important, plus le chenal se comble rapidement, entraînant une avulsion. Cette dépendance a été conrmée et quantiée expérimentalement [START_REF] Bryant | Experimental study of avulsion frequency and rate of deposition[END_REF][START_REF] Ashworth | Relationship between sediment supply and avulsion frequency in braided rivers[END_REF][START_REF] Reitz | Flooding and ow path selection on alluvial fans and deltas[END_REF].

Inuence des ux sur la morphologie des cônes alluviaux

En plus de contrôler la dynamique de croissance des cônes alluviaux, les ux d'eau et de sédiment ont aussi une inuence sur leur morphologie, notamment sur leur pente.

Le ux d'eau : Le ux d'eau, Q w , est dicile à mesurer sur le terrain. En eet, les chenaux sont souvent éphémères et les mesures eectuées de façon épisodique lors de mission de terrain [START_REF] Stock | Controls on alluvial fan long-proles[END_REF][START_REF] Gaurav | Morphology of the kosi megafan channels[END_REF][START_REF] Métivier | Geometry of meandering and braided gravel-bed threads from the bayanbulak grassland, tianshan, pr china[END_REF]. Pour suppléer ce manque d'information, l'aire du bassin versant alimentant le cône est souvent utilisée comme substitut pour le débit d'eau. En utilisant ce proxy, les observations de terrain montrent que la pente des cônes alluviaux diminue lorsque l'aire drainée (i.e. le débit d'eau) augmente [START_REF] Drew | Alluvial and lacustrine deposits and glacial records of the upper-indus basin[END_REF][START_REF] Denny | Alluvial fans in the Death Valley region, California and Nevada[END_REF][START_REF] Saito | Slope of alluvial fans in humid regions of japan, taiwan and the philippines[END_REF]Blair and McPherson, 2009, Fig. 1.8). Cette observation est conrmée par les modèles numériques et expérimentaux de Densmore et al. (2007[START_REF] Rohais | Sedimentary record of tectonic and climatic erosional perturbations in an experimental coupled catchment-fan system[END_REF] qui mettent en évidence une diminution de la pente des cônes lorsque l'intensité des précipitations augmente.

Le ux de sédiment : La mesure de ux de sédiment, Q s , sur le terrain s'avère être encore plus compliquée. En eet, la rivière transporte des sédiments principalement en période de crue ce qui rend les mesures particulièrement diciles et dangereuses. De plus, à l'heure actuelle, la mesure du ux de sédiment s'eectue par échantillonnage ponctuel, ce qui ne permet d'avoir qu'une estimation locale du ux de sédiment [START_REF] Meunier | Flow pattern and sediment transport in a braided river: The torrent de st pierre(french alps)[END_REF][START_REF] Liu | Measuring bedload in gravel-bed mountain rivers: averaging methods and sampling strategies[END_REF]. Dans les cônes [START_REF] Ikeda | Experimental study of heterogeneous sediment transport[END_REF], [START_REF] Stock | Controls on alluvial fan long-proles[END_REF] ont montré qu'il existe une corrélation entre la diminution du ux de sédiment en aval et celle de la pente des cônes. Cette diminution de la pente des cônes en aval a donc pour conséquence de générer un prol en long concave. Ces observations sont conrmées par des modèles expérimentaux dans lesquels une augmentation du ux de sédiment entraîne une augmentation de la pente [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Ashworth | The relationship between channel avulsion, ow occupancy and aggradation in braided rivers: insights from an experimental model[END_REF]. Cette augmentation de la pente n'est cependant pas linéaire. Il est nécessaire de bien comprendre les lois de transport qui contrôlent la mise en mouvement des sédiments pour pouvoir quantier l'inuence de ce ux sur la pente des cônes [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF].

Le rapport des ux Q s /Q w : D'un point de vue expérimental, l'inuence des ux d'eau et de sédiment sur la géométrie des cônes est souvent exprimée en terme de rapport de ces ux, Q s /Q w . La pente des cônes augmente quand le rapport des ux augmente (Le [START_REF] Le Hooke | Geometry of alluvial fans: Eect of discharge and sediment size[END_REF]Paola et al., 1992a;[START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Clarke | An experimental investigation of autogenic behaviour during alluvial fan evolution[END_REF][START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF]. Cette dépendance de la pente au rapport des ux a été initialement proposée par Parker et al. (1998a).

Les auteurs s'appuient sur l'équation de conservation de la masse et trois relations internes qui décrivent l'écoulement et le transport de sédiment : la résistance du uide, en utilisant une relation de Chézy [START_REF] Paola | The inuence of transport uctuations on spatially averaged topography on a sandy, braided uvial fan[END_REF], une loi de transport des sédiments [START_REF] Meyer-Peter | Formulas for bed-load transport[END_REF], en aval provoque une diminution de la pente. Le graphique modié de [START_REF] Ikeda | Experimental study of heterogeneous sediment transport[END_REF] illustre cet eet dans un chenal, les pentes obtenues sont approximativement celles atteintes par les cône alluviaux. D'après [START_REF] Stock | Controls on alluvial fan long-proles[END_REF].

la
que la rivière s'élargisse vers l'aval (Parker et al., 1998a).

En utilisant ces trois relations, les auteurs suggèrent qu'il existe une fonction f telle que, [START_REF] Glover | Stable channel proles[END_REF][START_REF] Henderson | Stability of alluvial channels[END_REF]Parker, 1978a,b;[START_REF] Seizilles | Width of laminar laboratory rivers[END_REF]. Il est donc nécessaire de contraindre la largeur de la rivière pour estimer le ux de sédiment à la surface des cônes. [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF] ont mis en place une expérience formant un cône alluvial conné dans un plan vertical. Dans cette expérience, pour s'aranchir du problème de la largeur de la rivière, le dépôt se construit entre deux plaques de Plexiglass distantes de 2.5 cm. La largeur de l'écoulement est ainsi bien contrainte. De plus, pour éviter les relations empiriques imposées par la nature turbulente de l'écoulement, le uide utilisé est un mélange visqueux d'eau et de glycérol de masse volumique ρ = 1210 kg m -3 et de viscosité η = 40 mPa s.

S = f Q s Q w (1.
Ce uide permet de générer un écoulement laminaire caractérisé par un nombre de Reynolds Re ≈ 10. Finalement, pour simplier la mesure de la loi de transport, les grains utilisés sont mono-disperse. Il s'agit de billes de verre de diamètre d s = 477 µm et de densité ρ s = 2485 kg m -3 (Fig. 1.11).

Pour étudier l'inuence des diérents paramètres toutes les expériences ont été réalisées avec les mêmes grains tout en faisant varier indépendamment les débits de uide et de sédiment.

Lorsque le ux de sédiment qui alimente le cône est très faible, la pente du cône est contrôlée par le seuil de mise en mouvement des sédiments. Ainsi, [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF] Comme nous l'avons vu, la pente du cône est généralement exprimée en fonction du rapport des ux (Parker et al., 1998a;[START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF]. Cette relation rend compte de l'augmentation de la pente du cône avec le débit de sédiment et de sa diminution avec le débit d'eau. Cependant, cette relation ne permet pas l'existence d'une pente seuil, atteinte lorsque le ux de sédiment devient nul [START_REF] Seizilles | Width of laminar laboratory rivers[END_REF]. Il est donc nécessaire de découpler les eets des débits d'eau et de sédiment pour comprendre leur inuence respective sur la pente des cônes.

Concernant l'inuence de la taille des grains sur la morphologie des cônes alluviaux, l'augmentation de la pente des cônes alluviaux avec la taille des grains est bien documentée, que ce soit dans le milieux naturel ou expérimentalement [START_REF] Blissenbach | Relation of surface angle distribution to particle size distribution on alluvial fans[END_REF][START_REF] Ikeda | Experimental study of heterogeneous sediment transport[END_REF].

Lorsqu'un cône est composé de plusieurs tailles de grains, les observations de terrain et expérimentales montrent qu'un mécanisme de ségrégation permet de séparer les sédiments grossiers des sédiments ns [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF]. La position de ce front de ségrégation dépend des paramètres d'entrée (Q s , Q w et d s ), ainsi que de la proportion de sédiment n et grossier (Paola et al., 1992a)

q s = q 0 (θ -θ c ),
(1.7) où q 0 est un ux de sédiment caractéristique qui dépend des caractéristiques physiques du grain et du uide. Un suivi de la trajectoire des grains, par analyse vidéo, montre que les grains se déplacent majoritairement dans la direction de l'écoulement, mais qu'il existe aussi une composante transverse de se déplacement [START_REF] Seizilles | Cross-stream diusion in bedload transport[END_REF]. Les grains se comportent individuellement comme des marcheurs aléatoires, ce qui génère un ux diusif transverse à la direction de l'écoulement. Ce ux de sédiment diusif est caractérisé par une longueur de diusion l d .

En considérant la loi de transport (Eq. 1.7) et la longueur de diusion déterminées expérimen- et al., 2013). Dans ce cas, les caractéristiques géométriques de la rivière (pente, largeur et profondeur) sont décrites par des lois d'échelles, qui permettent de relier ces critères morphologiques à la taille des sédiments que transporte la rivière ainsi qu'à son débit [START_REF] Glover | Stable channel proles[END_REF][START_REF] Henderson | Stability of alluvial channels[END_REF][START_REF] Seizilles | Width of laminar laboratory rivers[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF]. Dans le but d'appliquer cette théorie aux cônes alluviaux, nous construisons un cône alluvial avec un débit de sédiment très faible an d'être dans les conditions les plus proche possibles du seuil de mise en mouvement des sédiments. Pour faciliter la compréhension des phénomènes physiques, nous utilisons un sédiment mono-disperse ainsi qu'un uide visqueux. À l'aide de cette expérience nous mettons en évidence l'inuence du débit uide et de la taille des sédiments sur la morphologie du cône alluvial.

34

Chapter 2. Fan near the threshold of sediment motion 2.1 Alluvial river at the threshold of sediment motion When a river ows on an alluvial bed, its geometry spontaneously evolves towards a stationary shape [START_REF] Lacey | Stable channels in alluvium[END_REF]. This equilibrium shape is controlled by the threshold of sediment motion [START_REF] Glover | Stable channel proles[END_REF][START_REF] Henderson | Stability of alluvial channels[END_REF][START_REF] Seizilles | Width of laminar laboratory rivers[END_REF]. When sediment grains are immersed in a river, two forces act on them: a normal force, induced by gravity and a tangential force induced by gravity and ow-induced stress τ (Fig. 2.1). When the river is at equilibrium, it is assumed that everywhere across the bed, sediments are at the threshold of motion (Eq. 2.1). From [START_REF] Seizilles | Width of laminar laboratory rivers[END_REF].

µ τ θ c (ρ s -ρ)gd s 2 + sin 2 ψ = µ 2 cos 2 ψ, (2.1)
where µ is the friction coecient, τ the ow-induced stress, θ c the critical Shields parameter (Appendix A), ρ and ρ s the uid and sediment density, g the acceleration of gravity and ψ the bed-angle in the transverse direction. For a laminar ow, within the shallow-water approximation, the ow-induced stress reads,

τ = ρgDS, (2.2)
where S is the longitudinal slope of the channel, and D its depth. The shallow-water approximation involves that cos ψ ≈ 1, Eqs. (2.1) and (2.2) together form a rst-order dierential equation:

SD L 2 + ∂D ∂y 2 = µ 2 , (2.3)
where L is a characteristic length which depends on sediment and uid physical properties,

L = θ c (ρ s -ρ)d s µρ .
(2.4) A solution of Eq. (2.3) gives the cosine cross-section of the channel, D = µL S cos Sy L .

(2.5)

To go further, we need to consider the mass and momentum balance for the uid,

Q w = channel U Ddy.
(2.6)

Within the shallow-water approximation, the uid friction at the bottom of the channel balances the gravity. As a consequence the momentum balance leads to Poiseuille velocity prole,

U = gSD 2 3ν , (2.7)
where ν is the uid kinematic viscosity. Finally, by combining the shape of the cross-section (Eq. 2.5), the water mass balance (Eq. 2.6) and the momentum balance (Eq. 2.7), we obtain Chapter 2. Fan near the threshold of sediment motion two scaling laws linking a viscous river width, W h , and slope, S h , to the water discharge and sediment characteristics,

S h = µ 4g 9ν 1/3 θ c µ ρ s -ρ ρ d s 4/3 1 Q 1/3 w (2.8) W h = π µ 9ν 4g 1/3 µ θ c ρ (ρ s -ρ)d s 1/3 Q 1/3 w (2.9)
This threshold theory holds for laboratory and natural rivers in its turbulent version which reads,

W H = π µ 3/4 θ c µ ρ s -ρ ρ d s -1/4 3 2 3/2 K(1/2) √ Cf g 1/4
Q w ,

(2.10)

where K(1/2) ≈ 1.85 is the elliptic integral of the rst kind, and C f is Chézy's coecient of uid friction [START_REF] Devauchelle | Longitudinal prole of channels cut by springs[END_REF][START_REF] Gaurav | Morphology of the kosi megafan channels[END_REF]Métivier et al., 2017, Fig. 2.2).

Then, to compare the slope of our experimental fan to the one of an alluvial river at the threshold of sediment motion, we need to produce an experimental fan fed with a single channel and a low sediment discharge. We use a 80 cm-wide and 50 cm-long tank to produce our experimental fan. At the back of the tank a 15 cm-high vertical wall simulates the mountain front. In the center of this wall, a 2.5 cm-wide tilted channel directs sediment and uid mixture toward the tank. The opposite side of the tank is bounded by a trench to evacuate uid. To prevent uid accumulation at the outlet of the fan, the fan is built on a 60 cm-wide and 30 cm-long shelf covered with a 200 µm sieve mesh. This facility allows us to build a fan with an opening angle of 180°(Fig. 2.3). The uid discharge is kept constant using a header tank and we monitor its value using an electromagnetic owmeter (Kobold, MIK-5NA10AE34R). A conveyor-belt supplies sediment into the channel, 2.2. Experimental set-up 37 this feeder-system is set on a scale (Ohaus, Explorer 35) to measure the sediment discharge.

Experimental set-up

Every experimental measurement is automatically saved to a database (details of the experiment monitoring are presented in Appendix B).

To avoid empirical closing relations implied by turbulent ow, we use a viscous uid which is a mixture of water (40 %) and glycerol (60 %). This uid is characterized by a density ρ = 1150 kg m -3 and an associated kinematic viscosity ν = 7 10 -6 m 2 s. These physical properties are set by the composition of the mixture (Appendix B.2). To ensure constant uid discharge we regularly measure the density of the uid. We oset the water evaporation, surveyed through regular density measurement, by adding water to the mixture. The resulting ow is laminar with a Reynolds number of about 50. The uid is mixed with a well-sorted uniform sediment composed of corundum grains with a median diameter d 50 = 415 µm and a density of ρ s = 3900 kg m -3 (Fig. 2.4a).

The grain size is measured using image analysis and the density using a pycnometer. This sediment is also dened by a characteristic friction coecient µ = 0.7 and a critical Shields number θ c = 0.14 (Fig. 2.4b, measurement method is described in Appendix A).

= q 0 (θ-θ c ), q 0 = 162 ± 5 g s -1 m -1 , θ c = 0.14± 0.005.
In order to remain near the threshold of sediment motion and quantify the inuence of the uid discharge on the fan slope, we perform six experiments. The sediment discharge, Q s , remains xed at a low value, whereas the uid discharge, Q w , varies between experiments (Table 2.1).

Fan growth

At the beginning of the experiment the surface of the experimental set-up is bare. We start the uid and sediment feed at the same time and keep them constant during all the experiment.

An axisymmetrical deposit spontaneously forms very rapidly. To monitor the fan evolution, we acquire top-view pictures using a camera (Canon EOS 100D with a lens: Canon 28-105 mm f/3.5-4.5 APO Macro 0.5m/1.6ft Ultrasonic) xed above the tank center. We record an image every ten minutes (Fig. 2.5). The growth of our fan is similar to the one of [START_REF] Van Dijk | Autocyclic behaviour of fan deltas: an analogue experimental study[END_REF] and [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]; the fan is built by alternating sheet ows and channelized ows. However, we observe that the period where sheet-ow dominates rarees with time so that the fan becomes mostly fed by a single channel. This channel avulses with time and ows across all the fan surface maintaining its axisymmetry (Fig. 2.5). The mechanism of avulsion is driven by the sediment deposition and its frequency correlated to the sediment discharge [START_REF] Bryant | Experimental study of avulsion frequency and rate of deposition[END_REF][START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]. Using high frequency pictures, we observe the lling of the channel, which leads to an avulsion (Fig. Using the still image we acquired, we can measure the evolution of the fan radius. To do so, we develop an algorithm that automatically locates the fan toe (Appendix C.1). Because the fan color is well dened, we can detect its boundary by applying a threshold. Then we measure the average radius from the apex to the boundary. For this, we approximate the fan boundary by a semi-circle. The observed variability between the measured distance and this average radius is less than 5%. After a transient of about three hours, the fan radius evolves linearly with the cubic root of time (Fig. 2.7).

h cm 10 h 20 h 30 h 40 h 50 h

This evolution agrees with mass balance of the fan. As we assume our deposit is perfectly semi-conical, its volume V reads

V = πSR 3 6 , (2.11)
where R is the fan radius and the aperture angle equals π. Assuming that all the sediment injected is trapped into the fan, we nd

V = Q s t ρ s (1 -λ)
.

(2.12) Using Eqs.(2.11) and (2.12), we recover the result of [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF],

R = 6 Q s ρ s (1 -λ)πS 1/3 t 1/3 . (2.13)
The growth rate of the fan depends primarily on the sediment discharge, the aperture angle of the fan surface to calculate its volume. We then compare this value to the volume of sediment injected into the fan (Eq. 2.12). We measure the volume of our six experimental fans every hour, and we represent this volume as a function of the volume injected into the fan (Fig. 2.9). Both volumes are linearly correlated. This then enables us to calculate the value of the porosity as the slope of the line that passes through the experimental points is equal to 1 -λ. With this method, we estimate the porosity of the deposit, λ ≈ 0.4. we do not observe a signicant variation of the slope between the centerline slope and the slope of the fan anks (Fig. 2.10b). As a consequence, we assume that the fan is axisymmetric and that its slope is the same as the one of the river that feeds it.

To study the dynamics of the fan growth, we acquire DEMs every hour and plot the timeevolution of the spatially averaged prole (Fig. 2.11a). To the rst order, the slope remains constant during the fan growth. This constant slope is reached after a transient of about 10 hours (Fig. 2.11b). At the beginning of the experiment, the slope is higher and then it reaches its equilibrium value. Once equilibrium is achieved, we still observe a small, within 5%, variability of the slope. One possible explanation for this variability, is the alternation of sheet ows and channelized ows [START_REF] Van Dijk | Autocyclic behaviour of fan deltas: an analogue experimental study[END_REF]. When the ow is channelized, the local uid discharge is higher consequently, the slope shallows. Hence, to the rst order, the fan growth is self similar.

As a consequence, we conclude that an experimental fan fed with uniform sediment and constant uid and sediment discharges is characterized by its slope only, and that this slope is spatially and temporally constant. Therefore, understanding the dynamics of a fan growth reduces to understanding what controls its slope.

Threshold fan

To compare our experimental measurements to the threshold theory, we measure the slope of our experimental fan and the width of the river that feeds it. An experimental fan diers from an experimental river by the fact that it can not be formed without sediment deposition so that grains in the feeding channel are never exactly at the threshold of motion. Yet, we force the channel to remain close to the threshold of sediment motion by imposing a very low sediment discharge (0.2 g min -1 , approximately 25 grains per second). To verify that the river that built the fan is near the threshold of sediment motion, we rst compare its width to the theoretical width of an alluvial river at equilibrium, W h (Fig. 2.12). To do so, we manually locate, on top view pictures, the banks of the river. We measure the river width on 10 positions along its course and average them. We repeat the measurement each hour during the fan construction for our six experiments. The error bars in Fig.

(2.12) represent the variability of the channel width over space and time. The red line is the theoretical width calculated using Eq. (2.9). 10 1 10 0 Water discharge, Q w (L min 1 ) 2 × 10 0 3 × 10 0 River width, W (cm) Figure 2.12: Width of the feeding channel as function of the water discharge, red line threoritical width Eq. (2.9).

As predicted by the equilibrium theory, the width of the channel scales as the water discharge to the power one third [START_REF] Seizilles | Width of laminar laboratory rivers[END_REF]. Looking carefully, however, we note that the width of the channel is slightly smaller than the width of a threshold channel. This overestimation is about 10%, and it is of the same order as the width variability. Consequently, to the rst order, we assume that the width of the river accords with the equilibrium theory. Now, we focus on the fan slope. In the previous section (Section. 2.3) we demonstrated that the slope of the fan stays constant during its growth, and made the assumption that the slope of the fan is the one of the river that builds it. As a consequence, to compare the slope of the fan to the theoretical one (Eq. 2.8), we represent the space and time averaged slope as a function of the uid discharge (Fig. 2.13). The red line is the slope we calculated using Eq. (2.8). We observe that our experimental slopes follow the trend of the theory, they evolve with the uid discharge power minus one third. Yet, the theory underestimate the slope by a factor of approximately two (red dashed line in Fig. 2.13). Seizilles de Mazancourt (2013) and [START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF] observe the same underestimation for experimental rivers [START_REF] Seizilles De Mazancourt | Forme d'équilibre d'une rivière[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF]. To understand the origin of this oset from the theory, we look at the scaling relationship between the slope and the water discharge,

10 1 10 0 Water discharge, Q w (L.min 1 ) 10 2 10 1 Fan slope, S S h 2S h
S h = µ 4g 9ν 1/3 θ c µ ρ s -ρ ρ d s 4/3 1 Q 1/3 w . (2.14)
In this equation the values of the acceleration of gravity, g, the friction coecient, µ, the sediment density and diameter, ρ s and d s , are well determined. The uid characteristic, ρ and ν, are monitored during the run, and their variability is less than 5%. The uid discharge Q w , can be overestimated because we neglect the water inltration, and because we consider that all the uid injected in the experiment ows through one single channel. As mentioned earlier, the ow occurs sometimes in two channels, or as a sheet-ow. Because the uid discharge inuence on the slope is to the power one third, we would need to divide it by eight to resolve this underestimation. This seems unreasonable. The last parameter of Eq. (2.14), is the critical Shields number θ c . Despite the fact that we carefully estimated this parameter using an independent and specic set-up (Appendix A), this measurement remains a matter of debate [START_REF] Charru | Erosion and deposition of particles on a bed sheared by a viscous ow[END_REF][START_REF] Andreotti | Les milieux granulaires, Entre uide et solide[END_REF]. In Eq. (2.14), the critical Shields number is raised at the power four over three, so an error on its measurement could have a high inuence on the value of the threshold slope. The critical Shields number, θ c , is also involved in the theoretical estimation of the river width at the equilibrium but, in contrast to the slope, it is only raised at power one third (Eq. 2.9). As a consequence, an error on the measurement of the critical Shields parameter has a much smaller inuence on the width than on the slope.

Conclusion

In this chapter we propose to use the theoretical framework developed for alluvial rivers to study the relation between the fan slope and the uid discharge Q w and grain size d s [START_REF] Seizilles | Width of laminar laboratory rivers[END_REF]. We generate six experimental alluvial fans fed with low sediment discharge in order to be as close as possible to the threshold of sediment motion. This procedure enables us to produce a fan mainly fed by a single-thread channel. This channel spontaneously avulses, to maintains the fan axisymmetry. For constant input parameters (Q w , Q s and d s ), the slope of our experimental fan also remains constant. We nd that for a given sediment size, despite an uncertainty on the prefactor, the slope of the fan is controlled by the water discharge and scales with the water discharge to the power minus one third. The slope of the fan is controlled by the river that builds it. As a consequence, the aperture angle of the fan does not inuence the fan slope. The aperture angle of the fan and the imposed sediment discharge control the growth rate of the fan.

In natural systems, however, alluvial fans are composed of a variety of sediments. Thus we can not transpose our results directly to natural fans. To go further we need to investigate the inuence of a non-uniform sediment mixture, and to quantify the impact of the sediment discharge on the dynamics of fan growth. [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF] and [START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF] have

shown that the formation of an alluvial fan fed with bimodal sediments leads to the formation of segregated and imbricated fans. The next chapter elaborates on these previous works, and explores the relation between the composition of the sediment mixture and the fan prole.

The result of [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF] suggests that the sediment discharge causes a departure from threshold, in particular near the fan apex where the sediment discharge is the highest. In chapter 4, we extend this experimental set-up to quantify the inuence of the sediment discharge on a complete fan.

Chapter 3

Bimodal laboratory fan

Abstract

Using laboratory experiments, we investigate the growth of an alluvial fan fed with two distinct granular materials. Throughout the growth of the fan, its surface maintains a radial segregation, with the less mobile sediment concentrated near the apex. Scanning the fan surface with a laser, we nd that the transition between the proximal and distal deposits coincides with a distinct slope break. A radial cross section reveals that the stratigraphy records the signal of this segregation. To interpret these observations, we conceptualize the fan as a radially symmetric structure that maintains its geometry as it grows. When combined with slope measurements, this model proves consistent with the sediment mass balance and successfully predicts the slope of the proximal-distal transition as preserved in the fan stratigraphy. While the threshold channel theory provides an order-of-magnitude estimate of the fan slopes, driven by the relatively high sediment discharge in our experimental system, the actual observed slopes are 3-5 times higher than those predicted by this theory.

Introduction

When a river leaves a mountain range to enter lowlands, it hits shallow slopes and loses valley connement. This abrupt change causes it to deposit its sedimentary load into an alluvial fan [START_REF] Bull | The alluvial-fan environment[END_REF][START_REF] Rachocki | Alluvial fans: a eld approach[END_REF][START_REF] Blair | Alluvial fans and their natural distinction from rivers based on morphology, hydraulic processes, sedimentary processes, and facies assemblages[END_REF][START_REF] Harvey | Alluvial fans: geomorphology, sedimentology, dynamicsintroduction. a review of alluvial-fan research[END_REF][START_REF] Blair | Processes and forms of alluvial fans[END_REF]. As the river builds this sedimentary structure, its bed rises above the surrounding land, and its channel becomes unstable. At this point, either the river erodes its banks to migrate laterally, or, during a large ood event, it overows, and in a process referred to as avulsion, establishes a new course for its channel [START_REF] Field | Channel avulsion on alluvial fans in southern arizona[END_REF][START_REF] Slingerland | River avulsions and their deposits[END_REF][START_REF] Sinha | The great avulsion of kosi on 18 august 2008[END_REF]. In both cases, the river constantly explores new paths to ll up hollows in the deposit surface and preserve its radial symmetry. The resulting deposit acquires the conical shape which characterizes alluvial fans.

As the rst sedimentary archive along the river's course, an alluvial fan records the history of its catchment [START_REF] Hinderer | From gullies to mountain belts: a review of sediment budgets at various scales[END_REF]. Indeed, the geometrical reconstruction of a fan provides an estimate of its volume which, through mass balance, yields the average denudation rate of the catchment [START_REF] Kiefer | Gravity-based mass balance of an alluvial fan giant: the arcas fan, pampa del tamarugal, northern chile[END_REF][START_REF] Jayko | Late quaternary denudation, death and panamint valleys, eastern california[END_REF][START_REF] Jolivet | Unbalanced sediment budgets in the catchmentalluvial fan system of the kuitun river (northern tian shan, china): Implications for mass-balance estimates, denudation and sedimentation rates in orogenic systems[END_REF][START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF]. Furthermore, when the river transports multiple grain sizes, it usually deposits the coarser sediment (gravel) near the fan apex, and the ner sediment (sand) at its toe. This segregation produces a gravelsand transition front which moves forward and backward as the fan adjusts to external forcing.

In radial cross section, this series of progradations and retrogradations appears as a boundary between lithostratigraphic units, a pattern often interpreted as the signature of tectonic or climatic events (Paola et al., 1992a;[START_REF] Clevis | Numerical modelling of drainage basin evolution and three-dimensional alluvial fan stratigraphy[END_REF][START_REF] Charreau | The neogene xiyu formation, a diachronous prograding gravel wedge at front of the tianshan: Climatic and tectonic implications[END_REF][START_REF] Whittaker | Decoding downstream trends in stratigraphic grain size as a function of tectonic subsidence and sediment supply[END_REF][START_REF] Dubille | Rapid grain size coarsening at sandstone/conglomerate transition: similar expression in himalayan modern rivers and pliocene molasse deposits[END_REF].

To interpret the morphology and stratigraphy of an alluvial fan, we need to understand how it translates the input signal (e.g., water and sediment discharges) into its own geometry (e.g., its size, downstream slope and stratigraphy). For instance, [START_REF] Drew | Alluvial and lacustrine deposits and glacial records of the upper-indus basin[END_REF] observed that the lower the water discharge Q w , the steeper the fan slope. More recent observations point at the inuence of the sediment discharges Q s on the slope, often in the form of the ratio Q s /Q w . In general, the slope steepens when this ratio increases (Parker et al., 1998a,b). At rst sight, the shape of an alluvial fan is well approximated by a cone, but a closer look often reveals a steeper slope near the apex (Le [START_REF] Le Hooke | Geometry of alluvial fans: Eect of discharge and sediment size[END_REF]Blair, 1987a;[START_REF] Blair | Processes and forms of alluvial fans[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF]. Possible explanations for this include the decrease in sediment discharge caused by deposition (transport hypothesis), or the downstream ning of the sediment (threshold hypothesis) [START_REF] Blissenbach | Relation of surface angle distribution to particle size distribution on alluvial fans[END_REF][START_REF] Rice | The nature and controls on downstream ning within sedimentary links[END_REF][START_REF] Stock | Controls on alluvial fan long-proles[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF]. In practice, the variations of grain-size, slope and sediment discharge along a fan are correlated. When the sediment is broadly distributed in size, these variations are smooth, whereas a bimodal distribution generates a segmented fan [START_REF] Bull | Geomorphology of segmented alluvial fans in western Fresno County, California[END_REF][START_REF] Williams | Aspects of alluvial fan shape indicative of formation process: A case study in southwestern california with application to mojave crater fans on mars[END_REF].

Only seldom do eld measurements allow us to separate the various parameters aecting the morphology of a fan, making it dicult to isolate their respective inuence. One way around this problem is to use laboratory experiments, where small alluvial fans can be easily produced under well-controlled conditions [START_REF] Schumm | Experimental uvial geomorphology[END_REF][START_REF] Parker | Progress in the modeling of alluvial fans[END_REF][START_REF] Paola | The unreasonable eectiveness of stratigraphic and geomorphic experiments[END_REF][START_REF] Clarke | Experimental alluvial fans: Advances in understanding of fan dynamics and processes[END_REF].

When water and sediment are injected onto the bottom of a tank, a deposit spontaneously forms around its inlet. The formation of this deposit is remarkably similar to that of natural fans;

in particular a network of migrating and avulsing channels distributes radially the sediment across the fan surface. As the forcing parameters vary, the deposit responds by adjusting its morphology. [START_REF] Muto | Autogenic response of uvial deltas to steady sea-level fall: Implications from ume-tank experiments[END_REF], for example, showed that a base level fall induces upstream channel entrenchment, terrace abandonment, and fan progradation.

The sediment discharge Q s determines the growth rate of an experimental fan. Indeed, mass balance requires that the fan volume increase in proportion to the sediment input. Thus, as a consequence of the symmetry, the radius of the fan increases as (Q s t) 1/3 , where t is the time elapsed since the beginning of the experiment [START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF][START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF].

Avulsions occur more frequently as the sediment discharge increases, showing that the internal dynamics of an experimental fan adjusts to the forcings [START_REF] Bryant | Experimental study of avulsion frequency and rate of deposition[END_REF][START_REF] Ashworth | Relationship between sediment supply and avulsion frequency in braided rivers[END_REF][START_REF] Clarke | An experimental investigation of autogenic behaviour during alluvial fan evolution[END_REF][START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]. This adjustment allows the fan to maintain its conical shape which, at rst order and for a single grain size, is characterized by its slope only.

Even in simplied experiments (constant inputs, single grain size), there is no clear consensus about the mechanism by which a fan selects its own slope. Most investigators observed that a low water discharge, a high sediment discharge, and coarse grains all contribute to a steeper fan (Le [START_REF] Le Hooke | Geometry of alluvial fans: Eect of discharge and sediment size[END_REF][START_REF] Clarke | An experimental investigation of autogenic behaviour during alluvial fan evolution[END_REF]. However, the respective inuence of water and sediment discharges on the slope remains debated. [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF], [START_REF] Van Dijk | Autocyclic behaviour of fan deltas: an analogue experimental study[END_REF] and [START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF] hypothesized that the slope is a function of the dimensionless ratio [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF] propose that all three parameters act independently.

Q s /Q w . In contrast,
In their experiment, a fan composed of uniform sediment grows between two parallel plates that conne it to the vertical plane. They found that the ow maintains the deposit surface near the threshold of motion. As a result, a lower water discharge causes the fan to steepen. The sediment discharge perturbs the fan prole only moderately, by steepening the slope in proportion to its intensity. As the sediment is deposited along the fan, the slope returns to its threshold value as it approaches the toe; the associated curvature in this process being proportional to the sediment input.

50

Chapter 3. Bimodal laboratory fan Accordingly, the downstream curvature of an alluvial fan composed of uniform sediment can be interpreted as a signature of spatial variation in sediment transport. However, one can wonder what happens when the fan is composed of non-uniform sediment? When the grain size is broadly distributed, downstream ning can also aect the fan prole. This phenomenon occurs in ume experiments, where large grains concentrate near the inlet [START_REF] Paola | Downstream ning by selective deposition in a laboratory ume[END_REF][START_REF] Smith | The gravel-sand transition: ume study of channel response to reduced slope[END_REF]. In the experiment of [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF], the fan builds its upper part out of large grains, and deposits the smaller ones near its toe. Consequently, the proximal slope is signicantly steeper than the distal one, a signal whose form is similar to the curvature induced by deposition. We should also expect that this segregation would also appear in the fan's stratigraphy, a process that, to our knowledge, has not been previously investigated in laboratory experiments.

Here, we investigate the impact of a bimodal sediment on the morphology and stratigraphy of an alluvial fan. To do so, we generate a laboratory fan fed with a mixture of two granular materials (Section. 3.2). Our experiment generates a segregated deposit, similar to the laboratory fan of [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]. We rst analyze its morphology, describing the growth of each part of the deposit independently . We then relate the spatial distribution of the sediment to the proximal and distal slopes (Section. 3.3). Based on these observations, and appealing to the threshold-channel theory, we propose a geometrical model to describe the fan deposit (Section. 3.4). Producing experimental alluvial fans has become common in geomorphology [START_REF] Schumm | Experimental uvial geomorphology[END_REF][START_REF] Bryant | Experimental study of avulsion frequency and rate of deposition[END_REF][START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Ashworth | Relationship between sediment supply and avulsion frequency in braided rivers[END_REF][START_REF] Van Dijk | Autocyclic behaviour of fan deltas: an analogue experimental study[END_REF][START_REF] Clarke | An experimental investigation of autogenic behaviour during alluvial fan evolution[END_REF][START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF][START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF][START_REF] Clarke | Experimental alluvial fans: Advances in understanding of fan dynamics and processes[END_REF]. Here we use a setup similar to that of, for example, [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF] to generate a radially symmetric fan over a horizontal basal surface (Fig. 3.1). In our experiments, however, a bimodal sediment mixture allows the fan to form a segregated deposit, visualized by color.

Experimental set-up

The tank we use to produce alluvial fans is 2 m-wide, and more than 5 m-long. Its bottom 3.2. Experimental set-up 51 is covered with a black rubber tarpaulin. At the back of the tank, a 30 cm-high, vertical wall simulates the mountain front against which the fan leans. To prevent ow concentrations, large pebbles (∼ 5 cm) are placed along the base of this back-wall. The three other sides are bounded by trenches to evacuate water (Fig. 3.1). It is noted that, even with these trenches, the surface tension maintains a 0.5 cm-deep sheet of water over the base of the tank. Assuming that this standing water aects only the base of the fan, we nd that it represents less than 1% of its volume. Based on this simple calculation, we hereafter neglect its inuence in our analysis and interpretation.

To ensure constant inputs of water and sediment into the experiment, we use a constant-head tank to supply the water, and an Archimedes screw to supply the grains. The uxes of water and sediment merge in a funnel, which directs them toward the tank. Before reaching the fan, water and sediment ow through a 10 cm-wide, wire-mesh cylinder lled with pebbles. This device reduces the water velocity and homogenizes the mixture (Fig. 3.1). The mixture we used is composed of black coal and white silica grains, the colors of which are easily distinguished. The coal grains are larger and lighter than the silica grains (Table 3.1). To quantify the mobility of these grains, we measure their respective transport laws in independent experiments (Appendix A). We nd that both transport laws, for pure coal and pure silica, exhibit an unambiguous threshold, below which there is no transport (Fig. 3.2). This threshold is about 0.34 N m -2 for coal, and 0.52 N m -2 for silica. Beyond this threshold, the sediment ux appears proportional to the distance to threshold, with a proportionality constant of 2.4 10 -5 m 2 s -1 for coal and 4.8 10 -6 m 2 s -1 for silica. As a result, the same shear stress τ induces a larger ux of coal than silica, at least when the two species are unmixed. In other words, despite their larger size, the coal grains are more mobile than the silica ones. This, of course, is due the rst being lighter than the latter. To formalize this density-induced reversal of mobility, we need to introduce the Shields parameter θ, which is the ratio of the shear stress over the grain's weight [START_REF] Shields | Anwendung der ahnlichkeits mechanik und der turbulenz-forschung auf die geschiebebewegung[END_REF]:

θ = τ (ρ s -ρ)gd s , (3.1)
where ρ is the density of water, ρ s is the density of sediment, g is the acceleration of gravity, and we approximate the grain size d s with its median value d 50 . For our sediments, the denominator in Eq. (3.1) is larger for silica than for coal, indicating that the density dierence prevails over 52 Chapter 3. Bimodal laboratory fan grain size to govern the mobility of our grains. This is in contrast to the experiments of [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF], where the mobility dierence is driven by grain size. When expressing the threshold for transport in terms of the Shields parameter, we nd θ c = 0.19 for coal, and θ c = 0.25 for silica (Table 3.1). These values reinforce the mobility contrast induced by density. 3.1).

When dierent grains are mixed, the shear stress exerted on each species depends on the mixture composition [START_REF] Wilcock | Surface-based transport model for mixed-size sediment[END_REF][START_REF] Houssais | Bedload transport of a bimodal sediment bed[END_REF]. The shear stress required to move the larger grains in a mixture is lower than for large grains alone, because the smaller ones cause them to protrude into the uid. Conversely, small grains in a mixture require a higher shear stress because they are shielded from the ow by neighboring large grains [START_REF] Einstein | The bed-load function for sediment transportation in open channel ows[END_REF]. For grains of dierent densities but uniform size, exposure and hiding are negligible [START_REF] Viparelli | Downstream lightening and upward heavying: Experiments with sediments diering in density[END_REF]. There exist no universal transport law accounting for all these phenomena, and deriving an empirical one for our mixture would be a daunting task.

We thus use the transport laws of Fig. (3.2) to account for dierential transport and estimate the mobility of our grains, although this is certainly a rough approximation. If it holds, at least qualitatively, we expect the rivers that build our experimental fans to segregate the sediment based on grain mobility, by depositing silica while transporting coal further downstream. 2012) also produced about 5 channels, although only one of them was active at a time. Both congurations occur in the eld [START_REF] Weissmann | Glacially driven cycles in accumulation space and sequence stratigraphy of a stream-dominated alluvial fan, san joaquin valley, california, usa[END_REF][START_REF] Hartley | Large distributive uvial systems: characteristics, distribution, and controls on development[END_REF]. In our experiments, bedload appears as the dominant transport mode, although a small amount of ne coal is suspended, and gets deposited on the banks. The width of our channels varies between about 1 and 2 cm. Assuming they share the total water discharge evenly, the typical Reynolds number of their ow is above 500, suggesting that, most of the time, they are turbulent. They avulse regularly to maintain the radial symmetry of the fan.

During an avulsion, overbank ow occurs temporarily, a phenomenon also observed by [START_REF] Bryant | Experimental study of avulsion frequency and rate of deposition[END_REF] and [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]. Our experiment stops when the deposit reaches the sides of the tank, typically after 3 to 4 hours.

As it grows, the fan deposits the silica grains upstream of the coal grains. Accordingly, the apex of the fan is mostly composed of silica, whereas coal constitutes most of its toe. The boundary between the two types of sediment follows the path of channels, thus adopting a convoluted shape. To explore the inuence of the sediment composition on the morphology of the fan, we varied φ, the volumetric proportion of silica in sediment mixture, from 25% to 80%

over ve experiments (Table 3.2).

3.3 Self-similar growth of a segmented fan During each run, we track the evolution of the fan surface with a camera (Nikon D90 with a wide-angle lens Nikon AF DX Fisheye-Nikkor 10.5 mm f/2.8G ED) xed above the center of the tank. We record an image every minute (Fig. 3.3a). The exact location of the boundary between silica and coal varies signicantly during a run. For a run, however, this boundary appears at a constant location relative to the fan length. This fraction depends on the composition of the sediment mixture (Table 3.3). To conrm this observation, we manually locate the fan toe on 26 pictures, 10 minutes apart from each other (Fig. 3.3a). From these individual measurements, we estimate the average radius R c of the fan with an accuracy of about 6% on each picture. We then rescale each picture with the corresponding value of R c , thus normalizing the size of the fan to one. Finally, we average all the normalized pictures of an experimental run (Fig. 3.3b).

By construction, the average picture shows a fan of radius one. It also conrms that the fan is radially symmetric, and reveals a somewhat blurred but localized transition between the silica and coal deposits. This observation suggests that the fan preserves the spatial distribution of coal and silica as it grows. To verify the self-similarity of the fan growth, we analyze the evolution of its geometrical properties. To do so, we manually locate the silica-coal transition and the fan toe (Fig. 3.3a).

We then calculate the average distance R s from the apex to the transition. The boundary of the silica deposit being more convoluted than the toe, the standard deviation of R s is about 19%.

Both distances increase in proportion to the cube root of time (Fig. 3.4). Following [START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF] and [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF], we interpret this observation as a direct consequence of mass balance. Indeed, the total mass M of the deposit increases linearly with time:

M = Q s,m t (3.2)
where Q s,m is the total mass ux of sediment. To express this relation in terms of volumes, we need to measure the packing fraction λ of our sediment mixture. In general, this quantity 3.3. Self-similar growth of a segmented fan 55 depends on the composition of the mixture. To estimate it, we measure the packing fraction of pure silica, of pure coal, and of a 50 % silica-coal mixture (red dots, Fig. 3.5). The three values are similar, with a mean of 55% ± 1%. Accordingly, we approximate the packing fraction of the entire deposit with this value, regardless of the composition of the sediment mixture. This approximation introduces an error of less than 5 %. We now dene the volume discharge of sediment Q s , such that

Q s,m = Q s (1 -λ) (φρ silica + (1 -φ)ρ coal ) (3.
3) and substitute Q s for Q s,m in Eq. (3.2). The mass balance then reads

V = Q s t (3.4)
where V is the total volume of the deposit. In a self-similar fan, any distance scales like the cube root of the fan volume; in particular, both R c and R s increase in proportion to (Q s t) 1/3 . Our experimental fans conform to this scaling, thus supporting the hypothesis of a self-similar growth. A direct consequence of this self-similarity is that the relative location of the transition, dened by the ratio R = R s / R c , remains constant throughout growth (R = 0.62 ± 0.04 for run 2, other runs are presented in Table 3. 3,Figs. 3.3b and 3.4). This self-similarity means that, as it grows, the fan preserves its structure, which can therefore be extrapolated from the nal deposit. A few minutes after the experiment stops, all the surface water has drained away from the fan, leaving the entire deposit emergent. At this point, we scan the deposit's surface with a laser to measure its topography (OptoEngine MRL-FN-671, 1 W, 671 nm). A line generator converts the beam into a laser sheet (60°opening angle, 1 mm thick), the intersection of which with the fan surface is recorded by a camera attached to the laser, about 2 m above the tank bottom (Sick Ranger E50, 12.5 mm lens). The precision of the measurement is better than 1 mm in every direction. Using the digital elevation model (DEM) of our experimental fan, we compute the nal volume of our fans to check the total packing fraction of the deposit (blue dots, Fig. 3.5). Despite some dispersion, we nd that the packing fraction of our deposit is about 54% ± 2%, close to the value estimated independently.

The elevation contours of the DEM are well approximated by concentric circles, another indication of radial symmetry (Fig. 3.6). This property suggests that we can compute the radiallyaveraged prole of the fan with minimal loss of information [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]. To do so, we interpolate the DEM along 34 radii, 5°apart from each other, at the end of each run (Fig. 3.6). For each run, the resulting proles are similar to each other, and dier from the mean by less than 7% (Fig. 3.7a). The average fan prole is steeper near the apex than at the toe and can be approximated by two segments of uniform slope. Natural fans sometimes feature a similarly segmented prole [START_REF] Bull | Geomorphology of segmented alluvial fans in western Fresno County, California[END_REF][START_REF] Blair | Processes and forms of alluvial fans[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF]. When we plot the downstream slope of this average prole as a function of the distance to the apex, the transition appears as a decreasing sigmoid curve (Fig. 3.7b). To evaluate the location of the transition, and the extension of the transition zone, we t a hyperbolic tangent to the slope prole (Fig. 3.7b for run 2, other runs in Table 3.3). For run 2, we nd that the slope plateaus to a value of about 0.29 near the apex, and to about 0.10 near the toe. We dene the location of the transition as the inection point of the sigmoid, which occurs at 55% ± 9% of the total fan length (Fig. 3.7b). The slope thus breaks where the sediment turns to coal, suggesting that these transitions are closely related (Table 3.3, R ≈ 0.62 ± 0.04). The location of the transition depends on the mixture composition (Table 3.3). We now dene the extension of the transition zone as the characteristic length of the sigmoid. For run 2, we nd that the transition between the two segments of the fan occurs over a length of 32% of the total fan length. This value is almost independent of the sediment mixture (about 30% ± 3% on average for all runs). [START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF] found a comparable value (about 22%) for natural and laboratory fans.

To investigate the relation between the slope break and the silica-coal transition, we now turn our attention to the internal structure of the deposit. After the water and sediment supplies have been switched o, the fan remains intact, and we can cut it radially to reveal a vertical cross section (Fig. 3.8). Silica and coal appear segregated, in accordance with the top-view pictures of the fan (Table 3.3), and with the experiments of [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]. Silica concentrates near the apex, in the upper part of the deposit, whereas coal concentrates at the fan toe. The location of the silica-coal transition uctuates, and generate an intricate stratigraphy that combines segregation at the fan scale, and stratication near the transition. The transition zone shows alternating layers of silica and coal, which extend over about one third of the crosssection area. In natural fans, such stratications result from uctuations of the sediment and water discharges, but this mechanism cannot be invoked in our experiments (Paola et al., 1992a;[START_REF] Clevis | Numerical modelling of drainage basin evolution and three-dimensional alluvial fan stratigraphy[END_REF][START_REF] Whittaker | Decoding downstream trends in stratigraphic grain size as a function of tectonic subsidence and sediment supply[END_REF]. Dry granular ows can also generate a similar pattern (Makse et al., 1997a,b). In our case, the succession of channel avulsions is another possible mechanism. Our observations do not allow us test these hypotheses.

Mass balance

Based on our laboratory observations, we propose a rst-order geometrical model of an alluvial fan fed with a bimodal mixture of sediments. We consider a radially symmetric structure, which grows by expanding itself without changing its geometry. A consequence of these assumptions is that the geometry of the fan, at any time, is entirely determined by a xed, two-dimensional template of its cross section (Fig. 3.9). The simplest possible template consists of two triangles with a common side. The proximal triangle denes the geometry of the silica deposit, and the distal one represents the coal deposit. Three dimensionless parameters dene this template: the proximal slope S s , the distal slope S c and the relative location of the transition R = R s /R c .

Map view Cross-section

Silica Coal R s R c R c R s H s H c S s S c S t
Figure 3.9: Representation of an alluvial fan (template). Silica: orange; coal : green.

The geometry of the template sets the proportion of silica and coal in the deposit. As a consequence, mass balance relates the three parameters that dene the fan template to the 3.4. Mass balance 59 composition of the sediment mixture injected in the experiment, φ. Indeed, since the sediment discharge is constant, and assuming the deposit is fully segregated and the packing fraction is constant, we should have

V s V s + V c = φ , (3.5)
where V s is the volume of silica in the deposit, and V c that of coal. For a self-similar fan, this relationship holds at any time. The silica deposit is composed of two half-cones sharing their base. Its volume reads

V s = π 6 R 2 s H s , (3.6)
where H s is the elevation of the fan apex. To calculate the volume of coal in the deposit, we rst evaluate that of a truncated half cone with slope S c , radius R c , and height H c (the elevation of the transition). We then withdraw the volume of the lower cone of the silica deposit. The resulting volume reads

V c = π 6 R 2 c + R c R s H c .
(3.7)

The proximal and distal slopes are simply those of the corresponding right triangles:

S s = H s -H c R s and S c = H c R c -R s .
(3.8)

Using the four above equations, we nally relate the composition of the sediment mixture to the geometry of the fan, as a function of the slope ratio and the transition location:

φ = (1 -S)R 4 -SR 3 -R 2 (R + 1) ((1 -S)R 3 -1) , (3.9)
where we have dened the ratio of proximal slope to distal slope S = S s /S c . Equivalently we may express the composition of the sediment mixture as a function of the slope ratio and the slope of the transition:

φ = 1 -S t 1 + S t ((SS t ) 2 + 3SS t + 2) , (3.10)
where we have dened the ratio of transition slope to proximal slope S t = S t /S s .

If the template is a reasonable representation of the fan geometry, the location and the slope of the transition and the two surface slopes of the deposit should adjust to the composition of the Chapter 3. Bimodal laboratory fan sediment input, according to Eqs. (3.9, 3.10). To evaluate this model, we measure the geometry of the fan at the end of every experimental run (Table 3.3). Using the radially averaged prole we rst t, using a linear regression, the proximal and distal slopes and calculate their ratio. Then, we estimate the location of the transition using the position of the inection point (Section. 3.3). We nd that, for all runs, the proportion of silica in the deposit, as deduced from our measurements through Eqs. (3.9, 3.10), matches the composition of the sediment mixture (Fig. 3.10). At rst order, we can thus represent our experimental fan as a radially symmetric, fully segregated structure which preserves its shape as it grows. These features determine the dynamics of the fan, and the geometry of its deposit. This model, however, involves two free parameters: the proximal and distal slopes. These are selected by the fan itself, by a mechanism that remains to be understood. Each deposit is built by a collection of channels, which select their own slope according to the composition of the bed, and to their sediment and water discharges. On the DEM of our experimental fans, the channels are virtually invisible, showing that their downstream slope is that of the fan (Fig. 3.6). It is thus reasonable to assume that the deposit inherits the slope of the channels that build it. The way a river selects its morphology is still a matter of debate, but it has been recently pointed out that most laboratory rivers, including those owing over an experimental fan, remain near the threshold for sediment transport [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF][START_REF] Seizilles | Width of laminar laboratory rivers[END_REF][START_REF] Reitz | Diusive evolution of experimental braided rivers[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF]. Assuming a channel is exactly at threshold yields a theoretical relationship between its water discharge and its slope [START_REF] Glover | Stable channel proles[END_REF][START_REF] Henderson | Stability of alluvial channels[END_REF]. Could this theory inform us about the slope of our fans?

Returning to our experimental fans, we nd them enmeshed in a collection of channels owing radially (Fig. 3.11). These channels sometimes bifurcate downstream, but do not recombine as they would in a braided river. We would like to compare their slope to the prediction of the pictures per run). We rst divide the surface of each fan into ve concentric bins, where we count the active channels and measure their widths (at least two cross sections per channel and per bin, Fig. 3.11). We then average the number of channels, and their width, over experimental runs.

The resulting quantities depend on the time of their measurement, and on the distance from the apex, r. Further averaging over time yields radius-dependent quantities, whereas averaging over distance yields time-dependent quantities (Fig. 3.12).

When plotted as a function of radius, the width of the channels varies between about 1 and 2.5 cm, with no clear trend (Fig. 3.12a). The variability of the width is much larger in the proximal part of the fan than in its distal part. When plotted as a function of time, we nd that the width is more consistent, with a relative variability of about 10% around a mean value of 1.3 cm (Fig. 3.12b). Overall, the channels appear reasonably homogeneous in size, suggesting that they share the total water discharge evenly.

The number of channels n c varies between 5 and 6 across the fan (Fig. 3.12c). As expected for a radially oriented structure, we count fewer channels near the apex. We also nd fewer channels near the toe, although the poor color contrast of the coal-dominated areas probably bias our count. This variability compares with the disparity we observe between runs. The number of channels is nearly constant over time (Fig. 3.12d). Hereafter, we choose n c = 5.5, and divide the total water discharge accordingly.

We now wish to compare the slope of our experimental fans with the threshold theory, applied to the characteristic channel dened above. This theory assumes that the combination of gravity and ow-induced shear stress maintains the channel bed at the threshold of motion [START_REF] Glover | Stable channel proles[END_REF][START_REF] Henderson | Stability of alluvial channels[END_REF][START_REF] Seizilles | Width of laminar laboratory rivers[END_REF]. As a result, the width, depth and slope of the channel are set by its water discharge. In particular, according to the simplest version of this theory [START_REF] Devauchelle | Longitudinal prole of channels cut by springs[END_REF][START_REF] Gaurav | Morphology of the kosi megafan channels[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF], the equilibrium slope reads

S H = g µ 3 θ c µ ρ s -ρ ρ d s 5 1/4 2 3/2 K(1/2)n c 3 C f Q w , (3.11)
where µ is Coulomb's coecient of friction (Table 3.1), ν = 10 -6 m 2 s -1 is the kinematic viscosity of water, K(1/2) ≈ 1.85 is the elliptic integral of the rst kind, and C f is Chézy's coecient of uid friction. The Chézy coecient C f depends on the bed roughness and the ow Reynolds number. For simplicity, we approximate C f with a constant value of 0.02 [START_REF] Moody | Friction factors for pipe ow[END_REF][START_REF] Chow | Open channel hydraulics[END_REF]. Since we imposed the same water discharge during all experimental runs, and found the number of channels n c to be relatively constant, the slope corresponding to the threshold theory depends on the sediment only. We nd S Hs ≈ 0.042 for silica, and S Hc ≈ 0.023 for coal, using

Eq. (3.11).

Intuitively, we expect that, all things being equal, the fan slope increases with sediment discharge. Previous observations support this intuition, but there is no consensus yet about its physical origin, which involves the response of a single channel to sediment transport and its destabilization into multiple threads [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Ashworth | Relationship between sediment supply and avulsion frequency in braided rivers[END_REF]. We do not nd any correlation between sediment discharge and slope in our experiment (Fig. 3.13).

Even after normalizing our measurements according to the threshold theory, the data points appear segregated according to the sediment species: the mean slope of the silica deposit is about S s /S Hs = 5.6 ± 2.0, whereas we nd S c /S Hc = 2.9 ± 1.5 for coal (S s ≈ 0.23 and S c ≈ 0.068).

The surface slopes of the two fan segments are thus signicantly higher than predicted by the threshold theory.

A possible cause for this departure from the threshold channel could be the bimodal mixture we use. To assess this hypothesis, we have produced an experimental fan with pure silica ( Q s ≈ 0.014 L min -1 , Q w ≈ 2.6 L min -1 ). We found that, like its bimodal counterparts, its slope was approximately ve times higher than predicted by the threshold-channel theory (S s = 0.2).

Another possible explanation is the inltration of surface water into the deposit. Indeed, based on Eq. (3.11), a lower water discharge induces a steeper channel. Measuring this leakage would be experimentally challenging. Finally, the breakdown of the threshold-channel theory could result from sediment transport, since active channels must be above threshold [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF]. In their one-dimensional experiment, [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF] have shown that the higher the sediment input in their experiment, the more slope departs from its threshold value. Again, we cannot evaluate quantitatively this hypothesis in our experiments.

The proximal and distal slopes seem independent from sediment discharge (Fig. 3.13). For lack of a physical interpretation, we now treat this observation as an empirical fact, and attribute a xed value to the ratio of proximal slope to distal slope: S = S s /S c = 3.4 ± 1.0. Substituting this value in Eqs. (3.9, 3.10), the mass balance relates, without any additional parameters, the composition of the sediment mixture to the location and the slope of the transition (Fig. 3

.14).

Despite signicant uncertainties, which probably reect the rudimentary mass balance we used, our observations agree with this semi-empirical relationship.

In principle, one could use Fig.

Conclusion

Using a laboratory experiment, we generated alluvial fans fed with a bimodal sediment. Five or six active channels deposit their sediment load to form a radially-symmetric fan. The heavier sediment (silica) concentrates around the apex, whereas the lighter one (coal) get deposited near the toe. The location of silica-coal transition uctuates over about 30% of the total fan length.

A radial cross section of the deposit reveals a similar segregation: two superimposed triangles make up the stratigraphy of the fan. The lowest triangle is mostly coal, whereas the upper one, located near the apex, is mostly silica. The transition between the two parts of the fan uctuates to produce strata, which extend over 30% of the total fan length. As a rst approximation, we may represent this transition with a straight line, and treat the fan structure as two imbricated deposits. Combining this geometric model with mass balance, we nd that the fan preserves this structure as it grows, with a precision of about 15%. This observation suggests that our laboratory fans act essentially as sieves, which segregates the sediment they are fed with. This process controls the geometry of the resulting deposit. As a consequence, we can use the nal geometry of our laboratory fans to infer the composition of the sediment input. In practice, a topview picture of the deposit suces to do so. Alternatively, measuring the slope of the transition in the stratigraphy, even if the latter is incomplete, also suces.

Natural fans often exhibits a sharp transition from gravel to sand [START_REF] Blair | Processes and forms of alluvial fans[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF]. Like in our experiments, this front divides the fan prole into two segments. The proximal segment, composed mainly of gravel, is steeper than the distal one, composed mainly of sand. [START_REF] Bull | Geomorphology of segmented alluvial fans in western Fresno County, California[END_REF] and Blair (1987a) found natural fans featuring three segments bounded by two successive transitions. Again, the size of the deposited sediment changes abruptly at each front. These observations suggest that the segregation mechanism at work in 3.5. Conclusion 65 our experiment can repeat itself to generate nested deposits. A natural extension of our work would be to enrich the sediment mixture with additional grain sizes (or densities) to produce fans with multiple segments. We would expect these fan to sort sediment species based on their mobility, and reduce their slope downstream, as observed on the surface of many natural fans [START_REF] Stock | Controls on alluvial fan long-proles[END_REF]. In other words, the structure of an alluvial fan should reect the composition of its sediment input. For instance, in principle, one could infer the grain-size distribution of the sediment input from a DEM of the fan.

In practice, however, secondary processes such as weathering, runo, and aeolian erosion reworks the surface of most natural fans, thus hampering our ability to infer their history from their present state [START_REF] De Haas | Debris-ow dominance of alluvial fans masked by runo reworking and weathering[END_REF]. To circumvent this issue, one can either reconstruct geometrically the paleosurface of the fan, or use its stratigraphy. Indeed, even partial access to the internal structure of the fan could reveal the slopes of the transitions in the stratigraphy, and thus the grain-size distribution of the input.

In our experiments, the inputs of water and sediment were constant. In general, this is not true for natural fans, and the interpretation we propose here does not apply in its present, oversimplied, form. The self-similar model we propose here thus cannot account for climatic and tectonic signals. However, the fundamental hypothesis upon which it relies, namely that the fan sorts the sediments based on their mobility and adjusts its own slope accordingly, might still hold when the inputs uctuate. If so, our geometrical model might be extended to account for these uctuations. This is the subject of present work.

Our experiments also suggest that the process by which an alluvial fan distributes grain sizes in its deposit, although a primary control on its structure, may not be the most puzzling component of its machinery. The way it selects its slope remains a challenging problem, which we have circumvented here by tting a parameter to our observations (Le [START_REF] Le Hooke | Geometry of alluvial fans: Eect of discharge and sediment size[END_REF][START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Stock | Controls on alluvial fan long-proles[END_REF][START_REF] Van Dijk | Autocyclic behaviour of fan deltas: an analogue experimental study[END_REF][START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF][START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF]. Indeed, the threshold theory can only provide us with a rst-order estimate for the slope of a channel. We need to understand how a channel adjusts its slope to its sediment load.

Recent investigations have shown that, provided the sediment discharge is low enough, one can produce stable active channels in laboratory experiments [START_REF] Seizilles | Width of laminar laboratory rivers[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF]. If this method works for a laboratory fan as well, it might generate a single-channel fan.

This would be a simpler experimental tool to investigate the relationship between the slope of a fan and the intensity of its sediment input.

68

Chapter 4. Departure from threshold: nite sediment discharge

Inuence of sediment transport on alluvial river

In chapter 2, we demonstrate that when a river does not transport sediment, its shape is maintained by a force balance between gravity and the ow-induced shear stress. When the river transports sediment, grain are advected on average along the ow, but a transverse sediment ux appears. Because of the bank slope, gravity pulls the moving grains away from the banks toward the center of the channel (Parker, 1978b, Fig. 4.1). Consequently, the river widens. For a river to reach and maintain an equilibrium shape, sediments must then be transported from the center of the channel to the banks. Looking at the motion of grains in a laboratory channel, [START_REF] Seizilles | Cross-stream diusion in bedload transport[END_REF] demonstrate that this motion follows a random walk in the transverse direction (Fig. 4.1). Sediments diuse from the center of the bed to the banks. At equilibrium, the inward gravity ux balances the outward diusive ux.

The gravity ux, q g , is proportional to the surface concentration of moving grains, n, and to the bank slope, such as,

q g = αnV s ∂D ∂y , (4.1)
where α is a dimensionless coecient, V s the grains velocity, D the channel depth and y the transverse direction. The diusive ux, q d , can be measured experimentally [START_REF] Seizilles De Mazancourt | Forme d'équilibre d'une rivière[END_REF][START_REF] Seizilles | Cross-stream diusion in bedload transport[END_REF]. It reads

q d = -l d V s ∂n ∂y , (4.2)
where l d is a diusive length, which depends on the sediment shape and uid characteristic.

Depth

Moving grains concentration q g q d q g q d Gravity flux

Diffusive flux 2). In this phase portrait, each value of ñ0 , that denes a solution of Eq. (4.6), is represented by a semi-circular trajectory (Abramian et al.,prep,Fig. 4.2a). These solutions are comprised in a narrow region bounded by two asymptotic solutions (red line and green dot in Fig. 4.2a).

The rst asymptotic solution corresponds to a river at the threshold of sediment motion D = µ t everywhere across the bed (Glover and Florey, 1951, Section. 2.1). In this case, the river cross-section has a cosine shape (Eq. 2.5). The second one is a xed point which corresponds to an innitely wide river, D = µ t + (Seizilles de Mazancourt, 2013; Abramian et al., prep). In the phase space, the distance between these two asymptotic solutions is equal to , thus proportional to the river slope. Consequently, when the slope becomes small, solutions of Eq. (4.6) tend toward the threshold of sediment motion. In this limit, the river cross section can be decomposed in two components. The banks are represented by two half cosines (Eq. 2.5). They are at the threshold of sediment motion and do not contribute to sediment transport. The central part of the channel is at and slightly above the threshold of sediment motion. Sediment transport occurs only in this at section.

In the limit of this assumption, and using the sediment and uid mass balances we dene a rst-order approximation for the slope of a river that transports sediment (Abramian et al., prep),

S S h = 1 + Q s Q * s 1/3 , (4.7)
where Q * s is a characteristic sediment ux: Q * s = 4q 0 l d /3α, with q 0 the characteristic sediment discharge per unit of river width measured with the transport law q s = q 0 (µ -µ t ) (Appendix A).

We now suggest to compare this theory (Eq. 4.7) to the slope of an experimental fan when the sediment discharge increases. ñ0 . The red lines represent the rst asymptotic solution, where D = µ t , the river cross-section is a cosine. The green dot is the xed point, it correspond to the second asymptotic solution, where D = µ t + , the corresponding river is at. The blue and orange lines represent two solutions for dierent ñ0 .

Fan prole

To quantify the inuence of the sediment discharge on a fan morphology, we use the same experimental set-up as the one described in chapter 2. The sediment and uid characteristics are also the same. We perform 14 experiments with constant uid Q w and sediment Q s discharges.

Table . 4.1 lists the values of Q w and Q s used in each experiment.

We use the moiré technique previously described (Chapter 2 and Appendix C.2) to measure the slope of our experimental fan. To compare the slope for dierent sediment discharge, we normalize the fan elevation, h, and the distance from the apex, r, by the radius, R, of the fan.

Qualitatively, we observe that, for a constant uid discharge, the fan slope increases with the sediment discharge (Fig. 4.3). When the sediment discharge becomes small, the fan slope reaches a minimum value. These observations qualitatively agree with the theory (Section. 4.1). The river that builds the alluvial fan deposits its load along the fan. As a consequence, the sediment 4.2. Fan prole 71 Table 4.1: Experimental parameters for the 14 runs. The error on the sediment discharge corresponds to the sediment deposited in the inlet channel, which is geometrically estimated.

Fluid discharge

Sediment discharge Sediment discharge Fluid discharge Sediment discharge

Q w (L min -1 ) Q s (g min -1 ) Q s (g min -1 ) Q w (L min -1 )
Q s (g min -1 ) 0.13 ± 0.0084 1.68 ± 0.048 0.82 ± 0.024 0.49 ± 0.0056 3.08 ± 0.12 0.73 ± 0.02 4.9 ± 0.15 0.33 ± 0.02 0.68 ± 0.013 4.83 ± 0.14 8.1 ± 0.28 0.74 ± 0.016 6.18 ± 0.26 0.77 ± 0.019 13 ± 0.36 1.22 ± 0.028 discharge decreases downstream. Following the decrease of sediment discharge, the slope of the fan should diminish near the fan toe [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF]. The proles of Fig. (4.3) do not exhibit a clear concavity, even though they merge near the fan toe. In our experiments, the sedimentary load is entirely deposited into the fan, since the sediment discharge vanishes at the toe. Consequently, the slope near the fan toe should be the equilibrium slope S h (Chapter 2).

To test this assumption, we extract the fan slope near its toe. To do so, we extract the slope where the dimensionless radius of the fan, r/R, lies between 0.9 and 1. We then compare these distal slopes to the threshold slopes reach for the corresponding water discharge (Fig. 4.4a).

Accordingly to the results of chapter 2, we nd that the slope at the fan toe is approximately equals to twice the threshold slope. For the four experiments involving the higher sediment discharge, however, we measure deeper slopes than the one predicted. A possible explanation of this dierence is the short duration of the experiments. In fact, we observe in chapter 2 that the time required to reach the threshold slope is approximately 10 hours. When we impose a high sediment discharge the fan reaches the tank boundary before 10 hours. Consequently, we stop the experiment.

Due to sediment deposition along the fan, we only know the sediment discharge near the fan apex. Accordingly, to quantify the inuence of sediment discharge on the fan slope, we consider the proximal slope of our fans. However, the presence of the channel at the inlet of the fan disturbs the prole, which can be reversed or at (Fig. 4.3). To have the best measurement of the proximal slope of our fan, we extract it where the dimensionless radius, r/R, lies between 0.1 and 0.4. Our two sets of experiments involve dierent water discharge (Table . 4.1). Consequently, their equilibrium slopes, S h , dier (Chapter 2). To investigate the departure from this threshold slope, we represent the proximal slope of our fans normalized by the equilibrium slope, S/S h , as a function of the sediment discharge injected into the experiment (Fig. 4.4b). Figure (4.4b) shows that the data from the two sets of experiments collapse on the same trend. This means that the inuence of the uid discharge is totally captured by the threshold slope S h . The uid discharge therefore only controls the threshold slope. Contrary to previous study, here we separate the inuence of uid and sediment discharges [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Van Dijk | Autocyclic behaviour of fan deltas: an analogue experimental study[END_REF][START_REF] Clarke | An experimental investigation of autogenic behaviour during alluvial fan evolution[END_REF][START_REF] Powell | Varying discharge controls on timescales of autogenic storage and release processes in uvio-deltaic environments: Tank experiments[END_REF]. We then t Eq. (4.7) on the data to estimate the threshold slope, S h , and the characteristic sediment discharge Q * s . We nd that the characteristic sediment discharge is about 0.7 g min -1 and the threshold slope is 1.9 times the theoretical one. The value of the characteristic sediment discharge depends on the sediment characteristic (d s , ρ s ) and the diusive length l d . In agreement with the observation of chapter 2, the threshold slope is two times higher than the theoretical one, hence S/S h = 1.9, when Q s = 0.

Despite this underestimation of about 2, the slope of the fan departs from threshold following the theoretical prediction of Eq. (4.7).

The sediment deposition along the fan generates a concave up prole. Consequently, the volume of our experimental fan cannot be approximated by a semi-cone anymore. The volume of the fan is the volume of sediment injected into the experiment,

V = Q s0 t 1 -λ (4.8)
To quantify the departure from threshold, we dene a volume ratio

V = V V h = 6Q s0 t (1 -λ)πS h R 3 . (4.9)
This volume ratio correspond to the fan volume divided by the volume of a threshold fan, V h built with the same uid discharge and composed of the same sediment. In equation (4.9), we note that the volume ratio depends on the aperture angle of the fan (here, this angle is equal to π).

With these experiments, we have quantied the inuence of sediment discharge on the slope (Eq. 4.7). Due to sediment deposition along the fan, the slope of the fan shallows along its radius following Eq. (4.7). Using this equation, coupled with mass balance, we can calculate the theoretical fan prole and volume numerically. The next section details this reasoning.

Mass balance

The fan grows from sediment deposition. This mechanism is described by the sediment mass balance:

(1 -λ)πr ∂h ∂t

+ ∂ ∂r (Q s ) = 0, (4.10)
where h is the fan elevation. The relation between the slope of a river and its sediment discharge (Eq. 4.7) allows us to dene the sediment transport equation along the fan

Q s Q * s = S S h 3 -1. (4.11)
To solve Eqs. (4.10) and (4.11), we need to dene three boundary conditions:

At the inlet, the sediment discharge is controlled by the experimental input, Q s (0, t) = Q s0 .

We assume that all sediments are trapped into the fan, consequently, at the fan toe the sediment discharge vanishes, Q s (R, t) = 0.

Finally, by denition, the fan elevation vanishes at its toe, h(R, t) = 0.

Because the fan grows with time, its radius evolves so we can not dene R a priori. The domain over which we must solve the mass balance equation has a free boundaries. One way around this problem is to rescale variables. Assuming the fan grows self-similarly and using the denition of the volume ratio (Eq. 4.9), we dene a similarity variable

R = 6Q s0 (1 -λ)πS h V 1/3 t 1/3 . (4.12)
Chapter 4. Departure from threshold: nite sediment discharge Accordingly, we dene a dimensionless radius (4.13) which varies between 0 and 1. We also dene a dimensionless elevation

X = r R ,
H(X) = h(r, t) S h R , (4.14)
and a dimensionless sediment discharge where

χ(X) = Q s (r, t) Q * s . ( 4
P = Q s0 Q * s V
, and

H = -(χ + 1) 1/3 .
(4.17)

These dierential equation describe the radial evolution of the dimensionless sediment ux χ (Eq. 4.16), and the dimensionless fan elevation H (Eq. 4.17 All the parameters that inuence the fan prole are included into one parameter P. Indeed, the inuence of the uid discharge, the fan aperture angle, and the grain size and density are 4.3. Mass balance 75 included into the volume ratio V. The grain size and density also control the characteristic sediment discharge Q * s and nally, P is proportional to the sediment discharge injected at the fan apex Q s0 .

We numerically solve Eqs. (4.16) and (4.17), using the python libraries: scipy.integrate.ode. This libraries allows to solve an equation system where we impose the boundary conditions and the integrating step. By denition, we only know the boundary conditions at the fan toe, χ(1) = 0 and H(1) = 0. Consequently, we integrate the two ODE from X = 1 to X = 0 using a negative step. The integration gives the spatial evolution of χ and H. To study the inuence of the P parameter on the fan prole, we repeat the integration for four specic values of P, .

These theoretical fan proles are represented in Fig. (4.5). Similar to our experimental proles, we nd that when the sediment discharge increases, the slope increases. On these theoretical proles, particularly the one of the highest value of P (green prole in Fig. 4.5), we can observe the concavity caused by sediment deposition. We also notice that, at the fan toe, the proles merge and the local slope reaches the threshold slope S h . Equations (4.16) and (4.17) cannot be solved analytically. We can, however, investigate two asymptotic regimes: when the sediment discharge vanishes and when the sediment discharge tends toward innity. Equation (4.17) is non-linear. To solve it, we need to linearize it. At very low sediment discharge we propose that χ = ξχ, Now, it is possible to solve Eqs. (4.16) and (4.19) using the boundary conditions. We nally obtain that when the sediment discharge vanishes:

P = Q s0 Q * s → V = 1. (4.20)
As expected, when the sediment discharge vanishes, the volume of the fan is the volume of a threshold fan.

When the sediment discharge tends toward innity, we cannot linearize Eq. (4.17). To nd the asymptotic solution, we redene χ and H such as:

     χ = P α χ H = P β H, (4.21)
where the exponents α and β are to be determined. Injecting these news variables into Eqs. (4.16) and (4.17), we nd

     P α χ = 2 X P β+1 (X H -H), P β H = -( χ P α + 1) 1/3 . (4.22)
When the sediment discharge injected as the inlet is high, χ P α 1, we can now determine the value of the two exponents α and β,

       α = β + 1, β = α 3 . (4.23)
We obtain α = 3/2 and β = 1/2. Now, Eqs. (4.16) and (4.17) read

         P 1/2 χ = 2 X P 3/2 (X H -H), H = -χ + 1 P 3/2 1/3 , (4.24)
with the boundary conditions χ(1) = 0, H(1) = 0, and P 3/2 χ(0

) = Q s0 Q * s .
The last of these boundary conditions implies that

V = χ0 3/2 Q s0 Q * s 1/3 . (4.25)
This relation determines the evolution of the volume ratio, V, when the sediment discharge is high. The evolution of the volume ratio, V, is proportional to the dimensionless sediment discharge to the power one third. The prefactor of this power law, however, is unknown. The only way to estimate its value is to solve Eq. (4.24) numerically. Doing this we obtain χ0 ≈ 0.47.

Conclusion 77

Quantitatively, these two asymptotic solutions show that, when the sediment discharge is ten times lower than the characteristic sediment discharge, its inuence on the fan slope is negligible.

In contrast, when the sediment discharge is 100 times higher than the characteristic one, we can approximate the fan slope by the slope of its apex, despite the decrease of slope along the fan.

We now propose to compare our experimental measurements to the theoretical departure from threshold. To do so, we measure the experimental volume ratio of our fan. The volume of the fan, V , is measured using the DEM. We then divide these values by the volume of the corresponding threshold fan, V h (Eq. 2.11). In this last equation, we assume that the slope is the threshold slope, S h , reached for a given uid discharge (0.33 or 0.73 L min -1 ). We calculate these slopes using (Eq. 2.8). In accordance with chapter 2 and Section. 4.2, we multiply the threshold slope by the prefactor estimated experimentally (Fig. 4.4). To estimate the volume of the threshold fan, we also need the radius of the fan, R, at the end of the experiment. We measure this radius using top-view pictures (Appendix C.1). Finally, we represent the experimental volume ratio as a function of the dimensionless sediment discharge for each experimental fans (Fig. 4.6). We use the characteristic sediment discharge estimated by the t (Fig. 4.4). Figure (4.6) suggest that if we know the water discharge and sediment characteristics of a fan, we can estimate the sediment discharge based on its volume or its slope.

Conclusion

Using a simplied experimental set-up and a rst-order theoretical analysis developed for alluvial rivers (Abramian et al., prep), we assess the inuence of sediment transport on an alluvial fan prole. Not surprisingly, the fan slope increases with the sediment discharge. Based on 14 experiments involving dierent uid and sediment discharges, we are able to determine the evolution of the sediment discharge along the fan radius (Eq. 4.11). This equation coupled with mass balance allows to determine the fan prole. The slope of the theoretical fan proles is steeper near the apex than near the toe. At the toe, because we assume all the sediment is deposited onto the fan, the slope reaches the threshold slope, S h . These results are in agreement with the results of [START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF] for a one-dimensional fan.

In principle, if a natural fan presents a concave-up prole, the measurement of its slope near the toe allows to determine the uid discharge that built it (Eq. 2.8). If we know the characteristic sediment discharge, the ratio between the slope near the apex and the slope at the toe then gives a direct access to the sediment discharge (Eq. 4.7).

The asymptotic solution demonstrates that, when the sediment discharge, Q s , is ten times lower than the characteristic sediment discharge, Q * s , its inuence on the fan slope is negligible. It is then a reasonable assumption to consider that the fan is at the threshold of sediment motion. In this case, if we know the characteristic sediment discharge Q * s , we can infer the water discharge that build the fan by measuring the fan slope. It is also possible to estimate the maximum sediment ux that contributes to the fan building. In fact, the sediment ux should be lower than ten times the characteristic sediment discharge.

In this model, we can use the nal shape of a fan, thank to its self-similar growth, to determine the uxes that built it. Strictly speaking, this works only if the input parameters stay constant during the fan history. When forcing vary during the growth, the slope of the fan varies

Variation of the water discharge

In the previous chapters, we have shown that the slope of the fan depends strongly on the water discharge. When the water discharge increases, the slope of the fan decreases. A remarkable consequence of an increase of the water discharge is incision near the apex of the fan [START_REF] Weissmann | Factors controlling sequence development on quaternary uvial fans, san joaquin basin, california, usa[END_REF][START_REF] Guerit | Reconstruction of the sediment trans-port conditions in the urumqi alluvial fan (northeast tian shan, china)[END_REF][START_REF] Cesta | Timing and nature of alluvial fan development along the chajnantor plateau, northern chile[END_REF]. We propose to use mass balance to determine the conditions necessary to cause an incision.

Incision criterion

We consider a semi-conical fan at the threshold of sediment motion. Following the notations dened in Fig. ( 5.1), we dene its volume,

V = πH a R 2 f 6 , (5.1)
and its slope,

S H = H a R f (5.2)
Figure 5.1: A fan at the threshold of sediment motion.

Using Eqs. (5.1) and ( 5.2), we nd that the elevation of the fan, H a , reads

H a = 6S 2 H V π 1/3 , (5.3)
where V is the volume of the deposit. Because the fan is at the threshold of sediment motion, the sediment discharge impacts the growth rate of the fan only, whereas the water discharge controls its slope. To be closer to natural system, we chose to use the expression of the threshold slope determined for turbulent river,

S H = g µ 3 θ c µ ρ s -ρ ρ d s 5 1/4 2 3/2 K(1/2) 3 C f Q w .
(5.4)

Where K(1/2) ≈ 1.85 is the elliptic integral of the rst kind, and C f is Chézy's coecient of uid 5.1. Variation of the water discharge 81 friction. The Chézy coecient C f depends on the bed roughness and the ow Reynolds number [START_REF] Devauchelle | Longitudinal prole of channels cut by springs[END_REF][START_REF] Gaurav | Morphology of the kosi megafan channels[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF].

An incision occurs when the fan apex elevation, H a , decreases. To investigate the variation of the water discharge necessary to generate incision, we derive Eq. ( 5.3) to obtain the time evolution of the fan apex elevation Ḣa , Ḣa

H a = 1 3 2 Ṡ S + V V .
(5.5)

Incision occurs when Ḣa < 0, consequently the condition to generate incision reads,

2 Ṡ S + V V < 0 (5.6)
To express this condition as a function of the water discharge, we derive Eq. ( 5.4). In this equation, we assume that all the parameters, except the water discharge, are constant with time.

We obtain

Ṡ S = - 1 2 Qw Q w .
(5.7)

Finally, using mass balance V = Q s , we infer the condition necessary to generate incision:

Qw Q w V Q s > 1.
(5.8) Inequality (5.8) is a necessary condition for incision. Before testing numerically the validity of this criterion, we discuss its meaning.

The rst remark concerning this criterion is that, for fans fed with the same water, Q w , and sediment, Q s , discharges, the bigger the fan is, the lower the variation of the water discharge has to be to generate incision. This observation can be expressed in terms of time. The volume of the fan corresponds to the integral of the sediment discharge over the period of the fan activity,

P , V = P 0 Q s dt (5.9)
When the sediment discharge is constant, the volume of the fan equals the sediment discharge times the period of activity, V = Q s P . In such conditions, the incision criterion read,

Qw

Q w P > 1.

(5.10)

Consequently, if two fans are subjected to the same variation of water discharge, the older one (i.e. the biggest) may be incised whereas the younger (i.e. the smallest) is not.

Finally, if we impose an abrupt increase of the water discharge characterized by after a few cycle, as the volume of the fan increases.

To illustrate this incision criterion, we now vary the water discharge during the fan construction and observe the resulting numerical fan stratigraphy. All equations are made dimensionless and we now use arbitrary units to represent the results of the model. In our model, we assume that the fan is axisymmetric. When the river begins to incise its fan, the axisymmetry is not maintained anymore and mass balance, as we have written it, breaks down. Thus, we stop the simulation when the incision starts.

Numerical simulation

We present two examples, in both cases we keep the sediment discharge constant and we impose an increase of the water discharge from 1 to 3 (Fig. 5.3). The only dierence between these two example is the time, T , over which the variation occurs. When T is smaller (i.e. when the water discharge varies faster), we observe an incision (Fig. 5.3b).

For each value of the water discharge, we calculate the fan apex elevation H a , using Eq. ( 5.3), and its radius R f , using Eq. (5.2), and we represent the corresponding fan prole (Fig. 5.4).

When the apex elevation becomes lower than the previous one, the river has incised the fan apex. Consequently we stop the simulation (Fig. 5.4b).

To test our incision criterion, we calculate its value at each time during the simulation (Eq. 5.8). We then represent its evolution (Fig. 5.5). For the example without incision this criterion does not exceed the value of one. This is in accordance with the stratigraphy of the fan where layers do not overlap (Fig. 5.4a).

In gure (5.4), we observe that the layers making up the fan stratigrahy get closer to each other, near the fan apex, when the water discharge increases. This is a signature of a slope decrease. To study the evolution of the slope when the water discharge decreases, we represent its evolution for both cases (Fig. 5.6).

In the rst case, the slope of the fan decreases without generating an incision (Fig. 5.6a).

At the end of the rst simulation, the slope of the fan, is lower than the one at which incision occurs in the second simulation. When incision occurs, the slope of the fan is 0.078, this slope is reaches at T = 37. In the case without incision the slope of 0.078 is reached earlier, at T = 35. Consequently, the fan is smaller and the change of the slope is not enough to generate incision. This observation conrms that a decrease of slope does not always generate incision but can be contemporaneous of an aggradation. The more important criterion to generate incision is the balance between the rate at which the slope decreases and the volume of the fan (i.e. its age).

We now propose to numerically investigate the inuence of the variation of the sediment composition, following the result of chapter 3.

Variation of the sediment-mixture composition

Variations of the sediment-mixture composition can result from changes in the climatic conditions or in the lithology of the drainage basin upstream of the fan. (Paola et al., 1992a;[START_REF] Marr | A two-diusion model of uvial stratigraphy in closed depositional basins[END_REF]. To investigate the inuence of such changes, we consider the growth of a fan composed of bimodal sediment. We assume that the two types of sediment have dierent sizes, and therefore dierent mobilities. Consequently, the deposit is segregated on the fan morphology and stratigraphy [START_REF] Paola | Downstream ning by selective deposition in a laboratory ume[END_REF][START_REF] Smith | The gravel-sand transition: ume study of channel response to reduced slope[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF][START_REF] Delorme | Self-similar growth of a bimodal laboratory fan[END_REF]. We propose to describe the stratigraphy of a fan subjected to variation of the sediment mixture composition. 5.2. Variation of the sediment-mixture composition 85

Mass balance

We consider a fan composed of gravel (in orange) and sand (in green). The gravel deposit lies above the sandy one (Fig. 5.7). Two deposits thus form the fan, one composed of gravels and one composed of sands. Both deposits are characterized by a volume (V g and V s ), a slope (S g and S s ), an apex elevation (H a and H t ), and a radius (R t and R f ). To introduce our analysis, we rst consider the evolution of a one dimensional fan. Then we will extend our results to a three dimensional radially symmetric fan. In both case, we consider that the slope of both deposits is constant and it is dened by the threshold of sediment motion. We do not know a priori, the location of the transition between gravel and sand, dened by the function h t . The volume of the gravel deposit, however, reads (5.11) where h(r) = H a -S g r.

V g = Rt 0 (h -h t ) dr,
(5.12)

We can also dene the gravel slope,

S g = (H a -H t ) R t .
(5.13) Finally, mass balance reads,

dV g dt = Q g , (5.14)
where Q g is the gravel ux. The derivative of Eq. (5.11) combined with Eqs. (5.12), (5.13) and

(5.14), relates the amount of sediment injected into the fan to its growth,

S g Q g = Ḣa (H a -H t ).
(5.15)

Chapter 5. Varying forcing Similarly, we dene the volume of the sand deposited in the fan,

V s = Rt 0 h t dr + R f Rt h dr, (5.16)
where

h(r) = H t -S s (r -R t ).
(5.17)

We also dene the slope of its surface,

S s = H t (R f -R t )
.

(5.18)

Using mass balance, its growth reads:

dV s dt = Q s , (5.19)
where Q s is the sand discharge. Using Eqs. (5.16), (5.17), (5.18), and (5.19), we nally obtain the relation between the volume of sand injected in the experiment and the growth of the sand deposit,

S s Q s = Ḣt H t + S s S g ( Ḣa -Ḣt )H t .
(5.20)

We can now dene the evolution of the fan apex elevation and the evolution of the gravel-sand transition elevation,

           Ḣa = S g Q g (H a -H t ) Ḣt = S S s Q s -Ḣa H t H t (S -1) , (5.21)
where S = S g /S s is the ratio of the proximal over the distal slope. We rst solve these equations is steady state.

Steady state

To solve the system of equation ( 5.21), we need to dene two initial conditions, that is the initial geometry of the fan. The only parameters that matter for the fan growth are H a and H t . We choose their initial values arbitrarily, respecting one condition: the apex of the fan must be higher than the transition (H a > H t ).

We nd that our numerical fan, built with constant forcing, resembles the one of our bimodal laboratory fan (Figs. 3.8 and 5.8a). The two deposits are segregated and the transition line is approximately straight. However, at the beginning of the growth, the slope of the transition diers from the nal one. This is caused by the initial conditions we impose. The evolution of fan apex and the location of the transition, show that after a some time, the stratigraphy reaches a self-similar regime. Regardless the initial conditions, when forcing are constant, the growth of the fan becomes self-similar.

This observation suggests a self-similar solution. To nd it, we postulate that H t is propor-

tional to H a , H t = αH a , (5.22)
where α is a dimensionless coecient of proportionality. Using this denition, we can rewrite Eqs. (5.15) and (5.20),

   S g Q g = Ḣa H a (1 -α) S s Q s = Ḣa H a α 2 + 1 S (α -α 2
) .

(5.23) Using Eqs. (5.23), we can calculate α:

α = -1 + 1 + 4 (S -1) (φ -φ 2 ) 2 φ (S -1) , (5.24)
where,

φ = Q g Q g + Q s , (5.25)
is the proportion of gravel in the sediment mixture. The self-similar solution can be analytically calculated, we nd that the coecient of proportionality α depends on the composition of the sediment mixture, φ, and the ratio of slopes S.

To show that, regardless of initial conditions, the fan always evolves into a self-similar geometry, we represent the phase portrait of the fan dynamics, where solutions of Eqs. (5.21) are represented as lines for dierent initial values of H a and H t . Eqs. (5.21) have no physical solution when the initial value of H t is larger than H a (Fig. 5.9). We nd that all trajectories in the phase space tend toward the line corresponding to self-similar growth, which slope is α. For a natural fans, this result suggests that regardless of its history, when forcing are constant, the fan growth should become self-similar. We can now turn to the more interesting case, when the climatic, tectonic or lithological forcing vary, and ask whether these variations can be recorded in the fan's stratigraphy?

Inuence of the forcing on the stratigraphy

To investigate this question, we extend this model to a radially symmetric fan, closer to natural system. In this geometry, the derivation of the evolution equation is slightly more dicult than for the one-dimensional fan. They are, however, very similar to their one-dimensional counterpart.

We therefore present their nal form only,

                                 Q g = π 2 Ḣa R 2 t Q s = π 2 Ḣt (R 2 f -R 2 t ) + Ṙt H t R f S g = H a -H t R t S s = H t R f -R t .
(5.26) 5.2. Variation of the sediment-mixture composition 89 Similarly, we can link the evolution of the fan apex elevation and the evolution of the gravel front to the input parameters,

             Ḣa = 2 S 2 g Q g π (H a -H t ) 2 Ḣt = S 2 S 2 s Q s /π -Ḣa H 2 t (S -1) + H a H t H 2 t (S 2 -3 S + 1) + H a H t (2 S -1)
.

(5.27)

In chapter 3, we showed that the location of the gravel-sand transition depends on the composition of the sediment mixture, φ. We therefore investigate the inuence of a variation of this forcing on the fan stratigraphy.

To do this, we build a fan with a constant water discharge Q w , and we vary the sedimentmixture composition φ. The slope of the gravel and sand deposits remains constant. We assume that the avulsion frequency is high enough to allow the variation to be record in the overall deposit. Our rst numerical experiment consists in varying the sediment mixture from an equal proportion of sand and gravel to a mixture richer in sand. Hence φ varies from 0.5 to 0.05. Figure (5.10) represents the fan cross section and the evolution of the mixture during the fan's growth. At the beginning of the experiment, as well as at its end, we keep the mixture composition constant.

During the transient, we decrease linearly the proportion of gravel. The resulting stratigraphy shows an aggradation and progradation of the gravel front following by an aggradtion and retrogradation of this front.

To the contrary, when the gravel proportion in the mixture increases, φ varies from 0.5 to 0.95, the slope of the transition changes signs, but we do not observe an alternate upstream and downstream migration of the gravel-sand transition (Fig. 5.11). The gravel front progrades during all the simulation and its elevation increases and then decreases without generate incision.

We now simulate a catastrophic event, such as a landslide, which brings coarse sediment into the river during a brief period of time. To exemplify this we suddenly multiply by three the amount of gravel into the mixture, hence φ evolves from 0.5 to 0.75 to mimic a catastrophic input of gravel in the system. Then, after this catastrophic event, the mixture rapidly returns to its original value (Fig. 5.12). The resulting stratigraphy, shows a sharp decrease of the transition slope, during the event. When the mixture returns to its initial composition, the transition slope returns to its original value. This simulation raises the question of time scale. [START_REF] Paola | Downstream ning by selective deposition in a laboratory ume[END_REF] propose that the time over which a variation has to occur to be recorded in the stratigraphy, has to be higher than a characteristic time which depends on the size of the fan. This assumption should be tested numerically using a systematic study where we should vary the duration of the change and the size of the fan.

Conclusion 91

Conclusion

Using the results of chapters 2 and 3, we numerically investigated the evolution of an alluvial fan built under varying forcing. First, we explicit the condition necessary to generate an incision. A balance between the volume of the fan (i.e. its age) and the rate of the slope decrease determines the occurrence of incision. Second, we observe the stratigraphic record of a variation of the sediment composition. We notice that the signal in the stratigraphy depends on the forcing.

With these simplied models, we propose some tools to better understand the relationships between paleo-variations of the water discharge or of the sediment composition and the inner structure of the alluvial fan.

For simplicity, we vary only one parameter at the same time: the water discharge or the sediment composition. However, in nature, an increase of the water discharge should cause an increase of the sediment discharge. In the case where we vary the water discharge, we assume that the sediment discharge does not inuence the slope of the fan and we keep it constant even though the water discharge increases. Consequently, the slope of the fan is only determined by the threshold of sediment motion. On contrary, we propose that changes of the sediment mixture composition occur without change of the water discharge, therefore the slope of both deposit stay the same. To improve these models, we should consider the inuence of the sediment discharge on the fan slope, using the results of chapter 4.

Finally, this chapter only presents preliminary results. We now have to perform systematic studies, to test the validity of these models and nd their limitations. The record, in the fan stratigraphy, of changes of the forcing seems strongly dependent on the size of the alluvial fan [START_REF] Paola | Downstream ning by selective deposition in a laboratory ume[END_REF]. The next step is therefore, to determine scaling laws that link the intensity of the change to the system dimension.

Chapter 6

Perspectives: exploring natural systems Les résultats présentés dans les chapitres précédents sont obtenus dans des conditions simpliées.

Cependant, ils permettent de mettre en évidence les relations d'échelles entre la morphologie des cônes et les paramètres d'entrée. Dans nos expériences, l'écoulement est laminaire, ce qui n'est pas le cas des rivières naturelles où l'écoulement est turbulent. Néanmoins, dans le cadre expérimental, il semble que la morphologie des rivières laminaires et turbulentes est souvent contrôlée par les mêmes mécanismes physiques [START_REF] Malverti | Small is beautiful: Upscaling from microscale laminar to natural turbulent rivers[END_REF][START_REF] Chatanantavet | Physically based model of downstream ning in bedrock streams with lateral input[END_REF]. De plus, les cônes naturels ont une histoire plus complexe que les cônes expérimentaux.

Ils sont généralement construit dans des zones tectoniquement actives et leur morphologie peut donc être modiée par l'action des failles. Les ux qui les alimentent varient également au cours du temps. Enn les sédiments qui les composent présentent le plus souvent une granulométrie étendue. Les résultats présentés précédemment ne peuvent donc pas être appliqués directement aux objets naturels. Cependant, ils peuvent servir à identier les paramètres à étudier sur le terrain dans l'espoir de déterminer les paramètres d'entrée ayant permis la formation des cônes examinés.

Dans ce chapitre de perspectives, nous présentons donc des clés pour utiliser nos résultats expérimentaux, dans le cadre d'objets naturels. Dans un premier temps, nous nous intéressons à la pente de cônes alluviaux naturels. Nous montrons que l'étude du prol en long des cônes alluviaux permet de proposer une estimation des ux les ayant construits. Dans une seconde partie nous montrons, à partir de l'exemple des cônes du piedmont Nord du Tianshan, Chine, comment le critère d'incision permet d'étudier les variations de ux d'eau au cours de la formation d'un cône. 94 Chapter 6. Perspectives: exploring natural systems

Fan slope

In the previous chapters we have investigated three kinds of alluvial fans characterized by a radial prole that depends on the external forcing:

A linear prole, when the sediment discharge is low enough (Chapter 2).

A segmented prole, when the sediment that feeds the fan is bimodal (Chapter 3).

A concave prole, when the sediment discharge is signicantly higher than the characteristic one (Chapter 4).

To investigate whether the knowledge gained from experiments can be used to extract information about the dynamics of natural fans, we now look at three natural examples. Two of them, the Bayingol and the Guertu fans, are located in the Tianshan, China. The third one, the Dog Canyon fan, is located at Oliver Lee Park, New Mexico, USA.

As these fans have been previously investigated, data sets are available [START_REF] Poisson | Impact du climat et de la tectonique sur l'évolution géomorphologique d'un piémont-Exemple du piémont Nord du Tian Shan depuis la n du Pléistocène[END_REF][START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF][START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF][START_REF] Métivier | Geometry of meandering and braided gravel-bed threads from the bayanbulak grassland, tianshan, pr china[END_REF].

In their study of the Dog Canyon fan, [START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF] characterize the channel geometry, the particle size and the elevation at dierent positions along the fan. The data is available through Figshare 1 .

For the Chinese fans, [START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF] and [START_REF] Métivier | Geometry of meandering and braided gravel-bed threads from the bayanbulak grassland, tianshan, pr china[END_REF] provide information about grain size and water discharge. To access the slope of these fan, we use Digital Elevation Models (ALOS-2, AW3D30 2 ). These DEMs have a horizontal resolution of approximately 30 m

(1 arcsec). Using these DEMs we extract the fan elevation along six radius. We then measure the local slope for each prole and average them. Using this procedure we nally compare the averaged slope of the fan, to our model.

Bayingol fan

The rst fan we study is located in the South-West part of the Bayanbulak Grassland (Tianshan, China). This intramontane sedimentary basin stands at an elevation of about 2500 m above sea level. The basin is surrounded by sloping meadows, enclosed in the Tianshan mountains, which provide water to the main river, the Kaidu River [START_REF] Métivier | Geometry of meandering and braided gravel-bed threads from the bayanbulak grassland, tianshan, pr china[END_REF].

The apex of the Bayingol Fan stands at approximately 2600 m above sea level and its toe at a mean elevation of 2400 m. The fan radius is about 20 km. Consequently, the slope of the fan is approximately 0.01 (Fig. 6.1). The Bayingol Fan is fed by a braided river, the Bayingol River, a tributary to the Kaidu River. Its bed is mostly composed of gravel, with a median grain size d 50 = 2.8 cm. The sediment is quite uniform suggesting that the slope of the fan should not be segmented (Chapter 2). To investigate more precisely the slope of the fan, we extract six radial proles from the DEM. We choose to extract proles on the western part of the fan to avoid the active river (Fig. 6.1a).

We nd that the radial proles resemble each other, which meaning that the fan is radially symmetric (Fig. 6.1b). Consequently, the average slope of these six proles is a good approximation of the overall fan slope. By representing this average slope as a function of the distance 1 http://dx.doi.org/10.6084/m9.figshare.1197211 2 http://www.eorc.jaxa.jp/ALOS/en/aw3d30/index.htm from the fan apex, we notice that this slope is reasonably constant along the fan. The Bayingol fan slope is equal to 0.01 with a standard deviation of 20% (Fig. 6.1c).

Following our experimental results, a fan with a constant slope should be close to the threshold of sediment motion (Chapter 2). Accordingly, its slope depends only on the sediment characteristic and the water discharge. We now use the relation between the slope of a fan and the water discharge, in its turbulent version, to estimate the water discharge that built this fan, [START_REF] Métivier | Geometry of meandering and braided gravel-bed threads from the bayanbulak grassland, tianshan, pr china[END_REF] have measured the water discharge, Q w , in the individual threads that compose the Bayingol River. We compare these measurements to our estimate from the fan slope (Fig. 6.2). It corresponds to the lower discharges measured in the eld. This suggests that the slope of a fan built by a braided river is controlled mostly by the threshold slope of the smaller threads, that is, the steeper ones.

S H = g µ 3 θ c µ ρ s -ρ ρ d s 5 1/4 2 3/2 K(1/2) 3 C f Q w . ( 6
However, this result has to be considered carefully, as the equation we use does not take the number of threads that compose the river into account. If the ow distributes itself evenly among the threads, though, we can approximate the total discharge by multiplying the water discharge, estimated using Eq. (6.1), by the number of thread that composed the river.

Dog Canyon Fan This fan is located at Oliver Lee State Park near Alamogordo, New Mexico.

Dog Canyon drains the Sacramento Mountain range, which makes up the eastern boundary of the Tularosa Basin [START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF]. This alluvial fan is an example of a bimodal alluvial fan, composed of gravel and sand. [START_REF] Miller | Generalized sorting prole of alluvial fans[END_REF] have collected the particle size along the fan (Fig. 6.3). The fan prole presents a relatively well-dened slope break located at the transition from a gravel-dominated bed to a sand-dominated bed. The slope upstream of the gravel-sand transition is constant and steeper than the one downstream of the transition. This structure is typical of bimodal fan [START_REF] Delorme | Self-similar growth of a bimodal laboratory fan[END_REF].

The fan prole is similar to the one we observe in our experiments (Chapter 3). Accordingly, we propose to use the method developed in chapter 3 to decipher the composition of the sediment mixture, φ = V gravel V sand + V gravel , that built this fan. According to Eq. (6.2),

φ = (1 -S)R 4 -SR 3 -R 2 (R + 1) ((1 -S)R 3 -1) , (6.2)
the composition of the sediment mixture can be calculated knowing only the relative position, R = R gravel /R fan , of the transition and the ratio of slopes, S = S gravel /S sand . To estimate this values we represent the evolution of the slope as a function of the distance from the fan apex, and we t these data with a sigmoid to estimate the value of S sand , S gravel , and R (Fig. 6.4).

The two plateaus of the sigmoid give the value of the proximal and distal slopes, and the inexion point the location of the transition (Fig. 6.4). We nd that the upper slope is 0.04, the lower slope 0.01 and the relative location of the transition R = 0.49. Introducing these values in Eq. ( 6.2), we obtain φ = 0.16. We thus expect the initial mixture of sediment to be composed of 16% of gravel. Because we cannot access the fan stratigraphy, we cannot assure that the sediment composition did not change during the fan construction. From the fan surface, we can only obtain the composition of the sediment mixture involved in the more recent period of the fan activity. One way to estimate the evolution of φ during the fan growth could be to investigate the stratigraphy of the fan. In the startigraphy, the migration of the gravel front should result from variations of φ (Chapter 5). Guertu fan This fan is located on the northern side of the Tianshan mountains. This range is separated from the Junggar Foreland Basin by series of Est-West-oriented folds (Fig. 6.5). A dozen of large rivers drain the northern side of this range. They originate from glaciers in the hinterland mountains and ow northwards to the Junggar Basin. These rivers have built and incised several alluvial fans (Fig. 6.5). Tectonic and climatic studies show that the cycle of fan aggradation-incision is climate-driven and linked to changes in water and sediment discharges [START_REF] Poisson | Impact du climat et de la tectonique sur l'évolution géomorphologique d'un piémont-Exemple du piémont Nord du Tian Shan depuis la n du Pléistocène[END_REF][START_REF] Poisson | Holocene hydrological changes inferred from alluvial stream entrenchment in north tian shan (northwestern china)[END_REF][START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF]. These fans are made up by conglomerates [START_REF] Guérit | Caractéristiques morpho-sédimentaires des cônes alluviaux et reconstitution de leurs paléo-ux d'eau et de sédiments: Exemples naturels du piedmont nord du Tian Shan (Xinjiang, Chine) et modélisation expérimentale[END_REF]. The Urumqi River (10 in Fig. 6.5), is the only river which sediment size and discharge are documented [START_REF] Liu | Erosion rates deduced from seasonal mass balance along the upper urumqi river in tianshan[END_REF]. The median grain size is about ≈ 20 mm.

Because of the geomorphological similarity between these fans, we assume that the Urumqi River shared its characteristics with the other rivers of the range piedmont. From [START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF].

To investigate the inuence of water and sediment discharges on the slope of a fan, we chose the western Guertu fan because it is likely to be less aected by fault activity. We thus analyze the Pleistocene fan fed by the Guertu River (1 in Fig. 6.5). This fan was active between 300 kyr and 12 kyr, and then incised during the last deglaciation [START_REF] Poisson | Holocene hydrological changes inferred from alluvial stream entrenchment in north tian shan (northwestern china)[END_REF]. The Gertu fan apex stands at an elevation of approximately 1000 m above sea level, and its toe at an elevation of 400 m. The fan radius is approximately 25 km. Consequently, its mean slope is about 0.025.

To describe more accurately the shape of the fan radial prole, we extract six proles on both sides of the entrenched river (Fig. 6.6a). These proles are gently concave up (Fig. 6.6b).

The three eastern proles cross the entrenched river, so we use only, the three western ones to measure the averaged local slope. Figure (6.6c), shows that the fan slope decreases with the distance from the fan apex. On the proximal part of the fan, where r/R lies between 0.1 and 0.5, the slope is constant and about 0.027. Then the slope slowly decreases to reach a minimum 6.1. Fan slope 99 of about 0.017 at the fan toe, where r/R is between 0.9 and 1. This slope variation resembles the one induced by a decreases of the sediment discharge along the fan [START_REF] Stock | Controls on alluvial fan long-proles[END_REF]. If this is true, following the results of chapter 4, the slope reaches at the fan toe should be the threshold slope, S H , and the ratio of the slope at the fan apex over the slope at the fan toe, can be used to estimate the sediment discharge that fed the fan (Eq. 4.7).

This equation was derived for fans fed by a laminar river and thus does not apply directly to natural fans. To adapt it to turbulent rivers, we use the relation, proposed by [START_REF] Seizilles De Mazancourt | Forme d'équilibre d'une rivière[END_REF], between the slope and the sediment discharge of a turbulent river:

S = S H 1 + 3 2 3/2 K(1/2) Q s Q * s . (6.3)
To estimate the absolute sediment discharge, we need to measure the characteristic sediment discharge, Q * s , for this type of sediment. Despite this unknown parameter, we use the ratio of the slope at the fan apex over the slope at the fan toe to estimate, through Eq. ( 6.3), the relative sediment discharge, Q s /Q * s . Assuming this relation holds, we nally obtain that the sediment discharge that fed this fan is approximately one and a half times the characteristic sediment discharge. To convert this value into an absolute discharge we need to measure the characteristic sediment discharge, Q * s , for this sediment. Using these three natural examples, we have shown that the scaling laws we nd experimentally, can in principle be used to estimate the order of magnitude of the input parameters implied in the construction of natural fans. However, we have to consider these estimations carefully. First, we study only the fan surface. Therefore these estimations can only reect the last stage of fan growth. To investigate the variation of the forcing during the fan's growth, we should explore the fan cross-section. Observed changes of slope in the fan stratigraphy could then be used to reconstruct the history of discharge variations. Second, we assume that the fans are built near the threshold of motion. Third, our scaling laws only apply to a fan fed by a single-thread 100 Chapter 6. Perspectives: exploring natural systems river. However, each thread of braided river seems to follow scaling laws that are similar to single-thread rivers [START_REF] Gaurav | Morphology of the kosi megafan channels[END_REF]. Consequently, as we discussed for the Bayingol fan, we need to characterize the morphology of the individual threads to improve our understanding of the fan dynamics.

One of the most remarkable eects of the increase of water discharge is the fan apex incision [START_REF] Weissmann | Factors controlling sequence development on quaternary uvial fans, san joaquin basin, california, usa[END_REF][START_REF] Guerit | Reconstruction of the sediment trans-port conditions in the urumqi alluvial fan (northeast tian shan, china)[END_REF][START_REF] Cesta | Timing and nature of alluvial fan development along the chajnantor plateau, northern chile[END_REF]. Using the results of chapter 5, we now propose to estimate the rate of discharge variation that generated incision of the northern Tianshan Pleistocene fans.

Incision criterion

We can use the fans of the northern side of the Tianshan mountain to test our incision criterion (Chapter 5). Indeed, the Pleistocene fans have been incised during the last deglaciation, approximately 12 kyr ago [START_REF] Poisson | Impact du climat et de la tectonique sur l'évolution géomorphologique d'un piémont-Exemple du piémont Nord du Tian Shan depuis la n du Pléistocène[END_REF][START_REF] Poisson | Holocene hydrological changes inferred from alluvial stream entrenchment in north tian shan (northwestern china)[END_REF][START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF]. This incision induced the formation of smaller fans further downstream, the Holocene fans (Fig. 6.5). [START_REF] Guerit | Denudation intensity and control in the chinese tian shan: new constraints from mass balance on catchment-alluvial fan systems[END_REF] have investigated these two series of fans. They have estimated the sediment discharge that built them, by reconstruction of their volume. They also measured the water discharge of the rivers that feed the Holocene fans (Table 6.1). Consequently, we can use these two data sets together with the denition of the incision criterion, to propose a lower bound for the rate of variation of the water discharge induced by the last deglaciation. We assume that all of these Pleistocene fans are incised at the same time 12 kyr ago during the last deglaciation. During the deglaciation, the glaciers melted inducing an increase of the water discharge. Accordingly, using a stream-erosion model, [START_REF] Poisson | Holocene hydrological changes inferred from alluvial stream entrenchment in north tian shan (northwestern china)[END_REF] estimated that during the early Holocene, when incision starts, the Kuitun River discharge was twice the present one. This evolution of water discharge follows the general climatic trend in central Asia.

We assume that all the rivers that drain the northern Tianshan evolve similarly during the early Holocene. And we estimate that the fraction of the sediment load deposited on the fan is 17% of the total sediment ux. This value corresponds to the proportion of sediment transported as 6.3. Conclusion 101 bedload [START_REF] Liu | Measuring bedload in gravel-bed mountain rivers: averaging methods and sampling strategies[END_REF]. Using Eq. (5.8) and the data of [START_REF] Guérit | Caractéristiques morpho-sédimentaires des cônes alluviaux et reconstitution de leurs paléo-ux d'eau et de sédiments: Exemples naturels du piedmont nord du Tian Shan (Xinjiang, Chine) et modélisation expérimentale[END_REF], we calculate a minimum rate of variation for the water discharge during the early Holocene. To generate incision, this rate must be at least, (6.4) where Qs, Q w and V are the sediment discharge, water discharge and volume of the fan just before incision, at the end of the Pleistocene. Following the result of [START_REF] Poisson | Holocene hydrological changes inferred from alluvial stream entrenchment in north tian shan (northwestern china)[END_REF],

Qw = Q s Q w V ,
we estimate that the water discharge at this time is at least the actual one. Finally, we measure the minimum rate of water discharge variation, needed to generate incision, for each fan of the northern Tianshan, using Eq. ( 6.4) and Table (6.1). Assuming that all the fans were incised simultaneously, the regional minimum rate of water discharge variation, corresponds to the maximum value of Qw calculated. We nd that the regional minimum rate of the water discharge variation is approximately 2381.1 m 3 yr -2 (Table 6.1).

To compute the increase of water discharge necessary to incise these fans we need an estimate of the period, T , over which the perturbation occurred. Based on the paleo-levels of the Dzungar Lac, [START_REF] Poisson | Impact du climat et de la tectonique sur l'évolution géomorphologique d'un piémont-Exemple du piémont Nord du Tian Shan depuis la n du Pléistocène[END_REF] suggests that the degalciation causes an increase of the water discharge from 12 kyr to 6 kyr. Accordingly, we assume that the maximum value of T is 6 kyr. Assuming that the water discharge increase linearly during this period ∆Q w = Qw T , we nd that an increase of about 1% of the water discharge is sucient to generate incision. This low value may explain why all the fans of the northern Tianshan were incised simultaneously.

This example illustrates how the incision criterion could be used to determine the rate at which the water discharge has changed during a climatic event. This criterion is based on the assumption that the sediment discharge is small enough to neglect its inuence on the fan slope.

In chapter 4, we showed it is true only if the sediment discharge is ten times lower than the characteristic sediment discharge, Q * s . Otherwise, we would have to take into account its contribution to the fan slope.

Conclusion

In this chapter, we discuss the application of our experimental results to natural fans. We do this in order to highlight the parameters that we need to measure on the eld to reconstruct the paleo-uxes that built the fan.

Our analysis of three natural systems in China and the United States of America shows that the scaling laws, we determine experimentally, give a reasonable order of magnitude of the paleo-uxes that built the fan. However, these estimation have to be considered carefully for several reasons. First, in natural system the ow is turbulent. We therefore need to adapt scaling laws that were derived for laminar ows. Second, natural fans form under varying conditions.

Consequently, without access to the fan stratigraphy, we can only infer input parameters that contributed to the last stage of the fan growth. Third, our scaling laws are based on the assumption that the fan is built by a single-thread river. However, in the eld a lot of fans are built by braided rivers.

In order to go further and estimate, with a higher accuracy, the parameters and uxes that control the dynamics of natural fans we need to acquire specic eld data. First, we should characterize the sediment that enter the fan area to estimated the characteristic sediment ux Q * s . Second, we also need to quantify and characterize the number of threads that feed the fan to introduce them in our calculations. Finally, to reconstruct the evolution of the input parameters, surface measurements are not enough. We need to study the fan stratigraphy to describe the evolution of the slope and the position of the gravel front. Enn, dans le troisième cas, lorsque le ux de sédiment injecté à l'apex du cône devient important, la pente du cône s'éloigne de sa valeur seuil proportionnellement à (1 + Q s /Q * s ) 1/3 . Nous avons pu mettre en évidence que le ux de sédiment, Q s , était cependant négligeable si sa valeur était dix fois inférieure au ux de sédiment caractéristique, Q * s . La valeur de ce ux de sédiment caractéristique dépend de la taille des sédiments ainsi que de leur longueur de diusion.

Conclusions et perspectives

Ce ux peut être mesuré expérimentalement en suivant la trajectoire des grains ou alors estimé en ajustant la loi de transport à l'échelle de la rivière, Eq. (4.11), sur les données expérimentales.

Ces expériences nous ont permis d'identier les mécanismes physiques qui contrôlent la pente des cônes alluviaux. Toutefois, ces expériences sont réalisées dans un contexte simplié et s'appuient sur des approximations. Leur application directe aux objets naturels n'est donc pas immédiate.

La première simplication est l'utilisation d'un uide visqueux pour obtenir un écoulement laminaire. Lorsque l'écoulement est turbulent, les mécanismes qui déterminent la géométrie de la Perspectives Distribution continue de taille de grains En s'appuyant sur les résultats du chapitre 3, nous pourrions sans doute extrapoler le comportement d'un cône alluvial composé de deux tailles de grains à un cône composé d'un distribution continue de taille de grains. Lorsque le cône est composé de deux types de sédiment de mobilité diérentes, nous avons pu observer que les sédiments formant le dépôt étaient ségrégés. Les grains les plus mobiles sont déposés en aval tandis que les grains les moins mobiles se déposent à proximité de l'apex du cône.

En considérant que ce même mécanisme de ségrégation a lieu lorsque le mélange sédimentaire est composé de plus de deux types de sédiments, on peut imaginer que le cône formé présente un prol composé de plusieurs segments linéaires (Fig. 6.7). Chacun de ces segments étant associés à un type de sédiment, sa pente est contrôlée par le seuil de mise en mouvement des sédiments du lit de la rivière. En utilisant une représentation simplié de la géométrie de ces cônes imbriqués, il serait alors possible à partir du prol du cône de reconstituer la distribution de sédiment ayant alimenté le cône. Enn, en passant à la limite de dépôt innitésimaux, nous devrions pouvoir représenter la formation d'un cône alluvial alimenté par une distribution continue de taille de grains. Ce mécanisme de ségrégation, tel que nous l'avons observé au chapitre 3, engendre aussi une ségrégation verticale. Dans le cas où la composition du mélange sédimentaire n'a pas varié au cours du temps, une coupe verticale du cône devrait nous permettre d'observer la présence des grains les plus ns à la base du cône.

Variation des forçages

Pour vérier l'enregistrement dans la stratigraphie d'une variation des forçages, nous avons conçu un dispositif expérimental préliminaire à une dimension dans le cadre du stage de licence de Cécile Bidaut (Fig. 6.8). Ce dispositif nous permet d'avoir accès à la stratigraphie du cône au cours de 106 Conclusions et perspectives sa formation. Le cône est conné dans un chenal en Plexiglass de 20 cm de haut, 1.5 m de long et 2.5 cm de large. Nous avons utilisé deux types de sédiments caractérisés par une mobilité et une couleur diérente : des grains de corindon noir (d s = 800 µm, ρ s = 3900 kg m -3 ) et des grains de plastique multicolore (d s = 800 µm, ρ s = 1500 kg m -3 ). La diérence de mobilité entre ces deux types de sédiments est déterminée par leur diérence de densité. Le corindon, plus lourd, se dépose en premier. Nos expériences préliminaires, ont permis de vérier que les dépôts sont ségrégés en fonction de la mobilité des grains. 5 cm q s , q w 5 cm

Cônes extra-terrestres

Nous avons montré dans le chapitre 6, que l'étude des prols des cônes naturels pouvait nous donner une indication de l'ordre de grandeur des ux les ayant construits. Or, on observe sur d'autres planètes des dépôts sédimentaires qui semblent avoir été mis en place par des écoulements liquides, en particulier sur Mars [START_REF] Moore | Large alluvial fans on mars[END_REF]. Il semblerait donc que l'on puisse Pour illustrer cette idée nous nous intéressons au Peace Vallis fan, situé au nord du cratère Gale [START_REF] Williams | Martian uvial conglomerates at gale crater[END_REF][START_REF] Palucis | The origin and evolution of the peace vallis fan system that drains to the curiosity landing area, gale crater, mars[END_REF]. [START_REF] Palucis | The origin and evolution of the peace vallis fan system that drains to the curiosity landing area, gale crater, mars[END_REF] ont extrait des prols de ce cône à partir du DEM HCN an d'en étudier le pente (Fig. 6.10). Nous proposons donc d'utiliser ces données pour interpréter la morphologie du cône à la lumière de nos résultats.

On observe que dans la partie supérieure du cône les trois prols sont confondus (Fig. 6 En ce qui concerne le ux de sédiment, de la même façon que sur le cône de la Guertu (Chapitre 6), nous proposons d'utiliser le rapport de la pente proximal sur la pente distale pour estimer le débit de sédiment en fonction du débit de sédiment caractéristique, The shield has two separate channels. Each uses four of the Arduino pins to drive or sense the motor (direction, pulse width modulation, brake and current sensing). Combining these two channels, we can drive one bipolar stepper motor. We only need to turn on each channel alternatively. To do so we use the following Arduino sketch 3 . The motor speed is inversely proportional to the delay we impose between the activation of each channel. over more than four hours. The mass of sediment decreases linearly with time, which means that the sediment discharge is constant. In this system, the sediment discharge is controlled by the draining of the hopper. Contrary to uid draining, the ow of granular solids through orices does not depend on the height of grains in the hopper [START_REF] Beverloo | The ow of granular solids through orices[END_REF]. To ensure that, in our system, the sediment discharge does not depend on the mass of sediment in the hopper, The uid discharge has to stay constant during our experiment. We observe that after a long period however, the uid discharge decreases. This decrease is caused by water evaporation, which increases the uid density and viscosity. The discharge can be estimated using Poiseuille's law to represent the ow in the pipe connecting the tank to the experiment, Q w = πr 4 gzρ 8ηL ,

S H = g µ 3 θ c µ ρ s -ρ ρ d s 5 1/4 2 3/2 K(1/2) 3 C f Q w . ( 6
Q s Q * s = S S H 2 -1 2 2/3 K(1/2) 3 . ( 6
(B.1)
where r is the tube radius, z the tank elevation, and L the pipe length. ). We nd that the data accord with each other, suggesting that the discharge variation is indeed caused by evaporation. As a consequence, to keep the uid discharge constant during the run, we regularly measure the density of our mixture, and, when necessary, we add water to compensate for evaporation.

B.3 Experiment monitoring

During the experiment, we regularly record the uid discharge, the mass of sediment in the hopper and we acquire top-view pictures. Because the experiment runs continuously for more than one day, we automate these acquisitions using Python codes. The libraries pySerial5 and pyFirmata6 allow us to control the Arduino, and we use the gphoto7 library to control the camera. The following library (called pyCone.py) contains the necessary functions to control our devices. Using these functions, we can automatically acquire the uid discharge, the sediment discharge and take pictures of our experiment. We choose to save all of our experimental data in a database using MySQL. The following code allows to measure the uid discharge from the owmeter every ten seconds and take picture every ve minutes. To have homogeneous lighting for our picture, a light panel is automatically switched on before the picture is taken.

  Figure 1.1 : Cône alluvial Tian Shan, Chine. a) Vue de face. b) Photo aérienne Google Earth.

Figure 1

 1 Figure 1.2 : Classication des diérents types de cône. a) Cône de débris, Grotto canyon, Vallée de la Mort, USA. b) Cône chenalisé en tresse, la Kosi, Inde. c) Cône à écoulement méandriforme, le Taquari, Brésil. Images Landsat issues de Google Earth.

Figure 1

 1 Figure 1.3 : (a) Image Landsat du cône de la Kosi (Inde) montrant la localisation de (b) un prol topographique radial (Nord -Sud), et (c) un prol topographique transverse (Est -Ouest).

1. 2 .

 2 Reconstitution des paléo-ux 17 technique nécessite d'avoir accès à deux types d'informations ; (a) une cartographie 3D du dépôt, et (b) l'âge du dépôt.

Figure 1

 1 Figure 1.4 : Mesure du volume de sédiment stocké dans le cône alluvial Holocène de Kuitun (Tian Shan, Chine). A) Modèle numérique de terrain de la topographie actuelle du cône, données SRTM3. B) Topographie de la base du cône Holocène estimée à partir de la pente du cône Pléistocène. C) Diérence d'altitude entre A) la topographie actuelle et B) la base du cône. Cette diérence correspond à l'épaisseur de sédiments stockée pendant l'Holocène. D'après Jolivet et al.(2014)

  Figure 1.5 : Exemple de dispositifs expérimentaux utilisés. a) L' Experimental Earthscape Facility (XES) Basin (Kim et al., 2006). b) Le système couplé bassin versant cône alluvial utilisé par Rohais et al. (2012). c) Le dispositif utilisé par Clarke et al. (2010).

Figure 1 . 6 :

 16 Figure 1.6 : Relation entre la taille des sédiments et la pente d'un cône alluvial, Aubrey Clis, Arizona. En ordonnée, taille des sédiments et pente locale, en abscisse distance par rapport à l'apex du cône. D'après Blissenbach (1952).

Figure 1 . 7 :

 17 Figure 1.7 : Relation entre la fraction de sable et la pente de cônes. Données issues de Stock et al. (2008).

Figure 1

 1 Figure 1.8 : Relation entre la pente du cône et l'aire du bassin versant (utilisée comme proxy pour le débit d'eau). D'après Saito and Oguchi (2005).

  Figure 1.9 : Modèle conceptuel de l'inuence du ux de sédiment sur la pente des cônes alluviaux. Hypothèse de Drew (1873) : la diminution du ux de sédiment Q s (illustrée par les èches)

Figure 1

 1 Figure 1.12 : Expériences utilisées pour caractériser les mécanismes physiques, agissant à diérentes échelles, qui permettent de comprendre la morphologie des cônes alluviaux. a) Suivi de la trajectoire des grains, les lignes bleues représentent la trajectoire d'un grain, D'après Abramian et al. (prep). b) Caractérisation du seuil de mise en mouvement des sédiments, loi de transport à

Figure 2 . 1 :

 21 Figure 2.1: Schematic representation of a laboratory river. Both the depth D and the velocity U are functions of the transverse coordinate y. The river is invariant in the streamwise direction x.

2. 1 .

 1 Figure 2.2: Lacey's law and threshold theory in laboratory experiments. Green: single-thread channels; blue: threads from braided rivers. Red line corresponds to the threshold theory. Shaded area and dashed lines indicate uncertainty about the parameters in experiments. From Métivier et al. (2017).

  Figure 2.3: Schematic representation of the experimental set-up used.

Figure 2

 2 Figure2.4: a) Probability density function of sediment diameter. b) Transport law q s = q 0 (θ-θ c ), q 0 = 162 ± 5 g s -1 m -1 , θ c = 0.14± 0.005.

Figure 2 . 5 :

 25 Figure 2.5: Top view pictures of the experimental fan at dierent time.

Figure 2

 2 Figure 2.6: High frequency pictures, 10 min apart, channel lling from the distal to the proximal part.

Figure 2

 2 Figure 2.7: Evolution of the fan radius with time.

Figure 2 . 8 :

 28 Figure 2.8: Topography acquisition. White dot: detection of the fan contours. Magenta dashed line: average radius

Figure 2

 2 Figure 2.9: Volume of sediment injected into the experiment versus volume of the deposit. Red line: linear t.

Figure 2

 2 Figure 2.10: Variation of the fan slope at dierent angles. a) Fan proles at dierent angles. Red line: average prole. b) Mean slope of the fan as a function of the angle. Red line: mean slope of the fan.

Figure 2

 2 Figure 2.11: Evolution of the fan slope. a) Fan prole at dierent time. Six hours separate two consecutive lines. b) Slope of the fan as a function of time.

Figure 2 .

 2 Figure 2.13: Slope of the fan as function of the water discharge, red line threshold slope, red dashed line = 2S h .

  Figure 3.1: Experimental set-up. (a) Front-view picture. (b) Top-view representation.

  Figure 3.2: (a) Cumulative density function of the grain size. Orange: silica, green: coal. (b) Transport laws. Volumetric ux per unit width, as a function of dimensional shear stress. Dashed lines correspond to Eq. (A.10) tted to the data (method in Appendix A, coecients in Table3.1).

Figure 3

 3 Figure 3.3: Top-view pictures of an experimental fan (run 2). (a) Time evolution. Green dashed line indicates fan toe, R c . (b) Average of rescaled pictures. The 26 pictures are each 10-minutes apart. Dashed lines indicate silica-coal transition (orange) and fan toe (green). After rescaling, the fan length is one. Transition between silica and coal occurs at dimensionless distance R from apex.

Figure 3

 3 Figure 3.4: Evolution of the radial fronts of the silica (orange) and coal (green) deposits in run 2.

Figure 3 . 5 :

 35 Figure 3.5: Packing fraction of the deposit as a function of the composition of the sediment mixture. Blue dots calculated from experiment. Red dots measured independently. The red dashed line is the mean packing fraction measured independently.

Figure 3

 3 Figure 3.6: Digital elevation model of an experimental fan (run 2). Black lines: elevation contours 15 mm apart from each other. White dashed lines indicate the bounds used for averaging (only two sample radii 5 degree apart are represented for clarity).

Figure 3

 3 Figure 3.7: (a) Fan proles at dierent angles (run 2). Gray: individual proles; magenta: average prole. (b) Average downstream slope (magenta). Fitted hyperbolic tangent (dashed gray). Inection point (gray dot) and boundary of the transition area (vertical dashed gray line).

Figure 3

 3 Figure 3.8: Average radial prole (magenta line), superimposed on radial cross section, for run 2. Dashed lines: slope of the silica (orange) and coal (green) deposits. White dashed line indicates silica-coal transition. Scale is for the picture. h and r are respectively the fan elevation and the distance to the apex.

Figure 3

 3 Figure 3.10: Proportion of silica inferred from the geometry of the deposit, after Eq. (3.9) (blue) and after Eq. (3.10) (green), as a function of the composition of the sediment input. Red line: perfect agreement.

  Figure 3.11: Top-view of an experimental fan superimposed with measurement bins (white), and channels cross sections (blue).

Figure 3

 3 Figure 3.12: Evolution of active channels for all the runs. Channel width as a function of the dimensionless radius (a) and time (b). Number of channels as a function of dimensionless radius (c) and time (d). Black dashed line: average. Shaded area: variability over experimental runs.

Figure 3

 3 Figure 3.13: Slope normalized by the threshold slope, calculated with Eq. (3.11), as a function of the sediment discharge. Dashed lines: average slopes.

  (

t

  Figure 3.14: Relative position of the transition R (blue) and dimensionless transition slope S t (green), as a function of the composition of the sediment input. Dot: experimental measurements. Dashed line: Eqs. (3.9, 3.10) with S = 3.4.

Figure 4 . 1 :

 41 Figure 4.1: Conceptual model of sediment ux in the transverse direction y of a river.

  Eq. (4.3) leads to n = n 0 e Dα/l d , (4.4) 4.1. Inuence of sediment transport on alluvial river 69 where l d , can be measured in a laboratory channel. The surface concentration of moving grain, n, depends on the force exerted by the ow on the sediment bed (Lajeunesse et al., 2010a dimensionless coecient. Finally, combining Eqs. (2.3), (4.4) and (4.5) leads to D2 + D 2 -µ t e -D/ = ñ0 , (4.6) where we introduce a dimensionless depth D = D S/L, a dimensionless position ỹ = y S/L, and we dene = S l d /αL and ñ0 = n 0 d 2 s /β. The value of ñ0 depends on the uid and sediment uxes (Abramian et al., prep). This equation has no simple analytical solution. To study the solution of Eq. (4.6), Abramian et al. (prep) represent the phase portrait ( D, D ) for dierent values of ñ0 at constant (Fig. 4.

Figure 4

 4 Figure 4.2: a) Phase portrait, b) River cross-section, solutions of Eq. 4.6 for dierent values of

Figure 4

 4 Figure 4.3: Normalized fan proles for ve experiments (Q w = 0.33 L min -1 , Q s varies from 0.13 to 13 g min -1 ).

Figure 4

 4 Figure 4.4: Slope of the fan as a function of the sediment discharge. a) Distal slope. Colored areas, threshold slope S h multiply by a prefactor of two to be in accordance with the result of chapter 2. b) Normalized proximal slope. Red dashed line, Eq. (4.7) with Q * s and S h tted on the data.

  .15) Injecting these news variables into the mass-balance and the sediment-deposition equations (Eqs. 4.10 and 4.11), we obtain two ordinary dierential equations χ = 2XP(XH -H), (4.16)

Figure 4

 4 Figure 4.5: Numerical fan prole with dierent values of P.

  Combining Eqs. (4.17) and (4.18), we only keep the rst order terms to obtain the linearized equation H = -1.

  w = 0.33 L.min 1 Q w = 0.73 L.min 1

Figure 4

 4 Figure 4.6: Evolution of the volume ratio as a function of the sediment injected. Blue and orange dots: experimental data (Q w =0.33 L min -1 and Q w =0.73 L min -1 ). Red line: numerical solution showing two asymptotic solutions (green dashed lines). The coecients are the ones of (Fig. 4.4).

  chapitres précédents, nous avons étudié des cônes construits en maintenant constant les paramètres d'entrée (Q s , Q w et d s ). Dans ce cas, les cônes croissent de façon auto-similaire et acquièrent une forme d'équilibre contrôlée par ces paramètres. La pente est contrôlée, au premier ordre, par le débit d'eau et la taille des sédiments (Chapitre 2). Lorsque le système est composé de plusieurs taille de grain, on observe alors la formation de cônes imbriqués (Chapitre 3). Enn, lorsque le ux de sédiment devient important, la pente d'un cône s'éloigne de la pente seuil (Chapitre 4). En se basant sur ces résultats, nous proposons maintenant d'étudier un cône dont les paramètres d'entrée varient au cours de sa formation. Nous proposons de reconstruire l'histoire des ux ayant alimentés le cône en observant la stratigraphie du dépôt. Dans ce chapitre, nous explorons les conséquences des variations de deux paramètres d'entrée: le debit d'eau et la composition du mélange sédimentaire. Pour étudier l'inuence d'une variation du débit d'eau, nous nous plaçons dans le cas le plus simple: un cône au seuil composé d'un sédiment uniforme. Nous considérons donc que le ux de sédiment est susamment faible pour ne pas modier le prol du cône, et utilisons le formalisme présenté au chapitre 2 pour prédire le prol du cône. La deuxième partie se consacre à l'inuence de la composition du mélange sédimentaire. Nous utilisons les résultats obtenus au chapitre 3 pour prédire la géométrie des dépôts. En résolvant les équations de conservation de la masse, nous montrons comment des variations du débit d'eau ou de la composition du mélange de sédiment peuvent s'enregistrer dans la stratigraphie du cône. 80 Chapter 5. Varying forcing

0 Figure 5

 05 Figure 5.2: Schematic example of a varying water discharge. ∆Q w is the dierence between the minimum and the maximum water discharge, P the period of fan activity, and T the time over which the variation occurs.

  Figure 5.3: Evolution of the forcing in our simulations. a) The variation is too weak to induce incision. b) The change is fast enough to induce incision. Dots: time at which we stop the simulation.

  Figure 5.5: Evolution of the incision criterion. a) The criterion remains below one. b) The criterion reaches one.

Figure 5 . 7 :

 57 Figure 5.7: Schematic representation of the fan with denitions. Orange hatched area: gravel deposit. Green hatched area: sand deposit. Red line: gravel-sand transition.

Figure 5

 5 Figure 5.8: Numerical results. a) Stratigraphy of the deposit. Orange: gravel deposit. Green: sand deposit. Layers are separated by a constant amount of time. Red line: gravel-sand front. b) Evolution of the elevation of the fan apex and transition between gravel and sand during growth.

Figure 5

 5 Figure 5.9: Phase portrait for equation (5.21). Red dashed line: self-similar solution (Eq. 5.24). Shaded area: Eqs. (5.21) have no solution.

Figure 5

 5 Figure 5.10: a) Stratigraphy of a bimodal fan, time increases by 10 between two lines. b) Evolution of he proportion of gravel, φ.

Figure 5

 5 Figure 5.11: a) Stratigraphy of a bimodal fan, time increases by 10 between two lines. b) Evolution of the proportion of gravel, φ.

  Figure 6.1: a) Digital elevation model of the Bayingol Fan. White line: fan boundaries, and colored lines: prole line. b) Radial proles of the fan. c) Average slope of the fan as a function of the dimensionless distance from apex, red line: mean value.

  .1) Using this equation, with K(1/2) ≈ 1.85, C f ≈ 0.1, µ ≈ 0.7, θ c ≈ 0.04, ρ s = 2650 kg.m -3 , ρ = 1000 kg.m -3 and d s = d 50 = 0.028 m, we nally estimate a water discharge about 0.11 m 3 s -1 .

Figure 6

 6 Figure 6.2: Distrubition of water discharge owing through the threads that compose the Bayingol River. Orange line: water discharge estimated from the slope of the fan. Data from Métivier et al. (2016).

  Figure 6.3: Prole of the Dog Canyon fan. Main channel prole from smoothed DGPS data. Inset shows aerial image of fan with the entire fan outlined in black, main channel denoted by orange line, and secondary channel denoted by yellow line. From Miller et al. (2014).

Figure 6

 6 Figure 6.5: Present-day morpho-sedimentary map of the northern side of the Chinese Tianshan.

  Figure 6.6: a) Digital elevation model of the Guertu fan. White line: fan boundaries; Colored lines: prole lines. b) Radial proles of the fan. c) Average slope of the fan as a function of the dimensionless distance from apex. Red lines: mean slope near the apex and near the toe.

  Figure 6.7 : Représentation schématique d'un cône alluvial composé de cinq catégories de sédiments, caractérisés par une mobilité croissante vers l'aval.

Figure 6

 6 Figure 6.8 : Cône alluvial en 1D alimenté par deux types de sédiments. Exagération verticale ×2. Photographie : Cécile Bidaut.

Figure 6

 6 Figure 6.9 : Expérience préliminaire de construction d'un cône alluvial lorsque les paramètres d'entrée varient. En haut, évolution du débit d'eau (bleu) et de sédiment (orange). En bas, photographies du cône avant (gauche) et après (droite) l'augmentation du débit d'eau.

  .10B).Le dépôt est donc axisymétrique. En outre, ces prols sont similaires aux prols que l'on observe expérimentalement ou sur les cône terrestres. Il semble donc légitime d'utiliser la méthode présentée au chapitre 6 pour estimer les ux ayant permis la construction de ce cône. En représentant la pente du cône en fonction de la distance à l'apex,[START_REF] Palucis | The origin and evolution of the peace vallis fan system that drains to the curiosity landing area, gale crater, mars[END_REF] montrent que la pente du cône diminue progressivement de 3% à l'apex, jusqu'à 1%, à la transition entre le cône supérieur et le cône inférieur (Fig.6.10C). Une diminue de la pente peut être engendrée soit par une diminution de la taille des sédiments vers l'aval, soit par une diminution du ux sédimentaire.Concernant la taille des sédiments nous utilisons les estimations eectuées par[START_REF] Williams | Martian uvial conglomerates at gale crater[END_REF] et[START_REF] Palucis | The origin and evolution of the peace vallis fan system that drains to the curiosity landing area, gale crater, mars[END_REF]. En analysant les images acquises par le MSL Curiosity rover, qui s'est posé au pied du cône (Fig.6.10A),[START_REF] Williams | Martian uvial conglomerates at gale crater[END_REF] estiment que la taille des sédiments varie entre 2 et 40 mm. Le modèle hydraulique s'appuyant sur la géométrie des chenaux utilisé

Figure 6 .

 6 Figure 6.10 : (A) Le Peace Vallis fan, sur lequel on observe les lignes de niveau (marron). Les lignes colorées représentent l'emplacement des trois prols longitudinaux, prol ouest en rouge (W), prol central en vert (C) et prol est en bleu (E). UF, représente le cône supérieur, LF le cône inférieur et CS la surface cratérisée. L'étoile jaune représente la zone d'atterrissage du MSL Curiosity rover. (B) Prols du cône le long des trois segments représentés sur la gure A (extrait à partir du DEM HCN). La ligne pointillée marque la limite entre les cône supérieur et inférieur. (C) Pente moyenne du cône calculée tous les 100 m. Pour les prols orientaux et centraux, le prol tracé en noir à été reconstruit en prenant en compte l'érosion post dépôt. D'après Palucis et al. (2014).

. 5 )

 5 Pour estimer le débit d'eau à partir de l'équation (6.5), nous avons besoin des caractéristiques des grains et de l'écoulement. La majorité de la surface martienne est constituée de basalte, nous considérons donc que la densité des sédiments ρ s est égale à 3000 kg m -3 . Pour les raisons explicitées précédemment nous choisissons une taille de grain égal à 40 mm. L'accélération de la gravité g sur Mars est égale à 3.7 m s -2 . Le coecient de friction µ, est un paramètre géométrique qui ne doit pas dépendre des conditions environnementales. Les observations de Atwood-Stone and McEwen (2013) ont conrmées que ce paramètre n'est pas modié par la faible gravité martienne, nous prenons donc µ = 0.7. Enn, nous prendrons des valeur de θ c = 0.04 et C f = 0.1[START_REF] Williams | Martian uvial conglomerates at gale crater[END_REF][START_REF] Palucis | The origin and evolution of the peace vallis fan system that drains to the curiosity landing area, gale crater, mars[END_REF].Finalement, nous obtenons un débit d'eau de l'ordre de 0.8 m 3 s -1 . Cette valeur est du même ordre de grandeur (bien que cinq fois inférieure) que celle estimée par[START_REF] Palucis | The origin and evolution of the peace vallis fan system that drains to the curiosity landing area, gale crater, mars[END_REF], qui proposent un débit de 3.7 m 3 s -1 .

Figure

  Figure B.3: Stepper motor, model 42SH33-4AM.

Figure B. 4 :

 4 Figure B.4: Arduino motor shield R3

  r i a l . a v a i l a b l e ( )> 0 ) { d e l a y L e n g t h = S e r i a l . p a r s e F l o a t ( ) ; / / r e

  Fig. (B.5a) shows the evolution of the mass of sediment in the hopper as a function of time,

Fig

  Fig. (B.5b) represents the sediment discharge as a function of the mass in the bin . The sediment discharge stays constant. However, we note that there is a slight increase of discharge when the hopper is almost empty. This increase is about 2% (Fig. B.5b). Therefore we only need to ensure that the hopper never empties to keep the sediment discharge constant. Finally, to make our experiment reproducible, we calibrate the sediment discharge. For a xed distance between the hopper outlet and the belt, we nd that the sediment discharge evolves linearly with the inverse of the delay imposed to the stepper motor (Fig. B.6).

Figure

  Figure B.8: Variation of the uid discharge caused by the evaporation of the water from the mixture. The shaded area corresponds to the uid discharge measured with an electromagnetic owmeter, dots are calculated using Eq. (B.1).

  # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # Co n n e c t i o n t o A r d u i n o d e f c o n n e c t _ a r d u i n o ( ) : port_base_name = ' / d e v /ttyACM ' p o r t _ s u f f i x _ l i s t = [ ' 0 ' , ' 1 ' , ' 2 ' , ' 3 ' , ' 4 ' ] b o a r d _ c o n n e c e d = F a l s e f o r p o r t _ s u f f i x i n p o r t _ s u f f i x _ l i s t : p o r t = port_base_name + p o r t _ s u f f u t i l . I t e r a t o r ( b o a r d ) i t . s t a r t ( ) # n e c e s s a i l e d t o c o n n e c t t o A r d u i n o ! ' r e t u r n b o a r d# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # FLOWMETER # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # M e a s ur e t e n s i o n d e f g e t _ t e n s i o n ( b o a r d , pin_number , n b _ m e a s u r e s = 1 0 ) : b o a r d . a n a l o g [ pin_number ] . e n a b l e _ r e p o r t i n g ( ) t i m e . s l e e p ( . 1 ) t e n s i o n = [ ] f o r n i n r a n g e ( n b _ m e a s u r e s ) : t e n s i o n = t e n s i o n + [ b o a r d . a n a l o g [ pin_number ] . r e a d u r n mean ( t e n s i o n )* 5 . # v o l t s # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # C o n v er t t e n s i o n i n t o d i s c h a r g e d e f t e n s i o n 2 d i s c h a r g e ( t e n s i o n ) : r a n g e _ i n t e n s i t y = [ 4 e -3 , 2 0 e -3] # Amp r a n g e _ d i s c h a r g e = [ 0 , 1 . 0 0 1 ] # L/ min r e s i s t a n c e = 2 1 7 . # ohm i n t e n s i t y = t e n s i o n / r e s i s t a n c e # a m p e r e r e t u r n r a n g e _ d i s c h a r g e [ 0 ] + ( i n t e n s i t yr a n g e _ i n t e n s i t y [ 0 ] ) * ( r a n g e _ d i s c h a r g e [ 1 ]r a n g e _ d i s c h a r g e [ 0 ]) / ( r a n g e _ i n t e n s i t y [ 1 ] -r a n g e _ i n t e n s i t y [ 0 ] )# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # r e c o # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # Camera c o n f i g ur a t i o n d e f param ( i s o =0 , s h u t t e r s p e e d= 2 0 , a p e r t u r e= 2 1 ) : #D e f a u l t p a r a m e t e r s o s . s y s t e m ( ' g p h o t o 2 --s e t -c o n f i g -i n d e x i s o=%i '%i s o ) o s . s y s t e m ( ' g p h o t o 2 --s e t -c o n f i g -i n d e x s h u t t e r s p e e d=%i '%s h u t t e r s p e e d ) o s . s y s t e m ( ' g p h o t o 2 --s e t -c o n f i g -i n d e x a p e r t u r e=%i '%a p e r t u r e ) s y s t e m ( ' g p h o t o 2 --get -c o n f i g i s o=%i '%i s o ) o s . s y s t e m ( ' g p h o t o 2 --get -c o n f i g s h u t t e r s p e e d=%i '%s h u t t e r s p e e d ) o s . s y s t e m ( ' g p h o t o 2 --get -c o n f i g a p e r t u r e=%i '%a p e r t u r e )# # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # # f .PortComm= s e r i a l . S e r i a l ( ) s e l f . PortComm . t i m e o u t = 5 s e l f . PortComm . p o r t=com # s e t s t h e p a r a m e t e r s f o r a c q u i s i t i o n # c h e c k e d f o r t h e Ohaus 5 0 and t h e E x p l o r e r f . PortComm . p a r i t y = s e r i a l . PARITY_NONE s e l f . PortComm . s t o p b i t s = s e r i a l . STOPBITS_TWO s e l f . PortComm . b y t e s i z e = s e r i a l . SEVENBITS s e l f . O r d r e = "P\ r \ n " f . PortComm . p a r i t y = s e r i a l . PARITY_NONE s e l f . PortComm . s t o p b i t s = s e r i a l . STOPBITS_TWO s e l f . PortComm . b y t e s i z e = s e r i a l . SEVENBITS s e l f . O r d r e = " S I \ r \ n " s e l f . I l e n g t h = 1 9 e l i f s e l f . b a l== ' E x p l o r e r 2 2 ' : p r i n t s e l f . b a l s e l f . PortComm . b a u d r a t e = 9 6 0 0 s e l f . PortComm . p a r i t y = s e r i a l . PARITY_NONE s e l f . PortComm . s t o p b i t s = s e r i a l . STOPBITS_TWO s e l f . PortComm . b y t e s i z e = s e r i a l . SEVENBITS s e l f . O r d r e = "P\ r " f . PortComm . p a r i t y = s e r i a l . PARITY_NONE s e l f . PortComm . s t o p b i t s = s e r i a l . STOPBITS_ONE s e l f . PortComm . b y t e s i z e = s e r i a l . EIGHTBITS s e l f . O r d r e=" " i f s e l f . PortComm . i s O p e n ( ) : s e l f . PortComm . c l o s e ( ) s e l f . PortComm . o p e n ( ) d e f G e t W e i g h t ( s e l f ) : " " " r e t r i e v e s t h e c u r r e n t w e i g h t o f t h e b a l a n c e " " " i f s e l f . O r d r e< >" " : s e l f . PortComm . f l u s h I n p u t ( ) s e l f . PortComm . w r i t e ( s e l f . O r d r e ) b y t e s r e a d= s e l f . PortComm . r e a d ( s e l f . I l e n g t h ) e l s e : s e l f . PortComm . f l u s h I n p u t ( ) b y t e s r e a d= s e l f . PortComm . r e a d l i n e

SV

  f r o m pyCone i m p o r t * f r o m d a t e t i m e i m p o r t n n e c t i o n d a t a b a s e c o n n = MySQLdb . c o n n e c t ( h o s t=" l o c a l h o s t " , u s e r=" manip " , p a s s w d=" manip " , db=" e x p e r i m e n t a l _ f a n " ) c u r s o r = c o n n . c u r s o r ( ) # C o n n e c t i o n A r d u i n o b o a r d = c o n n e c t _ a r d u i n o ( ) # i n i t i a l i s a t i o n t 0 = t i m e . t i m e ( ) B.3. Experiment monitoring 125 d=d a t e t i m e . t o d a y ( ) p r i n t d . s t r f t i m e ( "%d/%m/%y , % H:%M:%S " ) r a c t e r i s t i c s r h o s = s y s . a r g v [ 5 ] d s = s y s . a r g v [ 6 ] # Camera c o n f i g u r a t i o n i s o =s y s . a r g v [ 7 ] s h u t t e r s p e e d=s y s . a r g v [ 8 ] a p e r t u r e=s y s . a r g v [ 9 ] # # S a v e e x p e r i m e n t c h a r a c t e r i s t i c i n d a t a b a s e s q l = " i n s e r t i n t o manip v a l u e s (% s , %s , %s , %s ) " c u r s o r . e x e c u t e ( s c h a r g e m e a s u r e m e n t v a l t i m e=t i m e . t i m e ( ) T = v a l t i m et 0 t e n s i o n = g e t _ t e n s i o n ( b o a r d , 0 ) d i s c h a r g e = t e n s i o n 2 d i s c h a r g e ( t e n s i o n ) i t c h on l i g h t b o a r d . d i g i t a l [ 3 ] . w r i t e ( 1 ) b o a r d . d i g i t a l [ 2 ] . w r i t e ( 1 ) # C o n f i g u r a t i o n c a m e r a param ( i n t ( i s o ) , i n t ( s h u t t e r s p e e d ) , i n t ( a p e r t u r e ) ) # p i c t u r e name o u t p u t="%s /%s_%s " % ( f o l d e r , fname , i ) # t a k e s p i c t u r e command=" g p h o t o 2 --c a p t u r e -image-and-download --f i l e n a m e %s " % ( o u t p u t ) o s . s y s t e m ( command ) v a l t i m e=t i m e . t i m e ( ) T = v a l t i m es o r . e x e c u t e ( s q , ( i n t ( code_manip ) , o u t p u t , T) ) c o n n . commit ( ) #t i m e s l e e p b e f o r e n e x t l o o p t i m e . s l e e p ( i n t ( i n t e r ) ) j = j + 1 e n d = d a t e t i m e . t o d a y ( ) p r i n t e n d . s t r f t i m e ( "%d/%m/%y , % H:%M:%S " ) c o n n . c l o s e ( ) The next code allows to measures and records the mass of sediment on the scale. This measurement allow us to estimate the sediment discharge. f r o m p y l a b i m p o r t * f r o m pyCone i m p o r t * i m p o r t t i m e i m p o r t s e r i a l i m p o r t MySQLdb # C o n n e c t i o n d a t a b a s e c o n n = MySQLdb . c o n n e c t ( h o s t=" l o c a l h o s t " , u s e r=" manip " , p a s s w d=" manip " , db=" e x p e r i m e n t a l _ f a n " ) c u r s o r = c o n n . c u r s o r ( ) # # a r g u m e n t s code_manip=s y s . a r g v [ 1 ] t s t e p= f l o a t ( s y s . a r g v [ 2 ] ) # t i m e s l e e p b e t w e e n two m e a s u r e m e n t s #c o n n e c t i o n t o b a l a n c e a=l a b . b a l a n c e ( ' E x p l o r e r 3 5 ' , ' / d e v / ttyUSB0 ' ) a _ w e i g h t = [ ] t = [ ] t 0=t i m e . t i m e ( ) #l o o p t o m e a s u r e w e i g h t u n t i l we s t o p t h e e x p e r i m e n c page slope of coal deposit S g page slope of gravel deposit S h page threshold slope for laminar ow S H page threshold slope for turbulent ow S s page slope of silica or sand deposit s page volume of silica or sand deposit L 3 V page volume of a fan divided by the corresponding threshold fan W page channel width m W h page threshold width for laminar ow m W H page threshold width for turbulent ow m X page dimensionless fan radius

  Les cônes alluviaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Reconstruction des paléo-ux . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction Les surfaces continentales sont le lieu d'interactions complexes entre les phénomènes tectoniques et climatiques. Leur évolution est contrôlée par l'action de l'érosion et du dépôt de sédiment.

	Table des matières
	Introduction
	1 État de l'art
	1.1

  Clarke et al. (2010) (Fig. 1.5c). Ces dispositifs sont conçus pour comprendre l'inuence des paramètres tels que les ux (Q w et Q s ) et la taille des grains (d s ). Le cône se construit sur une surface plane qui n'est pas mobile. L'angle d'ouverture latérale de ces dispositifs peut varier d'un modèle à l'autre et atteindre 180°. 1.3.1 La tectonique et le climat Les variations tectonique et climatique impactent la formation des cônes alluviaux en modiant les conditions aux limites à diérentes échelles. Tout d'abord, à l'échelle régionale, la tectonique et le climat contrôlent l'espace disponible pour la mise en place des cônes alluviaux. En eet, comme nous l'avons vu précédemment, les grands cônes alluviaux se construisent dans des espaces créés par l'action des failles et/ou la diminution du niveau de base

). Ce dispositif est constitué de deux boites : Une première boite en érosion qui peut être surélevée pour simuler une surrection tectonique. Cette boite est placée sous un brumisateur pour simuler la pluie. Les sédiments érodés dans cette boite sous l'action de la pluie sont transportés jusqu'à une deuxième boite, en sédimentation, dans laquelle ils se déposent. Cette boite en sédimentation peut être abaissée pour simuler une subsidence.

Enn, il existe de nombreux systèmes expérimentaux conçus sans la possibilité de faire varier l'espace disponible pour la mise en place du dépôt, par exemple ceux de

[START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF]

20 Chapitre 1. État de l'art et

Inuence respective des paramètres d'entrée sur la pente du cône C'est précisément dans le but d'isoler l'inuence des diérents paramètres d'entrée (d s , Q w et Q s ) sur la pente du cône que

  cette section, nous avons montré que la pente des cônes alluviaux est liée aux paramètres d'entrée (d s , Q w et Q s ). La pente des cônes alluviaux augmente avec la taille des grains ainsi que

	Van Dijk et al. 2009 Whipple et al. 1998 Clarke et al. 2008 Powell et al. 2011
	10 1
	Slope, S
	6 × 10 2
	4 × 10 2
	10 2 Dimensionless discharges, Q s /Q w	10 1
	Figure 1.10 : Pente du cône en fonction du rapport des ux. Modié d'après Van Dijk et al.
	(2009).
	le débit de sédiment. Au contraire, elle diminue avec l'augmentation du débit d'eau. Cependant,
	l'inuence de ces trois paramètres n'est généralement pas quantiée indépendamment. Pourtant,
	l'étude des résultats de Clarke et al. (2010), suggère que l'inuence du débit d'eau et de sédiment
	doit être considérée individuellement et non pas sous forme de rapport.
	1.3.4

Dans

  montrent que lorsqueFigure 1.11 : Dispositif expérimental et dénition des variables ; q s et q w les ux par unité de surface de sédiment et d'eau, h(x, t) la hauteur du cône, et L(t) la longueur maximum du cône. D'après Guerit et al. (2014). le ux de sédiment tend vers zéro, la pente du cône tend vers une valeur seuil qui est contrôlée, au premier ordre, par la taille des sédiments d s et qui varie en proportion de 1/

	Chapitre 1. État de l'art
	cônes alluviaux est contrôlée par les ux d'eau et de sédiment et la taille des sédiments. L'idée
	est donc de reconstituer les paléo-ux à partir de la pente des cônes alluviaux. Pour résoudre ce
	problème inverse il est indispensable de quantier l'inuence des ux d'eau et de sédiment et de
	la taille des grains sur la pente du cône.
	√	Q w . Dans ces
	conditions, le ux de sédiment contrôle uniquement la vitesse de croissance du cône. Lorsque
	le ux de sédiment augmente, le prol du cône est modié. Les sédiments déposés le long du
	cône ajoutent une déviation quadratique au prol linéaire de celui-ci. Cette déviation est alors
	proportionnelle à q s .
	Cette étude met ainsi en évidence une hiérarchie entre les diérents paramètres qui contrôlent
	la pente du dépôt. En eet, lorsque le ux de sédiment est faible, celui-ci peut être négligé, au
	premier ordre, et la pente du cône est contrôlée par la taille des sédiments et le débit d'eau. Par
	contre, plus le ux de sédiment augmente, plus la pente s'éloigne de sa valeur seuil. Les sédiments
	étant déposés le long du cône, cet écart au seuil se caractérise par la formation d'un prol concave.
	La forme de ce prol est interprétée comme la signature du transport sédimentaire. Pour aller
	plus loin, il serait cependant nécessaire de tester cette théorie sur un cône complet.
	Les observations présentées dans ce chapitre nous permettent de dénir le cadre dans lequel
	s'incrit notre étude, cela nous amène donc à présenter les objectifs de cette thèse.
	1.4 Objectifs de la thèse
	Pour comprendre l'évolution des surfaces terrestres, il est nécessaire de reconstituer les mou-
	vement de masse à la surface de la Terre. De par leur position à l'interface entre les zones
	d'érosion et les zones de dépôts, les cônes alluviaux apparaissent comme un témoin primordial
	des transferts de masse à la sortie des reliefs continentaux. Les reconstitutions de paléo-ux
	sédimentaire se font, à l'heure actuelle, par reconstruction de volume. Cette technique permet
	d'estimer l'ordre de grandeur des ux ayant construits les cônes, cependant la période d'activité
	du cône, nécessaire à cette estimation, est source d'incertitude. Grâce aux observations de terrain
	et à l'utilisation de modèles expérimentaux les études antérieures ont montrées que la pente des

Table 2 .

 2 1: Experimental parameters for the six runs.

	Run	Fluid discharge	Sediment discharge
		Q w (L min -1 )	Q s (g min -1 )
	1	0.9 ± 0.01	0.2 ± 0.002
	2	0.8 ± 0.01	0.2 ± 0.002
	3	0.4 ± 0.01	0.2 ± 0.002
	4	0.3 ± 0.01	0.2 ± 0.002
	5	0.25 ± 0.01	0.2 ± 0.002
	6	0.2 ± 0.01	0.2 ± 0.002

Table 3 .

 3 1: Physical characteristics of the sediment. The measurement method is presented in Appendix A. The friction coecient µ is the tangent of the angle of repose.

		Density	Grain size
		ρ s (kg m -3 )	d 50 (µm) d 90 (µm)
	Silica	2650 ± 50	130	200
	Coal	1500 ± 50	400	800
		Critical Shields	Friction coecient
		θ c	µ
	Silica	0.25 ± 0.02	0.42 ± 0.04
	Coal	0.19 ± 0.008	0.58 ± 0.04

Table 3 .

 3 2: Experimental parameters for the ve runs. bottom of the tank, it forms a half-cone deposit. Initially, a sheet ow spreads uniformly over this sediment body. After a few minutes, the ow connes itself into distinct, radial, channels (typically ve or six). All these channels appear to transport sediment simultaneously. The experiment ofReitz and Jerolmack (

	Run	Water discharge	Sediment discharge	Silica fraction
		Q w (L min -1 )	Q s (L min -1 )	φ
	1	2.6 ± 0.1	0.019 ± 0.001	0.5±0.05
	2	2.6 ± 0.1	0.045 ± 0.001	0.5±0.05
	3	2.6 ± 0.5	0.027 ± 0.001	0.25±0.02
	4	2.4 ± 0.1	0.027 ± 0.001	0.25±0.02
	5	2.6 ± 0.1	0.020 ± 0.001	0.8±0.08
	An experimental run begins with an empty tank. When the mixture of water and sediment

Table 3 .

 3 3: Geometrical characteristics of the experimental fans, measured at the end of each run. The errors on R are due to uctuations of the silica-coal transition.

	Run	Slope ratio	Transition location	S t /S s
		S	R	S t
	1	3 ± 0.3	0.56 ± 0.07	0.37 ± 0.08
	2	2.9 ± 0.1	0.55 ± 0.09	0.36 ± 0.08
	3	4 ± 0.6	0.41 ± 0.1	0.57 ± 0.12
	4	4.6 ± 0.6	0.39 ± 0.1	0.61 ± 0.13
	5	3.3 ± 0.3	0.83 ± 0.2	0.12 ± 0.03

Table 6 .

 6 1: Geomorphological characteristics and water and sediment uxes of the northern Tianshan alluvial fans. Q s is the total sediment ux, including suspended load and dissolved load. V is the volume of sediment in the fan.

	River	Q s Pleistocene V Pleistocene Q w Holocene	Qw
		×10 5 m 3 yr -1	×10 9 m 3	×10 8 m 3 yr -1 m 3 yr -2
	Kuitun	5.1	23.4	6.3	1169.2
	Anjihai	2.5	11.5	3.1	571.5
	Jingou	1.1	5.1	3.2	590.2
	Manas	1.6	7.3	12.8	2381.1
	Taxi	0.6	2.6	2.3	445.7
	Hutubi	2.8	12.6	4.8	900.1
	Sautun	1.4	6.6	3.1	563.3
	Toutun	1.4	6.2	2.3	448.2
	Ürümqi	0.9	4.2	2.4	431.1

  Au cours de cette thèse nous avons montré que la morphologie des cônes alluviaux est contrôlée par les paramètres d'entrée qui les alimentent (Q w , Q s et d s ). À l'aide de modèles physiques simpliés, nous avons montré l'importance de comprendre les phénomènes physiques agissant à l'échelle du grain et de la rivière pour étudier le comportement d'un cône alluvial (Fig.1.12).En utilisant ces modèles expérimentaux, nous avons déterminé les lois d'échelles qui relient les caractéristiques géométriques d'un cône alluvial aux paramètres d'entrée qui les alimentent.

Expérimentalement, nous avons produit trois types de prols caractéristiques des cônes naturels :

(1) un prol linéaire, (2) un prol segmenté et (3) un prol concave. Nous avons pu montrer que chaque type de prol se forme dans des conditions expérimentales particulières.

Dans le premier cas, lorsque le ux de sédiment est très faible, celui-ci n'inuence pratiquement pas la pente du cône et le prol radial est linéaire. La pente du cône atteint une valeur seuil qui est contrôlée par le débit d'eau et la taille des sédiments. Pour un écoulement laminaire, la pente croît de façon inverse à la racine cubique du débit d'eau. Pour un écoulement turbulent, la morphologie de la rivière est déterminée par les mêmes mécanismes, seul l'exposant de la loi d'échelle est modié, la pente du cône évolue proportionnellement à l'inverse de la racine carrée du débit d'eau.

Dans le second cas, l'injection d'un mélange sédimentaire dont les sédiments présentent des mobilités diérentes produit un prol segmenté. Chaque segment de ce prol dépend du seuil de mise en mouvement du sédiment composant le lit de la rivière. Nous montrons qu'il est possible, en étudiant la géométrie du dépôt, de reconstruire le mélange sédimentaire ayant conduit à la construction d'un prol segmenté.

  .6) Dans le cas du Peace Vallis fan, la pente proximale est trois fois supérieure à la pente distale, ce qui donne un ux de sédiment 14 fois supérieur au ux de sédiment caractéristique, Q * s . Pour estimer le ux de sédiment absolue il est nécessaire de quantier ce ux caractéristique. Celui-ci dépend de la taille des grains et de la longueur de diusion. Des expériences sont actuellement en cours dans l'équipe pour dénir la relation entre la longueur de diusion et les caractéristique de l'écoulement et des sédiments.

à seuil à l'échelle de la rivière. e) Formation d'un cône alluvial en laboratoire.

ses avulsions. Le cône alluvial hérite donc sa pente de la rivière qui le forme. Lors de la formation des cônes alluviaux les sédiments se déposent, le ux de sédiment diminue donc vers l'aval. La pente de la rivière alimentant le cône s'ajuste à cette diminution du ux de sédiment, générant un prol concave. Pour vérier que la pente du cône suit l'évolution du ux de sédiment, nous étudions la pente d'un cône expérimental construit par une rivière (Fig.1.12e).Ces mécanismes et le passage d'une échelle à l'autre seront décrits plus en détail tout au long de ce manuscrit.

Chapitre 1. État de l'art

Conclusions et perspectives rivière restent valables. Seul le prol de vitesse est modié, celui-ci fait intervenir un coecient de frottement, C f , qui dépend de la rugosité du fond ainsi que de la vitesse et de la taille de l'écoulement. En intégrant ce prol de vitesse turbulent sur la largeur de la rivière il est donc possible de retrouver les équations de régime pour un écoulement turbulent.En outre, dans la nature, de nombreux paramètres, que nous n'avons pas pris en compte, peuvent inuencer la dynamique d'un cône : l'activité tectonique, qui peut modier la pente de la rivière, la distribution étendue de taille des sédiments transportés, la végétation.Enn, toutes nos expériences ont été réalisées en maintenant constant les paramètres d'entrée alimentant le cône. De la sorte, le cône croît de façon auto-similaire. Or ce n'est souvent pas le cas dans la nature où les cônes se construisent sur des échelles de temps qui peuvent aller jusqu'à la centaine de milliers d'années.Pour étudier l'inuence de l'évolution de ces forçages pendant de la construction du cône nous avons donc développé un modèle fondé sur la conservation de la masse et l'hypothèse du seuil de transport. Celui-ci nous a permis d'observer la forme des enregistrements sédimentaires dans la stratigraphie du cône, en lien avec des variations de forçages. De nouveau, ce modèle est simplié : nous ne faisons varier qu'un seul forçage à la fois et négligeons l'inuence du ux de sédiment sur la pente du cône. Celle-ci est donc déterminée uniquement par la taille des sédiments et le débit d'eau. Dans la nature, une augmentation du débit d'eau génère souvent une augmentation du ux de sédiment. Il serait donc intéressant de compléter ce modèle en incluant l'inuence du ux de sédiment sur la pente du cône, étudiée au chapitre 4.Enn, dans un dernier chapitre, nous avons montré comment les résultats obtenus expérimentalement pourraient être utilisés pour étudier les cônes naturels. Évidemment, la complexité des objets naturels nécessite d'adapter nos résultats. Nous avons cependant mis en évidence l'importance de caractériser les sédiments qui composent le cône, an de quantier les lois de transport qui contrôlent leur mise en mouvement. Les rivières qui forment les cônes alluviaux sont souvent des écoulements en tresse, or nos résultats expérimentaux, à l'exception du chapitre 3, reposent sur l'hypothèse d'un cône alimenté par un chenal unique. Pour reconstruire les ux à partir de la pente du cône, il est donc nécessaire de décrire avec précision le nombre et les caractéristiques morphologiques des chenaux qui composent la rivière alimentant le cône. Enn, du fait de la longue période de temps sur laquelle le cône se construit, celui-ci peut enregistrer les variations des ux dans sa stratigraphie. Pour reconstruire l'évolution de ces ux il serait intéressant d'avoir accès à la stratigraphie du cône, soit à l'aide de tranchées naturelles (incision de la rivière) ou articielles, soit en utilisant des techniques d'imagerie du sous-sol, qui permettraient d'étudier la géométrie des strates (pente et courbure).

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Remerciements

Chapter 4. Departure from threshold: nite sediment discharge accordingly. One way to investigate the variations of input parameters and the slope response to these variations is to study the fan stratigraphy.

Chapter 4

Departure from threshold: nite sediment discharge Comme nous l'avons vu précédemment, l'augmentation du ux de sédiment a plusieurs conséquences sur la morphologie et la dynamique de croissance d'un cône alluvial. Premièrement, l'augmentation du ux de sédiment augmente la fréquence d'avulsion [START_REF] Bryant | Experimental study of avulsion frequency and rate of deposition[END_REF][START_REF] Sinha | On the controls of uvial hazards in the north bihar plains, eastern india[END_REF][START_REF] Ashworth | Relationship between sediment supply and avulsion frequency in braided rivers[END_REF][START_REF] Assine | River avulsions on the taquari megafan, pantanal wetland, brazil[END_REF][START_REF] Reitz | Flooding and ow path selection on alluvial fans and deltas[END_REF]. Deuxièmement, en considérant la conservation de la masse, un ux de sédiment plus important accroît également la vitesse de croissance du cône [START_REF] Reitz | Experimental alluvial fan evolution: Channel dynamics, slope controls, and shoreline growth[END_REF]Powell et al., 2012, Section. 2.3). Troisièmement, une augmentation du ux de sédiment modie le prol en long du cône [START_REF] Whipple | Channel dynamics, sediment transport, and the slope of alluvial fans: Experimental study[END_REF][START_REF] Stock | Controls on alluvial fan long-proles[END_REF][START_REF] Guerit | Laboratory alluvial fans in one dimension[END_REF]. Dans ce chapitre, nous nous intéressons à cette dernière inuence. Dans le chapitre 2, nous avons montré que la pente d'un cône proche du seuil de mise en mouvement des sédiments atteint une pente critique, S h , qui est contrôlée par la taille des sédiments, d s , et le débit de uide, Q w . Pour comprendre comment l'augmentation du ux de sédiment éloigne la pente de cette valeur seuil, nous utilisons les travaux d 'Abramian et al. (prep) concernant la forme d'équilibre d'une rivière transportant des sédiments. Nous montrons ensuite comment l'étude de la pente d'un cône alluvial permet de remonter au ux de sédiment qui l'a construit. The normal force is proportional to the volume of the grain d 3 s , and the density dierence (ρ -ρ s ):

where d s is the grain diameter, g the acceleration of gravity, ρ and ρ s the densities of the uid and of the sediment respectively and α a shape factor. The tangential force depends on the ow-induced shear stress, τ , and the grain surface, d 2 s , according to

where β is another shape factor.

When the grain is at equilibrium, the ratio of the tangential force over the normal force is equal to the friction coecient, µ (de Coulomb, 1821)

When the ratio of the tangential force over the normal force is higher than µ, the grain is set into motion. In the context of uvial geomorphology, this Coulomb's law is expressed in term of the Shields number [START_REF] Shields | Anwendung der ahnlichkeits mechanik und der turbulenz-forschung auf die geschiebebewegung[END_REF], θ = τ (ρ s -ρ)gd s .

(A.4)

The critical Shields number corresponds to the threshold of sediment motion. It is related to the friction coecient, µ, through the shape factors [START_REF] Seizilles De Mazancourt | Forme d'équilibre d'une rivière[END_REF]:

(A.5)

The critical Shields number depends on the grain and uid characteristics, Re * , (Fig. A.2). A lot of studies, tried to experimentally measure this number, but at present there is no reliable experimental curve [START_REF] Shields | Anwendung der ahnlichkeits mechanik und der turbulenz-forschung auf die geschiebebewegung[END_REF][START_REF] Mantz | Incipient transport of ne grains and akes by uids-extended shield diagram[END_REF][START_REF] Yalin | Inception of sediment transport[END_REF][START_REF] Charru | Erosion and deposition of particles on a bed sheared by a viscous ow[END_REF][START_REF] Loiseleux | Onset of erosion and avalanche for an inclined granular bed sheared by a continuous laminar ow[END_REF][START_REF] Ouriemi | Determination of the critical shields number for particle erosion in laminar ow[END_REF][START_REF] Ouriemi | Sediment dynamics. part 1. bed-load transport by laminar shearing ows[END_REF][START_REF] Aussillous | Investigation of the mobile granular layer in bedload transport by laminar shearing ows[END_REF] Shields number, Particule Reynolds number, Dimensionless particule diameter, . From [START_REF] Julien | Erosion and sedimentation[END_REF].

The critical Shields number θ t and the particle Reynolds Re * depend both on the uid shear stress as a consequence [START_REF] Andreotti | Les milieux granulaires, Entre uide et solide[END_REF] propose to use another dimensionless number, to characterize the uid and the sediment: the Archimedes number Ar. This dimensionless number represents the ratio of gravitational an inertial stress over viscous stress.

Near the threshold, the Archimedes number is related to the particle Reynolds number

A.2 Experimental set-up

To calibrate the transport laws of our sediments, we use an independent set-up similar to that of Q s , Q w 5 cm Since the Reynolds number is below 500 in our ume (even when the uid is water), we may assume that the ow is laminar. For simplicity, we also assume that the bed is at, consequently the ow-induced shear stress is the same everywhere. The shear stress acting on the sediment thus follows Poiseuille's law:

where W is the width of the channel, and ν the viscosity of the uid. We then compute the Shields parameter, which represents the ratio of the ow-induced shear stress τ to gravity:

During each experiment we maintain constant uid and sediment discharges then, to calibrate a transport law, we reproduce this experiment with dierent sediment discharges. For each experiment, we calculate the Shields number using Eq. (A.8). We nd that below a critical value θ c , which correspond to a critical shear stress τ c , the sediment ux vanishes. Above this threshold, the sediment ux appears proportional to the departure from the critical Shields parameter (Fig. A.4):

For the silica: The belt and pulley system The conveyor belt and the motor are linked by a belt and pulley system. This allows for mechanical power, torque, and speed to be transmitted across axes. We 1 http://docs-europe.electrocomponents.com/webdocs/133e/0900766b8133e904.pdf 2 https://store.arduino.cc/arduino-motor-shield-rev3

B.2 Fluid discharge

The uid we use is a mixture of water and glycerol. The physical properties, density and viscosity of this mixture depend on its composition (Shankar and Kumar, 1994, Fig To measure the average radius of the fan, we calculate the distance from the apex of the fan to each pixel of the boundary. Because we assume our fan is axisymmetric, we consider that the mean radius is a good approximation of the fan size. We treat the variability of this radius as an error.

1 http://scikit-image.org/docs/dev/api/skimage.html 130 Appendix C. Measurements

C.2 Fan morphology acquisition C.2.1 Moiré pattern

To measure the morphology of our experimental fan, we use a phase unwrapping technique [START_REF] Bruning | Digital wavefront measuring interferometer for testing optical surfaces and lenses[END_REF][START_REF] Brémand | A phase unwrapping technique for object relief determination[END_REF]. A video-projector projects a pattern, composed of sinusoidal fringes with a grating spacing p and an incidence angle i, on the experiment ( On the contrary, when the surface is irregular, the phase is shifted proportionally to the height of the surface,

The phase expression, Eqs. (C.1) and (C.2), are modulo 2π. Consequently, the reconstruction of the surface map requires phase unwrapping procedure [START_REF] Limare | Optical method for measuring bed topography and ow depth in an experimental ume[END_REF]. Using a set of phase-shifting pictures, this procedure allows the removal of jumps larger than π between adjacent pixels (Limare et al., 2011, Fig. C.3 ).

This method allows for image-based measurements over a continuous surface at high frequency. We use the Light3D commercial software package to measure our fan topography every hour during an experimental run.

C.2.2 Calibration

To measure precisely the elevation of our experimental fan, we need to calibrate and estimate the accuracy of our measurement system. To do so, we scan the surface of a milk bath, and we use it 2 https://fr.ids-imaging.com/store/products/cameras/usb-2-0-cameras/ueye-se.html To correct this optical deformation, we apply a numerical correction to the measured data.

To do so, we estimate the lens-induced deformation at dierent elevations throughout the surface extension of the experimental set-up. We measure the elevation of reference surfaces (milk surface), at nine dierent elevations from 7 to 132 mm above the bottom of the tank, and compare the digital elevation models to the real elevation of the plane. In We can see that the residual error is mostly accounted by a polynomial of order two, and that a higher one does not improve the correction. Consequently, we t the deformed surfaces by a second-order polynomial. This function has 6 coecients which have to be determined.

The aim of our method is to determine the coecients of the polynomial for each elevation.

To do so, we t the polynomial function to the dierent planes and calculate the coecients. We Appendix C. Measurements where P xy is the coordinates vector (P xy = [1, x, y, xy, x 2 , ...]) of each Z point, Z meas is the elevation given by the DEM and Z true is the real elevation of the plane (i.e. h).

We then estimate the evolution of these coecients with the elevation, h. Appendix D

List of variables

The units are indicated to introduce the dimension of the variable.

Morphologie des cônes alluviaux par Pauline Delorme

Résumé

Les cônes alluviaux sont des dépôts sédimentaires qui se mettent en place à la sortie des reliefs en érosion. Cette position leur confère un rôle d'archive sédimentaire. Leur géométrie est déterminée par les trois paramètres clés que sont le débit d'eau, la taille des sédiments et le débit de sédiment. Dans ce manuscrit, nous présentons trois séries d'expériences qui permettent de quantier indépendamment l'inuence de chacun des paramètres d'entrée sur la morphologie du cône. Nous montrons que le prol longitudinal du cône est contrôlé par la rivière qui le construit. Lorsque le ux de sédiment est faible, la rivière alimentant le cône est au seuil de mise en mouvement des sédiments. En conséquence, sa pente, et donc celle du cône, dépendent uniquement de la taille des sédiments charriés et du débit d'eau. Cette pente augmente avec la taille des sédiments et diminue avec le débit d'eau. Les cônes composés de sédiment de mobilité diérentes permettent d'illustrer l'inuence de la taille des sédiments sur leur pente. Dans nos expériences, à l'instar des cônes naturels, le dépôt est ségrégé: les sédiments les moins mobiles se déposent en amont du cône, où la pente est plus forte. Cette ségrégation nous permet d'estimer la composition du mélange initial à partir de la géométrie du dépôt. Enn, nous avons montré que lorsque le débit sédimentaire augmente, la pente du cône s'élève au dessus de sa valeur seuil. Dans ce cas, puisque la rivière dépose les sédiments, le débit sédimentaire diminue vers l'aval, ce qui induit un prol concave. Nous montrons ainsi que la pente proximale du cône enregistre la valeur de ce débit sédimentaire. Finalement, nous présentons des résultats préliminaires permettant de prendre en compte l'enregistrement de la variation des forçages dans la stratigraphie du cône, et envisageons des pistes pour appliquer ces observations aux cônes naturels. Mots clefs: cône alluvial, rivière, seuil de mise en mouvement, archive sédimentaire Abstract Alluvial fans are sedimentary deposits growing at the outlet of drainage areas. This position makes them valuable sedimentary records. Three key parameters determine their geometry: the water discharge, the sediment size and the sediment discharge. In this manuscript, based on three series of experiments, we quantify independently the inuence of each parameter on the structure of the deposit. We nd that the morphology of the fan is controlled by the river that builds it. For a low sediment discharge, the river that feeds the fan is at the threshold of sediment motion. As a result, its slope, and therefore that of the fan, depend only on the size of the sediment and on the water discharge. The slope increases with sediment size and decreases with water discharge. Experimental alluvial fans composed of bimodal sediment, like to their natural counterparts, are segregated. The less mobile sediments are deposited rst, near the fan apex, where the slope is steeper. This segregation allows us to estimate the composition of the sediment mixture from the fan geometry. When the sediment discharge injected at the apex increases, the slope of the fan departs from its threshold value. Sediments are deposited along the fan, thus causing the sediment discharge to decrease downstream, generating a concave prole. As a consequence, the proximal slope of the fan records the sediment discharge. Finally, we present preliminary results showing how the stratigraphy of the fan records variations of the forcing (sediment and water discharges). We then discuss the applicability of these observations to the natural fans. Key words: alluvial fan, river, threshold of motion, sedimentary record