Anaïs Abramian

Self-organization of sediment transport in alluvial rivers by

Keywords: Geomorphology, granular matter, statistical physics, instabilities, sediment transport

An alluvial river builds its own bed with the sediment it transports. The channel bounds the ow, which in turns deforms the channel through erosion and deposition. This coupling between ow and sediment transport selects the shape and the size of the river. In this manuscript, we investigate it using laboratory experiments.

The rst ingredient of this coupling is gravity, which pulls the moving grains towards the center of the channel, thus continually eroding the banks. However, due to the roughness of the bed, the trajectory of a moving grain uctuates across the stream. The bedload layer is therefore a collection of random walkers which diuse towards the less active areas of the bed. In a river at equilibrium, this diusion counteracts gravity to maintain the banks.

When gravity and diusion are out of balance, their interaction causes an instability. Indeed, if an initially at bed of sediment is perturbed with longitudinal streaks, the owinduced shear stress is weaker where the ow is shallower. Therefore, bedload diusion induces a sediment ux towards the crests of the perturbation. This positive feedback induces an instability which can generate new channels. We suggest that this mechanism could initiate the braiding of alluvial rivers. amies, aussi, Morgane, Lauranne, Aline, pour votre soutien sans faille ! Merci évidemment à mes deux parents, qui, depuis toujours me soutiennent et m'encouragent dans tous mes choix : vous m'avez fait conance dès mon plus jeune âge, et c'est vous qui m'avez donné le goût d'apprendre. Merci à ma petite soeur Sophie : je t'aime et je t'admire énormément.

Auto-organisation du transport sédimentaire dans les rivières alluviales par Anaïs Abramian Mots-clés : géomorphologie, matériaux granulaires, physique statistique, transport de sédiments, instabilités.

Résumé

Les rivières alluviales construisent leur propre lit avec les sédiments qu'elles transportent. En eet, l'écoulement qui les parcourt entraîne et dépose des sédiments, déformant ainsi le lit qui le conduit. Ce couplage sélectionne la forme d'équilibre d'une rivière.

Dans cette thèse, nous étudions l'inuence du transport sédimentaire sur la forme et la stabilité d'une rivière alluviale. Pour ce faire, nous reproduisons des rivières en laboratoire en laissant s'écouler un liquide visqueux sur un lit granulaire. L'aspect du chenal ainsi formé dépend des débits de liquide et de sédiment injectés en entrée.

A l'aide de ces expériences, nous mettons en évidence les deux mécanismes qui contrôlent l'équilibre d'une rivière. D'abord, la gravité entraîne les grains vers le centre du chenal. Ce mécanisme érode continuellement les berges de la rivière, et tend donc à l'élargir. Cependant, les collisions d'un grain avec le lit dévient sa trajectoire dans la direction transverse à l'écoulement. Les grains se comportent ainsi comme des marcheurs aléatoires, qui, collectivement, diusent vers les berges de la rivière. A l'équilibre, cette diusion compense la gravité, et xe ainsi la forme de la rivière.

Lorsque la diusion prend le dessus sur la gravité, elle peut induire une instabilité. En eet, si on perturbe un lit sédimentaire avec des stries longitudinales, le cisaillement uide est plus faible là où l'écoulement est moins profond. Par conséquent, les grains diusent depuis les creux de la perturbation vers ses crêtes. Cette rétroaction déstabilisante pourrait générer de nouveaux chenaux, et expliquer la formation des rivières en tresses.

Remerciements

Merci Eric, merci Olivier. Vous m'avez proposé un sujet de thèse passionnant et vous m'avez accordé votre conance tout au long de ces trois années de thèse. Grâce à votre enthousiasme, votre exigence et votre patience, j'ai appris ce qu'était le métier de chercheur dans les meilleurs conditions. Vous m'avez transmis le plaisir que vous avez à travailler, et vous m'avez donné envie de continuer dans la recherche, en tout cas pour les quelques années à venir.

Merci également à Pascale Aussillous, qui a rapporté ce manuscrit avec beaucoup de bienveillance : c'était toujours un vrai plaisir de te croiser lors de ma thèse. Valérie Vidal a également rapporté mon travail : merci pour tes questions et tes remarques qui ont amélioré ce manuscrit, ainsi que le soin que tu as eu à relire ma thèse en détails. Je t'ai rencontrée dès mon M1 pour faire un projet expérimental sur le bruit des volcans . C'est sans aucun doute lors de ce projet que mon attrait pour la géophysique a débuté. Merci à Philippe Gondret, qui m'a suivie depuis mon M2 en tant que membre de mon comité de thèse, puis de mon jury de thèse ; merci à Philippe Claudin, dont les remarques sont toujours très intéressantes ; et enn merci Claude Jaupart, qui veille sur tous les doctorants du laboratoire avec attention et bienveillance, et qui est l'une des raisons pour laquelle nous nous sentons si bien en DFG.

Les permanents du laboratoire n'y sont pas pour rien non plus ! Merci à François (Métivier), le chef de la géomorpho, pour ta bonne humeur et ta disponibilité. Tu assures aussi bien lors de l'organisation de notre mission de terrain en Chine, qu'à la tête de l'UFR. Merci à Edouard (Kaminski) avec qui j'ai eu un immense plaisir à enseigner les TPs et les TDs et qui m'a appris beaucoup sur l'enseignement à l'université. C'était extrêmement agréable de pouvoir proposer des nouveautés, d'en discuter et, parfois, de les tenter. Merci à tous les autres membres permanents de DFG : ce labo est très chaleureux grâce à votre bonne humeur permanente. J'ai également enseigné avec des enseignants chercheurs d'autres laboratoires de l'IPG et j'y ai toujours pris un très grand grand plaisir. Je remercie tout particulièrement la team TP dont font partie Alex (Fournier) et Pascal (Godefroy), sans qui ces TPs ne fonctionneraient pas si bien. Enn, Zarie (Rouas) fait marcher cette UFR comme sur des roulettes, ce qui a rendu l'enseignement à STEP d'autant plus agréable.

Évidemment, la bonne ambiance qui règne au sein de DFG est aussi due aux doctorants. Merci aux vieux qui m'ont accueillie chaleureusement : Adrien, Alberto, Samuel, Loïc, etc. Merci aux jeunes qui rendent la pause café très longue conviviale : Cyril(s), Valentin, Viviane, Audrey, Alexandra. Enn, je remercie tout particulièrement Pauline, avec qui j'ai beaucoup interagi sur le travail, et pas seulement ! C'est toujours un plaisir de discuter avec toi.

Merci aux chercheurs hors du laboratoire qui m'ont suivie tout au long de ce travail de thèse : Sylvain Courrech du Pont, membre de mon comité de thèse, Pierre-Yves Lagrée, souvent en visite à l'IPG, qui a toujours de bons conseils. Merci à l'école doctorale, qui m'a permis de faire un séjour à l'étranger durant l'été 2017. J'ai fait partie, pendant 3 mois, de l'équipe de Shmuel Rubinstein (Harvard University, Cambridge, USA). Thanks Shmuel : you hosted me in the best conditions, and gave me some of your passion on mechanical instabilities. Thanks to Emmanuel (Virot) : I learnt a lot with you, and your references to all these old, fancy papers are always awesome. Thanks also to the PhD students at the SMR lab : Lisa, Ryan, Sam, etc., whom I spent also a very nice conference trip in Israel.

Merci à ceux que j'ai rencontré avant ma thèse, et qui m'ont donné envie de faire de la physique. Je pense notamment à Jeremie Palacci, ainsi qu'à deux de mes profs de prépa qui ont beaucoup compté lors de mon cursus : Emmanuel Angot et Marie-Pierre Rey-Nony.

Vos cours resteront des exemples pour moi.

Mes amis de toute part ont été formidables lors de ces trois ans, ainsi que le jour de ma soutenance. Merci à tous pour cette journée que vous m'avez faite passer. Merci aux anciens de l'ENS : Clara, Laura, car sans vous ma licence et mon master (et mes vacances) auraient été bien diérents ! Vous avez été aussi très présentes lors de ces 3 années. Merci Helmy pour nos discussions passionnantes, entre autre sur la physique. Merci à Valentin pour tout ce que tu as fait le jour de ma soutenance. Pendant ma thèse, j'ai eu l'occasion à plusieurs reprises de recroiser des personnes du M2 ; je pense en particulier à Pierre (Barrat-Charlaix) et à nos conversations interminables. Enn, merci à mes plus vieilles

Introduction générale

Une rivière alluviale, contrairement à une rivière à fond rocheux, s'écoule sur une épaisse couche de sédiments (Fig. 1). Lorsqu'elle creuse son lit, elle entraîne, transporte, dépose des sédiments, et façonne ainsi sa propre forme [START_REF] Leopold | River channel patterns: braided, meandering, and straight[END_REF]). Cette forme s'ajuste rapidement aux forçages extérieurs, qu'ils soient anthropiques, climatiques ou tectoniques, ce qui la rend constamment mobile. Par exemple, la rivière Kosi, qui prend sa source dans l'Himalaya et qui coule dans la région densément peuplée du Bihar en Inde, a changé régulièrement de lit au cours des siècles passés (Fig. 2, [START_REF] Gole | Inland delta building activity of Kosi river[END_REF]).

Ces modications brutales du cours d'une rivière, appelées avulsions, sont régulièrement observées dans les deltas ou sur les cônes alluviaux, où les sédiments sont déposés en masse par la rivière. Du point de vue géologique, les rivières jouent un rôle fondamental dans le cycle de formation et d'érosion de la croûte terrestre (cycle géologique). Elles incisent les reliefs, concentrent la majorité des ux de matière sur les continents et façonnent la surface continentale [START_REF] Allen | Earth Surface Processes[END_REF].

De l'amont vers l'aval, la rivière est successivement un lieu d'érosion, de transit, puis de dépôt. Sa charge sédimentaire est le résultat de l'interaction, soit chimique soit mécanique, entre l'eau et les roches. Selon sa charge sédimentaire, elle acquiert diérentes morphologies le long de son parcours. Une morphologie en tresses est associée à un transport de sédiments intense, tandis qu'une rivière à chenal unique se trouve généralement en plaine et transporte moins de sédiments [START_REF] Métivier | Alluvial Landscape Evolution: What Do We Know About Metamorphosis of Gravel-Bed Meandering and Braided Streams? Gravel-Bed Rivers: Processes, Tools[END_REF].

La rivière achemine ensuite la plupart de cette charge sédimentaire vers les océans (Anderson & Anderson 2010). Cependant, une partie des sédiments transportés par la rivière peut aussi s'accumuler à la sortie des chaînes de montagnes et former des dépôts stratiés sur des épaisseurs qui peuvent atteindre plusieurs kilomètres. L'analyse statigraphique de L'objectif de cette thèse est donc d'étudier l'inuence de l'apport en sédiments sur la morphologie, la dynamique et la stabilité d'une rivière. Pour ce faire, nous nous appuyons sur des expériences de laboratoire en conditions simpliées : lit sédimentaire composé d'une taille unique de grains et écoulement laminaire.

Dans le premier chapitre, nous présentons une synthèse des connaissances actuelles sur la morphologie des rivières et la modélisation du transport sédimentaire. En nous appuyant sur des observations de terrain, nous relions la morphologie des rivières à leur débit d'eau.

Ensuite, nous décrivons des études qualitatives concernant l'inuence du transport de sédiments sur la forme d'une rivière. Enn, nous présentons les modèles utilisés en transport sédimentaire.

Chapitre 1

État de l'art 1.1.3 Seuil de mise en mouvement : nombre de Shields Lorsqu'un uide s'écoule sur une couche de sédiments, il applique une force tangentielle à la surface du lit (Fig. 1.3a). Cette force s'exprime en fonction du cisaillement uide τ ,

f t = ατ d 2 s , (1.3)
où d s est la taille d'un grain, et α est un coecient sans dimension qui dépend de la forme du grain et de la nature de l'écoulement.

Cette force tend à déloger un grain tandis que son poids le retient. Ce dernier, normal à la surface du lit, s'écrit,

f n = β(ρ s -ρ f )gd 3 s , (1.4)
où ρ f , ρ s et g correspondent à la densité du uide, à celle du grain et à la gravité respectivement. Le coecient β dépend de la forme des grains : β = π 6 pour une sphère, ou β = 1

µ = f t f n = ατ β(ρ s -ρ f )gd s . (1.5)
Au seuil d'entraînement des grains, ce rapport est le coecient de frottement µ t . Si µ > µ t , l'écoulement est capable d'entraîner un grain.

En géomorphologie uviale, on utilise traditionnellement le nombre de Shields θ (Shields 1936) :

θ = τ (ρ s -ρ f )gd s = β α µ , (1.6)
Au seuil d'entraînement, le nombre de Shields prend la valeur

θ t = β α µ t , (1.7)
où θ t est appelé nombre de Shields critique.

Les expériences montrent que le nombre de Shields critique dépend au premier ordre de l'écoulement au voisinage du lit et de la taille des grains (Shields 1936[START_REF] Julien | Erosion and sedimentation[END_REF][START_REF] Bungton | The legend of AF Shields[END_REF][START_REF] Paphitis | Sediment movement under unidirectional ows: an assessment of empirical threshold curves[END_REF]. Cette dépendance peut s'exprimer en fonction du nombre de Reynolds de grain, déni par, Re s = u s d s ν .

(1.8) où u s est la vitesse du uide proche des grains, et ν est la viscosité cinématique du uide.

Des modèles semi-empiriques permettent d'avoir une expression du nombre de Shields critique en fonction du nombre de Reynolds de grain selon le régime turbulent ou laminaire (Fig. 1.4). Contrairement au cas d'un lit plat discuté dans le paragraphe 1.1.3 (Fig. 1.3a), une rivière forme un chenal dont la profondeur D varie en fonction de la coordonnée transverse y (Fig. 1.3b). Dans cette géométrie, le poids apparent du grain présente une composante tangentielle en plus de sa composante normale. Dans ces conditions, l'expression des forces normale et tangentielle appliquées sur un grain à la surface du lit devient

f 2 n = [β(ρ s -ρ f )gd 3 s cos φ] 2 ,
(1.9)

f 2 t = [ατ d 2 s + β(ρ s -ρ f )gd 3 s S] 2 + [β(ρ s -ρ f )gd 3 s sin φ] 2 , (1.10)
où φ est l'angle du lit par rapport à l'horizontale, dans la direction transverse à l'écoulement (Fig. 1.3b). La pente du chenal dans le sens de l'écoulement S est généralement comprise entre 10 -6 et 10 -1 , et a donc une contribution négligeable sur le poids d'un grain.

La condition de seuil s'écrit donc ατ β(ρ s -ρ f )gd s 2 + sin 2 φ = µ 2 t cos 2 φ .

(

La contrainte τ est induite par l'écoulement, qui lui-même dépend de la forme du chenal.

Il nous faut son expression pour fermer le système.

Dans un premier temps, nous considérons que l'angle φ est susamment faible pour utiliser l'approximation de Saint-Venant : nous négligeons le transfert de quantité de mouvement dans la direction transverse à l'écoulement, car nous considérons la largeur de la rivière bien supérieure à sa profondeur. C'est le cas de la plupart des rivières naturelles, pour lesquelles les rapports d'aspects varient entre 10 et 100 (Fig. 1.6). A l'aide d'un bilan de forces sur une section de la rivière, on déduit que la contrainte correspond au poids de l'eau projeté dans la direction de l'écoulement, τ = ρ f gDS , (1.12) où D est la profondeur du chenal, qui dépend de la coordonnée transverse y.

Régime turbulent

Dans le cas turbulent, on peut utiliser une relation empirique pour avoir une expression de la contrainte exercée par l'écoulement sur le lit :

τ = c f ρ f U 2 , ( 1
U = gDS c f . (1.19)
Dans ces conditions, les lois d'échelles de la largeur et de la pente deviennent Certains sont constants dans la nature, comme la densité de l'eau (ρ f ≈ 1000 kg/m

S = I gµ 3 c f 1 4 L 5 4 Q -1 2 w , (1.20)
W = πL S = π I µ 2 t c f gL 1 4 Q 1 2 w , (1.21
Q w = chenal U D dy = 4 9 µ 3 g ν L 4 S 3 (1.24)
Par conséquent, la théorie du seuil conduit aux deux lois d'échelle (Seizilles et al.

2013) :

S = µ t 4g 9ν

1 3 L 4 3 Q -1 3 w , (1.25)
W = πL S = π µ t 9ν 4gL 1 3 Q 1 3 w . ( 1

Conclusion

La théorie du seuil prédit correctement la forme en cosinus des rivières expérimentales dont les berges sont au seuil de mise en mouvement. Elle rend compte également, au premier ordre, de l'évolution de la largeur des rivières naturelles en fonction de leur débit (Fig. Le débit de sédiments résulte d'un échange continu entre les grains entraînés et déposés.

L'ensemble des grains mobiles forment une couche dont l'épaisseur est de l'ordre de la taille d'un grain d s . On peut dénir ainsi une densité surfacique de grains en mouvement n, et le ux de sédiments par unité de largeur s'écrit alors Cette vitesse peut s'exprimer en fonction du nombre de Shields,

q s = nv , (1.28
u s ∼ 18 θ V s , (1.31
v = c v V s θ , (1.33)
où c v est un coecient empirique, de l'ordre de 0.4 (Seizilles et al. 2014).

Concentration des grains en mouvement

La conservation de la masse de sédiments dans la couche charriée nous donne

∂n ∂t + ⃗ v ⋅ ⃗ ∇n = ṅe -ṅs , (1.34)
où ṅe et ṅs correspondent aux taux d'érosion et de sédimentation, respectivement (Fig. Les grains ont également une vitesse v donnée par l'équation (1.33). Proche du seuil,

lorsque θ ≈ θ t , on considère v ∼ α v V s , (1.40) où α v est égal à c v θ t .
Le ux de grains dans le sens de l'écoulement s'écrit donc, Supposant qu'une rivière s'ajuste de sorte que les grains qui composent son lit soient au seuil d'entraînement, sa section forme un cosinus (paragraphe 1.1.4). Cette théorie, connue sous le nom de la théorie du seuil, a été testée expérimentalement en conditions laminaires (paragraphe 1.1.5). Elle explique, au premier ordre, la loi de Lacey, selon laquelle la largeur d'une rivière naturelle est proportionnelle à la racine de son débit d'eau (Fig. 1 Enn, nous nous intéressons à la stabilité d'une rivière à chenal unique lorsqu'elle transporte des sédiments. Le chapitre 5 présente ainsi une instabilité associée à la diusion transverse du ux sédimentaire.

q s = α v V s n , (1.41
q s = α n α v d 2 s (θ -θ t ) . ( 1
σ 2 y (x) = 1 N N i=1 y i (x)

Introduction

At rst order, the water discharge of a river controls its shape and its size, as discussed in the rst chapter of this manuscript (section 1.1). Assuming its bed is at the threshold of motion, a river is bound to acquire a cosine cross section at equilibrium (Henderson 1961). This theory, known as the threshold theory, gives a physical interpretation of the empirical Lacey's law, according to which the width of an alluvial river is proportional to the square root of its discharge [START_REF] Lacey | Stable channels in alluvium[END_REF][START_REF] Métivier | Laboratory rivers: Lacey's law, threshold theory, and channel stability[END_REF].

By denition, this theory cannot take sediment transport into account. Field observations, however, indicate that this process inuences a river's shape (section 1.2). Specifically, rivers generally accommodate additional bedload by widening their channel. More remarkably, intense sediment transport may split the river course into a braid. Although these features have been observed previously in the laboratory, the relation between the shape of a river and its sediment discharge remains unclear and there is no theoretical framework to explain quantitatively these observations.

To address this problem, we choose to use laboratory experiments. We start by investigating the coupling between the shape of an erodible bed and its sediment transport in an idealised geometry: a sediment bed conned between two walls (ume). In this setup, a viscous uid shears the sediment bed, composed of a single size of grains. We use a viscous uid to keep the ow laminar and escape the complications of turbulence. The bed adopts a dierent cross section and downstream slope, depending on the uid and sediment discharges that we inject into the ume (section 2.3).

Our aim, then, is to model the bed's equilibrium. To do so, we track travelling grains to understand how they distribute themselves onto the bed (section 2.4). We then propose a model that couples the bed elevation and the local ux of grains (section 2.7).

Experimental setup

The experimental setup, sketched on gure 2.1, is a ume (length 1 m, width 3 cm) lled with plastic sediment grains (density ρ s = 1520 g/L). Grains have a non-spherical irregular shape; their size varies slightly from grain to grain, although the distribution of particle size is centered around a mean diameter d s = 830 µm (Fig. 2.4b, section A.1).

At the inlet of the channel, we inject a mixture of glycerol (40%) and water (60 %) (density ρ f ≈ 1160 g/L, viscosity ν ≈ 10 cP) at a constant discharge (Fig. 2.3).

Meanwhile, a conveyor belt brings grains from a tank to the channel and delivers a sediment discharge proportional to its pulley velocity. The pulley is controlled by a stepper motor, the velocity of which depends on the delay between two steps. This feeder stands on a scale which continously measures the sediment mass injected into the ume (Fig. 2.2a). The sediment input is stable and reproducible (Fig. proportional to the delay between two motor steps (Fig. 2.4b). Delorme (2017) describes this conveyor system in details.

As the uid ows above the sediment layer, it entrains some of the supercial grains.

The ux of grains is heterogeneous across the channel and seems higher in the center.

Conjointly, the bed adjusts progressively its shape and its downstream slope, and curves near its center. This is likely due to the connement of the uid between two walls, that makes the ow-induced shear stress vary across the channel.

After approximately one day, there is no visible change of the bed elevation and of the downstream slope; the bed seems to have reached steady state. This equilibrium is dynamical, meaning that sediment grains are constantly travelling along the ow. The sediment ux is more intense in the center of the channel, where the ow is deeper. On both sides, the bed forms banks made up of virtually immobile grains.

We perform 5 experimental runs, where we maintain the uid discharge 1 L/min, while varying the sediment discharge Q s (Table 2.1). For all these runs, we observe the same phenomena, although the bed seems to reach a slightly dierent shape.

Q w around Run Q w [L/min] Q s [g/
In particular, when the sediment discharge is higher, the bed seems atter in the center and the travelling grains are more numerous there.

In the following, we investigate how the bed evolves when the sediment discharge varies and how the sediment ux is distributed across the stream. To do so, we measure the bed cross section and the local sediment ux (sections 2.3 and 2.4). Based on these measurements, we then deduce the transport law (section 2.5) and the diusion length of our grains d (section 2.6). Finally, we model the relation between the bed elevation and the sediment ux.

Cross section 2.3.1 Bed elevation

At equilibrium, the active sediment bed reaches a shape which seems invariant in the ow direction x. Upon the interruption of the ow and sediment input, bedload transport stops almost immediately and the bed morphology freezes, while the uid slowly drains out of the channel. Once the sediment bed is dry, we can measure its shape based on the deviation of the laser sheet (Fig. 2.1). In this section, we present the calibration and the measurement of the dry bed elevation.

Laser sheet

A laser sheet is placed on a rail above the ume. With a stepper motor, we can translate it automatically over a distance of 20 centimetres in the downstream direction x (Fig. 2.1).

Its translation velocity depends on the pulley velocity and is about 2 mm/s. We control its velocity by an Arduino card connected to a computer. A camera placed above the ume translates with the laser and records its projection onto the bed surface, at a frame rate of 50 fps (Fig. 2.5a). The coordinate of this deviation depends on the bed elevation h through h = x laser tan θ laser , (2.1) where θ laser is the angle of the laser with respect to the horizontal, and x laser is the deviation of the laser line (Fig. 2.6). The coecient of proportionality in this relation, tan θ laser , is measured by a calibration, that we describe in the next paragraph.

Calibration

To calibrate our laser scanner, we swap the sediment ume for a tube lled with milk.

The latter combines a horizontal and white diusive surface, the height of which we can vary easily by adding or removing milk.

The laser sheet is projected onto the milk surface (Fig. 2.5a). We rst detect its position with image analysis. Figure 2.5b shows the intensity of the laser along a downstream line (white, dashed line of 2.5a); 0 corresponds to a black pixel, and 100 to a bright color (green, here). The intensity shows a peak, which corresponds to the location of the laser line. A t of the peak with a second-order polynomial allows us to locate the maximum of intensity.

Repeating this process, we extract the position of the entire laser line : x laser (y).

We then translate the laser over 20 centimetres in the downstream direction, and compute the average of the laser location ⟨x laser ⟩ (red, dashed lines of Fig. 2.7a).

Then, we sequentially raise the surface elevation by adding exactly 25.0 cL of milk per step. This addition of milk results in a level change of ∆h = 0.57 ± 0.08 cm (the total surface of the basin is 4.39 ⋅ 10 2 cm 2 ). This change of level moves the intersection of the laser sheet with the milk surface. We iterate our image analysis and locate the new laser deviation for dierent elevations of the milk surface. This procedure yields the relation between the surface elevation h and the averaged laser location (Fig. 2.7c):

h [cm] = (0.0031 ± 0.0002)⟨x laser ⟩ [pixels] (2.2)
which corresponds to a laser angle θ laser of 28.3 ○ .

We check the cross-stream atness of the surface by looking at the uctuation of x laser around its average for each step (Fig. 2.7b). We observe a curvature of the milk surface of less than 1 pixel of laser deviation, corresponding to an uncertainty on the height of 80 µm. This curvature might be due to surface tension, or to lens-induced distortion.

Finally, we can also measure the downstream slope of our laser scanner. To do so, we plot the averaged laser location ⟨x laser ⟩ as a function of the laser scanner position (Fig.

2.7d

). The downstream slope is less than 2 pixels in 20 cm, which corresponds to a slope of 0.3 . It is the slope of the scanner path with respect to the horizontal. In the following, we will assume that our scanner is perfectly horizontal, and treat this value as our uncertainty on slope measurements ( 2.3.2).

Bed cross section

Based on the calibration relation between the laser location and the surface elevation (equation (2.2)), we measure the elevation of the sediment bed, once the channel is dry, at the end of an experimental run.

To do so, we translate the laser sheet over a streamwise distance of 20 centimetres to scan the bed and obtain a movie with the laser deviation due to the bed elevation. First, we reduce the noise due to the bed roughness by averaging the movie over 10 consecutive frames.

Then, we locate the laser line x laser on each averaged frame with the procedure described above (Fig. 2.8a and b). The bed elevation h(y) is then proportional to the laser deviation.

Finally, we average the bed elevation on all the frames and obtain the prole showed on Each color corresponds to a run of table 2.1.

Downstream slope

At equilibrium, the bed cross section is invariant in the x-direction. However, the bed lowers continuously downstream, meaning that it has reached a downstream slope S. The corresponding shift of the laser line allows us to measure this slope.

When we plot the mean elevation of the bed as a function of the laser streamwise position, we nd a continuous increase (Fig. 2.9). The proportionality coecient of this curve is the downstream slope of the bed. In the case of the experimental run 2 (Tab. 2.1), S = (7.0 ± 0.3) ⋅ 10 -3 .

(2.

3)

The uncertainty corresponds to the slope of the scanner path, measured during the calibration (section 2.3.1).

We measure the equilibrium slope of the sediment bed for each experimental run. We observe that the latter increases with the sediment input (Fig. 2.10). This indicates that the bed self adjusts to the sediment input.

Flow depth

The laser scanner allows us to measure both the bed elevation h and its downstream slope S. However, we also need the ow depth to get a complete picture of the ume's cross section.

As we know both the bed elevation and the ume width, the position of the ow surface depends on the water discharge Q w , which reads:

Q w = ∬ u dy dz . (2.4)
where u is the ow velocity. Consequently, we rst calculate the ow velocity as a function of the surface elevation.

As the Reynolds number in our experiments remains around 10, the ow is laminar.

The downstream invariance of the ow then simplies the Navier-Stokes equations. Also, the ow is at steady state, which removes the dependence of the equation with time.

Finally, the ow velocity satises the Stokes equation, which takes the form of a Poisson equation in this geometry, η∇ 2 u = -ρ f gS .

(2.5) Equation (2.5) involves only parameters that we have measured, such as the slope S. We now supplement equation (2.5) with two boundary conditions. First, the velocity vanishes on the sediment bed and on the side walls (no-slip condition, u = 0). Secondly, the shear stress vanishes on the ow surface (free-surface condition, ∂ y u = 0).

To calculate the ow velocity u, we now solve equation ( 2 nite-elements software FreeFem++ (Hecht 2012). We use our experimental bed elevation as a boundary. We supplement it with two vertical side walls and x the surface elevation.

We can then compute the corresponding velocity eld inside this domain (Fig. 2.11a).

Then, the integration of the ow velocity across the stream yields the uid discharge for a given surface elevation. We iterate this process for dierent surface elevations. To deduce the ow surface in our ume experiment, we interpolate the elevation which corresponds to our uid discharge input (Fig. 2.11b).

In the case of run 2 displayed in gure 2.11, we nd the surface elevation at 1.06 ± 0.03 cm, which corresponds to a maximum depth of 0.79 ± 0.03 cm.

We measure the ow depth for each experimental run. Here again, we observe that the sediment bed adjusts its depth to the sediment input. In particular, it decreases as a function of the sediment discharge, although this trend remains of the order of the measurement uncertainty (Fig. 2.12).

Sediment ux

As the bed reaches its equilibrium shape, sediment transport distributes itself into a steady prole across the stream. In this section, we explain how we measure the sediment- ux prole across the ume.

Bedload movies

A camera (Canon 700D) placed above the ume records the motion of the particles over the bed with a frequency of 50 frames per second.

The camera eld of view covers an area of 6 centimetres in the sreamwise direction, and 3.5 centimetres in the cross-stream direction (slightly more than the entire channel width), with a resolution of 1280 × 1024 pixels. A LED light panel illuminates this area enough to x the time exposure at 0.01 second for a f/8 lens aperture.

The camera's memory allows us to record only 10-minutes movies. To improve the statistical relevance of our measurements, we record 10 sequences of 10 minutes for each run. Here, since our grains are colored, we track them with a colorimetric method. The image processing consists in three steps: (i) color conversion of the images, (ii) centre detection and (iii) trajectories reconstruction. In the next section, we provide details for each step.

HSV color system

Each frame is recorded in the RGB (Red, Green, Blue) color system, in which red, green and blue light are mixed to reproduce colors (Fig. 2.13a). However, the relationship between the amounts of red, green, and blue light and the resulting color is not intuitive.

The human eye may, for example, nd two colors close whereas they do not have similar coordinates in the RGB system (Table 2.

2). A transformation enables us to convert colors

from the RGB to the HSV color system.

Unlike RGB, HSV (Hue, Saturation, Value) color system separates the color information from the image intensity and lightness (Fig. 2.13b). The hue (H) of a color refers to which pure color it resembles (Fig. 2.13b). For example, the two colors that we considered previously, close in tints, have a similar hue (Tab. 2.2). More generally, all tints, tones and shades of a pure color have the same hue. The saturation (S) of a pixel encodes how white it is. A pure blue is fully saturated (S = 1), whereas tints of blue have saturations less than 1. White has a saturation of 0. The value (V) of a pixel, also called its lightness, describes how dark its color is. A value of 0 is black, and an increasing lightness moves the color away from black.

We now apply this conversion to our bedload movies. For a single frame, we compute the histogram of the hue (Fig. 2.14). We distinguish 6 major peaks which correspond to classes of grains (blue, red, orange, green, etc.). To track a class of grains, we dene a hue range. For example, the blue grains are in the range 081 (Fig.

Grain positions

Once we have chosen the range of hues, we discriminate pixels based on their saturation, which we require to be higher than 0.3. This criterion gives a map, characterised by clusters of pixels, and corresponding to individual grains of this class (Fig. 2.15). We detect the local maxima of this criterion, which gives us the positions of the corresponding grains.

To avoid detecting twice the same grain, we dene a minimum distance between two maxima. The number of detected grains depends on this parameter (Fig. 2.16a). We x it at 10 pixels, which corresponds to a grain size. We also convolve the criterion picture with a Gaussian lter to reduce the noise. Here again, the number of detected grains depends on the size of this lter, which we x at 10 pixels (Fig. 2.16b).

Dependence of the detection on the ow depth

Our tracking method is based on top-view movies. However, the ow depth varies across the channel. We have to ensure that the number of particles our algorithm detects is not biased by the uid depth.

To do so, we use an independent experiment. It is a at sediment bed, covered by a uid layer. No particle is moving. We simply detect, at rest, a certain number of grains as a function of the uid depth. We start from a uid depth of 30 mm and we sequentially reduce the uid depth by removing some uid with a syringe. Then, we plot the number of detected grains as a function of the uid depth (Fig. 2.17).

For a transparent liquid, there is no bias. The number of detected particles varies by less than ±5% (Fig. 2.17, in blue). This means that even if the uid depth is not constant across the channel, we are able to track particles without signicant bias, provided the

Grains trajectories

The method described above yields, for each frame, the positions of grains belonging to a same class. We now need to identify these grains on two successive frames to build their trajectories.

The simplest way to reconstruct a trajectory is to link each particle to its nearest neighbour in the next frame. This method, however, does not yield the most likely set of particle couples: the optimal set depends on the distribution of all the particles. Consequently, we use an equivalent of the Hungarian algorithm, which optimises a global cost function based on all particles positions (Munkres 1957, Heyman & Ancey 2014). We provide the corresponding codes in appendix A (section A.4).

With this procedure, we sometimes loose track of some particles, either because they collide with another particle and they appear as a single large particle, or because they are covered by travelling particles. We thus allow a certain number of frames over which a grain can disappear (typically 10 frames). Table 2.3 summarizes the parameters we choose for the whole procedure.

Then, we record the trajectories' coordinates and the time of their appearance. Figure 2.18 shows 15 trajectories from run 2. This enables us to check if the algorithm works, but it is dicult to do so for all trajectories (typically 30000 per experimental run).

In the next section, we compute the sediment ux based on these trajectories. We will also show how we evaluate the reliability of our measurements.

Q w = 1.1 L/min, Q s = 0.

Sediment-ux prole

Based on the trajectories of individual grains, we now compute the local sediment ux.

To do so, we dene an imaginary line perpendicular to the ume cross section (one of the dashed grey lines in Fig. 2.19b). Each time a trajectory crosses this line, we record the position y cross of the crossing. We then attribute to this position the value s, which is equal to +1 if the trajectory crosses the line in the downstream direction. Conversely, if a trajectory crosses this line in the upstream direction, s = -1. We then iterate this procedure for 100 imaginary lines (Fig. 2

.19b).

Based on these quantities, we dene the cumulative distribution of crossings over the ume cross section,

C(y) = y cross,i <y s i . (2.6)
According to this denition, this function starts from 0 on the left wall, increases with the cross-stream coordinate, and reaches its maximum on the other side of the channel.

As we tracked a class of grains, and not all the travelling grains, we normalise this distribution with the input sediment discharge that we measure independently with the scale (Fig. 2

.19c).

The derivative of the cumulative distribution provides us with the sediment-ux prole across the channel (Fig. 2.19d): q s (y) = dC(y) dy .

(2.7)

However, we rst need to smooth out the cumulative distribution to reduce the noise which the numerical derivation exaggerates. Thus, we x the number of bins N bins on which we interpolate the distribution (Fig. 2.19c, points). The aspect of the sediment prole then depends on the number of bins. For N bins = 200, the standard deviation on q s varies from 0.03 to 0.1, depending on the location in the channel (Fig. 2.20a). Conversely, a small number of bins ensures a good precision, but a low resolution (Fig. 2.20c). We nd that N bins = 40 is a good compromise between precision and resolution (Fig. 2.20b). It corresponds to a bin size of about 0.8 mm (approximately one grain size).

Overall, xing the bin size of the interpolation sets the number of particle crossings we record per bins (Fig. 2.20d). The more crossings we record (typically by recording long movies), the more accurate our measurement.

To evaluate the precision of our sediment-ux measurement, we select an area of the prole in which the sediment ux is homogeneous. Then, we measure the sediment ux for several bin sizes. As the sediment ux is homogeneous in the selected area, the measurement of q s should be equal. In practice, the ux slightly varies from one bin to another one due to experimental noise. We dene the relative error as the ratio between the standard deviation of these measurements std(q s ) and the mean sediment ux ⟨q s ⟩. As expected, the relative error decreases with the average of the number of crossings per bin N crossings (Fig. 2.21). We nd that the measurement is accurate within an error of 10 % if the number of crossings is higher than 3000, which is true near the center of the channel (with

N bins = 40). A t provides: std(q s ) ⟨q s ⟩ = 6.2 N crossings . (2.8)
In the following, we will use this formula to evaluate the error of our sediment-ux measurement, based on the number of crossings we use in each bin.

To further check if our method yields reasonable results, we measure the sediment ux on an uniformly-moving bed. To do so, we set an immobile dry sediment bed and we translate the camera at constant velocity (2 mm/s). Consequently, in the referential of the camera, all the grains move downstream uniformly. Then, we track the grains during 5 minutes. We therefore expect the sediment-ux prole to be homogeneous across the channel.

We obtain the cumulative distribution of gure 2.22a and use it to calculate the sediment-ux prole across the channel (Fig. 2.22b). As expected, the prole is approximately constant across the channel, and this ensures that there is no bias induced by the lighting conditions or the lens-induced distortions.

Figure 2.19 summarizes our combined measurements of cross section and sediment-ux prole for an experimental run. We nd that the shape of the sediment bed is correlated with its sediment-ux prole. Sediment transport is more intense around the center of the channel, and virtually vanishes near the banks. Due to viscous stress on the side walls, we expect less friction on the bed near the banks. This might explain why transport is weaker there. To investigate this hypothesis, we need to calculate the shear stress exerted by the ow on the bed. We also need to characterise our grains: to measure their transport law and to evaluate their diusion across the stream. We will do that in the next two sections, based on the measurements we just presented.

Transport law

In sections 2.3 and 2.4, we described our measurements of the cross section, the ow velocity and the sediment-ux prole. In this section, we infer the transport law of our grains from these measurements.

Flow-induced shear stress

The transport law is the relation between the density of entrained grains and the Shields parameter, which is proportional to the shear stress the ow exerts on the sediment bed (section 1.3). We have already measured the local intensity of transport with the sedimentux prole. We now need to compute the ow-induced shear stress.

In 2.3.3, we used nite elements to calculate the velocity eld of the ow u(y, z) in the channel's cross section. From this ow-velocity eld, we deduce the shear stress τ exerted by the ow on the bed:

τ = η e n ⋅ ∇u , (2.9)
where η is the viscosity, and e n is the unit vector normal to the bed surface. Figure 2.23a presents a typical shear-stress prole. As expected, this quantity varies across the stream (Fig. 2.23a): it is higher near the center of the channel and it vanishes at its corners, because spatial derivatives of the velocity vanish at this point (∂ x u = ∂ y u = 0).

In most experimental studies, the ow-induced shear stress is calculated using the Chapter 2 : Self organisation of sediment transport in a ume shallow-water approximation, according to which τ = ρ f gDS. To test the validity of this approximation in laboratory umes, we compare the shear stress deduced from equation (2.9) to the prediction of the shallow-water approximation (Fig. 2.23b). In the center of the ume, the shallow-water approximation overpredicts the shear stress by about 15%.

Near the side walls, this approximation further breaks down. It predicts a nite shear stress, whereas, in reality, the friction with the walls causes it to vanish.

Transport law

From the shear stress τ , we now calculate the Shields number by

θ = τ ∆ρgd 3 s . (2.10)
The prole of the Shields number is thus proportional to the shear-stress prole and varies across the channel. Near the banks, the Shields number is small and the sediment transport vanishes. Away from the banks, the Shields number increases, then crosses its critical value, and transport begins. Transport increases towards the center of the channel.

Therefore, a single experiment allows us to cover a wide range of Shields number θ and of sediment ux q s . We can now plot the sediment ux as a function of the Shields number for any cross-stream coordinate y in an experimental run (Fig. If we plot all the runs together, we obtain the gure 2.24. All runs gather around a common transport law, indicating that this transport law is a local relationship. It depends on the grains and the viscous uid, only.

Finally, we t a transport law on all these runs, and we nd q s = q 0 (θ -θ t ) , with θ t = 0.169 ± 0.001 (2.11) q 0 = (5.9 ± 0.2) ⋅ 10 4 grains/m/s .

(2.12) This transport law does not account for the eect of the bed cross-stream slope on the threshold of grains entrainment. This eect is usually embedded with the force ratio µ, linked to the Shields number by ( 1.1.4, equation (1.13)):

µ = µ t θ t θ 2 + (∂ y h) 2 .
(2.13)

As our sediment bed does not transport sediment over its entire width, we measure the sediment ux only in an area where the bed cross-stream slope remains smaller than 10 -2 (computed with nite elements) for run 1: Q w = 1.12 L/min, Q s = 0.65 g/min. This yields the transport law of grains with a single experiment. b. Sediment ux as a function of the Shields parameter for each run of table 2.1. All experiments merge on a single transport law. Black line: t function q 0 (θ -θ t ). We nd q 0 = 5.9 ⋅ 10 In conclusion, our experimental setup enables us to measure a transport law with a single experimental run. When the sediment input is higher, the range of sediment ux on the bed is larger and thus yields a transport law over a broader range of shear stress.

Comparison with the shallow-water approximation

Most experimenters measured the transport law with the shallow-water approximation.

This approximation neglects cross-stream diusion of momentum and, by denition, does not take into account the side walls. It predicts a uniform sediment ux on a at sediment bed, sheared by a ow of depth D, and yields an expression for the local shear stress:

τ = ρ f gDS . (2.14)
To compare our transport law to its shallow-water counterpart, we calculate the shear stress with equation (2.14). This expression involves the downstream slope S of the sediment bed, which we measured directly in our experiments, and the ow depth D. Assuming a one-dimensional Poiseuille ow, the ow depth depends on the uid discharge Q w as follows:

D = 3Q w ν gSW 1 3 , (2.15)
where W denotes the width of the channel. We can thus calculate the shallow-water shear stress for each experimental run, and the corresponding Shields number.

Regarding the sediment ux q s , it reads straightforwardly

q s = Q s W , (2.16)
by assuming a homogeneous sediment ux across the bed.

We report on gure 2.25 the transport law calculated with the shallow-water approximation. Each experimental run yields a data point. We t these points by equation (2.11) and nd θ t = 0.15 ± 0.01 and q 0 = (2.4 ± 0.6) ⋅ 10 4 grains/m/s, or, equivalently, q 0 d 2 s V s = 1.7 ± 0.5.

This corresponds to a relative dierence for the critical Shields number of about 6% and for the coecient q 0 of 60%. According to gure 2.23, the shallow-water approximation over-estimates the shear stress and therefore the Shields number, which explains the lower slope of the transport law.

Overall, our experiments conrm the classical assumption of bedload transport: (i)

there is a local transport law (all curves are superimposed on gure 2.24), (ii) there is a well-dened threshold. Furthermore, they reveal the eect of the side walls in a conned channel, and the limits of the shallow-water approximation. However, we did not investigate the inuence of the cross-stream slope of the bed, although this inuence plays a role in the denition of the threshold ( 1.1.4). In the next section, we characterize bedload diusion of our grains.

Bedload diusion

According to Seizilles et al. (2014), bedload grains collide with the bed as they travel downstream. The resulting deviations induce cross-stream diusion. A cross-stream ux q d appears, which is oriented towards less active areas,

q d = -d ∂q s ∂y , (2.17)
where d is a diusion length.

In our experimental ume, sediment transport is more intense in the channel center, resulting in a strong transverse gradient of concentration of bedload grains. As a result, bedload diusion must play a role in the dynamical equilibrium of the bed. over bigger grains (830 µm). We now proceed to measure the diusion length for our grains.

To do so, we use the grains' trajectories of 2.4.2. By shifting the trajectories so they all start from the same point (Fig. 2.26a), we nd that the grains disperse across the stream as they travel downstream. We measure the cross-stream variance of the particle positions Repeating this procedure for dierent experimental conditions (experimental runs of table 2.1), we nd that the diusion length remains constant as the intensity of sediment transport varies (Fig. 2.26c, inset).

These results conrm the diusive behaviour of our bedload grains and provide us with the intensity of the cross-stream ux induced by diusion. This ux is oriented from the most active part of the channel (the center) towards the less active ones (the side walls).

In the following, we investigate how this ux controls the bed's shape.

Bed equilibrium: Boltzmann distribution

As discussed in the previous section, the presence of a gradient of sediment ux in our experiments induces a diusive ux of particles oriented from the center towards the banks. This ux carves the bed near the channel center while depositing grains near the walls. However, as the bed cross section reaches steady state, another mechanism must oppose the diusive ux.

As rst proposed by Parker (1978a), gravity likely induces such a ux, because of the cross-stream slope of the bed. We expect gravity to pull grains down the slope and bring them back to the center (Parker 1978a, Kovacs & Parker 1994[START_REF] Seizilles | Forme d'équilibre d'une rivière[END_REF]). Assuming that gravity induces a cross-stream ux in proportion to the cross-stream slope ∂h ∂y and to the local sediment ux q s , we get:

q g = -γq s ∂h ∂y , (2.19)
where γ is a coecient in the range 0.11 [START_REF] Chen | Sediment transport on arbitrary slopes: Simplied model[END_REF]). This gravity ux vanishes when the bed is horizontal, for instance at the center of the channel.

In steady state, we expect that gravity counteracts diusion. Following this assumption, mass balance requires that the divergence of the transverse ux vanish. As the sediment bed is invariant along the ow direction (section 2.3), this translates into

∂q y ∂y = 0 , (2.20)
where q y is the sum of gravity and diusion uxes. This implies that q y is constant across the channel.

Since the cross-stream ux vanishes at the side walls, it must therefore vanish across the entire ume. Consequently, we get

q d = -q g .
(2.21)

Replacing the expressions of q d and q g (equations (2.17

q s ∝ e -h λ , (2.23)
where λ = d γ. The coecient of proportionality of this law depends on the limits of integration, which we will measure in our experiments.

If the above scenario holds, we therefore expect that sediment transport q s and bed elevation h self-organize so that q s decreases exponentially with h. This exponential distribution is reminiscent of the equilibrium governing the density of Brownian particles suspended in a liquid. These particles, for example colloids, are small enough to be sensitive to thermal agitation and their collisions with the molecules of the surrounding liquid make them diuse. Eventually, they sediment and reach a steady-state distribution. The competition between gravity, pulling the particles down, and diusion, which favors mixing, prompts them to follow a Boltzmann distribution. As a result, the density of particles decreases exponentially with the elevation. In this case, the exponential decay is the length

k B T mg,
where k B is the Boltzmann's constant, T the temperature of the surrounding uid, and m the mass particle. This length has been measured by [START_REF] Perrin | Les atomes[END_REF] to demonstrate the existence of atoms in uids, as suggested by [START_REF] Einstein | On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat[END_REF]. Since then, this exponential distribution has been applied to a wide range of systems in statistical physics According to this prediction, the density of bedload particles should follow a Boltzmann distribution. In the next section, we use our measurements to test this scenario in our experiments.

Boltzmann distribution in laboratory umes

Boltzmann law

Based on physical arguments, we proposed in the previous section that the shape of the sediment bed is dictated by the equilibrium between a diusive ux, due to grains collisions with the rough bed, and a gravity ux, induced by the cross-stream slope of the bed. If this scenario holds, we expect an exponential relationship between the sediment ux and the bed elevation. In our ume experiments, we measure both the bed elevation and the sediment ux at steady state (sections 2.3 and 2.4). We can use these measurements to evaluate this scenario.

We rst test the Boltzmann distribution on a single experimental run. To do so, we just need to plot the logarithm of the sediment ux q s as a function of the bed elevation h or conversely. In practice, the spatial resolution of our sediment ux measurement is lower than that of the bed elevation. To compare the two data sets, we therefore project the bed elevation onto the same grid using an interpolation (Fig. 2.27a and b). We can then plot the bed elevation h as a function of the logarithm of the sediment ux log(q s ). Figure 2.27c presents the result for the run 1 of table 2.1. In this space, the Boltzmann equilibrium predicts a straight line. We can t these data with a linear function, and propose a rst estimate for λ = 160 ± 10 µm.

If the theoretical scenario of section 2.7 holds, the Boltzmann length should not depend on the experimental conditions. Therefore, we now compare this distribution against all experimental runs of table 2.1.

Figure 2.28a shows the bed elevation as a function of the logarithm of the local sediment ux for all our experimental runs. Although the latter seem to have the same slope for sucient high sediment uxes (typically > 10 -1 grains/cm/s), each run appears to follow its own Boltzmann distribution. This is because the reference for the bed elevation is arbitrary: in our setup, it depends on the downstream slope and on the thickness of the sediment layer.

A way to set our elevation reference is to choose the lowest point of the bed h min as the origin, for each run (Fig. 2.28b). This has the inconvenient of basing the reference on a single measurement point. The advantage, however, is that it rids our bed elevation proles from this arbitrary value.

The resulting curves seem to accord with equation (2.23) for sediment uxes in the range of [10 -1 , 3 ⋅ 10 1 ] grains/cm/s. However, this accordance is less obvious for lower sediment uxes (< 10 -1 grains/cm/s). We suspect that this might be due to slow sediment motion, called creep [START_REF] Houssais | Onset of sediment transport is a continuous transition driven by uid shear and granular creep[END_REF]. We investigate this point in the next paragraph.

Creep

During our analysis, we do not discriminate mobile grains from static ones: we count all grains in our ux measurements. However, the grains caught in the static supercial layer are submitted to the collisions with moving grains [START_REF] Houssais | Onset of sediment transport is a continuous transition driven by uid shear and granular creep[END_REF]. Depending on their depth in the bed, they slowly move downstream, introducing an additional, small streamwise ux. We suggest this additional ux biases our measurements of low sediment uxes. In this section, we try to evaluate its intensity in our experiments.

In our bedload movies, we dierentiate visually the moving layer, scrolling downstream, from the static bed remaining in the background. To extract the latter automatically, we average the movies over a number of frames N f . Then, tracking grains with the procedure described in section 2.4 enables us to get the trajectories of the background particles. From trajectories, we deduce the particles velocities.

The distribution of their downstream velocity is slightly shifted away from zero (Fig.

2.29a

). Its average depends on the number of frames over which we average (Fig. 2 This range of velocities corresponds to a ux of intensity: (2.26)

q s,creep = ⟨v x ⟩ d 2 s , ( 2
They perform experiments with dierent shear stresses, above and below the threshold.

In this range of shear stress, they found a creep velocity of particles on the bed between 10 -1 and 10 -3 d s /s, which corresponds to

1.5 ⋅ 10 -6 < v creep V s < 1.5 ⋅ 10 -4 .
(2.27)

In order of magnitude, our measurement is comparable with that of Houssais et al. (2015). In all our experimental runs, the creep-induced ux is more than two and a half orders of magnitude smaller than the highest bedload ux q max . It may, nonetheless, explain the departure of our measurements from the Boltzmann distribution near the ume's side walls.

To visualize the comparative order of magnitude of creep in our experiments, we materialize the limit where the sediment ux reaches the order of magnitude of the creep with a vertical dashed line on gure 2.28b. In the following, we do not consider any more our sediment-ux measurement below this limit.

Finally, we check the cross-stream motion of the background particles. We found that the distribution of their cross-stream velocity v y , averaged over a movie, is centred around zero (Fig. 2.29a). Also, this holds regardless the number of frames over which we average (Fig. 2.29b). The fact that it is not exactly zero may be due to a small, misalignment of the camera with the ume.

Boltzmann length

We now take into account the creep motion of particles into our experimental verication of the Boltzmann distribution. After having translated the Boltzmann law of each experimental run by choosing a reference for the bed elevation (Fig. 2.28b), we remove the small sediment uxes from our data (typically < 10 -1 grains/cm/s: right hand-side of the gray limit, Fig. 2.28b). The data points then gather around a straight line, as predicted by our model, over about 3 orders of magnitude for the sediment ux. The coecient of proportionality should be the inverse of the Boltzmann characteristic length λ. We now estimate this length more precisely in our experiments.

In choosing h min as the reference for the bed elevation, we set the proportionality coecient of equation (2.23). The minimum elevation corresponds to the center of the sediment bed. At the center, the sediment ux reaches its maximum q max . We thus expect all the runs to follow:

q s = q max e -(h-h min ) λ .
(2.28)

To check equation (2.28) against our observations, we now plot hh min as a function of the logarithm of q s q max (Fig. 2.30). We then deduce the value of λ by tting each data set independently (Fig. 2.30, inset). We observe that the Boltzmann length remains around the same value and we thus t all experimental data together. A linear t yields:

λ = 120 ± 4 µm , (2.29)
and thus, λ = (0.144 ± 0.005) d s .

(2.30)

Equipped with this measure of λ, we can now estimate the parameter γ, which accounts for the gravity-induced ux of sediment. By denition of λ, it is the ratio between the Normalized bed elevation as a function of q s q max for each experimental run (Tab. During all our analysis, we set the number of bins N bins over which we calculate the derivative of the cumulative sediment distribution. With this procedure, we deduce the sediment-ux prole (section 2.4). We now test the sensitivity of the Boltzmann length λ to N bins . We still observe the Boltzmann law for any bin size N bins . However, for a very small number of bins, the data are too sparse to t them correctly. For N bins high enough, the Boltzmann length λ does not vary signicantly (Fig. 2.31).

Conclusion

In this chapter, we investigated the coupling between ow and sediment transport in the simplest possible conguration: a ume with a xed width. Even then, a sediment bed sheared by a ow is not a one-dimensional system, but rather generates a curved bed at equilibrium. Simultaneously, the coupling between ow and sediment transport adjusts the sediment-ux prole.

To characterise the bed's equilibrium, we measured its cross section and its local sediment ux. We then identied two mechanisms involved in this equilibrium. First, a diusive ux brings grains from the center of the channel towards the side walls. Secondly, gravity pulls grains down the slope. The balance between these two mechanisms governs the bed's equilibrium.

The signature of this gravity-diusion equilibrium is the Boltzmann distribution, which predicts that the sediment ux decreases exponentially with the bed elevation. The bed then adjusts its shape and its slope to maintain this relation.

However, natural rivers are not conned. They have an additional degree of freedom: they adjust their own width. In the next chapter, we investigate if a sediment bed also self-organises towards a gravity-diusion equilibrium when we remove the side walls. To do so, we build an experimental setup to reproduce laboratory rivers that transport sediment.

Chapter 3

Self organisation of sediment transport in an active laboratory river

Introduction

In the previous chapter, we investigated the coupling between ow and sediment transport in a ume. When a sediment bed is conned between two walls and sheared by a viscous ow, it adopts a curved shape. We showed that, at equilibrium, the bed elevation and the sediment transport adjust to balance cross-stream diusion and gravity. This equilibrium translates mathematically into the exponential decay of the sediment ux with the bed elevation (Boltzmann distribution).

Could this adjustment mechanism control the cross section of an active river ? The adjustment of this conned sediment bed seems indeed analogous of that of a natural river transporting sediment. The latter diers from the ume only by its ability to adjust its own width.

To investigate the equilibrium shape of an active river, we aim at reproducing them in the laboratory. To our knowledge, only Ikeda et al. Here, we combine a viscous uid and low-density grains to prevent the growth of bedforms. As a result, our experimental river generates a stable single-thread channel (section 3.2). With time, it reaches steady state and exhibits a well-dened cross section, while the distribution of moving grains adjusts to its elevation.

In this chapter, we propose that this equilibrium is governed by the diusion-gravity balance introduced in chapter 2. To evaluate this hypothesis, we perform the same measurements in the laboratory rivers as in the ume: we measure the bed cross section and the sediment-ux prole (sections 3.3 and 3.4, respectively).

Laboratory rivers : experimental setup

To generate active rivers in the laboratory, we use an inclined plane (90 × 190 cm), covered with a layer of sediment (Fig. 3.1). We use the same sediment than in the ume experiment (Ch. 2). Accordingly, all grains are made of the same plastic (density ρ s = 1520 g/L), and have the same size (d s = 830 ± 80 µm, section A.1).

At the beginning of an experiment, we atten the dry sediment bed with a rake and carve a straight channel into it, from the inlet to the outlet. We inject at the inlet of this downstream, entrained by the ow.

During the rst hour of the experiment, the ow spreads over the entire bed and forms an almost uniform sheet of uid. After a few hours, the ow carves a path, usually along the initial one. During this transient, the laboratory river continuously extracts more grains than it deposits, and thus erodes the bed. As a result, the sediment ux in this channel is higher than the sediment discharge we impose (Fig. 3.4). Erosion is often heterogeneous along the ow: it is usually higher downstream. Gradually, the sediment ux reaches steady state, so that it is equal to the input sediment discharge Q s everywhere (Fig. 3.4).

At equilibrium, the ow forms a straight, single-thread channel of a few centimetres in width (Fig. 3.2). This equilibrium is dynamical: grains are constantly dislodged and deposited by the ow, but the sediment discharge is constant and the river bed does not evolve with time. The moving grains indicate that the bed is above the threshold of entrainment, unlike channels with a vanishing sediment discharge (Seizilles et al. 2013).

We x the slope of the frame before an experiment, but the layer is thick enough that the laboratory river can adjust its slope according to its sediment and uid discharges. If the frame's slope is steeper than the equilibrium slope of the river, the ow incises the bed and forms a gorge to reach equilibrium. Conversely, if the initial slope of the frame is lower than the equilibrium slope of the river, the sediment is deposited at the inlet, forming an alluvial fan. We adjust the slope of the frame as close as possible to its expected value.

Overall, the equilibrium shape of the laboratory river is mainly controlled by the input parameters: the uid and sediment discharges. To investigate the inuence of the sediment discharge on the river's shape, we perform a series of experimental runs, where we keep the uid discharge around 1 L/min and we vary the sediment input (Table 3.1). For each run, we measure the bed's cross section and the sediment-ux prole along the cross section. We describe these measurements in the following sections.

River cross section

For a moderate sediment discharge, our rivers exhibit a well-dened cross section over a distance of approximately 1.5 meters downstream of the inlet. In this section, we describe how we measure the river's cross section. This measurement diers from that of chapter 2 (section 2.3) in that laboratory rivers are not strictly invariant in the streamwise direction.

Consequently, we cannot scan the topography over more than one centimetre. Moreover, the size of the plane precludes the use of a milk bath to calibrate the bed-elevation measurement.

That being so, we still use an inclined laser placed in an area above the channel. This area is far enough from the inlet and outlet to avoid perturbations due to the entry and Run exit conditions. The laser sheet is projected onto the channel in the eld of view of a camera (Canon 700 D). We choose the angle θ laser of the laser to focus in an area where the channel is fairly straight.

Q w Q s
To measure the laser angle θ laser , which depends on the experimental run, we place an aluminium scale in the eld of view of the camera. Its shape is known accurately (machined piece): each step is 1.0 cm wide and 2.0 mm high (Fig. 3.5, left). The laser sheet crosses the stairs and is deviated further at each step. We rst locate the laser deviation x laser with the same method as in 2.3.1 (Fig. 3.5a, dashed red line). This method provides us with a relation between the scale elevation h and the position of the laser line x laser . For example, in the case of run 4 (Tab. 3.1, Fig. 3.5b):

h [m] = (3.451 ± 0.002) ⋅ 10 -5 x [pix] . (3.1)
The coecient of proportionality is the tangent of the laser angle, tan θ laser , which corresponds to a laser angle of θ laser = 27.6 ○ .

We then remove the scale to let the laser intersect the channel's bed. As the experiment is running, the uid still ows in the river. Therefore, the laser line is deviated by the uid to the position x ′ laser (Fig. 3.6). Although the laser and the camera do not move, we record a movie to average the laser position with time. We then locate x ′ laser with the method

θ ′ laser = π 2 -arcsin 1 n sin π 2 -θ laser , (3.2)
where n, the refractive index of the water-glycerol mixture, is 1.41 at 20 ○ C. This yields θ ′ laser = 38.9 ○ .

Then, we stop the uid and sediment inputs. As a result, bedload stops almost immediately, and water slowly drains out of the plane, leaving a dry river bed. Now, the laser line crosses the dry river bed at the coordinate x laser (Fig. 3.6). Based on the laser angle θ laser , we thus measure the bed elevation h with equation (3.1) calibrated on the scale. In practice, we average spatially the bed elevation by scanning the channel over about one centimetre in the streamwise direction. The channel is too sinous to average over a longer distance.

Finally, based on the wet and dry laser locations (x ′ laser and x laser respectively), we calculate the ow depth D using geometrical relations:

D = x laser -x ′ laser 1 tan θ laser -1 tan θ ′ laser (3.3)
with our measurements of θ laser and θ ′ laser .

We show on gure 3.7c an entire cross section for the run 4 of table 3.1. As we are not able to average the ow depth downstream, it is noisier than the bed elevation. The resulting uncertainty on the ow depth is approximately ∆D = 0.5 mm .

(3.4) Figure 3.8 shows the cross section of 3 experimental rivers produced with the same ow discharge, for dierent sediment discharges. In the absence of sediment input, the river resembles the cosine shape of the threshold theory (section 1.1). As we increase the sediment discharge, the river widens and attens.

From the river cross sections, we extract the channel width W , maximum depth D max , and aspect ratio W D max . Figure 3.9 shows the variation of these 3 quantities with the sediment discharge for a xed uid discharge (Tab. 3.1).

We nd that the width increases of about 50% when the sediment discharge varies from 0 to 0.77 g/min. Meanwhile, the depth seems to decrease, although the variability of our measurements is comparable with the amplitude of this trend. The aspect ratio of the laboratory river increases with sediment discharge, by a factor of about 2.

These preliminary observations show that the intensity of the sediment discharge inuences not only the shape of a river but also its size. To put these observations on more quantitative grounds, we now turn our attention to the transverse prole of sediment ux.

Sediment-ux prole

We track the motion of the grains entrained by the ow using the method described in section 2.4. This colorimetric method enables us to track independently blue, red and orange grains. For each run, we record one hour of bedload movie. Table 3.1 indicates the corresponding number of frames and trajectories.

From the trajectories, we deduce the cumulative sediment ux. We normalize this cumulative distribution with the input sediment discharge Q s (Tab. 3.1, column 3). We then calculate the local sediment ux by averaging the cumulative distribution over 40 bins and deriving the resulting curve (as in 2.4.3).

As a result, we measure both the topography and the sediment ux in our laboratory rivers, for dierent sediment discharge Q s (Fig. 3.8). The area below a sediment-ux prole corresponds to the total sediment discharge, and naturally increases with it. We observe that the entire shape of the sediment-ux prole also varies with Q s . In particular, its maximum increases, although more slowly.

On gure 3.10, we plot this maximum sediment ux q max as a function of the sediment input Q s . As one might intuitively expect, it increases with the sediment discharge.

However, it seems to saturate at large sediment discharges. This suggests that the river accommodate a higher sediment discharge by widening its channel, rather than increasing the intensity of the sediment ux. We observe the same behaviour for the average of the sediment-ux prole as a function of the sediment discharge (Fig. 3.10, orange).

Overall, the cross section and the sediment-ux prole of a river appear to be correlated (Fig. 3.8). At equilibrium, the river adjusts its shape and its sediment ux to the total sediment input. Specically, the sediment ux is more intense near the center of the river, where the latter is deeper. It virtually vanishes towards the banks, as in the ume experiment (Ch. 2, Fig. 2. 19). Accordingly, we suggest that our laboratory rivers follow the same Boltzmann distribution, as in the ume experiment (section 2.8). In the next section, we test this hypothesis.

Boltzmann distribution in a laboratory river

In chapter 2, we found that a sediment bed conned between two walls and sheared by a viscous ow adopts a steady-state shape controlled by the equilibrium between diusion and gravity (section 2.8). This equilibrium translates mathematically into an exponential relationship between the bed elevation and the local sediment ux:

q s ∝ e -h λ , (3.5)
where λ is the characteristic length of the Boltzmann distribution. In this section, we test this scenario in our laboratory rivers.

Boltzmann distribution

We start by plotting the raw bed elevation as a function of the sediment ux for each experimental run of table 3.1 (Fig. 3.11a). The data gather around straight lines in the log-lin space, and thus each run seems to accord with equation (3.5). This suggests that a laboratory river self-organises to reach a steady state where gravity compensates cross-stream diusion.

In this setup, the camera and the laser move independently, instead of being xed on the same rail, like in the ume experiment. As a consequence, the reference for the bed elevation is arbitrary: it depends on the relative position between the laser line and the camera. Since the prefactor of an exponential distribution is just the reference elevation, each run appears to follow its own distribution.

We t each distribution and measure the corresponding characteristic length λ (Fig.

3.11b

). Although the latter increases slightly with the sediment discharge, it remains close to the value we measured in the ume experiment (section 2.7, λ = 0.14 d s ). We now want to compare the Boltzmann distribution in a river to the one we measured in our ume. To do so, we need to normalise each data set.

Comparison with the ume experiment

To superimpose each data set, we could use the normalisation of our measurements in the ume, by dividing the sediment ux by its maximum q max and subtracting to the bed elevation its minimum h min ( 2.8.3). However, our data are noisier than those of the ume experiments. Therefore, it is not appropriate to base the data normalisation on one data point only. We choose to normalise our data by their averages: ⟨h⟩ and ⟨q s ⟩.

We thus plot h -⟨h⟩ as a function of the logarithm of q s ⟨q s ⟩. We compare it with the following distribution:

q s ⟨q s ⟩ = e -(h-⟨h⟩) λ , (3.6)
with the value of λ we measured in the ume experiment : λ = 0.144 d s ( 2.8.3).

All the data gather around this law, regardless of the sediment discharge (Fig. 3.12).

The equilibrium of a river is thus governed by the same mechanisms as a conned bed: the balance between gravity and diusion. In addition, the characteristic Boltzmann length seems to depend on the sediment grains and the viscous liquid only, as the one we measured in the ume works for the river.

Unfortunately, the cross section of a laboratory river is often not perfectly symmetric, likely due to the sinuosity of its course. This explains the dispersion of the data around this law, especially for run 4, for example (Fig. 3.12). This dispersion is independent of our measurement noise (typical error bars are plotted for two data points). This suggests that the measurement of the Boltzmann characteristic length is more accurate with a ume experiment.

More generally, we expect that these mechanisms hold regardless of the nature of the ow (laminar or turbulent). Indeed, we did not do any assumptions on the ow regime to derive the Boltzmann distribution (section 2.7). In particular, grains should also diuse in a turbulent stream, even if the intensity of the diusion changes (and thus the diusion length). If conrmed, a ume experiment is then the ideal experimental setup to calibrate the Boltzmann law of natural rivers. One could use a conned bed of sampled sediment grains, sheared by water. This would enable us to measure the Boltzmann characteristic length of a natural river.

Inuence of the sediment discharge on the river's width

In the previous section, we showed that, at equilibrium, a laboratory river balances cross-stream diusion and gravity. As the river is free to adjust its width, we suggest that this mechanism also selects the river's width. We now focus on the inuence of the sediment discharge on the width of a river.

We thus produce 20 laboratory rivers, but we measure only their width (which demands less time than measuring the sediment prole). We visualize their planform geometry by dying the mixture of water and glycerol. Based on a color threshold, we measure the area of the river (Fig. 3.13a). Then, we get its curvilinear length with the function skeletonize() in the Scikit Image Python library (Fig. 3.13b). We then dene its average width as the ratio between the area and the curvilinear length.

In accordance with eld observations (Ch. 1, section 1.2), the width of our laminar rivers depends on their sediment discharge: the width increases with the sediment discharge (Fig. 3.14). In chapter 4, we propose a simplied model to explain the inuence of the sediment discharge on the river's shape.

Planform geometry

When we further increase the sediment input above about 1.6 g/min, the river slowly destabilizes and splits its course to form a braid, consisting of interlaced channels (Fig.

3.15)

. Some of them are active, other are immobile and sometimes abandoned. If we impose a sediment discharge much higher than this approximative threshold, the river destabilizes more rapidly, typically in a few tens of minutes.

This threshold above which a laboratory single-thread channel destabilizes into braids is in accordance with the existence of a critical slope (or aspect ratio) above which the river forms braids. This could also explain why single-thread channels are sparse at the laboratory: there exists only a small range of sediment discharge for which a single-thread channel is stable.

Although braided rivers move constantly, we can dene a dynamical equilibrium for which the sediment output equals the input [START_REF] Meunier | Sediment transport in a microscale braided stream: From grain size to reach size[END_REF]). However, we did not check this criterion in our experiments yet.

Conclusion

In this chapter, we successfully produced single-thread, active rivers in the laboratory by introducing a viscous uid and sediment at the inlet of a tilted sediment layer. The ow carves its bed in the sediment and, over time, adopts a well-dened cross section.

By analogy with a conned sediment bed (Ch. 2), we measured the cross section and the sediment-ux prole of these laboratory rivers. These measurements show that the sediment ux decreases exponentially with the bed elevation, regardless of the sediment input. This relationship indicates that the equilibrium of a river is governed by the balance between cross-stream diusion and gravity, like that of the sediment bed in a ume.

This statistical equilibrium probably controls the shape of a river's cross section, which depends on the sediment input. Specically, a river widens and attens with its sediment discharge. This eect qualitatively accords with eld observations (section 1.2), suggesting that laboratory rivers are controlled by the same mechanism than their natural counterparts.

Our observations also support that of [START_REF] Stebbings | The shapes of self-formed model alluvial channels[END_REF] about the transient of a laboratory river. In particular, he observed that a river (i) accommodates its additional bedload by widening and (ii) destabilizes into braids for larger sediment discharges. This suggests that rivers are stable for small to moderate sediment discharges, and might explain why active, single-thread channels in the laboratory are so sparse in the literature.

In the following chapters, we will further investigate the inuence of the sediment discharge on the shape of a single-thread channel. We will propose a physical model, that we will test against our experiments (Ch. 4). Then, we will focus on the stability of a single-thread channel (Ch. 5).

cm

Introduction

In chapter 3, we described how we maintain single-thread rivers which transport sediment in the laboratory. At a given uid discharge, we observe that the shape of these laboratory rivers depends on the sediment input. When the sediment input increases, the river widens, while becoming atter and shallower (Fig. 3.8).

To understand how a river adjusts its shape to the sediment input, we investigated its equilibrium at the grain scale by tracking bedload grains. We demonstrated that the river's equilibrium is governed by the balance between the cross-stream diusion of grains, due to the bed roughness, and gravity, that pulls grains towards the center. The signature of this balance is the Boltzmann distribution: the sediment ux decreases exponentially with the bed elevation (section 3.5).

In this chapter, we relate this grain-scale mechanism to the river's morphological parameters, such as its width, its aspect ratio, or its slope. To do so, we need to model the ow. In particular, we must evaluate the shear stress exerted by the ow on the river's bed. However, we do not know a priori the river cross section, and therefore the ow boundaries. These boundaries evolve through the coupling between the ow and the bed. Therefore, the calculation of the ow inside this domain depends on the shape of the domain. In particular, we are not able to solve analytically the Stokes equation in an arbitrary domain.

We start with the simplest possible model for the ow: the shallow-water approximation. This approximation provides us with a local expression for the shear stress that depends only on the depth D of the river.

Supplemented by a model for the transport law, which relates the local sediment ux to the local shear stress, we derive a rst-order dierential equation for the river's depth (section 4.2). Finally, we test this model against our experimental proles (sections 4.5 and 4.6).

Equation for the depth of an active river

In this section, we derive a dierential equation for the river's depth D. To do so, we use three equations that couple the ow depth D, the local sediment ux q s , and the ow-induced shear stress τ :

• The Boltzmann distribution relates the local sediment ux to the bed elevation, and thus the ow depth;

• The transport law relates the local sediment ux to the ow-induced shear stress;

• A ow model should link the ow-induced shear stress to the cross section of the river.

We rst detail these relations, and then derive the dierential equation that results from their combination.

Boltzmann distribution

At equilibrium, an active laboratory river self-organises to balance cross-stream diusion and gravity (section 3.5). As a result, its local sediment ux q s depends on its bed elevation h through, q s = q max exp h minh λ ,

where h min is the minimum elevation of the river, that we identify with its value at the center. This distribution implies that sediment transport reaches its maximum at the center (q s (h = h min ) = q max ), and decreases exponentially with elevation, towards the banks.

This expression came up naturally with the experiments of chapters 2 and 3, as it is easier to measure the elevation h of the bed when the river is dry, than the river depth (uncertainty on the elevation of ∆h = 0.1 d s compared to ∆D ≈ d s , section 3.3). However, noting that (Fig. 4.1)

h -h min = D max -D , (4.2)
the Boltzmann distribution (4.1) can be rewritten in terms of the river depth, D,

q s = q max exp D -D max λ , (4.3)
where we identify D max with the depth of the river at the center.

Equation (4.3), provides us with a rst relation between the ow depth and the local streamwise sediment ux. In turn, the sediment ux depends on the ow-induced shear stress, through a transport law. This is what we discuss in the next section.

Transport law

In chapter 2, section 2.5, we expressed the sediment ux as a function of the Shields parameter :

q s = q 0 (θ -θ t ) , (4.4)
with q 0 = 5.86 ⋅ 10 4 grains/m/s. However, this relation holds for a at bed, whereas the cross section of a river is not at.

In chapter 1, 1.1.3, we introduced the force ratio µ, which takes into account the crossstream slope on the threshold of motion. The parameter µ is the ratio of the tangential to the normal force exerted on a sediment grain,

µ 2 = (ατ ) 2 (β∆ρgd s ) 2 + ∂D ∂y 2 = µ t θ t θ 2 + ∂D ∂y 2 . (4.5)
The second term, D ′2 , takes into account the cross-stream slope in the force balance on a grain. As it provides a reasonable prediction of a river shape at the threshold of motion ( 1.1.4), we propose to generalize the transport law (4.4) to an inclined bed by replacing the Shields parameter θ with the force ratio µ,

q s = q µ (µ -µ t ) , (4.6)
with q µ = q 0 θ t µ t . This relation accounts for the local intensity of the ux, but does not say anything about its direction. We are not aware of any experimental observation that would support this heuristic hypothesis.

Equation (4.6) relates the shear stress to the cross-stream slope of the bed, and thus the derivative of the ow depth. Now, we need to approximate the ow-induced shear stress τ , which appears in the expression of θ, to close our system. In the following, we use the simplest possible approximation of the ow: the shallow-water approximation.

Flow-induced shear stress

We begin our analysis with a simple expression for the shear stress, given by the shallow-water approximation,

τ = ρ f gDS , (4.7)
where D is the depth of the river, and S its downstream slope.

According to this approximation, we neglect the transverse gradient of the ow velocity, and thus the transverse diusion of momentum. With this expression, the ow-induced shear stress is thus a local function of the depth. However, it remains coupled to the rest of the system, through the two quantities D and S.

Dierential equation for the depth of a river

We now combine equations (4.3), (4.6) and (4.7) to obtain a dierential equation on the depth of the river. According to the shallow-water approximation, the ratio µ reads:

µ = DS L 2 + ∂D ∂y 2 , (4.8)
where

L = θ t µ t ∆ρ ρ d s (4.9)
is the characteristic length, dened in 1.1.4, of the order of the grain size (equation

(1.14)).
On the other hand, equations (4.6) and (4.3) combine into µµ t = q max q µ exp D -D max λ .

χ = q max q µ = µ t θ t q max q 0 . (4.13)
In the rest of the present section, we will use only dimensionless variables. To simplify the notations, we omit the symbol ˜up to the end of section 4.

χ = D max -µ t , (4.14)
which is the dimensionless transport law in the center of the river, as χ is proportional to the maximum sediment transport q max . Accordingly, we rewrite equation (4.11) as:

( √ D 2 + D ′2 -µ t ) exp µ t -D s = χ exp -χ s . (4.15)
Finally, equation (4.15) involves two coecients, χ and s , which depend implicitly on the sediment and uid discharges of the river. In the following, we estimate their typical values in our laboratory rivers.

The slope parameter s

We call s the slope parameter as it is proportional to the downstream slope of the river, S. Unfortunately, we were not able to directly measure the slope of our laboratory rivers. As a rough estimate, we can use the slope of a threshold channel with equation (1.25) of 1.1.5, that we rewrite here:

S = µ t 4g 9ν 1 3 L 4 3 Q -1 3 w , (4.16)
which gives S = 4 ⋅ 10 -3 , and thus, s ≈ 10 -2 .

(4.17)

For a channel at the threshold of grains entrainment, the slope depends only on the uid discharge Q w ( 1.1.5), whereas it departs from its threshold value when the channel transports sediment. Therefore, we expect that a river transporting sediment reaches a higher slope, which would increase the value of s , while maintaining its order of magnitude.

We will propose a more accurate estimate of s in section 4.5. At this point, we will only keep in mind that s is a small parameter.

The transport parameter χ

The parameter χ is proportional to q max (equation (4.13)). For a river at the threshold, q max = 0 and we thus expect that χ = 0. For an active laboratory river, χ depends on the sediment-ux prole. For example, in the run 3 of table 3.1 (section 3.5), q max = 2190 ± 100 grains/m/s, and thus χ = 0.18 ± 0.03 . In section 3.4, we showed that the maximum sediment ux q max depends on the total sediment discharge Q s . It increases with the latter, until it seems to saturate around q max ≈ 4000 grains/m/s.

In the following, we investigate how equation (4.15) behaves mathematically. We will return to our experiments in section 4.5.

the center of the channel (y = 0). We dene δ as

D = D max -δ = (µ t + χ) -δ , (4.22) D ′ = -δ ′ , (4.23)
with δ > 0. We now assume that δ << D max .

Introducing these quantities in equation (4.21), we have, at rst order, a dierential

equation on δ, δ ′2 = 2 (µ t + χ) 1 - χ s δ = Aδ , (4.24)
where we dene

A = 2 (µ t + χ) 1 - χ s . (4.25)
For χ < s , A > 0 and thus δ ′ is negative, which corresponds to a convex bed. For χ > s , the solution is not physically acceptable. We therefore consider the positive case only.

Equation (4.24) then integrates into

δ = A 4 y 2 , (4.26)
which gives us the evolution of the depth and of the cross-stream slope near the channel's center. This provides us with two starting values of y and D for the integration, provided that χ < s :

y start = 4δ A , (4.27)
D start = µ t + χ - A 4 y 2 start . (4.28)
We use these expressions as initial conditions for the numerical integration.

This method enables us to solve equation (4.15) numerically for a couple of parameters ( s , χ). In appendix B, we evaluate this method by calculating the numerical error. We nd that the relative error of our numerical solutions is less than 10 -11 . In the following paragraphs, we discuss the numerical proles obtained with this method.

Proles

Inuence of the slope parameter s First, we set the value of χ to 0.1, and therefore the river depth to D max = µ t + χ (equation (4.14)). We then vary s , keeping it larger than χ. For large values of s , the river shape resembles a cosine (the threshold channel). When s decreases, the river widens and a at segment grows around its center (Fig. As we know the maximum sediment ux (q max = q µ χ), we can compute the sedimentux prole of these cross sections using equation (4.3). The sediment-ux prole also varies with s (Fig. 4.2b). Although its maximum value remains constant, its smallest value, at the banks, increases with s . For very low values of s , it virtually vanishes at the banks.

Inuence of the transport parameter χ

Now, we x s = 0.2, which sets the river downstream slope S. We chose this unrealistically large value for s to separate graphically the proles corresponding to dierent transport parameters. We then vary χ, which makes the river maximum depth vary. The ratio χ s still remains smaller than 1.

Compared to the threshold channel, where D max = µ t , our active rivers widen and grow a at segment around their center (Fig. 4.3a). The river depth increases from µ t to µ t + s when we increase χ from 0 to s .

The sediment-ux prole also varies with χ (Fig. 4.3b). For χ = 0.1, the channel is close to the threshold, and transports virtually no sediment. When χ increases, the sediment transport increases and its prole widens. The parameter χ, which is the maximum sediment transport, varies from 0 to s .

As expected, these results are in accordance with these of Seizilles (2013). Qualitatively, these cross sections also accord with our experiments. In particular, when a river transports more sediment, it widens by growing a at segment near its center. However, in the laboratory, we do not x χ and s but rather the uid and the sediment discharges. We thus can not compare directly these cross sections with our experiments now.

Phase portrait

In the previous section, we solved numerically equation (4.15) for given values of the parameters χ and s . To get a broader view on the behaviour of equation (4.15), it is useful to plot its phase portrait, which is a geometrical representation of all its solutions in the phase plane (D, D ′ ). Although the phase portrait does not provide us with analytical solutions, it provides us with qualitative insight. On gure 4.4, we plot the contours of the following function:

ξ(D, D ′ ) = ( √ D 2 + D ′2 -µ t ) exp D -µ t s , (4.29)
which corresponds to the left-hand side of equation (4.15). To each value of ξ corresponds a river, which takes the form of a trajectory in the phase plane. The resulting gure is the phase portrait of equation (4.15). As the function ξ depends on s only, the phase portrait is set by the value of s . For clarity, we rst chose s = 1, although it is an unrealistically ). Each trajectory in the phase portrait corresponds to a prole plotted below in the same color. Full, black lines: river at the threshold (χ = 0) and separatrix (χ = 0.99 s ). Black point: xed point, corresponding of an innitely-wide, at river (χ = s ). Red line: example of an actual river between the two boundaries. Each trajectory corresponds to a distinct value of D max between µ t and µ t + s . White, dashed line: trajectory corresponding to an unrealistic river. Each cross section is labelled by its value of χ.

0 t t + t + s Depth, D -t 0 t Slope, D' = s = 0.
large value.

In this phase portrait, trajectories are bounded on the left by the threshold channel which satises equation (1.13),

D 2 + D ′2 = µ 2 t , (4.30)
and is therefore a circle of radius µ t in the phase plane (Fig. 4.4). This trajectory corresponds to ξ = 0 . Each trajectory inside this circle corresponds to a river below the threshold and does not transport sediment. The parameter ξ is negative in this domain.

We now place a few river proles in this phase portrait. We chose, for example, χ = 0.5 s , and compute numerically the river prole D(y), that we plot in the (D, D ′ ) plane.

We obtain the red trajectory in the phase plane (Fig. 4.4). The left bank of the river corresponds to the point of coordinates (D = 0, D ′ = -µ t -ξ e -µt s ). From this left bank, the depth and the cross-stream slope simultaneously increase. In the center, D ′ vanishes and the trajectory thus crosses the D-axis in D max = µ t + χ. The depth then decreases, as the cross-stream slope increases to reach the right bank of the channel (Fig. 4.4). we expect s to be about 10 -2 ( 4.2.4). This suggests that the domain of our laboratory rivers is very thin. The xed point should consequently lie virtually on the threshold circle.

For any value of the slope parameter s , however, the transport parameter χ should be less than the slope parameter, χ < s , (4.36) With dimensional quantities, this criterion translates into:

q max S < λq 0 L θ t µ t . (4.37)
This means that, if we x the downstream slope S of the river, the maximum sediment ux is bounded regardless of the water discharge.

In the next section, we test these qualitative predictions against our experimental observations.

Experimental phase portrait

We now test the shallow-water model against our laboratory rivers (Ch. 3) by plotting our experimental cross sections in the phase plane. To do so, we need to plot the local cross-stream slope D ′ of a river cross section as a function of its dimensionless depth DS L. However, we could not measure the downstream slopes S of our laboratory rivers. To bypass this issue, we use the transport law to estimate the slope of a river:

q s = q µ (µ -µ t ) . (4.38)
In the channel center, using the shallow-water approximation, this transport law relates the sediment ux q max to its depth D max :

q max = q µ D max S L -µ t . (4.39)
We can estimate the channel downstream slope based on the above expression:

S = L D max µ t + q max q µ , (4.40)
We calculate the downstream slope of our experimental rivers with equation (4.40) (Tab. 4.1). We nd that it remains close to the slope of the threshold channel ( 4.2.4), and, as expected, slightly increases with sediment discharge. Now that we have an estimate of the downstream slope of a laboratory river, we can plot its trajectory in the phase plane. Figure 4.6a shows the phase portrait of an experimental run (Run 3, Tab. 3.1). All points gather around the circle of radius µ t , which corresponds to the threshold channel. They also accumulate near the maximum depth, where D ′ vanishes, which corresponds to the neighbourhood of the xed point.

According to the shallow-water theory, these points should lie between the threshold channel and the xed point. To locate the xed point in this phase portrait, we need to estimate the slope parameter s , which is proportional to the slope (section 4.4). As expected, s is of the order of 10 -2 , and the separatrix collapses almost perfectly with the threshold channel (Fig. 4.6a). Within the measurement uncertainty, we recover the qualitative behaviour of a river transporting sediment: it sets its bed near the threshold of motion.

An independent way to nd where the trajectory crosses the D-axis is to estimate the transport parameter χ = q max q µ . For run 3, we have χ = 0.18 ± 0.03. We report this value on gure 4.6a and nd that it lies near the xed point, although the uncertainty on the experimental trajectory is much larger than the size of its theoretical phase portrait.

If our theoretical scenario holds, all trajectories should gather near the threshold circle.

To verify that in our laboratory rivers, we now plot the phase portrait of all our experimental runs (Fig. 4.6b). Despite signicant noise, all the trajectories indeed gather around this circle, showing that all rivers self-organise close to the threshold, regardless of their sediment discharge.

A remarkable consequence of this hypothesis is that the trajectory of a river in the phase plane (DS L, D ′ ) should be a circle. Therefore, in the absence of any sediment-ux measurement, we can plot it in the plane (D, D ′ ) and adjust its aspect ratio to t a circle: the tting parameter on the D-axis should be proportional to its slope S. In principle, this method would allow us to measure the downstream slope of a river, based on its cross section only.

We test this method against our laboratory rivers. We start by plotting directly the cross-stream slope as a function of the dimensional depth D. It forms a trajectory that we t by an ellipse (Fig. 4.7):

D ′ + DS L 2 = µ 2 t . (4.41)
We deduce the adjustable values of the slope S and of the friction coecient µ t . For example, in the case of the experimental run 3 (Tab. 3.1), we nd µ t = 0.9 ± 0.2 and S = 0.008 ± 0.002.

We iterate this method for all our runs, and compare this measurement of the slope with the one measured with the sediment ux (equation (4.40)). We obtain gure 4.8.

These two methods yield comparable results, with a relative error of 13% in average.

The fact that a river is close to the circle of radius µ t in the phase plane thus provides us with a method to measure the slope of a river based on its cross section only. Moreover, it supports qualitatively the shallow-water approximation of a river. According to its phase portrait, a river self-organises close to the threshold of grain entrainment. Moreover, we measured the slope parameter s and the transport parameter χ of all our laboratory rivers in the present section (Tab. 4.1). These are the only two parameters that are involved in equation (4.15). In principle, provided that χ < s , we should be able to compute numerically the shape of a river based on the experimental measurements of χ and s . However, we nd that χ is always larger than s in our laboratory rivers, although we are able to measure both quantities with sucient precision (Tab. 4.1). This can be due to our assumptions on the transport law (equation (4.6)), which has never been established experimentally. Another possible failure of our theory is that the shallow-water approximation does not provide an accurate estimation of the shear stress, especially on the river's banks. Despite this discrepancy between theory and observations, the shallow-water approximation accords qualitatively with our experiments. However, this theory is based on two quantities, χ and s , that we can measure, but that we cannot x in an experiment. Now, we investigate the inuence of the parameters we can control directly in our experiment: the uid and the sediment discharges, Q w and Q s . In the next section, we thus investigate the evolution of the river's prole as a function of the input parameters.

Regime relations 4.6.1 Numerical regime relations

In this section, we investigate the evolution of a river's prole as a function of the input parameters: the uid and the sediment discharges (Q w and Q s respectively). To do so, we use numerical simulations to relate χ and s to Q w and Q s .

In this aim, we generate dimensionless river proles D(ỹ) for dierent couples of ( s ,χ), keeping χ < s (section 4.3 and 4.4), and using a numerical method dierent from the one discussed in 4.3.1. This new method is faster and less sensitive to precision issues (Appendix B). Given a numerical river prole, we can calculate its characteristics (width, aspect ratio, uid and sediment discharges) as follows.

For each numerical prole, we compute straightforwardly the dimensionless width and depth: The Boltzmann distribution then yields the sediment ux prole:

W = 2 max( ỹ) ,
qs = q s q µ = χ exp D -D max s . (4.44)
from which we deduce the total sediment discharge Q s by integrating qs over the river prole. In dimensional form, Q s reads:

Q s = q s dy = L S q µ qs dỹ . Knowing that s = λS L, the dimensionless sediment discharge Qs is thus:

Qs = Q s Q * s = 1 s qs dỹ , (4.46) where Q * s = q µ λ.
The uid discharge of a given river cross section, in dimensional form, is:

Q w = U D dy , (4.47)
where U is the vertically-averaged ow velocity. Assuming a one-dimensional Poiseuille ow prole, we get U = gD 2 S 3ν. In dimensionless form: where we dene

Qw = Q w Q * w = 1 3 s D3 dỹ , (4.48) 0 2 4 Sediment discharge, Q S /Q * S 0 1 2 Fluid discharge, log(Q S /Q * S ) a 0 2 4 Sediment discharge, Q S /Q * S 0 1 2 Fluid discharge, log(Q S /Q * S ) b s = 5 s = 3 s = 1 = 3 = 1
Q * w = gLλ 3 3ν . (4.49)
We thus have expressions for the width, the depth, the uid and the sediment discharges associated to a given river prole.

We now investigate the characteristics of a large number of numerical proles. To generate these proles while covering a uniform range, we randomly pick values of s and χ in given intervals (still with χ < s ). The randomness of this procedure rids us of the a priori denition of the number of proles: the longer the simulations, the more data points we obtain.

We calculate the dimensionless sediment and uid discharges of each prole with equation (4.46) and (4.48) respectively. Figure 4.9a shows the dimensionless uid discharge

Q w Q * w as a function of the dimensionless sediment discharge Q s Q * s for 100 proles.
Based on these values, we build a triangular grid (alpha-shaped Delaunay triangulation)

and associate a set of parameters ( s , χ, W , D) to each node of this triangulation. Interpolating these values linearly on each triangle, we can then plot the contours of any parameter.

Specically, gure 4.9b shows the contours of s and χ in this space. This enables us to relate the couple of parameters (Q w , Q s ) that we control in the laboratory to the parameters (χ, s ) that explicitly appear in equation (4.15). We nd that, if we want to reach large values of the sediment and uid discharges, we need to choose a large s , and χ as close as possible to s .

To explore larger ranges of s and χ, and increase the accuracy of the interpolation, we iterate this process for a few thousands of numerical proles. We thus obtain a wide range of river proles and can investigate not only the relation between (Q w , Q s ) and ( s , χ), but also the relation between any couple of parameters. In the following, we discuss successively the evolution of the aspect ratio, the width, the slope and the maximum sediment ux of a river as a function of its sediment discharge (regime relations).

We start by investigating the evolution of the aspect ratio as a function of the sediment discharge (Fig. 4.10a). On gure 4.10a, the triangular mesh indicates the extent of the domain covered by our numerical solutions. The shape of this domain is restricted by the numerical accuracy. Indeed, large values of sediment discharge require χ to be very close to s . For example, a sediment discharge Qs = 30 corresponds to a value of χ = (1 -10 -15 ) s , which is of the order of the numerical accuracy of a oating point number in Python. Thus, larger sediment discharges cannot be reached with our current numerical method.

Computing the contours of dimensionless uid discharges in the ( W D, Qs ) plane, we obtain curves that predict the evolution of a river's aspect ratio as a function of its sediment discharge for a given uid discharge (Fig. 4.10a).

In our laboratory experiments, we maintain the uid discharge around 1 L/min and the characteristic uid discharge is

Q * w = 7.9 ⋅ 10 -11 m 3 /s . (4.50)
As a result, the dimensionless uid discharge is about 10 5 , which corresponds to the magenta contour in gure 4.10a.

To compare this prediction to our experiments, we now place our data in this gure.

In our experimental conditions, the characteristic sediment discharge is

Q * s = 1.5 grains/s , (4.51)
or, equivalently,

Q * s = 2.6 ⋅ 10 -2 g/min . (4.52)
Normalizing the sediment discharge to this value, we plot the aspect ratio of our laboratory rivers as a function of the dimensionless sediment discharge (Fig. 4.10a, magenta dots). The resulting data lie signicantly below the theoretical prediction, showing that the shallow-water theory overestimates the aspect ratio for a given uid discharge by a factor of about 2.

We now investigate how the width of the river evolves as a function of the sediment discharge according to the shallow-water theory (Fig. 4.10b). Here, the width is normalized with respect to its value at the threshold W th for the same uid discharge (equation (1.26)), so that it is equal to one for a vanishing sediment discharge (Fig. 4.10b, black square).

Adding our experimental data, we nd again that the shallow-water model overestimates the width of a river by a factor of 2.

Repeating the same procedure for the slope, we nd that the shallow-water model yields a reasonable order of magnitude (Fig. 4.11a). However, we did not measure directly the slope of our laboratory rivers, but rather estimated it based on the maximum sediment ux, using the shallow-water approximation (section 4.5). The agreement between theory and experiments might be therefore biased.

Finally, our numerical simulations show that, according to the shallow-water theory, the maximum of the sediment-ux prole q max saturates with the sediment discharge for a uid discharge higher than Q * w (Fig. 4.11b). This evolution accords qualitatively with our experimental observations (section 3.4). However, once again, the theory fails to predict the experimental data quantitatively.

Overall, we have had a limited success when comparing our experiments to the model prediction. In the next paragraph, we relate our model to the analytical approximation of the shallow-water theory proposed by Seizilles (2013).

Asymptotic river

On gure 4.10a, we observe that the isolines of uid discharges decreases seem to converge towards an upper limit, meaning that the shallow-water model predicts a maximum aspect ratio for a given sediment discharge.

We speculate that this limit corresponds to the separatrix we observe in the phase portrait of equation (4.15) (section 4.4). Indeed, when χ get closer to s , its trajectory approaches asymptotically the separatrix. The resulting asymptotic river is close to the threshold at its banks, and transports sediment uniformly near its center (Fig. 4.12), as proposed previsouly by [START_REF] Seizilles | Forme d'équilibre d'une rivière[END_REF]. We now compute the aspect ratio of this asymptotic river as a function of its sediment discharge.

We name W T the width of the at segment where the river transports sediment (Fig. 4.12). The total sediment discharge then reads

Q s = q max W T = q 0 θ t µ t s W T . ( 4

.53)

As s = λS L (equation (4.12)), we get, .12 Asymptotic river. In brown: sediment bed. In blue: ow. W T is the width over which the river transports sediment uniformly. W H is the width of a river at the threshold of motion. W is the total width of the river. threshold, the total width of the river reads

Q s Q * s = W T S L , (4.54) with Q * s = q µ λ = q 0 θ t µ t λ .
W = L S π + Q s Q * s . (4.56)
In addition, the dimensionless depth of this channel is equal to µ t + s , and thus, its depth writes

D max = L S µ t + λ ≈ L S µ t .
(4.57)

The ratio of equation (4.57) and (4.56) then yields the aspect ratio (Seizilles 2013):

W D max = 1 µ t π + Q s Q * s . (4.58)
This equation is plotted on gure 4.10 (dashed black line). We observe that the trend collapses with the upper-bound limit for the aspect ratio, which conrms that the model predicts a maximum aspect ratio for a given sediment discharge.

The same reasoning can be followed for the width and the slope of the asymptotic river (dashed black lines in Fig. 4.10b and 4.11a, respectively).

Conclusion

In this chapter, we derived a dierential equation for the depth of the river (equation (4.15)) assuming the simplest possible model for the ow: the shallow-water approximation.

This equation involves two parameters, χ and s , which are linked to the maximum of the sediment ux q max and the downstream slope of the river S, respectively.

The dierential equation (4.15) reproduces qualitatively our experimental observations on a river cross section, provided χ < s . In particular, as a river transports more sediment, it grows a at, active segment near its center (Fig. 4.3).

To get an overview of the solutions to this equation, we then investigated the phase portrait of equation (4.15). We found that acceptable rivers are bounded by two limit trajectories: the threshold channel, and an innitely at and wide river, corresponding to the xed point in the phase portrait. The value of s sets the size of this physical-rivers domain. The theory also predicts that the transport parameter is bounded by the slope parameter: χ < s .

For small values of s , the two limits almost collapse and the phase portrait is reduced to the circle formed by the threshold channel. This is in accordance with our experimental observations: all our laboratory experiments form a circle in the phase plane (DS L, D ′ ), indicating that they self-organise close to the threshold of sediment transport.

However, by measuring the parameters χ and s in our laboratory rivers, we systematically found χ > s . We thus cannot predict the river's shape using the shallow-water theory. Despite this discrepancy, we proceeded to relate numerically s and χ with our control parameters: the uid and sediment discharges Q w and Q s , respectively. We obtained regime relations for the aspect ratio, the width, the slope and the maximum sediment ux of a river as a function of its input parameters. These regime relations accord qualitatively with our observations. Specically, the maximum sediment ux of a river saturates, meaning that the latter accommodates a higher sediment discharge by widening its channel. Unfortunately, the numerical regime relations fail to agree quantitatively with our observations.

We suspect that this discrepancy is due mainly to the shallow-water approximation. As discussed in chapter 2, this approximation predicts the shear stress with an error of more than 15% (section 2.5). However, in the phase portrait, the river trajectories are conned in a very thin domain and therefore are very close to each other. As a consequence, any error on the ow-induced shear stress, and thus on the force ratio µ, causes a drastic change in the river's shape.

The estimation of the shear stress could be improved by taking into account twodimensional eects in the ow. This is not an easy task, though: as we do not know the shape of the river, this would be a moving-boundary problem. If we address this problem numerically with nite elements, it would involve technical diculties associated to the evolution of the mesh as the boundary evolves towards the stationary shape.

Overall, this model supports that a river (i) self-organises close to the threshold of motion, (ii) widens to accommodate a larger sediment discharge rather than increase the intensity of the sediment ux. When the sediment input becomes too high, we observed that a laboratory river destabilizes into braids (section 3.7). In the light of the present chapter, we suggest that it cannot accommodate its shape any more to remain close to the threshold. This prompts us now to investigate the stability of an active river in the next chapter.

Chapter 5

Streaks induced by bedload diffusion

Abstract

A uid owing over a granular bed can move its supercial grains, and eventually deform it by erosion and deposition. This coupling generates a beautiful variety of patterns such as ripples, bars and streamwise streaks. Here, we investigate the latter, sometimes called sand ridges or sand ribbons. We perturb a sediment bed with sinusoidal streaks, the crests of which are aligned with the ow. We nd that, when their wavelength is much larger than the ow depth, bedload diusion brings mobile grains from troughs, where they are more numerous, to crests. Surprisingly, gravity can only counter this destabilising mechanism when sediment transport is intense enough. Relaxing the long-wavelength approximation, we nd that the cross-stream diusion of momentum mitigates the inuence of the bed perturbation on the ow, and even reverses it for short wavelengths. The diusion of momentum thus opposes that of entrained grains to select the most unstable wavelength. This instability might turn single-thread alluvial rivers into braided channels.

Introduction

When water ows over a granular bed with enough strength, it dislodges some of the supercial grains and entrains them downstream (Shields 1936[START_REF] Einstein | Bed load transport as a probability problem[END_REF], Bagnold 1973). As long as the ow-induced force is comparable to their weight, the entrained grains remain close to the bed surface, where they travel with the ow, until they eventually settle down. In steady state, the balance between entrainment and settling sets the number of travelling grains, which thus depends primarily on the ow-induced shear stress (Charru Bedload transport is often heterogeneousit scours away the bed somewhere, and deposits the entrained material elsewhere (Exner 1925). The ow then adjusts to the deformed bed, and alters the distribution of erosion and deposition. This uid-structure interaction generates bedforms through various instabilities (Seminara 2010, Charru et al.

2013).

Current ripples are iconic underwater bedforms found in streams, on beaches and sometimes in the sedimentary record [START_REF] Allen | Sedimentary structures, their character and physical basis[END_REF] In the above examples, the sediment grains travel along the force that entrains them. This is certainly true at rst order, but bedload particles slide and roll over a rough bed, which makes them waver erratically around their average trajectory (Nikora et al. To test this scenario, we investigate the stability of a at sediment bed sheared by a laminar ow. We begin with the shallow-water approximation (section 5.4), and extend our analysis to a full two-dimensional Stokes ow (section 5.5). We then consider a stream covered with a rigid lid, which might facilitate measurements in a laboratory experiment (section 5.6). Finally, we look for this instability in previous experiments (section 5.7).

Bedload instability

Base state

We consider an innitely-wide, at granular bed sheared by a free-surface, laminar ow (gure 5.1). A small slope S drives the uid along x, the streamwise direction, but we will neglect its eect on the weight of a grain later on. We further assume that the size of a grain, d s , is much smaller than the ow depth D. In steady state, the shear stress the uid exerts on the bed, τ , is the projection of its weight on the streamwise direction:

τ = ρgDS (5.1)
where ρ is the density of the uid, and g the acceleration of gravity.

To entrain a grain, the uid needs to pull it with enough strength to overcome its weight. Mathematically, this happens when the ratio of these two forces, θ, exceeds a threshold value, θ t . Shields (1936) dened this ratio as

θ = τ (ρ s -ρ)gd s , (5.2)
where ρ s is the density of a grain. below threshold, the bedload layer is empty (n = 0).

After equation ( 5.3), a ow near threshold can only entrain a sparse bedload layer.

Then, the velocity of the travelling grains is that of the uid near the bed, which is proportional to shear stress in a laminar ow (Seizilles et al. 2014). As a consequence, the average velocity in the bedload layer is proportional to Stokes' settling velocity, which reads

V s = (ρ s -ρ)gd 2 s 18η (5.4)
where η is the viscosity of the uid. The sediment ux, q s , results from the collective motion of the bedload grains:

q s = α v nV s , (5.5)
where α v is a dimensionless coecient. In a laminar ow, Seizilles et al. Equations (5.1) to (5.5) represent a uniform base state, both in the downstream direction x and in the cross-stream direction y. In the following, we add a perturbation to it to introduce bedload diusion.

Bedload diusion

Heterogeneity drives diusion. To introduce some of it in our system, we now carve streamwise streaks into the granular bed, in the form of a sinusoidal perturbation of amplitude h and wavelength λ (gure 5.1). The uid ow and the sediment bed remain invariant along x, and our system is now two-dimensional.

To illustrate the mechanism of the bedload instability, we consider, in this section, that the amplitude of the perturbation is much smaller, and its wavelength much longer, than the ow depth. With these assumptions, we expect the shallow-water approximation to yield a reasonable estimate of the shear stress τ , and therefore of the Shields parameter θ.

Both are then proportional to the local ow depth, Dh, and therefore of lesser intensity at the crest of the perturbation. Mathematically, θ = ρDS

(ρ s -ρ)d s 1 - h D .
(

According to equation (5.3), the bedload layer is thus denser in the troughs than on the crests. Its density reads

n = n 0 - α n θ 0 d 2 s h D (5.7)
where n 0 and θ 0 are the density of moving grains and the Shields parameter in the base state, respectively. Like equation (5.3), from which it is derived, the above equation only holds above threshold, that is, when

h D ≤ θ 0 -θ t θ 0 . (5.8)
This condition sets the maximum amplitude the perturbation h can reach before the following analysis breaks down.

Following Seizilles et al. (2014), we now treat the bedload grains as independent random walkers. As they travel downstream at the average velocity α v V s , their crossstream velocity uctuates around zero. We represent this process by a series of random sideways steps, the amplitude of which is a fraction of the grain size. Statistically, the accumulation of these steps generates a diusive ux, q d , towards the less populated areas of the bedload layer. As long as their density remains low, the moving grains do not interact with each other, and bedload diusion obeys Fick's law:

q d = -α v d V s ∂n ∂y , (5.9)
where we expect the diusion length, d , to be a fraction of the grain size. Seizilles et al.

(2014) found d ∼ 0.03 d s in their laminar experiment, and we are not aware of any other measurement of this empirical parameter.

The stage is now set for the bedload instability: diusion brings grains from the troughs, where bedload is more intense, to the less populated crests, thus furrowing further into the bed (gure 5.2). This mechanism, however, is countered by gravity, to which we now turn our attention.

Stability

As a grain travels over a slanting bed, gravity deects its trajectory downwards. For an arbitrary slope, the mathematical formulation of the resulting cross-stream ux is intricate, and its experimental evaluation challenging (Kovacs & Parker 1994). Fortunately, for a linear stability analysis, we may content ourselves with a rst-order formulation valid for shallow slopes, and near the threshold of motion. Accordingly, we assume that the crossstream, slope-induced ux, q g , is proportional to the density of moving grains, and to the cross-stream slope of the bed:

q g = -γα v V s n ∂h ∂y , (5.10)
where γ is another empirical parameterthe last we will need. Based on the observations of Yamasaka et al. (1987) in a wind tunnel, Chen et al. (2009) proposed that γ lies in the range 0.1 -1. Most probably, this value depends on the Reynolds number at the grain's scale, and on the relative density of the grain. To our knowledge, it has never been measured in a laminar ow.

The gravity ux, q g , brings bedload grains from crest to trough, and thus hinders the instability driven by the diusion ux, q d . Mass balance mediates this competition through the Exner equation:

∂h ∂t + d 3 s φ ∂ ∂y (q d + q g ) = 0 (5.11)
where φ and t are the packing fraction of the sediment bed and time, respectively. For randomly packed spheres, the packing fraction is about φ ∼ 0.6 [START_REF] Bernal | Packing of spheres: co-ordination of randomly packed spheres[END_REF].

The bedload density equation (5.7), the cross-stream ux equations (5.9) and (5.10), and the Exner equation form a closed system, which the slope-induced ux makes non-linear. We consider an innitesimal perturbation of the bed, and expand our equations up to rst order. When we combine its four equations, the resulting linear system takes the familiar form of the heat equation:

∂h ∂t - q s0 d 3 s φ γ - d D θ 0 θ 0 -θ t ∂ 2 h ∂y 2 = 0 (5.12)
where q s0 is the bedload intensity of the base state. The diusivity of this equation depends on the base state, and indeed can change sign. Its expression straightforwardly conveys the character of the two cross-stream uxes. The slope-induced ux, represented by γ, tends to increase the diusivity, and thus stabilises the system. Conversely, the diusive ux, represented by d , destabilises the bedas anticipated, bedload diusion drives the instability.

We now introduce the transport number, Tr, as the ratio of the two terms of the diusivity in equation (5.12):

Tr = γD d θ 0 -θ t θ 0 . (5.13)
Combining this denition with the bedload transport law (equation (5.5)) recasts the transport number into a dimensionless sediment ux:

Tr = γDd 2 s α v α n d V s q s0 θ 0 . (5.14)
Literally, Tr depends not only on the sediment ux of the base state, but also on the Shields parameter, θ 0 . Near the threshold, however, θ t becomes a good approximation for θ 0 ; the transport number then bears its name rightly.

When bedload is intense in the base state, the transport number is large, and so is diusivity in equation (5.12). Sediment transport therefore stabilises the bed. Conversely, a vanishing sediment ux brings the transport number arbitrarily close to zero. Eventually, the eective diusivity in equation (5.12) becomes negative, and bedload diusion destabilises the bed.

This surprising result can be understood as follows. After the shallow-water approximation, the density of moving grains is simply proportional to the local depth (equation (5.7)). As a consequence, its cross-stream gradient does not depend on the intensity of bedload in the base state. Conversely, the slope-induced ux is proportional to n 0 , and therefore to the downstream sediment ux in the base state. Intense bedload thus favours the stabilising mechanism, without aecting bedload diusion.

To formalise this discussion, we now perform the linear stability analysis of equation (5.12), an elementary undertaking that will serve as a guide for the next section. We rst introduce the following characteristic time:

T = φD 3 θ 0 α n α v V s d d s (5.15)
which, when lengths are rescaled with D, makes equation (5.12) dimensionless. Next, we express the bed perturbation as a sinusoidal wave of dimensionless amplitude ĥ:

h D = Re ĥ exp ik y D + σ t T , (5.16)
where k and σ are the dimensionless wavenumber and growth rate of the perturbation, respectively. Substituting this expression in equation (5.12) yields the dispersion relation of the instability:

σ = k 2 (1 -Tr) .
(5.17)

When the base state transports enough sediment to bring the transport number above one, the growth rate is negative for any wavenumber, indicating that all perturbations of the bed decay (gure 5.3). This oversimplied model transitions to instability drastically:

all wavelengths suddenly become unstable as the transport number is reduced below one.

Short-wavelength perturbations grow innitely fast, and equation (5.12) is then ill-posed.

In reality, of course, streamwise streaks will never grow innitely fast. This mathematical exaggeration is only due to some rash assumption. In the next section, we attempt to

x our model by dropping the shallow-water approximation.

Regularisation by the ow 5.5.1 Stokes ow

Bedload diusion reinforces streamwise streaks provided (i) bedload transport is weak, and (ii) the ow exerts a lower shear stress on the crests than in the troughs. When the wavelength of the perturbation is much longer than the ow depth, the shallow-water approximation ensures that the latter condition is fullled. Otherwise, this approximation unduly maintains the bedload instability. In this section, we replace the latter with a more accurate representation of the ow.

We still consider a streamwise-invariant, laminar ow, but its downstream velocity u is now a function of y and z (gure 5.2). This Stokes ow then obeys Poisson's equation:

η ∇ 2 u = -ρgS (5.18)
where η is the viscosity of the uid, and the Laplacian operator ∇ 2 applies to cross-stream coordinates only. We further assume that the ow does not slip along the bed surface, and that the uid surface is free of constraint. Mathematically,

u = 0 at z = h (5.19)
and ∂u ∂y = 0 at z = D .

(5.20) Solving Poisson's equation with the above boundary conditions yields the shear stress the ow exerts on the bed surface:

τ = η e n ⋅ ∇u at z = h (5.21)
where e n is the unit vector normal to the bed surface.

Like in section 5.4, the ow entrains supercial grains as bedload, thus altering the bed surface through erosion and deposition. Here, however, this coupling takes the form of a free-boundary problem in two dimensions, since the Exner equation transforms the domain over which Poisson's equation is to be solved. Fortunately, linearising makes its stability analytically tractable.

Let us rst consider the velocity prole of the base state, u 0 . Like the bed itself, it is invariant along the streamwise and cross-stream direction, and Poisson's equation reduces to the direction normal to the bed, z. Its solution is half the classical parabola of the Poiseuille ow:

u 0 = 9 θ 0 V s D d s 2 z D - z D 2 , (5.22)
which provides us with a velocity scale for the ow, namely θ 0 V s D d s .

The velocity prole of the base state equates the source term of Poisson's equation.

Therefore, the velocity perturbation is a solution of Laplace's equation or, equivalently, it is the real part of some analytical function of the complex coordinate y + iz. This velocity prole vanishes at the surface of the bed (equation (5.19)), the elevation of which is perturbed by sinusoidal streaks (equation (5.16) at t = 0). In addition, its normal derivative also vanishes at the surface of the ow or, equivalently, it is symmetrical with respect to this surface (equation (5.20)). At rst order, we nd:

u = u 0 -18 θ 0 V s D d s Re ĥ cosh(k) cosh ik y + iz D + k .
(5.23)

Figure 5.4 shows the iso-velocity lines of the rst-order ow, that is, the contours of the above expression. When the wavelength of the perturbation is much larger than the ow depth (λ ≫ D or, equivalently, k ≪ 1), the Stokes ow accords qualitatively with the shallow-water approximation: its contours constrict above troughs, indicating intense shear stress on the deepest parts of the bed. This distribution of shear stress favours the instability.

The iso-velocity lines tell a dierent story for short wavelengths (k ≫ 1). As the space between streaks gets narrower, the bulk of the ow cannot squeeze in between crests any more. The latter then nd themselves fully exposed to the ow, from which they shelter the troughs. As a result, shear stress is more intense on the crests than in the troughs, and bedload diusion will combine with gravity to wear the streaks down. We thus anticipate that the screening induced by the diusion of momentum across the ow, which is the signature of Laplace's equation, will provide us with the regularisation we lacked in section 5.4.

Dispersion relation

To turn the above reasoning into a quantitative expression, we rst need to calculate the shear stress on the sediment bed. Deriving equation (5.23) with respect to z, and evaluating the result on the bed yields the bed shear stress, and therefore the Shields parameter, at rst order:

θ = θ 0 1 -(1 -k tanh k)
Re ĥ exp iky D .

(5.24)

According to this expression, the perturbation of the Shields parameter is either in phase or in antiphase with the bed perturbation, depending on wavelength (gure 5.4). For long wavelengths (k ≪ 1), the Shields parameter approaches its shallow-water expression (equation (5.6)); it then reaches its maximum above a trough, and the bed is unstable. Conversely, for short wavelengths (k ≫ 1), equation (5.24) reduces to θ ≈ θ 0 1 + k Re ĥ exp iky D ,

which reverses the phase of the Shields parameter with respect to the bed perturbation.

The shear stress then concentrates on the crests, thus stabilising the bed.

Equipped with a new expression for the Shields parameter, we now return to the stability analysis of section 5.4. We substitute equation (5.6) for equation (5.24), and reuse the expression of the bedload uxes, which the new ow model leaves unaected (equations (5.3), (5.9) and (5.10)). Injecting these equations into the mass-balance equation (5.11) yields a new dispersion relation:

σ = k 2 (1 -Tr) -k 3 tanh k . (5.26)
This expression equals its shallow-water counterpart (equation 5.17 Figure 5.5a shows the trajectory of the most unstable mode on a stability map. As the transport number crosses its critical value, the bedload instability appears with a vanishing wavenumber, which quickly increases. The wavenumber of the most unstable mode then increases towards its maximum value k max , which it reaches when the base state transports no sediment (Tr = 0). The largest of all most unstable wavenumber, k max , is the positive solution of the following equation:

3k tanh(k) + k 2 1 -tanh 2 k = 2 .
(5.27) Numerically, k max ≈ 0.813. Returning to dimensional quantities, we nd that k max translates into a remarkably universal ratio. When the intensity of bedload vanishes, the most unstable wavelength, λ max , is simply proportional to the ow depth, regardless of any other parameter:

λ max D = 2π k max ≈ 7.73 . (5.28)
This result suggests a recipe for an experimenta hypothetical one for now. To identify the bedload instability unambiguously, one need only reduce the bedload ux until the wavelength of the instability reaches its minimum, which should be the above value. Unfortunately, laboratory experiments seldom abide by idealised theories; we will return to this question in the discussion (Section 5.7).

In addition to a wavelength, the most unstable mode also has a growth rate, σ max ≈ 0.300, which we calculate by injecting k max into the dispersion relation, and sending the transport number to zero. Like the wavenumber of the most unstable mode, the growth rate is at its highest when the base state transports no sediment (Tr = 0, gure 5.5c). This statement, counter-intuitive with dimensionless quantities, becomes preposterous in physical units: After equation (5.15), the dimensional growth rate also reaches its maximum when bedload transport vanishes. Since it is bedload that feeds the instability, we would expect its absence to turn o the growth of the streaks. This apparent paradox is due to the threshold of the sediment transport law (equation (5.3)), which brings the linear regime to its end.

End of the linear regime

As the ow-induced shear stress approaches the threshold, bedload vanishes from the base state, and so does the eciency of the gravity-induced ux. This, however, does not aect the sensitivity of bedload, which responds as strongly as ever to any perturbation of the ow. Therefore, while the stabilising mechanism vanishes, the unstable coupling between the ow and the bed persiststhe bedload instability then thrives. Such full-blown growth, however, will quickly meet its end. The linear expansion of the transport law holds as long as the perturbed bed remains above the threshold. This gets more demanding as the base state itself approaches the threshold (gure 5.6). We dene h max as the amplitude of the perturbation when the Shields parameter hits the threshold.

Mathematically, after equation (5.24), h max reads:

h max = d γ Tr 1 -k tanh k .
(5.29)

For any value of the transport number Tr, we calculate numerically the wavenumber k of the most unstable mode, and inject the result in the above expression (gure 5.6). We nd that, as the intensity of bedload vanishes (Tr → 0), so does the maximum amplitude of the most unstable mode. Our linear stability analysis then holds for imperceptible streaks only.

Once the perturbation has outgrown h max , the instability enters a non-linear regime, to which the present analysis grants no access. We speculate, however, that the perturbation could keep growing beyond this point, since its driving mechanismactive troughs and frozen crestsshould survive the end of linearity. (5.30)

Rigid lid

Apart from the factor of two in the above expression, the lid does not aect our system, at least according to the shallow-water approximation. This factor of two does not even propagate beyond equation (5.3) where, by virtue of the normalisation by n 0 and θ 0 , it retreats into the denition of the base state. We are thus left with equation (5.17) again, that is, the same dispersion relation as in section 5.4.

Based on the above reasoning, we expect the lid to aect the bedload instability only marginally, if at all. To check this formally, we now repeat the two-dimensional stability analysis of section 5.5, with a new boundary condition at the ow surface. Specically, we replace the free-boundary condition with a no-slip one. At rst order, we nd the following velocity eld: We rst disqualify the experiment of Charru et al. (2004a), the driving lid of which precludes the instability (section 5.6.2). Next, we need to consider the side walls that conne the ow in an actual channel. In principle, we should treat them as no-slip boundaries, and account for them even in the base state. This, however, will produce a base state that varies in the cross-stream direction, thus complicating tremendously the analysis.

u = 18 θ 0 V s D d s z D 1 - z D -Re ĥ sinh k sinh ik y + iz D + k , ( 5
Instead, we will content ourselves with a less ambitious analysis which, we hope, will yield an order-of-magnitude interpretation of laboratory experiments. Accordingly, we replace the side walls with free-sleep boundaries for the uid, and impervious boundaries for the sediment. Both conditions preserve the homogeneity of the base state, to the cost of mathematical rigour.

An expected consequence of the side walls is to select acceptable modes (gure 5.10).

For the perturbation to full the boundary conditions at the side walls, its wavenumber needs to belong to a discrete set of values:

k = nπ D W (5.36)
where n is an integer, and W is the width of the channel. Substituting this expression into the dispersion relation for an open channel, equation (5.26), we nd that no instability can grow in a too narrow channel. As the transport number vanishes, the rst mode (n = 1)

is unstable only when W D ≳ 2.26 (gure 5.10a). The higher the mode, the wider the channel needs to be to accommodate its growth.

Considering all acceptable modes in a given experiment, we can identify the fastest growing one, and treat it as the dominant mode (gure 5.11). In the parameter space made up by the transport number and the aspect ratio, the unstable domain is tiled with individual modes. When the transport number is below one, the order of the most unstable mode increases with the aspect ratio of the channel. Overall, the unstable domain is bounded by the same curve as the rst mode.

We now wish to place the laboratory experiments of Ouriemi et al. Preliminary though it may be, gure 5.11 suggests that laboratory channels are usually too narrow for the bedload instability to ourish.

Conclusion

The linear stability analysis we have presented in this paper identies a new instability associated to bedload transport, caused by the cross-stream diusion of the travelling grains. This instability produces bed streaks aligned with the ow, in the absence of any secondary currents. Because it can only grow near the threshold of sediment transport, the unstable perturbation quickly enters a non-linear regime which we have not investigated.

The bedload instability is sensitive to boundary conditions; it persists in a rectangular pipe, but disappears when the ow is driven by a travelling lid.

Despite the simplicity of its driving mechanism, we could not nd any report of this instability appearing in a laboratory experiment. This, of course, raises the question of its actual existence, but we suspect that the basic reason of its absence from the literature is that the aspect ratio of laboratory channels is usually too small to accommodate its growth. We speculate that this might not be fortuitous, as experimenters often wish the sediment bed to be invariant in the cross-stream direction.

At this point, we can only imagine an ideal setup, specially designed to observe the bedload instability. In the light of the present paper, it would involve a transparent rectangular pipe, with an aspect ratio of a few tens. The sediment and uid discharges would be kept constant until the streaks, visualised with a laser beam, emerge. In such an experiment, the measurement of the critical transport number would yield an estimate of the parameter γ, which encapsulates the inuence of gravity on cross-stream bedload transport.

Cross-stream uxes of sediment shape the channel of alluvial rivers and, most probably, select their size [START_REF] Ikeda | Stable width and depth of straight gravel rivers with heterogeneous bed materials[END_REF]. A reliable theory of bedload diusion, tested against laboratory experiments, would therefore help us understand their morphology. Once such framework is set, it will become a matter of numerical routines to explore it beyond linearity. A natural question to ask, then, will be whether the bedload instability grows as far as to split a broad channel into smaller ones, thus generating the precursor of a braided river [START_REF] Stebbings | The shapes of self-formed model alluvial channels[END_REF] Ce mode présente un domaine stable, où la moindre perturbation appliquée sur le lit décroît, et un domaine instable, pour lequel une perturbation du lit s'amplie (Fig. 5.10).

La limite entre ces deux domaines correspond à une ligne de stabilité marginale, où une perturbation ne s'amplie ni ne diminue. Cette ligne pourrait ainsi correspondre à des formes stationnaires du lit, où le ux transverse de grains s'annule sur le prol entier (q y = 0). Cette solution linéarisée est très approximative: une meilleure description devrait tenir compte des berges, où la profondeur s'annule, qui est donc inaccessible par une théorie linéarisée. Nous l'utilisons ici à titre d'illustration.

Cette limite de stabilité dépend du nombre de transport, que l'on peut relier au débit de sédiments par :

Tr = D W Q s Q * s , ( 5

Conclusion

An alluvial river selects its own shape through the coupling between ow and sediment transport. Accordingly, it adjusts its width and slope to its water and sediment discharges.

In this manuscript, we used simplied experiments to investigate this coupling in the laboratory.

We rst investigated this coupling in an idealised geometry: that of a sediment bed conned in a laboratory ume and sheared by a viscous ow. We found that the sediment bed selects its shape and its sediment-ux prole to reach equilibrium. By tracking travelling grains on the bed, we identied two mechanisms involved in this equilibrium. First, grains are pulled towards the center by gravity. Second, they collide with the rough bed and thus behave as random walkers. As they are more numerous in the channel center, they diuse towards the banks, which are the less populated areas. The balance between these two uxes governs the bed's equilibrium. The signature of this equilibrium is the exponential decay of the sediment ux with the bed elevation. This relation is analogous to the Boltzmann distribution that governs the density of Brownian particles suspended in a liquid. Although this statistical equilibrium applies to a wide range of systems, it had never been observed in the context of sediment transport.

We then produced laboratory rivers which transport sediment. For a moderate sediment input, they form a single-thread channel at equilibrium. We showed that these channels adjust their cross section and their downstream slope to satisfy the Boltzmann distribution, like in the conning ume. Their shape depends on their uid and sediment discharges. Specically, at a xed uid discharge, they widen and atten with the sediment discharge.

The Boltzmann distribution alone, however, does not predict the equilibrium shape of these rivers.

To understand how a river adjusts its shape to the sediment discharge, we modelled the ow with the shallow-water approximation. This simple model predicts that rivers remain slightly above the threshold of grains entrainment. We checked this prediction in our laboratory rivers by plotting their trajectories in the phase plane. We found that, regardless of their sediment discharge, they all gather around the circle that corresponds to the threshold channel. [START_REF] Phillips | Self-organization of river channels as a critical lter on climate signals[END_REF] showed that the ow-induced shear stress on the bed of a natural river is seldom more than 10% above the critical value, suggesting that natural rivers also self-organise to maintain their bed near the threshold. However, two rivers may exert a comparable shear stress on their bed, while adopting signicantly distinct cross sections. Indeed, according to our shallow-water model, all trajectories lie in a narrow domain of the phase plane, such that they are very close to each other. Inside this domain, however, a river's prole varies from the cosine shape to an innitely-wide, at river. It means that the river's shape is extremely sensitive to shear stress variations. In the range of sediment discharges we explored, the aspect ratio of our laboratory rivers doubles, while the shear stress varies by less than 20%.

When the aspect ratio of our laboratory rivers reaches about ten, the channel destabilizes into a braid. This observation prompted us to investigate theoretically the stability of a sediment bed sheared by a viscous ow. In doing so, we identied a new instability, caused by bedload diusion. This instability generates streamwise streaks, provided sediment transport is weak enough. If the streaks grow enough to reach the ow surface, this instability would split a wide channel into smaller ones. We suggest that it is a similar instability that turns our single-thread rivers into braids. This instability, however, has not been observed in the laboratory yet.

Perspectives

Bedload instability in the laboratory

The bedload instability presented in chapter 5 suggests that a at sediment bed sheared by a viscous ow is unstable, provided the sediment ux is weak. This instability should produce streaks aligned with the ow, but this pattern has never been observed in the laboratory (section 5.7). Here, we briey expose how a laboratory setup could produce these streaks.

The ideal setup would consist of a transparent rectangular pipe, with an aspect ratio of a few tens, half lled with sediment grains. Using a pipe, rather than a an open channel, rids us of other bedforms, such as chevrons or bars, and should boost the growth of the streaks (Fig. 5.11b). At the inlet, we would inject a viscous uid, such as a mixture of glycerol and water, at a constant discharge. Meanwhile, we would inject sediment at a constant discharge using the sediment feeder of our ume experiment (section 2.2). The diculty, however, would be to produce an homogeneous base state.

To do so, we would initially impose a high sediment discharge, to bring the transport number above one. According to our theory, the bed should then be stable. Over time, this procedure should produce an homogeneous base state. We would then decrease suddenly the uid discharge, which would lessen the sediment ux almost immediately. This process would bring the transport number below one, so that the system should cross the instability threshold. Hopefully, streaks should then appear, at the wavelength predicted by the theory (Fig. 5.11b).

Apart from supporting our theory -or proving it wrong-, observing these streaks could also provides us with an independent measurement of the Boltzmann length we have introduced in this manuscript, λ. Indeed, the threshold of the instability is set by the transport number, which is inversely proportional to λ. In addition, it would inform us about the non-linear evolution of these instability, and thus about the ability of the streaks to reach the ow surface, and to produce a braid.

Beyond the shallow-water approximation

The bedload instability underlines the limitations of the shallow-water approximation (Ch. 5). These limitations are probably responsible for the failure of our shallow-water theory to predict the equilibrium shape of a river (Ch. 4). To improve this theory, we are now convinced that we need to account for the diusion of momentum across the stream.

We should then replace the shallow-water approximation with the Stokes equation in two dimensions. Here, we explain how we would address this problem numerically.

We would rst dene an initial cross section, to which we assign a xed slope. With nite elements, we would then solve the Stokes equation in this domain (for instance, using FreeFem). As this is not an equilibrium shape yet, the cross-stream ux of sediment should be nite. We would then evolve the bed based on the sediment mass balance. Considering that the river is invariant in the streamwise direction, the mass balance keeps the cross section area constant. Re-meshing this new cross section would enable us to compute the new ow, and to repeat the same procedure. After a few iterations, the river should converge to its equilibrium prole.

As a rst attempt, we would run this numerical simulation for a ume, where our measurements are more accurate than in our laboratory rivers. We suspect, however, that the re-meshing procedure might generate instabilities. In addition, the convergence of this numerical scheme is still uncertain.

Natural rivers

The results of this manuscript are based on laboratory experiments where the ow is laminar. Natural rivers, however, are turbulent. Still, we expect that bedload grains collide with the rough bed and diuse across the ow in turbulent ows as well. Turbulence could aect the intensity of this diusion, and thus the diusion length, but the basic mechanism should hold. Therefore, we expect that the equilibrium between gravity and diusion, and thus the Boltzmann distribution, should hold in natural rivers.

To test this scenario against eld data, we need both the sediment ux and the bed elevation across a river. In most cases, however, only the total sediment discharge is measured in the eld. To our knowledge, only Liu et al. (2008) measured both the cross section and the sediment-ux prole in a natural river: the Urumqi river, China (Fig. Using these data, we now test the Boltzmann distribution,

q s = q max e -h λ , (5.38)
by plotting the bed elevation h as a function of the sediment ux (Fig. 5.14). In the semi-log space, the data gather around the straight line of slope 1 λ, where λ = 0.046 m.

If we compare this value to the median grain size, we nd: λ = 2.1 d 50 .

(5.39)

This preliminary example suggests that bedload diusion not only occurs in natural rivers, but also controls their shape. If it is so, this mechanism would also select the slope of a river. In turn, this slope controls the morphology of sedimentary structures, such as alluvial fans, that rivers sometimes build by depositing sediment at the outlet of mountain should be constant along the prole of a river (right-hand side of equation (4.15)). In this section, we check how accurately our numerical method fulls this evolution.

To do so, we compute a river cross section for ( s , χ) = (0.5, 0.25), which corresponds to ξ = 0.1516 (equation (B.1)). We obtain both the depth of the river D and its cross-stream slope D ′ . Then, we calculate the value of ξ num with (left-hand side of equation (4.15)):

ξ num (y) = ( D 2 (y) + D ′2 (y) -µ t ) e -D(y) s (B.2)
for the whole prole. Accordingly, we dene the relative numerical error as, err(y) = ξ num (y) -ξ ξ

(B.3) Figure B
.1a shows its value along a river prole. The error is slightly higher on the banks, and its mean value is 3.4 ⋅ 10 -16 , which corresponds to the oating point precision in Python. Now, we evaluate the integral of this error along the prole as a function of the ratio χ/ s , which varies between 0 and 1. First, we x χ = 0.1 (which is equivalent to xing a maximum river depth), and we calculate all the cross sections for χ/ s from 0 to 1 (Fig.

B.1b

). Then, we x s = 0.01 (which is equivalent to x a phase portrait), and we vary χ s from 0 to 1 (Fig.

B.1c

). In all cases, we nd an error smaller than 10 -11 , which supports the use of this numerical method to compute river proles.

B.2 Another numerical method to solve equation (4.15)

Although the numerical method used in section 4.3 is accurate, it is slow and sensitive to low values of s (typically < 0.01). This method is based on the integration of an ordinary dierential equation on D with the solver ode in Python, that we can not parametrize.

Thus, to accelerate the computation of a prole, we developed another numerical method, which is also less sensitive to the precision issues. This enabled us to calculate thousands of proles in a few seconds and for broader ranges of the parameters χ and s . We used it to generate the abacuses (section 4.6).

To do so, we introduce new notations, based on the phase portrait (section 4.4). Specif- ). However, it seems to impact only its neighbourhood, in a range of ±10 mm, as suggested by our previous measurements. The hole then only locally distort the ridge of the stability landscape.

We now investigate how the stability landscape changes on a same can, without manufactured hole. We use a commercial can and expect it to have defects due to the fabrication process or the transportation. We turn it sequentially to probe the amplitude of the ridge for 10 dierent positions in its mid plane (around 360 ○ ). We observe that the amplitude of the ridge for a given axial load varies of about 20% with the poker position ( So far, to ensure that this discrepancy is not due to the asymmetry of a Coke can lid, we probed the can in its mid plane with and without its lid (Fig. C.8b). We did not observe any dierence, suggesting that a defect on the top of the can does not inuence the stability landscape at all.

C.6 Conclusion

During this project, we investigated the eect of a defect on the stability of Coke cans.

We developed a non-destructive procedure to measure accurately the buckling load of a can in which we drilled a hole. By probing the can close to this hole, we measure the decrease of the ridge in its stability landscape without damaging it. The amplitude of this ridge converges towards the buckling load of the can, that we independently measure by

Boltzmann distribution of sediment transport

A. Abramian, 1 O. Devauchelle, 1, * G. Seizilles, 1, 2 and E. Lajeunesse The coupling of sediment transport with the flow that drives it allows rivers to shape their own bed. Cross-stream fluxes of sediment play a crucial, yet poorly understood, role in this process. Here, we track particles in a laboratory flume to relate their statistical behavior to the self organization of the granular bed they make up. As they travel downstream, the transported grains wander randomly across the bed's surface, thus inducing cross-stream diffusion. The balance of diffusion and gravity results in a peculiar Boltzmann distribution, in which the bed's roughness plays the role of thermal fluctuations, while its surface forms the potential well that confines the sediment flux.

When water flows over a layer of solid grains, the shear stress it exerts on the sediment's surface entrains some of the grains as bedload [1,2]. Eventually, the flow deposits the traveling grains downstream [3,4]. The balance of entrainment and deposition deforms the sediment bed [5], thus changing the flow and the distribution of shear stress. This coupling, through various instabilities, generates sand ripples in streams [6,7], rhomboid patterns on beaches [8], alternate bars in rivers [9] and, possibly, meanders [10][11][12][13].

More fundamentally, the coupling of water flow and sediment transport is the mechanism by which alluvial rivers choose their own shape and size, as they build their bed out of the sediment they carry [14][15][16][17]. To do so, however, rivers need to transport sediment not only downstream, but also across the flow [18,19]. On a slanted bed, of course, gravity will pull traveling grains downwards; it thus diverts the sediment flux away from the banks of a river [20,21]. What mechanism opposes this flux to maintain the river's bed remains an open question. Here, we suggest the inherent randomness of sediment transport plays a major role in the answer.

The velocity of bedload grains fluctuates as they travel over the rough bed, and the bedload layer constantly exchanges particles with the latter [22], thus calling for a statistical description of bedload transport [23,24]. At its simplest, this theory involves a population of noninteracting grains traveling, on average, at velocity V x , close to the grain's settling velocity [4,[START_REF] Lajeunesse | Gravel-bed Rivers: Gravel Bed Rivers and Disasters[END_REF]. If n is the surface density of traveling grains, the downstream flux of sediment reads q s = nV x . As long as sediment transport is weak, the traveling grains do not interact significantly, and their average velocity V x can be treated as a constant. Both the streamwise and cross-stream velocities, nonetheless, fluctuate significantly [22,[START_REF] Seizilles | [END_REF].

A little-investigated consequence of these fluctuations is the cross-stream dispersion they induce [27,[START_REF] Aussillous | Proceedings of the National Academy of Sciences[END_REF]. Indeed, as it travels downstream, a grain bumps into immobile grains like a ball rolling down a Galton board. The random deviations so induced turn its trajectory into a random walk across the stream [START_REF] Seizilles | [END_REF][START_REF] Samson | [END_REF]. We thus expect a cross-stream, Fickian flux to bring traveling grains towards the less populated areas of the bed (lower n). Mathematically,

q d = -d ∂q s ∂y (1)
where q d is the fluctuation-induced Fickian flux, y is the [START_REF] Seizilles | [END_REF]. To our knowledge, neither the cross-stream flux of grains q d , nor its consequences on the bed's shape, have been directly observed.

To measure the Fickian flux q d , we set up a particletracking experiment in a 3 cm-wide flume {Fig. 1(a), detailed experimental methods in Sup. Mat.}. We inject into the flume a mixture of water and glycerol (density ρ = 1160 g L -1 , viscosity η = 10 cP) at constant rate (Tab. I). We use a viscous fluid to keep the flow laminar (Reynolds number below 250). Simultaneously, and also at constant rate, we inject sieved resin grains (median diameter d s = 827 µm, density ρ s = 1540 g L -1 ). After a few hours, the sediment bed reaches its equilibrium shape.

This equilibrium, however, is a dynamical one: the flow constantly entrains new grains, and deposits other ones onto the bed. A camera mounted above the flume films the traveling grains through the fluid surface, at a frequency of 50 fps [Tab. I, Fig. 1(b), Sup. Movie]. Although made of the same material, the grains are of different colors, which allows us to locate them individually on each frame. We then connect their locations on successive frames to reconstruct their trajectories with a precision of 0.1 d s [30,Sup. Mat.].

The resulting trajectories are mostly oriented downstream, as expected, but they also fluctuate sideways, like in previous bedload experiments [22,[START_REF] Seizilles | [END_REF]27]. These fluctuations cause them to disperse across the stream as they travel downstream [Fig. 2(a)]. We now distribute all our trajectories into 25 logarithmically-spaced bins, according to their travel length xx 0 (x 0 is the starting point of each trajectory), and calculate the variance σ 2 y for each bin [Fig. 2(b)]. We find that, for trajectories longer than a few grain diameters, the cross-stream variance increases linearly with the travel distance. Seizilles et al. [START_REF] Seizilles | [END_REF] interpreted a similar relationship as the signature of a random walk across the stream (they also found that the auto-correlation of the cross-stream trajectories decays exponentially). Accordingly, we now fit the re- lation σ 2 y = 2 d (xx 0 ) to our trajectories (beyond 3d s downstream of their starting point). To estimate d , we treat the above relation as the reduced major axis of our data set: 2 d = std(σ 2 y )/std(xx 0 ), where the standard deviation is over bins, that is, over the data points of Fig. 2(b). Using our entire data set (typically 3 × 10 4 trajectories per run), we get d = (0.024 ± 0.002) d s .

(

) 2
where the uncertainty is the expected standard deviation of d . This value is close to previous measurements in pure water [START_REF] Seizilles | [END_REF], although it is most likely affected by the physical properties of the fluid and of the grains. That diffusion expresses itself through a length scale, as opposed to a diffusion coefficient, betrays its athermal origin: it is the driving (here, the flow) that sets the time scale. This property relates bedload diffusion to the diffusion induced by shearing in granular materials and foams [31][32][33].

Although fluctuations disperse the traveling grains across the stream, most grains travel near the center of the channel [Fig. 1 flume's wall. At equilibrium, we expect gravity to counteract this flux; for this to happen, the bed's cross section needs to be convex. In fact, the bed cannot maintain a flat surface: in this configuration, the fluid-induced shear stress would be weaker near the sidewalls, and so would the intensity of sediment transport. Bedload diffusion would then bring more grains towards the bank, thus preventing equilibrium. In the following, we investigate this coupling.

We use an inclined laser sheet to measure the elevation profile of the bed [Fig. 1(a), Sup. Mat]. The laser source is fixed on a rail which allows it to scan the flume over 20 cm. We evaluate the tilt of the rail by scanning a tub of still milk; it is less than 0.03 %. At the end of an experimental run, we switch off the fluid input; this brings the bed to a standstill in a matter of seconds. We then let the fluid drain out of the flume, and use the laser scanner (i) to measure the bed's downstream slope S (Tab. I) and (ii) to spatially average the bed's cross-section, h(y) [Fig. 3(a)]. We find that the sediment bed is convex for all our experimental runs. Its surface gently curves upwards near the center of the flume, and steepens near the walls. This observation indicates that the sediment bed has spontaneously created a potential well to confine the traveling particles in its center.

Unfortunately, measuring the flow depth based on the deflection of the laser beam proved imprecise. Instead, we used finite elements to solve the Stokes equation in two dimensions [34,Sup. Mat.], namely:

η∇ 2 u = ρgS , (3)
where u is the streamwise velocity of the fluid. We further assume that the free surface is flat, that the viscous stress vanishes there (∂u/∂z = 0, where z is the vertical coordinate) and that the fluid does not slip at the bed's surface (u = 0). Knowing the bed's downstream slope S, we then adjust the elevation of the water surface to match the fluid discharge. In addition to the flow depth, this computation provides us with the velocity field of the flow [Fig. 3(a)], and thus the intensity of the viscous stress τ that the fluid exerts on the bed [Fig. 3(b)]. We find that, like the sediment bed, the viscous stress varies across the flume; it reaches a maximum at the center of the channel, and vanishes where the bed's surface joins the walls-as expected for a laminar flow. We now wish to relate the flow-induced stress to sediment transport. To measure the latter, we divide the flume's width into 50 bins, and count the trajectories that cross a constant-x line within each bin, per unit time. This procedure yields a sediment-flux profile (Sup. Mat.). Repeating it for 10 different lines across the channel, we obtain an average sediment-flux profile, q s (y) [we keep only data points for which the relative uncertainty is less than one, Fig. 3(c)]. In accordance with the distribution of trajectories in Fig. 1(b), the sediment flux appears concentrated around the center of the flume. It vanishes quickly away from the center, much before the fluid-induced stress has significantly decreased.

Following Shields, we now relate the sediment flux to the ratio of the fluid-induced stress to the weight of a grain, θ [35]:

θ = τ (ρ s -ρ)gd s ( 4
)
where g is the acceleration of gravity. The Shields parameter is an instance of the Coulomb friction factor; strictly speaking, on a convex bed like that of Fig. 3(a), its expression should include the cross-stream slope, ∂h/∂y [17]. In our experiments, however, we find that this correction is insignificant where sediment transport is measurable. Accordingly, we content ourselves with the approximate expression of Eq. ( 4). Plotting the intensity of the sediment flux as a function of the force driving it, in the form of the Shields parameter θ, shows a well-defined threshold [Fig. 4(a)]: no grain moves when the fluid-induced stress is too weak to overcome its weight, but the sediment flux increases steeply past this threshold. This emblematic behavior, apparent in a single experimental run, is confirmed by the superimposition of our five experimental runs [Fig. 4(a)]. Indeed, within the variability of the measurements, the five corresponding transport laws gather around a common relation, which we may treat as linear above the threshold Shields stress θ t [START_REF] Seizilles | [END_REF]:

q s = q 0 (θ -θ t ) (5)
where q 0 is a constant of order (ρρ s )g/η. Fitting this transport law to our complete data set, we get q 0 = 544 ± 48 grains s -1 cm -1 and θ t = 0.167 ± 0.003, where the uncertainty is the standard deviation over individual runs. These values correspond to a typical transport law in a laminar flow [2,[START_REF] Seizilles | [END_REF].

The local intensity of the flow-induced stress controls the local flux of sediment-just as expected. More surprisingly, perhaps, the sediment bed needs to adjust its shape so that, in total, the flume conveys the sediment discharge that we impose at the inlet. We suggest that it does so by balancing the Fickian flux, q d , which pushes the traveling grains away from the flume's center, with the gravity-induced flux, q g , which pulls them towards the lowest point of the bed's surface. As a first approximation, we may assume that the latter is proportional (i) to the cross-stream slope of the bed and (ii) to the local intensity of the downstream flux of sediment, q s . Mathematically,

q g = -α q s ∂h ∂y (6)
where α is a dimensionless constant. Although conducted in air, the experiments of Chen et al. [21] suggest that it should be of order unity or less. At equilibrium, the gravity-induced flux q g needs to match the Fickian flux q d . Adding Eqs. ( 6) and (1) yields the Boltzmann equation, which we readily integrate into an exponential distribution:

q s (y) = q 0 exp - h(y) λ B , (7)
where q 0 is an integration constant, and λ B = d /α is the characteristic length of the distribution. Distinctively, this distribution relates two quantities (q s and h) that depend on the space coordinate y, but the latter does not explicitly appear in its expression. This, however, does not make it a local relationship: unlike the transport law of Eq. ( 5), it features an integration constant which depends on the sediment and water discharges of each experiment. These properties, typical of a Boltzmann distribution, appear when plotting the bed elevation as a function of the sediment discharge [Fig. each experiment, the data points trace twice the same line in the semi-logarithmic space, as they go from one side of the channel to the other, but the position of this line depends on the experimental run.

To bring all our experiments into the same space, we now divide Eq. ( 7) by its geometrical mean. This rids us of the integration constant q 0 , and turns the distribution of sediment transport into q s (y)

q s g = exp - h(y) -h a λ B , (8)
where • g and • a are the geometric and arithmetic means, respectively. Within the variability of our observations, the data points from all experimental runs gather around a straight line, which we interpret as Eq. ( 8). Fitting the characteristic length λ B to our entire data set, we find λ B = 0.10±0.01 mm, where the uncertainty is the standard deviation over individual runs. More tellingly, this value corresponds to λ B = (0.12 ± 0.02) d s ,

showing that the characteristic length compares with the grain size. Returning to the definition of λ B , we find that the constant α in Eq. ( 6) is about 0.2, in agreement with previous estimates [21].

Although temperature plays no role here, the structure of Eq. ( 8), as well as its derivation, makes it a direct analog of the Boltzmann distribution, where the crossstream deviations of the grains' trajectories play the role of thermal fluctuations. Pursuing this analogy, we suggest that the scale of λ B is inherited from the roughness of the underlying granular bed, which we believe causes the cross-stream deviations-a mechanism reminiscent of, but somewhat simpler than, the shear-induced diffusion observed in granular flows [31][32][33]. To support this hypothesis, however, we would need more experiments with different grains and fluids.

The familiarity of the Boltzmann distribution should not obscure the peculiarity of the phenomenon we report here. We naturally expect that random walkers will distribute themselves in a potential well according to this distribution; what is remarkable here, however, is that the system spontaneously chooses the shape of the potential well to match the transport law. This is possible only because the sediment bed is made of the very particles that roam over its surface.

A practical consequence of this self-organization is that sediment transport cannot be uniform across a flume, thus prompting us to reevaluate the transport laws measured in this classical set-up. (If we were to assume uniformity in our experiments, we would underestimate q 0 by a factor of two.) In the context of dry granular flows, the traditional rotating-drum experiment has been challenged on similar grounds [36].

Much remains to be done to understand how the bed builds its own shape. To do so, we will have to drop the equilibrium assumption. A first step in that direction was to demonstrate theoretically that the cross-stream diffusion of sediment could generate a distinctive instability, but the associated pattern has not been observed yet [37]. More generally, the consequences of bedload diffusion on the morphology of rivers, and ultimately on that of the landscapes they carve, belong to uncharted territory. We report here the experimental methods used to produce the results presented in the main document. More details can be found in reference [1] a , Ch.

I. GRAINS AND FLUID

In this article, all experiments involve the same irregular resin grains, a blast media manufactured by Guyson 1 [Fig. 1(a)]. Seizilles et al. [2] used the same material to produce laboratory rivers. The grains are made of urea resin, the density of which is 1520 kg m -3 (manufacturer's value). Using a pycnometer, we find an average density of ρ s =1488 kg m -3 ; we use this value in our numerical calculations.

The manufacturer provides a grain-size range of [650, 800] µm. Using the software ImageJ 2 , we analyze a picture similar to Fig. 1(a) to measure the grainsize distribution of Fig. 1(b). Representing each particle as a sphere, we find an average grain diameter of 0.81 ± 0.19 mm (standard deviation), and a median diameter of d s = 0.82 mm (the value we use in the paper).

The advantage of resin over quartz or glass is its low density, which reduces the settling velocity of the grains, and therefore slows them down as they travel in the bedload layer [3]. This slowness facilitates the grain-tracking procedure described in Sec. II. We estimate this settling velocity with Stokes' formula for the settling velocity of a sphere:

V s = (ρ s -ρ)gd 2 s 18ρν , (1)
the value of which depends on the fluid's density ρ and viscosity ν.

To further slow down our grains (and to prevent turbulence), we use a mixture of water (60 %) and glycerol (40 %) to entrain the grains. As water can evaporate over the course of an experimental run, we measure the density of our mixture with an oscillating U-tube densimeter (Anton Paar 3 DMA 5000 M) at the beginning of each run, and before each measurement. We compensate for evaporation by adding water to the fluid to maintain its density near ρ = 1161 g L -1 with an accuracy of about 1.3 g L -1 (standard deviation over experiments). This ensures that its viscosity remains near ρν = (1.02±0.03)×10 -2 Pa s -1 .

Based on the above values, we find that the settling velocity of our grains is about V s = 1.0 cm s -1 , with a relative uncertainty of about 50 %, mainly due to the dispersion of the grain size. We thus expect the traveling grains to move at about 12 times their own length per second; this frequency is a lower bound for the frame rate of the movies used for grain tracking.

II. GRAIN TRACKING

Most of our grains are white, but some of them come in a distinct hue: mainly red, orange or blue (a few percent of the total for each color) [Fig. 1(a)]. We use this property to track them using top-view movies of the traveling grains (Fig. 2), recorded with a Canon4 700D SLR camera, fitted with a 250 mm macro lens. This camera allows us to reccord movies at 50 color frames per second. Each frame is a 1280 × 1024 array; at this resolution, the grain size d s is about 10 pixels. To locate individual grains on a frame, we first decompose the image array into its hue, saturation and value. We then select pixels whose saturation is more than 0.3, to ensure that their hue is well-defined. We then choose a hue range (for instance [0 • , 64 • ] for blue grains) and pick the pixels that fall into this range [Fig. 2(a)]. When submitted to these criteria, a movie frame yields an array of booleans, where clusters corresponds to blue grains. We smooth this array by convolving it with a 10-pixel Gaussian filter, and locate the maximums of the resulting frame using the peak_local_max function of the Scikitimage 5 Python library, requiring a minimum distance of 10 pixels between peaks.

Figure 2(b) shows the result of this grain-detection procedure for the movie frame shown in Fig. 1 (main document). In total, we find 74 blue grains in this frame. Assuming each of them occupies an area d 2 s , we estimate that blue grains make up about 1.8 % of the sediment. This value, of course, depends on the definition of "blue" (the hue range).

At this point, we can only speculate that the uncertainty on the position of each grain resulting from the detection procedure lies between a pixel (about 0.1d s ) and d s . To estimate it more accurately, we first need to track the grains through a series of movie frames, thus reconstructing their trajectories. 5

scikit-image.org

To track a particle over two successive frames, we use a variant of the Hungarian algorithm [4,5], implemented in the Munkres 6 Python library. We often loose track of a traveling particle as it drifts near a bed particle of the same color; we thus allow a particle to disappear during 10 successive frames before assigning its position to a new trajectory.

Figure 1(b) (main document) shows some trajectories measured with the above procedure, but the movie provided as supplemental material gives a better sense of the result. Before counting traveling grains to measure a sediment flux (Sec. III), we first turn to immobile grains to estimate the uncertainty of our tracking procedure. We select, among all the trajectories of run #1, those which lasted at least 1 minute (500 frames), and extend downstream over less than 10 pixels (about d s ). There are 101 such trajectories, which we assume correspond to immobile grains belonging to the sediment bed. On average, their standard deviation is 0.41 px along the stream, and 0.52 px across the stream; we therefore estimate that the uncertainty on the position of each grain is about one pixel.

III. SEDIMENT FLUX

To translate a collection of trajectories into a sediment flux, we first draw N lines = 10 evenly spaced lines across the stream, and find where they intersect the trajectories. We record the cross-stream coordinate y of each intersection, as well as its direction [upstream (+1) or downstream (-1)]. A typical movie records between 4,000 and 12,000 intersections during its ∆t ≈ 10 min duration. Summing the directions of these intersections, and dividing the result by the number of cross-stream lines, and by the movie duration, yields the discharge corresponding to a collection of trajectories [Fig. 3(a)]. For the experimental run #1, we have recorded 6 movies in about an hour, and tracked the blue particles in each of them. This provides us with six independent measurement of the discharge for the same experimental run. Although these measurements fluctuate around their average, they do not show any clear trend, indicating that the system is in steady state. The amplitude of the fluctuations (about 10 %) matches the expected deviation for a normally distributed number of intersections [vertical error bars in Fig. 3(a)].

Averaged over the 6 movies of run #1, the discharge of blue grains is about 0.63 grains s -1 , while the sediment input for this run is 42.4 grains s -1 . The ratio of the two yields an estimate of the proportion of blue grains: about 1.5 %, to be compared with the value of 1.8 % found in Sec. II. This mismatch might be due to the difference between the numerical proportion of a class of grains, and the proportion of visible area it occupies on the bed's surface. To bypass this issue, we normalize the sediment flux profiles of Fig. 3 (main document) with the total sediment input.

We now use the location of the intersections between trajectories and cross-stream lines to calculate the crossstream profile of the sediment flux. Before summing the directions of the intersections, we first distribute them into 19 even-sized bins according to their position across the stream (only 16 were populated during run #1). The discharge through a bin, divided by the bin size, yields an estimate of the local sediment flux for each movie [Fig. 3(b)]. Like with the total discharge, we find that the sediment flux profile fluctuates around the average, but show no significant trend.

Finally, we now have six independent measurements of the sediment flux trough N bin = 19 bins across the stream; we may thus probe the statistics of the sediment flux measurement [Fig. 3(c)]. For each bin, we count the average number of recorded crossings, N cross , and its standard deviation (both quantities are with respect to the 6 independent measurements). The statistical significance of such a small number of independent measurements is certainly questionable, but we nonetheless find that the classical square root relationship between standard deviation and sample size tolerably fits our data:

σ[N cross ] ≈ 4.1 N cross , (2)
where the standard deviation, σ, and the mean, • , are with respect to the 6 independent measurements. We further assume that N cross is proportional to the local sediment flux:

N cross ≈ q s ∆t N lines W N bin ( 3
)
where W is the width of the channel. Equations ( 2

IV. CROSS SECTION

At the end of an experimental run, we switch off the water and sediment inputs. The moving grains settle down and stop within seconds of the run's end. We then let the fluid drain out of the channel for a few minutes, and scan the bed's surface with a laser sheet [Fig. 1(a), main document]. The intersection of the laser sheet with the bed appears in the view range of the camera used to track the traveling particles [Fig. 4(a)]. The bed deforms this intersection in proportion to its own elevation h, allowing us to reconstruct the bed's cross-section, after identifying its position on the picture, x laser with the maximum of green light intensity [Fig. 4(b)].

To calibrate this device, we first scan a channel filled with still milk, the whiteness of which generates a sharp laser line. The laser source and the camera are attached to a translating carriage, whose position is controlled with a step motor. We check that this 20-cm translation is horizontal, up to a slope of about 0.03 % with respect to the milk's surface; we hereafter treat this value as the precision on the downstream slope measurement [Fig. 4(c)]. Next, we add a fixed volume of milk into the channel, thus elevating the bath's surface by a known amount (0.57 ± 0.08 cm), and scan its surface again. We repeat this procedure four times; the resulting shift of the laser line yields the conversion factor for the bed elevation: which corresponds to a laser inclination of 28.3 • . The position of the laser line on the milk's surface varies across the channel by less than a pixel, and the resulting precision of the bed elevation measurement is below 30 µmless than a grain diameter. The roughness of the bed therefore dominates the uncertainty on h. To average out some of this variability, we average our bed profiles over the 20 cm-long excursion of the translating carriage.

V. STOKES FLOW

To produce the sediment transport law of Fig. 4(a) (main document), we need to estimate the shear stress induced by the fluid on the sediment bed. We do so by solving numerically the two-dimensional Stokes equation above the measured bed surface, using the software FreeFem++ 7 [6].

In dimensionless form, the Stokes equation [equation (3), main document] reads

∇ 2 u * = 1 , (5)
where u * is the downstream velocity field made dimensionless with W 2 gS/ν, and space coordinates are made dimensionless with W . For a given flow depth and bed profile, solving Eq. ( 5) numerically yields a velocity field and a dimensionless discharge Q w * , with

Q w * = νQ w gSW 4 = u * dy dz , (6)
where integration is over the (dimensionless) cross section of the flow. Unfortunately, we were not able to measure the elevation of the fluid surface with satisfactory precision using the laser scanner, mostly because of the fluctuation induced by traveling grains. Instead, we used the following procedure to estimate it. We first assume an elevation for the water surface, and generate a triangularmesh approximation of the flow cross-section [Fig. 5(a)]. We then solve Eq. ( 5) using finite elements and compute the dimensionless discharge corresponding to this water elevation. Repeating these steps yields an approximate relation between discharge and flow depth [Fig. 5(b)].

We use an electromagnetic flow meter to measure the fluid input into the channel, with a relative precision below 2 % (Kobold 8 Mik 0.5-10 L min -1 ). The fluctuations of the discharge during an experimental run are of the order of this precision. Considering the relative uncertainty on the slope S and that on the viscosity ν, we may estimate the dimensionless fluid discharge, Q w * , with a relative precision of less than 10 % [red error bars on Fig. 5(b)]. The numerical relation of Fig. 5(b) then allows us to estimate the flow depth with an uncertainty of 0.2 mm (less than 4 %). We may now evaluate how much the uncertainty on the flow depth affects the shear stress.

Turning our attention to Fig. 5(c), we find that a 3mm change of the flow depth affects the shear stress by up to 40 %; we therefore expect that the uncertainty on the computed shear stress, inherited from the uncertainty on the discharge, slope and viscosity, be less than 4 %hardly a concern. Conversely, the actual variability of the bed elevation (about one grain diameter), would affect the shear stress by about 15 %. This estimate appears as an error bar in Fig. 3(b) (main document).

  Malgré l'évolution constante de son cours, une rivière favorise le développement d'une société. Les populations s'implantent près de ses berges pour s'approvisionner en eau courante ou faciliter les transports de marchandises et de voyageurs. De ce fait, l'Homme cherche à domestiquer les cours d'eau pour les mettre en valeur et protéger les territoires environnants. Pour contrôler la forme et la dynamique d'une rivière, on peut avoir recours à diérents ouvrages. Par exemple, un système de digues restreint la rivière Kosi au bord ouest du cône alluvial et empêche ses avulsions catastrophiques (Sinha 2008, Mishra et al. 2008). Un immense barrage a également été construit dans les années 1960 à la frontière indonépalaise. Il alimente un réseau de canaux d'irrigations sur une zone de 250 km 2 et fournit

Figure 1

 1 Figure 1 Photographie de l'Ain à Saint-Maurice-de-Gourdans, Port Galland (France).Le lit de cette rivière alluviale est composée de galets, dont la taille est comprise entre 1 et 100 mm. L'échelle approximative est donnée par les baigneurs.

Figure 2

 2 Figure 2 Evolution du lit de la rivière Kosi (Inde). a. Vue satellite du cône alluvial de la rivière Kosi, qui se jette dans le Gange. Encadré en haut à droite : situation géographique du cône, au nord-est de l'Inde (carte générée via le site d-maps.com). Crédits : Google Earth. b. Evolution du cours de la Kosi depuis le 18ème siècle, d'après Gole & Chitale (1966).

Sommaire 1 . 1 20 Chapitre 1 :Figure 1 . 1 ( 1 . 1 )S ∝ Q -1 2 w.( 1 . 2 )

 112011111212 Figure 1.1 Exemple d'une rivière alluviale : rivière Kaidu (Tian-Shan, Chine) vue de haut. Crédits : Google Earth.

Figure 1 . 2 Figure 1 . 3

 1213 Figure 1.14

Figure 1 . 4

 14 Figure 1.4 Valeur seuil du nombre de Shields θ t en fonction du nombre de Reynolds de grain, Re s . Points : données expérimentales. Lignes : modèle empirique. D'après Julien (1995), Bungton (1999).

Figure 1 . 5

 15 Figure 1.5 Etude expérimentale de la morphologie d'une rivière de laboratoire en régime laminaire. D'après Seizilles et al. (2013). a. Rivière expérimentale vue de haut. La rivière (colorée en bleu) s'écoule de la gauche vers la droite. b. Section d'une rivière expérimentale en condition laminaire. Marron : lit de sédiments. Bleu : écoulement. Ligne rouge : théorie du seuil. c. Largeur des rivières en fonction de leur débit Q w . Bleu : points expérimentaux. Ligne rouge : théorie du seuil (équation (1.26)). d. Rapport d'aspect des rivières en fonction de leur débit Q w . Bleu : points expérimentaux. Ligne rouge : théorie du seuil (équation (1.17)).

) où I est un coecient déni par I = π 2 -π 2 cos 3 2

 223 x dx ≈ 1.75 .

(1. 22 )

 22 Cette relation est en accord avec la loi empirique de Lacey (équation (1.1)) pour les données de terrain (Fig.1.2a). Cependant, le pré-facteur dépend de nombreux paramètres.

. 26 )

 26 Ainsi, la largeur et la pente d'équilibre d'une rivière s'adaptent à son débit. L'équation (1.26) est l'analogue de la loi de Lacey en régime laminaire, avec un exposant 1/3 (au lieu de 1/2 en turbulent). Cette expression est en bon accord avec les mesures expérimentales de Seizilles et al. (2013) (Fig. 1.5).

Figure 1 . 7

 17 Figure 1.7 Rivière Wiliam vue de haut (Canada). La rivière s'écoule de la gauche vers la droite. Crédits : Google Earth. Encadré en haut à droite : localisation géographique de la rivière au Canada (carte générée via le site d-maps.com).

Figure 1 . 8

 18 Figure 1.8 Evolution de la morphologie de la rivière William (Canada) le long de son cours. D'après Smith & Smith (1984). A. Évolution du transport de sédiments en fonction de la distance longitudinale. B et C. Caractéristiques des grains (taille et granulométrie) en fonction de la distance longitudinale. D. Largeur de la rivière le long de son cours.

Figure 1 . 9

 19 Figure 1.9 Rapport d'aspect de rivières naturelles en fonction de leur débit de sédiments. Rivières graveleuse en Idaho. Données de King et al. (2004) compilées par Métivier & Barrier (2012).

Figure 1 .

 1 Figure 1.10 Transition méandre-tresse du Rhône (Suisse) du à un apport élevé en sédiments. Coordonnées : 46 ○ 18'19.38"N, 7 ○ 38'25.96"E. Crédits : Google Earth.

Figure 1 .

 1 Figure 1.11 Transition méandre-tresse de la rivière Kaidu (Tian Shan, Chine) du à un apport élevé en sédiments. Coordonnées : 43 ○ 3'55.67"N, 84 ○ 21'37.72"E. Crédits : Google

Figure 1 .Figure 1 .

 11 Figure 1.12 Transition tresses-méandres. D'après Leopold & Wolman (1957) et Carson (1984). Points remplis : rivières en tresses. Points vides : rivières en méandres. Ligne pleine : critère de Leopold & Wolman (1957) (équation (1.27)). Lignes en pointillées : corrections eectuées par Carson (1984).

Figure 1 .

 1 Figure 1.15 Transport par charriage sur un lit granulaire cisaillé par un écoulement uniforme et stationnaire. Dénition du taux d'érosion et du taux de dépôt. D'après Lajeunesse et al. (2017).

Figure 1 .

 1 Figure 1.16 Lois de transport expérimentales. a. Expérience et modèle de Charru et al. (2004a). b. Expérience de Seizilles et al. (2014). En bleu : points expérimentaux. En rouge : équation (1.41).

1. 4 .Figure 1 .

 41 Figure 1.17 Diusion transverse des grains. D'après Seizilles et al. (2014). a. Grains transportés par charriage sur un lit de sédiments. Trajectoires renormalisées par la position initiale. b. Dispersion des grains en fonction de leur position longitudinale. Bleu : points expérimentaux. Rouge : relation linéaire ajustée sur les données.

1. 5

 5 Organisation du manuscrit Dans le chapitre 2, nous commençons par étudier expérimentalement l'inuence du transport de sédiment sur une rivière dont nous xons la largeur. Nous nous concentrons ainsi sur le couplage écoulement-transport à l'échelle du grain. Nous utilisons un canal dans lequel un écoulement visqueux entraîne un lit sédimentaire. Nous mesurons le transport sédimentaire et la forme du lit, ce qui nous amène à modéliser leur interaction. Le chapitre 3 confronte ce modèle à des rivières de laboratoire dont la largeur n'est plus xée. Nous utilisons un plan incliné, recouvert d'une couche épaisse de sédiments sur lequel un uide visqueux s'écoule. Une rivière de quelques centimètres de large se forme spontanément. Cette forme dépend des ux d'eau et de sédiments que l'on injecte à l'entrée. Dans le chapitre 4, nous identions le mécanisme par lequel la rivière sélectionne sa taille en fonction des paramètres de forçage, comme les débits d'eau et de sédiments. Nous développons un modèle théorique que nous comparons ensuite à nos données expérimentales.

Figure 2 . 1

 21 Figure 2.1 Setup of the ume experiment fed by plastic sediment grains and a mixture of water and glycerol. A camera and a laser sheet are xed on a rail placed above the channel, allowing them to translate over 20 cm in the downstream direction (x-direction).

Figure 2 . 2

 22 Figure 2.2 Measurement of the sediment input. a. Mass of sediment remaining in the tank, as a function of time. The slope of this curve corresponds to the sediment discharge delivered by the conveyor belt. b. Grain-size distribution of our plastic sediment.

Figure 2 . 3 Figure 2 . 4

 2324 Figure 2.3 Details of the ume experiment. Left. Flume inlet. Grains delivered by the sediment feeder fall into a funnel which directs them into the channel. Center. Detail of the ume inlet. Sediment grains fall from the funnel. They initially stay at the surface of the ow, and gradually settle on the bed. They are then entrained by the ow as bedload.Right. Flume outlet. Grains are separated from the uid by a sieve. The funnel diameter is 10 cm and the ume width is 3 cm.

Figure 2 . 5

 25 Figure 2.5 Laser line projected onto the milk surface. a. Laser sheet on the milk surface, seen from the top. The laser line along the streamwise direction allows us to align the laser sheet with the bed. Dashed red line: location of the laser maximum intensity. Dashed white line: y = 570. b. Image intensity along the white dashed line of a. (y = 570).Fit

  Fitof the value by a 2-order polynomial function near the rough maximum. Laser width : 20 pixels.

Figure 2 . 6

 26 Figure 2.6 Laser deviation x laser as a function of the bath or the bed elevation h: Top and side view.

Figure 2 . 7 Figure 2 . 8

 2728 Figure 2.7 Calibration of the laser scanner with a milk bath. a. Laser location for dierent bath elevations. Dashed red lines : Location of the laser intensity peak. b. Laser location across the channel: deviation from its mean value. Each color corresponds to a bath elevation. c. Calibration curve: elevation of the milk surface as a function of the laser location. Blue points: measurements. Red dashed line: linear t. d. Cross-stream average of the laser line as a function of its downstream position.

  Fit of the intensity by a second-order polynomial. Laser width : 20 pixels. c. Sediment bed elevation measured by the deviation of the laser sheet. The distance between the ume bottom and the minimum of the bed elevation is arbitrary.

  gure 2.8c. This method allows us to measure the bed elevation with an accuracy of ∆h = 80 µm = 0.1 d s . The associated image analysis procedure is quick (less than one hour of computer time for each run).

Figure 2 . 9

 29 Figure 2.9 Downstream-slope measurement. Example of run 2 (Tab. 2.1). In blue: sediment bed elevation along the streamwise direction. Dashed red line: linear t.

Figure 2 .

 2 Figure 2.10 Downstream slope of sediment beds as a function of the sediment discharge.

Figure 2 .

 2 Figure 2.11 Flow numerical computation. Example of run 2 (Tab. 2.1). a. Velocity eld in the channel cross section for a given ow depth. b. Flow depth as a function of the uid discharge. Blue points: numerical computation. Red point: experimental value.

2. 4 . 2

 42 Particles tracking Dierent methods of image processing have been previously developed to track bedload particles. Some experimenters tracked particles manually (Charru et al. 2004a). Since then, computerized methods have much improved as they benetted from the development of particle image velocimetry. To track black tracers in a crowd of white grains, one can use a threshold on the light intensity (Seizilles et al. 2014). When the moving grains are not contrasted enough with the underlying bed, some experimenters used a method based on background subtraction (Heyman & Ancey (2014)).

Figure 2 .

 2 Figure 2.13 Illustration of the RGB (left) and HSV (right) color systems. Credits: Wikipedia Creative Commons.

Figure 2 .

 2 Figure 2.14 Histogram of hues from the picture of gure 2.15a. Selected hue range: [0, 81], corresponding to blue hues. The colorbar indicates the hue values for a saturation of 0.5 and a value of 0.8.

Figure 2 .

 2 Figure 2.15 Grain detection in our experimental ume. a. Image of the channel seen from the top. Black circles: detected blue grains. b. Detection criterion on the image. Black circles: location of the maxima.

Figure 2 .

 2 Figure 2.16 Dependence of the number of detected grains on the tracking parameters. a. Number of detected grains as a function of the minimum distance between two maxima. b. Number of detected grains as a function of the Gaussian-lter size. The dashed red lines correspond to the values we chose.

Figure 2 .

 2 Figure 2.17 Number of detected grains as a function of the uid depth. Blue: transparent uid. Green: uid with blue dye.

Figure 2 .Figure 2 .

 22 Figure 2.18 Selection of 15 trajectories of sediment grains. a. Channel seen from the top. White: selection of grains trajectories. b. Streamwise position of particles as a function of time. Horizontal segments correspond to deposited grains.

  34 g/min. a. Cross section. Brown: sediment. Blue: ow. b. Selection of grain trajectories. Grey dashed lines: crossing lines where the cumulative sediment ux is computed. c. Cumulative sediment ux. In blue: raw data. Magenta points: data distributed into 30 bins. d. Sediment-ux prole, calculated from the binned cumulative sediment ux.

Figure 2 .

 2 Figure 2.20 Dependence of the sediment-ux prole on the number of bins. Left: Sediment-ux prole for dierent numbers of bins. Shaded areas show the measurement uncertainty. a. Number of bins N bins = 200. b. N bins = 40. c. N bins = 10. d. Number of particle crossings per bin by varying the number of bins. N bins = 10 (red), 40 (green), 200 (blue).

Figure 2 .

 2 Figure 2.21 Relative error of the sediment-ux measurement as a function of the number of particle crossings. Red line: Fit by the inverse square root of the number of particle crossings (equation (2.8)).

Figure 2 .

 2 Figure 2.22 Sediment ux of a bed translated uniformly. a. Cumulative distribution of grains crossings across the channel. The origin corresponds to the left-hand wall of the channel, and 3 cm on the right-hand wall. b. Sediment-ux prole calculated from the cumulative prole. Dashed line: mean sediment ux.

Figure 2 .

 2 Figure 2.23 Flow-induced shear stress. Example of run 2 (Tab. 2.1). a. Flow-induced shear stress on the bed across the channel. Points: shear stress computed with nite elements based on the experimental bed elevation. Full line: shear stress predicted by the shallow-water approximation. b. Ratio between the ow-induced shear stress computed with nite elements and the shear stress calculated with the shallow-water approximation.

  2.24). With this single run experiment, we have a rst estimation of the Shields number at the threshold: θ t ≈ 0.15. Comparatively, Seizilles et al. (2014) found θ t = 0.125 for a particle Reynolds number of Re s = 25 and Charru et al. (2004a) measured 0.16 for Re s = 0.3.

Figure 2 .

 2 Figure 2.24 Transport law. a. Sediment ux as a function of the Shields parameter

  Seizilles et al. (2014) measured d = 9.5 µm for plastic grains of size 344 µm, entrained by water. In our ume, we use a more viscous uid (mixture of water and glycerol) owing

σ 2 y

 2 for dierent streamwise positions x (Fig.2.26a). As expected for random walkers, this

Figure 2 .

 2 Figure 2.26 Bedload diusion. a. Selection of 15 grain trajectories. The trajectories are shifted so that their origin coincide. As grains travel downstream, trajectories drift away from the center. b. Distribution of grains' trajectories along the vertical dashed lines of panel a (with the same color code). c. Variance of cross-stream coordinates as a function of the streamwise position of grains. Each color corresponds to an experimental run (Tab. 2.1). Dashed black line: Linear t of experimental data, which yields d = (0.0254 ± 0.0002) d s . Inset: diusion length, tted by independent t for each run, as a function of the sediment discharge of the run. Dashed black line: diusion-length value computed by the t on multiple experiments.

  )

  at the equilibrium (Kramers 1940, Royall et al. 2005, Piazza & Parola 2008).

Figure 2 .

 2 Figure 2.27 Boltzmann distribution for a single experiment. Run 2: Q w = 1.1 L/min, Q s = 0.34 g/min. a. Bed elevation. Each point is interpolated onto the grid of the sediment-ux prole. b. Sediment-ux prole. c. Interpolated bed elevation as a function of the local sediment ux. Dashed red line: ane t (λ = 160 ± 10 µm).

  Figure 2.29 Creep characterization. a. Distribution of streamwise velocity v x (in green) and cross-stream velocity v y (in blue) of particles in the background of the moving layer of grains. We average movies over N f = 100 images. b. Average background velocities for run 1. Mean streamwise velocity ⟨v x ⟩ (in green) and mean cross-stream velocity ⟨v y ⟩ (in blue) as a function of the number of frames over which the movie is averaged. The dashed blue line corresponds of the mean of ⟨v y ⟩. Error bars correspond to the standard deviations of distribution divided by the square root of occurrences.

  .25) which lies in the range [0.1, 0.01] grains/cm/s.We now compare our estimates to the measurements of Houssais et al.(2015). They characterize creep in a closed, annular ume, that contains a granular bed (particle diameter d s = 1.5 mm, density ρ s = 1190 kg/m 3 ) immersed in a uid (density ρ f = 1050 kg/m 3 , viscosity η = 72.2 mPa s). They nd a settling velocity V s of V s = 0.0024 m/s .

Figure 2 .

 2 Figure 2.30 Boltzmann distribution of sediment transport in our ume experiments.

  2.1).Two points are plotted with their error bars (estimated with equation (2.8)). Dashed black line: linear t with λ = (0.144 ± 0.005) d s . Inset: each run is individually tted and gives a value of λ. We plot λ d s as a function of the sediment discharge of the run. Black dashed line: λ computed with the t of the multiple experiments (main panel).

Figure 2 .

 2 Figure 2.31 A linear t of the above gure give a value of λ as a function of the bin size for run 1 of table 2.1.

( 2 . 31 )

 231 Yamasaka et al.(1987) measured this parameter in an inclined wind tunnel. They found a range of 0.1 -1 for γ, depending on the grain size, which accords with the theoretical predictions of Chen et al.(2009) in turbulent streams. To our knowledge, this parameter had never been measured in a laminar ow.

  (1998) successfully maintained an active single channel in the laboratory, yet cut in half with a side wall to maintain it straight. Actually, maintaining an active single-thread channel in the laboratory has proven challenging. Indeed, most experimental channels turn into braids, likely due to the growth of unstable bedforms (Schumm et al. 1987, Paola et al. 2009). Some experimenters prevent the destabilisation of their channel by adding cohesive sediment (Paola et al. 2009), or by growing riparian vegetation (Tal & Paola 2007). The requirement of these ingredients, however, remains a matter of debate (Métivier & Barrier 2012).

Figure 3 . 1

 31 Figure 3.1 Laboratory river: experimental setup. Q w and Q s are the ow and the sediment discharges, respectively. A laser sheet is projected onto the channel and is deviated by the bed topography. A camera placed above the bed displays the channel seen from the top.

Figure 3 . 2 Figure 3 . 3

 3233 Figure 3.2 Top view of a laboratory river. It corresponds to the experimental run 2 of table 3.1 (Q w = 1.0 L/min and Q s = 0.42 g/min).

Figure 3 . 4

 34 Figure 3.4 Evolution of the sediment discharge at one half of the river's length. Blue points correspond to the sediment discharge measured with a movie sequence of 10 to 13 minutes (section 3.4). Red dashed line: decreasing exponential function tted to the data. The decay time is about one hour.

Figure 3 . 5

 35 Figure 3.5 Measure of the laser angle and calibration of bed-elevation measurement. The laser sheet is projected onto a calibration scale. Each step is 2.0 mm high and 1.0 cm wide a. Field of view of the camera, from the top. Detection of the position of the laser (red dashed line). The resolution of the image is 6.62⋅10 -6 m/pixels. b. Measurement of the laser angle (calibration curve). Blue points: Measurement. Red dashed line: t by an ane function. The coecient of proportionality is tan θ laser .

Figure 3 . 6 Figure 3 . 7

 3637 Figure 3.6 Laser deviated by the ow x ′ laser . Flow from right to left. When the channel

Figure 3 . 8

 38 Figure 3.8 Bed elevation and sediment-ux prole for dierent runs. The sediment input increases downwards (0, 0.15 and 0.77 g/min). In the same order, these proles correspond to runs 1, 4 and 5. Dashed lines: measurement uncertainty calculated with equation (2.8).

Figure 3 . 9 Figure 3 .

 393 Figure 3.9 Dependence of the morphology of laboratory rivers with respect to the sediment discharge. a. Width. b. Maximum depth. c. Aspect ratio.

Figure 3 .

 3 Figure 3.11 Experimental distribution of the sediment ux in laboratory rivers. a. Bed elevation as a function of local sediment ux. Each color corresponds to a run. b. Characteristic length of the Boltzmann distribution, λ, normalized by the grain size d s for each run. The dashed black line corresponds to the value measured in the channel experiment (section 2.7).

3. 6 .Figure 3 .

 63 Figure 3.12 Experimental Boltzmann distribution in laboratory rivers. Each color corresponds to an experimental run of table 3.1. The black dashed line corresponds to the law measured in the ume experiment. Two indicating error bars are plotted for run 4 (calculated with equation (2.8)).

Figure 3 .Figure 3 .

 33 Figure 3.13 Image analysis for the width measurement of a laboratory river (top view). We use a colorimetric detection. a. Threshold on blue and measure of the area of the channel (red). b. Detection of the curvilinear length (red dashed line) of the channel.

Figure 3 .

 3 Figure 3.15 Planform geometry of a braided laboratory river. Input parameters: Q w = 1.0 L/min and Q s = 1.7 g/min.

Figure 4 . 1

 41 Figure 4.1 Sketch of a river and notations. h is the bed elevation, and h min its minimum. D is the depth of the river, and D max its lowest elevation. Brown: sediment bed. Blue: ow.

(4. 10 )

 10 Finally, equation (4.8) and (4.10) yield a dierential equation on D(y), the river's depth. We introduce the dimensionless depth D = DS L and the dimensionless coordinate ỹ = yS L. These quantities depend on the downstream slope S of the river and thus, implicitly, on its sediment and uid discharges. With these notations, we get

Figure 4 . 2

 42 Figure 4.2 Numerical resolution of equation (4.15). We x χ = 0.1 and vary s . For each value of s : Top. River cross sections. Brown: sediment. Blue: ow. Dots represent the end of each prole. Bottom: sediment-ux proles. Dots represent the end of each prole.

Figure 4 . 3

 43 Figure 4.3 Numerical resolution of equation (4.15). We x s = 0.1 and vary χ. For each value of χ: Top. River cross sections. Brown: sediment. Blue: ow. Dots represent the end of each prole. Bottom: sediment-ux proles. Dots represent the end of each prole.

Figure 4 . 4

 44 Figure 4.4 Phase portrait of equation (4.15). The colorbar indicates the values of the parameter ξ (equation (4.29)). Each trajectory in the phase portrait corresponds to a prole plotted below in the same color. Full, black lines: river at the threshold (χ = 0) and separatrix (χ = 0.99 s ). Black point: xed point, corresponding of an innitely-wide, at river (χ = s ). Red line: example of an actual river between the two boundaries. Each trajectory corresponds to a distinct value of D max between µ t and µ t + s . White, dashed

Figure 4 .

 4 Figure 4.4 shows trajectories for dierent values of χ. When D max approaches µ t + s , the river widens and features a at segment, as we remarked in the previous paragraph.When D max = µ t + s , the trajectory is a xed point, meaning that D and D ′ are constant. Physically, this corresponds to an innitely wide and at river for which

(4. 33 )

 33 This xed point is dened by∂ y D = 0 , (4.34) ∂ yy D = 0 .

(4. 35 )

 35 When D max > µ t + s , trajectories correspond to unphysical rivers (Fig.4.4, white dashed line), which never reach a bank. Consequently, all acceptable rivers lie between the threshold trajectory and the xed point. A separatrix bounds this domain (black line where χ = 0.99 s , Fig.4.4).The size of this domain is set by s , since the function ξ(D, D ′ ) depends on s only (equation (4.29)). To evaluate how this domain changes with s , we draw the phase portrait for dierent values of s (Fig.4.5). When s decreases, the xed point approaches the threshold trajectory and the domain of acceptable channels narrows. In our experiments,

Figure 4 . 5

 45 Figure 4.5 Phase portraits of equation (4.15) with dierent values for s . Colorbars indicate the values of ξ (equation (4.29)). The black, full lines bounds the region where trajectories correspond to physically acceptable rivers.

Figure 4 . 6

 46 Figure 4.6 Experimental phase portraits. a. Single experiment: run 3 of table 3.1. Black line: theoretical phase portrait for s = 0.02. The separatrix collapses onto the threshold circle, and the phase portrait is virtually reduced to the threshold circle. Blue points: experimental data. Blue area: measurement uncertainty. Magenta square: D max = µ t + χ.

Figure 4 . 7

 47 Figure 4.7 Measure of the slope S and of the friction coecient µ t by tting a river's phase portrait. Blue points: experimental data for run 3 (Tab. 3.1). Red line: t by an ellipse.

Figure 4 . 8

 48 Figure 4.8 Comparison of two methods to measure the slope of a river. y-axis: slope calculated with equation (4.40).x-axis: slope measured by the t of a river's phase portrait.

( 4

 4 .42) Dmax = max( D) .

Figure 4 . 9

 49 Figure 4.9 Illustration of the method to generate an abacus. a. Blue Points: 100 numerical proles for s in [0.01, 10] and χ s in [0.5, 1]. Grey lines: Triangulation. The isolated point is due to the alha shape (α = 0.3 here). b. Contours of s and χ according to the uid and sediment discharges.

Figure 4 .

 4 Figure 4.10 Regime relations for the width and aspect ratio of a river. a. Aspect ratio of a river as a function of its dimensionless sediment discharge Q s Q * s at a xed uid discharge. Grey: triangulation. Blue lines: contours of the dimensionless uid dischargesQ w Q * w . Magenta line: contour of Q w Q * w for our experimental conditions (Q w = 1 L/min).Magenta points: experimental data. Black square: Threshold theory. Black dashed line: asymptotic river described in 4.6.2. b. Normalised width of a river as a function of its dimensionless sediment discharge Q s Q * s at a xed uid discharge. Grey: triangulation. Blue lines: contours of Q w Q * w . Magenta line: contour of Q w Q * w , which corresponds to our experimental conditions (Q w = 1 L/min). Blue points: experiments at Q w = 1 L/min. Magenta points: experimental data. Black square: Threshold theory. Black dashed line: asymptotic river described in 4.6.2. Triangulation downsampled by a factor of ten.

Figure 4 .

 4 Figure 4.11 Regime relations for the slope and the maximum sediment ux of a river. a. Downstream slope of a river S as a function of its dimensionless sediment discharge Q s Q * s at a xed uid discharge. Grey: triangulation. Blue lines: contours of the dimensionless uid discharge Q w Q * w . Magenta line: contour of Q w Q * w , which corresponds to our ex- perimental conditions (Q w = 1 L/min). Magenta points: experimental data. Black square: Threshold theory. Black dashed line: asymptotic river described in 4.6.2. b. Maximum sediment ux q max of a river as a function of its dimensionless sediment discharge Q s Q * s at a xed uid discharge. Grey: triangulation. Blue lines: contours of the dimensionless uid discharge Q w Q * w . Magenta line: contour of Q w Q * w , which corresponds to our ex- perimental conditions (Q w = 1 L/min). Magenta points: experimental data. Triangulation downsampled by a factor of ten.

( 4 .Figure 4

 44 Figure 4.12 Asymptotic river. In brown: sediment bed. In blue: ow. W T is the width over which the river transports sediment uniformly. W H is the width of a river at the threshold of motion. W is the total width of the river.

  et al. 2004b, Lajeunesse et al. 2010). Accordingly, the sediment ux resulting from their collective motion, called bedload transport, is usually expressed as a function of shear stress (Meyer-Peter & Müller 1948).

  , Coleman & Melville 1994). They result from the inertia of the ow, which concentrates shear stress just upstream of their crest (Kennedy 1963, Charru 2006, Charru et al. 2013). Nascent ripples make the most of this mechanism by orienting their crest across the ow. At least initially, they do not involve any crossstream sediment ux. By contrast, the oblique crest of alternate bars diverts the water ow to induce the cross-stream bedload ux that makes them unstable (Parker 1976, Colombini et al. 1987, Devauchelle et al. 2010a, Andreotti et al. 2012). Although less common, streamwise streaks materialise cross-stream bedload more neatly their crest remains aligned with the stream as they grow (Karcz 1967, Colombini & Parker 1995, McLelland et al. 1999). To initiate a cross-stream ux of sediment, these bedforms use a subtle peculiarity of turbulence. When streamwise ridges perturb their boundary, turbulent ows generates transverse, counter-rotating vortices (Colombini 1993, Vanderwel & Ganapathisubramani 2015). Over a granular bed, these slow secondary currents transport sediment across the primary ow to accumulate it in upwelling areas, thus reinforcing the ridges that brought them about. A similar phenomenon occurs when grains of dierent sizes make up the bed, the heterogeneous roughness of which then plays the role of ridges (McLelland et al. 1999, Willingham et al. 2014).

Figure 5 . 1

 51 Figure 5.1 Sediment bed perturbed by longitudinal streaks. A layer of uid (blue) ows over a granular bed (brown). The reference frame is inclined with respect to gravity (downstream slope S). The vertical grey line with a diamond marker symbolises a plumb line.

(2014) found α v ∼ 0. 4 .

 4 Most authors relate bedload directly to the Shields parameter with a sediment transport law (Meyer-Peter & Müller 1948). Combining equations (5.3) and (5.5), we nd that bedload transport is proportional to the distance to threshold. The specic expression of this law, however, is still debated, and is likely to depend on the particle Reynolds number (Ouriemi et al. 2009). Here, we choose a simple law that compares reasonably with near-threshold, laminar experiments (Charru et al. 2004b, Seizilles et al. 2014).

Figure 5 . 2

 52 Figure 5.2 Mechanism of the bedload instability. a: Bed elevation (brown) and free surface ow (blue). b: Distribution of the corresponding downstream sediment ux. Red arrows indicate bedload diusion.

Figure 5 . 3

 53 Figure 5.3 Dispersion relation of the bedload instability, in a free-surface ow. Dashed lines: shallow-water approximation (equation (5.17)). Solid lines: two-dimensional Stokes ow (equation (5.26)).

Figure 5 . 4

 54 Figure 5.4 Stokes ow above streamwise streaks. Bed elevation (brown) and iso-velocity contours (blue scale). Below the red dashed line, the shear stress is higher than in the base state. a: long wavelength (kD = 1.4). b: short wavelength (kD = 4.2).

Figure 5 . 5

 55 Figure 5.5 Stability map of the bedload instability, for a free surface ow. a: Dispersion relation. Darker areas correspond to faster growth rate. Red line: most unstable wave number for a given value of the transport number. b: Growth rate of the most unstable wavenumber. c: Maximum growth rate as a function of the transport number.

Figure 5 . 6

 56 Figure 5.6 Saturation of the bedload instability. Red line: amplitude of the bed perturbation at the transition to the non-linear regime, h max , as a function of the transport number. Inset: transition criterium, the amplitude of the perturbation of the Shields parameter reaches θ 0 -θ t .

Figure 5 . 7

 57 Figure 5.7 Dispersion relation of the bedload instability, in a conned ow. Dashed lines: shallow-water approximation (equation (5.17)). Solid lines: two-dimensional Stokes ow (equation (5.32)).

. 31 )= k 2 ( 2 -Figure 5 . 8 Figure 5 . 9 5 . 7 Figure 5 .

 31225859575 Figure 5.8 Stability map of the bedload instability, for a conned ow. a: Dispersion relation. Darker areas correspond to faster growth rate. Red line: most unstable wave number for a given value of the transport number. b: Growth rate of the most unstable wavenumber. c: Maximum growth rate as a function of the transport number.

  (2007), of Lobkovsky et al. (2008) and of Seizilles et al. (2014) on the stability diagrams of gure 5.11. Unfortunately, we can only do this approximately since, for instance, Ouriemi et al. (2007) used a round pipe, and let the sediment discharge decreases slowly until the bed reached the threshold for bedload transport (table 5.1). The transport number in their experiment thus ranges from zero to a nite value. Lobkovsky et al. (2008) also let the sediment discharge decrease, although in a rectangular, closed channel. Finally, Seizilles et al. (2014) used an open channel, and fed it with a constant input of sediment, thus approaching more closely the theory presented here.Table5.1 summarises our estimates for the parameters of these experimentsmany of them are uncertain. In addition, the parameter γ, which accounts for the gravity-induced

Figure 5 .Table 5 . 1

 551 Figure 5.11 Stability map for the bedload instability in a nite-width channel. Blue scale indicates growth rate of the fastest-growing mode. Red boxes show the range of laboratory experiments. a: open channel. b: xed lid

  , Métivier et al. 2017). 5.9 Forme d'équilibre d'une rivière Cette étude souligne les limites de l'approximation d'eau peu profonde (shallow water ), qui est insusante pour décrire correctement l'instabilité aux courtes longueurs d'onde. De ce fait, nous pensons qu'elle est également insusante pour prédire la forme d'équilibre d'une rivière active, ce que nous avions déjà soupçonné dans le chapitre 4. Il faudrait donc prendre en compte la diusion transverse de quantité de mouvement dans l'écoulement et résoudre l'équation de Stokes à deux dimensions. Cependant, ce problème est dicile, car nous ne connaissons pas a priori la forme de la rivière, et donc la limite du domaine de résolution de l'équation. Cette limite fait partie de la solution du couplage entre le lit érodable et l'écoulement. Malgré tout, nous pouvons utiliser la théorie linéarisée de l'article ci-dessus pour approcher l'écoulement de Stokes à deux dimensions. Dans la section 5.7, nous étudions l'inuence d'une largeur nie sur la sélection de longueur d'ondes des stries (Fig. 5.10). Nous mettons en évidence l'existence de modes discrets susceptibles de se développer ou non selon le rapport d'aspect du chenal, déni comme le rapport entre sa largeur et sa profondeur. En particulier, le premier mode correspond à un rapport d'aspect d'une demi longueur d'onde. Si l'on imagine que notre rivière à l'équilibre présente un prol convexe, ce mode serait le plus approprié pour la représenter (Fig. 5.12a).

. 37 )Figure 5 .

 375 Figure 5.12 Illustration of the equilibrium shape of a river with a two-dimensional ow. a. Mode 1 of the instability (shifted). b. Aspect ratio of laboratory rivers as a function of dimensionless sediment discharge. Blue points: laboratory rivers (Tab. 3.1). The red line corresponds to the stability domain limit for the rst mode.

5. 13 )

 13 . Liu et al. (2008) measured the intensity of bedload 8 km downstream of the source, at an elevation of 3300 m, where gravels make up most of the stream's bed (the median grain size is d 50 = 21.5 mm). They surveyed the river every day during two ow seasons (approximately 3 months) from June to August (2005 and 2006). To measure the bedload, they used a Helley Smith sampler of inlet 30×15 cm (Helley & Smith 1971).

Figure 5 .

 5 Figure 5.13 Bed cross section and sediment ux of the Urumqi river (China). a. River cross section. Brown: sediment bed. Blue: water. b. Sediment-ux prole across the channel.

  ranges. During their formation, these fans gradually accumulate the sediment deposition in stratigraphic layers, recording the history of climate changes. By measuring the slopes of the layers that make up the fan, we could in principle estimate the paleo-uxes of water and sediment (Delorme et al. 2018). With our shallow-water model, we had a limited success to predict this relation. An improved theory, however, would allow us to establish a physically-based relation between the slope of an alluvial fan, and the discharges of the river that built it. Finally, the measurements of Liu et al. (2008) should encourage us to collect sedimentux proles across alluvial rivers, instead of the total sediment discharge. We could then evaluate how the Boltzmann length varies with the grain size and how a broad distribution

Figure 5 .Figure A. 4

 54 Figure 5.14 Boltzmann distribution in the Urumqi river. Bed elevation as a function of the logarithm of the sediment ux. Red dashed line: linear t.

Figure B. 1

 1 Figure B.1 Estimate of the relative numerical error. a. Error along a cross section prole for s = 0.5 and χ = 0.25. In blue: relative numerical error. Red, dashed line: mean of the relative numerical error. b. Relative numerical error as a function of χ s for χ = 0.5. c. Relative numerical error as a function of χ s for s = 0.5.

Figure C. 2 Figure C. 8

 28 Figure C.2 Investigation of a can stability by probing its side with a poker. Experiments of Virot et al. (2017). a. Force-displacement curve for seven cans: poker force F P as a function of the poker displacement D P for dierent axial loads F A . Inset: axial load measurement. b. Three-dimensional representation of these independent curves for 89 cans in the space (F A , F P , D P ). c. Three-dimensional representation of these curves probed on the same can without make it buckle. d. Stability landscape of shell buckling. The red point (s. b.) corresponds to the spontaneous buckling, and the brown point (m. b.) to the minimum buckling.

  Fig. C.8a). If we perform the same measurements twice, we observe that these variations are remarkably reproducible (Fig. C.8a). These measurements suggest a local correlation between the defect of a can and the amplitude of the stability-landscape ridge.

1 1

 1 Institut de Physique du Globe de Paris, 1 rue Jussieu, 75238 Paris, France 2 Now at Naxicap Partners, 5-7 rue de Monttessuy, 75007 Paris, France (Dated: May 10, 2019)

FIG. 1 .

 1 FIG. 1. (a) Experimental setup and notations. Two Plexiglas panels confine the flume laterally. The x axis is aligned with the flow. (b) Part of the camera's field of view (background picture) with superimposed grains trajectories (red lines). Dots and arrows indicate beginning and end of trajectories, respectively. Data from experimental run #1.

FIG. 2 .

 2 FIG. 2. Cross-stream dispersion of the traveling grains. (a) 34 trajectories from run #1, with starting point shifted to origin. (b) Cross-stream variance of shifted trajectory. Dashed black line: linear relation with diffusion length d fitted to data [Eq. (2)]. Inset: d fitted independently to individual runs. Error bars show uncertainty.

  FIG. 3. (a) Average cross-section of the flume during run #1. Beige: sediment layer. Blue colormap: downstream flow velocity calculated with finite elements. (b) Flow-induced shear stress on the bed. (c) Sediment flux measured by grain tracking. Error bars indicate measurement uncertainty (Sup. Mat.).

FIG. 4 .

 4 FIG. 4. (a) Local sediment transport law. Marker types indicate individual experimental runs. Solid line: experimental run #1. Dashed black line: Equation (5) with q0 = 544 grains s -1 cm -1 and θt = 0.17. (b) Distribution of sediment flux with respect to bed elevation. Colors and markers similar to (a). Dashed black line: Bolztmann distribution [Equation (8)] with λB = 0.10 mm.

1

  Institut de Physique du Globe de Paris, 1 rue Jussieu, 75238 Paris, France 2 Now at Naxicap Partners, 5-7 rue de Monttessuy, 75007 Paris, France (Dated: May 10, 2019)

  2.

FIG. 1 .

 1 FIG. 1. (a) Sediment grains. (b) Shaded histogram: grainsize distribution of a sample containing 1084 grains. Red dashed line: normal distribution with the mean and standard deviation of the sample.

FIG. 2 .

 2 FIG. 2. Grain detection. (a) Histogram of pixel hue for the frame shown in (b). Selected range (black arrow) corresponds to blue grains. (b) Movie frame from run #1 (Tab. I, main document). Dark circles: detected blue grains. Actual picture is broader than shown.

FIG. 3 .

 3 FIG. 3. Sediment flux measurement for run #1. (a) Total discharge of blue grains, for 6 independent movies. Horizontal error bars correspond to movie duration. The experiment was started the day before time on horizontal axis. Vertical error bars: uncertainty based on Eq. (2). Dashed black line: average over movies. (b) Flux of blue particles. Colors correspond to (a). Dashed black line: average over movies. Black error bars: uncertainty based on Eq. (2). (c) Standard deviation of the number of detected crossings, as a function of the number of detected crossings (blue dots). Dashed black line: fitted square-root relation.

  ) and (3) yield an approximate formula for the uncertainty on the sediment flux [error bars on Figs. 3(a) and 3(b), and on Fig. 3 (main document)].

hFIG. 4 .

 4 FIG. 4. Bed-elevation measurement. (a) Laser sheet projected on the channel bed (top view), after smoothing with a 4-pixel Gaussian filter. Orange dashed line: location of peak intensity. (b) Green intensity of pixels along the white, dashed line of (a) (y = 570 pixels). Orange dashed line: fit of the laser peak with a second-order polynomial. (c) Cross-stream average of the sediment bed elevation measured as a function of the position of the laser carriage (green line). Dashed black line: affine fit.

FIG. 5 .

 5 FIG. 5. Finite-elements simulation of the Stokes flow for run #1. (a) Mesh used to calculate the flow field of Fig. 3(a) (main document). (b) Dependence of the flow depth on the dimensionless discharge. Solid line: numerical relation. Blue points: deepest and shallowest simulations in (c). Red square: best estimate for run #1; error bars show uncertainty on dimensionless discharge, and resulting uncertainty on depth. (c) Influence of the maximum flow depth on the dimensionless shear stress.

1

  Photographie de l'Ain à Saint-Maurice-de-Gourdans, Port Galland (France).2Evolution du lit de la rivière Kosi (Inde). . . . . . . . . . . . . . . . . . . . .1.1 Exemple d'une rivière alluviale : rivière Kaidu (Tian-Shan, Chine) vue de haut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Corrélation empirique entre la morphologie des rivières et leur débit d'eau. 1.3 Bilan des forces s'exerçant sur un grain au niveau de la surface du lit. . . . 1.4 Valeur seuil du nombre de Shields θ t en fonction du nombre de Reynolds de grain, Re s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Etude expérimentale de la morphologie d'une rivière de laboratoire en régime laminaire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Rapport d'aspect W D en fonction du débit d'eau. . . . . . . . . . . . . . . 1.7 Rivière Wiliam vue de haut (Canada). . . . . . . . . . . . . . . . . . . . . . . 1.8 Evolution de la morphologie de la rivière William (Canada) le long de son cours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Rapport d'aspect de rivières naturelles en fonction de leur débit de sédiments. 1.10 Transition méandre-tresse du Rhône (Suisse) du à un apport élevé en sédiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11 Transition méandre-tresse de la rivière Kaidu (Tian Shan, Chine) du à un apport élevé en sédiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Transition tresses-méandres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Inuence of sediment transport in laboratory channels (Stebbing's experiments). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 Théorie du seuil pour les rivières de laboratoire. . . . . . . . . . . . . . . . . 1.15 Transport par charriage sur un lit granulaire cisaillé par un écoulement uniforme et stationnaire. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Lois de transport expérimentales. . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Diusion transverse des grains. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Setup of the ume experiment fed by plastic sediment grains and a mixture of water and glycerol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Measurement of the sediment input. . . . . . . . . . . . . . . . . . . . . . . .

  de frottement solide µ t . Au centre du chenal, la pente s'annule, et la profondeur maximale est donc D max = µ t L S (Seizilles et al. 2013).

									(1.14)
	Cette longueur, qui ne dépend que de la nature des sédiments et du uide, est de l'ordre
	de la taille du grain.
	Sur les berges, la profondeur s'annule (D = 0). D'après l'équation (1.13), la pente vaut
	donc le coecient Une solution de l'équation diérentielle (1.13) est donc
		D(y) =	µ t L S	cos	Sy L	,	(1.15)
	ce qui prédit une largeur de rivière égale à
				W =	πL S	.
	W D max	=	π µ t	≈ 4 et	W ⟨D⟩	=	π 2 2µ t	≈ 7
									Cette expres-
	sion est valable aussi bien en régime turbulent qu'en régime laminaire (Glover & Florey
	1951, Henderson 1961, Seizilles et al. 2013).

En accord avec l'approximation de Saint-Venant, on peut considérer que cos φ ≈ 1, et la condition de seuil (équation

(1.11)

) devient ainsi

[START_REF] Glover | Stable channel proles[END_REF]

DS L

2

+

∂D ∂y 2 = µ 2 t (1.13) où l'on dénit la longueur caractéristique L = θ t (ρ s -ρ f )d s µ t ρ f . (1.16) Pour une largeur de rivière donnée, une unique pente d'équilibre est possible, et inversement. Il est donc impossible d'imposer à la fois la pente et la largeur de la rivière : c'est elle qui acquiert sa pente d'équilibre. Enn, le rapport d'aspect d'une rivière au seuil est constant, (1.17) À notre connaissance, la forme en cosinus prédite par la théorie du seuil (équation (1.15)) n'a pas été testée expérimentalement en régime turbulent. Cependant, comme cette forme ne dépend pas de la nature de l'écoulement, Seizilles et al. (2013) ont mis en place un dispositif expérimental pour reproduire au laboratoire des rivières miniatures en régime laminaire. Il s'agit d'un lit incliné, couvert de sédiments, à l'entrée duquel on injecte un liquide visqueux. Au bout de quelques heures, un chenal se forme spontanément et chaque grain qui compose son lit est au seuil de mise en mouvement. En mesurant la section de leurs rivières expérimentales, Seizilles et al. (2013) ont démontré la validité de la théorie du seuil dans le cas laminaire (Fig. 1.5). 1.1.5 Largeur d'équilibre d'une rivière au seuil Connaissant la forme de la rivière au seuil, il nous reste à déterminer la vitesse de l'écoulement pour calculer son débit. Cette expression varie selon le régime (laminaire ou turbulent) de l'écoulement.

Table 1 .

 1

	Brahmaputra	15 000	1670
	Amazone	480 000	1000
	Danube	6 500	67
	Kosi	2 036	43
	1 Estimations du débit d'eau moyen et du débit de sédiments à l'embouchure
	de diérentes rivières. D'après Milliman & Meade (1983) et Sinha (2009).
	borne inférieure (Fig. 1.6).
	Une cause probable de ce désaccord entre la théorie du seuil et les données de terrain
	est que la plupart des rivières naturelles transportent des sédiments. Les sédiments qui

1.2a). Cependant, les mesures de terrain montrent que le rapport d'aspect d'une rivière, déni comme le rapport entre sa largeur et sa profondeur, varie de manière signicative d'un site à l'autre. Les valeurs sont dispersées et la théorie du seuil semble seulement prédire une Rivière Débit d'eau (m 3 /s) Débit de sédiments (10 6 t/an) composent le lit des rivières alluviales ne sont donc pas au seuil de mise en mouvement, mais sont transportés par l'écoulement. Dans la section suivante, nous proposons un aperçu des connaissances concernant l'inuence du transport sédimentaire sur la morphologie des rivières, en nous fondant sur des observations de terrain et de laboratoire. 1.2 Inuence du transport de sédiments sur la morphologie d'une rivière Bien que la théorie du seuil prédise au premier ordre la largeur des rivières naturelles, elle sous-estime en général leur rapport d'aspect. Cet écart entre théorie et observations de terrain peut provenir de l'inuence du transport sédimentaire dans les rivières, qui n'est pas prise en compte dans la théorie du seuil. En eet, en incisant les reliefs, les rivières entraînent avec elles de la matière sous forme de sédiments. A l'échelle du globe, on estime qu'elles déchargent chaque année 13.5 milliards de tonnes de sédiments vers les océans (Milliman & Meade 1983). La quantité de sédiments dépend principalement de l'aire du bassin versant, de sa topographie et de la lithologie du sol et varie selon les rivières (Tableau 1.1). Cette quantité évolue également en fonction du climat (Mueller & Pitlick 2013). La taille et la composition des sédiments résultent des processus d'altération physicochimiques du sol. Les grains de petite taille comme les limons et les argiles sont assez légers pour être retenus en suspension dans l'écoulement. Les sables et les graviers, eux, sont transportés près du lit et avancent par bonds successifs. Ce mode de transport, appelé charriage, contrôle au premier ordre la forme du lit. C'est donc celui que l'on étudiera dans ce manuscrit.

  1.2.1 Largeur de la rivièreDe nombreuses observations de terrains suggèrent que le ux de sédiments exerce un contrôle fort sur la morphologie d'une rivière. L'exemple le plus frappant est sans doute celui de la rivière William, qui se situe dans l'état du Saskatchewan, au Canada[START_REF] Smith | William River: An outstanding example of channel widening and braiding caused by bed-load addition[END_REF]). Cette rivière forme initialement un chenal étroit et profond, puis atteint un champs de dunes au niveau du lac Athabasca, où elle se charge en sédiments (Fig. 1.7). En l'espace de 25 km, son débit solide est multiplié par 40, sa largeur passe de 50 à 300 mètres, et sa forme évolue d'un chenal unique vers une tresse (Fig. 1.8).

	Les observations de terrain compilées par Métivier & Barrier (2012) viennent compléter
	celles de Smith & Smith (1984). Elles suggèrent également que le rapport d'aspect d'une
	rivière augmente avec le débit de sédiments (Fig. 1.9).

Parker (1978a)

est le premier à proposer un modèle pour expliquer l'élargissement d'une rivière lorsqu'elle transporte des sédiments. Les grains qui composent le lit de la rivière sont entraînés par l'écoulement et voyagent dans la direction de la force qui s'exerce sur eux. Lorsqu'ils se trouvent au centre du chenal, cette force est parallèle à l'écoulement.

En revanche, sur les berges, cette force possède une composante transverse à l'écoulement, à cause de la pente du lit. Par l'action de leur poids, la trajectoire des grains est donc légèrement déviée vers le centre du chenal (Seizilles (2013), Ch. 5). Ce mécanisme érode les berges du chenal qui s'élargit progressivement. Si rien ne compense ce mécanisme, la rivière devrait donc s'élargir indéniment. Or, sur le terrain, on observe que les rivières aqcuièrent une largeur d'équilibre. Pour résoudre ce paradoxe, souvent nommé paradoxe de Parker, Parker propose diérents mécanismes capables de compenser l'érosion des berges (Parker 1978a;b).

Dans une rivière sableuse, où la suspension des sédiments est prédominante, Parker (1978a) suggère que la diusion de particules du centre du chenal vers les berges induit un ux de grains, transverse à l'écoulement, capable de contrebalancer l'érosion des berges.

Dans les rivière graveleuses, cependant, les sédiments ne sont pas transportés en suspension, mais sont charriés par l'écoulement et restent près du lit de la rivière.

Parker (1978b)

propose donc un autre mécanisme pour équilibrer l'eet de la gravité : la diusion de quantité de mouvement de l'écoulement, liée à la turbulence. Cette diusion aurait pour eet de diminuer la contrainte exercée par l'écoulement au niveau des berges. Cette contrainte pourrait atteindre une valeur inférieure à la contrainte seuil d'entraînement des grains.

Dans ce cas, les berges ne transporteraient aucun grain, et le ux gravitaire s'annulerait.

Cependant, ces mécanismes sont diciles à caractériser sur le terrain ou au laboratoire et n'ont jamais été testés expérimentalement. Dans ce manuscrit, nous nous inspirerons de ces modèles pour prédire la forme d'équilibre d'une rivière de laboratoire (Ch. 2 et 3).

Table 1 .

 1

		Régime laminaire	Régime turbulent
		Charru et al. (2004b) Lajeunesse et al. (2010)
	V s	(ρ s -ρ) g d 2

1.15).

Une particule entraînée par l'écoulement rebondit sur le lit avec une fréquence caractéristique, de l'ordre de l'inverse du temps que met une particule à sédimenter d'une taille 2 Expressions du taux d'érosion, de déposition et de la vitesse moyenne des grains selon le régime d'écoulement.

(1.35)

Chaque fois que la particule atteint le lit, elle a une probabilité nie de se faire piéger. Si l'on suppose cette probabilité uniforme, le taux de déposition est proportionnel au nombre de grains mobiles et à la fréquence de rebond,

ṅd = c d n V s d s , (

1

.36) où c d est un coecient que l'on peut déterminer empiriquement (Charru et al. 2004a). Le taux d'érosion est proportionnel au nombre de grains au repos par unité de surface, de l'ordre de 1 d 2 s , divisé par un temps caractéristique d'érosion. Ainsi, s (θ -θ t ) , (1.38) où c e est un coecient déterminé empiriquement (Charru et al. 2004a).

cônes alluviaux, qui se trouvent généralement à la sortie des chaînes de montagnes (Delorme 2017). Ces zones de dépôts sédimentaires sont utilisées pour reconsti- tuer le climat passé des continents, selon les paléo-ux d'eau et de sédiments. Pour mesurer les paléo-ux d'une rivière à partir des sédiments qu'elle a déposés, il faut estimer le vo-

  .3).

	leur inuence, à grande échelle, sur la morphologie d'une rivière.
	Pour ce faire, nous avons choisi une approche expérimentale. Seizilles (2013) a montré
	que l'on peut étudier le couplage écoulement-transport en utilisant des rivières expérimen-
	tales en conditions laminaires. Ces conditions d'écoulement permettent de s'aranchir des
	dicultés liées à la turbulence, et des lois empiriques qui y sont associées. L'objectif est
	donc (i) de former des rivières au laboratoire, (ii) d'étudier la sensibilité de ces rivières à
	Pourtant, le rapport d'aspect d'une rivière naturelle, déni comme le rapport entre ses paramètres d'entrée : son débit d'eau et, surtout, de sédiments.
	sa largeur et sa profondeur, est généralement sous estimé par cette théorie. Les données
	de terrain suggèrent que le transport de sédiments, dont le rôle est négligé par la théorie
	du seuil, contrôle également la forme des rivières (1.2). Lorsqu'une rivière transporte des
	sédiments, elle a tendance à s'élargir, et donc à accroître son rapport d'aspect. Cependant,
	jusqu'à présent, l'inuence du débit de sédiments sur la forme d'une rivière n'est comprise
	qu'empiriquement.
	Un des objectifs de ce manuscrit est de comprendre comment une rivière ajuste sa
	largeur au transport sédimentaire. Cette étude permettrait par exemple de mesurer la
	charge sédimentaire d'une rivière uniquement à partir d'images satellites, s'aranchissant
	ainsi des dicultés de mesures de débits de sédiments sur le terrain (Mouyen et al. 2018).
	Cette méthode s'appliquerait également à des rivières diciles d'accès (Smith et al. 1996).
	Enn, elle pourrait être extrapolée pour étudier les processus d'érosion à la surface d'autres
	planètes ou satellites, comme Titan, où des chenaux et des motifs d'érosion uviatile ont
	été observés (Jaumann et al. 2008).
	Un autre paramètre qui dépend du débit sédimentaire d'une rivière est sa pente longi-
	tudinale, c'est-à-dire dans la direction de l'écoulement (section 1.2). La pente d'une rivière
	contrôle celle des lume des dépôts et donc la pente du cône. Ainsi, mieux comprendre l'inuence du débit de
	sédiments sur la pente d'une rivière est primordial pour évaluer l'ampleur des changements
	climatiques passés et estimer les transferts de masses sur les continents.
	Lorsqu'il est susamment élevé, le débit de sédiments d'une rivière peut déstabiliser
	son cours pour former des tresses (paragraphe 1.2.2). Un autre objectif de ce manuscrit
	sera d'étudier la stabilité d'une rivière à chenal unique vis à vis du forçage sédimentaire.
	Toutes ces problématiques sont liées au couplage entre le transport sédimentaire et
	l'écoulement (section 1.3). Malgré le nombre de publications dans ce domaine, ce couplage
	écoulement-transport et ses conséquences sur la morphologie d'une rivière sont encore mal
	compris aujourd'hui. Cette problématique constitue la base de ce manuscrit. Il s'agit de
	comprendre les mécanismes physiques à l'échelle du grain, de les modéliser, et d'étudier

Table 2 .

 2 1 Experimental parameters for the 5 runs.

			min] Re s Tracking duration [frames]	Tracking colors
	1	1.12	0.65	0.90	300803	Blue, red
	2	1.11	0.34	0.91	204978	Blue, red
	3	1.13	0.33	0.84	372266	Blue
	4	0.87	0.37	0.83	543528	Blue, red, orange
	5	0.83	0.74	0.75	208289	Blue

  2.14).

	Red	Green	Blue	Hue
	0.211	0.149	0.597	68	○
	0.495	0.483	0.721	60.5

○

Table

2

.2 Dierent colors and their coordinates in the rgb system. Corresponding hue.

Table 2

 2

.3 Tracking parameters.

  4 grains/m/s and a critical Shields number of θ t = 0.169.

	Dimensionless sediment flux, q s d 2 s /V s	0.10 0.0 0.1 0.2	0.15 Shields parameter, Method described in 2.5.2 Shallow-water approximation	0.20

Figure 2.25 Comparison of transport-law measurements. In blue: method described in 2.5.2. In red: transport law calculated with the shallow-water approximation. (Fig. 2.19). Its eect is thus negligible.

  Figure 2.28 Boltzmann distribution for multiple experiments. Each color corresponds to a run of table 2.1. a. We plot the raw data of sediment ux and bed elevation, without normalization. b. The data are translated with a reference for the bed elevation h min .

	Normalized occurrence	50 100 0	0.6 Background velocity, v/d s [1/s] [cm] 0.02 0.00	0.02 a	a 10 2 Number of averaged frames, N f 10 1 10 3
			Bed elevation, h	0.4
				0.2	10 3	10 2 Sediment flux, q s [grains/cm/s] 10 1 10 0	10 1	10 2
			[cm]			b
			Bed elevation, h h min	0.0 0.2	10 3	10 2 Sediment flux, q s [grains/cm/s] 10 1 10 0	10 1	10 2
							0.65 0.34	Qs [g/min] 0.33 0.37	0.74
							.29b),
	but remains between [0.01, 0.001] d s /s, which corresponds to
							6 ⋅ 10 -5	<	⟨v x ⟩ V (2.24)

s < 6 ⋅ 10 -4 .

  Contents 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Laboratory rivers : experimental setup . . . . . . . . . . . . . . .

3.3 River cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Sediment-ux prole . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Boltzmann distribution in a laboratory river . . . . . . . . . . .

3.5.1 Boltzmann distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Comparison with the ume experiment . . . . . . . . . . . . . . . . 3.6 Inuence of the sediment discharge on the river's width . . . . 3.7 Planform geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3 .

 3

				Tracking duration	Number of trajectories
		[L/min]	[g/min]	[frames]	of length > d s
	1	1.00	0	-	-
	2	0.99	0.42	218595	30747
	3	1.00	0.33	437622	22501
	4	0.87	0.15	530172	28890
	5	0.98	0.77	324582	55741

1 experimental parameters for the 5 runs.

Table 4 .

 4 1 Values of the transport parameter χ, of the slope parameter s , and of the parameter ξ for the laboratory rivers of table 3.1.

	Run	S	s	χ	ξ
	1	0.0047 ± 0.0008 0.011 ± 0.002	0	0
	2	0.010 ± 0.002	0.026 ± 0.006 0.19 ± 0.04 2.8⋅10 -4
	3	0.009 ± 0.002	0.022 ± 0.005 0.18 ± 0.05 7.8⋅10 -4
	4	0.006 ± 0.001	0.014 ± 0.003 0.11 ± 0.03 2.7⋅10 -4
	5	0.008 ± 0.001	0.020 ± 0.004 0.20 ± 0.05 6.9⋅10 -5

b. Multiple experiments. Each color corresponds to a run in table 3.1.

  Based on laboratory observations, Charru et al.(2004b) suggested that the number of grains the uid dislodges from the bed, per unit surface and time, is proportional to the distance to threshold, θθ t . The bedload layer is fed by this constant input. Conversely, it loses a fraction of its population through settling. When moving grains are too sparse to interact, the settling rate is proportional to the number of moving grains per unit area, n(Aussillous et al. 2016). At equilibrium, the density of moving grains thus reads α n is, like θ t , a dimensionless, empirical parameter (Lajeunesse et al. 2010). For illustration, θ t ∼ 0.1 and α n ∼ 0.01 are typical values for these parameters in a laminar ow (Seizilles et al. 2014). Equation (5.3) is valid only above threshold, that is, when θ ≥ θ t ;

	n =	α n s d 2	(θ -θ t )	(5.3)

where

TABLE I .

 I Experimental parameters. Run #1 serves as an example in all figures. coordinate, and d is the diffusion length, which scales with the amplitude of the trajectory fluctuations. Tracking resin grains in a water flume, Seizilles et al. found d ≈ 0.03 d s (d s is the grain size)

	Run Sediment input Fluid input Slope Tracking time
	#	[grains s -1 ]	[L min -1 ]	%	[min]
	1	42.4	0.83	0.88	93
	2	37.4	1.12	0.79	100
	3	21.3	0.87	0.77	181
	4	19.7	1.13	0.69	68
	5	19.2	1.11	0.71	124
	cross-stream

MW d'électricité par an aux deux pays. Ces installations ont permis d'accroître l'activité agricole de cette région et ont contribué à son développement. Cependant, l'aménagement de la Kosi a fortement impacté sa dynamique. Les sédiments qu'elle a déposés ont surélevé son lit et fragilisé ses digues. Suite à une forte mousson, le 18 août 2008, une brèche s'est formée et la rivière a brutalement abandonné son lit au prot d'un ancien tracé, inondant des villes, des villages et des terrains agricoles (Sinha 2008; 2009). Elle a détruit ainsi des centaines de milliers d'habitations et a entraîné le déplacement de plus d'un million de personnes. En résumé, l'attractivité d'un euve s'accompagne d'une urbanisation ainsi que d'une concentration des populations et des activités dans des zones à risque d'inondation. C'est

www.anton-paar.com

global.canon/en/

pypi.org/project/munkres/

We thank F. Métivier, D.H. Rothman and A.P. Petroff for insightful discussions. We are also indebted to P. Delorme for the sediment feeder, and to J. Heyman for his suggestions on particle tracking. O.D. and E.L. were partially funded by the Émergence(s) program of the City of Paris, and the EC2CO program, respectively.

Chapter 2

Self organisation of sediment transport in a flume

Simple solutions

Equation (4.15) is a non-linear ordinary dierential equation. We are not aware of any general analytical solution. However, it presents two trivial solutions.

When χ vanishes, which means there is no sediment transport, equation (4.15) becomes

which is the equation at the threshold of motion ( 1.1.3). The solution is then the cosine prole derived by Henderson (1961) ( 1.1.4, equation (1.15)).

On the other hand, an innitely wide and at river is also a solution of equation (4.15).

Indeed, if we take D ′ (y) = 0 and D ′′ (y) = 0 along the entire prole, we nd D = µ t + s .

(4.20)

Numerical solutions

Apart from the two simple solutions mentioned in 4.2.5, we are not aware of any other analytical solutions of equation (4.15). As a consequence, we approach its solutions numerically. In this section, we describe our numerical method, and discuss its solutions for dierent values of the transport parameter χ and the slope parameter s .

Numerical method

To solve numerically equation (4.15), we need the expression of D ′ as a function of D.

The derivative of D is, after equation (4.15):

(4.21)

A positive sign means that the depth increases with the cross-stream coordinate y. We expect that in a real river, the depth decreases from the center to the banks (convex bed).

Therefore, we retain the negative solution only.

We supplement equation (4.21) with initial conditions at the channel center (y = 0). In the center, the depth is D(y = 0) = D max = µ t + χ and the cross-stream slope of the river vanishes (D ′ = 0). Now, we integrate this equation with the ode function of the library scipy.integrate in Python. However, at the beginning of the integration, D ′ vanishes, and we cannot run the integration directly. Therefore, we need to expand equation (

A.1 Grain size

In the experiments of sections 2.2 and 3.2, we use the same plastic grains. They are sold by the company Guyson and made of a urea-based thermoset resin. According to the manufacturer, their density is 1520 kg/m

3 . Using a pycnometer, we found ρ s = 1488 kg/m 3 .

To produce these grains, the manufacturer crushes the initial material. This process yields angular, irregular grains, which are then sieved. The manufacture provides an approximate granulometry (mean diameter in [650, 800] µm). To measure the gain size accurately, we take a top-view picture of grains spread over a black surface. Using a threshold on the image intensity, we measure their areas (Fig. A.1a). Then, we divide the square root of this area by π 4, which provides us with an estimation of the grain diameters. We obtain the cumulative distribution of grain diameters of gure (A.1b) and then the distribution of grain diameter by deriving the latter. We dene the grain diameter d s as the mean diameter and therefore d s = 827 ± 86 µm .

A.2 Friction coecient

The static friction coecient µ t of a granular material is usually measured from the angle of repose of a conical heap [START_REF] Jaeger | Physics of the granular state[END_REF]. However, in our case, the interstitial uid may inuence the packing stability of the heap. Therefore, we measure the angle of repose of an immersed heap.

To do so, we use an independent experimental setup, consisting in a tank lled with a mixture of water and glycerol (Fig. Here, we content ourself with the mean value of the heap's slope, which provides us with a friction coecient of µ t = 0.8 ± 0.1 . This method yields an estimate of the friction coecient. It does not take into account the packing rearrangement of the heap over long times. Our laboratory rivers, however, ow during tens of hours, and their bed has time to rearrange. We expect then a larger angle of repose when the packing is denser, although we were not able to check it.

In section 4.5, we suggested another method to evaluate the friction coecient of our immersed grains, based on the cross section of a laboratory river. For run 3 (Tab. 3.1), we found µ t = 0.85 ± 0.05. This value is compatible with the heap method.

A.3 Mixture of glycerol and water

The viscous uid used in our experiments (sections 2.2 and 3.2) is a mixture of water and glycerol, the density and the viscosity of which increase with the proportion of glycerol (Fig.

A.4a). Working with the same uid, Delorme (2017) observed variations of the density over time, due to the evaporation of water (Fig. ,p. 120). As explained in section 2.2, we regularly measured the density of the uid during an experimental run, and compensated for evaporation by adding water to the mixture.

A.4 Tracking of particles

We provide here the library we used to track our colored grains in sections 2.2 and 3.2. from pylab import * from munkres import Munkres, print_matrix from numpy.random import rand from scipy.cluster.vq import kmeans,vq from scipy import ndimage from skimage.feature import peak_local_max import cv2 import json import matplotlib.colors as colors self.x = mummy [1] self.y = mummy [2] def addPoint( self, X ) :

def mummify( self ):

The ratio µ varies from its value at the banks, µ banks , where it is the smallest (closest to the threshold), to its value at the center of the river's channel, µ center , where it reaches a maximum (where transport is the most intense), with

The angle φ is linked with µ through equation (B.6):

(B.9)

Once we have µ and φ, we deduce the depth and the cross-stream slope with equation (B.5). To get the river cross section, we need the cross-stream coordinates y which correspond to the depths. We use:

By integrating equation (B.10), we deduce the cross-stream coordinates y, and then the whole prole of a river, for given values of ( s , χ).

Overall, this numerical method does not require any solver as ode, but only an integration with a basic function trapz in Python.

B.3 Numerical codes B.3.1 Method with ode

We provide here the code corresponding to the method described in section 4.3.1. This method is based on an implicit method to solve the dierential equation (4.15).

B.3.2 Method used for the regime relations

We provide here the code corresponding to the method described in section B.2, and which we used in section 4.6 to generate abacuses.

We rst dene an array of µ which is between µ bank and µ center (equation (B.8)). We then calculate the corresponding array of φ with equation (B.9), and deduce the depth with equation (B.5). Finally, we have the derivative of y, that we integrate with trapz (equation (B.10)). We then have the dimensionless prole D(y). The response of the can depends on the axial load. Figure C.2a shows the evolution of the poker force F P as a function of its displacement D P . For a weak axial force (typically smaller than 400 N), the poker force increases with its displacement, as it enters into the can. For intermediate axial forces (between 400 and 700 N), it decreases and then increases again. In these two regimes, the poker can enter deeply into the can without triggering the buckling.

If we now clamp the can with an axial load higher than approximately 700 N, its response to poking changes drastically and the curve F p (D P ) is surprising. As the poker enters into the can, the force it applies on it increases, decreases, and then vanishes. At this point, we hear a snap and the can buckles, showing a simple dimple. The moment when the poker force vanishes means that it does not touch the can any more. The shell then accelerates toward the center of the cylinder, away from the poker.

Making the can buckle with the poker on the side is equivalent to triggering the instability with a nite-amplitude perturbation. This suggests that this instability is intrinsically non-linear, and may explain why the linear stability analysis of the can fails to predict the buckling events. This bifurcation is reminiscent of the transition towards turbulence in a pipe ow ( They measured the evolution of the poker force as a function of its displacement for any axial forces. Then, they combined these curves in a three-dimensional phase space (F A , force increases. Although the end of this ridge seems to tend towards a buckling force, it is experimentally dicult to probe it, as the can buckles before.

If the poker stops before the lake, the can does not buckle while we probe it. In this case, the ridge is reversible, and leaves no plasticity in the can. Consequently, we can compute the whole stability landscape of a can just in probing it, by stopping the poker just after it reaches its maximum force. When the poker is back to its origin, Virot et al. Virot et al. (2017) showed that all cans have the same general landscape. However, they speculated that the part of the landscape close to the critical axial load should depend on the can, and on its defects. It should be smoothly distorted there.

To test this scenario, we now introduce intentionally a defect in the can and investigate how it alters the stability landscape in the next section.

C.3 Stability landscape of a defected can

We start by drilling a hole of diameter 1 mm in the can's shell, at mid-height. Then, we use the same custom-made bi-axial machine to probe a commercial Coke can (7.5 .oz.).

We initially compress the can by lowering the upper plate of the machine, while recording continuously the axial force and displacement of this plate (Fig.

C.3).

While the can is clamped by the axial plates, the poker advances slowly towards it (typically 2 mm/s). The can is placed such that the poker reaches the surface 5 mm below the hole (Fig. C.3). The poker force increases with its displacement, reaches a maximum and then decreases, as it was the case in the experiments of Virot et al. (2017). We manually stop the poker just after its maximum, and return it to the origin. This does not damage the can. We obtain one curve of the landscape stability ( Red dashed line: second-polynomial t.

low displacements of the poker (typically 50 µm), because the perturbation needed for the can to buckle becomes too small. This time, the stability landscape seems to converge towards a critical axial force more drastically than a can without hole. Is this converging load related to the buckling load of the can ? We investigate this question in the next section.

C.4 Buckling load prediction

In section C.3, we measured the stability landscape of a can with a hole. We showed that the ridge of the stability landscape is much sharper and converges towards an axial load which is of the order of can's buckling loads. This procedure is reversible, and we did not damage the can. Therefore, we can now measure the buckling load of the shell.

To do so, we keep the poker away from the can and we compress axially the can by lowering the upper plate of the machine at a constant velocity (5 mm/min). The axial load increases until it reaches the can's buckling load. Simultaneously, a dimple appears in the neighbourhood of the hole and we hear a snap sound. The can is now irreversibly damaged.

We now compare the buckling load that we measured to the convergence of the stability landscape. On gure C.5, we plot in blue the projection of the can's ridge in the plan (F A , F P ). We also place on the F A -axis its buckling load (blue star). The ridge seems to converge towards the buckling load. We then t the ridge with a second-order polynomial

C.5 Spatial inuence of the hole

We now turn the can to probe it slightly away from the hole with the procedure of section C.4. For example, if the poker reaches the can's surface also below the hole, but 5 mm on its side, the landscape ridge still converges towards the buckling load of the can (Fig. C.6). As a result, we predict the can buckling load with an accuracy of 5 % (Fig.

C.5b

). At which distance from the hole can we still predict the buckling load of a can, based on the convergence of its ridge ?

To answer this question, we probed the can sequentially around the hole (from -50 mm to 50 mm around). We track the amplitude of the ridge for dierent axial loads, without damaging the can. We then normalise it by its value at an axial force of F A = 600 N (Fig.

C.7a

). We nd that the ridge quickly decreases and converges when we probe it just below the hole, as expected (vanishing distance to the hole, Fig. C.7a). At 5 mm away from the hole, we still observe that the ridge decreases signicantly with the axial load. The convergence is slightly smoother but is enough to predict accurately the buckling load (Fig.

C.6). Further 10 mm, however, the ridge does not converge and our procedure to predict the buckling load does not work any more. This seems that a can acquires a peculiar stability landscape for each point of its shell.

To estimate how this landscape is distorted by the hole, we probed a can before and after The inuence of the hole, however, is only local. We thus have to probe near the main defect of the shell, which is possible only because we introduced it intentionally.

Though, in the light of our analysis, we suggest the following procedure if we do not know where the main defect of the can is. We x the axial compression load, and we probe the can on the whole surface, with a resolution of 10 mm×10 mm. We speculate, that, for a sucient high compression, the lowest value of the maximum poker force corresponds to the main defect's neighbourhood. We could then predict the buckling load of the can by tracking the convergence of the ridge at this point. • Ch. : Bertrand Stoeth "La rhodanie"

• Ch. : Vasantha Yogananthan, "The Crossing" (Madhubani, Bihar, Inde, 2014)

• Ch. : Bertrand Stoeth "La rhodanie"

• Ch. 5 : Vasantha Yogananthan, "Ramayana Schooltrip" (Valmiki Nagar, Bihar, Inde, 2014)