Bedforms in a turbulent stream: ripples, chevrons and antidunes

TL;DR: In this paper, a linear stability analysis of a flat bed is performed to study the asymptotic behaviors of the dispersion relation with respect to the physical parameters of the problem, showing that the same instability produces ripples or chevrons depending on the influence of the free surface.

Abstract: The interaction between a turbulent flow and a granular bed via sediment transport produces various bedforms associated to distinct hydrodynamical regimes. In this paper, we compare ripples (downstream propagating transverse bedforms), chevrons and bars (bedforms inclined with respect to the flow direction) and anti-dunes (upstream propagating bedforms), focusing on the mechanisms involved in the early stages of their formation. Performing the linear stability analysis of a flat bed, we study the asymptotic behaviours of the dispersion relation with respect to the physical parameters of the problem. In the subcritical regime (Froude number $\fr$ smaller than unity), we show that the same instability produces ripples or chevrons depending on the influence of the free surface. The transition from transverse to inclined bedforms is controled by the ratio of the saturation length $L_{\rm sat}$, which encodes the stabilising effect of sediment transport, to the flow depth $H$, which determines the hydrodynamical regime. These results suggest that alternate bars form in rivers during flooding events, when suspended load dominates over bed load. In the supercritical regime $\fr>1$, the transition from ripples to anti-dunes is also controlled by the ratio $L_{\rm sat}/H$. Anti-dunes appear around resonant conditions for free surface waves, a situation for which the sediment transport saturation becomes destabilising. This resonance turns out to be fundamentally different from the inviscid prediction. Their wavelength selected by linear instability mostly scales on the flow depth $H$, which is in agreement with existing experimental data. Our results also predict the emergence, at large Froude numbers, of 'anti-chevrons' or 'anti-bars', i.e. bedforms inclined with respect to the flow and propagating upstream.

Summary

1. Introduction

• Alluvial rivers often develop bedforms, which result from the unstable interaction between bed, sediment transport and water flow.
• Ripples and dunes migrate downstream, and their crest is orthogonal to the flow direction (Fig. 1a).

2. Sediment transport

• In order to predict the emergence of bedforms from a flat sedimentary bed, one needs to model erosion and deposition of particles.
• In the saturated regime, the concentration φ is almost homogeneous and equal to φb, which is selected by the mass conservation across the interface between static and mobile grains: ϕ↓ = Vfall φb = ϕ↑(u∗).
• The time needed for a particle to settle down over the flow thickness H is H/Vfall.
• The sediment flux relaxes towards its saturated value ~qsat = qsat~t over a typical distance Lsat along the direction of the flow: Lsat ( ~t · ~∇ ) ~q = ~qsat − ~q (2.20) The relaxation equation (2.20) presents two interesting limits.
• When the inner layer invades the whole system, the basal shear stress becomes sensitive to the free surface waves induced by the bedforms.

4. Transition from transverse to inclined bedforms

• The dispersion relation derived above is first analysed in its simplest version, that is for a vanishing Froude number and a sediment transport model independent of the bed slope.
• Two values of Lsat/H have been chosen, which are on both sides of the transition between ripples and chevrons.
• On the contrary, above the transition, the growth rate is larger in the presence of a free surface, thus showing its destabilizing effect on inclined patterns.
• At the linear order, the effect of the bed slope on the sediment transport can be represented by an apparent threshold that depends on the longitudinal slope by a factor 1+∂xZ/µ, where µ is the avalanche slope.
• In other words, for any set of secondary parameters, the linear instability presents a transition controlled by the dominant parameter Lsat/H .

5. Transition to anti-dunes

• The authors have previously analysed the limit of a vanishing Froude number, for which the free surface is almost flat, but still plays an important role through the pressure field.
• As the actual flux is spatially delayed with respect to the saturated flux, the saturation transient shifts the point at which the flux is maximum downstream of the crest.
• In the next sub-section, the authors investigate the consequences of these features for anti-dunes.
• In the lower part of the diagram, one observes a narrow region of wavenumbers close to the resonant conditions for which the growth rate is very large and the propagation speed negative.
• For these data, the ratio H/z0 is also large and they appear in dark points in the lower part of the diagram.

6. Summary and perspectives

• The authors have shown that ripples and chevrons result from the same instability, although in different hydrodynamical regimes.
• The transition between ripples and chevrons is mainly controlled by the parameter Lsat/H .
• The authors results also predict the emergence, at large enough Froude numbers, of ‘anti-chevrons’ or ‘anti-bars’, i.e. bedforms inclined with respect to the flow and propagating upstream, whose experimental evidence is an interesting goal.
• The authors thank L. Langevin for permission to use his sand pattern photo.
• This work has benefited from the financial support of the Agence Nationale de la Recherche, grant ‘Zephyr’ (#ERCS07 18).

Figures

Figure 16. Component Ax of the shear stress in phase with a transverse bedform (α = 0) as a function of the Froude F and of the rescaled wavenumber kH . Antidunes form in the region where Ax < 0 (light gray). The dotted line corresponds to the resonance predicted by the inviscid equations and the dashed line to the actual resonance curve.

Figure 15. Rescaled amplitude ∆ of deformation of the free surface above a transverse bedform (α = 0) as a function of the Froude F and of the rescaled wavenumber kH . The dashed line corresponds to the resonance predicted by the inviscid equations and the dotted line to the actual resonance curve. The anti-dune region is shown in light gray. The dark gray region corresponds to strong resonance (∆ > 5).

Figure 17. (a) Growth rate σ as a function of the rescaled wavenumber kH and of the angle α for Lsat/H = 5, F = 1 and H/z0 = 102 in the limit u∗/uth → 1. Isocontours for L2sat σ/Q = −150,−100,−50,−25,−10,−4,−1, 0,−0.2, 0.2, 0.4, 0.421, 1.5. Two local maxima can be observed (•). (b) Propagation speed c in the same conditions. Isocontours for Lsat c/Q = −75, 0, 1, 5, 10, 25, 50, 100

Figure 6. Maximum growth rate (a) and corresponding mode angle (b) computed over all possible α, for each wavenumber kH . Curves obtained in the limit F → 0, u∗/uth → ∞, and for Lsat/H = 2 −4, 2−2, 20, 22, 24, 26, 28. The dashed line corresponds to Lsat/H = 1/4 (Fig. 5a) and the dotted line to Lsat/H = 4 (Fig. 5b).

Figure 11. Pattern angle αm of the most unstable mode as a function of Lsat/H in the limit F → 0. The grey solid line corresponds to the limit u∗/uth → ∞ (for any value of µ−1). The thin black solid line corresponds to the limit µ−1 = 0 and u∗/uth = 1. The three other curves are computed with a finite avalanche slope µ = tan(32◦), for u∗/uth = 1 (dotted line), for u∗/uth = 2 (dotted-dashed line) and for u∗/uth = 4 (dashed line).

Figure 19. Relation between Froude number F and anti-dune wavenumber k, rescaled by the flow thickness H . The symbols correspond to experimental data collected by Recking et al. 2009. The gray level of the points encodes the value of H/z0 in a logarithmic scale. White corresponds toH/z0 = 10 1 and black to H/z0 = 10 4. The solid line is the predicted wavenumber at the smallest value of H/z0 for which there is an instability. The dotted line is the predicted wavenumber for H/z0 = 10 2. In both cases, we considered the limit u∗/uth → 1. The curves obtained in the other limit, u∗/uth → ∞ are almost indistinguishable.

Table 1. Some of the experimental and field data from the literature. H is the flow depth. d is the typical sediment size (e.g. d50). U is the mean velocity. R = UH/ν is the flow Reynolds number. The saturation length has been computed with Lsat = 10d for bedload (Fourrière et al. 2010) and with as Lsat = HU/Vfall for transport in suspension, where Vfall is the particle settling velocity (Claudin et al. 2011). Notes: (a,b) Transport mode is not specified in the experiment description of Chang et al. (1971). Suspension is expected in case (a) as the sediments are plastic pellets with specific gravity of 1.05. Bedload is expected for the sand grains (case b). (c) Both pattern are observed together. (d) In Lanzoni (2000b), a bimodal sediment mixture is used. Only ripples are observed when partial transport occurs, i.e. below the threshold of motion for the largest particles; Bars are observed above it. (e) These data correspond to the portion of the river displayed in figure 1c. (f) Suspension is observed during flood events.

Figure 4. Diagram of the hydrodynamic regimes as a function of the Froude number F and the wavelength normalized by the flow thickness, kH . Black thick line: resonance curve predicted by the inviscid theory. Grey schematics: qualitative shape of the free surface with respect to the undulations of the bottom. Roman text: hydrodynamical regimes. Italic text: relevant mechanism dominating hydrodynamics.

Figure 5. Growth rate σ as a function of the rescaled wavenumber kH and of the angle α for u∗/uth → ∞ and F → 0. (a) Data computed for Lsat/H = 1/4. The most unstable mode (•) corresponds to ripples (α = 0). Isocontours for L2sat σ/Q = −5,−2, 0, 10−6, 10−5, 10−4, 10−3, 10−2, 10−1, 0.2. (b) Data for Lsat/H = 4. The influence of the free surface on the basal shear stress drives the most unstable mode towards higher α values. Isocontours for L2sat σ/Q = −75,−25,−10,−1,−0.1, 0, 0.005, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35. The dotted lines correspond to the maximum over all possible values of α of growth rate, for a given kH .

Figure 1. (a) Subaqueous ripples (Zion National Park, USA, 37◦16’N 112◦57’W); photo credit: B. Andreotti. The crests are transverse to the flow (from top to bottom). (b) Chevrons formed by backswash on the beach (Honokai Hale, Hawaii, 21◦21’N 158◦08’W); photo credit: L. Langevin. (c) Antidunes (California, USA, 35◦41’N 121◦17’W); photo credit: B. Andreotti. The sea lion gives the typical scale. (d) Alternate bars in the Loire river (France, 47◦24’N 0◦22’W); photo credit: Digital Globe.

Figure 9. Erosive limit: incision instability. (a) Growth rate σ as a function of the rescaled wavenumber kH and of the angle α for Lsat/H = 10 7 in the limit F → 0 and u∗/uth → 1. The isocontours are taken close to 1, for LsatH σ/Q − 1 = −104,−103,−102,−101,−100. (b) Wavenumber km (solid line) and angle π/2− αm (dashed line) of the most unstable mode as a function of Lsat/H , for F → 0. The dotted line is a guide for the eyes and is a power law of exponent −0.3.

Figure 12. (a) Ratio ∆ of the free surface amplitude to the pattern amplitude, as a function of the rescaled wavenumber kH and of the angle α for u∗/uth → ∞ and F = 0.8. Isocontours: L2sat σ/Q = −10−4,−10−3,−2, 0, 10−3, 0.04, 0.12, 0.16. (b) Growth rate σ as a function of the rescaled wavenumber kH and of the angle α for u∗/uth → ∞ and F = 0.8. There are two local maxima of the growth rate of comparable amplitude, represented by black points (•). Isocontours: L2sat σ/Q = 10 −3, 10−2, 0.1, 0.5, 0.75, 1, 2.

Figure 18. Characteristics of the most unstable mode as a function of Lsat/H for three Froude numbers: F = 0.9 (dotted lines), F = 1 (solid lines) and F = 1.5 (dashed line). H/z0 is kept constant and equal to 102. The curves are obtained in the limit u∗/uth → 1. (a) Propagation speed cm. Antidunes correspond to cm < 0. (b) Angle αm of the wavenumber with respect to the flow direction. Transverse bedforms correspond to αm = 0. (c) Wavenumber km rescaled by the flow thickness. (d) Growth rate σm rescaled using the saturation length Lsat and the reference flux Q.

Figure 10. Growth rate σ as a function of the rescaled wavenumber kLsat for u∗/uth → 1 and F → 0. Comparison between the case with (solid line) and without a free surface (dashed and gray lines). The solid and gray lines correspond to the maximum over the angle α of the growth rate. The dashed line is for α = 0 (transverse ripples). (a) Lsat/H = 2 −4 (b) Lsat/H = 4.

Figure 8. Schematic explanation of the chevrons instability mechanism in the limit of vanishing kH . Black (resp. grey) arrows show the modulation of the longitudinal (transverse) flow. (a) Bedforms almost transverse to the flow (α close to 0). Gravity is dominant so that the longitudinal fluid motion is down-slope. (b) Bedforms almost aligned with the flow (α close to π/2). The longitudinal slope effect is negligible. Longitudinal fluid motion is controlled by the modulation of flow thickness, which induces a weakening (resp. an increase) of the basal friction upstream (resp. downstream) of the crest. Therefore, the modulation of the longitudinal fluid motion is up-slope.

Figure 14. Schematic view of anti-dunes and scketch of the instability mechanism. The saturation length Lsat represents the spatial lag between the shear stress and the sediment flux.

Figure 13. Wavelength km and angle αm of the most unstable mode as a function of Lsat/H , in the limit u∗/uth → ∞. The solid line corresponds to F = 0.8 (transitions at Lsat/H ≃ 0.2, at ≃ 0.3 and at ≃ 2.9 102 marked by ◦) and the dotted line to the limit F → 0 (solid line in Fig. 7). The gray zone indicates the simultaneous existence of two growth rate maxima of comparable amplitude.

Figure 3. Schematic of bedforms at an angle α with respect to the flow. (a) Plane wave. (b) Alternate bars (guided waves in a channel of width W ). The x-axis is in the direction of the flow (black arrows), and y is transverse to it. z is the third coordinate, perpendicular to the bed mean plane. We note Z(x, y) the bed profile.

Figure 2. Sediment concentration profile φ(z) for different sedimentation velocities Vfall. (a) At small sedimentation velocity, the particles are suspended by turbulent fluctuations and occupy the entire flow. (b) Transitional case. (c) Conversely, at high sedimentation velocity, the sediment flux is concentrated near the bed, in a transport layer whose thickness is proportional to d.

Figure 7. Angle αm and wavenumber km of the most unstable mode as a function of Lsat/H , for F → 0. The solid line corresponds to the limit u∗/uth → ∞ (transitions at Lsat/H ≃ 1.0 and at ≃ 2.9 102 marked by ◦) and the dotted line to the limit u∗/uth → 1 (transitions at Lsat/H ≃ 1.6).

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Bedforms in a turbulent stream: ripples, chevrons and
antidunes
Bruno Andreotti, Philippe Claudin, Olivier Devauchelle, Orencio Durán,
Antoine Fourrière
To cite this version:
Bruno Andreotti, Philippe Claudin, Olivier Devauchelle, Orencio Durán, Antoine Fourrière. Bedforms
in a turbulent stream: ripples, chevrons and antidunes. Journal of Fluid Mechanics, Cambridge
University Press (CUP), 2011, 690, pp.94 - 128. �10.1017/jfm.2011.386�. �hal-00661656�

Under consideration for publication in J. Fluid Mech.
1
Bedforms in a turbulent stream:
ripples, chevrons and antidunes
By B R U N O A N D R E O T T I , P H I L I P P E C L A U D I N,
O L I V I E R D E V A U C H E L L E
†
, O R E N C I O D U R
´
A N
A N D A N T O I N E F O U R R I
`
E R E
Lab oratoire de Physique et M´ecanique des Milieux H´et´erog`enes,
(PMMH UMR 7636 ESPCI – CNRS – Univ. Paris Diderot – Univ. P. M. Curie),
10 rue Vauquelin, 75231, Paris Cedex 05, France.
†
Department of Earth, Atmospheric an d Planetary Sciences, Massachusetts Institute of
Technology, 77 Massachusetts Avenue, Cambridge MA 02139-4307, USA
(Received 15 September 2011)
The interaction between a turbulent flow and a granular bed via sediment transport
produces various bedforms associated to distinct hydrodynamical regimes. In this pa-
per, we compare ripples (downstream propagating transverse bedforms), chevrons and
bars (bedforms inclined with respect to the flow dire c tion) and anti-dunes (upstream
propagating bedforms), focusing on the mechanisms involved in the early stages of their
formation. Performing the linear stability analysis of a flat b ed, we study the asymp-
totic behaviours of the dispersion relation with respect to the physical parameters of the
problem. In the subcritical r egime (Froude numbe r F smaller than unity), we show that
the same instability produces ripples or chevrons depending on the influence of the free
surface. The tra nsition from transverse to inclined bedforms is controled by the ratio of
the sa turation length L
sat
, which encodes the stabilising effect of sediment transport, to
the flow depth H, which determines the hydrodynamical regime. These results suggest
that alternate bars for m in rivers during flooding events, whe n suspended load domi-
nates over bed load. In the supercritical r e gime F > 1, the transition from ripples to
anti-dunes is also controlled by the ratio L
sat
/H. Anti-dunes appear around resonant
conditions for free surface waves, a situation for which the sediment transport saturation
becomes destabilising. This resonance turns o ut to be fundamentally different from the
inviscid prediction. Their wavelength sele cted by linear instability mostly scales on the
flow depth H, w hich is in agreement with existing experimental data. Our results also
predict the emergence, at large Froude numbers, of ‘anti-chevrons’ o r ‘anti-bars’, i.e.
bedfo rms inclined with respect to the flow and propaga ting upstream.
1. Introduction
Alluvial rivers often develop bedforms, which result from the unstable interaction
between bed, sediment transport and water flow. The velocity field is perturbed by
the pr esence of bedforms. In tur n, the fluid motion induces sediment transport which,
through erosion and dep osition, deforms the bed (Julie n 1998 ). Depending on the
flow depth and velocity, a s well as on the sediment properties, various patterns can
be observed (Ashley 1990). Ripples and dunes migrate downstream, and their crest
is orthogona l to the flow directio n (Fig. 1a). The wavelength λ of ripples is much
smaller than the water depth H, so that the flow around thes e bedforms is not in-

2 B. Andreotti, P. Claudin, O. Devauchelle, O. Dur´an and A. Fourri`ere
Figure 1. (a) Subaqueous ripples (Zion National Park, USA, 37
◦
16’N 112
◦
57’W); photo credit:
B. Andreotti. The crests are transverse to the flow (from top to bottom). (b) Chevrons
formed by backswash on the b each (Honokai Hale, Hawaii, 21
◦
21’N 158
◦
08’W); photo credit: L.
Langevin. (c) Antidunes (California, USA, 35
◦
41’N 121
◦
17’W); photo credit: B. And reotti. The
sea lion gives th e typical scale. (d) Alternate bars in the Loire river (France, 47
◦
24’N 0
◦
22’W);
photo credit: Digital Globe.
fluenced by the water surface. By contrast, dunes have a typical length compara-
ble to, or larger than H, and thus interact with the free surfac e. Although dunes
and ripples were prev iously seen as two distinct modes of the same linear instability
(Richards 1980, Sumer & Bakioglu 1984, McLean 1990), it is now proposed that dunes
result from the coa rsening of ripples, through the non-linear increase of their wavelength
(Raudkivi & Witte 1990, Raudkivi 20 06, Fourri`ere et al. 2010). Such a coarsening has
been reported in numerous e xperimental studies (Mantz 1978, Gyr & Schmid 1989, Baas 1994,
Coleman & Me lville 1994a, Baas 1999, Robert & Uhlman 2001, Coleman et al. 2003, Venditti et al. 2005b,
Langlois & Valance 200 7, Rauen et al. 2008).
Unlike dunes and ripples, the crest of what is called ‘chevrons’ or ‘rhomboid patter n’
(Fig. 1 b) forms an angle α with the flow direction (Woodford 1935, Chang & Simons 1970,
Morton 1978, Karcz & Kersey 1980, Ikeda 1983, Daerr et al. 2003, Deva uchelle et al. 2010a).
Chevrons migrate downstream and interact with the free surface. Alternate bars in rivers
and channels can be thought of as the superimposition of two rhomboid patterns with
opposite angles (Fig. 1d). The boundary conditions at the bank select their transverse
wavenumber, the channel acting as a wave guide for the chevrons insta bility. Several sta-
bility analyses have been devoted to these bedforms, related to meandering and braiding
in rivers (e.g., Callander 1969, Parker 1976, Fredsøe 1978, Tubino et al. 1999). Flume
experiments have also been performed to reproduce alternate bars at the laboratory sc ale
(e.g. Chang et al. 197 1, Schumm & Khan 1972, Ikeda 1983, Fujita & Muramoto 1985 ,
Lisle et al. 1991, Lisle et al. 1997, Lanzoni 2000a, Lanzoni 2000b). Interestingly, these
bars are observed when the submergence, i.e. the ratio between the water depth H and
the grain size d, is mo de rate and the Froude number F is below unity (the Froude number
compares inertia with gravity). Some figures of Fujita & Muramoto 1985 suggest that
alternate bars result from a linear instability, but La nz oni 2000a reports the emergence

Bedforms in a turbulent stream: ripples, chevrons and antidunes 3
of ripples fir st. Unfortunately, few contributions describe the early stages of the bed
evolution.
Antidunes form in supercritical flows, that is for a Froude number larger than unity, and
migrate upstream (Fig. 1c). They have received much exp e rimental and theoretical atten-
tion (e.g . Raudkivi 1966, Ikeda 1983, Alexander et al. 2001, Carling & Shvidchenko 2002),
either on their own (Parker 1975, Kubo & Yokokawa 2001, Colombini & Stocchino 2005)
or in association with other bedforms (K ennedy 1963, Reynolds 1965, Engelund 1970,
Hayashi 1970, Huang & Chiang 2001, Colombini 2004, Colombini & Stocchino 2008). Step-
pool sequences (Chin 1999, Curran 2007, Le nzi 2001, Weichert, et al., 2008, Whittaker and Jaeggi, 1982)
and cyclic steps (Parker & Izumi 2000, Kostic et al. 2010) are extreme forms of an-
tidunes, the growth of which has caused the flow to cros s periodically the transition
from subcritical to supercritical regimes.
The pr imary linear instability ca using ripples and chevrons results from the phase la g
between sediment flux q and bed topography Z (Kennedy 1963, Hayashi 1970, Parker 19 75,
Richards 198 0, Engelund & Fredsøe 1982, McLean 1990). This phase lag has two main
contributions (Andreotti et al. 2002). First, there is a hydrodynamical effect. For a
wavelength λ sufficiently small with r e spect to the water depth H (ripples), fluid iner-
tia causes a phase advance of the basal shear stress τ, which is destabilis ing. However,
when λ compares with H, the confinement of the flow can cause a phase delay of τ with
respect to Z. The second contribution to the phase lag between Z a nd q is related to
the relaxation of the s ediment flux towards equilibrium. This stabilizing e ffect involves
a length scale L
sat
, the s o-called sa turation length. Its value depends on the mode of
sediment transport. For bedload it is about ten times the grain diameter d in a turbulent
flow (Fourri`ere et al. 2010). For suspended transport, the saturation le ngth scales with
the water depth (Claudin et al. 2011). It corresponds to the distance a grain travels
horizontally before settling down.
In sections 2 and 3, we g eneralize the stability analysis of Fourri`ere et al. (2010)
to non-transverse patterns, in order to investigate the emergence of chevrons and bars,
and we determine the growth rate σ associated to a bedform of wavenumber k = 2π/λ
and of angle α. Section 4 is devoted to the analysis of the problem in the limit of a
vanishing Froude number (F → 0). We show that the ratio between the flow depth H
and the saturation le ngth L
sat
controls the trans ition from tr ansverse patterns (ripples)
to inclined patterns (chevrons and bars). Finally, section 5 is devoted to finite Froude
numbers effects and in particular to the transition from ripples to antidunes.
2. Se diment transport
In order to predict the emergence of bedforms from a flat sedimentary bed, one needs
to model erosion and deposition of particles . The aim of this section is to show tha t the
different modes of sediment transport can be descr ibed within the very same theoretical
framework. This has already been discusse d in a series of pa pers investigating the re-
laxation of sediment transport towards equilibrium (Parker 197 5, Sauermann et al. 20 01,
Andreotti et al. 2002, Charru 2006, Andreotti & Claudin 2007, Andreotti et al. 2010, Fourri`ere et al. 2010,
Claudin et al. 2011). We propose here a summa ry of these results.
2.1. Transport regimes
Figure 2 shows that the two modes of underwater se diment transport, namely bedload
and suspended load, can be distinguished by the profile of the grain concentration φ in
the mobile phase. Neglecting inertia, the particle velocity equals that of the fluid. In
turbulent flows, the water velocity fluctuations tend to homogenise the sediment concen-

4 B. Andreotti, P. Claudin, O. Devauchelle, O. Dur´an and A. Fourri`ere
Figure 2. Sediment concentration p rofile φ(z) for different sedimentation velocities V
fall
. (a)
At small sedimentation velocity, the particles are suspended by turbulent fluctuations and oc-
cupy the entire flow. (b) Transitional case. (c) Conversely, at high sedimentation velocity, the
sediment flux is concentrated near the bed, in a transport layer whose thickness is proportional
to d.
tration φ. This effect is balanced by gravity, which tends to settle the particles down
at a velocity V
fall
. In the low concentration limit, the sedimentation flux is therefore
ϕ
↓
= V
fall
φ. The simplest model taking these two effects into account is:
∂
t
φ + ~u ·
~
∇φ =
~
∇ · (D
~
∇φ) + V
fall
∂
z
φ (2.1)
where D is an effective turbulent diffusion coefficient and ~u the velocity field. For the sake
of simplicity, we consider here D as a constant proportional to the mean flow velocity
U (or equivalently, to the shear velocity u
∗
) and to the flow thickness H, which controls
the typical turbulent mixing leng th: D = βHU . In the homogeneous steady state –
called the saturated state in this context – the upward diffusive flux is balanced by the
downward sedimentation flux:
D∂
z
φ = −V
fall
φ . (2.2)
Introducing the basal concentration φ
b
, this equation integrates into
φ = φ
b
exp

−
V
fall
βU
z
H

(2.3)
This expression provides a good approximation to expe rimental data (Rouse, 193 6;
Vanoni, 1946; van Rijn, 1984b). The dimensionless parameter V
fall
/U controls the tran-
sition between suspended transport and bed load. At small sedimentation velocity, tur-
bulent fluctuations are more efficient than gravity so that the conce ntration profile is
homogeneous (Fig. 2a). Sediment transport takes place over the entire flow thickness H.
Conversely, at large sedimentation velocity, gravity c oncentrates the moving particles at
the surface of the bed (Fig. 2 c). In this limit, the sediment transport layer is limited by
the grain diameter d.
2.2. Turbulent suspension
We now consider more specifically the asymptotic regime V
fall
/U ≪ 1 , which corresponds
to turbulent suspension. The basal concentration and thus the overall transport are
governed by the rate o f entrainment ϕ
↑
of grains from the s tatic sand bed (or ‘pick-up
function’). As this entrainment results from the hydrodynamical drag on the grains , ϕ
↑
is controlled by the basal shear stress τ
xz
or equivalently by the shear velocity u
∗
≡
p
τ
xz
/ρ
f
. Empirical measurements show that ϕ
↑
is a growing function of u
∗
above a
threshold u
th
∝ V
fall
(van Rijn, 1984a), typically proportional to (u
2
∗
−u
2
th
) (Partheniades,