We use the erosion–deposition model introduced by

Alluvial rivers transport the sediment that makes up their bed. From a
mechanical standpoint, the flow of water applies a shear stress on the
sediment particles and entrains some of them downstream. When the shear
stress is weak, the particles remain close to the bed surface as they travel

Bed load transport is inherently random

Altogether, the combination of these stochastic processes results in a
downstream flux of particles. Fluvial geomorphologists measure this flux by
collecting moving particles in traps or Helley–Smith samplers

An alternative approach to sediment flux measurements is to follow the fate
of tracer particles. In November 1960,

The dispersion of the tracers, expressed as the variance of their location,
results from the randomness of bed load transport.

As the particle continues its course, collisions deviate its trajectory. In
this intermediate regime, the variance increases nonlinearly with time

With time, tracers settle back on the bed, where they can remain trapped for
a long time. How the distribution of resting times influences the long-term
dispersion of tracers remains unknown. The data collected by

The variability of the stream discharge further complicates the
interpretation of field data. Bed load transport occurs when the shear stress
exceeds a threshold set by the grain size. Most rivers fulfill this condition
only a small fraction of the time, making sediment transport highly
intermittent

Laboratory experiments under well-controlled conditions isolate these two
effects. For instance,

In a recent paper,

In most rivers, sediment is broadly distributed in size. This likely
influences the dispersion of bed load tracers

For moderate values of the shear stress, the concentration of moving
sediments is small, and we can neglect the interactions between particles.
The erosion–deposition model introduced by

To investigate the dispersion of bed load particles, we consider some of
them to be marked (Fig.

When subjected to varying flow and sediment discharges, the bed of a stream
accumulates or releases sediments

For steady and uniform transport, the surface concentration of moving
particles,

Combining Eq. (

Granular bed sheared by a steady and uniform flow. The bed is a mixture of marked (red) and unmarked (white) grains.

Complemented with initial and boundary conditions, Eqs. (

A single parameter controls Eqs. (

In the next section, we numerically solve Eqs. (

Laboratory measurements of bed load often use top-view images

In summary, the proportions of mobile and static tracers,

To study the evolution of the tracer concentration, we solve
Eqs. (

Evolution of the tracer concentration (

The early evolution of the plume depends on initial conditions. In most field
experiments, tracers are deposited at the surface of the river bed when the
flow stage is low and sediment is motionless

With these initial conditions, the evolution of the plume follows two
distinct regimes. At early times, the flow gradually dislodges tracers from
the bed and entrains them in the bed load layer. During this entrainment
regime, only a small proportion of the tracers move. Consequently, the plume
develops a thin tail in the downstream direction
(Fig.

With time, the plume moves downstream and spreads both upstream and
downstream. As a result, the concentration rapidly decreases to small levels.
The plume becomes gradually symmetrical and tends asymptotically towards a
Gaussian distribution (Fig.

To better illustrate this evolution, we introduce the mean position of the
plume of tracers:

After a characteristic time of the order of

Next, we establish the equivalence between diffusion and the long-time behavior of the tracers.

The diffusion at work in Eqs. (

To formally establish the equivalence between diffusion and the long-time
behavior of the plume, we follow a reasoning similar to the one developed for
chromatography

By multiplying Eq. (

We interpret this formal derivation as follows. In the reference frame of the plume, a tracer at rest on the bed moves backward, while a tracer entrained in the bed load layer moves forward. At long timescales, the proportions of tracers in each layer equilibrate. Consequently, the probability that a tracer will be entrained and move forward equals that of deposition. In the reference frame of the plume, the exchange of particles between the bed and the bed load layer is thus a Brownian motion driving the linear diffusion of the plume.

In the next section, we investigate the evolution of the location, the size, and the symmetry of the plume as it propagates downstream.

Concentration, defined as the number of tracers per unit of area, depends on
the area over which it is measured. Its value is meaningful when the
measurement area is much larger than the distance between particles and much
smaller than the plume. During the entrainment regime, the plume develops a
thin tail containing only a small proportion of tracers. Measuring the
concentration profile during this regime is thus challenging. To our
knowledge, only

Multiplying Eq. (

We now focus on the variance of the plume. Multiplying Eq. (

We follow a similar procedure to derive the skewness of the plume.
Multiplying Eq. (

Equations (

As discussed in Sect.

Anomalous diffusion arises from heavy-tailed distributions of either the
step length or the waiting time

With time, the plume enters the diffusive regime. Its velocity and its
spreading rate relax towards constants while its skewness decreases
(Fig.

The transition between the entrainment and the diffusive regime occurs when
the skewness reaches its maximum value. Equating the skewness estimated from
Eqs. (

The asymptotic regimes (Eqs.

Our description of the plume of tracers is based on the assumption that
sediment transport is in steady state. This hypothesis is often satisfied in
laboratory flumes

The intermittency of bed load transport influences the propagation of tracers
in several ways. First of all, sediment transport during a flood modifies the
structure of the bed

If this assumption holds, the simplest way to account for bed load
intermittency is to assume that the river alternates between two
representative stages: (1) a low-flow stage during which tracers are immobile
and
(2) a flood stage characterized by a representative sediment flux

In practice, evaluating the intermittency factor requires continuous
monitoring of the river discharge and a correct estimate of the entrainment
threshold.

Here, we suggest another way to circumvent the intermittency of sediment
transport. Plotting the plume variance, (

The entrainment regime corresponds to small traveled distances. In this
regime, both the size of the plume and its asymmetry increase with traveled
distance (Fig.

After the plume has traveled over a distance roughly equal to the flight
length, its skewness reaches a maximum value and starts decreasing. This
change in dynamics indicates the transition towards the diffusive regime.
Equations (

Equating the skewness estimated from Eqs. (

When expressed in terms of the distance traveled by the plume, the asymptotic
regimes are insensitive to the intermittency of bed load transport. They are
thus a robust test of our model and can help us interpret field data. Let us
assume that a dataset records the evolution of a plume of tracers released in
a river over a distance long enough to explore both the entrainment and the
diffusive regime. During the diffusive regime, the skewness decreases with
the traveled distance. A fit of the data with
Eq. (

According to Sect.

We used the erosion–deposition model introduced by

Our model captures in a single theoretical framework the transition between
two asymptotic regimes: (1) an early entrainment regime during which the
plume spreads nonlinearly and (2) a late-time relaxation towards classical
advection–diffusion. The latter regime is consistent with previous
observations

When expressed in terms of the distance traveled by the plume, the asymptotic regimes are insensitive to the intermittency of bed load transport in natural streams. According to this model, it should be possible to estimate the particle flight length and the average bed load transport rate from the evolution of the variance and the skewness of a plume of tracers in a river.

No data sets were used in this article.

The authors declare that they have no conflict of interest.

It is our pleasure to thank Pascal Allemand, David John Furbish, Colin Phillips, Douglas Jerolmack, and François Métivier for many helpful and enjoyable discussions. This work was supported by the French national program EC2CO-Biohefect/Ecodyn//Dril/MicrobiEn, “Dispersion de contaminants solides dans le lit d'une rivire”. Edited by: Patricia Wiberg Reviewed by: two anonymous referees