A new particle-based reduced-complexity model to simulate sediment transport and channel morphology in steep streams in presented. The model CAST (Cellular Automaton Sediment Transport) contains phenomenological parameterizations, deterministic or stochastic, of sediment supply, bed load transport, and particle entrainment and deposition in a cellular-automaton space with uniform grain size. The model reproduces a realistic bed morphology and typical fluctuations in transport rates observed in steep channels. Particle hop distances, from entrainment to deposition, are well fitted by exponential distributions, in agreement with field data. The effect of stochasticity in both the entrainment and the input rate is shown. A stochastic parameterization of the entrainment is essential to create and maintain a realistic channel morphology, while the intermittent transport of grains in CAST shreds the input signal and its stochastic variability. A jamming routine has been added to CAST to simulate the grain–grain and grain–bed interactions that lead to particle jamming and step formation in a step-pool stream. The results show that jamming is effective in generating steps in unsteady conditions. Steps are created during high-flow periods and they survive during low flows only in sediment-starved conditions, in agreement with the jammed-state hypothesis of Church and Zimmermann (2007). Reduced-complexity models like CAST give new insights into the dynamics of complex phenomena such as sediment transport and bedform stability and are a useful complement to fully physically based models to test research hypotheses.

The morphodynamics of steep gravel-bed rivers is characterized by complex
feedbacks between sediment supply and storage

The step-pool morphology is commonly encountered in mountain catchments at
slopes grater than

A hypothesis on step stability was proposed by

Physically based modeling of flow and sediment transport in steep mountain
streams in mobile bed conditions is impractical because (a) the flow field
over the rough bed is very complex, (b) single-grain mobility is impossible
to solve, and (c) long-term simulations are required to develop a dynamically
changing channel bed. An alternative to fully physically based modeling is
that of reduced-complexity models. Instead of solving differential equations
of flow and sediment transport, reduced-complexity models formulate physically meaningful
local flow–grain interaction rules with very few parameters in a cellular
automaton space. The reduced-complexity framework has also been applied
successfully in fluvial geomorphology

In this paper we present a new reduced-complexity stochastic model for
step-pool streams based on grain–grain and grain–bed interactions: CAST (Cellular Automaton
Sediment Transport). CAST simulates a generic fluvial channel on a
cellular-automaton domain, where the bed is formed by an arrangement of
particles like in a sandpile model

The paper objectives are (a) to present a new reduced-complexity model that
simulates bed load transport and channel morphology at the grain scale and to
test the effect of different parameters and stochastic forcing on the model
outcomes, and (b) to explore the effect of jamming on sediment transport and step
formation, in comparison with the framework of the jammed-state
hypothesis of

CAST operates in 2-D cellular-automaton space, which is a rectangular grid of
constant length (

Particles in the model domain can be either on the bed or in motion, i.e.,
part of the bed matrix or of the transport matrix (Fig.

Particles enter the system with a specified input rate

Diagram of the model. The space is discretized in a longitudinal
dimension

The first parameter of CAST is the particle input rate

Particles are transported as bed load along the channel with a constant
velocity (see Eq.

Particles move preferentially directly to the downstream cell (

Sediment flux in the model,

The key process in CAST is the particle entrainment which is considered to be
dependent on the local bed topography and on the flow conditions. The degree
of exposure of particles on the bed has been shown to strongly influence
sediment entrainment and transport especially in steep streams

The effect of local topography on entrainment is accounted for by calculating
the local relative exposure

Deterministic and stochastic parameterization of entrainment in
CAST.

Entrainment is based on

Figure

Conceptually, the value of

Particles in transport (i.e., belonging to the

The relation between particle deposition and bed surface is modeled using the
relative exposure matrix

CAST needs one boundary condition for the lateral banks and one for the
downstream boundary at the channel outlet. The boundary condition for the
banks is deposition when a moving particle in the

In order to avoid long simulation times required to fill the channel with
particles, we start every simulation from an initial slope, slightly less
than the equilibrium slope, with random noise. The model in not sensitive to
this initial condition – i.e., the final equilibrium slope is only a function
of the chosen set of parameters (mainly the input rate

CAST operates in two modes, with and without dynamic jamming. The rough-bed
model without jamming (CAST

For every time step, the computation sequence is as follows. (1) Sediment
input enters the system in the first cross section. (2) Particles in
transport move one cell downstream (straight, left, or right) – if they collide
with other particles or with one of the banks, then they deposit; otherwise, they
remain in transport. In the case of CAST

To set up the model, we have run a set of simulations using CAST

Values of the parameters used in the steady-state simulations.

We analyzed the results in terms of the following:

Storage volume

Specific sediment flux

Mean relative exposure

Particle hop distance HD: the step length of a single particle from the point it is entrained (or enters the channel) to the point it is deposited or exits the channel.

The main outputs of CAST

One of the advantages of reduced-complexity models like CAST is that it is possible to
track the movement of every single particle in the system and thus to compute
all particle step lengths (measured from entrainment to deposition). This is
an important quantity which, since Einstein's probabilistic theory on bed
load transport

With the steady-state simulations we explored the effect of changing input
rate and entrainment parameter on the model outcomes. These two parameters
are important because they can be linked to the jammed-state diagram
parameters of

CAST

Probability density functions of simulated particles hop distances (red dots) fitted with an exponential distribution (blue line) for four different parameter sets.

Selected CAST

Some of the simulations, characterized by low input rate and high entrainment
probability (

The stochastic simulation of 20 realizations of each of the 27 parameter sets
showed that the mean storage volume

The values of four key variables for the 27 simulations (parameter
combinations), averaged over the 20 realizations, are shown in Fig.

Specific sediment flux

The mean particle hop distances (

The analysis above showed that CAST

Jamming simulations were run with the same parameter sets of the steady-state
rough-bed model case. Three different situations occur:

When particle activity is too low (low sediment transport) the jamming threshold is rarely (often never) exceeded.

When particle activity is too high (high sediment transport) jamming is occurring too often in time and space, and the storage volume of the system keeps increasing because of the large amount of particles depositing upstream of the step structures. As a result an equilibrium channel is never reached.

When particle activity is in between the two previous situations, jamming is occurring at a rate which allows the formation of steps and maintains an approximately equilibrium channel.

Comparison between simulations without jamming (RBM) and with
jamming (JM) for

The first situation represents the rough-bed case discussed previously. The
second one represents a case which is very unlikely to happen in river
systems where fluvial sediment transport is rarely going to exceed the
jamming threshold and certainly not for very long periods of time (e.g., only
during large flood events). For the purpose of this study we focus on the
last situation where jamming is effectively creating steps. When particles
are jammed and instantly deposited, they trigger a deposition process which
is propagating upstream, since the values of relative exposure

We show the effect of adding jamming to the model by comparing simulations
having the same parameter sets and same initial conditions (

In the simulations presented in the previous sections we used a stochastic parameterization for particle entrainment and a constant sediment supply. To explore the effect of stochasticity on the model results, in the next two sections we quantify the effect of stochasticity in entrainment and sediment supply explicitly.

The entrainment probability in CAST can be parameterized as a deterministic
or stochastic process (Sect.

The comparison for a simulation with

Comparison between stochastic entrainment (

This analysis support our choice of modeling the entrainment as a stochastic process. This is not only more physically reasonable because the process of particle displacement is random per se, but it is not possible to obtain a realistic rough-bed morphology in a reduced-complexity model like CAST without a stochastic parameterization of particle entrainment.

The effect of stochasticity in the input rate

The effect of stochasticity in the input rate is much smaller than that of
entrainment. The distributions of sediment flux (Fig.

Comparison between constant input rate (“Const” without jamming and
“Const

Unsteady simulations with four consecutive floods.

These results highlight that the variability and the fluctuations observed in
the sediment output variables of the model do not depend on the variability
of the sediment input but are instead a function of the internal dynamics of
the system given by the local grain–grain and grain–bed interactions. In
other words, CAST acts as a shredding filter of the input forcing

Although jamming is effective in generating a step-like morphology under
certain steady-state sediment input and entrainment conditions, we recognize
that step formation is an intermittent process in which flow variability in
time is important. Typically step-pool sequences are partially or totally
destroyed during large flood events and then reworked and stabilized during
the following low-flow periods
(e.g., Lenzi, 2001; Turowski et al., 2009; Molnar et al., 2010). We show the
effects of changing flow conditions by simulating four consecutive floods of
equal magnitude (Fig.

Values of the parameters used in the unsteady simulations.

The temporal pattern of storage volume and sediment flux in the unsteady
simulations is shown in Fig.

Unsteady simulations with four consecutive floods.

The specific sediment flux when the input is constant (Fig.

The unsteady flow also has impacts on bed roughness in CAST. The time series
of the standard deviation

Unsteady simulations with four consecutive floods. Empirical cumulative distribution functions of specific sediment flux. Blue lines identify the rough-bed model and red the jamming model. Solid lines identify the constant input case (Case I) and dashed lines the variable input case (Case II).

The same can be inferred from the longitudinal profiles of bed elevation of
the simulations with jamming (Fig.

Unsteady simulations with four consecutive floods. Time series of the
standard deviation of relative exposure

To quantify this effect directly on step formation, we introduce step density

Unsteady simulations with four consecutive floods. Longitudinal
profiles of bed elevation computed at the end of every high-flow period (left
column:

Unsteady simulations with four consecutive floods. Time series of step
density with

The CAST model without jamming, CAST

CAST assumes a stochastic description of sediment transport, following and
corroborating recent research

The stochastic parameterization of CAST does not assume a priori any
probability distribution for particle hop distances, and yet they turn out to
be well fitted by an exponential distribution, in agreement with previous
theoretical and field studies

Finally, CAST also reproduces the shredding effect sometimes visible in
sediment transport

We showed with CAST

We did not observe any specific wavelength of step occurrence, as usually
predicted by hydraulic-based theories on step formation

Our modeling approach has by definition some simplifications and limitations
which we think can be improved in future research. First, the uniform size of
the sediment prevents us from specifically modeling any grain-size effect that
might indeed be very important in steep-channel dynamics. We partially
incorporated these effects in the stochastic parameterization of entrainment:
the fact that, for the same value of relative exposure

Furthermore, we did not account for the transfer of momentum that could
happen when a particle is deposited and thus enhances the probability of
entrainment of the surrounding grains (i.e., “collective entrainment” as in

In conclusion, in our opinion the strongest limitation of the current model is the absence of sediment sorting and other grain-size effects. All these phenomena will be incorporated in the next version of the model, which will have different grain-size fractions.

We presented a new particle-based reduced-complexity model, CAST (Cellular Automaton Sediment Transport), that simulates bed load transport and changes in channel morphology, including the processes of jamming and step formation. The model simulates grain–grain and grain–bed interactions with uniform-size particles and can have stochastic or deterministic parameterizations for sediment input rate and particle entrainment. With only a few parameters, it is possible to simulate channels with different sediment supply and flow conditions. At steady state, CAST can reproduce a realistic bed morphology and typical fluctuations in transport rates whose memory features are consistent with previous experimental data. Moreover, particle hop distances are well fitted by exponential distributions, in agreement with field observations. One of the main results is the role played by stochasticity both in the entrainment and in the input rate. A stochastic input rate does not change the final outcome of the model compared to a constant input having the same mean. However, if the entrainment is modeled deterministically, the resulting channel does not have the typical variable bed roughness encountered in real fluvial systems.

The dynamic effect of particle jamming was added to test under which conditions steps are formed and remain stable in steep channels. The effect of jamming has been tested in unsteady simulations where the entrainment probability and the input rate have been changed to simulate a sequence of high-flow and low-flow periods. CAST generates step structures during high-flow periods that survive during low flows in simulations with sediment-starved conditions, in agreement with the jammed-state hypothesis. Our results support the jammed-state hypothesis as a framework to explain step formation and stability, and, more generally, they show the potential of reduced-complexity models at a grain scale with stochastic parameterizations. We are of the opinion that models such as CAST can give new insights into the dynamics of complex phenomena like sediment transport and step formation and can be useful to test research hypotheses in fluvial geomorphology.

Matteo Saletti acknowledges the support of the SNSF for funding his PhD (grant number