In most sediment transport models, a threshold variable dictates the shear
stress at which non-negligible bedload transport begins. Previous work has
demonstrated that nondimensional transport thresholds (

Despite over a century of quantitative study (Gilbert, 1914), it often
remains challenging to predict gravel transport rates to much better than an
order of magnitude because of the complexity of grain interactions with the
flow and the surrounding grains (e.g., Schneider et al., 2015; Nitsche et
al., 2011; Rickenmann, 2001; Wilcock and Crowe, 2003; Chen and Stone, 2008).
Predictive models for complex systems often derive utility from their
simplicity, as is the case with the widely-used Meyer-Peter and
Müller (1948) transport equation, as modified by Wong and Parker (2006):

Threshold of motion data from both field and experimental studies. A
power law regression to these data gives

Thresholds of motion for gravel often span an order of magnitude or more
(Fig. 1). Variability in

Although thresholds of motion may dynamically evolve over time, I suggest
several reasons why an assumption of constant

In order to review previous work in an organized manner, I categorize factors
affecting

Grain controls are physical characteristics of individual clasts that
influence

Sand content is a related GSD bed state control: increasing sand content of
alluvial bed surfaces has been shown to decrease gravel thresholds of motion
(e.g., Curran and Wilcock, 2005; Iseya and Ikeda, 1987; Jackson and Beschta,
1984). Wilcock and Crowe (2003) explicitly incorporated this sand dependence
into their transport model:

Many studies have explored the bed state control of stabilizing structures formed by coarse grain clusters (e.g., Church et al., 1998; Strom and Papanicolaou, 2009). Other bed state controls include the degree of overlap, interlocking and imbrication among grains, and bed compaction or dilation (e.g., Parker, 1990; Wilcock and Crowe, 2003; Sanguinito and Johnson, 2012; Buscombe and Conley, 2012; Mao, 2012; Kirchner et al., 1990; Marquis and Roy, 2012; Powell and Ashworth, 1995; Richards and Clifford, 1991; Ockelford and Haynes, 2013). By combining experimental data and a numerical model, Measures and Tait (2008) show that increasing grain-scale bed roughness tends to shelter downstream grains, reducing entrainment. These factors show that grains can move from less stable to more stable configurations, even if grain size does not change. Coarse grain clusters can also enhance bed stability by increasing surface roughness, tending to deepen potential grain pockets.

Flow characteristics influencing

Recent work also suggests that sediment transport can affect

Recking (2012) compared bed load monitoring records from steep natural
channels (

It should be noted that these reference stress equations describe reference
stresses for the

Feedback between channel morphology and bedload transport defines mountain
river morphodynamics. The Exner equation of sediment mass conservation
quantifies how transport changes correspond to topographic changes (Paola and
Voller, 2005):

The overall goal of the present work is to understand and model possible
feedbacks among thresholds of motion, changes in transport rate, and the
morphological evolution of channels. First, I hypothesize that variability in
gravel

The paper is organized as follows. First, I propose a conceptual model for
how

The

Section 1.1 shows that a great many variables and processes influence

Consistent with the form of most bedload transport equations (e.g., Eq. 1),

In the case of a channel reach undergoing net erosion (

Mechanistically,

These

While the above discussion makes the case that

A limitation of Eq. (8) is that, for dimensional consistency, the units of

Flume experiment data (Johnson et al., 2015).

The flume experiments used to calibrate

The experiments started with mixed-size sediment screeded flat. Initially,
all surface sizes were observed to be mobile (and therefore above thresholds
of motion). At the beginning no sediment was fed into the upstream end
(

To quantify thresholds of motion from these experimental data (Fig. 2)
requires a transport model. The Wilcock and Crowe (2003) “Surface-based
Transport Model for Mixed-Size Sediment”, abbreviated as W&CM, is used
for two main reasons. First, the model can, at least in principle, account
for the effects of changing surface GSD on

A key variable in the W&CM is

The hiding function exponent

Note that Eq. (16) is slightly modified from the exact Wilcock and
Crowe (2003) version by replacing

In this section, the W&CM is used to calculate best-fit thresholds of
motion. Next, the

The experimental data are used to determine

While

Interestingly, Fig. 4 shows that the hiding function exponents determined
using the nonlinear multiple regressions for

Data points are based on power-law fits to exponent

Best-fit models (Eqs. 4, 8 and 10) compared to experimental
constraints. The periods of upstream sediment supply (

Figure 5 compares experimentally constrained thresholds of motion to several
predictions of these trends. First, I test whether surface sand fraction
(Eq. 4) can explain the evolution of

Best-fit threshold evolution models.

Various

Model fits using

Next, an idealized morphodynamic model demonstrates how the proposed

For a given discharge this allows both

Two model variations are compared: in the “Exner-only” morphodynamic model,

Away from equilibrium, rates of change of bed elevation along a river profile
should depend not only on the sediment flux at a given channel cross section,
but also on the average velocity at which grains move downstream. This
control has occasionally been ignored in previous models of profile
evolution. In my model, it is crudely incorporated by assuming that average
bedload velocity is a consistent fraction of water velocity, broadly
consistent with previous findings that bedload velocities are proportional to
shear velocity (e.g., Martin et al., 2012). The modeling timestep is set to
be equal to the time it takes sediment to move from one model node (bed
location) to the next, and is adjusted during simulations. While this
approach makes the temporal evolution of channel changes internally
consistent within the model, timescales for model response will still be much
shorter than actual adjustment times in field settings because flood
intermittency is not included (the model as implemented is always at a
constant flood discharge). In addition, the upstream sediment supply is
imposed in the model, while in natural settings hillslope-floodplain-channel
coupling could greatly affect

Morphodynamic model parameters.

Table 2 provides parameters used for morphodynamic modeling. Although the
highly simplified model is not intended for quantitative field comparisons,
variables

Profile evolution, comparing the morphodynamic responses of models
with and without threshold evolution. The initial condition is an equilibrium
channel with

Figures 6 and 7 compare how longitudinal profiles respond to an increase in
sediment supply, for both the Exner-only (constant

I measure an equilibrium timescale (

Slope and critical shear stress evolution, for sediment supply
increases (which correspond to Fig. 6 models) and decreases by factors of 5.
As in Fig. 6,

Over a

Equilibrium timescales are quite sensitive to

Morphodynamic model sensitivity to sediment supply perturbations and

Finally, Fig. 9 shows that the spatial as well as temporal evolution of

Spatial and temporal evolution of morphodynamic slopes, for the same
models shown in Fig. 6. Slope is initially at equilibrium and responds to the

In this section, the dependence of

As described in Sect. 2.1, previous work on sediment supply-dependent
thresholds of motion includes Recking (2012), who proposed high sediment
supply (

Comparison of experimental and best-fit model constraints on

The

The evolution of

The experiments suggest that

In Fig. 8c, the Exner

In the experiments, average slopes changed very little in response to changes
in sediment supply and transport disequilibrium, while grain size and bed
roughness changed much more (Fig. 2; bed roughness is presented in detail in
Johnson et al., 2015). Because the W&CM already accounts for grain size
changes in determining experimental

An implication of

Next, I argue that

State variables are integral to many disciplines, including control systems
engineering and thermodynamics. Thermodynamic state variables include
temperature, pressure, enthalpy and entropy. By definition state variables
are path-independent (Oxtoby et al., 2015). For example, temperature (

Channel morphodynamics can be described by a similar framework of state and
path variables. Analogous to heat, the cumulative discharges of both water
and sediment are path variables that drive bed state evolution. Channel
morphology can be described by numerous bed state variables, including but
not limited to surface GSD, slope, width, depth, bed roughness, surface grain
clustering, interlocking, overlap, imbrication, and finally

Entropy is the state variable perhaps used most often to characterize channel
systems (e.g., Chin and Phillips, 2007; Leopold and Langbein, 1962;
Rodriguez-Iturbe and Rinaldo, 1997). Entropy can provide a closure for
underconstrained sets of equations, by assuming that geomorphic systems
inherently maximize their entropy at equilibrium (Kleidon, 2010; Chiu, 1987).
A limitation of some maximum-entropy landscape models is that
physically based surface processes are not always explicitly modeled, making
them less useful for predicting landscape responses to environmental
perturbations, even if they can create reasonable equilibrium morphologies
(Paik and Kumar, 2010). In contrast to entropy, state variable

I suggest that landscape evolution models could incorporate subgrid-scale
channel feedbacks by treating

Calculations of best-fit

I propose a new model in which feedback causes

After empirically calibrating three model parameters, the

Finally,

Data sources for Fig. 1 are given in its caption. An additional data table
(data set “grl53586-sup-0002-supplementary.xls”) that is provided with
Johnson et al. (2015) contains data on Fig. 2. Data and matlab code to make
figures in the paper based on experimental calibrations are available through
Figshare (

I thank Alex Aronovitz for conducting the flume experiments, Wonsuck Kim for
aiding in the experimental design, Lindsay Olinde for helpful discussions,
and Mike Lamb and Jeff Prancevic for sharing their