Alluvial and transport-limited bedrock rivers constitute the majority of fluvial systems on Earth. Their long profiles hold clues to their present state and past evolution. We currently possess first-principles-based governing equations for flow, sediment transport, and channel morphodynamics in these systems, which we lack for detachment-limited bedrock rivers. Here we formally couple these equations for transport-limited gravel-bed river long-profile evolution. The result is a new predictive relationship whose functional form and parameters are grounded in theory and defined through experimental data. From this, we produce a power-law analytical solution and a finite-difference numerical solution to long-profile evolution. Steady-state channel concavity and steepness are diagnostic of external drivers: concavity decreases with increasing uplift rate, and steepness increases with an increasing sediment-to-water supply ratio. Constraining free parameters explains common observations of river form: to match observed channel concavities, gravel-sized sediments must weather and fine – typically rapidly – and valleys typically should widen gradually. To match the empirical square-root width–discharge scaling in equilibrium-width gravel-bed rivers, downstream fining must occur. The ability to assign a cause to such observations is the direct result of a deductive approach to developing equations for landscape evolution.

Mountain and upland streams worldwide move clasts of gravel

Geomorphologists commonly separate rivers into two broad categories based on
the factor that limits their ability to change their long profile:
detachment-limited and transport-limited

Here we present a new derivation for transport-limited gravel-bed river long-profile evolution that is based on relationships derived from theory, field work, and experimentation. We argue that developing this deductive approach – considering specific process relationships – is essential to advancing fluvial geomorphology and landscape evolution.

Much past work has focused on an inductive “stream-power” based formulation
for detachment-limited river incision, in which the erosion rate is
proportional to the drainage area (as a proxy for geomorphically effective
discharge) and channel slope (e.g.,

Writing a set of equations to describe the long-profile evolution of
transport-limited gravel-bed rivers, in contrast, is aided by an extensive
history of study that can be directly applied to models of long-profile
evolution. This includes open-channel flow and flow resistance that can be
applied to sediment-covered channels

Here we link sediment transport and river morphodynamics to develop equations
to describe gravel-bed river long profiles and, as a necessary extension,
their tightly coupled width evolution. Our approach is complementary to a
recent set of relations for alluvial river long profile shapes developed by

Our approach is outlined as follows: first, we generate fully coupled
equations of gravel transport and fluvial morphodynamics to describe how
channel long profiles change. Second, we investigate how the governing
equations for gravel-bed rivers differ when we assume a channel with a
self-formed equilibrium width vs. an externally set width.
Third, we derive both analytical and numerical solutions for the case of an
equilibrium-width channel, which is nearly ubiquitous in nature

We consider gravel-bed rivers to exist in one of two states:
equilibrium-width and fixed-width. In the first, we assume that the
channel-forming (i.e., bank-full) shear stress on the bed remains a constant
ratio of the critical shear stress that sets the threshold for initiation of
sediment motion

Our primary focus here is on equilibrium-width rivers, which are common
throughout the world

Schematic block diagram of sediment transport through a reach of a transport-limited river. Variables are defined in the text and in Appendix A. The balance of sediment input, sediment output, and uplift determine whether the river bed at each point downstream will rise, fall, or remain at a constant elevation.

We split our derivations into sections on equilibrium-width
(Sect.

We derive an equation for the evolution of the long profile of an
equilibrium-width gravel-bed river that lies within a valley whose shape is
arbitrary (although at least as wide as the channel) and may evolve through
time. We first state a modified Exner equation for the conservation of
bed-load sediment discharge (

Equation (

Following this definition of a sediment continuity equation, we take several
steps towards developing a simple formulation for the total discharge of
sediment through the river,

Towards this eventual goal, our second step is to define bed-load sediment
discharge per unit width,

While the

Basal shear stress induces a drag force on the grains and drives sediment
transport. To compute this basal shear stress (

The drag force on sediment grains induced by basal shear stress is resisted
by the submerged weight of the grains. The ratio of these forces defines the
Shields stress:

In equilibrium-width gravel-bed rivers, the dimensionless basal shear stress
at the channel-forming discharge is assumed to be maintained as a constant
multiple of the dimensionless critical shear stress for initiation of
sediment motion

The channel-forming discharge, also termed the geomorphically effective
discharge, is equivalent to the bank-full flow in a self-formed gravel-bed
river with gravel bars and banks.

Substituting

It may be counterintuitive that sediment discharge per unit width increases with grain size. This is a result of the equilibrium-width argument. Channel geometry adjusts to maintain a constant excess basal shear stress regardless of grain size. However, larger grains have a greater vertical dimension: many small grains rolling or sliding along the bed will displace less mass than a single larger grain.

Equation (

For a self-formed gravel-bed channel, channel depth must satisfy
Eq. (

Next, we compute mean water flow velocity (

Water discharge per unit width can be computed by multiplying

The final equation that we require to obtain channel width (

Finally, channel width (

We express this equation first in terms of magnitudes,

Equation (

Hydraulic geometry adjustment in an equilibrium-width gravel-bed river causes
bed-load sediment discharge to be independent of grain size. Sediment
discharge per unit width increases with grain size as

In this derivation, we hold

While

We combine Eqs. (

Uplift and subsidence (

Equation (

If the width of the river is externally known and is identical to the width
of the valley, another solution is possible. To produce this solution, we
first simplify the Exner equation to its one-dimensional form for the case in
which

Combining this form of the Exner equation with the

Equation (

To formulate the differential equation for long-profile evolution of a
transport-limited gravel-bed river of arbitrary width, we combine our
transport relationship (Eq.

In addition to the variable space–time dependencies listed in
Eqs. (

Two analytical solutions are presented here to help build intuition into the shape of gravel-bed river long profiles. The most generally applicable of these, for an equilibrium-width gravel-bed river that is neither aggrading nor incising in an area with no tectonic activity, is presented first. This solution is a power law that relates measurable hydrologic and landscape parameters to river long-profile shape. The second analytical solution is for a fixed-width river that adds the additional assumptions that width, discharge, and grain size are held constant. This solution provides an equilibrium transport slope.

In order to analytically solve special cases of the provided equations for
river channel long-profile evolution, we need a way to write
Eq. (

Based on observations

These equations are continuum idealizations of a river with a tributary network. Real rivers experience discrete jumps in water discharge at tributary junctions. The smooth curves of water discharge vs. down-valley distance produced by these relationships, in contrast, are beneficial for building intuition.

Solutions to Eq. (

In order to develop an analytical solution to Eq. (

For such a no-uplift steady-state condition to persist over geologic time
requires a constant input of sediment from the hillslopes. This may be
reasonable for a river that reaches an equilibrium long profile much more
rapidly than the surrounding landscape evolves and its relief changes. It is
also useful as a benchmark for numerical solutions
(Fig.

Applying Eq. (

In order to generate an analytical solution for a fixed-width gravel-bed
river, starting from Eq. (

To solve more general cases of Eqs. (

When

The sediment-transport formulation that we present in
Eq. (

The appropriate value of

We normalize

From this relationship, we can assign values to the following parameters from

We make the modification that contributing area must be raised to a power,

In the case of Eq. (

These assumptions produce a convex long profile because as drainage area
increases, sediment supply increases more strongly than water discharge. A
straightforward solution is to adjust

Steady-state numerical model outputs with steady uplift (base-level
fall), subsidence (base-level rise), or neither. These numerical solutions
are formulated following Eq. (

An approximate value for the gravel persistence exponent,

Figure

The small value of

The slope–area concavity index defined in Eqs. (

Equation (

Transport-limited river-channel long profiles evolve in response to water and
sediment inputs

These distinct modes of response can help us to distinguish whether the river
is responding primarily to changes in water and/or sediment supply, or to
changes in base level. This may help to disentangle the effects of climate –
often related to water and sediment supply

Uniform changes in the input sediment-to-water discharge ratio, in the
absence of changes in the uplift rate (or equivalently, the rate of base-level
change), determine the steepness index of a channel, but do not affect its
concavity (Eqs.

As the sediment-to-water discharge ratio increases, a steeper channel is
required to mobilize the sediments, and as a result, the channel steepness
index (

Transient long profiles from numerical model runs and their response
times to external forcings. Each fine gray line in panel sets 1 and 2,
corresponding to black dots on panel set 3, represents 30 000 years with an
intermittency of

Changes in the rate of base-level rise or fall, including those caused by
tectonic subsidence or uplift, modify the concavity but not the steepness of
a transport-limited gravel-bed river long profile
(Fig.

The analytical solution (Eq.

Concavity changes uplift (or subsidence) rates increase when
compared to a characteristic alluvial response rate.

Even though uniform changes in the water-to-sediment discharge ratio cannot
impact long-profile concavity on their own, they can (through volume balance)
influence the degree to which rates of uplift (or subsidence)
changes do. Uplift or subsidence add or remove material from the
bed of the river, and changes in concavity are the river's response to
redistribute sediment discharge to balance these local sources or sinks of
sediment. If the sediment discharge of the river is large compared to the
amount of material moved by uplift or subsidence, then only a small
adjustment of concavity is necessary to balance this source (uplift) or sink
(subsidence) and maintain steady-state topography. A river carrying very
little sediment, however, will have to dramatically change its long-profile
concavity in order to reach steady state. Therefore, the steady-state
long-profile concavity (Fig.

Covarying tectonic uplift (or base-level fall) and the input
sediment-to-water supply ratio produces a range of channel long
profiles

In order to compare both sediment discharge and uplift using a dimensionless
parameter, we define a characteristic alluvial response rate (

We note that

Dividing the tectonic uplift (or subsidence) rate (

Rivers also exhibit a transient response to changes in base level at a rate
that is proportional to the alluvial response rate,

In the above section, we have separated the effects of
tectonics and climate as concavity and steepness responses, respectively. Our
concavity changes derived from theory and their causes are generally
consistent with the broad range of concavities and causes thereof synthesized
by

Section

It has long been recognized that river channel width scales with discharge to
the

Equation (

Combining Eqs. (

The range of physically permissible values for the exponent that relates
drainage area to discharge,

In an equilibrium-width self-formed gravel-bed river channel, the common field observation that river channel width is proportional to the square root of water discharge may be explained by a combination of the direct impact of river discharge on channel width and by downstream fining of bed-material sediment.

Using standard values of

We have produced equations to describe the long-profile evolution of
transport-limited gravel-bed rivers by combining the Exner equation for
conservation of volume, the

Our derivation brings to light several significant relationships that may aid
further efforts to understand river long profiles: (1) the sediment-transport
formula for an equilibrium-width (

In this paper, we have derived a physics-based expression for the long-profile evolution of transport-limited gravel bed rivers, whose parameters are determined by theory, experimentation, and field work. We hope that this approach to understanding gravel-bed rivers provides forward momentum towards a more formal treatment of sediment transport and fluvial morphodynamics in river long-profile analysis and landscape evolution. Furthermore, by combining our derivation with other observations, we predict relationships among valley morphology, coarse-sediment production and evolution, and the power-law scaling between drainage area and geomorphically effective floods. While rivers are complex, we hope that these connections with broader pieces of the geomorphic puzzle can provide a path to build a better theory of fluvial system change and landscape evolution.

The GitHub repository at

In the canonical Exner equation, the one-dimensional negative divergence in
sediment flux is stated to be proportional to aggradation or incision, as
given in Eq. (

In order to understand the evolution of a valley, one can first rewrite
Eq. (

This equation corresponds to Fig.

In order to solve this equation, we require a relationship that links the
down-valley sediment discharge,

Our equations for sediment transport follow the river, but
geomorphic evolution occurs along valley networks. In some
cases

Our main goal is to understand the evolution of valley networks, as channels
perform the geomorphic work but valleys are the geomorphic units that evolve
and constitute the broader landscape. In alluvial systems, valley geometries
are not always identical to those of the channel networks that occupy them
(Fig.

Down-valley discharge of water and sediment in each cross section is
equal, and equals the down-channel discharge. Here, this is demonstrated
geometrically, and results from discharge per unit width decreasing as the
flow becomes more oblique to the valley cross section line, but the fraction
of the total cross-valley line occupied by river channel increases in an inversely
proportional manner. This remains true even for up-valley flow, which must be
balanced with down-valley flow for water to be able to move from the upstream
to the downstream ends of the valley. The direct down-valley discharge of
water or sediment per unit width is given by

For an angle

This geometric argument contains one mathematical caveat. Where a channel
flows directly across the valley (i.e., in the

By considering a continuity solution rather than a geometric one, it is
possible to reason that our treatment of the above caveat is correct. In a
system at steady state, each valley cross section must transmit downstream as
much discharge as it is provided by the cross section immediately upstream.
Therefore, discharge through every cross section must be equal, regardless of
channel orientation. If this is the case, then a channel segment directed in
line with the valley must transmit just as much water and sediment as a
channel segment that is at an oblique angle to the valley axis. As a result,

Finally, while sediment (or any) discharge remains path-independent, the
magnitude of this discharge is path-dependent. A more sinuous river course
decreases channel slope (Eq.

The full

Including explicit directionality is not common when representing fluvial
geomorphology mathematically. Slope,

Equation (

Time-averaged values of

For an implicit solution, the terms inside the square brackets, plus

A Neumann boundary condition of sediment-discharge input is used to set the slope at the upstream boundary
using a “ghost-point” approach. This is solved for a defined

The general discretization of Eq. (

ADW wrote the paper, coupled the equations, and wrote the accompanying computer program. TFS motivated the paper through field studies; checked and revised the text, figures, and equations; supervised the project; and acquired funding for the research. Both authors contributed to the direction of the study and the discussion of its implications for fluvial geomorphology.

The authors declare that they have no conflict of interest.

This study was funded in large part by the Emmy-Noether-Programme of the Deutsche Forschungsgemeinschaft (DFG) grant no. SCHI 1241/1-1 awarded to Taylor F. Schildgen. Field and lab observations and discussions with Stefanie Tofelde, who revised an early version of the paper, helped to motivate the work. Conversations on channel concavity with Kelin Whipple and Greg Tucker stimulated initial thoughts on Sect. 5 of this paper, Chris Paola commented on an early form of the derivation, and Sam Holo pointed out an error in our treatment of sinuosity that appeared in the Discussions paper. Comments from Rebecca Hodge and one anonymous reviewer helped us to improve the paper. Edited by: Tom Coulthard Reviewed by: Rebecca Hodge and one anonymous referee