Models of detachment-limited fluvial erosion have a long history in landform evolution modeling in mountain ranges. However, they suffer from a scaling problem when coupled to models of hillslope processes due to the flux of material from the hillslopes into the rivers. This scaling problem causes a strong dependence of the resulting topographies on the spatial resolution of the grid. A few attempts based on the river width have been made in order to avoid the scaling problem, but none of them appear to be completely satisfying. Here a new scaling approach is introduced that is based on the size of the hillslope areas in relation to the river network. An analysis of several simulated drainage networks yields a power-law scaling relation for the fluvial incision term involving the threshold catchment size where fluvial erosion starts and the mesh width. The obtained scaling relation is consistent with the concept of the steepness index and does not rely on any specific properties of the model for the hillslope processes.

Fluvial incision is a major if not dominant component of
long-term landform evolution in orogens. When modeling fluvial
erosion, restriction to the detachment-limited regime
considerably simplifies the equations. Here it is assumed that
the erosion rate at any point of a river can be predicted from
local properties such as discharge and slope, while sediment transport
is not considered.
The generic differential equation for the topography

Concerning the fluvial incision term

Equation (

While widely used and in principle simple, all models of the type
described by Eqs. (

If local transport is not considered, the scaling problem leads
to a canyon-like topography, where the width of the valleys decreases
with mesh width.
This behavior is illustrated in Figs.

Fluvial equilibrium topographies computed for identical parameter values on grids with different spacing (

Profiles through the topographies shown in Fig.

Relief increases with decreasing grid spacing because the smallest
catchment size that can be resolved is

The independence of river steepness of resolution is, however, lost as soon
as local transport comes into play. Figure

River segments in equilibrium with uplift for different mesh
widths

The reason for the increasing channel steepness is that the local
transport is conservative, so the river not only has to incise
into the rock at its bed but also has to remove the material coming
from the hillslopes. Regardless of the model used for local transport,
a flux of

This scaling issue has been known for more than 25 years, and two
approaches have been suggested to overcome the problem.

While straightforward at first sight, this scaling approach is not
free of problems. The channel width in general increases in the downstream
direction so that equilibrium river profiles are no longer consistent
with Eq. (

In order to overcome this problem,

Thus there seems to be no completely satisfactory
solution of the scaling problem so far. Several contemporary modeling
studies

Other recent approaches navigate around the scaling problem by neglecting
the flux of material from the hillslopes into the rivers.
The recently presented landform
evolution model TTLEM

The simple example considered in the previous section involves a dependence on
grid spacing

Let us start from the simplest approach to distinguish channel sites
from hillslopes by defining a threshold
catchment size

Flow pattern of the central region of Fig.

If the size of this area was the same for each river site,
rescaling the fluvial erosion rate (Eq.

In the following, numerically obtained equilibrium drainage networks are
analyzed in order to find out how

Figure

Eroded area

The increase in

Both the number of river segment sites and the number of channel head sites decrease
with an increasing threshold

Ratio of total area eroded by all river segments to total area eroded by all channel head sites as a function of the fluvial threshold

This result suggests that the dependency of

Equations (

Black axes: eroded area as a function of the fluvial threshold. Colored axes: cumulative distribution of the catchment sizes.

The black dashed line in Fig.

The relation to the catchment-size distribution (Eqs.

The power-law parameters

Parameter values of the power-law relation between eroded area and fluvial threshold (Eq.

Eroded area

In addition, Table

Table

Parameter values of the power-law relation between eroded area and fluvial threshold (Eq.

These results suggest defining the values

In order to estimate

Let us first return to the example of parallel rivers considered in Fig.

It should be noted that this example is not related to the approach to estimate

Numerical results for the scenario considered in Fig.

The second example refers to the scenario considered in Fig.

The mean steepness index

Mean steepness index

It may be surprising that the example of fluvial incision and hillslope diffusion considered
in the previous section yields a mean steepness index greater than 1, although the scaling concept was developed in order to preserve channel steepness. The concept is, however, based on a generic hillslope process where the direction of transport follows a hypothetic fluvial equilibrium pattern and turns into fluvial erosion at a given threshold catchment
size

This is, however, a real property of the hillslope process here, and it is not the goal of the scaling approach to remove it. The concept presented here aims at removing the dependence on the resolution and providing the way in which values of the erodibility should be interpreted. Here it is suggested that they should be considered in combination with a fluvial threshold

In turn, the residual dependence of channel steepness on resolution is a problem, in particular because it is not clear whether it converges in the limit

Nevertheless it is important to keep the difference between detachment-limited
erosion and pure bedrock incision in mind. Here it is assumed that the ability
of the river to take up particles and carry them away concerns
both the riverbed and material coming from adjacent hillslopes.
If we, conversely, assume that all material coming from the hillslopes
is instantaneously removed by the river without any consequences, there
is no feedback of the hillslopes to the rivers, and Eq. (

The results of this study have consequences for scaling relations in coupled
models of rivers and hillslopes.

This study presents a simple scaling relation for the fluvial
incision term in landform evolution models involving detachment-limited
fluvial erosion and hillslope processes. In order to avoid a
dependence of the simulated topographies on the spatial resolution
of the grid, the fluvial incision term must be multiplied by a scaling
factor depending on the ratio of the threshold catchment size

All codes and computed data can be downloaded from the FreiDok data repository 155182

The author declares that there is no conflict of interest.

The author would like to thank the two anonymous reviewers for their thorough consideration and for their very constructive suggestions to improve the readability of the paper. The author would also like to thank Wolfgang Schwanghart for the editorial handling.

This paper was edited by Wolfgang Schwanghart and reviewed by Taylor Perron and two anonymous referees.