Debris flows regularly traverse bedrock channels that dissect steep landscapes, but our understanding of bedrock erosion by debris flows and their impact on steepland morphology is still rudimentary. Quantitative models of steep bedrock channel networks are based on geomorphic transport laws designed to represent erosion by water-dominated flows. To quantify the impact of debris flow erosion on steep channel network form, it is first necessary to develop methods to estimate spatial variations in bulk debris flow properties (e.g., flow depth, velocity) throughout the channel network that can be integrated into landscape evolution models. Here, we propose and evaluate two methods to estimate spatial variations in bulk debris flow properties along the length of a channel profile. We incorporate both methods into a model designed to simulate the evolution of longitudinal channel profiles that evolve in response to debris flow and fluvial processes. To explore this model framework, we propose a general family of debris flow erosion laws where erosion rate is a function of debris flow depth and channel slope. Model results indicate that erosion by debris flows can explain the occurrence of a scaling break in the slope–area curve at low-drainage areas and that upper-network channel morphology may be useful for inferring catchment-averaged erosion rates in quasi-steady landscapes. Validating specific forms of a debris flow incision law, however, would require more detailed model–data comparisons in specific landscapes where input parameters and channel morphometry can be better constrained. Results improve our ability to interpret topographic signals within steep channel networks and identify observational targets critical for constraining a debris flow incision law.

Debris flows are effective at transporting coarse sediment

Topographic signatures of geomorphic processes, which we define as quantitative connections between processes and the morphology of a landform, can be used to infer the presence and rates of geomorphic processes from topographic data

Examples of channel profiles from the San Gabriel Mountains, California, USA, along with best fit curves of the form

Past work demonstrates that the length of the channel network upstream of the debris flow to fluvial transition zone, which we roughly associate with

These findings underscore the need to develop a quantitative framework that can be used to explore topographic signatures generated by debris flow erosion, assess the sensitivity of topographic signatures to climatic and tectonic forcing, and ultimately interpret these signatures to gain process-based insights about the evolution of steep landscapes. In particular, there is a need to understand the relative importance of fluvial and debris flow processes in setting the location and form of the morphologic transition associated with

In landscape evolution models, fluvial erosion is modeled based on empirical relationships between local terrain attributes such as slope and drainage area, readily computed from a digital elevation model (DEM)

Several subsequent studies have improved our understanding of the grain-scale processes that control debris flow incision rates and their relationship to bulk flow properties that are more amenable to measurement and model simulation.

Here, we address this gap by developing a nonlocal modeling framework to predict the evolution of a 1D channel profile eroded by both fluvial and debris flow processes. In this framework, the routing of debris flows down the channel profile as well as estimates of spatial and temporal variations in bulk debris flow properties are represented either through a process-based model that relies on a set of partial differential equations

In the proposed 1D model framework, which is designed to simulate longitudinal channel profiles, the rate of change in elevation,

Fluvial erosion is computed using the threshold-stochastic stream power incision model presented by

We propose a general formulation that can be used to estimate the erosion rate attributable to debris flows,

We also present a reduced-complexity routing algorithm, which closely follows the methodology presented by

The initial step in computing the erosion rate attributable to debris flows is to determine the runout path of the debris flow as well as its bulk properties at different points along that path. The process-based debris flow routing model is based on a set of conservation laws for mass and momentum within a depth-averaged framework. This particular model formulation was chosen because it provides sufficient complexity to enable exploration of the links between flow properties and the morphology of the resulting channel profile. The governing equations represent the flow of a two-component mixture, solids suspended in a Newtonian fluid

We assume that the ratio of pore fluid pressure to total basal normal stress decays with time since the debris flow entered the model domain,

The friction coefficient is a function of the inertial number

The governing equations are solved numerically on a grid with uniform spacing. We use a first-order, shock-capturing finite-volume method with a Harten–Lax–van Leer contact (HLLC) approximate Riemann solver

Debris flows enter the domain through the upper boundary, which can be conceptualized as the channel head, and are routed down the channel profile. We define a series of 20 ghost cells above the uppermost grid cell that effectively extend the model domain for the purpose of initializing a debris flow. Elevations of each ghost cell are determined by assuming that the slopes of all ghost cells are equal to the slope at the uppermost grid cell. Debris flows are initiated from a static pile of debris defined on the ghost cells. This procedure provides some time for debris to begin to flow before it enters the model domain, similar to what might be expected for debris flows that initiate in a colluvial hollow or gully upstream of a channel head. In nature, we expect debris flow volume to vary with drainage area as sediment is entrained and deposited along the runout path

At each grid cell in the model domain (i.e., excluding ghost cells), the debris flow incision rate is computed using Eq. (

We use a series of empirical relationships defined by

To begin, we specify debris flow volume passing through each grid cell as a function of upstream drainage area (

In this study, we fix all model parameters within a given simulation. As such, the channel profiles that develop can be thought of as reflecting the morphology of a channel shaped by the repeated impacts of a characteristic debris flow. Future studies could explore the effects of debris flows characterized by a distribution of parameters to better reflect natural variations in flow properties.

When using empirical relationships to determine flow properties along the debris flow runout path, we can derive an approximate analytical solution for the slope of the upper, debris-flow-dominated reach of the channel at steady state. We begin by considering debris flow erosion as quantified by Eqs. (

Solving for slope as a function of drainage area, we obtain

Our numerical experiments have two goals, which are treated in turn. First, we assessed which erosion laws, as defined by different values of

We explored model behavior for different values of

A landscape evolution model designed to simulate the evolution of channel longitudinal profiles in response to both debris flow and fluvial erosion should produce steady-state channel profiles that are well described by Eqs. (

We therefore assessed model performance for different

We assessed performance of the landscape evolution model with different values of

All simulations began with an initial condition determined by the analytical solution for a steady-state fluvial channel, specifically

We performed sensitivity analyses using both the process-based and empirical routing models to explore how the topographic signature of debris flow incision is likely to be expressed in different settings. Motivated by the results of our numerical experiments to constrain

To perform the sensitivity analysis with the process-based routing model, we used a Latin hypercube sampling strategy to select

We performed a qualitative sensitivity analysis by visually examining model output using colored scatter plots and also performed a quantitative global sensitivity analysis using the PAWN method

At large drainage areas, modeled profiles exhibit a power law scaling between slope and drainage area that is expected based on the fluvial incision law (Figs.

Numerical experiments using the process-based routing model to determine which slope (

Numerical experiments using the empirical routing model to determine which slope (

Modeled profiles exhibited the expected power law scaling between slope and drainage area at high-drainage areas where fluvial incision dominated debris flow incision. The coefficient of determination (

Numerical experiments using the empirical routing model that highlight the importance of the volume–area scaling exponent

More generally, the extent to which modeled channel profiles exhibit a decrease in slope at small drainage areas depends on

Two defining characteristics of the simulated steady-state channel profiles, the near-constant slope that they approach near the channel head and the minimum drainage area at which there is a power law scaling between slope and drainage area, can be summarized by the following two metrics:

Model parameters.

PAWN sensitivity indices of the process-based model.

Scatter plots summarizing results of the sensitivity analysis with the process based debris flow routing model. Sensitivity of

Steady-state longitudinal profiles produced by the process-based model for various parameter combinations. Variations in

Results from the

Scatter plots summarizing results of the sensitivity analysis with the empirical debris flow routing model. The relationship between

Steady-state longitudinal profiles produced by the empirical routing model for various parameter combinations. Variations in

The sensitivity of steady-state long-channel profiles to changes in rock uplift rate leads to power law relationships between

Simulations indicate that

PAWN sensitivity indices of the empirical model.

There is a power law relationship between

Results indicate that many members within the proposed family of debris flow incision laws, as formulated by Eqs. (

Here, we assess different debris flow incision laws based on their ability to reproduce a general pattern in slope–area data (i.e., Eq.

In general, the proposed empirical and process-based approaches for estimating bulk debris flow properties along the channel profile do not appear to result in different model behavior (Figs.

Model results help clarify the roles played by debris flow and fluvial erosion processes in setting longitudinal profile form in the upper channel network. Changes in parameters related solely to fluvial erosion do not have a strong influence on

Simulations, assuming

Model results support previous observations indicating that the morphology of channel profiles in debris-flow-dominated landscapes may provide constraints on erosion rates in steady-state landscapes (Figs.

Simulations indicate increases in erosion rate, or equivalently rock uplift rate in a steady-state landscape, lead to an increase in

We describe a model that provides a framework for exploring the effect of episodic debris flows on channel longitudinal profiles. The model also comes with several limitations. We assume that all debris flows initiate at the channel head, whereas debris flow initiation locations in natural landscapes will be more varied. Past work highlights the role that network structure, specifically the number of debris flow initiation locations upstream of a given channel reach, may play in controlling channel form

The landform evolution model presented here may serve as a basis for future studies that aim to test or validate potential debris flow incision laws, incorporate debris flow incision into 2D landscape evolution models, or explore how the upper channel network responds to tectonic or climatic perturbations. The process-based routing model may be best suited for modeling 1D channel profiles where changes in flow volume can be neglected and debris flow constituents are sufficiently well known to allow for estimates of the model parameters, thereby minimizing the number of numerical experiments needed to characterize model behavior. The empirical debris flow routing algorithm provides an efficient framework for investigating the effects of different debris flow bulking relationships and exploring large parameter spaces. It is also particularly promising for application in 2D landscape evolution models given its simplicity relative to process-based debris flow routing models and its ability to connect slope and drainage area, which are readily available in nearly all landscape evolution models, with bulk debris flow properties relevant to debris flow incision.

We present a novel framework for incorporating erosion by debris flows into a model for channel profile evolution. We propose two methods to estimate debris flow runout and bulk debris flow properties (e.g., depth, velocity) throughout the channel network, one based on a process-based debris flow routing model and the other based on an empirical routing approach. Combined with a geomorphic transport law describing the relationship between debris flow depth, channel slope, and a debris flow incision rate, we are able to quantify spatial variations in the debris flow incision rate throughout the channel network. We explore the performance of a family of potential debris flow incision laws by comparing the form of modeled longitudinal channel profiles with those typically observed in debris-flow-dominated landscapes. Results demonstrate that a debris flow incision law based on flow depth, slope, and debris flow passage time can reproduce the relationship between slope and drainage area that has been interpreted as a topographic signature of debris flows, given general constraints on empirical exponents that relate flow depth and local channel slope to the incision rate. Since a large subset of the proposed family of erosion laws is capable of reproducing this topographic signature of debris flows, additional criteria and more precise bounds on poorly constrained model parameters are needed to test and validate debris flow erosion laws. Simulations indicate that both

The parameterization for the stochastic stream power model is not tuned to any particular landscape or geographic region, but relies on values and relationships that are based on typical values reported by

Scaling relationships that relate drainage area and channel width are often derived from data that include drainage areas greater than those modeled in this study. Recall that in this study we use the term channel to broadly refer to a concentrated axis of erosion along valley bottoms. Since debris flows initiate and traverse channels at low-drainage areas, we quantified channel width at drainage areas less than

Estimates of channel width, estimated from a high-resolution lidar-derived digital elevation model, as a function of drainage area for a portion of the San Gabriel Mountains, USA. The area burned in the 2016 Fish Fire and experienced a series of debris flows during the first rainy season following the fire that scoured valleys and channels to bedrock in many places. For comparison, relationships between channel width and drainage area as determined by

We fit a line,

The analytical solution for steady-state channel slope indicates that slope is a power law function of drainage area with an exponent,

The process-based routing model does not directly account for downstream changes in debris flow volume. When using the empirical routing model, for example, we prescribe debris flow volume as a function of drainage area according to

Steady-state channel profiles using the process-based routing model where we parameterize an increase in debris flow frequency with drainage area,

Here, we report results for the PAWN sensitivity analysis when using erosion laws with

PAWN sensitivity indices for the empirical model:

PAWN sensitivity indices for the empirical model:

Tables below provide details on the value or range of model parameters used in different numerical experiments.

Stochastic stream power model parameters.

Parameters used when running numerical experiments with the process-based routing model to constrain

Parameters used when running numerical experiments with the empirical routing model to constrain

Parameters used in the sensitivity analysis with the process-based routing model.

Parameters used in the sensitivity analysis with the empirical routing model.

Model code is available on HydroShare at

The initial idea for the study arose from conversations among LAM and SWM. The model code was written by LAM. WS extracted slope and area data from the San Gabriel Mountains. All authors contributed to the design of numerical experiments and interpretation of results. LAM performed the numerical experiments and wrote the paper with input and editing from all authors.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This material is based upon work supported by the National Science Foundation under grant no. 1951274. Jason Kean, Francis Rengers, Leslie Hsu, Ryan Gold, and Janet Carter provided comments as part of U.S. Geological Survey review. We would like to thank Alexander Densmore and an anonymous reviewer for providing reviews that improved the manuscript.

This research has been supported by the National Science Foundation (grant no. 1951274).

This paper was edited by Jean Braun and reviewed by Alexander Densmore and one anonymous referee.