scholarly journals Transport-limited fluvial erosion – simple formulation and efficient numerical treatment

2020 ◽  
Vol 8 (4) ◽  
pp. 841-854
Author(s):  
Stefan Hergarten
Keyword(s):  
Power Law ◽  
Constitutive Law ◽  
Tectonic Geomorphology ◽  
Simple Relationship ◽  
Stream Power ◽  
Simple Formulation ◽  
Fluvial Erosion ◽  
Empirical Results ◽  
Numerical Effort ◽  
River Profiles

Abstract. Most of the recent studies modeling fluvial erosion in the context of tectonic geomorphology focus on the detachment-limited regime. One reason for this simplification is the simple relationship of the constitutive law used here – often called stream-power law – to empirical results on longitudinal river profiles. Another no less important reason lies in the numerical effort that is much higher for transport-limited models than for detachment-limited models. This study proposes a formulation of transport-limited erosion where the relationship to empirical results on river profiles is almost as simple as it is for the stream-power law. As a central point, a direct solver for the fully implicit scheme is presented. This solver requires no iteration for the linear version of the model, allows for arbitrarily large time increments, and is almost as efficient as the established implicit solver for detachment-limited erosion. The numerical scheme can also be applied to linear hybrid models that cover the range between the two end-members of detachment-limited and transport-limited erosion.

scholarly journals Transport-limited fluvial erosion – simple formulation and efficient numerical treatment

2020 ◽  
Author(s):  
Stefan Hergarten
Keyword(s):  
Power Law ◽  
Linear Models ◽  
Constitutive Law ◽  
Tectonic Geomorphology ◽  
Stream Power ◽  
Simple Formulation ◽  
Fluvial Erosion ◽  
Empirical Results ◽  
Numerical Effort ◽  
River Profiles

Abstract. Most of the recent studies modeling fluvial erosion in the context of tectonic geomorphology focus on the detachment-limited regime. One reason for this simplification is the direct relationship of the constitutive law used here – often called stream-power law – to empirical results on longitudinal river profiles. Another, not less important reason lies in the numerical effort that is much higher for transport-limited models than for detachment-limited models. This study proposes a simple formulation of transport-limited erosion that is as close to empirical results on river profiles as the stream-power law is. As a central point, a direct solver for the fully implicit scheme is presented. This solver requires no iteration for the linear version of the model, allows for arbitrarily large time increments, and is almost as efficient as the established implicit solver for transport-limited erosion. The numerical scheme can also be applied to linear models between the two extremes of detachment-limited and transport-limited erosion.

scholarly journals A stream-power law for glacial erosion and its implementation in large-scale landform-evolution models

2021 ◽  
Author(s):  
Stefan Hergarten
Keyword(s):  
Sediment Transport ◽  
Large Scale ◽  
Numerical Models ◽  
Stream Power ◽  
Glacial Erosion ◽  
Simple Formulation ◽  
Landform Evolution ◽  
Fluvial Erosion ◽  
Numerical Effort ◽  
Long Time

Abstract. Modeling glacial landform evolution is more challenging than modeling fluvial landform evolution. While several numerical models of large-scale fluvial erosion are available, there are only a few models of glacial erosion, and their application over long time spans requires a high numerical effort. In this paper, a simple formulation of glacial erosion which is similar to the fluvial stream-power model is presented. The model reproduces the occurrence of overdeepenings, hanging valleys, and steps at confluences at least qualitatively. Beyond this, it allows for a seamless coupling to fluvial erosion and sediment transport. The recently published direct numerical scheme for fluvial erosion and sediment transport can be applied to the entire domain, where the numerical effort is only moderately higher than for a purely fluvial system. Simulations over several million years on lattices of several million nodes can be performed on standard PCs. An open-source implementation is freely available as a part of the landform evolution model OpenLEM.

scholarly journals Modeling glacial and fluvial landform evolution at large scales using a stream-power approach

2021 ◽  
Vol 9 (4) ◽  
pp. 937-952
Author(s):  
Stefan Hergarten
Keyword(s):  
Sediment Transport ◽  
Large Scale ◽  
Numerical Models ◽  
Stream Power ◽  
Glacial Erosion ◽  
Simple Formulation ◽  
Landform Evolution ◽  
Fluvial Erosion ◽  
Numerical Effort ◽  
Long Time

Abstract. Modeling glacial landform evolution is more challenging than modeling fluvial landform evolution. While several numerical models of large-scale fluvial erosion are available, there are only a few models of glacial erosion, and their application over long time spans requires a high numerical effort. In this paper, a simple formulation of glacial erosion which is similar to the fluvial stream-power model is presented. The model reproduces the occurrence of overdeepenings, hanging valleys, and steps at confluences at least qualitatively. Beyond this, it allows for a seamless coupling to fluvial erosion and sediment transport. The recently published direct numerical scheme for fluvial erosion and sediment transport can be applied to the entire domain, where the numerical effort is only moderately higher than for a purely fluvial system. Simulations over several million years on lattices of several million nodes can be performed on standard PCs. An open-source implementation is freely available as a part of the landform evolution model OpenLEM.

scholarly journals Tectonic geomorphology at small catchment sizes – extensions of the stream-power approach and the <i>χ</i> method

2016 ◽  
Vol 4 (1) ◽  
pp. 1-9 ◽  
Author(s):  
S. Hergarten ◽  
J. Robl ◽  
K. Stüwe
Keyword(s):  
Empirical Relationship ◽  
Tectonic Geomorphology ◽  
Stream Power ◽  
Simple Method ◽  
Small Catchment ◽  
The Best Approximation ◽  
Fluvial Incision ◽  
Catchment Size ◽  
Almost All ◽  
River Profiles

Abstract. Quantitative tectonic geomorphology hinges on the analysis of longitudinal river profiles. The model behind almost all approaches in this field originates from an empirical relationship between channel slope and catchment size, often substantiated in the form of the stream-power model for fluvial incision. Significant methodological progress was recently achieved by introducing the χ transform. It defines a nonlinear length coordinate in such a way that the inherent curvature of river profiles due to the increase of catchment sizes in the downstream direction is removed from the analysis. However, the limitation to large catchment sizes inherited from the stream-power approach for fluvial incision persists. As a consequence, only a small fraction of all nodes of a digital elevation model (DEM) can be used for the analysis. In this study we present and discuss some empirically derived extensions of the stream power law towards small catchment sizes in order to overcome this limitation. Beyond this, we introduce a simple method for estimating the adjustable parameters in the original χ method as well as in our extended approaches. As a main result, an approach originally suggested for debris flow channels seems to be the best approximation if both large and small catchment sizes are included in the same analysis.

scholarly journals Constraining the Stream Power Law: a novel approach combining a Landscape Evolution Model and an inversion method

2013 ◽  
Vol 1 (1) ◽  
pp. 891-921
Author(s):  
T. Croissant ◽  
J. Braun
Keyword(s):  
Power Law ◽  
Erosion Rate ◽  
Landscape Evolution ◽  
Evolution Model ◽  
Drainage Area ◽  
Inversion Method ◽  
Stream Power ◽  
Time Step ◽  
Novel Approach ◽  
River Profiles

Abstract. In the past few decades, many studies have been dedicated to our understanding of the interactions between tectonic and erosion and, in many instances, using numerical models of landscape evolution. Among the numerous parameterizations that have been developed to predict river channel evolution, the Stream Power Law, which links erosion rate to drainage area and slope, remains the most widely used. Despite its simple formulation, its power lies in its capacity to reproduce many of the characteristic features of natural systems (the concavity of river profile, the propagation of knickpoints, etc.). However, the three main coefficients that are needed to relate erosion rate to slope and drainage area in the Stream Power Law remain poorly constrained. In this study, we present a novel approach to constrain the Stream Power Law coefficients under the detachment limited mode by combining a highly efficient Landscape Evolution Model, FastScape, which solves the Stream Power Law under arbitrary geometries and boundary conditions and an inversion algorithm, the Neighborhood Algorithm. A misfit function is built by comparing topographic data of a reference landscape supposedly at steady state and the same landscape subject to both uplift and erosion over one time step. By applying the method to a synthetic landscape, we show that different landscape characteristics can be retrieved, such as the concavity of river profiles and the steepness index. When applied on a real catchment (in the Whataroa region of the South Island in New Zealand), this approach provide well resolved constraints on the concavity of river profiles and the distribution of uplift as a function of distance to the Alpine Fault, the main active structure in the area.

scholarly journals Tectonic geomorphology at small catchment sizes – extensions of the stream-power approach and the χ method

2015 ◽  
Vol 3 (3) ◽  
pp. 689-714 ◽  
Author(s):  
S. Hergarten ◽  
J. Robl ◽  
K. Stüwe
Keyword(s):  
Empirical Relationship ◽  
Tectonic Geomorphology ◽  
Stream Power ◽  
Simple Method ◽  
Small Catchment ◽  
The Best Approximation ◽  
Fluvial Incision ◽  
Catchment Size ◽  
Almost All ◽  
River Profiles

Abstract. Quantitative tectonic geomorphology hinges on the analysis of longitudinal river profiles. The model behind almost all approaches in this field originates from an empirical relationship between channel slope and catchment size, often substantiated in form of the stream-power model for fluvial incision. A significant methodological progress was recently achieved by introducing the χ transform. It defines a nonlinear length coordinate in such a way that the inherent curvature of river profiles due to the increase of catchment sizes in downstream direction is removed from the analysis. However, the limitation to large catchment sizes inherited from the stream power approach for fluvial incision persists. As a consequence, only a small fraction of all nodes of a DEM can be used for the analysis. In this study we present and discuss some empirically derived extensions of the stream power law towards small catchment sizes in order to overcome this limitation. Beyond this, we introduce a simple method for estimating the adjustable parameters in the original χ method as well as in our extended approaches. As a main result, an approach originally suggested for debris flow channels seems to be the best approximation if both large and small catchment sizes are included in the same analysis.

scholarly journals Constraining the stream power law: a novel approach combining a landscape evolution model and an inversion method

2014 ◽  
Vol 2 (1) ◽  
pp. 155-166 ◽  
Author(s):  
T. Croissant ◽  
J. Braun
Keyword(s):  
Power Law ◽  
Erosion Rate ◽  
Landscape Evolution ◽  
Evolution Model ◽  
Drainage Area ◽  
Inversion Method ◽  
Stream Power ◽  
Time Step ◽  
Novel Approach ◽  
River Profiles

Abstract. In the past few decades, many studies have been dedicated to the understanding of the interactions between tectonics and erosion, in many instances through the use of numerical models of landscape evolution. Among the numerous parameterizations that have been developed to predict river channel evolution, the stream power law, which links erosion rate to drainage area and slope, remains the most widely used. Despite its simple formulation, its power lies in its capacity to reproduce many of the characteristic features of natural systems (the concavity of river profile, the propagation of knickpoints, etc.). However, the three main coefficients that are needed to relate erosion rate to slope and drainage area in the stream power law remain poorly constrained. In this study, we present a novel approach to constrain the stream power law coefficients under the detachment-limited mode by combining a highly efficient landscape evolution model, FastScape, which solves the stream power law under arbitrary geometries and boundary conditions and an inversion algorithm, the neighborhood algorithm. A misfit function is built by comparing topographic data of a reference landscape supposedly at steady state and the same landscape subject to both uplift and erosion over one time step. By applying the method to a synthetic landscape, we show that different landscape characteristics can be retrieved, such as the concavity of river profiles and the steepness index. When applied on a real catchment (in the Whataroa region of the South Island in New Zealand), this approach provides well-resolved constraints on the concavity of river profiles and the distribution of uplift as a function of distance to the Alpine Fault, the main active structure in the area.

River profiles from digital elevation models – limitations and new ideas

2020 ◽  
Author(s):  
Stefan Hergarten ◽  
Jörg Robl
Keyword(s):  
Tectonic Geomorphology ◽  
Limiting Factor ◽  
Mesh Width ◽  
Tectonic History ◽  
Digital Elevation ◽  
New Ideas ◽  
Channel Slope ◽  
River Profiles ◽  
Made In

&lt;p&gt;Longitudinal river profiles have been a central if not even the most important subject in tectonic geomorphology since the 1950s. During the last decades, considerable progress has been made in unraveling the tectonic history from river profiles. Going along with the rapidly increasing availability of DEMs, however, scientists try to derive more and more information from the topography. So the quality of the DEM is still a limiting factor in many studies. In particular, local channel slopes are strongly affected by the DEM. Several approaches have been proposed in order to reduce the errors and to distinguish specific features such as knickpoints from noise of the DEM.&lt;/p&gt;&lt;p&gt;In this study we use DEMs with a mesh width of 1 m obtained from airborne laser scans and reduce their resolution artificially in order to analyze the effect of the mesh width on the accuracy of river profiles systematically. Based on the results, we present an idea how the errors in channel slope could be reduced with focus on narrow valleys. Going beyond the majority of the previously published approaches, our idea does not only take into account the elevation along the river profile, but also the curvature of the topography in direction normal to the valley floor.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;

Evaluating the effect of variable lithologies on rates of knickpoint migration in the Wutach catchment, southern Germany

2020 ◽  
Author(s):  
Andreas Ludwig ◽  
Wolfgang Schwanghart ◽  
Florian Kober ◽  
Angela Landgraf
Keyword(s):  
Transient Response ◽  
Stream Power ◽  
Main Trunk ◽  
Southern Germany ◽  
Level Change ◽  
Base Level ◽  
Bulk Parameter ◽  
River Profiles ◽  
Topographic Evolution ◽  
Headward Erosion

&lt;p&gt;The topographic evolution of landscapes strongly depends on the resistance of bedrock to erosion. Detachment-limited fluvial landscapes are commonly analyzed and modelled with the stream power incision model (SPIM) which parametrizes erosional efficiency by the bulk parameter K whose value is largely determined by bedrock erodibility. Inversion of the SPIM using longitudinal river profiles enables resolving values of K if histories of rock-uplift or base level change are known. Here, we present an approach to estimate K-values for the Wutach catchment, southern Germany. The catchment is a prominent example of river piracy that occurred ~18 ka ago as response to headward erosion of a tributary to the Rhine. Base level fall of up to 170 m triggered a wave of upstream migrating knickpoints that represent markers for the transient response of the landscape. Knickpoint migration along the main trunk stream and its tributaries passed different lithological settings, which allows us to estimate K for crystalline and sedimentary bedrock units of variable erodibility.&lt;/p&gt;

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