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.

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 Short communication: Analytical models for 2D landscape evolution

2021 ◽  
Author(s):  
Philippe Steer
Keyword(s):  
Directed Acyclic Graph ◽  
Landscape Evolution ◽  
Numerical Models ◽  
Analytical Models ◽  
Uplift Rate ◽  
Stream Power ◽  
Time Step ◽  
Power Equation ◽  
Novel Approach ◽  
Unique Approach

Abstract. Numerical modelling offers a unique approach to understand how tectonics, climate and surface processes govern landscape dynamics. However, the efficiency and accuracy of current landscape evolution models remain a certain limitation. Here, I develop a new modelling strategy that relies on the use of 1D analytical solutions to the linear stream power equation to compute in 2D the dynamics of landscapes. This strategy uses the 1D ordering, by a directed acyclic graph, of model nodes based on their location along the water flow path to propagate topographic changes in 2D. I demonstrate that this analytical model can be used to compute in a single time step, with an iterative procedure, the steady-state topography of landscapes subjected to river, colluvial and hillslope erosion. This model can also be adapted to compute the dynamic evolution of landscapes under either heterogeneous or time-variable uplift rate. This new model leads to slope-area relationships exactly consistent with predictions and to the exact preservation of knickpoint shape throughout their migration. Moreover, the absence of numerical diffusion or of an upper bound for the time step offer significant advantages compared to numerical models. The main drawback of this novel approach is that it does not guarantee the time-continuity of the topography through successive time steps, despite practically having little impact on model behaviour.

scholarly journals Short communication: Analytical models for 2D landscape evolution

2021 ◽  
Vol 9 (5) ◽  
pp. 1239-1250
Author(s):  
Philippe Steer
Keyword(s):  
Directed Acyclic Graph ◽  
Landscape Evolution ◽  
Numerical Models ◽  
Analytical Models ◽  
Uplift Rate ◽  
Stream Power ◽  
Time Step ◽  
Power Equation ◽  
Novel Approach ◽  
Unique Approach

Abstract. Numerical modelling offers a unique approach to understand how tectonics, climate and surface processes govern landscape dynamics. However, the efficiency and accuracy of current landscape evolution models remain a certain limitation. Here, I develop a new modelling strategy that relies on the use of 1D analytical solutions to the linear stream power equation to compute the dynamics of landscapes in 2D. This strategy uses the 1D ordering, by a directed acyclic graph, of model nodes based on their location along the water flow path to propagate topographic changes in 2D. This analytical model can be used to compute in a single time step, with an iterative procedure, the steady-state topography of landscapes subjected to river, colluvial and hillslope erosion. This model can also be adapted to compute the dynamic evolution of landscapes under either heterogeneous or time-variable uplift rate. This new model leads to slope–area relationships exactly consistent with predictions and to the exact preservation of knickpoint shape throughout their migration. Moreover, the absence of numerical diffusion or of an upper bound for the time step offers significant advantages compared to numerical models. The main drawback of this novel approach is that it does not guarantee the time continuity of the topography through successive time steps, despite practically having little impact on model behaviour.

scholarly journals Landscape evolution models using the stream power incision model show unrealistic behavior when <i>m</i> ∕ <i>n</i> equals 0.5

2017 ◽  
Vol 5 (4) ◽  
pp. 807-820 ◽  
Author(s):  
Jeffrey S. Kwang ◽  
Gary Parker
Keyword(s):  
Steady State ◽  
Landscape Evolution ◽  
Drainage Area ◽  
Uplift Rate ◽  
Stream Power ◽  
River Incision ◽  
Vertical Incision ◽  
Small Scales ◽  
Insight Into

Abstract. Landscape evolution models often utilize the stream power incision model to simulate river incision: E = KAmSn, where E is the vertical incision rate, K is the erodibility constant, A is the upstream drainage area, S is the channel gradient, and m and n are exponents. This simple but useful law has been employed with an imposed rock uplift rate to gain insight into steady-state landscapes. The most common choice of exponents satisfies m ∕ n = 0.5. Yet all models have limitations. Here, we show that when hillslope diffusion (which operates only on small scales) is neglected, the choice m ∕ n = 0.5 yields a curiously unrealistic result: the predicted landscape is invariant to horizontal stretching. That is, the steady-state landscape for a 10 km2 horizontal domain can be stretched so that it is identical to the corresponding landscape for a 1000 km2 domain.

scholarly journals Scaling and similarity of a stream-power incision and linear diffusion landscape evolution model

2018 ◽  
Vol 6 (3) ◽  
pp. 779-808 ◽  
Author(s):  
Nikos Theodoratos ◽  
Hansjörg Seybold ◽  
James W. Kirchner
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Degrees Of Freedom ◽  
Landscape Evolution ◽  
Initial Conditions ◽  
Evolution Model ◽  
Model Parameters ◽  
Stream Power ◽  
Linear Diffusion ◽  
Characteristic Scales

Abstract. The scaling and similarity of fluvial landscapes can reveal fundamental aspects of the physics driving their evolution. Here, we perform a dimensional analysis of the governing equation of a widely used landscape evolution model (LEM) that combines stream-power incision and linear diffusion laws. Our analysis assumes that length and height are conceptually distinct dimensions and uses characteristic scales that depend only on the model parameters (incision coefficient, diffusion coefficient, and uplift rate) rather than on the size of the domain or of landscape features. We use previously defined characteristic scales of length, height, and time, but, for the first time, we combine all three in a single analysis. Using these characteristic scales, we non-dimensionalize the LEM such that it includes only dimensionless variables and no parameters. This significantly simplifies the LEM by removing all parameter-related degrees of freedom. The only remaining degrees of freedom are in the boundary and initial conditions. Thus, for any given set of dimensionless boundary and initial conditions, all simulations, regardless of parameters, are just rescaled copies of each other, both in steady state and throughout their evolution. Therefore, the entire model parameter space can be explored by temporally and spatially rescaling a single simulation. This is orders of magnitude faster than performing multiple simulations to span multidimensional parameter spaces. The characteristic scales of length, height and time are geomorphologically interpretable; they define relationships between topography and the relative strengths of landscape-forming processes. The characteristic height scale specifies how drainage areas and slopes must be related to curvatures for a landscape to be in steady state and leads to methods for defining valleys, estimating model parameters, and testing whether real topography follows the LEM. The characteristic length scale is roughly equal to the scale of the transition from diffusion-dominated to advection-dominated propagation of topographic perturbations (e.g., knickpoints). We introduce a modified definition of the landscape Péclet number, which quantifies the relative influence of advective versus diffusive propagation of perturbations. Our Péclet number definition can account for the scaling of basin length with basin area, which depends on topographic convergence versus divergence.

scholarly journals Divide mobility controls knickpoint migration on the Roan Plateau (Colorado, USA)

Geology ◽  
2020 ◽  
Vol 48 (7) ◽  
pp. 698-702 ◽  
Author(s):  
Wolfgang Schwanghart ◽  
Dirk Scherler
Keyword(s):  
Colorado River ◽  
Drainage Area ◽  
Lagrangian Model ◽  
Stream Power ◽  
Migration Rates ◽  
Tectonic History ◽  
Area Loss ◽  
Temporal Aspects ◽  
History Of ◽  
River Profiles

Abstract Knickpoints in longitudinal river profiles are proxies for the climatic and tectonic history of active mountains. The analysis of river profiles commonly relies on the assumption that drainage network configurations are stable. Here, we show that this assumption must be made cautiously if changes in contributing area are fast relative to knickpoint migration rates. We studied the Parachute Creek basin in the Roan Plateau, Colorado, United States, where knickpoint retreat occurs in horizontally uniform lithology so that drainage area is the sole governing variable. In this basin, we identified an anomalous catchment in the degree to which a stream power–based model predicted knickpoint locations. The catchment is experiencing area loss as the plateau edge is eroded by cliff migration in proximity to the Colorado River. Model predictions improve if the plateau edge is assumed to have migrated over the time scale of knickpoint retreat. Finally, a Lagrangian model of knickpoint migration enabled us to study the kinematic links between drainage area loss and knickpoint migration and offered constraints on the temporal aspects of area loss. Modeled onset and amount of area loss are consistent with cliff retreat rates along the margin of the Roan Plateau inferred from the incisional history of the upper Colorado River.

3D lithological structure in a steady state model drives divide migration

2021 ◽  
Author(s):  
Emma Graf ◽  
Simon Mudd ◽  
Florian Kober ◽  
Angela Landgraf ◽  
Andreas Ludwig
Keyword(s):  
Steady State ◽  
Landscape Evolution ◽  
Geophysical Methods ◽  
Drainage Area ◽  
Stream Power ◽  
K Value ◽  
Model Simulations ◽  
Rock Structure ◽  
The Difference ◽  
Future Landscape

&lt;p&gt;Predicting future relief is a longstanding challenge in the field of geomorphology. Past denudation and incision rates can be reconstructed and modelled from field data such as thermochronometers, cosmogenic nuclides or optically stimulated luminescence, whereas future rates are then, by definition, fully unknown. Predicting future landscape evolution is further complicated by the dynamic nature of drainage networks, as well as the necessity of constraining properties such as erodibility in order to make sensible predictions. One of the few constraints available for future landscape properties is the underground stratigraphy imaged by wells or geophysical methods. The 3D rock structure will eventually be exhumed and can be utilised to constrain the future states of model simulations.&lt;/p&gt;&lt;p&gt;In this contribution, we present a landscape evolution model capable of ingesting 3D lithologic information and adapting to alternative channel networks, and demonstrate it using a study area in the Swiss Jura Mountains. The model calculates local relief using steady state solutions of the stream power incision model, and also quantifies hillslope relief using a very simple critical slope gradient where hillslope angles are set to a critical value on pixels that have a small drainage area. Further, drainage divides are allowed to migrate to minimize sharp breaks in relief across drainage divides.&lt;/p&gt;&lt;p&gt;We calibrate the values of erodibility, K, for each lithological unit by extracting ranges of apparent K value from the present-day landscape based on drainage area and gradient along the drainage network. This is then further refined by i) using a Monte Carlo approach to create combinations of K based on these ranges, and ii) comparing the real and model landscape for each combination with the aim to minimise the difference between the two. We then run selected model simulations of future base level fall and potential drainage reorganisation events, highlighting the effects of i.) spatially variable erodibility and ii.) lateral changes of the main channel axis on divide migration.&lt;/p&gt;

scholarly journals Bedload transport controls bedrock erosion under sediment-starved conditions

2015 ◽  
Vol 3 (3) ◽  
pp. 291-309 ◽  
Author(s):  
A. R. Beer ◽  
J. M. Turowski
Keyword(s):  
Erosion Rate ◽  
Landscape Evolution ◽  
Bedload Transport ◽  
Minor Importance ◽  
Stream Power ◽  
Catchment Scale ◽  
Data Set ◽  
Erosion Processes ◽  
Bedrock Erosion ◽  
Mountainous Landscape

Abstract. Fluvial bedrock incision constrains the pace of mountainous landscape evolution. Bedrock erosion processes have been described with incision models that are widely applied in river-reach and catchment-scale studies. However, so far no linked field data set at the process scale had been published that permits the assessment of model plausibility and accuracy. Here, we evaluate the predictive power of various incision models using independent data on hydraulics, bedload transport and erosion recorded on an artificial bedrock slab installed in a steep bedrock stream section for a single bedload transport event. The influence of transported bedload on the erosion rate (the "tools effect") is shown to be dominant, while other sediment effects are of minor importance. Hence, a simple temporally distributed incision model, in which erosion rate is proportional to bedload transport rate, is proposed for transient local studies under detachment-limited conditions. This model can be site-calibrated with temporally lumped bedload and erosion data and its applicability can be assessed by visual inspection of the study site. For the event at hand, basic discharge-based models, such as derivatives of the stream power model family, are adequate to reproduce the overall trend of the observed erosion rate. This may be relevant for long-term studies of landscape evolution without specific interest in transient local behavior. However, it remains to be seen whether the same model calibration can reliably predict erosion in future events.

scholarly journals eSCAPE: Regional to Global Scale Landscape Evolution Model v2.0

2019 ◽  
Author(s):  
Tristan Salles
Keyword(s):  
Landscape Evolution ◽  
Iterative Algorithms ◽  
Sedimentary Basins ◽  
Global Scale ◽  
Evolution Model ◽  
Mathematical Representation ◽  
Stream Power ◽  
Geological Time ◽  
Surface Evolution ◽  
Source To Sink

Abstract. eSCAPE is a Python-based landscape evolution model that simulates over geological time (1) the dynamic of the landscape, (2) the transport of sediment from source to sink, and (3) continental and marine sedimentary basins formation under different climatic and tectonic conditions. eSCAPE is open-source, cross-platform, distributed under the GPLv3 license and available on GitHub (escape-model.github.io). Simulated processes rely on a simplified mathematical representation of landscape processes – the stream power and creep laws – to compute Earth's surface evolution by rivers and hillslope transport. The main difference with previous models is in the underlying numerical formulation of the mathematical equations. The approach is based on a series of implicit iterative algorithms defined in matrix form to calculate both drainage area from multiple flow directions and erosion/deposition processes. eSCAPE relies on PETSc parallel library to solve these matrix systems. Along with the description of the algorithms, examples are provided and illustrate the model current capabilities and limitations. eSCAPE is the first landscape evolution model able to simulate processes at global scale and is primarily designed to address problems on large unstructured grids (several millions of nodes).

scholarly journals River profile response to normal fault growth and linkage: an example from the Hellenic forearc of south-central Crete, Greece

2017 ◽  
Vol 5 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Sean F. Gallen ◽  
Karl W. Wegmann
Keyword(s):  
Profile Analysis ◽  
Normal Fault ◽  
Drainage Area ◽  
Uplift Rate ◽  
Stream Power ◽  
River Incision ◽  
South Central ◽  
History Of ◽  
River Profiles ◽  
Fault Growth

Abstract. Topography is a reflection of the tectonic and geodynamic processes that act to uplift the Earth's surface and the erosional processes that work to return it to base level. Numerous studies have shown that topography is a sensitive recorder of tectonic signals. A quasi-physical understanding of the relationship between river incision and rock uplift has made the analysis of fluvial topography a popular technique for deciphering relative, and some argue absolute, histories of rock uplift. Here we present results from a study of the fluvial topography from south-central Crete, demonstrating that river longitudinal profiles indeed record the relative history of uplift, but several other processes make it difficult to recover quantitative uplift histories. Prior research demonstrates that the south-central coastline of Crete is bound by a large ( ∼  100 km long) E–W striking composite normal fault system. Marine terraces reveal that it is uplifting between 0.1 and 1.0 mm yr−1. These studies suggest that two normal fault systems, the offshore Ptolemy and onshore South-Central Crete faults, linked together in the recent geologic past (ca. 0.4–1 My BP). Fault mechanics predict that when adjacent faults link into a single fault the uplift rate in footwalls of the linkage zone will increase rapidly. We use this natural experiment to assess the response of river profiles to a temporal jump in uplift rate and to assess the applicability of the stream power incision model to this setting. Using river profile analysis we show that rivers in south-central Crete record the relative uplift history of fault growth and linkage as theory predicts that they should. Calibration of the commonly used stream power incision model shows that the slope exponent, n, is  ∼  0.5, contrary to most studies that find n  ≥  1. Analysis of fluvial knickpoints shows that migration distances are not proportional to upstream contributing drainage area, as predicted by the stream power incision model. Maps of the transformed stream distance variable, χ, indicate that drainage basin instability, drainage divide migration, and river capture events complicate river profile analysis in south-central Crete. Waterfalls are observed in southern Crete and appear to operate under less efficient and different incision mechanics than assumed by the stream power incision model. Drainage area exchange and waterfall formation are argued to obscure linkages between empirically derived metrics and quasi-physical descriptions of river incision, making it difficult to quantitatively interpret rock uplift histories from river profiles in this setting. Karst hydrology, break down of assumed drainage area discharge scaling, and chemical weathering might also contribute to the failure of the stream power incision model to adequately predict the behavior of the fluvial system in south-central Crete.

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