scholarly journals Dimensional analysis of a landscape evolution model with incision threshold

2020 ◽  
Vol 8 (2) ◽  
pp. 505-526
Author(s):  
Nikos Theodoratos ◽  
James W. Kirchner
Keyword(s):  
Dimensional Analysis ◽  
Governing Equation ◽  
Dimensionless Parameter ◽  
Landscape Evolution ◽  
Initial Conditions ◽  
Uplift Rate ◽  
Stream Power ◽  
Linear Diffusion ◽  
Scaling Properties

Abstract. The ability of erosional processes to incise into a topographic surface can be limited by a threshold. Incision thresholds affect the topography of landscapes and their scaling properties and can introduce nonlinear relations between climate and erosion with notable implications for long-term landscape evolution. Despite their potential importance, incision thresholds are often omitted from the incision terms of landscape evolution models (LEMs) to simplify analyses. Here, we present theoretical and numerical results from a dimensional analysis of an LEM that includes terms for threshold-limited stream-power incision, linear diffusion, and uplift. The LEM is parameterized by four parameters (incision coefficient and incision threshold, diffusion coefficient, and uplift rate). The LEM's governing equation can be greatly simplified by recasting it in a dimensionless form that depends on only one dimensionless parameter, the incision-threshold number Nθ. This dimensionless parameter is defined in terms of the incision threshold, the incision coefficient, and the uplift rate, and it quantifies the reduction in the rate of incision due to the incision threshold relative to the uplift rate. Being the only parameter in the dimensionless governing equation, Nθ is the only parameter controlling the evolution of landscapes in this LEM. Thus, landscapes with the same Nθ will evolve geometrically similarly, provided that their boundary and initial conditions are normalized according to appropriate scaling relationships, as we demonstrate using a numerical experiment. In contrast, landscapes with different Nθ values will be influenced to different degrees by their incision thresholds. Using results from a second set of numerical simulations, each with a different incision-threshold number, we qualitatively illustrate how the value of Nθ influences the topography, and we show that relief scales with the quantity Nθ+1 (except where the incision threshold reduces the rate of incision to zero).

scholarly journals Dimensional analysis of a landscape evolution model with incision threshold

2020 ◽  
Author(s):  
Nikos Theodoratos ◽  
James W. Kirchner
Keyword(s):  
Dimensional Analysis ◽  
Governing Equation ◽  
Dimensionless Parameter ◽  
Landscape Evolution ◽  
Initial Conditions ◽  
Uplift Rate ◽  
Stream Power ◽  
Linear Diffusion ◽  
Scaling Properties

Abstract. The ability of erosional processes to incise into a topographic surface can be limited by a threshold. Incision thresholds affect the topography of landscapes and their scaling properties, and can introduce non-linear relations between climate and erosion with notable implications for long-term landscape evolution. Despite their potential importance, incision thresholds are often omitted from the incision terms of landscape evolution models (LEMs) to simplify analyses. Here, we present theoretical and numerical results from a dimensional analysis of an LEM that includes terms for threshold-limited stream-power incision, linear diffusion, and uplift. The LEM is parameterized by four parameters (incision coefficient and incision threshold, diffusion coefficient, and uplift rate). The LEM's governing equation can be greatly simplified by recasting it in a dimensionless form that depends on only one dimensionless parameter, the incision-threshold number Nθ. This dimensionless parameter is defined in terms of the incision threshold, the incision coefficient, and the uplift rate, and it quantifies the reduction in the rate of incision due to the incision threshold relative to the uplift rate. Being the only parameter in the dimensionless governing equation, Nθ is the only parameter controlling the evolution of landscapes in this LEM. Thus, landscapes with the same Nθ will evolve geometrically similarly, provided that their boundary and initial conditions are normalized according to appropriate scaling relationships, as we demonstrate using a numerical experiment. In contrast, landscapes with different Nθ values will be influenced to different degrees by their incision thresholds. Using results from a second set of numerical simulations, each with a different incision-threshold number, we qualitatively illustrate how the value of Nθ influences the topography, and we show that relief scales with the quantity Nθ + 1 (except where the incision threshold reduces the rate of incision to zero).

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 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.

The interaction between uplift and landscape evolution in central and northern Qilian Shan: Insights from numerical modeling

2020 ◽  
Author(s):  
Yifei Li ◽  
Huai Zhang ◽  
Zhen Zhang
Keyword(s):  
Tibetan Plateau ◽  
Landscape Evolution ◽  
Great Influence ◽  
Arid Climate ◽  
Structural Deformation ◽  
The Tibetan Plateau ◽  
Lateral Growth ◽  
Uplift Rate ◽  
Fault Propagation

&lt;p&gt;The Qilian Shan, located in the northeastern margin of the Tibetan Plateau, is characterized by intensive Cenozoic structural deformation with rapid lateral growth due to the continuous Indo-Asian continental collision. Both low-temperature thermochronological dating and geological mapping suggest that the major emergence of Cenozoic Qilian Shan occurred in the Miocene. The central and northern Qilian Shan uplift successively, and deformation has passed away from the adjacent Hexi Corridor Basin into the Gobi-Alashan. The regional landform shows a low-relief surface in the Qilian Shan hinterland and high steep relief in the northern range front.&lt;/p&gt;&lt;p&gt;The rivers rising in the hinterland of the Qilian Shan, i.e., the Shule River (SL), Beda River (BD), and Hei River (HE), are flowing across the northern range front. It is noteworthy that the development of these rivers is within the context of the in-sequence fault propagation pattern with the lifespan of ~3 Ma. When combined with the differential topographies between hinterland and range front, this kind of river drainage pattern inevitably has abundant geodynamical significances, mainly in terms of the long-term coupling between tectonic and surficial processes. To date, the dynamic conditions in shaping the aforementioned tectonic landscape features remain unknown and are critical in revealing the lateral growth of the NE Tibetan Plateau. A series of landscape evolution models are conducted based on thick-skinned Qilian Shan structural wedge. The wavelength of mountains is constrained by the critical wedge theory.&lt;/p&gt;&lt;p&gt;Our results show that the in-sequence fault propagation together with the arid climate since the Miocene contributes to the low-relief topography in the hinterland of Qilian Shan. The front regions with rapid uplifting rates cut off rivers. Thus, sediments from the hinterlands cannot be directly carried out by rivers. The intermountain areas receive sediments from the adjacent uplift regions, resulting in an increased elevation. Because of the long-term average arid climate, the river incision is limited. For most areas, it is difficult to form transversal rivers immediately that cut through mountains and carry sediment out of the plateau. With the northeastward in-sequence fault propagation, the transversal rivers finally formed with headwaters within the hinterland of Qilian Shan, such as the SL, BD and HE. The broad consistency of landforms, in turn, strongly favors the geological conclusion that faults in the central and northern Qilian Shan were activated sequentially. The rapid uplift rate in the active range front is tested in the range of 0.6-1.0 mm/a. It is found that this rate is insensitivity to the drainage and landscape evolution pattern. However, the background uplift rate has a great influence on the elevation of the plateau and is positively correlated. The current topography of &gt;4000 m in the hinterland of Qilian Shan is controlled by a background uplift rate of ~0.2mm /a.&lt;/p&gt;

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

2018 ◽  
Author(s):  
Nikos Theodoratos ◽  
Hansjörg Seybold ◽  
James W. Kirchner
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Characteristic Length ◽  
Initial Conditions ◽  
Model Parameters ◽  
Length Scale ◽  
Characteristic Length Scale ◽  
Stream Power ◽  
Linear Diffusion ◽  
Characteristic Scales

Abstract. Scaling and similarity of fluvial landscapes can reveal fundamental aspects of the physics driving their evolution. Here we perform dimensional analysis on 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 a previously defined characteristic length scale, but introduce new characteristic height and time scales. We use these characteristic scales to 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 re-scaled 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 re-scaling a single simulation. This is orders of magnitude faster than performing multiple simulations to span multi-dimensional parameter spaces. The characteristic length, height, and time scales 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.

A graphical method to interpret how incision thresholds influence topographic and scaling properties of landscapes

2020 ◽  
Author(s):  
Nikos Theodoratos ◽  
James W. Kirchner
Keyword(s):  
Graphical Method ◽  
Numerical Models ◽  
Drainage Area ◽  
Stream Power ◽  
Simple Method ◽  
Linear Diffusion ◽  
Scaling Properties ◽  
Curvature Lines ◽  
Straight Lines ◽  
The Relationship

&lt;p&gt;Theoretical analysis of the governing equations of numerical models can reveal relationships between topographic properties, such as drainage area, slope, and curvature, in simulated landscapes. These relationships are testable predictions; they can diagnose whether real-world landscapes could potentially arise from similar mechanisms. For example, the stream-power incision model is consistent with drainage area and slope data that plot as straight lines on logarithmic axes.&lt;/p&gt;&lt;p&gt;Here we graph theoretical relationships between topographic curvature and the steepness index, which depends on drainage area and slope. These relationships plot as straight lines for steady-state landscapes that have evolved according to a model that combines stream-power incision, linear diffusion, and uplift. Further, they link topography (drainage area, slope, and curvature) to characteristic length scales of the landscape, which depend on the competition between the processes of incision, diffusion, and uplift.&lt;/p&gt;&lt;p&gt;Adding an incision threshold to the model changes the relationship between the steepness index and topographic curvature. We examine these changes graphically and we show that they shed light on how incision thresholds influence topographic and scaling properties of landscapes. Specifically, we present a graphical method that consists of plotting steepness index&amp;#8211;curvature lines and of tracing their intersections with each other and with the coordinate axes. This simple method reveals both how topography and process competition are influenced by the incision threshold, and how these influences vary within a given landscape and across different landscapes.&lt;/p&gt;

A morphologically-consistent expression for the transition from stream-power-law regime to a debris-flow regime

2020 ◽  
Author(s):  
Odin Marc ◽  
Hussain Alqattan ◽  
Sean Willett
Keyword(s):  
Steady State ◽  
Debris Flow ◽  
Landscape Evolution ◽  
Drainage Area ◽  
Slope Gradient ◽  
Stream Power ◽  
Denudation Rates ◽  
Fluvial Incision ◽  
Critical Slope

&lt;p&gt;&amp;#160;Many long-term landscape evolution models are currently combining equations describing the evolution of the surface under fluvial incision (using the so-called stream-power incision model) and hillslope transport (often modeled as linear diffusion). Some models combine these two terms (e.g., Fastscape) and implicitly contain a transition from hillslope to fluvial processes dependent on the ratio of the diffusive and fluvial erosional parameters, D and K respectively (Perron et al., 2009). Other models require as input a hillslope-fluvial transition length (e.g., DAC) and apply hillslope erosion from the ridge-top to this lengthscale and fluvial incision only downstream of it. Still, in both cases the influence of non-linear processes such as landslide and debris-flow on this transition are not accounted.&lt;/p&gt;&lt;p&gt;We have analyzed the scaling between slope gradient and drainage areas in LIDAR-derived high-resolution DEM for &gt;30 catchments, with apparent steady-state morphology, and where long-term denudation estimates, E, were estimated from cosmogenic nuclides . The catchments span different lithology, climate and denudation rates from ~0.05 to ~3 mm/yr but show a consistent pattern where substantial portion of upstream channels exhibit slope gradient roughly constant with drainage area, and transition towards a negative scaling between slope and area (characteristic of fluvial processes) after a critical drainage area, A&lt;sub&gt;c.&lt;/sub&gt; Previous work (Stock and Dietrich, 2003) suggested the portion with constant slope may be dominated by erosion due to debris-flow processes, maintaining the channel at a critical slope, S&lt;sub&gt;df&lt;/sub&gt;.&lt;/p&gt;&lt;p&gt;Here we show that both S&lt;sub&gt;df&lt;/sub&gt;, and A&lt;sub&gt;c&lt;/sub&gt;, are strongly correlated to the long-term denudation, E. Further, we find that S&lt;sub&gt;df&lt;/sub&gt; seems to saturate at a critical slope angle, S&lt;sub&gt;c&lt;/sub&gt; , near 40&amp;#176; when denudation rates reach about 1mm/yr consistent with predictions for the slope of a non-linear diffusive hillsllopes (Roering et al., 2007). Combining this expression with the empirical model for the steady-state slope of Stock and Dietrich, 2003, and enforcing the consistency with a stream-power-law downstream we find that the steady state values for S&lt;sub&gt;df&lt;/sub&gt; and A&lt;sub&gt;c&lt;/sub&gt; can be fully expressed as analytical functions of E, K, D and S&lt;sub&gt;c&lt;/sub&gt;. We assess the validity of these expressions with independent estimate of K and D extracted from local channel steepness and hilltop curvature.&amp;#160;&lt;/p&gt;&lt;p&gt;As the impact of debris flow on landscape morphology seems ubiquitous on landscape with more than 0.1 mm/yr of erosion, the classical landscape evolution formulation may need to be upgraded to correctly represent steady-state morphology of the upstream part of catchment (&lt;span&gt;i.e.&lt;/span&gt;, &lt;1km&lt;sup&gt;2&lt;/sup&gt;). Even if it still lack physical basis, we propose a formulation that adequately represent the steady state morphology from ridge to large drainage area. We show that it yield a new definition of Chi that may be better match the morphology of channel approaching ridges and we also discuss how to implement this new-steady state formulation in landscape evolution models.&lt;/p&gt;

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

2017 ◽  
Author(s):  
Jeffrey S. Kwang ◽  
Gary Parker
Keyword(s):  
Steady State ◽  
Landscape Evolution ◽  
Drainage Area ◽  
Uplift Rate ◽  
Stream Power ◽  
Channel Network ◽  
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 = vertical incision rate, K = erodibility constant, A =  upstream drainage area, S = 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; indeed, this ratio has been deemed to yield the “optimal channel network.” Yet all models have limitations. Here, we show that when hillslope diffusion (which operates only at 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 1 m2 horizontal domain can be stretched so that it is identical to the corresponding landscape for a 100 km2 domain.

scholarly journals Graphically interpreting how incision thresholds influence topographic and scaling properties of modeled landscapes

2020 ◽  
Author(s):  
Nikos Theodoratos ◽  
James W. Kirchner
Keyword(s):  
Peclet Number ◽  
Power Laws ◽  
Dimensionless Number ◽  
Stream Power ◽  
Linear Diffusion ◽  
Scaling Properties ◽  
Péclet Number ◽  
Straight Lines ◽  
And Diffusion ◽  
Characteristic Scales

Abstract. We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM) with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes, but not in landscapes with diffusion. We illustrate the response of these curvature–steepness-index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use, and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.

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