A very efficient O(n), implicit and parallel method to solve the stream power equation governing fluvial incision and landscape evolution

Geomorphology ◽  
2013 ◽  
Vol 180-181 ◽  
pp. 170-179 ◽  
Author(s):  
Jean Braun ◽  
Sean D. Willett
Keyword(s):  
Landscape Evolution ◽  
Stream Power ◽  
Parallel Method ◽  
Fluvial Incision ◽  
Power Equation

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

<p> 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.</p><p>We have analyzed the scaling between slope gradient and drainage areas in LIDAR-derived high-resolution DEM for >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<sub>c.</sub> 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<sub>df</sub>.</p><p>Here we show that both S<sub>df</sub>, and A<sub>c</sub>, are strongly correlated to the long-term denudation, E. Further, we find that S<sub>df</sub> seems to saturate at a critical slope angle, S<sub>c</sub> , near 40° 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<sub>df</sub> and A<sub>c</sub> can be fully expressed as analytical functions of E, K, D and S<sub>c</sub>. We assess the validity of these expressions with independent estimate of K and D extracted from local channel steepness and hilltop curvature. </p><p>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 (<span>i.e.</span>, <1km<sup>2</sup>). 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.</p>

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 Modeling Short-Term Landscape Modification and Sedimentary Budget Induced by Dam Removal: Insights from LEM Application

2020 ◽  
Vol 10 (21) ◽  
pp. 7697
Author(s):  
Dario Gioia ◽  
Marcello Schiattarella
Keyword(s):  
Landscape Evolution ◽  
Dam Removal ◽  
Sediment Flux ◽  
Short Term ◽  
Open Problems ◽  
Landscape Modification ◽  
Natural Processes ◽  
Base Level ◽  
Geomorphic Response ◽  
Fluvial Incision

Simulation scenarios of sediment flux variation and topographic changes due to dam removal have been investigated in a reservoir catchment of the axial zone of southern Italy through the application of a landscape evolution model (i.e.,: the Caesar–Lisflood landscape evolution models, LEM). LEM simulation highlights that the abrupt change in base level due to dam removal induces a significant increase in erosion ability of main channels and a strong incision of the reservoir infill. Analysis of the sediment dynamics resulting from the dam removal highlights a significant increase of the total eroded volumes in the post dam scenario of a factor higher than 4. Model results also predict a strong modification of the longitudinal profile of main channels, which promoted fluvial incision upstream and downstream of the former reservoir area. Such a geomorphic response is in agreement with previous analysis of the fluvial system short-term response induced by base-level lowering, thus demonstrating the reliability of LEM-based analysis for solving open problems in applied geomorphology such as perturbations and short-term landscape modification natural processes or human impact.

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 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).

Closure to “Applicability of Unit Stream Power Equation”

1978 ◽  
Vol 104 (7) ◽  
pp. 1095-1103
Author(s):  
Chih Ted Yang ◽  
John B. Stall
Keyword(s):  
Stream Power ◽  
Power Equation
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