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 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.

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 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 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).

scholarly journals Spatio-temporal dynamics of sediment transfer systems in landslide-prone alpine catchments

2019 ◽  
Author(s):  
François Clapuyt ◽  
Veerle Vanacker ◽  
Fritz Schlunegger ◽  
Marcus Christl ◽  
Kristof Van Oost
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Landscape Evolution ◽  
Sediment Budget ◽  
Spatial And Temporal Scales ◽  
Sediment Fluxes ◽  
Downstream Distance ◽  
Denudation Rates ◽  
Decadal Scale

Abstract. Tectonic and geomorphic processes drive landscape evolution over different spatial and temporal scales. In mountainous environments, river incision sets the pace of landscape evolution, and hillslopes respond to channel incision by e.g. gully retreat, bank erosion and landslides. Sediment produced during stochastic landslide events leads to mobilisation of soil and regolith on the slopes that can later be transported by gravity and water to the river network. Quantifying sediment storage and conveyance requires an integrated approach accounting for different space and time scales. To better understand mechanisms and spatial and temporal scales of geomorphic connectivity in mountainous environments, we characterised the sediment cascade of the Entle River catchment located in the foothills of the Central Swiss Alps. We quantified sediment fluxes over annual, decadal and millennial time scales using respectively UAV-SfM techniques, classic photogrammetry and in situ produced cosmogenic radionuclides. At the annual scale (2013–2015), the sediment budget of the Schimbrig earthflow is roughly in equilibrium, despite the fact that we measured intense sediment redistribution on the hillslopes. At the decadal scale (1962–1998), Schwab et al. (2008) reported episodes of sediment export that were not directly related to increased geomorphic activity on the hillslopes. At the millennial scale, catchment-wide denudation rates show a positive relationship with downstream distance or drainage area, when ignoring landslide-affected catchments. The latter are characterised by a negative relationship between denudation rates and downstream distance, along with high variability in denudation rates. The high denudation rates that we measured in the earthflow-affected Schimbrig catchment are illustrative for its high rates of geomorphic activity in comparison to adjacent areas. Our data show that the elevated denudation rates of the landslide-affected catchments are not necessary traceable when analyzing long-term sediment fluxes of the wider geographic area, as the landslide-affected catchments are often only a small fraction of the total catchment. The multi-temporal assessment of sediment fluxes indicates that (1) landslides can provide local sediment pulses, and mobilise material that becomes available for further mobilisation and transport when hillslopes and channels are connected. (2) Connection and disconnection cycles occur at decadal time scale. (3) Phases of high geomorphic activity at the catchment scale are episodic over thousands of years. Consequently, one single landslide has not necessarily an impact on the long-term sediment budget of first-order catchments. Rather, it is the cumulated effect of multiple landslides which are intermittently connected to the channel network at the decadal scale that may regulate sediment fluxes at the regional scale over the millennial time scale.

scholarly journals Modelling soil detachment capacity by rill flow with hydraulic variables on a simulated steep loessial hillslope

2018 ◽  
Vol 50 (1) ◽  
pp. 85-98
Author(s):  
Nan Shen ◽  
Zhanli Wang ◽  
Qingwei Zhang ◽  
Hao Chen ◽  
Bing Wu
Keyword(s):  
Rill Erosion ◽  
Slope Gradient ◽  
Stream Power ◽  
Flow Discharge ◽  
Erosion Models ◽  
Physically Based ◽  
Critical Slope ◽  
Steep Slopes ◽  
Hydraulic Variables ◽  
Rill Flow

Abstract Modelling soil detachment capacity by rill flow with hydraulic variables is essential to understanding the rill erosion process and developing physically based rill erosion models. A rill flume experiment with non-erodible flume bed and small soil samples was conducted. Seven flow discharges and six steep slope gradients were combined to produce various flow hydraulics. The soil detachment capacity increases with the increase in slope gradient and flow discharge. The critical slope gradients of 21.26 and 26.79% cause the detachment capacity to increase at a slow pace. The soil detachment capacity can be defined by a power function of flow discharges and slopes. The contribution rates of slope gradient and flow discharge to soil detachment capacity are 42 and 54%, respectively. The soil detachment capacity increases with shear stress, stream power and unit stream power; the increase rates of these parameters are greater under gentle slopes than steep slopes. Stream power is the superior hydrodynamic parameter describing soil detachment capacity. The linear model equation of stream power is stable and reliable, which can accurately predict soil detachment capacity by rill flow on steep loessial hillslopes. This study can help to sufficiently clarify the dynamic mechanism of soil detachment and accurately predict soil detachment capacity for steep loessial hillslopes.

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).

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