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Earth Surface Dynamics An interactive open-access journal of the European Geosciences Union
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Volume 6, issue 3 | Copyright
Earth Surf. Dynam., 6, 779-808, 2018
https://doi.org/10.5194/esurf-6-779-2018
© Author(s) 2018. This work is distributed under
the Creative Commons Attribution 4.0 License.

Research article 25 Sep 2018

Research article | 25 Sep 2018

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

Nikos Theodoratos1, Hansjörg Seybold1, and James W. Kirchner1,2 Nikos Theodoratos et al.
  • 1Dept. of Environmental Systems Science, ETH Zurich, Zurich, 8092, Switzerland
  • 2Swiss Federal Research Institute WSL, Birmensdorf, 8903, Switzerland

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.

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We perform dimensional analysis on a frequently used landscape evolution model (LEM). Defining characteristic scales in a novel way, we significantly simplify the LEM and develop an efficient numerical modeling approach. Our characteristic scales are physically meaningful; they quantify competitions between landscape-forming processes and are related to salient properties of landscape topography. Dimensional analyses of other LEMs may benefit from our approach in defining characteristic scales.
We perform dimensional analysis on a frequently used landscape evolution model (LEM). Defining...
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