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Discussion papers
https://doi.org/10.5194/esurf-2019-80
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/esurf-2019-80
© Author(s) 2020. This work is distributed under
the Creative Commons Attribution 4.0 License.

Submitted as: research article 09 Jan 2020

Submitted as: research article | 09 Jan 2020

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This discussion paper is a preprint. It is a manuscript under review for the journal Earth Surface Dynamics (ESurf).

Dimensional analysis of a landscape evolution model with incision threshold

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

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).

Nikos Theodoratos and James W. Kirchner
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Nikos Theodoratos and James W. Kirchner
Nikos Theodoratos and James W. Kirchner
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Short summary
We non-dimensionalized a commonly used model of landscape evolution. Whereas the original model included four parameters, we obtained a dimensionless form with a single parameter. Working with this form saves computational time and simplifies theoretical analyses.
We non-dimensionalized a commonly used model of landscape evolution. Whereas the original model...
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