Abstract
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).
Highlights
In the uppermost parts of drainage basins, water is not flowing over the ground surface or is flowing too weakly to incise into it
We perform a dimensional analysis of an landscape evolution models (LEMs) that includes terms describing stream-power incision, linear diffusion, and uplift (Eq 1)
The LEM assumes that incision is limited by a threshold, that there is no incision at points √with drainage area A and slope |∇z| such that the quantity A|∇z| is below a threshold value √θ and that this threshold reduces incision at points with A|∇z| > θ
Summary
In the uppermost parts of drainage basins, water is not flowing over the ground surface or is flowing too weakly to incise into it. If τ is the shear stress that water exerts on the bed and τθ is a critical value of shear stress (equivalently, τ and τθ could refer to stream power), the rate of incision is zero for τ ≤ τθ , and it can be described by a term of the form k(τ − τθ )α, for τ > τθ , where k and α are constants (e.g., Howard, 1994) Including such incision terms in LEMs changes the topographic properties of the landscapes that are synthesized; for example, it leads to decreased drainage densities, more convex hillslopes, and steeper slopes (e.g., Howard, 1994; Tucker and Bras, 1998; Perron et al, 2008). We hypothesize that these three scales are reasonable choices even after adding an incision threshold to the LEM, and we test this hypothesis by applying these scales and examining the resulting rescaled equations
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