Scale Invariance in Landscape Evolution Models Using Stream Power Laws

Abstract:

Landscape evolution models (LEM) commonly utilize stream power laws to simulate river incision with formulations such as E = KA^mSⁿ, where E is a vertical incision rate [L/T], K is an erodibility constant [L^1-2m/T], A is an upstream drainage area [L²], S is a local channel gradient [-], and m and n are positive exponents that describe the basin hydrology. In our reduced complexity model, the landscape approached equilibrium by balancing an incision rate with a constant, uniform, vertical rock uplift rate at every location in the landscape. From our simulations, for a combination of m and n, the landscape exhibited scale invariance. That is, regardless of the size and scale of the basin, the relief and vertical structure of the landscape remained constant. Therefore, the relief and elevation profile of the landscape at equilibrium were only dependent on the coefficients for erodibility and uplift and an equation that described how upstream area, A, increased as the length of a stream increased. In our analytical 1D models, we utilized two equations that described upslope area, (a) A = Bl, where B is the profile width [L], and l is the stream length from the ridge [L] and (b) A = Cl^h, Hack’s Law, where C is a constant [L^2-h] and h is a positive exponent. With these equations, (a) m = n and (b) hm = n resulted in scale invariance. In our numerical 2D models, the relationship between A and l was inherent in the actual structure of the drainage network. From our numerical 2D results, scale invariance occurred when 2m = n. Additionally, using reasonable values from the literature for exponents, n, m and h, resulted in singularities at the ridges in the landscape, which caused truncation error. In consequence, the elevation of the ridge increased as the number of grid cells in the domain increased in the numerical model, and the model was unable to converge. These singularities at the ridges appeared when (a) m ≥ n and (b) hm ≥ n in the analytical model and 2m ≥ n in the numerical model. Here we present (1) 1D analytical solutions and (2) 2D numerical solutions that demonstrate scale invariance in LEMs and (3) the consequences of the singularity in 2D LEM numerical simulations. These results will help provide insight about the structure and dynamics of landscapes and drainage networks and shed light on geomorphological empirical relationships.