Bridging Pointillism and Realism: Convergence of Discrete Particle-based and Continuum Mechanics-based Models of Hillslope Evolution

Abstract:

A two-dimensional, discrete particle-based, hillslope-valley cross-section model incorporates stochastic processes for physical weathering, diffusion-like transport, base-level lowering, rock failure, and the production and decay of $^{10}$Be. Each process comprises a series of Poisson arrivals, in continuous time, of binomial processes acting on discrete particles composing a rectangular grid, in which each grid cell contains bedrock, soil, rocky debris, or air. These spatially discrete processes have continuum analogs, i.e., geomorphic transport laws (GTLs), which can therefore be expressed in terms of parameters of the discrete model. The latter is thus calibrated, the model's pixellated, pointillist-like rendering comes into focus, and model predictions can be tested in the real world, specifically the hillslopes of the Oregon Coast Range. Given a random-walk model of diffusive transport, analytical expressions for the flux law and hillslope profile are derived from the expected value of transport distance as a function of slope gradient. The predicted flux law and slope profile are identical to those predicted by the nonlinear diffusive transport law from the literature when the critical slope parameter is unity. Whereas the analytical expression predicts diverging (i.e., very large) flux rates and nearly planar slopes at gradients that are smaller than observed, the model simulates diverging flux and planar slopes at steeper gradients, similar to those corresponding to critical slopes greater than one, e.g., values similar to 1.25 (see figure). Sampling of simulation events reveals that those steeper gradients arise from ``trapping" of particles in micro-topographic pits. The discrete particle-based model therefore recapitulates the nonlinear diffusive transport law previously derived and demonstrated by others and, moreover, predicts that ``extra-steep" slopes result from emergent micro-topography.