Info on the numerical model...

The numerical model videos each have associated input files. The input parameter files are bit difficult to decipher, but do give numerical values to all of the variables read into the model, in case you're interested in looking at the actual numbers. Note that instead of uplifting every grid node at a time step, I simply dropped the outlet cell a specified amount. Another bit of clutter I added to the model is to allow a variable number of iterations between calculations of a given process. As far as I can tell, varying the iterations between calculations does not enhance lateral migration. Note that grid spacing changes the on screen resolution (I ran some simulations on smaller grids...), but the actual viewed image size is constant between runs. I make no claims as to the plausibility of this model. The point of this modeling is to get some visual idea of landscape stability, with very simplified rules for processes. I think it is useful to start with very simple models, and see how well they do...Whether the model yields a realistic result is a somewhat qualitative endeavor. As Prof. Randall Barnes once quoted in a geostatistics class, "All models are wrong, some are useful..." (after George Box).

The basic erosion law is:

z(i+1) = z(i) + Uplift - StreamEros - Diffusion - Landslide - FlowCurv, where

StreamEros = k1 * Q^m * S^n, Q is flow through a cell (upstream area * rainfall rate), S is slope magnitude

Diffusion = k2 * (local slope in steepest descent direction) [erodes cell, adds erosion to lower cell; not an entirely defensible implementation of diffusion, but I just wanted a purely slope dependent transport condition]

Landslide = k3 * (slope-slope_crit) [no deposition of eroded material...]

FlowCurv = k4 * (1-dot product{flow direction into and out of cell))

The 'k's are constants of proportionality. I included these additional terms because each of them seems like a possible source of lateral migration. The additional terms beyond stream erosion in general increase the time required to reach a stable network, but the landforms still achieve stable forms at long erosional periods. The boundary conditions for the numerical runs mimic those in my physical sandbox. I chose m = 0.25 and n = 1 in the stream erosion law because upstream area vs slope plots from sandbox elevation fields yield m/n ~ 0.25 (actually, they're somewhat lower, ~0.15). Note: The relief in the numerical landform is set by k1(very sensitive to k1!), uplift, and m/n. The most dynamic run is R9, with some very exciting movement happening. However, the total relief developed in this run is only ~2 cm, nearly an order of magnitude less than any of the sandbox runs. In a recent chat (March?, 2001) with Dave Furbish, we discussed the potential destabilizing effect of directional diffusion (increases for a given slope aspect; meant to capture microclimate effects, preference of burrowing critters for southwest facing slopes, or any process that may affect diffusion because of aspect...). Dave is 99% sure asymmetric diffusion will generate self-sustaining migration. I put in a simple diffusive erosion term that increases to a maximum when aligned with a given aspect direction, and decreased to nil at or beyond 90 degrees to that direction. R29 and R30 are a couple of runs where I varied the strength of the directional diffusive coefficient. I don't see any significantly self-sustaining oscillations or lateral migration that is substantially different from the 'stream erosion only' case. Sorry Dave!