The rate of much natural phenomena goes with the gradient of a variable such as chemical diffusion (concentration), heat conduction (temperature), and slope dependent sediment transport (elevation). These relationships can be characterized by simple differential equations. In this lecture and exercise, we will work to show a simple numerical (approximate solution) to the diffusion erosion problem. Much literature in physics, engineering, math, and geoscience is dedicated to these problems. I reviewed the very basics in an appendix to my dissertation. THis is a useful reference to the content in the lecture
Documentation (Arrowsmith, 1995).
Slope dependent sediment flux (transport rule)
A very simple rule for sediment transport is that the movement rate of material (sediment flux, qs) goes with the local slope (delta elevation over delta distance). We will assume that we are examining a profile of a hillslope over which this is the only way materials move. Soil creep, biogenic processes (burrowing, other animal induced disturbances), rainsplash, meteorite impact, etc. are processes which might drive sediment flux this way.
Basic idea of slope dependent sediment transport emphasizing biogenic driver (Dibiase, 2006)
Slope dependent sediment transport. k is the rate constant.
If we take our hillslope and divide it into small pieces, we can treat each like a little bank. The balance that is in each bank is the difference between what goes in (qs in) and what goes out (qs out). Δqs is simply then the difference between them. And, the change in sediment transport (Δqs) over that very small element (Δx) determines the change in elevation with time Δh / Δx This video tries to explain simply:
Pulling it together
Given the slope dependent transport rule and continuity, we can substitute -kΔh / Δx for Δqs:
Numerical approximations for diffusion and set up in Excel
Quick review on gradients and indices
Before we get started on approximating the diffusion equation, let's remind ourselves about gradients ("slopes"). In the figure below, the left side shows a curve in black which we are trying to approximate. This can be done at any place along the curve. Our first location is at xi (i is called the index) and then we go a short distance Δx to xi+1. If we use just a linear extrapolation, the error is not too bad. On the right side of the figure, we add in a position at xi-1. Now we can try to approximate going forward or backward from xi. Or, more interestingly, we can go from xi-1 to xi+1 and then center at xi, but having gone a distance of 2Δx.
Basic difference equations for implementation
We set up the basic equations on "nodes"--these will ultimately be cells in our Excel spreadsheet). We treat time just as another coordinate (index is l) and march forward, estimating the future elevation as a forward difference from now and we estimate the slope for our transport rate using a centered difference (index is i).
Narration of equations
Calculation nodes detail
Put it all together in Excel. Don't forget to keep it clean following Tufte's inspiration:
Background presentation including analytical solutions
Here is a ppt and pdf that includes the graphics from above as well as a couple of analytical solutions (exact integrations for set initial and boundary conditions) for the diffusion equation.