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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).

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 (*q _{s in}*) and what goes out (

Given the slope dependent transport rule and continuity, we can substitute *-kΔh / Δx* for Δq_{s}:

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 *x _{i}* (

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*).

Put it all together in Excel. Don't forget to keep it clean following Tufte's inspiration:

- Implement the finite difference model for diffusion erosion in a spreadsheet. You should use one tab that is easy to read and interact with ("Interface") and then use a second tab that does all of the calculations ("Model Calculations"). You should have a chart that shows the initial and final forms and is not full of chart junk. Your model should have at least 30 space steps and at least 200 time steps.
- Using an idealized ramp-step initial form (what we have been demonstrating), make plots showing
the initial form and final forms for:
*age*= 1 ka, and*k*= 1 m^{2}/kyr*age*= 5 ka, and*k*= 1 m^{2}/kyr*age*= 10 ka, and*k*= 1 m^{2}/kyr*age*= 1 ka, and*k*= 10 m^{2}/kyr.

Discuss the effects of varying*age*and*k*on the final form. HINT: look at the product*age * t*. - Try different initial profiles. What are some other forms
that might be interesting to see what happens as they change shape? Change the bold numbers in the Initial Elevations column and show the development over time of
**TWO**different initial profiles of your choice. Show the profiles at time zero, an intermediate age, and a fairly old and well degraded age. What is the overall character of the change with time? Where does deposition occur and where does erosion occur? Indicate these zones on your plots. As the form ages, how well can you tell what the initial form was?

Last modified: September 29, 2015