Exploring diffusion for hillslope changes using a spreadsheet

I became obsessed with diffusion erosion modeling in my PhD work. It is a simple (certainly oversimplified) way to think about how hillslopes may change over time in the absence of mass wasting, debris flow, and fluvial processes. There is a lot to say about it, but I wanted to capture a few items I recently developed.

Here is an explanation and assignment on the topic in my Computers in Geology class: Lecture 8: Exploring diffusion using Excel.

One of the challenges that I have had in some applications is that the computational "space" was too small in the spreadsheet, given that it is fixed. Of course this is not a problem if one dynamically determines the number of time steps for example based on a stability criterion and you do it with a for or while loop in something like Matlab. So, when I was helping Emily Apel with her senior thesis recently, I built her a big spreadsheet (seemed easier given the limited time that she had.

Here is the original spreadsheet with only 27 space steps and 191 time steps. It is good for teaching and quick demos: LINK to Spreadsheet.

Above is the screen cap of the main interface page where the user just changes the bold cells and watches the calculations in real time.

Above is the screen cap of the Model Calculation Space tab which shows the compuational engine with its fixed elevation boundary conditions and explicit centered in space and forward in time finite differences.

Here is the big spreadsheet with 250 space steps and 1000 time steps: LINK.

And, here is a video that I built to explain the general activity for Emily Apel, but it may be useful for others. It explains the two spreadsheets that are linked above.

One of the cirtical concepts that is accessible in both of these spreadsheets is the opportunity explore not only initial step models, but also continuously displaced scenarios.

Here are a few other blog posts and recent publications which might be of interest as well:

• One dimensional morphological modeling of transport and production- limited fault scarps
• Updated review of fault scarp analysis
• Exploring the Topographic Evolution of Cinder Cones
• Some of my and my colleague's papers on these topics:
• Arrowsmith, J R. and Rhodes, D. D., Original forms and initial modifications of the Galway Lake Road scarp formed along the Emerson Fault during the 28 June 1992 Landers, California, earthquake, Bulletin of the Seismological Society of America, 84, 511-527, 1994.
• Arrowsmith, J R., Pollard, D. D., and Rhodes, D. D., Hillslope development in areas of active tectonics, Journal of Geophysical Research, 101, B3, 6,255-6,275, 1996. Correction: Journal of Geophysical Research, 104, B1, 805, 1999.
• Arrowsmith, J R., Rhodes, D. D., and Pollard, D. D., Morphologic dating of scarps formed by repeated slip events along the San Andreas Fault, Carrizo Plain, California, Journal of Geophysical Research, 103, B5, 10,141--10,160, 1998.
• Hilley, G. E., Arrowsmith, J R., Amoroso, L., The role of the interaction of normal faults and fractures on scarp morphology, Geophysical Research Letters, 28, p. 3,777-3,780, 2001.
• de Michieli Vitturi, M. and Arrowsmith, J R., Two dimensional nonlinear diffusive numerical simulation of geomorphic modifications to cinder cones, Earth Surface Processes and Landforms, doi:10.1002/esp.3423, 2013.
• Wei Zhanyu, Arrowsmith, J R., He, H., Evaluating fluvial terrace riser degradation using LiDAR-derived topography: An example from the northern Tian Shan, China, Journal of Asian Earth Sciences, Volume 105, Pages 430-442, 2015.
• Xu, J., Arrowsmith, J R, Chen, J., Schoenbohm, L. M., Li, T., Yuan, Z., Evaluating young fluvial terrace riser degradation using a nonlinear transport model: With application to the Kongur Normal Fault in the Pamir, northwest China, Earth Surface Processes and Landforms, DOI: 10.1002/esp.5022, 2020.