Homework #6.5 earth surface processes Humphrey 2012,

The first question is for those of you who didn’t completely finish question 4 in the last homework.

Question 1, Redo question 4 from last week, using only diffusion creep (no sheetwash).

The above figure is a cross-section of two gulleys in the Laramie Basin, separated by a remnant pile of erodable material 5m high, with dimensions as above. Model the behavior of the interfluve, under the process of just rainsplash. Assume the erosion rate, C, is as in previous problems. Assume points A and B are the locations of small streams. Point A is eroding downward at 1m per 5000years, but B is not eroding. What happens to the interfluve and streams over time? Specifically, when will the stream at A capture the stream at B? (Hint, what you need to do is change the Boundary Condition [BC] at the location of A. The way to do this is to move the elevation of A down the appropriate amount with each time step).

Notes: the boundaries are located at A and B, not at the edges of the picture! (since A and B are specified elevations, they make good BCs, whereas the edges of the picture are undefined)

Question 2, Revisit question 2 of previous homework. The code is exactly the same, but use more Sheetwash. I didn’t make the sheetwash coefficient large enough in the last homework to show what I wanted to demonstrate.

The height of the scarp is 2 meters.

This week we assume a whole suite of processes are operating over time; some purely diffusional, and some that depend on distance from the divide (you can think: rainslash and sheetwash). Assume; that in this desert environment, the time evolution of this feature can be approximately described by a simple addition of the difference equations we developed in class:

………. Eqn A

where Z is the elevation of “bin” i at time t, and the x length of bins is Dx. Note if D_process is zero, this is the same as the pure rainsplash equation. The time step size is Dt, and the governing rate coefficient for the distributed process is C_process (assume a value of 5x10^-12m²s^-1) and for the slope length process is D_process (assume a value of 5x10^-12ms^-1). Note; in the equation all the quantities on the right are known at time “now” (or t) and only the new elevation at t+1 appears on the left as the unknown. Calculate the time evolution of this (2D) feature for 10,000yrs into the future, and report the elevation of the node at the top of the scarp. Assume the nodes at x=0 and at x=10 remains at a constant elevation.

Note you may have to take a slightly smaller time step, if you get oscillations.

Question 3,

Reproduce the argument for Stoke’s Law for settling velocity, and therefore reproduce the Stoke’s equation (minus the value of the coefficient). You might find it useful to follow the following line of logic:

a. Gravity force on particle

b. Total drag force on particle due to surface shear stress

c. A scale for the viscous shear stresses

d. Balance gravity and viscous drag (F=ma) to get Stoke’s Law (with an incalculable constant)