Homework #6.1 earth surface processes Humphrey 2013,

We are now going to revisit the homework of last week.  See last week’s homework if you need to be reminded of solution techniques. Below is a profile through a small fault (earthquake) scarp, which has broken the surface of a desert landscape.

h5graphic

The height of the scarp is 2 meters. 

Question 1,  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 Dprocess 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 Cprocess (assume a value of 5x10-12m2s-1) and for the slope length process is Dprocess (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.

 Question 2,  We will now try to apply a different Boundary Condition at the points x=0 and x=10.  The BC we will apply is that the sed flux (qsed) is zero at these two points.  To make the flux zero, all we need to do is make the slope zero.  So that our question is to repeat question 1, but with zero flux entering and exiting the diagram.

 The easiest way to change the BC to a zero slope condition is to add an extra node to the left of the zero node (and for the right BC, an extra node past the right end of the problem).  For the constant elevation BC in question 2, you made the nodes on the boundaries, at x=0 and at x=10, constants.  Now make them variable, but add an extra (dummy) node just outside the boundary (you will have 2 extra nodes compared to question 1).  At each time step you need to add an extra equation that sets this dummy node equal to the node that just node inside the boundary (not equal to the boundary node but one node inside the boundary node, and note this node will change with time!). This will force the slope at the boundaries to remain zero (note that this makes Zi+1 – Zi-1 zero [ie the slope is 0] if the boundary node is i).  It also allows you to calculate a curvature at the interfluve and to calculate the erosion of the node at the interfluve (left boundary).  Extend the time out to 20,000 years, and report the amount of erosion at the interfluve.