Slide 24 of 66
Notes:
In the case of fluvial erosion, the equation considered earlier for bedrock fluvial channels is that the rate of fluvial downcutting is a power function of gradient, S, and contributing area, A, which in a 2-D landscape is proportional to distance from the divide. That is ?z/?t|c = -Ec = Kt (Kz Sh xg - ?c). When solved for gradient as a function of distance from the divide (top equation). This equation involves two additive factors, Ec/Kt, and ?c. In general one of these two terms will dominate. Case A is where the first term dominates and Case B is where the second term dominates. For Case A the fluvial gradient will be nearly linearly proportional to erosion rate, since h~0.7-1.0, but for threshold channels gradient is independent of erosion rate.
The equations for gradient as a function of distance for mass wasting and for fluvial erosion can be combined if it is assumed that there is a handover distance, x0, from the divide where mass wasting erosion becomes less important than fluvial erosion. In solving for x0 it is assumed that the gradients for fluvial and mass wasting erosion at the handover location are equal and that the erosion rate for each process is also equal and, therefore, half of the overall erosion rate E, giving the bottom equation for the handover distance (solved in this case for sloped dominated by linear diffusion).
