1. Fluvial Landscape Erosion Kirsten Meeker, Bjorn Birnir, Terence Smith, George Merchant www.cs.ucsb.edu/~kmeeker/erosion.htm

2. Modeling fluvial landscape evolution using a simple system of nonlinear conservation equations produces stages of evolution and stochastic results with characteristic statistical properties.

3. Conservation Equations

4. Flow Equations

5. Boundary Conditions upper boundary (ridge) h = 0 q w = q s = 0 lower boundary (absorbing body of water) H = h 0 = h Lateral boundaries (infinite extent) periodic

6. System Properties Ill-posed problem Shocks develop in water flow Results vary widely with initial conditions Large Fourier components (smallest spatial scale) grow fastest, all modes grow exponentially Nonlinearities saturate, producing colored noise Statistical measures are invariant – width function

7. Numerical Methods Water Equation: Forward-time center space scheme with upwind differencing, explicit O(  x,  t)

8. Sediment Equation: Crank-Nicholson scheme, implicit O(  x 2,  t 2 )

9. Can be expressed in matrix form as Ax=b Solved using preconditioned biconjugate gradient method diag(A) used as preconditioner

10. Parallel Program Partitioning

11. References Towards an elementary theory of drainage basin evolution: I. The theoretical basis. Terence R. Smith, Bjorn Birnir and George E. Merchant Computers & Geosciences 23(8), 811-822 Towards an elementary theory of drainage basin evolution: II. A computational evaluation. Terence R. Smith, George E. Merchant and Bjorn Birnir Computers & Geosciences 23(8), 823-849 The scaling of fluvial landscapes. Björn Birnir, Terence R. Smith and George E. Merchant, Computers & Geosciences 27(10), 1189- 1216