Fluvial Landscape Erosion Kirsten Meeker, Bjorn Birnir, Terence Smith, George Merchant

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

Conservation Equations

Flow Equations

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

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

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

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

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

Parallel Program Partitioning

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), 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), The scaling of fluvial landscapes. Björn Birnir, Terence R. Smith and George E. Merchant, Computers & Geosciences 27(10),