Landscape Erosion Kirsten Meeker

Outline Plan Progress Verification Performance

Plan Analyze sequential code Select parallel tools and partitioning Convert in stages, preserving functioning of whole simulation Stochastic PDE’s, individual results are a function of random parameters including numerical noise Success of results are measured by statistical parameters “Clean” maintainable, portable code Improve performance, currently hours to days

Decisions Maintainability: Use MPI for portability on clusters Investigate solver libraries: PETSc Modify functions to use only needed input parameters, to try to eliminate use of global Params struct Performance: Use row-wise partitioning Consider writing data to disk from each processor then reassembling result off-line Try to eliminate multiple passes over grid

Progress Converted main and water surface routines Created a set of utility functions: scatter, gather comm_to_local, local_to_comm print_grid

Verification of Initial Elevation

Verification of Initial Water

Verification of Final Elevation

Performance Vs. Grid Size

Performance Vs. Number of Processors

Conclusions Too much unnecessary data messaging Cell structure has 17 values, only 3 needed! Reduce message size and cache hits Water algorithm is fine-grained 4 passes over grid means 4 border exchanges Landscape erosion is a SOC

Landscape Erosion Kirsten Meeker

Outline Model Behavior Conclusions

Model Equations

Ill-Posed Results vary widely with initial conditions

Noise driven Initial surface randomly perturbed Substrate and rainfall constant Shocks develop in water flow Singularities in sediment flow: waterfalls or rapids

Dynamic Behavior Large Fourier components (smallest spatial scale) grow fastest, all modes grow exponentially Nonlinearities saturate, producing colored noise

Invariant statistical measures Width function Statistically self-similar if there exists a scaling  = 0.5 during channel formation  = 0.7 mature landscape Agrees with data from Ethiopia, Somalia, Saudi Arabia (badland conditions)

Conclusion Bridge between stochastic and deterministic modeling physically-based PDE model random walk models Channel formation is a Brownian process Mature landscape is diffusion driven by quenched noise - driven interface in a random media Combination is analogous to directed percolation networks