Moving boundary problems in earth-surface dynamics Damien Kawakami, Vaughan R. Voller, Chris Paola

1. Moving boundary problems in earth-surface dynamics Damien Kawakami, Vaughan R. Voller, Chris Paola NSF, National Center for Earth-surface Dynamics, University of Minnesota, USA.

2. What is NCED? A National Science Foundation Science and Technology Center NCED develops integrated models of the physical and ecological dynamics of the channel systems that shape Earth’s surface through time, in support of river management, environmental forecasting, and resource development

3. Examples of Sediment Fans Badwater Deathvalley 1km How does sediment- basement interface evolve

4. Sediment Transport on a Fluvial Fan Sediment transported and deposited over fan surface From a momentum balance and drag law it can be shown that the diffusion coefficient n is a function of a drag coefficient and the bed shear stress t Sediment mass balance gives when flow is channelized n= constant when flow is “sheet flow” A first order approx. analysis indicates n  1/r (r radial distance from source)

5. An Ocean Basin Swenson-Stefan

6. Limit Conditions: Constant Depth Ocean q=1 Enthalpy solution angle of repose h a L s(t) A “Melting Problem” driven by a fixed flux with Latent Heat L Track of Shore Line NOT

7. Limit Conditions: A Fixed Slope Ocean A Melting Problem driven by a fixed flux with SPACE DEPENDENT Latent Heat L = gs q=1 h a b similarity solution s(t) g = 0.5 Enthalpy Sol.

8. The Desert Fan Problem -- A 2D Problem A Stefan problem with zero Latent Heat

9. A two-dimensional version (experiment) • Water tight basin -First layer: gravel to allow easy drainage -Second layer: F110 sand with a slope ~4º. • Water and sand poured in corner plate • Sand type: Sil-Co-Sil at ~45 mm • Water feed rate: • ~460 cm3/min • Sediment feed rate: ~37cm3/min

10. The Numerical Method -Explicit, Fixed Grid, Up wind Finite Difference VOF like scheme fill point The Toe Treatment r P E Square grid placed on basement .05 grid size Flux out of toe elements =0 Until Sediment height > Downstream basement At end of each time step Redistribution scheme is required To ensure that no “downstream” covered areas are higher Determine height at fill : Position of toe

11. Experimental Measurements • Pictures taken every half hour • Toe front recorded • Peak height measure every half hour • Grid of squares • 10cm x 10cm

12. Observations (1) • Topography • Conic rather than convex • Slope nearly linear across position and time • bell-curve shaped toe

13. Observations (2) • Three regions of flow • Sheet flow • Large channel flow • Small channel flow • Continual bifurcation governed by shear stress

14. y – y(x,t) = 0 On toe height at input fan with time

15. Moving Boundaries on Earth’s surface A number of moving boundary problems in sedimentary geology have been identified. It has been shown that these problems can be posed as Generalized Stefan problems Fixed grid and deforming grid schemes have been shown to produce results in Reasonable agreement with experiments Improvements in model are needed Utilize full range of moving boundary numerical technologies to arrive at a suite of methods with geological application Use large scale general purpose solution packages

16. Full sim sol Will give a q=-1 at x =0 a consrant q on s=2lam t to ½ And eta = 0 at s