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Moving boundary problems in earth-surface dynamics , Vaughan R. Voller

Moving boundary problems in earth-surface dynamics , Vaughan R. Voller NSF, National Center for Earth-surface Dynamics, University of Minnesota, USA. Input From Chris Paola, Gary Parker, John Swenson, Jeff Marr, Wonsuck Kim, Damien Kawakami. What is NCED?. A National Science Foundation

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Moving boundary problems in earth-surface dynamics , Vaughan R. Voller

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  1. Moving boundary problems in earth-surface dynamics , Vaughan R. Voller NSF, National Center for Earth-surface Dynamics, University of Minnesota, USA. Input From Chris Paola, Gary Parker, John Swenson, Jeff Marr, Wonsuck Kim, Damien Kawakami

  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. Two Problems of Interest Shoreline Fans Toes

  5. 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)

  6. An Ocean Basin Swenson-Stefan

  7. 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

  8. 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.

  9. Limit Conditions: Sea-Level Change Very Steep Angle of Repose q=1 b Reaches Steady State Position s = 1/(dL/dt) s(t) b=0.1 b=1 Enthalpy Sol. dL/dt = 0.1

  10. a L(t) b s(t) Limit Conditions: Sea-Level Change Finite Angle of Repose v n An enthalpy like fixed grid Solution can be constructed

  11. The concept of an “Auto-Retreat” To stay in one place the flux to the shore front Needs to increase to account for the increase in the accommodation increment with each time step NOT possible For flux to increase So shoreline moves landward Auto-retreat

  12. a L(t) b s(t)

  13. “Jurassic Tank” A Large Scale Exp. ~1m Computer controlled subsidence

  14. XES basin (“Jurassic Tank”) Subsidence Mechanism

  15. How does shore line move in response to sea-level changes Swenson et al can be posed as a generalized Stefan Problem

  16. Numerical Solution 1-D finite difference deforming grid Base level (n calculated from 1st principles) Measured and Numerical results

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

  18. 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

  19. 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

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

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

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

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

  24. Front Perturbations: An Initial Model Example shows a “numerical experiment” of sediment filling of a deep constant depth ocean with persistent (preferred) channelization Solution of Exner with Simplified Swenson-Stefan condition and Spatially changing diffusion coefficient Next change Diffusion field with time

  25. 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

  26. Full sim sol

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