National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface 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

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface What is NCED? 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 A National Science Foundation Science and Technology Center

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface 1km Examples of Sediment Fans How does sediment- basement interface evolve Badwater Deathvalley

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Fans Toes Shoreline Two Problems of Interest

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Sediment mass balance gives Sediment transported and deposited over fan surface From a momentum balance and drag law it can be shown that the diffusion coefficient is a function of a drag coefficient and the bed shear stress  when flow is channelized = constant when flow is “sheet flow” A first order approx. analysis indicates  1/r (r radial distance from source) Sediment Transport on a Fluvial Fan

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface An Ocean Basin Swenson-Stefan

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Limit Conditions: Constant Depth Ocean q=1   L A “Melting Problem” driven by a fixed flux with Latent Heat L s(t) angle of repose Enthalpy solution Track of Shore Line NOT

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Limit Conditions: A Fixed Slope Ocean q=1   A Melting Problem driven by a fixed flux with SPACE DEPENDENT Latent Heat L =  s s(t)  Enthalpy Sol. similarity solution 

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Limit Conditions: Sea-Level Change Very Steep Angle of Repose q=1 s(t)  Enthalpy Sol. Reaches Steady State Position s = 1/(dL/dt)   dL/dt = 0.1

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Limit Conditions: Sea-Level Change Finite Angle of Repose v n An enthalpy like fixed grid Solution can be constructed s(t)   L(t)

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface 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

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface s(t)   L(t)

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface “Jurassic Tank” A Large Scale Exp. ~1m Computer controlled subsidence

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface XES basin (“Jurassic Tank”) Subsidence Mechanism

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface How does shore line move in response to sea-level changes Swenson et al can be posed as a generalized Stefan Problem

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Base level Measured and Numerical results ( calculated from 1 st principles) Numerical Solution 1-D finite difference deforming grid

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface The Desert Fan Problem -- A 2D Problem A Stefan problem with zero Latent Heat

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface 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 cm 3 /min Sediment feed rate: ~37cm 3 /min

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface The Numerical Method -Explicit, Fixed Grid, Up wind Finite Difference VOF like scheme Flux out of toe elements =0 Until Sediment height > Downstream basement fill point PE The Toe Treatment Square grid placed on basement At end of each time step Redistribution scheme is required To ensure that no “downstream” covered areas are higher r Determine height at fill : Position of toe.05 grid size

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Pictures taken every half hour –Toe front recorded Peak height measure every half hour Grid of squares 10cm x 10cm Experimental Measurements

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Observations (1) Topography –Conic rather than convex –Slope nearly linear across position and time –bell-curve shaped toe

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Observations (2) Three regions of flow –Sheet flow –Large channel flow –Small channel flow Continual bifurcation governed by shear stress

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface y –  (x,t) = 0 On toe height at input fan with time

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface 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 Front Perturbations: An Initial Model Next change Diffusion field with time

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface 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

National Center for Earth-surface Dynamics Modeling physical and ecological dynamics of channel systems that shape Earth’s surface Full sim sol