Validation and Verification of Moving Boundary Models of Land Building Processes Vaughan R. Voller National Center for Earth-surface Dynamics Civil Engineering, University of Minnesota Wax LakeSolid Crystal Growing in undercooled melt Tetsuji Muto, Wonsuck Kim, Chris Paola, Gary Parker, John Swenson, Jorge Lorenzo Trueba, Man Liang

Fans Toes Shoreline MovinG Boundaries in the Landscape

1km Examples Badwater Deathvalley Sediment Fans Sediment Delta

sediment h(x,t) x = u(t) bed-rock ocean x shoreline x = s(t) land surface  

An Ocean Basin The Swenson Analogy: Melting vs. Shoreline movement Swenson et al, Eur J App Math, 2000

Physical Process Isolate Key Phenomena Experiment Phenomenological Assumptions Model Approximation Assumptions Numerical Solution Validation: If assumptions for Analytical solution are consistent with Physical assumptions In experiment Can VALIDATE phenomenological assumptions Verification: Comparison of numerical and analytical predictions VERIFY Numerical Approach The Modeling Paradigm Limit Case Assumptions Analytical Solution

CASE OF CONSTANT BASE LEVEL and Bed Rock The delta progrades into standing water. The rate of progradation slows in time as deeper water is invaded. The bedrock-alluvial transition migrates upstream. Slide from MUTO and PARKER---Muto Experiments

Experiments and image analysis by Tetsuji Muto and Wonsuck Kim, In slot flume

q0q0 h A mathematical model based on the Swenson Stefan Analogy with Fixed base slope and sea level Note 4 conditions 2 for the 2 nd order equations 2 for the 2 moving boundaries

Similarity Solution

q0q0 h To develop numerical solution write problem in terms of Total Sediment Balance (enthalpy). Then there is NO need to treat shoreline conditions making for an easier numerical solution “Latent Heat” Amount of sediment that needs to be provided To move shoreline a unit distance (L = 0 in sub-aerial) Numerical Solution

q k=k-1 k-1 k i-1i i+1 ONLAP CONDITION h q q On-lap update—if Update on-lap node flag 1<L<0

Physical Process Isolate Key Phenomena Experiment Phenomenological Assumptions Model Approximation Assumptions Numerical Solution Validation: If assumptions for Analytical solution are consistent with Physical assumptions In experiment Can VALIDATE phenomenological assumptions The Modeling Paradigm Limit Case Assumptions Analytical Solution

seawardlandward Experiments Analytical Solution Get Fit by choosing diffusivity Bed porosity fixed at 30% Experiment vs. Analytical: VALIDATION Two Consistency Checks 1. Compare physical and Predicted surfaces A little more concaved than we would like (experiment may be better modeled by Non-linear diffusion) 2. Across a range of experiments best fit diffusivity should scale with water discharge Reasonable

Physical Process Isolate Key Phenomena Experiment Phenomenological Assumptions Model Approximation Assumptions Numerical Solution The Modeling Paradigm Limit Case Assumptions Analytical Solution Verification: Comparison of numerical and analytical predictions VERIFY Numerical Approach

NUMERICAL VS. ANALYTICAL: Verification

An Interesting Limit Case q0q0 No- on-lap A horizontal fluvial surface coinciding with sea level

In a Two-Dimensional plan view this limit case gets a little more interesting

Current: Towards a CAFÉ Delta Model (Voller, Paola, Man-Ling) The simulation shows a “particle” solution of the filling model. This is based on the introduction, probabilistic movement, and deposition of particles in the domain. IT can be shown that this is a solution of the discrete equations associated with a Finite Element Model of the governing equations. Cellular RULES can be introduced by linking the probability of particle movement to the path taken. Thereby modeling channels and vegetation. Can make physical arguments that a suitable Background model is the filling of a thin-cavity (Hele-Shaw cell) CAFÉ—Background deterministic (PDE) model solved with Finite Elements Superimposed with a Cellular (rule based Model)

Some Examples Uniform Probs High Middle Prob High Edge Efi Research Question: How is CADFE model based on a “normal” PDE Related to a “fractional derivative PDE”

Saltwater intrusion occurs when saltwater from the Gulf moves into areas that have formerly been influenced by freshwater. As saltwater intrudes into a fresh marsh, the habitat will be altered as the plants and organisms that once thrived in the freshwater marsh cannot survive in saltwater. If the intrusion of saltwater is gradual enough, plants and organisms that can survive in a saltwater habitat begin to invade and grow, eventually establishing a brackish marsh. If saltwater vegetation does not replace the freshwater plants, the area will become exposed mud flats, and they are likely to revert to open water. This process is common in an abandoned delta lobe where the discharge of the river decreases or even in areas of the modern delta where freshwater is diverted or maintained within existing channels.