Validation and Verification of Moving Boundary Models of Land Building Processes Vaughan R. Voller

1. 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 Tetsuji Muto, Wonsuck Kim, Chris Paola, Gary Parker, John Swenson, Jorge Lorenzo Trueba, Man Liang Wax Lake Solid Crystal Growing in undercooled melt

2. MovinG Boundaries in the Landscape Shoreline Fans Toes

3. Sediment Delta Examples Badwater Deathvalley Sediment Fans 1km

4. land surface shoreline ocean x = u(t) x = s(t) a sediment h(x,t) bed-rock b x

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

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

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

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

9. A mathematical model based on the Swenson Stefan Analogy with Fixed base slope and sea level q0 h Note 4 conditions 2 for the 2nd order equations 2 for the 2 moving boundaries

10. Similarity Solution

11. Numerical Solution “Latent Heat” Amount of sediment that needs to be provided To move shoreline a unit distance (L = 0 in sub-aerial) 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 q0 h

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

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

14. Reasonable Experiment vs. Analytical: VALIDATION Get Fit by choosing diffusivity Bed porosity fixed at 30% 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 Analytical Solution Experiments seaward landward

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

16. NUMERICAL VS. ANALYTICAL: Verification

17. An Interesting Limit Case q0 No- on-lap A horizontal fluvial surface coinciding with sea level

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

19. Current: Towards a CAFÉ Delta Model (Voller, Paola, Man-Ling) CAFÉ—Background deterministic (PDE) model solved with Finite Elements Superimposed with a Cellular (rule based Model) Can make physical arguments that a suitable Background model is the filling of a thin-cavity (Hele-Shaw cell) 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.

20. 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”

21. 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.