Effect of Subsidence Styles and Fractional Diffusion Exponents on Depositional Fluvial Profiles

1. Effect of Subsidence Styles and Fractional Diffusion Exponents on Depositional Fluvial Profiles Vaughan Voller: NCED, Civil Engineering, University of Minnesota Liz Hajek: NCED, Geosciences, Pennsylvania State University Chris Paola: NCED, Geology and Geophysics, University of Minnesota

2. Objective: Model Fluvial Profiles in an Experimental Earth Scape Facility --flux In long cross-section, through sediment deposit Our aim is to predict steady state shape and height of sediment surface above sea level for given sediment flux and subsidence sediment deposit subsidence

3. One model is to assume that transport of sediment at a point is proportional to local slope -- a diffusion model In Exner balance sediment deposit This predicts a surface with a significant amount of curvature subsidence

4. --flux BUT -- experimental slopes tend to be much “flatter” than those predicted with a diffusion model Referred to as “Curvature Anomaly ” Hypothesis: The curvature anomaly is due to “Non-Locality”

5. A possible Non-local model: sediment flux at a point x at an instant in time is proportional to the slope at a time varying distance up or down stream of x up down Two parameters: “locality weighting” “direction weighting” (balance of up to down stream non-locality) 1 -1 up-stream only down-stream only

6. ~3m Consider the following conceptual model Y Y Y Y max channel length In experiment surface made up of transient channels with a wide range of length scales Assumption flux in any channel (j) crossing Y—Y Is “controlled” by slope at current down-stream channel head --a NON-LOCAL MODEL with

7. Consider the following conceptual model Y Y flux across a small section controlled by slope at channel head Y Unroll Y max channel length representative Flux across at x is then a weighed sum of the current down-stream slopes of the n channels crossing Y-Y

8. A Finite Difference Form i-1 i i+1 i+n-1 i+n-2 x Order and Weight channels by down-stream distance from x Flux at x is weighted sum of down-stream slopes With appropriate power law weights Recovers right-hand Caputo Fractional Derivative Provides a finite difference form for Exner

9. So with non-local channel model problem to solve is alpha close to 1 moves to single local weight at x Smaller alpha more uniform dist. of weights

10. Use the finite difference solution of Shows that a small value of alpha (non-locality) will reduce curvature and get closer to the behavior Seen in experiment

11. A little more analysis: A general linear subsidence problem Analytical solution sediment subsidence rate/2 With negative sub. rate slope Can get negative curvature For alpha<1 With positive sub. rate slope Much harder to “flatten profile” By decreasing alpha No Negative curvature Other “flattening models” e.g., non-linear diff

12. Conclusions * A non-local channel concept has lead to a fraction diffusion sediment deposition model * With locality factor alpha ~0.25 (1 is local) model comes close to matching “flatness” of XES * But the non-local model introduces additional degrees of freedom-- this makes it easier to fit * The conceptual model helps BUT we still do not know how to independently determine the value of the locality factor alpha or direction factor beta * The theoretical appearance of a negative curvature for a negative sloping subsidence (not seen in other models) suggests a experiment that may go a long way to validating our proposed non-local deposition model