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

Objective: Model Fluvial Profiles in an Experimental Earth Scape Facility --flux sediment deposit subsidence 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 One model is to assume that transport of sediment at a point is proportional to local slope -- a diffusion model In Exner balance This predicts a surface with a significant amount of curvature

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

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) up-stream only down-stream only 1

~3m YY 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 Consider the following conceptua l model Y Y max channel length

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

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

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

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

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 Other “flattening models” e.g., non-linear diff No Negative curvature

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