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

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inq --flux)(mmh

)/( smm

sediment deposit

subsidence

)(mmx x

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

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inq)(mmh

)/( smm

sediment deposit

subsidence

0)(, with

,

0

hqdx

dhk

dx

dhk

dx

dq

dx

d

in

)(mmx x

One model is to assume that transport of sediment at a point is proportional to local slope -- a diffusion model

dx

dhkq

In Exnerbalance

This predicts a surfacewith a significant amount of curvature

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)(mmh

)/( smm

--fluxinq

x)(mmx

BUT -- experimental slopestend to be much “flatter” thanthose predicted with a diffusion model

Hypothesis:

The curvature anomaly isdue to

“Non-Locality”

Referred to as “Curvature Anomaly ”

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x

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

downor upinsatnat * slopekq up

down

Two parameters: “locality weighting”

“direction weighting” (balance of up to down stream non-locality)

local)(10 locality)-zero (

11 up-stream only down-stream only

1

-1

~3m

YY

In experiment surface made up oftransient channels with a wide range of length scales

Assumption flux in any channel (j) crossing Y—YIs “controlled” by slope at current down-stream channel head

--a NON-LOCAL MODEL with

x

Consider the following conceptual model

Y

Y

downjj sq

x

max channel length

10,1

n

j

downjjx sWkq

1

Y

Y

Consider the following conceptual model

xY

Y

downjj sq

x

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 sectioncontrolled by slopeat channel head

max channel length

Unroll

i-1ii+1i+n-1 i+n-2nq nW

1nW

2W

1W

x

1 xn 1nq 2q

1q

A Finite Difference Form

n

j

jijij

n

j jx x

hhWk

dx

dhkq

1

12

1

Flux at x is weighted sum of down-stream slopes

Provides a finite difference form for Exner

0

0.1

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0.3

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)(lim

0 xd

hdqx

x

With appropriate power law weights

Recoversright-hand CaputoFractional Derivative

Order and Weightchannels by

down-stream distance from x

So with non-local channel model problem to solve is

0

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0.3

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0)(,)(

with

,)(

0

*

*

hqxd

hdk

xd

hdk

dx

d

in

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0

alpha close to 1 moves to single local weight at x

Smaller alpha more uniform dist. of weights

1 xn

10,1

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0

0)(,)(

with

,)(

0

*

*

hqxd

hdk

xd

hdk

dx

d

in

Shows that a small value of alpha (non-locality) will reduce curvature and get closer to the behaviorSeen in experiment

Use the finite difference solution of

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=1

=0.25

XES10

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-1

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0

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1

1.5

2

0 0.2 0.4 0.6 0.8 1

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-1

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0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

0)1(,)(

with),()(

0

2

hqxd

hdqmx

xd

hd

dx

d m

A little more analysis: A general linear subsidence problem

Analytical solution

122 )1()2(

)()1(

)3(x

qx

mh

m

sediment

subsidence rate/2

25.0

1

2,4 qm 2,4 qm

With negative sub. rate slopeCan get negative curvatureFor alpha<1

With positive sub. rate slopeMuch harder to “flatten profile”By decreasing alpha

Other “flattening models” e.g., non-linear diff

dx

dh

dx

dhkq

N

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

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=1

=0.25

XES10

* 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

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* 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

n

j

jijijx x

hhWkq

1

12