Effect of Subsidence Styles and Fractional Diffusion Exponents on Depositional Fluvial Profiles...
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Transcript of Effect of Subsidence Styles and Fractional Diffusion Exponents on Depositional Fluvial Profiles...
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
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-10 -8 -6 -4 -2 0
)(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
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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
-2
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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
-2
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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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1
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2
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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