non-local transport in channel networks vaughan voller civil engineering university of minnesota
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Non-local Transport in Channel Networks Vaughan Voller Civil Engineering University of Minnesota. source. coolgeology.uk.com. sink. and help from his “collective”. Work with. - PowerPoint PPT PresentationTRANSCRIPT
Non-local Transport in Channel Networks Vaughan VollerCivil EngineeringUniversity of Minnesota
Tetsuji Muto, Wonsuck Kim, Gary Parker, John Swenson, Jorge Lorenzo Trueba, Man Liang Matt Wolinsky, Colin Stark, Andrew Fowler, Doug Jerolmack, Dan Zielinski
and help from his “collective”
source
sink
Vamsi Ganti, Chris Paola, Efi Foufoula
Work with
coolgeology.uk.com
It is a truth universally acknowledged1 that the transport of sediment in the landscape can have a significant effect on the safety and wellbeing of humankind
J. Austen 1813
Upland Debris Flows Ocean Delta Building
Wonsuck Kim, EOS 2010
Let us first look at modeling deltas
hhwa.dot.gov
1km
Sediment Fans
Examples of Sediment Deltas
Water and sediment input
Main characteristic: Channels (at multiple scales) transporting material through system
A simple model for sediment transport in delta
sediment flux
Land
Wateradvancingshore-line
profile view
water
land
bed-rock
water
land
bed-rock
Sediment Transport and Diffusion 101
x
)(xhDriver--sediment flux [volume/length-time]
qdx
dhqslope ;~
A linear diffusion modelgood place to start (Paola 92)
inq
outq
Balance of flux across x
x
h
xx
q
x
qq outin
Exner balanceBed-rocksubsidence
A LOCAL model-local slope
Diffusion solution “too-curved”
~3m
“Jurasic Tank” Experiment at close to steady state
How well does this model work ?
Lxdx
dh
dx
d 0,
subsidence
diffusion
~3m
Clear separation between scale of heterogeneity and domain. An REV can be identified
Volume over which average properties can be applied globally.
Heterogeneity occurs at all scalesUp to an including the domain.REV can not be identified
--scales are power-law distributed
But in Experiment
Lxdx
dh
dx
d 0,Is this equation valid
Assumption in Model
Suggests a “non-local” Model
An aside: A simple example of non-local transport: A block sliding on an oil film down an inclined
At equilibrium: down-slope weigh balanced by up-slope shear-stress
velocityLOCAL slope
Now consider a series of 3 rigidly connected blocks on different slopes
At equilibrium: velocity of system
1
2
3
33
1
3*
23
1
2*
13
1
1*
1
cos
cos
cos
cos
cos
cosslopeslopeslopev
ii
ii
ii
Then Velocity of Block 1
Or A WEIGHTED SUM OF UPSTREAM SLOPESVELCOITY DEPENDS ON NON-LOCAL SLOPES
~3m
Clear separation between scale of heterogeneity and domain. An REV can be identified
Volume over which average properties can be applied globally.
Heterogeneity occurs at all scalesUp to an including the domain.REV can not be identified
--scales are power-law distributed
But in Experiment
Lxdx
dh
dx
d 0,Is this equation valid
Assumption in Model
Suggests a “non-local” Model
How can we conceptualize a non-local model?
Cannels arriving at X-X have a distribution of up stream lengths
Flux in a given channel is controlled by slope at channel head
X X
n
i xiXxiX dx
dhWq
1 )1(
~
Or in limit
dd
dhxq
x
X 0
~
One possible set of power law weights gives
A second aside: Fractional Derivatives
Basic Calculus
Cauchy Repeated integral
Generalize to non-integer case
The 1-alpha integral of the first derivative is the alpha order fractional derivative
Definition by analogy
Integral of second derivative is first derivative
nth integral of n-1th is 1st derivative
How can we conceptualize a non-local model?
Cannels arriving at X-X have a distribution of up stream lengths
Flux in a given channel is controlled by slope at channel head
X X
n
i xiXxiX dx
dhWq
1 )1(
~
Or in limit
dd
dhxq
x
X 0
~
One possible set of power law weights gives
But in reality Heterogeneity occurs at all scalesUp to an including the domain.REV can not be identified
Non-local sediment transport
Model
XX dx
dhq ~X
0
1
suggests we need a non-local model, e.g.,
dd
dhxq
x
X 0
~
flux ~ local slope A “weighted” sum of upstream slopes
Flux at a point depends on slopes at up-stream locations--information is forwardly propagated
Can do the opposite –have a downstream dependence--backward propagation
n
i xiXxi
x
X dx
dhWd
d
dhxq
1 )1(
1
~)(~
Weighted sum of downstream slopes
measure of locality10 (local)
0
1
dd
dhxd
d
dhxq
x
x
X
1
0
)(2
1
2
1~
So for general non-local treatment we model flux as
Locality direction 11 (all up-stream)
Introduce some nomenclature
dd
dhx
dx
hd x
0)1(
1
Can be interpreted as the integral of the 1st derivativeth)1(
The Caputo fractionalDerivative of order alpha
)(2
1
2
1*
xd
hd
dx
hdK
dx
d
Then non-local governing transport equation has the form
In scaled domain 10 x
0,10;
1
1
xxdx
d
Note
10,)1()(
xxd
d
xd
d
downstream dependenceupstream dependence and/or
Application a source to sinksediment transport model
hill-slope
delta
weathering-erosion
upliftsubsidence
by-pass transport
deposition-burial
Key variable sediment flux
sm
mq
2
3
The Sediment Cycle
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0 1
A first order model Mass Balance Model (divergence of flux)
Eliminate by-pass -region
erosion/uplift
deposition/subsidence
normalize domain
)(xh
Model with Exner Equation
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
erosion-uplift
depo.-sub.
divergence of fluxExner mass-balance deposit thickness
above datum
erosion deposition
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
Arrive at a solution by assuming flux in Exner can be modeled by diffusion (worked for delta)
dx
dhq i.e., sediment flux ~ slope
21
2221
0;2
xx
h
1;
2
)1(21
2221
xx
h
Easy-solution
Consistent with field and lab But----Surfaces may be too-”curved”
Uplift:
Sub:
Predicts a concaved-down erosional profile with maximum elevation at the upstream domain boundary x =0 and a concave-up depositional profile with a minimum elevation at the downstream boundary x=1
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
Arrive at a solution by assuming flux in Exner can be modeled by diffusion (worked for delta)
dx
dhq i.e., sediment flux ~ slope
21
2221
0;2
xx
h
1;
2
)1(21
2221
xx
h
Easy-solution
Predicts a concaved-down erosional profile with maximum elevation at the upstream domain boundary x =0 and a concave-up depositional profile with a minimum elevation at the downstream boundary x=1
Consistent with field and lab But----Surfaces may be too-”curved”
erosion deposition
At this point I can go one of two ways:
1. I could add more physics, and features to provide a better match with reality but with more parameters n
--OR2. Explore the model space of this simple construct and see how much it might be able to inform about possible system behaviors
I am in the modeling school that believes
1
1~
nQ
Understanding Parameters
So I will do 2
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
Arrive at a solution by assuming flux in Exner can be modeled by diffusion (worked for delta)
dx
dhq i.e., sediment flux ~ slope
21
2221
0;2
xx
h
1;
2
)1(21
2221
xx
h
Easy-solution
Predicts a concaved-down erosional profile with maximum elevation at the upstream domain boundary x =0 and a concave-up depositional profile with a minimum elevation at the downstream boundary x=1
Consistent with field and lab But----Surfaces may be too-”curved”
erosion deposition
2q
0h
2)(
xd
hd
dx
d
Consider—non-local depositional system with down-stream dependence beta=-1
1)1()2(
2)(
xxh
Can be fit to observations
5.01Voller and Paola JGR 2010
Before After
—sub. rate2
But the BIG question remains Is this non-local model physically meaningful ?
Some good evidence—
Channels scales are known to be fractal(power-law scaling)
pdf’s --e.g., measured sed. transport at a point over time is thick tailed
But no direct measure of locality value alpha
Also--Can we extend the treatment to the hillslope? (YES-- Vamsi Ganti et al. JGR 2010) And what is the effect of the Locality direction (beta)?
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
)(2
1
2
1)(
xd
hd
dx
hdxq
To answer last question let us return to our combined erosion-depositional system
use a general non-local model for flux
And exam role of Beta for fixed alpha (0.7)
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0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
)(2
1
2
1)(
xd
hd
dx
hdxq
To answer last question let us return to our combined erosion-depositional system
And us a general non-local model for flux
First Beta = 1—only upstream locality
Control-information from upstream
Correct shape and max location for fluvial surfaceIn erosional domain
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
)(2
1
2
1)(
xd
hd
dx
hdxq
To answer last question let us return to our combined erosion-depositional system
And us a general non-local model for flux
Correct shape and max location for fluvial surfaceIn erosional (hillslope) domain
But incorrect shape in depositional domain minimum elevation not at sea-level !
Beta = 1—only upstream locality
Control-information from upstream
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
)(2
1
2
1)(
xd
hd
dx
hdxq
To answer last question let us return to our combined erosion-depositional system
And us a general non-local model for flux
Correct shape and min location for fluvial surfaceIn depositional domain
Now try Beta = -1—only downstream locality
Control-information from downstream
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
)(2
1
2
1)(
xd
hd
dx
hdxq
To answer last question let us return to our combined erosion-depositional system
And us a general non-local model for flux
Correct shape and mix location for fluvial surfaceIn depositional domain
But incorrect shape in erosional domain maximum elevation not at continental divide !
Now try Beta = -1—only downstream locality
Control-information from downstream
0)(,0)0(;0,1 21
21 hqx
dx
dq
0)1(,0)(;1,1 21
21 qhx
dx
dq
To answer last question let us return to our combined erosion-depositional system
IN fact Only physically reasonable solutionsUNDER FRAC. DER. MODEL OF NON-LOCALITY Require that locality points upstream inThe erosional domain but needs to point Downstream in the depositional domain.
Transport controlled by upstream features inerosional regime but controlled by downstreamfeatures in depositional domain
depositionxd
hd
erosiondx
hd
xq
,1)(
,1
)(
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Is there a distinguishing feature between these regimes that may explain this switch inThe direction of transport (flow of information) ----
Erosional domainConverges informationdown-stream
Depositional domainDiverges information down-stream
A win-win
The MATH is suggesting something interesting about nature
If confirmed this could have important consequences for our understanding earth-surfaceDynamics and transport in channel systems--
If invalidated might require rethinking and assessment of current non-local transport models
Direction matters in non-local systems
To end of a Philosophical note---
validation
observation
validation
physical description/hypothesis
mathematical construct
mathematical construct
physical description/hypothesis
observation ?
Both approaches offer valid methods for advancement of our understanding
Data Driven
Theory Driven
Math Modeling can be used in two ways
e.g., laminar-turbulent transition
e.g., relativity
Delta growth withChannel formation
Sediment transport rulesCoupled to simplified Shallow water solver.
Man Liang—
With Paola and Voller
water
land
bed-rock
Sediment Transport and Diffusion 101
dx
dhqslopeq ;~
x
)(xhDivergence of flux across x
Diffusive Flux
This divergence of flux can be balanced by
subsidence
inq
outq
dx
dh
dx
d
bed-rock
x
h
xt
h
or surface rise
inq
outq
x
h
xx
q
x
qq outin
Diffusive Exner Equation
A One D Experiment mimicking building of delta profile, Tetsuji Muto and Wonsuck Kim Sediment and Water Mix introduced into a slot flume (2cm thick) with a fixedSloping bottom and fixed water depth
0qshore-line and sediment/rock boundary moves in response to sediment input
)t(sx)t(s,x
h
t
hbash
2
2
h(t)
Can model with a diffusion equation (in terms of sediment height h) between two moving boundaries—the shoreline Ssh and the sediment/rock Sba
Exhibits Closed Form Solution !
Experiment vs. Analytical: VALIDATION
experimental
analytical
Jorge Lorenzo Trueba, et al J. Fluid Mech. (2009), vol. 628, pp. 427–443
Position mm
Tim
e s