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

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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