Direction and Non Linearity in Non-local Diffusion Transport Models

F. Falcini , V. Ganti V.1,2, R. Garra , E. Foufoula Georgiou, C. Paola, V.R. Voller

University of Minnesota

1km

Sediment Fans

Hydrological Examples of interest: Sediment Depositional Deltas

Water and sediment input

Main characteristic: Channels (at multiple scales) transporting and depositing sediment through and on system

~3m

“Jurasic Tank” Experiment system for building deltas

Multi-scaled channelized surface with heterogeneities

A “Two-D Porous Media” ?

An experimental system: Sediment input into standing water, in a subsiding basin

Will first look at non-local and non-linear effects in this system

A first order mass balance model: Geometry and Governing Equation

A first order mass balance model: Geometry and Governing Equation

Piston subsidence

q –input sed-flux balances subsidence

A first order mass balance model: Geometry and Governing Equation

q –input sed-flux balances subsidence

A first order mass balance model: Geometry and Governing Equation

10,1 xdx

dq

0)0(,1)0( hq

)(xh

Assume a diffusion model

dx

dhq

221 )1( xh

Sediment elevation above datum

Diffusion solution “too-curved”When compared to experiment

~3m

Compare with experiment

q –input sed-flux balances subsidence

10,1 xdx

dq

0)0(,1)0( hq

)(xh

One solution use a non-linear diffusion model

01,1

1

dx

dh

dx

dhqq linnon

1)1(~ xh

One proposed Improvement is a non-linear model Posma et al., 2008

313.01)1(~ xh

Better Comparison with Experiment ------BUT

Required value of beta much smaller than expected theoretical value 6.01

q –input sed-flux balances subsidence

10,1 xdx

dq

0)0(,1)0( hq

)(xh

use a non-local model

1)1(~ xh

Another solution is a non-local model Voller and Paola, 2010

01,)(

xd

hdqq locnons

Note: same form as non-linear

1

)()1(

1

)( x

dd

dhx

xd

hd

Where: The right hand Caputo derivative isInterpret as weighted sum of down--stream local slopesWhy RIGHT HAND ?

313.01)1(~ xh

Get the identical Comparison with Experiment ------BUT

Difficult to know how to obtain value for alphaBut non-locality is clearly in system soexpected to be less than 1

q –input sed-flux balances subsidence10,1 x

dx

dq

0)0(,1)0( hq)(xh

use a NLNL model

1)1(~ xh

Motivates development of a non-local non-linear NLNL model

Note: same form as non-linear

0,1

)()1(

1 1 11

x

NLNLs d

d

dh

d

dhxq

The weighted sum of down-stream non-linear slopes down

The same form again !But now The non-local “dilutes” the non-linearity---To obtain same fit the non-local allows for a weaker non-linearity

Current work withFede Falcini and others

1)1(~ xh

But now The non-local “dilutes” the non-linearity---To obtain same fit the non-local allows for a weaker non-linearity

0.313 1.000 0.313 0.333 0.938 0.313 0.400 0.781 0.313 0.602 0.519 0.313 0.667 0.469 0.313 1.000 0.313 0.313

A range of alpha andbeta values canachieve fit

Including values of beta in theoretical range

What about direction in Non-local models?

Consider a simple source to sinksediment transport model

hill-slope

delta

weathering-erosion

upliftsubsidence

by-pass transport

deposition-burial

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 datumExpected profile shape

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

Now model this combined erosion-depositional system with a fractional model

use a general non-local model for flux

And exam role of for fixed alpha (0.7)

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direction

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

And us a general non-local model for flux

First gamma = 1—only upstream non-locality

Control-information from upstream

Correct shape and max location for fluvial surfaceIn erosional domain

Now model this combined erosion-depositional system with a fractional model

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

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 !

First gamma = 1—only upstream non-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

And us a general non-local model for flux

Correct shape and min location for fluvial surfaceIn depositional domain

Now try gamma = -1—only down-str. non 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

And us a general non-local model for flux

Correct shape and max location for fluvial surfaceIn depositional domain

But incorrect shape in erosional domain maximum elevation not at continental divide !

Now try gamma = -1—only down-str. non 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

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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Voller et al, GRL 2012

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

The Direction of flow of information matters in non-local systems

Direction and Non Linearity in Non-local Diffusion Transport Models

Although difficult to quantify there is sufficient Physical evidence to suggest that Non-locality is present inSediment transport systems

“toy” models presented here have shown that

1

1

1

)1()1(

)1(

xx

xNon-linear

Non-localNLNL

Non-locality Dilutes apparent Non-linearity

)(2

1

2

1)(

xd

hd

dx

hdxq