National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

MOVING BOUNDARY PROBLEMS ON THE EARTHS SURFACEV.R. Voller+, J. B. Swenson*, W. Kim+ and C. Paola+

+ National Center for Earth-surface Dynamics University of Minnesota, Minneapolis*Dept. Geological Sciences and Large Lake Observatory, University of Minnesota-Duluth

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

Ganges-Brahmaputra Delta “growth” of sediment delta into oceanGrain Growth in Metal Solidification

From W.J. Boettinger

m

10km

Commonality between solidification and ocean basin formation

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

An Ocean Basin

Melting vs. Shoreline movement

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

pressurizedwater reservoir

to water supply

solenoidvalve

stainless steelcone

to gravel recycling

transport surface

gravel basement

rubber membrane

experimental deposit

Experimental validation of shoreline boundary condition

~3m

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

Experimental validation of shoreline boundary condition

dt

dS)su(

21

dx)x(dt

dZ)su()t,s(q

dtds

S)su( f2u

s

blsff

eXperimental EarthScape facility (XES)

Flux balance at shoreline

Flux base subsidence slope

Calculated frontvelocity from

exp. measurment of RHS

measured

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

Limit Conditions: A Fixed Slope Ocean

q=1

s(t)

similarity solution

q

2

)(erfe

)(erf1,t2s

22/1

21

21

2

21

21

0

5

10

15

20

25

0 100 200 300

Time

sh

ore

line

0if),x(LH

2

2

xt

H

Enthalpy Sol.

A Melting Problem driven by a fixed flux with SPACE DEPENDENT

Latent Heat L = s

dt

dss

x)t(sx0,

xt s2

2

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

A Further Limit Solution—No sediment storage in the fluvial domain

s(t)

Let diffusivity LARGE in analyticalsolution

qt2s

From previous analytical or

Simple Geometry

s(t)

A even more simple versionAssume a “cliff” face at shoreline

Can model 2-D problem like polymer filling

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

A Monte-Carlo (lattice-Boltzmann) Polymer Filling Algorithm

“Dump” total flux in “gate” cell and redistribute excess over amount requiredfor filling to neighboring cells in ratios proportional tocoefficients of the discretization of

0)k( Iteration can be written in the form of Lattice Boltzmann iterations

Voller: To be published in JCP 2005

0)k( Note:

Can be used to accountfor effects of channels

PW E

]0,1F[MAX]0,1F[MAX]1,F[MINF E21

W21

PNEWP

In One-D iterations could look like

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

High K(1) chanalized surface

Low K(0.05) few channels

Simulation of shoreline motion into a variable depth ocean with variable channelization

t=50

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

t=100

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

t=150

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

t=200

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

t=250

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

t=300

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

WHY Build a model

Stratigraphy and Shoreline

-30-20-10

0102030405060

0 10 20 30

Models can predict stratigraphy“sand pockets” = OIL

The Poe

Shoreline position is signature of channels

National Center for Earth-surface Dynamicsan NSF Science and Technology Center

www.nced.umn.edu

s(t) s(t)

z(t)

Further Work:--Include Ocean Level Rise

0

5

10

15

20

25

30

35

40

45

50

0 20 40 60 80 100

shoreline

sea-level

geometric – modelof shoreline movementwith changing sea level

NOTE: REVERSE ofshoreline!

q=1