Moving boundary problems in earth-surface dynamics
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Moving boundary problems in earth-surface dynamics , Vaughan R. Voller NSF, National Center for Earth-surface Dynamics, University of Minnesota, USA. Input From Chris Paola, Gary Parker, John Swenson, Jeff Marr, Wonsuck Kim, Damien Kawakami. What is NCED?. A National Science Foundation

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Moving boundary problems in earth surface dynamics vaughan r voller

Moving boundary problems in earth-surface dynamics

, Vaughan R. Voller

NSF, National Center for Earth-surface Dynamics,

University of Minnesota, USA.

Input From

Chris Paola, Gary Parker, John Swenson, Jeff Marr,

Wonsuck Kim, Damien Kawakami


Moving boundary problems in earth surface dynamics vaughan r voller

What is NCED?

A National Science Foundation

Science and Technology Center

NCED develops integrated models of the physical and ecological dynamics of the channel systems that shape Earth’s surface through time, in support of river management, environmental forecasting, and resource development


Moving boundary problems in earth surface dynamics vaughan r voller

Examples of Sediment Fans

Badwater Deathvalley

1km

How does sediment-

basement interface

evolve


Moving boundary problems in earth surface dynamics vaughan r voller

Two Problems of Interest

Shoreline

Fans Toes


Moving boundary problems in earth surface dynamics vaughan r voller

Sediment Transport on a Fluvial Fan

Sediment transported and

deposited over fan surface

From a momentum balance and

drag law it can be shown that

the diffusion coefficient n

is a function of a drag coefficient

and the bed shear stress t

Sediment mass balance gives

when flow is channelized n= constant

when flow is “sheet flow”

A first order approx. analysis indicates n  1/r

(r radial distance from source)


Moving boundary problems in earth surface dynamics vaughan r voller

An Ocean Basin

Swenson-Stefan


Moving boundary problems in earth surface dynamics vaughan r voller

Limit Conditions: Constant Depth Ocean

q=1

Enthalpy solution

angle of repose

h

a

L

s(t)

A “Melting Problem” driven by a fixed flux with Latent Heat L

Track of Shore Line

NOT


Moving boundary problems in earth surface dynamics vaughan r voller

Limit Conditions: A Fixed Slope Ocean

A Melting Problem driven by a

fixed flux with SPACE DEPENDENT

Latent Heat L = gs

q=1

h

a

b

similarity solution

s(t)

g = 0.5

Enthalpy Sol.


Moving boundary problems in earth surface dynamics vaughan r voller

Limit Conditions: Sea-Level Change Very Steep Angle of Repose

q=1

b

Reaches Steady State Position

s = 1/(dL/dt)

s(t)

b=0.1

b=1

Enthalpy Sol.

dL/dt = 0.1


Moving boundary problems in earth surface dynamics vaughan r voller

a

L(t)

b

s(t)

Limit Conditions: Sea-Level Change Finite Angle of Repose

v

n

An enthalpy like fixed grid

Solution can be constructed


Moving boundary problems in earth surface dynamics vaughan r voller

The concept of an “Auto-Retreat”

To stay in one place the flux to the shore front

Needs to increase to account for the increase in the

accommodation increment with each time step

NOT possible

For flux to increase

So shoreline moves landward

Auto-retreat


Moving boundary problems in earth surface dynamics vaughan r voller

a

L(t)

b

s(t)


Moving boundary problems in earth surface dynamics vaughan r voller

“Jurassic Tank” A Large Scale Exp.

~1m

Computer controlled subsidence


Moving boundary problems in earth surface dynamics vaughan r voller

XES basin (“Jurassic Tank”) Subsidence Mechanism


Moving boundary problems in earth surface dynamics vaughan r voller

How does shore line move in response to sea-level changes

Swenson et al can be posed as a generalized Stefan Problem


Moving boundary problems in earth surface dynamics vaughan r voller

Numerical Solution

1-D finite difference deforming grid

Base level

(n calculated from 1st principles)

Measured and Numerical results


Moving boundary problems in earth surface dynamics vaughan r voller

The Desert Fan Problem -- A 2D Problem

A Stefan problem with

zero Latent Heat


Moving boundary problems in earth surface dynamics vaughan r voller

A two-dimensional version (experiment)

  • Water tight basin

    -First layer: gravel to allow easy drainage

    -Second layer: F110 sand with a slope ~4º.

  • Water and sand poured in corner plate

  • Sand type: Sil-Co-Sil at ~45 mm

  • Water feed rate:

  • ~460 cm3/min

  • Sediment feed rate: ~37cm3/min


Moving boundary problems in earth surface dynamics vaughan r voller

The Numerical Method

-Explicit, Fixed Grid, Up wind Finite Difference VOF like scheme

fill point

The Toe Treatment

r

P

E

Square grid

placed on

basement

.05 grid size

Flux out of toe elements =0

Until Sediment height >

Downstream basement

At end of each time step

Redistribution scheme is required

To ensure that no

“downstream” covered areas are higher

Determine height at fill : Position of toe


Moving boundary problems in earth surface dynamics vaughan r voller

Experimental Measurements

  • Pictures taken every half hour

    • Toe front recorded

  • Peak height measure every half hour

  • Grid of squares

  • 10cm x 10cm


Moving boundary problems in earth surface dynamics vaughan r voller

Observations (1)

  • Topography

    • Conic rather than convex

    • Slope nearly linear across position and time

    • bell-curve shaped toe


Moving boundary problems in earth surface dynamics vaughan r voller

Observations (2)

  • Three regions of flow

    • Sheet flow

    • Large channel flow

    • Small channel flow

  • Continual bifurcation governed by shear stress


Moving boundary problems in earth surface dynamics vaughan r voller

y – y(x,t) = 0

On toe

height at input

fan with time


Moving boundary problems in earth surface dynamics vaughan r voller

Front Perturbations: An Initial Model

Example shows a “numerical experiment”

of sediment filling of a deep constant depth

ocean

with persistent (preferred) channelization

Solution of Exner with

Simplified Swenson-Stefan condition and

Spatially changing diffusion coefficient

Next change

Diffusion field with time


Moving boundary problems in earth surface dynamics vaughan r voller

Moving Boundaries on Earth’s surface

A number of moving boundary problems

in sedimentary geology have been

identified.

It has been shown that these problems

can be posed as Generalized Stefan

problems

Fixed grid and deforming grid schemes have been shown to produce results in

Reasonable agreement with experiments

Improvements in model are needed

Utilize full range of moving boundary numerical technologies to arrive at

a suite of methods with geological application


Moving boundary problems in earth surface dynamics vaughan r voller

Full sim sol


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