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Moving boundary problems in earth-surface dynamics , Vaughan R. Voller

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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

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

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)

Swenson-Stefan

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

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.

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

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

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

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

Swenson et al can be posed as a generalized Stefan Problem

1-D finite difference deforming grid

Base level

(n calculated from 1st principles)

Measured and Numerical results

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

-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

- Pictures taken every half hour
- Toe front recorded
- Peak height measure every half hour
- Grid of squares
- 10cm x 10cm

- Topography
- Conic rather than convex
- Slope nearly linear across position and time
- bell-curve shaped toe

- Three regions of flow
- Sheet flow
- Large channel flow
- Small channel flow
- Continual bifurcation governed by shear stress

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 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

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