A Linear Stability Analysis of

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A Linear Stability Analysis of
Saltwater as Applied to Land Sea Ice

Presented to

Associate Director

Office of Science


Katherine Roddy

Research Alliance in Math and Science

Computational Earth Sciences Group

Mentor: Katherine J. Evans (Kate)


August 13, 2008

Oak Ridge, Tennessee

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Outline


Outline


Motivations


Background and the model


Governing equations





Preliminary results


Ongoing Work


Video


Questions



http://www.cen.ulava.ca

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Motivations: Climate Research



Small flows research: serve as sub
grid scale for global climate model



Today’s state of the art models:
20 km spacing



GLIMMER: GENIE Land Ice Model
with Multiply Enhanced Regions



Fortran 95



GLINT



three dimensional dynamics



Modeling key component of global
picture: field operations expensive,
difficult



Melting of land ice shelf:
catastrophic effects for sea level
world wide

http://www.bbc.co.uk

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Motivations: Physical Application


Cracks in land sea ice


Saltwater fluid dynamics
inside crack: vertical box
model


Effects of global warming
accelerated:


convection provides
positive
feedback

to melting


CNN 7/30/2008: “Ice Sheet
breaks loose off Canada”


http://www.sethwhite.org

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Background and the Model


Basic Fluid Flow


Vertical cavity with horizontally imposed temperature gradient


Counterclockwise, buoyancy
-
driven, shear flow



L

h

y

x

Cold Wall

Hot Wall

(0,0)

(0,1)

T = 0

T = 1

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Background and the Model


The Stability Problem


Conduction regime, primary shear flow


Perturbation to bifurcation:


Critical Rayleigh number:
Ra

=
Gr

*
Pr


Critical Grashof number at specified Prandtl number


Critical wavenumber


Convection regime, secondary flows



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Background and the Model


Solution Variables


Grashof number,
Gr
: fluid parameter


Wavenumber,
m
: spatial scale







Larger
m

Smaller
m

m

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Background and the Model


Previous Work


Infinite vertical cavity with horizontally imposed temperature
gradient


Air as fluid with application to double
-
paned windows


Other fluids: oceanography, metallurgy, atmospheric science,
vulcanology



Investigators

Fluid

Pr

Gr*

m*

A

Batchelor, 1954 (theory)

air

0.73

18700

n/a



Vest & Arpaci, 1969 (theory)

air

0.71

7800

2.65

33.33

Vest & Arpaci, 1969 (experiment)

air

0.71

8700

2.74



Bergholz, 1978 (theory)

air

0.71

8920

2.76

33.33

Bergholz, 1978 (theory)

pure water

6.7

15300

1.40

37.04

Bergholz, 1978 (theory)

pure water

6.7

12000

2.00

25

Ruth, 1979 (theory)

air

0.70

8041.422

2.810



Ruth, 1979 (theory)

water

7.0

7868.426

2.767



McBain & Armfield, 2003 (theory)

air

0.7

8041.4222

2.8098



McBain & Armfield, 2003 (theory)

water

7

7868.4264

2.7671



Evans, to be submitted (theory)

air

0.71

8040

2.80



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Background and the Model


Mentor’s Work


Mentor: Katherine J. Evans (Kate)


To be submitted (Journal of Fluid Mechanics): Linear Stability
Analysis as a Temporal Accuracy Benchmark



Investigators

Fluid

Pr

Gr*

m*

A

Batchelor, 1954 (theory)

air

0.73

18700

n/a



Vest & Arpaci, 1969 (theory)

air

0.71

7800

2.65

33.33

Vest & Arpaci, 1969 (experiment)

air

0.71

8700

2.74



Bergholz, 1978 (theory)

air

0.71

8920

2.76

33.33

Bergholz, 1978 (theory)

pure water

6.7

15300

1.40

37.04

Bergholz, 1978 (theory)

pure water

6.7

12000

2.00

25

Ruth, 1979 (theory)

air

0.70

8041.422

2.810



Ruth, 1979 (theory)

water

7.0

7868.426

2.767



McBain & Armfield, 2003 (theory)

air

0.7

8041.4222

2.8098



McBain & Armfield, 2003 (theory)

water

7

7868.4264

2.7671



Evans, to be submitted (theory)

air

0.71

8040

2.80



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Background and the Model


Current Study


Infinite vertical cavity with horizontally imposed temperature
and
salinity

gradients


L

h

y

x

Cold, Dilute Wall

Hot, Concentrated Wall

(0,0)

(0,1)

T = 0

C = 0

T = 1

C = 0.5

C. Petrich, et. al.

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Governing Equations


Nondimensionalized Navier
-
Stokes equations, continuity equation,
conservation equations for temperature and concentration


Linear Stability Analysis: eigenvalue problem



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Preliminary Results: Pure Water


MATLAB figures


Pure water:
Pr

= 7,
Le

= 1,
Ns

= 1, and concentration buoyancy
term = 0.


Gr

= [7500 : 100 : 11500],
m

= [1.65 : 0.1 : 3.65]

stable region

unstable region

stable region

unstable region

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Critical Grashof number: 7870


Critical wavenumber: 2.75


(water, Pr = 7, A =

)



Investigators

Fluid

Pr

Gr*

m*

A

Batchelor, 1954 (theory)

air

0.73

18700

n/a



Vest & Arpaci, 1969 (theory)

air

0.71

7800

2.65

33.33

Vest & Arpaci, 1969 (experiment)

air

0.71

8700

2.74



Bergholz, 1978 (theory)

air

0.71

8920

2.76

33.33

Bergholz, 1978 (theory)

pure water

6.7

15300

1.40

37.04

Bergholz, 1978 (theory)

pure water

6.7

12000

2.00

25

Ruth, 1979 (theory)

air

0.70

8041.422

2.810



Ruth, 1979 (theory)

water

7.0

7868.426

2.767



McBain & Armfield, 2003 (theory)

air

0.7

8041.4222

2.8098



McBain & Armfield, 2003 (theory)

water

7

7868.4264

2.7671



Evans, to be submitted (theory)

air

0.71

8040

2.80



Preliminary Results: Pure Water

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Addition of salinity gradient to model:


Gr
* = 7880,
m
* = 2.85


Expect similar
m
: similar fluid properties


Expect higher
Gr
: salt stabilizing











Preliminary Results: Saltwater

Pr

= 7,
Le

= 100,
Ns

= 0.16

Gr

= [7000 : 100 : 11000]

m

= [1.85 : 0.1 : 3.85]

unstable region

stable region

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Ongoing Work


Examine third variable: growth rate,



Incorporate mentor’s research


Run nonlinear solution method (JFNK Algorithm) Fortran code
to compare results


Time as benchmark


Final state solution vs. development of secondary flows



K. J. Evans

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Questions

CNN, Wednesday, July 30, 2008:

“Ice Sheet breaks loose off Canada”












Video
(2:16)


http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html
-

cnnSTCVideo

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References


G. K. Batchelor. Heat transfer by free convection across a closed cavity between vertical boundaries
at different temperatures.
Q. Appl. Math.
, 12(3):209
-
233. 1954.


R. F. Bergholz. Instability of natural convection in a vertical fluid layer.
J. Fluid Mech.,

84:743
-
768.
1978.


K. J. Evans. Linear stability analysis as a temporal accuracy benchmark. To be submitted, 2008.


The MathWorks Inc. MATLAB 7.4. Licensed under Dartmouth College, 2007.


G.D. McBain and S.W. Armfield. Natural convection in a vertical slot: accurate solution of the linear
stability equations. Sydney, NSW, July 2003. Eleventh Computational Techniques and Applications
Conference, Fifth International Congress on Industrial and Applied Mathematics.



GLIMMER. <http://glimmer.forge.nesc.ac.uk>


S. Mergui and D. Gobin. Transient double diffusive convection in a vertical enclosure with
asymmetrical boundary conditions.
J. Heat Transfer
, 122:598
-
602. 2000.


C. Petrich, P. J. Langhorne and T. G. Haskell. Formation and structure of refrozen cracks in land
-
fast
first
-
year sea ice.
J. Geophys. Res.
, 112. 2007.


D.W. Ruth. On the transition to transverse rolls in an infinite vertical fluid layer
--
a power series
solution.
Int. J. Heat Mass Transfer
, 22:1199
-
1208. 1979.


C. M. Vest and V. S. Arpaci. Stability of natural convection in a vertical slot.
J. Fluid Mech.
, 36:1
-
15.
1969.


J. A. C. Weidman and S. C. Reddy. A MATLAB differentiation matrix suite.
ACM Transactions on
Mathematical Software.
26(4):465
-
519. 2000.



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Acknowledgments

The author would like to thank research mentor Katherine J. Evans
(Kate), without whom this project would not have been possible.

The Research Alliance in Math and Science program is sponsored by
the Office of Advanced Scientific Computing Research, U.S.
Department of Energy.

The work was performed at the Oak Ridge National Laboratory, which
is managed by UT
-
Battelle, LLC under Contract No. De
-
AC05
-
00OR22725. This work has been authored by a contractor of the U.S.
Government, accordingly, the U.S. Government retains a non
-
exclusive, royalty
-
free license to publish or reproduce the published
form of this contribution, or allow others to do so, for U.S. Government
purposes.

Finally, the author would like to recognize George Seweryniak from the
U.S. Department of Energy, sponsor of the RAMS program.



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Questions

Thank you for the opportunity
to speak today.


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