# Presentation: 467547

## 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**

##  Outline

-  Outline

**Outline**
**Motivations**
**Background and the model**
**Governing equations**

**Preliminary results**
**Ongoing Work**
**Video**
**Questions**

- http://www.cen.ulava.ca

## 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

## 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

## Background and the Model	Basic Fluid Flow

**Vertical cavity with horizontally imposed temperature gradient**
**Counterclockwise, buoyancy-driven, shear flow**

## Background and the Model	The Stability Problem

- 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**

## Background and the Model	Solution Variables

**Grashof number, *****Gr*****: fluid parameter**
**Wavenumber, *****m*****: spatial scale**

- Larger *m*

- Smaller *m*

*m*

## Background and the Model	Previous Work

- 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 |  |

## Background and the Model	Mentor’s Work

- 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 |  |

## Background and the Model	Current Study

- Background and the Model	Current Study

**Infinite vertical cavity with horizontally imposed temperature **_**and salinity**_** gradients**

- C. Petrich, et. al.

## Governing Equations

**Nondimensionalized Navier-Stokes equations, continuity equation, conservation equations for temperature and concentration**
**Linear Stability Analysis: eigenvalue problem**

## 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

## Preliminary Results: Pure Water

**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: Saltwater

**Addition of salinity gradient to model:**
***Gr****** = 7880, *****m****** = 2.85**
**Expect similar *****m*****: similar fluid properties**
**Expect higher *****Gr*****: salt stabilizing**

***Pr***** = 7, *****Le***** = 100, *****Ns***** = 0.16**
***Gr***** = [7000 : 100 : 11000]**
***m***** = [1.85 : 0.1 : 3.85]**

- unstable region

- stable region

## 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

## 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

- CNN, Wednesday, July 30, 2008: “Ice Sheet breaks loose off Canada”Video (2:16) _[http://www.](http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html#cnnSTCVideo)__[cnn](http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html#cnnSTCVideo)__[.com/2008/WORLD/americas/07/30/canada.arctic.ice.](http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html#cnnSTCVideo)__[ap/index](http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html#cnnSTCVideo)__[.html - ](http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html#cnnSTCVideo)__[cnnSTCVideo](http://www.cnn.com/2008/WORLD/americas/07/30/canada.arctic.ice.ap/index.html#cnnSTCVideo)_

## 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.

## 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.

## Thank you for the opportunity to speak today.Any questions?

- Thank you for the opportunity to speak today.Any questions?

- http://i110.photobucket.com