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

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.

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

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  1. A Linear Stability Analysis of Saltwater as Applied to Land Sea Ice Presented to Associate Director Office of Science Katherine RoddyResearch Alliance in Math and ScienceComputational Earth Sciences Group Mentor: Katherine J. Evans (Kate) August 13, 2008 Oak Ridge, Tennessee

  2. Outline Outline Motivations Background and the model Governing equations • Preliminary results • Ongoing Work • Video • Questions http://www.cen.ulava.ca

  3. 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 Motivations: Climate Research http://www.bbc.co.uk

  4. 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

  5. Background and the Model Basic Fluid Flow Vertical cavity with horizontally imposed temperature gradient Counterclockwise, buoyancy-driven, shear flow T = 0 T = 1 L h Cold Wall Hot Wall y x (0,0) (0,1)

  6. 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

  7. Background and the Model Solution Variables • Grashof number, Gr: fluid parameter • Wavenumber, m: spatial scale m Larger m Smaller m

  8. 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

  9. 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

  10. Background and the Model Current Study Infinite vertical cavity with horizontally imposed temperature and salinity gradients T = 0 C = 0 T = 1 C = 0.5 L Cold, Dilute Wall Hot, Concentrated Wall h y x (0,0) (0,1) C. Petrich, et. al.

  11. Governing Equations • Nondimensionalized Navier-Stokes equations, continuity equation, conservation equations for temperature and concentration • Linear Stability Analysis: eigenvalue problem

  12. 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] unstable region unstable region stable region stable region

  13. Preliminary Results: Pure Water • Critical Grashof number: 7870 • Critical wavenumber: 2.75 (water, Pr = 7, A = )

  14. 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

  15. 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

  16. 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 Questions

  17. 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.

  18. 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.

  19. Thank you for the opportunity to speak today.Any questions? Questions http://i110.photobucket.com

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