Managed by UT-Battelle for the Department of Energy 1 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

Managed by UT-Battelle for the Department of Energy 2 Outline  Outline  Motivations  Background and the model  Governing equations  Preliminary results  Ongoing Work  Video  Questions

Managed by UT-Battelle for the Department of Energy 3 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

Managed by UT-Battelle for the Department of Energy 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”

Managed by UT-Battelle for the Department of Energy 5 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 = 0T = 1

Managed by UT-Battelle for the Department of Energy 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

Managed by UT-Battelle for the Department of Energy 7 Background and the Model Solution Variables  Grashof number, Gr: fluid parameter  Wavenumber, m: spatial scale Larger mSmaller m m

Managed by UT-Battelle for the Department of Energy 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 InvestigatorsFluidPrGr*m*A Batchelor, 1954 (theory)air n/a  Vest & Arpaci, 1969 (theory)air Vest & Arpaci, 1969 (experiment)air  Bergholz, 1978 (theory)air Bergholz, 1978 (theory)pure water Bergholz, 1978 (theory)pure water Ruth, 1979 (theory)air  Ruth, 1979 (theory)water  McBain & Armfield, 2003 (theory)air  McBain & Armfield, 2003 (theory)water  Evans, to be submitted (theory)air 

Managed by UT-Battelle for the Department of Energy 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 InvestigatorsFluidPrGr*m*A Batchelor, 1954 (theory)air n/a  Vest & Arpaci, 1969 (theory)air Vest & Arpaci, 1969 (experiment)air  Bergholz, 1978 (theory)air Bergholz, 1978 (theory)pure water Bergholz, 1978 (theory)pure water Ruth, 1979 (theory)air  Ruth, 1979 (theory)water  McBain & Armfield, 2003 (theory)air  McBain & Armfield, 2003 (theory)water  Evans, to be submitted (theory)air 

Managed by UT-Battelle for the Department of Energy 10 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.

Managed by UT-Battelle for the Department of Energy 11 Governing Equations  Nondimensionalized Navier-Stokes equations, continuity equation, conservation equations for temperature and concentration  Linear Stability Analysis: eigenvalue problem

Managed by UT-Battelle for the Department of Energy 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] stable region unstable region stable region unstable region

Managed by UT-Battelle for the Department of Energy 13  Critical Grashof number: 7870  Critical wavenumber: 2.75 (water, Pr = 7, A =  ) InvestigatorsFluidPrGr*m*A Batchelor, 1954 (theory)air n/a  Vest & Arpaci, 1969 (theory)air Vest & Arpaci, 1969 (experiment)air  Bergholz, 1978 (theory)air Bergholz, 1978 (theory)pure water Bergholz, 1978 (theory)pure water Ruth, 1979 (theory)air  Ruth, 1979 (theory)water  McBain & Armfield, 2003 (theory)air  McBain & Armfield, 2003 (theory)water  Evans, to be submitted (theory)air  Preliminary Results: Pure Water

Managed by UT-Battelle for the Department of Energy 14 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

Managed by UT-Battelle for the Department of Energy 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

Managed by UT-Battelle for the Department of Energy 16 Questions CNN, Wednesday, July 30, 2008: “Ice Sheet breaks loose off Canada” Video (2:16) - cnnSTCVideo - cnnSTCVideo

Managed by UT-Battelle for the Department of Energy 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):  R. F. Bergholz. Instability of natural convection in a vertical fluid layer. J. Fluid Mech., 84:  K. J. Evans. Linear stability analysis as a temporal accuracy benchmark. To be submitted,  The MathWorks Inc. MATLAB 7.4. Licensed under Dartmouth College,  G.D. McBain and S.W. Armfield. Natural convection in a vertical slot: accurate solution of the linear stability equations. Sydney, NSW, July Eleventh Computational Techniques and Applications Conference, Fifth International Congress on Industrial and Applied Mathematics.  GLIMMER.  S. Mergui and D. Gobin. Transient double diffusive convection in a vertical enclosure with asymmetrical boundary conditions. J. Heat Transfer, 122:  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.,  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:  C. M. Vest and V. S. Arpaci. Stability of natural convection in a vertical slot. J. Fluid Mech., 36:  J. A. C. Weidman and S. C. Reddy. A MATLAB differentiation matrix suite. ACM Transactions on Mathematical Software. 26(4):

Managed by UT-Battelle for the Department of Energy 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- 00OR 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.

Managed by UT-Battelle for the Department of Energy 19 Questions Thank you for the opportunity to speak today. Any questions?