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inverting for ice crystal orientation fabric

inverting for ice crystal orientation fabric. Kristin Poinar Inverse theory term project presentation December 7, 2007 The University of Washington. outline. Background Stress and strain rate data Glen’s Flow Law Inverse Problem Improvement on Glen’s Flow Law

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inverting for ice crystal orientation fabric

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  1. inverting forice crystal orientation fabric Kristin Poinar Inverse theory term project presentation December 7, 2007 The University of Washington

  2. outline • Background • Stress and strain rate data • Glen’s Flow Law • Inverse Problem • Improvement on Glen’s Flow Law • Reconstruction of crystal orientation fabrics • Exploration • Results

  3. ^ z ^ x how does an ice sheet deform? h (thickness) Deviatoric stress [Paterson] Background Inverse Problem Exploration Results

  4. how does ice deform? Glen’s Flow Law (Glen’s flow law has a few problems: ) “semi-empirical” A0 depends on many things Deviatoric shear stress in flow direction (x) Shear strain rate in direction of flow (x) empirically-determined exponent (from 1.2 - ???) Background Inverse Problem Exploration Results

  5. theoretical improvement on flow law • A0 is not a universal constant: varies with • ice grain size • impurity content • ice crystal orientation (“fabric”) • How can the flow law be written to take the crystal orientation fabric into account? Background Inverse Problem Exploration Results

  6. theoretical improvement on flow law • Crystal orientation fabric vector: Direction of ice crystal axes (perpendicular to basal plane) • c = • Slip vector: Direction of maximum shear stress on basal plane • m = • c and m define G • G = • G = [Paterson] Background Inverse Problem Exploration Results

  7. new flow law using fabrics [Azuma (1994)] define shear stress in terms of G: The old A0 is now in terms of G: it includes crystal orientation fabric! - N E W F L O W L A W - (DATA) (Physics) (MODEL) Background Inverse Problem Exploration Results

  8. the inverse problem • Four parameters: cx, mx, cz, mz • Nonlinear problem • Solve 240 times, independently at 10 m depth intervals from 0 to -2400 m (DATA) (Physics) (MODEL) Background Inverse Problem Exploration Results

  9. insert a fabric signal into the strain rate LGM fabric : strain rate increases by factor of ~2.5 arbitrary increases in strain rate, just because Depth (m) Volcanic ash impurities: very slight strain rate increase Log( Strain Rate ) (yr-1) Background Inverse Problem Exploration Results

  10. Depth (m) Strain rate: ~ 10-4 /year Strain rate error: 7 x 10-6 /year add noise Background Inverse Problem Exploration Results

  11. calculate G = . Log( Elements of G ) G (bed) Depth (m) (surface) Background Inverse Problem Exploration Results

  12. solve for c (crystal fabric axis) and m (slip) • Crystal fabric • cx ~ 4 x 10-5(green) • cz ~ 0.005 (blue) • Slip direction • mx ~ 5 x 10-4(red) • mz ~ 0.02 (black) • Is there any change with depth? • Strain rates we added are NOT recoverable log( Magnitude ) (bed) Depth (surface) Background Inverse Problem Exploration Results

  13. what about error? • Error is 10 orders of magnitude larger than the result! (mx) (mz) (cx) (cz) (bed) Depth (surface)

  14. the error limits the fit (bed) Depth (surface)

  15. Crystal fabric cx ~ 4 x 10-5(green) cz ~ 0.005 (blue) Slip direction mx ~ 5 x 10-4(red) mz ~ 0.02 (black) Crystal fabric: axis points in direction normal to an ice crystal’s basal plane (averaged over all crystals) c = z, approximately Slip direction axis points in direction of maximum shear stress on the basal plane m = z, approximately fabric orientation conclusion m is more strongly vertical than c Background Inverse Problem Exploration Results

  16. recap • Data: strain rate • Model: G = • Stress and strain rate data • Glen’s Flow Law • Inverse Problem • Improvement on Glen’s Flow Law • Reconstruction of crystal orientation fabrics • Results • Ice crystals are oriented ~vertically • Maximum shear stress is ~vertical • Weak conclusions due to large error • Crystal orientation fabrics are not reconstructible (Physics) (MODEL) (DATA) • Crystal fabric • cx ~ 4 x 10-5 • cz ~ 0.005 • Slip direction • mx ~ 5 x 10-4 • mz ~ 0.02

  17. Achterberg, A. et al., “First Year Performance of the IceCube Neutrino Telescope,” Astropart. Phys. 20, 507 (2004). Alley, R. “Flow-law hypotheses for ice-sheet modeling,” J. Glac., 38 129 (1992). Azuma, N., “A flow law for anisotropic ice and its application to ice sheets,” EPS 128, 601-604 (1994). Harper, J.T. et al., “Spatial variability in the flow of a valley glacier,” J. Gphys Res., 106 B5 8547-62 (2001). Mase, G.T. and G.E. Mase, Continuum Mechanics for Engineers, CRC Press (1999). Pettit, E. and E.D. Waddington, “Anisotropy, abrupt climate change, and the deep ice in West Antarctica,” NSF proposal 0636795 (2006). Paterson, W.S.B., The Physics of Glaciers, Pergamon Press (1981). Price,P.B. et al, “Temperature profile for glacial ice at the South Pole,” PNAS 99 12, 7844-7 (2002). references Background Motivation Exploration Results

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