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 Stress and strain rate data Glen’s Flow Law Glen’s Flow Law Inverse Problem Improvement on Glen’s Flow Law Improvement on Glen’s Flow Law Reconstruction of crystal orientation fabrics Reconstruction of crystal orientation fabricsExplorationResults

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

how does ice deform? Glen’s Flow Law (Glen’s flow law has a few problems: ) “semi-empirical”A 0 depends on many things Shear strain rate in direction of flow (x) Deviatoric shear stress in flow direction (x) empirically-determined exponent (from ???) Background Inverse Problem Exploration Results

theoretical improvement on flow law A 0 is not a universal constant: varies with ice grain size ice grain size impurity content impurity content ice crystal orientation (“fabric”) ice crystal orientation (“fabric”) How can the flow law be written to take the crystal orientation fabric into account? Background Inverse Problem Exploration Results

theoretical improvement on flow law Background Inverse Problem Exploration Results 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 = G = [Paterson]

new flow law using fabrics define shear stress in terms of G: [Azuma (1994)] The old A 0 is now in terms of G: it includes crystal orientation fabric! - N E W F L O W L A W - Background Inverse Problem Exploration Results (DATA)(Physics)(MODEL)

the inverse problem Four parameters: c x, m x, c z, m z Nonlinear problem Solve 240 times, independently at 10 m depth intervals from 0 to m (DATA)(Physics)(MODEL) Background Inverse Problem Exploration Results

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

add noise Strain rate: ~ /yearStrain rate error: 7 x /year Depth (m) Background Inverse Problem Exploration Results

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

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

what about error? Error is 10 orders of magnitude larger than the result! (c x ) (c z ) (m x (m x ) (m z ) (bed) Depth (surface)

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

fabric orientation conclusion Crystal fabric c x ~ 4 x (green) c x ~ 4 x (green) c z ~ (blue) c z ~ (blue) Slip direction m x ~ 5 x (red) m x ~ 5 x (red) m z ~ 0.02 (black) m z ~ 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 m is more strongly vertical than c m is more strongly vertical than c Background Inverse Problem Exploration Results

recap Data: strain rate Model: G = Stress and strain rate data Stress and strain rate data Glen’s Flow Law Glen’s Flow Law Inverse Problem Improvement on Glen’s Flow Law Improvement on Glen’s Flow Law Reconstruction of crystal orientation fabrics Reconstruction of crystal orientation fabricsResults Ice crystals are oriented ~vertically Ice crystals are oriented ~vertically Maximum shear stress is ~vertical Maximum shear stress is ~vertical Weak conclusions due to large error Weak conclusions due to large error Crystal orientation fabrics are not reconstructible Crystal orientation fabrics are not reconstructible (DATA) (Physics)(MODEL) Crystal fabric c x ~ 4 x c z ~ Slip direction m x ~ 5 x m z ~ 0.02

references 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., (1992). Azuma, N., “A flow law for anisotropic ice and its application to ice sheets,” EPS 128, (1994). Harper, J.T. et al., “Spatial variability in the flow of a valley glacier,” J. Gphys Res., 106 B (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 (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, (2002). Background Motivation Exploration Results