How does ice deform under stress?  Bring 2 questions about each paper to class on Monday April 06.

• Alley, R.B. 1992. Flow law hypotheses for ice-sheet modeling. J. Glaciol. 38(129), 245-256.
• Budd, W.F. and T.H. Jacka. 1989.  A review of ice rheology for ice sheet modelling.  Cold Regions Science and Technology 16. 106-144.

• Paterson, W.S.B. 1994.  The physics of Glaciers, Chapter 5, 78-102. Structure and deformation of ice.

• Paterson, W.S.B. 1994. The physics of Glaciers, Appendix 1, 410-419. Stress and strain

• Azuma, N. 1994. A flow law for anisotropic ice and its application to ice sheets. Earth and Planetary Science Letters 128, 601-614.

• Thorsteinsson, T., and E.D. Waddington. 2003. Spatial and temporal scales of anisotropic effects in ice-sheet flow. Annals of Glaciology 37, 40-48.

• Paterson, W.S.B. 1994 The physics of glaciers, 3rd ed.  Chapter 11 - Steady flow 238-247.

Solving the  momentum equations for all stress components can be difficult and time-consuming, so it is advantageous to know when simpler approximations can be used. The Shallow Ice Approximation (SIA) allows us to assess when and if various stress terms are negligible.

Ed's notes on Shallow Ice Approximation (SIA)

Depth-age relations are essential for interpretation of ice-core paleoclimate records.  To find the age of layers at various depths, we can track ice particles along their trajectories; all particles of the same age comprise an internal layer, and where that layer intersects an ice core, it can be used to date the ice there.

Dansgaard, W., and S.J. Johnsen.  1969.  A flow model and a time scale for the ice core from Camp Century, Greenland.  Journal of Glaciology 8(53), 215-223.

is a classic paper about tracking particles in 1-D.  We will use concepts form this paper to develop particle paths and internal layers in 2-D flowbands.

Several papers by Niels Reeh in 1988-89 introduced formulations of ice flow that are useful for layer tracking.

Reeh, N. 1988. A flow-line model for calculating the surface profile and the velocity, strain-rate, and stress fields in an ice sheet. Journal of Glaciology 34(116), 46-54.

Reeh, N. and W.S.B. Paterson. 1988. Application of a flow model to the ice-divide region of Devon Island Ice Cap, Canada.  Journal of Glaciology 34(116), 55-63.

Reeh, N. 1989. The depth-age profile in the upperpart of a steady-state ice sheet.  Journal of Glaciology 35(121), 406-417.

When the shape of the horizontal-velocity profile with depth can be characterized, Ed's notes on Velocities, Paths, and Layers provide a generic framework to calculate isochrones in a flowband for accumulation studies, for divide-migration studies, and for ice-core analyses

For a more recent treatment of particle paths and layers, Fred Parrenin and Richard Hindmarsh are using coordinate transformations to turn particle paths into straight lines, and to derive analytical solutions for particle paths and isovjhrones.  If you are interested, check out

Parrenin, F., R.C.A. Hindmarsh, and F. Remy. 2006. Analytical solutions for the effect of topography, accumulation rate and lateral flow divergence on isochrone layer geometry. Journal of Glaciology 52(177), 191-202.

They used the method to reconstruct layers in

Parrenin, F., and R.C.A. Hindmarsh. 2007. Influence of a non-uniform velocity field on isochrone geometry along a steady flowline of an ice sheet. Journal of Glaciology 53(183), 612-622.

Forces exerted on structures by creeping snow can be an important engineering and environmental consideration.  See -

• McClung, D. 1982. A one-dimensional analytical model for snow creep pressures on rigid structures. Can. Geotech J. 19(4), 401-412.

The basal boundary condition is very important for temperate glaciers, and for ice sheets that reach the pressure melting temperature at their beds. For fundamentals of sliding, see Van der Veen, Chapter 4.

• Van der Veen, C.J. 1999. Basal sliding.  in Fundamentals of glacier dynamics, Balkema, Rotterdam, Chapter 4, p.66-102.

For discussion of the impact of basal water pressure,

• Anderson, R.S. et al. 2004. Strong feedbacks between hydrology and sliding of a small alpine glacier. JGR 109 F03005, doi: 10.1029/2004JF00120.

The theory of response of glaciers to climate change was first developed by J.F. Nye in the 1960s.  For a discussion of response times, see

• Johannesson, T., C.F. Raymond, and E.D. Waddington. Time-scale for adjustment of glaciers to changes in mass balance. 1989. Journal of Glaciology 35(121), 355-369.

For extension to include lapse-rate feedback,

Harrison, W.D., D.H. Elsberg, K.A. Echelmeyer, and R.M. Krimmel. On the characterization of glacier response by a single time constant.  Journal of Glaciology 47(159), 659-664.