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Geos 300 Homework 11: Estimating a glacial surface profile using the perfectly plastic model of ice deformation (using Excel)

The basal shear stress of a glacier (t) is proportional to the sine of the angle of the ice surface (a) times the acceleration of gravity (g), times ice density (r), times thickness (h):

t   =  rgh(sina).                                                                      (1)

Adopting the small angle approximation, i.e. sina = tana, equation (1) can be written:

t =  rgh|Dh/Dx+Db/Dx|                                                           (2)

where h is the thickness of the ice, b is the bed elevation, and x is horizontal distance. If the ice thickness and the bed elevation are represented by a sequence of values h_i and b_i, where i = 1,2,3…. then equation (2) can be written in terms of the ice thickness and bed elevation at a point i and the thickness and bed elevation at the next point, i+1, downice:

t =  rgh_i/Dx|h_i – h_i+1+ b_i – b_i+1|                                                 (3)

If one prescribes a value for t, Equation (3) can be used to solve for the glacier shape. Equation (3) is first applied at the terminus, where b_i, b_i+1, and h_i+1 (= 0) are known. Equation (3) is a quadratic equation for h_i with solution

h_i = ((h_i+1+ b_i+1 – b_i) + sqrt((h_i+1+ b_i+1 – b_i)²+4tDx/rg))/2         (4)

Given the solution for h_i, one can then apply equation (3) at the next point upice, and so on up the valley. The file “glacialprofile.xls” does this for you given values that you input for t, r, g, and b_i for i = 1,2,3…

Using the procedure described above and the spreadsheet model, reconstruct a valley glacier of your choice. Elevation data can be extracted from Google Earth.

• ENTER upice distance in column A about every 1 km, starting at the terminus, up the main stream.
• DETERMINE centerline valley-floor elevations at each step and ENTER them in column B.
• DETERMINE lateral moraine elevations at the same steps as far upvalley as possible, and ENTER them in column I.
• VARY the Effective Basal Shear Strength (column C) as required to match near-terminus ice thickness to the moraines.
• USE a 1.0 bar Effective Basal Shear Stress within the mountains.
• GRAPH your longitudinal profiles.
• Briefly DISCUSS your results.