8.1. Water flowing along a glacier bed must warm up as the ice thins and the pressure melting point increases. Water flowing up an adverse bed slope must warm up more rapidly, as the ice is thinning more rapidly. The energy needed to warm the water comes from viscous dissipation. Determine how steep the bed slope can get, relative to the surface slope, without exceeding the amount of viscous energy available. Obtain a numerical value for the constant of proportionality between the two slopes.

8.2. The discharge in a horizontal subglacial conduit with a circular cross section is 0.025 m3 s—1. The water pressure in the conduit is 1.5 MPa and the hydrostatic pressure in the adjacent ice is 2.0 MPa. The Manning roughness of the conduit is 0.1 m—1/3s and the viscosity parameter, B, is 0.16 MPa a1/3. Determine the pressure gradient in the conduit, the radius of the conduit, the water velocity in the conduit, and the melt rate on the conduit walls (or closure rate).

8.3. An esker splits as shown in Figure P2. Stratigraphic relations suggest that the branch around the end of the ridge is younger. Explain why the esker changed course, and estimate the basal shear stress at the time of the change in course. Assume that the glacier had a parabolic profile, h = ■sfcx. Assume further that water flow down the potential gradient could be maintained even though some water might be forced to refreeze to keep the temperature at the pressure melting point.

8.4. Consider a glacier with a parabolic profile, h = where x is the horizontal coordinate in meters and h is the surface elevation. Assume that the glacier is 2 km long and is on a horizontal bed. It is drained by a circular conduit at the bed. Calculate and plot the height of the hydraulic grade line as a function of distance from the terminus for discharges of 0.015 m3 s— 1, a winter

Outwash fans

Contour interval 10 m

Figure P2.

discharge, and 1.0 m3s-1, a summer discharge. Use a channel Manning roughness of 0.1 m-1/3s and ice viscosity parameter, B = 0.06 MPa a1/3. Assume that the conduit is at atmospheric pressure within 50 m of the margin. (As the integration has to be carried out numerically, you might want to write a short program to do the calculations.)

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