The basal boundary condition

The constant of integration may be evaluated by using the boundary condition $ = $o on z = 0. In other words, we presume that the temperature gradient at the bed, $o, is known or can be estimated. Making these substitutions in Equation (6.18) yields ec = $o. Thus, replacing ec with $o and $ with d0/dz in Equation (6.18) yields:

This is a solution for the temperature gradient as a function of elevation above the bed.

The requirement that the temperature gradient in the basal ice be known is fundamentally unavoidable. However, this is not as serious a problem as one might, at first, expect. In the steady state, $o is adjusted so that all of the heat coming from within the Earth, the geothermal flux, can be conducted upward into the ice. Thus, if the geothermal flux can be estimated, $o can be calculated because the constant of proportionality between the two, the thermal conductivity of ice, K, is known.

To clarify the physical processes by which $o is adjusted, consider a non-steady-state situation in which $ o is too low. Some of the geothermal heat would then remain at the ice-rock interface where it would warm the ice. Because the temperature decreases upward in the glacier, the ice being colder than the Earth's interior, such warming would increase $o until all of the heat could be conducted upward into the ice, thus tending to re-establish the steady state. (For the moment, we neglect basal melting.)

Geothermal heat is produced by radioactive decay in the crustal rocks as well as by residual cooling of the mantle and core. Numerous measurements of the geothermal flux have been made, so we have a fair idea of its magnitude in different geological terranes. Geophysi-cists use the heat flow unit, or HFU, to describe this flux: 1 HFU is

Table 6.1. Geothermal fluxes in some geological terranes in which glaciers are or were found

Heat flux

Table 6.1. Geothermal fluxes in some geological terranes in which glaciers are or were found

Heat flux

Basal gradient

Locality

HFU

mW m 2

Km-1

Reference

Canadian Shield

0.8

33

0.0151

World average

1.2

50

0.0226

East Antarctica

1.21

50

0.0226

Budd etal, 1971

Baffin Bay

1.35

56

0.0255

West Antarctica

1.41

59

0.0264

Budd etal., 1971

1 Estimated.

1 Estimated.

1 ^cal cm-2 s-1. In glaciology, however, it is more common to use W m-2. The world-wide average geothermal flux is 1.2 HFU or 50 mW m-2. This corresponds to a temperature gradient in basal ice of 0.0226 K m-1. The gradient in the underlying rock will normally be somewhat different as the thermal conductivity of the rock will not be the same as that of the ice. In general, geothermal fluxes are highest in volcanic terranes, high in geologically young terranes, and lowest in geologically ancient terranes. A few examples of geothermal fluxes in glaciated areas are given in Table 6.1.

In the discussion above, we asserted that knowledge of ft o was "fundamentally unavoidable". It is true, of course, that a boundary value problem such as this could be solved with some other basal boundary condition, such as the basal temperature. (This will be left as an exercise for the reader.) However, as the basal temperature is one of the quantities that we are particularly eager to determine, and as basal temperatures are much harder to estimate from existing data than are basal temperature gradients, choosing fto as the basal boundary condition is the only logical choice in most situations.

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