Z t 2 e

where & is the period of the variation (in this case, radians per year) (Carslaw and Jaeger, 1959, p. 65). If the seasonal variation in temperature at the surface is sinusoidal, the temperature profile at any given time during the year can be calculated from Equation (6.31). Some profiles for representative times are shown in Figure 6.8.

Now at any given depth, 0max(z,t) and 0min(z,t) occur when — z^K = ±tt/2, respectively. Thus, using Equation (6.31) to obtain 0(z,t) at these two times, and solving for z yields:

From this we find that z = 10m if k = 16 m2 a-1, a value appropriate for unpacked snow, and A is 1.2% of 0r. In ice, with k = 37.2 m2a—1, z = 15.2 m. Thus, temperatures measured at a depth of 10 m in snow and firn or 15 m in ice should closely approximate the mean annual temperature.

The most serious assumption in this calculation is that accumulation and ablation can be ignored. In the accumulation area, accumulating snow insulates the surface, reducing 0 r. This is probably not too serious a problem. However, in the ablation area, as noted earlier, there is not only the insulating effect of snow during the fall and winter, but also warming by percolating meltwater in the late spring or early summer and then lowering of the ice surface later in the summer. Combined, these processes result in 15 m temperatures that are likely to be warmer than the mean annual temperature, as noted.

Equation (6.32) can also be used to calculate effects of temperature oscillations over longer time spans. For example, oxygen isotope variations in ice cores have revealed temperature cycles during the Pleistocene that have a range ~5 °C and a period of ~2000 years. These cycles were discovered by Dansgaard and Oeschger (1989) and are called Dansgaard-Oeschger cycles (see also Dansgaard et al., 1993). Their cause is not yet understood. With k = 38 m2a-1, a value appropriate for ice at -10 °C, and a more liberal A of 0.050r, z = 466 m. Thus, such cycles would affect basal temperatures significantly only near the margin where the ice was less than ~500 m thick.

The speed, w, of propagation of a temperature maximum or minimum resulting from such a sinusoidal cycle of temperature is (2«k)1/2, and the time lag between the maximum at the surface and the maximum at some depth, z, is thus, z/w. In the case of the Dansgaard-Oeschger cycles, the speed would be about 0.49 ma-1, and the signal would reach a depth of 466 m after about 950 years.

Let us now consider the temperature profile in the firn area some distance from an ice divide, a problem studied by Robin (1970). We will restrict the problem to two dimensions; assume that strain heating is negligible; and ignore conduction as K is low in firn, while the advective terms are significant. Equation (6.13) then becomes:

d0/dt may be thought of as being composed of two parts, a thickening or thinning of the ice sheet with time, and climatic change, thus:

Here, ezzH represents the change in thickness of the ice sheet by flow (or vertical strain), bn represents thickening by accumulation, and the difference between them is the net change in surface elevation. Multiplication by the lapse rate, X, or rate of decrease in temperature with increasing elevation yields the resulting change in temperature at the glacier surface. (As ezz is normally compressive, or negative, in an accumulation zone, ezzH will be a negative number; bn is positive. If (ezzH + bn) turns out to be negative, representing net thinning, multiplication by the negative lapse rate yields a positive d0/dt.) To this is added any change in temperature due to secular climatic change, d0o/dt.

d0 dt d0 d0

d0 d0o

Figure 6.9. Interpretation of z terms in Equation (6.34). The expression u ■ dQ/dx represents the change in temperature at the glacier surface as the ice flows to lower elevations; dQ/dx can thus be replaced with aX, where a is the surface slope of the glacier. Finally, w is equated with the accumulation rate, bn. Making these substitutions in Equation (6.33) yields:

dQo dQ

The meaning of the terms in Equation (6.34) can be clarified by reference to Figure 6.9. A particle of snow deposited at A has moved to B after n years, and is buried under nbn meters of new accumulation. In the absence of conduction, it is still at the temperature at which it was deposited at A. The surface above B was at C when the snow was deposited at A and, owing to the lapse rate, snow then accumulating at C was nuaX degrees warmer than that which was accumulating at A.In addition, the ice sheet has thinned by an amount n(ezzH + bn) over the intervening years, and the surface is now at D, which is n(ezzH + bn)X degrees warmer than C. Finally, there may have been secular climatic warming at a rate dQo/dt, so snow at D is n dQo/dt warmer than it would be otherwise. Thus the surface at D is:

d t warmer than the firn at B. To obtain the temperature gradient from B to D, divide by nbn and cancel the ns, thus:

d z bn which, with minor manipulation, can be shown to be the same as Equation (6.34).

When one is far from the edge of an ice sheet, it is very difficult to determine whether the ice sheet is thickening or thinning; that is, whether (ezzH + bn) is positive or negative. One can measure bn in snow pits; the problem is to measure ezz without a stationary base upon which to establish a survey point. Furthermore, such observations would span

Table 6.2. Values of parameters in Equation (6.34) for Site 2 in Greenland and Byrd Station in Antarctica

Site 2 Byrd Station us, m a 1

0.001 15 0.031

0.000 24 0.018

Note that de/dz is measured below 150 m depth.

only a short time interval. However, suppose we can measure u, a, k, bn, and de/dz and have reason to believe that deo/dt is negligible. Then, Equations (6.34) or (6.35) can be solved for (ezzH + bn). Two examples are shown in Table 6.2. The results, a 0.031 m a-1 thickening rate at Site 2 in Greenland and a 0.018 m a-1 thinning rate at Byrd Station in Antarctica, are surprisingly reasonable.

While potentially providing a sensitive measure of the state of health of an ice sheet, this technique is probably not especially useful because moderately deep boreholes are needed to obtain de/dz, and deo/dt is not known well enough. However, the derivation of Equations (6.34) and (6.35) serves to emphasize that, in general, as one moves away from the divide, temperature gradients near the surface of an ice sheet become positive; that is, the temperature decreases with depth (decreasing z). We now turn our attention to a more sophisticated model that enables us to investigate such temperature distributions deep in the ice and far from a divide.

0 0