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Figure 14.15. Theoretical response of the terminus of South Cascade Glacier to a sinusoidal perturbation of amplitude b, period, T, and frequency, w. Curves shown are the phase lag, <p, the time lag <p/w, and the amplitude of the response |H|/fr|. (After Nye, 1963b, p. 107, Figure 9. Reproduced with permission of the author and the Royal Society, London.)

The curve of y in Figure 14.15 is the phase lag between the variation in budget and the response of the terminus. For example, for an oscillation in mass balance that has a period of 100 years, the phase lag is approximately 110°. This means that the maximum thickness of the glacier at the terminus (and hence the maximum extent of the glacier) would occur (110/360) ■ 100 = 31 years after the maximum in the mass balance. This latter number can be read from the curve of y/a>, using the inner scale on the left side of the figure. Thus y/rn is the time lag between the maximum accumulation rate and the maximum thickness. For variations in budget with very long periods, the phase lag decreases, but the time lag does not change appreciably. For example, for an oscillation with a period of 1000 years, the time lag is ~43 years. Conversely, for oscillations with a period of only 1 year, which would represent the seasonal cycle from winter accumulation to summer melt, y = 90° so the time lag is 1/4 year. In other words, the maximum thickness does not occur when the rate of snow fall is a maximum, but rather at the end of the accumulation season when accumulation gives way to melt.

The curve of \H\/b1 shows the change in thickness of the glacier at the terminus, expressed in terms of the perturbation in accumulation. For a perturbation with a period of 100 years, the increase in thickness here would be about 100 times the amplitude of the perturbation. Thus, a perturbation with an amplitude of 0.1 m would produce a change in thickness of ~10 m.

The ultimate objective of an analysis such as this might well be to solve the inverse problem, namely, given a history of advance and retreat of a glacier, to deduce the mass balance history and thus to learn something about the climatic changes that produced the fluctuations. Nye (1965b) did this for South Cascade Glacier and for Storglaciaren with mixed results. He concluded that the records of terminus position of the two glaciers were not sufficiently well known to accurately deduce annual changes in net budget, but that coarser features of the records yield net balance figures that are in agreement with decadal means of recent observations.

It is of interest to use the data in Figure 14.14 to estimate the respective time scales from Equations (14.19). As South Cascade Glacier averages about 800 m in width, tC = 47 years and tD = 33 years. T. Johannesson (written communications dated November 7 and 14, 1996) suggests that time scales calculated in this way, however, are likely to be maximum estimates because many perturbations do not cover the entire glacier and thus are advected and diffused over the glacier more rapidly. Nevertheless, the relative magnitudes should be correct. Because tD < tC, disturbances should be damped by diffusion before a significant unstable response is generated. T. Johannesson (written communication dated December 23, 1995) finds that this is generally the case, and thus argues that diffusion cannot be neglected.

As with Storglaciaren, it is difficult to estimate tv for South Cascade Glacier because, again, there is a riegel beneath the middle of the ablation area. However, it appears that 100 < h omax < 200mand b(i0) = 5ma-1, so 20 < tv < 40 years. Using detailed field data to evaluate be, Ge, andH, Harrison et al. (2001) obtained a value of tVH of ~36 years. These values are reasonably consistent with those for tC and tD obtained above, particularly considering that the latter are likely to be maximum values. This is somewhat unusual, however, as Johannesson et al. (1989) find that tv is usually significantly longer than tC or tD, as noted earlier. The value of tVH is also consistent with Nye's estimate of 43 years. For comparison, the 1/r0 time scale for South Cascade Glacier is about 15 years.

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