## Further study of the response time

Johannesson et al. (1989) have studied the question of response times and of conditions at the terminus in greater detail. They identify three possible natural time scales that might be used in the analysis of glacier responses:

m3m 1

Here, tC and tD are time constants for propagation or diffusion* of a disturbance over the length of a glacier. In effect, they are measures of the time required to establish the general shape of the new thickness profile, h1(x, t). As the size of such a disturbance decreases with time, the rate of propagation or diffusion also decreases. Therefore, as with 1/Y0, tC and tD are measures of the time required for the processes to proceed about 2/3 of the way to completion. Similarly, as we shall show below, tV is the time required for accumulation (or loss) of about 2/3 of the volume (per unit width), V1, required to re-establish an equilibrium geometry after a perturbation in mass balance that adds a volume of ice b1l0 to the glacier every year.

Johannesson et al. find that tV is usually appreciably longer than tC or tD. This means that perturbations in ice thickness are spread out over the glacier by propagation and diffusion rather quickly in comparison with the time needed for accumulation of the additional mass. Clearly, a glacier cannot be considered to have returned some specified fraction of the way to a new equilibrium state until the necessary additional mass has accumulated (or been lost).

In an extension of the Nye theory, tV would be viewed approximately as follows: at any given time after a perturbation in mass balance, the mean perturbation in thickness, averaged over the length of the glacier, would equal the perturbation at the terminus, h1(l0,t), multiplied by some function of the conditions in the datum state, of the magnitude of the perturbation, b1 , and of time, thus:

Remember that l0 is the position of the terminus in the datum state. Once a new equilibrium geometry has been attained, at t the increase in volume of the glacier would be obtained by multiplying Equation (14.20) by the length of the glacier, thus:

However, once a new steady state has been attained, the annual mass gain resulting from the perturbation, b1l0, must equal the flux past the old terminus position, ub0(£0)-h1, thus:

Eliminating h1 from these two equations yields:

* The n2 term in Equation (14.19b) comes from the Fourier solution of the diffusion equation (T. Johannesson, written communications dated November 7 and 14, 1996).

Whence, from Equation (14.19c):

Wb0(i0)

Thus, by this approach, tV turns out to be sensitive to the unknown sliding speed at the terminus. In addition, the function, f, is highly sensitive to the details of the variations in c0 and D0, especially near the terminus (Johannesson et al., 1989, p. 364 and appendices).

Johannesson et al. have developed a much simpler geometrical argument to estimate tV. Consider the situation in Figure 14.9 in which an advance of a glacier by an amount Al is illustrated graphically by cutting the glacier at its point of maximum thickness and sliding the lower part forward by Al. Then, the increase in volume of the glacier is approximately Al-h0max. Detailed numerical modeling suggests that this is a good approximation to the response of a real glacier when the dynamical properties of the glacier are the same in the initial and final state, and thus influence the initial and final profiles in the same way. Now, rather than equate the annual mass gain resulting from the perturbation with the flux past the old terminus position, as in Equation (14.22), we equate it with the mass loss over the new part of the glacier, Al, thus:

where bt is the mean net balance rate over Al which may be approximated by b(lo), the net balance rate at the terminus (a negative quantity) if Al is small. Therefore:

Whence, from Equation (14.19c):

Thus, tV can be estimated, quite easily, from knowledge of the thickness of the glacier and the net balance rate at the terminus.

We noted above that tv is a time constant in the same sense that 1 /y0 is. Let us now demonstrate this. Immediately after a permanent change in balance rate, b1, the rate at which additional mass, V1, is acquired by the glacier, dV1/dt, is B1 = b1l0. However, as the glacier becomes longer (by an amount hi(t)), some of the additional annual input is lost through ablation in the new part of the terminus region. Thus:

Now from Equation (14.26) or Figure 14.9, hi = V1/h0max, so:

Ui h0max

Comparing this with Equation (14.15), it is clear that h0max/\b(i0)\ is the analog of 1/y0.

When calculating tV in practice, the three-dimensional geometry of the glacier must be taken into consideration. Thus, in the case of a glacier like Storglaciaren that has a number of overdeepened basins in its longitudinal profile, h0max needs to be replaced by an appropriate longitudinally averaged thickness. In addition, the terminus of Storglaciaren is constrained between bedrock and morainal highs so that its width is about half the average width of the glacier (Figure 12.9). Accordingly, Equations (14.25) to (14.27) need to be generalized to three dimensions. For example, if we write Equation (14.25) as:

where A0 is the initial area of the glacier and W(i0) is the width of the terminus, Equation (14.19c) becomes:

b Aq

V1 must now be estimated based on the glacier geometry. For instance, by analogy with Figure 14.9, one might consider that the new geometry could be approximated by (mentally) sliding forward the central part of the glacier of width W(i0). V1 is then h0max W(i0) hi, where h0max is a mean thickness over this central part. Inserting this in Equation (14.29) and using Equation (14.28) then yields:

Harrison et al. (2001) have pointed out that the formulation of tv in Equation (14.19c) ignores the normal increase in bn with elevation. Owing to this increase, perturbations that result in thickening or thinning of a glacier are effectively amplified. This positive feedback, known as the Bodvarsson effect (Bo3varsson, 1955), can lead to significant underestimates of the response time. In extreme cases, it may lead to unstable (or runaway) growth or shrinkage of a glacier.

To include this effect, Harrison et al. suggest adding a term to Equation (14.27), thus:

His now a thickness scale, not necessarily equal to h0 max, be is an effective balance rate at or just below the terminus, and Ge is the effective budget gradient. The subscript H is added to tv to distinguish this time scale from that of Johannesson et al. (Readers should refer to Elsberg et al. (2001) and Harrison et al. (2001) for a rigorous derivation of Equation (14.31) and definitions of be, Ge, and H. Suffice it to say here that the theory is grounded in the reference-surface approach to mass balance discussed briefly in Chapter 3 (p. 25).) It will be readily seen that the added term has the desired effect. Both Ge and H are positive, so subtracting GeH from |be| in the denominator increases tvH and in extreme cases may make tVH negative - the unstable response.

Unfortunately, be, Ge, and H are not easy to determine. To evaluate be one needs detailed balance rate data from the terminus that can be extrapolated into the area below the terminus and also a map showing bed elevations in the terminus area. In an application of their theory to South Cascade Glacier in the state of Washington (USA), Elsberg et al. (2001) found that setting be « 0.75 b(l0) yielded a good approximation to the true value. Likewise, rigorously H = dV/dA and neither A Vnor AA are measured routinely. Finally, an average value of Ge may not be appropriate; again Elsberg etal. (2001) found that setting Ge equal to 0.9 times the specific balance gradient half way between the terminus and the equilibrium line worked well on South Cascade Glacier, but they caution that these approximations may not be appropriate for other glaciers.

An important characteristic of Equation (14.31) appears if one notes that since be is a balance rate near the terminus, and bn = 0 at the equilibrium line, then be/Azt^e « Ge, where Azt^e is the elevation difference between the terminus and the equilibrium line. Then the ratio of the two terms in the denominator of Equation (14.31) is:

GeH H

Clearly, tVH becomes large as this ratio approaches unity, and the system is unstable if it exceeds unity. Azt^e will be smaller on glaciers that are relatively flat, and may approach H on such glaciers. Thus glaciers with low slopes may be expected to have longer response times, a characteristic that does not appear in Johannesson et al.'s formulation.

For purposes of illustration, let us put some realistic numbers into some of these equations. The ablation rate on the lower part of the tongue of Storglaciaren averages ~1.3 m a-1 and the mean thickness over the central region of the glacier is between 100 and 150 m, so tV is ~ 100 years. Rigorous estimates of Ge and be probably could be obtained for Storglaciaren, but as the necessary calculations have not been done let us use the approximations that Elsberg et al. found appropriate for South Cascade Glacier. Then, be « -0.98 m a-1 and Ge « 0.003 a-1 (Schytt, 1968, and unpublished data), so tVH ranges from ~150 to ~300 years, depending on the value of H. For comparison, numerical modeling (Brugger, 1992) suggests a response time of ~80 years, while field measurements show that about 2/3 of Storglaciaren's retreat from its Little Ice Age maximum position, which it reached in 1910, took place in ~45 years (Holmlund, 1987). The sizeable difference between tV and tVH probably reflects, in part, the fact that the denominator in Equation (14.31) is a small difference between two numbers that are large (compared with their difference) and that have large uncertainties. However, the result serves to emphasize the potential importance of the Bo 5 varsson effect. The more rapid response observed is likely to be a consequence of two factors: (1) b(l0) was probably higher (more negative) when the glacier extended to lower elevations, and (2) the change from Little Ice Age conditions was hardly a small perturbation. In any case, all of these times are substantially longer than the 1 /5r0 («13 years) time scale mentioned above. This is in part because diffusion is neglected in the latter, as noted, and in part because the 1 /5r0 time scale does not allow enough time to accumulate or lose the required mass, and thus violates conservation of mass.

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