J. Oerlemans - SIMPE GACIER MODES Figure 1. The slope of a glacier determines to a large extent its sensitivity to climate change. 1. A slab of ice on a sloping bed The really simple glacier has a uniform width, rests on a bed with a constant slope s (figure 2) has a constant balance gradient β, and behaves perfectly plastically in a global sense. So the specific balance can be written as (1.1) Here h is surface elevation and E the equilibrium-line altitude. The total mass budget of the glacier is zero when (1.2) 2 (H m + b! E) s (1.3) In eq. (1.3) H m is the mean ice thickness. Note that variations in the surface elevation do not play a role, because the balance profile is linear. It is also obvious that the equilibrium length does not depend on the balance gradient. Hm can be related to the slope of the bed by assuming perfect plasticity! g sh m = " b = b - sx x Figure 2. Geometry of the simple glacier model; the slope of the bed is constant and the equilibrium line horizontal. Solving for the glacier length yields: = equilibrium line " b dx =! " (H + b # sx # E) dx = h=b+h Altitude b =! (h " E) (1.4) -1-
Here! is the corresponding yield stress. Eliminating H m from eqs. (1.3)-(1.4) then leads to = 2 $! & s " gs + b % # E ' ) (1.5) ( The solution is illustrated in figure 3. The value of b! E has been set to 5 m,! /("g) to 1 m. For reference the solution for constant ice thickness (1 m) is also shown. For the full solution ice thickness increases with decreasing slope, which implies an upward shift of the equilibrium point (intercept of equilibrium line and glacier surface). So the dependence of on the bed slope now becomes stronger, especially for small values of s. It is clear that there is no solution for a flat bed, unless the equilibrium line is allowed to slope upwards. In a more realistic treatment the ice thickness not only depends on the slope, but also on the glacier length itself. This point will be considered later. The simple model can also be used to make an order-ofmagnitude estimate of climate sensitivity. From eq. (1.5) we find (km) 1 8 6 4 2 full solution H m = 1 m.2.4.6.8.1.12.14 Slope of bed Figure 3. Equilibrium length of a glacier on a bed with a constant slope. d =!2 /s (1.6) de So glaciers on a bed with a smaller slope are more sensitive in an absolute sense (figure 1). The fractional change in glacier length is 1 d de = $! & " gs + b % # E ' ) ( #1 (1.7) This quantity actually decreases with decreasing slope. Now we try to make a first-order estimate of the dependence of glacier length on air temperature. Assuming that the equilibrium line follows an isotherm (which admittedly is not a very good approaximation), we can write de dt fa =! 1 ", (1.8) where T fa is the ambient air temperature and γ is the temperature lapse rate (< ). Eqs. (1.6) and (1.8) can be combined to give: d = 2 dt fa! s (1.9) - 2 -
Figure 4 shows the result. arger valley glaciers typically have mean slopes between.1 and.2, implying that a 1 K temperature rise would lead to a 1 to 3 km decrease in glacier length. These figures appear reasonable, and we can conclude that the simple glacier model provides an interesting first-order description of the relation between climate change and glacier response. 2. Varying width Many glaciers have wider accumulation basins and narrow tongues, which will affect the sensitivity of their length to climate change. This can be investigated with two coupled basins of different width. Figure 5 illustrates the idea. A glacier tongue of constant width is nourished by an upper basin of length ub, also of constant, but different, width. To find the general solution we have to consider two cases, namely! ub and > ub. For! ub the solution is given by eq. (1.3): d/dt fa (km K -1 ) -2-4 -6-8 large valley glaciers.5.1.15.2.25 Slope of bed Figure 4. Relation between glacier length and air temperature as a fucntion of the bed slope. = 2 (H m + b! E). (2.1) s When the mass budget of the upper basin is positive, will be larger than ub. The length of the glacier is then determined by ub " (H + b! sx! E) dx + # " (H + b! sx! E) dx = (2.2) ub In this expression ξ is the width of the glacier tongue scaled with the width of the upper basin. For most glaciers! < 1. Evaluating the integrals yields: Figure 5. A perfectly plastic ice sheet on a originally flat bed with isostatic adjustment. The dashed curve refers to the case without adjustment of the bed.! 1 2 "s2 + "(H m + b! E) + + (1!")(H m + b! E) ub! 1 2 s(1!") 2 ub = (2.3) With E' = E! b! H m (E' < ), the solution is: =! 1 ) s E' + # E'2 +2s 1!" " (E' ub + 1 s 2 & + $ ub )' % 2 * + ( 1/ 2,. -. (2.4) An example is shown in figure 6. The upper basin has a length of 5 km. Other parameter values are: b = 2 m, s =.1, H m = 1 m. is plotted as a function of E for three different values of ξ. For! =.25, the width of the upper basin is four times that of the glacier tongue, which is - 3 -
not an unusual situation. Clearly, in this case the glacier length increases rapidly when the equilibrium line sinks below 185 m. In terms of the climate sensititivy d/de, a clear maximum exists when the glacier starts to form the tongue. When the equilibrium line sinks still further, d/de will approach the value of the original model glacier of uniform width (! = 1). When the lower part of the glacier is wider than the upper part the opposite is seen. The sensitivity is at a minimum when the budget of the upper basin is just positive. In conclusion we can state that glaciers with a narrow tongue are the ones that are most sensitive to climate change, especially when the glacier front is just below the upper basin. 3. Response times (km) 2 15 1 5! =4! =1! =.25 14 15 16 17 18 19 2 E (m) Figure 6. Glacier length as a function of the equilibrium line altitude, depending on the geometry as prescribed by ξ. Earlier estimates of glacier response time were based on the theory of kinematic waves (Nye, 196). However, this yielded values that, as we know now, are far too large (Van de Wal and Oerlemans, 1995). Estimates of response times roughly fall in two classes: in one case the estimate is based on the notion that ice velocity is the most relevant quantity, because it determines how fast mass can be transported from one place on the glacier to another (e.g. Oerlemans, 21); in the other case (referred to as volume time scale), the mass balance conditions (notably the balance gradient or the ablation rate at the snout) are assumed to be most relevant (e.g. Jóhannesson et al., 1989). There is a relation of course, because glaciers with a higher balance gradient normally flow faster. The concept of an e-folding respons time formally relates to a linearized system (because then the approach to a new equilibrium is exponential in time). However, glaciers are not linear systems, implying for instance that the response time depends on the magnitude of the climatic change impose to the glacier. This is undesirable, because response time should be a physical property of a glacier, and be independent of forcing or glacier history. However, when changes in the forcing are sufficiently small, the concept of an e-folding response time is useful. A linear response quation can be written as d' dt = 1 (ce' "') (3.1)! Here and E are glacier length and equilibrium-line altitude relative to reference values, c is the climate - 4 -
sensitivity, and τ is the response time. The solution reads (for constant E and initial state ' (t = ) = ): ' (t) = (1! e!t /" ) ce' (3.2) This implies that at t =! about 2/3 of the response has been accomplished (Figure 7). In the literature one sometimes encounters the term reaction time, loosely defined as the time it takes before a glacier snout reacts to a relatively sudden change in the climate. It should be noted that this is a vague concept, because the reaction time may depend on the glacier history in an untransparent way. A distinction should be made between response times for glacier volume and glacier length. When the specific balance changes, the volume will respond immediately. For glacier length it may take a bit more time, in particular when the changes are more apparent at higher parts. Numerical experimentation has indeed shown that response times for volume are typically 3% shorter than response times for length. [note: the volume time scale referred to above can be a time scale for glacier length; the word volume her refers to the way the response time is estimated!]. ' (nondimensional) 1.8.6.4.2 '(t) 1 2 3 4 5 6 time (nondimensional) 1-e -1 Figure 7. Solution of the linear response equation for a constant forcing. 4. A simple dynamic response time From a general point of view the adjustment of the glacier length to a change in the equilibrium-line altitude requires a decrease or increase in the mass transfer down-glacier. The speed at which this can be accomplished is determined by the characteristic ice velocity u. The adjustment has to find its way down to the glacier front, so seems to be an appropriate length scale. Thus we may estimate a response time τ in the following way:! " c u, (3.1) where c is a dimensionless constant of order unity. It is not difficult to work out the implications of eq. (3.1) for the analytical glacier model. With reference to figure 2 the volume flux at the equilibrium line is Altitude h = b + H b = b - sx equilibrium line H E U E =! / 2 # (H " sx " E') dx (3.2) x Here H E and U E are the ice thickness and the mean ice velocity at the equilibrium line. Evaluating the integral - 5 -
yields H E U E =! # $ 2 (H m " E') + s % 4 2 & ' ( (3.3) H m now is the mean thickness of the glacier in the accumulation zone. If H m is assumed to be identical to the overall mean ice thickness the condition for equilibrium reads (eq. (1.3)): H m! E' = s 2 (3.4) Combining eqs. (3.3)-(3.4) gives U E = 3! s 2 8 H E (3.5) We then find for the response time! " c u = c 1 = 8c 1 H E U E 3# s = c 2 H m # s (3.6) In the subsequents steps in eq. (3.6) constants have been absorbed in the c i s. In the last step it has been assumed that the thickness at the equilibrium line is proportional to the mean thickness of the glacier (H m ). No we use a result from numerical experimentation with a plane-shear glacier model to relate the mean ice thickness to the slope and glacier length (Oerlemans, 21; chapter 6): 1/ 2 " µ % H m = $ ', where (3.7) # 1 +! s& µ! 9 m; "! 3 Note that for a slope going to zero the mean ice thickness becomes proportion to 1/2, which is exactly true for perfectly plastic or a Vialov ice sheet. Combining eqs. (3.6) and (3.7) now yields! = c 2 µ 1/ 2 " s (1 + # s) 1/ 2 1/ 2 = c 3 " s (1 + # s) 1/ 2 1/ 2 (3.8) Response time (a) 4 35 3 25 2 15 1 A few remarks can be made about eq. (3.8). First of all, the response time is inversely proportional to the balance gradient β, which is a direct consequence of higher ice velocities associated with larger mass turnover. Then τ decreases with glacier length, with seems to be at odd with our intuition. However, large glaciers normally have small slopes, and the result is a m uch weaker dependence of τ on 5 5 1 15 2 25 3 ength (km) Figure 7. Response times for a set of 169 glaciers (Oerlemans, 25) as a function of glacier length. - 6 -
. How to determine the constant c 3? A possibility is to match the simple estimate of the response time described here with results from numerical flow line models (unfortunately for a small set of 6 glaciers). The result is c 3 = 13.6 m 1/ 2 (Oerlemans, 25). It is interesting to see how eq. (3.8) works out for the set of 169 glaciers used in Oerlemans (25). In figure 7 the response time is plottes as a function of length. Clearly, there is no relation. The fastest glaciers in this sample are steep mountain glaciers, or glaciers with a very fast mass turnover (like Franz-Josef glacier, New Zealand). The slowest glaciers are found on Svalbard; some have a very small slope and moderate mass turnover. References and further reading Jóhannesson T., C.F. Raymond and E.D. Waddington (1989): Time-scale for adjustment of glaciers to changes in mass balance. Journal of Glaciology 35, 355-369. Nye J.F. (196): The response of glaciers and ice sheets to seasonal and climatic changes. Proceedings of the Royal Society of ondon A256, 559-584. Oerlemans J. (21): Glaciers and Climate Change. A.A. Balkema Publishers, 148 pp. ISBN 926518137 Oerlemans J. (25): Extracting a climate signal from 169 glacier records. Science 38, 675-677; 1.1126/science.11746 Van de Wal, R.S.W. and J. Oerlemans (1995): Response of valley glaciers to climate change and kinematic waves: a study with a numerical ice-flow model. Journal of Glaciology 45 (137), 142-152. - 7 -
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