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An Empirical Scheme for Estimating the Dynamics of Unmeasured Glaciers W. F. Budd and I. F. Allison INTRODUCTION in the Course O

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An Empirical Scheme for Estimating the Dynamics of Unmeasured Glaciers W. F. Budd and I. F. Allison INTRODUCTION in the Course O Snow and Ice-Symposium-Neiges et Glaces (Proceedings of the Moscow Symposium, August 1971; Actes du Colloque de Moscou, août 1971): IAHS-AISH Publ. No. 104, 1975. An empirical scheme for estimating the dynamics of unmeasured glaciers W. F. Budd and I. F. Allison Abstract. An attempt is made to place the experience of field glaciologists on a numerical foundation. A summary of empirical data on ablation variation with elevation and latitude is presented with a scheme for estimating the balance of a glacier. Empirical data on the velocity, surface slope, and ice thickness are used to construct a graph of total flow as a function of surface slope and velocity. The data are interpreted in terms of the flow law of ice, the equations of motion and the glacier dimensions. The total flow and slope at the glacier equilibrium line provides an estimate of the glacier thickness and mean velocity at this position. As an example the calculations are carried out for a hitherto unmeasured glacier on Heard Island. These estimates are compared with the results of measurements made during February-March 1971. Résumé. Une tentative est faite pour donner une base numérique à l'expérience des glaciologues de terrain. On présente un résumé de données de terrain sur la variation de l'ablation avec l'altitude et la latitude et une règle pour estimer le bilan d'un glacier. Des données sur la vitesse, la pente et l'épaisseur de glace sont utilisées pour établir un graphe du débit solide total en fonction de la pente de la surface et de la vitesse en surface. Les données sont interprétées en rapport avec la loi de fluage de la glace, les équations du mouvement et les dimensions du glacier. Le débit solide total et la pente du glacier à la ligne d'équilibre permettent une estimation de l'épaisseur du glacier et de sa vitesse moyenne à ce niveau. Comme exemple, des calculs sont faits pour un glacier de l'île Heard n'ayant pas été jusqu'ici l'objet de mesures. Ces estimations sont comparées aux résultats de mesures faites en février-mars 1971. INTRODUCTION In the course of collecting data from measurements on glaciers for evaluating the applicability of a numerical modelling programme it became apparent that the data tended to group into systematic variations. These systematic variations of glacier features — such as: accumulation-ablation, velocity, ice thickness, and volume flux — became useful guides to the unknown features of other glaciers for which some of the measurements are not yet available. The success to which these guides can predict the unknown features of existing glaciers may be expected to be an indicator of the degree to which we can estimate future or past properties of glaciers. In any scheme of generalization for which only the most important relevant variables are considered while the less important are ignored, a certain amount of scatter in the predictions must result. The practical usefulness of the general scheme then depends on this scatter being small compared with the overall variation. The use of additional local knowledge for individual cases then needs to be called on to make a closer approach to the final estimate. In the absence of any data at all even coarse estimates, or orders of magnitude are useful for the planning and preparation of field work on the glacier. The question, then, which we wish to answer is: given just the information on the position, and elevation of a glacier, how well can we expect to estimate its physical characteristics': The following scheme attempts to give an answer to this. There is not sufficient space here to give the derivation of the nomographs used. These will be An empirical scheme for estimating the dynamics of unmeasured glaciers 247 discussed in detail elsewhere. Hence just the results and their use are outlined, with the recent application to a glacier on Heard Island as an example. REQUIREMENTS The first and most easily obtained information about a glacier is usually an elementary map showing the geographical position, an outline of the surface plan and some surface elevations. (1) If we know how the net accumulation and ablation on the average vary with elevation, we can estimate the distribution of net balance (a) over the area. From this by integrating the balance over the area as a function of the distance along the central flowline (x) we can obtain the variation in the net cross-section flux (0 along the glacier, that would be required for steady-state balance, as follows: Qx = j J a ây àx o yi X = [a Ydx o where y is the surface width at x and a is the mean balance over the width. The flux Q has its maximum Qp at the firn line where a = 0. The error in estimating Q is generally proportionally least near this maximum position where its variation with x is small. If Fis the mean velocity over the cross-section, Z is the maximum depth, and Y the mean cross-section width, then the flux is given by Q=VYZ (2) Thus if we have a relation connecting the velocity, the ice thickness and the cross-section shape it becomes possible to determine the thickness and velocity from the flux. In a similar way it would be possible to extend this along the glacier to obtain the profile of thickness and velocity throughout. This information is then sufficient to construct a numerical model of the glacier and calculate other features such as vertical velocities, strain rates, particle paths, trajectories, ages, travel times, response times, etc. Our problem then reduces to the determination of: (1) accumulation-ablation versus elevation and (2) the velocity and ice thickness from the flux. ACCUMULATION Accumulation is one of the most difficult elements to estimate because of its highly irregular variation on both the large geographical scale and the local topographical scale. If the precipitation of the locality is known its variation with elevation can often be estimated with regard to local topography. When such information is lacking the accumulation rates in neighbouring regions must serve as a guide. Failing this, world and local precipitation maps, e.g. Hann (1939) and Landsberg (1970), give a useful indication of the large-scale variability, as well as the mean values from the information of existing meteorological stations. Used in conjunction with the variation of precipitation with elevation these data provide a reasonable estimate of accumula­ tion rate. The variation of precipitation and accumulation of snow with elevation also depends on the locality. This has been discussed by a number of authors; for example, 248 W. F. Buddand I. F. Allison European Alps: Uttinger (1951), Hoinkes (1954), Lliboutry (1965); North Atlantic Coasts: Ahlmann (1948);Scandanavia: Ostrem and Liestol (1964); Canada: Ostrem (1965); Northern America: Meier and Post (1962); General and World coverage: Hann (1939), Landsberg (1970). Typical average precipitation gradients range from —0.3-1.5 m yr-1km _1 but the local conditions need to be considered in each individual case. ABLATION Ablation, being less dependent on wind effects than accumulation, is an easier element to estimate. Figure 1 shows a collection of annual mean ablation versus elevation 10 15 Ablation (m/yr) FIGURE 1. Ablation rate variation with elevation and latitude. curves for different latitudes in the northern and southern hemispheres from a number of different sources. This compilation will be discussed in detail by Budd in a paper in preparation. A list of the glaciers and sources are given in the Appendix. The variations of ablation rate appear quite systematic, but there is a scarcity of data from the southern hemisphere as yet. Although this is partly due to the lack of land masses in the south it is expected that the diagram will be clarified by further measurements An empirical scheme for estimating the dynamics of unmeasured glaciers 249 from South America, New Zealand, Heard Island and Kerguelen and New Guinea. The mean variation with latitude has been discussed by Haefeli (1962) and also Meier and Post (1962). This variation can largely be explained in terms of variations in summer temperatures. For example, the mean variation of summer temperatures with latitude is shown in Fig. 2 from Hann (1939). These explain the large difference between the northern and southern hemisphere ablation rates. The tempera­ ture/latitude variation, however, is quite irregular locally, as seen for example from the summer temperature charts (cf. Hann, 1939, Abb. 28 and 29). s >» . / /V C=^5^\ 20 //' ' Ju> // \\\ : f /•' ^ // \\\ w-' / / \\ > - I / e - 1 /Jan ur * "/ / I ' \\ ' a;. s ! V\\ -• E fi / / \\ \ \-' / \\ / «0 \\ .• \ N 8Q 40 40 80 S Latitude (° ) FIGURE 2. Temperature variation with latitude (Hann, 1939). The net temperature effect has been summarized by Krenke and Khodakov (1966) as shown in Fig. 3. The dependence of ablation on temperature has also been discussed by Hoinkes (1955), Hoinkes and Rudolph (1962), Hoinkes et al. (1968) and Lang (1967). The relation can largely be understood in terms of the different effects of radiation on ice, depending on whether it is at or below the melting point, together Summer mean temperature FIGURE 3. Ablation rate variation with summer mean temperatures (Krenke and Khodakov, 1966). with the annual and diurnal variations of the temperature of the ice. A detailed account of this will be given elsewhere. Here we may use the ablation-temperature relation together with an elevation-temperature relation (see, for example, Hann, 1939) to obtain an ablation-elevation relation as illustrated by Fig. 1. With this 250 W. F. Budd and I. F. Allison ablation-elevation information and the knowledge of the location and elevation profile of the glacier a reasonable estimate can be made of the ablation rates for the glacier.
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