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 / - 1 /Jan ur e
* "/ / 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. Taken together with the accumulation the net balance versus elevation can be calculated and by integrating over the area of the glacier the accumulation and ablation estimates may then be adjusted to make the total net balance of the glacier zero. The errors due to such a steady-state assumption are usually smaller than the other errors here.
VELOCITY AND ICE THICKNESS From the available measurements of velocity, ice thickness, surface slope, and cross- section shapes for large-scale averages of glaciers, discussed by Budd and Jenssen (1975), the surface slope a multiplied by a shape factor «Sj , has been plotted against the centreline surface velocity V, in Fig. 4. The shape factor Si has been estimated from Nye's (1965) results (compare Fig. 3 of Budd and Jenssen on p. 264). Iso- thickness curves have been drawn on this plot from 50 m to 700 m. These ice thickness values are again the maximum (centreline) thicknesses. In addition the product VZ, which represents the central flux rate per unit width ($) has been calculated and these isolines superimposed on the graph. From this graph, if we know the flux rate , the surface slope a, and the approximate cross-section shape, we can read off the values of Z and V for maximum ice thickness and velocity. Since: Q V Y V we have only to estimate Vand Fin terms of F and Y to be able to use Q, with Fig. 4 to determine FandZ. The values of Y in terms of Y for a variety of shapes are given in Table 1.
TABLE 1. Ratio of mean width F, to surface width Y, shape factor S2
Cross section: Triangle Parabola Ellipse Rectangle
2_ S2 = YlY: 1 0.50 0.67 0.77 1.0
From Nye's (1965) results the values of the mean cross-section velocity F to the maximum surface velocity V for different width to depth ratio, for a parabola, are given in Table 2.
TABLE 2. Ratio of mean velocity to maximum velocity, shape factor S3 Tv — — Width-depth ratio ^- 1 2 3 4
Shape factor S3 = V/V 0.674 0.652 0.612 0.578 0.443
Since the flux is given by:
Q=YVZ = S2 YS3 VZ Q =VZ S,S3Y Since initially we do not know Z or the shape, a first guess can be made for the shape factors which can be used to obtain a first guess Z from Fig. 4. This can then be An empirical scheme for estimating the dynamics of unmeasured glaciers 251
2 3 4 10 30 30 40 10O 200 300 Velocity m/yr
FIGURE 4. Ice thickness and flux variation with velocity and surface slope.
used to correct the shape factor, which usually only makes a slight difference, and the new Z is read off more precisely. Further repetition may be carried out if required.
APPLICATION Heard Island is an isolated island of about 400 km2 in the southern Indian Ocean at latitude 53° S, rising to 2745 m in its centre and covered with steep glaciers reaching to sea level (Fig. 5). Because it is situated close to the Antarctic Convergence it is expected that the glaciers on Heard Island may be sensitive indicators of climate changes. Although some qualitative information was available on these glaciers (see Budd and Stephenson, 1970), no measurements had been made of accumulation, ablation, ice thickness or velocity. An opportunity arose for Antarctic division personnel to visit the island in conjunction with a French expedition for about 6 weeks from 26 January to 10 March 1971. During this visit one of us (LA.) carried out some initial measurements on the glaciers. These measurements will be discussed in detail in a separate report. They included measurements of surface elevation, velocity and strain rate across the Vahsel Glacier near the firn line (Fig. 6). In addition ablation measurements were carried out on the glacier and continuous meteorological observations of temperature, pressure, wind speed and direction, sunshine, precipitation, cloud cover, humidity, and visibility were made at a base station during the period. A gravity profile was also carried out across the movement line of Vahsel Glacier for ice thickness determinations. In order to gauge the magnitudes to be expected for these various features to be measured, some calculations were made according to the above scheme. The ablation rate was estimated from the curves of Figs. 1 and 2 using information on the temperatures from weather records of the expeditions of the ANARE 1948-54 from the Commonwealth Bureau of Meteorology (1950-57). The accumulation was estimated from the average precipitation at the base station over 1948-54, plus an increase with height of 0.6 m/yr-1 km-1. (Ahlmann found values of 0.4 and 0.6 m/yr-1 km-1 for similar island glaciers in high northern latitudes.) The net 252 W. F. Budd and I. F. Allison
73*is' yy?o'
73*15' FIGURE 5. Map of Heard Island.
I / ^? ùim - Kilometres
Heights in métros FIGURE 6. Vahsel Glacier showing stake line and strain grid. An empirical scheme for estimating the dynamics of unmeasured glaciers 253 accumulation, ablation and balance as calculated are shown in Table 3 together with the elevation areas and the flux across the firn line, namely: 6 3 1 QF=16.7x 10 m yr"
Assuming a shape factor S2 = 0.7 f = 0.7x3 km VZ = Q/Y* 7.9 x 103m2 yr"1
For the shape factor S3, since the glacier is very wide (compared with its depth) we take:
S3 = 0.5
Then
TABLE 3. Calculation of net balance and flux
Elevation Precipitation Ablation Balance Area Flux (m) (myr" l) (m yr"1 ) (m yr-1 ) (km2) (m3 yr-1 xlO6) 0 2.9 -7.0 -4.1 3.9 -10.9 100 3.0 -4.5 -1.5 5.9 - 4.4 250 3.1 -3.1 -0.0 3.6 + 2.1 400 3.2 -2.0 + 1.2 2.2 + 3.8 600 3.3 -1.0 +2.3 2.0 + 5.5 1000 3.5 -0.3 +3.2 1.9 + 6.6 2200 4.2 _ +4.2 Mean flux across fiin line Qp ~ 16.7 x 106 m3 yr '
The surface slope is always difficult to specify precisely. Since the graph of Fig. 4 was derived from large-scale averages we take a large-scale average slope of 12° =; 21%. The shape factor Si is initially taken as 5] = 1. From Fig. 4 with Sa = 21% and VZ = 15.9 x 103 m2 yr"1 we read off Z ~ 80 m and V ~ 140 rn/yr. This low value of Z does not substantially alter the assumed shape factors. The expected errors are about ±25%, judging from the scatter in the original data represented by Fig. 4. The uncertainty also increases towards the edge of the curves of Fig. 4 because,the data are more sparse there.
MEASUREMENTS Table 4 lists the measured values at the firn line compared to the expected,values.
TABLE 4. Comparison between estimated and measured values
Parameter Ablation during Feb. Velocity Ice thickness
Symbols and units a m of ice V m yr"1 Z m Expected values 0.6 140 80 Measured values 0.65 ± .05 117 ± 38 73 ± 15
The maximum measured velocity and thickness agree well with the expected value. Errors in the ice thickness determinations from the gravity values are high because of 254 W. F. Budd and I. F. Allison the unknown influence of the ice-rock moraines. The strain grid measurements at the firn line and the repeated levelling across it confirm the position of the firn line and the position of maximum flux. Further across the glacier it appears that the velocity becomes higher (perhaps up to 250 m/yr) and the ice thickness less. This seems to be a result of the higher surface slopes there. The elevation contours as shown on the map were found to be somewhat in error.
CONCLUSIONS It seems that the above general scheme for estimating unknown features of glaciers is certainly good to the order of magnitude and possibly within ±25% for velocity and ice thickness. Some similar calculations carried out for other glaciers on Heard suggest the values obtained here may be fairly typical — indicating that the glaciers there are generally broad, thin, but rather fast moving. Hence it is expected that they would have relatively short response times to climate variations. The usefulness of the nomographs presented would be increased by the addition of more data, especially in regions where the information is at present lacking.
REFERENCES Ahlmann, H. W. (1948) Glaciological research on the North Atlantic coasts. R.G.S. Research Series, No. 1. Budd, G. M. and Stephenson, P. J. (1970) Recent glacier retreat on Heard Island. ISAGE Symposium SCAR I ASH Pub. No. 86, 449-58. Budd, W. F. and Jenssen, M. J. D. (1975) Numerical modelling of glacier systems. This volume. Commonwealth Bureau of Meteorology (1950-57) Meteorology: Heard and Macquarie Islands 1948-54. ANARE Reports, Series D, 1-7. Hann, J. (1939) Lehrbuch der Météorologie, 1. 5th Ed, revised by R. Sirring, Keller, Leipzig. Haefeli, R. (1962) The ablation gradient and the retreat of a glacier tongue. Symposium of Obergurgl. IUGG I ASH Pub. No. 58, 49-59. Hoinkes, H. (1954) Neue Niederschlagszahlen aus den zentralen Ôtztaler Alpen. 49-50 Jahresbericht des Sonnblick-Vereins fur die Jahre 1951-1952, 19-27. Springer, Wien. Hoinkes, H. (1955) Measurements of ablation and heat balance on Alpine Glaciers. /. Glaciol, 2, 497-501. Hoinkes, H. and Rudolph, R. (1962) Variations in the mass balance of Hintereisferner 1952-1961, and their relation to variations of climatic elements. Symposium of Obergurgl. IUGG IASH, Pub. No. 58, 16-28. Hoinkes, H. et al. (1968) Glacier mass budget and mesoscale weather in the Austrian Alps 1964 to 1966. By H. C. Hoinkes, F. Howarka, and W. Schneider. General Assembly of Berne. IUGG IASH Pub. No. 79, 241-54. Krenke, A. H. and Khodakov, V. G. (1966) The connection between surface melting of glaciers and air temperature. (In Russian) Akad. Nauk SSSR, Institut Geographii, Materialy glatsiologicheskikh Isseldovanii. Chronicles, Discussions, 12, 153-64. Landsberg, H. E. (1970) World Survey of Climatology. Elsevier, New York. Lang, H. (1967) Uber den Tagesgang im Gletscherabfluss. 9. Internationale Tagung fiir Alpine Météorologie. Verôffentlichungen der Schweizerischen Meteorologischen Zentralanstalt. No. 4, 32-8. Lliboutry, L. (1965) Traité de Glaciologie, 2. Paris, Masson et Cie. Meier, M. F. and Post, A. S. (1962) Recent variations in mass net budgets of glaciers in Western • North America. Symposium of Obergurgl. IUGG IASH Pub. No. 58, 63-77. Nye, J. F. (1965) The flow of a glacier in a channel of rectangular, elliptic or parabolic cross section./. Glaciol, 5,661-90. Ostrem, G. (1965) Mass balance studies on glaciers in Western Canada 1965. Geograph. Bull., 8, No. 1,81-107. Ostrem, G. and Liestol, O. (1964) Glaciological investigations in Norway, 1963 (In Norwegian). Norsk Geografisk Tidsskrift Bd. XVIII, 1961-1962, H. 7-8, 281-340. Uttinger, H. (1951) Zur Hohenabhângigkeit der Niederschlagsmenge in den Alpen. Archiv fur Météorologie, Geophysik und Bioklimatologie, Ser. B, Bd. II, H. 4, 360-82. An empirical scheme for estimating the dynamics of unmeasured glaciers 255 APPENDIX Sources of ablation data Lat. Publication No. Glacier N. Author (Abbreviated) 1. Khumbu 28 Muller(1964) Unpublished manuscript 2. Mir Samir 35 Gilbert etal. (1969) /. Glac. 8, 52, 55. 3. Bazhin 35 Corbel (1962) Neiges et Glaciers A. Colin No. 311.224pp. 4. Elena 0 Whittowe?a/. (1963) /. Glac. 4,35,581-616. 5. Fedchenko 39 Schul'ts(1962) Lednik Fedchenko Tashkent. 6. Petroff 42 Avsiuk(1958) IASH 47, 72-104. 7. Dinwoody 43 Meier and Post (1962) IASH 58, 63-77. 8. St. Sorlin 45 Corbel (1962) Neiges et Glaciers A. Colin No. 311.224pp. 9. Gorner 46 Renaud (1952) J. Glac. 2, 11,54-7. 10. Rhône 46 Mercanton (1916) Vermessungen am Rhonegletscher N. Oschr. 52, 1-190. 11. Hintereisferner 47 Blumke and Hess Wiss. Erg. H. Z. AV. 1(2). (1899) 12. Aletsch 46.5 Kasser(1954) IASH 39,339. 13. Saskatchewan 52 Meier (1960) USGS Prof. Pap. 351. 14. Peyto 52 Ostrem(1966) Geog.Bull. 81-107. 15. South Cascade 48 Meier and Tangborn J. Glac. 5, 41, p. 547. (1965) 16. Place 50 Ostrem(1966) Geog.Bull. 81-107. 17. Nigardshreen 62 Ostrem(1963) Norsk geogr. tidsskr. 18, 156-202. Ostrem and Liestol Norsk geogr. tidsskr. (1964) 18,281-340. 18. Seward-Malaspina 60 Meier and Post (1962) IASH 58, 63-77. Sharp (1951) Geol. Soc. Am. Bull. 62, 726-44. 19. Hoffel 64 Ahlmann(1948) Roy. Geog. Soc. Res. Ser. 1. 20. Sôrbreen 71 Fitch etal. (1962) IASH 58, 207. 21. Burroughs 69 Taylor (1963) /. Glac. 4,36,731-52. 22. Decade 69.5 Stanley and Hodgson N. Baffin I. Field Rep. (1968) 1967. 23. Greenland 70 Bauer (1959, 1968) Medd. Gr