CALCULATIONS OF SLIP OF NISQUALLY GLACIER ON ITS BED : NO SIMPLE RELATION OF SLIDING VELOCITY TO SHEAR STRESS

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1 CALCULATIONS OF SLIP OF NISQUALLY GLACIER ON ITS BED : NO SIMPLE RELATION OF SLIDING VELOCITY TO SHEAR STRESS Mark F. MEIER U.S. Geological Survey, Tacoma, Washington ABSTRACT Measurements on Nisqually Glacier, Mounth Rainier National Park, Washington, at a profile fixed in space, give 22 yearly values of surface velocity (w s ), surface slope (a), and changes in ice thickness (/;) During this period ( ), u s, a and h changed continuously by large amounts. These data were used to calculate values of shear stress (T) on the bed and slip on the bed («&) according to standard procedures. Correction for the longitudinal gradient in longitudinal normal stress was made. The result is that «& is not a simple, single-valued function oft. This implies either (1) the velocity distribution within a valley glacier cannot be calculated this way, or (2) none of the existing theories of glacier sliding which relate sliding velocity to shear stress can be valid. RÉSUMÉ Des mesures effectuées sur le Nisqually Glacier, Parc National du Mont Rainier, Washington, à un profil fixe dans l'espace, a donné 22 valeurs annuelles de la vitesse superficielle (n B ), de la pente superficielle (a) et des modifications de l'épaisseur de glace (h). Durant la période , n s,xe h ont changé continuellement de quantités notables. Ces données ont été utilisées pour calculer des valeurs de la tension tangentielle (T) sur le lit et le glissement sur le lit (/is) suivant les méthodes habituelles. Une correction a été faite pour le gradient logitudinal des tensions longitudinales normales. Le résultat en est que nt> n'est pas une fonction simple, à valeur unique de T. Ceci implique soit que: 1) la distribution de la vitesse dans un glacier de vallée ne peut pas être calculée de cette façon; 2) aucune des théories existantes du glissement des glaciers basées sur une relation entre la vitesse de glissement et la tension tangentielle, n'est valide. All theories of the sliding of a glacier on its bed involve calculation of an effective shear stress at the base of the ice and the use of some functional relation between this shear stress and the velocity or the rate of shear strain. Two assumptions are usually involved: 1. «(, =/(r) for a given channel shape and roughness, and 2. T can be calculated from a simple formula, such as a = fpgh sin a, to a useful degree of approximation. In these expressions, ui> is the velocity of sliding on the bed, T is the shear stress on the bed,/is a channel shape factor, p is the mean density of the glacier, g is the acceleration due to gravity, h is the ice thickness, and x is the surface slope. The purpose of this paper is to show that this simple approach may not be valid. The two assumptions may be tested by calculating the stress and sliding velocity on the bed of a glacier which has varied its thickness and speeds by marked amounts over a period of time. Nisqually Glacier in Mount Rainier National Park, Washington, has been the subject of a long continued observational program (Johnson, 1960), and sufficient data are now available to make these calculations at a single transverse profile for each year from 1943 to The results arc of particular interest because at this 49

2 profile the glacier changed in thickness by a factor of 1/3, in surface slope by a factor of 2, and in surface speed by a factor of more than 10. Thus bed slip velocities can be calculated over a wide range of stresses and surface speeds, while the channel shape and bed roughness presumably remained constant. In effect, the procedure is equivalent to a comparison of 22 ice streams with different thicknesses and surface slopes but equal channel configurations. Nisqually Glacier is a fast-moving valley glacier on the south side of Mount Rainier, Washington. It is about 6.5 km long, and descends in altitude from 4,360 to 1,410 m (1966) above sea level (fig. 1). It was in a state of continued thinning and recession from the late 19th century until From 1946 to 1951 the terminus continued to thin and retreat but a slow " wave" of thickening progressed down glacier. From 1953 to 1966 the terminus advanced (first through stagnant ice until 1961 and thence over bare ground) at an average rate of about 40 m/yr. Measurements reported here were made at " Profile 2," 1.01 kmup-glacier from the 1966 terminus, at an altitude of about 1,830 m, where the glacier is about 610 m wide (fig- 1). The surface altitude of the glacier at this profile was determined by yearly stadia surveys in late summer, and by topographic maps constructed in 1941 and 1946 by plane table surveys and in 1951, 1956, 1961, and 1966 by aerial photogrammetry. Surface slopes were determined from the maps every 5 years, with values for most of the intermediate years determined by ground photography. Velocities were determined each summer by surveying the locations of large marked rocks or (beginning in 1963) welded steel tetrahedra. The rocks and tetrahedra were marked or cmplaced high above Profile 2 and were measured as they traveled down to and below the profile so that the longitudinal gradient of surface velocity could also be determined. The velocity data reported are averages for two-year intervals centered on September of each year; in other words the " 1950" velocity is the average speed of the ice past Profile 2 from September 1949 to September 1951, and this velocity value is compared with slope and thickness data obtained in September These slope, thickness, and velocity data are presented in figure 2. The surface altitude measurements are relatively precise (standard error < 2 m). The surface slope could be determined from maps to within a standard error of 0.1 or less ; the slope measurements from photographs have an accuracy of < o.5 (in , no data on slope were available, so values were estimated using a curve of oh/ôt and standard errors of 1 were assumed). The velocity surveys are probably correct to within a few percent (1 m or so per year), but some of the rocks may have slid on the surface ice. Most of the velocity markers were large (more than 1 m in diameter), and none of these were seen to be perched on pedestals, so it is believed that errors due to this problem are of the order of a metre or two. The estimated standard errors in u, are < 7 m/yr. The actual calculation of ut, and T involved an additional complication. Changes in thickness of Nisqually Glacier have been measured precisely from year to year, but the total thickness has never been measured. The mean thickness at Profile 2 for one year (1961) was calculated as follows: the net mass balance and the rate of change in height over the surface of the glacier below the profile was measured, the results were integrated over the whole glacier surface below the profile to get the ice discharge through the profile, and the discharge was divided by the mean surface velocity (assuming a parabolic cross section) to get the mean thickness. Another computation, less straightforward and less accurate, gave results which agreed with the above to within 30 percent: by measuring the speed of propagation of kinematic waves at different thicknesses for several different years, and assuming that the speed of wave propagation is proportional to the velocity and thus to the thickness raised to a constant power, the mean thickness was calculated. Because of the uncertainty in thickness, all bed slip and stress 50

3 NI SQUALLY GLACIER Profite 3 Profile 2 Contour interval 100 m. V Profile 1 Fig i Topographic map of Nisqually Glacier, Mount Rainier, Washington, as of

4 -,1850 ' , Average over 250-m. widlh in mid-channel_\ 1820 error SURFACE ELEVATION H " I J 1 L I I I 1 I I I I I I I I I I I I SLOPE A x Measured from maps Measured from photographs \ Value used in calcula Iion^ 1 with - 9h/ dt I960 Fig. 2 Variations in slope, surface elevation, and surface velocity at Profile 2, Nisqually Glacier. Velocity values show maximum velocity on the centreline; mean velocity on transverse profile is approximately 2/3 as great. 52

5 values were calculated for the minimum (127 m in 1961) and maximum (157 m in 1961) bounds of thickness. The results were similar in form, although different in absolute value. The basic data on stone or tetrahedra locations were plotted on a time-distance diagram. From this it was possible to determine the surface velocity (w s ) as a function of time at Profile 2; all values were adjusted to mean velocities on a transverse profile. In addition, the first and second derivatives of velocity with longitudinal distance were determined. The transverse profile of velocity approximated curves for a parabolic channel shape (Nye, 1965, p. 681), from which the shape factor/(nye, 1965, p. 679) was calculated (*). The shear stress was then calculated according to the approximate formula T = fpgh sin a The bed slip was calculated occording to the standard formula (Nye, 1952, p. 84) 6 ç n+l\ which is derived from an integration of the Glen flow law (Glen, 1955) in the form ê = (T/B)" where is the rate of shear strain, and B (^ 2) and n (ç^ 3) are empirical constants. The results of these calculations arc plotted as graphs of ut> and u s against r (figs. 3, 4). Separate curves were plotted for maximum and minimum thicknesses. The Weertman theory of bed slip (1957) predicts a parabolic dependence of«;, on r. However, the resulting curves are complex, with several loops. The curves are similar in form with the same number of loops. Thus several values of ui, (or u t ) appear to be possible for a single value of x. The most remarkable aspect of these results is the slight (and irregular) variation of r associated with large variations of u/,. The values plotted in figure 3 are not derived independently; a calculation of x was involved in the calculation of «6. In order to show that the basic difficulty is not a quirk of the calculation, T and u s are plotted infigure4. These two quantities are entirely independent but the graphs also show the loops and the narrow range oft for large variations in velocity. If u s were due entirely to sliding, it would be proportional to T 2 according to the Weertman theory. If u s were due entirely to differential movement within the ice, it would be proportional to r n h where n is about 3 (Nye, 1952, p. 85). Obviously the results confirm neither relation. A more sophisticated calculation of H& and shear stress was made, considering the known longitudinal strain rates and calculated longitudinal stresses. The shear stress was calculated from f* T ' = fpgh sin a- (d<j x jôx)dy oj where a xx is the longitudinal normal stress, x a longitudinal coordinate axis, and y a coordinate axis directed perpendicular to the surface (Glen and Weertman, 1963, p. 8; Lliboutry, 1965, pp ). By assuming that it was constant with depth, a xx could be calculated from the Glen flow law (with modification of the coefficient B to account B (*) Since the glacier changed in thickness by a factor of 1/3, the ratio of width to depth at Profile 2 changed from about 3.1 to about 4.2, causing a change in/from about 0.57 to about

6 100 r L i r SOi* '-f fe-, ft T -100 r 1^1 r...j z 50! t in bars Fig. 3 Bed slip (ub) as a function of shear stress at the bed (r). The upper graph (dots) was calculated for minimum possible thickness (127 m in 1961); lower graph (circles) was calculated for maximum possible thickness (157 m in 1961). Short lines indicate standard errors. Numbers beside each point are last two digits of year of observation.

7 for uniaxial normal stress and strain rate) and known values of ôu s lôx. Values for duslùx were calculated for 50-m reaches above and below Profile 2; the computation of ôoxx/ôx was taken over a 50-m reach centered on the profile. The resulting correction to «6 was small, but the correction to r was more important; the two terms in the equation for T' were of the same order of magnitude (fig. 5) (*). This correction does not 100 j- 3 * o... J.57 «*S6. S5»SB»59 cr ' C 50! 1 > 52. o J i 50 MS»18 o o T 1 in bars Fig. 4 Surface velocity (««) as a function of shear stress at the bed (T). Dots were calculated for minimum possible thickness; circles were calculated for maximum possible thickness. Standard errors in T are same as in figure 3. Standard errors in Us average 4 m/yr and in no case exceed 7 m/yr. (*) Fortuitously, the longitudinal strain rate du s ßx at Profile 2 was usually almost zero (<0.06 yr" 1 ), with compressing flow (up to 0.20 yr~ l ) above the profile and extending flow (up to 0.08 yr" 1 ) below it. Thus there was a negligible correction to be applied to «6, and the contribution of a X x to the octahedral (or "effective") shear stress was also small. However, the gradient do X xldx ranged between 0.19 and 1.90 bars per 100 m. 55

8 remove the discrepancy, but the estimated standard errors make the results somewhat inconclusive. A proper calculation would involve transverse stresses and strain rates also, but this calculation is impossible with the data available for Nisqually Glacier. Surface velocity vectors are essentially parallel in the vicinity of Profile 2, suggesting that transverse normal strain rates arc small. 100:,}- -> r ~" _ - I» _:r; VT-;.'; ".._i. so 3"! 50 f' in bars Fig. 5 Bed slip () as a function of corrected shear stress at the bed (r'), for minimum possible thickness. Short lines indicate estimated standard errors. No computations were made for , because the velocity data were too sparse to adequately define gradients. Many approximations and unknowns are involved in these calculations. Stresses from far above or far below Profile 2 were not considered; these stresses should not have influenced the stress gradient term directly, but stress gradient is a non-linear function of strain rate gradient and non-zero longitudinal stress level would have some effect on the computation of stress gradient. The " shape factor"/is only an approximation, and it was derived from an analysis which assumes no sliding on the bed. The Nisqually Glacier is rather irregular in shape, and one must make rather drastic assumptions about the symmetry of the channel and the unimportance of transverse stresses. Perhaps the assumption that a xx is constant with depth is unjustified. Perhaps the ice is markedly anisotropic near the bed, or there is a variable discontinuity (a zone of strong shearing) in the flow profile. Perhaps the value of slope should have been averaged over a great distance, not just measured in the vicinity of the profile. Perhaps the effect of water pressure at the bed or the roughness of the channel changed from year to year. Although this is a long list of possible complications, it does not seem likely that they could alter the major result: figure 5 docs not show a simple parabolic or cubic dependence of u\, on T', even allowing for relatively large changes in the values. The absolute values of the results presented in figures 4 and 5 arc not accurate, but the relative variations from point to point are known to a degree of precision as high as, or higher than, most other calculations of bed slip in observed glaciers. The fact that these relative variations indicate a certain inconsistency or difficulty in the method has 56

9 an important implication. One or both of the following statements must be true: (1) the velocity distribution within a typical valley glacier cannot be calculated by an integration of the simple stress equation and the Glen flow law, and/or (2) none of the existing theories of glacier sliding which relate sliding velocity to shear stress are valid. Thus a simple test for bed slip theories will be hard to devise, and more sophisticated calculation procedures should be used for many glacier flow problems. REFERENCES GLEN, J. W., ( 1965): The creep of polycrystalline ice: Proceedings Royal Society, Ser. A, Vol. 288, pp GLEN, J. W., and WEERTMAN, J., (1963): Discussion of paper by Lliboutry: International Union of Geodesy and Geophysics, Bull. International Association of Scientific Hydrology, Vol. 8, No. 2, June 1963, pp JOHNSON, ARTHUR, (1960): Variation in suface elevation of the Nisqually Glacier Mt. Rainier, Washington: International Union of Geodesy and Geophysics, International Association of Scientific Hydrology, Pub. 19, pp LLIBOUTRY, L., (1965): Traité de Glaciologie: Masson and Cie., Paris, 1,040 p. NYE, J. F., (1952): The mechanics of glacier flow: Journal ofglaciology, Vol. 2, No. 12, pp NYE, J.F., (1965): The flow of a glacier in a channel of rectangular, elliptic or parabolic cross-section: Journal of Glaciology, Vol. 5, No. 41, pp WEERTMAN, J., (1957): On the sliding of glaciers: Journal of Glaciology,Vol. 3, No. 21, pp