R. I. FERGUSON Department of Geography, University of Hull, Hull, England

Sinuosity of Supraglacial Streams

ABSTRACT streams begin to develop at the interface be- tween saturated snowpack and underlying ice. Measurements of the flow and channel form This means that channels may already be well of supraglacial streams, mostly perennial, on a established when they become apparent at the Swiss glacier are analyzed for comparison surface, and assessment of their age is further with normal alluvial . Channel incision complicated by the survival of older channels. depths can be explained by a simple model The widespread assumption that supraglacial taking into account frictional melting and channels are somehow obliterated in winter is direct ablation. Stream widths are proportional hard to justify, since once cut in ice they can to the square root of discharge. Discriminant only be destroyed by ablation of that ice. The and regression analyses support the theory that few abandoned channels to be seen on glaciers maximum develop in streams with in the European Alps take a long time to disap- the highest initial rates of power expenditure pear, possibly years rather than months despite per unit bed area. their small size and higher ablation rates than in the Arctic. In addition, resurveying of OF SUPRAGLACIAL streams in the same position on one glacier in STREAMS successive summers has revealed too great a Channels formed in glacier ice by surface similarity in channel plan to be coincidental meltwater streams have long been familiar to (Fig. 1). These observations indicate the rele- explorers and field glaciologists but have only vance to temperate valley glaciers of the views recently attracted the attention of fluvial of workers in the Arctic (Glen, 1941, p. 72; geomorphologists. Leopold and Wolman's Holmes, 1955, p. 16; Nobles, 1960, p. 51) that (1960) observation that supraglacial although some new channels are formed each are similar in form to alluvial ones, and their year and others are destroyed, most are re- recognition that meandering of these sediment- occupied perennially. There may however be free streams implies a hydrodynamic origin, zonal differences in the proportion of annual stimulated recent research. The possibility of streams, and in the dominant processes encour- confirming other ideas about normal rivers, or aging channel formation and stream diversion. generating new ones, by studying these rapidly In northwest Greenland, Nobles found old adjusting channels in a relatively uniform envi- channels were only shallowly incised at the ronment has formed the goal of subsequent time of maximum melt and were thus prone work on their geometry (Zeller, to drainage disruption, but that the main effect 1967; Knighton, 1972) and the channel changes of saturated slush flows was to clear existing during its development (Dozier, 1970). With channels rather than either to initiate new the same general aim, this paper discusses the ones, as suggested by Ewing (1970, p. 135) from results of an investigation of the relation be- Canadian experience, or to inhibit all channel- tween supraglacial stream sinuosity and possible ling, as on Scandinavian valley glaciers, ac- controlling factors. cording to Stenborg (1968, p. 27-28). In Switzerland, too, drainage changes appear to Streams seem to occur on glaciers and icecaps be most frequent at the start of the ablation in almost all areas experiencing appreciable season, but generally affect only short stretches surface ablation. They form in unfissured hol- of the better established perennial streams. lows where meltwater from a sufficiently large drainage area is concentrated, but are not The writer measured channel geometry at 20 necessarily initiated on bare ice. When ablation stations on three different streams on the Lower is at its peak in late spring and early summer, Arolla glacier, Switzerland, in 1967. Sudden- extensive winter snow cover remains, and injection salt-dilution discharge measurements

Geological Society of America Bulletin, v. 84, p. 251-256, 3 figs., January 1973 251

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02 0-4 Sine of glacier slope

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0 0-04 0 08 Stream discharge (m3s_1) Figure 2. Relations between width, discharge, and local glacier slope for stations on three streams (different symbols), Lower Arolla glacier, 1967.

lated dominant discharges correlate strongly with channel width (Kendall's r = 0.89) and the approximate relation W = 1.9 Q1/2 fits the data well (Fig. 2), indicating widths less than half those expected at similar discharges in alluvial streams. Significant negative rank cor- relations exist between either discharge or width and local glacier or channel slopes, but partial correlation suggests that width and slope are only indirectly related via discharge (TWQ.S = 0.89, TWS.Q = -0.08) and that the S-Q link reflects not so much the erosional development of concave long profiles as the greater probability on gentler slopes of large Figure 1. Plans of channel on Lower Arolla glacier, uncrevassed drainage areas. September 1966/July 1967. Over-all shortening is due It is generally assumed that supraglacial to compressive ice flow. streams erode by frictional melting of their channels, but this is not the only process at at some of these showed that a high proportion work. Some mechanical probably takes of the meltwater produced by surface ablation place, assisted by preferential melting along runs off in channels, implying fairly low ice crystal boundaries; more important, direct permeability, and that although discharge evi- radiation through the flowing water can melt dently fluctuates diurnally it is fairly constant the channel bed, as noted by Holmes (1955, through each afternoon. Measured or interpo- p. 34) and Dozier (1970, p. 100). The thermal

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part of stream erosion can be quantified. The within such areas does not easily form new potential energy loss rate per unit length of channels because of its lesser volume and the channel is r = QpgS/P where S is the sine of frequent divergence of flow around cobbles the local glacier slope, P the channel sinuosity, and boulders. Channels in bare ice, though, and p the mass density of the water in the often possess strikingly regular meanders, testi- stream; in stable conditions with no change in fying to the near-homogeneity of the material stream kinetic energy or in ice and water in which they are cut. Glacier structure and temperatures (=0°C) this must all be dissi- surface topography control the existence and pated in boundary erosion. If stream width is shape of drainage basins—usually long and assumed invariant then the vertical incision thin in the Alps because of medial moraines— rate through channel melting is i = pgQS/ but the quasi-lithology of the ice has little PicrWP where p; and a are the density and the apparent effect on channel form or direction; latent heat of melting of the ice of the glacier the streams on the Lower Arolla glacier flow surface. The constants are such that the expo- almost perpendicularly to both dip and strike nential increase in discharge with distance of the annual foliation. Since channels are con- because of this increment is negligible: dQ/ds tinually cutting down into the ice, possibly by — 10-5 or 10~6£), according to slope, in SI a total of several meters through one summer, units, and any perceptible downstream rise in their plan and cross-sectional form have con- discharge must be due to overland flow and siderable freedom to adjust in response to local radiant bed melting. The additional potential variations in the distribution of energy dissipa- energy loss through incision is equally in- tion. The nonvertical incision which permits significant. meander development without sediment depo- sition is evident in the typical undercutting at The net rate of channel incision relative to bends, and in the overhangs and terraces pre- the ice surface can be written as i — (1 — k)a served in the walls of deep . where a is the surface lowering (ablation) rate and \ the relative efficiency of radiant bed It is proposed here that stream sinuosity melting, which presumably falls from close to depends on the power expenditure rate in rela- unity for very shallow channels to zero for tion to stream size and ice erodibility. In the deep narrow canyons. It is clear that if i > a underlying general view of erosional meander the channel must get progressively deeper development, the extent to which an inherent despite the accompanying decrease in this meandering tendency is realized depends on explains the rapid development of channels in the ratio between bank resistance and erosive spring, for although meltwater is abundant, power, the latter indicated by parameters such ice lowering is relatively slow because of pro- as average shear stress or the rate of power tection by the remaining snow cover. Channels expenditure per unit boundary area. In ap- with / < a will deepen relative to the surface, proximately balanced conditions, erosion is but at a declining rate as they asymptotically neither so restricted that the channel remains approach an equilibrium depth whose ratio to straight, nor so great that it becomes braided; width will be higher the closer i is to a. Substi- instead meandering develops, amplifying itself tution of measured values of channel variables through flow and cross-section asymmetry but for the Lower Arolla glacier gives estimated restricted and eventually halted by the reduc- values of i in the range 0.3 to 1.3 X 10-6 m tion in energy gradient and increase in resist- s~l, the upper limit being close to the average ance as sinuosity rises. In near-uniform ma- summer ablation rate. The highest estimate terials, as here, the highest sinuosities should turns out to apply to the most deeply incised develop in the channel reaches with the greatest channel reach, while the lowest one corresponds initial areal power expenditure rate. This rate to the shallowest channel, that of a temporary is obviously unknown, but can be estimated by stream on the steep glacier snout. Further QS/W if the ratio between width and wetted qualitative support for this incision model is perimeter is roughly constant between chan- provided by the observation on several Alpine nels and neither changes progressively during glaciers, including the Lower Arolla, of streams channel evolution. The validity of assuming which become deeply incised after flowing that measurements of supraglacial stream form from a bare-ice area to one with a debris and flow at a particular time indicates equi- mantle thick enough to reduce surface lowering librium conditions depend on channel age, the considerably. What meltwater is produced rate of channel adjustment, and the stability

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of dominant discharge. Dozier found the s:.nu- osity of a new channel continued to increase all summer, while Knighton reports a reduction, associated with a decline in peak discharges and implying equilibrium had already been attained. In the present study most channel reaches were on perennial streams, but no doubt some scatter in the results is attributable to differing stages of adjustment and fluctua- tions in the surface melt rate, particularly since measurements were not all made at the same time. The 1967 data provide an inconclusive test of the sinuosity hypothesis proposed above. The rank correlations between P and QS or qS 0 05 0 1 0-2 0 5 (where q — Q/W) are positive but only the Sine of glacier slope former reaches the 0.05 significance level, and Figure 3. ., stream width, and gla- although all partial correlations between P ard cier slope, 82 stream reaches, Lower Arolla glacier, Q or q given and P and S given Q or q, are 1967/1969/1970. Crosses indicate low sinuosity, dots positive, none is particularly strong. A larger high sinuosity. Best-fit disciiminant line shown is sample of less comprehensive measurements on W = 0.098 the same glacier in 1969 and 1970 is, however, available. No discharge figures are included, and direction of the predicted relations. Two but the excellent width-discharge correlation points from the more detailed results (Fergu- already mentioned permits use of width as a son, 1971) may be noted. First, the very similar surrogate; and although sinuosity was not exponents of width and slope in the equation measured or estimated in 1969, all reaches just given and the closeness of the discriminant were visually classed as of high or low sinuosity. line's slope to —1 (Fig. 3) show that the as- A similar classification in 1967 turned out to sumed equilibrium extent of meander develop- be nonoverlapping, with a separating value of ment is indeed an increasing function of the P = 1.15. Whether the pooled data can be WS product, and thus of the areal power treated as a random sample when up to three expenditure QS/ W in view of the dependence l12 traverses along each channel are included is of Won Q . Secondly, substitution of channel debatable, but channel adjustment is possibly for glacier slope reduces or removes the signifi- rapid enough, and ablation sufficiently variable, cance of the discriminant analysis and the for channel form to be an essentially inde- simple and multiple regressions for sinuosity. pendent response to different controlling condi- This too is in agreement with theory, since the tions each year. equilibrium sinuosity indicates the stage by which an initial power excess has been dissi- Multiple regression analysis of the combined pated; some residual positive correlation should data (n = 82) can be used to obtain a prediction (and does) remain because of the increase in equation for a zero-mean dummy variable, resistance, and thus required channel slope, representing sinuosity class, in terms of the with sinuosity for a given channel size. logarithms of W and S. This discriminant function indicates in the present case that high Although these results support the proposed sinuosity is associated with high slope at a theory, they should be regarded as tentative. given width and vice versa (Fig. 3), as ex- Similar findings from a larger and more accu- pected, and the associated multiple and partial rately measured sample embracing several gla- correlations are all significant at the 0.001 level. ciers and a wider range of stream size would be The use of the numerical sinuosity estimates more convincing. Extension of the techniques for the 1967 and 1970 samples, and logs of all of discriminant analysis of channel pattern and variables, both S and WS, but not W, turn out regression analysis of sinuosity to alluvial rivers to be good predictors of P in simple regressions, would necessitate inclusion of at least one and the multiple regression P = 1.79 erodibility variable. The findings of several S0187 has highly significant partial and multiple previous studies can be interpreted as con- correlations. These results confirm the existence sistent with the present theory, but an over-all

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test using published data is impossible because channel pattern [Ph.D. thesis]: Univ. Cam- of differences in the choice or definitions of bridge. variables—differences reflecting the complica- Glen, A. R., 1941, A sub-arctic glacier cap: The tions added by the sediment factor, the greater West Ice of North East Land: Geog. Jour., v. 98, p. 65-76, p. 135-146. variability of discharge, and the ambiguity Holmes, G. W., 1955, Morphology and hydrology between low-erosion and high-erosion (braided) of the Mint Julep area, southwest Greenland: low-sinuosity streams. This complexity empha- Arctic, Desert, Tropic Information Center, sizes the advantages of studying supraglacial U.S. Air University Pub. A 104B, 50 p. streams, but at the same time cautions against Knighton, A. D., 1972, Meandering habit of supra- uncritical extrapolation of their behavior. glacial streams: Geol. Soc. America Bull., v. 83, p. 201-204. ACKNOWLEDGMENT Leopold, L. B., and Wolman, M. G., 1960, meanders: Geol. Soc. America Bull., v. 71, Some of the field work reported here was p. 769-794. carried out during the tenure of a Natural Nobles, L. H., 1960, Glaciological investigations Environment Research Council research stu- on the Nunatarssuaq ice ramp, northwestern dentship. Greenland: U.S. Army SIPRE Tech. Rept. 66, 57 p. Stenborg, T., 1968, Glacier drainage connected with ice structures: Geog. Annaler 50A, p. REFERENCES CITED 25-53. Dozier, J., 1970, Channel adjustments in supra- Zeller, J., 1967, Meandering channels in Switzer- glacial streams: Arctic Inst. North Amer. Res. land: Internat. Assoc. Sci. Hydrology Pub. Paper 57, p. 69-117. 75, p. 174-186. Ewing, K. J., 1970, Supraglacial streams on the Kaskavvulsh Glacier, Yukon Territory: Arctic Inst. North Amer. Res. Paper 57, p. 121-167. MANUSCRIPT RECEIVED BY THE SOCIETY JUNE 12, Ferguson, R. I., 1971, Theoretical models of river 1972

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