E l e c t r o n i c   r e p r i n t



Andrzej Kostrzewski and Zbigniew Zwolinski


The principal fluvial process at work in supraglacial channels is thermal erosion. It is of primary importance in river channels of the cold zone (Jahn 1970), as is also the case in the supraglacial stream on the Ragnar glacier under study. When analysing its downstream hydraulic geometry, it is worth noting that the rate of variations in the stream width b is more than twice faster than that of its mean depth f (Fig. 4a, Table 4). This means that the channel widens rather than deepens downstream. This is also confirmed by the width/depth ratio F (Fig. 4b, Table 4), which according to Schumm (1960) accounts not only for changes in the channel shape, but also provides information about the nature of material building it.

Fig. 4. Downstream hydraulic geometry of a supraglacial stream channel, Ragnarbreen, Spitsbergen

Table 4. Exponents of the downstream hydraulic geometry of the channel of a supraglacial stream, Ragnarbreen, Spitsbergen

Parameter Exponent Value
W b 0.45
D f 0.21
V m 0.34
P x 0.41
A ex1 0.66
F ex2 0.23
R ex5 0.25
Re ex6 0.59
Fr ex7 0.24
V* ex8 0.12
Ω ex9 1.04
n ex10 -0.18
ff ex11 -0.44
τ ex12 0.25

Note: Exponents ex1 ... ex12 have been calculated from exponential equation after Zwolinski (1989)

This pattern of the width W and depth D parameters is indicative of a greater contribution of vertical rather than lateral thermal erosion. It results from the fact that the measurements were carried out in the other half of the ablation season, when the glacier surface had undergone a radical superficial ablation transformation: its last winter snow cover had already melted away. Thus, glacier ice was deprived of a protection shielding it against the sun, and the channel could develop faster on the horizontal rather than vertical plane. In consequence, while the stream water had a higher temperature than the glacier ice, it melted its banks faster than its bed. Despite the slight depth of the active channel (2-4 cm), the stream incision was quite considerable, and ranged from a few to over 30 cm. The incision was the result of that particular ablation season, because the mean ablation rate under the channel bed could amount to about 0.5 cm day-1. Hence, it cannot be a prepared channel dating back to earlier ablation seasons.

The b > f relation is claimed by Carlston (1969) to be typical of minor rivers. Moreover, in the opinion of Rhodes (1987), the high relation b/f = 2.14 indicates that a stream carries no load of clastic material, and the cohesion of its bank material (in this case, ice) displays low variability. These conclusions conform fully to the observations of the supraglacial stream in question.

The velocity exponent m is contained between the values of the width and depth exponents, b and f respectively (Fig. 4a, Table 4). The increasing velocity of water flow is stimulated by the flat channel morphology (hence, low resistance), the growing discharge volume, and the lack of clastic sediments. Park (1981) adds a stream's convex long profile to these controls. In the case of the stream under study, the glacier surface was also convex. Since m > f, the stream has an increasing competence down its course (Wilcock 1971). In alluvial rivers this manifests itself in increased transport of suspended material (Leopold, Maddock 1953). The stream in question, however, did not carry clastic material. The result of its increasing competence and lack of sediments was a slightly greater change in its power Ω than its water discharge Q, because the exponent for Ω ex9 = 1.04 (Fig. 4c, Table 4). Such a pattern of the parameters is conditioned by the diminishing resistance down the stream, as is confirmed by hydraulic geometry relationships involving the Manning n and Darcy-Weisbach ff coefficients (Fig. 4f, Table 4). The diminishing values of resistance and friction down the long profile of supraglacial streams are their characteristic feature (Dozier 1976, Ferguson 1973, Knighton 1981, Park 1981).

The rate of variations in the velocity controls the rate of change in the channel cross-section area, which increases downstream, ex1 = 0.66 (Fig. 4c, Table 4). It follows from the discharge continuity equation, and hence it can also be calculated from the exponential equation ex1 = b + f (Zwolinski 1989). The increasing velocity V and depth D result in growing Reynolds Re and Froude Fr numbers (Fig. 4d, Table 4). The ice bed of the supraglacial stream was smooth and devoid of bed forms, and that is why the rate of change in the Froude number Fr was not considerable. High values of this number (Fr > 1.55) indicate shooting flows in the stream, which Knighton (1981) and Parker (1975) claim may cause its channel to meander. Although the rate of depth D variations is low, it strongly influences the Reynolds number Re (cf. Table 2 and Table 3), which shows a downstream increase in turbulence (cf. Fig. 4d, Table 1.

The growing rate of depth D variation also means an increase in the rates of variation in the hydraulic radius R, shear velocity V* and shear stress τ (Fig. 4e, Table 4). Worth noting are similar variation rates of depth D and the hydraulic radius R (Fig. 4a, e, Table 4), as well as in width W and the wetted perimeter P (Fig. 4a, b, Table 4), which has also been observed in alluvial channels (cf. Richards 1982, Zwolinski 1989).

When comparing the analysed segment of the supraglacial stream against the model b-f-m diagram for downstream hydraulic geometry proposed by Rhodes (1987), it fits the third area distinguished on it. This is an area in which channels are characterised by an increasing b/f ratio, an increasing velocity exponent m, a decreasing m/ex1 ratio, an increasing competence, or m > f, and an increasing m/f ratio. For Ragnarbreen supraglacial stream, the respective values are 2.14, 0.34, 0.52, 0.34 > 0.21, and 1.61.

The third area on Rhodes' b-f-m diagram includes rivers from a variety of morphoclimatic environments, e.g. an ephemeral stream in an arid climate, a stream in a loess area, a stream flowing into the ocean, an alpine stream, a pro-glacial stream, and even a river in an urbanised area. Such a wide variety of environments is a convincing proof that discharge is the dominant factor controlling downstream hydraulic geometry. Local or regional influences do not play any significant role in the formation of a river channel down its course. Similar conclusions have been drawn by Hey (1978), Park (1977) and Rhodes (1987), who state that the character and steadiness of variations in a river's long profile point to the increasing discharge as the determinant of the shape and size of its channel.

However, it might be useful to keep in mind Park's (1981) opinion that the development patterns of supraglacial streams and alluvial rivers clearly diverge in the context of hydraulic geometry relations. He only sees supraglacial streams as similar to pro-glacial rivers (quoting the example of the White River, after Fahnestock 1963) because of the similarity in their hydrological regimes. The studies of Rhodes (1987) allow the thesis about a different development of supraglacial and alluvial streams to be rejected. Thus, Leopold and Wolman's (1960) earlier claim that hydraulic mechanisms operating in a river channel are basically the same should still be regarded as valid. Marston (1983) and Richards (1982) also come to this conclusion.