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


HYDRAULIC GEOMETRY
OF A SUPRAGLACIAL STREAM,

RAGNARBREEN, SPITSBERGEN

Andrzej Kostrzewski and Zbigniew Zwolinski

VARIABILITY AND RELATIONS OF GEOMETRIC AND HYDRAULIC PARAMETERS

The range of variability of hydraulic geometry parameters of studied supraglacial stream is shown in Table 1. Worth noting is the fact that the measured range of water discharge 0.00348-0.01551 m3 s-1 resembles that recorded for other supraglacial streams, e.g. on the glaciers Arolla (Switzerland): 0.003-0.1 m3 s-1 (Ferguson 1973) and 0.002-0.045 m3 s-1 (Park 1981), Austre Okstinbreen (Norway): 0.002-0.052 m3 s-1 (Knighton 1981) and 0.005-0.027 m3 s-1 (Knighton 1985), and Österdalsisen (Norway): 0.005-0.02 m3 s-1 (Knighton 1972). The width range of the Ragnarbreen stream 0.19-0.49 m also fits the data for the above supraglacial streams: 0.2-0.6, 0.13-0.42, 0.15-0.5 and 0.2-0.6 m. This fact justifies the comparison of the results of the Spitsbergen observations with those made of streams on other glaciers, and hence making generalisations.


Table 1. Variability of parameters of the hydraulic geometry of a supraglacial stream on Ragnarbreen glacier, Spitsbergen


Parameter min. mean max. Unit
Q 0.00348 0.01041 0.01551 m3 s-1
W 0.19 0.31 0.49 m
D 0.0191 0.0283 0.0373 m
V 0.757 1.162 1.607 m s-1
A 0.0046 0.00881 0.0138 m
P 0.2384 0.3706 0.5398 m
F 7.68 11.66 23.05 -
R 0.0176 0.0236 0.031 m
Fr 1.55 2.22 2.91 -
Re 8168 15558 23287 -
Ω 2.36 7.45 10.97 N s-1
V* 0.122 0.141 0.162 m s-1
τ 14.92 20.07 26.36 N m2
n 0.016 0.0214 0.0291 -
ff 0.0682 0.1296 0.2286 -

The spatial analysis of individual parameters shows that most of them tend to increase perceptibly down the long profile of the studied stream fragment (Fig. 3). This is an effect of the growth in discharge and hence in the area of the stream catchment, or, as Park (1981) puts it, of the increasing sources of supply. It is indicative of the permanence of the ablation process throughout the catchment area. The only parameters that decrease downstream are the Manning resistance coefficient n and the Darcy-Weisbach friction factor ff, which is closely associated with an increase in the water discharge, and especially with its velocity.


Fig. 3. Patterns of selected parameters in the long profile of a supraglacial channel, Ragnarbreen, Spitsbergen


When analysing the patterns of all the measured, and hence calculated, parameters of the channel cross-section, they can be seen to change irregularly (Fig. 3), either diminishing or growing. The changes occur in some places of the long profile and repeat themselves for each parameter, even though their direction and range may vary. The changes are readily visible in e.g. the two simplest parameters, viz. width W and depth D. As in Park's (1981) research, variations in the depth of the Ragnarbreen stream are more regular than those in its width. In his comment on the irregularity of geometric and hydraulic parameters, Knighton (1981, 1985) accounts for them by changes in channel morphology, especially overdeepening in bends. However, the fragments of the supraglacial stream selected for measurements were straight reaches. Thus, his explanation is not satisfactory in this case, although it cannot be discarded altogether, because such situations had also been observed at places other than the cross-section sites. Variations in the parameters of the stream may be due to tributaries flowing into the main channel, changing local gradients, and cracks in the glacier. On the basis of measured (not calculated) parameters W, V and A, we can trace the influence of these factors on the channel of a supraglacial stream.

The pattern of perpendicular cracks in the glacier has a marked influence on the middle part of the long profile (300-400 m). It caused the channel width to more than double, while the water flow velocity diminished only slightly.

The largest area of the channel cross-section A to be measured was the penultimate one (680 m). This was the effect of the least local gradient, which amounted to 0.0673 at that place. It was responsible for the increased channel width there and a marked decrease in the water flow as compared with the preceding cross-section.

Worth noting is the fact that the other calculated parameters have identical or very similar patterns (with one exception) to those of the three measured ones, viz. W, D and V. The three, however, show no resemblance in their patterns down the long profile. The correspondence looks as follows:

  1. the parameters corresponding to channel width W are wetted perimeter P and width-depth ratio F,
  2. those corresponding to the channel cross-section area A are water discharge Q and stream power Ω (Fig. 3b); there is also some similarity in the patterns of width W, and hence of wetted perimeter P and width-depth ratio F,
  3. directly proportional to flow velocity V are the Froude Fr and Reynolds Re numbers, and while the Manning resistance coefficient n and the Darcy-Weisbach friction factor ff are inversely proportional (Fig. 3c), and
  4. four parameters form a distinct group, namely mean depth D, hydraulic radius R, shear velocity V* and shear stress τ (Fig. 3d); their patterns are almost identical, especially of the last three, and the parameter to which they correspond most closely is the flow velocity V.
Of the four groups mentioned above, group (a) contains geometric parameters defining the size and shape of the channel. The second (b) embraces geometric and hydraulic parameters which not only define the dimensions of the channel, but also its energy potential. The last two groups (c) and (d) include hydraulic parameters defining the nature and conditions of water flow in the channel. Through a natural selection, using the criterion of the similarity of patterns of all the parameters under analysis, a division has been obtained which systematises the significant statistical relations holding in the supraglacial stream involved. The strength of these relations is presented in a correlation matrix (Table 2), while all those which are significant at the α < 0.05 level are listed separately in Table 3. There are 38 significant relations among the 105 pairs of geometric and hydraulic parameters studied. Notable in the correlation matrix is the fact that correlation coefficients range from almost 0 (Fr-τ) to 1 (R-τ). Such a range is indicative of differences and on the other hand of similarities in the patterns of parameters measured downstream; of differences and similarities in their responses to changes in the discharge; and of relationships and lack of relationships among them. Most of the significant and non-significant correlations are obvious enough to require no explanation. The importance of others has been discussed with regard to alluvial channels, e.g. W-P, D-R, Q-Ω or n-ff (cf. Richards 1982, Zwolinski 1989), and confirmed with regard to the channel of a supraglacial stream. Thus, they can be treated as regularities independent of the environmental conditions in which a river channel has developed.


Table 2. Correlation matrix for geometric and hydraulic parameters of the channel of a supraglacial stream, Ragnarbreen, Spitsbergen


Parameters Q W D V A P F R Fr Re Ω V* τ n ff
Q 1.00
W 0.58 1.00
D 0.57 -0.19 1.00
V 0.72 0.09 0.41 1.00
A 0.86 0.74 0.52 0.29 1.00
P 0.65 0.99 -0.08 0.14 0.81 1.00
F 0.17 0.87 -0.63 -0.10 0.32 0.81 1.00
R 0.71 0.03 0.97 0.44 0.69 0.14 -0.45 1.00
Fr 0.54 0.28 -0.06 0.88 0.13 0.28 0.28 -0.00 1.00
Re 0.83 0.04 0.77 0.90 0.51 0.13 -0.32 0.78 0.59 1.00
Ω 0.99 0.68 0.47 0.67 0.89 0.74 0.30 0.63 0.55 0.75 1.00
V* 0.71 0.02 0.98 0.45 0.68 0.13 -0.46 1.00 0.02 0.80 0.63 1.00
τ 0.71 0.03 0.97 0.44 0.69 0.14 -0.45 1.00 -0.00 0.78 0.63 1.00 1.00
n -0.35 -0.17 0.19 -0.80 0.06 -0.15 -0.26 0.16 -0.97 -0.46 -0.36 0.14 0.16 1.00
ff -0.45 -0.21 0.06 -0.85 -0.06 -0.21 -0.23 0.02 -0.96 -0.55 -0.45 -0.00 0.02 0.98 1.00

Table 3. Correlations at the α < 0.05 level between geometric and hydraulic parameters of a supraglacial stream channel, Ragnarbreen, Spitsbergen


Parameters r α R2
R - τ 1.000 0.00000 100.00
R - V* 0.999 0.00000 99.80
V* - τ 0.999 0.00000 99.80
W - P 0.994 0.00000 98.76
Q - Ω 0.990 0.00000 98.11
n - ff 0.985 0.00000 96.98
D - V* 0.975 0.00000 95.12
D - R 0.974 0.00000 94.90
D - τ 0.974 0.00000 94.90
Fr - n -0.965 0.00000 93.15
Fr - ff -0.960 0.00000 92.07
V - Re 0.896 0.00044 80.36
A - Ω 0.885 0.00066 78.37
V - Fr 0.881 0.00077 77.54
W - F 0.870 0.00108 75.60
Q - A 0.864 0.00126 74.68
V - ff -0.845 0.00208 71.41
Q - Re 0.825 0.00327 68.14
P - F 0.812 0.00437 65.86
A - P 0.808 0.00471 65.24
V - n -0.798 0.00569 63.63
Re - V* 0.797 0.00580 63.47
R - Re 0.784 0.00732 61.41
Re - τ 0.784 0.00732 61.41
D - Re 0.766 0.00980 58.66
Re - Ω 0.746 0.01314 55.72
P - Ω 0.742 0.01391 55.13
W - A 0.738 0.01479 54.48
Q - V 0.723 0.01812 52.28
Q - V* 0.714 0.02050 50.91
Q - R 0.711 0.02112 50.57
Q - τ 0.711 0.02112 50.57
A - R 0.691 0.02702 47.70
A - τ 0.691 0.02702 47.70
A - V* 0.680 0.03066 46.18
W - Ω 0.679 0.03089 46.09
V - Ω 0.673 0.03309 45.24
Q - P 0.651 0.04146 42.39

Park (1981) has found that the flow velocity increased down a supraglacial stream, and accounts for it by the increase in the gradient of the glacier surface, which has a convex profile, and primarily by the growth in discharge and the size of the channel. We might add, on the basis of the correlation matrix and the spatial analysis of the parameter patterns, that the velocity V changes in accordance with the Froude Fr and Reynolds Re numbers, and inversely to the Manning resistance coefficient n and the Darcy-Weisbach friction factor ff (Fig. 3c). The increase in the dimensions of the channel implies decreasing resistance in the water flow. A smooth stream-bed devoid of bed forms is reflected in high and growing values of the Froude number Fr. With the increase in depth down the stream, turbulence of the flowing layer of water grows too, which causes the Reynolds number Re to go up. All these factors are clearly involved in the increase in velocity V in the supraglacial stream.