Equipotential surfaces in a glacier

In a permeable porous medium, water flows in the direction of the negative of the maximum gradient of the potential, $, where $ is defined by:

Here, $o is a reference potential, Pw is the pressure in the water, pw is the density of water, g is the acceleration of gravity, and z is the elevation above some datum level such as sea level.

To gain some appreciation for this concept, consider the situation in a lake (Figure 8.2). Let $ = $1 at point 1 on the lake surface. Moving down a distance Az to point 2 increases Pw by pwgAz but decreases the third term on the right in Equation (8.1) by the same amount. Therefore $2 = $ 1 and there is no flow between points 1 and 2. However, if the lake

Figure 8.3. In a steady state, closure of a cylindrical conduit, u, is balanced by melt, m.

surface slopes gently towards the outlet, moving horizontally at constant z = z2 from point 2 to point 3 will result in a decrease in Pw, and hence in Flow will then be toward the position of lower Pw, which also is a position of lower $. In other words, it is not the gradient in Pw that controls the direction of flow, but the gradient in $.

To determine the potential field in a glacier from Equation (8.1), we must determine Pw everywhere. Pw is not hydrostatic because the water is moving, and most of it is a long way from the surface through many small passages.

In general, the pressure in the ice, Pi, is different from that in the water, and the ice deforms as a result of this pressure difference. Pw rarely exceeds Pi significantly, but it can be much less than Pi. Passages may thus increase in size slightly at very high water pressures, and they decrease in size rapidly at low pressures. In addition, as noted, heat generated by viscous dissipation melts conduit walls, enlarging passages (Figure 8.3). In a steady state, the rate of closure of passages by creep of ice, u, is equal to the melt rate, m, so the net rate of increase in size of the passages, r = m — u, is 0. (Although mathematically untidy, note that we have defined positive u as being inward, while positive m is outward. This simplifies some of the later equations.)

Let us assume thatthe flow ofice can be representedby se = (a e / B)n, that ice is incompressible and isotropic, and that the passages are circular in cross section. We further define the pressure causing creep closure, Pc, by:

(Figure 8.3). To a good approximation, Pi = pig(H — z), where His the elevation of the ice surface above the datum level (Figure 8.4). Then:

(Nye, 1953). This relation will be derived in Chapter 12 (Equation (12.22)). In the derivation it is assumed that ae = (l/v^a^where a^is the radial stress deviator. In other words, other components of the devi-atoric stress tensor, and hence of the strain rate tensor, are assumed to be negligible (see Equation (2.10)). Thus, there can be no deformation of the ice other than that resulting from the presence of the passage. In the present application, in which this assumption is clearly violated, we add a multiplying factor, K, which is approximately 1. K equals 1 if ae = (1/*Jl)a^. Rearranging and substituting for Pc and Pi, we can rewrite Equation (8.2) as:

Ice surface

Figure 8.4. Coordinate axes used in discussion of conduit closure.

(If u < 0, implying that the passage is opening as a result of water pressures in excess ofthe ice pressure, | u | must be used, and the sign of the second term adjusted accordingly, but u < 0 is rare in nature.) In excess of a couple of kilometers from the glacier terminus, parameters in the last term on the right change relatively little along a tunnel. Thus, combining Equations (8.1) and (8.4) and taking the derivative with respect to an arbitrary direction, s, yields:

ds ds ds

To determine the orientations of equipotential planes in the glacier, we make use of the fact that if s lies in such a plane, 9$/9s = 0, so:

Our objective is to define the dip of this plane, ¡3. The dip in some horizontal direction, x, will be dz/dx because z is the vertical coordinate of the plane (Figure 8.5). Therefore, multiply Equation (8.6) by ds/dx and rearrange, thus:

Pw - Pi d s d x or, inserting numerical values for the respective densities, letting a = dH/dx, the slope ofthe glacier surface, and noting that tan p = dz/dx:

Thus, the equipotential planes dip upglacier (note the minus sign) with a slope of about 11 times the slope of the glacier surface (Figure 8.6), a result first obtained by Shreve (1972), and water entering the glacier through a moulin should reach the bed of the glacier some distance downglacier from the entry point. In support of this, Iken and Bindschadler (1986) found that pressure fluctuations in boreholes drilled to the bed of Findelengletscher, just below the firn edge, did not correlate with marked variations in stream flow entering nearby moulins. Rather, the pressure fluctuations seemed to reflect a slower delayed input of water through the snow cover above the firn edge.

dx dx dz

Figure 8.5. Sloping conduit showing how ds and p are defined.

Figure 8.5. Sloping conduit showing how ds and p are defined.

50 m L Approximate scale 100m

Figure 8.6. Longitudinal section of a glacier showing upglacier-dipping equipotential surfaces and the theoretical directions of englacial water flow. Inset shows dimpling of an equipotential surface and consequent diversion of flow in smaller passages toward the main conduit. (After Hooke, 1989, Figure 1.)

Rigorously, the equipotential surfaces are defined only within the water passages, but with suitable caution, they can be treated as though they were defined throughout the glacier. Because Pw < P{ under normal conditions, O is slightly lower in the conduit than in the surrounding ice. Thus, the equipotential surfaces are dimpled in the vicinity of the conduits. As O decreases downglacier, the dimples point upglacier as shown in Figure 8.6. With the use of the theory presented below, it can be shown that the difference between Pi and Pw increases with increasing conduit size, so dimples around larger conduits are larger. Thus, water in smaller conduits flowing normal to equipotential surfaces will be deflected toward the larger ones. This strengthens the tendency of the conduit system to evolve toward an arborescent pattern.

Alternative derivation of equipotential-plane dip

Consider the situation in Figure 8.7. There is a conduit along the bed between points 1 and 2. We wish to determine under what conditions the upglacier slope of the hill will be parallel to an equipotential plane so that water in the conduit will not flow. The ice pressure at (1) is Pi1 = pig(h1 + h2 + Ah), and that at (2) is Pi2 = pigh2. In the absence of water flow and conduit closure, the pressure in the water at (1), Pw, would be the sum of Pi2 plus the hydrostatic head in the conduit, pwgh1. If Pi1 > Pw, the conduit will begin to close and water will be forced out over the hill. Thus, the condition we seek is Pi1 — Pw, or:

Solving this for h1, dividing by Ax, noting that a — - Ah/Ax and tan P — h1 /Ax, and inserting numerical values for the densities leads directly to Equation (8.7) Q.E.D.

Figure 8.7. Sketch illustrating alternative derivation of dip of equipotential planes in a glacier.
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