## Melt rates in conduits

Let us now consider the rate of melting of conduit walls, following Shreve (1972). The total amount of energy available per unit length of conduit, As, per unit time is:

Some of this energy must be used to warm the water to keep it at the pressure melting point as ice thins in the downglacier direction. The rest is available to melt ice, thus:

m As (2rnr p1 L) + pw Cw C --- pi g As Q = Q — As

Here, r is the radius of the conduit, L is the latent heat of fusion, Cw is the heat capacity of water, and C is the change in the melting point per unit of pressure (see Equation (2.2)). As you will see from inspection of the terms and the dimensions of the various quantities in them, the first term on the left is the energy used to melt tunnel walls, and the second is the energy needed to warm the water to keep it at the pressure melting point. Here, we have implicitly taken the positive s-direction to be upglacier, in the direction opposite to that of the water flow. Thus, both 3 $/3s and 3(H - z)/3s are positive.

It is common to define k = pwCwC. Inserting numerical values (pw = 1000 kg m-3, Cw = 4180 J kg-1K-1, and C = 0.074 x 10-6 KPa-1)

we find that k = 0.309 and that it is dimensionless. If we assume that the water is saturated with air, and adjust C accordingly, k = 0.410.

Then, using Equation (8.5) and dividing by As yields:

or solving for m:

2rnrpi L

It is interesting to insert some numbers into this equation to get a sense of the magnitude of m. Consider a horizontal tunnel so dz/d s = 0. Suppose the tunnel has a diameter of 0.5 m and that it is under a glacier with a surface slope of 0.01. We now need a relation between Q and the tunnel roughness. The Gaukler-Manning-Strickler equation is one of two that are commonly used for such calculations. It is:

Here, v is the mean velocity over the tunnel cross section, R is the hydraulic radius of the tunnel, or the cross-sectional area divided by the perimeter (so R = r/2 in circular tunnels), S is the nondimensional headloss:

Pwg d s which is approximately equal to the glacier surface slope, and n' is known as the Manning roughness coefficient. For smooth channels, n' may be as low as 0.005 m—1/3 s, but studies of floods, called jokulhlaups, resulting from drainage of ice-dammed lakes through subglacial conduits yield values ranging from 0.08 to 0.12 m—1/3 s (Bjornsson, 1992). A still higher value was obtained from dye-trace experiments on Storglaciaren; flow velocities there suggested n' & 0.2 m—1/3 s (Seaberg et al., 1988; Hock and Hooke, 1993). Where roughness elements on the tunnel walls and floor are large in comparison with the tunnel size, n' will be higher; this is probably responsible for the relatively high value from Storglaciaren. Choosing an intermediate value of 0.1 m—1/3 s, Equation (8.13) gives a mean velocity of about 0.25 m s—1, or Q & 0.05 m3 s—1, and Equation (8.14) gives d$/ds & 98 N m—3. Whence m & 0.22 m a-1. This may not seem like a lot but, volumetrically, the amount of ice melted in a year is 2.6 times the size of the original conduit.

A consequence of this melting and the resulting inward flow of ice towards the conduit is that structures such as foliation in the ice are also bent inward. A beautiful example of this is shown in Figure 8.8.

Some heat may also be advected into the glacier in water originating at the glacier surface and entering the englacial conduit system by way of moulins. The melt rate from such water, ms, is:

2^rpi L

(Shreve, 1972). Here, 3 9/3s is the rate at which the water cools as it flows through the conduit. If we assume that 3 9/3s ~ 0.1 K km-1 and use the discharge in the previous example, ms ~ 0.08 m a-1. Thus, this is a heat source that cannot be neglected. A possible mitigating factor, however, is that ice crystals are often carried in streams on a glacier surface. Thus, some of the heat would be used to melt these crystals rather than the conduit walls. It is not clear how the energy will be partitioned in this situation. One would also expect most of this heat to be consumed in the moulin itself or in the first few hundred meters of flow in an englacial passage.

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