## References

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Journal of Geology, Vol. 68, No. 6, p. 601–25.

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Harrison, W.D.
1975[b]. Temperature measurements in a temperate glacier.
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Hobbs, P.V.
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Taylor, P. Unpublished.A large thermal coring drill for temperate glaciers.

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Frank, F.C.
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Mae, S
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## Appendix A Shape of air columns in veins

When gas enters a vein, its shape must satisfy the requirements of mechanical equilibrium. Qualitatively, the gas fills the central portion of the vein, excluding the liquid to fillets at the three vertices. Details of the shape can be worked out approximately by assuming that the pressure difference across the liquid-gas interface is everywhere the same. The interface between the gas-filled central part of a vein and the liquid in the fillets is cylindrical and the corresponding pressure difference is easily calculated as a function of the cross-sectional shape of the gas column. The interface at the end of a gas column is doubly curving, but the pressure difference fan be estimated using the hydraulic radius of the gas column cross-section (Baer, 1972, p. 441). Equating these two pressure differences determines the shape of the gas column cross-section. The width averaged over all viewing directions is 1.91r, where *r* is the root-mean-square radius. The difference between the gas and liquid pressures, the capillary pressure, is given by 2.4γLV/r where γLV is the liquid-gas surface energy.

This configuration is not in thermal equilibrium. Because the pressure in the gas is larger than that in the liquid, those parts of the vein wall exposed to the gas must be colder than those adjacent to the liquid near the vertices. Whatever the details, melting and refreezing would cause a rounding of both vein and air-column cross section. Since heat would be transported only locally, there should not be much change in total vein cross-sectional area. A limiting cross-sectional shape would be circular with the liquid excluded almost entirely, in which ease the width of the air column would be 2r. Although the shape of the end of the gas column could still be complex, it would probably be close to hemispherical, which would give capillary pressure 2γLV/r. Because of the small scale of the veins, it is likely that the local redistribution of heat occurs rapidly. If the vein liquid contains im purities, the process could he slowed, but a similar shape change would occur as long as impurities can diffuse along the vein walls.

## Appendix B Adiabatic relaxation

When the vein pressure or impurity content is changed, melting or freezing will occur at the vein walls, and the latent heat will change the temperature of the surrounding ice. If the process is adiabatic and the liquid content is small, then

after equilibrium has been re-established. *A* is the vein cross-sectional area, *T* is the temperature, and the sub scripts 1 and 2 refer to values before and after the change, *h, ρ*_{i}
and *c*_{i}
are respectively the Specific latent heat of fusion, the density and the specific heat capacity of pure ice. *l* is the vein length per unit bulk volume, and *p* is the bulk density of the ice. Also *p* ≈ *p*_{i}
When phase change takes place elsewhere than in the veins, *c*_{i}
must be replaced by an effective bulk heat capacity c’ of the ice surrounding the veins.

For the special case in which the vein water is continuously replaced by water of a given impurity content and pressure, *T*
_{2} is determined by Equation (4a) when vein curvature is neglected. If *T,* represents the *in situ* temperature, which is known, the vein area change from *in situ* conditions can be calculated directly from Equation (B-1).

A more complicated case occurs when the pressure change is specified, but the impurity content per unit length of vein *C*_{i}
remains constant. Equations (4) then give

where *p* is the vein pressure, and vein curvature has been neglected. This is used to eliminate *T*
_{2} — *T,* in Equation (B-1) with the result

where *ρ*
_{i} = *C*_{l}/A_{l}
, is the initial temperature depression from impurities. This assumes that all the liquid is in the veins; otherwise *c*
_{i} is replaced by *c*
^{1}. If the ice is line enough *(I* very large) equilibrium on pressure release is restored by freezing some of the relatively abundant vein water, with a small change in vein area and a temperature change approaching *— β*(*p*_{2}—p_{i}). In coarse ice, equilibrium is restored mainly by lowering the vein temperature by concentration of impurities, with a large change in vein area and a small temperature change.

The time for adjustment of the vein size can be roughly estimated by assuming the vein to be cylindrical and to be surrounded by an infinite medium of ice. The rate of heat flow is then calculated by standard analytical methods (Carslaw and Jaeger, 1950). Equilibrium will be reached when the total heat flow per unit length of vein reaches of *pc*#x2019; (*T*
_{2} — *T*
_{1})/*l*. The result is not very sensitive to vein size. If the liquid is confined to the veins, equilibrium is reached on the time scale of seconds to a few minutes, depending upon grain size. The assumption that the process is adiabatic, or that little heat conducts in from the surface before its completion, is therefore reasonable for locations which are more than several grain diameters into the core.

## Appendix C Heat conduction in to sample

If the fractional liquid-water content of a temperate ice sample is small, and if the liquid is confined to the veins, the thermal diffusion equation is

where *T* is temperature, *A* is the vein cross-sectional area, and *l* is time. *K* = 2.1 W m^{−1} deg^{−1} is the thermal conductivity of ice, *c*
_{1} = 2.1 × 10^{3}J kg^{−1} is its specific heat capacity, ρi = 0,92 Mg m^{−3} is its density, and *h* = 3.34 × 10^{5} J kg^{−1} is its specific latent heat, ρ is the bulk density of the sample (ρ ≈ ρi) and *l* is the total vein length per unit volume. *lA* is the total liquid content per unit volume, and the second term on the right side expresses the fact that it may change with temperature. *A* can be eliminated with the help of Equations (4), which give

In the special case that the impurity content per unit length of vein *C*
_{l} remains constant, Equations (C-1) and (C-2) give

where the effective heat capacity *c* is given by

.

Here *θ’ = δ — — β p—*
*T* the combined temperature-lowering by impurities and curvature, and λ = ;/2(C_{i}’)^{½}. An effective thermal diffusivity *κ* can also be defined as κ = K/*pc*.

In the applications considered, the temperature depression *θ* due to impurities is the major contribution to *θ’;* this gives *θ* ≈ *θ’* and λ ≈ = 0. Since C_{l} = *Aθ* by definition from Equation (4b), the effective heat capacity may be written in the alternative forms

The first of these is the form given by Harrison (1972); *lC*
_{l} is identified as the temperature depression by im purities when the last of the solid phase disappears upon melting, and this single parameter determines the behavior.

If *θ* is constant, *δA/δt* is found from Equation (4a) with *θ* constant. Then

where *θ’ — Θ* is the temperature lowering due to curvature.

The numerical solution of the diffusion equation with temperature-dependent heat capacity is straight forward by a method similar to that described in Carslaw and Jaeger (1959, chapter 18). Once the temperature is found, the vein area can be found from Equation (C-2). Alternatively, the vein area can be found directly by eliminating *T* rather than *A* in Equations (C-1) and (C-2).

## Appendix D Statistical analysis of blockage of veins by bubbles

Consider a unit area of some reference plane. There are *n* veins crossing this area. We want to know how-many of these veins can actually transport water.

Let *p* be the probability that any given network link represented by a segment of vein between two four-grain intersections is not blocked by any bubbles (or other type of blockage). Then it is clear that only *pn* of the links crossing the unit area on the plane are unblocked. In addition, some of these *pn* unblocked links cannot transport water because all of the paths leading to (hem from above or below (he plane are blocked. Let *Q(p)* be the probability that a given link crossing the plane is cut off from distant veins below the plane. In order lo transport water, the link must be connected to the vein network both above and below the reference surface. Therefore, the effective number of veins crossing the surface is

In order to estimate *p,* we consider the average spacing *s* of bubbles along veins. It is reasonable to assume that the probability of finding a single bubble touching a vein in a very short length interval is proportional to the average number of bubbles per unit length of vein (*I/s)* and that the probability of finding more than one bubble for such a short length interval is zero. In this case the probability of finding *k* bubbles on a link of length *b* is given by Poisson’s formula. In particular the probability of no blockage or *k* = o is given by

Fig. D1.
*Simplified topological model of the vein network.*

To estimate *Q(p)* consider access to successively more remote groups of four-grain intersections denoted as levels 1, 2, 3, etc. in the simplified topological model of the vein network shown in Figure D-1. For the first level it is clear that *Q*
_{1} = (1—*p*)^{3}. To get *Q*_{2}
consider a single one of the links leading to level 1, Starting along this path, access to level 2 is blocked if this link to level 1 is blocked (probability (1 —*p*) or if this link is unblocked (probability *p)* but the triplet of veins joining it to level 2 is blocked (probability *Q*
_{i}. Thus *Q*
_{2} = [(1 — *p*) + *pQ*
_{1}]^{3} where the third power arises because there are three possible starting paths. Successive application of this argument gives

Fig. D2.
*Fig. D-2. Fraction of veins capable of transporting liquid across reference plane *(n_{e}/n) *Versus the ratio of mean spacing of four-grain intersections (b) to the mean spacing (s) of blockages in veins.*

It was found that the sequence *(Q*_{m}) converges fairly rapidly, which is consistent with the obvious requirement that when *p* does not depend on position, local connectedness of the network implies large-scale connectedness as well. The topological model used in this calculation neglects the fact that some of the paths lead hack on themselves and that some cross back over the reference plane. Because these complications should become most important at the more distant levels of four-grain intersections and *{Q*_{m}} converges fairly rapidly, the limit of the sequence is probably a reasonable estimate of *Q* for the real network.

Together, Equations (D-1), (D-2), and (D-3) define the relationship between *n*
_{e}, *n*, *b* and *s*. This is plotted in Figure D-2. One finds that *n*_{e}/n is essentially zero (< 10^{−3} for *b/s* 1.2.