Introduction
Snow deposited on polar ice sheets is transformed to firn and then to ice through processes which involve both densification and recrystallization. Densification is achieved through the elimination of pore space, while recrystallization involves the change of crystal shape and size, and the production and growth of crystal bonds.
In near-surface regions, diurnal and seasonal variations produce layers with different properties. Since pore spaces in firn are in contact with the atmosphere, no process in firn is totally free of atmospheric influence. However, the annual temperature wave is mostly damped out by 10 m depth, and the effects of shorter-term temperature variations and other surface processes are significant to only a few meters depth. As a first approximation, distinct layers are assumed to develop only in the upper few meters. As these layers are buried, they entera relatively isothermal region extending from about 10 m depth to the firn - ice transition. The rate at which densification and recrystallization occur in t h i s region is ultimately related to the mean annual temperature and accumulation rate, and to the structure of the firn entering the region.
This paper reports the results of studies on the firn structure of a 50 m core taken from Dome C, East Antarctica, during the 1973-79 austral summer. Dome C camp is located on an ice divide at 74°30`S, 123°10'E, elevation 3 240 m. Measurements by Bolzan (unpublished) during the 1978-79 and 1979-80 austral summers indicate that the 10 in temperature is about -54.3°C. The mean annual accumulation rate is 34 kg m-2 a-1 (Palais 1980) and the firn-ice transition is at about 100 m depth (Reference Raynaud, Duval, Lebel and LoriusRaynaud and others 1979).
The 75 mm diameter core was drilled by personnel from the Polar Ice Coring Office in Lincoln, Nebraska, USA, and was shipped to Columbus where i t was studied during the fall and winter of 1979-80.
Core description
Visual inspection of the core, as well as examination in transmitted light, reveals distinct layering. Individual layers range from one to several tens of millimeters in thickness. The core is mainly composed of coarse-grained firn with high light transmissivity, but 5 to 10% of the core consists of thin, finegrained layers of low transmissivity. Contacts between layers are abrupt. Although more distinct at the surface, layers are still well defined at So m. At no point is there any clear indication of regular spacing corresponding to annual layering, or any other regularity. There are also small variations in transmissivity and grain-size within the coarse firn, but these variations were not as pronounced as those between coarse and fine firn, and were not closely investigated.
Results
Density measurements
The densities of 1 m long core sections were measured, as well as the densities of superjacent fine and coarse firn layers. The density variation with depth of the 1 m long firn sections defines an “average” density profile and is given by
for fine firn the density varies with depth according to
while for coarse firn
where z is the depth in meters.
The correlation coefficients for the empirical fits to the density data given by Equations (1),(2),(3),, and are 0.995, 0.989, and 0.996 respectively. The data for fine and coarse firn, along with the analytical fits, are plotted in Figure 1. Firn ages were calculated using the average density profil e. From Figure 1 it is apparent that (1) coarse firn is.initially less dense than fine firn, (2) coarse firn densifies more rapidly than fine firn of equal age, causing the difference in density between coarse and fine firn at a given age to decrease with time, and it can be shown that (3) coarse firn densifies more rapidly than fine firn of equal density.
Fig.1. Density as a function of depth for coarse grained and fine-grained firn. Solid lines are empirical fits to the data described in the text.
Crystal-size measurements
Crystal cross-sectional areas for fine and coarse firn are plotted in Figure 2. For details of the measurement procedure see Reference AlleyAlley (1980).
Fig.2. Crystal cross-sectional area as a function of time for coarse-grained and fine-grained firn.
Reference GowGow (1969) and Reference Stephenson and ŌuraStephenson (1967) observed that crystal area increases linearly with time in the isothermal region; that is
where A i s the mean cross-sectional area at time t, A0 is the extrapolated mean cross-sectional area at time zero, and K is the crystal-growth rate, usually expressed in mm2 a-1 . From Figure 2 we see that the Dome C data are in good agreement with this relation. A linear least-squares fit gives a growth rate of about 4.2 × 10-4 mm2 a-1 for coarse firn and 1.2 × 10-3 mm2 a - 1 for fine firn, a growth rate nearly three times as large.
Reference GowGow (1969) has found that the crystal-growth rate can be related to the mean annual temperature by
where K0 is a constant, R is the gas constant, T is the mean annual temperature in kelvins, and E is the activation energy for the crystal-growth process. Data reported inReference Gow Gow (1969) on growth rates and temperatures for five sites in Greenland and Antarctica fix K0 at 6.75 × 107 mm2 a-1 and the activation energy at 47.0 × 103 J mol-1. For an average firn temperature of -54.3°C, Equation (5)gives a growth rate of 4.2 × 10-4 mm2 a-1, in good agreement with the value obtained for coarse firn in this study.
The crystals studied by Reference GowGow (1969) were taken without regard to stratigraphy or structure, and so reflect the average firn properties at a particular site. At Dome C, the average firn properties are very nearly those of coarse firn, which comprises 90 to 95% of the firn core. We see that the crystal-growth process for firn at widely scattered polar sites is typical of coarse firn at Dome C, even though the typical crystal cross-sectional area varied widely from site to site in Gow's study. At South Pole, for example, crystal sizes measured by Gow near the surface are about the same as those of near-surface fine-firn crystals at Dome C. However, fine-firn Dome C crystals proceed to grow twice as rapidly, and do so in firn more than 3°C colder.
Sphericity measurements
The sphericity of a crystal in thin section is the diameter of the largest inscribed circle divided by the diameter of the smallest circumscribed circle. Regular polyhedra with many faces have relatively large sphericities, while very elongate or irregular crystals have low sphericities. Figure 3 shows examples of sphericity for some typical crystal shapes.
Fig.3. Sphericities of some typical grain shapes
Average sphericities were measured for each sample Reference Alley(Alley 1980) and the variation of sphericity with depth for fine and coarse firn is plotted in Figure 4. For fine-grained firn, sphericity decreases steadily with depth to 45 m; then i t appears to increase. No such regular behavior is seen for coarsegrained firn. However, we see that crystals i n coarse firn tend to be less spherical than crystals in fine firn at the same depth. Possibly the large scatter in the coarse-firn data i s due to the inclusion of different types of firn under the label “coarse”. It is possible that careful study might allow the distinction of different types of coarse firn at depth, based on sphericity, grain size, and other characteri stics. Such a distinction, however, was beyond the scope of the present study
Fig.4. Sphericity as a function of depth for coarse grained and fine-grained firn.
Surface area measurements
Based on probability theory, Reference Smith and GuttmanSmith and Guttrnan (1953) developed a method to estimate specific areas of crystal boundary (here Sc is the total ice-ice interface area per unit volume) and internal free surface (here Sf is the total ice-air interface area per unit volume). For a random surface in any three dimensional system:
where S i s the specific area in m2 m-3 (either Sc or Sf),L is the total length of line drawn with arbitrary spacing on the surface in m, and N is the number of intersections between the lines and either free surfaces or crystal boundaries seen on the plane of the section. Details and results of specific area measurements are published in Reference AlleyAlley (1980). Some values are given in Table I.
The ratio of the specific area of crystal boundary to total specific surface area for a sample, β = Sc/(Sc +Sf), is also the ratio of crystal-bond area to total crystal-surface area for the average crystal in that sample. From Table I, we see that in generale increases with density. However, even though the density of fine firn is larger than that of coarse, both have similar β values.
Crystal-bond characteristics
The crystal bond is that surface along which two crystals make contact. The radius r of the average bond in each sample was determined using the method of Reference FullmanFullman (1953) and Reference KryKry (1975). The radius R of the average crystal in each sample was calculated from measured crystal cross-sectional areas by assuming that crystals were spherical. Values of r, R, and r/R are listed in Table I. The ratio r/R is a measure of the relative size of bond and crystal. From Table I we conclude that bonds in coarse firn are larger relative to their average crystal size and that bond radius and the relative bond size increase more rapidly with time for fine firn.
Since crystal bonds are larger in coarse firn than in fine, but the ratio of crystal bond to total surface areaβ is nearly the same, it follows that an average crystal in fine firn participates in more bonds than a crystal in coarse firn. Again assuming spherical crystals, the total surface area per crystal At can be calculated from measured cross sectional areas. The product β x At is the average area per crystal involved in bonding. For bonds of circular cross-section, the number of bonds per crystal is then
Values of Nb, are listed in Table I. The number of assumptions involved in this calculation is considerable, so that the results obtained must be treated as rough approximations (Reference KryKry 1975). We tentatively conclude that crystals in fine firn are bonded to more nearest neighbors than crystals in coarse firn. The number of bonds per crystal may increase somewhat with depth for coarse firn, but any trends are poorly defined.
Discussion
Reference GowGow (1969, Reference Gow1975) treated densification of firn as being analogous to the unconfined sintering of ceramics, a process in which a ceramic powder heated near its melting point densifies in the absence of a confining stress. The driving force for unconfined sintering is provided by the large free energy of crystal surfaces relative to the free energy of the lattice structure that forms the interior of the crystal (surface free energy). This driving force varies inversely with the radius of curvature of grain surfaces, so that an increase in the radius of curvature of nearly spherical grains from 1 to 10μm decreases the rate of sintering by a factor of 10 (Reference BudworthBudworth 1970). Based on this argument, one would expect that fine-grained firn would densify more rapidly than coarse-grained firn of equal density.
TABLE I VARIATION OF FINE AND COARSE FIRN PARAMETERS WITH DEPTH
A possible explanation for the observed densification behavior may be the way in which fine and coarse firn structures respond to load. Crystals in coarse firn are joined by relatively wide necks to a few (3 or 4) neighbors. This configuration i s probably the result of extensive diagenesis due to the large temperature gradient in the near-surface region. Crystals in coarse firn exhibit a strong vertical shape orientation near the surface (Reference AlleyAlley 1980), perhaps as a result of strong vertical vapor and heat transport. This structure is far from closest packing and might be expected to under go significant particle rearrangement under an applied load.
Crystals i n fine firn tend to be more spherical and are joined by relatively narrow necks to many (6 or 7) neighbors. This configuration may result from wind-packing, followed by limited diagenesis due to the large temperature gradient in the near surface zone. Unlike coarse firn, fine-firn crystals exhibit no shape orientation close to the surface (Reference AlleyAlley 1980). This structure is more nearly closest packed and might be expected to be more stable under an applied load, as particles are less able to undergo extensive rearrangement.
Theoretically, several distinct mechanisms may operate simultaneously during the densification process. These include diffusion mechanisms such as volume, surface, and grain-boundary diffusion, mechanisms involving the mechanical deformation of particles such as creep and plastic flow, the transport of mass by evaporation and condensation, and particle rearrangement as grains slide past each other in response to an applied stress. The overall densification rate is then a sum of rates due to the various operative mechanisms, with different mechanisms possibly being dominant at different stages of the sintering process. We may ask then, what is the dominant mechanism for the densification of fine and coarse firn and is the same mechanism dominant for both fine and coarse firn?
Reference CobleCoble (1970) has derived expressions for the densification rates due to lattice diffusion and grain-boundary diffusion mechanisms. He explicitly incorporated the effect of load and assumed that closed spherical pores are distributed uniformly throughout the material and that the diffusional flux is spherically symmetric. For the grain-boundary diffusion mechanism, he also assumes that three grain boundaries pass through each pore (see Reference Wilkinson, Ashby and KuczynskiWilkinson and Ashby 1975). We have used these models to calculate densification rates for fine and coarse firn with one modification. Since the overburden is supported only at crystal-crystal contacts, we have taken the applied load to be the effective stress, Pb, acting across a crystal bond or neck:
where Pa is the overburden.
The results of our calculations indicate that both diffusion-controlled models predict that fine firn should densify more rapidly than coarse firn, in agreement with earlier qualitative considerations, but contrary to observed behavior. We conclude, at least within the context of Coble's models, that diffusion-controlled mechanisms are not dominant for both fine and coarse firn.
Wilkinson and Ashby have considered the pressure sintering process when densification is controlled by mechanical creep. They assume that the distribution of pores and grains in_the compact is uniform and that the strain rate, 
, for the fully densified material under a uniaxial stress,σ,is given by a power law
They have constructed densification models for the initial stage of sintering, where discrete grains are connected by well-defined necks, the intermediate stage of sintering, for which pores are cylindrical and isolated, and the final stage of sintering, characterized by isolated, spherical pores.
We have taken n = 3 inEquation (9) and calculated densification rates for the initial stage and the intermediate stage, the models most likely to describe the densification process over the depth interval we are considering. We have considered the case where the applied load is the overburden and have also let the applied load be the effective neck stress. We find that for either case, Wilkinson and Ashby's creep-controlled densification models predict that coarse firn densifies more rapidly than fine firn, as observed. However, for flow-law exponents greater than one, predicted densification rates for either fine or coarse firn increase with depth, making these models unacceptable. However, if a Newtonian viscous creep mechanism were operating at the temperatures (~ -54.0°C) and applied neck stresses (~1 bar) occurring here, then the Wilkinson and Ashby creep models would be viable candidates for the dominant densification mechanism.
However, it is difficult to assess the relative magnitude of the different densification models discussed above due to uncertainties in the values of the relevant parameters that appear in the theories, such as the exponent n and the factor A in the flow law. We tentatively conclude that diffusion mechanisms do not dominate the densification process for both fine and coarse firn in the near-surface region at Dome C, while the status of a power-law, mechanical- creep mechanism as the dominant process in the densification of fine or coarse firn is wore uncertain. We may expect particle rearrangement to play a significant role in the densification of coarse firn, while for fine firn the more stable structure may restrict the dominant densification mechanisms to diffusion and evaporation-condensation.
Currently no unified theory exists which explains all of the observed phenomena of densification and recrystallizatio in polar firn. More empirical data are needed to define trends in crystal-size and shape parameters and crystal-bond characteristics. Theoretical work is also needed, especially to quantify the contribution of grain boundary slip to the densification process, before such a unified theory can begin to emerge.
Acknowledgements
This work was supported by US National Science Foundation grant DPP-76-23428. Special thanks are due to Ron Coffman for technical assistance during the core analysis and to Robert Tope for drafting the diagrams.
