FORMATION OF GLACIAL

Here we consider the transformation of to ice and normal grain growth. The deformation of ice is treated later. The processes that depend on temperature generally follow an Arrhenius relation, where the rate is ∝ exp(−c/T ), where c is a constant, and T is temperature. This means that when it gets colder, these processes slow down rapidly.

A Transformation of Snow to Ice

The transformation of snow to ice occurs in the top layer of the and ice sheets. The depth range depends mostly on temperature. The density of the snow as it falls on the surface depends on the weather conditions (see Table ??). In calm conditions the density ρ ' 100 kg m−3. If it is windy, there is breaking of the corners of snowflakes, and the density is more like ρ ' 300 kg m−3. After the snow has fallen on the surface there are three processes that are all active together and work to transform the snow to ice. Those are: 1) Packing and/or settling, 2) thermodynamic processes, and 3) deformation under load. The packing involves further breaking of the snowflakes (wind-blown surface layer), and settling of the snow . The thermodynamic processes all aim at minimizing the free energy. This is achieved by reduc- ing surface area which reduces free energy. Since a sphere has the smallest surface area for a given volume, this process makes the snow crystals round. Figure 1 shows a schematic of how the snow crystals get round by thermodynamic processes. The higher the curvature is, the less stable the grain. Larger grains grow and smaller ones disappear. When these processes have acted on the snowpack, the grains are “mostly” spheres of nearly equal size, and the density is ρ ∼ 550 kg m−3. The speed of these thermodynamic processes is highly dependent on temperature (as long as the temperature is below 0◦C).

Sintering. Compressing particles into a coherent body (Uvarov and Isaacs, 1986).

Under load, sintering occurs. During sintering the spheres are “glued” together where they touch, see Figure 2. When air is trapped inside bubbles, the grains are glued together where they touch. We call this glacier ice, and the density is ρ = 830 kg m−3. −3 Under load there is continuous deformation, and ice slowly reaches ρi ∼ 917 kg m . The bubbles change from triangular shape, at triple junctions, to spherical, but the flow of ice can make them elliptical.

1 Density vs. depth A simple emperical relation for the evolution of density with depth is,

ρ = ρi − (ρi − ρs) exp(−Cz), (1)

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Air 2

Snowflake

1 b a

Figure 1: Rounding of a snowflake. 1. Molecular diffusion: a) In the ice (volume diffusion), and b) on the surface (surface diffusion). 2. Vapor diffusion: Higher vapor pressure at concave (odds) than convex parts.

Bonds Air bubbles

Figure 2: During sintering the snow crystals are “glued” together where they touch.

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0

50

100

150

Depth (m) 200

250

300 0 200 400 600 800 1000 Density (kg m−3)

Figure 3: Density profile according to Equation 1, with C =0.0275 m−1 which fits the data from Byrd, Greenland (Paterson, 1994).

where ρi is ice density, ρs is the density at the surface, C is a constant, and z is depth. Figure 3 shows the theoretical density profile given by Equation 1. More realistic models have to include the different processes responsible for the densification, such as Grain Boundary Sliding, Dislocation Creep, and Lattice- and Boundary diffusion (Arthern and others, 2000). At Dye 3, Greenland, the depth to the firn-ice transition is 65-70 m and the age is 100 a. There the accumulation rate is 490 kg m−2 a−1, and surface temperature -19 ◦C. In Antarctica, at the site of the Vostok , the corresponding numbers are 95 m and 2500 a, where the accumulation is only 22 kg m−2 a−1, and the surface temperature really cold, -57 ◦C. In Iceland, firn (ρ ∼ 550 kg m−3) is produced during the summer. Glacier ice is formed at 20-30 m depth, and takes less than 10 years.

2 Depth Hoar Under conditions that produce strong vertical temperature gradient, and thus a strong gradient of vapor pressure, such as in autumn when the surface can cool rapidly while the deeper layers are relatively warm, large ice can form at depth. These crystal, having an average grain size of 2 to 5 mm and densities of 100 to 300 kg m−3, are called depth hoar, see Figure 4. Depth hoar crystals are formed by sublimation, and only develop in unconsolidated snow. The strong vapor pressure causes vapor to rise and condense to form depth hoar crystals in the upper layers. Figure 5 shows a schematic of how depth hoar forms.

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Figure 4: Hoar crystal from a deep-freeze chest. This cup shape is also commonly seen in depth hoar. Magnification is 29X (LaChapelle, 1992, Fig. 63).

Very cold surface Depth hoar

Empty space heat moisture in some cases Warm ice Temperature

Figure 5: Formation of depth hoar.

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B Grain Growth

Crystals grow by grain-boundary migration, which is driven mainly by the curvature of grain boundaries and differences in stored energy between grains (Alley and others, 1986). Assuming that the growth rate is controlled by the interfacial free energy of the grain boundaries leads to an equation for normal grain growth (Cole and others, 1964), where the growth of the mean crystal diameter increases with time (Gow, 1971; Alley and others, 1986) according to a parabolic growth law 2 2 D − D0 = k · t, (2) where t is time, and D0 is the mean crystal diameter at t = 0. The grain growth factor is Q k = k exp(− ), (3) 0 RT where T is the temperature, Q is activation energy for grain-boundary self-diffusion, R the gas constant, and k0 is a constant that depends on impurity concentration. Typical values for ice are −9 2 −1 −1 k0 =8.2 · 10 m s and Q = 40 kJ mol (Alley and others, 1986). Figure 6 show the theoretical evolution of crystal size, with numbers that are similar to those found at Byrd station, Greenland.

1.5

1.4

1.3

1.2 Diameter (mm) 1.1

1 0 20 40 60 80 100 Time (years)

Figure 6: Crystal size as a function of age according to Equation 2, with k = 120 × 10−4 mm2 a−1.

Snow to Ice Problems −1 −3 1 Find ρ at 40 m depth if C =0.025 m , ρs = 300 kg m . 2 Find the type of snowflakes that form at a) T = -10◦C and excess vapor density of 0.2, b) T = -15◦C and excess vapor of 0.4, and c) T = -2◦C and excess vapor of 0.01. Figure in “The Handbook” for instance.

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3 How much would there be in a 50 cm by 40 cm by 100 cm fish-tank filled with (1 m3 = 1000 l):

– Water – Settled snow – – New snow – Glacier ice

4 Discuss the difference at the same depth, if two sites are at the same temperature, but one has 3× the accumulation rate. What will be the same (similar) and what will be different?

Grain Growth Problems • How large will a crystal be after 50 years if the initial size is 1 mm in diameter and the growth rate is k = 120 · 10−4 mm2 a−1

Alley, R. B., J. H. Perepezko and C. R. Bentley. 1986. Grain growth in polar ice: I. Theory. Journal of Glaciology, 32(112), 413–424.

Arthern, R. J., D. P. Wienbrenner and E. D. Waddington. 2000. Densification of water ice deposits on the residual north polar cap of mars. Icarus, 144, 367–381.

Gow, A. J. 1971. Depth-time-temperature relationships of ice crystal growth in polar glaciers. CRREL Research Report 300 , pages 1–19.

LaChapelle, E. R. 1992. Field Guide to Snow Crystals. International Glaciological Society, Cam- bridge.

Paterson, W. S. B. 1994. The physics of Glaciers. Pergamon, 3rd edition.

Uvarov, E. B. and A. Isaacs. 1986. The Penguin Dictionary of Science. Penguin Books, England, sixth edition.

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