Лёд и Снег • 2014 • № 4 {128)

УДК 551.322:548.2

Steady-state size distribution of air bubbles in polar ice

© 2014 г. V.Ya. Lipenkov1, A.N. Salamatin2

1Arctic and Antarctic Research Institute, St.-Petersburg; 2Kazan (Volga Region) Federal University


Установившееся распределение пузырьков воздуха по размерам в рекристаллизационном льду

В.Я. Липенков1, А.Н. Саламатин2

Арктический и Антарктический научно-исследовательский институт, Санкт-Петербург;

2Казанский (Приволжский) федеральный университет

Статья принята к печати 18 сентября 2014 г.

Air bubbles, geometrical properties, ice core, paleoclimate reconstruction, polar ice, size distribution.

Геометрические свойства, ледяной керн, палеоклиматическая реконструкция, пузырьки воздуха, распределение по размерам, рекристаллизационный лёд.

Between the close-off depth and the bubble-to-hydrate transition zone in polar ice sheets, the geometrical properties of air bubbles, such as number concentration and size of bubbles, are mainly controlled by firn temperature and ice accumulation rate prevailing during the snow to ice transformation [3], and by the bubble compression in the course of bubbly ice densification [13]. This implies that the data on the bubble properties can be used for reconstruction of the past climate change. On the basis of our earlier studies of bubbly ice densification and the new measurements of air bubbles in the Antarctic ice cores, we have developed a theory of bubble evolution in polar ice and propose an inverse procedure for bubble size conversion to specified conditions at the close-off depth. Both outcomes of the research contribute to elaboration of the new paleoclimatological tool based on the bubble properties.

List of symbols

b accumulation rate c ice porosity (volume fraction of bubbles) F(r) probability density function of bubble-size (r) distribution

f(r, p) probability density function of bubble-size (r)

and pressure (p) distribution h depth

L total length of pores per unit mass of ice l size of ice grains (grain edge length) N number of bubbles after their disintegration per

unit mass of ice p gas pressure in a bubble pl load pressure r radius of a bubble rp radius of pores and elongated bubbles s coefficient of variation of bubble radii r T temperature t time

U volume of a unit cell V total gas content of ice v volume of a bubble y number of pores per grain a length/radius ratio of cylindrical bubbles

volume/edge length ratio of polyhedron chosen for ice grain approximation ro compression rate of a bubble p, density of pure ice Z rd/r ratio Z' rc/r ratio X mean value of variable x

o(x) standard deviation of variable x


c close-off characteristic d characteristic at the end of bubble disintegration trans characteristic at the beginning of the bubble-to-

hydrate transition zone 0 standard conditions (STP).


Polar ice cores have for a long time been recognized as an invaluable source of data on past climate and atmosphere history. Traditionally most of the pa-leoclimatic information comes from the analyses of water isotopes, entrapped atmospheric air, and soluble and insoluble impurities in ice. However, some of the physical properties of ice itself can also be used as a supplementary and independent source of pa-leoclimatic data. In particular, the geometrical properties of air inclusions in ice are thought to store information about past temperature and accumulation rate. Earlier studies [1] revealed climate-related variations in air-bubble sizes and number concentrations at different depths in the Vostok ice core. The updated experimental profiles of air bubble properties at Vostok ([1, 12] and newly obtained data) are shown in Fig. 1 along with the ice isotope record [10]. The data demonstrate that the sizes (number concentrations) of bubbles are smaller (greater) in ice formed under the full glacial conditions (the Last Glacial Maximum or LGM) than in Holocene ice.

The link between bubbles and climate can be understood if one takes into account that bubble sizes and number concentrations are primarily controlled by grain sizes at the pore close-off depth, as was first proposed by A. Gow [8] and quantitatively demonstrated in subsequent publications [3, 12, 28]. The grain size at this depth is in turn a function of the grain-growth rate (depends mainly on the firn temperature), and the age of ice at pore close-off controlled by densification process (which is temperature and accumulation dependent [21]). Accordingly, the geometrical properties of bubbles at any depths between the pore close-off and the bubble-to-hydrate transition (500-1250 m at present time at Vostok [12]) are mainly controlled by firn temperature and ice accumulation rate prevailing during the snow to ice transformation [3], and by the bubble compression in the course of bubbly ice densification [13, 19]. Hence, before using the bubble size record for paleoclimate reconstruction, one has to reduce the bubble sizes measured at different depths to the close-off conditions. With this in mind, we develop here a model which describes the evolution of the bubble-size distribution with depth and allows us to reduce the sizes of compressed air bubbles to the original conditions prevailing at the close-off depth during bubble formation.

Formation of bubble ensembles in polar ice

In dry snow, a structural re-arrangement of ice grains by linear-viscous boundary sliding is a dominant mechanism of densification [5]. When snow reaches a relative density of about 0.6-0.7, the coordination number of grains (number of contacts per grain) approaches 6-7, thus, making the sliding impossible. Deformation of grains by power-law creep allows further densification of firn leading to further increase in the number of contacts per grain and growth of the average contact area [7, 22, 23]. As a result, when firn reaches a relative density of 0.8, the pores are reduced to thin cylinders going along the ice-grain edges, whereas the grains are similar by shape to equilibrium space-filling polyhedrons of the type of Kelvin's or Williams' tetrakaidecahedron [14].

Owing to curvature and surface tension the cylindrical pores in ice are unstable with respect to their disintegration into a number of spherical bubbles (see e.g., [25]). The plastic deformation of grains, the ice-grain growth, and the disintegration (pinch-off) of pores are the three processes that acting simultaneously control the dynamics of pore closure and air trapping in firn [6].

Because of percolation and sometimes sealing effects, the air becomes isolated from the atmosphere

few meters above the depth at which zero open porosity is observed from the porosity measurements done on small samples of firn [15, 29]. However, regarding macro-scale in-situ properties of polar firn, both the air isolation in terms of pressure and the actual pore closure occur simultaneously at the same depth level hereafter referred to as «close-off depth» and denoted by hc. The amount of air trapped at this level determines the air content of ice. Thus, the firn porosity at which the air is isolated and the close-off depth can be estimated from the air content of recent ice and the experimental porosity/density profile, provided the atmospheric pressure and the firn temperature are known [15].

Further development of bubble ensemble below hc involves post-closure disintegration of isolated pores and elongated bubbles and compression of all air inclusions in plastically deforming ice matrix driven by the pressure lag between the two phases. By definition, the disintegration alone does not affect the volume concentration of air in ice, but tends to increase the number of bubbles. The process persists until the length/radius ratio, a, of all cylindrical bubbles is reduced to about 2n [25]. In the case of Vostok this condition is satisfied at about 160-170-m depth (60-70 m below pore close-off). However, «isometric» bubbles (i.e. bubbles with a < 2n) represent up to 50% of the total bubble population already at 105 m and up to 80% at 110 m, whereas a of remaining elongated inclusions at 110-m depth rarely exceeds 4n. Thus, the number of bubbles in sinking ice is mainly determined within a narrow depth interval of the first tens of meters below hc, where radius rp of elongated inclusions still remains within a few percent of pore radius rpc at the close-off depth.

Based on the above consideration, we postulate that at the end of bubble disintegration the number of isometric bubbles per unit mass of ice, N, is proportional to the length/radius ratio of firn pores at the close-off depth:

N = LJarVc, (1)

where Lc is the total length of pores per unit mass of ice at the close-off depth. With this assumption, it was deduced [3] that the number of bubbles in polar ice is linked to the grain size at the close-off depth as

N = G/Q, N ( \ y_ 1.5 ii > C 0.5 (2)

aP, I Cc

Here y is the number of pores per grain, and the value of is specified by the type of regular polyhedrons chosen for ice-grain approximation, pt is the density of pure ice; a is assumed to be a constant parameter which refers to a preferred wavelength of bubble surface per-

Fig. 1. Variations of the geometrical properties of air bubbles with depth in the Vostok ice core:

a — the deuterium record from the Vostok ice core indicating the local surface temperature change [10]. The 8D axis is inverted to facilitate comparison with the experimental profile of the number concentration of air bubbles in ice; b — number of normal bubbles and microbubbles in 1 g of ice, N. Vertical bars indicate variability of the normal bubble concentrations within 9 cm thick ice core layer; c — mean radii (r) of normal bubbles (measured) and microbubbles (calculated using the data on microbubbles at 183 m depth). Vertical bars for (r) of normal bubbles as in b. The solid curve was calculated from the data on normal bubbles at 183 m depth a

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