Basic notations, assumptions and relations

Fast post-nucleation conversion of air inclusions to air hydrates in ice sheets, especially rapid in the B-type clathration, makes the probability of finding a "semi-bubble" (an air inclusion partly transformed to air hydrate) extremely low. With this in mind, let us consider (see Fig. 1) a single initially spherical air bubble which converts to air hydrate in infinite ice matrix containing only non-nucleated air bubbles and complete air-hydrate crystals. The current volumes, Vb and 14, of the air and the hydrate for both possible types of post-nucleation conversion are linked to the equivalent-sphere radius r_b of the bubble and to the external radius r_h of the whole bubble-hydrate complex (semi-bubble):

At the very moment of the air-hydrate nucleation, bubble pressure is close to the load pressure _P1 in ice (Reference Lipenkov, Salamatin and DuvalLipenkov and others, 1997; Reference Salamatin, Lipenkov and DuvalSalamatin and others, 1997), but almost immediately, as an initial air-hydrate layer (or crystal) forms on the bubble wall, the pressure drops to the hydrate dissociation pressure p_d (Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others, 1998b). Based on the considerations presented in the introduction, we assume that in the following compression stage of the conversion the free gas phase is in equilibrium with the hydrate.

Hence, in accordance with Van der Waals and Plat-teeuw’s theory (Reference Van der Waals and Platteeuw1959)

where and are mole fractions of nitrogen and oxygen in the air bubble and the hydrate crystal growing on its walls, , and and are the respective dissociation pressures of pure N₂- and O₂-hydrates expressed in MPa after Reference MillerMiller (1969) as functions of temperature T (in K):

The process of post-nucleation conversion of a single bubble to an air hydrate in an ice sheet is quite short, and T is considered hereafter as a constant parameter.

The air in bubbles downward along the transition zone becomes more enriched with nitrogen (Reference IkedaIkeda and others, 1999). As for air hydrates, even at the beginning of the transition, their N₂/O₂ ratio does not fall below 2. Natural predominance of nitrogen with its mole mass M_N, close to the oxygen mole mass MO₂ allows us to use in mass-balance calculations the gas state equation for standard atmospheric air, disregarding the minor (3-4%) changes in its mole mass and density ρ (or pressure p) caused by the gas composition variations:

where z_g and R_g are the non-ideal compressibility coefficient (depending on thermodynamic conditions) and the gas constant. For the same reasons, the densities of the air (mainly nitrogen) and water components ρ_{h a} and ρ_hw in the clathrate hydrate phase at the dissociation pressure are assumed to be constant and are taken after Hondoh (Reference Hondoh1989) and Hondoh and others (Reference Hondoh, Anzai, Goto, Mae, Higashi and Langway1990).

All the main physical characteristics and parameters used in this study, and their symbols and values, are presented in Table 1.

Mass-balance equations

The mass of nitrogen and oxygen in the air bubble and the hydrate, growing on its wall, changes due to the gas diffusive transport between the developing bubble-hydrate complex and the surrounding sources and sinks of air molecules, i.e. other air and hydrate inclusions distributed in the ice. Consequently, the equation of the total gas mass balance in the semi-bubble takes the form (normalized by the factor Ma/(zgR_gT)):

where t is the time. The overall surface mass fluxes of nitrogen q_N2 and oxygen qO₂ from neighboring air and hydrate inclusions to the semi-bubble are normalized by the factor M_aρi/M_w (M_w is the mole mass of water), i.e. expressed as the equivalent ice-volume flow rate of the same number of water molecules. Parameters p_ha and p_iw, defined as

are the pressures at which the densities of air and water molecules in free gaseous phases are equal to those in air hydrate (_ρha) and in ice (_ρi), respectively.

Correspondingly, the total mass balance of nitrogen is described by the relation:

The outer fluxes of the air constituents are consumed on the hydrate interfaces with ice and air by the clathration process for which the surrounding ice serves as an infinite source of water molecules. The volume deformation rate (divergence) of the ice, ω_i, at the contact with the bubble-hydrate complex is determined by the ice mass-balance equation:

The above consideration does not depend on the actual form of the growing hydrate and its location in (or around) the bubble. In the A-type conversion process, however, water molecules have to move through the hydrate shell toward the bubble (see Fig. 1), and the bubble air diffuses in the opposite direction toward the ice to form new quantities of air hydrate at both phase boundaries. As a result, the divergence of the spherical hydrate layer, ω_h, differs from cu_h and the bubble-air mass-balance equation gives:

where parameter dha, introduced by Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others (1998b), can be interpreted as the mass fraction of the air hydrate formed of the initial bubble air at the interface with the ice due to the diffusion of the air through the hydrate shell. The value 1 -d_ha is the relative quantity of the air hydrate formed at the interface with the gas phase due to the diffusion of water. In the case of the B-type conversion the hydrate crystal is a continuum incompressible structure without any cavities, and ω_h = 0.

Equations (2)-(4) generalize the earlier developed model (Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others, 1998b) of the conversion of an air bubble to clathrate air-hydrate crystal in ice under the basic assumption that the gas components in the bubble-hydrate system evolving under polar-ice-sheet conditions are in thermodynamic equilibrium, and the bubble pressure after nucleation is equal to the hydrate dissociation pressure.

Air-constituents diffusion through ice

Nitrogen- and oxygen-mass exchange between coexisting air bubbles and hydrates in ice sheets is the diffusive mass transfer (Reference IkedaIkeda and others, 1999), resulting in considerable fractionation of the air components. The driving forces are the gradients of mole concentrations of nitrogen and oxygen in the ice matrix. At the phase boundaries their values are proportional to the corresponding partial pressures of the equilibrium gaseous (vapor) phases. Thus, at the ice-hydrate interface the concentrations of the air constituents dissolved in the ice are determined by the dissociation pressures p_dN2 and p_dO2 multiplied by the mole fractions of the respective gas components in the hydrate crystal, while at the air-ice interface the concentrations are related to the corresponding partial fractions of the bubble pressure. These concentrations are the same in the vicinity of semi-bubbles undergoing different types of conversion, provided that the air pressure in them is equal to the dissociation pressure p_d. In non-nucleated bubbles the pressure is identified with the load pressure p1 which in the transition zone is much higher than the dissociation pressure, thus causing the air diffusion.

Actually we have also to distinguish the gas composition of the reference bubble-hydrate system from the mean gas compositions of the surrounding non-nucleated bubbles and coexisting completely formed hydrate crystals , which fractions in the total inclusion number concentration N₀ are n_h and 1 – n_b, respectively. Hence, for instance, the background mole concentration of nitrogen in ice is determined by the averaged partial pressure induced by both groups of inclusions:

where the averaging parameter 7 ranges from 0 to 1, following the value n_b.

The difference between the latter value and the equilibrium "nitrogen-vapor" pressure at the external surface of the growing hydrate crystal is the driving force of the nitrogen mass transfer. The same considerations are valid for the pressure in the case of oxygen.

The choice of parameter 7 is not obvious and will be discussed in section 4. Here, at the stage of simulating the conversion of a single reference bubble to air-hydrate crystal, the pressures and should be regarded as given model parameters. Thus, γ just links them to the averaged gas compositions of neighboring air bubbles and hydrates, so its particular value is not important. As a reasonable approximation we take γ ≈ n_b.

To evaluate the diffusive mass fluxes, we use the conventional cell-model approximation for diluted multiphase media illustrated in Figure 1. A spherical semi-bubble of type A or B with growing air hydrate is confined to a spherical ice cell of a specific volume related to one inclusion, that is of radius

The averaged, background concentrations are assumed on the cell boundary, at r = r_c. Gas solubility in ice is very low, and the total amount of air dissolved in the ice is negligibly small (Reference IkedaIkeda and others, 1999). For the latter reason, the nitrogen and oxygen diffusion through the spherical ice layer toward the developing bubble-hydrate complex is a quasi-stationary, spherically symmetric process. The obvious solution of the corresponding boundary-value cell problem yields:

Here, by definition, and are the self-diffusion (permeation) coefficients of pure nitrogen and oxygen in ice at a given (fixed) temperature T and dissociation pressures and , respectively.

It is also worth noting that volume concentrations of bubbles and hydrates within transition zones of ice sheets are very small and τ_b, τ_h ≪ τ_c. In such a case the cell-model approach is not crucial: the diffusive fluxes in Equations (5) and their analogues practically do not depend on the cell size.

Hydrostatic compression

After air-hydrate nucleation, the pressure in the bubble abruptly decreases to the dissociation pressure p_d. The difference between the load pressure in the ice matrix p1 and p_d is resolved in plastic compressive deformations of the ice and the hydrate around the bubble. A mathematical description of non-linear creep flow and pressure relaxation in bubbly media (Reference Salamatin and DuvalSalamatin and Duval, 1997) allowed us to construct an approximate model for hydrostatic compression (or decompression) of the air bubble converting to hydrate crystal (Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others, 1998b) in case of the A-type transformation.

The ice and clathrate air hydrate are described as non-linear viscous isotropic materials with similar polynomial flow laws relating effective strain rates e₀ and deviatoric stresses τ₀:

where a_s and _μsl, _μs2 are the exponent (creep index) and the rheological coefficients (constants) at the reference temperature T*, and Q_s is the activation energy. Subscript "s" corresponds to ice (s = i) or to hydrate (s = h).

As a final result, the ice and hydrate volume deformation rates, ωi and ω_h, introduced in Equations (4), are linked to the pressure drop – p_d by the following relation: _Pl

The integrals Ψ_s(ω), s = i, h, are approximately evaluated and written in the form:

The ice flow law (Equation (6)) was confirmed in special studies of the bubbly-ice densification process in polar ice sheets (Reference Lipenkov, Salamatin and DuvalLipenkov and others, 1997; Reference Salamatin, Lipenkov and DuvalSalamatin and others, 1997). In mechanical tests (Reference Duval and CastelnauDuval and Castelnau, 1995) and in laboratory experiments on the air-hydrate formation (Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others, 1998b) performed at relatively high temperatures, ice appeared to be somewhat softer than in natural polar conditions. We use here, however, the in situ ice rheology as determined by Reference Lipenkov, Salamatin and DuvalLipenkov and others (1997) for the Vostok region. In accordance with Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others (1998b), air hydrate, when treated as a plastic material, was found to be at least one order harder than ice. This is in good agreement with the direct experimental measurements by Reference Stern, Kirby and DurhamStern and others (1996) performed for methane clathrate hydrates. Plausible values of the rheological parameters in Equations (6) and (7) estimated by Reference Lipenkov, Salamatin and DuvalLipenkov and others (1997) and Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others (1998b) are given inTable 1.

Equations (4) and (7) for the idealized hard spherical hydrate shell around the bubble in the A-type conversion process represent the limiting case of the lowest rates of the bubble compression and, hence, the bubble-to-hydrate transformation. At the same time, taking co_h = 0, instead of Equation (4b) for the B-type conversion, we assume that the growing hydrate crystal is freely squeezed into the bubble and does not hinder the bubble-compression and ice-deformation processes. Thus, we come to the upper bound of the conversion rates.

Initial conditions

The system of simultaneous Equations (1)–(7) needs to be supplemented with additional relations which determine the initial values of characteristics describing the conversion of a reference bubble to clathrate air hydrate after the hydrate-crystal nucleation on the bubble wall at t = 0. Let us designate by subscript "0" the bubble volume V_b0 (bubble radius τ_b0) and the bubble (load) pressure p₁₀ just before the nucleation; the gas composition is identified with , . Due to relatively high rates of air and water diffusion through clathrate hydrate almost immediately after nucleation, the hydrate crystal, growing on the bubble wall, reaches the size at which the initial supersaturation is removed and the bubble pressure drops from p₁₀ to the dissociation pressure p_d. Gas composition of the bubble also changes from to . This momentary process is too fast to be noticeably influenced by the air diffusion from neighboring inclusions and compressive deformation of the ice. Thus, the mass-conservation law for the bubble air (Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others, 1998b) determines the initial volumes of the air bubble and the hydrate crystal after its nucleation:

where p_d corresponds to the initial bubble-air composition which, being in equilibrium with the hydrate crystal (see Equations (1)), satisfies the following transcendental equation:

The load pressure in the ice matrix does not remain constant as the transforming bubble-hydrate complex sinks into the ice-sheet thickness:

The rate of loading L_p is proportional to the ice accumulation rate b, that is L_p ∼ g_Pib (g is the gravity acceleration), but should be corrected to account for the thinning of the overlying ice stratum in the vertical direction due to global ice-sheet dynamics (e.g. Reference Salamatin, Lipenkov and DuvalSalamatin and others, 1997).

Preliminary notes on model parameters

The most uncertain but principal parameters of the model (Equations (1-10)) are the permeation (self-diffusion) coefficients of nitrogen and oxygen in ice. There are as yet no reliable data on their values reported and available from direct measurements, especially at the extreme conditions typical of polar regions. However, preliminary estimations can be made. The mechanism of the diffusion in ice is thought to be an interstitial one and for water molecules was studied by Reference Goto, Hondoh and HigashiGoto and others (1986). Additional experiments were conducted to measure the diffusion coefficients of hydrogen (Reference Strauss, Chen and LoongStrauss and others, 1994) and noble gases (Reference Satoh, Uchida, Hondoh and MaeSatoh and others, 1996). Reference IkedaIkeda and others (1999) extrapolated these data to N₂ and O₂ at 263 K by calculating the potential barriers of the gas-water molecular interactions, which are the proxy for the activation energy of the gas molecule jumps in ice lattice. They also estimated the concentrations of N₂ and O₂ in ice from the Van der Waals diameters of the gas molecules using the gas solubility measurements by Reference Satoh, Uchida, Hondoh and MaeSatoh and others (1996). Combining these results, we arrive at a rough prediction of the diffusive permeation coefficients of N₂ and 0 ₂ in ice at 263 K for dissociation pressures and , respectively: and . Thus is noticeably greater than . The in situ diffusive transport of air molecules in ice was indirectly estimated by Reference Uchida, Mae, Hondoh, Lipenkov and DuvalUchida and others (1994c) from the growth rate of air-hydrate crystals in the Antarctic ice sheet. They found the air permeation coefficient to be of the order of 10⁻⁷ to 10⁻⁶ mm² a⁻¹ (∼10⁻²⁰ to 10⁻¹⁹ m² s⁻¹) at 233 K and 10 MPa. This agrees with Reference IkedaIkeda and others’ (1999) data if the difference in temperature is taken into account.

Recent observations of the gas fractionation in Vostok ice cores (Reference IkedaIkeda and others, 1999) imply that the typical time τ_E of the diffusive gas-mass exchange between the air bubble transforming to air-hydrate crystal and its neighbors, coexisting bubbles and hydrates, at the beginning of the transition zone at 220 K could be comparable with the compression (air-hydrate formation) time τ_c . Hence, for n_b ∼ 1 and for the time-scale ratios discussed in the introduction, at the beginning of the transition, τ_E should be 10⁵ times larger than the diffusion time τ_D in air hydrate. The gas-mass exchange time-scale τ_E for an air-hydrate crystal can be derived from Equations (3) and (5), as

In accordance with Reference Salamatin, Hondoh, Uchida and LipenkovSalamatin and others (1998b)

where the diffusive mass-transfer coefficient in air hydrate D for the Vostok transition zone at 220 K is about 10⁻² mm² a⁻¹. Thus, the corresponding values of the self-diffusion coefficients of the air constituents in ice should be of the order of 10"⁷ mm² a-1 (∼10-21 m² s⁻¹). The latter estimate is consistent with the above predictions by Reference Uchida, Mae, Hondoh, Lipenkov and DuvalUchida and others (1994c) and Reference IkedaIkeda and others (1999) and is confirmed below (seeTable 1) by model constraining.

The present-day environmental conditions of the air-hydrate crystal growth on bubble wall after its nucleation at different depth levels (h) at the beginning, in the middle and at the end of the transition zone at Vostok are presented in Table 2. Direct optical-microscope observations of the Vostok ice cores by Reference LipenkovLipenkov (1989) showed that, in spite of the hydrostatic compression, the mean radius of air bubbles within the transition zone remains substantially constant, slightly varying from 0.06-0.065 mm in the upper part to 0.045-0.05 mm at the end. This becomes possible due to the preferential, selective clathration of bubbles of minimal, "critical" size: τ_b ∼ 0.02-0.025 mm. The number fraction of such bubbles rapidly decreases with depth, thus increasing the initial dimensions of air inclusions involved in clathration. Our choice of the initial bubble radii (τ_b0) inTable 2 follows this tendency.

The rate of loading L_p has been estimated in accordance with the mean ice-accumulation rate at Vostok station, b « 24 mm a⁻¹, determined over the past 22 years (Reference Barkov and LipenkovBarkov and Lipenkov, 1996).

The number concentration of inclusions N₀ is taken equal to 0.8 mm⁻³ (τ_c ∼ 0.67 mm) in all simulations.

Computational experiments and discussion

The time-scale of the gas-mass exchange, τ_E, given by Equation (11), depends on the bubble number fraction n_b and is proportional to the squared radius of the air bubble converting to air-hydrate crystal. It has minimal values in the upper part of the transition zone, where the air-hydrate formation (compression) time τ_c reaches its maximum and where air bubbles have initial atmospheric gas composition and dominate in the inclusion ensemble (n_b ≈ 1). Accordingly, first computational tests showed that at the beginning of the transition zone the initial gas composition of a growing hydrate, determined by Equations (8) and (9), rapidly changes (due to the selective diffusion of oxygen from the neighboring bubbles) to a certain new one, distinguished as , , which further remains constant during the formation and development of the crystal unless the air composition in the neighboring bubbles itself is changed. At the end of the transformation, when the amount of air in the bubble becomes small, the ratio between the righthand sideflux terms in Equations (2) and (3) is equal to

Substituting and for and in Equations (5), we obtain a theoretical estimate for the permeationcoefficients ratio valid at the beginning of the transition:

For present-day conditions at Vostok station, at about 650 m depth the load pressure is Pl « 5.6 MPa, and the dissociation pressures at temperature T « 220 K are and . For atmospheric air trapped in bubbles , while, in accordance with Reference IkedaIkeda and others (1999), the N₂/O₂ ratio in air hydrates is . Therefore, Equation (12) predicts

This relationship is hereinafter fixed in all computer simulations. It is additionally validated in section 4. However, it should be noted that this estimate is rather sensitive to feasible changes in depth (load pressure) and N₂/O₂ ratios. For instance, at the 600 m level, where Pl ≈ 5.1 MPa, the ratio would be 2.5 for the same degree of the fractionation and would range from 1.8 to 2.6 if varies within ±5% limits.

The post-nucleation transformation of an air bubble to air-hydrate crystal in polar ice sheets, being mainly controlled by the bubble compression, does not noticeably depend on the gas-mass exchange between the semi-bubble and the neighboring air and hydrate inclusions. On the other hand, the latter process is primarily responsible for the total size and the growth rate of the air hydrates after the conversion. Approximately, at the beginning of the transition, where n_b ≈ 1, the bulk air-mass flux from bubbles to hydrates is determined by the initial supersaturation, _P1 – p_d, and the mean size of the air hydrates is related to their mean age. Ice-core analysis shows that the corresponding radius, τ_h, of the hydrate crystals at 700 m depth at Vostok is about 0.036 mm (V_h = 1.95 × 10⁻⁴ mm³). They are mainly formed below 620 m, and their mean age can be estimated as 1.5–2ka at the present-day accumulation rate of about 24 mm a⁻¹. The diffusive permeation coefficients of nitrogen and oxygen presented inTable 1 are obtained (with accuracy not worse than ±10%) to satisfy the latter constraint: , , corresponding to a temperature of 220 K (dissociation pressures and ). Figure 2 shows the described process of air-hydrate crystal growth and the temporal changes in the gas composition of the developing bubble-hydrate complex at the beginning of the transition zone at Vostok. The limiting difference between types A and B of the post-nucleation transformation of an air bubble to clathrate air hydrate can be clearly seen.

Fig. 2. Air-bubble and air-hydrate characteristics vs tine simulated for the beginning of the transition zone at Vostok (600m depth), (a) Volumes, Vh. Vh. (b) Nitrogen mole fractions, . A and B indicate type of bubble-to-clathrate conversion.

The deduced estimates of the self-diffusion coefficients of the air components in ice agree well with the preliminary ones discussed in the previous subsection. Based on these results obtained for 220 K as well as on the predictive calculations performed by Reference IkedaIkeda and others (1999) for 263 K, we can attempt to evaluate the self-diffusion activation energy Q_d defined by the conventional relation:

where subscript "s" designates either oxygen (s = O₂) or nitrogen (s = N₂), and permeation coefficient D_s* corresponds to the reference temperature T* and linearly depends on pressure (gas concentration). In general, we should have assumed different activation energies for the air components, but limited confidence in the extrapolation procedures of Ikeda and others (Reference Ikeda1999) at 263 K inclines us not to do this. Finally, on average, taking into account the difference in dissociation pressures at different temperatures, we obtain Q_d = 50±3Jmol⁻¹, which is very close to the self-diffusion activation energy of water interstitial in ice (Goto and others, Reference Goto, Hondoh and Higashi1986). This estimate is sufficiently reliable to account for relatively small temperature variations within the transition zones, for instance, 220-225 K at Vostok. In the case of Uchida and others (Reference Uchida, Mae, Hondoh, Lipenkov and Duval1994c), at 233 K and 10MPa, the N₂- and 02-permeation coefficients would be 5.7 × 10-7mm2 a-1 (1.8 × 10⁻²⁰m² s⁻¹) and 1.6 × 10⁻⁶mm^{2 a-1} (5.2× 10⁻²⁰m²s-1^), respectively, which corresponds to the lower bound of the range estimated in the cited study.

Figures 3 and 4 present the principal features of the air-hydrate formation process in the middle and at the end of the transition zone for present-day (Holocene) climatic conditions (seeTable 2). From Salamatin and others (Reference Salamatin, Hondoh, Uchida and Lipenkov1998b) the rate of bubble shrinking increases with depth (see Figs 2a-4a) following the increase in the initial relative supersaturation of bubbles, p10/pd, from 1.5 to 2.3. The values of the typical compression time τ_c given inTable 2 refer to the moment at which the volume of the disappearing air bubble becomes 100 times smaller than that of the growing hydrate. The time-scale of the air-hydrate formation decreases from 1300 a (or 200 a) at the beginning to 50 a (or 3 a) at the end of the transition zone for the conversion of type A (or B). On the other hand, the typical time τ_E for the gas-composition variation in a growing air hydrate, calculated in Table 2 in accordance with Equation (11), increases considerably with depth from 500 to 5000 a. As mentioned, this is for two reasons: (1) larger bubbles transform to air hydrates at greater depths, and (2) the decrease in bubble number fraction reduces the background concentration of air constituents dissolved in the ice. As a result, first air hydrates which appear at the beginning of the transition at Vostok acquire a large quantity of air from surrounding bubbles and completely change their gas composition due to the selective diffusion of oxygen during the period of their formation (see Fig. 2b). However, even in the middle of the transition zone (Fig. 3b), and especially at its end (Fig. 4b), the gas composition of the growing air-hydrate crystal remains practically constant in the course of the transformation, while the main changes in the composition of air hydrates formed in the middle of the transition take place in the post-conversion period. The continuous gas-mass exchange between the air hydrates and coexisting bubbles along with nucleation of new air hydrates play the principal role in the gas fractionation (Ikeda and others, Reference Ikeda1999) within the transition zone. This phenomenon is discussed in the next section.

Fig. 3. Air-bubble and air-hydrate characteristics vs time simulated for the middle of the transition zone at Vostok (800m depth), (a) Volumes, Vh. Vh. (b) Nitrogen mole fractions, . A and B indicate type of bubble-to-clathrate conversion.

Fig. 4. Air-bubble and air-hydrate characteristics vs time simulated for the end of the transition zone at Vostok (1100 m depth), (a) Volumes, _Vb,_Vh. (b) Nitrogen mole fractions, . A and B indicate type of bubble-to-clathrate conversion. ( c) Compressive deformation rates of the air-hydrate coat (ωh) and surrounding ice (ωi) in case A.

Another peculiarity of the post-nucleation conversion of air bubbles to air hydrates can be distinguished in Figures 2a-4a. Because of the high supersaturation of bubbles, a large quantity (30-60%) of air in them almost immediately after nucleation is converted to hydrate crystals. Figure 4c shows that for the A-type transformation the compressive deformation rates of the ice and the hydrate layers, surrounding the bubbles, reach their maximum values at this moment, these values being especially high in the deeper part of the transition zone. This may cause destruction of the hydrate shells, followed by further recrystallization of different pieces. At the same time, at high supersaturation the conversion of type B can also start with simultaneous nucleation of several hydrate crystals in a single air bubble. Thus, the probability of finding such "broken" (split into separate parts) air-hydrate crystals is proportional both to the local supersaturation of bubbles and to the rate of the hydrate nucleation in a unit of ice volume. The latter tends to zero as the number of bubbles diminishes at the end of the transition zone, decreasing the above probability. This may explain the experimental observation that air-hydrate crystals of several similar-sized splinter-like pieces (see Fig. 5) are common for the Vostok ice cores mainly in the middle of the transition zone.

Fig. 5. Clathrate air-hydrate inclusion composed of several splinter-like pieces in the fresh Vostok ice core recovered from 850 m depth.

Simplified mass-balance model

The averaged nitrogen and oxygen mass fluxes through the ice matrix toward air hydrates and bubbles are direct analogues of Equations (5):

where and are the mean radii of air-hydrate crystals and bubbles, respectively, assumed to be much less than the cell radius r_c (see Fig. 1).

The mass conservation of air components implies that and determines the parameter 7 as

and determines the parameter γ as

For the transition zone at Vostok the ratio and does not differ much from unity. Hence, the approximation γ ≈ n_b used in the previous section seems reasonable.

It is important to remember that the typical time τ_E of the gas-composition variations in a single air-hydrate crystal significantly exceeds the time-scale of the bubble-to-hydrate conversion (τ_c) throughout the transition zone, except maybe for its beginning at Vostok (see Table 2). This means that the gas composition of a newly formed air hydrate , at least in the deeper part of the transition zone, is initially close to that of the air bubble .

Thus, the mass-balance equation for nitrogen in gaseous (bubble) phase can be written in the following form:

which yields

Combining this equation with that written for oxygen, we obtain the principal relation which determines the rate of the bubble-gas composition changes (gas fractionation) in the transition zone:

To construct a complete model of the gas fractionation, Equation (14) should be considered together with the analogous one for which would include the additional gas-mass exchange term proportional to the rate of air-hydrate nucleation dn_b/dt. To solve this system of simultaneous equations, we would need to know the kinetics of the nucleation itself. The latter problem is beyond the scope of this paper, although its solution would allow us to constrain the gas-fractionation model by the direct Raman spectra measurements (e.g. Reference IkedaIkeda and others, 1999) and to infer more valid estimates of the self-diffusion coefficients . It is proposed that these questions be studied in future. Nevertheless, even at the present stage of our knowledge it is useful to examine Equations (13) and (14) on the basis of scale analysis to derive the typical time τ_F of the bubble-gas fractionation in transition zones of ice sheets:

Naturally it has the same structure and is of the same order as τ_E at the beginning of the transition.

Let τ _Z be the typical time of the hydrate nucleation covered by the transition zone. The principal conclusion can be deduced at this stage: a noticeable gas fractionation within the transition zone man ice sheet takes place only τF ≪ τ_Z, i.e when the rate of the air-hydrate nucleation is much less than the gas fractionation rate.

Model validation on Vostok data

We can easily estimate τ_F from Equation (15) for conditions in the transition zone at Vostok (see Tables 1 and 2). For the depth range 800-1100 m the gas-fractionation time for bubbles τ_F ∼ 300 a. Further, in accordance with Reference Salamatin, Lipenkov, Barkov, Jouzel, Petit and RaynaudSalamatin and others (1998a), the ice age is 28 ka at the beginning of the transition zone (550 m) and 94 ka at the end (1250 m). This 700 m stratum was formed in the cold glacial period when ice-accumulation rate was approximately half the present rate. Hence, the age of the ice layer that has now reached the end of the transition zone was about 60 ka at the beginning. Soon after that (approximately 10 ka), the Holocene warming started, with increased ice-accumulation rates, and the conversion of all bubbles to air hydrates took about 30-35 ka. This estimate of T_Z for central Antarctica is much larger than r_E and is two orders of magnitude larger than that of τ_F.

Thus, at Vostok, the bubble-gas fractionation in the deeper part of the transition zone, as n_b decreases, becomes a quasi-stationary process. Its principal limiting step is the nucleation of air hydrates. It determines the rate of the mean air-hydrate gas-composition evolution from its initial oxygen-enriched state to that close to atmospheric air. The gas composition in air bubbles simply follows the changing load (bubble) pressure to correspond with the gas composition of air hydrates and would be constant if the latter two factors remained unchanged. Mathematically this means that the lefthand side of Equation (14) becomes negligibly small compared with the gas-mass fluxes and .

Therefore,

and we come directly to the following relation:

which is an analogue of Equation (12) at the end of the transition zone.

Laboratory Raman spectroscopic studies by Reference IkedaIkeda and others (1999) showed that below 850 m the gas composition of air hydrates in the Vostok ice cores is close to atmospheric: Thus, if we fix the self-diffusion coefficients ratio , as determined before, then it is possible to use Equation (16) to evaluate the N₂/O₂ ratios in air bubbles. In the depth range 900-1100 m, the load pressure varies from 7.8 to 9.6 MPa, while the dissociation pressures are and . This leads to an N₂/O₂ ratio in bubbles of 9.6-10.9. The direct measurements by Reference IkedaIkeda and others (1999) give . The obvious agreement between the theoretical estimates and the experimental data (see Fig. 6) tends to support the validity of the developed model.

Fig. 6. Comparison of direct measurements of N₂/O₂ ratios in air bubbles and air hydrates in the Vostok ice core (Reference IkedaIkeda and others, 1999), with theoretical prediction of the N₂/O₂ ratio in air bubbles at the end of the transition zone (bold line). Error bars correspond to the standard deviations of the measurements performed in the different inclusions of the same ice sample. Dashed line indiates atmospheric N₂/O₂ ratio (3.7).

Discussion: application to Greenland ice-sheet conditions

It might be interesting to apply the above theory to the analysis of air-hydrate formation conditions in the Greenland ice sheet, studied through the GRIP and GISP2 projects. The transition zone in the Greenland ice cores is found at 1000-1500 m depth (Reference Pauer, Kipfstuhl, Kuhs and ShojiPauer and others, 1999) with ice ages of about 5.3-9.4 ka (Reference MeeseMeese and others, 1997). It covers the Holocene period, with rather stable temperatures around 240 K and ice-accumulation rates close to 250 mm ^ (Reference Cuffey and ClowCuffey and Clow, 1997). Thus, the duration of the total phase transition from air bubbles to air hydrates (the typical time of nucleation) is τ_z ∼ 4 ka, i.e. one order of magnitude less than in the Vostok region in Antarctica.

To understand whether a noticeable gas fractionation could occur in central Greenland, we should estimate the corresponding time-scale τ_F, using Equation (15). The depth profiles of the hydrate number concentration and total gas-volume concentration along the GRIP ice core (Reference Pauer, Kipfstuhl and KuhsPauer and others, 1997,Reference Pauer, Kipfstuhl, Kuhs and Shoji1999) allow us to calculate the mean bubble radius . The dissociation pressure , the self-diffusion coefficient , and for load pressures Pi ∼ 10 MPa we get τ_F ∼ 200–400 a. Thus, the gas-fractionation rates (time-scales τ_F and τ_E) in central Greenland for present-day (Holocene) climatic conditions are comparable with those at Vostok, but at much (10 times) higher nucleation rates. As a result the number of oxygen- enriched hydrates and the mean degree of gas fractionation sharply decrease at the beginning of the transition zone. The variance of the gas composition (N₂/0₂ ratios) in the air hydrates increases as τ_E becomes comparable with τ_z in the deeper part of the transition zone. Therefore, we come to the following conclusion. A significant gas fractionation, on average, is ruled out for air hydrates in the Greenlandice cores and might be detected only (1) as oxygen enrichment in rare hydrate crystals formed at the very beginning of the transition, or (2) as nitrogen enrichment in rare hydrate crystals newly formed at the end of the transition zone. Indeed, N₂ / 0 ₂ ratios as low as 2.63 and as high as 4.49 were observed in the air hydrates by Reference Pauer, Kipfstuhl and KuhsPauer and others (1997). More evident (systematic and statistically valid) nitrogen enrichment might be expected in the air bubbles in the deepest part of the transition zone, but such data are so far not available.