Syngenetic processes

Process 1a has been investigated by Matsuo and Miyake (1966) who considered the gas content of a primary ice cover composed of horizontally elongated crystals with a random c-axis orientation. We will refer to this layer as the transition layer. It is reasonable to assume that the water trapped between these initial crystals will contain a large amount of dissolved air. There are also probably a number of air bubbles that are initially trapped as well. Matsuo and Miyake also suppose that atmospheric gas diffuses into this layer and obtain, from the diffusion equation, the following expression for the concentration of gas in the ice:

where C is the content of a given gas in the ice at time t with time t = 0 taken at the beginning of freezing and C_∞ is the content at t = ∞, C_w is the concentration of the same gas in the water, V_t is the volume of the transition layer within the limits of area s, and β is the absorption coefficient which is equal to 0.340, 0.346, and 0.76 cm min^–1 for nitrogen, oxygen, and CO₂, respectively (at a temperature of 0° C). Assuming that the two principal components of the gas entrapped in the ice are nitrogen and oxygen, we can write the following equation for the gas porosity of the ice v_i

where i is the number of the process.

It is obvious that s/V_t = h_t where h_t is the thickness of the transition layer. In this case, on the basis of Equation (1) we have

Here C_ΣW = C_Nw+C_ow and the coefficient β can be taken as approximately 0.342 cm min^–1 or 2.34 x 10^–4 cm d^–1.

In Equations (1) and (3), C_∞ and V _∞ are unknown. Matsuo and Miyake did not determine them and their determination would require quite long experiments. Judging from the observations made by Savel’yev (1963) during three months in the Arctic, v _∞ can be more than 0.050. Our own observations made during a 101 d period on the White Sea gives V_∞ greater than 0.112.

Process 1b is caused by the release during freezing of a part of the gases that were initially dissolved in the sea-water. During freezing the salinity and density of the upper layer of the sea of thickness z are raised because of the rejection of salt by the growing ice sheet. This causes what Zubov (1945) has termed the vertical winter circulation. Assuming that this process is basically one-dimensional, if as a result the salinity of the portion of the water column involved in the vertical circulation sz can be raised by dS_w over the area s during the time dt, then as a consequence the solubility of some gas can be lowered by dL. In this case a gas with a volume s dL can be released from the water column sz, and the gas concentration in the ice layer that has formed dh will be

If the increase in the depth of the winter circulation layer is neglected in comparison with the value of Z, we can determine z from

where dM is the mass of ice in layer dh, dm is the mass of salts included in it, S is its salinity, — dm_w and — dM_w are the decreases in the masses of salt and of water within the limits of the layer z, and S_W is the salinity of the water. The mass changes dM and —dM_w can be expressed in terms of dh, z, s, the ice density S, and the increase in the water density dρ:

and

Substituting these values in Equation (5), we obtain

and from Equations (4) and (6)

Let us now consider the concentration within the ice of the two main gases that are dissolved in the sea-water by writing Equation (7) as

and

Because of Equation (2) we have

Here V _1b, is the ice porosity which is formed as the result of process 1b and

Note that Equation (8) includes the ice density which is itself connected to the porosity by means of the relation

where δ_p is the density of pure ice. Therefore instead of Equation (8) we have

As has been pointed out by Tsurikov (1976), the ratio of the salinities of ice and of water (S/Sw) is related to the rate of growth of the ice cover (w = dh/dt) in the following manner

Therefore

We will use the empirical formulae for the nitrogen and oxygen saturation of sea-water developed by Fox (see Zubov, 1957) in which the results are expressed in dimensionless units. These are

and

where θ is the temperature of the water and [Cl] _w its chlorinity. Newer equations are available for the dependence of the oxygen solubility on temperature and chlorinity; for instance those of Green and Carrit (UNESCO, 1973). However they are complicated and inconvenient to use. Also for the present purposes the differences between the values of L ₀ calculated by the formulae of Fox and of Green and Carrit are negligible (for θ = 0°C, they do not exceed 0.8% of the quantities themselves, which does not have a great effect on our approximate calculations).

The water temperature at the lower surface of the ice is at the freezing point τ, the exact value of which is dependent on the water composition. The dependence between these two quantities can be simply expressed by using a formula of Miyake (1939)

Taking this into account, we can rewrite Equations (14) and (15) as

from which we obtain

To obtain the density of sea-water we will use the empirical equation of Knudsen (Forch and others, 1902)

where σ₀ is the relative density for 0°C, σ_d is the relative density of distilled water, and A(θ) and B(θ) are functions of temperature. σ₀ can be expressed as a function of the chlorinity of the sea-water by

and σ_d, A(θ), and B(θ) vary with temperature as

Bccause in the case we are considering θ = τ and τ is associated with the chlorinity of the water via Equation (16), we can solve Equations (19) to (22) neglecting the terms in [Cl]_w with powers higher than the third. Differentiating the resulting equation we obtain

By means of Equations (18) and (23), the values of dL _N/dρ, dL _O/dρ, and dL_Σ /dρ given in Table I can be calculated. The results obtained by graphically interpolating the derivatives (Fig. 1) can then be substituted in either Equation (11) or Equation (13) to determine the ice porosity v _Ib.

Table I.The derivatives dL _N/d _p , dL _o/d _p , and dL _Σ/d _p

Fig. 1

Process ic, i.e. the gas release from the sea bottom, can be treated by assuming that the rate of release is constant

Hence in the time interval dt a gas volume equal to dVs is released and will accumulate at the base of the ice sheet. In addition

ana from Stefan’s (1891) analysis of ice growth

and

where k_i is the thermal conductivity of ice, θ is the temperature of the upper boundary of the ice, and A is the quantity of heat necessary for the formation of 1 cm³ of sea ice at a temperature equal to the freezing point of sea-water τ. From Equations (24) and (25) we now have

Here the value A is equal to the density of sea ice δ multiplied by the latent heat of pure ice λ and by a factor which gives the mass of pure ice in 1 g of sea ice. According to Tsurikov (1975), this factor is equal to or larger than 0.1. Hence A ≥ 0.1 λδ and

It should be noted that if in this equation τ = θ, i.e. if the temperature gradient is equal to zero, ice growth stops.

Now consider the porosity of some layer, say h₂ — h₁ . In this case we must integrate Equation (27) within the limits of 0 to v_Ic and h₁ to h₂ . As a result we have

where Δh ² = h₂ ²—h₁ ² . Now taking Equation (10) into account we have

and if we wish to consider the porosity of the complete thickness of the ice cover, then h₁ = 0 and

Epigenetic processes

Process 2a (i.e. the substitution of air for a part of the brine) occurs during the course of ice melting, when closed brine pockets change into interconnected channels that extend through the ice sheet. The process should be analysed starting from a knowledge of the volume of substituted brine. The relative volume of brine in the complete thickness of the ice cover is equal to the ratio of the brine volume V _b to the total ice volume considered

By analogy, the relative volume of brine in the portions of the ice above water-level and below water-level must be

where Z_u and z₀ are draft and free-board of the ice relative to sea-level. Because V _b0 = V _b — V _bu, according to the last three equations we have

and from Equation (30) we obtain

From Archimedes Principle, if the ice is not covered with snow, it follows that

where M and M _w are the masses of ice and displaced water respectively and g is the acceleration due to gravity. The assumption about the absence of snow on the ice is reasonable because the snow cover usually melts before significant melting starts within the ice itself. We then rewrite Equation (32) as

where

As the brine in the above-water part of the ice is replaced by air we can rewrite Equation (31) as

and then substituting from Equation (33)

Next taking Equation (10) into account we obtain

which, when solved as a quadratic equation, becomes

Tsurikov (1976) concluded that the formation of through channels occurs only after the relative brine volume on every horizon of the submerged portion of the ice reaches a value of at least 0.044. Then assuming that v _bu is equal or greater than its minimum, we can obtain an estimate of the maximal value for v_2a from Equation (34). First we note that the density of pure ice δ_ρ only changes from 0.917 (at o°C) to 0.922 (at —23°C) and if we take an average value of δ_ρ = 0.92 the possible error. from this source will not be more than 0.3%. Also, although the density of sea-water varies with temperature and composition, if we consider its value to be ρ =1.00 the possible error due to this is less than 3%. Taking these approximate values into account, Equation (35) can be written as

If from the two roots of Equation (36) we select the one which gives the increase of v_2a that depends on the increase in v _b as in Equation (34), we obtain

Process 2b considers the release upon further freezing of gas that is dissolved in brine trapped within sea ice. During the course of freezing, sea-water with a mass of ρV_W turns into pure ice with mass δ_p V _p and brine with mass ρ _b V _b. From this we can write

where V _w is the volume of the frozen water, and V, V _p, and V _b are the volumes of the sea ice, pure ice, and brine respectively. During this process some of the gas that was initially dissolved in the unfrozen sea-water having a concentration of C_w will still remain in the brine where the gas concentration has now reached saturation concentration L and some will come out of solution resulting in gas-filled pores within the ice. Therefore

where C is the relative gas content of the ice.

From Equation (38) we have

and, as has been demonstrated by Tsurikov and Tsurikova (1972), the relative mass of brine in the ice ρ _b V _b/δV is equal to the ratio of the chlorinity of sea ice to that of the brine

Now, substituting the last two equations in Equation (39), we have

and, taking Equation (2) into account

where C _NW and C_ow are the concentrations of nitrogen and oxygen in the sea-water before it freezes. Now denoting C_Nw+C_ow ⁼ C_Σw and substituting the value of δ as obtained from Equation (10) into Equation (41) we obtain

This equation includes the densities of brine ρ _b, sea-water ρ, and pure ice δ_ρ all of which are variables. However the possible error resulting from taking ρ _b = 1.10 cannot be more than 7% ( ρ _b = 1.00 at —0.2 ° C and 1.19 at —23°C). The errors resulting from assuming ρ = 1.00 and δ_ρ = 0.92 are even less, as stated earlier 0.3 and 3.0% respectively. Hence as an approximation we can obtain from Equation (43)

I know of no determinations of the solubility of gases in the brine formed from freezing sea-water. Therefore to obtain values of L _N and L ₀ we have used Equation (17) even though this equation can only be used with confidence to chlorinity values corresponding to sea-water ([Cl] _b < 20‰) and to temperatures above —2.0° C. Values of L _N and L ₀ so calculated are presented in Table II and are positive only to some limiting [Cl]b value which appears to lie in a range between 83 and 86‰. This corresponds to a freezing temperature of between — 8.4 and — 8.8°C. Furthermore, from Table II we can see that the value of L_Σ/L_N changes only slightly. Therefore taking its value as 1.56 we can simplify Equation (44) to

Values of L _N and L _O for different values of the chlorinity of the brine are given in Figure 2.

Table II.Limits of brine saturation by nitrogen and oxygen

Fig. 2

Process 2c is concerned with the formation of pores that are filled with water vapour. As ice temperatures decrease during a period of ice growth, ice gradually freezes out of the brine so that the brine composition will change to the specific composition that is in equilbrium with ice at the new temperature. The volume of ice that is formed will be more than the volume of brine that it has replaced. As a result strains will occur in the ice causing stresses which can exceed the strength of the ice and result in the formation of cracks. In the opinion of Savel’yev (1963) these cracks quickly refreeze.

In the course of the following rise of temperature dθ, a part of the ice that forms the walls of the brine pockets will be dissolved and the volume of brine will increase by dV _b, a volume which is less than the volume of melted ice dV_p . If cracking of the ice took place on the previous cooling cycle and the cracks refroze, a void with the volume

can appear on the upper ends of the brine pockets. These voids are quickly filled with water vapour as a result of the partial evaporation of ice and brine.

It is obvious that V _p = M _p/δ_P. Also if we neglect the relatively small mass of crystallized salts and assume that M = 1, we have M _p + M _b = 1 and hence V _P = (1—M _b)/δ_p, so

and

Then instead of Equation (46) we can write

or, taking Equation (40) into account,

This equation can be integrated between the limits 0 and V_2c or ([Cl]_b)₀ and ([Cl]_b)₂, taking into account that d[Cl] = 0 in as much as the chlorinity of the ice is not dependent on temperature. As a result we have

where

Because δ_ρ ≈ 0.92 Mg m ^-3 and ρ ≈ 1.10 Mg m^-3 we can write

Exogenetic processes

Of the two exogenetic processes we mentioned earlier, no way has been found to develop a quantitative formulation of process 3a, therefore we will only discuss process 3b.

Process 3b which is concerned with the migration of air inclusions across the snow-ice interface can be investigated using a theoretical relation developed by Kheysin and Cherepanov (1969). In considering the rate of migration (dζ/dt) of spherical air bubbles through ice they obtained

where k_i and k_a are the thermal conductivities of ice and air, D is the diffusion coefficient for water vapour in air, and γ is the density of saturated water vapour. According to Kheysin and Cherepanov the values of these parameters when applied to Equation (49) are k_i = 54 x 10^–4 cal g^-1 cm^-1 s^-1, k_a = 5.84 x 10^-3 cal g^-1 cm^-1 s^-1, D = 22 cm² s^-1 and δ = 0.9 g cm^-3. If the snow is quite fresh we can consider that λ = 80 cal g^-1 and we can obtain values for

from table 132 in Zubov (1957). In Table III we have calculated the migration rate of air bubbles as a function of temperature for a constant temperature gradient of 1 deg/cm. For comparison the migration rate of the air bubbles is two orders of magnitude smaller than for brine pockets (values also given in Table III). Hence the motion of air inclusions from snow into ice, although possible, is clearly not of major importance.

Table III.The rate of migration of air bubbles and brine pockets in the ice with the temperature gradient I/deg cm^–1