## Introduction

A knowledge of the total porosity of sea ice is important in interpreting its physical properties. It has been shown, both theoretically and experimentally, that the mechanical, thermal, and electrical properties of sea ice vary with the relative volume of brine existing in the ice (Weeks and Assur, 1967; Schwarz and Weeks, 1977). However, the air or gas volume in the ice is also important. This is particularly true for low-salinity sea ice, such as multi-year ice, where the gas volume may make up a major portion of its total porosity.

The gas volume of sea ice can be calculated given the density, salinity, and temperature of the ice. Generally, Assur’s (1958) phase equilibrium table (table III, p. 124) is used to perform such calculations. This table gives the relative masses of the different components in “standard sea ice” at temperature intervals of 2 deg. These data are based primarily on the experimental determinations of Nelson and Thompson (1954) and Ringer (1928). Here the expression standard sea ice is used to refer to sea ice of such a composition that the relative concentrations of ions in its melt water are the same as in normal sea-water. From the density of the various components, the theoretical, or gas-free, density of the sea ice can then be calculated and compared to the measured density to determine the actual gas volume present in the ice. However, since the table is based on standard sea ice having a salinity of 34.325‰, it is first necessary to reduce the data to the appropriate ice salinity. This is done by multiplying the masses of all the components, except pure ice, by *S*
_{i}/34.325 where *S*
_{i} is the ice salinity in parts per thousand. The mass of pure ice is then 1.000 g minus the total mass of brine and precipitated salts. Even though these calculations are straight-forward, they are both time-consuming and prone to arithmetic errors. In fact they are rarely performed and estimates of the amount of gas present in sea ice and of the total void volume of the ice are rarely obtained.

In the present paper, equations are derived from which the gas volume and brine volume can be directly calculated given the ice salinity, temperature, and density. The equations cover the temperature range from −2 °C to −30 °C and consider the presence of solid salts. Equations are also given to calculate sea-ice density and the variation of gas and brine volume of a sea-ice sample, if the temperature of the sample is changed from the temperature at which the ice density was originally obtained.

## Previous work

Only a few investigators have advanced methods for calculating the density and gas volume in sea ice. Zubov (1945) calculated the density of gas-free sea ice and provided a table of values that could be used at temperatures of −23 °C and warmer. However, his work was done prior to Nelson and Thompson’s (1954) experiments on the chemistry of sea-ice brine. Zubov also did not take into account the presence of solid salts. In 1960, Anderson (1960) published a table giving the density of gas-free sea ice at 14 selected temperatures in the temperature range between 0 and −50 °C. This work was based on Assur’s (1958) phase relations. As the exact method of calculation is not given, it is not clear whether Anderson considered the presence of solid salts in the ice. The only relation available that calculates the gas volume of sea ice directly was developed by Schwerdtfeger (1963). However, it is only valid at temperatures warmer than −8.2 °C, the Na_{2}SO_{4} · 10 H_{2}O crystallization temperature. He also neglected the presence of solid salts and assumed that the volume of brine was equal to that of pure water. For warm sea ice these are reasonable approximations. At the conclusion of this report comparisons are made between the results of these previous authors and the present results.

It should also be noted here that Assur’s (1958) brine volume table and the brine volume equations based on this table derived by Frankenstein and Garner (1967) assume a constant ice density of 0.926 Mg/m^{3}. Consequently, these reports should be used with caution in gas volume calculations. The brine volume obtained from these sources should be multiplied by *ρ*/0.926. where *ρ* is the gas-free theoretical density, or better still, the measured ice density in Mg/m^{3} at the temperature of interest, to determine the brine volume for a given sea ice density.

## Derivation of equations

A sample of sea ice is composed of pure ice, brine, solid salts, and gas. In determining the gas volume of sea ice we are interested in the relative masses and volumes of the different components at a given ice salinity, temperature, and bulk density.

In the derivations to follow, *m*
_{
l
}, *ρ*
_{
l
}, and *V*
_{
l
} are the mass, density, and volume of component *l* where the subscripts a, b, i, s, and ss denote the component air, brine, pure ice, salt, and solid salts, respectively. *M* is the bulk mass, *V* the bulk volume, and *ρ* the bulk density (these should not be confused with the gas-free theoretical mass, volume, and density). The terms and denote the mass of salt in the brine and mass of salt in the solid salts, respectively. The mass of gas in the ice is assumed to be negligible.

### Brine volume

The salinity of the ice *S*
_{i} is defined as

The salinity of the brine *S*
_{b} is defined as

From these two relations an equation for the brine volume of sea ice can be derived. It should be noted that the brine salinity and the relative amounts of salt in the brine and solid salts are unique functions of ice temperature via phase relations.

From Equation (1):

and

or

where

Solving for from Equation (3) and noting from Equation (2) that

we obtain

which, because *M* = *ρV*, can also be written as

where *V*
_{b}/*V* is the relative brine volume. If we neglect the salt present as solid salt, *k* = 0, and Equation 4 reduces to

a relation utilized earlier by Weeks and Lofgren (1967).

Because the brine density can be approximated (Zubov, 1945; Cox and Weeks, 1975) by

and because *S*
_{b} and *k* are unique functions of temperature, the relative brine volume can be expressed as

where *S*
_{i} is in parts per thousand, *ρ* is in Mg/m^{3}, and

Values for *F*
_{1}(*T*) are given in Table I and plotted against temperature in Figure 1. *S*
_{b} and *k* are determined from Assur’s (1958) phase equilibrium table. The lines through the data in Figure 1 are least-squares curves of the form

where *T* is the ice temperature (°C). Coefficients for the curves are given in Table II (p. 312).

Table I. Values for *S*
_{b}, *k*, *C*, *F*
_{1}(*T*), and *F*
_{2}(*T*) at different temperatures

Fig. 1. Function F_{1}(T) versus temperature. Curves determined by method of least squares.

Table II. Coefficients for functions *F*
_{1}(*T*) and *F*
_{2}(*T*) determined by method of least-squares

### Solid salt volume

Also, via the phase relations, at any given temperature the mass of solid salts (if solid salts are present) is proportional to the mass of brine, that is

where *C* is a function of temperature. The relative volume of solid salts *V*
_{ss}/*V* is then equal to

where *ρ*
_{ss} is the average solid salt density assumed to be constant at 1.5 Mg/m^{3}.

### Pure ice volume

The mass of pure ice is equal to

or from Equation 8

Since

we have

Solving for the relative pure ice volume *V*
_{i}/*V* we obtain:

### Air volume

The relative air volume *V*
_{a}/*V* is equal to

Substituting Equations (9) and (10) into (11) we obtain

Equation (12) can be simplified by defining

substituting for (*V*
_{b}/*V*) from Equation (4), and recalling the definition of *F*
_{1}(*T*) given in Equation (6). The resulting relation is

Values for *F*
_{2}(*T*) are given in Table I and plotted against temperature in Figure 2. *C* is determined from Assur’s (1958) phase equilibrium table. The lines through the data in Figure 2 are also least-squares curves of the same form as Equation (7).

Fig. 2. Function F_{2}(T) versus temperature. Curves determined by method of least squares.

Coefficients for these curves are given in Table II. In determining the value of *F*
_{2}(*T*) at a given temperature, the pure ice density was calculated from (Pounder, 1965);

### Sea-ice density

The sea-ice density can be found from Equation (14) by solving for *ρ*:

Unlike the calculated densities given in Schwerdtfeger (1963) and Anderson (1960), this equation considers the presence of both air and solid salts in the ice. A comparison between the different density estimates is given later in this paper.

### Gas volume at another temperature

The gas volume and brine volume of sea ice are determined by measuring the volume and mass of the ice sample at a given temperature and the salinity of the melted sample. If tests are conducted at a temperature different from the temperature at which the bulk density was determined, the air and brine volume must be recalculated.

Equations are now derived to calculate the air and brine volumes at a temperature that differs from the temperature at which the ice density was determined. Primed variables denote the initial value of the different variables.

It is initially assumed that the brine and air pockets are inter-connected. From Equation (5) we know that

Because the mass and bulk salinity of the ice do not change with temperature,

By re-arranging Equation (14), substituting for *ρ* based on Equation (5), and defining

it can be shown that

Also dividing both sides of Equation (16) by *V* and noting that

we obtain for the new relative brine volume

From Equation (18) we can also conclude that

Substituting Equation (19) into Equation (20) and solving for *V*
_{a}/*V* we obtain for the new relative air volume

If the volume of the sample is not known at the new temperature, one can assume that the change in the external dimensions of the sample is due to the volumetric change in the ice matrix, that is

In the temperature range where many investigations are carried out

If the brine and gas pockets are not connected, a change in gas volume may also take place if the ice is warmed, that is, as less dense ice is melted to dilute the brine. In this case, the relative volume of the gas pockets remains the same, but voids are produced in the brine cavities. By setting *V*′_{a}/*V*′ to zero in Equation (21) we can estimate the increase in the relative gas volume in the brine cavities. Adding this to the original relative gas volume, we obtain

and in many applications

If the brine and air pockets are not connected and the ice is cooled, the relative air volume will not change. However, due to phase changes, some brine will be lost from the sample as a result of brine expulsion.

## Discussion

As methods for calculating the gas content of sea ice invariably contain a calculation of the density of gas-free sea ice, it is interesting to compare the density estimates developed in this report with previously published estimates. Table III gives such a comparison. As can be seen, the agreement between the present results and those of Anderson (1960) is very good (the present values are at most +0.000 3 Mg/m^{3} higher with a modal difference of only +0.000 1 Mg/m^{3}). This agreement is, however, not surprising in that both Anderson’s and our calculations are based on Assur’s phase equilibrium tables. The agreement essentially verifies both sets of calculations. When comparisons are made with Schwerdtfeger’s (1963) results for sea ice warmer than −8.2 °C, much larger differences are found. Within this temperature range these differences vary from +0.000 9 Mg/m^{3} at warm temperatures (−2 °C) and low salinites (1.0‰) to +0.015 4 Mg/m^{3} at −8°C and 20‰. The agreement with the earlier values of Zubov (1945) is better than with Schwerdtfeger, with the largest difference being +0.005 8 Mg/m^{3}.

Table III. Calculated densities (in Mg/m^{3}) of gas-free sea ice at different temperatures and salinities as given by different authors

It is also interesting to use the relations that have been developed to examine the importance of considering the presence of solid salts in making gas volume calculations. For sea ice warmer than −8.2 °C, Schwerdtfeger (1963) calculated the relative air volume from

where *ρ* is in Mg/m^{3}, *S*
_{i} in ‰, and *T* in °C. As mentioned earlier he neglected the presence of solid salts and assumes that the volume of brine in the ice is equal to the volume of pure water. If we neglect the presence of solid salts in our equations, Equation (12) simplifies to

The results from Equations (23) and (24) are compared with the results from Equation (12) in Table IV. The air volume of sea ice of different assumed salinities, densities, and temperatures are calculated using these three equations.

Table IV. Air volume (‰) calculations from three methods for sea ice having different densities, salinities, and temperatures

All the calculated values at a given salinity, density, and temperature are in close agreement. This indicates that, even for sea ice below the NaCl · 2H_{2}O eutectic temperature, neglecting the presence of solid salts in air volume calculations is a reasonable assumption. For warm, high density, high salinity sea ice (0.930 Mg/m^{3}), Schwerdtfeger’s value differs by 1.8‰, or nearly 100%. However, the brine volume of this ice is relatively much greater (85.1‰) and there is little difference in the total ice porosity.

## Conclusions

Equations have been derived to calculate quickly the air volume of sea ice considering the presence and absence of solid salts. The results show that neglecting the presence of solid salts is a reasonable assumption. However, in this era of computers and sophisticated hand calculators, we recommend considering the presence of solid salts and the use of Equation (12) to calculate the air volume of sea ice. A program for making such calculations on a Hewlett-Packard 41C calculator is available from the authors.

Earlier we mentioned that very few direct determinations of the gas content of sea ice have been undertaken. This might lead the reader to question the usefulness of the calculations outlined here because of a lack of available comparisons between calculated gas porosities and measured gas porosities. Fortunately such a comparison has now been completed for first-year sea ice (Nakawo, 1983) indicating a very good agreement between the calculated and measured gas porosities. Inasmuch as the calculation procedure we have developed is very rapid, and the measurements required to utilize it are simple, it is hoped that gas volume will, in the future, join brine volume and specimen temperature as part of the sample description for all physical property tests on sea ice.