Introduction

The d.c. resistivity of sea ice is influenced by many factors including the salinity, temperature, and age of the ice as well as the size, distribution, and geometry of the brine cells. Basically, the d.c. resistivity of sea ice may be studied either by standard d.c. measurements of small ice samples obtained from cores cut from the ice sheet, or by geoelectric sounding techniques. The former method has the advantage that the electric field direction is well defined with respect to the geometry of the brine cells. However, it has the disadvantage that significant brine drainage can occur before the measurement can be performed, and this leads to erroneous results. The latter technique of geoelectric sounding overcomes this problem since the resistivity is measured as a function of the electrode spacing (i.e. the depth of penetration) and therefore the ice is measured in situ. Unfortunately, however, due to the complex nature of sea ice, interpretation difficulties usually hamper the amount of information which can be derived from the geoelectric sounding results. With investigations of this kind, one obtains values for both apparent resistivity and apparent ice thickness. In this note, we extend the usual interpretation of geoelectric sounding results on sea ice to obtain values of in-situ resistivity for directions both parallel and perpendicular to the brine cells of the sea-ice strata. In addition, using a general conductivity mixing formula in conjunction with the results obtained for the resistivity parallel to the brine cells, we calculate values of “average” brine cell lengths for relatively warm sea ice.

Field Results: Pond Inlet, N.W.T.

The present work is based on the results of a series of Schlumberger geoelectric soundings performed on first-year sea ice near Pond Inlet, N.W.T. in the spring of 1972 (Reference Cowan, Cowan, Timco, Kohnen and ThyssenCowan and others, unpublished ; Reference Thyssen, Thyssen, Kohnen, Cowan and TimcoThyssen and others, 1974; Reference KohnenKohnen, [ 1976]). In soundings of this kind, a current is introduced into the ground by two electrodes, and using non-polarizable electrodes, the potential difference is measured between two points centred about the mid-point of the line connecting the current electrodes. The outer electrodes are then expanded about the mid-point to obtain greater depth of penetration into the ice. The measured resistivity ρ is then determined by

where V is the potential difference, I is the current, L is the current-electrode spacing, and d is the voltage-electrode spacing. The measured resistivity is plotted against L/2 on log–log paper to evaluate the resistivity and thickness of the ice cover.

Twenty-two soundings were made spaced at 1 km intervals across first year ice from Pond Inlet on northern Baffin Island to a southern point on Bylot Island. These soundings were repeated at intervals of c. 2 weeks (16–19 May, 3–6 June, and 15–21 June) in order to obtain information on the change in resistivity with time (i.e. temperature). In addition, on 8 June (i.e. immediately after the 3–6 June traverse), a pit c. I m deep was dug into the unrafted ice to measure the apparent resistivity perpendicular to the top surface of the ice.

Interpretation of the results indicated that the ice could be considered as a two-layer case: (1) a low resistivity ρ _u layer for the upper c. ¼ of the ice sheet; and (2) a higher resistivity ρ _a layer for the lower c. ¾ of the ice sheet (Reference Thyssen, Thyssen, Kohnen, Cowan and TimcoThyssen and others, 1974; Reference KohnenKohnen, [1976]). The values for these layers averaged over the 22 soundings for each traverse are listed in Table I. In addition, both the average temperature T and brine volume ν _b of the lower ice layer are listed in this table. The latter was calculated from (Reference Frankenstein and GarnerFrankenstein and Garner, 1967)

where the salinity S of the ice at Pond Inlet was measured to be c. 5‰ relatively independent of depth in the lower layer (Reference KohnenKohnen, [1976]). For the resistivity measurement perpendicular to the ice surface, twenty measurements yielded ρ _v ≈ 229 Ω m (Reference Cowan, Cowan, Timco, Kohnen and ThyssenCowan and others, unpublished). For more details regarding the experimental aspects and general results of this investigation, including a general discussion on the factors influencing the resistivity of sea ice, the reader is referred to the articles by Reference Thyssen, Thyssen, Kohnen, Cowan and TimcoThyssen and others (1974) and Reference KohnenKohnen ([1976]).

Microstructure of Sea Ice

When sea-water freezes, the resulting ice sheet shows distinct features which are characteristic of solids grown from impure melts. In the upper surface layer of the sheet, a large number of the ice crystals grow with their c-axis aligned vertically. However, as the sheet thickens, a process of geometric selection takes place whereby crystals with favoured orientations eliminate those in other orientations by cutting them off from the melt. This results from the fact that the growth-rate is more rapid along the (0001) basal plane than in the direction perpendicular to it (Reference Hillig, Doremus, Doremus, Roberts and TurnbullHillig, 1958). This anisotropic growth-rate has the net effect that with increasing depth, the orientations of the c-axes of the crystals change from being predominantly vertical to predominantly horizontal, such that from a depth of c. 40 cm to the bottom of the ice sheet, the c-axes are strictly horizontal. During growth, liquid brine is trapped and contained in thread-like pockets or cells which are aligned in very thin layers parallel to the (0001) basal plane (i.e. perpendicular to the c-axis), and at intercrystalline boundaries. These brine inclusions within the ice crystals produce a pronounced substructure which is readily observable in thin sections (Fig. 1). In general, the size and shape of the thread-like brine pockets are a complex function of the ice salinity and temperature. For sea ice, the concentration of the brine in the ice is determined by the phase diagram of salts and water, which because of the many kinds of salt present is quite complex (Reference AssurAssur, 1958). With changes in temperature, the liquid inclusions change size. If the ice cools, some water freezes out of the brine pockets. This decreases the size of the brine cells and increases the salt concentration in the cells to the equilibrium value. If the ice warms, the brine wall dissolve some ice, thereby increasing the volume of the cell. At the higher ice temperatures close to melting, this increase in volume often leads to an interconnection of the brine pockets in the vertical direction. Typical diameters of these brine cells are on the order of 0.05 mm (Reference AndersonAnderson, 1960; Reference PounderPounder, [ ^c 1965]). The lengths of the brine cells are much more variable than the diameters, and are not well known. Quoted values for the brine cell lengths of relatively warm ice are typically on the order of 2–3 cm (Reference PounderPounder, 1965; Reference AddisonAddison, 1969).

Fig. 1. Photomicrograph of a vertical cross-section of a thin sheet of NaCl ice showing both the laminar microstructure of the thread-like brine cells and the air pockets. The grid is 1 cm on a side.

Anisotropy of Sea Ice in the Lower Layer

From the foregoing discussion it is evident that the lower layer of sea ice may be treated as a relatively pure ice matrix enclosing long, thin, spheroidal or cylindrical-shaped pockets of brine which are all aligned in the vertical direction. Since the ice matrix has a resistivity much higher than that of the brine inclusions, the resistivity in this layer would be highly anisotropic. Clearly, the resistivity of this layer would be dependent upon the direction of the current relative to the brine cells, such that the resistivity in the vertical direction would be much lower than the resistivity in any horizontal direction. At this stage, for a proper analysis of the resistivity data, we must examine the nature of the ice studied to determine if, on a macroscopic scale, the resistivity would be the same in any horizontal direction; i.e. to determine if there is a preferred horizontal alignment of the c-axes of the ice crystals in the lower layer. Recently, for example, Reference Campbell and OrangeCampbell and Orange (1974) have found evidence of horizontal electrical anisotropy in sea ice off the Canadian Arctic Islands, and Reference Gow and WeeksGow and Weeks (1977) have found strongly-preferred crystal orientation in the lower layer of sea ice near Narwhal Island in the Beaufort Sea. To check for the possible existence of horizontal electrical anisotropy in the present case of the Pond Inlet ice, two independent Schlumberger profiles were performed at each station such that the profiles were perpendicular to one another but centred at the same mid-point. This was done at each of the first eight stations on the 16–19 May traverse. With this technique, one samples essentially the same volume of ice at each station, but from two different (perpendicular) directions. If a strong alignment of c-axis orientation in the horizontal plane exists in the ice, then a large discrepancy would be found between the two apparent resistivity values. This was not found to be the case. The average difference between the two resistivity values was on the order of 5%, which was within the experimental accuracy of the tests.

For the Pond Inlet ice considered in this analysis, we therefore have the case of transverse isotropy, i.e. resistivity independent of direction in the horizontal plane but different perpendicular to the plane. Situations similar to this are frequently encountered in geophysical prospecting with the difference that in mineral prospecting, the resistivity along the stratified layer is usually less than the resistivity perpendicular to it. In spite of this difference, we can still apply the general formulation for this situation as discussed by Reference Bhattacharya and PatraBhattacharya and Patra (1968). In the case of microscopic anisotropy, the measured(apparent) resistivity ρ _a determined using the Schlumberger configuration is given by

where, as shown in Figure 2, α is the dip angle of the electrically isotropic (El) planes from the horizontal, ϕ is the angle made by the point of observation and the strike direction of the El planes, λ is the coefficient of anisotropy, and ρ _m is the mean resistivity.

Fig. 2. Geometry for Equation (3) showing the air–ice surface and a vertical section through the ice: X’ γ’ is air-ice boundary surface; XY is electrically isotropic (EI) planes; α is dip of El planes from horizontal; ϕ is angle made by point of observation and the strike direction of the EI planes.

Defining ρ _p as the resistivity parallel to the electrically isotropic planes (i.e. the horizontal resistivity), and ρ ₁ as the resistivity along the anisotropic direction (i.e. the vertical resistivity which is parallel to the long direction of the brine cells), λ and ρ _m are given by

For the Schlumberger measurements on the surface of the ice, α = 0 and thus the resistivities are all measures of the mean resistivity, i.e.

However, for the experiment on the wall in the ice pit, with the array of electrodes vertical, α = π/2 and ϕ = π/2, and the apparent resistivity measured in the vertical direction on the wall is (from Equation (3))

Thus, from Equations (4) and (5), and using ρ _m = 136 Ω m (i.e. the mean resistivity of the lower layer for the second traverse, Table I), and ρ _v = ρ _p = 229 Ω m (Reference Cowan, Cowan, Timco, Kohnen and ThyssenCowan and others, unpublished), we get λ = 0.59 for this ice. This value of λ for the ice on all three traverses and the value for the average mean resistivity ρ _m for each traverse (Table I) can then be used to get the resistivity parallel to the brine strata ρ ₁ and the resistivity perpendicular to the brine strata ρ _p from Equations (4) and (5). These values are listed along with the temperature and brine volume of this layer in Table II. Note that the resistivity decreases with increasing temperature and brine volume. This is a reflection of the changes in size and geometry of the brine cells with temperature. This suggests that by applying a suitable mixing formula, it should be possible to obtain some general information on the dependence on temperature and brine volume of the size and geometry of “average” brine cells in sea ice.

Conductivity Model: Systems Containing Dispersed Spheroids

In choosing a way to describe the electrical properties of sea ice, it is clear that due to the nature and complexity of sea ice, several approximations and simplifications must necessarily be made. Ideally, we would like to use a model which treats the electrical resistivity using some prescribed form of statistical distributions for the brine-cell lengths, widths, and spatial distribution. Unfortunately, due to the lack of such a theory as well as insufficient information on the physical properties of brine cells, this is not yet possible. In the past, however, there have been several theories which have been used successfully to calculate the apparent resistivity of a two-phase heterogeneous system. These are based on the premise of constant size and shape of the inclusions, and the assumption that the heterogeneous system can be represented electrically by an equivalent network consisting of linear elements. Several such treatments date back to the times of Maxwell, Poisson, and Lord Rayleigh. For the present case of sea ice which may be regarded as a mosaic of brine pockets in an ice matrix, we choose as a model that of conducting spheroids in a poorly-conducting matrix (Fig. 3). Clearly, such a model is very highly idealized and should not be interpreted as a rigorous representation of sea-ice structure. Nevertheless, it is felt that such an approach does provide useful information with regard to the typical characteristics of brine cells in sea ice. Similar treatments have been used successfully to describe both the mechanical properties of sea ice (Reference AssurAssur, 1958; Reference Weeks, Assur and KingeryWeeks and Assur, 1963 Reference Weeks and Assur1968) and the thermal conductivity (Reference Anderson and WashingtonAnderson, 1958), and to estimate the electrical relaxation time of an ice-brine mixture (Reference Hoekstra and CappillinoHoekstra and Cappillino, 1971).

Fig. 3. Schematic cross-sectional representation of the prolate spheroid conductivity model of brine cells for Equations (8)) and (9).

Following Reference FrickeFricke (1924) and Reference BeekBeek (1967), the resistivity ρ of a system of the type shown in Figure 3 is given by

where greatly simplifyfor the present case σ_i is the conductivity of the ice matrix, σ_b is the conductivity of the brine inclusions, and ν _b is the brine volume in parts per thousand. The depolarization factor A_a for the case of prolate spheroids (with the field parallel to the long axis of the spheroid) is

where b is the radius of an average brine cell and l is the length of a brine cell. Thus, the reisistivity of such a two-phace heterogeneous systems depends upon the electrical conductivity of the matric and inclusions, and both the relative amount and axial-ratio (geometry and size) of the inclusion. In the present case, we can greatly simplify Equation (8) and (9) since σ_b ⪢ σ_i, 10^–3 v_b ≈ 0.05 and l ⪢ b (i.e. A_a ⪡ 1). Thus, Equation (8) becomes

and Equation (9) becomes (after Reference BeekBeek, 1967)

For sea ice, we can make reasonable estimates for the brine volume as well as the conductivity of both the ice matrix and the brine cells. Moreover, typical values of the radii of brine cells in sea ice have been determined petrologically. Thus, using the values of the in-situ resistivity for the direction parallel to the brine cells ρ ₁ we can use this formulation to calculate values of brine cell lengths for the model.

Application Using Average Values for Sea Ice

To apply this to the present case, we first choose overall average values for ρ ₁ and ν _b from the 66 Schlumberger soundings in order to check the general applicability of these equations for sea ice. From Table II,

and . To obtain a value for the conductivity of the ice matrix, we note that the matrix is itself not pure ice since it contains small amounts of solid salts, impurities, and trapped air. As such, a representative value for σ_i would be obtained by measurement of either low-salinity sea ice or very cold sea ice. Independent measurements on both warm, low-salinity sea ice (Reference Pounder and LittlePounder and Little, 1959) and on very cold sea ice measured perpendicular to the long axis of the brine cells (Reference Dichtel and LundquistDichtel and Lundquist 1951) yield d.c. conductivity values of 3 × 10⁻⁵ S/m;hence σ_i = 3 × 10⁻⁵ S/m. To obtain a value for σ_b, we know that from the salinity (5‰) and brine volume (c. 63‰) of the Pond Inlet ice, the concentration of salt in the brine pockets is on die order of 80 g/l. For electrolytic solutions of this concentration, the conductivity σ_b is c. 5 S/m (Reference Hoekstra and CappillinoKeller and Frischknecht, 1966). Since the dimensions and spacing of the brine cells vary considerably, they must be analysed statistically. This has been done by petrological studies for both the spacing and width of the ice platelets, and also for the radius of the brine cells (Reference AndersonAnderson, 1960; Reference Weeks, Assur and KingeryWeeks and Hamilton, 1962). For sea ice of brine volume 63‰, typical brine-cell radii are b ≈ 0.046 mm (Reference AndersonAnderson, 1960). Then using Equations (10) and (11) with , v_b = 63 ‰ σi = 3 × 10^–5 S/m, σ_b = 5 S/m, and b = 0.046 mm, we get ῑ ≈ 1.7 cm. This says that, from the electrical point of view, the value of the electrical resistivity of sea ice parallel to the brine-cell direction would be the same as that measured experimentally if the ice consisted of brine cells of 1.7 cm length and 0.046 mm radius and brine volume of 63 ‰; that is, electrically they are equivalent. Thus, in a manner similar to taking a statistical value for b, we can call ī the “average” brine cell length. This calculated value for ῑ of 1.7 cm is in excellent agreement with the values of brine-cell lengths of 2−3 cm quoted by Reference PounderPounder ([ ^c 1965]) and Reference AddisonAddison (1969). Certainly it is reassuring that this indirect calculation yields a value which agrees well with the brine cell dimensions obtained from direct petrological measurements.

The Temperature Dependence of the Size of the Brine Cells

We can extend this treatment by making use of the information gained on the three separate traverses across the ice at Pond Inlet. From Table II, ρ ₁ (v _b = 40‰) = 104 Ω m, ρ ₁ (v_b = 67‰) = 80 Ω m, and ρ ₁ (v _b= 82‰) = 45 Ω m. Recall that each one of these values is an average of 22 soundings. These values for ρ₁ are plotted as a function of temperature and brine volume in Figure 4, with a smooth curve drawn through to connect the experimental points.

Fig. 4. Graph of resistivity ρ ₁ parallel to the brine-cell direction versus brine volume ν _b and temperature T . Solid curve: experimental points each representing 22 soundings; dashed curve: resistivity values calculated from the mixing formula (Equations (10), (11)) for various “average” brine-cell lengths l.

For this calculation, we make the reasonable approximation that in the temperature range –6.5° C to –3° C, both σ_i and σ_b, are independent of temperature. Further, we adopt the model of brine volume dependence for the radius b of the brine cells advanced by Reference AndersonAnderson (1960), i.e.

where ω is the ice platelet width and c is the average cell spacing along the brine layer. For sea ice, ω is approximately 0.046 cm and c/ω is c. 0.5 (Reference AndersonAnderson, 1960). Using this equation and σ_i = 3 × 10^–5 S/m and σ_b = 5 S/m in Equations (10) and (11), we calculate ρ ₁ as a function of brine volume for various constant brine-cell lengths. These are plotted in Figure 4. In this case for constant l,ρ ₁ is essentially independent of brine volume. Thus, this model suggests that the conductivity parallel to the length of the brine cells changes only slightly if only the widths of the brine cells increase with increasing brine volume. Certainly, this result seems reasonable. Thus, the fact that experimentally ρ ₁ does change with temperature and brine volume (Table II) indicates that the brine-cell lengths do change in this temperature range. Using the intersection points of the curves calculated from the above model and the smooth curve drawn through the experimental ρ ₁ values (Fig. 4), we arrive at the temperature (or brine volume) dependence of the “average” brine cell lengths as shown in Figure 5. In this case, the results suggest that the brine-cell lengths increase only slightly for c. 30‰ < ν _b < c. 70‰ but then rise sharply for ν _b > c. 70‰. This indicates that the rate of extension of the brine cells is much more rapid for temperatures above — 3.5°C. That is, in this higher temperature region, there is a “runaway” condition of brine-cell enlargement and coalescence to form long continuous brine channels. These channels would lead to the partial desalination of sea ice as discussed by Reference BenningtonBennington (1967) and Reference Eide and MartinEide and Martin;(1975). Moreover, a calculation of (I/l)( Δl/Δν _b)/(I/_b)(Δ _b / Δν_b ) indicates that this ratio is less than one for brine volumes less than c. 7o‰. This would suggest that for ν_b < c. 7o‰ the relative change in the brine-volume (or temperature) dependence of the brine cells is greater in width than in length. That is, with increasing temperature, the brine cells become more spherical in shape before they increase in length and interconnect to form continuous networks throughout the (warmer) sea ice. This general behaviour agrees qualitatively with that observed from petrological studies of brine cells.

Fig. 5. Graph of “average” brine-cell lengths l and “average" number N of brine cells versus brine volume ν_b and ice temperature T.

“Average” Number of Brine Cells in Sea Ice

Using the model of spheroidal-shaped brine cells, we can calculate the temperature dependence of the “average” number N of brine cells in the sea ice by simply dividing the brine volume fraction of the ice by the volume of an average brine cell, i.e.

This function is plotted in Figure 5. In this case, N ≈ 900 (cm)^–3 for ν _b < c. 7o‰ and drops sharply for brine volumes above this value. The sharp drop in the curve simply reflects the fact that the brine cells are much larger and interconnected in this warmer temperature region.

Discussion

Although the present analysis of the in-situ resistivity of sea ice in terms of its microstructure yields encouraging results, it is clear that since the model employed is highly idealized, the results of such a treatment must be used with reservation. Several refinements and additions would be desirable in order to obtain more detailed and reliable results. Although the data used in this study represent 66 geoelectric soundings of 22 each over three different ice temperatures, more data of this nature over a wider range of ice temperatures would be invaluable. In addition, Schlumberger profiles on a vertical wall of sea ice for each temperature sampled would yield important information on the functional dependence of λ on temperature. Refinements to the model include incorporating a statistical variation in the size of the brine cells, and noting that the geometry of the brine cells change from long, thin needles at the colder temperatures to conducting discs for very warm temperatures (close to melting). The latter is easily accomplished by choosing a model of conducting ellipsoids for the brine cells. Further, if a great deal of experimental data were available, it would be worthwhile to try to refine the value for σ_i and to consider the relatively small temperature dependence of both σ_b and σ_i in this temperature region.

In this first attempt at calculating average brine cell lengths from the in-situ resistivity data for sea ice, the results are encouraging. The calculated value for an average brine cell length in sea ice of 1.7 cm is in good agreement with that observed petrologically. Moreover, the derived temperature dependence of the shape and geometry of the brine cells agrees at least qualitatively with direct petrological observations. Since this aspect of sea ice is understandably complex, petrological studies have not determined the quantitative changes in brine-cell length with temperature.' As such, the present treatment of interpreting the d.c. resistivity of sea ice in terms of its microstructure yields a useful method for evaluating these changes.