The Editor,
Journal of Glaciology
Sir, Dielectric relaxation in temperate glaciers: comments on Dr P. W. F. Gribbon’s paper
Reference GribbonGribbon (1967) measured the electrical capacitance between two wires laid on the surface of a glacier and related the results to changing ice properties with depth. The theory of the experiment (in section 3·1 of his paper) and the derivation of relaxation frequency from “Cole–Cole” plots are in error and we show here how a better formulation of the problem raises doubts about his conclusions.
We may determine the capacitance C per unit length between two infinitely long cylinders of radius a, separation h, and distance d from the plane boundary between two dielectrics of relative permittivity ϵ 1 and ϵ 2 by simple electrostatic theory (see Fig. 1). To a good approximation
Fig. 1. Geometry of the electrodes
valid for positive values of d greater than, or equal to a.
The vacuum capacitance per unit length between the cylinders isC vac = πϵ 0/ln(h/a), so we define an apparent permittivity by the ratio of the measured capacitance to the vacuum capacitance.
1. The Effect of the Glacier Surface
For cylinders close to the boundary, that is d much less than h, Equation (1) reduces to
Let the boundary be the surface separating the air from a homogeneous glacier and consider two experimental situations:
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I. The wire electrodes resting on, or above, a wet melting glacier as in Gribbon’s measurements from Greenland. Then ϵ 1 = 1 and ϵ 2 = ϵ.
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II. The wire electrodes resting on, or melting into, a cold névé as in France. Then invert Figure 1. and ϵ 1 = ϵ refers to the névé and ϵ 2 = 1 refers to the air above.
In neither case can the capacitance be represented by a constant geometrical factor times the glacier permittivity except that when the cylinders are in contact with the boundary, d = a and the apparent permittivity
in both (I) and (II). This is the same value as is obtained when the cylinder lies symmetrically in the boundary surface.
Now suppose the glaciers behave as ideal Debye dielectrics with properties tabulated below.
Table I Parameters for Three Types of Ice or Névé
The wire radius a is given as 4×10−4 m, and let us consider a separation h of 80 m.
The Cole–Cole diagrams of the apparent permittivity, ϵ app, for raised wires, for buried wires, and for several heights d have been plotted and the results can be summarized as follows:
(I) Raised wires (Fig. 2)
The low frequency points fall on the arc of a circle. This is a Maxwell–Wagner polarization phenomenon. A circle results since at frequencies less than one-tenth of f r the dielectric has constant permittivity and conductivity. The low frequency limit of capacitance
Fig. 2. Cole–Cole plots of the real and imaginary parts of apparent permittivity for wires raised above the surface of a warm wet glacier
which is the capacitance between the wire and its image in a perfect conductor.
An arc of another circle may be made to fit the high frequency values, but the apparent relaxation frequency f m (where ϵ″ is a maximum), is higher than the true value, and if no allowance is made for the conductivity (see later) then
which gives apparent relaxation frequencies shown in Table II.
Table II. Apparent Relaxation Frequencies for Wires 80 m apart over Pure Ice with Negligible Conductivity and a True Relaxation Frequency of 8.0 kHz
(II) Buried wires (Fig. 3)
The apparent permittivity ϵ app = (ϵ+1)/2 for wires buried within 1 cm of the surface and the apparent relaxation frequency is the same as that of the homogeneous glacier to a good approximation.
Fig. 3. Cole–Cole plots of the real and imaginary parts of apparent permittivity for wires buried beneath the surface of a cold névé
2. Detection of Discontinuities within the Glacier
Because of the uncertainties described in the previous section, we now imagine the wire electrodes to be buried at such a depth in the glacier that the air surface does not affect the capacitance, and we consider instead a surface of discontinuity between upper and lower layers of ice. Let ϵ 1 refer to the surface layer, and ϵ 2 refer to deep ice with the properties tabulated in column 3 of Table I. For d much greater than a,
Taking the radius of the wire to be 4×10−4 m and the separation 80 m as before, then Figure 4 shows a Cole–Cole plot of apparent permittivity for a wet (Greenland) surface and Figure 5 for the cold névé surface (as in France).
Fig. 4. Cole–Cole plots of the real and imaginary parts of apparent permittivity for warm wet ice overlying deep pure dense ice. Superimposed is a plot for deep ice alone
Fig. 5. Cole–Cole plots of the real and imaginary parts of apparent permittivity for cold névé overlying dense warm ice
The result to be noted is that even if the ice layer is only 1 m below the wires, the apparent permittivity is essentially that of the surface layer, for high frequencies. However, at low frequencies the apparent conductivity tends towards the conductivity of the deep ice and relaxation frequencies deduced from either Cole–Cole plots or the variation of conductivity will be wrong.
3. Effect of Conductivity
With a high d.c. conductivity σ, the high-frequency values fall onto an arc which gives a false impression of the static permittivity. Following Reference GränicherGränicher and others (1957) the value obtained is
where
If the maximum value of ϵ″ occurs at a frequency f m then
and the observed value of f m will differ from f r as shown in Table III.
Table III. Frequency f m for Maximum Loss Factor ϵ″ for the Deep Ice in Table 1, but Various Conductivities
For measurements covering a wide frequency range the best value of the relaxation frequency would be determined by plotting the conductivity against frequency and obtaining the frequency for which σ = (σ s+σ ∞)/2 where σ s and σ ∞ are the static and high-frequency limiting values.
4. Interpretation of Gribbon’s Results
The general result of this discussion is that it is possible to obtain information only about the surface layers of the glacier. For Gribbon’s wires lying near the surface and melting into the glacier in places, situations (1) and (II) could occur in parallel. Then we expect the apparent relaxation frequencies to be higher than the true values, and the f m and α values (his figures 6 and 7) to be random, depending on surface configurations.
However, Gribbon’s f m values are lower than currently accepted values for pure ice (Reference Auty and ColeAuty and Cole, 1952) and snow (Reference Ozawa and KuroiwaOzawa and Kuroiwa, 1958) and impurities tend to increase the relaxation frequency (Reference GränicherGränicher and others, 1957) so we suppose that this is due to his method of obtaining f m in the presence of high d.c. conductivity.
High conductivity snow exhibits an increased static permittivity (Reference Ozawa and KuroiwaOzawa and Kuroiwa, 1958) and this cannot be fitted to the Debye equations. However, the calculations given here for ideal snows are altered only slightly, and it may be found that α > 0 at high frequencies, and lower f m values may be obtained than with the ideal Debye snow.
Probably, Gribbon’s figures 3 and 4 do not show pronounced surface effects. If we determine the relaxation frequency by the conductivity method the results are both higher than 15 kHz and a more lengthy analysis suggests that the relaxation frequency is greater than 20 kHz for Gribbon’s figure 5.
J. G. Paren
Scott Polar Research Institute, Cambridge, England 15 September 1967
