1. Introduction
Theoretical studies of heat transmission through floating covers of even pure ice are considerably complicated by the continuous formation of ice at the ice-water interface. In the absence of warm water currents below, as long as a temperature gradient exists at the lower boundary, the ice must grow. The three equations of basic importance in specifying the thermal conditions existing in an ice cover of uniform thermal conductivity k, specific heat c, density p, and thickness h at time t are
where θ is the temperature, J is the heat flux at a depth 0 ≤ x ≤ h, L is the latent heat of formation, and dh/dt is the rate of advance of the lower boundary.
Numerous works, e.g. Reference KolesnikovKolesnikov (1958), which have appeared since Reference StefanStefan’s (1891) classical solutions to the ice-growth problem, have, on the whole, resulted in unwieldy expressions requiring many unsatisfactory approximations to permit practical applications. A clear quantitative (and qualitative) picture of thermal occurrences in a floating ice cover, especially if non-uniform, is best given by electrical analogue methods. This technique has been discussed by Reference SchwerdtfegerSchwerdtfeger (1964), in a design which involves the usual electrical analogues for equations (1.1) and (1.2), and a novel automatic switching device to simulate freezing and hence to provide an adjustable electrical analogue for latent heat and equation (1.3).
When an analogue computer of the above type is not available or when short calculations are required, it is still desirable to have suitable analytical procedures at hand. Reference StefanStefan (1891) derived a simple solution for ice of negligible specific heat, which although infrequently acknowledged, is often used. It states that the ice thicknesses h 1 and h 2 before and after a time interval t 2−t 1, during which the surface temperature is given by θ 0, are given by
Stefan himself discussed the two serious limitations of this equation which both result from neglecting the specific heat of the ice. These are that no allowance is made for the variable energy content of an ice cover nor for the time taken for a change in surface temperature to modify the temperature gradient at the ice-water interface. The assumption of zero specific heat of course implies infinite thermal diffusivity, k = k/pc, under which conditions equation (1.1) shows all temperature changes to occur instantaneously throughout the ice cover, which in turn results in a constant uniform temperature gradient. Observations reported by Reference BarnesBarnes (1928, p. 28) to support his simple, Stefan-type equation actually show that the effect of finite thermal diffusivity is important even in the history of ice whose thickness is only of the order of a millimetre. Barnes made rapid measurements on St. Lawrence River ice formed in areas cleared by icebreakers and ferries. Invariably, the theoretically calculated ice thickness exceeded the observed value and, although no comment was given, this error progressively increased for greater ice thicknesses.
As a result of the high values for the specific heat of sea ice (Reference SchwerdtfegerSchwerdtfeger, 1963[b]) especially at temperatures near to the freezing point, Stefan’s growth solution for ice of constant specific heat c:
where θ 1, and θ 2 are the ice surface temperatures at times t 1 and t 2 respectively, and the other symbols are as in equation (1.4), is not sufficiently accurate in general. In the original paper, equation (1.5) is merely quoted, and appears to be partly empirical.
Because Stefan’s general equation is inapplicable to ice in general, particularly sea ice to which most work on floating ice is directed, this paper will discuss the energy content of an inhomogeneous ice cover and the rate of transfer of heat through this medium, with a view of deriving a two-stage correction to Stefan’s basic equation (1.4).
2. Changes of Heat Content of A Growing Ice Cover
Figure 1 shows the temperature distribution in an ice cover initially of thickness h, before and after an additional thin layer of thickness Δh has been formed, assuming the idealized case of continued uniformity of temperature gradient, which would occur only in the case of very slow growth. It is seen that the temperature of the ice at a distance x below the surface is given by:
Temperature changes during ice growth
where θ 0 the ice surface temperature, and θ F, the freezing temperature, that of the ice-water interface. The change in temperature of the ice at depth x after freezing of the layer Δh is
The change in heat content of a volume Δx, in a cylinder of unit cross-sectional area, at a depth x is thus given by:
where c s x is the specific heat of the sea ice at a depth x. The specific heat of sea ice, or any ice frozen from water containing soluble impurities, was shown by the author (Reference SchwerdtfegerSchwerdtfeger, 1963 [b]) to be given by
where θ and σ are the temperature and salinity of the ice, α is a constant relating temperature to the equilibrium salinity of the concentrated mother solution, L i and c i are the latent and specific heats of pure ice and c w is the specific heat of water. Substitution of equation (2.1) in (2.4) gives the specific heat c s x as a function of x, so that ΔQ i can be integrated through the ice cover to give the total change in heat content for unit increase in ice thickness,
The integration leads to
The term
At the same time, the heat involved in the freezing of new ice only, is simply given by
where L s is the latent heat of sea ice. The author has previously shown (1963[b]) that:
where s is the salt content of the parent sea-water. Hence
It can immediately be seen that as θ 0→θ F, Q i/Q F→o, and as θ 0→−∞, Q i/Q F→+∞ substantiating Stefan’s observation that equation (1.4) holds most accurately for ice surface temperatures near to the freezing point.
Using values for the specific heat calculated by the author (1963[b]), Q i/Q F, the ratio of heat involved in cooling the ice cover to that connected with simultaneous freezing, may be calculated. The graph in Figure 2 shows these values as a function of salinity and temperature for ice frozen from sea-water having a freezing point of −1.8° C.
Corrections to Stefan’s simple ice-growth equation. Ratio of contributions to surface heat flux from ice-cover cooling Qi to that for new ice forming QF as a function of we surface temperature for various salinities
It is clear from Figure 2 that the latent heat due to freezing at the bottom of the ice cover is comparable in magnitude to the heat content change taking place in the cover as a whole. The first step towards an improved form of equation (1.4) must therefore be to replace the latent heat L by its“effective” value
3. The Rate of Transfer of Energy Through An Ice Cover
Solution of the thermal diffusion equation (1.1) shows that the time taken for a given degree of completion of a temperature change initiated by a sudden discontinuity in surface temperature is directly proportional to the square of the ice thickness, and inversely so to the diffusivity, i.e.
The author has shown (Reference SchwerdtfegerSchwerdtfeger, unpublished) that a similar relation holds for an ice cover whose surface temperature is continually changing with time.
On considering sea ice, it is difficult to include the diffusivity changes in any practical application, as the time lags in temperature change are most conveniently determined from maxima or minima in the growth rate and temperature curves. This means that a significant change in the diffusivity occurs during a single interval of observation. However, the mean value of the diffusivity in the time interval between a minimum and the next maximum, or between a maximum and a minimum, in the ice temperature remains approximately constant, unless the temperatures averaged over the same times themselves show a marked change. Hence equation (3.1) is written:
The general application of equation (3.2) requires an expression for the mean thickness h over the time interval t θ . It is usually possible to write:
where h o is the initial ice thicknessFootnote * and dh/dt is the mean rate of increase during the time t θ . Therefore
In order to obtain values for χ, the set of observations of temperature, heat flux and ice growth shown in Table 1 and Figures 3 and 4, were obtained on annual sea ice in Hudson Bay near Churchill, Manitoba, between January and May 1961. As well as showing rate of ice growth, the graph in Figure 3 also shows the heat flux at a depth of 20 cm. in the ice cover. The time axes in Figures 3 and 4 use single capital abbreviations for the months. Although the ice growth curve necessarily lacks detail, owing to the limited number of observations, the main features are visible and similar on both curves. The predominant minimum and maximum in the heat flux, occurring between 8 February and 9 February, and 22 February and 23 February respectively, are reproduced particularly clearly on the growth curve, but other minor trends also appear to be followed. The ratios of the two minima and the two maxima are approximately 80 and 70 cal. cm.−3 respectively. As these values are reasonable approximations to the latent heat of formation plus cooling of the existing ice, it is reasonable to assume that the information from the graph is a suitable basis for further deduction.
Ice thickness and Rate of Growth at Button Bay
Ice surface heal flux and growth rate of sea ice in Hudson Bay near Churchill, Manitoba, January to April 1961
Ice temperatures at various depths in degrees Centigrade below the sea-water temperature of −1.65°C. Data from Hudson Bay near Churchill, Manitoba, January to April 1961
Tables II and III below, summarize the method of calculating χ and time delays from the observations, reports on which have appeared by Reference Schwerdtfeger and PounderSchwerdtfeger and Pounder (1963) and Reference SchwerdtfegerSchwerdtfeger (1963 [a]), the latter describing the heat-flux measurements.
Summary of Data on Effect of Surface Flux on Growth Rate
In Table II we are thus led to a numerical value for what might be termed the“lag coefficient” χ = (1.7±0.3)×10−3 days cm.−2, which is applied in Table III.
When the surface temperature, rather than the heat flux at a depth of 20 cm., was compared with the rate of growth of ice, χ was found to be equal to 1.2±0.2, and the values for t θ calculated by means of equation (3.4) were found to be 13±2 and 18±3 days respectively. The lower value of χ in this case is due to the higher value of the diffusivity when the ice above the flux meter influenced the mean value. The two alternatives for calculating t θ are certainly in satisfactory accord.
The above conclusions are supported by the records showing ice temperature as a function of depth and time. In Figure 4 the ice temperatures in the Hudson Bay ice cover being discussed are shown at five points separated by 25 cm. along a vertical axis in the cover. It is seen that the time for similar characteristics became longer between the lower levels where the temperatures are higher and the mean diffusivity is lower. Similarly the time lag for transmission of’ a minimum in temperature was greater than for a maximum for the same reason. This is of course also why the mean value for χ was less for the entire cover than for the portion below the flux meter at the 20 cm. level. More extensive observations would permit the determination of the lag coefficient for an ice cover as a function of surface temperature. In general, of course, mean surface temperature and hence the lag coefficient tend to increase toward the end of the winter.
4. Conducted Surface Heat Flux and the Latent and Specific Heats of Sea Ice
The same Hudson Bay observations provide empirical estimates for the latent and specific heats of sea ice for comparison with theoretical values such as that given by equation (2.7). The daily heat-flux totals at the 20 cm. level are shown in Figure 3. Table III shows that the flux measurements on 30 January and 19 March are reflected in terms of ice growth on 11 February and 6 April respectively. A graph of the ice thickness such as may be obtained from Table 1, shows that the heat lost between the former dates caused the growth of 21.3 cm. of’ new ice and the cooling of an average of 97.8 cm. of established ice below the flux meter. The total flux of heat lost between 30 January and 19 May may be calculated as 1514.5 cal. cm.−2. These quantities satisfy Stefan’s simple ice growth equation with an effective latent heat L = 77.1 cal. cm.−3, or for a density of 0.915 g. cm.−3, L = 77.6 cal. g.−1.
This experimental result may now be compared to theoretically calculated values. A salinity profile, reported by the author (1963[b]), indicated a mean salinity of 5‰ for the ice cover. Equation (2.7) shows the latent heat of formation of this ice from sea-water of 30‰ salinity to be 65.8 cal. g.−1. Figure 2 indicates that 10.6 cal. g.−1 were lost by cooling of the older established ice, whose average surface temperature was −9.0° C. These two thermal quantities total 76.4 cal. g.−1, being in good agreement with the experimentally determined 77.6 cal. g.−1. Assuming that the mean salinity of the cover was estimated to the nearest part per thousand, a little less than 5 per cent uncertainty is attached to the theoretically calculated quantity.
5. Ice Surface, Temperature, Growth and Thermal Conductivity
The necessary modifications to enable a practical application of Stefan’s simple ice growth equation (1.4). have been separately tested in the preceding sections. The full procedure will now be illustrated for the same Hudson Bay data which provided some of the necessary empirical information. In calculating the freezing exposure, i.e.
The fully modified form of Stefan’s equation (1.4) is
In the case of the data at hand
From these figures we obtain a thermal conductivity of k = 4.87×10−3 cal. cm.−1 sec.−1°C.−1 for the sea-ice cover. The chief sources of error are in the effective latent heat terni, some 5 per cent, and the uncertainty in the temperature transmission time, which leads to a possible 5 per cent error in the freezing exposure. The value of the thermal conductivity is thus only certain to within about 7 per cent.
It may be noted that the thermal conductivity of sea ice of 5‰ salinity and a temperature of −5.5° C. (the mean temperature of the cover) has been given as 4.7×10−3 c.g.s. units on theoretical grounds by the author (Reference SchwerdtfegerSchwerdtfeger, 1963[b]). This is well within the limits of the experimental determinations.
Ice Surface Temperature
6. Conclusion
The two examples given in Sections 4 and 5 show to what degree the thermal properties of an inhomogeneous ice cover can be accounted for by relatively simple techniques. It should be stressed that this discussion has taken no account of the fact that the heat content of upper and lower ice layers respond at different times to a change in surface temperature. Because of this, the modified Stefan equation (5.1) is more accurate when covering longer periods of time. Although it would be possible to calculate the time distribution of heat energy from different depths, as distinct from that of the actual freezing process, more accurate information on the temperature and specific-heat profiles than would normally be available, or calculable, would be required. Given such information there would be ample jusgification for the use of an analogue computer. In many practical cases however the modified Stefan equation (5.1) will provide an adequate answer.
Acknowledgements
A number of discussions with Dr. U. Radok have materially influenced the form of this paper.
