Equation 1 is fitted to the data by using finite difference
approximations to the derivatives. Central differences were used to
calculate 
, that is, 
, where 
is the ith value of U for a particular
thermistor. The second spatial derivative 
is approximated by 
, where
is the value of T for thermistor j at a particular time.
The time difference 
used is 1 hour, and the spatial
difference 
is the
thermistor spacing 100mm. The resulting numerical estimates of
and 
are fitted with a least squares straight line, the slope of
which is 
when is
treated as the independent variable.
Values after day 110 are not used, as they are affected by solar
heating. Temperatures above
-5 
C are not used, as models of thermal conductivity suggest
that it varies rapidly with temperature in this range. Over 24,000
points remain available for fitting.
In the usual simple least squares method, the independent (x)
variable is error free. Measurement errors in x bias the fitted
slope downwards (Fuller, 1987), by a factor which is the fractional
relative measurement error in x. We estimate that this factor is the
same for both variables and
, varying from about 10% near the
top of the ice, to 20% near the bottom. We thus fit two least squares
slopes, one with as the x
variable and the other with as the
x variable; the geometric
mean of the two values of k obtained is an unbiased estimate of the
value of k. The uncertainty in this value of k is estimated to be
about 3%. The main source of this uncertainty is the uncertainty in
estimating the measurement errors (that is, in estimating the bias);
the least squares fitting procedure gives 95% confidence limits on
the fitted (biased) k values that are several orders of magnitude
lower than 3%.
A typical set of data and the two fitted least squares straight lines
are plotted in Fig. 2, for the thermistor at 700mm depth. The
residuals of one of these fits are plotted against spatial temperature
gradient in Fig. 3. An increase in variance is apparent for higher
temperature gradients. This is a feature of all thermistors, and is
apparent in Fig. 4, which shows R 
values versus depth, plotted
separately for data with temperature gradients above and below
15 Cm 
, as well as for data with all gradients present. R
is the index of fit (Chatterjee and Price, 1991), and R is the
correlation coefficient. R is a measure of how much of the
variance is accounted for by the linear fit, with a value near one
indicating that most of the variance is accounted for and that the
data is very linear in nature. Lower R values are consistently
seen for the higher temperature gradients, indicating higher variance
in this data.
In Fig. 5 are the geometric mean (unbiased) values of k obtained for
each thermistor, plotted against depth. Circled values have R 
, and are less reliable. Included are values for pure ice from
Yen (1981) for comparison, with an indication of the spread of values
obtained experimentally. Temperatures for the pure ice values were
obtained by averaging the temperature for each thermistor. Also shown
are the fitted unbiased k values obtained when the salinity is taken
to be 6.5 
. The R
values for these fits are shown in Fig. 4, and show increasing
variance as depth increases, due largely to the overall linear
temperature gradients established in the ice, giving smaller
temperature changes near the bottom of the ice, and hence a smaller
signal to noise ratio.
The low value for k found at 200m might be due to the higher salinities typically found at this depth.
Table 1 gives fitted values of k and R for each thermistor, as
plotted in Figs. 4 and 5.
Values obtained for k are generally on or below the ranges reported
for pure ice, consistent with the model predictions of Schwerdtfeger
(1963). However, there is a trend towards increased values of k at
greater depths (that is, higher temperatures) -- the reverse of the
trend for pure ice.
The increased variance seen when 
Cm suggests fitting k values separately for
Cm and Cm . The results are plotted in Fig. 6,
and reveal no significant change in k with temperature gradient.
Circled values have R , and should not be given much weight.
The above results suggest that some mechanism other than conduction is contributing to heat flow through sea ice, and giving nonlinear heat flow effects. These become apparent at higher temperatures (when brine volumes are larger) and at higher temperature gradients (when convective effects might be larger). We suggest that the migration of brine in sea ice is a likely candidate.
The work of Lytle and Ackley (1996) also considers the convective
contributions to heat flow through sea ice. They find that brine
convection occurs at very small temperature gradients, apparently in
contradiction to the present work. However, their study was conducted
on second-year ice, and convection through interconnected brine
channels occurs even at very small temperature gradients when sea ice
is warmed to near -1.8 C. Our study is on first-year ice, which
does not have the initial matrix of last year's ice riddled with many
brine channels found in Lytle and Ackley's study. They also had a
slush layer over the ice, due to surface flooding of the ice and snow
pack, and convection was between this slush layer and the underlying
ocean.