Geoscience Reference
In-Depth Information
Fig. 4.8
Analytic solutions of
the Stefan
'
s model and Zubov
'
s
model, which accounts for the
atmospheric surface layer
buffering and snow cover. The
cases of snow-free ice and
effective snow insulation are
shown
fl
flux from water to the ice is small in lakes with weak winter circulation but in the presence
of signi
cant currents or heating from lake bottom the model can be badly biased.
In semi-empirical modi
cations of the model, the Stefan
'
is coef
cient is replaced with
reduced coef
cient a*,
½ ≤
a*/a
≤
1. The ranges of the ratio a*/a have been suggested as,
e.g., 0.5
0.85 (Nolan 2013). The surface temperature needs to
be normally replaced by air temperature when it is below the freezing point, i.e. the
freezing-degree-days are expressed as
-
0.8 (Ashton 1989) or 0.6
-
S
a
ð
t
Þ
¼
R
t
.
0
max 0
;
T
f
T
a
ð
t
0
Þ
Example 4.4.
(a) Using the air temperature data for Oymakon, East Siberia (the coldest location in the
Northern Hemisphere) from the winter of 1981
-
1982 (the coldest one for this region
in the period 1971
s model provides the ice thickness of 2.7 m (Kirillin
et al. 2012). This is an upper limit for the thickness of seasonal lake ice because the
Stefan
-
2000), Stefan
'
'
s model generally overestimates the ice growth rate.
(b) For a 100-day cold season with the mean air temperature
10
°
C below the freezing
-
point, we have S
a
= 1000
s model gives the ice thickness of
105 cm, which can be reduced in empirical modi
°
C d, and the Stefan
'
cation to 52.5 cm for a* =
½
a.
p
H
(c) Ice growth can reach the bottom when
, where H is the lake depth. Assuming
that the ice salinity is zero, the water salinity develops as S
w
(t)H(t)=S
w
(0)H(0) that
lowers the freezing point. Starting with S
w
(0) = 1
a
‰
and H(0) = 1 m, at H(t)=1cm
the salinity is 100
‰
and the freezing point well below 0
°
C (for sea-water, it would
be
6.1
°
).
-
4.3.2.1 Atmospheric Surface Layer as a Buffer
A snow-free ice sheet is considered. In Stefan
is model it is implicitly assumed that all heat
conducted to the ice surface can be taken to the atmosphere. But this is not necessarily
true. Therefore a more realistic approach is to add an atmospheric surface layer to the
Stefan
'
'
s model
(Barnes 1928; Zubov 1945). Here the linear surface heat balance
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