Geoscience Reference
In-Depth Information
q
L
f
dh
dt
¼ k
T
f
T
a
b
0
h
þ b
h
s
þ
b
1
; b
¼
k
ð
4
:
43
Þ
k
s
Compared with Eq. (
4.40
), the presence of snow cover has brought the term
ʲ
h
s
to the
denominator. The parameter
ʲ
*
10 describes the insulation ef
ciency of snow as
compared with ice. If
ʲ
h
s
= constant, the solution (Eq.
4.41
) applies with b
1
replaced by
b
1
+
h
s
. However, in general snow thickness depends on time, and Eq. (
4.42
) cannot be
analytically solved; in fact, also the snow conductivity (and therefore
ʲ
ʲ
) depends on time
determined by metamorphic processes in the snow cover. Since
ʲ ≫
1, even a thin snow
cover dominates over the atmospheric surface layer buffering.
Example 4.5.
Let us add a constant snow insulation thickness h
s
*=
ʲ
h
s
to the Stefan
'
s
p
. It is easy to see that the ice thickness
model. The snow-free Stefan solution is
h
0
¼ a
in the presence of snow is h = h
0
−
h
s
* for h
s
*
≪
h
0
,andh
→
½
h
0
for h
s
*
h
0
.
→
c case of snow-covered ice is obtained by assuming that snow accumulation is
proportional to the ice thickness. Snow and ice thickness are in general positively cor-
related, and although the proportionality is a quite limited assumption, it provides an
overall view of the in
A speci
uence of snow cover during the whole ice growth season. The
solution (
4.41
) can be utilized. Denoting h
s
=
fl
ʻ
h, the denominator in Eq. (
4.43
) becomes
cient a
2
is divided by (1 +
(1 +
ʻʲ
)h + b
1
, and, consequently, in the solution the coef
ʻʲ
):
s
h
2
ðÞþ
a
2
þ
b
1
h
ðÞ
¼
1
þ kb
S
a
ðÞ
b
0
t
½
b
1
ð
4
:
44
Þ
Requiring that only congelation ice forms, we must have
k
.
1
=
3
(no
fl
flooding), and
since
. Therefore ice thickness evolution over the winter
season is reduced by a factor of about
ʲ
*
10, we have
1
þ kb
.
4
½
when the insulation of snow is most ef
cient
(Fig.
4.8
). This result also agrees with the fact that empirical modi
cations a* of the
growth law coef
cient are at lowest about a*
a/2.
*
4.3.2.3 Heat Flux from the Lake Body
The heat
flux from the water body can be also examined using analytic tools (Barnes
1928; Lepp
fl
ranta 1993). It is independent of the ice except that in the melting season,
when the solar heating of the water depends on the snow and ice thickness. In analytic
modelling, this
ä
fl
flux is given as a prescribed function of time. The mean water
ice heat
-
5Wm
−
2
; e.g., 2
5Wm
−
2
in deep Lake P
fl
rvi (Shirasawa
et al. 2006) based on measurements and less than 1 W m
−
2
in shallow Lake Vanajavesi
based on model tuning (Yang et al. 2012); both lakes are from southern Finland. The
in
flux has been estimated as 1
ää
j
ä
-
-
flux on the annual maximum ice thickness was up to about 5 cm. The
source of the heat is in the water body itself, bottom sediments, or sun. The
fl
uence of the heat
fl
rst and
second sources are signi
cant especially in the early ice season, while the third one can be
large in spring and summer.
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