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interior at depth depending on the light attenuation distance and heat conduction, and then
there is, due to a positive feedback in that diffusion is slower in water than in ice, drop in
the daily conduction distance to 11 cm.
Consider the heat content of the upper layer of depth H
ʺ
−
1
+ d), taken as the
representative depth of supraglacial lakes. In the beginning of the summer, the heat
content above 0
2(
*
°
C liquid water reference is U
¼
ˁ
(cT
-
z¼H
dU
dx
¼
k
@
T
Q
0
þ
Q
T
þ
ð
6
:
8
Þ
@
z
where the last term is the bottom heat
flux, denoted below by Q
b
. The summer energy gain
is used for increasing the temperature and eventual melting. The main part of the heat gain
is due to the penetration of solar radiation into the ice. The surface absorption of solar
radiation is to a large degree compensated by heat loss due to net terrestrial radiation and
sublimation; the bottom heat
fl
2Wm
−
2
, corresponding to a
fl
flux is small and stable, Q
b
* -
Cm
−
1
.
During the time period t, the vertically integrated melt becomes
temperature gradient of
1
°
-
t
Q
0
þ
Q
b
þ
Q
T
cTH
L
f
H
þ
ð
:
Þ
¼
6
9
q
L
f
where the overbar stands for ti
me
av
er
aging
.
The criterion whether a supraglacial lake
forms is thus H
[
0
, or, since Q
0
þ
Q
b
Q
T
, we have an approximate criterion
Q
T
t
q
T
Tmin
[
ð
6
:
10
Þ
cH
100 W m
−
2
,
where T
Tmin
is the minimum monthly mean winter temperature. With QT
T
*
ʺ
−
1
t
*
60 days and
*
1 m, we have H
*
7 m and T
Tmin
>
-
38.8
°
C.
Example 6.4
. In the Aboa station, western Dronning Maud Land, the winter temperature of
the surface layer of the ice sheet is
7 m, solar heating by 100 Wm
−
2
in
20
°
C. Taking H ¼
-
60 days gives melt water equivalent of H
. This is approximately as observed in
the vicinity of the station. Further south 200 km, in the Svea Research Station, the annual
mean air temperature is about
¼ 0
:
83 m
°
é
n 1994), and assuming that the
amplitude of the annual cycle is there similar to that in Aboa, T
Tmin
would lie in the range
from
-
20
C (Isaksson and Karl
C in Svea, high enough to satisfy the criterion H
.
Next, consider the depth scale of the lake in more detail. The distribution of solar
energy with respect to depth depends on the transparency of the medium. At depth z, the
solar power absorption is QTκ
T
ʺ
30 to
25
°
[
0
-
-
H
j
1
, i.e. the optical thick-
ness, as one natural depth scale. Furthermore, in time t the porosity pro
exp(
-
ʺ
z), which suggests
le
m
¼
mð
z
Þ
can be
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