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temperature. Considering the linearized heat
fl
ux k
0
+ k
1
(T
a
−
T
f
), in April
May k
0
-
increases from 4 to 100 W m
−
2
and k
1
*
C
−
1
is rather stable (Table
4.1
).
Consequently, the direct physical link of melting to degree-day is weak, but it is clear that
as the spring progresses, air temperature tends to increase and ice thickness tends to
decrease, and the correlation between these two quantities follow.
As the
20 W m
−
2
°
first model, all melting can be included in the ice thickness de
ned as ice
volume per unit area. We have, from Eq. (
4.36a
,
b
):
Q
T
þ
Q
w
0
d
dt
¼
q
L
f
Q
0
þ
1
e
j
h
½
ð
1
m
Þ
h
ð
4
:
52
Þ
50 W m
−
2
, albedo must fall
below 0.75 for the melting season to begin. Using the linear approximation for the surface
heat
200 W m
−
2
and the other terms sum to
Since Q
s
*
* -
fl
flux, we have the integrated form
Z
t
Q
T
þ
k
1
T
a
T
f
þ
Q
w
dt
0
h ¼ h
0
q
L
f
k
0
þ
1
e
j
h
ð
4
:
53
Þ
0
where h
0
is the ice thickness in the beginning of the melting period. In the melting season
the net radiation is increasing strongly, and therefore k
0
and Q
T
must be taken as functions
of time. The
first part is the integrated radiative energy, while the second part gives the
positive-degree-day formula, which alone is often used for simple modelling of melting.
Example 4.9.
Equation (
4.53
) can be simpli
ed for an approximate model. If there are no
signi
cant heat sources in the water body or sediments, we can assume QT
w
*
Q
T
exp
(
-
ʺ
h) and de
ne k
0
*=k
0
*+Q
T
. Also Tf
f
=0
°
C can be assumed. Thus
Z
t
dt
0
h ¼ h
0
q
L
f
k
0
þ
k
1
T
a
0
It is seen that melting consists of two factors, one dominated by net radiation and one
by degree-days. We can estimate that in the melting season (take April in southern
Finland) k
0
*
25 W m
−
2
and k
1
*
15 W m
−
2
C
−
1
. Taking T
a
*
0, Q
T
*
°
3
°
C, we have
2.0 cm day
−
1
h
0
−
25 days. However,
in the melting season both net radiation and air temperature steadily increase. If this
increase is nonlinear, the estimated length of the melting period will be affected. Non-
linearity is particularly true in the case of solar radiation, since both incoming solar
radiation and albedo increase with the time (see Lepp
h
×
t, and thus 50 cm thick ice would melt in
*
*
ä
ranta 2014). From the air tem-
C
−
1
. In high Arctic
perature term we obtain the degree-day coef
cient as 0.57 cm
°
C
−
1
Canadian lakes, the degree-day coef
cient is around 1.5 cm
°
(Mueller et al. 2009),
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