Geoscience Reference
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where
ʴ
is declination,
ʵ
=23
°
27
′
is the inclination of ecliptic,
ˑ
is year angle starting from
January 1st (0
ˑ
0
is its value at the spring equinox. Assuming that the net
terrestrial radiation is constant, zero upcrossing of net radiation takes place at a
≤ ˑ ≤
2
ˀ
), and
fixed solar
zenith angle. The rate of change of declination is
365d
180
d
#
¼
360
d
d
dt
¼
2
p
p
d
d
# #ð Þ
1
sin
e
cos
365d
q
ð
4
:
34
Þ
2
½
sin
e
sin
ð# #
0
Þ
This equation gives the change of declination in degrees per day.
Example 4.2.
Assume that the zero upcrossing of the net radiation balance is at the time of
spring equinox. Then
day
−
1
. When the latitude
changes by 3.9
°
, Eqs. (
4.33
and
4.34
) show that the day of zero upcrossing of net radiation
balance changes by 10 days. At
ˑ − ˑ
0
= 0, and we have d
ʴ
/dt = 0.39
°
ˑ − ˑ
0
= 50 days, May 10th, the change has decreased to
day
−
1
.
Melting takes place at the top and bottom boundaries and in the interior of the ice sheet.
According to recent
0.27
°
ranta et al. 2010), boundary
melting and surface melting are of the same magnitude and the exact levels depend on the
weather conditions during the melting season. Due to terrestrial radiation losses, i.e.
the cool skin phenomenon, at the top surface, it is common to have QT
o
<0<Q
o
+ Q
T
in
the melting season. Solar radiation warms the water beneath the ice and a part of this will
return to ice bottom to add on the bottom melting.
Typical to the melting season is the strong heterogeneity, which appears as a patchy
surface with snow, bare ice and wet spots (Fig.
4.2
). The patchiness develops due to
positive feedback of melting to albedo: once albedo starts to decrease, melting further
lowers the albedo and speeds up the decay process. Also, in shallow areas, the water
beneath ice warms faster and melts ice from the bottom, as observed in melting starting
from lake shorelines. Thus, melting of ice is a vertical
field data (Jakkila et al. 2009; Lepp
ä
horizontal process.
Top and bottom melting decrease the thickness of ice, while internal melting increases
the porosity of the ice. Therefore, a melting lake ice sheet needs to be described by the
evolution of its thickness, temperature and porosity, {h = h(t), T = T(z, t),
-
ν
ν
=
(z, t)}. For
freshwater ice, the melting of ice obtained from:
z¼h
q
L
if
dh
0
Q
w
þ
k
@
T
0
dt
¼
Q
ð
4
:
35a
Þ
@
z
z¼0
;
0
0
¼ max Q
0
þ
k
@
T
0
0
0
¼ 0
;
if T
0
¼ 0
C
;
Q
if T
0
\
0
C
Q
;
ð
4
:
35b
Þ
@
z
Q
I
¼
@
@
z
k
@
T
@
z
Q
T
e
Kz
ð
4
:
35c
Þ
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