Geoscience Reference
In-Depth Information
4.3.2 Congelation Ice
Stefan
s model considers a congelation ice sheet (Fig.
4.7
). Latent heat released in the
growth at the ice bottom is conducted away through the ice. The assumptions are the
following: (i) Thermal inertia of ice is ignored, i.e., quasi-steady approach; (ii) Penetration
of solar radiation into the ice is ignored, (iii) There is no heat
'
flux from water to ice; and
(iv) The surface temperature is a known function of time, T
0
= T
0
(t). The assumptions
(i
fl
le is linear (see Eq.
4.24
), and with the
assumption (iii) ice growth is obtained from the bottom boundary condition (Eq.
4.25c
)as
a simple separable equation:
(i-ii) mean that
the ice temperature pro
-
q
L
f
dh
dt
¼ k
T
f
T
0
ð
4
:
37
Þ
h
In this quasi-steady model the heat conduction immediately adjusts to the surface
temperature. With the assumption (iv), Eq. (
4.37
) integrates into (see also Fig.
4.8
):
p
h
2
ð
0
Þþ
a
2
S
ð
t
Þ
h
ðÞ
¼
s
2k
q
L
f
Z
t
dt
0
ð
4
:
38
Þ
T
f
T
0
ð
t
0
Þ
a ¼
;
S ¼
0
where a is a
cient, which depends on the thermal properties of ice, and S is the
sum of freezing-degree-days. The theoretical value of a is 3.3 cm
fixed coef
C
−
1/2
day
−
1/2
; this
°
coef
cient will be called Stefan
'
s coef
cient below.
is model is the assumption (iv), since the surface tem-
perature is rarely known. Especially, when there is snow on the ice, the air temperature
can be quite different from the ice surface temperature. In principle, monitoring the surface
temperature would make it possible to produce a good estimator for the ice thickness
evolution. Another weakness is that the assumptions (i)
The weakest point in Stefan
'
(i)-(iii) all tend to bias the solution
-
upward (Lepp
ranta 1993). Ignoring thermal inertia produces a little more growth com-
pared with nature when the air temperature decreases toward middle winter. Solar radi-
ation is weak in the ice growth season except in cold arid mid-latitude climate. The heat
ä
Fig. 4.7
A schematic picture of Stefan
'
is model. Latent heat is released at the bottom in ice growth,
and this heat is conducted through the ice to the atmosphere. For the conduction, the surface
temperature must be below the freezing point
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