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and therefore net radiation becomes stronger than the degree-days in the ice breakup. For
comparison, river ice breakup is dominated by melting and erosion at the bottom of the ice
(Shen and Yapa 1985).
A special case of ice decay is sublimation. Even though latent heat
fluxes are usually
small in cold environment, in arid cold climate the situation can be different (e.g., Huang
et al. 2012). Ice decay due to sublimation is
fl
dh
dt
¼
Q
e
qð
L
f
þ
L
e
Þ
¼
q
q
C
e
q
a
q
0
ð
Þ
U
a
ð
4
:
54
Þ
10
−
3
. Thus for C
e
*
10
−
3
,
The saturation speci
c humidity at 0
°
C is 3.8
×
10 m s
−
1
and relative humidity of 50 %, the sublimation rate is 2 mm d
−
1
. For
50 days, the loss of ice by sublimation would be
U
a
*
10 cm.
*
4.3.6 Multi-year Ice
The analytic growth and melting models can be combined for the equilibrium multiyear
ice thickness. The classical Stefan
'
is model is used for ice growth, and the summer melting
is taken as
D
h ¼ Q
0
þ
Q
T
þ
Q
w
t
s
, where t
s
is the length of summer;
'
summer
'
is de
ned
as the period when the net heat
flux is positive. Assuming that not all ice melts in summer,
the thickness of ice after the nth summer is
fl
q
h
n
1
þ
aS
h
n
¼
D
h
;
n
1
ð
4
:
55
Þ
constant (inde-
pendent of ice thickness). At equilibrium, h
n+1
= h
n
= h
e
, where h
e
is the equilibrium
thickness,
The growth of ice depends on ice thickness, but we may take
ʔ
h
≈
h
e
¼
h
1
ðD
h
Þ
2
2
D
h
ð
4
:
56
Þ
The condition of multi-year ice is trivial: h
1
>
ʔ
h. When h
1
− ʔ
h is small, h
e
≈
h
1
− ʔ
h,
h
e
h
1
=ð
2
D
h
Þ
and when h
1
≫ ʔ
h, we have
.
Example 4.10.
m, we have h
e
= 3.75 m, but changing the
summer melt to 1 m gives us h
e
= 1.5 m, illustrating the high sensitivity of the equilibrium
thickness to summer melt. Dropping h
1
to 1 m would give h
e
= 0.75. Net radiation can be
easily increased by decreasing the summer albedo, while the sensitivity of the winter
growth to freezing-degree-days is
d
h ¼
2
a
2
=
h
e
For h
1
= 2 m and
ʔ
h =
½
ð
Þd
S
. E.g., if h
e
*
3 m, then
ʴ
h
*
3.7 cm
°
·
°
for a change of 100
C
day in the freezing-degree days, corresponding to a 0.5
C
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