Geoscience Reference
In-Depth Information
Averaged over the growth season, the heat conduction
equation is written as
W
-
BT
G
is the net longwave flux
F
LW
derived from an exter-
nally specified downwelling flux
F
LW
↓
minus the upwelling
flux
F
LW
↑
given by the Stefan-Boltzmann (SB) equation lin-
earized around 0
°
c:
F
LW↑
=
A
+
BT
G
, with
A
and
B
the value
and derivative of the equation at 0
°
c (320 W m
-2
and 4.6 W
m
-2
k
-1
, respectively).
W
is defined by
W
=
F
LW↓
-
A
so that
the net flux is
F
LW
= F
LW↓
-
(
A
+
BT
G
)
=
W - BT
G
.
the heat conduction equation is complemented by an
annual mean surface energy budget in which the heat loss
during ice growth is balanced by heat gain over the melt
season:
W
−
BT
G
=
kT
G
/
h
.
(1)
Here
h
and
T
G
are the growth season mean ice thickness and
surface temperature,
k
is the thermal conductivity of sea ice
(2 W m
-1
c
-1
), and the conductive heat flux
F
cond
is approxi-
mated as
−
k
z
≈ −
kT
G
/
h
,
where the seasonal average
of (
T/h
) is approximated by the ratio of the seasonal averages
of
T
and
h
(variables and values are also given in table 1).
T
G
/
W
G
(
W
−
BT
G
) +
W
M
S
=
0
.
(2)
Table 1.
Definitions of Variables Used in Text
Variable
S
, the average melt season net surface energy flux, is the sum
of
F
SW
and
F
LW
= F
LW↓
- A
, since melting is assumed to occur
at 0
°
c (here
F
LW↓
refers to the downwelling longwave flux
averaged over the melt season, which is, of course, different
from
F
LW↓
used in the definition of
W
, which is averaged
over the ice growth season). Also, the heat capacity is ig-
nored so that the ice warms up to 0
°
c at the start of the sunlit
portion of the year.
the implications of (1) and (2) can be understood by
plotting contours of constant
S
and
W
in the (
h
,
T
G
) plane, as
shown in Figure 2. For fixed
W
,
F
cond
® 0 as
T
G
®
W / B
since
the net longwave flux from the ice is zero when
T
G
= W / B
.
With a nonzero surface temperature
F
cond
® 0 requires
h
®
∞
since a small net surface flux must be matched by a small
conductive flux, which requires large
h
. (1) and (2) can be
combined to obtain
S
= -
kT
G
/ h
⋅ (τ
G
/
τ
M
)
, so that contours of
constant
S
are straight lines emanating from the origin. the
intersection of
S
and
W
contours at point
A
represents a valid
climate state in which energy balance and flux continuity are
both satisfied.
the dashed curves in figure 2 show the sensitivity of
h
and
T
G
to 20 W m
-2
changes in
W
and
S
. It is clear that
h
is considerably more sensitive to a decrease in
S
(compare
h
values at
A
and
B
) than to an equivalent decrease in
W
(compare
h
at
A
and
C
). the larger sensitivity to
S
can be
understood as a consequence of the fact that
T
= 0 during ice
melt, so that the ice cannot compensate for changes in the
imposed energy flux by adjusting its summer temperature.
Instead, a reduction in
S
must be compensated by a reduction
in the net winter surface flux
W - BT
G
, which must become
smaller as
S
is reduced. As stated above, a small net flux
means a small conductive flux which requires large
h
. Hold-
ing
W
fixed while reducing net flux by δ
S
means moving to
the right along the
W
curve in figure 2, on which
h
increases
without bound as the net flux approaches zero.
on the other hand, if
S
is held fixed while
W
is reduced,
the net winter flux (the left hand side of (1)) must remain
Definition
constant Value
If Any
A
upwelling longwave radia-
tion at 0°c
320 W m
-2
B
derivative of Stefan-
Boltzmann equation
at 0°c
4.6 W m
-2
k
-1
F
LW
,
F
SW
,
and
F
SL
surface net longwave, short-
wave, and turbulent fluxes
upwelling and
downwelling fluxes
F
¯,
F
h
mean sea ice thickness
h
D
sea ice thickness
calculated from surface
fluxes by (4)
k
thermal conductivity of sea
ice
2 W m
-1 °
c
-1
L
latent heat of fusion
3.34 × 10
5
J kg
-1
N
ice growth season
longwave atmospheric opti-
cal depth
3
R
annual mean surface energy
budget residual in W m
-2
S
net surface heat flux
during the melt season
Ŝ
S
+
R
(τ
M
+
τ
G
) / τ
M
T
G
mean surface temperature
for ice growth season
V
®
sea ice velocity
W
net surface heat flux
during ice growth season
a
surface albedo
r
L
∇⋅
V
®
in (6)
γ
r
sea ice density
917 kg m
-3
s
Stefan-Boltzmann
constant
5.67 × 10
-8
W m
-2
k
-4
τ
G
length of ice growth season
8 months
τ
M
length of ice melt season
4 months
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