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Figure 2.
Solid curves are isolines of
W
and
S
in the (
h,T
) plane derived from equations (1) and (2) for
W
= -162 W m
-2
and
S
= 44 W m
-2
assuming that the ice melt season (MJJA) is half as long as the ice growth season (SondJfMA). the
intersection of the isolines at
A
represents a valid climate state satisfying energy balance and surface flux continuity.
dashed curve through
C
is the isoline for
W
= -182 W m
-2
. dashed line through
B
is the isoline for
S
= 24 W m
-2
.
the same to satisfy (2). unlike the
S
reduction case, winter-
time temperature can adjust to the reduction in
W
, and the
increase in
h
is just enough to keep the same conductive flux
despite the lower temperature. the required increase in
h
is
proportional to the decrease in
T
, δ
h / h =
δ
T / T
, so no singu-
larity is encountered when
h
increases to balance a decrease
in
W
.
In reality,
F
LW↓
is strongly dependent on
T
G
, as longwave
be found using (2) with the above formula for
W
, resulting
in a sensitivity of
W
to
S
. Alternatively, (1) and (2) can be
recast in terms of
F
E
and
n
= (
N
+ 2)/
N
to express the win-
tertime net flux as
F
LW
=
F
E
- (
A
+
BT
G
)/
n
º
W
~
-
BT
G
/
n
.
W
~
-
BT
G
/
n
can then be substituted for
W
-
BT
G
in (1) and (2),
with
W
~
= -78 W m
-2
(the value used by t92) when
W
=
-162 W m
-2
as in figure 2.) using
N
= 3 as given by t92 (for
the winter season) and t
G
= 2t
M
, since melting typically oc-
curs from May to August, we have d
W
= 0.75d
S
. taking this
feedback into account, the
h
value for a 20 W m
-2
decrease
in
S
lies between points
B
and
D
on the dashed line in figure
2, an increase which does not qualitatively change the results
of the analysis.
radiation emitted by the surface warms the overlying atmos-
phere and increases its longwave emission. this dependence
is documented by
Key et al.
[1996] and used by
Lindsay
[1998] (hereinafter referred to as l98) to infer
F
LW↓
from
T
G
. the implied dependence of
W
on
T
G
means that a reduc-
tion in
S
leads to a lower value of
W
, which leads to a larger
increase in
h
for a decrease in
S
than would otherwise occur.
the dependence of
W
on
T
G
is incorporated into the version
of (1) and (2) presented by t92 through a simple atmosphere
in radiative equilibrium with the underlying surface. In this
atmosphere, the surface downwelling longwave flux is
F
LW↓
=
F
E
+
(
N
3. ExPEctEd tHIcknESS SPrEAd duE to
lonGWAVE flux ErrorS
the sensitivity of
h
to errors in surface energy fluxes can
be examined by solving (1) and (2) for
h
assuming errors of
order d
F
in the annual mean surface energy flux:
N
+ 2
)
F
LW↑
, an externally specified portion
F
E
plus a
contribution due to the upwelling surface flux in which
N
is the longwave optical depth of the atmosphere. using
this formula the change in
W
for an imposed change in
S
is
δ
W =
(
N
/ 2)(τ
M
/
τ
G
)δS. (this result is obtained by setting
W
=
F
E
+
N
t
G
t
M
h
=
-
k
·
W
(
1 + d
F
/
W
)
S
(
1 + d
F
/
S
)
B
·
1 +
.
(3)
¾
N
+2
(
A
+
BT
G
) -
A
, from which the sensitivity of
W
to
T
G
can be determined. the sensitivity of
T
G
to
S
can then
In a multimodel ensemble of climate simulations, model er-
rors in energy fluxes contribute to the range d
F
of model-
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