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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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