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The consistency between thickness and flux ranges is not
definitive, given the highly simplified form of the diagnostic
equation. one important feedback mechanism not discussed
in section 2 is the latent heat transport associated with ice
motion. The effective heat flux due to ice transport is given
approximately by Ñ×
L
r
hV
®
, where
L
is the latent heat of fu-
sion, r is ice density, and
V
®
is ice velocity. Assuming that
the ice density and thickness are spatially uniform and ice
motion is independent of thickness, the heating rate becomes
(r
L
Ñ×
V
®
)
h
, and the heat energy added per year is g
h
, where
g = (t
M
+ t
G
)(r
L
Ñ×
V
®
) [
Bitz and Roe
, 2004]. the term g
h
can
be added to the left-hand side of (2), which can then be com-
bined with (1) to produce a modified equation for the varia-
tion of
T
and
h
along an isoline of
S
:
is a need to balance the upwelling surface energy flux dur-
ing ice growth with a conductive flux through the ice that
depends on ice thickness. this dependence means that the
small upwelling flux associated with cold surface tempera-
ture must be accompanied by large ice thickness. Second,
since the melt season temperature is held to the freezing
point, the system cannot regulate its energy gain in summer,
and the melt season gain must be compensated by growth
season loss. thus, low summer insolation must be balanced
by low longwave radiation in winter, leading to very thick
ice. While the equations used to formalize this assessment
are severely simplified, the basic conceptual insight may still
be helpful in understanding the large spread of sea ice in the
20c3M simulations.
the approximately inverse relationship between
Ŝ
and
h
D
in figure 3 is analogous to the derivation of uncertainty in
global climate sensitivity derived by
Roe and Baker
[2007]
[see also
Bitz
, this volume]. In their analysis, the inverse re-
lationship is between climate sensitivity (the change in glo-
bal temperature for a unit change in radiative forcing) and
the factor 1-
(if
, where
(if
is the feedback factor representing the
net effect of climate feedbacks. they show that this inverse
relationship produces a long-tailed distribution in climate
sensitivity for a normally distributed range of
(if
values. the
same derivation applied to the
S
-
h
curve in figure 3 would
show that a large model ensemble with normally distributed
S
values would have positively skewed
h
values, so one
might always expect to see a few models with extreme thick-
ness.
Randall et al.
[2007] note that 20c3M sea ice simula-
tions do not show dramatic improvement over the preceding
generation of climate models despite considerable model
development, and they ascribe the lack of improvement to
persistent problems in the atmosphere and ocean component
models. But our sensitivity analysis suggests a role for in-
trinsic sea ice sensitivity, since a relatively large reduction
in flux uncertainty is required to reduce thickness uncer-
tainty. Hopefully, improvements in physical understanding
and model construction will narrow the range of
S
and
Ŝ
values. However, the close association between
S
and sur-
face temperature poses a challenge to such model improve-
ment, since the implied sea ice-albedo feedback will amplify
errors in the surface energy budget caused by inadequacies in
parameterization.
the potential implications of our analysis for anthro-
pogenic climate change simulations can be understood by
regarding d
F
in (3) as the longwave flux increase due to
anthropogenic greenhouse gas emissions. following the
logic of section 4.1, the primary effect of a greenhouse gas-
induced flux increase is to shift the
Ŝ
value to the right in
figure 3b. the resulting decrease in
h
is nearly proportional
to
h
2
(if
h
µ1/
Ŝ
then d
h
µ-
h
2
d
Ŝ
), so that thick ice thins faster,
kT
+
S
t
M
t
G
h
+ g
h
2
= 0
.
(6)
Instead of the straight lines in figure 2, isolines of
S
are
downturned parabolas, and
h
®¥ only if
T
® - ¥ so the ex-
treme ice thickness discussed in section 2 is avoided. Physi-
cally, extreme thickness is avoided because rapid ice growth
due to low insolation leads to thicker ice which, for the same
ice velocity, means an increase in ice transport.
the effect of sea ice motion cannot be easily determined
from the 20c3M simulation archive, although it is included
implicitly in (4) through the budget residual
R
. the effect of
ice motion is to prevent low
Ŝ
values, since low
S
leads to
thicker ice and more ice export, thus higher
R
compensates
lower
S
. despite this compensation, low values of
S
do oc-
cur in the ensemble. furthermore, figure 6 shows that the
dominant balance in most models is between longwave and
shortwave fluxes, leaving a smaller role for motion. For the
two models in which longwave loss dramatically exceeds
shortwave gain, sensible heat flux makes a stronger contri-
bution to the budget deficit than the residual term. The role
of ice export was examined by
Hibler and Hutchings
[2002]
(discussed by
Bitz and Roe
[2004]). they found that the as-
sumption of thickness-independent motion is reasonable for
h
< 2 m but fails for thicker ice, as strong thick ice resists
motion induced by wind stress.
discussions of uncertainty in sea ice simulations such as
those of
Randall et al.
[2007],
DeWeaver
[2007], and
Kattsov
et al.
[2005] typically address ensemble spread in sea ice by
discussing uncertainties in the numerous processes which
affect the ice, like cloud radiative properties, surface albedo,
and the stability of the upper ocean. the analysis presented
here is intended to complement such discussions by consid-
ering the inherent sensitivity of sea ice to surface flux er-
rors. In the simple diagnostic framework, the sensitivity can
be understood as a consequence of two factors: first, there
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