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of the true f - for the CMIP3 models because CCSM3 has
among the highest Arctic climate sensitivity of any CMIP3
model [ Bitz et al. , 2008].
Figure 3 shows examples of distributions from equations
(9) and (10) that arise from doubling CO 2 . The parameters
used for the distribution of DT T (D T o = 1.2°C, f - = 0.65, and
s f = 0.13) are derived from recent climate models as dis-
cussed by Roe and Baker [2007]. The parameters used for
the distributions of D h are f 0.21 , estimated from CCSM3
(see section 3.3); ' h o 1 m, chosen to give a peak in the
distributions at a little over 1 m; and sf f = 0.1 for the nar-
rowest and s f = 0.21 for the slightly broader distribution.
I do not know the correct values for f - and s f that repre-
sent the mean and uncertainty of the influence of ice-albedo
feedback on ice thickness from current models. My esti-
mate of uncertainty in f from just one model is bound to be
much smaller than the range of f across models. Presum-
ably, the main factors that give rise to different values of f
across models are difference between open ocean and sea
ice albedos and how the ocean-ice heat flux is partitioned
between lateral and basal melt. However, Figure 3 shows
that even a very large sf f gives a narrow distribution for D h .
5. INFLUENCE OF THE MEAN STATE ON
UNCERTAINTY IN FUTURE THICKNESS
Because the present-day thickness north of 70°N in the
CMIP3 model differs by more than a factor of 3 (see Fig-
ure 1), an estimate of the uncertainty caused by errors in
the mean state is in order. Sea ice is stabilized primarily by
the inverse relation between net sea ice growth and thick-
ness, which Bitz and Roe [2004] called the growth-thickness
feedback process. On an annual mean basis and provided
the climate conditions are not too anomalous, sea ice experi-
ences net melt (growth) when the ice exceeds (falls below)
its equilibrium thickness. This leads to an adjustment pro-
cess, which was described by Untersteiner [1961, 1964],
that yields an equilibrium thickness. The growth adjustment
can be considered analogous to the blackbody-radiative ad-
justment process that causes the planet to reach an equilib-
rium temperature.
When the climate is perturbed, such as by increasing CO 2 ,
this adjustment process acts to damp the response some-
what. However, for sea ice, the damping is a strong function
of thickness itself. In other words, D h o is a strong function of
h , while as explained above D T o is nearly a constant. Bitz and
Roe [2004] calculated the dependence of D h o on h for an ide-
alized coupled atmosphere and ice slab without ice-albedo
feedback using the formulation given by Thorndike [1992]:
With an uncertainty of 100% of f - (s f = 0.21), ice-albedo
feedback still only has a rather modest influence on uncer-
tainty in D h because f is so small. In contrast, the feedbacks
that influence global mean temperature give an f that is more
than 3 times larger. Even a small uncertainty in f is important
as f approaches 1 because 1 - f appears in the denomina-
tor of equations (4) and (5). Thus a more important issue is
whether I have underestimated f - . I have let f - (s f = 0.21),
which is the feedback factor I computed for CCSM3 in the
previous section. This is unlikely to be an underestimate
kn w Bh 2
Bn w k A n w D 2
1
1
n s
hB n w
kn w Bh
' A
' h o
n w
(11)
where the parameters and variables are defined in Table
3. The hat over D h o is added to emphasize that this ideal-
ized model lacks many processes that I had lumped into D h o
above. The term in brackets in equation (11) has a fairly
weak thickness dependence, so its h can be replaced with a
constant - , and the leading dependence on h is parabolic:
' h o | q rh 2
(12)
where q and r are independent of h .
Bitz and Roe [2004] considered how ice export might al-
ter equation (12). Hibler and Hutchings [2002] (updated by
Hibler et al. [2006]) argue that export increases with thick-
ness for ice thickness between about 0 and 4 m. Bitz and Roe
[2004] reasoned that this sensitivity of export on the mean
state is likely to enhance the sensitivity of D h o to h among
Figure 3. Distributions from equations (9) and (10) for D h (with
D h o = -- 1 m, f - = 0.21 , and s f = 0.1 (black line) and s f = 0.21
(grey line)) and D T (dashed line, with D T o = -- 1.2°C, f - = 0.65 , and
s f = 0.13 ). The distributions for D h are much narrower because
f - is much smaller for h than for T .
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