Some Aspects of Uncertainty in Predicting Sea Ice Thinning - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 68

which is the case for all but a few tiny areas. In most of the
Arctic, f varies between about 0.1 and 0.3, with an average
of 0.21 north of 70°N. The error in my estimate of f that re-
sults from variability in the mean thickness north of 70°N is
0.02. The closer f is to one, the closer the system is to experi-
encing a runaway feedback. Although f indicates ice-albedo
feedback is positive, it is not very big.

back factor that affects global temperature change is about
3 times larger than the one I estimated for the influence of
ice-albedo feedback on sea ice thickness.
A number of studies have attempted to estimate what por-
tion of f in equation (5) results from the influence of ice-
albedo feedback on global mean temperature, not to be
confused with the influence of ice-albedo feedback on sea
ice thickness. Bony et al. [2006] summarize estimates of ice-
albedo feedback in three studies and find a mean ice-albedo
feedback factor of about 0.12 with a range of about -0.03 to
0.4. (Note that I have used a different definition for feedback
than Bony et al., so I have had to convert their estimates to
match my definition.) For reference from CCSM3, I find the
feedback factor on global mean temperature for all feedbacks
is 0.57, and for ice-albedo feedback alone it is 0.24. (There
is no reason to expect that ice-albedo feedback should have
the same feedback factor for global mean temperature as it
does for ice thickness.)
Roe and Baker [2007] also pointed out that it is possible to
compute the uncertainty in DT T that results from uncertainty
in f . Specifically, the uncertainty can be related to the prob-
ability density function D T (D T ) that the global temperature
change is DT T . And the distribution in DT T can be related to a
distribution in f by

4. INFLUENCE OF ICE-ALBEDO FEEDBACK ON
UNCERTAINTY IN FUTURE THICKNESS

In a recent landmark paper, Roe and Baker [2007] showed
that much of the uncertainty in climate model predictions of
future global mean warming could be estimated analytically
from the uncertainty in climate feedbacks. I shall borrow
heavily from their work to show that, in contrast, uncertainty
in ice-albedo feedback has very little influence on the un-
certainty of 21st century ice thickness trends among CMIP3
models.
In the equation of climate sensitivity for global tempera-
ture change (D T ):

' T o
1 f

' T

(5)

D T ' T D f f df

d ' T

(7)

D T o derives from assuming a blackbody planet, in which
blackbody radiation emitted by the planet stabilizes the cli-
mate by cooling (or warming) the planet when the planet
exceeds (or falls below) its equilibrium temperature and f is
the feedback factor (this time for DT T not D h ). Thus the most
basic negative feedback process for stabilizing temperature
is normally excluded from f , and, instead, its influence is
contained in the reference climate sensitivity D T o . For the
blackbody planet assumption, D T o can be estimated from the
first term in a Taylor's series expansion:

Roe and Baker [2007] assumed a normal distribution for f ,
with mean f - and variance s f ,

exp f f 2
2V f

1
— 2SV f

D f f (8)

and then computed the resulting distribution for DT T ,

' T 2 exp 1 f ' T o ' T 2

1
— 2SV f

' T o

D T ' T

(9)

1

2V f

w

w T V T 4

' R

(6)

' T o

T E

With the same basic relation between ice thickness and
feedback as with global mean temperature and feedback, by
analogy the distribution for D h is

where s T 4 is the Steffan-Boltzmann law, T E is the effective
radiative temperature of the planet, and ' R | 3.2 W m 2 is
an estimate of the change in the top of atmosphere outgo-
ing longwave radiation when CO 2 is doubled. Because is T 4
is well approximated by a tangent line, the first term in the
Taylor's series is a good approximation. In other words,
the temperature dependence of D T o is easily neglected, and
equation (6) gives D T o = 1.2°C, which is fairly accurate for
a wide range of temperatures.
Roe and Baker [2007] note that f in equation (5) is on av-
erage about 0.65 for recent climate models. Hence the feed-

' h 2 exp 1 f ' h o ' h 2

1
— 2SV h

' h o

(10) D h ' h

2V f

Here, I use this equation to represent the distribution of
the thickness change averaged north of 70°N owing to ice-
albedo feedback, so all variables in equation (10) are consid-
ered averaged north of 70°N as well.