Some Aspects of Uncertainty in Predicting Sea Ice Thinning - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 70

Table 3. Definitions for the Idealized Analytic Model
Variable Description Unit of Measure

h annual mean ice thickness variable

thickness change for
idealized model

variable

D h o

D A radiative forcing 22.6 W m -2

A s T f with T f = 273 K 320 W m -2

B 4s T 3

f

4.6 W m -2

k thermal conductivity 2 W m -1 K -1

n w , s optical depth for winter

or summer

2.5 or 3.25

D atmospheric heat transport 100 W m -2

models with ice in motion, which is the case for nearly all
CMIP3 models. Neglecting the influence of the mean state
on export here gives a conservative estimate for the the un-
certainty in thickness change because of uncertainty in the
mean state.
Given a distribution for h , D h ( h ), the distribution of D h o is

Figure 4. Distributions from equations (14) and (15) for D h o

(dashed line) and D h (solid line) with - = - 1.8 m, s h = 0.77 m,
f = 0.21 , and s f = 0 .

D h o ' h o D h h dh
d ' h o

(13)

1

D h ' h D f f D h o ' h 1 f 1 f df

(16)

If h is assumed to be normally distributed with mean - and
variance s h 2 , then using equation (12) and equation (8) with
f replaced by h gives

f

[see, e.g., Springer , 1979]. A numerical solution to this in-
tegral is shown in Figure 5 with the same parameters as in
the previous paragraph except s f = 0.21. Uncertainty in f
increases slightly the probability of greater thickness change
at the expense of decreasing the probability of the peak.

' h o q r h

2

1

D h o ' h o | exp

2V h 2 r V h 2S' h o

(14)
Now if D h o is the only source of uncertainty in D h , then

D h ' h D h o ' h o d ' h o

d ' h D h o ' h 1 f 1 f (15)

Figure 4 shows examples of distributions from equations
(14) and (15) that arise from doubling CO 2 but without any
uncertainty in f (hence s f = 0). Again, I use these equations
to represent the distribution of thickness change averaged
north of 70°N, so I have taken averages north of 70°N that
give - = 1.8 m and s h = 0.77 m from the CMIP3 models for
1950-2000 (see Figure 1) and f = 0.21 from CCSM3 (see
section 3.3). The distribution for D h is influenced by ice-
albedo feedback such that it is broader and the thickness
change is larger than for D h o .
It is possible to compute the distribution of D h with un-
certainty in both D h o (via h ) and f by computing the ratio of
distributions. The result is the Mellin convolution

Figure 5. Distributions from equation (16) for D h with s f = 0.21
(grey line) and with s f = 0 (black line, which is identical to the

black line in Figure 4). Both lines have - = - 1.8 m, s h = 0.77 m,
and f = 0.21