What Is the Trajectory of Arctic Sea Ice? - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 174

Thus the two-dimensionality of the trajectory is fairly ro-
bust. nevertheless, it is not straightforward to give physical
interpretations to the principal components in equation (1).
Therefore we take a slightly different approach, as follows.

3.2. Thin Ice and Thick Ice

Motivated by the results in the previous section, we form
linear combinations of the bins g 1 through g 7 that have sim-
ple physical interpretations, at the expense of accounting for
less than the maximum possible variance. The combinations
are simply thin ice (g 1 + g 2 ) and thick ice (g 3 + g 4 + g 5 + g 6 +
g 7 ). Unlike the principal components, which are uncorre-
lated, thin ice and thick ice have nonzero covariance, and
thus the total variance cannot be neatly partitioned into thin
and thick contributions. The sum of var(g 1 ) + … + var(g 7 ) is
300 × 10 -4 . The covariance matrix of the first two principal
components is

� �

69 0
0 224

× 10 − 4 ,

Figure 3. Trajectory of Arctic sea ice in thin/thick space from PIO-
MAS model output. Axes give the concentration of each ice type.
All years (1958-2005) are shown in gray. The annual cycles go
counterclockwise. Monthly values are shown for the years 1966
(circles) and 2005 (squares).

while the covariance matrix of thin ice (first row/column)
and thick ice (second row/column) is

� �

69 − 50
− 50 113

× 10 − 4 .

Thus thin ice and thick ice are negatively correlated, but
their variances (69 and 113 × 10 -4 ) capture a reasonably large
fraction (61%) of the total variability.
figure 3 shows the trajectory of Arctic sea ice in thin/thick
space. The axes give the concentration of each ice type. All
years (1958-2005) are shown in gray. The annual cycles go
counterclockwise. Monthly values for january to December
are shown for the years 1966 and 2005. The distance from a
point on the trajectory to the line thin + thick = 1 is the frac-
tion of open water. Consider the physical sources and sinks
of thin ice and thick ice throughout the year. During the win-
ter, thin ice grows thicker (sink of thin ice, source of thick
ice). This can be seen in january through April as the total
ice concentration remains close to 1 but the trajectory moves
toward the upper left. May is a transition month, when thick
ice reaches a maximum. from june through September,
thick ice decreases. This melting of thick ice is a source of
thin ice, which is offset to some extent by the melting of
the thin ice itself. In 1966 the thin ice increased from june
to September (being replenished by melting thick ice more
quickly than its own melt rate), whereas in 2005 the thin
ice decreased from june to September (being replenished
by melting thick ice more slowly than its own melt rate).

notice that the total ice concentration in September 1966
was about 0.8 while in September 2005 it was 0.6. October
and november are freeze-up months, with rapid growth of
thin ice and hardly any increase in thick ice. December is
another transition month, when full winter conditions are
reached, with a total ice concentration close to 1 again.
The gradual shift over time toward less thick ice and lower
total ice concentration in September is clearly evident in fig-
ure 3 (more so than in the principal component trajectory of
figure 2). What are the factors driving this transition? They
can be separated into two broad categories: external forcing
and internal dynamics. External forcing refers to (for exam-
ple) increasing air temperature that melts more ice; internal
dynamics refers to the response of the system to perturba-
tions in its state. for example, if the system has multiple sta-
ble equilibrium states, a small perturbation could knock the
system from one basin of attraction into another, sending it
on a course toward less ice even when normal forcing condi-
tions are restored. Several investigators have found multiple
stable states in sea ice models [e.g., Flato and Brown , 1996;
Hibler et al. , 2006; Merryfield et al. , this volume], which we
discuss in more detail later. These were physical models, in
which processes such as ice growth and melt were explicitly