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Figure 16. Temporal evolution of (a) single-realization standard deviation of A , computed for each year n from values
A n- 10 to A n as in (13) and subsequently averaged over 10 4 realizations, for the three parameter sets considered in the top
row of Plate 1 (thick curves), or six CCSM realizations (thin curve) and (b) integral time scale t n for the same three-
parameter sets, also computed from 10 4 realizations.
circulation and by Carpenter and Brock [2006] in the context
of ecosystem regime shifts, and thus could provide a poten-
tial “early warning mechanism” for the approach of a regime
threshold. As an idealized test of this approach, t n was com-
puted from 10 4 -realization ensembles, summing from m =
n - 10 to m = n + 10 , for each of the three cases considered
in Figure 16a. The results, plotted in Figure 16b, show that
t » 2 years throughout the 20th century, reflecting the auto-
correlation time scale of the forcing H n . In the cases b = 3 ´
10 -12 W m -4 and b = 2 ´ 10 -12 W m -4 in which the saddle-node
bifurcation is realized, t n increases substantially over the 2
decades or so preceding to the bifurcation point, whereas lit-
tle effect is seen for the case b = 1 ´ 10 -12 W m -4 in which
the saddle-node bifurcation is not realized. (This increase in
integral time scale with increasing b could explain why the
single-realization standard deviation computed for a mov-
ing window according to (13) is less sensitive to b than the
ensemble standard deviation: in the former case, increased t
leads to a larger variance fraction at periods excluded by the
11-year window. This will tend to compensate any tendency
for overall variance to increase with b .)
This result suggests that similar early warning signals
for the collapse of summertime Arctic ice extent via such
a bifurcation could exist, although the detection of trends
in integral time scale from a single realization of the ran-
dom process is a challenging problem of time series anal-
ysis (computing t n as above even from six realizations of
CCSM3 or our stochastic model does not yield an obvious
signal). An interesting direction of future research would be
the development of statistical measures of proximity to a bi-
furcation in which trends are most easily detected [ Held and
Kleinen , 2004].
4. SENSITIVITy TO PARAMETERIZATIONS
The major simplifications adopted in section 3 enable
analytical tractability but are imperfect, as is inevitable in
studies of this type. In order to assess sensitivity to these
simplifications, this section examines the effects of changes
that make the parameterizations arguably more realistic,
though more complex.
A significant simplification in section 3 is the “two-season
approximation” implied by selecting March ice thickness
and September ice extent as dependent variables. Consider
in particular equation (5), which posits that annual mean
ocean shortwave absorption increases proportionally to Sep-
tember open water area. While such a relation is plausible
based on Figure 5, it does not allow for ice cover in other
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