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the upper is described during the period of increasing
—
,
and the lower is described during the period of decreasing
—
. For comparison the dashed curves describe evolution of
A
n
for the same time-dependent
—
but without fluctuations
(s
0
= 0), and the dotted curves describe ideal hysteresis cy-
cles determined from the equilibrium solutions.
One aspect of Figure 15 that is immediately evident is
that the numerical solutions exhibit a separation between
the ascending-
—
and descending-
—
branches in all three
cases, even though multiple equilibria are present only for
the larger two values of
b
. This is because of the finite time
required for solutions of (6) - (8) to adjust to changes in
H
.
Such a separation depends on the rate of change of
—
and is
reduced if
—
varies more slowly than in the case illustrated
here. The overall separation of the two branches does, how-
ever, increase with
b
, in accordance with the increasingly
hysteretic character of the equilibria.
A second aspect of Figure 15 that bears mention is that the
H
fluctuations greatly broaden the range in
—
over which
the ascending and descending branches separate. This is
because in the presence of fluctuations the transitions be-
tween equilibria become “blurred,” an effect which is also
evident in Plate 1. An alternative view is that in the presence
of fluctuations multiple equilibria can exert a vestigial influ-
ence beyond the parameter range in which they are formally
present through ephemeral trapping of solution trajectories,
as discussed by
Monahan
[2002b].
realization is available, and other fluctuation measures must
be considered. Figure 16a plots for each year a single-reali-
zation standard deviation computed from
A
for that year and
the 10 preceding years, i.e.,
n
å
m
=
n
-
M
é
ë
n
=
M
-
1
i
(
A
m
,
i
- <
A
>
n
,
i
)
2 1/ 2
,
s
(13)
with
M
= 10, where <> denotes an average over years
n
-
M
to
n
, for
b
= 1, 2, and 3 ´ 10
-12
W m
-4
. Other parameters
are set as in the top row of Plate 1, so that in each case
A
vanishes near 2040 if
H
fluctuations are absent; recall that
multiple equilibria occur for the larger two values of
b
. To
damp statistical noise, this fluctuation measure is averaged
over each realization of a 10
4
-member ensemble; that is, we
plot
¾
s
n
. The resulting values (thick curves) increase signifi-
cantly, by a factor between 5 and 10, from the beginning of
the 20th century to the interval 2020 to 2050 where most
abrupt transitions occur (see Plate 1). However, the peak
values are only weakly sensitive to
b
, in contrast to the en-
semble standard deviations s
n
(not shown), which increase
by »40% between the lowest and highest values of
b
. (A
similar result is obtained if the temporal standard deviation
(13) is computed with the linear trend removed.) Thus there
is no clear signal that a bifurcation is being approached, al-
though in all cases the temporally increasing variance does
point to an accelerating decline in sea ice cover which may
or may not be characterized by a transition between multiple
equilibria.
For comparison, the thin curve in Figure 16a shows
¾
3.5. Are the Simulated Abrupt Transitions Predictable?
s
n
calculated from six realizations of CCSM3. Its general
behavior is similar to that of the thick curves, except that the
result is statistically noisier, as expected with many fewer
realizations. (Results obtained using six realizations of the
simple model exhibit comparable noise.) The CCSM3 stand-
ard deviations noticeably exceed those of the simple model
prior to about 1980. This is likely due to atmospheric forcing
variability, which is not accounted for in the simple model
and is comparatively most important in this epoch when
OHT forcing is relatively small (Figure 4).
A second potential indicator of an impending bifurcation
is the integral time scale t
n
, which can be obtained from the
correlation matrix via
When multiple equilibria are present, as in Figure 9, for
example, rapid transitions from relatively large
A
to
A
» 0
can result from an incremental increase in forcing
H
. In such
cases, there is relatively little prior indication from the gen-
eral magnitude of
A
that a seasonally ice-free Arctic is im-
minent. This raises the question of whether other aspects of
the time series for
A
might indicate more clearly that such a
threshold is being approached.
One potential candidate for such an indicator is the ensem-
ble standard deviation of
A
,
n
= (
N
−
1
)
−
1
N
ª
«
¬
i
=
1
(
A
n
,
i
−
A
n
)
2 1
/
2
,
e
V
(12)
n
=
1
2
m
cor
(
A
n
,
A
m
)
where
N
is the number of realizations,
i
denotes an individ-
ual realization, and the overbar denotes an ensemble aver-
age. This is suggested by the result, noted in section 3.3,
that when multiple equilibria are present the modeled sea
ice response to OHT fluctuations amplifies dramatically as
the bifurcation point
H
c
is approached, as discussed by
Held
and Kleinen
[2004] and
Carpenter and Brock
[2006]. In the
case of the actual climate system, of course, only a single
W
(14)
and is a measure of the temporal persistence of fluctuations
in
A
. (For an AR(1) process, t is equal to the characteristic
autocorrelation time scale.) An increase in this quantity as
a random process approaches a saddle-node bifurcation is a
generic behavior, as discussed by
Kleinen et al.
[2003] and
Held and Kleinen
[2004] in the context of the thermohaline
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