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the time when mean
A
is decreasing most rapidly and abrupt
decreases occur most frequently, a result that is revisited in
the next section.
Dependence on
b
is illustrated by considering values
b
= 1 ´
10
-12
W m
- 4
and 3 ´ 10
-12
W m
- 4
, in addition to the CCSM3
value of 2 ´ 10
-12
W m
- 4
. To facilitate comparison,
F
is ad-
justed in the non-CCSM3 cases so that, in the absence of
OHT fluctuations, the transition from finite
A
to
A
= 0 (with
or without hysteresis) occurs near year 2040 as in the default
case. These values are
F
= 2.55 m in the low-
b
case, which
lies well within the single-equilibrium regime I in Figure 11,
and
F
= 3.8 m in the high-
b
case, which lies well within the
multiple-equilibrium regime II.
With s
0
= 0.6 W m
-2
, the CCSM3 default, it is seen in the
top row of Plate 1 that the probability and timing of abrupt
transitions is comparable for all three values of
b
, even though
the decrease in mean
A
occurs somewhat more rapidly for
higher
b
than for lower
b
. Thus, it appears that in this instance
the primary influence governing abrupt transitions is the rel-
atively large OHT fluctuations, together with the increased
sensitivity of
A
to changes in
H
as the transition to
A
e
= 0 is
approached, as discussed in section 3.3 and illustrated in Fig-
ure 10. However, for reduced OHT variability as simulated
using lesser values of s
0
(second and third rows of Plate 1),
the frequency of abrupt transitions becomes much more sen-
sitive to
b
. For example, for s
0
= 0.15 W m
-2
, 4 times smaller
than the value characterizing CCSM3, abrupt transitions are
entirely absent for
b
= 1 ´ 10
-12
W m
-4
, whereas for
b
= 3 ´
10
-12
W m
-4
,
p
abrupt
exceeds 0.4 near year 2040. Not surpris-
ingly, as s
0
decreases, the range of times over which abrupt
decreases occur becomes increasingly narrow. Also of note
is the even greater sensitivity to s
0
of the abrupt increases,
which have become very infrequent even for s
0
= 0.3 W m
-2
.
Finally, the bottom row of panels, for which s
0
= 0, shows
that in our simple model some OHT variability is essential
for abrupt decreases to occur for the parameters considered.
(For the case
b
= 3 ´ 10
-12
W m
-4
, however, the abrupt de-
crease threshold is nearly met, so that for slightly larger
b
an
abrupt decrease would occur near 2040.)
3.4.2. Vacillation between equilibria in a warmer control
climate.
In a nonlinear system having multiple equilibria
that is forced stochastically, fluctuations in forcing can po-
tentially trigger transitions between stable equilibria, even
when the ensemble mean forcing is stationary. This phe-
nomenon has been discussed in the context of simple mod-
els of oceanic thermohaline circulation, e.g., by
Monahan
[2002a, 2002b], and in the sea ice context by
Flato and
Brown
[1996]. In the present context, one might consider
a situation in which warming has stabilized at some future
date, and climate, though warmer than at present, is station-
ary. Such a situation can be represented by assigning a fixed
value ensemble mean
—
in equation (4).
Figure 14 illustrates such a scenario with constant
—
=
8 W m
-2
with other parameters, including the standard devia-
tion s
0
= 0.6 W m
-2
of the stochastic component of forcing,
set to their CCSM3 default values. This value for
—
is real-
ized in CCSM3 at around 2030 (Figure 4a) and lies within
the range of
H
for which multiple equilibria are present (Fig-
ure 9b). In the 200-year time series shown in Figure 14a,
A
n
appears to vacillate between very small or zero values
and somewhat larger values in the range 2 ´ 10
6
km
2
~
A
n
~
4 ´ 10
6
km
2
. This impression is borne out by the probability
density for
A
n
, computed from a 10
4
-year time series, which
is clearly bimodal (Figure 14b).
Such bimodality persists when
—
and s
0
are varied some-
what about the values assigned in Figure 14. With
—
fixed,
for example, the value of
A
n
at which the upper lobe of
p
(
A
n
)
peaks increases as s
0
increases, exceeding 5 ´ 10
6
km
2
for
s
0
= 3 W m
-2
. Conversely, when s
0
is reduced at constant
—
, the value of
A
n
characterizing the peak of the upper lobe
of
p
(
A
n
) decreases toward the
A
n
equilibrium value of about
2 ´ 10
6
km
2
. If s
0
= 0.6 W m
-2
is fixed instead, this bimodal-
ity persists for
—
in the range 6.3 W m
-2
~
—
~
8.8 W m
-2
,
Figure 14.
(a) A 200-year time series of
A
n
for CCSM3 parameter values and fixed
¯
= 8 W m
-2
characterizing CCSM3
climate near 2030. (b) Probability density of
A
n
for a 10
4
-year continuation of this time series.
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