Multiple Equilibria and Abrupt Transitions in Arctic Summer Sea Ice Extent - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 160

3.3.2. Dependence on M (0) , M (0) , and T * . The dependence
of hysteretic transition properties A e ( H c ) and H c on M (0) +
M (0) and T * is illustrated in Figure 13, where F and b are
fixed at CCSM3 values and, as in Figure 11, the solid con-
tours indicate A e ( H c ) and the dashed contours H c . Again,
the zero contours of these quantities divide the plane into
three regimes, denoted as I¢, II¢, and III¢. Regime II¢, which
contains the CCSM3 case (indicated again by the asterisk),
is analogous to regime II in Figure 11; that is, a hysteretic
transition from positive A e to A e = 0 occurs for a critical H c
> 0. Similarly, regime I¢ is analogous to regime I in Figure
11, in that as H increases A e approaches A e in a continuous
manner. Finally, regime III¢ is analogous to III in Figure 11
in that the transition from A e to A e occurs at negative H , so
only the A e equilibrium is present when H > 0. In regime II¢,
the magnitude of the abrupt transition increases as T * and
M (0) + M (0) decrease, as does the critical OHT value H c .

role, and (2) to illustrate possible additional consequences
of multiple equilibria for Arctic sea ice evolution, includ-
ing vacillation between multiple stable equilibria, as well as
hysteretic behavior that occurs when a period of warming is
followed by a gradual return to a cooler climate.

3.4.1. Abrupt transitions in a warming climate. To obtain
a statistical description of Arctic sea ice evolution under
climatic warming as simulated by (6) - (8), ensembles con-
sisting of 10 4 realizations of the random process e n , which
governs the fluctuating component of the synthetic OHT
time series H n described by (3) and (4), are considered. For
default values of the parameters, these synthetic time series
resemble statistically the CCSM3 OHT time series, as dem-
onstrated in Figure 4. To assist in determining the relative
importance of OHT fluctuations and multiple equilibria in
contributing to abrupt transitions, sensitivity is considered
to the scale factor s 0 of OHT fluctuations and to the ocean
shortwave absorption parameter b that substantially governs
whether multiple equilibria are present (see Figure 11).
Results for these various cases are illustrated in Plate 1.
We consider first that for which s 0 and b are set to their
CCSM3 default values, shown in the middle panel of the
topmost row in Plate 1. Here and in the other panels, en-
semble mean A n is indicated by the thick blue curve, and a
range of 1 ensemble standard deviation s A above and be-
low the mean is indicated by the thin blue curves. The thick
red curve indicates the probability p abrupt that a given year
lies within an interval of abrupt transition in A as defined
in section 1, and the thin red curve the probability of abrupt
increase as defined by an analogous criterion. In this panel,
the probability of an abrupt decrease occurring in the en-
semble of seven CCSM3 21st century simulations subject
to the SRES A1b forcing scenario is additionally indicated
by the green histogram. Overall, the temporal distribution
and frequency of the abrupt decreases in CCSM3 and in the
synthetic time series agree reasonably well. One notable dif-
ference is that the latter feature some abrupt increases asso-
ciated with large swings in A n driven by OHT fluctuations,
whereas in CCSM3 no such increases exceeding the abrupt
transition threshold occur. These increases tend to be paired
with subsequent decreases, particularly after 2050 or so,
which together compose partial, temporary recoveries in ice
extent. One such event is visible near year 2065 in Figure
7a, although in this instance the abrupt transition threshold
is not exceeded.
A further result pertaining to the synthetic time series in
this panel is that, as warming progresses, the decrease in A n
below a particular threshold can be hastened or delayed by a
decade or more for realizations lying within ±s A of the mean.
It is also seen that is A tends to be largest at approximately

3.4. Numerical Experiments

In this section numerical solutions to (6) - (8), forced by
synthetic OHT time series H n obtained from (3) and (4), are
described. Objectives are (1) to examine statistical proper-
ties of sea ice evolution and abrupt transitions under climatic
warming as simulated by (6) - (8), with a view toward deter-
mining the extent to which multiple equilibria may play a

Figure 13. Properties of the hysteretic transition from A e > 0 to A e >
0, as functions of parameters M ( s 0 + M ( b 0 and T * , with F and b fixed
at CCSM3 values. Contour definitions and intervals are as in Figure
11. The asterisk denotes CCSM3 values, and regimes I¢, II¢, and III¢
are described in the text.