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�
(
H
) =
1
A
n
+
1
−
A
n
A
±
F
−
(
1
+
T
∗
/
2
)
wH
T
±
√
−
+
T
±
e
2
[
F
−
(
1
+
T
∗
/
2
)
wH
]
1
±
ª
¬
1
−
Z
(
H
)
,
(A1)
=
1
−
Z
,
(A6)
H
+
T
∗
wH
/
2
e
e
where
where
T
e
and
Z
are given by (A1) and (A2). From (A1) and
(A6) it can be deduced that in the physical regime where
T
e
³ 0 and 0 £
Z
£ 1,
A
e
is always stable, and
A
e
is always
unstable. The third branch
A
e
is deduced to be stable by simi-
lar means.
Z
(
H
)
≡
4
wbA
max
T
∗
(
M
(
s
)
0
+
M
(
b
)
0
+
wH
)
−
2
(
F
−
wH
)
T
∗
wH
[
F
−
(
1
+
T
∗
/
2
)
wH
]
2
.
(A2)
Clearly,
Z
³ 0 for
H
³ 0 , while
Z
£ 1 is necessary for either
branch to be a physical solution. This implies a critical value
for
H
, corresponding to a saddle node bifurcation, above
which (A2) ceases to be real valued:
Acknowledgments.
Greg Flato, Slava Kharin, Eric DeWeaver,
and three anonymous reviewers are thanked for suggesting im-
provements to earlier versions of this paper.
�
REFERENCES
ª
«
¬
1
+
F
+ (
M
(
s
)
+
M
(
b
)
0
+
2
bA
max
T
∗
2
)
2
H
c
=
F
w
]
]
0
1
−
,
wbA
max
T
∗
Arzel, O., T. Fichefet, and H. Goosse (2006), Sea ice evolution over
the 20th and 21st centuries as simulated by current AOGCMs,
Ocean. Modell.
,
12
, 401-415.
bitz, C. M., and W. H. liscomb (1999), An energy-conserving
thermodynamic model of sea ice,
J. Geophys. Res.
,
104
, 15,669-
15,677.
bitz, C. M., P. R. Gent, R. A. Woodgate, M. M. Holland, and
R. Lindsay (2006), The influence of sea ice on ocean heat uptake
in response to increasing CO
2
,
J. Clim.
,
19
, 2437-2450.
björk, G., and J. Söderkvist (2002) Dependence of the Arctic
Ocean ice thickness distribution on the poleward energy flux in
the atmosphere,
J. Geophys. Res.
,
107
(C10), 3173, doi:10.1029/
2000JC000723.
budyko, M. l. (1966), Polar ice and climate, in
Proceedings of the
Symposium on the Arctic Heat Budget and Atmospheric Circu-
lation
,
Rand Corp. Res. Memo.
, RM-5233NSF, edited by J. O.
Fletcher, pp. 3-22, Rand Corp., Santa Monica, Calif.
budyko, M. l. (1974),
Climate and Life
,
Int. Geophys. Ser.
, vol.
18, 508 pp., Academic, New york.
Carpenter, S. R., and W. A. brock (2006), Rising variance: A lead-
ing indicator of ecological transition,
Ecol. Lett.
,
9
, 311-318.
Flato, G. M., and R. O. brown (1996), Variability and climate sen-
sitivity of landfast Arctic sea ice,
J. Geophys. Res.
,
101
, 25,767-
25,777.
Flato, G. M., and Participating CMIP Modelling Groups, (2004),
Sea-ice and its response to CO
2
forcing as simulated by global
climate models,
Clim. Dyn.
,
23
, 229-241.
Gildor, H., and E. Tziperman (2000), Sea ice as the glacial cycles'
climate switch: Role of seasonal and orbital forcing,
Paleoocea-
nography
,
15
, 605-615.
Gildor, H., and E. Tziperman (2001), A sea ice climate switch
mechanism for the 100 kyr glacial cycles,
J. Geophys. Res.
,
106
,
9117-9133.
Guckenheimer, J., and P. Holmes (1983),
Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields
, 459 pp.,
Springer, New york.
Held, I. M. (2005), The gap between simulation and understanding
in climate modeling,
Bull. Am. Meteorol. Soc.
,
86
, 1609-1614.
]
]
(A3)
where z º 1 -
T
*
/2. Corresponding values
A
e
are obtained by
substituting (A1) into (9b):
)(
M
(
s
)
0
+
M
(
b
)
ª
«
¬
e
(
H
) =
A
max
1
−
(
T
∗
/
T
±
0
+
wH
/
2
)
1
+ (
T
∗
/
T
e
)
wH
/
2
e
A
±
.
(A4)
Conditions for these solutions to be physically realiz-
able, in addition to
H
£
H
c
, are
T
e
and
A
e
³ 0. If
A
e
(
H
c
) =
A
e
(
H
c
) > 0, then the saddle node bifurcation is realized as in
Figures 9, 12b, and 12d. In this case,
A
e
is nonnegative on the
interval
H
c
£
H
£
H
c
, where
c
=
F
−
(
M
(
s
)
+
M
(
b
)
)
T
∗
−
wbA
max
w
(
1
+
T
∗
/
2
)
H
−
0
(A5)
is obtained by setting
A
e
= 0. Otherwise, if
A
e
(
H
c
) < 0, the
saddle node bifurcation is not realized, as in Figures 12a and
12c. In this instance,
A
e
is always negative and hence un-
physical, and
A
e
is nonnegative for
H
£
H
c
, with
H
c
given by
the right-hand side of (A5).
The third equilibrium solution
T
e
,
A
e
, given by (10a) and
(10b), exists for
H
c
£
H
£ ¥ when
A
e
(
H
c
) > 0 (multiple equi-
librium case), and for
H
c
£
H
£ ¥ when
A
e
(
H
c
) £ 0 (single
equilibrium case).
The local stability of these solutions can be deduced by
considering the response to small perturbations from equilib-
rium, e.g., by letting
A
n
=
A
e
(1 + e) with |e|®0; the solution
is asymptotically stable (i.e., attracting) if -2 < (
A
n
+1
-
A
n
)/
e
A
e
< 0 and unstable otherwise. Carrying out this procedure
analytically leads to
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