Geoscience Reference
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unstable equilibrium between two stable ones. (In our non-
linear empirical model, the tipping point for September is
actually the entire hypotenuse of the gray triangle in figure
8). The transition across a tipping point could be precipitated
by a particular event, or it could be gradual. Lindsay and
Zhang [2005] hypothesized that the thinning of Arctic sea
ice since 1988 was triggered by the export of older, thicker
ice out of the Arctic basin in the late 1980s and early 1990s,
driven by atmospheric circulation patterns associated with
high values of the Arctic Oscillation (AO) and the Pacific
Decadal Oscillation (PDO). The thinning continued in the
following years, even though the AO and PDO returned to
near normal levels, because the ice-albedo feedback was
then exerting a larger influence because of the increase in
summer open water and thin ice. The abrupt reductions in
future Arctic sea ice in CCSM3 simulations were triggered
by “pulse-like” events of ocean heat transport into the Arctic
Ocean [ Holland et al. , 2006b, this volume], followed by a
similar increase in the influence of the ice-albedo feedback
due to more open water. Winton [2006] examined two cli-
mate models that both became seasonally ice free gradually
as the air temperature increased, but further warming caused
an abrupt loss of sea ice year-round in one of the models
(MPI) because of the ice-albedo feedback. The other model
(CCSM3) lost all of its sea ice more gradually, driven prima-
rily by the ocean heat flux. Thus we have examples of both
abrupt and gradual changes between states.
A simple model that exhibits a tipping point is d z /d t =
- kz ( z - a )( z - b ) + f( t ) where k > 0 is a constant, 0 < a < b are
constants, and f( t ) is the external forcing. (We are think-
ing of z as the September sea ice extent). first consider
f = 0. The two stable equilibria are z = 0 and z = b . They are
separated by the unstable equilibrium point (tipping point)
z = a . Any trajectory that starts with z > a converges to
z = b ; any trajectory that starts with z < a converges to z =
0. The constant k (with units 1/time) determines the rate of
convergence. Large k means strong internal dynamics (short
response time of the system); small k means weak internal
dynamics (long response time). now turn on the external
forcing f. Clearly f can be formulated to kick the system
back and forth between its two regions of attraction, either
randomly or cyclically, suddenly or gradually. If f is of the
form A sin(ω t ) and we nondimensionalize time by t′ = kt then
the nondimensional external forcing is F′ = ( A / k )sin((ω/ k ) t ′).
The ratio A / k is the strength of the external forcing relative
to the strength of the internal dynamics. The ratio ω/ k is
the response time of the system relative to the period of the
external forcing. If A / k is small, the system will not cross
the tipping point; if A / k is large, it will cross back and forth
regularly. Suppose instead that F′ is a random process with
mean zero and standard deviation S . Then S / k is analogous
to A / k , and the autocorrelation time scale of F′ is analogous
to the period 2π/ω. If S / k is small, the system will fluctuate
about one of the stable equilibrium points. At larger values
of S / k , the system will make occasional, gradual transitions
across the tipping point, sometimes flipping back and forth
several times before approaching one of the stable equilib-
rium points. At still larger values of S / k , the transitions are
abrupt. In the limiting case of large A / k or S / k , the system is
dominated by the external forcing, and the internal dynamics
become negligible.
We are led to the following thoughts about tipping points:
(1) A tipping point may be approached or crossed suddenly
because of an externally forced event, or the transition may
be more gradual as the external forcing changes gradually.
(2) When the system is near a tipping point, it is more sen-
sitive or susceptible to being nudged into a new state by
small perturbations. (3) Crossing a tipping point is not an
irreversible event. The system can be driven back into the
previous state if the external forcing changes course. (4) The
trajectory of the system is determined by a balance between
the amplitude and period of the external forcing and the re-
sponse time of the system (internal dynamics).
5. SUMMARy AnD COnCLUSIOnS
We have defined the trajectory of sea ice to be the path
in phase space of the ice thickness distribution, (g 1 (t),…,
g n (t)), where g k (t) is the fractional area of sea ice in bin k at
time t. We analyzed the 48-year monthly output of an ice-
ocean model with seven bins and found that the first two
principal components of the trajectory account for 98% of
the variance: the trajectory is essentially two-dimensional.
Simplifying the ice thickness distribution into thin ice, thick
ice, and open water, we constructed a linear model with em-
pirically determined periodic coefficients that matches the
mean annual cycle of the 48-year ice-ocean model output.
The linear model was found to be stable. We then modi-
fied the linear model by adding a quadratic term to simulate
enhanced melting of thin ice in summer when the amount
of open water is large, i.e., a crude ice-albedo feedback.
The nonlinear model was found to have two stable annual
cycles, one with ice-free summers and one with perennial
summer ice, qualitatively similar to the results of Flato and
Brown [1996]. The annual cycle with ice-free summers re-
sembles the late 21st century annual cycles of the CCSM3
model projection.
The projection of an ice-free Arctic in summer is not new,
nor is the idea that Arctic sea ice may have multiple sta-
ble states. Twenty-eight years ago, Parkinson and Kellogg
[1979] ran a sea ice model forced by atmospheric warming
commensurate with a doubling of CO 2 . They found that a 5°C
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