Multiple Equilibria and Abrupt Transitions in Arctic Summer Sea Ice Extent - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 148

abrupt, so that an infinitesimal change in forcing or a suf-
ficiently large finite perturbation can bring about finite (and
possibly large) climate response. bifurcation theory pro-
vides a mathematical framework for representing this di-
chotomy [e.g., Guckenheimer and Holmes , 1983] and for
describing sudden changes as shifts between multiple equi-
libria. While such shifts may sometimes appear to occur in
complex models, to apply the tools of bifurcation theory
generally requires substantial simplification, with “essen-
tial” aspects of the climate system for a particular problem
distilled to a relatively simple set of equations. Such an ap-
proach can yield physical insights and enables a thorough
exploration of parameter dependences, although clearly
care must be taken both in formulation and in interpreting
results.
One aspect of the climate system that has been thoroughly
studied in this manner is the response of the North Atlan-
tic meridional overturning circulation to changes in surface
freshwater flux [e.g., Stommel , 1961; Titz et al. , 2002]. bi-
furcation studies of conceptual models, in conjunction with
numerical experiments involving circulation models, have
indicated that increased freshwater flux into the North At-
lantic can cause a sudden “switching off” of the overturn-
ing, leading to a rapid cooling in this region [e.g., Stocker
and Wright , 1991; Rahmstorf , 1995; Manabe and Stouffer ,
1999]. For such abrupt changes to occur requires positive
feedback, which in this instance is provided by advective
and convective processes [ Rahmstorf , 1999].
Another component of the climate system with potential
to change rapidly is sea ice. In this instance, positive feed-
back is provided by the absorption of sunlight by low-albedo
open water when high-albedo ice cover retreats, promoting
further ice melt and inhibiting freezing. The possibility that
Arctic sea ice in particular might abruptly retreat under
anthropogenic warming has received increasing attention
[ Lindsay and Zhang , 2005; Winton , 2006], in part because
of the unexpectedly rapid shrinkage of summer ice cover in
recent years [e.g., Stroeve et al. , 2007], as well as recent cli-
mate model results that exhibit very rapid declines in 21st
century summer ice extent under projected anthropogenic
forcings [e.g., Holland et al. , 2006a].
This paper has three main parts. The first, in section 2, is
a brief review of several previous investigations that have
examined multiple equilibria of sea ice. The second, in sec-
tion 3, is a case study in which we attempt to represent the
complex behavior of the CCSM3 climate model, which in-
cludes sudden decreases in 21st century summer Arctic ice
extent, with a simple set of equations. The third part, in sec-
tion 4, demonstrates how seemingly subtle changes in the
parameterizations used in section 3 can lead to qualitative
differences in behavior, illustrating some of the challenges

inherent in such simplified analyses of the climate system.
Discussion and conclusions are provided in section 5.

2. MUlTIPlE SEA ICE EqUIlIbRIA:
A bRIEF OVERVIEW

An early quantitative argument that Arctic climate might
have two stable regimes, one ice covered and the other ice
free, was given by Budyko [1966, 1974] based on energy
balance considerations. Since than, a number of investiga-
tions have examined the possibility that multiple sea ice
equilibria can exist because of positive ice-albedo feedback
and that the associated bifurcation structure can be obtained
from a simplified set of equations. A brief review of these
efforts follows.

2.1. Planetary Energy Balance Models

North [1975, 1984] considered a simple energy balance
climate model consisting of an ordinary differential equation
for equilibrium surface temperature as a function of latitude,
with balances between solar heating, outgoing longwave
radiation and poleward atmospheric heat transport repre-
sented. (The model was originally devised to study ice sheet
inception, although North [1984] recognized its applicability
to Arctic sea ice.) Surface albedo takes on two values, cor-
responding to ice or a less reflective earth surface, depending
on temperature. As a solar constant parameter Q increases,
the equilibrium latitudinal extent of ice cover diminishes un-
til a critical value is reached. At this point a further infini-
tesimal increase in Q causes the system to jump to a second,
ice-free equilibrium. These stable equilibria coexist over a
range of Q and are connected by an unstable equilibrium
solution branch representing a small ice cap [ North , 1984,
Figure 1]. This instability was described as the small ice cap
instability or SICI.

2.2. Local Thermodynamic Models

A second approach that has yielded simplified equations
whose equilibria can be studied considers the thermody-
namics of the atmosphere-ocean-ice system locally, ignor-
ing horizontal nonuniformities. Thorndike [1992] examined
such balances under seasonal forcing. The warming influ-
ences of solar radiation and poleward atmospheric and ocean
heat transport and the cooling influence of upward longwave
radiation were included, taking albedo changes into account.
The resulting equations have two stable equilibria, one rep-
resenting perennial ice cover and the other ice free, that co-
exist over a range of poleward heat transports [ Thorndike ,
1992, Figure 9]; a third equilibrium, ice free in summer