Sea Ice–Albedo Feedback and Nonlinear Arctic Climate Change - Arctic Sea Ice Decline: Observations, Projections, Mechanisms, and Implications - page 109

of sea ice in an ocean mixed layer, thereby simulating rather
than parameterizing the albedo temperature relationship, and
found that seasonally ice-free states were unstable. Under
increased forcing, Thorndike's “toy” model transitions di-
rectly from annually ice-covered to annually ice-free states
inducing a large and abrupt increase in surface temperature.
The Arctic sea ice cover has been in decline since the 1950s
[ Vinnikov et al ., 1999]. This decline is more pronounced in
the summer, and recent years have produced striking record
minima [ Stroeve et al ., 2005]. Some researchers have noted
that nonlinear behaviors such as thresholds and tipping points
may be associated with this decline [ Lindsay and Zhang ,
2005; Serreze and Francis , 2006]. The goal of this paper
is to assess the potential for nonlinearity of Arctic climate
change in the Intergovernmental Panel on Climate Change
(IPCC) fourth assessment report (AR4) climate models. In
section 2 we demonstrate potential nonlinearities in a simple
model and develop a strategy for assessment. In section 3 we
examine 21st century simulations for signs of nonlinearity.
Section 4 continues this search by examining two strongly
forced experiments as they become annually ice free. Sec-
tion 5 shows that the nonlinear behavior of one of these ex-
periments is similar to the EBM SICI. Section 6 explores the
stabilizing effect of ocean surface fluxes and atmospheric
heat transport on the sea ice with special GCM experiments
designed to illuminate the climate response to sea ice region
changes. Section 7 summarizes and discusses the results.

perature, that this drop would resemble a cliff. However, the
seasonal cycle and other variability allow sampling of vari-
ous ice-cover states at any given long-term mean tempera-
ture, smoothing the relationship. For simplicity, let us take
this smoothed section to be linear and call it the ramp. The
slope of the ramp depends on the drop in albedo between its
endpoints and the temperature range over which the drop is
experienced. The drop in planetary albedo, the albedo above
the atmosphere, will be less than the jump in surface albedo
because only part of the insolation reaches and interacts
with the surface. There may also be changes in atmospheric
properties with temperature that impact the planetary albedo
drop. Gorodetskaya et al . [2006] have used satellite sea ice
cover and shortwave data to estimate the albedo drop for
Northern Hemisphere sea ice regions. They obtain a 0.22
planetary albedo change for a 100% change in sea ice cover.
This is roughly half the surface albedo difference between a
typical sea ice cover and seawater.
The balance expressed in (1) is depicted schematically in
Plate 1. The steepness of the albedo ramp, the drop divided
by the ramp temperature range, impacts the character of the
nonlinearity. In particular, if the ramp is so steep that, as
warming occurs, the extra shortwave absorption exceeds
the extra loss of energy from outgoing longwave radiation
(OLR), the total feedback will be positive, and there will be
unstable transitions between ice-covered and ice-free states.
This is an example of the slope stability theorem of energy
balance models (see Crowley and North [1991, pp. 18-19]
for an elementary discussion). We can form an expression
for the critical ramp temperature range, ΔT C , between stable
and unstable solutions:

2. ElEMENTARy ARCTIC ClIMATE dyNAMICS

The potential for a nonlinear relationship between ice
albedo and temperature to generate nonlinear climate change
can be demonstrated with a very simple energy balance
model. Consider the energy balance at the top of an isolated
polar atmosphere:

DT C = SΔa/B. (2)

A larger range is needed for stabilization when the insola-
tion and the albedo drop are large and when the longwave
damping is small.
Plate 1 shows schematic examples of subcritical and su-
percritical shortwave absorption profiles. When the albedo
ramp steepness is supercritical, the total feedback is posi-
tive in the ramp temperature range so it will contain only
unstable equilibria. As a result, the ramp range becomes a
forbidden zone, inaccessible with any forcing. As forcing is
slowly varied, these temperatures are skipped leading to a
discontinuity in polar temperature. Since the polar tempera-
ture contributes to the global mean temperature, it would
also have a (much smaller) temperature discontinuity when
the ramp steepness is supercritical.
If we insert the insolation at the North Pole (173 W m −2 ),
the Gorodetskaya et al . [2006] planetary albedo drop, and
a satellite-estimated OLR damping value (1.5 W m −2 k −1

A + BT = S[1 - a(T)]. (1)

The model represents a balance between absorbed short-
wave radiation, insolation (S) times a planetary coalbedo
(1 − a), and parameterized outgoing longwave radiation with
a linear dependence on surface temperature, T. The model
is isolated in the sense that the atmospheric heat transport
convergence is held fixed, bundled with the longwave in-
tercept into A. The nonlinearity of the model comes from
the nonlinear dependence of α on T. At very low mean tem-
peratures, where snow never melts, albedo is insensitive to
temperature. The same is true at high mean temperatures
where there is no ice. Between these flat sections, there is a
drop from snow to seawater albedos. One might expect, on
the basis of the liquid/ice transition occurring at a fixed tem-