Modeling Abrupt Climate Change as the Interaction Between Sea Ice Extent and Mean Ocean Temperature Under Orbital Insolation Forcing - Abrupt Climate Change: Mechanisms, Patterns, and Impacts - page 68

climate states). It can be shown that in both models, the
system regains bistability when the insolation drops through
the same threshold value.
If SIO captures some real aspect of the relationship be-
tween sea ice extent and the temperature of the ocean it
covers, it would be that the two variables synchronously
oscillate as a single dynamic system, consistent with the
mechanism proposed by Saltzman et al. [1981] and de-
scribed by equations (A2a) and (A2b) in Appendix A. As
shown there, mean ocean temperature (y) is linearly related
to the time integral of sea ice extent (x), while x itself is the
solution to a nonlinear differential equation. Following Saltz-
man et al. [1981], x(t) is the motion of the sea ice edge
relative to an “ equilibrium ” value and y(t), whose time de-
rivative is proportional to x(t), is the change in mean ocean
temperature around that which corresponds to the same
equilibrium.
A goal of this investigation is to gain some qualitative
understanding of how different parts of the climate system
interact with each other to create a burst of abrupt climate
change. We have shown that the use of low-order climate
models helps greatly, especially to interpret the results of the
much more elaborate ECBILT-CLIO. In no experiment or set
of experiments are the results inconsistent across models,
and in no case was there evidence of propagating errors
influencing ECBILT-CLIO ' S stability for the crucial pair of
variables sea ice extent and mean ocean temperature. In this
sense, we usually worked in reverse to the usual modality,
using the simple models to check the reliability of the
EMICs. The ability of the simple models to highlight the
usually counterintuitive nonlinear interactions that would
escape even a detailed analysis of any EMIC output is their
most important virtue. Physically, sea ice appears as a logical
choice for the abruptness of climate change because of its
effectiveness as ocean heat insulator and rapid response to
temperature changes. These are brought about by the varying
orbital insolation and ocean water temperature, whose be-
havior we have shown is tied (at least in the model) to sea ice
by virtue of being the two components of the system

where x represents the climate variable of interest, f(x,t)=V (x)+

M(t), and V ′ (x) is the derivative of a two-well potential V(x)=

′

bx 4 ). M(t) is the (zero mean) external (orbital)
forcing. The white noise x(t) is Gaussian and such that
< ξ (t) ξ (s)> = D δ (t s), D being the noise level. The cubic
nonlinearity in V

(cx + ax 2

(x) is a damping (negative feedback) term that
originates the pitchfork bifurcation that sustains the bistable
system [Gitterman, 2005; Saltzman, 2002]. Here we assume
these states to be the stadial/interstadial states. Parameters a, b,
and c are positive constants that allow different mean climate
states to be simulated by altering the geometry of the two-well
potential (unless otherwise stated, we use c = 0 and a = b~1).
We choose M(t) to be the summer insolation function at 65°N
[Laskar et al., 2004] and integrate equation (A1) numerically
using Euler

′

smethod[Moss and McClintock,1989].Compu-
tationally, it is convenient to write M(t)=Am(t)fortheforcing
function, where A is a parameter that denotes the level of inso-
lation

'

fluctuation and m(t) is the normalized M(t). Equation
(A1) is the Newtonian equivalent of a small ball jumping
randomly between two contiguous symmetric wells separated
by a barrier [Gitterman, 2005]. The height of the barrier is a 2 /4b
or 0.25. This implies that in the absence of noise, the amplitude
of the external forcing A less than the threshold value 0.25 will
not induce jumps in the system. The noise (which we de ne as

internal forcing) simulates stochastic forcing by weather and
plays the important role of exciting jumps across the barrier in
either direction, which hence occur at random times only if A=
0. If A

0 but small, and below threshold, jumps can occur that
are modulated by M(t). With one well signifying the cold
stadial state and the other the interstadial, the model is a
heuristic yet useful representation of a bistable climate system
[Imkeller, 2001; Timmermann and Lohmann, 2000].

≠

A2. NOISY VAN DER POL OSCILLATOR

Simple energy considerations show that in glacial times,
the interaction among ocean temperature, sea ice distribution,
and sea-atmosphere greenhouse gas

uxes can support non-
linear, self-sustained, relaxation-type thermal oscillations in
which the sea ice edge advances and retreats with a millennial-
scale period [Saltzman et al., 1981]. Saltzman and Moritz
[1980] estimated the fundamental period of such thermal
oscillator as lying between 300 and 3000 years. From the
previous Langevin model, it is easy to build a

' s free

(nonlinear) oscillation.

APPENDIX A: MODELS

“ noisy ” version

A1. LANGEVIN BISTABLE EQUATION

of Saltzman s oscillator: We add a new coupled variable y(t)

to model (A1) to represent zero-mean mean ocean tempera-
ture variations, while the variable x(t) now represents zero-
mean variation in sea ice extent, and write

'

Our simplest model is a stochastic differential equation of
the Langevin type [e.g., Gitterman, 2005; Imkeller, 2001;
Timmermann and Lohmann, 2000] given by

d x
d t ¼ f ð x ; t Þþξð t Þ ;

d x
d t ¼ − y þ c þ ax − bx 3

ðA1Þ þ ξð t Þ ðA2aÞ