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 64

Figure 6. Two of many possible independent realizations that can be obtained with the van der Pol oscillator sea ice
oscillator (SIO) (equations (A2a) and (A2b) in Appendix A) reproduce the general structure of the NGRIP data when using
a fundamental period of 1500 years in equation (A2a). Frequency modulation distorts this period and creates sidebands
with power ~10 to ~1 kyr that control the duration of the warming episodes. Strong-amplitude modulation creates the two
cold intervals around 65 and 18 ka, where insolation is low. The time series are robust against 10% changes in the
controlling parameters (see also Figure 7). The insolation curve is the summer monthly mean at 65°N [Laskar et al., 2004].

“ waveform that is typical of the Antarctic surface

temperature [Shackleton et al., 2000]. Over long time scales,
the benthic

triangular ” observed between methane-synchronized temperature prox-

ies from Greenland and Antarctica [EPICA Comm. Mem-
bers, 2006; Blunier and Brook, 2001]. Saikku et al. [2009]
con

18 O proxy is known to be equally sensitive to
both global ice volume and deep ocean temperature [Shack-
leton, 2000], but here it is reasonable to infer that since
global ice volume does not change rapidly at the millennial
scale of the D-O, the benthic record is likely to be a relatively
accurate proxy of deep ocean temperature variation (J. Sver-
inghaus, personal communication, 2010). It is important to
emphasize that both the simple SIO model and ECBILT-
CLIO reproduce the phase relationship and

δ

rm that the deep ocean undergoes contemporaneous
temperature changes with those in proxies of air surface
temperature in Antarctica, while SST and surface air temper-
ature follow the changes seen in Greenland proxies. But
these intriguing relations between

18 O planktonic and ben-
thic records are shown in SIO to be the consequence of the
coupling between sea ice extent and oceanic temperature
[Saltzman and Moritz, 1980] that can be described as fol-
lows: Regionally, mean ocean temperature begins to increase
as sea ice reaches its maximum extent, eventually (several
hundred years later) reaching a peak at which sea ice begins
to retreat. As sea ice rapidly retreats, the ocean cools until it
becomes cold enough that sea ice begins to grow back. This
nonlinear dance between rapid sea ice response and slow
buildup of temperature of the ocean underneath is a relaxa-
tion oscillation described by the van der Pol equation (see
Appendix A), and the two variables play a similar role to that
of displacement and velocity in a simple linear pendulum
oscillation, only that here the oscillation is a self-sustained,
nonlinear one.

δ

”
observed in core MD95-2042 as shown in Figures 8 and 9.
It is important because the reason(s) for the peculiar contrast-
ing shape of waveforms in this and in other records of
shallow and deep ocean temperatures [Saikku et al., 2009]
are not understood and remain a mystery. However, from the
data and the model, it is easy to show that the key relation-
ship between the two variables is readily demonstrable: the
time integral of the planktonic (SST or air surface tempera-
ture proxy) time history is proportional to the benthic (deep
oceanic temperature proxy) time history. More physically
relevant, deep ocean temperature is

“ waveforms

/2 shifted with respect
to sea ice extent variation. This is the same relationship

π