Geoscience Reference
In-Depth Information
d
y
d
t
¼ Ω
lution is 3° by 3°, and there are 20 unevenly spaced vertical
levels in the ocean. The coupled model includes realistic
topography and bathymetry [Campin and Goosse, 1999].
CLIO is a primitive equation and includes a relatively so-
phisticated parameterization of vertical mixing [Goosse and
Fichefet, 1999]. A three-layer sea ice model, which takes into
account sensible and latent heat storage in the snow-ice
system, simulates the changes of snow and ice thickness in
response to surface and bottom heat
2
x
−
Am
ð
t
Þ
:
ðA2bÞ
The new equation (A2b) adds a new variable to the forced
Langevin equation (A1). Without the noise term and with A =
0, the system (A2a) and (A2b) transforms into a canonic van
der Pol equation on the unknown
θ
(t)=(3b/a)
1/2
x(t), giving
d
2
θ
d
t
2
−
a
ð1−θ
d
d
t
þ Ω
2
2
Þ
θ ¼ 0
:
ðA3Þ
flux. There are three main
boundary conditions in ECBILT-CLIO that can be easily kept
fixed (continental ice extent (described through albedo extent);
orbital parameters (eccentricity, tilt, and precession); and
greenhouse gas concentrations), and since each can be chosen
to represent either PI or LGM conditions, 2
3
independent
combinations can be run with these simple choices of mean
climate state (Figure A1). The results for
Equations (A2a) and (A2b) include positive feedback due to
sea ice albedo and greenhouse gases (lumped into parameter
a) and negative feedback that limit sea ice extent (cubic non-
linearity proportional to parameter b).
Ω
is the natural free
circular frequency of the undamped, linearized system. It is
important to state that the stochastic component in equation
(A2a) makes the oscillator
“
almost intransitive
”
[Lorenz,
1976], which thus exhibits the bistable behavior of equation
(A1) plus the free oscillation. The Am(t) term in the right-
hand side of equation (A2b) produces frequency modulation
of the periodic relaxation, creating much of the characteristic
pattern observed in Greenland
'
s ice cores. Equations (A2) or
(A3) represent self-sustained, relaxation oscillations such as
found in many nonlinear systems in physics, engineering,
biology [Saltzman, 2002; Gitterman, 2005], and oceanic cir-
culation [e.g., Marchal et al., 2007]. Other nonlinear oscilla-
tors with noncircular orbits in phase space can certainly
produce similar behavior; however, the van der Pol oscillator
is probably the simplest one to do so and thus is favored by
many scientists interested in modeling relaxation oscillations.
In practice, two free nondimensional parameters A and a
are adjusted to best fit the data, and b is used to control the
amplitude range of
θ
. The period T =2
π
/
Ω
trades off with A,
so that as T increases, A must decrease in order to maintain a
satisfactory fit to the data. Noise level controls the relative
effect of the insolation forcing through SR.
fixed insolation
forcing show bistable behavior for
“
intermediate
”
mean cli-
mates. Though the intermediate cases are only crudely repre-
sentative of the conditions existing during the last ice age, in
terms of global temperature, the results in Figure A1 show
temperature variability consistent with the range of Arctic
climates estimated from the ice core records. The ECBILT-
CLIO results also show that both LGM and PI boundary
conditions produce stable climates without D-O-like oscilla-
tions, in qualitative agreement with ice core data (Figure 1).
An alert reviewer pointed out that Friedrich et al. [2010]
made a series of modeling experiments with ECBILT and
concluded that the D-O-like oscillations are likely to be an
artifact of the code. They show that the oscillations disappear
(actually they become damped but are still visible) if the
“
is shut off. This is not very surpris-
ing, however, as in our experience, D-O-like oscillations in
ECBILT-CLIO fade if the full topography of the ice sheets is
used as a boundary condition, or the water column
Hudson Strait out
ow
”
s temper-
ature gradient is very small or zero (D-O-like instability in
the code seems to require fresh and cold shallow water
overlying salty and warm deeper ocean), or whenever sea
ice retreats, allowing the deep ocean to cool (e.g., Figure 9).
D-O-like oscillations also peter out and disappear south of
Newfoundland and North of the Antarctic Ocean because the
sea ice cover is an important control of deep-ocean warming
(e.g., Figure 2), and thus, the D-O-like oscillations in the
code are mostly a polar region phenomenon. The point being
that the example of Friedrich et al. [2010] lacks generality
because it cannot rule out the possibility that some other
“
'
A3. ECBILT-CLIO
We use the global, atmosphere-ocean-ice coupled climate
model of intermediate complexity, EBILT-CLIO (version 3).
The atmospheric component, ECBILT, is a spectral T21 global
quasi-geostrophic model with simple parameterization for
the diabatic heating due to radiative fluxes, the release of
latent heat, and the exchange of sensible heat with the surface
[Opsteegh et al., 1998]. The model contains a full hydrolog-
ical cycle, which is closed over land by a bucket model for
soil moisture. Synoptic variability associated with weather
patterns is explicitly computed. The ocean component CLIO
is a free-surface ocean general circulation model coupled to a
thermodynamic-dynamic sea ice model. The horizontal reso-
made to ECBILT-CLIO somewhere in its thousands of
lines of code, for instance, by changing diffusivity coeffi-
cients or its response to fresh water hosing [e.g., Knutti et al.,
2004] will fully restore the D-O-like oscillations. We certainly
agree that computer simulations should not be taken as
proof of natural mechanisms, yet in some cases, models can
x
”
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