On the (Nonlinear) Causes of
Abrupt Climate Change During
the Last Ice Age
J.A. Rial
Wave Propagation Lab, University of North Carolina-Chapel Hill
The Astronomical Theory of the
Ice Ages
•
•
•
Precession
Tilt
Eccentricity
The Milankovitch
Periodicities
Eccentricity
Tilt or Obliquity
41,000years
Eccentricity
100,000years
Precession of the
Equinoxes
21,000years
Evidence of Climate's Nonlinear
Response to Astronomical
Forcing
•
•
•
Frequency shifts
Frequency (or phase) modulation
Fast warming, slow cooling and
'self
-similarity'
'self-similarity'
Frequency shift at the Mid-Pleistocene Transition
Frequency Modulation
Tilt
Eccentricity
Rial,J.A., Science,
Science, 285 (1999)
Saw-tooth self-similarity: Fast Warming-Slow Cooling
DansgaardOeschger
oscillations
Rial et al., Climatic Change, 2004
A “toy” model to visualize the
sawtooth asymmetry and FM
Wave Propagation Lab, UNC-Chapel Hill
(No external forcing
required)
Sawtooth
waveform
Frequency
Modulation
Ice Volume and Temperature
Fitting the long-period records
Modeling the Greenland (GRIP)
Short-Period Time Series
Greenland ice core
w
w
w
w
w
w
today
100,000
yrs ago
W: Abrupt warming (10oC or more)
FREQUENCY DEMODULATION OF GRIP REVEALS A 2.75ky 'CARRIER'
Modulating Phase or "Intelligence"
Best Fitting Periods 72.1ky and 41ky
Modulating Phase (LP filtered)
1/75 ~ 1/41-1/95
1/72.1 = 1/41 - 1/95
(combination tone)
1/35 ~ 1/19-1/41
Heinrich events->
1
The thermal oscillator has the
form of a Van der Pol nonlinear
equation
Wave Propagation Lab, UNC-Chapel Hill
The Van der Pol Equation
(Self-exciting oscillations with a limit cycle)
d2x/dt2 + v(x2-1)dx/dt + ωo2 x = 0
Is equivalent to
dx/dt = y - v (x3/3 -x)
dy/dt = - ωo2x
Wave Propagation Lab, UNC-Chapel Hill
In a simplified climate model
An energy balance equation
CT dT/dt = -αL -βT + Q(1+ε coswt)
Coupled to a logistic growth equation for the sea ice
CLdL/dt = T/α - (a L3/3 -bL)
Become a Van der Pol equation for the sea ice extent
d2L/dt2 + v(L2-1) dL/dt + ωo2 L = G(1+ε coswt)
…and with G=0, it gives Saltzman's nonlinear thermal
oscillator
CT, CL, Q, α, β, a, b are positive constants
Saltzman's NONLINEAR THERMAL OSCILLATOR
ICE AGE ATMOSPHERE
L(t)
0
T(t)
SEA ICE
OCEAN
θ(t)
Saltzman's NONLINEAR THERMAL OSCILLATOR
ICE AGE ATMOSPHERE
L(t)
0
T(t)
SEA ICE
OCEAN
θ(t)
Saltzman's NONLINEAR THERMAL OSCILLATOR
ICE AGE ATMOSPHERE
L(t)
0
T(t)
SEA ICE
OCEAN
θ(t)
Saltzman's NONLINEAR THERMAL OSCILLATOR
ICE AGE ATMOSPHERE
L(t)
0
T(t)
SEA ICE
OCEAN
θ(t)
Saltzman's NONLINEAR THERMAL OSCILLATOR
ICE AGE ATMOSPHERE
L(t)
0
T(t)
SEA ICE
OCEAN
θ(t)
The self-sustained, relaxation oscillation of the thermal oscillator
Forcing Saltzman's Oscillator
with Milankovitch Cycles
Linear, forced by cosine
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Animation decompressor
are needed to see this picture.
Nonlinear, forced by cosine
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Time (arbitrary units)
Nonlinear, forced by simulated Insolation
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Time (arbitrary units)
GRIP vs VdPol Model
VdPol Model is Robust
GRIP vs VdPol Model
GRIP
Kyears B.P.
Prediction
GRIP vs VdPol Model
Kyears B.P.
Concluding Remarks
•
A simplified climate model describing the nonlinear oscillations of sea
ice driven by deep ocean temperature closely reproduces the GRIP
data. A natural free period of 2.75ky retrieved form the data by
demodulation appears to be the free period of the sea
ice/atmosphere/ocean system.
•
Frequency modulation of this free oscillation by the insolation (tilt and
precession) generates time-series features consistent with many of
the puzzling features of the GRIP time series.
•
The model is deterministic, and it can be chaotic.
•
Though the origin of the 2.75ky period is not resolved, it is
commensurate with predictions made by Saltzman's sea ice
oscillator. We shall use GCMs to understand the origin and physics of
this period.
Research supported by NSF grant ATM-0241274
Wave Propagation Laboratory,
University of North Carolina-Chapel Hill
Ice core data from Greenland (millennial scale)
Dansgaard-Oeschger
Oscillations.
-16
FM in the D/O
Relative Temp (oC)
FM in the D/O
-6
-12
-16
Carrier: 2.7ky
Modulator: 7.5ky
The Paleoclimate Time Series
A Complex tale of
•
•
•
•
Fast warming-Slow cooling
Frequency Modulation
Non-Stationarity
Chaos and Order
Sawtooth
Mid-Pleistocene Transition
FM
The abrupt warming episodes of the last Ice Age
…and the Dow Jones crash of 1987
What does self-similarity mean?
The abrupt warming episodes of the last Ice Age
…and the Dow Jones crash of 1987
What does self-similarity mean?
Fast Warming - Slow Cooling
and The Younger Dryas (YD) event
Milankovitch Forcing and Ice Core Data
~ 21ky
0
-10
Tilt 41ky
-20
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QuickTime™ and a
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Time (arbitrary units)
Global temperature over the last Millennium ∆T (oC)
1
0
… and during the last Ice Age
Normal abrupt climate change, or CO2 warming?
Time series from the Antarctic Ice cap
Today
400,000 years ago
The self-sustained, relaxation oscillation of the thermal oscillator
1
2
4
3
4
1
3
2
Frequency modulation (FM) in millennial-scale climate series
7.5ky
1/74 ~ 1/41-1/95
1/37 ~ 1/19-1/41
Deep-sea
sediment
Ice core
Sea-surface
Temp