Download Glacial Cycles Richard McGehee Seminar on the Mathematics of Climate Change

Glacial Cycles
Glacial Cycles
Richard McGehee
Temperatures in the Cenozoic Era
Seminar on the Mathematics of Climate Change
School of Mathematics
October 7, 2009
Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (2008)
Glacial Cycles
18O
Glacial Cycles
in Foraminifera Fossils During the Past 4.5 Myr
18O
2.5
2.5
3
Benthic Data (δ18O
O)
Benthic Data (δ18
8O)
3
35
3.5
4
4.5
5
3.5
4
4.5
5
5.5
‐4500
in Foraminifera Fossils During the Past 1.0 Myr
5.5
‐4000
‐3500
‐3000
‐2500
‐2000
‐1500
‐1000
‐500
0
time (Kyr)
Lisiecki, L. E., and M. E. Raymo (2005), A Pliocene-Pleistocene stack of 57 globally distributed benthic d18O
records, Paleoceanography,20, PA1003, doi:10.1029/2004PA001071.
Glacial Cycles
Recent (last 400 Kyr) Temperature Cycles
Vostok Ice Core Data
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‐900
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‐400
‐300
‐200
‐100
0
time (Kyr)
Lisiecki, L. E., and M. E. Raymo (2005), A Pliocene-Pleistocene stack of 57 globally distributed benthic d18O
records, Paleoceanography,20, PA1003, doi:10.1029/2004PA001071.
Glacial Cycles
What Causes Glacial Cycles?
Widely Accepted Hypothesis
The glacial cycles are driven by the variations in the Earth’s orbit
(Mil k it h Cycles),
(Milankovitch
C l ) causing
i a variation
i ti in
i incoming
i
i solar
l
radiation (insolation).
This hypothesis is widely accepted, but also widely regarded as
insufficient to explain the observations.
The additional hypothesis is that there are feedback
mechanisms that amplify the Milankovitch cycles. What these
feedbacks are and how they work is not fully understood.
J.R. Petit, et al (1999) Climate and atmospheric history of the past 420,000 years from the Vostok ice core,
Antarctica, Nature 399, 429-436.
1
Glacial Cycles
Glacial Cycles
Solar Forcing (Hays, et al)
Hays, et al, Science 194 (1976), p. 1125
http://en.wikipedia.org/wiki/Milankovitch_cycles
Glacial Cycles
Glacial Cycles
Hays Summary
Climate Response, Hays, et al
Increasing
contribution
Forcing
Response
precession
eccentricity
obliquity
obliquity
eccentricity
precession
The explanation is that there are nonlinear feedbacks.
Is there another explanation?
Three different temperature proxies from sea sediment data.
Hays, et al, Science 194 (1976), p. 1127
Hays, et al, Science 194 (1976), p. 1127
Glacial Cycles
Glacial Cycles
Climate Response (Zachos, et al)
Zachos Summary
Increasing
contribution
Power spectrum of
climate for the last 4.5
Myr. Note the peaks at
41Kyr and 100 Kyr.
Forcing
Response
precession
obliquity
obliquity
eccentricity
eccentricity
precession
Nonlinear effects?
Forcing is defined as the maximum insolation at latitude 65° N.
Is there another definition of forcing?
Zachos, et al, Science 292 (2001), p. 689
Zachos, et al, Science 292 (2001), p. 689
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Glacial Cycles
Glacial Cycles
Ice Albedo Feedback Model
Heat Balance
R
∂T
= Qs ( y ) (1 − α ( y,η ) ) − ( A + BT ) + C (T − T )
∂t
insolation
albedo
outward
radiation
heat
transport
y = sine of latitude
T(y) = annual mean temperature
Qs(y) = annual mean insolation
Q = global annual mean insolation
This equation has a stable equilibrium consisting of polar ice caps.
The latitude of the equilibrium ice boundary and the equilibrium global annual
mean temperature are functions of the parameters.
Historical Overview of Climate Change Science, IPCC AR4, p.96
K.K. Tung, Topics in Mathematical Modeling, Princeton (2007), Chapt 8
http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf
Glacial Cycles
Glacial Cycles
Ice Albedo Feedback Model
Ice Albedo Feedback Model
∂T
R
= Qs ( y ) (1 − α ( y,η ) ) − ( A + BT ) + C (T − T )
∂t
Idea
s( y) =
Instead of solar forcing (maximum insolation at 65 N latitude), use the global
annual mean temperature predicted by the model.
Using Kepler’s Laws, we can compute:
Q=
2π
2 ⌠
s( y) = 2 ⎮
π ⌡0
1−
(
2 ⌠
⎮
2π
π 2 ⌡0
1−
(
1 − e2
)
2
1 − y 2 sin β cos γ − y cos β d γ
Note that Q, the global annual mean insolation depends only on the
semimajor axis and the eccentricity.
Ka 2
Note that s(y), the insolation distribution by latitude, depends only on the
obliquity.
1 − e2
)
2
1 − y sin β cos γ − y cos β d γ
2
Ka 2
Q=
Note that the effect due to precession disappears when averaged over a year.
a = semimajor axis
e = eccentricity
β = obliquity
Glacial Cycles
Glacial Cycles
Global Annual Average Insolation
Global Annual Average Insolation
Q=
Laskar:
Ka 2
1− e
Q=
2
Ka 2
1 − e2
Eccentricity
0.06
0.05
eccentriicity
0.04
0.03
0.02
0.01
0.00
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0
time (Kyr)
Note periods of about 100 Kyr and 400 Kyr.
Semi major axis does not change much:
.005% corresponding to .01% change in global average insolation
J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth,
Astronomy & Astrophysics 428, 261–285.
As e varies between 0 and 0.06, (1-e2)-1/2 varies between 1 and 0.0018,
or about 0.2%. (Twenty times the effect due to a.)
J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth,
Astronomy & Astrophysics 428, 261–285.
3
2 ⌠
⎮
s( y) =
Glacial Cycles
Glacial Cycles
Relative Insolation Function
Relative Insolation Function
2π
π 2 ⌡0
1−
)
(
2
1 − y 2 sin β cos γ − y cos β d γ
1.3
Obliquity
22
24.5
1.2
green = obliquity of
22.0°
24.5
1.1
1
23.5
23.0
red = obliquity of
24.5°
22.5
relatve insolation
obliquity (degrees)
24.0
0.9
0.8
0.7
22.0
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‐100
0
0.6
time (Kyr)
0.5
Note period of about 41 Kyr.
0.4
J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth,
Astronomy & Astrophysics 428, 261–285.
0
Glacial Cycles
0.1
0.2
0.3
0.4
0.5
0.6
sine(latitude)
0.7
0.8
0.9
1
Glacial Cycles
Global Annual Mean Temperature
Annual Mean Insolation
(as a function of latitude)
Qs ( y )
15
y = sine of latitude
Computed global
mean temperature for
extremes in
eccentricity and
obliquity
Q = gglobal annual mean insolation
(depends primarily on eccentricity)
s(y) = relative insolation as a function of latitude
(depends only on obliquity)
14.8
14 6
14.6
14.4
14.2
14
We can use the ice-albedo feedback model to compute the equilibrium ice
line, the global mean temperature, and the polar mean temperature as
functions of eccentricity and obliquity.
13.8
beta=24.5
13.6
beta=22
e=0.00
e=0.05
Glacial Cycles
Glacial Cycles
Polar Annual Mean Temperature
Conclusions
‐17.5
Computed polar mean
temperature for
extremes in
eccentricity and
obliquity
1. Precession doesn’t
matter.
‐18
2. Obliquity is more
important than
eccentricity.
‐18.5
‐19
‐19.5
beta=24.5
‐20
3. Polar temperatures
vary twice as much
as global
temperatures.
beta=22
e=0.00
e=0.05
4
Glacial Cycles
Glacial Cycles
14.8
Conclusions
obliquity
obliquity
Increasing
contribution
eccentricity
14.2
14
eccentricity
precession
14.4
13.8
Annual Global
Mean
Temperature
The data are
not duplicated
by either
method, but
note the
spectral
similarities.
precession
When the usual definition of forcing is replaced by the predictions of the icealbedo feedback model, the relative effects due to the Milankovitch cycles
agree with the observed data.
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0
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0
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0
time (Kyr)
2.5
3
data
Benthic Data (δ
δ18O)
Response
3.5
4
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5
5.5
‐4500
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time (Kyr)
580
560
Insolation
at 65° N at
summer
solstice
540
Qday (W/m2)
Model Prediction
GMT (°C)
14.6
ice-albedo
model
520
500
480
460
440
420
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time (Kyr)
ice-albedo
model
Power
Glacial Cycles
0
Heat Balance
obliquity →
0.01
0.02
Power Spectra
0.03
0.04
0.05
0.06
frequency 1/Kyr
data
Power
Note
dominance of
the obliquity
signal.
Climate and the Earth’s Glacial Cycles
0.00
eccentricity
↓
0.01
0.02
0.03
0.04
0.05
0.06
frequency 1/Kyr
precession →
Power
Insolation
at 65° N at
summer
solstice
0
0.01
0.02
0.03
0.04
0.05
frequency 1/Kyr
0.06
Historical Overview of Climate Change Science, IPCC AR4, p.96
http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf
Climate and the Earth’s Glacial Cycles
Glacial Cycles
Not Explained By Ice Albedo Feedback
Temperatures in the Cenozoic Era
R
∂T
= Qs ( y ) (1 − α ( y,η ) ) − ( A + BT ) + C (T − T )
∂t
insolation
albedo
outward
radiation
heat
transport
The observed amplitude of temperature variation is about
5 times higher than that predicted by the model.
Other feedback mechanisms (e.g. greenhouse gases) clearly
matter.
The long-term trends are also not in the model and are not
explained by Milankovitch cycles.
Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (2008)
5
Glacial Cycles
More from Zachos
A. Power spectrum of climate for
the last 4.5 Myr. Note the peaks at
41Kyr and 100 Kyr.
B. Power spectrum of climate for
the period 25 Myr bp to 20.5 Myr
bp. Note the new peak at 400 Kyr
and the “split” peaks at 126Kyr and
95 Kyr.
Zachos, et al, Science 292 (2001), p. 689
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