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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 1
Topics in Applied Mathematics:Introduction to the Mathematics of Climate
Mondays and Wednesdays 2:30 – 3:45http://www.math.umn.edu/~mcgehee/teaching/Math5490-2014-2Fall/
Streaming video is available athttp://www.ima.umn.edu/videos/
Click on the link: "Live Streaming from 305 Lind Hall".Participation:
https://umconnect.umn.edu/mathclimate
Math 5490
Is the globe warming?
What determines the Earth’s temperature?
What is the role of human activity?
How big is the problem?
Math 5490
Anthropogenic Warming
Math 5490 9/8/2014
Is the globe warming?What determines the Earth’s temperature?
What is the role of human activity?
How big is the problem?
Anthropogenic Warming
Math 5490 9/8/2014
Global Mean Temperature
Technical Summary, IPCC AR4, p.37http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_TS.pdf
Is the Globe Warming?
Anthropogenic Warming
Math 5490 9/8/2014
Petit, et al, Nature 399 (June 3 1999), pp.429-436
Antarctic Temperature Data
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000
year
delta
deg
C
Is the Globe Warming?
Anthropogenic Warming
Math 5490 9/8/2014
-10-8-6-4-2024
050100150200250300350400450
kiloyear bp
tem
pera
ture
Petit, et al, Nature 399 (June 3 1999), pp.429-436
Antarctic Temperature Data
Is the Globe Warming?
Anthropogenic Warming
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 2
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.
Benthic 18O Data
Is the Globe Warming?
2.5
3
3.5
4
4.5
5
5.5
‐4500 ‐4000 ‐3500 ‐3000 ‐2500 ‐2000 ‐1500 ‐1000 ‐500 0
Benthic D
ata (δ
18O)
time (Kyr)
Anthropogenic Warming
Math 5490 9/8/2014
Is the Globe Warming?
Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (2008)
Anthropogenic Warming
Math 5490 9/8/2014
Is the globe warming?
Yes, but, from a geologic perspective, not so much.
What determines the Earth’s temperature?What is the role of human activity?
How big is the problem?
Anthropogenic Warming
Math 5490 9/8/2014
Why isn’t the Earth a Snowball?
4 (1 )T S
where T = surface temperature (Kelvin)
S = solar influx (W/m2)
α = albedo (reflectivity)
σ = the Stefan-Boltzmann constant
For current values , 255 18 0T K C F
Heat Balance
solar incoming (visible)outgoing (infrared)
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Math 5490 9/8/2014
What Determines the Earth’s Temperature?
Gary Stix, Scientific American September 2006, pp.46-49
Anthropogenic Warming
Math 5490 9/8/2014
Heat Balance
Historical Overview of Climate Change Science, IPCC AR4, p.96http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 3
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Petit, et al, Nature 399 (June 3 1999), pp.429-436
100
150
200
250
300
350
400
‐450 ‐400 ‐350 ‐300 ‐250 ‐200 ‐150 ‐100 ‐50 0
atmos CO
2(ppm
)
time (Kyr)
Atmospheric CO2 (Vostok data)
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Petit, et al, Nature 399 (June 3 1999), pp.429-436
Atmospheric CO2 (Vostok data)
100
150
200
250
300
350
400
‐450 ‐400 ‐350 ‐300 ‐250 ‐200 ‐150 ‐100 ‐50 0
atmos CO
2(ppm
)
time (Kyr)
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Pam Martin, University of Chicago, 2010
Atmospheric CO2 & Temperature (Vostok data)
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
‐10‐8‐6‐4‐2024
68
1012
150 200 250 300 350 400 450
δT (°C)
CO2 (ppm)
Current conditions are well outside the range recorded in the ice core data.
Vostok Data
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Vostok Data
‐10‐8
‐6‐4‐202468
1012
150 200 250 300 350 400 450
δT (°C)
CO2 (ppm)
Extrapolate linear regression to 400 ppm CO2.
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
http://www.carleton.edu/departments/geol/DaveSTELLA/Carbon/long_term_carbon.htm
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 4
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Rainwater containing dissolved CO2 falling on silicate rocks replaces a silicon atom with a carbon atom, ultimately producing calcium carbonate (limestone) and silicon dioxide (quartz). For
example, calcium silicate (Wollastonite):
3 2 3 2CaSiO CO CaCO SiO
Under volcanic conditions, the carbon atom is replaced by a silicon atom, completing the long term carbon cycle.
3 2 3 2CaCO SiO CaSiO CO
Silicate Weathering
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (2008)
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
http://www.carleton.edu/departments/geol/DaveSTELLA/Carbon/long_term_carbon.htm
Math 5490 9/8/2014
Is the globe warming?
Yes, but, from a geologic perspective, not so much.
What determines the Earth’s temperature?
Physics! Solar heating plus the carbon cycle.
What is the role of human activity?How big is the problem?
Anthropogenic Warming
Math 5490 9/8/2014
Anthropogenic WarmingWhat Determines the Earth’s Temperature?
The Short-Term Carbon Cycle
Couplings Between Changes in the Climate System and Biogeochemistry, IPCC AR4, p.513http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH07.pdf
Math 5490 9/8/2014
Anthropogenic WarmingWhat is the Role of Human Activity?
http://www.climate.noaa.gov/images/about_climate/greenhouse_maunaloa.jpg
The Mauna Loa Observatory
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 5
Anthropogenic WarmingWhat is the Role of Human Activity?
http://scrippsco2.ucsd.edu/graphics_gallery/mauna_loa_record/mauna_loa_record.html
Math 5490 9/8/2014
Anthropogenic WarmingWhat is the Role of Human Activity?
100
150
200
250
300
350
400
‐450 ‐400 ‐350 ‐300 ‐250 ‐200 ‐150 ‐100 ‐50 0
atmos CO2(ppm
)
time (Kyr)
Math 5490 9/8/2014
Anthropogenic WarmingWhat is the Role of Human Activity?
Carbon isotope ratios and atmospheric oxygen depletion indicate that the increase in atmospheric CO2 comes from burning fossil fuels.
Changes in Atmospheric Constituents and in Radiative Forcing, IPCC AR4, Chap. 2, p.138
http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH02.
pdf
Math 5490 9/8/2014
Anthropogenic WarmingWhat is the Role of Human Activity?
Models with and without human activity.
Understanding and Attributing Climate Change, IPCC AR4, Chap. 9, p.684
http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_Ch09.
pdf
Math 5490 9/8/2014
Is the globe warming?
Yes, but, from a geologic perspective, not so much.
What determines the Earth’s temperature?
Physics! Solar heating plus the carbon cycle.
What is the role of human activity?
How big is the problem?
Anthropogenic Warming
Math 5490 9/8/2014
http://nobelprize.org/nobel_prizes/peace/laureates/2007/
The IPCC Fourth Assessment Report
Anthropogenic Warming
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 6
Prediction Methodology
Global Climate Projections, IPCC AR4, p.753http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH10.pdf
Anthropogenic WarmingThe IPCC Fourth Assessment Report
Math 5490 9/8/2014
Global Mean Temperature Predictions
Summary for Policy Makers, IPCC AR4, p. 14http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_SPM.pdf
Anthropogenic WarmingThe IPCC Fourth Assessment Report
Math 5490 9/8/2014
Surface Temperature Predictions
Technical Summary, IPCC AR4, p. 72http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_SPM.pdf
Anthropogenic WarmingThe IPCC Fourth Assessment Report
Math 5490 9/8/2014
The Last Interglacial Period
Technical Summary, IPCC AR4, p. 58http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_SPM.pdf
Global average sea level was likely between 4 and 6 m higher during the last interglacial period, about 125,000 years ago, than during the 20th
century, mainly due to the retreat of polar ice. Ice core data suggest that the Greenland Summit region was ice-covered during this period, but reductions in the ice sheet extent are indicated in parts of southern
Greenland. Ice core data also indicate that average polar temperatures at that time were 3°C to 5°C warmer than the 20th century because of
differences in the Earth’s orbit. The Greenland Ice Sheet and other arctic ice fields likely contributed no more than 4 m of the observed sea level
rise, implying that there may also have been a contribution from Antarctica.
Anthropogenic Warming
Math 5490 9/8/2014
How big is the problem?
-10-8-6-4-2024
050100150200250300350400450
kiloyear bp
tem
pera
ture
Petit, et al, Nature 399 (June 3 1999), pp.429-436
Antarctic Temperature Data
Anthropogenic Warming
Math 5490 9/8/2014
How big is the problem?
Vostok Data
Anthropogenic WarmingHow big is the problem?
‐10‐8‐6‐4‐202468
1012
150 200 250 300 350 400 450
δT (°C)
CO2 (ppm)
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 7
USA in the Ice Free
Earth
Anthropogenic WarmingHow big is the problem?
Clarence Lehman, University of Minnesota, 2006
Math 5490 9/8/2014
The Modern Ice Free EarthSea level rises 63 meters.
Anthropogenic WarmingHow big is the problem?
Math 5490 9/8/2014
Clarence Lehman, University of Minnesota, 2006
Is the globe warming?Yes, but, from a geologic perspective, not so much.
What determines the Earth’s temperature?Physics! Solar heating plus the carbon cycle.
What is the role of human activity?We are adding CO2 to the atmosphere.
How big is the problem?Not big. Yet. ...
Anthropogenic Warming
Summary
Math 5490 9/8/2014
Energy Balance
Conservation of Energy
temperature change ~ energy in – energy out
short wave energyfrom the Sun
long wave energyfrom the Earth
Everything else is detail.
Math 5490 9/8/2014
Energy BalanceStefan‐Boltzmann Law
4F Tpower flux (W/m2) temperature (K)
Stefan-Boltzmann constant8 2 45.67 10 W/m K
Math 5490 9/8/2014
Energy BalanceStefan‐Boltzmann Law
4F Tpower flux (W/m2) temperature (K)
Stefan-Boltzmann constant8 2 45.67 10 W/m K
Examplesurface temperature of the Sun: 5780K
power flux: 5.67x10-8 x (5780)4 = 6.33x107 W/m2
total solar power output: 6.33x107 x 4π(rS)2 , where rS = radius of the sun = 6.96x108 m
total solar output: 3.85x1026 W200 nanoseconds = time it takes for the sun to produce
the equivalent of the annual global electricity productionhttp://astronomybythecosmos.com/tag/sun/
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 8
Energy BalanceInsolation
Solar flux at a distance r from the sun:
27 27 2
2
6.33 10 4 6.33 10 W/m4
S Sr rFr r
rS = 6.96x108 mr = 1.5x1011 m
21368 W/mF
2 6
17
W, radius of Earth 6.37 10 m
1.74 10 WE EF r r
F
Power intercepted by the Earth:
Math 5490 9/8/2014
Energy BalanceInsolation
Solar flux at a distance r from the sun:27 2
7 22
6.33 10 4 6.33 10 W/m4
S Sr rFr r
rS = 6.96x108 mr = 1.5x1011 m
21368 W/mF
2 6
17
W, radius of Earth 6.37 10 m
1.74 10 WE EF r r
F
Power intercepted by the Earth:
Biologically Stored Energytotal coal reserves: 1015 kgenergy content: 3x107 J/kg
total energy in coal reserves: 3x1022 J= 2 days of insolation
Math 5490 9/8/2014
Energy BalanceInsolation
Simple ModelAssume that Earth is a perfectly thermally conducting black body.
Global Average Insolationintercepted flux: F = 1368 W/m2
Earth cross-section: πrE2surface area: 4πrE2
average flux: 1368/4 = 342 W/m2 = Q
4
1 41 4 8342 5.67 10
279K 6 C 43 F
Q T
T Q
Dynamics
4dTR Q Tdt
heat capacity stable equilibrium
Math 5490 9/8/2014
Energy BalanceAlbedo
Not all the insolation reaches the surface. Some is reflected back into space.
The proportion reflected is called the albedo, denoted α .For Earth, α ≈ 0.3 .
1 41 4 80.7 0.7 342 5.67 10255K 18 C 0 F
T F
Simple ModelAssume that Earth is a perfectly thermally conducting black body,
but only 70% of the insolation is absorbed.
Dynamics
4(1 )dTR Q Tdt
stable equilibrium
Math 5490 9/8/2014
Energy BalanceAlbedo
Not all the insolation reaches the surface. Some is reflected back into space.
The proportion reflected is called the albedo, denoted α .For Earth, α ≈ 0.3 .
1 41 4 80.7 0.7 342 5.67 10255K 18 C 0 F
T F
Why isn’t the Earth a snowball?Simple Model
Assume that Earth is a perfectly thermally conducting black body,but only 70% of the insolation is
absorbed.
Math 5490 9/8/2014
Energy Balance
Planck’s Function
3
22 1( , )
1h kThB Tc e
flux density
frequency
temperature
Planck’s constant
speed of light Boltzmann’s constant
Black Body Radiation
wave number c
Raymond T. Pierrehumbert, Principles of Planetary Climate, Cambridge University Press, 2010.
Math 5490 9/8/2014
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Math 5490 9/8/2014
Richard McGehee, University of Minnesota 9
Energy BalanceInsolation vs. OLR
Raymond T. Pierrehumbert, Principles of Planetary Climate, Cambridge University Press, 2010.
OLR = Outgoing Longwave Radiation
insolation
OLR
visibleinfrared
Math 5490 9/8/2014
Energy BalanceGreenhouse Gases
Raymond T. Pierrehumbert, Infrared radiation and planetary temperature, Physics Today, 2011.
Absorption Spectra
Math 5490 9/8/2014
Energy BalanceGreenhouse Effect
How to learn more.
Math 5490 9/8/2014