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

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

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

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

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

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

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

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

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