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Math 5490 10/13/2014 Math 5490 October 13, 2014 Topics in Applied Mathematics: Isotopes as Climate Proxies Introduction to the Mathematics of Climate Mondays and Wednesdays 2:30 3:45

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 1

Topics in Applied Mathematics: Introduction to the Mathematics of Climate

Mondays and Wednesdays 2:30 – 3:45

http://www.math.umn.edu/~mcgehee/teaching/Math5490-2014-2Fall/

Streaming video is available at

http://www.ima.umn.edu/videos/

Click on the link: "Live Streaming from 305 Lind Hall". Participation:

https://umconnect.umn.edu/mathclimate

Math 5490

October 13, 2014

Isotopes as Climate Proxies

How do we know the past climates?

Math 5490 10/13/2014

Isotopes as Proxies

Summary

mass balance phase 2 δ2 phase 1 δ1 total δ0 p q

1 2

p q     

1 2

q p           equilibrium fractionation ε Math 5490 10/13/2014

Isotopes as Proxies

Biology Matters

2 2 2 3 3 3 3 3 3

CO H O H CO H HCO HCO H CO Ca CO CaCO

      

        atmosphere

  • cean

foraminifera Temperature dependent fractionation occurs at every step. The result: the δ18O in foram shells is about +30‰ compared with the surrounding water (depending on temperature). (δ18O)/dT ≈ ‐0.25 ‰/ ⁰C (Reference: Pierrehumbert’s book)

And then there’s carbon.

Math 5490 10/13/2014

and is complicated.

Isotopes as Proxies

Biology Matters and is yet still more complicated.

Fractionation is about ‐25‰.

2 2 6 12 6 2

6CO 6H O C H O 6O    photosynthesis δ1 = δ13C δ2 = δ13C

2 1

0.025     Result: Plants, animals, coal, and oil are all lighter in 13C than inorganic carbon.

Math 5490 10/13/2014

Isotopes as Proxies

Math 5490 10/13/2014

Carbon 14

Nitrogen neutron Carbon 14 proton Carbon 14 Nitrogen electron neutrino

Carbon 14 is created by cosmic rays in the upper atmosphere. Carbon 14 decays in the biosphere with a half‐life of about 6000 years. Carbon 14 is the basis of carbon dating. Good for about 50,000 years. At 60,000 years, it is down to 2‐10 of its original level.

http://en.wikipedia.org/wiki/Carbon-14

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 2

Isotopes as Proxies

Math 5490 10/13/2014

Carbon 14 The rate of production of carbon 14 in the past is largely

  • unknown. It depends on cosmic events producing neutron

reaching the upper atmosphere and it depends on variations in the Earth’s magnetic field. Carbon 14 is also fractionated by biology, so the concentration in the atmosphere depends on biological 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

Isotopes as Proxies

Math 5490 10/13/2014

Isotopes as Proxies

Math 5490 10/13/2014

1982 – 2002

δ13C: ‐7.6‰ to ‐8.1 ‰ Recent History

Isotopes as Proxies

Math 5490 10/13/2014

1982 – 2002

atmospheric C02: 340 ppm to 372 ppm Recent History

Isotopes as Proxies

Math 5490 10/13/2014

1 gigatonne (Gt) = 109 metric tonnes = 1012 kilograms (kg) = 1015 grams = 1 petagram (Pg) 1 tonne = 1000 kg = 2205 lb = 1.102 tons Example: current fossil carbon emissions: about 9 GtC/yr atomic wt of carbon: 12 atomic wt of oxygen: 16 molecular wt of CO2: 44 (= 12 + 16 + 16) Current emissions of CO2: (44/12)x9 = 33 GtC02/yr How do gigatonnes convert to ppm in the atmosphere?

Carbon Measurements

Isotopes as Proxies

Math 5490 10/13/2014

Atmospheric pressure at sea level: 14.7psi = (14.7psi)/((6.45cm2/in2)(2.205lb/kg)) = 1.03 kg/cm2 Mass of atmosphere above one square meter: 10300 kg Surface area of Earth: 5.10x1014m2 Total mass of atmosphere: (1.03x104)(5.10x1014) = 5.25x1018 kg = 5.25x106 Pg Approximate composition of atmosphere: N2: 80% molecular wt 14x2 = 28 O2: 20% molecular wt 16x2 = 32 Mean molecular wt of atmosphere: 0.8x28 + 0.2x32 = 28.8

Carbon Measurements

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 3

Isotopes as Proxies

Math 5490 10/13/2014

Total mass of atmosphere: 5.25x106 Pg Mean molecular wt of atmosphere: 28.8 g/mole Number of moles in atmosphere: (5.25x1021g)/(28.8g/mole) = 1.82x1020 moles Number of moles in one millionth of the atmosphere: 1.82x1014 Atomic wt of C: 12 g/(mole of CO2) Mass of 1 ppm CO2: (12 g/mole)(1.82x1014 mole/ppm) = 2.19x1015 g = 2.19 Pg Actual conversion:

1 ppm by volume of atmosphere CO2 = 2.13 Gt C

Source: CDIAC (Carbon Dioxide Information Analysis Center)

cdiac.ornl.gov/pns/convert.html

Carbon Measurements

Isotopes as Proxies

Math 5490 10/13/2014

Atmospheric Carbon

850 Gt 750 Gt 700 Gt 800 Gt

Isotopes as Proxies

Math 5490 10/13/2014

1982 – 2002 increase in atmospheric CO2 340 to 372 ppm 724 GtC to 792 GtC : 68 GtC emissions 6 GtC/yr for 20 yr : 120 GtC About 57% stayed in the atmosphere. δ13C: -7.6‰ to -8.1 ‰ coal?: δ2 air: δ1 air: δ0

Isotopes as Proxies

δ13C Budget

p = 724/792 = 0.914 q = 68/792 = 0.086 724 GtC δ13C: -7.6‰

1 2 2 1

( ) ( 8.1 0.914 ( 7.6)) / 0.086 13.4 p q p q                  68 GtC 792 GtC δ13C: -8.1‰ coal?: δ13C = -13.4‰ Other processes are important.

Math 5490 10/13/2014

Isotopes as Proxies

Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (2008) Math 5490 10/13/2014

Paleocence‐Eocene Thermal Maximum (PETM)

http://static.palaeontologyonline.com/Figure1.jpg

Isotopes as Proxies

Paleocence‐Eocene Thermal Maximum (PETM)

Math 5490 10/13/2014

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 4

http://maya-gaia.angelfire.com/mammals_rise_title_thm.jpg

Isotopes as Proxies

Paleocence‐Eocene Thermal Maximum (PETM)

Math 5490 10/13/2014

Isotopes as Proxies

Zachos, et al, Science 292 (2001), p. 689 Math 5490 10/13/2014

Isotopes as Proxies

Zachos, et al, Science 292 (2001), p. 689 Math 5490 10/13/2014

Isotopes as Proxies

Math 5490 10/13/2014 Zachos, et al, Science 292 (2001), p. 689

Sharp decrease in δ13C, interpreted as massive oxidation

  • f organic carbon

Sharp decrease in δ18O, interpreted as large increase in temperature

Paleocence‐Eocene Thermal Maximum (PETM)

Isotopes as Proxies

Sharp decrease in δ18O, interpreted as a rapid increase in temperature. Sharp decrease in δ 13C, interpreted as massive oxidation of sequestered organic carbon.

‐3 ‐2 ‐1 1 2 3 ‐1 ‐0.5 0.5 1 1.5 2 2.5 3 ‐56 ‐55.8 ‐55.6 ‐55.4 ‐55.2 ‐55 ‐54.8 ‐54.6 ‐54.4 ‐54.2 ‐54 d13C d18O Myr

Site 690

d18O d13C

Math 5490 10/13/2014

Paleocence‐Eocene Thermal Maximum (PETM)

Isotopes as Proxies

Math 5490 10/13/2014

Paleocence‐Eocene Thermal Maximum (PETM)

‐3 ‐2 ‐1 1 2 3 ‐1 ‐0.5 0.5 1 1.5 2 2.5 3 ‐56 ‐55.8 ‐55.6 ‐55.4 ‐55.2 ‐55 ‐54.8 ‐54.6 ‐54.4 ‐54.2 ‐54 d13C d18O Myr

Site 690

d18O d13C

δ1 = 1.9, δ0 = -0.8

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 5

??: δ2

  • cean +

atmos: δ1

  • cean +

atmos: δ0

Isotopes as Proxies

p δ13C = 1.9‰

1 2 1 2 1 2 1 1 2 1

(1 ) ( ) p q q q q q                        δ13C = -0.8‰ If organic carbon was oxidized, δ2 = -25‰. Math 5490 10/13/2014 q δ13C = ??

1 2 1

0.8 1.9 0.10 25 1.9 q             

δ13C Budget

Math 5490 9/3/2014

Sigman & Boyle, Nature 207 (2000), p.860

Preindustrial Carbon

Isotopes as Proxies

  • cean and atmosphere:

8000 + 700 + 600 = 39300 GtC Let’s say 40000. ??: δ2

  • cean +

atmos: δ1

  • cean +

atmos: δ0

Isotopes as Proxies

p δ13C = 1.9‰ 40000 GtC δ13C = -0.8‰ If organic carbon was oxidized, δ2 = -25‰.

Math 5490 10/13/2014

q δ13C = ??

1 2 1

0.8 1.9 0.10 25 1.9 q             

δ13C Budget

If x Gt of organic carbon was oxidized, 0.10 0.90 4000 4444 40000 x q x x x      It would take about 4400 Gt of organic carbon to produce the PETM δ13C spike.

Math 5490 9/3/2014

Sigman & Boyle, Nature 207 (2000), p.860

Preindustrial Carbon

Isotopes as Proxies

Where did 4400 Gt of

  • rganic carbon come

from? Known coal reserves today: 1000 Gt Preindustrial terrestrial carbon: 2100 Gt 4400 Gt is a lot. ??: δ2

  • cean +

atmos: δ1

  • cean +

atmos: δ0

Isotopes as Proxies

p δ13C = 1.9‰ 40000 GtC δ13C = -0.8‰ If methane was oxidized, δ2 = -60‰.

Math 5490 10/13/2014

q δ13C = ??

1 2 1

0.8 1.9 0.44 60 1.9 q             

δ13C Budget

If x Gt of organic carbon was oxidized, 0.044 0.956 1760 1800 40000 x q x x x      It would take about 1800 Gt of methane to produce the PETM δ13C spike.

Isotopes as Proxies

Math 5490 10/13/2014

PETM vs Today

Can we burn 4000 – 5000 Gt of carbon? Might be difficult: only 1000 Gt of known coal deposits, less of

  • il.

But we are getting better at finding and extracting it. Do we have to burn that much carbon, since we are burning it faster than the deep ocean can absorb it?

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 6

Isotopes as Proxies

Math 5490 10/13/2014

PETM vs Today

Lee R. Kump, The Last Global Warming, SCIENTIFIC AMERICAN, July 2011, p 57

Heat Imbalance

Math 5490 10/13/2014

James Hansen, et al, Earth’s Energy Imbalance: Confirmation and Implications, SCIENCE 308 (2005), p. 1431 Average Insolation: 342 Watts per square meter Yearly Insolation: 342 W yr m‐2 Current Heat Imbalance: 0.85 ± 0.15 Wm‐2

Heat Imbalance

Math 5490 10/13/2014 Hansen, et al, SCIENCE 308 (2005), Table S1

Warm air 1ºC: 0.32 W yr m-2 or 8 hours Warm land surface 1ºC: 0.7 W yr m-2 or 18 hours Warm ocean 1ºC to depth of 1 km 93 W yr m-2 or 3.3 months Melt enough ice to raise sea level 1 m (assuming ice temperature -10ºC and sea temperature 15ºC) 9.3 W yr m-2 or 10 days Melt all ice on Earth: 650 W yr m-2 or 23 months Assume all insolation goes toward warming.

Heat Imbalance

Math 5490 10/13/2014 Hansen, et al, SCIENCE 308 (2005), Table S1

Warm air 1ºC: 0.32 W yr m-2 or 3.8 months Warm land surface 1ºC: 0.7 W yr m-2 or 8.4 months Warm ocean 1ºC to depth of 1 km 93 W yr m-2 or 93 years Melt enough ice to raise sea level 1 m (assuming ice temperature -10ºC and sea temperature 15ºC) 9.3 W yr m-2 or 9.3 years Melt all ice on Earth: 650 W yr m-2 or 650 years Assume global heat imbalance of 1 W m-2.

Heat Imbalance

Math 5490 10/13/2014 Hansen, et al, SCIENCE 308 (2005), Table S1

Warm air 1ºC: 0.32 W yr m-2 or 3.8 months Warm land surface 1ºC: 0.7 W yr m-2 or 8.4 months Warm ocean 1ºC to depth of 1 km 93 W yr m-2 or 93 years Melt enough ice to raise sea level 1 m (assuming ice temperature -10ºC and sea temperature 15ºC) 9.3 W yr m-2 or 9.3 years Melt all ice on Earth: 650 W yr m-2 or 650 years Assume global heat imbalance of 1 W m-2.

Heat Imbalance

Math 5490 10/13/2014 Hansen, et al, SCIENCE 308 (2005), Table S1

Warm air 1ºC: 0.32 W yr m-2 or 3.8 months Warm land surface 1ºC: 0.7 W yr m-2 or 8.4 months Warm ocean 1ºC to depth of 1 km 93 W yr m-2 or 93 years Melt enough ice to raise sea level 1 m (assuming ice temperature -10ºC and sea temperature 15ºC) 9.3 W yr m-2 or 9.3 years Melt all ice on Earth: 650 W yr m-2 or 650 years Assume global heat imbalance of 1 W m-2.

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Math 5490 10/13/2014 Richard McGehee, University of Minnesota 7

Heat Imbalance

Math 5490 10/13/2014 Hansen, et al, SCIENCE 308 (2005), Table S1

Claim: During the glacial cycles, the average heat imbalance has been a fraction of 1 W m-2. Proof: A heat imbalance of 1 W m-2 would raise the sea level 100 meters in 930 years. But it took 10,000 years. The imbalance was more like 0.1 W m-2. Corollary: We might have a problem with a heat imbalance of 1 W m-2.