of 36 /36
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 8, 2014

Math 5490 - University of Minnesotamcgehee/teaching/Math5490-2014-2...2014/10/08  · Math 5490 October 8, 2014 Isotopes as Climate Proxies How do we know the past climates? Math 5490

  • Author
    others

  • View
    3

  • Download
    0

Embed Size (px)

Text of Math 5490 - University of Minnesotamcgehee/teaching/Math5490-2014-2...2014/10/08  · Math 5490...

  • 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 5490October 8, 2014

  • Isotopes as Climate Proxies

    How do we know the past climates?

    Math 5490 10/8/2014

  • Isotopes as Proxies

    What is this? δ18O (‰)

    Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J. 2 (2008)

    Math 5490 10/8/2014

  • http://eo.ucar.edu/staff/rrussell/climate/paleoclimate/sediment_proxy_records.html

    Ocean Sediment Cores

    Isotopes as Proxies

    Math 5490 10/8/2014

  • 18O as a Climate Proxy

    Foraminifera absorb more 18O into their skeletons when the water temperature is lower and when more 18O is in the water.

    Thus higher concentrations of 18O in foraminifera fossils indicate lower ocean temperatures and higher glacier volume.

    The isotope 16O preferentially evaporates from the ocean and is

    sequestered in glaciers, leaving the heavier isotope 18O more highly concentrated in the ocean. Thus

    oceanic concentration of the isotope 18O is higher during glacial periods.

    Isotopes as Proxies

    Math 5490 10/8/2014

  • What is this? δ18O (‰)

    ‰ : “per mil,” “per thousand”1000‰ = 100% = 110‰ = 1% = 0.01

    1‰ = 0.1% = 0.001

    18O: Oxygen 18: 8 protons 8 electrons 10 neutrons17O: Oxygen 17: 8 protons 8 electrons 9 neutrons16O: Oxygen 16: 8 protons 8 electrons 8 neutrons

    Most of the oxygen atoms on Earth are 16O. About 1 in 500 atoms is 18O. About 1 in 2500 is 17O.

    There are other oxygen isotopes, but they are unstable.

    Isotopes as Proxies

    Math 5490 10/8/2014

  • What is this? δ18O (‰)

    ExampleGiven a sample of calcium carbonate (CaCO3) from a foraminifera fossil,

    suppose that the ratio of 18O atoms to 16O is r = 0.002013 = 2.013‰. How would we report this finding?

    How would we measure it in the first place?The instruments measure the difference between two samples. Typically,

    one measures the difference between the sample of interest and a standard sample. A common standard is something called “Vienna

    Standard Mean Ocean Water” (VSMOW), for which the ratio of 18O atoms to 16O is s = 0.0020052. Then

    18 0.002013O 1 1 0.00390.0020052

    r s rs s

    So we would report

    δ18O = 3.9 ‰

    Isotopes as Proxies

    Math 5490 10/8/2014

  • What is this? δ18O (‰)Conversely

    The 18O content of a sample is reported asδ18O = 3.9 ‰

    using the VSMOW standard.What is the proportion of 18O in the sample?

    18 181 O, (1 O)

    0.0020052(1 0.0039) 0.002013

    r r ss

    r

    Noter is the ratio of 18O to 16O. The proportion of 18O in the sample is

    Isotopes as Proxies

    0.002013 0.0020081 1.002013

    rr

    For small values of r, these are approximately equal.

    Math 5490 10/8/2014

  • 1Pierrehumbert, Principles of Planetary Climate, Cambridge U Press, New York, 2010.2http://en.wikipedia.org/wiki/%CE%9413C

    Isotopes Ratio Standard Source

    D:H 0.0001558 VSMOW Pierrehumbert1

    13C:12C 0.0112372 PDB Wikipedia2

    18O:16O 0.0020052 VSMOW Pierrehumbert1

    18O:16O 0.0020672 VPDB Pierrehumbert1

    Standards:VSMOW: Vienna Standard Mean Ocean WaterPDB: Pee Dee BelemniteVPDB: Vienna Pee Dee Belemnite

    Common Standards

    Isotopes as Proxies

    Math 5490 10/8/2014

  • What does δ18O (‰) tell us?

    r1 = ratio of 18O:16O in vaporr2 = ratio of 18O:16O in liquid

    At equilibrium,r1 = f r2

    where  f  is the fractionation factor.  (depends a lot on temperature)

    FractionationExample: Evaporation of Water

    liquid

    vaporevaporationcondensation

    Isotopes as Proxies

    Math 5490 10/8/2014

  • r1 = ratio of 18O:16O in vaporr2 = ratio of 18O:16O in liquid

    r1 = f r2

    FractionationWhat about δ?

    21 21 21

    1 1 1 1 1fsr fr f

    s s s

    Note that the standard drops out.f  is usually close to 1, so let

    1 2 2 21 1 1 Since  ε and  δ are typically small,  εδ is even smaller, so

    1 2

    Isotopes as ProxiesWhat does δ18O (‰) tell us?

    1f often expressed as ‰

    Math 5490 10/8/2014

  • δ18O(water) = δ2 f = 0.99 = 1+εδ18O(vapor) = δ1       ε = ‐0.01 = ‐10‰

    Example: Evaporation of Water

    waterδ = δ0

    dry air

    afterbefore

    waterδ2

    air + vaporδ1

    δ18O(water) = δ0δ18O(vapor) is undefined

    1 2

    2 0 1 0 00, 0.01

    The 18O content of the vapor is 10‰ less than that of the ocean. 

    Isotopes as ProxiesWhat does δ18O (‰) tell us?

    Assume only a small amount of vapor forms.

    Math 5490 10/8/2014

  • ocean

    glaciers

    afterbefore ocean

    common hydrogen:  1H = H    heavy hydrogen:  2H = DVSMOW  D:H  =  0.0001558 = 0.1558 ‰   (very small)

    Assumptionscurrent ocean:  δD = 0 current glaciers: δD = -420 ‰

    2% of all the water is in glaciers.Question

    If all the glaciers melted, what would the deuterium content of the ocean become?

    Isotopes as ProxiesExample: Melting Glaciers

    Math 5490 10/8/2014

  • Isotope ratios

    Let  M be the total number of hydrogen (and deuterium) atoms in the ocean and glaciers  (usually computed in moles).

    Let  p = 0.02  be the proportion of water in the glaciers,and let  q = 0.98  be the proportion of water in the oceans.Let  xi denote moles of deuterium and  yi denote moles of 

    hydrogen, according to this table:

    Isotopes as ProxiesExample: Melting Glaciers

    Glaciers Oceans TotalD x1 x2 x0H y1 y2 y0

    1 1 1 2 2 2 0 1 2 1 2( ) ( )r x y r x y r x x y y

    glaciers oceans total

    Math 5490 10/8/2014

  • Isotopes as ProxiesExample: Melting Glaciers

    1 1 1 2 2 2

    1 1 2 2

    glaciers oceansratio

    molesx r y x r y

    x y pM x y qM

    11 1

    1 1

    22 2

    2 2

    1 1

    1 1

    pM r pMy xr r

    qM r qMy xr r

    1 2

    1 2 2 1 1 2 1 2 1 2 1 21 20

    1 2 2 1 2 1

    1 2

    1 2 1 20

    2 1

    (1 ) (1 )1 1(1 ) (1 )

    1 1

    1

    r pM r qMx x r r p r r q r p r r p r q r r qr rr pM qMy y r p r q p r p q r q

    r r

    r p r q r rrr p r q

    Solve for moles

    Solve for combined ratio

    Math 5490 10/8/2014

  • Isotopes as ProxiesExample: Melting Glaciers

    1 2 1 20

    2 11r p r q r rr

    r p r q

    Since  r1 and  r2 are very small, a good approximation is

    0 1 2

    0 1 2

    0 1 2

    (1 ) (1 ) (1 )r pr qr

    s ps qs

    p q

    1 20.02 0.98 0.42 0p q Recall

    0 0.02 ( 0.42) 0.0084

    If all the glaciers melted, the deuterium content of the ocean would decrease by about  8.4 ‰

    Math 5490 10/8/2014

  • phase 2

    phase 1

    total

    Isotopes as ProxiesMore Generally

    Phase 1 Phase 2 TotalRare isotope x1 x2 x0Common isotope y1 y2 y0

    1 1 1 2 2 2 0 1 2 1 2( ) ( ) so 1i i i

    r x y r x y r x x y yx y r

    Isotope ratios

    Math 5490 10/8/2014

  • Isotopes as ProxiesSame Computation

    1 1 1 2 2 2

    1 1 2 2

    glaciers oceansratio

    molesx r y x r y

    x y pM x y qM

    11 1

    1 1

    22 2

    2 2

    1 1

    1 1

    pM r pMy xr r

    qM r qMy xr r

    1 2

    1 2 2 1 1 2 1 2 1 2 1 21 20

    1 2 2 1 2 1

    1 2

    1 2 1 20

    2 1

    (1 ) (1 )1 1(1 ) (1 )

    1 1

    1

    r pM r qMx x r r p r r q r p r r p r q r r qr rr pM qMy y r p r q p r p q r q

    r r

    r p r q r rrr p r q

    Solve for moles

    Solve for combined ratio

    Math 5490 10/8/2014

  • Isotopes as Proxies

    1 2 1 20

    2 11r p r q r rr

    r p r q

    Since  r1 and  r2 are very small (rare isotope assumption),a good approximation is

    0 1 2

    0 1 2

    0 1 2

    (1 ) (1 ) (1 )

    r pr qr

    s ps qs

    p q

    Math 5490 10/8/2014

    Same Computation

  • oceanδ2

    glaciersδ1

    afterbefore oceanδ0

    What about 18O changes?

    Isotopes as ProxiesExample: Melting Glaciers

    Math 5490 10/8/2014

    Assumptionscurrent ocean:  δ18O = 0 current glaciers: δ18O = -50‰

    2% of all the water is in glaciers.

    0 1 2 0.02 ( 0.05) 0.001p q

    If all the glaciers melted, the 18O content of the ocean would decrease by about  1 ‰

  • phase 2r2

    phase 1r1

    totalr0

    Isotopes as ProxiesFractionation

    Assume that fractionation is at equilibrium.

    Math 5490 10/8/2014

    1

    2

    0 1 2

    r frr pr qr

    fractionation:

    mass balance:

  • Isotopes as ProxiesFractionation

    Math 5490 10/8/2014

    10 1 2

    2

    r f r pr qrr

    0 1 2 2 2 2

    2 1

    0 0

    ( )

    1

    r pr qr pfr qr pf q r

    r r fr pf q r pf q

    If  f is close to  1:

    2 1

    0 0

    111 1 1 1

    1 1

    ffpf q p p p q

    pf q pf q

    r rp qr r

  • Isotopes as ProxiesFractionation

    Math 5490 10/8/2014

    1 0 2 0(1 ) (1 )r q r r p r

    delta notation

    1 0 2 0

    1 0 0 2 0 0

    (1 )(1 ) (1 ) (1 ) (1 ) (1 ) (1 )1 1 1 1

    i ir ss q s s p s

    q q p p

    1 0 2 0q p

    ε and δ0 are both small, so εδ0 is even smaller, so it can be ignored.

  • Example: Deuterium in Water Vapor

    waterδ0

    dry air

    afterbefore

    waterδ2

    air + vaporδ1

    Isotopes as Proxies

    Math 5490 10/8/2014

    A small amount (compared to the ocean) of water vapor forms.What is the deuterium content of the vapor?

    1 0 2 0

    1 0

    0 10 (0.08) 1 0.08

    q p p qq

    Assumptionscurrent ocean:  δ0 = δD = 0 fractionation: ε = -80‰

    Under these assumptions, the deuterium content of the vapor is less than that of the ocean by 80 ‰.

  • Everything depends on temperature.

    Isotopes as Proxies

    Math 5490 10/8/2014

    Temperature (°K)

    Temperature (°C)

    Temperature (°F)

    δ18O

    273 0 32 -11.7‰290 17 62 -10.1‰350 77 170 -6.0‰

    Pierrehumbert, Principles of Planetary Climate, Cambridge U Press, New York, 2010

    Water Evaporation Fractionation Factors for 18O

  • Everything depends on temperature.

    Isotopes as Proxies

    Math 5490 10/8/2014

    Formulae from Gerrit Lohmann, 2007

    Water Evaporation Fractionation Factors 

    ‐0.12

    ‐0.1

    ‐0.08

    ‐0.06

    ‐0.04

    ‐0.02

    0

    0 10 20 30 40 50 60 70 80 90 100

    fractio

    natio

    n ‐1

    ⁰C

    18O

    D

  • Example: Deuterium in Rain

    vaporδ0

    afterbefore

    rainδ2

    vaporδ1

    Isotopes as Proxies

    Math 5490 10/8/2014

    40% of the water vapor condenses to rain.What is the deuterium content of the rain and of the remaining vapor?

    1 0 2 0

    1 0

    2 0

    0.6 0.40.08 (0.09) 0.4 0.1160.08 (0.09) 0.6 0.026

    q p p qqp

    Assumptionsvapor before:  δ0 = δD = -80‰ fractionation: ε = 90‰

    Remaining vapor:  δD = -116‰Rain: δD = -26‰

  • Example: Deuterium in Rain

    vaporδ0

    afterbefore

    rainδ2

    vaporδ1

    Isotopes as Proxies

    Math 5490 10/8/2014

    Repeat60% of the remaining water vapor condenses to rain.

    1 0 2 0

    1 0

    2 0

    0.4 0.60.116 (0.1) 0.6 0.1760.116 (0.1) 0.4 0.076

    q p p qqp

    Assumptionsvapor before:  δ0 = δD = -116‰ fractionation: ε = 100‰

    Remaining vapor:  δD = -176‰Rain: δD = -76‰

  • Example: Deuterium in Rain

    vaporδ0

    afterbefore

    rainδ2

    vaporδ1

    Isotopes as Proxies

    Math 5490 10/8/2014

    Repeat again80% of the remaining water vapor condenses to rain.

    1 0 2 0

    1 0

    2 0

    0.2 0.80.176 (0.105) 0.8 0.2600.176 (0.105) 0.2 0.155

    q p p qqp

    Assumptionsvapor before:  δ0 = δD = -176‰ fractionation: ε = 105‰

    Remaining vapor:  δD = -260‰Rain: δD = -155‰

  • Example: Deuterium in Snow

    vaporδ0

    afterbefore

    rainδ2

    vaporδ1

    Isotopes as Proxies

    Math 5490 10/8/2014

    This time it snows.90% of the remaining water vapor condenses to snow.

    1 0 2 0

    1 0

    2 0

    0.1 0.90.260 (0.11) 0.9 0.3590.176 (0.11) 0.1 0.249

    q p p qqp

    Assumptionsvapor before:  δ0 = δD = -260‰ fractionation: ε = 110‰

    Remaining vapor:  δD = -359‰Rain: δD = -249‰

  • Isotopes as Proxies

    Math 5490 10/8/2014

    And So It Goes

    fractionationfractionation

    Pierrehumbert, Principles of Planetary Climate, Cambridge U Press, New York, 2010

  • Isotopes as Proxies

    Math 5490 10/8/2014

    Vostok and Dome C Differ

    Pierrehumbert, Principles of Planetary Climate, Cambridge U Press, New York, 2010

  • Isotopes as Proxies

    Math 5490 10/8/2014

    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 ocean

    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: Pierre Humbert’s book)

    And then there’s carbon.

    and is yet still more complicated.

  • Isotopes as Proxies

    Math 5490 10/8/2014

    Biology Matters

    and is yet still more complicated.

    Fractionation is about ‐25‰.

    2 2 6 12 6 26CO 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. 

  • Isotopes as Proxies

    Math 5490 10/8/2014

    Zachos, et al, Science 292 (2001), p. 689

  • Isotopes as Proxies

    Math 5490 10/8/2014

    Paleocene-Eocene Thermal Maximum (PETM)

    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

    0

    1

    2

    3

    ‐1

    ‐0.5

    0

    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