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/
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Click on the link: "Live Streaming from 305 Lind Hall".Participation:
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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 2
11 1 1 1 1
fsr fr fs 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 x0
H 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 x0
Common 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