Math 5490 9/24/2014 Budykos Model Math 5490 Suggested Reading - PDF document

Math 5490 9/24/2014 Budyko’s Model Math 5490 Suggested Reading 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 Hoffman & Schrag, Snowball Earth , K.K. Tung, Topics in Mathematical Modeling , P RINCETON U NIVERSITY S CIENTIFIC A MERICAN , January 2000, 68-75 P RESS , 2007, Chapter 8 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles The Big Picture During the last 5 million years the Earth has seen fairly regular cycles of advancing and retreating glaciers. What causes them? Why did they change a million years ago? http://www.snowballearth.org/when.html Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles The Big Picture Temperatures in the Cenozoic Era http://www.snowballearth.org/when.html Hansen, et al, Target atmospheric CO2: Where should humanity aim? Open Atmos. Sci. J . 2 (2008) Math 5490 9/24/2014 Math 5490 9/24/2014 Richard McGehee, University of Minnesota 1

Math 5490 9/24/2014 Glacial Cycles Glacial Cycles 18 O as a Climate Proxy The isotope 16 O preferentially evaporates from the ocean and is 18 O in Foraminifera Fossils During the Past 4.5 Myr sequestered in glaciers, leaving the heavier isotope 18 O more highly 2.5 concentrated in the ocean. Thus 3 oceanic concentration of the isotope Benthic Data ( δ 18O) 18 O is higher during glacial periods. 3.5 4 Foraminifera absorb more 18 O into their 4.5 skeletons when the water temperature is lower 5 and when more 18 O is in the water. 5.5 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 Thus higher concentrations of 18 O in foraminifera time (Kyr) fossils indicate lower ocean temperatures and higher glacier volume. 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. Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Recent (last 400 Kyr) Temperature Cycles 18 O in Foraminifera Fossils During the Past 1.0 Myr Vostok Ice Core Data 2.5 3 Benthic Data ( δ 18O) 3.5 4 4.5 5 5.5 ‐ 1000 ‐ 900 ‐ 800 ‐ 700 ‐ 600 ‐ 500 ‐ 400 ‐ 300 ‐ 200 ‐ 100 0 time (Kyr) J.R. Petit, et al (1999) Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Lisiecki, L. E., and M. E. Raymo (2005), A Pliocene-Pleistocene stack of 57 globally distributed benthic Antarctica, Nature 399, 429-436. d18O records, Paleoceanography ,20, PA1003, doi:10.1029/2004PA001071. Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Heat Balance What Causes Glacial Cycles? Widely Accepted Hypothesis The glacial cycles are driven by the variations in the Earth’s orbit (Milankovitch Cycles), causing a variation in incoming solar radiation (insolation). This hypothesis is widely accepted, but also widely regarded as insufficient to explain the observations. The additional hypothesis is that there are feedback mechanisms and/or triggering mechanisms that amplify the Milankovitch cycles. What these feedbacks are and how they work are not fully understood. Historical Overview of Climate Change Science , IPCC AR4, p.96 http://ipcc-wg1.ucar.edu/wg1/Report/AR4WG1_Print_CH01.pdf Math 5490 9/24/2014 Math 5490 9/24/2014 Richard McGehee, University of Minnesota 2

Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Eccentricity Earth’s Orbit Kepler’s First Law: The orbit of every planet is an ellipse with the Sun at one of the two foci. Johannes Kepler (1571-1630) Eccentricity = c/a John Imbrie & Katherine Palmer Imbrie, Ice Ages: Solving the Mystery, Harvard Univ. Press, 1979. Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Eccentricity Eccentricity Perihelion: 91.5 Aphelion: 94.5 Change in radius: 3/93 = 3.2% Perihelion: 91.5x10 6 mi Aphelion: 94.5x10 6 mi Change in insolation: 6.4% Semimajor axis: 93x10 6 mi Six percent less insolation Eccentricity: 1.5/93 = 0.016 in the southern winter than the northern winter. 6.4% of 342 W/m 2 = 22 W/m 2 Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Global Annual Average Insolation Global Annual Average Insolation K   26 4 10 Watts Specific angular momentum (angular momentum per unit mass): Solar output: Solar intensity at distance r from the sun:    r   2 2 1 m s K     Q t 2 2 Wm Total annual solar input:    r t 4 P   P   Kr 2 dt Kr 2  dt Kr 2 Kr 2   2 E  E  E d   E    Joules r 2 2 Cross section of Earth: m      2  4  r t 4 4 2 E 0 0 0 Kr 2 E 2 W  Global solar input: Kr 2   r t 4 E Mean annual solar input: Watts  P 2 Total annual solar input ( P = one year (in seconds)): Mean annual solar intensity on the Earth’s surface:  P P Kr 2 K   1 = Kr 2 Kr 2 dt   E  2 E dt E W m   Joules        P r 2 P r t 2  r t 2 2 4 8  4 4 E 0 0 Math 5490 9/24/2014 Math 5490 9/24/2014 Richard McGehee, University of Minnesota 3

Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Global Annual Average Insolation Planetary Motion   Gm m x  x d x n 2  i j j i Kepler’s Third Law: m dt i  i 2 3  x x  j 1 a  a = semimajor axis P  j i 3 2 j i  Derived from Kepler:   e = eccentricity e 2  a 2 1 Isaac Newton 1642-1727 Mean annual solar intensity: K Ka ˆ a Ka ˆ 3 2 1 2 2 The orbits of all the planets can be  2 = = Wm  computed (both forward and backward P   8 e 2 e 2 1 1 in time) for billions of years. Jacques Laskar (1955-) Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Global Annual Average Insolation Eccentricity ˆ Ka 2 0.06 Laskar:  e 2 1 0.05 0.04 eccentricity 0.03 0.02 0.01 0.00 ‐ 1000 ‐ 900 ‐ 800 ‐ 700 ‐ 600 ‐ 500 ‐ 400 ‐ 300 ‐ 200 ‐ 100 0 time (Kyr) Note periods of about 100 kyr and 400 kyr. The effect due to eccentricity is more significant, but not that much: As e varies between 0 and 0.06, (1- e 2 ) -1/2 varies between 1 and Semi major axis does not change much: 1.0018, or about 0.2%. (Twenty times the effect due to a .) 0.005% corresponding to .01% change in global average insolation J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & Astrophysics 428 , 261–285. Astrophysics 428 , 261–285. Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Obliquity Obliquity 24.5 24.0 obliquity (degrees) 23.5 23.0 22.5 22.0 ‐ 1000 ‐ 900 ‐ 800 ‐ 700 ‐ 600 ‐ 500 ‐ 400 ‐ 300 ‐ 200 ‐ 100 0 time (Kyr) Note period of about 41 Kyr. J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & Astrophysics 428 , 261–285. http://upload.wikimedia.org/wikipedia/commons/6/61/AxialTiltObliquity.png Math 5490 9/24/2014 Math 5490 9/24/2014 Richard McGehee, University of Minnesota 4

Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Precession Precession Index 0.06 0.04 0.02 0.00 ‐ 0.02 ‐ 0.04 ‐ 0.06 ‐ 1000 ‐ 900 ‐ 800 ‐ 700 ‐ 600 ‐ 500 ‐ 400 ‐ 300 ‐ 200 ‐ 100 0 time (Kyr) index = e sin ρ , where e = eccentricity and ρ = precession angle (measured from spring equinox) Note period of about 23 Kyr. J. Laskar, et al (2004) A long-term numerical solution for the insolation quantities of the Earth, Astronomy & Astrophysics 428 , 261–285. http://earthobservatory.nasa.gov/Library/Giants/Milankovitch/milankovitch_2.html Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Eccentricity John Imbrie & Katherine Palmer Imbrie, Ice Ages: Solving the Mystery, Harvard Univ. Press, 1979. http://en.wikipedia.org/wiki/Milankovitch_cycles Math 5490 9/24/2014 Math 5490 9/24/2014 Glacial Cycles Glacial Cycles Daily Average Insolation at Summer Solstice at 65° N Daily Average Insolation at Summer Solstice at 65° N Insolation at a point on the Earth’s surface 580 K                                      I r 560 , , , , , cos cos cos cos sin sin sin sin cos    r 2 4 540 520 ( φ , γ ) = (latitude, longitude) W/m^2 500 ( r , θ ) = position of Earth in orbital plane 480 β = obliquity angle 460 ρ = precession angle 440 Daily average insolation at latitude φ at summer solstice 420 ‐ 1000 ‐ 900 ‐ 800 ‐ 700 ‐ 600 ‐ 500 ‐ 400 ‐ 300 ‐ 200 ‐ 100 0 Kyr      2 e 1 sin 1       2  I e      Q       d  , , , cos cos cos sin sin    2  e 2 2 0 1 Math 5490 9/24/2014 Math 5490 9/24/2014 Richard McGehee, University of Minnesota 5