Math 5490 10/1/2014 Glacial Cycles Math 5490 October 1, 2014 - PDF document

Math 5490 10/1/2014 Glacial Cycles Math 5490 October 1, 2014 Topics in Applied Mathematics: Who was Milankovitch? Introduction to the Mathematics of Climate Mondays and Wednesdays 2:30 – 3:45 Milutin Milankovitch was a Serbian mathematician and professor at the http://www.math.umn.edu/~mcgehee/teaching/Math5490-2014-2Fall/ University of Belgrade. Streaming video is available at http://www.ima.umn.edu/videos/ Milutin Milankovitch In 1920 he published his seminal work on Click on the link: "Live Streaming from 305 Lind Hall". 1879-1958 the relation between insolation and the Participation: Earth’s orbital parameters. https://umconnect.umn.edu/mathclimate In 1941 he published a book explaining his entire theory. His work was not fully accepted until 1976. Math 5490 10/1/2014 Glacial Cycles Glacial Cycles What happened in 1976? Solar Forcing (Hays, et al) Hays, Imbrie, and Shackleton, “Variations in the Earth's Orbit: Pacemaker of the Ice Ages,” James D. Hays Science 194 , 10 December 1976. John Imbrie “It is concluded that changes in the earth's orbital geometry are the fundamental cause of the succession of Quaternary ice ages.” Nicholas Shackleton Hays, et al , Science 194 (1976), p. 1125 Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Spectral Analysis of the Milankovitch cycles. Climate Response, Hays, et al Laskar’s computations Spectra 0.06 0.05 eccentricity 0.04 Eccentricity 0.03 0.02 0.01 0 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 Kyr 24.5 obliquity (degrees) 24.0 23.5 Obliquity 23.0 22.5 22.0 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 Kyr 0.06 0.04 precession index 0.02 Precession 0 ‐ 0.02 Three different temperature proxies from sea sediment data. ‐ 0.04 ‐ 0.06 ‐ 2000 ‐ 1800 ‐ 1600 ‐ 1400 ‐ 1200 ‐ 1000 ‐ 800 ‐ 600 ‐ 400 ‐ 200 0 Hays, et al , Science 194 (1976), p. 1125 Kyr Math 5490 10/1/2014 Math 5490 10/1/2014 Richard McGehee, University of Minnesota 1

Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Summer Solstice 65 ° N Milankovitch vs. Climate 580 540 W/m 2 500 precession obliquity 460 eccentricity? 420 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 Kyr precession obliquity eccentricity? Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Hays, et al, Summary Climate Response (Zachos, et al) Forcing Response precession eccentricity Increasing obliquity obliquity contribution Power spectrum of climate for the last 4.5 eccentricity precession Myr. Note the peaks at 41Kyr and 100 Kyr. Hays’ explanation is that there are nonlinear feedbacks. Are there other explanations? Hays, et al , Science 194 (1976), p. 1127 Zachos, et al , Science 292 (2001), p. 689 Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Zachos, et al, Summary Spectral Analysis of the Climate Data 2.5 Forcing Response 3.0 3.5 δ 18 O precession obliquity 4.0 4.5 Increasing obliquity eccentricity 5.0 contribution 5.5 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 Kyr eccentricity precession 41 Kyr 23 Kyr 100 Kyr power Nonlinear effects? Other explanations? 0.00 0.01 0.02 0.03 0.04 0.05 0.06 frequency (1/Kyr) Zachos, et al , Science 292 (2001), p. 689 eccentricity obliquity precession Math 5490 10/1/2014 Math 5490 10/1/2014 Richard McGehee, University of Minnesota 2

Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Zachos Summary (Revised) Spectral Analysis of the Climate Data 2.5 If we assume that glaciation depends on annual average insolation 3.0 instead of insolation at summer solstice, then forcing and response 3.5 δ 18 O 4.0 are aligned. Conclusion Remains: 4.5 The Milankovitch cycles 5.0 5.5 “pace” the Earth’s climate. ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 Kyr Forcing Response Exactly how is not so clear. 100 Kyr 41 Kyr 23 Kyr precession obliquity power Increasing obliquity eccentricity contribution 0.00 0.01 0.02 0.03 0.04 0.05 0.06 frequency (1/Kyr) eccentricity precession eccentricity obliquity precession Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Something’s Missing Something’s Missing Lisiecki ‐ Raymo Stack 24.5 2.5 obliquity (degrees) 24.0 3 3.5 23.5 obliquity d18O data 4 23.0 4.5 22.5 5 22.0 5.5 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 ‐ 6000 ‐ 5000 ‐ 4000 ‐ 3000 ‐ 2000 ‐ 1000 0 Kyr Kyr Lisiecki ‐ Raymo Stack 2.5 100 Kyr 41 Kyr 23 Kyr 3 obliquity 3.5 power spectrum power eccentricity? d18O 4 climate precession 4.5 5 0.00 0.01 0.02 0.03 0.04 0.05 0.06 5.5 ‐ 6000 ‐ 5000 ‐ 4000 ‐ 3000 ‐ 2000 ‐ 1000 0 frequency (1/Kyr) Kyr Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Last Million Years is Different What’s up with the Last Million Years? 100 Kyr 41 Kyr 23 Kyr 100,000 Year Problem: Why did the eccentricity signal become obliquity power eccentricity? -5 to 0 Myr so dominant during the last million years? precession 400,000 Year Problem: If the last million years is dominated by eccentricity, what happened to the 400,000 year cycle? A transition 0.00 0.01 0.02 0.03 0.04 0.05 0.06 frequency (1/Kyr) occurred about one million years ago: the -5 to -1 Myr amplitude eccentricity increased and the dominant period changed from 41 kyr to 100 Kyr 41 Kyr 23 Kyr 100 kyr. obliquity power -1 to 0 Myr eccentricity? data precession 0.00 0.01 0.02 0.03 0.04 0.05 0.06 frequency (1/Kyr) Math 5490 10/1/2014 Math 5490 10/1/2014 Richard McGehee, University of Minnesota 3

Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Budyko’s Model Budyko’s Model  d     1 h T T y dy ( ) ( ) surface temperature sin(latitude) ice line dt 0 stable equilibrium η *  T         R Qs y y A BT C T T ( )(1 ( , )) ( ) ( )  t 6 4 2 heat capacity OLR insolation albedo heat transport 0 h ( η ) reduces to ‐ 2  d   ‐ 4       T T h ( ) ( ) dt c ‐ 6 ‐ 8 ‐ 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Budyko’s Model Budyko’s Model   d d  h   h  e  ( ) ( , , ) dt dt The function h , and hence the equilibrium solution η * , depends on all The eccentricity e and the obliquity β are given by Laskar as functions of time. the parameters of the Budyko model.  T         R Qs y y A BT C T T ( )(1 ( , )) ( ) ( ) 24.5 0.06  t obliquity (degrees) 0.05 24.0 eccentricity In particular, η * depends on Q and s ( y ) , which depend on the 0.04 23.5 0.03 23.0 eccentricity e and the obliquity β . 0.02 22.5 0.01 0 22.0 Q    Q e ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 0 Kyr Kyr  e 2 1 Therefore, the stable equilibrium ice line is a function of time:     2 2 2            s y  y 2 y d , 1 1 sin cos cos  2      t e t t *( ) *( ( ), ( )) 0     e * *( , ) McGehee & Lehman, A Paleoclimate Model of Ice-Albedo Feedback Forced by Variations in Earth’s Orbit, McGehee & Lehman, A Paleoclimate Model of Ice-Albedo Feedback Forced by Variations in Earth’s Orbit, SIAM J. A PPLIED D YNAMICAL S YSTEMS 11 (2012), 684–707. SIAM J. A PPLIED D YNAMICAL S YSTEMS 11 (2012), 684–707. Math 5490 10/1/2014 Math 5490 10/1/2014 Glacial Cycles Glacial Cycles Budyko’s Model Budyko’s Model d  0.94    h e ( , , ) dt sin(latitude) 0.93 stable equilibrium η * ( e ( t ), β ( t )) ice line 0.92 6 0.91 4 ‐ 5500 ‐ 5000 ‐ 4500 ‐ 4000 ‐ 3500 ‐ 3000 ‐ 2500 ‐ 2000 ‐ 1500 ‐ 1000 ‐ 500 0 2 Kyr 0 h ( η ) ‐ 2 ‐ 4 ‐ 6 GMT ‐ 8 ‐ 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 η McGehee & Lehman, SIAM J. A PPLIED D YNAMICAL S YSTEMS 11 (2012), 684–707. Math 5490 10/1/2014 Math 5490 10/1/2014 Richard McGehee, University of Minnesota 4