Glacial Cycles: from Aristotle to Hogg and Back to Budyko

Transcription

1 Glacial Cycles: from Aristotle to Hogg and Back to Budyko Richard McGehee School of Mathematics University of Minnesota Climate Change Summer School Mathematical Sciences Research Institute July 28, 2008

2 Question: What can mathematicians do about climate change? Answer: Look at Models. What kind?

3 Not this kind: Giselle Heidi Klum

4 These kinds: Conceptual Predictive

5 A Conceptual Model

6 A Predictive Model Add epicycles. Ptolemy s Universe

7 Epicycles are Predictive With the data available to the Greeks, epicycles fairly accurately predict planetary motions. In particular, the model explains retrograde motion.

8 Enhanced Predictive Model

9 Conceptual Models with Fewer Epicycles Nicolaus Copernicus Heliocentric + Epicycles Tycho Brahe Geoheliocentric Model + Epicycles

10 Epicycles Epicycles Trigonometric Polynomials Advantages: Data-driven Accurate predictions Consistent t with a variety of conceptual models Disadvantage: Associated conceptual models become ridiculously complicated.

11 Conceptual Model Kepler s 1 st Law Johannes Kepler

12 Kepler s Model Kepler s Laws 1. Planets move on ellipses with the sun at a focus. 2. Planets sweep out equal areas in equal time. 3. The square of the period is proportional to the cube of the semimajor axis. Advantages: Elegant (Occam s razor) Predictive No epicycles Disadvantage: Still doesn t explain everything.

13 Along Comes Newton F = ma Equations of Planetary Motion 2 d xj Gmm i j j = 2 3 j i dt i j xi xj ( ) m x x x 3, j = 1, n j Sir Isaac Newton No Epicycles! No words! Only equations.

14 Planetary Motion Jacques Laskar Observatoire de Paris Jacques Lakar, et al, 2004, Astronomy and Astrophysics 428,

15 What about Glacial Cycles? kiloyear bp temperature e -10 The Vostok temperature data cries out for power spectrum analysis (epicycles). But first, let s look more at the Earth s orbit. Petit, et al, Nature 399 (June ), pp

16 Earth s Heat Balance Historical Overview of Climate Change Science, IPCC AR4, p.96

17 What Causes Glacial Cycles? Milankovitch Hypothesis The glacial cycles are driven by the variations in the Earth s orbit (Milankovitch Cycles), causing a variation in incoming solar radiation (insolation), triggering glacial cycles.

18 Eccentricity

19 Obliquity

20 Precession

21 Insolation Global Annual Average Insolation Solar intensity at distance r from the sun: Angular momentum: k 1 ( ) = r( t) 2 Q t = 2 Ω= r θ Q t k θ = Ω () 1 Mean annual solar input ( T = one year): 1 T k T 2π k Q = Q t dt = dt = T 0 TΩ 0 TΩ 1 1 ( ) θ

22 Insolation Kepler s Third Law: 32 T = k a 2 a = semimajor axis Mean annual solar input: Derived from Kepler: ka 3 Q = Ω 32 1 e = k aω e = eccentricity Finally: Q = ka e 2

23 Laskar: From Aristotle to Hogg Insolation Q ka 2 5 = 2 1 e δa δq Wm 2 What about eccentricity?

24 Eccentricity Note periods of about 100 kyr and 400 kyr. ( ) 12 0 < e< < 1 e < δq Wm 2 2 Zachos, et al, "Trends, Rhythms, and Aberrations in Global Climate 65 Ma to Present". Science 292 (2001), 687.

25 Obliquity Note period of about 41 kyr. Zachos, et al, "Trends, Rhythms, and Aberrations in Global Climate 65 Ma to Present". Science 292 (2001), 687.

26 Precession Note period of about 23 kyr. Zachos, et al, "Trends, Rhythms, and Aberrations in Global Climate 65 Ma to Present". Science 292 (2001), 687.

27 Solar Forcing

28 Solar Forcing (Hays, et al) Hays, et al, Science 194 (1976), p. 1125

29 Climate Response, Hays, et al Three different temperature proxies from sea sediment data. Hays, et al, Science 194 (1976), p. 1127

30 Hays, et al, Summary 1) Three indices of global climate have been monitored in the record of the past 450,000 years in Southern Hemisphere ocean-floor sediments. 2)... climatic variance of these records is concentrated in three discrete spectral peaks at periods of 23,000, 42,000, and approximately 100, years. These peaks correspond to the dominant periods of the earth's solar orbit, and contain respectively about 10, 25, and 50 percent of the climatic variance. Hays, et al, Science 194 (1976), p. 1131

31 Hays, et al, Summary 3) The 42,000-year climatic component has the same period as variations in the obliquity of the earth's axis and retains a constant phase relationship with it. 4) The 23,000-year portion of the variance displays the same periods (about 23,000 and 19,000 years) as the quasiperiodic precession index. 5) The dominant, 100,000-year climatic component has an average period close to, and is in phase with, orbital eccentricity. Unlike the correlations between climate and the higher-frequency orbital variations (which can be explained on the assumption that the climate system responds linearly to orbital forcing), an explanation of the correlation between climate and eccentricity yprobably requires an assumption of nonlinearity. Hays, et al, Science 194 (1976), p. 1131

32 Hays, et al, Summary 6) It is concluded that changes in the earth's orbital geometry are the fundamental cause of the succession of Quaternary ice ages. 7) A model of future climate based on the observed orbital-climate relationships, but ignoring anthropogenic effects, predicts that the long-term trend over the next seven thousand years is toward extensive Northern Hemisphere glaciation. Hays, et al, Science 194 (1976), p. 1131

33 Solar Forcing

34 Recent History (Cenozoic) Zachos, et al, Science 292 (2001), p. 689

35 Climate Response A. Power spectrum of climate for the last 4.5 Myr. Note the peaks at 41Kyr and 100 Kyr. B. Power spectrum of climate for the period 25 Myr bp to 20.5 Myr bp. Note the new peak at 400 Kyr and the split peaks at 126Kyr and 95 Kyr. Zachos, et al, Science 292 (2001), p. 689

36 Are Milankovitch Cycles Sufficient? A Milankovitch Skeptic Climate variability in this range of periods is difficult to distinguish from a form of random walk with small superimposed deterministic elements. Evidence that Milankovitch forcing controls the records, in particular the 100 ka glacial/interglacial, is very thin and somewhat implausible, Carl Wunsch Carl Wunsch, 2004, Quantitative estimate of the Milankovitch-forced contribution to Observed Quaternary climate change, Quaternary Science Reviews 23 (2004),

37 What about CO 2? CO 2 lags temperature in the Vostok data. Hansen, et al, 2007, Phil. Trans. R. Soc. A 365,

38 Does CO 2 Provide the Feedback? dt c = S() t + G( C) σt 4, dt dc dt = V ( W0 + WC 1 ) + β( Cmax C) max ε,0. dt dt Andrew Hogg Hogg 2008, Geophysical Research Letters 35.

39 Hogg s Model dt c = S ( t ) + G ( C ) σt 4, dt dc dt = V ( W0 + WC 1 ) + β( Cmax C) max ε,0. dt dt weathering volcanos CO2 outgassing 2π S() t = S + Si sin i Γi G( t) G Alnl C = + C0 insolation greenhouse forcing Should weathering depend on temperature?

40 Does CO 2 Provide the Feedback? Hogg 2008, Geophysical Research Letters 35.

41 and Back to Budyko M. I. Budyko, The effect of solar radiation variations on the climate of the Earth, Tellus XXI (1969), Mikhail Ivanovich Budyko

42 and Back to Budyko Ice Line Model Recall Tung s lecture: dt K = Qs( y) 1 α ( T)( y) I T y + H T y dt ( ) ( )( ) ( )( ) The annual global average insolation is Q. The annual average insolation as a function of latitude θ, where y = sinθ, is Qs(y). Q is largely determined by the eccentricity, but s(y) is determined from a combination of the other orbital elements. What about s(y,t)?

43 and Back to Budyko What kind of Models? Three suggestions: 1. Hogg s model with weathering as a function of CO 2 and temperature. 2. Budyko s model with insolation as a function of latitude and geologic time. 3. Budyko s model with a Hogg-like CO 2 feedback.