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    • CommentRowNumber1.
    • CommentAuthorJohn Baez
    • CommentTimeNov 21st 2010

    I started a page

    Milankovitch cycle

    which includes links to some online references that try to use stochastic resonance to explain how these cycles cause ice ages. I still need to explain some of the main problems with these attempts. These problems are nicely summarized by the Wikipedia article.

    I find this stuff to be very fun and tantalizing…

  1. here is the latest from Hansen and Sato published 5 days ago:

    Milankovic climate oscillations help define climate sensitivity and assess potential human-made climate effects. We conclude that Earth in the warmest interglacial periods was less than 1°C warmer than in the Holocene and that goals of limiting human-made warming to 2°C and CO2 to 450 ppm are prescriptions for disaster. Polar warmth in prior interglacials and the Pliocene does not imply that a significant cushion remains between today’s climate and dangerous warming, rather that Earth today is poised to experience strong amplifying polar feedbacks in response to moderate additional warming.
    Deglaciation, disintegration of ice sheets, is nonlinear, spurred by amplifying feedbacks. If warming reaches a level that forces deglaciation, the rate of sea level rise will depend on the doubling time for ice sheet mass loss. Gravity satellite data, although too brief to be conclusive, are consistent with a doubling time of 10 years or less, implying the possibility of multi-meter sea level rise this century. The emerging shift to accelerating ice sheet mass loss supports our conclusion that Earth’s temperature has returned to at least the Holocene maximum. Rapid reduction of fossil fuel emissions is required for humanity to succeed in preserving a planet resembling the one on which civilization developed

    tantalizing and scary

    • CommentRowNumber3.
    • CommentAuthorJohn Baez
    • CommentTimeJan 25th 2011

    I can never be 100% sure whether Hansen is striving for scientific truth or striving to scare the hell out of us.

    • CommentRowNumber4.
    • CommentAuthorRobert Smart
    • CommentTimeJan 25th 2011

    This reminds me of a previous case. Hansen et al got a paper published and the associated press release was mentioned in Barry Brook’s blog. It said (or was reported to have said) that the rate of ice discharge from Antarctica had doubled in 5 years. My response still seems true:

    “Ice discharge from Antarctica has already doubled in the past five years.” People (including me) have stopped believing all these super-scary tidbits. Now all they do is bring the whole story into disrepute (even if true!). Also the statement is carefully phrased. As we’ve seen across the Northern Hemisphere lately: warmer seas means more moisture means more snow. So increasing discharge could still be compatible with increasing rather than decreasing Antarctic ice. Time for Global Warming campaigners to clean up their act, because the current line is playing into the hands of the coal industry.

    The conversation then proceeded to decide that the paper is actually about the net loss of ice. But in that case maybe the net loss of ice was nearly zero 5 years ago, so a doubling is meaningless. In fact it is always meaningless to talk about percentage changes, even 100%, in quantities which are the relatively small difference between two very big numbers. Of course it is done all the time: “Company profit is up 25% since last year”.

    • CommentRowNumber5.
    • CommentAuthorDavidTweed
    • CommentTimeJan 25th 2011
    • (edited Jan 25th 2011)

    I wouldn’t say it’s meaningless at all. You certainly need to be aware of whether the inferences being drawn from percentage figures are appropriate, which often varies depending upon knowing the other context including often other numbers. (For example, I’d be much more interested in the innate variability of net ice loss than the fact it’s a difference between two big numbers when trying to figure out if it’s important.)

    • CommentRowNumber6.
    • CommentAuthorJohn Baez
    • CommentTimeJan 26th 2011
    • (edited Jan 26th 2011)

    It’s not meaningless, but one wants more information. Luckily we have some of that information at Sea level rise:

    Indications that the West Antarctic Ice Sheet is losing mass at an increasing rate come from the Amundsen Sea sector, and three glaciers in particular: the Pine Island, Thwaites and Smith Glaciers:

    Data reveals they are losing more ice than is being replaced by snowfall. Total ice discharge from these glaciers increased 30% in 12 recent years, and the net mass loss increased 170% from 39 ± 15 Gt/yr to 105 ± 27 Gt/yr. The melting of these three glaciers alone is now contributing an estimated 0.24 millimetres per year to the rise in the worldwide sea level (see the article by Jenny Hogan above).

    More generally, there has been substantial increase in Antarctic ice mass loss in the ten years 1996-2006, with glacier acceleration a primary cause:

    In 1996 the net mass loss was 78 ± 78 gigatons/year. By 2006 this had risen to 153 ± 78 gigatons/year.

    I was more talking about things like the chapter near the end of Hansen’s book where he describes in vivid detail a ’runaway greenhouse’ scenario where Earth becomes like Venus. Nobody I know thinks this is likely - the conditions under which this could occur have been studied.

    Anyway, it would be great if someone could add the abstract of the Staffan mentioned to the wiki, with a link.

    • CommentRowNumber7.
    • CommentAuthorDavidTweed
    • CommentTimeJan 26th 2011
    • (edited Jan 26th 2011 by John Baez)

    Done: note here to update the ref once paper becomes finalised version.

    • CommentRowNumber8.
    • CommentAuthorJohn Baez
    • CommentTimeJan 26th 2011

    • CommentRowNumber9.
    • CommentAuthorJohn Baez
    • CommentTimeJul 15th 2011
    • (edited Jul 15th 2011)

    An interesting issue related to Milankovitch cycles:

    Although small, Ceres and Vesta gravitationally interact together and with the other planets of the Solar System. Because of these interactions, they are continuously pulled or pushed slightly out of their initial orbit. Calculations show that, after some time, these effects do not average out. Consequently, the bodies leave their initial orbits and, more importantly, their orbits are chaotic, meaning that we cannot predict their positions. The two bodies also have a significant probability of impacting each other, estimated at 0.2% per billion year. Last but not least, Ceres and Vesta gravitationally interact with the Earth, whose orbit also becomes unpredictable after only 60 million years. This means that the Earth’s eccentricity, which affects the large climatic variations on its surface, cannot be traced back more than 60 million years ago. This is indeed bad news for Paleoclimate studies.

    I’m not sure why it’s bad news for paleoclimate studies, since I’ve never heard of anyone trying to correlate climate with Milankovitch cycles before the current glacial cycles began, say roughly at the start of the Pleistocene, about 2.6 million years ago.

    By the way, this picture of Ceres from the Hubble Telescope makes me wonder why we paid so much money for the darn thing:

    • CommentRowNumber10.
    • CommentAuthorTim van Beek
    • CommentTimeJul 15th 2011

    Because climate scientists assumed that the inbound solar radiation was reasonably stable over the last 60 or 65 million years and that the plaeoclimate temperature record is the best indication of a high climate sensibility?

    • CommentRowNumber11.
    • CommentAuthorNathan Urban
    • CommentTimeJul 15th 2011

    Climate scientists do sometimes correlate ancient climates with Milankovitch cycles, for example with the Eocene hyperthermals (also here). The cycles can’t be calculated absolutely from astronomical data for reasons discussed above, but rather have to be inferred from proxies. This also provides a common means of dating ocean cores, cyclostratigraphy, which counts cycles past some reference stratum, for example this paper on the PETM; also discussed somewhat in my recent preprint on the PETM.

    • CommentRowNumber12.
    • CommentAuthorTim van Beek
    • CommentTimeJul 15th 2011

    A somewhat related question: In order to get past the undergraduate level, I think I need access to research papers. Is there something similar to the arXiv in climate science, or are there research journals with free access? Or are a lot of people publishing their papers on their own website like you do?

    • CommentRowNumber13.
    • CommentAuthorNathan Urban
    • CommentTimeJul 15th 2011

    No, there’s nothing that the geosciences community has rallied around like the physics community did with the arXiv. I find that annoying. I suppose I could put my papers on the arXiv myself, as a few people in climate science do, but I haven’t bothered since hardly anyone in my community looks there for e-prints. You mostly have to look for authors’ personal web sites. There are a few journals out there with free access, but they’re not common.

    • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeJul 16th 2011

    Eocene Milankovitch cycles - cool! But that’s still a long way from 60-70 million years ago; I’d be pretty shocked (though delighted) if people were trying to correlate late-Cretaceous Milankovitch cycles to climate cycles. But maybe the unpredictability kicks in enough to be problematic even 30 million years ago?

    Nathan: it’s not too much work to put your papers on the arXiv, and if you do, you may start a trend. It’s got to happen someday. And even if climate scientists don’t look there yet, other scientists do. So, more people would see your papers.

    • CommentRowNumber15.
    • CommentAuthorTim van Beek
    • CommentTimeJul 16th 2011

    Yes, please start this trend.

    • CommentRowNumber16.
    • CommentAuthorFrederik De Roo
    • CommentTimeJul 16th 2011
    • (edited Jul 16th 2011 by John Baez)

    Yes, it would be nice to have freely accessible preprints from climate science. What would be the right arxiv-section, btw?

    • CommentRowNumber17.
    • CommentAuthorJohn Baez
    • CommentTimeJul 16th 2011
    • (edited Jul 16th 2011)

    The perfect arXiv section does not exist, but some of these will do the job:

    Atmospheric and Oceanic Physics; Biological Physics; Chemical Physics; Computational Physics; Data Analysis, Statistics and Probability; Fluid Dynamics; General Physics; Geophysics; Instrumentation and Detectors.

    • CommentRowNumber18.
    • CommentAuthorMartin Gisser
    • CommentTimeJul 17th 2011

    Tim, regarding access to research papers:

    Do you have any connections left to your old alma mater? I’m lucky one prof and one Akademischer Rat are still there, so I got a computer network account as “guest scientist”. They even offer a VPN, so I can read from home anything from Science to Journal of Functional Analysis.

    • CommentRowNumber19.
    • CommentAuthorJohn Baez
    • CommentTimeJul 18th 2011

    Tim: I can get a certain number of research papers for you.

  2. I actually found several references both in Environmental Science/Physics and on Arxiv, so it would be good if Climate science could be added there :-)

    For arXiv i get the ones I selected in my news feed reader (google right now).

    I also have access to my alma mater here in Stockholm, and their library are having many publications/books as e-books, which i actually like a lot even though I miss writing proofs in the margins :-)

    • CommentRowNumber21.
    • CommentAuthorTim van Beek
    • CommentTimeJul 18th 2011

    Martin asked:

    Do you have any connections left to your old alma mater?

    No, I have basically stopped communication ten years ago.

    John wrote:

    Tim: I can get a certain number of research papers for you.

    That’s nice, but I’d need to learn what people are publishing about in general, so I don’t know what papers I should be interested in. Speaking of which, an arXiv wouldn’t be that useful to me right now. In hep-th I can find out myself which papers are interesting to me and which are not.

    I also have access to my alma mater here in Stockholm, and their library are having many publications/books as e-books, which i actually like a lot even though I miss writing proofs in the margins :-)

    I could try to get into the TU Munich. The big library, the Bayrische Staatsbibliothek, is open to everybody, though. But its a one hour trip away from my apartment. When I was there to get an ID card, the clerk looked sternly at me and asked “You do know that we don’t have Hollywood movies here, right? We have books about science only.”

    • CommentRowNumber22.
    • CommentAuthorJohn Baez
    • CommentTimeJul 18th 2011

    That’s nice, but I’d need to learn what people are publishing about in general, so I don’t know what papers I should be interested in.

    You can freely read abstracts of papers in most journals… I know, that’s not nearly as good as being able to read papers, but it’s one way to get a feeling for what people are publishing about. For example, it’s fun to read abstracts of papers in Journal of Climate.

    • CommentRowNumber23.
    • CommentAuthorMartin Gisser
    • CommentTimeJul 18th 2011
    • (edited Jul 19th 2011 by John Baez)

    Tim said:

    No, I have basically stopped communication ten years ago.

    Actually me too. (E.g. I dropped the PhD TA/RA post for they didn’t allow me a decent computer in my office (and my thesis would not be accepted written in cuneiform on clay tablets) - meanwhile almost all of my internets dreams came true (incl. places like this here!)…). But then, when I returned (and found the library open) I got greeted quite friendly.

    Your story of the Munich library bureaucrat sounds soo familiar and typical. What a paradise a U.S. or Finnish library is!

    • CommentRowNumber24.
    • CommentAuthorTim van Beek
    • CommentTimeJul 18th 2011

    The Staatsbibliothek has got a happy ending, while everybody frowned at me after realizing that I was not a student of one of the Munich universities, in the end I got what I wanted: “Causal Nets of Operator Algebras”, by Baumgärtel + Wollenberg.

    I half expected a body check at the entrance - after all it is possible to get access to really valuable books there, although you only get to look at them, not to lend them - and was prepared to use phrases like “algebraic quantum field theory” and “Doplicher-Roberts reconstruction theorem”. But that turned out to be unnecessary.

    • CommentRowNumber25.
    • CommentAuthorJohn Baez
    • CommentTimeJul 24th 2011

    I’ve been looking at some key early papers on Milankovich cycles. I added a bit to Milankovitch cycle, as follows:

    This piece of work reviews evidence for the Milankovitch explanation of glacial cycles, at least up to 1989:

    Abstract: Spectral analysis of geological records show periodicities corresponding to those calculated for the eccentricity (400 and 100 ka), the obliquity (41 ka) and the climatic precession (23 and 19 ka). It is precisely the geological observations of this bi-partition of the precessional peak, confirmed to be real in astronomical computations, which was one of the most impressive of all tests for the Milankovitch theory. Concerning the question of whether or not the observed cycles account for most of the climatic variability having periods in the range predicted by the astronomical theory, substantial evidence (from cross-spectral analysis, coherency analysis, and modelling) is provided that, at least near frequencies of variation in obliquity and precession, a considerable fraction of the climate variance is driven in some way by insolation changes accompanying changes in the earth’s orbit. The variance components centered near a 100 ka cycle dominate most ice volume records and seem in phase with the eccentricity cycle, although the exceptional strength of this cycle needs a non-linear amplification by the glacial ice sheets themselves and associated feedbacks.

    As the insolation spectra change with latitudes and with the type of parameters considered, the diversity of the spectra of different climatic proxy data recorded in different places of the world over different periods is used to better understand how the climate system responds to the insolation forcing. This study of the physical mechanisms involved is also achieved through the analysis of the log-log shapè of the geological records and through the comparison, in frequency domain, between simulated climatic time series and proxy data.

    The evidence, both in the frequency and in the time domain, that orbital influences are felt by the climate system, implies that the astronomical theory may provide a clock with which to date Quaternary sediments with a precision several times greater than is now possible.

    • CommentRowNumber26.
    • CommentAuthorJohn Baez
    • CommentTimeJul 24th 2011

    According to the above article, a 1976 paper by Hays, Imbrie, and Shackleton was important in finding spectral peaks at 100 ka, 42 ka, 23 ka, and 19 ka in data from a composite deep-sea core from the Indian Ocean. Berger writes:

    By the 1970s, the grounds upon which the first strong test of the Milankovitch theory was going to be based, were settled. Judicious use of radiometric dating and other techniques gradually clarified the details of the time scale (Broecker et al., 1968); better instrumental methods came on the scene for using oxygen isotope data as ice age relics (Shackleton and Opdyke, 1973); ecological methods of core interpretation were perfected (Imbrie and Kipp, 1971); global climates in the past were reconstructed (CLIMAP, 1976); and astronomical calculations were checked and refined (Vernekar, 1972; Berger, 1976b). Unfortunately it is not possible in this paper to give credit to all scientists responsible for the new picture of ice ages and their cause (refer to Imbrie and Imbrie, 1979; Berger, 1980a, 1988; Imbrie, 1982). Scientists were thus ready to show that the climatic features in the geological record go through precisely the same rhythms as do the orbital parameters of the earth, with sufficiently close links in time (in phase) to confirm a relationship of cause and effect.

    In August 1975, at the climate conference of the World Meteorological Organization in Norwich, the spectral characteristics of two time series independently built – one in geology and one in astronomy – were compared, for the first time in the history of climate investigation. The joint efforts of Hays, Imbrie, and Shackleton (published in full in Science, December 1976) demonstrated that geologic spectra contain substantial variance components at many frequencies similar to those obtained by Berger from astronomical computations (published in Nature, January 1977).

    The principal feature of the variance spectra obtained from a composite deep-sea core from the Indian Ocean (RCl1-120 and E49-18), spanning the past 468 ka, was a characteristic red noise signal over periods ranging from about 100 ka to some thousand years. Superimposed on this signal were several peaks representing significant responses at orbital frequency (Fig. 1). On both isotopic (ice volume) and temperature radiolarian spectra, peaks appeared for cycles roughly 100 ka, 42 ka, 23 ka, and 19 ka long.

    The obliquity signal with a period of about 42 ka was gratifyingly consistent and detectable for the past 300 ka. The related climatic variations lagged about 9 ka behind the rhythm of the changing tilt of the earth. This consistency was less, however, in the older part of the record, because of a presumed less accurate time scale. This lead Hays et al. (1976) to adjust the time scale slightly (the ’tune-up’ being well within the uncertainties of the dating) to obtain the same phase relationship going back to 425 ka BP.

    A 23 ka cycle, corresponding with the precession of the equinoxes, also appeared strongly in the geological record. Although the uncertainties about dating produced relatively greater uncertainties in the phase relationships for this higher frequency, and the earth’s orbit was almost circular between 350 ka and 450 ka, the tune-up adjustments still showed that the climatic factors change closely in step with the wobble over the whole period. The most delicate and impressive of all tests for the Milankovitch theory, however, came with a closer examination of the astronomical solution. The precession of the equinoxes was split into two frequency components with periods of 23 ka and 19 ka. This was calculated both by Berger (1977) from his theoretical investigation of the astronomical theory and observed by Imbrie in the Hays and Shackleton records (1976).

    Surprisingly, however, the dominant cycle in Hays et al.’s data had a consistent period of 100 ka, with the coldest periods of ice age activity coinciding dramatically with the periods of near-circularity of the earth’s orbit. This is exactly the opposite of what Milankovitch claimed, because in his theory eccentricity could only modulate the size of the precession effect, but the conclusion is compatible as far as the total energy effect is concerned (lower eccentricity, lower total energy received from the sun).

    As the 100 ka year cycle is extremely weak in the insolation data set, with obliquity and precession sharing almost entirely the total variance, one of the mysteries which remained when trying to relate paleoclimates to astronomical insolations was to find how this cycle could be enhanced to become the dominant cycle observed in geological records.

    The paper by Hays, Imbrie and Schackleton is:

    From several hundred sediment cores studied stratigraphically by the CLIMAP project, they selected two (RC11-120 and E49-18) from the Indian ocean and measured:

    1) δ 18O\delta^{18}O, the oxygen isotopic composition of planktonic foraminifera,

    2) Ts, an estimate of summer sea-surface temperatures at the core site, derived from a statistical analysis of radiolarian assemblages,

    3) percentage of Cycladophora davisiana, the relative abundance of a radiolarian species not used in the estimation of Ts.

    Identical samples were analyzed for the three variables at 10-cm intervals through each core.

    Here are their findings for RC11-120:

    and for E49-18:

    After some processing described in the paper, here are the power spectra for these data:

    • CommentRowNumber27.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011 by John Baez)
    Hmm... according to Berger, the effect on insolation of the eccentricity cycle of 100k is very weak compared to the other cycles (tilt, precession). The effect is also different. A decrease in eccentricity actually lowers total annual insolation by a small amount. Tilt & precession do not change the total insolation over a year, while they do change the distribution over the seasons.

    Actually, the eccentricity cycle of 413ka is also stronger than the 100ka cycle. Unfortunately, the last Ma of ice ages reflect the 100ka cycles much better than the 413ka cycle. To deepen the mystery, the spectrum of all the astronomical variables (including the 413ka period) show up in the spectrum derived from geology, which can't be a coincidence!

    But essentially, what Berger does is advanced curve fitting. Comparing spectra. He only touches briefly on possible physical mechanisms. Hays, Imbries and Shackleton even seem to make a virtue out of ignoring physics, and state "In contrast, we postulate a single, radiation-climate system which transforms orbital inputs into climatic outputs. We can therefore avoid the obligation of identifying the physical mechanisms of climatic response and specify the behavior of the system only in general terms".

    How confusing this is. Does later literature explain why these recurring ice ages are paced by the 100k cycle, and not by any of the others? I have seen various attemps to explain how the astronomical variables influence climate, but I'm not sure if these are solid or mainly conjectural. It seems to me that these explanations are more convincing for tilt and precession than for excentricity. But at least in this last million year the 100ka excentricity period has been the main effect.
    • CommentRowNumber28.
    • CommentAuthorNathan Urban
    • CommentTimeJul 27th 2011

    No, it’s still not conclusively known why the glacial interglacial cycle follows a 100 ky cycle, although there are theories.

    • CommentRowNumber29.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011)

    There is stochastic resonance, which is a great seductive idea. However, I noticed that some of the experts on the subject writes that “Although it was brilliant, subsequent data did not support this idea as an explanation for the ice ages. ” (Gammaitoni et al.) They do not comment further on what data they are referring to. It would be interesting to see them.

    Are there other theories that could explain the 100ka?

    • CommentRowNumber30.
    • CommentAuthorJohn Baez
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011)

    Marcel wrote:

    A decrease in eccentricity actually lowers total annual insolation by a small amount.

    Yes. Thanks to a conversation we had on the blog, Greg Egan did a nice calculation of this, which I’ll reproduce here, in preparation for putting it on the wiki. It takes advantage of the fact that both gravity and solar radiation obey an inverse-square law!

    Here is his calculation:

    The angular velocity dθdt=Jmr 2\frac{d \theta}{d t} = \frac{J}{m r^2}, where JJ is the constant orbital angular momentum of the planet and mm is its mass, so if the radiant energy delivered per unit time to the planet is dUdt=Cr 2\frac{d U}{d t} = \frac{C}{r^2} for some constant CC, the energy delivered per unit of angular progress around the orbit is

    dUdθ=Cr 2dtdθ=CmJ.\frac{d U}{d \theta} = \frac{C}{r^2} \frac{d t}{d \theta} = \frac{C m}{J}.

    So the total energy delivered in one period will be U=2πCmJU=\frac{2\pi C m}{J}.

    How can we relate the orbital angular momentum JJ to the shape of the orbit? If you equate the total energy of the planet, kinetic 12mv 2\frac{1}{2}m v^2 plus potential GMmr -\frac{G M m}{r}, at its aphelion r 1r_1 and perihelion r 2r_2, and use JJ to get the velocity in the kinetic energy term from its distance, v=Jmrv=\frac{J}{m r}, when we solve for JJ we get:

    J=m2GMr 1r 2r 1+r 2=mbGMaJ = m \sqrt{\frac{2 G M r_1 r_2}{r_1+r_2}} = m b \sqrt{\frac{G M}{a}}

    where a=12(r 1+r 2)a=\frac{1}{2} (r_1+r_2) is the semi-major axis of the orbit and b=r 1r 2b=\sqrt{r_1 r_2} is the semi-minor axis. But we can also relate JJ to the period of the orbit, TT, by integrating the rate at which orbital area is swept out by the planet, 12r 2dθdt=J2m\frac{1}{2} r^2 \frac{d \theta}{d t} = \frac{J}{2 m} over one orbit. Since the area of an ellipse is πab\pi a b, this gives us:

    J=2πabmT.J = \frac{2 \pi a b m}{T}.

    Equating these two expressions for JJ shows that the period is:

    T=2πa 3GMT = 2 \pi \sqrt{\frac{a^3}{G M}}

    So the period depends only on the semi-major axis; for a fixed value of aa, it’s independent of the eccentricity.

    If we agree to hold the orbital period TT, and hence the semi-major axis aa, constant and only vary the eccentricity of the orbit, we have:

    U=2πCmJ=2πCbaGMU=\frac{2\pi C m}{J} = \frac{2\pi C}{b} \sqrt{\frac{a}{G M}}

    Expressing the semi-minor axis in terms of the semi-major axis and the eccentricity, b 2=a 2(1e 2)b^2 = a^2 (1-e^2), we get:

    U=2πCGMa(1e 2)U=\frac{2\pi C}{\sqrt{G M a (1-e^2)}}

    So to second order in ee, we have:

    U=πCGMa(2+e 2)U = \frac{\pi C}{\sqrt{G M a}} (2+e^2)

    The expressions simplify if we consider average rate of energy delivery over an orbit, which makes all the grungy constants to do with the gravitational dynamics go away:

    UT=Ca 21e 2\frac{U}{T} = \frac{C}{a^2 \sqrt{1-e^2}}

    or to second order in ee:

    UT=Ca 2(1+12e 2)\frac{U}{T} = \frac{C}{a^2} (1+\frac{1}{2} e^2)
    • CommentRowNumber31.
    • CommentAuthorJohn Baez
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011)

    One thing we should check is whether perturbations in the Earth’s orbit actually hold the semi-major axis roughly constant and only change the eccentricity, as Greg assumes in his calculation!!!

    Assuming this, I put some numbers into Greg’s formula; again I’ll copy what I wrote to here, as a warmup for putting it on the wiki. (Some reformatting needs to be done to transfer something from the blog to the wiki, but this forum uses the same format as the wiki.)

    Here’s what I wrote:

    We can now work out how much the actual changes in the Earth’s orbit affect the amount of solar radiation it gets! According to Wikipedia:

    The shape of the Earth’s orbit varies in time between nearly circular (low eccentricity of 0.005) and mildly elliptical (high eccentricity of 0.058) with the mean eccentricity of 0.028.

    The total energy the Earth gets each year from solar radiation is proportional to

    11e 2\frac{1}{\sqrt{1-e^2}}

    where ee is the eccentricity. When the eccentricity is at its lowest value, e=0.005e = 0.005, we get

    11e 2=1.0000125\frac{1}{\sqrt{1-e^2}} = 1.0000125

    When the eccentricity is at its highest value, e=0.058e = 0.058, we get

    11e 2=1.00168626\frac{1}{\sqrt{1-e^2}} = 1.00168626

    So, the change is about

    1.00168626/1.0000125=1.001673731.00168626/1.0000125 = 1.00167373

    In other words, a change of merely 0.167%.

    That’s puny! And the effect on the Earth’s temperature would naively be even less!

    Naively, we can treat the Earth as a greybody. Since the temperature of a greybody is proportional to the fourth root of the power it receives, a 0.167% change in solar energy received per year corresponds to a percentage change in temperature roughly one fourth as big. That’s a 0.042% change in temperature. If we imagine starting with an Earth like ours, with an average temperature of roughly 290 kelvin, that’s a change of just 0.12 kelvin!

    As always, you shouldn’t trust my arithmetic. But the upshot seems to be this: in a naive model without any amplifying effects, changes in the eccentricity of the Earth’s orbit would cause temperature changes of just 0.12 °C!

    This is much less than the roughly 5 °C change we see between glacial and interglacial periods. So, we have some explaining to do.

    And of course one place to start is to realize that the Earth has not always had glacial periods. Something is happening now that didn’t always happen.

    • CommentRowNumber32.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011 by John Baez)

    Great calculation!

    To get from this to a calculation of annual insolation over a period of eccentricity change we want to make the assumption that the major semi axis of the orbit does not change, or at least that we know how it changes. Do we know this?

    (PS: Oh, this place has good latency, you do get a reply even before you have asked the question!)

    • CommentRowNumber33.
    • CommentAuthorJohn Baez
    • CommentTimeJul 27th 2011

    Marcel wrote:

    There is stochastic resonance, which is a great seductive idea. However, I noticed that some of the experts on the subject writes that “Although it was brilliant, subsequent data did not support this idea as an explanation for the ice ages. ” (Gammaitoni et al.) They do not comment further on what data they are referring to. It would be interesting to see them.

    It would be great if you could look up that data and see what it was. I think one should always be careful, reading scientific papers, when someone says “that idea didn’t work” without saying exactly why. In physics, at least, a number of good ideas have temporarily been ignored because of “folk wisdom” saying they don’t work.

    Are there other theories that could explain the 100ka?

    I don’t know! I’m going to China for two weeks, and I won’t have much time to work on this stuff until I get back. So, it would be great if you (or anyone else with good library access) did a little investigation of this question. A good place to start is this review article:

    • CommentRowNumber34.
    • CommentAuthorJohn Baez
    • CommentTimeJul 27th 2011
    • (edited Jul 27th 2011)

    Do we know this?

    I don’t personally know this. I’m sure someone does (or thinks they do). I really want to get ahold of Milankovitch’s original book, which is supposedly very self-contained and good.

    I believe it was written in Serbian. It has a somewhat hair-raising story:

    It took him about 2 years to write this book, and the manuscript was submitted to print four days before Germany started bombing Yugoslavia. The bombing destroyed the printing factory and, and all copies of the book — except one copy that Milanković took for himself, on the first day of printing.

    I will tell this story in more detail in This Week’s Finds

    Now a version is available in German under the title Kanon der Erdebestrahlung und Seine Anwendung auf das Eiszeitenproblem. It was also translated into English in 1969 under the title Canon of Insolation of the Ice-Age Problem by the Israel Program for Scientific Translations, and published by the U.S. Department of Commerce and the National Science Foundation, Washington, D.C.

    • CommentRowNumber35.
    • CommentAuthorNathan Urban
    • CommentTimeJul 27th 2011

    For a theory of the 100 ky cycle, you may want to look at Peter Huybers’ publications. He has done a lot of recent innovative work in this area, which won him, in part, a MacArthur award. For example, in this paper he proposes that it is a chaotic response to the obliquity cycle, and that it can flip chaotically between 40 and 100 ky.

    • CommentRowNumber36.
    • CommentAuthorJohn Baez
    • CommentTimeJul 27th 2011

    Thanks, Nathan! I still am way behind when it comes to learning all this stuff.

    • CommentRowNumber37.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 27th 2011
    • (edited Jul 28th 2011 by John Baez)

    Here is what I got from reading Peter Huybers. If I missed something important, please kick me.

    In this paper, Huybers proposes the following: The eccentricity period does not matter, that’s a red herring. Throw it away, don’t think about it. Its all about tilt (obliquity). The glaciation 40ky period indeed corresponds to the dominant obliquity period. Sometimes, this period gets doubled or tripled, giving periods of 80 ky or 120 ky. These two cases he subsumes under the common name 100\sim100 ky.

    The problems he wants to solve is: Why does doubling and tripling occur, and why do we get sequences of single periods (as in early Pleistocene) , and sequences of repeated periods (as in late Pleistocene and Holocene). That is, following a single period, we will most likely get a single period, and following a 100\sim 100ky period, we will most likely get a 100\sim100 ky period.

    As a preliminary and as supporting evidence he makes one strange observation about the δ 18O\delta^{18}O data. The curve γ\gamma describing this, which presumably also measures the total amount of ice on the planet, has the following strange property: The derivative γ\gamma\prime is negatively correlated to γ\gamma, with a time lag of 9 ky. The long time lag is curious, and he does suggest several possible physical explanations for this correlation, without chosing his favourite. It seems (though it’s not stated absolutely clearly in the article), that this correlation is mainly valid in the ablation phase, that is during de-glaciation.

    An interesting fact is that the correlation persists through the change from 40ky to 100\sim100 ky periods. This seems to imply that the physics of ablation is the same in both cases. I think that Huybers takes this as evidence that the forcing is also the same. Since excentricity can’t be blamed when the period is 40ky, excentricity can’t be guilty even when the period is 100ky.

    Based on this observation, Huybers writes down a time dependent difference equation, whose time dependence codifies a periodic forcing, with period 40 ky. He states that the difference equation has chaotic behaviour, but also that there exists a periodic orbit with period 40ky. Close to this periodic orbit, there are solutions with periods 80ky. These solutions alternate between 40ky of mild glaciation and 40 ky of severe glaciation. There are also cycles of period 120 ky (I don’t know if they are also supposed to be close to the basic periodic orbit) , passing through 40ky of interglacial, 40 ky of mild glacial and finally 40 ky of full glacial. So it seems that we can produce sequences of cycles close to these periodic orbits which reproduce the geological record. Unfortunately I did not entirely understand the mathematics of this part of the article.

    The system is chaotic, so we should expect random shifts between the various periodic orbits. There is no particular external reason for the shift from 40ky to 100ky.

    • CommentRowNumber38.
    • CommentAuthorJohn Baez
    • CommentTimeJul 28th 2011

    This is fascinating, Marcel! Regardless of whether Huyber’s theory is right or not, his article sounds like it’s full of useful clues. Thanks for summarizing it. I’ll read it fully when I have time and start blogging about this and other papers we’ve been looking at here. I’m afraid I need to write to really think hard about things.

    • CommentRowNumber39.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 28th 2011
    • (edited Jul 31st 2011 by John Baez)

    I’m still a little confused about the variations of the semi-major axis, but not as much as yesterday. The authority on the relevant celestial mechanics seems to be Laskar. There are three relevant papers with him as author or first named co-author:

    1. Laskar, J., The chaotic motion of the solar system. A numerical estimate of the chaotic zones, Icarus 88 (1990), 266-291.

    2. Laskar, J., Joutel, F. and Boudin, F., Orbital, precessional and insolation quantities for the Earth from -20 Myr to + 10Myr, Astronomy and Astrophysics 270 (1993), 522-533.

    3. Laskar, J., Gastineau, M., Joutel, F., Robutel, P., Levrard, B. and Correia, A.C.M., A long term numerical solution for the insolation quantities of Earth. Astronomy and Astrophysics 428 (2004), 261-285.

    In [2] section 4.1, they state that for the purpose of computing insolation, they neglect the secular variations of the semi major axis. There is a diagram for the variation of the semi-major axis in [3] (fig 11) which seems to justify this. In [3] section 7, they state that an early version of this fact was known to Laplace and Lagrange!

    In these papers they describe a rather complete computation of orbit elements, obliquity and precession for several tens of millions of years. There is some uncertainties, for instance due to the tides on Earth. In [2] 7.2 it is taken into account that the magnitude of the tides might change during an ice age. This is further discussed in [3] 4.2, but I’m not entirely convinced that they really really know by how much. They do seem pretty convinced that their calculations are correct from -20 My to present. If you try to extend the calculations to earlier times, you will eventually be up against chaos. In [3] section 11 they state that it is hopeless to try to get to the mesozoic, before - 65My.

    The best place to start reading these papers is probably [3], especially because of the interesting diagrams and tables.

    • CommentRowNumber40.
    • CommentAuthorzskoda
    • CommentTimeJul 30th 2011

    Proper Latin spelling is Milutin Milanković, and in Serbian cyrillic spelling Милутин Миланковић.

    • CommentRowNumber41.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 30th 2011
    • (edited Jul 31st 2011 by John Baez)

    Imbrie J. and Imbrie J. Z., Modeling the Climatic Response to Orbital Variations Science 207 (1980) 943-953 seems interesting. The point of view is that to an ice sheet, we are all day-flies. Everything else except the Milankovitch cycles takes place on much shorter timescales. Therefore one could try to model the ice sheets as a very slow dynamical system only involving astronomical forcing. Later, one could model climate with the state of the ice sheet as a boundary condition.

    They formulate such a model, favourably compare it to data, and also to more complex previous models, especially Calder’s.

    One important remark is that increase and decrease of ice sheets is not symmetric, but that decrease is faster. The state of the climate is given by a single real parameter y. The state yy will later be compared to the oxygen isotope data δ 18O\delta^{18}O. They write down a time dependent differential equation for y in the following way. First they define a “forcing function”

    x(t)=Obliquity+α (Excentricity) sin( (Precession) ω).x (t)= \text{Obliquity} + \alpha \text{ (Excentricity) } \sin(\text{ (Precession) } -\omega).

    Here the eccentricity, precision and obliquity are functions of time given externally by celestial mechanics. The parameters α\alpha and ω\omega are free, time independent parameters. The differential equation depends on two further parameters T cT_ c and T wT_ w, expressing how fast ice increases in cool times respectively decreases when it’s warm. The equation is (equation (2), p 949, third column)

    dydt={1T c(xy) ifxy, 1T w(xy) ifyx. \frac{d y}{d t}= \left\{ \begin{aligned} \frac 1{T_c} (x-y) & if x\ge y ,\\ \frac 1{T_w} (x-y) & if y\ge x . \end{aligned} \right.

    The game is to determine the parameters α,ω,T c,T w\alpha,\omega,T_ c,T_ w (and presumably y(0)y (0)) so that the resulting curve y(t)y (t) is as similar as possible to the observed δ 18O(t)\delta^{18}O (t) data.

    If T cT_ c and T wT_ w did agree, this equation would be a linear differential equation. Because they do not agree, we get some strange kind of non-linear differential equation.

    Also note that this forcing does not take into account that excentricity changes total annular insolation. There seems to be at least three schools in this game. The pure obliquists think that the effects of eccentricity are too small to play a role for the ice sheets, and that the important forcing is the tilt of the axis of the Earth. I suspect Huybers of being a pure obliquist. The equation above is a moderate obliquist approach, since we forget about the direct insolation effects of the eccentricity, but at least we do rememeber that the eccentricity modulates the effects of the precession.

    They make certain choices of the parameters involved (p 950, third column), and make a rather convincing comparison with data from two different ice cores (figure 8). As we go back in time, the comparison gets less impressive, and beyond -350 ky it’s not so good (p 951, first column).

    At the end they do notice that the model doesn’t do very well on the 100ky periodicity. Figure 10 in this paper seems to provide very strong arguments for the third school, the eccentrics. I would like to see a more recent version of this diagram.

    • CommentRowNumber42.
    • CommentAuthorJohn Baez
    • CommentTimeJul 31st 2011
    • (edited Jul 31st 2011)

    Thanks for all your comments, Marcel! They’re very helpful.

    I corrected your spelling from “excentricity” to “eccentricity” throughout. This means that the third school of Milankovic cycle theorists are now called “eccentrics”. Since an “eccentric” also refers to someone who is “unconventional and slightly strange”, this might seem a bit insulting to members of the third school. But since there are fewer people in this third school, perhaps that’s okay as long as everyone has a good sense of humor.

    I have not been able to get this equation of yours to display correctly. I don’t believe \begin{cases} works here or on the wiki. I’ve tried to correct it to this, but it still doesn’t seem to work:

    $$

    \frac{dy}{dt}=

    \begin{aligned}

    \frac 1{T_c} (x-y) & \textrm{if} x\geq y ,\\

    \frac 1{T_w} (x-y) & \textrm{if} y\geq x .

    \end{aligned}

    $$

    I’ll keep working on it.

    • CommentRowNumber43.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 31st 2011
    • (edited Jul 31st 2011)

    I’m an eccentric at heart. Thanks for correcting me!

    The equations looked fine on my machine yesterday, they don’t now. One thing I discovered by experimenting is that you can’t write T_c, you have to make a space between “_ “and “c”. On the other hand, T_1 is fine. Weird.

    • CommentRowNumber44.
    • CommentAuthorJohn Baez
    • CommentTimeJul 31st 2011

    Okay, I got it working nicely. This:

    $$

    \frac{d y}{d t}= \left\{

    \begin{aligned}

    \frac 1{T_c} (x-y) & if x\ge y ,\\

    \frac 1{T_w} (x-y) & if y\ge x .

    \end{aligned} \right.

    $$

    produces this:

    dydt={1T c(xy) ifxy, 1T w(xy) ifyx. \frac{d y}{d t}= \left\{ \begin{aligned} \frac 1{T_c} (x-y) & if x\ge y ,\\ \frac 1{T_w} (x-y) & if y\ge x . \end{aligned} \right.

    Unfortunately I’m not sure which of the changes I made were strictly necessary. I added the \left\{ and \right. to get a nice looking parenthesis at left of the cases. This forum and the wiki have a curious feature, namely that a word like if, written without spaces, will automatically come out roman like this:

    if if

    Unfortunately this means that $$\frac{dy}{dt}$$ gives

    dydt \frac{dy}{dt}

    while $$\frac{d y}{d t}$$ gives the more attractive

    dydt\frac{d y}{d t}
    • CommentRowNumber45.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 31st 2011
    • (edited Jul 31st 2011)

    I still get a “broken image” for the display formula for the forcing function. I’ve on windows right now, and tried out firefox, explorer and chrome. Also, there is no spacing between “if” and the symbols in the aligned formula. (That doesn’t really bother me, just wanted you to know in case it displays right on your system).

    • CommentRowNumber46.
    • CommentAuthorJohn Baez
    • CommentTimeJul 31st 2011

    The only problem I see in the formula for the forcing function is a slight shortage of space between ’if’ and the inequalities. I can easily fix that. I don’t understand the other problems you’re having! I use Firefox.

    • CommentRowNumber47.
    • CommentAuthorJohn Baez
    • CommentTimeJul 31st 2011
    • (edited Jul 31st 2011)

    Here in Beijing I’m somehow not in the mood for deep thought about Milankovic cycles, but I’m having a lazy day in the hotel, recovering from an excess of tourism, so I did accomplish a little: I moved some material from this forum to the wiki.

    So, now there is a section called

    Milankovitch cycle: The effect of changes in eccentricity

    which computes how changes in eccentricity (with semi-major axis held fixed) affect the global annual insolation of the Earth.

    Also, there is a section called

    Milankovitch cycle: Technical papers

    which summarizes a number of technical papers. This is largely borrowed from what Marcel has written here.

    • CommentRowNumber48.
    • CommentAuthormarcel bokstedt
    • CommentTimeJul 31st 2011
    • (edited Jul 31st 2011)

    Have a nice evening in Peking, don’t worry about the Milankovic cycles, they are utterly patient and will wait for you. I’m soon going to Peking myself for ten moths starting in September (teaching algebraic topology at Tsinghua).

    There is something fundamentally strange about Itex. I’m not making anything up, here are the results I get from my machine:

    Typing T_c inside itex: T cT_c gives a broken link.

    Typing T_1 inside itex: T 1T_1 gives the wanted result.

    Typing T_ c inside itex: T cT_ c (here a space is inserted) also gives the wanted result.

    Typing:

    a=

    \begin{cases}

    b if a=b\

    c if a=c

    \end{cases}

    inside itex:

    a={bifa=b cifa=c a= \begin{cases} b if a=b\\ c if a=c \end{cases}

    actually works for me, and gives what I wanted (except for spacing).

    And here is something utterly strange:

    x(t)=Obliquity+α (Excentricity) sin( (Precession) ω).x (t)= \text{Obliquity} + \alpha \text{ (Excentricity) } \sin(\text{ (Precession) } -\omega).

    which is pasted from your correction above gives a broken image symbol, but

    x(t)=Obliquity+α(Excentricity) sin( (Precession) ω).x (t)= \text{Obliquity} + \alpha \text{(Excentricity) } \sin(\text{ (Precession) }-\omega).

    which I just typed in on my keyboard works correctly on the preview, although I can’t see a difference between the codes in the two lines. In both cases the code looks like

    x (t)= \text{Obliquity} + \alpha \text{ (Excentricity) } \sin(\text{ (Precession) } -\omega).

    What is going on here??? What do other people see on their machines???

    • CommentRowNumber49.
    • CommentAuthorJohn Baez
    • CommentTimeAug 1st 2011

    I’m soon going to Peking myself for ten moths starting in September (teaching algebraic topology at Tsinghua).

    Interesting! Liang Kong just found out I was in Beijing, and invited me to give a talk at Tsinghua - but it’s too late. I see Tsinghua is on the west side of Beijing, like where I’m staying. Unfortunately all the really pleasant neighborhoods I’ve found so far are on the east. But it’s an immense city and I’ve only scratched the surface.

    I’ll copy your question about itex over to the category “Technical”. That’s the category for discussions of the technical infrastructure of the Forum and the Wiki. They’re both run by Andrew Stacey, and putting a conversation in the category “Technical” means he’ll see it.

    • CommentRowNumber50.
    • CommentAuthorIsaac Held
    • CommentTimeAug 2nd 2011

    Marcel and John,

    With regard to Imbrie and Imbrie (1980) the problem is that it requires the equilibrium response to which one relaxes asymmerically to be much larger than the actual response, given the parameters they get from their fit – ie their equilibrium climate sensitivity seems implausibly huge. I mentioned this in an old paper.