Milankovitch Cycles

James Hansen has a new paper (a draft for review), “Paleoclimate Implications for Human-Made Climate Change.” We’ll discuss it in a future post. There’s a so-called “review” by Martin Hertzberg at WUWT in which he claims that Hansen fails to understand the Milankovitch cycles. But it’s Hertzberg whose understanding is a failure.

Hertzberg quotes Hansen:

Earth orbital (Milankovic) parameters have favored a cooling trend for the past several thousand years, which should be expected to start in the Northern Hemisphere (NH). For example, Earth is now closest to the sun in January, which favors warm winters and cool summers in the Northern Hemisphere.

Then Hertzberg says:

Those statements are typical of the misunderstanding in the popular literature of the Milankovic cycles. Since we are now further from the sun in the NH summer, he argues that the NH should get less solar insolation in the NH summer thus “favoring the growth of glaciers and ice-caps in the NH”. So why then we may ask are we now in an Interglacial Warming? What Hansen fails to realize is that when we are further from the Sun in NH summer we move more slowly in orbit, and are therefore exposed to the summer sun for a longer period of time.

Sorry, Martin, Hansen is right and you’re wrong.

First let’s examine the issue of further-during-summer and longer-duration-summer.

The energy I received from the sun at any moment is inversely proportional to the square of the distance from the sun r

I = S/r^2.

The total energy E received throughout the year is the integrated momentary energy over time

E = \int I ~dt = \int (S/r^2) ~dt.

One of the properties of planetary orbits is that the planet’s longitude in its orbit \lambda increases at a rate d\lambda/dt which is also inversely proportional to the square of the distance r

d\lambda / dt = h / r^2,

where h is a constant, the specific angular momentum of the planet in its orbit (the angular momentum per unit mass). This is an equivalent statement of Kepler’s second law of planetary motion.

This means that the infinitesimal change in longitude is related to the infinitesimal change in time

(r^2/h) ~d\lambda = dt.

If we substitute this expression for dt in our annual-total energy integral we get

E = \int (S/r^2) ~dt = \int (S/r^2) (r^2/h) ~d\lambda = \int (S/h) ~d\lambda.

When we integrate time over an entire year, that’s the same as integrating longitude over a full circle. But the longitude integral is of (S/h), which is a constant. So the integral itself is a constant. In other words, it doesn’t depend at all on whether the earth is closest to the sun during northern hemisphere winter, or summer, or spring or fall or whenever. It’s always the same. And that’s true not only for the earth as a whole, but for any specific location on earth. If your location is further from the sun during summer (so a summer day gets less solar energy), summer also lasts longer (you have more summer days), and those two competing factors cancel each other out. Exactly.

The changing relationship between the seasons and when we’re closest to the sun is the precession cycle, one of the Milankovitch cycles. We’ve just seen that this particular Milankovitch cycle has no effect at all on total annual solar insolation, either for the planet or any single location on it. If that were the whole story, then this particular Milankovitch cycle would have no effect on ice sheet growth or decay.

But it’s not the whole story. Suppose you live in the extreme north, at latitude 65 deg.N. Suppose further that the precession cycle is such that you’re further from the sun during your summer, and closer during winter. Then midsummer day won’t be as warm as it would otherwise — so you can expect less ice melting. Also, midwinter day won’t be as cold as otherwise, so you can expect the warmer winter air to hold more moisture and produce more snowfall. Less midsummer melt, more midwinter snowfall, both tend to increase ice accumulation and make ice sheets grow. If the precession cycle is reversed, you’ll have warmer midsummer and more ice melt, colder midwinter and less snowfall, both of which tend to decrease ice accumulation and make ice sheets shrink. That’s why the precession cycle actually does affect ice sheet growth and decay. Even though you get exactly the same total solar energy over the entire year, the precession cycle affects when you get that energy during the year.

Right now, in the northern hemisphere we’re closer to the sun in winter and further in summer — just the right conditions to increase ice growth and cool the planet. The situation is opposite in the southern hemisphere, but since there’s so little land (compared to ocean) in the southern hemisphere there’s a lot less land-based ice as well. So the present condition of the precession cycle favors the increase of glacial ice, especially in the northern hemisphere, just as Hansen said.

Hertzberg reveals his failure to understand when he asks, “why then we may ask are we now in an Interglacial Warming?” We’re not. We’re in an interglacial warm but not warming. And we’re in that interglacial warm because 12,000 years ago, when we came out of the last glaciation, the precession cycle was opposite its present condition — the northern hemisphere was closest to the sun during summer and furthest during winter, exactly the condition which tends to make ice sheets waste away. The present state of the precession cycle is tending to reverse that, just as Hansen said.

Hertzberg also said “What Hansen fails to realize is that when we are further from the Sun in NH summer we move more slowly in orbit, and are therefore exposed to the summer sun for a longer period of time.” On the contrary, I’m confident that Hansen is fully aware of that, and fully aware of the fact that it’s through the timing of incoming solar energy, not its yearly total, that the precession cycle affects the growth and decay of ice sheets. But apparently, Hertzberg is not so aware.

In fact Hertzberg is so confused about the whole issue that he says:

So 10,000 years ago the earth was further from the sun during NH winter and it spent a longer time on the winter half of the orbit, thus both effects re-enforced each other to give us a marked Glacial Cooling. (Actually the peak in that Glacial Cooling occurred several thousand years earlier than 8,000 BC.)

On the contrary, 10,000 years ago the precession cycle so tended to warm the earth that that was near the peak of interglacial warmth. Until now … but the present warmth isn’t due to Milankovitch cycles.

When it comes to the influence of Milankovitch cycles on glaciation, Hertzberg couldn’t be more confused. His assertion that it’s Hansen who is confused about the issue is just an example of the Dunning-Kruger effect.


I’ve posted about Milankovitch cycles before. Those who want more detail can find some in these posts:

Wobbles, part 1

Wobbles, part 2


82 responses to “Milankovitch Cycles

  1. Tamino, I have a question about the Milankovitch cycles I hope you can clarrify. I understand that we are at, or near the minimum of summer insolation at 65 degrees north in our current cycle. But I have heard it claimed both that the Milankovitch cycle will continue to result in cooler summers (where it not for global warming), and that on the contrary, northern summers are now warming because we have passed the current cycle’s minimum. Do you know the transition date from a reducing summer insolation to an increasing summer insolation in the NH? Also, do you know how long it will be before NH summer insolation next goes below the minimum value it will reach (has reached?) in this cycle?


    [Response: I don’t know the precise upcoming date, but I do know that we have not yet reached the minimum of summer insolation at 65N in the N.Hem, it has yet to come. Claims that we’re on the upswing of Milankovitch influence are just wrong, we’re still on the downswing (but a lot nearer minimum than maximum). If you’re intensely interested, you can get data for orbital parameters and insolation for the past 5 million years from Berger & Loutre, and for the past 50 million and future 20 million years from Laskar et al.]

  2. Great argument Tamino.
    You lost me a bit with the integrals, but the whole shows how we realy have to rely on the experts and not think we know better. The climate scientists are not lying to us. We should be paying attention and we’re not!

    • Ricki, think of the integrals as sums–that is all they are. Calculus like most math breaks down to addition and subraction. Think of it like this: We’re breaking the ellipse into pieces that represent equal angles in the orbit. Tamino has shown that all the pieces are equal, so equal angles receive roughly equal angles even though it takes different times to traverse them.

  3. What I understand Hertzberg is saying is that when the NH is further from the Sun in the summer (as now) then there are two opposing factors effecting the amount of solar energy the NH receives in the summer:
    A) greater distance from the Sun reduces the amount of energy
    B) longer summer (because of Kepler’s Second Law) increases the amount of energy.

    Hertzberg is claiming that effect B) is greater than effect A) and that therefore Hansen’s claim that, “the NH should get less solar insolation in the NH summer” in such circumstances, is wrong. Hertzberg states “Exposure time in this case is more significant tha[n] daily insolation caused by our further distance during the NH summer” and claims that in 2010 the “summer” period (equinox to equinox) lasted 7.1 days longer than the winter.

    Tamino shows that the total solar energy received per year is not dependent on such factors, but I cannot see that you address the particular point of whether the effect B) overwhelms effect A). That is, you do not show that the energy received from the Sun, in the NH summer further from the Sun situation, is more or less than in the NH closest to the Sun situation.

    Of course, Hertsberg does not show this either – he merely makes the assertion that effect B) is “more significant” than A) without any supporting calculation, so he can hardly expect to be taken seriously.

    It also seems to me that the effect B) might be entirely irrelevant: it is just a red herring designed to throw dirt at Hansen in a way that isn’t easy to refute, since it is not based on any calculation (other than the extra 7.1 days). But it would be interesting to see graphs of daily insolation at 65N for the “furthest” and “closest” situations – I assume someone has done them?

    For me, Hertsberg confirms himself as a rather nasty fruitcake when he continues, talking about the Vostok core, and makes the utterly ridiculous claim that, “the current increase in CO2 … is merely the continuation of a natural process that has nothing whatever to do with human activity” whilst making false insinuations about “propagandists and scientists”. Indeed, it is he who is the propagandist.

    [Response: When computing the total energy during summer (or any season defined by earth’s longitude along its orbit rather than a constant time duration) the two factors exactly cancel, and the energy is unaffected by the precession cycle.

    But it’s not just total energy that counts, it’s *when* the energy arrives — is it more intense at midsummer (like 12000 years ago) or more evenly distributed (like now). “More evenly distributed” is, I believe, what Hansen is referring to when he says “less solar insolation in the NH summer.” After all, he has both the mathematical and astronomical skillz, and this isn’t idle speculation — it’s standard in the literature, and insolation at 65N during midsummer is the “standard” index for correlating Milankovitch cycles with glacial cycles.

    And of course there’s the Milankovitch cycle of obliquity, which *does* affect annual (and seasonal) total insolation for a given latitude (but not for the globe as a whole).

    As for the current CO2 increase being a natural process that has nothing to do with human activity, that’s one of those total wingnut ideas, dumber than a bag of hammers.]

    • Not that it really matters in the context of Hertzberg’s poor understanding, false claims, and laughable insults, but does anyone else get 7.1 days?

      Taking the equinoxes as being
      20.03.2010 17:32 and
      23.09.2010 03:09

      I get
      7.55 using the sidereal year (365.256 days)
      7.56 using the tropical year (365.242 days)
      7.54 using the anomalistic year (365.259 days).

    • Thanks for that. I’m still struggling somewhat, but I take your point about the importance of MIDsummer insolation.
      In your final equation, the solar energy received, E, is directly proportional to lambda, that is the angle of longitude that the planet sweeps through, whatever that angle is (ie not just 360 degrees for a whole year or 180 for a summer). Is that correct?

      [Response: Yes. You’ve received half the solar energy you’ll get during the year when you’ve swept out an angle of 180 degrees, one quarter of the year’s energy for an angle of 90 degrees, etc.

      I’m not sure that MIDsummer insolation is “the” controlling factor, but it seems to be the most commonly used quantity.]

  4. since there’s so little land (compared to ocean) in the southern hemisphere there’s a lot less land-based ice as well.

    Um, I think the Antarctic ice cap would like to disagree.

    [Response: Right you are. But during glaciation, the northern hemisphere ice sheets are huge — and they’re the ones that grow and decay, since the Antarctic ice sheet is so stable (it’s still there in spite of the present interglacial).]

  5. Steve Metzler

    Tamino, hi,

    So glad you weighed in on this. The Dunning-Kruger is so strong in the collective WUWT readership that sometimes you feel it might bolt from the stable and go on stampede.

    Hansen (though formally educated as an astronomer, as they love to point out) was doing climate science before most of those idiots were born. And yet, they won’t lend him the slightest shred of credibility. Everything the poor guy says must be wrong, as in their eyes he is the ultimate embodiment of ‘warmist’ evil. Any attempt to point out that these armchair scientists who post their pseudo-scientific drivel on WUWT are talking out their rear ends is met with a circling of horses; nay, a stampede!

    • The Dunning-Kruger is so strong in the collective WUWT readership that sometimes you feel it might bolt from the stable and go on stampede.

      Luckily Judith Curry is there to give it a new home.

  6. It seems to me that the only “argument” Mr. Hertzberg has is common sense. His WUWT/SPPI article shouts at me: “don’t bother with actual science, here’s my kitchen knowledge! And a nice graph! Go buy my book, here’s an absurdly nonsensical chapter taken from it!” There’s no deep analysis involved, there’s only cursory calculations which may or may not be relevant (they are not). It sounds plausible, it appeals to common sense, it’s easy to grasp. And it’s so wrong that my head begins to hurt.

    A very quick glance at the actual science tells that insolation at northern latitudes (65N) in summer seems like a good indicator of ice age cycles – and that has been going down (colder) at the same time when Hertzberg’s bizarro world has been warming because its northern latitudes are “exposed to the summer sun for a longer period of time”. Did I say my head hurts?

    I could almost call “common sense” more like “common cognitive prejudices and biases in thinking.” It’s easy to fall prey to them. Common sense and intuition are tools of understanding, and not very good at that, at least by themselves. As one delves deeper into the ways of nature – into science – somehow they become gradually less and less useful. Nature’s imagination overwhelmingly surpasses our own; her ways are not intuitively obvious. That’s why we need careful reasoning, analysis and evaluation of evidence. Actual science, that is.

  7. “The situation is opposite in the southern hemisphere, but since there’s so little land (compared to ocean) in the southern hemisphere there’s a lot less land-based ice as well. ” The latitude of the Southern continents is also inportant. Very little of the land lies between 60-80S, the sort of latitudes we can expect snow to lie during spring and have a serious impact.

  8. That is Martin Hertzberg the combustion scientist I presume.

  9. You know, we need another term than Dunning-Kruger for the sort of person who is so utterly ignorant that they don’t even realize they are ignoranant AND is so ideologically blinkered that they cannot learn. Dunning and Kruger’s research ended on a hopeful note–people could learn. Hertzberg and his fellow ideologues are stupid.

    Hertzberg has been at this for years. He’s one of the “experts” who convinced ultra-lefty Alexander Cockburn that current warming is due to a “natural cycle”. Hertzberg’s real expertise: he’s a retired explosives and combustion scientist. It clearly ain’t celestial mechanics.

  10. Thanks Tamino. Well said.

  11. Aahh, but but Tamino, you used math, formula, weird symbols and logic. That’s not fair!

  12. Your mathematical statement of Kepler’s second law is correct but you seem to consider r to be constant. It’s not, because the earth’s orbit is elliptical.
    Say the orbital distance varies by a factor of 2, smallest in ‘summer’, and the seasons are defined by the time taken to sweep an arc of 90 degrees. With that degree of orbital variation, summer would last a quarter of the time of winter.
    Just be glad the eccentricity is not so large

    [Response: No I don’t consider (or suggest) r to be a constant. What I’m saying (and many others have confirmed) is that the amount of time spent at a certain distance has the opposite r-dependence from the instantaneous insolation at a certain distance (because the angular velocity has the same r-dependence as instantaneous insolation), so when the energy is summed in a *longitude* integral, the integrand is constant. Hence the total integrated energy throughout a single orbit is unaffected by the precession cycle.

    This isn’t my speculation, it’s standard well-known Milankovitch theory.]

    • Jeebus, dude, did you even bother to read what Tamino wrote? If we ignore the atmosphere, what determines equilibrium temperature is the energy flux. It is lower in the Northern Hemisphere. It doesn’t matter how long the Northern Hemisphere hangs there. The equilibrium temprature remains lower. Likewise, the winter is warmer. Yes, if you throw in the atmosphere and oceans, , it complicate things a wee bit, but the basic physics is radiative.

    • ‘Doubtful’ is doubtful of Kepler’s 2nd law. Let me know how that works out for you.

  13. Hertzberg has popped up before spreading misinformation:

  14. Apparently by shear coincidence, I was looking at Huybers 2006b this weekend. Readers might enjoy a glance at his letter in Science:

    Click to access Huybers_EarlyPleistoceneSummer.pdf

    [Response: Yes, it’s an excellent (and thought-provoking) paper. It’s Huybers’ suggestion for why the obliquity cycle seems to impact glaciation more strongly than the precession cycle, which it does — especially during the Pliocene and early Pleistocene (before the mid-Pleistocene transition).

    But it’s not the whole story, because integrated summer insolation (a la Huybers) doesn’t vary with the precession cycle but glaciation does (even in the Pliocene and early Pleistocene, although only weakly). I also have a “pet theory” along similar lines, that the driving factors might include what I call “melting insolation,” which is the total insolation over threshold for some “melting threshold” value (the idea is exactly analogous to that of “cooling degree days”).]

  15. Let’s see if I’ve got this right. (I’m one of your math-challenged readership, I’m afraid, so following the discussion about the ‘longitude integral’ induces uncertainty–or, better, feelings of uncertainty–for me.)

    Following that discussion, when you get to:

    If that were the whole story, then this particular Milankovitch cycle would have no effect on ice sheet growth or decay. . .

    you could have added “. . . and Hertzberg would be correct.” Yes?

    [Response: No, because he maintains that the duration factor overwhelms the proximity factor, when in fact they exactly cancel. But he is correct that the duration effect is opposite to the proximity effect.]

    Of course, as you go on to point out, it’s not the whole story–unsurprisingly, as despite the misapplied sneer it’s Hertzberg, not Hansen, who is writing “in the popular literature.” (As we all know, Milankovitch theory has been reviewed quite thoroughly by now by much better analysts than Herzberg.)

    Case in point would be the final quote, the one beginning “so 10,000 years ago. . .” At the top of his discussion he argues that orbital speed compensates for solar intensity; at the bottom, that the two reinforce each other. (To make this claim, of course, he ignores the point that the shorter summer is sufficiently compensated by more intense insolation–resulting in no annual change, as your discussion shows.) But logically, you can’t have it both ways–and it doesn’t take great mathematical acumen to see that.

    In that sense, it appears to be a classic of denialist doublethink. Argue the case for summer when you can use it to cast doubt, even if you have to make an unmotivated change of subject to do it. Then argue the case for winter when you can use it to make an apparent case for cooling. (Then don’t bother to check your facts–or hope your readers don’t. IMO, the bit about the “earlier” “peak in global cooling” suggests it was more the latter than the former.)

    But then, it’s all about the FUD–and this topic is confusing enough as is. When Kaufman et al 2009 came out, I had some online discussions with a usual ally–a pretty astute guy–about the effects of the various Milankovitch cycles, and particularly the precession cycle, trying to get this issue straight. It wasn’t easy.

    • Thanks, Tamino. As an addendum, I found this correction note in Science regarding some technical issues with the proxy data in Kaufman 2009. It’s interesting in its own right, and has a nice graph of reconstructed Arctic temperature showing both original and corrected versions.

      Click to access Correction_and_Clarification.pdf

      • I guess I’d add, also, that it wasn’t clear to me from the quoted portion that Hertzberg’s argument was that ‘duration rules.’ I didn’t check out the original, which presumably would have disabused me about what he was saying; thoroughness would have called for that, but life is short. . .

  16. Tamino, thanks for the tips. From Berger and Loutre, I found out that the insolation in June at 60 degrees north will be the same in a thousand years as it is currently, and increase thereafter, indicating that the minimum has not arrived yet, but will in the next thousand years. Unfortunately with thousand year time steps it does not narrow it down further.

    I’ll take this opportunity to thank you for your work on this blog. You are consistently the clearest, most informative and most logical expositor of the science on any of the science blogs I have read.

  17. Do we need to start a fund to promote stamp collecting, bridge and other hobbies for all these emeritus professors so they don’t all go so…emeritus on us? This is some world class stupid here. This guy is basing his entire argument on a fundamental, demonstrable misunderstanding.

    As I said above, just think of how this would work with a big rock–no atmosphere or oceans. The radiation flux determines the equilibrium temperature. The radiation flux decreases as the inverse square. Lower temperature in summer means less melting. Yes, the oceans and atmosphere do take time to warm up, but this doesn’t change the fundamental argument. This idjit would have us freezing in the interglacials and walking around in shorts in the glacials. THIS is weapons-grade stupid.

  18. I’m going to see if I can start something here. Every time I see someone debunk some nonsense at WUWT, as Tamino has done here, I am going to post the following comment from a WUWT mod:

    “This is a BLOG rather than a science journal. Though some entries may contain errors at least here they are not hidden. … bl57~mod”

  19. Daniel J. Andrews

    What surprises me is not that he thinks Hansen has made a relatively simple mistake that is found in popular literature, but his inference that this simple mistake is believed and cited by scientists writing articles in the peer-reviewed journals–AND isn’t challenged by any of the experts who read these articles. In other words, scientists are collectively incompetent or corrupt.

    On second thought, can’t say that surprises me after all–it is just more of the same idiocy we hear from the conspiracy minded. Nothing new under the sun, sigh.

  20. Presumably you have used circular formulae to simply the mathematical represenation, but it does imply a constant orbital radius.
    I quite accept that the total integrated energy through one orbit is unaffected if the average orbital distance remains constant. And the four seasons are defined by the orbital time of perihelion, aphelion and the pionts of Aries, all of which are spaced by orbital spans of 90 degress and thus receive equal amounts of energy. However, the timing of crossing the points of Aries does not remain fixed with varying orbital eccentricity. The duration/length of the seasons varies accordingly. Therefore the average seasonal energy flux must also vary. What the effect of this averaging is can be debated.

    [Response: No, I have not used circular formulae to simplify the mathematical representation, and I have not implied in any way that the orbital distance remains constant. This is your misunderstanding, not my misimplication.

    As I have said repeatedly now, the point is that the changing earth-sun distance during the year introduces two competing effects — closer to the sun means more intense insolation but also that it’s more brief — and because of Kepler’s 2nd law those two factors exactly balance each other. It makes the time integral of a time-varying quantity equal the longitude integral of a constant, so the average seasonal energy flux is unaffected by changes in the timing between perihelion and and the seasons, i.e., by the climatic precession angle. This is very well-established straightforward mathematics, it’s in the literature, I didn’t make it up or “discover” it.]

    • Gavin's Pussycat

      Doubtful, tamino’s derivation is exact for arbitrary (even very large) eccentricities. It is really obvious.

      Yes, there is a (small, for the Earth) dependence of the annual (or seasonal) total energy on eccentricity for a given period (i.e., semi-major axis) through h. But that was not the argument.

  21. Tamino – I’ve been reading Hansen’s draft and some of the papers it references – in particular a few of his papers (with and without coauthors) from 2007. What seems new is that it’s looking at deep ocean temperature proxies as a record of global past climate, but the conclusions they reach (the 1 degree C above pre-industrial for previous interglacials, and the danger etc of going beyond) seem to be essentially the same as he’s been saying for quite some time. Of course the Milankovitch statement in particular was nothing at all new and quite beside the main point.

    The claim I found most interesting was the statement that acceleration of icesheet melting ought to be exponential with doubling time of order 5 to 10 years. I don’t see a lot of support for that claim in this paper or the previous ones he cites; it seems to be mostly conjecture, though with a few pieces of paleoclimate evidence to back it up. I’m wondering if anybody else has come across a real physical basis for Hansen’s claim of exponential growth in melt rate?

    • I reread the scientific reticence article a couple of weeks ago and all the sudden “5 meters” is ubiquitous.

      I’m a piker at searching scholar, but a few months ago I was looking for anything topping the horizon that might indicate papers based on nonlinear melting are in the pipeline, and could find no hint of it.

      Is anybody seriously working on modeling nonlinear melting? When scientists go to the ice sheets, they talk about it a lot because what they are seeing is not their father’s linear melting, but that is about as far as it goes.

      In the Copenhagen Diagnosis there are references to Delta Committee, 2008 and WBGU, 2006. Don’t know if they’re incorporating nonlinear melting.

    • That’s a good question and one whose answer I’d like to see. It’s not clear to me why Earth’s ice melt, like so many of these properties of the climate which are sought, ought to follow any simple arithmetic formula when looking at intermediate time periods.

      I suspect we will need to wait at least for another decade of GRACE measurements to really sort out the matter.

  22. What this says about Hertzberg is that he doesn’t bother to read anything before opening his yap (like alot of the posters at WUWT) – pseudointellectualism pure and simple. For those interested I recommend Ice Ages and Astronomical Cycles by Muller and MacDonald – everything you wanted to know about the theory (and alternate theories) of Milankovitch cycles.

  23. For those who are interested in more details about Milankovitch cycles and how they affect climate, I’ve posted on the topic before. See the UPDATE at the end of this post for links.

  24. Well, I leave you to decide why you used circular formulae, your error is not my problem.
    The average seasonal flux for a given eccentricity of orbit is indeed fixed, but it is not fixed as the eccentricity varies.
    This is trivial mathematics, and is well established.

    [Response: I’m well aware of the variation with eccentricity, in fact I posted about it over three years ago (it’s linked in the update to this post).

    But I never contradicted that. You, however, claimed I had said or implied that earth-sun distance was constant, which I never did. The error is yours, and if you can’t admit that you have a much more serious problem that failure to understand Milankovitch cycles.]

    • @Doubtful:

      It’s just a matter of making a simple change of variables in an integral. Maybe Tamino could have written I(t), lambda(t) and r(t) to emphasize the time dependence, but the result is quite obvious already as written.

      • “It’s just a matter of making a simple change of variables in an integral. ”
        Well you know that

        I know that

        And it would have avoided a silly distraction . . .

        But . . .

  25. Tamino,
    Your coverage of Milankovitch cycles (here and Wobbles 1 and 2) has provided some of the most enjoyable study for me, and it continues.

    Your response to the first commenter
    “we have not yet reached the minimum of summer insolation at 65N in the N.Hem, it has yet to come”
    challenges my understanding of the calculations provided by Laskar’s online calculator.

    I used Laskar’s calculator for monthly insolation at 65N for June, July, and August from 10,000 years before now to 10,000 into the future.

    I get the lowest 65N insolation value in June, 1,000 years ago (see ****), suggesting the minimum has passed. However, if I consider midsummer to be July, the minimum is at 3,000 year from now. I infer July is the standard for “midsummer”. Correct?

    My other assumption is that in Laskar’s calculator, the solstices are fixed at Jun and Dec. Do you know if that’s a correct assumption?

    thanks you,

    June 65N
    -10.000 506.363742
    -9.000 500.714265
    -8.000 493.772719
    -7.000 486.292833
    -6.000 479.092985
    -5.000 472.746640
    -4.000 467.577032
    -3.000 464.114729
    -2.000 462.230155
    -1.000 461.877584 ****
    0.000 462.983378
    1.000 465.139231
    2.000 467.944510
    3.000 471.224713
    4.000 474.364436
    5.000 477.051338
    6.000 479.072308
    7.000 479.999597
    8.000 479.741965
    9.000 478.465355
    10.000 476.103581

    July 65N

    -10.000 505.706734
    -9.000 505.958295
    -8.000 504.251348
    -7.000 500.701707
    -6.000 495.861936
    -5.000 490.153264
    -4.000 484.121336
    -3.000 478.407560
    -2.000 473.267764
    -1.000 468.979617
    0.000 465.776296
    1.000 463.587154
    2.000 462.408846
    3.000 462.258302 **
    4.000 462.918520
    5.000 464.244046
    6.000 466.128633
    7.000 468.184252
    8.000 470.188604
    9.000 471.908311
    10.000 473.014601

    Aug 65N
    -10.000 394.638391
    -9.000 400.038867
    -8.000 404.531441
    -7.000 407.479964
    -6.000 408.620837
    -5.000 407.804215
    -4.000 405.302359
    -3.000 401.364807
    -2.000 396.480208
    -1.000 391.089980
    0.000 385.603747
    1.000 380.365025
    2.000 375.750149
    3.000 371.929670
    4.000 369.191528
    5.000 367.632005
    6.000 367.316617
    7.000 368.214847
    8.000 370.200981
    9.000 372.919733
    10.000 376.117377

    [Response: I used the Berger & Loutre data, which includes a calculation of July and January insolation at 65N and S, and at 15N and S. As for Laskar’s online calculator, I haven’t checked but I see no reason to doubt that you did it right.

    In any case, clearly we’re near minimum now (by any number of measures) and it’s mistaken to believe that there’s any significant increase going on — especially enough to cause the current global warming.]

    [Response 2: I tried the online calculator from Laskar, and computed insolation on June 21 (the most common date I’ve seen in the paleo literature for Milankovitch effects). It reaches its minimum not 1000 years ago, but about 480 years in the future.]

    • Thank you. I got the same result now that I knew what to look for. Very helpful. And I agree with your first response in that exact timing of the minimum is not important in assessing the claim that warming is orbit-driven. My interest however, is in considering the optimum time that insolation is effective in deglaciation. I wonder if looking at July and August peaks would show a stronger correlation to the onset of deglaciation than June. Since rising CO2 is also a factor, insolation may be more effective after the northern hemisphere peak in photosynthesis. I’ll treat this as my assignment.

    • Gavin's Pussycat

      > My other assumption is that in Laskar’s calculator, the solstices
      > are fixed at Jun and Dec. Do you know if that’s a correct
      > assumption?

      I don’t think it matters. Even if it would be wrong for the current civilian calendar, you want to do this computation for days that are fixed relative to the solstices / equinoxes, i.e., the seasonal cycle.

      • Thank you, GP. I’m now more confident that when I enter “Jun 21” in Laskar’s calculator, I’m really saying “northern summer solstice” and not or drifting civil calendar date. However, new to me was that the minimum or maximum insolation dates change by thousands of years when you go from Jun to July. I’ve plotted Jun insolation values against Vostok temperatures and CO2 and observed visually a correlation with the onset of the last two interglacials, but not with the two earlier interglacials. Then I added the EPICA temperature data and got a good correlation with the last four interglacials. But now I’m eager to see if that correlation improves if I look at Jun, July, and Aug insolation values. My guess is that the resolution of the icecores is too coarse, and I should start with Greenland cores and try this with just the most recent termination.

        [Response: There’s more to it than just midsummer insolation — whether we define midsummer by the solstice, or some other point during summer. There’s much to be said for Huybers’ suggestion of using “summer heat,” or other possible metrics (perhaps my pet theory of “melting degree days”).

        And in fact the situation is, recently (by which I mean the last 800,000 years), rather complex. Before 800,000 years ago (before the “mid-Pleistocene transition”) glacial advance and retreat was almost entirely paced by the obliquity cycle of about 41,000 years rather than the precession cycle of around 18,000-23,000 years (it varies). In fact the Pleistocene before the mid-Pleistocene transition has been called the “41kyr world.” But the precession cycle is present in paleoclimate data even then (if only weakly), although it’s much stronger after the mid-Pleistocene transition. And, since the mid-Pleistocene transition the “cycle” (a term sometimes misused and often misinterpreted) is more like 100,000 years. It has been suggested that this is related to one of the eccentricity cycles, but the link is still tenuous.

        There’s a lot we don’t know about the timing of glacial cycles, especially after the mid-Pleistocene transition.]

      • So, “melting threshold” would be determined by the state of various Earth systems, e.g., extent of icesheets, greenhouse gas levels, ocean current modes (whether an ocean is outgassing or sequestering CO2), and others; and therefore, the threshold value would be changing over time. Also changing over time would be insolation and feedbacks that trigger deglaciation when they excede the threshold. I look forward to posts or papers on this.

  26. Thanks for debunking another spectacularly wrong post on WUWT, Tamino.

    At one point during his Goddard fiasco, I briefly suspected Anthony might actually give a damn about the soundness of his guests’ work. Silly me.

  27. David B. Benson

    It is uncertain just what value of insolation produces the hypothsized threshold effect for transitions between the three states interglacial/interstade/stade. The minimum we are in now (or, depending upon unknown factors, has passed or is yet to come) is the sole anomously warm minimum over the entire Quaternary. W.F. Ruddiman hypothezes that it is only anthropogenic CO2+methane which has kept the globe from entering the transition to massive ice sheets in the far north. Other than his interesting papers on this matter, a relevant source is Archer & Ganapolski’s “A Moveable Trigger?” (2005) which contains a reference to the original study of the three state transitions in paloclimate. Figure 3(b) of the cited figure uses June insolation at 65 degrees north. Are we agree that that particular value reached its minimum about 1000 years ago?

  28. Tamino,

    I was wondering about your view on the whole debate from the Ruddiman versus epica questions. Specifically about the whole debate with the proper analogue.

  29. Ummm … I’m sorry, but you have this effect exactly reversed. The most important consideration is not Earth’s total insolation, but the insolation at 65° N, which is the (rough) latitude at which most snow and ice forms in winter, hence the most important latitude for ice/albedo feedback. Consider that at midwinter, the Sun is below the horizon for 20 hours a day at that latitude, and suddenly it becomes clear that the effect of changing r (being closer or farther from the Sun) is mostly lost. Further, since the Sun’s altitude is very low when it is up, the incident radiation is quite low too (one must multiply by sin(alt) — not in your computations) — so yes, latitude does make a big difference.

    Obviously such conditions are bad for insolation, but consider how that changes as the length of the season changes. In winter, 65 N isn’t getting much sun regardless. But in the current situation (perihelion near NH winter solstice) winter snow season is near shortest possible number of days, while summer melt season is near longest possible number of days. This is ideal for reducing snow albedo during the year as a whole and warming the planet — which is why we are, in the current situation, in an interglacial rather than an ice age.

    [Response: I’ve never suggested what you think, in fact I’m fully aware of the importance of insolation at high latitudes where ice and snow are prominent. Also, this analysis is not specific to global average, it applies as well to insolation at a specific latitude. But that’s not the topic here, it’s Herzberg’s fautly comprehension.

    You can even read the articles linked to in the update, to find the dependence of insolation on geometrical factors (including the latitude of the observer).]

  30. Marion Delgado

    Why do I think WUWT is going to be on Coast to Coast AM soon :)

  31. Michael Hillinger

    I am an infrequent visitor to WUWT–just to see what the deniers are up to. I read the Hertzberg post and wanted to see how a real scientist would respond. Thank you for your extensive debunking. I’ll make sure to stop by frequently to see your thoughts. Please keep it up, I know it takes time to debunk such posts but it is invaluable to those of us who don’t have the background to do it.

  32. Tamino,
    I don’t know if you have seen this website before

    but it includes datasets for pretty much all climatic analyses’ including Gravity, ocean heat content, temperature, radiosonde, precipitation etc… You can also pick specific locations using it. I don’t know if you have seen it before but here is also an R script for extracting data from any dataset on the site:

    Hopefully this is of use to you.

  33. > TSTKTS
    > a retired explosives and combustion scientist.

    As an alternative to D/K or TSTKTS, and in the spirit of ‘hate the sin, not the sinner’ — I’d suggest:

    His argument is a petard.

    Rich in meanings, most of them wholly apt:

  34. Tamino,

    Thank you for the clear explanation that the overall power supplied (TOA) remains constant over a year despite the Milankovitch cycles for any given latitude.

    It is however, distributed temporarily in a different fashion depending on the position in the cycle. Thus we have more lower power input days balenced with fewer higher power input days.

    Given the apparent non-lineralities in the climate system surely this difference should mean that the outcomes will be different? Ice melting being one of them.

  35. This new article from Schmidt (from Hansen’s lab) et al. contains an interesting figure:

    Click to access gmd-4-33-2011.pdf

    Fig. 1. Insolation difference (2000–850 CE) (Wm−2) as a function
    of latitude and Julian day.

    • David B. Benson

      Thank you for the link.

    • What I suspect would be interesting is to note how this change in insolation at a given latitude between 850 asnd 2000 was achieved. An animation of how, and where it changed might be instructive.

  36. Okay, I’ve had something accepted that I am more proud of than anything else I’ve ever written.

    My research article, “Planet Temperatures with Surface Cooling Parameterized,” will appear in the peer-reviewed science journal Advances in Space Research.

    [Response: Congratulations! Is there a preprint available online?]

  37. Thanks, guys.

    They say I should have galley proofs in four weeks, and it should go up in a preliminary electronic version shortly after that.

  38. I hate to say it, but I suspect that the bozos are simply paid to come up with contrary nonsense and that there “belief” is proportional to the amount paid rather than their evidence. They’re just there to delay action not to improve our understanding.

  39. I have a question (vaguely related) about the cycles. What accounts for the generalized shape of the graph of temps: elevator up/stairstep down? Is it simply easier to melt ice than it is to freeze ice or does it have something to do with lack of symmetry in insolation?

    [Response: One possibility is that glacial wasting can happen mechanically, not just thermodynamically, so it’s susceptible to much more rapid decrease, while glacial accumulation is necessarily a slow process of accumulation.]

  40. This was mentioned in EOS
    “New Software to Analyze How Extremes Change Over Time
    New software called extRemes, developed by scientists at the National Center for Atmospheric Research, allows users to analyze extreme behavior in nature as weather patterns, climate, sea level, and other phenomena change over time.”
    They refer to “Stationarity is dead: Whither water management” by Milly et al. (2008) among others. Curious to hear more about how this is affecting practice in science and engineering generally.

  41. bobbyy53snake

    What Hansen fails to realize is that when we are further from the Sun in NH summer we move more slowly in orbit, and are therefore exposed to the summer sun for a longer period of time.

    Is Hertzberg really saying that the reason why we get warmer when moving slower in the NH summer is because the earth is unable to dodge as many sunbeams as it can when it moves faster in the winter??(wow!!)