Open Mind

Wobbles, part 2

December 2, 2007 · 12 Comments

In a previous post, we looked at the effect of changing eccentricity on the total amount of solar energy intercepted by earth throughout a year. Now I’d like to take a look at how precession and obliquity (the tilt of the earth’s axis) affect the distribution of incoming sunlight. Neither of these factors affects the total energy intercepted by the entire planet earth throughout the year. But they have a profound effect on how that energy is geographically distributed.

This post has a lot of equations; if that makes your eyes glaze over you can skip past them and look directly at the graphs (and read some of the discussion) below.

Earth’s rotation axis points toward the north pole of the sky, which is very near the star Polaris, the “north star.” But the axis of the earth doesn’t remain forever pointing in the same direction. Like a spinning top, the spin axis wobbles; in fact it desribes a large circle around the sky in a cycle of about 26,000 years. What’s important for climate is the relationship between the direction of our spin axis, and the point of perihelion, or closest approach to the sun in our annual orbit. Both the direction of the spin axis, and the direction of perihelion, change over time, so the “climatic precession” changes in a complex cycle, which shows a changing period, usually between 18,000 and 23,000 years. So, when an astronomer refers to the precession cycle, she generally means the movement of the spin axis alone with a cycle of about 26,000 years, but when a climate scientist refers to the precession cycle he generally means the relationship between the spin-axis direction and the perihelion direction, with a cycle of between 18,000 and 23,000 years.

If an observer is at latitude L, and the sun is at declination \delta (declination is the sun’s latitude on the celestial sphere), and the earth is at a distance r from the sun, then the total solar energy intercepted at that location during an entire day is

I = {S a^2 \over \pi r^2} [h \sin L \sin \delta + \sin h \cos L \cos \delta],

where S is the solar constant, a is the semi-major axis of earth’s orbit (basically, the average sun-earth distance), and h is the hour angle at sunset, given by

\cos h = \max[-1, \min[1, -\tan L \tan \delta]].

See for example Berger 1979, Long-term Variations of Daily Insolation and Quaternary Climate Changes, Journal of Atmospheric Sciences, 35, 2362-2367. If \lambda is the longitude of earth along its orbit, then

r = a(1-e^2) / [1 + e \cos (\lambda - \bar \omega)],

where \bar \omega is the longitude of perihelion, the point of closest approach to the sun. So we can say

I = {S [1 + e \cos (\lambda - \bar \omega)]^2 \over \pi (1-e^2)^2} [h \sin L \sin \delta + \sin h \cos L \cos \delta].

We can separate out the part which depends on the observer’s latitude L and the sun’s declination \delta, giving the geometric part as

G(L, \delta) = {1 \over \pi} [h \sin L \sin \delta + \sin h \cos L \cos \delta],

and write

I = {S [1 + e \cos (\lambda - \bar \omega)]^2 \over (1-e^2)^2} G(L,\delta).

Finally, the sun’s declination can be determined from

\sin \delta = - \sin \epsilon \sin \lambda,

where \epsilon is the obliquity of the earth, which is the angle between earth’s equator and its orbit (basically, the tilt angle of earth’s spin axis).

The northern hemisphere is in midsummer when earth’s longitude is \lambda = \frac{3}{2} \pi. At that time the insolation is

I_{mid} = {S [1 - e \sin \bar \omega]^2 \over (1-e^2)^2} G(L,\epsilon).

Hence midsummer insolation depends on the orbital parameters e and \bar \omega, and on the obliquity of the ecliptic \epsilon; the eccentricity, precession, and obliquity cycles.

How might these variables affect insolation? The precessional parameter \bar \omega varies throughout the entire circle in its cycle of about 19,000 to 23,000 years. Hence the quantity \sin \bar \omega can vary between -1 and +1. But the actual “precession effect” depends on the quantity e \sin \bar \omega. This is what is tabulated when precession is computed in a climate context. Eccentricity, for practical purposes, can vary from near 0 to a little above 0.05 (it gets a bit higher, but only rarely). Hence the precession parameter e \sin \bar \omega varies, for practical purposes, between -0.05 and +0.05.

When the longitude of perihelion \bar \omega is \frac{1}{2} \pi (or simply 90 degrees), closest approach to the sun occurs during northern hemisphere (NH) midwinter/southern hemisphere (SH) midsummer, and if eccentricity is e = 0.05 then the precession parameter is +0.05. Under this condition, the NH will have milder winters and milder summers, the southern hemisphere will have colder winters and hotter summers. We see that precession affects the earth’s two hemispheres oppositely, intensifying contrast between the seasons for one hemisphere while suppressing the seasonal contrast for the other. When perihelion occurs at NH midwinter/SH midsummer, the NH has milder seasons while the SH gets more extreme ones. Here’s a graph comparing NH midsummer (SH midwinter) insolation when the precession parameter takes the values -0.05 and +0.05:


We see that at high northern latitudes there’s a very strong difference — some 100 watts per square meter (W/m^2) extra heating on midsummer day! This can greatly affect melting of ice sheets at high northern latitudes. Here’s a similar comparison for NH midwinter/SH midsummer:


Because the two hemispheres are affected oppositely, whenever precession causes greater ice-sheet melting in the NH, it has the opposite effect in the SH. However, for nearly the last million years there’s been much more ice in the NH than the SH (because there’s more land in the NH), so the NH impact dominates; greater NH midsummer insolation tends to reduce global ice mass, reducing glacial extent. However, in the past there have been times when glaciation in the SH (particularly, Antarctica) was highly variable with the precession cycle, and to some extent at least precession causes a “see-saw” of ice mass between NH and SH (see Raymo et al. 2006, Science, 313, 492-495).

Since most ice melting occurs during midsummer, it’s the high-latitude midsummer insolation that is generally taken as the primary controller of glacial changes. And the fact is that 100 W/m^2 is a lot — it’s hard to imagine that such a change wouldn’t have a dramatic impact on glacial dynamics.

Precession isn’t the only factor in midsummer insolation; obliquity \epsilon is also important. Earth’s obliquity varies on a roughly 41,000-year cycle between about 22 and 24.5 degrees. Here’s a comparison of NH midsummer/SH midwinter insolation between obliquity 22 and 24.5 degrees:


We see that obliquity also brings greater midsummer insolation to high latitudes, as much as 60 W/m^2 — again, a considerable amount. One of the differences is that obliquity affects both hemispheres in the same way; more axial tilt makes the seasons more intense for both halves of the earth. Therefore obliquity never causes a “see-saw” of ice between the hemispheres, it simply causes more or less ice melt globally. That’s why, for some 2 million years (from about 3 to 1 million years ago), worldwide ice mass was dominated by the 41,000-year obliquity cycle, a time period which has been called “the 41-kyr world” (Raymo and Nisancioglu 2003, Paleoceanography, 18, 1011).

Precession affects midsummer insolation but has no effect on the annual average. Obliquity, however, affects not only midsummer warmth but annual average as well. It doesn’t affect it globally, only locally, so the average over the entire planet doesn’t change, but the annual average at a given latitude can change considerably. Here’s a comparison of annual average insolation at various latitudes, between obliquity 22 and 24.5 degrees:


Higher axial tilt brings more solar energy, averaged throughout the year, to the poles while giving less to the equator. At the poles, the planet can get up to 17 W/m^2 more annual average insolation, while the equator can get nearly 4 W/m^2 less. But as I said before, when this change is averaged over the entire surface of the globe, the losses exactly cancel the gains and the planet-wide change is zero.

Categories: Global Warming · climate change

12 responses so far ↓

  • Vixt // December 3, 2007 at 3:06 am

    Dumb question compared to all the elegant math, but I have to ask.

    Wouldn’t the regional changes in sunlight tend to affect the growth rates of the biomass, and thus also affect CO2 uptake? Couldn’t some CO2 trends be explained by changes in biomass activity during these periods?

    It seems like the wobble and incoming regional sunlight changes would do things to the regional biosphere too.

    [Response: That's a topic on which I know very little; I can only speculate.

    The greatest changes are at extreme latitudes, where there's the least biomass (at least on land). Certainly areas that are covered with massive ice sheets have little plant matter to be affected! But I would suppose changes in available sunlight would have some effect where there's enough plant matter, and I have heard that changes in the biosphere are an important part of the carbon-cycle changes associated with glaciation and deglaciation. Changes in sunlight could also affect oceanic life, but since nothing other than eccentricity changes will alter the global average, what one part of earth gains from precession/obliquity another part loses.

    But as I say, I'm only speculating.]

  • Marion Delgado // December 3, 2007 at 9:27 am

    I would add that plant cover does not respond in a linear fashion to changes in either sunlight or C02 due to variations in the vegetation and the status of other limiting conditions.

  • dhogaza // December 4, 2007 at 5:09 am

    My eyes *are* glazed over, but mostly because you’ve given McInyre “last post rights” in the other thread.

    I assume you’re aware he could post whatever lie he might chose to, there? And that he’s been further exposed as being willing to post lies, there?

    Better would be to just close the thread, and invite him to be a “guest blogger”, where he could be challenged.

    [Response: Since much of that thread consisted of criticism of the CA site, I think it's fair to let the "defense" have the last word. The critics certainly had ample opportunity to have their say. And I assure you there are *two* people still permitted to comment there; I'm the other.

    To other readers: please no more comments about the other comment thread on this or any other comment thread.]

  • Alan Woods // December 4, 2007 at 10:51 pm

    Seeing you’ll have the last say in said thread, perhaps you could do us the favour of poinitng out which of McIntyre’s assertions are the lies that dhogaza is having his self indulgent drama queen tantrum over?

  • Alan Woods // December 4, 2007 at 10:52 pm

    …otherwise I fail to see why you allow his posts to stand.

    [Response: If I consider McIntyre's final word (should he choose to submit it) objectionable, I'll object. Certainly nobody can honestly claim there was insufficient opportunity to speak his mind.

    Absolutely no more comments about that thread.]

  • Joel Shore // December 5, 2007 at 12:54 am


    Nice post. One thing I was wondering though: Although obliquity and precession changes do not change the total solar insolation hitting the earth, might they not still produce a net global forcing because they change the distribution of where and when the solar insolation hits the earth and the earth’s albedo is presumably not constant but varies from place to place and time to time…e.g., summer to winter (being higher, e.g., when and where there is ice and snow present)?

    Of course, once the ice sheets start to grow or shrink then the resulting change in albedo certainly produces a net forcing but I am wondering if a net forcing even exists before we consider any changes in the ice sheet distribution because, even though the insolation doesn’t vary, there are variations in the albedo that mean that the total solar insolation that is not returned to space varies. Is this true and, if so, is it significant or not?

    [Response: It is indeed true, and I've even calculated the net change in climate forcing due to redistribution of incoming solar energy to regions/times of different albedo (using ERBE data for the geographic and seasonal distribution of albedo). A change in obliquity from 22 to 24.5 degrees with no change in snow/ice distribution, increases earth’s albedo by about 0.0024, which decreases net global climate forcing (a change of about -0.83 W/m2). Obliquity increase also means less incoming solar energy on the oceans, which can cool the oceans and may lead to pre-deglaciation drawdown of CO2 from the atmosphere. My "pet" theory is that this may explain the small cooling which is sometimes observed just before the start of a deglaciation.]

  • Hank Roberts // December 5, 2007 at 3:43 pm

    Tamino, I expect you’re tired of me bringing up plankton, so to speak, but this is trawling not trolling (grin) so, one more time, because I think it vital. Skipping the oft-cited ‘Plankton Cooled a Greenhouse’ piece, some recent links:

    Agenda of the Workshop “Global Ocean Productivity and the Fluxes …
    “Marine primary production estimates from ocean color: a comparative study …. in an Ocean Global Circulation Model” (Buitenhuis, Le Quéré et al.) [PDF]

    The Dynamic Green Ocean Model: 6 Plankton functional groups … The following parameterisation of primary production is based on a qualitative …

    Dynamic Green Ocean Workshop 4, Villefranche-sur-Mer, France, Apr. 2006 (oral presentation). Uitz J. & H. Claustre, Primary production of phytoplankton …

  • ron lobeck // December 6, 2007 at 11:19 am

    the answer to my query probably lies in the maths, but I am unable to satisfy myself…if I shine a torch at right angles to a surface I get an intense spot of light( sunlight at equator) if I tilt the torch the same amount of light spreads out over a larger area(sunlight at higher latitudes) In the calculationand graphs showing the large difference in energy arriving at the higher latitudes does my simple analogy of light apply? Or is the energy difference calculated as that arriving at a square metre surface at right angles to the radiation ? I also understand that maximum obliquity is close to the perihelion giving warmer winters in the NH….?

    Clarification would be much appreciated.

    [Response: You're correct, when the light comes directly in from overhead (elevation 90 degrees) it's more intense. But there's another factor at work too. For an observer at one of the poles on midsummer's day, the sun is at a given elevation for a full 24 hours of the day -- the sun doesn't set. At the equator the sun around noon is nearly 90 degrees elevation, but no matter what the season there'll be 12 hours of day and 12 hours of night. That's why the poles at midsummer get the highest *daily* insolation of any point on earth.

    Of course, the poles also have to endure a full 6 months of uninterrupted darkness. That's why (as the final graph shows) when it comes to the *annual* insolation, it's the equator that gets the most, the poles get the least.]

  • ron lobeck // December 6, 2007 at 2:13 pm

    Thank you, but I’m still not clear: is the insolation (Radiation)calculated as arriving over 1 sq metre of the SURFACE of the earth or over an equal area at right angles to the surface? I hope you can understand what I’m trying to clear up in my mind! and thanks again for your excellent posts and patience. I have been curious about the obliquity etc since coming across the work of A vistorian astronomer called A W Drayson who in 1862 calculated that the pole of precession was not coincident with the pole of the ecliptic.he showed that after 2295AD the current decrease (in obliquity) will change to an increase an the strart of the next ice age will begin. How does this fit with current theory?calculations? I would be happy to provide a copy of the paper that refers to Professor Drayson’s work.

    [Response: Insolation is calculated as arriving over a unit area (like 1 m^2) of the surface of the earth if the planet were perfectly smooth (no hills or valleys).

    Precession and obliquity into the future are now calculated with very sophisticated computer programs which take into account the gravitational influence of the other planets.]

  • ron lobeck // December 6, 2007 at 3:16 pm

    Thanks again, how do Drayson’s figures fit with the sophisticated(complicated!) computer programs’ calculations of today? Is his figure of the year 2295 AD relevant,? are we approaching the point of maximum obliquity against perihelion for the NH? is so then it must have a very great influence on temperature/ latitude in the NH, where almost all the land exists and bearing in mind land heats up and cools down very quickly especially in the absence of water vapour…If you don’t know, could you refer me to someone who might have an answer?

    [Response: The year 2295 AD is not significant. Obliquity peaked about 9500 years ago, it's now on the decline; perihelion occurs not long after NH midwinter, making NH seasons milder than they would otherwise be.

    Changes in orbital parameters happen *very* slowly, so orbitally induced climate changes are very gradual too. Deglaciation typically takes at least 5000 years, and glaciation takes even longer. In fact orbitally induces climate changes are achingly slow compared to what we're seeing today.]

  • ron lobeck // December 6, 2007 at 6:25 pm

    Thanks again

  • door hinges // December 16, 2007 at 4:15 pm

    thank you,but it is still complex to me.

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