09 February 2015

The earth wobbles

The earth wobbles about in its rotation.  This was predicted long before it was observed, which is a story itself that I'll tell later.  For now, consider the earth and its rotation.  The north pole of the earth points towards the north star, and rotates once per day.  Open your right hand.  Your thumb points north, and when you close your fingers, they are moving in the direction of the earth's rotation.  With your arm making a right angle at the elbow, hand aiming away from your torso, you have an x-y coordinate system.  When you rotate your forearm, that moves your thumb in the x direction (positive or negative), when swing your arm forward/backward, that's the y direction.

The thing is, the earth (your thumb) doesn't always point in exactly the same direction.  There's a small bit of variation.  That's the wobble.  Since astronomers make their observations from the earth, it's very important to know exactly where the earth is pointing at any instant.  This lead (over 100 years ago) to the foundation of the IERS -- International Earth Rotation and Reference Systems Service.  Daily data from 1 January 1962 to (very nearly) the present are available at http://datacenter.iers.org/eop/-/somos/5Rgv/getTX/213/eopc04_08.62-now
Two things stand out to me in looking at this: There's a very slow tendency to increase x and y over time (increasing movement of the north rotational pole away from the original 0 point), and the more dramatic periodic variation.  The business of having slowly varying amplitude (size of the up and down) for the fast variations suggests a 'beat' is going on.  Namely, there are two different periodic variations going on.  When they're both at maximum, you get a large amplitude.  When they're at minimum, you've got a small amplitude.

The size of those variations, when we translate back to the earth, are seriously small.  As in, it's astonishing that it's possible to measure them, much less that it was done more than 100 years ago.  The size is around 0.1 arc seconds.  10 of those make 1 arc second.  60 arc seconds make 1 arc minute.  60 arc minutes make 1 degree.  90 degrees gets you from the equator to the pole.  Applying all that, 0.1 arc seconds makes for about 3 meters motion in the pole (about 10 feet), out of the about 10,000,000 meters equator to pole.

The periodic variations mean it's time for us to take a look at time series analysis.  For software, I use AnalySeries.  For a less technical introduction you could look at Grant Foster's book.  For the really low math introduction, see my Introductory Time Series Analysis blog post.  Our first rough cut at looking for the periodic variations is to look at the Blackman-Tukey spectrum of the data:
Warning: that vertical scale is logarithmic -- every horizontal line represents 10 times as much power as the one below it.  The horizontal axis tells us the frequency of the variation -- how many times per day it completes a full cycle. From 0.1 cycles per day (cpd) to 0.5 cpd, periods of 10 down to 2 days, I submit that we're looking at nothing more than noise.  Rather quiet noise, since the power is small.  And noise because no particular frequency stands out from the background.  If you squint hard, you might persuade yourself that near a frequency of 0.07 cycles per day (about 14 days period) something stands out a little.  But if you're squinting to see something, it probably isn't there.  Let's zoom in to where the power really is:

Maybe we can convince ourselves that the blips around 0.0055 and 0.008 cpd are standing up from the noise background. But ... still pretty close to squinting.  Particularly when we look near 0.0025 and see those two enormous peaks that are easily 100 times the noise background.

Those two peaks occur at periods of about 433 and 365 days (frequencies about 0.00274 and 0.00231 cycles per day). 365 days rings a bell, doesn't it?

Why would the year cause the earth's axis of rotation to move around?  Well, rotating things have to obey the law of conservation of angular momentum.  In the course of a year, we have the seasonal cycle of atmospheric pressure, winds, and ocean currents.  These all represent redistribution of angular momentum.  As these things shift, so does the earth's axis of rotation.  (And its rotation rate, see lod in the data set above -- lod = length of day.)

Leave the 433 days on hold for a moment.  Let's go back to that slow trend.  What might be its excuse?  Over 60 years ago, Walter Munk and Roger Revelle considered what would happen to the earth's rotation if/as Greenland and/or Antarctica melted.  The ice is at high latitudes, but when it melts, the water gets distributed all around the earth (change of angular momentum).  And Greenland is all in one narrow range of longitudes.  So as it melts the mass along that longitude decreases and the earth's axis of rotation drifts.  I haven't been able to find an online source for the original papers, but you can see the work in context in the National Academies Press book Sea Level Change (1990).

We've got excuses for 2 of the 3 striking features -- the slow trend due to melting Greenland and Antarctica, and the seasonal cycle in earth's orientation.  That leaves the 433 or so day cycle.  Nothing obvious to cause it.  14.2 months isn't anything that looks like climate.  This is the Chandler Wobble, and is the focus for that story I said I'd tell later.

Two classic references for understanding the earth's rotation are:
Munk and MacDonald, The Rotation of the Earth, 1960.
Lambeck, The Earth's Variable Rotation, 1980.
Both from Cambridge University Press.

The complete citation to the AnalySeries is:
Paillard, D., L. Labeyrie and P. Yiou (1996), Macintosh program performs time-series analysis, Eos Trans. AGU,  77:  379.
An electronic supplement of this reference is available at:
Upgrades of the software will be posted regularly at:

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