Characterizing Changes in the Angular Momentum of the Earth

Sloshing of the ocean’s waters, as exemplified by ENSO, can only be generated by a suitable forcing. Nothing will spontaneously slosh back and forth unless it gets the right excitation.  Some might hold a naive picture that simply the rotation of the earth can cause the sloshing, but we have to remember that this is a centrifugal force which is evenly directed downward with no changes over time. However, this last part — “no changes over time” — is only correct to zero-th order.  Two classes of mechanisms can disrupt this constant rotation rate.  First, the Chandler wobble of the earth’s axis causes a continuously changing angular momentum of a point of reference. That is a general wobble mechanism similar to the spinning of a top. The second class is of nonspecific events that can either speed up or slow down the rotation rate of the earth.  Note that the wobble may be a behavior that actually belongs to this class, as it may be hard to distinguish the specific mechanisms behind the change in rate. Both of these mechanisms have been measured and they both indicate a clear periodic signal of approximately 6 years.   The Chandler wobble was described in a previous SOI modeling post and it shows a strong average period of 6.45 years see Figure 1 below.

Fig 1: The Chandler Wobble leads to a continuous change of angular momentum of a point of reference, in this case the north pole. A phase shift occurs between 1920 and 1930, but otherwise the period is relatively fixed at 6.45 years, which is the beat frequency of the 433 day wobble and the calendar year.

Holme and Viron [1] measured the second mechanism by sensitive characterization of the length of day (LOD) data.  They only go back to 1964, but find a clear 5.9 year period in the LOD variations (see Figure 2 below).  The LOD delta is inversely related to an angular momentum change.

Fig 2 :  Holme and de Viron characterized the LOD variations and found a 5.9 year period.

So the obvious questions are whether the 6.45 year period and 5.9 year period are related and if they are, what causes the discrepancy?
Clues can be gained by looking at details of the Chandler Wobble data.  The angular momentum is provided as X and Y components, which allows us to look at the phase relationship between the two over time.   The wobble itself is yearly so that we can align the phase with a continuous growing yearly phase, which wraps at 2π boundary conditions.

Fig 3:  Alignment of the Chandler Wobble phase mapped against a constant rate theta increase.

You will have to open this image (Figure 3)  to see the areas of agreement and disagreement. There are around ten regions where the phase gets “jogged” temporarily.  This is enough to lose one year over the course of a four-year transient.  Figure 4 below shows the data folded on itself.

Fig 4:  The phase agrees with a constant growth except where transient jumps occur.

The jumps are well known in the Chandler Wobble literature and show up on the standard plot of X vs Y polar motion as transient poleward slips, see the yellow region in Figure 5 below. Ongoing research [2][3] characterizes these as jerks, which is a higher order acceleration term.

Fig 5: Transients in pole motion are evidenced by jogs in the circular motion.

It is entirely possible that these transients are responsible for changing the angular momentum from a 6.45 year period to the 5.9 year period shown in the LOD data. Observing 10 significant slips/jerks over the span of 100 years, indicates that a 10% change in period may occur as well.  Note that I haven’t worked out a good model for this and am simply going on intuition to what may be happening.

I am interested in understanding this discrepancy more as the SOI sloshing model requires an accurate characterization of the angular momentum forcing.  I have been using a periodic forcing of 6.45 years corresponding to the Chandler Wobble, but using a higher fidelity representation may push this down towards a 5.9 year period.  It is also possible that the 5.9 year period and 6.45 year period interfere with one another to form a beat pattern in the angular momentum forcing.

Update: A May paper describes possible associations between the lunar nodal cycle, Chandler wobble, and Arctic climate:

H. Yndestad, “The influence of the lunar nodal cycle on Arctic climate,” ICES Journal of Marine Science: Journal du Conseil, vol. 63, no. 3, pp. 401–420, Jan. 2006.

“Why are the lunar cycles so dominant? The polar movement is only 3–15 m, and the lunar nodal tide represents only a small fraction of daily sea-level changes, so why are there dominant lunar nodal cycles in the time-series? The answer lies in the fundamental difference between stationary and random cycles. Small changes in stationary cycles have great influence when they are integrated in time and space. Hence there would not be a fixed signal-to-noise ratio: the ratio, it would increase over time and space.”


[1] R. Holme and O. de Viron, “Characterization and implications of intradecadal variations in length of day,” Nature, vol. 499, no. 7457, pp. 202–204, 2013.  PDF

[2] A. Chulliat and S. Maus, “Geomagnetic secular acceleration, jerks, and a localized standing wave at the core surface from 2000 to 2010,” J. Geophys. Res. Solid Earth, vol. 119, no. 3, pp. 1531–1543, Mar. 2014.

[3] E. Bellanger, D. Gibert, and J.-L. Le Mouël, “A geomagnetic triggering of Chandler wobble phase jumps?,” Geophys. Res. Lett., vol. 29, no. 7, pp. 28–1, Apr. 2002.

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