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# Angular Momentum forcing for ENSO model

I am trying to understand what the significance is between the 6.4 year Chandler wobble period in angular momentum changes and the 5.9 year period in length-of-day (LOD) variations.

I wrote this up on my blog and will elaborate more if there is some interest http://contextearth.com/2015/05/25/changes-in-the-angular-momentum-of-the-earth/

I still haven't seen much by the way of research in characterizing ENSO as a forced sloshing model, using the angular momentum modulation factors as input. (Sometimes I have to wonder who makes the decision on what physics to use in climate models. )

Daniel's post on sloppy models has got me thinking of how to push these simplified models.

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1.

I am trying to understand what the significance is between the 6.4 year Chandler wobble period in angular momentum changes and the 5.9 year period in length-of-day (LOD) variations.

I don't know wHether this LOD variation (I have no access to the Holme de Viron article so I dont know where the data comes from) which you have is supposed to be the same as the LOD variations by the IERS - because these variations don't have a 5.9 ys period:

The 5.9 year period is also mentioned here so there could be a priori a mistake in the IERS data but then there are also other parameters at IERS called deps:

and dpsi:

which seem to have a similar period as the IERS-LOD variations however as it looks like with a lag. Unfortunately despite quite some emails to the IERS I couldn't find out what deps and dpsi and the new variables dX and dY are precisely.

The IERS curves show a rather strong irregularity in the years 1981-83. If I counted correctly your "jerk" is between the 1980 and 1981 node though. The IERS has also a "wobbling diagram":

and you can see a jerk, but from the plot it is not possible to deduce the year the jerk took place and I couldn't find a corresponding data file.

By the way are you sure that the Chandler wobble diagram in your image shows really angular momentum? In particular because the "jerk" is not really visible.

Not sure whether one can trust this autopilot instruments.

Comment Source:>I am trying to understand what the significance is between the 6.4 year Chandler wobble period in angular momentum changes and the 5.9 year period in length-of-day (LOD) variations. I don't know wHether this LOD variation (I have no access to the Holme de Viron article so I dont know where the data comes from) which you have is supposed to be the same as the LOD variations by the IERS - because these variations don't have a 5.9 ys period: ![lod](http://datacenter.iers.org/eop/-/somos/5Rgv/plotname/9/FinalsAllIAU2000A-LOD-BULA.jpg) The 5.9 year period is also mentioned <a href="http://phys.org/news/2015-04-gravitational-constant-vary.html">here</a> so there could be a priori a mistake in the IERS data but then there are also other parameters at IERS called deps: ![deps](http://datacenter.iers.org/eop/-/somos/5Rgv/plotname/7/FinalsAllIAU1980-DEPS-BULA.jpg) and dpsi: ![dpsi](http://datacenter.iers.org/eop/-/somos/5Rgv/plotname/7/FinalsAllIAU1980-DPSI-BULA.jpg) which seem to have a similar period as the IERS-LOD variations however as it looks like with a lag. Unfortunately despite quite some emails to the IERS I couldn't find out what deps and dpsi and the new variables dX and dY are precisely. The IERS curves show a rather strong irregularity in the years 1981-83. If I counted correctly your "jerk" is between the 1980 and 1981 node though. The IERS has also a "wobbling diagram": ![](http://datacenter.iers.org/eop/-/somos/5Rgv/plotname/9/FinalsAllIAU2000A-X_Y_PLAN-BULA.jpg) and you can see a jerk, but from the plot it is not possible to deduce the year the jerk took place and I couldn't find a corresponding data file. By the way are you sure that the Chandler wobble diagram in your image shows really <a href="http://en.wikipedia.org/wiki/Angular_momentum">angular momentum</a>? In particular because the "jerk" is not really visible. Not sure whether one can trust this <a href="http://en.wikipedia.org/wiki/Autopilot">autopilot instruments.</a> 
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2.

nad, Maybe I should cast it as the projection of the angular momentum is varying as the earth goes through a wobble. To a person at a fixed location on the earth, they will feel the inertial changes due to the wobble (if it is exaggerated enough). A similar effect would happen to a standing body of water. This is the absolute value of rdot, which is the rate of change of position of the pole as it goes through a wobble.

This period is 6.45 years for an extended interval. The jerks are not visible here as they only strongly reveal themselves as momentary shifts in the X and Y directions. So X and Y can shift with the absolute value not changing enough to see on this plot. Yet you do see it as a phase shift of X with respect to Y (or vice versa).

Comment Source:nad, Maybe I should cast it as the projection of the angular momentum is varying as the earth goes through a wobble. To a person at a fixed location on the earth, they will feel the inertial changes due to the wobble (if it is exaggerated enough). A similar effect would happen to a standing body of water. This is the absolute value of rdot, which is the rate of change of position of the pole as it goes through a wobble. ![rdot](http://imagizer.imageshack.us/a/img633/3521/2dUHua.gif) This period is 6.45 years for an extended interval. The jerks are not visible here as they only strongly reveal themselves as momentary shifts in the X and Y directions. So X and Y can shift with the absolute value not changing enough to see on this plot. Yet you do see it as a phase shift of X with respect to Y (or vice versa). 
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3.
edited May 2015

whoa!! IERS has a Gift animation: http://data.iers.org/plottool/data/FinalsAllIAU2000A-X_Y-BULA.gif where you can see all the dates. The animation which shows the wobbling is however wobbling too much (at least in my browser) so that one can hardly read of the dates. I see a jerk in 1973 (at the bottom of the pillar) but not in 1981.

Comment Source:whoa!! IERS has a Gift animation: <a href="http://data.iers.org/plottool/data/FinalsAllIAU2000A-X_Y-BULA.gif">http://data.iers.org/plottool/data/FinalsAllIAU2000A-X_Y-BULA.gif</a> where you can see all the dates. The animation which shows the wobbling is however wobbling too much (at least in my browser) so that one can hardly read of the dates. I see a jerk in 1973 (at the bottom of the pillar) but not in 1981.
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4.

I have taken their word for it that there is a 5.9 year oscillation in the LOD. They show that the delta LOD is +/- 0.13 msec, while the chart that nad presented shows ranges in delta LOD of several msecs. I was always assuming that this 5.9 year curve was the residual after filtering and compensating for the seasonal, tidal, and other shorter periodic forcings that also contribute to the varying LOD. So this in fact is a fairly subtle effect.

Yet I am still thinking that the Chandler Wobble with the 6.45 year period together with the occasional jerks somehow construct to generate a 5.9 year period as a second-order effect. The foundational question is whether a wobble can induce a modulated change in the measured rotation rate.

A preprint of their Nature paper: http://www.liv.ac.uk/~holme/nature_sub.pdf

Comment Source:I have taken their word for it that there is a 5.9 year oscillation in the LOD. They show that the delta LOD is +/- 0.13 msec, while the chart that nad presented shows ranges in delta LOD of several msecs. I was always assuming that this 5.9 year curve was the residual after filtering and compensating for the seasonal, tidal, and other shorter periodic forcings that also contribute to the varying LOD. So this in fact is a fairly subtle effect. Yet I am still thinking that the Chandler Wobble with the 6.45 year period together with the occasional jerks somehow construct to generate a 5.9 year period as a second-order effect. The foundational question is whether a wobble can induce a modulated change in the measured rotation rate. A preprint of their Nature paper: http://www.liv.ac.uk/~holme/nature_sub.pdf 
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5.

A preprint of their Nature paper: http://www.liv.ac.uk/~holme/nature_sub.pdf

Holmes and de Viron write:

The length-of-day (LOD) fluctuations from 1962 until 2012 are corrected for atmospheric and oceanic effects using assimilating general circulation models see supplementary material, Figure S1). This correction accounts for most of the variation at yearly and shorter periods. The remaining short-period signal is dominantly semi-annual; we therefore apply a 6-month running mean both to eliminate this signal and reduce shorter-period noise. Figure 1 shows that the data are well explained by a decadally varying signal and a constant 5.9-year periodic signal, ampli- tude 0.127 ms (determined iteratively – see methods section); the residual between the data and these two signals has a root mean square amplitude of less than 0.03 ms.

And in S1 it is written:

LOD observations, atmospheric and oceanic signals, and predictions. Here we show influence of removing atmospheric and oceanic predictions from the LOD timeseries, from the NCEP (28) and ECCO (kf80 version) (29) models respectively. We prefer the NCEP model prediction to that of the ERA40 re-analysis (30), thought possibly to better represent variation at longer periods, because of a significantly better fit of periods of variation around 1 year, allowing simple smoothing with a 6-month running average. The oceanic estimate is small (also for other oceanographic and hydrological predictions), probably less significant than the uncertainty in the atmospheric prediction.

So your delta LOD (Figure S3) seems to be some completely different curve. That is they seem to remove first the mostly annual variations via some atmospheric models and then they remove some "decadally varying signal" in order to get your curve via some interpolation procedure, which I don't fully understand:

Methods summary: The fit of the decadal trend and 5.9-year oscillation to the data was obtained iteratively. The decadal trend was fit with smoothing splines and subtracted from the data. From the initial fit to the residual, varying the period and seeking best fit, an oscillation of period 5.8 years and 0.12 ms amplitude was obtained (Figure S2 in supplementary information); this oscillation was then subtracted from the original data and the decadal trend redetermined. This two-stage process was repeated until convergence (4 stages) varying the spline knot spacing as necessary to allow good representation of the decadal variation; the fit in Figure 1 has a spacing of approximately 4 years.

I didn't see that they wrote where this decadal trend comes from. They try though to find a reason for the jerks:

sudden localised strong coupling could temporarily attach part of the fluid core to the mantle, this connection could result from a localised magnetic effect; one possible mechanism is flux expulsion, upwelling (vertical mo- tions) of fluid near the core surface leading to expulsion of toroidal magnetic field (not observable) into an electrically conducting mantle and its conversion into (observable) poloidal field at the core surface. Detailed modelling of this effect is outwith the scope of this letter, but in the methods section, we present scaling arguments suggesting that torsional motions of width 10^o, magnitude a fraction of a km yr^{-1} are sufficient to achieve the required angular momentum jump; a timescale for the transfer of angular momentum is of order 10 days, effectively instantaneous considering the 6-month running average of the data.

Frankly if this 5.9 year or as some others wrote (the citation list is incomplete in that draft) 6 year oscillation is really there then I could imagine that the jerks arise from the fact, that the earth may try to "get the two periods" into resonance namely a strictly 6 year period due to the annual oscillation around the sun and the 11 year period from the solar cycle.

I wrote if that oscillation is really there because I am still not happy about the data, in particular in Figure S1 (which is supposed to show the untreated LOD as I understood) their 1992 bump looks almost higher than the 1980 bump (around 2ms) and this is different from IERS (1980 average around 3ms, 1992 below 3ms).

Comment Source:>A preprint of their Nature paper: http://www.liv.ac.uk/~holme/nature_sub.pdf Thanks for the link. Holmes and de Viron write: >The length-of-day (LOD) fluctuations from 1962 until 2012 are corrected for atmospheric and oceanic effects using assimilating general circulation models see supplementary material, Figure S1). This correction accounts for most of the variation at yearly and shorter periods. The remaining short-period signal is dominantly semi-annual; we therefore apply a 6-month running mean both to eliminate this signal and reduce shorter-period noise. Figure 1 shows that the data are well explained by a decadally varying signal and a constant 5.9-year periodic signal, ampli- tude 0.127 ms (determined iteratively – see methods section); the residual between the data and these two signals has a root mean square amplitude of less than 0.03 ms. And in S1 it is written: >LOD observations, atmospheric and oceanic signals, and predictions. Here we show influence of removing atmospheric and oceanic predictions from the LOD timeseries, from the NCEP (28) and ECCO (kf80 version) (29) models respectively. We prefer the NCEP model prediction to that of the ERA40 re-analysis (30), thought possibly to better represent variation at longer periods, because of a significantly better fit of periods of variation around 1 year, allowing simple smoothing with a 6-month running average. The oceanic estimate is small (also for other oceanographic and hydrological predictions), probably less significant than the uncertainty in the atmospheric prediction. So your delta LOD (Figure S3) seems to be some completely different curve. That is they seem to remove first the mostly annual variations via some atmospheric models and then they remove some "decadally varying signal" in order to get your curve via some interpolation procedure, which I don't fully understand: >Methods summary: The fit of the decadal trend and 5.9-year oscillation to the data was obtained iteratively. The decadal trend was fit with smoothing splines and subtracted from the data. From the initial fit to the residual, varying the period and seeking best fit, an oscillation of period 5.8 years and 0.12 ms amplitude was obtained (Figure S2 in supplementary information); this oscillation was then subtracted from the original data and the decadal trend redetermined. This two-stage process was repeated until convergence (4 stages) varying the spline knot spacing as necessary to allow good representation of the decadal variation; the fit in Figure 1 has a spacing of approximately 4 years. I didn't see that they wrote where this decadal trend comes from. They try though to find a reason for the jerks: >sudden localised strong coupling could temporarily attach part of the fluid core to the mantle, this connection could result from a localised magnetic effect; one possible mechanism is flux expulsion, upwelling (vertical mo- tions) of fluid near the core surface leading to expulsion of toroidal magnetic field (not observable) into an electrically conducting mantle and its conversion into (observable) poloidal field at the core surface. Detailed modelling of this effect is outwith the scope of this letter, but in the methods section, we present scaling arguments suggesting that torsional motions of width 10^o, magnitude a fraction of a km yr^{-1} are sufficient to achieve the required angular momentum jump; a timescale for the transfer of angular momentum is of order 10 days, effectively instantaneous considering the 6-month running average of the data. Frankly if this 5.9 year or as some others wrote (the citation list is incomplete in that draft) 6 year oscillation is really there then I could imagine that the jerks arise from the fact, that the earth may try to "get the two periods" into resonance namely a strictly 6 year period due to the annual oscillation around the sun and the 11 year period from the solar cycle. I wrote if that oscillation is really there because I am still not happy about the data, in particular in Figure S1 (which is supposed to show the untreated LOD as I understood) their 1992 bump looks almost higher than the 1980 bump (around 2ms) and this is different from IERS (1980 average around 3ms, 1992 below 3ms).
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6.

Fun to speculate on the celestial origins of the 5.9 year cycle, but it may be nothing more than coincidence of numerology. 1/2 the jupiter cycle of 11.86 years. Near the 6 year beat frequency of anomalistic 27.55 day and draconic 27.21 day lunar months.

Comment Source:Fun to speculate on the celestial origins of the 5.9 year cycle, but it may be nothing more than coincidence of numerology. 1/2 the jupiter cycle of 11.86 years. Near the 6 year beat frequency of anomalistic 27.55 day and draconic 27.21 day lunar months.
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7.

Fun to speculate on the celestial origins of the 5.9 year cycle

Yes. Maybe my suspect is a completely unrealistic assumption. But it looks as if this has to stay speculative since I don't have any knowledge about the order of magnitudes of the magnetic interactions between earth and sun....and I have no idea where to get them from. I expect that there should have been some people who did the calculation about the two magnetic fields and the corresponding forcings, but where?

Comment Source:>Fun to speculate on the celestial origins of the 5.9 year cycle Yes. Maybe my suspect is a completely unrealistic assumption. But it looks as if this has to stay speculative since I don't have any knowledge about the order of magnitudes of the magnetic interactions between earth and sun....and I have no idea where to get them from. I expect that there should have been some people who did the calculation about the two magnetic fields and the corresponding forcings, but where?
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8.

The most interesting piece of numerology I have discovered recently is that if we assume that QBO is a 2.37 year period (on average very close), then this is exactly twice the fundamental Chandler Wobble (433 days) and is exactly a 2$\pi$ folded frequency of the lunar draconic (nodal) month of 27.21 days. In other words the beat frequency of the draconic month with the tropical (calendar) year, when shifted by 13*2$\pi$ reveals an aliased 2.3697 year period. About 13 draconic months fit into a year.

This triplet of coincidences perhaps suggests an origin of the oscillations.

Comment Source:The most interesting piece of numerology I have discovered recently is that if we assume that QBO is a 2.37 year period (on average very close), then this is exactly twice the fundamental Chandler Wobble (433 days) and is exactly a 2$\pi$ folded frequency of the lunar draconic (nodal) month of 27.21 days. In other words the beat frequency of the draconic month with the tropical (calendar) year, when shifted by 13*2$\pi$ reveals an aliased 2.3697 year period. About 13 draconic months fit into a year. This triplet of coincidences perhaps suggests an origin of the oscillations. 
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9.

the fundamental Chandler Wobble (433 days)

I've always wondered what this is supposed to be. Do you mean these oscillations ?:

2 $\pi$ folded frequency of the lunar draconic

What is a 2 $\pi$ folded frequency ?

Comment Source:> the fundamental Chandler Wobble (433 days) I've always wondered what this is supposed to be. Do you mean these oscillations ?: ![](http://datacenter.iers.org/eop/-/somos/5Rgv/plotname/10/FinalsDataIAU2000A-DX-BULA.jpg) >2 $\pi$ folded frequency of the lunar draconic What is a 2 $\pi$ folded frequency ? 
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10.

"Do you mean these oscillations ?:"

That's the Chandler wobble exactly. But that is within the inertial frame of reference of the rotating earth. If one subtracts out that motion, what is left is the tracing of the pole, which moves at the beat frequency of the Chandler wobble period of 433 days and the calendar period of 365 days, putting it at ~6.4 years.

"what is a 2 $\pi$ folded frequency ?"

This falls under the topic of signal aliasing.

Consider that for a given sampling rate one can't tell the difference between $sin(\omega_1t)$ and $sin((\omega_2 - n 2\pi)*t)$

so that the higher frequency will fold n-times onto the lower frequency.

example: Top, unaliased waveform. Bottom, aliased waveform

This is not only a sampling artifact but can physically manifest itself as a reinforcement of the higher frequency with a seasonal effect. This last part is extremely important and if that doesn't exist, then the effect is not real. Fortunately, many examples of this kind of behavior are known to exist in nature so it is a plausible explanation -- especially if it is something that can trigger a resonance such as QBO.

I discuss the aliasing topic more on my blog here

Comment Source:> "Do you mean these oscillations ?:" That's the Chandler wobble exactly. But that is within the inertial frame of reference of the rotating earth. If one subtracts out that motion, what is left is the tracing of the pole, which moves at the beat frequency of the Chandler wobble period of 433 days and the calendar period of 365 days, putting it at ~6.4 years. > "what is a 2 $\pi$ folded frequency ?" This falls under the topic of signal aliasing. Consider that for a given sampling rate one can't tell the difference between $sin(\omega_1t)$ and $sin((\omega_2 - n 2\pi)*t)$ so that the higher frequency will fold n-times onto the lower frequency. ![qbo](http://imagizer.imageshack.us/a/img820/3034/cxi.gif) example: Top, unaliased waveform. Bottom, aliased waveform This is not only a sampling artifact but can physically manifest itself as a reinforcement of the higher frequency with a seasonal effect. This last part is extremely important and if that doesn't exist, then the effect is not real. Fortunately, many examples of this kind of behavior are known to exist in nature so it is a plausible explanation -- especially if it is something that can trigger a resonance such as QBO. I discuss the aliasing topic more on my [blog here](http://contextearth.com/2014/06/17/the-qbom/)