Part B: Are Lunar Tides Responsible for Most of the Observed Variation in the Globally Averaged Historical Temperature Anomalies?

RETRACTION: The claim made in this blog post that the peak differential lunar force across the Earth's diameter (that is parallel to the Earth’s equator) produces an annually aliased signal with a period of 20.58 years is incorrect. The 384 day period in the peak differential lunar tidal force data that is used to establish 20.58 year bi-decadal period only exists for periods around the 4.53 year long term maxima in lunar tidal force. It turns out that the long term mean spacing between the short-term peaks in the differential tidal force is close to length of the Full Moon cycle, which is equal to 1.12743 tropical years. Hence, the 384 day spacing between peaks does not last long enough for the beat period of 20.58 years to physically meaningful. I would like to thank Paul Vaughan for pointing out this stupid mistake upon my part. In my next post, I will explain why the bi-decadal oscillation is more likely to be explained by a 20.85 tropical year period related to annual aliasing of the lunar tropical and anomalistic months. 
    
PART B: A Mechanism for the Luni-Solar Tidal Explanation 
PART A: Evidence for a Luni-Solar Tidal Explanation
[Please see the last post]

PART B: A Mechanism for the Luni-Solar Tidal Explanation 

A. Brief Summary of the Main Conclusions of Part A.

     Evidence was presented in Part A  to show that the solar explanation for the Quasi-Decadal and Bid-Decadel Oscillations was essentially untenable. It was concluded that the lunar tidal explanation was by far the most probable explanation for both features.

     In addition, it was concluded that observed variations in the historical world monthly temperature anomalies data were most likely determined by factors that control the long-term variations in the ENSO phenomenon.

     Further evidence was presented in Part A to support the claim that the ENSO climate phenomenon was being primarily driven by variations in the long-term luni-solar tidal cycles. Leading to the possibility that variations in the luni-solar tides are responsible for the observed variations in the historical world monthly temperature anomaly data 

     Copeland and Watts [1] did a sinusoidal model fit to the first difference of the HP smoothed HadCRUT3 global monthly temperature anomaly series and found that the top two frequencies in the data, in order of significance, were at 20.68 and 9.22 years.

     It is generally accepted that the ~ 9.1 - 9.2 year spectral feature is caused by luni-solar tidal cycles associated with the first sub-multiple of the 18.6 year Draconic cycle 9.3 (=18.6/2) = 9.3 years, possibly merged with the 8.85 year lunar apsidal  precession cycle, such that (8.85 + 9.3)/2 = 9.08 years . Hence the question really is: 

Can a plausible luni-solar tidal explanation be given for the 20.68 yr bi-decadal oscillation?

B.  A Potential Luni-Solar Tidal Mechanism  

     Wilson [2] has found that the times when Pacific-Penetrating Madden Julian Oscillations (PPMJO) are generated in the Western Indian Ocean are related to the phase and declination of the Moon. This findings provides observational evidence to support the hypothesis that the lunar tidal cycles are primarily responsible for the onset of El Nino events.  

     If this finding is confirmed by further study then it would reasonable to assume that changes in the level of generation of  PPMJO's is related the changes in the overall level of tidal stress acting upon the equatorial regions of the Earth. A good indicator of the magnitude of these tidal stresses is the peak differential luni-solar tidal force acting across the Earth's diameter, that is parallel to the Earth's equator.  

     The peak differential tidal force of the Moon (dF) (in Newtons) acting across the Earth's diameter (dR = 1.2742 x 10^7 m), along a line joining the centre of the two bodies, is given by:

 

where G is the Universal Gravitational Constant (= 6.67408 x 10^-11 MKSI Units),  M(E) is the mass of the Earth (= 5.972 x 10^24 Kg), m(M) is the mass of the Moon (= 7.3477 x 10^22 Kg), and R is the lunar distance (in metres) (N.B. that the negative sign in front of the terms on the right hand side of this equation just indicates that the gravitational force of the Moon decreases from the side of the Earth nearest to the Moon towards the side of the Earth that faces away from the Moon.)

Hence, the component of this peak differential lunar force (in Newtons) that is parallel to the Earth's equator is:

where R is the distance of the Moon and Dec(M) is the declination of the Moon.

     In like manner, the component of the peak differential tidal force of the Sun (in Newtons) acting across the Earth's diameter that is parallel to the Earth's equator is:

where Rs is the distance of the Earth from the Sun and Dec(S) is the declination of the Sun.

     The relatively rapid daily rotation of the Earth compared to the length of lunar month means that the effects upon the Earth of two differential tidal forces only changes slightly during any given single day. Hence, it is possible to define a slowly changing peak luni-solar differential tidal force acting across the Earth's diameter that is parallel to the Earth's equator, by simply adding each of the two forces above vectorially.

     The geocentric solar and lunar distances, solar and lunar declinations and Sun-Earth-Moon angles were calculated at 0:00, 06:00, 12:00, and 18:00 hours UTC for each day designated period (JPL Horizons on-Line Ephemeris System v3.32f 2008, DE-0431LE-0431 [3].) . This data was then used to calculate the peak differential luni-solar tidal force using the equations cited above. Figure 1a shows the calculated peak differential luni-solar tidal force for the period from Jan 1st 1996 to Dec 31 2015:

Figure 1a

     This plot shows that luni-solar differential tidal force reaches maximum strength roughly every 4.53 years (i.e every 60 anomalistic lunar months = 1653.273 days or every 56 Synodic lunar months = 1653.713 days), with the individual short term peaks near these 4.53 year maximums being separated by almost precisely 384 days (or more precisely 13 Synodic months = 383.8977 days). In order to emphasize this point, figure 1a is re-plotted in figure 1b for the time period spanning from 2000.0 to 2004.5:

Figure 1b.


C. Discussion

     What figures 1a and 1b show is that peak luni-solar differential tidal stress acting upon the Earth's equatorial regions reaches maximum strength roughly every a 4.53 years. This is very close to half the 9.08 year quasi-decadal oscillation. It also shows that around these 4.53 peaks in tidal stress, the individual peaks in tidal stress are almost precisely separated by 13 Synodic months.

     Wilson [4] has proposed that:

"The most significant large-scale systematic variations of the atmospheric surface pressure, on an inter-annual to decadal time scale, are those caused by the seasons. These variations are predominantly driven by changes in the level of solar insolation with latitude that are produced by the effects of the Earth's obliquity and its annual motion around the Sun. This raises the possibility that the lunar tides could act in "resonance" with (i.e. subordinate to) the atmospheric pressure changes caused by the far more dominant solar-driven seasonal cycles. With this type of simple “resonance” model, it is not so much in what years do the lunar tides reach their maximum strength, but whether or not there are peaks in the strength of the lunar tides that re-occur at the same time within the annual seasonal cycle."

     In essence, what Wilson [4] is saying is that we should be looking at tidal stresses upon the Earth that are in resonance with the seasons. (i.e. annually aliased). If we do just that, we find that the peaks in luni-solar differential tidal stressing every 13 synodic months (= 383.8977 days) will realign with the seasons once every:

(383.8977 x 365.242189) / (383.8977 - 365.242189) = 7516.06.07 days = 20.58 tropical years          

This is remarkable close to the 20.68 year bi-decadal oscillation seen by Copeland and Watts [1] in their sinusoidal model fit to the first difference of the HP smoothed HadCRUT3 global monthly temperature anomaly series.

   Hence, it is plausible to propose that the 9.08 year quasi-decadal oscillation and the 20.68 year bi-decadal oscillation  can both be explained by variations in the tidal stresses on the Earth's equatorial oceans and atmosphere caused by the peak differential luni-solar tidal force acting across the Earth's diameter that is parallel to the Earth's equator.

    Keeling and Whorf [5] gives support to this hypothesis by noting that the realignment time (or beat period) between half of a 20.666 tropical year bi-decadal oscillation and the 9.3 year Draconic cycle is simply 5 times the 18.6 year Drconic cycle:

(10.333 x 9.30) / (10.333 - 9.30) = 93.02 years = 5 x 18.6 tropical years

which is a well known seasonal alignment cycle of the lunar tidal cycles where:

1150.5 Synodic months = 33974.94253 days = 93.020 tropical years
1233.0 anomalistic months = 33974.76015 days  = 93.020 tropical years
1248.5 Draconic months = 33974.45667 days = 93.019 tropical years
which only about 7.3 days longer than precisely 93.0 tropical years.

Keeling Whorf [5] claimed that 93 period lunar tidal cycle is able to naturally re-produce the hiatus in the quasi-decadal oscillations of the rate-of-change of the smoothed global temperature anomalies that matched observed between 1900 and 1945.

APPENDIX

      It could be argued, however, that Keeling and Whorf's figure 03 [reproduced as figure 02 below] actually points a hiatus period between about 1920 and 1950's as this is the period over which the phase changes between the mean solar sunspot number and the peaks in their temperature anomaly curve:

Figure 2

    
Wilson [6] made a more accurate determination of the times at which the lunar-line-of-nodes aligned with the Earth-Sun line roughly once every 9.3 years {the blue line in figure 3 below] and when the lunar line-of-apse aligned with the Earth-Sun lineonce every 4.425 years [the brown line in figure 3 below]. They then used this to determine the 93 year cycle over which these two alignment cycles constructively and destructively interfered with each other [the red line in figure 3 below]. showing that the period of destructive interference actually extended from about 1920 to 1950's.        

Figure 3

Finally, Wilson [6] presented some data that showed that there was circumstantial evidence that the 93 year lunar tidal cycle does in fact influence temperature here on Earth.

    Wilson [6] found that "...when the Draconic tidal cycle is predicted to be mutually enhanced by the
Perigee-Syzygy tidal cycle there are observable effects upon the climate variables in the South Eastern part of Australia. Figure 4 below shows the median summer time (December 1st to March 15th) maximum temperature anomaly (The Australian BOM High Quality Data Sets 2010), averaged for the cities of Melbourne (1857 to 2009 – Melbourne Regional Office – Site Number: 086071) and Adelaide (1879 to 2009 – Adelaide West Terrace – Site Number 023000 combined with Adelaide Kent Town – Site Number 023090), Australia, between 1857 and 2009 (blue curve). 

     Superimposed on figure 4 is the alignment index curve from figure 3, (the red line). A comparison between these two curves reveals that on almost every occasion where there has been a strong alignment between the Draconic and Perigee-Syzygy tidal cycles, there has been a noticeable increase in the median maximum summer-time temperature, averaged for the cities of Melbourne and Adelaide. Hence, if the mutual reinforcing tidal model is correct then this data set would predict that the median maximum summer time temperatures in Melbourne and Adelaide should be noticeably above normal during southern summer of 2018/19."

Figure 4

 References

[1] Copeland, B. and Watts, A. (2009), Evidence of a Luni-Solar Influence on the Decadal and Bidecadal Oscillations in Globally Averaged Temperature Trends, retrieved at:
http://wattsupwiththat.com/2009/05/23/evidence-of-a-lunisolar-influence-on-decadal-and-bidecadal-oscillations-in-globally-averaged-temperature-trends/
[2] Wilson, I.R.G. (2016) Do lunar tides influence the onset of El Nino events via their modulation of Pacific-Penetrating MAdden Julian Oscillations?, submitted to the The Open Atmospheric Science Journal.

[3] JPL Horizons on-Line Ephemeris System v3.32f 2008, DE-0431LE-0431 – JPL Solar System Dynamics Group, JPL Pasadena California, available at: http://ssd.jpl.nasa.gov/horizons.cgi, Jul 31, 2013.

[4] Wilson, I.R.G. (2012), Lunar Tides and the Long-Term Variation of the Peak Latitude Anomaly of the Summer Sub-Tropical High Pressure Ridge over Eastern Australia., The Open Atmospheric Science Journal, 6, pp. 49-60.

[5] Keeling, CD.  and Whorf,  TP.  (1997), Possible forcing of global temperature by the oceanic tides.  Proceedings of the National Academy of Sciences., 94(16), pp. 8321-8328.

[6] Wilson, I.R.G. (2013), Long-Term Lunar Atmospheric Tides in the Southern Hemisphere, The Open Atmospheric Science Journal, 7, pp. 51-76.