Tuesday, December 20, 2016

A Direct Connection Between the Venus, Earth and Jupiter Tidal-Torquing Cycles (a Proposed Driver of Solar Activity) and the Long-Term Strength of Perigean Spring Tides



IMPORTANT CLAIMS:

1. The Synodic (phase) cycle of the Moon precisely re-synchronises with the times of the Extreme Perigean Spring tides (EPST) once every 574.60 topical years.

2. Jupiter precisely re-synchronises itself in a frame of reference that is rotating with the Earth-Venus-Sun line once every 575.52 tropical years.

3. The orientation of Jupiter to the Earth-Venus-Sun line produce the tidal torques that act upon the base of the convective layers of the Sun which are thought to be responsible for the periodic changes in the level of magnetic activity on the surface of the Sun (i.e. the Solar Cycle).

4. The period of time required for Jupiter to precisely re-align with the Earth-Venus-Sun line in a reference frame that is fixed with respect to the stars is 4 x 575.52 = 2302 years. This is the Hallstatt cycle that is intimately associated with the planetary configurations that are driving the VEJ Tidal-Torquing model for solar activity.

5. Hence, the repetition period for strongest of the Extreme Perigean Spring tides appears to match that of the planetary tidal-torquing forces that are thought to be responsible for driving the Solar sunspot cycle.

6. The Venus–Earth–Jupiter (VEJ) tidal-torquing model is based on the idea that the planet that applies the dominant gravitational force upon the outer convective layers of the Sun is Jupiter, and after Jupiter, the planets that apply the dominant tidal forces upon the outer convective layers of the Sun are Venus and the Earth. Periodic alignments of Venus and the Earth on the same or opposite sides of the Sun, once every 0.7993 sidereal Earth years, produce temporary tidal bulges on the opposite sides of the Sun’s surface. Whenever these temporary tidal bulges occur, Jupiter’s gravitational force tugs upon the tidally induced asymmetries and either slows down or speeds-up the rotation rate of plasma near the base of the convective layers of the Sun. The VEJ tidal-torquing model proposes that it is the variations in the rotation rate of the plasma in Sun’s lower convective layer, produced by the torque applied by Jupiter upon the periodic Venus–Earth (VE) tidal bulges that modulate the Babcock–Leighton solar dynamo. Hence, the model asserts that it is the modulating effects of the planetary tidal-torquing that are primarily responsible for the observed long-term changes in the overall level of solar activity.

What makes this simple VEJ tidal-torquing model so intriguing is the time period over which the Jupiter’s gravitational force speeds up and slows down the rotation rate of the Sun’s outer layers. Jupiter’s movement of 13.00 deg. per 1.5987 yr with respect to closest tidal bulge means that Jupiter will increase the rotation speed of the lower layers of the Sun's convective zone for 11.07 yrs. This is almost the same amount of time as to average length of the Schwabe sunspot cycle (11.1 ± 1.2 yrs, Wilson, 2011).

In addition, for the next 11.07 yrs, Jupiter will start to lag behind the closest tidal bulge by 13.00 deg. every 1.5987 yrs, and so its gravitational force will pull on the tidal bulges in such a way as to slow down the rotation rate of the outer convective layers of the Sun. Hence, the basic unit of change in the Sun’s rotation rate (i.e. an increase followed by a decrease in rotation rate) is 2 × 11.07 = 22.14 yrs. This is essentially equal to the mean length of the Hale magnetic sunspot cycle of the Sun, which is 22.1 ± 2.0 yrs (Wilson, 2011).

It is important to note that, the actual torques that are applied by Jupiter to the temporary tidal bulges induced by alignments of Venus and the Earth, vary in-phase with the observed 11 year sunspot cycle.

ARTICLE:

A. The 574.6 Year Cycle in Extreme Perigean Spring Tides


     Extreme Perigean Spring tides (EPST) occur when a New moon occurs at time when the Perigee of the lunar orbit points directly at the Sun or when a Full Moon occurs when the Perigee of the lunar orbit points directly away from the Sun (the latter are often called Extreme Super Moons). Figure 1 shows a schematic diagram an EPST occurring at a New Moon.

Figure 1.      




     EPST at New Moon re-occur once every 18 Full Moon Cycles (FMC) = 20.2937 tropical years (where one tropical year = 365.242189 days).

[Note: A FMC (= 411.78443029 days epoch J2000.0) is the the time required for the Sun (as seen from the Earth) to complete one revolution with respect to the Perigee of the Lunar orbit. In other words, it is the time between repeat occurrences of the Perigee of the lunar orbit pointing at the Sun. This value for the FMC is based upon a Synodic month = 29.53058885 days, an anomalistic month = 27.55454988 days - Epoch J2000.0.]  

     Figure 2, below, has as its initial starting point (T = 0.0 tropical years), a New Moon taking place at the precise time that the Perigee of the lunar orbit points directly at the Sun. In addition, this figure shows the number of days to (negative values on the y-axis) or from (positive values on the y-axis) a New/Full Moon for each of the EPST's that occur over the next 618.4 tropical years.

     Figure 2 shows that the point representing the New Moon at T = 0.0 tropical years is part of a triplet of points with the other two points occurring at  (-1.1274 tropical years, 1.64 days) and (+1.1274 tropical years, -1.64 days). Hence, a point starting at (-1.1274 tropical years, 1.64 days), reaches the x-axis at (573.4727 topical years, 0.0 days), leading to an overall repetition cycle of 574.600 tropical years.
   
Figure 2.



B. The Venus–Earth–Jupiter (VEJ) Tidal-Torquing Model 
     (A Proposed Driver of Solar Activity)


Quote from: Wilson, I.R.G.: The Venus–Earth–Jupiter spin–orbit coupling modelPattern Recogn. Phys., 1, 147-158

     "The Venus–Earth–Jupiter (VEJ) tidal-torquing model is based on the idea that the planet that applies the dominant gravitational force upon the outer convective layers of the Sun is Jupiter, and after Jupiter, the planets that apply the dominant tidal forces upon the outer convective layers of the Sun are Venus and the Earth. Periodic alignments of Venus and the Earth on the same or opposite sides of the Sun, once every 0.7993 sidereal Earth years, produces temporary tidal bulges on the opposite sides of the Sun’s surface (Fig. 3 – red ellipse). 

Figure 3.


      Whenever these temporary tidal bulges occur, Jupiter’s gravitational force tugs upon the tidally induced asymmetries and either slows down or speeds-up the rotation rate of plasma near the base of the convective layers of the Sun. The VEJ tidal-torquing model proposes that it is the variations in the rotation rate of the plasma in Sun’s lower convective layer, produced by the torque applied by Jupiter upon the periodic Venus–Earth (VE) tidal bulges that modulate the Babcock–Leighton solar dynamo. Hence, the model asserts that it is the modulating effects of the planetary tidal-torquing that are primarily responsible for the observed long-term changes in the overall level of solar activity. 

      It is important to note that tidal bulges will be induced in the surface layers of the Sun when Venus and the Earth are aligned on the same side of the Sun (inferior conjunction), as well as when Venus and the Earth are aligned on opposite sides of the Sun (superior conjunction). This means that whenever the gravitational force of Jupiter increases/decreases the tangential rotation rate of the surface layer of the Sun at inferior conjunctions of the Earth and Venus, there will be a decrease/increase the tangential rotation rates by almost the same amount at the subsequent superior conjunction. 

     Intuitively, one might expect that the tangential torques of Jupiter at adjacent inferior and superior conjunctions should cancel each other out. However, this is not the case because of a peculiar property of the timing and positions of Venus– Earth alignments. Each inferior conjunction of the Earth and Venus (i.e. VE alignment) is separated from the previous one by the Venus–Earth synodic period (i.e. 1.5987 yr). This means that, on average, the Earth–Venus–Sun line moves by 144.482 degrees in the retrograde direction, once every VE alignment. Hence, the Earth–Venus–Sun line returns to almost the same orientation with respect to the stars after five VE alignments of almost exactly eight Earth (sidereal) years (actually 7.9933 yr). Thus, the position of the VE alignments trace out a five pointed star or pentagram once every 7.9933 yr that falls short of completing one full orbit of the Sun with respect to the stars by (360−(360×(7.9933− 7.0000))) = 2.412 degrees (fig. 4). 

Figure 4.




     In essence, the relative fixed orbital longitudes of the VE alignments means that, if we add together the tangential torque produced by Jupiter at one superior conjunction, with the tangential torque produced by Jupiter at the subsequent inferior conjunction, the net tangential torque is in a pro-grade/retrograde direction if the torque at the inferior conjunction is greater/less than that of the torque at the superior conjunction.

Figure 5.





     What makes this simple tidal-torquing model so intriguing is the time period over which the Jupiter’s gravitational force speeds up and slows down the rotation rate of the Sun’s outer layers. 

     Figure 5 shows Jupiter, Earth and Venus initially aligned on the same side of the Sun (position 0). In this configuration, Jupiter does not apply any tangential torque upon the tidal bulges (the position of the near-side bulge is shown by the black 0 just above the Sun’s surface). Each of the planets, 1.5987 yrs later, moves to their respective position 1's. At this time, Jupiter has moved 13.00 deg. ahead of the far-side tidal bulge (marked by the red 1 just above the Sun’s surface) and the component of its gravitational force that is tangential to the Sun’s surface tugs on the tidal bulges, slightly increasing the rotation rate of the Sun’s outer layers.

     After a second 1.5987 yrs, each of the planets moves to their respective position 2's. Now, Jupiter has moved 26.00 deg. ahead of the near-side tidal bulge (marked by the black 2 just above the Sun’s surface), increasing Sun’s rotation rate by roughly twice the amount that occurred at the last alignment. This pattern continues with Jupiter getting 13.00 deg. further ahead of the nearest tidal bulge, every 1.5987 yrs. Eventually, Jupiter will get 90 deg. ahead of the closest tidal bulge and it will no longer exert a net torque on these bulges that is tangential to the Sun’s surface and so it will stop increasing the Sun’s rotation rate. 

      Interestingly, Jupiter’s movement of 13.00 deg. per 1.5987 yr with respect to closest tidal bulge means that Jupiter will get 90 deg. ahead of the closest tidal bulge in 11.07 yrs. This is almost the same amount of time as to average length of the Schwabe sunspot cycle (11.1 ± 1.2 yrs, Wilson, 2011). In addition, for the next 11.07 yrs, Jupiter will start to lag behind the closest tidal bulge by 13.00 deg. every 1.5987 yrs, and so its gravitational force will pull on the tidal bulges in such a way as to slow down the rotation rate of the outer convective layers of the Sun. Hence, the basic unit of change in the Sun’s rotation rate (i.e. an increase followed by a decrease in rotation rate) is 2 × 11.07 = 22.14 yrs. This is essentially equal to the mean length of the Hale magnetic sunspot cycle of the Sun, which is 22.1 ± 2.0 yrs (Wilson, 2011)." 


C. The 575.52 year Realignment Cycle for Jupiter in a Reference Frame that     is  Rotating with the Earth-Venus-Sun Line


The Movement of Jupiter with Respect to the Tidal-Bulge that is Induced in the Convective Layers of the Sun by Periodic Alignments of Venus and the Earth.

     The slow revolution of the Earth-Venus-Sun alignment axis can be removed provided you place yourself in a framework that rotates by 215.5176 degrees in a pro-grade direction [with respect to the fixed stars] once every 1.59866 years. In this rotating framework, Jupiter moves in a pro-grade direction (with respect to the Earth-Venus-Sun line) by 12.9993 degrees per [inferior conjunction] VE alignment.

     Figure 6 shows the position of Jupiter every VE alignment (i.e. 1.59866 years) in reference frame that is rotating with the Earth-Venus-Sun alignment line. This keeps the Earth and Venus at the 12:00 o'clock position in this diagram whenever the number of VE aligns is even and at the 6:00 o'clock position whenever the number of VE aligns is odd. In contrast, Jupiter starts out at JO and moves 12.9993 degrees every 1.59866 years, taking 11.07 years to move exactly 90 degrees in the clockwise (pro-grade) direction and 11.19 years to the position marked J7 (at roughly 91 degrees).

Figure 6.
 Also shown on this diagram is the position of Jupiter after 27, 28 and 29 VE alignments. This tells us that Jupiter completes exactly one orbit in the VE reference frame once every 44.28 years (= 11.07 years x 4), with the nearest VE alignment taking place at 28 VE alignments (= 44.7625 years) when Jupiter has moved 3.9796 degrees past realignment with its original position at JO.

     The following table shows how Jupiter advances by one orbit + 3.9796 degrees every 28 VE alignments until the alignment of Jupiter with the Earth-Venus-Sun line progresses forward by   13 orbits in the VE reference frame plus 51.7345 degrees. This angle (see * in table) is almost exactly equal to the angle moved by Jupiter in 4 VE aligns (i.e. 4 x 12.99927 degrees = 51.9971 degrees).   

 VE_multiple______Angle of______Orbits_+__Degrees   of 
12.9993_______Jupiter_______________________degrees

28_________363.9796_______1__+___3.9796____
56_________727.9592_______2__+___7.9592____
84________1091.9387_______3__+__11.9387___
112________1455.9183_______4__+__15.9183___
140________1819.8979_______5__+__19.8979___
168________2183.8775_______6__+__23.8775___
196________2547.8571_______7__+__27.8571___
224________2911.8366_______8__+__31.8366___
252________3275.8162_______9__+__35.8162___
280________3639.7958______10__+__39.7958___
308________4003.7754______11__+__43.7754___
336________4367.7550______12__+__47.7550___
364________4731.7345______13__+__51.7345__*

This means that Jupiter returns to almost exact re-alignment with the Earth-Venus-Sun line after:



(364 - 4) VE aligns = 360 VE aligns = 575.52 years 

[i.e. 12.9993 orbits of Jupiter in a retro-grade direction in the VE reference frame, falling 0.2625 degrees short of exactly 13 full orbits]


APPENDIX
The Planetary Connection to the 2300 Hallstatt Cycle

      It has long been recognised that there is a prominent 208 year de Vries (or Suess) cycle in the level of solar activity. The following blog post shows that the there is a 208.0 year de Vries cycle in the alignment between the times that Perigee of the Lunar orbit points directly at the Sun and the Earth's seasons provided that you measure the alignment in a reference frame that is fixed with respect to the Perihelion of the Earth's orbit:   

http://astroclimateconnection.blogspot.com.au/2016/05/there-is-natural-208-year-de-vries-like.html   

Its appearance, however, is intermittent. Careful analysis of the Be10 and C14 ice-core records show that the de Vries cycle is most prominent during epochs that are separated by about 2300 years (Vasiliev and Dergachev, 2002).  This longer modulation period in the level of solar activity is known as the Hallstatt cycle (Vitinsky et al., 1986Damon and Sonett, 1991Vasiliev and Dergachev, 2002).
  
Jupiter in a Reference Frame that is Fixed with Respect to the Stars

     Figure_A1 shows the orbital position of Jupiter, starting at (0,1), every 0.79933 years, over a period of 35.9699 years [i.e. just over three orbits of the Sun]. It is clear from this diagram that, in a reference frame that is fixed with respect to the stars,  the symmetry pattern perfectly re-aligns after moves roughly 24.26 degrees in a clockwise (pro-grade) direction. It takes Jupiter 71.9397 years (i.e. just over six orbits of the Sun or 45 VE aligns) to move 23.30 degrees in a clockwise (pro-grade) direction, to approach with one degree of producing a re-alignment of rotational symmetry.

Figure_A1


Re-aligning the Movement of Jupiter in the Rotating VE Reference Frame with its Movement in the Reference Frame that is Fixed with the Stars

    Figure_A2 shows the precise alignments Jupiter with the Earth-Venus-Sun line at 575.5176 years (360 VE aligns) and 1151.0352 years (720 VE aligns) in a frame of reference that is fixed with respect to the stars. Jupiter lags behind the VE alignments by 0.2654 degrees and 0.5251 degrees, respectively.

Figure_A2





  Figure_A3 shows the precise alignments of Jupiter with the Earth-Venus-Sun line at 1726.5528 years (1080 VE aligns) and 2302.0704 years (1440 VE aligns) in a frame of reference that is fixed with respect to the stars. Jupiter lags behind the VE alignments by 0.7876 degrees and 1.0502 degrees, respectively.

Figure_A3


  The important point to note is that after four precise Jupiter alignments of 575.5176 years (i.e. 4 x 575.5176 = 2302.07 years), the position of Jupiter advances from its initial position at JO (see figure 6 and figure_A1 above) by 24.2983 degrees. This angle is almost exactly the same as 24.26 degrees of rotation that is required to produce a re-alignment of the rotational symmetry of Jupiter, in the reference frame that is fixed with respect to the stars.

     Hence, the period of time required for Jupiter to precisely re-align with the Earth-Venus-Sun line in a reference frame that is fixed with respect to the stars is 2302 years. This is the Hallstatt-like cycle that is naturally found in the planetary configurations that are driving the VEJ Tidal-Torquing model for solar activity. 

References

Damon, P.E. and Sonett, C.P., 1991, “Solar and terrestrial components of the atmospheric 14C variation spectrum”, in The Sun in Time, (Eds.) Sonett, C.P., Giampapa, M.S., Matthews, M.S., pp. 360–388, University of Arizona Press, Tucson.

Vasiliev, S.S. and Dergachev, V.A., 2002, “The 2400-year cycle in atmospheric radiocarbon concentration: bispectrum of 14C data over the last 8000 years”, Ann. Geophys.20, 115–120.
http://www.ann-geophys.net/20/115/2002/

Vitinsky, Y.I., Kopecky, M. and Kuklin, G.V., 1986, Statistics of Sunspot Activity (in Russian), Nauka, Moscow



  

Thursday, May 19, 2016

Changes to the Earth's albedo appear to lead the SOI Index by approximately 6 months

Figures 1a shows the mean monthly apparent albedo anomalies from December 1998 to December 2014 as measured by ground-based earth-shine observations. The anomalies are calculated over the mean of the full period and positive anomalies are shown in red and negative in blue. The averaged standard deviation (error) of the monthly data is also indicated in the lower right corner.

Ref:
Palle, E., et al. (2016), Earth’s albedo variations 1998–2014 as measured
from ground-based earth-shine observations, Geophys. Res. Lett., 43, 
doi:10.1002/2016GL068025

Figure 1a and 1b



































Figure 1b shows the monthly Southern Oscillation Index (SOI) published by the Australian Bureau of Meterology (BOM) at:

ftp://ftp.bom.gov.au/anon/home/ncc/www/sco/soi/soiplaintext.html

The SOI has be shifted backward in time by 6 months.

Not withstanding the large error bars associated with the mean monthly apparent albedo anomalies, and the gap in the albedo anomaly data between June 2005 and December 2006, there appears to be rough correlation between the retarded SOI and the monthly apparent albedo anomaly.

If this correlation has any validity then I would predict that the Earth's mean monthly albedo anomaly will remain in negative territory from late 2014 till the end of 2015. 

The next few of years of data from the Earth-shine project should be very interesting if it does. 


Saturday, May 7, 2016

Moderate to strong El Nino events are triggered by the inter-annual variability in the lunar tides.

Moderate to strong El Nino events are triggered by long-term (i.e. inter-annual) variability in the lunar tides. Specifically, the timing of these events is directly related to 31/62 year Perigee/Syzygy lunar tidal cycle.
I do not have all the answers as to how this actually happens but the best answer that I can come up with is that slow forcings applied to the Earth by the lunar tides influences the formation and subsequent propagation of Madden-Julian Oscillations (MJO) along the Equatorial Indian Ocean and Pacific Oceans.
A MJO consists of a large-scale coupling between the atmospheric circulation and atmospheric deep convection. When a MJO is at its strongest, between the western Indian and western Pacific Oceans, it exhibits characteristics that approximate those of a hybrid-cross between a convectively-coupled Kelvin wave and an Equatorial Rossby wave. When a MJO moves from the western Indian Ocean into the western Pacific Ocean, it generally accelerates, becomes less strongly coupled to convection, and transitions into a convectively de-coupled (i.e. dry) Kelvin wave.
Periodically (i.e. roughly once every 4.5 years), the precise alignments of the lunar tidal forcings produce the right conditions that result an upsurge in the number and magnitude of what I call Pacific Penetrating MJO. These are MJO events that travel from the Eastern equatorial Indian Ocean, along the Equator, all the way into the Western Pacific Ocean, where they initiate Westerly Wind Bursts (WWB’s).
The spawning of these WWB’s takes place as the MJO event is transitioning from a hybrid-cross between, a convectively-coupled Kelvin wave and an Equatorial Rossby wave, and a convectively de-coupled (i.e. dry) Kelvin wave. The spawning of the WWB’s occurs in the Western Equatorial Pacific Ocean, somewhere between 60 deg E and 150 deg W longitude. The actual process involves the formation of a typhoon/cyclone pair straddling the equator which produces an intense WWB between the two intense low pressure cells.
The onset of El Nino event are marked by the weakening of the easterly trade winds associated with the Walker circulation. The actual drop off in easterly trade wind strength is always preceded by a marked increase in WWB’s in the western equatorial Pacific Ocean. The WWB’s help initiate an El Nino event by creating downwelling Kelvin waves in the western Pacific that propagate towards the eastern Pacific, where they produce intense localized warming, as well as by generating easterly moving equatorial surface currents which transport warm water from the warm pool region into the central Pacific.
The net result of the Moon’s involvement in the initiation of El Nino events means that:
El Niño events in New Moon epochs preferentially occur near times when the lunar line-of-apse aligns with the Sun at the times of the Solstices.
El Niño events in the Full Moon epochs preferentially occur near times when the lunar line-of-apse aligns with the Sun at the times of the Equinoxes.
For a full description of the meaning of Full and New Moon Epochs please read:

Are we looking in the wrong place for the connection between solar insolation and the world's mean temperature?

I propose that it is the lunar tidal modulation of the seasonal variations in the solar flux, coupled with the hemispherical asymmetry of the Earth’s surface properties (i.e. Nth Hemisphere – dominated by continents, St Hemisphere – dominated by oceans) that is responsible for setting the world's mean temperature.

On inter-annual time scales:

There are periodic slow downs in the Earth’s rotation rate every 13.66 days. These slows downs are caused by the passage of the lunar tidal bulge across the Earth’s equator once every half lunar tropical month = 13. 66 days. The ratio in the magnitude of the increase in Length Of Day (LOD) at one crossing with respect to the magnitude of the increase in LOD at the next is primarily governed by the orientation of the lunar line-of-apse with respect to the seasonal cycle as marked by the solstices and equinoxes (there is also a secondary effect caused by the 5 degree tilt of lunar obit with respect to the ecliptic). Please see:


On decadal time scales:

The climate variation is driven by the ratio in strength and frequency of El Nino to La Nina events – This is dominated by the 31/62 year lunar tidal cycle. For 31 years, El Ninos start in years when lunar line-of-apse is pointed at the Sun near the Winter/Summer solstices, then for next 31 years, El Ninos start in years when lunar line-of-apse is pointed at the Sun near the Vernal/Autumnal equinoxes.


On centennial to millennial time scales:

The climate variation appear to be correlated with variations in the overall strength of the Sun’s magnetic field as indicated by Be10 and C14 proxies. These primarily occur at 88.5 years (Gleissberg cycle), 208 years (de Vries Cycle), 354 years, 510 years, 708 years, 980 years (Eddy Cycle) and 2300 years (Hallstatt Cycle). It just so happens that these cycles in the Sun’s magnetic field strength are matched by the times at which the lunar line-of-apse points at the Sun at either the solstices or equinoxes of the Earth’s seasonal calendar (in a reference frame that is fixed with respect to the Earth’s orbit).

This produces a long term 21,000 year climate forcing which when coupled with the Milankovitch orbital forcing and the formation and melting of ice sheets at the Earth’s poles, produces the ice-age cycles.

There is a natural 208 year de Vries-like cycle in the Lunar Tidal Forces acting upon the Earth

1. The Defacto 13 year Lunar tidal cycle

There is a defacto 13 year lunar tidal cycle created by the difference 
between 31 perigee/syzygy cycle and the 18 year Saros cycle, such that:

13 years = 31 – 18 years

Now the reason for the 18.03 sidereal year Saros cycle is:

223 Synodic months______= 18.02931 sidereal years
239 anomalistic months___ = 18.02990 sidereal years
242 Draconic months______= 18.02941 sidereal years
[with closest alignment between Synodic and Draconic cycles,
though the anomalistic cycle is not far off]

while the reason for the 31 year cycle is:

383.5 Synodic months______= 31.005568 sidereal years
411.0 anomalistic months___ = 31.005401 sidereal years
416.0 Draconic months_____= 30.992708 sidereal years
[with closest alignment between Synodic and anomalistic cycles]

hence, the defacto 13 year cycle possibly comes about because:

160.5 Synodic months______= 12.976254 sidereal years
172.0 anomalistic months___= 12.9754963 sidereal years
174.0 Draconic months_____= 12.9632963 sidereal years
[with closest alignment between Synodic and anomalistic cycles]

2. The 208 year de Vries Cycle
Now 
16 x 13 years = 208 years
Thus, this would indicate that a possible reason for the ~ 208 de Vries cycle is:
16 x 160.5 = 2568.0 Synodic cycles________= 207.6200718 sidereal years
16 x 172.0 = 2752.0 anomalistic cycles_____= 207.6079414 sidereal years 

16 x 174.0 = 2784.0 Draconic cycles_______= 207.4127406 sidereal years

or with a drift of 3.0 Draconic cycles in roughly 208 years you get:
2787.0 Draconic cycles___= 207.6362457 sidereal years 
[with closest alignment between Synodic and anomalistic cycles]

Finally, if you allow for a slight drift of 4.5 Synodic cycles and 5.0 anomalistic and
Draconic cycles you get:

2572.5 Synodic cycles___= 207.98389 sidereal yrs_____= 207.99196 tropical yrs
2757.0 anomalistic cycles_= 207.98514 sidereal yrs____= 207.99321 tropical yrs
2792.0 Draconic cycles___= 208.00875 sidereal yrs____= 208.01682 tropical yrs

Of course, the drift required to realign the sidereal/tropical reference frame 
with the seasonal year of 4.5 synodic = 5.0 anomalistic = 5.0 Draconic cycles 
over 208 tropical years is not arbitrary since this is equivalent to:

132.887 days in 75970.375 days = 0.0017492 of a full cycle in 208 tropical years.

which for the Earth’s orbit is 0.6297133 degrees in 208.0 tropical years
or 10.90 ~ 11 arc second per tropical year!

This is just the (pro-grade) rate of precession of the perihelion of the Earth’s orbit 
which is currently about:

11.45 arc seconds per year – theoretical
11.62 arc seconds per year – observed epoch 2000.

CONCLUSION 
Hence, If we look at times where the perigee of the lunar orbit points at the 
Sun, they will realign with the seasons in (almost) precisely 208.0 tropical 
years or one de Vries Cycle, if you measure the alignment in a reference 
frame that is fixed with the Earth’s orbit (i.e. a frame that is tracking the 
perihelion precession of the Earth’s orbit).