I. The 11 year Schwabe and 22 year Hale Cycles.
The influence of planetary attractions on the solar tachocline
Dirk K. Callebaut, Cornelis de Jager and Silvia Duhau
Journal of Atmospheric and Solar-Terrestrial Physics 80 (2012) 73–78
Callebaut, Jager and Duhau stated the following [my bolding below]:
"So far the study of solar variability has identified five solar periodicities with a sufficient degree of significance (cf. the review by De Jager, 2005, Chapter 11). These periods are:
- The 11 years Schwabe cycle in the sunspot numbers. We note that this period is far from constant and varies with time, e.g. during the last century the period was closer to 10.6 years.
- The [22 year] Hale cycles of solar magnetism encompasses two Schwabe cycles and shows the same variation over the centuries.
- The 88 years Gleissberg cycle (cf. Peritykh and Damon, 2003). Its length varies strongly over the centuries, with peaks of about 55 and 100 years (Raspopov et al., 2004). The longer period prevailed between 1725 and 1850.
- The De Vries (Suess) period of 203–208 years, with a fairly sharply defined cycle length.
- The Hallstatt cycle of about 2300 years. An interesting new development (Nussbaumer et al., 2011) is the finding that Grand Minima of solar activity seem to occasionally cluster together and that there is a periodicity in that clustering. An example of such a cluster is the series of Grand Minima that occurred in the past millennium (viz. the sequence consisting of the Oort, Wolf, Sporer, Maunder and Dalton minima). This kind of clustering seems to repeat itself with the Hallstatt period."
They concluded that:
"It should be remarked in this connection that virtually none of the papers on planetary influences on solar variability succeeded in identifying these five periodicities in the planetary attractions."
"The challenge we face here is twofold: planetary influences should be able to reproduce at least the most fundamental of the five periodicities in solar variability, and secondly the planetary accelerations in the level of the solar dynamo should be strong enough to at least equalize or more desirably, to surpass the forces related to the working of the solar dynamo."
I believe that these statements are incorrect. In a series of posts,starting with the 11.1 year Schwabe and 22.2 year Hale cycles, I will address their first challenge by showing that the VEJ Tidal-Torquing model naturally produces all of the five periodicities that are seen in the solar data. There second challenge will be left to a later series of posts.
(N.B. Abreu et al 2012  have also addressed this question by showing that the planetary torques acting on a slightly aspheric (i.e. prolate ellipsoid) tachocline layer at the base of the Sun's convective layer, produce periodicities that match those of the 88 year Gleissberg cycle, the 208 year de Vries cycle, and the 3,200 year Hallstatt cycle. However, Abreu et al. 2012  model did not attempt to give an explanation for periods shorter than the 88 year Gleissberg cycle. In particular, their model does not provide a obvious explanation for the 11.1 year Schwabe and 22.3 year Hale cycles. The VEJ Tidal-torquing model is able to provide such an explanation.)
[Note: if you are not familiar with the VEJ Tidal Torquing model please see:
The Venus-Earth-Jupiter (VEJ) Tidal-Torquing Model is based upon the following set of simple principles:
- The dominant planetary gravitational force acting upon the outer convective layer of the Sun is that produced by Jupiter.
- Other than Jupiter, the two planets that apply the greatest tidal forces upon the outer convective layer 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 0.7997 sidereal Earth years, produces temporary tidal bulges on opposite sides of the Sun's surface layers (red ellipse in the schematic diagram below).
- Whenever these temporary tidal-bulges occur, Jupiter’s gravitational force tugs upon these tidally-induced asymmetries and either slows down or speed-up the rotation rate of plasma near the base of the convective layers of the Sun.
- What makes the VEJ Tidal-Torquing model intriguing, is the time period over which the Jupiter's gravitational pull speeds up and slows down the rotation rate of plasma near the base of the convective layers of the Sun, as Jupiter tugs on the tidal bulges.
[N.B. In the above diagram the planets are revolving in a clock-wise direction and the Sun is rotating in a clock-wise direction. Also, when near-side and far-side tidal bulges on the Sun's surface are referred to, it is with respect to the aligned planets Earth and Venus.]
The diagram above shows Jupiter, Earth and Venus initially aligned on the same side of the Sun (position 0). In this configuration, Jupiter does not apply any lateral torque upon the tidal bulges (The position of the near side bulge is shown by the black "0" just above the Sun's surface).
1.5993 years later, each of the planets move to their respective position 1's. At this time, Jupiter has moved 13.000 degree 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 plasma at the base of the convective layers of the Sun.
After a second 1.5993 years, each of the planets move to their respective position 2's. Now, Jupiter has moved 26.00 degrees ahead of the near-side tidal bulge (marked by the black 2 just above the Sun's surface), increasing the rotation rate at the base of the convective layers of the Sun by roughly twice the amount that occurred at the last alignment.
This pattern continues with Jupiter getting 13.000 degrees further ahead of the alternating near and far-side tidal bulges, every 1.5993 years. Eventually, Jupiter will get 90 degrees 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 rotation rate of the convective layers.
Interestingly, the Jupiter's movement of 13.000 degrees per 1.5993 years with respect to closest tidal bulge, means that Jupiter will get 90 degrees ahead of the closest tidal bulge in 11.07 years. This is almost the same amount of time as to mean length of the Schwabe Sunspot cycle (11.1 +/- 1.2 years) , .
In addition, for the next 11.07 years, Jupiter will start to lag behind the closest tidal bulge by 13.000 degrees every 1.5993 years, and so its gravitational force will pull on the tidal bulges in such a way as to slow the rotation rate of the convective layers down.
All together, there will be four periods of 11.07 years, with the gravitational force of Jupiter, increasing the Sun's rotation rate over the first and third periods of 11.07 years, and decreasing the Sun's rotation rate over the second and fourth periods of 11.07 years.
Hence, the basic unit of change in the Sun's rotation rate (i.e. and increase followed by a decrease) is 2 x 11.07 years = 22.14 years. This is essentially equal to the mean length of the Hale magnetic sunspot cycle of the Sun which is 22.1 +/- 2.0 yrs) , , .
However, the complete planetary tidal cycle is actually (4 x 11.07 years =) 44.28 years.
- The equatorial convective layers of the Sun are sped-up during ODD numbered solar cycles and slowed-down during EVEN numbered solar cycles . This could provide a possible explanation for the Gnevyshev−Ohl (G−O) Rule for the Sun .
- We proposed that it is the resultant variations in the rotation rate of the lower layers of the Sun's convective zone, produced by the planetary tidal-torquing of Venus, the Earth and Jupiter, that modulate the Babcock-Leighton solar dynamo. Hence, we claim that it is this modulation mechanism that is responsible for the observed long-term changes in the overall level of solar activity. In addition, this mechanism may be responsible for the torsional oscillations that are observed in the Sun's convective layer, as well.
 Abreu J. A., Beer J., Ferriz-Mas A., McCracken K.G., and Steinhilber F. Is there a planetary influence on solar activity? Astron & Astrophys., 2012, 548, A88
 Ian R. G. Wilson, Do Periodic Peaks in the Planetary Tidal Forces Acting Upon the Sun Influence the Sunspot Cycle? The General Science Journal, 2010.