Wilson I.R.G. and Sidorenkov N.S., Long-Term Lunar Atmospheric Tides in the Southern Hemisphere. Open Atmos Sci J 2013; 7: 51-76
How long does it take for the egg-shape of the lunar orbit to align with both the phase of the Moon and the annual seasonal cycle?
The 31/62/93/186-Year Perigee-Syzygy Lunar tidal Cycle
If you start out with a new moon at its closest point to the Earth [i.e. closest Perigee] around perihelion [i.e. at the start or end of the annual seasonal cycle], then the Line-of-Apse [representing the longest dimension of the egg-shape of the lunar orbit] must rotate seven times around the Earth with respect to the stars (i.e. 7 x 8.8506 = 61.954 ~ 62.0 tropical years) before a new moon reoccurs at closest Perigee at the same point in the annual seasonal cycle (e.g. perihelion).
Hence, if you start out with a new moon at closest perigee at or near the time of perihelion, you will get a Full Moon at closest perigee, on roughly the same day, 31 years later, and a New Moon at closest perigee, on roughly the same day, 62 years after the starting date.
How long does it take for the tilt of the lunar orbit to align with both the phase of the Moon and the annual seasonal cycle?
The 93/186-Year Draconic Cycle
If you start out with a new moon at the ascending node of the lunar orbit [i.e. the same point in the tilt of the lunar orbit] around Perihelion [i.e. at the start or end of the annual seasonal cycle], then the Line-of-Nodes [representing the tilt of the lunar orbit] must rotate five times around the Earth with respect to the stars (i.e. 5 x 18.5999 = 92.9996 ~ 93.0 tropical years) before it returns to a full moon at the descending node around perihelion.
This means that if you start out with a new moon at or near the time of perihelion that is close to one of the nodes of the lunar orbit and at perigee, 93 tropical years later you will have a full moon that is close to the opposite node and at perigee on roughly the same day of the year. This is true because:
1150.5 lunar synodic months = 33974.9425 days = 93.0203 tropical years
1233 lunar anomalistic months = 33974.7600 days = 93.0198 tropical years and
1248.5 draconic months =33974.4577 days = 93.0190 tropical years.
[N.B. The full tidal cycle is actually 186 years long since it takes this long for the Moon to return to a New Moon phase at a time when it returns to the same node, at perigee, and at perihelion].
How long does it take for the egg-shape and tilt of the lunar orbit to realign with both the phase of the Moon and the annual seasonal cycle?
In order to get a sense of the times when the 31/62/93/186-year lunar Perigee-Syzygy cycle and the 93/186-year lunar Draconic Cycle mutually reinforced each other, curves are plotted in Fig. (14) that indicate the strength of alignment between the two cycles between the years 1857 and 2024.
The blue curve in the above figure shows the angle between the line-of-nodes of the lunar orbit (i.e. the tilt of the lunar orbit) and the Earth-Sun line at the time of Perihelion (theta). The curve is derived in such a way as to highlight the southern summers where there is close alignment. This is done by plotting the function 1/(1+ theta).
Similarly, the brown curve in this figure shows the angle between the line-of-apse of the lunar orbit (i.e. the egg shape of the lunar orbit) and the Earth-Sun line at the time of perihelion (phi) plotted as the function - 1/(1 + phi). The functions represented by the blue and brown curves in this figure are used to generate the red curve.
The red curve is an alignment index that is designed to represent the level of reinforcement of the 93/186-year Draconic tidal cycle by the 31/62/93/186-year Perigee-Syzygy tidal cycle. This is done by plotting the values of the blue curve at times when there is a close alignment of the line-of-apse and the Earth-Sun line at perihelion (i.e. when phi < 16°).
The red curve shows that are two epochs 1872 to 1917 and 1973 to 2019, each lasting about 45 years, where there is a strong mutual reinforcement of the Draconic tidal cycle by the Perigee-Syzygy tidal cycle. Individual peaks in the mutual reinforcement occur roughly once every 9.3 years and comparable peaks in the two climate epochs are separated from each other by 93 years.
The Mutually Reinforcing Tidal Model
The fact that the Draconic tidal cycle is mutually enhanced by the Perigee-Syzygy tidal cycle has an observable effect upon the climate variables in the South Eastern part of Australia.
The figure below shows the median summer time (December 1st to March 15th) maximum temperature anomaly. This data is obtained from the Australian BOM High-Quality Data Set 2010, by taking the average 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), between 1857 and 2009 (blue curve).
Superimposed on this figure is the alignment index curve from the previous figure above (red curve).
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, the mutual reinforcing tidal model predicts that the median maximum summer-time temperatures in Melbourne and Adelaide should be noticeably above normal during the summer of 2018/19.