Sunday, December 15, 2013

Variations in the Earth's Climate on Decadal Time Scales and Proxigean Spring Tides

Richard Ray (2007) has made the bold claim that:

"Occasional extreme tides caused by unusually favorable alignments of the moon and sun are unlikely to influence decadal climate, since these tides are of short duration and, in fact, are barely larger than the typical spring tide near lunar perigee."

This post sets out to show that this claim is not completely true.

Richard Ray and David Cartwright (2007) have calculated the strengths and dates of the maximal lunar-solar tidal potentials over the period from 1 to 3000 A.D.  Thankfully, Prof. Ray has kindly made this data available upon request. The following arguments are based upon this data set which is known as the Ray-Cartwright Table.

Figure 1 below shows the total equilibrium ocean tides (T) caused by the lunar-solar tides between the years 2000 and 2010 A.D i.e.

T = (Vtot)/g

where Vtot is the total tidal potential due to the Sun and Moon, g is the acceleration due to gravity (= 9.82 m/s/s) and T is in cm.

Note: The terms "equilibrium ocean tide" and "tidal  potential" are used interchangeably in this post, however, both refer to the equilibrium ocean tide heights measured  in cm.

Figure 1

From figure 1 we can see that:

a) The total lunar-solar tidal potential (Vtot - green curve) is the sum of the lunar tidal potential (Vlun - red curve) and the solar tidal potential (Vsol - blue curve).

b) Vsol peaks once every year when the Earth is at or near perihelion (blue curve).

c) The largest values of Vlun occur whenever the subtended angle of the Sun and Moon (as seen from the Earth's centre) is either less than 9 degrees (i.e. close to New Moon) or greater than 171 degrees (i.e. close to Full Moon). This means that the largest values of Vlun occur very close to each New and Full Moon where they produce the Spring Tides (red curve).

d) The largest values of Vlun peak roughly once every 206 days when the spring tides occur at perigee. These tides are known as Perigean Spring Tides. The 206 year period is associated with the changing angle between the lunar line-of-apse and the Earth-Sun direction. This angle is determined by the combined motion of the Earth about the Sun and the precession of the lunar line-of-apse. The lunar line-of-apse takes 411.78 days to re-align with Earth-Sun line [note: 411.78/2 = 205.89 days].

e) Vtot (i.e. Vlun + Vsol - green curve) varies up and down between 55 and 62 centimetres once every every 206 days.

Hence, first impressions indicate that Ray (2007) and Ray and Cartwright (2007) correctly concluded that if you compare peak Perigean spring tides with typical Perigean spring tide that are adjacent in time, there is little or no difference in their relative strength on decadal time scales [e.g. compare  Perigean spring tides with total potentials that are greater than 60 cm in figure 1].

However, Ray (2007) and Ray and Cartwright (2007) have missed one important detail. The problem with their simple analysis is that it does not take into account the different ways in which the lunar tides can interact with the Earth’s climate system.

The most significant large-scale systematic variations upon the Earth's climate on an inter-annual to decadal time scale, are those caused by the annual 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 act in "resonance" with (i.e. subordinate to) the atmospheric 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 times do the lunar tides reach their maximum strength, but whether or not there are peaks in their strengths that re-occur at the same time within the annual seasonal cycle.

A good analogy is a child on a swing. If you consider the annual seasons as being the equivalent of the child on the swing as they slowly move back and forward then the lunar tides can be thought of as the hand of the person who pushes the swing. Clearly, the hand pushing the swing is most effective in imparting energy to the child on the swing if they give a push at the highest point of their motion. Similarly, peak lunar tides should have their greatest impact upon the seasonal swings of the climate system if they are applied at a specific point in the seasonal cycle e.g. the summer or winter solstices.

Figure 2 shows all of the total tidal potentials listed in the Ray-Cartwright Table that occur in the month of January between the years 1900 and 2010 A.D.

Figure 2

It is immediately evident from figure 2 that simply limiting the total tidal potentials to those that affect the Earth's climate system in January produces significant variations in the total tidal potential
on decadal time scales. Figure 2 shows that the peak equilibrium ocean tide (or total tidal potential) varies by +/- 7 %  either side of its mean peak value of 59 cm on a time scale of 4.425 years.

Note: The repetition cycle of 4.425 years is simply half the  time required for the lunar line-of-apse to precess once around Earth with respect to the stars. 

Even greater decadal variations in the total tidal potential are produced if we differentiate between those that occur at New Moon in January (Figure 3) from those that occur at Full Moon in January (Figure 4).

Figure 3

Figure 3 shows that the peak equilibrium ocean tide (or total tidal potential) at New Moon vary by +/- 13.5 %  either side of theirmean peak value of 55.5 cm, on a time scale of 8.85 years.

Figure 4

While figure 4 shows that the peak equilibrium ocean tide (or total tidal potential) at Full Moon vary bu the same amount over the same time scale of 8.85 years. However, the peak tidal potentials are shifted in phase by 180 degrees (equivalent to 4.425 years).

The effect of lunar phase on the magnitude of monthly variation in the total tidal potential on decadal time scales must be accounted for because at times near summer/winter solstice i.e. during the months of December or January and June or July, the tides induced by spring tides at New and Full Moon affect distinctly different parts of the planet.

Figure 5 shows the latitude of the sub-lunar point on the Earth's surface for each of the tidal potentials produced by the (near) New and (near) Full Moons that are displayed in figures 3 and 4.

Figure 5

We see that in figure 5 that the latitude of the sub-lunar points of all of the New Moons on the Earth's surface are between about 13 and 28 degrees South while the sub-lunar points of all of the Full Moons on the Earth's surface are between about 12 and 29 degrees North.

Note: Figure 5 shows that a clear 18.6 year sinusoidal variation in the latitude of the sub-lunar points tales place in each hemisphere. 

One way to correct the tidal potentials for the substantial differences in latitude between New and Full Moon is to multiply each potential by the cosine of the difference in latitude between its sub-lunar point and 23.5 degrees South. This give the approximate vertical tidal potential for each New and Full Moon event at a latitude of 23.5 degrees South (on the Earth's surface).

Figure 6

Figure 6 shows that the total equilibrium ocean tide corrected to a latitude of 23.5 degrees South. We can see from this figure that the tidal potentials at New Moon dominate total tidal potential. This means that the peak total equilibrium ocean tide (or peak total tidal potential) varies by +/- 13.5 %  either side of its mean peak value of 55.5 cm, on a time scale of 8.85 years.

Note: All the claims that are made in this post by the author also applies if the interaction window between the lunar tides and the Earth's climate occurs over a three month (seasonal) time period centred upon the winter solstice (May-Jun-Jul) or the summer solstice (Nov-Dec-Jan).

Hence, the claim by Richard Ray (2007) that:

"Occasional extreme tides caused by unusually favorable alignments of the moon and sun are 
unlikely to influence decadal climate, since these tides are of short duration and, in fact, are barely larger than the typical spring tide near lunar perigee."

is not completely true. 

Indeed if, as is most likely, the interaction between the lunar tides and Earth's climate primarily takes place over a monthly to season window then it clear from the above post that the total tidal potential can vary by at least +/- 13.5 %  either side of its mean peak value of 55.5 cm, on a time scale of 8.85 years.


Richard Ray(2007) also claimed that because of the short duration of each tidal event:

"A more plausible connection between tides and near-decadal climate is through “harmonic beating”
of nearby tidal spectral lines. The 18.6-yr modulation of diurnal tides is the most likely to be  detectable."

Note: Richard Ray is referring to the beat period  between the lunar Draconic month and the lunar 
Sidereal month known as the nodal period of lunar  precession:

(27.321661547 x 27.212220817)  = 6793.2277480 days

(27.321661547 - 27.212220817)
                                                      = 18.599 sidereal years

This claim may be partly true.


Ray, R.D., 2007, Decadal Climate Variability: Is 
There a Tidal Connection?, J. Climate20, 3542–3560.

Ray, R.D. and Cartwright, D. E., 2007, Times of peak astronomical
tides, Geophys. J. Int. (2007) 168, 999–1004

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