Guest Post by Willis Eschenbach
I’ve been listening to lots of stuff lately about tidal cycles. These exist, to be sure. However, they are fairly complex, and they only repeat (and even then only approximately) every 54 years 34 days. They also repeat (even more approximately) every 1/3 of that 54+ year cycle, which is 18 years 11 days 8 hours. This is called a “Saros cycle”. So folks talk about those cycles, and the 9 year half-Saros-cycle, and the like. The 54+ year cycle gets a lot of airtime, because people claim it is reflected in a sinusoidal approximately 54-year cycle in the for example the HadCRUT temperature records.
Now, I originally approached this tidal question from the other end. I used to run a shipyard in the Solomon Islands. The Government there was the only source of tide tables at the time, and they didn’t get around to printing them until late in the year, September or so. As a result, I had to make my own. The only thing I had for data was a printed version of the tide tables for the previous year.
What I found out then was that for any location, the tides can be calculated as a combination of “tidal constituents” of varying periods. As you might imagine, the strongest tidal constituents are half-daily, daily, monthly, and yearly. These represent the rotations of the earth, sun, and moon. There’s a list of the various tidal constituents here, none of which are longer than a year.
Figure 1. Total tidal force exerted on the Earth by the combination of the sun and the moon.
So what puzzled me even back then was, why are there no longer-period cycles used to predict the tides? Why don’t we use cycles of 18+ and 54.1 years to predict the tides?
Being a back to basics, start-from-the-start kind of guy, I reckoned that I’d just get the astronomical data, figure out the tidal force myself, and see what cycles it contains. It’s not all that complex, and the good folks at the Jet Propulsion Lab have done all the hard work with calculating the positions of the sun and moon. So off I went to JPL to get a couple hundred years data, and I calculated the tidal forces day by day. Figure 1 above shows a look at a section of my results:
These results were quite interesting to me, because they clearly show the two main influences (solar and lunar). Figure 1 also shows that the variations do not have a cycle of exactly a year—the high and low spots shift over time with respect to the years. Also, the maximum amplitude varies year to year.
For ease of calculation, I used geocentric (Earth centered) coordinates. I got the positions of the sun and moon for the same time each day from 1 January 2000 for the next 200 years, out to 1 Jan 2200. Then I calculated the tidal force for each of those days (math in the appendix). That gave me the result you see in Figure 1.
However, what I was interested in was the decomposition of the tidal force into its component cycles. In particular, I was looking for any 9 year, 18+ year, or 54.1 year cycles. So I did what you might expect. I did a Fourier analysis of the tidal cycles. Figure 2 shows those results at increasingly longer scales from top to bottom.
Figure 2. Fourier analysis of the tidal forces acting on the earth. Each succeeding graph shows a longer time period. Note the increasing scale.
The top panel shows the short-term components. These are strongest at one day, and at 29.5 days, with side peaks near the 29.5 day lunar cycle, and with weaker half-month cycles as well.
The second panel shows cycles out to 18 months. Note that the new Y-axis scale is eight times the old scale, to show the much smaller annual cycles. There are 12 month and 13.5 month cycles visible in the data, along with much smaller half-cycles (6 months and 6.75 months). You can see the difference in the scales by comparing the half-month (15 day) cycles in the top two panels.
The third panel shows cycles out to 20 years, to investigate the question of the 9 and 18+ year cycles … no joy, although there is the tiniest of cycles at about 8.75 years. Again, I’ve increased the scale, this time by 5X. You can visualize the difference by comparing the half-year (6-7 month) cycles in the second and third panels. At this scale, any 9 or 18+ year cycles would be very visible … bad news. There are no such cycles in decomposition of the data.
Finally, the fourth panel is the longest, to look for the 54 year cycle. Again, there is no such underlying sine-wave cycle.
Now, those last two panels were a surprise to me. Why are we not finding any 9, 18+, or 54 year cycle in the Fourier transform? Well … what I realized after considering this for a while is that there is not a slow sine wave fifty-four years in length in the data. Instead, the 54 years is just the length of time that goes by before a long, complex superposition of sine waves approximately repeats itself.
And the same thing is true about the 18-year Saros cycle. It’s not a gradual nine-year increase and subsequent nine-year decrease in the tidal force, as I had imagined it. Instead, it’s just the (approximate) repeat period of a complex waveform.
As a result, I fear that the common idea that the apparent ~60 year cycle in the HadCRUT temperatures is related to the 54-year tidal cycles simply isn’t true … because that 54 year repeating cycle is not a sine wave. Instead, looks like this:
Figure 3. The 54 year 34 day repetitive tidal cycle. This is the average of the 54-year 34-day cycles over the 200 years of data 2000-2200.
Now, as you can see, that is hardly the nice sine wave that folks would like to think modulates the HadCRUT4 temperatures …
This exemplifies a huge problem that I see happening. People say “OK, there’s an 18+ year Saros cycle, so I can divide that by 2. Then I’ll figure the beat frequency of that 9+ year cycle with the 8.55 year cycle of the precession of the lunar apsides, and then apply that to the temperature data …”
I’m sure that you can see the problems with that approach. You can’t take the Saros cycle, or the 54+ year cycle, and cut it in half and get a beat frequency against something else, because it’s not a sine wave, as people think.
Look, folks, with all the planets and moons up there, we can find literally hundreds and hundreds of varying length astronomical cycles. But the reality, as we see above, is not as simple as just grabbing frequencies that fit our theory, or making a beat frequency from two astronomical cycles.
So let me suggest that people who want to use astronomical cycles do what I did—plot out the real-life, actual cycle that you’re talking about. Don’t just grab the period of a couple of cycles, take the beat frequency, and call it good …
For example, if you want to claim that the combined tidal forces of Jupiter and Saturn on the sun have an effect on the climate, you can’t just grab the periods and fit the phase and amplitude to the HadCRUT data. Instead, you need to do the hard lifting, calculate the actual Jupiter-Saturn tidal forces on the sun, and see if it still makes sense.
Best regards to everyone, it’s still raining here. Last week, people were claiming that the existence of the California drought “proved” that global warming was real … this week, to hear them talk, the existence of the California floods proves the same thing.
In other words … buckle down, it’s gonna be a long fight for climate sanity, Godot’s not likely to show up for a while …
w.
THE USUAL: If you disagree with something that I or someone else said, please quote the exact words you disagree with, and tell us why. That way, we can all understand what you object to, and the exact nature of your objection.
CALCULATIONS: For ease of calculations, I downloaded the data for the sun and moon in the form of cartesian geocentric (Earth-centered) coordinates. This gave me the x, y, and z values for the moon and sun at each instant. I then calculated the distances as the square root of the sum of the squares of the xyz coordinates. The cosine of the angle between them at any instant is
(sun_x * moon_x + sun_y * moon_y + sun_z * moon_z) / (sun_distance * moon_distance)
and the combined tidal force is then
sqrt( sun_force^2 + moon_force^2 + 2* sun_force * moon_force * cos(angle))
DATA AND CODE: The original sun and moon data from JPL are here (moon) and here (sun), 20 Mb text files. The relevant data from those two files, in the form of a 13 Mb R “save()” file, is here and the R code is here.
EQUATIONS: The tidal force is equal to 2 * G * m1 * m2 * r / d^3, where G is the gravitational constant, m1 and m2 are the masses of the two objects, d is the distance between them, and r is the radius of the object where we’re calculating the tides (assuming that r is much, much smaller than d).
A good derivation of the equation for tidal force is given here.
Richard
Thanks for replying. I wasn’t suggesting you were meaning to be condescending you just talk like a “proof” sometimes when making a technical point ;).
<Tommy Flowers
I’m quite interested in the guy. I write scientific software for a living and work with a lot of really “geeky” guys. No one seems to have heard of him. Either that means they aint as geeky as I think they are (although I’m sure they are) or this guy has been really treated badly by history. His generosity to his coworkers when he got that rather modest prize speaks volumes of his character too.
As for retired, I have yet to meet an academic,with a passion for the subject, that has ever really retired. It sounds as if you are getting itchy feet.
cd says:
February 13, 2014 at 1:15 pm
“Thanks for replying. I wasn’t suggesting you were meaning to be condescending you just talk like a “proof” sometimes when making a technical point ;).”
Difficult to pitch it right without being face to face.
“<Tommy Flowers. I’m quite interested in the guy. I write scientific software for a living and work with a lot of really “geeky” guys. No one seems to have heard of him. Either that means they aint as geeky as I think they are (although I’m sure they are) or this guy has been really treated badly by history. His generosity to his coworkers when he got that rather modest prize speaks volumes of his character too."
The problems with dealing with those blank spaces on the map. You can get yourself into positions where what you do makes you invisible by necessity.
He was in such a position (as was Tutte). What they did with pattern analysis and simple circuitry blows your mind when you consider what they achieved. They figured out something neither of them ever saw, and unwrapped it in their mind. Saved in the process upwards of a few million lives into the bargain. And never even got a footnote in history (or nearly so – getting into the Royal Academy is not a small achievement but who ever even heard of them today).
"As for retired, I have yet to meet an academic, with a passion for the subject, that has ever really retired. It sounds as if you are getting itchy feet."
That curiosity will keep me looking, regardless of what I do.
Greg Goodman says:
February 13, 2014 at 12:46 am
“The question here is whether there could be an inter-annual or decadal scale horizontal displacement of water mass that could transport climatologically significant amounts of thermal energy.”
The short answer is that ocean currents transport water mass turbulently, whereas tides and other longwaves merely put water mass into an irrotational, coherent orbit of limited dimension. Thus there is scant basis for expecting any significant tidal heat transport or downward mixing outside the confines of coastal waters and estuaries. Why should anyone model such a physical implausibility?
1sky1 says:
February 13, 2014 at 5:02 pm
“The short answer is that ocean currents transport water mass turbulently, whereas tides and other longwaves merely put water mass into an irrotational, coherent orbit of limited dimension. Thus there is scant basis for expecting any significant tidal heat transport or downward mixing outside the confines of coastal waters and estuaries. Why should anyone model such a physical implausibility?”
I rather think you have never been to the Islands off Scotland and seen the tidal races that form there. Just where all that nice warm North Atlantic Drift is heading Northwards to the Arctic over the Greenland – Scotland ridge.
You might have a slightly different approach to Tides then.
agfosterjr says:
February 13, 2014 at 7:29 am
“Anyone foolish enough to think he can pop over to some bay with no nearby gauges and predict the tides, well for one thing, he has never tried it. –AGF”
Anyone presuming that I claimed such simply cannot read. FYI, I successfully predicted tides at a number of project sites, using as few as a dozen constituents
1sky1 says:
February 13, 2014 at 5:27 pm
“Anyone presuming that I claimed such simply cannot read. FYI, I successfully predicted tides at a number of project sites, using as few as a dozen constituents”
Want to give me a read for the Faroe Bank Channel? Percentage depth variation would be nice and a 120 year period if you can.
Willis
After cooling down a bit I read your response to me again. I get the feeling you thought I was being patronising and perhaps I was. I was only trying to explain what a 1D vector was and we seem to be talking cross-purposes. If I can try this again but in the context of your work:
Your magnitude is the “size” of a resultant vector from a vector operation between two vectors. If you’re only interested in the magnitude which you are then (right?) its still a 1D vector with magnitude and direction. In more precise terms:
Take a vector V with 3 elements: i, j, k (to avoid necessity of Cartesian convention as it could exist in any reference frame), its unit vector v and magnitude M:
V = vM
Now V has 2 elements: i,j and unit vector v (2 elements as well) with magnitude M:
V = vM
Now V with 1 element: i and unit vector v (1 element also) and magnitude M:
V = vM
In short, we now have split the vector into its directional component v and size M.
The 1D case is a special one as v is a unit vector v.i = -1/1, therefore:
V = vM = (-1*M) or M.
What you have done is found the magnitude of the 3D vector. Its implicit now that magnitude refers to the size of the vector as defined in that direction (the 3D vector), which can now be used as a unique 1D reference frame:
v.i is always equal to -1 or +1. Magnitude is always >=0.
You see V only has magnitude M in the given coordinate frame, and with direction -1/+1 ( a vector), which is only true in the reference frame defined by the 3D vector! In short your quantity cannot be described fully by magnitude only, without its directional component (M therefore is not a scalar).
Willis forgot to state that your quantity is expressed as a Force (Newtons) which is by definition a vector quantity.
This cannot be true.
Pressure is defined as force per area, newtons pre square meters.
Force being a vector does violate the principle of fpressure being a magnitude acting in all directions.
The same discussion can be carried out with Voltage and Tension.
conclusion: cd has quite obviously not conscidered the general case here.
cd says:
February 13, 2014 at 11:32 pm
“What you have done is found the magnitude of the 3D vector. Its implicit now that magnitude refers to the size of the vector as defined in that direction (the 3D vector), which can now be used as a unique 1D reference frame: ….In short your quantity cannot be described fully by magnitude only, without its directional component (M therefore is not a scalar).”
What you are basically describing is how you transform a fixed 3D Cartesian vector space into a single relative rotating reference space with a single magnitude and a single direction that is defined by the line as it rotates and along which that magnitude operates.
So you have swapped from an Inertial reference frame using parameters dictated by the JPL tables to an abstract Earth-Moon oriented rotating space.
In fact what you really want is a Earth spin axis/rotationally oriented rotating space so that you can properly observe the effects of both Sun and Moon somewhere on the Earth’s surface.
It’s all down to where you stand and what trajectory you follow as you watch the almost infinitely complex dance of the others.
Carbomontanus says:
February 14, 2014 at 12:48 am
http://answers.yahoo.com/question/index?qid=20081220120251AAtaYz4
What is the force of gravity in newtons ?
the equation used is
F = mg
where F is the force in newtons
m is the mass in kg
and g is the gravitational field intensity, which on earth is 9.8N/kg
=9.8N
Now run that by me again?
Carbomontanus
Force is a vector quantity, it has magnitude and direction. For example, an oblique force to a plane creates a different stress tensor than if applied normal to the plane.
Pressure is a scalar quantity (it doesn’t have direction), there is an equivalence with Force (units: N) because pressure is essentially force per area (units: N/A) but they are not the same type of quantity. In short, you can describe pressure completely by its magnitude, with force you need direction as the measure is for a particular reference frame, in a given direction.
Carbomontanus says:
February 14, 2014 at 12:48 am
What is the force of gravity in newtons ?
http://answers.yahoo.com/question/index?qid=20081220120251AAtaYz4
the equation used is
F = mg
where F is the force in newtons
m is the mass in kg
and g is the gravitational field intensity, which on earth is 9.8N/kg
=9.8N
Now run your answer by me again?
RichardLH
So you have swapped from an Inertial reference frame using parameters dictated by the JPL tables to an abstract Earth-Moon oriented rotating space.
That was the point of the comment to illustrate to Willis why he is referring 1) magnitude of a vector not a scalar quantity, that he seems to be assuming and 2) how his 3D case can be simplified to a 1D vector. Perhaps I misunderstood your original point which started all this.
I think for the purposes of his article an inertial reference frame is sound. I’m sure, as evidenced by the fact that the Moon is moving further from us and that the Earth’s spin is slowing down that this not correct and that there are other external factors affecting the system but again I think this is rather academic. I’m not arguing against the need for your proposed reference frame – but even that’s just for starters, where do you stop? I know the approach you’re proposing is used throughout geophysics, at the scales I work at there is never any need.
cd says:
February 14, 2014 at 2:54 am
“I know the approach you’re proposing is used throughout geophysics, at the scales I work at there is never any need.”
I think you are missing the important point I was trying to make.
The rotating vector that has been created is, in itself, following one of the parts of the Saros cycle it is attempting to demonstrate.
The Earth-Moon vector is one vector in the Saros cycle. The other is the Sun-Earth vector.
To be a treatment of what happens at the Earth’s surface (and hence be something that could or could not affect Climate) you need to swap to a Earth rotational space.
Then, and only then, do you need to add in the geography and fluidics.
Edit: Make that
…following the parts of…
RichardLH
Are you saying that we use a point on the Earth as a rotating reference frame. Are you then saying we measure the temporal shift in the magnitude of the gravitational pull from the Moon and the Sun in the vectors defined between our stationary point (on the surface) and the Moon and Sun (which of course are also changing relatively)?
Sorry Richard that should’ve been “fixed point” rather than “stationary point”.
1sky1: ” tides and other longwaves merely put water mass into an irrotational, coherent orbit of limited dimension. ”
That sounds a little like one Paul Vaughan’s science-like sound bites. Like most of Pauls comments it sounds impressive but does not actually convey anything useful. Perhaps you could rephrase it.
Ocean currents convey huge ammounts of thermal energy and a measured reduction in the flow of the gulf stream was a climate panic from an earlier generation (until they measured it again and it had gone back up).
What I’m trying to establish is whether harmonic interference patterns from the various fluctuations in lunar and solar forcing maybe modulating what is currently measured as “ocean currents”.
Now lunar distance whose direction of closest approach varies on an 8.85 year cycle is clearly a factor as is it’s declination angle, which varies in orientation over 18.6 years. There is also the repetition of 3D alignment of sun-moon-earth every 18.03 years.
What Willis attempted to calculate was the magnitude of the tidal force vector. That is once important factor , the other is it’s direction, the combined result of solar and lunar declination.
I say “attempted” because although he did a valid calculation for the individual sun and moon tide raising forces, he forgot that they each act in both directions (creating opposite bulges) so when he uses the (also incorrectly calculated) magnitude of the vector sum he is subtracting full moon luanr tides from the solar tide instead of adding.
I have provided corrections for his vector magnitude error and a modification to correctly add full-moon tides. He has adopted neither but if anyone wants, he have provided full Rcode to reproduce his graphs and spectra (which is exeamplary practice) and anyone can drop in the two-line mods I’ve posted.
Willis: “But that turned out not to be the case at all, as Figure 3 makes clear. There is no long, slow 54-year sine wave in the tidal data, that was just my misunderstanding.”
Sadly Willis you are talking way above your pay grade on all this. You have a very poor understanding of spectral analysis an how to interpret it’s results. Not only is the spectrum you posted wrong because you made a two mistakes in the maths , you still don’t get the reason long cycles, formed interaction for close short cycles, will not be seen in a Fourier spectrum.
I’m genuinely sorry that I’ve failed in my attempts to explain to you how that works.
Figure 3 has a small blip and 8.x years. That is a result of your getting the vector magnitude formula wrong. Once corrected it too disappears. None of which proves there are no long term variations arising from combined effects of the sub annual cycles.
The other problem with the spectral plots is that they are dominated by very strong diurnal and semi-diurnal tides and R scales accordingly. Any small amplitude peaks would not be visible the way it is plotted. That does not mean that a small tidal force acting in the same direction for 9y then back again could not be transporting significant amounts of heat.
This is where some experience in spectral analysis is useful. Just because there’s a FFT package for R does mean anyone knows how to use it. It is like the swamp of foetid misinformation that results from clicking on “linear trend” in excel without the slightest understand of whether it is a reasonable model for the data or whether it means anything at all. Most people seem to assume that there is some fundamental truth in a “trend” and it must correct “because the computer did it”.
So it was a good idea Willis starting this analysis and would give us some information if only he would correct his maths and replot the graphs. But he is drawing conclusions that are not justified on the basis of what is shown here.
Too much is being made of this 1D vector thing.
A scalar is a tensor of rank zero , a single quantity. A vector is rank 1, it requires two quantities to specify it , magnitude and direction. A matrix rank 2 etc.
Whatever coordinate system is used is largely irrelevant. A vector in three dimensional space can be represented as (x,y,z) or (r,theta,phi) etc. R in this case is an unsigned magnitude. The angular coords will determine if it is pointing ‘backwards’, not R.
Since we are not considering a one dimensional space I don’t see any point in discussing 1D vectors, though they could exist in a 1D space.
cd says:
February 14, 2014 at 3:53 am
“Are you saying that we use a point on the Earth as a rotating reference frame. Are you then saying we measure the temporal shift in the magnitude of the gravitational pull from the Moon and the Sun in the vectors defined between our stationary point (on the surface) and the Moon and Sun (which of course are also changing relatively)?”
Precisely that.
If we take, say, a point on the Earth’s surface like the Faroe‐Shetland Channel, then how the two vector sum plays out on that point over the whole of the Saros cycle and its multiples will likely to be of great importance.
By the way, you may also need to consider that most overlooked part of this whole picture, the Internal Tide. The thing that can move the thermocline up and down up 10’s of meters for the pitiful 0.3 meters of air/water interface we so avidly notice.
And how that interrupts (or not) the mixing flow of less than 0 degree C water with the greater than 6 degree C water across that particular interface or the 10 degree C differences it can cause at beaches locally and elsewhere and the like on a hourly basis.
I rather do think that you will not get that level of detail from a simple, JPL, plot but I’ve been wrong in the past.
Greg says:
February 14, 2014 at 4:31 am
“Since we are not considering a one dimensional space I don’t see any point in discussing 1D vectors, though they could exist in a 1D space.”
I sort of agree. In fact the 1D vector of which Willis only plotted the magnitude (i.e. the scalar part) tracks the mid point of the vector sum between the Moon and Sun.
As the actual trajectory of this vector in any case is not relative to any actual point here on the planet’s surface it is kinda irrelevant.
Greg says:
February 14, 2014 at 4:12 am
“This is where some experience in spectral analysis is useful. Just because there’s a FFT package for R does mean anyone knows how to use it. It is like the swamp of foetid misinformation that results from clicking on “linear trend” in excel without the slightest understand of whether it is a reasonable model for the data or whether it means anything at all. Most people seem to assume that there is some fundamental truth in a “trend” and it must correct “because the computer did it”. ”
Now you have touched on one of my “bete noir” points (pun).
The operation of long cycle frequency analysis if using an FFT in the presence of large proportions of noise and a short sample length.
What the noise does is effectively add a vertical line through which the FFT can track thus creating a band, not a line, for resolving. Once you get close to or below a single cycle sine wave (or cos) through that broad band of possible choices the number of those choices climbs dramatically. Such that any possible peak can be spread out over a very wide possible bandwidth.
The addition of noise and low cycle counts in the sample period in question makes FFTs fairly useless as a tool for longer wavelength analysis IMHO. I am sure you will disagree but I do have to make my point.
As to OLS trends, well that is mainly a tool of statisticians (cue quote originally about experts). If it is used, at least use it to determine a curve rather than a straight line. A continuous function not a discrete one. And then S-G rather than LOWESS.
It’s all rather like counting squares on a graph paper rather than switching to integrals 🙂
Linear Trend = Tangent to the curve = Flat Earth.
Greg
The point goes all the way back to Willis’ seemingly turning his eye up to the notion of a 1D vector. I tried to explain that what he was discussing was indeed the magnitude of a vector and how one could express a vector quantity in 1D (direction: -ve/+ve and magnitude); I wasn’t suggesting that what he was doing but Richard was talking at quite a high level. Willis then threw his toys of the pram because a mere pedant like me dare tell him something.
The angular coords…
Angular coords, let’s just stick to Cartesian coords (and yes I know there is a direct equivalence).
will determine if it is pointing ‘backwards’, not R.
Again, this has all been said.
Since we are not considering a one dimensional space I don’t see any point in discussing 1D vectors, though they could exist in a 1D space.
Agreed, it is immaterial, but with the caveat that a 1D reference frame can be defined in a 3D space; that was the point of my last post in order to help explain how a 1D vector might be used in the current context. Ultimately, Willis missed the point I made and seemed to be conflating, or at least assuming I was conflating, the directional component (+ve/-ve) with the magnitude.
Richard
Thanks for getting back. I understood the point you were making.
I just think given the distances from the Earth to the Moon and to the Sun, that any change in the magnitude and direction of the vectors would be so small (that between those form a surface point and those from the the center of gravity), that it would be just as easy to treat the Earth, and all points on its surface, as sharing a single point. I can see for local variations then your approach would be essential but Willis is looking at a global scale.
Again, one could go further, and say the magnitude of the vectors will be affected by local geology and crustal thickness. In which case you’ve just added complexity.
cd says:
February 14, 2014 at 5:10 am
“I can see for local variations then your approach would be essential but Willis is looking at a global scale.”
As am I. It is all about what you use as a reference frame. Everything on this planet moves relative to an Earth based rotational frame that revolves once a day (approx). So in order to understand how external influences, such as gravity from the Moon or Sun, can alter what happens here you need to resolve those factors into the reference frame in question. Until you do you are dealing with some abstract, non resolved rotational frame that has no bearing on how things affect (or don’t) things here on Earth.
Latitude has an important part to play here. In the same way that sunlight alters differently as you move to or from the Poles, so does gravity and the tides. 6 month daylight/night = 6 month high and 6 month low tides. 12 hour daylight = 2 * 6 hours high and low tides.
Month != Day.
If you have not even resolved that simple difference I don’t see how you can draw any meaningful conclusions from the rest.