Guest Post by Willis Eschenbach [Graphs updated to include error bars]
Inspired by the paper by the charmingly-named Maya Tolstoy discussed here on WUWT, I decided to see if tidal forces affect the timing of earthquakes and volcanoes. Dr. Tolstoy’s hypothesis is that tidal forces affect the timing of the subterranean eruptions … but she has only nine “events” (either eruptions or lava flows) to test her theory. On that thread I said I thought her hypothesis was wrong, but I hadn’t looked at the data.
I figured that IF, and it’s a big if, tidal forces are affecting volcanoes, they’d also affect earthquakes. So I decided to start by seeing if there is a relationship between the tidal forces and earthquakes by looking at as many earthquakes as I could find.
For the calculation of the tidal forces, I started by going to the marvelous JPL Horizons ephemeris. I set the variables as follows for the Sun. For the Moon I just changed the “Target Body”.
With the X, Y, and Z variables, I calculated the individual tidal forces from the sun and moon (see “TIDAL MATH” below), and added them as vectors to give the total tidal force. I calculated the tidal force on a “per kilogram” basis. Here is a sample of the results showing recent tidal forces:
Some comments on Figure 2. First, on a per-kilogram basis the forces are small. One grain of sand exerts a force of about 40-50 micronewtons downwards under earth’s gravity. I weigh about 70 kg, so the tidal forces make my weight vary at the equator (the Earth’s equator, not mine) by about 3 grains of sand … however, the total tidal forces are large because the earth has a very large mass.
Next, note that as you’d expect, the peaks in Figure 2 are not aligned with the calendar year. Instead they shift slowly through the calendar year over about an eight-year cycle. This means that we should not expect to see any annual variation in earthquakes by month. And this is the case for this dataset, monthly earthquake counts only vary by ±4% (not shown). In addition, note the rapidity of the changes. These cycle every lunar month, which is about twenty-eight days.
Having calculated the tidal forces, I got a database of all large (>5) earthquakes since 1900 from the US Geological Service. To examine the distribution of the data, I took a histogram of the tidal forces on the actual dates of the earthquakes, and I compared it to the full database of daily tidal forces during the same period. Figure 3 shows the results.
Figure 3. Distribution of tidal forces during earthquakes 1900-2007 (gold) compared to distribution of all daily tidal forces during the same period (red diagonal hatched).
As you can see, the answer is clearly NO. The histogram of the tidal force at the times of the earthquakes (gold) shows the same double-peaked distribution shown by the full tidal dataset (red hatched). There is no overall relationship between earthquakes and tidal forces.
Next, I wanted to examine volcanic eruptions. So I went to the Smithsonian Global Volcanism Program website and downloaded their eruption database. Using all confirmed eruptions with known dates back to 1800, I did the same thing with the eruptions that I did with the earthquakes. Figure 4 shows the results of that analysis:
Figure 4. Distribution of tidal forces during eruptions 1800-2013 (blue) compared to distribution of all daily tidal forces during the same period (red diagonal hatched). Errors adjusted to account for number of subsamples.
Once again, there is little difference between the two datasets. Yes, there is an exaggeration of the local peak of the tidal forces in the range 0.8 to 0.9 micronewtons (bottom scale), but given 95% confidence interval, that kind of variation is not unusual. Overall, volcanoes seem unaffected by tidal forces.
Now … why should this be the case, that the quakes and eruptions are NOT affected by the tidal forces? I mean, we know that the tidal forces cause tides in the ocean and in the atmosphere. And most importantly for this question, they also cause tides in the solid earth. These tides are on the order of about half a metre (a foot and a half) at the equator. So it seems logical that they would affect earthquakes and eruptions. My speculations about the reason they don’t seem to affect quakes and eruptions are as follows:
1. The tidal forces are always there, and are always rapidly changing. Vertical tidal forces go from local extreme to zero every six hours. As a result, any stable condition of the earth’s crust must be able to withstand the worst that the tides can do.
2. The forces basically affect all of any local area equally. The diameter of the earth is on the order of 13,000 kilometres (km) (8,000 miles). The earth tides are half a meter. Not half a kilometer. Half a metre. Figure 5 shows my drawing of how the tidal force operates on the earth. It is a stretching force that applies to land, sea, and air.
Figure 5. Tidal forces elongating a hypothetical planet and its ocean. The planet is free-falling into the sun, so there are no centripetal forces. Note that the planet is elongated as well, but this is not shown in the diagram because obviously, tides in the solid planet are much smaller than tides in the ocean. NOTE THAT THIS PLANET IS NOT THE EARTH.
Now, in Figure 5, the vertical motion due to tidal force is greatest along the line between the planet and sun. It goes to zero along the vertical plane that passes through the middle of the earth at a distance D from the sun. This is because the vertical tidal force is dependent on “r”, which varies from place to place and time to time on the actual earth (for the calculation see “TIDAL MATH” below).
As a result, any point on the earth goes from high vertical tidal displacement (for that point and time) to no vertical tidal displacement in six hours. Now, that six hours is a quarter of the circumference of the earth, which is about 10,000 km (6,200 mi). And over that distance of 10,000 km, we have a difference in elevation of half a metre. This is a vertical deflection of one part in twenty million … a very, very small amount
And that in turn means that per horizontal kilometre, the average difference in equatorial elevation due to tidal forces is five-hundredths of a millimetre, with a global maximum of about eight-hundredths of a millimetre. That small amount of deflection, one part vertical for each twenty million horizontal, means that the change in elevation is very, very gradual. And as a result, the entire local area is being affected pretty much equally.
Anyhow, that’s my explanation for the fact that although the earth is incessantly flexing from the tides, it doesn’t seem to affect the timing of earthquakes and eruptions as a whole. It’s because the flexing (by global standards) is both small and gradual.
2 AM … gotta go outside and see what the storm did. Raining all day here, and I’m happy about that …
Regards to everyone,
THE USUAL REQUEST: If you disagree with someone, please quote the exact words that you disagree with. That way, we can all see exactly what you are objecting to.
UNANSWERED QUESTIONS: Is there a tidal connection to the number of very small earthquakes (microseisms)? Do big earthquakes have a tidal connection? How about big eruptions? As with any investigation, each answer brings new questions … so please, don’t bust me for not answering all of them or assume I’m not aware of them.
TIDAL MATH: The tidal force operating on a one kg mass at a point at a perpendicular distance “r” as shown in Figure 5 is given by
T = 2 G * M * r / D^3
where T is tidal force (newtons), G is the gravitational constant, D and r are as in Figure 5 (metres), and M is the mass of the sun (kg).
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