Ed Fix has been back in touch about his solar activity simulation model. Ed couldn’t reveal too much last time around as the paper was pending publication in an Elsevier book. My thanks to Ed for being true to his word and returning here to the talkshop armed with a full explanation of his model and data. The spreadsheet and supporting info are here.
A couple of months back, David Archibald generated a bit of commotion with a post on WUWT about an as-yet-unpublished paper of mine. At the time, I said I’d talk more about it after the paper had actually been published.
The book _Evidence Based Climate Science_, Dr. Donald Easterbrook, ed. (Elsevier, 2011) has been published as an e-book (hardcover to follow in Sept), and a preview is available on Elsevier’s website, (http://www.elsevierdirect.com/ISBN/9780123859563/EvidenceBased-Climate-Science). So now I am prepared to talk about my paper, included as chapter 14 (beginning on page 335 in the preview). The paper’s title is “The Relationship of Sunspot Cycles to Gravitational Stresses on the Sun: Results of a Proof-of-Concept Simulation”.
This paper presents what I believe is a new approach to linking the motion of the sun around the barycenter of the solar system to the sunspot cycle. I consider this paper to be a progress report after the preliminary phase of a work in progress. This effort differs from earlier work in three main respects.
First, I used a signed sunspot cycle (in my paper, I called it “polarized”). I later learned that this approximately 22 year cycle is called the Hale cycle. A well-known solar physicist told me that it is not correct to plot it this way, because there is some overlap at the minimum between sunspots belonging to the preceding and following cycles. Of course, that same physicist’s well-known Cycle 24 predictions completely missed the long Cycle 23-24 minimum, and only predicted Cycle 24 to start increasing in early 2010 AFTER that had already happened; my model called it in 2008.
Second, this model uses only the <em>radial</em> component of the sun’s motion about the barycenter–that is, the distance of the barycenter from the center of the sun. Specifically, it uses the second derivative wrt time of the radial distance–that is the radial acceleration of the sun wrt the barycenter. This first iteration of the model ignores rotation completely, giving this version of the model some fundamental limitations. That’s why the sub-title says “proof-of-concept”, rather than “42”.
I plotted the radial velocity of the sun along with the polarized sunspots and noticed intriguing similarities. In a blinding flash of inspiration (or maybe indigestion) I realized that the second derivative of distance is acceleration, and that acceleration is simply force scaled by mass (F=ma). If that’s so, why not treat it like a force and see what happens.
The third difference between this approach and others is the use of a dynamic model. I used the simplest possible physical analog; a mass oscillating on a spring with damping. The force equation for that is: F=-kx-bv, where k is the spring constant (which defines the resonant period), x is the displacement from center, b is the damping coefficient, and v is the velocity of the mass. In this sunspot model, x represents the sunspot number, and v is the rate of change of the sunspot number.
There is a third term in the force equation for this application: “+z (d^2 r/dt)” (add a coupling constant, z, multiplied by radial acceleration — second derivative of r wrt time). This adds a forcing function. Now, a=F/m, where a is the rate of change of the rate of change of the sunspot number, and the sunspot number is:
I like the analogy of a small child bouncing a ball, where the child hasn’t yet mastered the finer points of ball-bouncing. His bouncing is likely to be erratic, he may bounce the ball at varying heights, miss the ball sometimes, or hit it on the way down instead of on the way up, etc. If we record the height of the ball at, say, 0.1 sec intervals, can we then build a model that duplicates the trajectory of the ball? We could probably build a reasonably good model of a bouncing ball, and derive the characteristics empirically–air friction, gravitational acceleration, elasticity and rebound damping, etc. However, if we don’t duplicate the forcing function of the child’s hand, the resulting model output will look nothing like the actual trajectory.
The equation was implemented on a spreadsheet (BIG spreadsheet). The time interval between calculations is 1 month, so t = t^2 = 1 in the equation, and v0 = x_t-1 – x_t-2. The equation I used for each iteration in the spreadsheet was:
To set the various coefficients, I started the model in April, 1749 with a value of -90 and rate of change of 0 9 (the approximate peak of Cycle 0) and let it oscillate from that point forward. I adjusted the coefficients until the model output matched the historical sunspot data. I was amazed to discover that it did, at least for a few cycles.
Between 1800 and about 1812, there was an unusually weak and long Cycle 5, followed by a long pause before the start of Cycle 6. Again, in 1890-1912, Cycle 13 Was unusually long and Cycles 13 and 14 were relatively weak, followed by a long pause before Cycle 15 started up. This model didn’t do that; there was obviously something going on that this simple model couldn’t model. So to simulate, in a sense, an aspect of the sunspot cycle generator that this model couldn’t follow, I clamped the output to 0 from 1807 to mid-1811, and again from Nov., 1904 to Sept., 1911, and let the model do its thing after the forced quiescence. In both cases, it started up and followed the subsequent sunspot cycles, with the exception that between 1811 and 1904, the model output is 180 degrees out of phase with the polarized sunspot cycle. To better show the correlation, in this figure I’ve flipped the polarity of the sunspot numbers during the 19th century
This model is one-dimensional; it only operates in the radial direction. However, the latest astronomical observations show that the solar system occupies at least two dimensions, and it rotates. My suspicion was that the rotation had flipped the polarity of the forcing function, rather than the polarity of the sunspots flipping. I’ve since modified that suspicion a bit. However, we do know that the Cycle 15 was in fact reversed in polarity from Cycle 14–that’s when Hale first observed the polarity reversal.
Remember the bouncing ball analogy. Up until this point, the model had always been started at -90 in 1749. What happens if I start the model at 0 in, say, 1500 and let it run? What happens to it in the 18th century and later? As the next figure shows, not much changes. Seeing this was my EUREKA moment.
As I see it, there are a couple of take-aways from this. First, there are two parts to the problem: the forcing function that activates the sunspot cycle, and the resonant characteristics of the sunspot generator itself. Concentrating on only the driving force, as many barycentrists do, or only on the internal mechanisms of the sun, as most solar physicists do, would cause you to miss crucial aspects of the overall cycle.
Second, I suspect there may be a couple of coincidences tied up in all of this. First coincidence; I believe the fact that Jupiter’s orbital period is nearly the same as the traditional Schwabe sunspot cycle is pure coincidence and probably inconsequential. The base period of the force produced by the sun’s acceleration toward and away from the barycenter is about 20 years–the synodic period of Jupiter and Saturn’s orbits. This is strongly modulated by the motions of Uranus and Neptune.
Second coincidence: the sun’s intrinsic resonant frequency is near the frequency of the driving force. I think of the barycenter’s movement as exciting the sunspot generator, rather than driving it. If you were to pull on the sun only once and never again, it might produce 2 or 3 weak eleven-year cycles, and then never spit out another sunspot. However, if the sun has it’s own intrinsic resonant frequency, and it’s excited by an outside quasi-periodic force, that force’s frequency needs to be near the sun’s resonant frequency, or nothing much will happen. Just speculating here, but if Jupiter or Saturn were in different orbits, we might not have much of a sunspot cycle at all.
So that’s it. Any comments? Unlike some in the climate science world, I can take criticism.