What Slow Fourier Transforms can tell us.
Guest essay by Stan Robertson, Ph.D., P.E.
On May 3, 2014, an article on WUWT by Willis Eschenbach entitled, The Slow Fourier Transform (SFT) was posted. As he noted, the amplitude of the Slow Fourier Transform components are in the same units as the fitted data, intervals of arbitrary length and irregular data can be used and periodicities rather than frequencies are automatically extracted. In addition to rediscovering a very useful mathematical tool, Willis went on to show that there were apparently no variations of temperature associated with solar cycle variations for several long term temperature records. Now my normal inclination would be to say that if Willis didn’t find any there probably aren’t many to be found. But, on the other hand, as I showed in an October 10, 2013 WUWT article entitled The Sun Does It: Now Go Figure Out How!, it does not take much of a temperature variation to represent a very significant solar contribution to ocean surface temperatures and heat content.
Several researchers, including Nir Shaviv (2008), Roy Spencer (see http://www.drroyspencer.com/2010/06/low-climate-sensitivity-estimated-from-the-11-year-cycle-in-total-solar-irradiance/) and Zhou & Tung (2010) have found that ocean surface temperatures oscillate with an amplitude of about 0.04 – 0.05 oC during a solar cycle. Using 150 years of sea surface temperature data, Zhou & Tung found 0.085 oC warming for each watt/m2 of increase of TSI over a solar cycle.
In my previous article, I showed that the changes of Total Solar Irradiance (TSI) over a solar cyle were too small, by at least a factor of 3.6, to cause temperature oscillations with an amplitude of 0.04 C. Since the variations of temperature considered were clearly associated with solar cycles, it seemed to me that the sun does something more to change ocean surface temperatures than just vary its TSI. But the whole idea would fall apart if there really are no significant variations of ocean temperature correlated with solar cycles. That motivated me to look in places where Willis had not and, in particular, to look at shorter and more recent temperature records that might be both more accurate and with better distribution over the ocean surfaces.
I downloaded the HADSST3 global sea surface temperature raw data (http://woodfortrees.org/plot/hadsst3gl ) and took a look at the data since 1954. This covers 60 years of data and about five and one half solar cycles. To get an idea of what sort of noise would be in these data, I fitted the sea surface temperatures to a cubic polynomial just to get rid of most of the systematic variations. The figure below shows a plot of the residuals for the last 60 years.
Figure 1 HADSST3GL residuals for the last 60 years
If we are looking for variations of about 0.04 C amplitude over the 5.5 solar cycles in the time period shown, then with apparently random variations of about 0.3 C amplitude in the record, the signal to noise ratio would be about 0.04 / 0.3 = 0.13. This would be a signal a long way down in the noise. So the question is, can we extract such a signal with a Slow Fourier Transform? To answer this question, I adopted Willis’ lovely SFT technique. I generated some test monthly data for a 60 year interval consisting of sine waves with a 10 year period plus monthly random noise in the range of +/-0.5 C. The slow FT results for waves with amplitude of 0.15 C, 0.1 C and 0.05 C would have signal to noise ratios of 0.3, 0.2 and 0.1, respectively. The results are shown in Figure 2.
Figure 2. Slow FT for test sine waves with 10 year period for a sixty year interval; 6 cycles.
As one might expect, the random variations would have both short period and long period apparent periodicities as shown in Figure 2. At a signal to noise ratio of 0.2 (blue line), or larger, the signal buried in the noise can be nicely extracted by the Slow FT. At a signal to noise ratio of 0.1, and none of the other curves to aid the eye, you might just have to believe that there might be a signal with a 10 year period. It is hardly bigger than the spurious noise peaks. Of course, there are much more sophisticated signal extraction processes than the Slow Fourier Transform. From comments that I have seen here on WUWT, there are some sharp readers around who could surely teach us some lessons. It might be expecting too much to see such a small signal in the noisy sea surface temperature data with an SFT method. But it is worth noting that in each of the test cases, the Slow FT peaks at 10 yr are smaller than the amplitudes that generated the test data by about ten to twenty percent with worse results at lower signal to noise ratios.
Since it is pretty clear that we will be looking for a small signal in a lot of noise, we probably ought to see where to look. A slow FT of the SIDC sunspot numbers for the years since 1954 shows a peak at 10.8 years as shown in Figure 3.
Figure 3. Slow FT for SIDC sunspot numbers 1954 – 2014
Now let’s have a look at the Slow FT for the sea surface temperature data. The average was subtracted to help suppress spurious long periods, but no smoothing was applied.
Figure 4 Slow FT for HADSST3gl sea surface temperatures
I leave it to the readers to decide whether or not there is a solar cycle signal in the HADSST3gl sea surface temperature record. Considering that the slow FT tends to understate the actual signal amplitude at low signal to noise ratios, I think that this might be a credible detection of a solar cycle driven temperature variation at a 10.4 year period with a signal to noise ratio of at least 0.065 C/ 0.3 C = 0.22.
For the remainder of this essay, I would like to extend and recapitulate some of my previous findings. The prevailing view in climate science is that the sun has contributed very little, if anything, to the warming of the last century. Finding that ocean temperatures are affected during solar cycles to a much larger degree than can be explained by the small changes of solar irradiance that reach the sea surfaces is a huge challenge to the prevailing view, but it rests on some bedrock physics. A detailed accounting for energy exchanges, including thermal energies is as fundamental as it gets.
I was able to account for the long term secular trends of both the sea surface temperature changes AND the ocean heat content since 1965 with a linearly increasing rate of surface heating. This involved numerically solving some heat transfer equations, including the absorption of solar energy, but it provided a simple, two parameter simultaneous fit to the sea surface temperature record AND the ocean heat content record. The two parameters found were a rate of increase of surface heat input of 0.31 watt/m2 per decade and an average thermal diffusivity of the upper oceans of 1 cm2/s. A fairly good fit to both trends was obtained as shown in Figure 5.
Figure 5. Measured and Calculated Sea Surface Temperature and Ocean Heat Content
A good fit was obtainable only for very narrow ranges of parameters. If the thermal diffusivity is taken to be too large, too much heat would be calculated for the ocean depths and surface temperatures would rise too little as the heat moves on to greater depths. If too small, the reverse occurs. If the input heating rate is too large, both rise too rapidly and if too small, both rise too little. The point of this exercise was to obtain a thermal diffusivity that could then be used to tell us how much surface temperature change could be produced by the changes of solar irradiance that occur during solar cycles. The answer is that the small variations of solar irradiance that reach the sea surfaces are far too small to produce temperature oscillations of even 0.04 C amplitude, much less the 0.065 watt/m2 amplitude suggested by Figure 4.
By the same computer program that I had used for my previous WUWT article, I have found that the amplitude of oscillating heat flux entering the ocean that would be required to produce surface temperature oscillations with the Figure 4 amplitude of 0.065 C would be 0.47 watt/m2 for thermal diffusivity of 1 cm2/s. How does this compare to the oscillating flux of solar radiation that reaches the sea surface? Let’s have a look at the solar irradiance changes over solar cycles. Figure 6 shows that TSI varies approximately sinusoidally over recent solar cycles with an amplitude of about 0.5 watt/m2 . (Thanks to Leif Svalgaard for TSI data.)
Figure 6 TSI variations for a few recent solar cyles.
As explained in my previous WUWT post, about 70% of one fourth of this amplitude, or 0.0875 watt/m2 enters the troposphere averaged over the earth area and day-night cycles. About
(160 watt/m2 /1365 watt/m^2) X 0.5 watt/m^2 = 0.0586 Watt/m2 is absorbed at the surface at wavelengths below 2 micron. About half the difference between the 0.0875 and 0.0586 watt/m2 reaches the surface at longer wavelengths and after scattering in the atmosphere. This give a solar TSI amplitude of 0.073 watt/m2 that is absorbed at the sea surface. This is about 6.4 times smaller than the 0.47 watt/m2 amplitude needed to drive surface temperature oscillations of 0.065 C. This result is in better agreement with the larger factors of 5 – 7 found by Shaviv (2008) ( see http://www.sciencebits.com/files/articles/CalorimeterFinal.pdf)
It is of some interest that my results were obtained without assuming any particular depth of an ocean mixing layer. For a thermal diffusivity of 1 cm2/s, the contribution to thermal gradients that vary with the solar cycle below the first ten meters would be much less than 0.001 C/m anyway. I saw no need to introduce a mixing zone with zero gradients and an arbitrary depth boundary.
This leaves us with a clear result that the TSI variations during solar cycles are not the direct drivers of the associated ocean temperature oscillations. Something else that varies with the solar cycles affects the amount of heat flux that penetrates the ocean surfaces. In my opinion, the most likely candidate would be cyclical variations of global cloud cover, but the mechanism that would control it is presently a research topic. Whatever the mechanism of the larger heating variations, it seems quite possible that it might be capable of producing long term secular trends under the control of the sun in addition to variations over solar cycles.
To examine this point, go back to the result shown in Figure 5. The heat flux required to account for the trends of increasing sea surface temperature and ocean heat content had to increase by 0.31 watt/m2/decade. Could this be due to greenhouse gases? CO2 is supposed to produce heating at a rate of about 3.7 watt/m2 per doubling period of its concentration. With concentration increasing at a rate of about 5% per decade, the doubling time would be about 14 decades. Since the heating effect is a logarithmic function of concentration, this would produce a linear heating at a rate of 3.7/14 = 0.26 watt/m2 per decade. This is certainly in the right ballpark to be part of the explanation of the apparent surface heating of the last few decades, however, when we recall that sulfate aerosols with negating effects would partially counter the CO2, it seems to me unlikely that CO2 is the entire explanation. Considering the similar period of rapid warming in the first half of the last century and the presently expanding and embarrassing pause of temperature increases, it seems to me that there is ample room for a significant solar contribution to the longer term warming periods. So I still think that the sun does a lot of it and I would still like to know how. Climate scientists would be well advised to spend some time trying to find out.
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All you folks who claim a large role in global temperatures for the variations in the sun’s strength over the sunspot cycle, here’s a graph for you:
What most people don’t realize is that out of the ~1360 W/m2 of total solar irradiance, the variation of the sun’s strength over a typical sunspot cycle is ± 0.5 W/m2. This is a variation of ± 0.04%, FOUR-HUNDREDTHS OF A PERCENT!!!
Now, it is theoretically possible that there is some amplification factor. Let’s be insanely generous and say that the variation is amplified by a factor of 10 … this still only gives us a variation of ± 0.4%, a minuscule four-tenths of a percent.
I find it highly unlikely that such a variation will make the slightest difference to the global temperature, whether in the short or the long run.
Finally, in the last half billion years the strength of the sun (using standard physics and the knowledge of stellar evolution) is estimated to have increased by 5%. IF the sun ruled the temperature we should have seen an increase in global temperature over that time of 5%, or about 15°C (27°F). Obviously, there is no sign of this in the geological record.
And it gets worse if we think that there is some mysterious amplification factor of ten regarding the solar strength … because then we should have seen an increase of about 150°C over that time, and that’s a joke.
w.
Willis Eschenbach It will not help. Now we begin to feel the effects of the solar impulse 2006.
Stephen Wilde says:
July 27, 2014 at 7:24 am
The claim being made here has nothing to do with ocean cycles.
One could well envisage that the temperature is driven by the variations in tree cover in China … I fear that postulating some unsubstantiated possibility does not advance the discussion.
So your claim is not that the sun is ruling things, but that some vague unspecified “ocean cycles” are in charge … if so, what is driving the ocean cycles? You’ve ruled out the sun driving the “ocean cycles” by pointing out that they are not in phase with each other.
The sun drives the daily cycles, and the annual cycles, and it is never true that sometimes the daily or annual temperatures are “in phase with solar activity and sometimes out of phase with it”. That’s what “driving” implies, that when sun goes up the temperature goes up and vice versa. If ocean cycles wander in and out of phase with the sun, that is prima facie evidence that one is NOT driving the other.
w.
What will be the extent of ice cover in April 2015?
http://oi62.tinypic.com/55jx40.jpg
By staring long enough at a waveform of noise, and with intense calculation, one can recover the Rorschach signal in said noise.
Visible waves in the stratosphere, cause the pushing warm air far to the north.
http://www.cpc.ncep.noaa.gov/products/intraseasonal/temp10anim.gif
My advice would. Solar activity is the weakest since 100 years. It is now a place for observation, because is going curiously.
I’ve had a bit of experience with noise and vibration control in the automotive industry. One must be careful not to simply compare the frequency content of a suspected input to that of an output to decide if the former is driving the latter. Unless a given dynamic system is being excited much below any of its inherent resonances, one would not expect the spectra to match. A given dynamic system will have its own frequency response function (FRF) to a given coupled input. In other words, if the spectrum of the coupled input was perfectly flat, the output response of the system will still be curvy and wiggly. The more degrees-of-freedom of the system, the more complex the curve shape will be. A single degree-of-freedom system, say, a single mass suspended by a spring, will have a single resonant frequency, which in the case of a mass on a spring is the square root of the spring stiffness k over the mass m. This simple single-degree-of-freedom FRF looks like the static response at frequencies far below the resonance, then climb to a peak (of infinity if there were absolutely no damping) at resonance, then starts falling again after the resonant point. It crosses back to the static response amplitude when the input frequency = 1.414 times the resonance, so any input at a frequency lower than this is amplified . At frequencies higher than this, the response starts dropping towards an asymptote of zero at infinite frequency. This is the range of attenuation.
Of course in reality, the only place where such perfect one-degree-of-freedom systems exist is in text books. The FRF of any real dynamic system will have a series on resonances and anti-resonances.
If you have an output, and you don’t know what the input is, I believe you’re better off looking at the coherence function between suspected input signals and the known output. Unfortunately, in my experiences, we essentially always knew what the input was, and we used machines to calculate coherence (which require multiple averaged response measurement), so I am not able to describe well the mathematics of such.
Ren, you linked to a Rossby Wave being propagated up into the stratosphere. Rossby waves are an instrinsic part of our atmosphere and demonstrate how Earth’s own climate and weather systems break into the stratosphere. These waves are often mistaken by solar enthusiasts as evidence of a top down solar mechanism. That is false. They have been clearly shown to be a bottom up phenomenon.
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCIQFjAA&url=http%3A%2F%2Fwww.rsmas.miami.edu%2Fusers%2Fisavelyev%2FGFD-2%2FRossby%2520waves.pdf&ei=uB_VU62BIdekyASwh4GYBA&usg=AFQjCNHAo23UtJpVqMnJkjI3Tr0adfenvA&bvm=bv.71778758,d.aWw
ren says:
July 27, 2014 at 7:43 am
Ren, I was going to skip this as I do virtually all of your comments, but I thought I could assist you in gaining some traction by pointing out that far too many of your comments are incomprehensible or wildly off topic.
For example the comment quoted above. What is the “It” that will not help? Science? The PDO? Praying for rain? Solar variations? Your sentence contains no meaning.
As a result, I generally can’t figure out what you might possibly be talking about, I am unwilling to guess, and I don’t have time to screw around with the back-and-forth of trying to extract your meaning from you.
Here is another example:
ren says:
July 27, 2014 at 8:20 am
What on earth is the point of this question? Are you looking for estimates? There are several blogs where people guess the future ice maxima or minima, but April? The maximum is usually in March, why April?
And in support of your curious question, you’ve posted an (uncited) image that appears to be from Cryosphere Today, no text, no citation, no reason given. That’s as useful to your question as a picture of a kitten, all you’ve done is irritate the reader by wasting our time.
And what does this have to do with putative solar cycle driven variations in temperature?
If you want people to comment on your ideas, the ideas need to be on point, on topic, clearly explained, and referenced or cited to something other than a random graphic with no provenance.
Just saying, if you want traction, your current methods are not assisting you in the slightest.
All the best,
w.
Pamela Gray says:
July 26, 2014 at 9:43 pm
Thanks for the kind words Gary. Weather and climate is my lifelong hobby and I am such a nerd about it. But I am not nearly as schooled as other professional and amateur informative scientists who contribute posts and comments here at WUWT. The honor goes to them as I am only a grasshopper. I have spent pleasurable hours and hours learning a great deal from the likes of Anth***, Willis, Bob Tisdale, and Leif.
===================
As long as it doesn’t go to your red-head, it’s all good 🙂
Willis Eschenbach says:
July 27, 2014 at 7:39 am
“This is a variation of ± 0.04%, FOUR-HUNDREDTHS OF A PERCENT!!!”
Poor argument. Plot similarly the global mean temperature on a Kelvin scale to see that the effect we are looking for is of tiny proportion as well, about +/- 0.2K in the ~60 year variation out of 300K.
Willis Eschenbach says:
July 27, 2014 at 7:51 am
“You’ve ruled out the sun driving the “ocean cycles” by pointing out that they are not in phase with each other.”
Non sequitur. In a system with lag, there is always as phase difference between input and output for periods shorter than the lag.
Willis Eschenbach
I show phenomena which science knows very little. Tell me why the lack of hurricanes in the Atlantic? What is “suddenly” happen?
Pamela Gray says: “The top of the atmosphere translates to 0.073 watt/m2 under clear sky conditions. However, over very long time spans we might have a convincing argument related to a building imbalance (as in more heat going in than going out) leading to long noisy cycles of warmth. But that imbalance begins to go the other way eventually. Leading to some very cold years, decades, and even longer before once again we climb back out of a cold spell. The oceans certainly have the capacity to absorb more heat than they give out (brrrr). ”
I think there is amble evidence already of what is the “noise” in the system. That is changes in the PDO, AMO, NAO, AO and etc. These cycles are not synchronized – which tends to introduce noise in the system. I think it is likely that these “oscillations” are caused by heat imbalances that happen over time.
I agree with many others who think solar output such as TSI vary over time and get amplified in various ways such that they have more of an impact that the energy alone will explain. Thus there are observations such as the one by the current essayist Stan Robertson who wrote above, “I think that this might be a credible detection of a solar cycle driven temperature variation at a 10.4 year period with a signal to noise ratio of at least 0.065 C/ 0.3 C = 0.22.”
Then in periods like the Maunder Minimum of around 1645 to 1710 or so, there is the theory that TSI gets to a “minimum” and stays flat during that period instead of continuing to dip. But is this what stars do? Is there a minimum – and then even as activity of the star declines, TSI stays flat? Observations from other stars don’t seem to be sufficient to answer this question (at least from what I could find). Some astronomers have been trying to determine how much the output of stars varies over time. However, the observations do suggest that there are similar stars to ours that probably vary more than the amount of variation that humans have thus far been able to measure. But the amount we have thus far been able to measure probably means little given that humans have had instruments capable of determining TSI only a relatively short time.
In a nutshell, then, I think changes in solar output including TSI have an impact on the earth’s climate. Variations in TSI are likely amplified through various means (i.e, changes in cloud cover). The other major input to changes in temperature are the various “oscillations” such as PDO, AMO, AO, NAO and etc. The oscillations vary over decades and multiple decades causing changes in energy absorbed and released from the oceans – thus having an impact on surface temperature and locations of winds, clouds and etc. They probably are modulated by heat imbalances that build up in ocean basins over time. The “oscillations” might also be influenced by solar changes – if solar output changes, this will impact how much heat builds up and where it builds up in the oceans.
FWIW, I Googled “ren comment bot” and discovered “Ren” was a handle used on the hackaday-dot-com site.
How are scientists like rabbits? They like to replicate.
kadaka, that makes sense. Artificial sentence structure. Changes in sentence syntax that appear to be automatic. Posting quotes without comment. Indeed we might have a “bot” in the belfry.
Willis mentions the extraordinary stability of temperature over aeons as a reason for not accepting a significant solar influence on climate.
The answer is that although the sun alters global cloudiness and thus the proportion of solar energy entering the oceans the subsequent shifts in the climate zones are the negative system response which keeps outgoing radiation equal to incoming radiation.
For a water planet the adjustment mechanism even works if the top of atmosphere insolation changes as a result of a large change in solar power such as that since the early faint sun.
It is the weight of the atmosphere on the ocean surface that determines the amount of energy that the oceans can retain from whatever incoming radiation arrives from the sun by setting the amount of energy required to achieve the phase changes of water.
The oceans then determine air temperatures.
bones says:
July 26, 2014 at 7:08 am
Leif Svalgaard says:
July 26, 2014 at 6:40 am
Stan, one example of how little change it takes in TSI to create clouds and rain is evident in the phenomenon of two rainy seasons in Tanzania/southern Kenya caused by the transit of the sun back and forth over the equator. The apparent transit angle is ~12 degrees on each side of the equator, however the rains are initiated before the extremes are reached – “long rains” starting in March-April and short rains in October-November. Therefore at ~5 degrees N and S would appear to be enough to initiate this. TSI on a square metre 5 degrees N or S is ~cos 5*TSI at the equator, i.e. is 0.99619 of the insolation at the equator. I believe this direct observation would present a measure of how sensitive “weather” is to small fluctuations in TSI.
Willis said:
“If ocean cycles wander in and out of phase with the sun, that is prima facie evidence that one is NOT driving the other.”
A logical non sequitur.
The sun determines global cloudiness which then affects the proportion of TOA insolation that gets into the oceans.
Internal system variability sets up cyclical movements within the oceans and there is no reason why that internal variability need be synchronous with the initial solar variations given the thermal inertia of the oceans, possible lunar influence and separate interactions between the individual ocean basins.
Furthermore, as per Willis’s own Thermostat Hypothesis it is the variable ocean/atmosphere response that eventually negates the initial solar effects on global cloudiness.
Less clouds skews ENSO towards more El Ninos compared to La Ninas and El Ninos are a cooling effect for the system overall even if the middle latitudes get a little warmer from more poleward zonal jets whilst the energy flows through the air more quickly on its way from the oceans to space.
The circulation changes in air and oceans are a therefore a negative system response to the initial solar induced changes in cloudiness.
Climate changes as observed so far (within the interglacial) seem not to involve significant changes in global temperature, merely changes in the distribution of available energy with the largest effects in middle latitudes as the climate zones shift to and fro latitudinally.
Even during Ice Ages there seem to be similar climate shifts on an approxomate 1000 to 1500 year time scale but they don’t cause Ice Ages to begin or end so solar variability causing cloudiness changes would appear to fit the bill both within Ice Ages and Interglacials.
Willis, try looking at this, http://climate4you.com/images/SunspotsMonthlySIDC%20and%20HadSST3%20GlobalMonthlyTempSince1960%20WithSunspotPeriodNumber.gif
In 1965, ’76, ’86, ’96, and ’08 there were solar minimums. Look up at the SST graph, and see the temp dips that correspond to the minimums. Yes, that’s wiggle-matching, but we are comparing real observations. Can you see the El Nino’s and La Nina’s in the SST graph? The SST anomalies in ’65 and ’76 show a clear negative temperature change during the solar minimums. There’s much more information in this graph than is apparent. I am not sure why you’re not picking up those temp drops at the solar minimums in your statistical work.
Most of the graphs from other datasets as seen here: http://climate4you.com/ (scroll down the page…) show the same temp drop features at the solar min marks. The point is, temps drop when solar activity drops off, the amount of the drop is dependent on the energetic strength of the particular cycle and lagged accumulated oceanic heat release that had built-up over many cycles.
At this point, because I really have other things to do today instead of spending all day on the comment section here, I’m going to return to this topic again in a week or two after finishing up my presentation on solar warming/cooling, where great attention will be given to how it happens. After which, Willis, I will welcome your point of view on the statistical aspects of my model, realizing many are dubious of solar-driven warming/cooling, including yourself.
There are lags in the system, and your stats work could help find the lag(s). David Stockwell determined a 2.75 year lag. Did anyone read David Stockwell’s papers on solar supersensitivity and accumulated sunspot anomaly? His stuff is here: http://landshape.org/enm/solar-supersensitivity-a-new-theory/ , papers here http://vixra.org/abs/1108.0004 and here http://vixra.org/abs/1108.0020 .
Just so you know, ren’s native language is not english….he does the best he can – he’s a smart guy who studies UV, cosmic arays, and the stratosphere. I just wish he was easier to understand!
Gary Pearse says:
July 27, 2014 at 11:41 am
Excellent point Gary.
Pamela Gray: “Indeed we might have a “bot” in the belfry.” Nice play on words. Ironic, too, since rén (人) means “person.”
Gary Pearse says:
July 27, 2014 at 11:41 am
Thanks, Gary. A couple comments on that. First, I find this:
Next, Kenya stretches from about 4°N to 4°S so it is directly on the equator.
Next, the swing of the sun is from 23.5°N to 23.5°S, not “~12 degrees on each side of the equator”. This makes the difference from when it is over the equator to the furthest point south equal at the TOA to cos(23.5) = 0.91, or a difference of 28 W/m2 between extremes.
Next, you’ve forgotten the variation in TSI caused by the varying distances from the sun. This increases the TSI when the sun is on one side of the Equator and decreases it when the sun is on the other side.
From the CERES data, here is the TOA monthly insolation for the Equator:
So while you may indeed be correct that the sun is causing the two rainy seasons, the annual variation in TSI in Kenya is about 50 W/m2 peak to peak, not the tiny amount you have calculated.
w.
PS—You say “however the rains are initiated before the extremes are reached”. In fact the rains start somewhere around the equinoxes (March & September), when the sun is directly over the equator.
“The solar cycle trend switches direction before and after cycle minimum, but
instrument degradation trend is expected to continue across the minimum.
• Trending over both declining and rising phases of the solar cycle can help clarify,
or at least bound, how much of the trend is due to instrument degradation. ”
https://www2.acd.ucar.edu/sites/default/files/heppasolaris/Pilewskie.pdf
Ren is fine… Ren Hoëk is rather adventurous and intelligent. As long as Stimpson J Cat (Stimpy) doesn’t show up, because that’s when Ren gets emotionally brittle. Then it WILL get very annoying. 😉