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.