The Solar Series: I Background | II: The notch filter | III: The delay (you are here) | IV: A new solar force? | V: Modeling the escaping heat. | VI: The solar climate model | VII — Hindcasting | VIII — Predictions
UPDATE: July 21 Thanks to Bernie Hutchins, David found a problem with the code, which means the notch no longer guarantees a delay. The delay still likely exists (see the other evidence in the references below) but this post, particularly figure 2 needs correction and updating. – Jo
Strap yourself in. The Notch in the Earth’s response to incoming solar energy means that every 11 years (roughly) the solar energy peaks, and at the same time the climate’s response to the extra energy changes. What on Earth is going on?
The thing about notch filters that is hard for anyone who isn’t an electrical engineer to understand is that it appears to start working before “the event” it is filtering out. This is obvious in the step response graph. That’s Figure 2 – which shows what happens where there is a sudden step up in solar radiation (the brown line). The blue line shows the temperature response, which paradoxically starts to “rise” before the step up even occurs. This makes no sense, of course. Electronic engineers know that a [non-causal] notch filter always means a delay. Something triggers the filter before the event begins and the notch filter responds after a delay.
By mathematical inference David shows that there
must be a delay. Later, by fitting the solar model to the measured temperatures, the delay is found to most likely be around 11 years (but is definitely between 10 and 20 years).
There is some empirical support for this from Lockwood and Froehlich’s paper, and also from Usokin, Archibald, Solheim, Soon, and Moffa-Sanchez.
The solar model also has a low pass filter, as well as the notch-delay combo. All the little flickering short rises and falls in solar radiation would not show up in Earth’s temperature. A low pass filter blends or smooths these short term cycles as the energy is absorbed by systems like the oceans. Only longer sustained changes in radiation make a difference, as heat either accumulates or dissipates over longer periods. The transfer function suggests (loosely) that changes lasting less than about 3 years make less and less impact on temperatures. — Jo
Building the Model Part 1: Deducing the Delay
We are building the solar model that would account for the recent global warming if it was associated almost entirely with solar radiation (notice that we didn’t say “caused”), and had no dependence on carbon dioxide. Here we assemble the first three parts of the model, a notch filter, a delay filter, and a low pass filter.
1 The Notch
In the previous post on exploring the data, we found that the most prominent feature in the empirical transfer function was the notch, which filters out the 11-year “hum” from the Sun.
The notch is a very curious fact. Solar radiation warms the Earth, providing nearly all the heat as incoming radiation—visible light, UV, infrared, and so on. So we’d expect the extra radiation from the Sun every 11 years to produce corresponding peaks in temperature here on Earth. Yet it doesn’t.
We have chosen to investigate what happens if the recent global warming was associated almost entirely with changes in solar radiation and has no dependence on carbon dioxide — the “solar assumption”. Obviously the solar assumption cannot be entirely true, and it is later discarded in the development of the solar model. It is only needed for finding the approximate parameters for the model, and it does not ultimately impact on whether the model is appropriate or not. This parallels the original development of the carbon dioxide theory, which temporarily assumed that carbon dioxide caused almost all of the global warming since 1800 — the “carbon dioxide assumption” — in order that the parameters of the carbon dioxide model could be found by curve-fitting it to the measured temperatures.
Using the solar assumption we curve-fitted a notch model to the measured temperatures, to find the approximate size and shape of the natural notch filter. It is more instructive to show the notch filter we eventually found by curve fitting the entire solar model (which contains a notch filter) to the measured temperatures, because that way we can build up the model’s transfer function piece by piece so that it matches the empirical transfer function. (The two notch filters are basically similar except for a different overall amplitude multiplier, that is, a vertical shift in the transfer function diagram.)
Ok, it’s notch shaped, and the notch is at 11 years, as we’d expect. (We are only concerned with the amplitudes, because we cannot adequately detect the phases of the sinusoids in the climate datasets.)
But what is much more interesting becomes apparent when the notch filter is portrayed in the time domain, as the step response. The step response of a system is what the output does when the input instantaneously steps up by one unit.
Notice that the step response starts several years before the step-up, which violates causality — it is impossible. In our universe, a response can only come after the corresponding stimulus. The non-causality of the step response of the notch filter in Figure 2 is not a fluke: in any electronic notch filter without an accompanying delay, the step response is blatantly non-causal. Notch filters by themselves are intrinsically non-causal.
2 The Delay
How we know there is a delay
When engineers design a filter whose transfer response has the desired shape of amplitude, but which is non-causal and therefore impossible, they simply include a delay with the filter. Adding a delay does not change the amplitude of the transfer function of the filter, it only changes the phase of the transfer function. The delay shifts the entire amplitude part of the step response to the right in diagrams such as Figure 2, without changing its shape, as if the time axis were replaced with a new time axis.
For example, the step response of the combination of the notch filter in Figure 2 and a 7 year delay is the blue line in Figure 2 shifted 7 years to the right. Just imagine sliding the blue line 7 years to the right — the dagger of the notch would move from year 0 to year 7. (Notice this would almost but not quite make the response causal, because the response would almost be zero before the stimulus begins. A delay of about 8 or 9 years would be sufficient to make it causal and therefore possible.)
The combined filter-delay combination can exist, so long as the response is shifted sufficiently that it is zero before the stimulus. For engineers designing filters, this means the combination is possible to build. (It is possible to use non-causal filters in simulations on a computer in the frequency domain, but they don’t exist in the real world!)
So given that we have detected a notch filter, and its step response is non-causal, one way that such a notch filter could possibly exist is that there is an accompanying delay sufficient to make it causal.
There do not appear to be any other possibilities. Adding any filter other than a delay filter would change the shape of the amplitude of the transfer function in Figure 2, which would change it from the notch filter we have detected into something else.
How much delay? Presumably as little as necessary, or at least this is what design engineers typically do. Given the notch filter portrayed in Figures 1 and 2, presumably the delay would be a bit larger than 8 years but presumably not a lot larger (like say 80 years).
To sum up, in the system whose input is TSI and whose output is temperature we have detected a notch filter whose transfer-function amplitude indicates that it cannot exist unless it is accompanied by a delay of several years.
Therefore we infer that there must be a delay of several years that is intrinsically part of the notch. The notch must be accompanied by a delay, because the notch on its own is physically impossible.
It is crucial to notice that this is a true delay, not due to a dissipative element like a store of ocean heat that declines at a rate governed by a time constant.
So let us add a delay filter into our solar model, in cascade with the notch filter. (“Cascade” means the output of one system is the input to the next system.) The delay is intrinsically due to or associated with the notch, so the notch and delay filters must always go together with nothing between them.
There is some supporting evidence for a delay
Later, when we fit the notch-delay solar model being developed here to the measured temperatures, we find that the delay is mostly likely around 11 years (but definitely between 10 and 20 years).
Lockwood and Froehlich’s famous 2007 paper showed that four solar indicators, including TSI, rose until about 1986 then started falling slightly, yet temperature kept on rising until the mid to late ‘90s. They interpreted this as evidence that the Sun is not a major influence on temperature. However 1986 + 11 = 1997, which is about when global warming stopped.
A strong correlation between the length of a solar cycle and the temperature during the following solar cycle was discovered in 1991 by Friis-Christainsen & Lassen. This correlation was verified and explored further in the last few years by David Archibald, and in 2012 Jan-Erik Solheim et al found that a lag of 11 years maximizes the correlation between solar cycle length and temperature. We will be doing a blog post on how this correlation fits in with the notch-delay solar model later in the series.
Willie Soon in 2009 found a ten year delay from TSI to sea surface temperature changes in the tropical Atlantic. Moffa-Sanchez et al in 2014 found a lag of around 12 years from TSI to North Atlantic surface temperatures over the last thousand years. Usoskin et al in 2004 found that the correlation coefficient between northern hemisphere temperature and reconstructed sunspot numbers from 850 AD was greatest when the temperature lagged the sunspot numbers by around 12 years. Greg Goodman finds a peak in the correlation between sea surface temperatures and sunspot numbers when the former lags by 10.1 years (though sunspot numbers would correlate a little like that with white noise).
3 The Low Pass Filter
The climate system is like a bucket of heat with a hole near the bottom of the bucket.
Unreflected TSI pours into the top of the bucket. About 30% of the TSI incident on the planet is reflected straight back to space by clouds, ice, snow, and so on, and does not heat the planet, but the other 70% warms the Earth. This figure of 30% is the “albedo” of the Earth.
The Earth emits energy to space as radiation across the electromagnetic frequencies, nearly all of it at infrared frequencies (that is, as “heat”). The amount of radiation is roughly proportional to the amount of heat in the climate system, or its temperature, just as the amount of water that spurts from a hole in a bucket is greater if there is a greater depth of water in the bucket.
The simply mechanics of the rates of heat input and output have the equation of a first order low pass filter, like an RC filter in electronics. This is a well-known observation in modeling the climate. We find from later fitting the notch-delay solar model to the measured temperatures that the time constant (or break frequency) is around 5 years, which agrees with what others have found.
While it is exaggerating to say that a low pass filter is “evident” in the empirical transfer function, one can imagine the amplitude of the empirical transfer function being equal to a product of:
- The amplitude of the transfer function of the notch filter in Figure 1.
- The amplitude of the transfer function of the low pass filter in Figure 3.
( The transfer function of two systems in cascade is the product of their individual transfer functions. We are ignoring phases here, so just multiply the amplitudes to get the amplitude of the combined system. The vertical scales on these graphs are logarithmic, so add the red lines on the graphs.)
The notch filter is obscuring the low pass filter in the empirical transfer function, so it is difficult to “see” the low pass filter. But a simple basic approach to heat accumulation tells us there must be a low pass filter, and it is at least compatible with the empirical transfer function. So we will include a low pass filter in our solar model, in cascade with the notch-and-delay filters.
In the next post we look at mechanisms that might cause the mysterious delayed correlation.
Notch-delay solar project home page, including links to all the articles on this blog, with summaries.
Archibald, David, http://www.davidarchibald.info/papers/Past-and-Future-of-Climate.pdf, 2010
Archibald, David, “Solar Cycles 24 and 25 and Predicted Climate Response”, Energy and Environment, Volume 17 No. 1, 2006, pages 29–35
Friis-Christainsen, E.; Lassen, K. ,(1991) “Length of the Solar Cycle; An Indicator of Solar Activity Closely Associated with Climate”, Science, , pp. 698-700
Lockwood, Mike; Froehlich, Claus, “Recent oppositely directed trends in solar climate forcings and the global mean surface air temperature”, Proceedings of the Royal Society, 2007
Moffa-Sanchez, Paola; Born, Andreas; Hall, Ian R.; Thornalley, David J.R.; Barker, Stehe, “Solar forcing of North Atlantic surface temperature and salinity over the past millennium”, Nature Geoscience, 2014, Supplementary Information
Solheim, Jan-Erik; Stordahl, Kjell; Humlum, Ole, “The long sunspot cycle 23 predicts a significant temperature decrease in cycle 24″, Journal of Atmospheric and Solar-Terrestrial Physics, 2012
Soon, Willie W.H., “Solar Arctic-mediated Climate Variation on Multidecadel to Centennial Timescales: Empirical Evidence, Mechanistic Explanation, and Testable Consequences”, Physical Geography, 2009, pp. 144-184.
Usoskin, I. G.; Schuessler, M.; Solanki, S. K.; Mursula, K., “Solar activity over the last 1150 years: does it correlate with climate?”, Proc. The 13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun, Hamburg, pp. 19 – 22, 2004
Usokin, I. G., M. Schuessler, S. K. Solanki, and K. Mursula 2005, Solar activity, cosmic rays, and the Earth’s temperature: A millennium-scale comparison, Journal of Geophysical Research, 110, A10102.
*July 21, I added the phrase [non-causal] in the intro which seemed overly complex on June 17, but is clearly important now. – Jo