A new way to extract a climate signal from weather noise: Seasonal lag

by David M. Barnett

Synopsis of  Global Warming as a System Response Theory Problem

The weather noise problem—how much warming?

Changes in weather can cause temperature changes from day to day by as much as 20°C.  The coldest day in mid-summer can be colder than the warmest day in mid-winter.  That is weather.  Yet we all know the difference between summer and winter, and that summer days typically are warmer than winter days.  That is our experience of climate.

It is said that the climate has warmed by a degree or so in the last 200 years. That is a very small number to extract from the day-to-day noise which can be an order of magnitude larger.  How can one extract that small secular temperature trend reliably? Even thermometer measurements at a single location are subject to perturbation by non-global-climate factors such as growing urbanisation or even the growth or removal of a stand of trees.

The warming attribution noise problem—is it CO2?

Many people believe that CO2, added to the atmosphere by burning fossil fuels since the industrial revolution, has increased the effective thermal resistivity between the earth’s surface and outer-space—the “greenhouse warming”—forcing a temperature rise to compensate.  But how does one disentangle this “greenhouse” hypothesis from other possible contributing causes such as land use changes or changes in insolation?  True ab initio modelling of the climate system is impractical due to the huge ranges of time and space scales that are important. How does one extract an attribution signal from the parametric noise?

Signal processing with a carrier wave

Ever since the invention of the magnetic tape-recorder signal engineers have known that it is much easier to extract a wanted signal from the noise, if the signal modulates a known high-frequency carrier wave.

As it happens, the seasonal shifts caused by the tilt of the earth’s axis and the earth’s orbit around the sun have a known period of one year, and the sun’s stimulation at any given location has a known phase.  The climate system processes this “carrier” signal input and produces a temperature output that can be measured.  For example, daily temperature measurements have been recorded at a location in Central England for more than two centuries.  The time series is available from BADC in the UK.

The climate signal

So what signal can we extract with period of one year?  Figure 1 shows a cosine of period 1-year fitted to the 4-year block 1772 to 1776. The blue cosine curve is the climate signal while the black curve is the daily weather noise. Unsurprisingly, the coldest time of year, according to climate, occurs in the last half of January. However, the day-to-day weather can vary greatly.  For example, late January 1773 had an unseasonably warm spell with mean daily temperatures as high as 9°C, despite this being the latter part of the “little ice age”.

Screen Shot 2023-02-26 at 12.53.26 PM

Figure 1: Least Squares fit of cosine with a period of one year to daily mean temperature data in a block of four-years

Seasonal lag and climate change

I have always been fascinated by the fact that the shortest day of the year is around 20th December, while the coldest time is typically towards the end of January.  The difference is the seasonal lag. In the case of 1772 to 1776 that coldest date (according to climate) was 19.5 days into January. That represents a lag of about 30 days which is close to  radians.

Why do the seasons lag behind the cycle of the length of daylight? The short answer is that the oceans store the heat.  It is analogous to an electrical RC filter (see Figure 2) where C is the ocean’s ability to store heat, and R is the effective resistance to heat escaping into space.  The larger R or C, the longer the phase lag.

Screen Shot 2023-02-26 at 12.56.04 PM

Now comes the magic.  The CO2 “greenhouse” hypothesis is that adding CO2 increases the effective thermal resistance, R.  In which case, the seasonal lag ought to increase along with the CO2.  The CR-filter theory also predicts that the difference between summer and winter should also increase, if thermal resistance is the sole cause of the increase in lag.

How seasonal lag changed between 1772 and 2005 in Central England

The daily mean temperature time series for Central England was divided into blocks of 4 years duration.  A cosine climate curve, with period one year, was fitted to each block (as in Figure 1, for example). The parameters for each block are Temperature Amplitude (i.e half the summer to winter temperature change), the temperature phase (expressed as the date in January of minimum climate temperature), and the Mean Temperature for the block.  These three parameters have been plotted on a graph whose abscissa covers the years 1772 to 2005.

There are considerable fluctuations in the parameters from 4-year block to 4-year block. I have therefore provided an “eye-guide” to the trends in the form of cubic curves least-squares fitted to each parameter series.  The results are summarised in Figure 3.

Screen Shot 2023-02-26 at 12.57.52 PM

Figure 3: Phase lag (date in January), temperature amplitude, and mean temperature of the 4-year blocks plotted for years 1772 to 2005, together with cubic trend lines fitted to the data.

Discussion of trends in the Central England temperature

First consider the seasonal phase lag.  The lag can vary quite wildly in a short interval.  Nevertheless, there seems to be a secular trend for the coldest period, from around 20 January in 1800 to around 25 January in 2005.

Note that while I have been expressing the phase in terms of the coldest period, the fit to the data is going to be dominated by what happens in spring and autumn when the temperature is changing most rapidly from week to week. Thus while the phase fit implies a climate coldest around 20 January, there could easily be an “unseasonal” warm spell then (as happened in 1773—see Figure 1).

Depth of cold at 20 January 1800 is a 31 day lag (approximately π/6 radians) from 20 December 1799 (shortest day). 200 years later the lag seems to have increased to about 36 days (approximately π/5 radians) by 2004. That is an apparent increase lag of about 5 days (approximately π/30 radians) in 200 years.

If the data can be modelled by Figure 2, then the phase lag trend is consistent with a combination of an increasing “insulation” effect, R, and the storage effect, C.

Can the phase lag be attributed solely to an increase in insulation?

Let us examine the hypothesis that the phase lag is due entirely to increasing insulation, and that the input flux amplitude, J, remained constant.

Between 1772 and about 1870, temperature amplitude between summer and winter had a declining trend from about 7.2 °C to about 6.4 °C. It was more-or-less constant after 1870.  The all-CO2 hypothesis would predict increasing temperature amplitude.  In fact, the decline in the early period cannot be matched assuming constant input flux, J.

What is surprising is that matching the temperature amplitude history requires J to decline between 1772 and 1870.

The contribution of changes in R to the change in seasonal lag is about 30%, while storage, C, must contribute the rest. One can speculate that the increase in C is due to an increase in the amount of ice-free sea during the summer.

Inferring other global warming parameters and sanity check 1870–2005.

It is also possible to estimate the maximum contribution R can make to warming and the maximum sensitivity of the temperature to CO2 (2°C per doubling).  However you will have to look at the mathematical reasoning in the full exposition: “Global Warming as a System Response Theory Problem” .

Robustness of the system response theory approach to limited geographical data

As a teenager I was an electronics hobbyist.  I would test an amplifier, say, by injecting a periodic signal at the input and then sample the response of the system at various points which did not necessarily have to be in the main signal path.  Often one could detect signal, even on the DC supply rails (which is one reason why large capacitors across the supply were needed to prevent feedback oscillations).  I mention this to highlight the fact that monitoring just a single point of a responsive system can provide useful information about the whole system.

Consider one of the pitfalls of the traditional approach which requires one to have good representation of the atmospheric temperature field over the whole globe and over extended periods of time.  Yet the sampling is heavily biassed to densely populated areas, only a few places have time series going back more than a few decades.  Even in those places that have extended time series, there is potential for inconsistencies between measurements taken decades apart (for example the growth of a stand of trees near the measuring station).

By contrast, a time series can yield seasonal phase data which is impervious to the inconsistencies in temperature base line across the decades. Further, even a single location may be representative a wide geographical area with varying terrain.

Concluding remarks

This is proof of concept discussion, using a single location’s dataset, with an unsophisticated analysis.  The inferences, such as maximum possible temperature sensitivity to CO2 are far from rigorous, but indicate some of the power of system response theory to tease out valuable climate physics from even limited observations.

Any serious attempt to model the climate should make use of as many features of the data as possible so as to reduce the danger of undetected hidden assumptions.  The systems response approach, and seasonal lag in particular, is a promising tool in this regard.  I note that some workers are beginning to use this approach in meteorology[1].  It does not seem too much of a leap to extend systems response theory to the global climate system.

Link to full manuscript Global Warming as a System Response Theory Problem

Biosketch:   Dr Barnett received a PhD in Physics in 1995 from the University of Texas at Austin.  His thesis: “Lyapunov Exponents of Many-Body Systems”.  He is presently and independent consultant based in the UK, and an Affiliate of the Institute for Advanced Physics.  His areas of research include:  The emergence of entropy and the arrow of time, The structure of water,  System response theory, Varying ways to idealise useful independent subsystems.

[1]. Milan Palusˇ, Dagmar Novotna ́ & Petr Tichavsky ́, GEOPHYSICAL RESEARCH LETTERS, 32, L12805 (2005) “Shifts of seasons at the European mid-latitudes: Natural fluctuations correlated with the North Atlantic Oscillation

via Climate Etc.


February 27, 2023 at 09:33AM

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