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The Spectrum of Average Global Temperatures

The Spectral Analysis of the Average Global Temperature

The record of global average temperatures shown below displays some interesting characteristic. One of these is the hint of a cycle.

In the above display the global average temperature appears to be fluctuating about a rising linear trend until about 1875. From 1875 to 1910 there is a declining trend and then from 1910 to 1945 there is a rising trend. From 1945 the average global temperature declined until about 1975. After 1975 the trend is upward until about 2000 and, although it is not shown in the graph, after that the trend has been downward. There is about a thirty year period for these trend episodes. This period from a minimum to a maximum or from a maximum to a minimum would be one half of a full cycle period. Thus the data indicate a full cycle period of about sixty years.

Spectral analysis is the method for identifying a cyclical period. The spectral analysis of the data from 1855 to 2003 is as shown. For additional information concerning the dynamics of temperature and spectral analysis see Spectrum of Global Temperature. Theory indicates that the spectrum should correspond to the sinc function; i.e., sinc(ω)=sin(ω)/ω.

The ratio of the peak heights for the peak at zero frequency and the next peak is about 1/5, more precisely 0.1936. The ratio for the sinc function is 2/(3π)=0.2122; not too bad of a correspondence.

The spectrum displays the sequence of peaks found previously from pure theory. The spectrum over a wider range of frequencies is shown below

The phenomenon of the peak at zero and the high values near zero is a manifestation of the fact that the average value is not zero. When the mean value is computed and subtracted from each datum the spectrum for the result is shown below.

As stated previously the period from a minimum to a maximum or from a maximum to a minimum is one half of a full cycle period. In the above spectrum the peak for a full cycle period at about 200 years corresponds to trend period of about 100 years. The data were not detrended so this peak may just represent the secular trend. The spectrum shows a peak for a full cycle period of 49 years, but this probably represents a cycle that has a period between 49 and about 55 years; roughly 52 years. This would correspond to a trend period of 26 years.

An extensive study of the spectra of time series relating to global temperature was carried out by Leonid B. Klyashtorin for the Russian Federation's Institute of Fisheries and Oceanography (FAO) in Moscow. The results were published in the FAO Fisheries Technical Paper No. 410, entitled "Climate change and long-term fluctuations of commercial catches: the possibility of forecasting." It was presented in Rome in 2001. Klyashtorin's study found a spectral peak for the global temperature anomalies for 1861 to 1998 for 55 years. This corresponds to a trend period of 27.5 years. For the spectral analysis of ice core temperatures over the 1421 year period from 552 to 1973 the dominant peak was for 54 years, corresponding to a trend period of 27 years. His analysis of the Atmospheric Circulation Index (ACI) which covered the Atlantic Eurasian region the dominant peak was for 50 years. He also analyzed indices for fluctuations in the anchovy mass and for tree ring temperature. For these the dominant peaks were 110 and 108, respectively, with secondary peaks at 51 and 56 years, respectively. Klyashtorin concluded that there is a climate cycle with average period of about 56 years. This corrsponds to a trend period of 28 years. This result came out of the analysis of long term series (c. 1500 years) as well as a shorter term series of about 150 years.

The full cycle period for the sunspot cycle is 22 years. This is much too short to be the source of the 25 to 30 half cycle period noted above.

Long Term Trend versus Short Term Cycles

The apparent linear trends in average global temperature lasting 25 to 30 years correspond to natural cycles in the Earth's climate system. Any forecasting of future average global temperatures must ignore these relatively short term trends. Roughly this would mean looking at the temperature at the midpoints of the trend periods. The rise in the temperature since 1990 was largely part of a natural cycle that is being reversed since 1998.

The graph below shows the average data points for the trend episodes. In the graph below these trend episode averages are repesented by the large red circles.

The slope of a line draw between the first and last averages is about 0.7°C per century. This is essentially the same as the growth rate determined by other methods.

Regression Estimates of the Long Term
Trend versus Short Term Cycles

The graph of the data indicates that the cycle is of a sawtooth form; i.e., piecewise linear. When regression analysis is applied the result is the following.

The coefficient of determination for this regression is 0.79.

In this regression the slopes of the trend lines in the different episodes are unconstrained. However the appearance of near equal slopes for the different upswings and for the different downswings is remarkable.

When the regression coefficients are constrained to give the same slopes on the upswings and to give the same slopes on the downswings the result is as shown below.

A continuation of the long term trend and the cycle of short term cycles means that the global temperature will be entering a downswing episode in the near future. Although spectral analysis gives trend period of about 26 years an inspection of the times between apparent turning points gives a period of about 32 years. A 26 year period would mean a downturn commencing in 2000 whereas a 32 year period would imply that the downturn commenced in 2006. Such a downturn would last until 2026 or 2038 depending upon whether the trend period is 26 or 32 years. During the downturn the average global temperature will decline by about 0.3°C.

(To be continued.)

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