Wavelet analyses of modern global temperature anomalies provides an excellent visualization tool of temperature signal characteristics and patterns over the past 150 years. Scafetta recognized key temperature oscillations of about 9, 20 and 60-years using power spectra of global surface temperature anomalies. There has been much discussion about the 60-year quasi-oscillation both in WUWT and publications.
Detrending the temperature time series and removing the 60-year underlying trend enables insights into the interplay of interannual and decadal scales. Wavelet analyses reveals these periodic signals have distinguished patterns and characteristics that repeat over time suggesting natural external and internal influences. Interannual wavelet patterns that consist of 9-year and 3 to 5-year quasi-oscillations are repeated and dominate over 70% of the instrumental record. The 3 to 5-year discontinuous breakouts are coincident to El Niño and La Niña events of the El Niño-Southern Oscillation (ENSO). A period of quiescence from 1925 to 1960 is devoid of most wavelet signals suggesting different or transitional climate processes.
Wavelet Analysis Method
Applying wavelet analysis on modern climate trends provides a visual tool demonstrating how key climate oscillations vary during the past 150 years. The method in this post uses various Loess high pass filters to initially detrend the HadCRUT4 temperature anomaly time series (download the data here). Detrending removes underlying longer-term trends and isolates shorter-term signals or oscillations for further evaluation. The KNMI Climate Explorer wavelet analysis is then applied to the detrended datasets. Wavelets were also evaluated for ocean, land, tropics northern hemisphere (NH) and southern hemisphere (SH) subsets. Although the results are quite interesting, only the global temperature evaluation is discussed.
Figure 1 illustrates the method used in this post. Figure 1b is the detrended or isolated time series of Figure 1a using a 20-year Loess high pass filter. Figure 1c is the wavelet analysis of the detrended data. This graph shows key wavelet patterns and how they vary over time.
Figure 1: a) HadCRUT4 global temperature time series. Blue line is annualized monthly temperature anomalies. Red line is 20-year Loess average. b) HadCRUT4 monthly (gray line) and annual (blue line) detrended using 20-year loess average. c) Morlet wavelet power spectrum of detrended HadCRUT4 monthly temperature anomalies after 20-year high pass filter. Y-axis is periodic signal, x-axis is time, and color bar is power (degree C squared). Power greater than ~3 are colored. Contours shown for all powers. The black line arc indicates the cone of influence; points outside are influenced by the boundaries of the time series.
The Morlet 6 wavelet analysis is used here which combines both positive and negative peaks into a single broad peakas described by Torrance. The Morlet wavelet is simply a sine wave multiplied by a Gaussian envelope. Wavelet analysis decomposes a timeseries into time and frequency space simultaneously. Information is obtained on both the peak power, or strength, of “periodic” signals within the time series and how this peak varies with time. In contrast, periodograms show the periodic signal and its peak power but not how the signal varies with time. Figure 2 shows both Morlet wavelets and corresponding periodograms for the HadCRUT4 timeseries after a 20 and 40-year loess high pass filter is applied.
Figure 2: a) and b) Morlet wavelet power spectrum for HadCRUT4 global temperature anomalies after a 20 and 40-year high pass Loess filter was applied. X-axis is time in years, y-axis is the periodic signal, and the color bar is power. Contours represent power and power contours greater than ~3 are colored. The black line arc indicates the cone of influence. c) and d) Periodograms of HadCRUT4 global temperature time series after a 20 and 40-year Loess high pass filter. Power is plotted on x-axis and the periodic signal is plotted on y-axis. The blue line denotes the 95% highest spectrum which is a power of 3.
The HadCRUT4 global time series after a 20-year high pass filter shows key periodic signals or peaks between 3-5 years as well as two distinct 9-year peaks. By definition, longer term signals greater than 20 years are filtered out in Figure 2a. By using a 40-year high pass filter that includes events shorter than 40 years, an additional 22-year periodic signal becomes apparent as shown in Figure 2b. Also, applying 60 year and greater high pass filters reveals the 60 to 70-year signal that has been extensively described by Scafetta and is not addressed in this post. This 60 to 70-year signal is starting to become visible as contours on the 40-year high pass filter wavelet plot and periodogram shown in Figures 2b and 2d. Note the 60 to 70-year signal is outside the cone of influence and subjected to edge effects.
The 22-year wavelet signal is dominant early in the timeseries, pre-1950. It becomes weaker in power and gradually dissipates after 1960. It does not appear to be present as a dominant key signal from 1960 to 1990. It is a continuous flat peak with a similar periodic signal between 20 and 22 years when present. Scafetta suggests this event has a solar origin. Because this 22-year signal is present mostly in the earlier HadCRUT4 data, it could also be associated with less reliable, noisier data in the record due to sparse data coverage as described by McLean.
The 9-year Wavelets are Dynamic over Time
The 9-year wavelet peaks are quite interesting, as they display quite a bit of character. Even though periodograms show a dominant 9.3-year peak, this peak changes in both power as well as periodic signal over 150 years. There are two dominant wavelet peaks which are mostly continuous for about 50 years each. The earlier 9-year wavelet peak occurs from about the years 1870 to 1920 and the later from about 1950 to 2000.
The two 9-year wavelet peaks have surprisingly similar curved shapes and transitional ends which are the result of changing to a reduced periodic signal of about 5-years. It is difficult to interpret the start (left edge) of the older 9-year wavelet because it is at the beginning of the data series and outside the cone of influence. The younger 9-year peak tapers and becomes very weak around the year 2000.
These 9-year events do have their own distinct characteristics. The older 9-year wavelet is slightly more discontinuous and appears to have a break around 1900. The younger 9-year wavelet is more continuous over the 50-year duration period.
These 9-year peaks are weak to absent during the years from about 1920 to 1950. As these peaks become weak towards these years, they appear to merge or transition with 5-year wavelet peaks and eventually disappear.
Discontinuous 3 to 5-year Wavelets Correspond to El Niño and La Niña Events
The 3 to 5-year wavelet peaks are discontinuous due to their short duration as shown in Figures 2a and 2b. These peaks occur frequently and correspond with key ENSO El Niño and La Niña episodes. Between 1950 to present, three discontinuous wavelet peaks increase in power about the 1998 El Niño episode, and around the La Niña episodes of 1976 and 1956. From 1870 to 1920, these 3 to 5-year peaks again increase in power about 3 more times. There is a dominant burst in power around 1900 and again about 1878, a well-known El Niño episode.
Importantly, these wavelet bursts are weak to absent from 1920 to 1950 during the same timeframe the 9-year wavelet peaks are absent. Torrence conducted the Morlet wavelet power spectrum on the NINO3 sea surface temperature time series. He also noted a relatively calm period of El Niño activity from 1920 to 1960, with increased El Niño activity shown by higher or larger power during 1880 to 1920 and 1965 to present. This is a relative observation as there have been weak El Niño events during this quiet period such as in 1944, 2010, and in 2016. The year 2016 is beyond the cone of influence and may eventually exhibit a wavelet appearance.
Climate Wavelet Cycles Separated by a Quiet Period Display Similar Patterns
For nomenclature simplicity, climate wavelet observations over the past 150 years have been termed Climate Wavelet Cycle 1 and Climate Wavelet Cycle 2 shown in Figure 3. Wavelet Cycle 1 occurs from 1870 to 1925. Wavelet Cycle 2 occurs from 1950 to 2000. A 25-year quiet period separates the cycles from about 1925 to 1950. Both wavelet cycles contain periodic signals or quasi-oscillations that show comparable wavelet patterns with minor exceptions. In general, the overarching patterns are similar in that both cycles contain a mostly continuous curved 9-year signal and discontinuous 3 to 5-year episodic signals. These wavelet cycles comprised over 70% of the interannual and decadal climate instrumental record.
Figure 3: Morlet wavelet plot of HadCRUT4 global temperature time series after 20-year Loess high filter pass. Same as Figure 2a with added annotation showing interpreted climate wavelet cycles with 9-year and 3 to 5-year wavelet patterns. El Nino events in red, La Nina in blue, both in gray.
The 9-year peaks are separated from the 3 to 5-year peaks for most of the time suggesting a unique process. Scafetta suggests this oscillation has a solar and lunar tidal origin. Wilson calculated a long-term lunar alignment tide index curve in his Figure 14 showing strong 9-year reinforced tidal cycles from 1973 to 2001 and previously from 1890 to 1917.
Another notable feature is the interaction across the interannual 3 to 5-year and 9-year boundaries. This suggests a transition between these events towards the quiet period from 1925 to 1950 when the 9-year wavelet peaks appear to merge with the 3 to 5-year wavelet peaks.
One unique dissimilarity between the two climate cycles is a dimming of the 9-year peak around 1900 in Climate Cycle 1. During this time a broad 5-year signal occurs suggesting prolonged El Niño/La Niña climate conditions. Another contrast between the cycles is the slightly higher occurrence of 5-year wavelet peaks in Climate Wavelet Cycle 1 versus Climate Wavelet Cycle 2. Some of these dissimilarities may also be due to noisier data in the earlier record of the HadCRUT4 data.
The quiescent period from 1925 to 1950 is very distinct. There are no 9-year oscillations and minor to weak 3 to 5-year wavelet peak breakouts. This important period from 1925 to 1950 represents a 25-year discontinuity or transition period where internal or external climate processes changed. Reinforced lunar tide peaks are weak and major volcanic activity is quiet. It was also a time of global climate warming.
Interplay of Ocean Oscillations and Wavelet Patterns
The Atlantic Multidecadal Oscillation (AMO) and Pacific Decadal Oscillation (PDO) changes from warm to cool phases were briefly examined to determine what, if any, relationship existed with the 9-year and 3 to 5-year wavelet patterns over time.
Quite interestingly, the transition of the AMO and PDO to warm and cool phases appear to correspond to terminations of the 9-year oscillations as shown in Figure 4. About 1925 the AMO begins to advance towards a warm phase around the same time the earlier 9-year peak begins to end. The shorter lived PDO began its decline to a cooler phase around 1950 and shortly thereafter the 9-year wavelet reappears. By 1956, the PDO was at its maximum cool phase as La Niña conditions prevailed. And once again, around the year 2000 the AMO is transitioning to a warm phase coincident with the termination of the later 9-year peak. Also, during 1900 as the AMO is descending to a cool phase there is a discontinuity in the 9-year event.
Figure 4: Morlet wavelet plot of HadCRUT4 global temperature time series after 20-year Loess high filter pass. Same as Figure 3 with added annotation showing AMO and PDO phase relationship with 9-year and 3 to 5-year wavelet patterns. El Nino events in red, La Nina in blue, both in gray.
The quiet period of about 25 years has unique natural internal and external processes. Both the AMO and PDO are in a warm phase. As mentioned earlier, this is also a quiet time for both major volcanic activity as well as reinforced lunar tidal peaks. Present day, Earth has been experiencing these same set of conditions since about 2000. The AMO is in a warm phase. There has been no significant major volcanic activity. And strong lunar tidal peaks ended around 2001 with a small peak around 2010. If the past 25-year quiescent period repeats itself, then Climate Wavelet Cycle 3 should begin around 2025 to 2030.
Periodic signals and quasi-oscillations present in temperature time series are not always constant over time. While this is no surprise, wavelet spectrum analysis provides an additional technique to help understand the interaction and transitional nature of these interannual and decadal patterns with time. There is a complex relationship between external and oceanic-atmospheric processes as demonstrated by the wavelet cycles and patterns of 9-year wavelet peaks and 3 to 5-year El Niño/La Niña episodes. Evaluation of detailed seasonal effects and incorporating the dynamic interactions between the ocean, land, NH and SH wavelet patterns may help understand the transitional nature of the climate quasi-oscillations from pole to pole.
Acknowledgements: Special thanks to Andy May and Donald Ince for reviewing and editing this article.
Special thanks to Geert-Jan v. Oldenborgh for delivering and maintaining the instruments used in this article.
Scafetta, N., (2013). Discussion on climate oscillations: CMIP5 general circulation models versus a semi-empirical harmonic model based on astronomical cycles. Earth-Science Reviews 126, 321-357.
Torrence C, Compo GP (1998). A practical guide to wavelet analysis. Bull. Amer. Met. Soc., 79(1): 62-78.
McLean, J., (2018). An Audit of the Creation and Content of the HadCRUT4 Temperature Dataset. Robert Boyle Publishing, 135.
Wilson, I.R., and Sidorenkov, N. S., (2013). Long-Term Lunar Atmospheric Tides in the Southern Hemisphere. The Open Atmospheric Science Journal. 7, 51-76.