The temperature values in Fig. 1 are not the only information about Earth’s past temperature behavior. The Earth has historically been experiencing regular temperature cycles. Table 2 summarizes the past five temperature epochs (Singer 2008 pp 51-59) in recorded history. Data summarized in that book show that many, diverse scientific studies have observed Earth temperatures cycling in about 1,500±500 year intervals. One ice core from the Vostok glacier in Antarctica observed 1,500 year temperature cycles (Lorius 1985) over more than 150,000 years. Other technical investigations like near shore sediment core analysis, studies of coral reefs, stalagmites, tree rings, iron filings, and fossilized pollen found the same 1,500 year temperature cycles (Singer pp 61-69). These studies infer effective or “proxy temperatures” which are not actual measured values but are temperatures changes that explain the variations seen in their data analyses. The cause of these 1,500 year cycles is believed to be solar activity (Singer 2008 pp 18-20, Braun 2005). There are additional results from climatologists and oceanographers supporting the 1,500 year cycles. The Gleissberg and DeVries-Suess cycles have been combined to estimate a 1,470 year cycle (Braun 2005) attributed to the sun. In addition, the Bond cycles (Bond 1997, Obrochta 2014) will be seen to closely correlate with the Cyclical Sine Model.
Table 2
Historical Earth temperature cycles
Epoch
|
Start Date
|
End Date
|
Midpoint Date
|
Unnamed Cold Period
|
750 BC
|
200 BC
|
475 BC
|
Roman Warming
|
200 BC
|
400
|
100
|
Dark Ages Cold
|
440
|
900
|
670
|
Medieval Warming
|
900
|
1300
|
1100
|
Little Ice Age
|
1300
|
1850
|
1575
|
During about the next two centuries the Earth will continue to experience higher temperatures and unusual things will happen. Areas away from the equator will experience increased migration and food production will migrate to those areas also. We will experience many of the changes mankind lived through in a period known as the Medieval Warming. Then, this warm, inviting place called Greenland was discovered by the Vikings who grew crops there to feed their animals (Lamb 1995). After our current warming the Earth will cool and experience a very cold period like The Little Ice Age when the Thames River in London froze solid at least 23 times (Duell 2013) and has not done so again since 1814. Londoners held “Frost Fairs” on the solid ice with even an elephant walking across. The good news is that we now have huge amounts of energy for use during very warm and very cold periods.
A cyclical model should have its repeated period in the range of 1,000-2,000 years to be consistent with the proxy data. The initial strategy for cyclical modeling is to start with an assumed sine wave amplitude for a cyclical temperature model and see where that leads. Cyclical behavior has long been modeled by a simple sine wave function like Eqn. 1 that can be applied for temperature as a function of time.
$$T\left(t\right)=A \text{s}\text{i}\text{n}(\frac{2\pi t}{Ƭ}+\phi )$$
1
To fit Earth temperature as a function of time to this sine wave we need the amplitude A, the period Ƭ, and the phase φ. Amplitude is the height of the sine wave above and below a center line. Period is the length of the wave for a single increase and decrease cycle. Phase is the placement of the entire wave along the time axis. A computerized method called nonlinear regression finds these values by minimizing the differences between data and model. Nonlinear regression programs try various combinations of the variables in an organized way with the goal of making the deviations between data and model as small as possible. Data points can be given different weights depending on various factors. In this work all data points were given equal weights except the final fit where data from 2014-2020 were given increased weights to bring them into better agreement. See Appendix 1 for more details about non-linear regression.
2.1 Single Sine Fit – 1850-2013 Data
We do not know how much average Earth temperature fell and rose from a mean value during the past five historical epochs. A base assumption is that it was the same for each past epoch and the present cycle. A first guess from the NOAA temperature for 2020 of 14.88° C. is that it may rise another 1° C. to a final value of 15.9° C. at the next maximum. The sine wave amplitude A would then be 2° C. to oscillate from a 13.9° C. standard baseline for the 20th century. For this case, the regression values would be only the period Ƭ and phase φ. This first regression used only the temperature data through 2013 to be on a par with the UNIPCC study. The regression results are shown in Table 3 and Fig. 5.
Table 3
Single Sine Cyclical Model Parameters 1850-2013 – Fixed Amplitude
Period, years
|
1,126
|
Amplitude, deg C.
|
2.00
|
Phase, radians
|
177.7
|
Data point weights
|
1
|
Next Max Temperature
|
15.9° C. in 2220
|
With the amplitude fixed at 2° C. every 1,126 years this heating and cooling cycle repeats itself. It is seen that the last five climate change cycles in recorded history agree reasonably well in timing with this model. To the naked eye the fit to measured temperatures also looks reasonable. The fit to the Little Ice Age, Medieval Warming, and the three other epochs in historical documents gives confidence that current Earth behavior agrees with its own history. To include these historic cycles, it was assumed that at the midpoint of each epoch (Table 2) the respective maximum or minimum temperature was reached. The single large data point for each epoch is shown in Fig. 5. All these data together yield the Single Sine Cyclical Model with fixed amplitude shown in Fig. 5 where the next maximum temperature of 15.9° C. is expected about 200 years from now in the year 2220 after a further temperature increase of another 1.38° C. from the 2013 value of 14.52° C. Total increase since 1850 would be about 2.44° C.
If you look closely at the measured temperatures compared to this single sine wave curve fit in Fig. 6, it is apparent that there is another cyclical variant present. The experienced eye of a nonlinear modeler notices that the data in Fig. 6 exhibit an oscillation. This regular trend is a strong indication that even though the fit looks reasonable in Fig. 5, improvements to the model must be made. Another sine wave needs to be added to the primary one. This second sine wave would have been needed even if a constant wave amplitude had not been assumed for this first fit. The oscillation is inherent in the temperature data itself as shown in Fig. 7 and Table 4 where a 66.6 year cycle oscillates about just a linear trendline of the data.
Table 4
Temperature data cyclicity about its trendline 1850-2013
Period, years
|
66.6
|
Amplitude, deg C.
|
.140
|
Phase, radians
|
107.8
|
2.2 Dual Sine Fit 1850-2013 Data
When a second sine wave is added to the first, the nonlinear regression results are shown in Fig. 8. For this fit all parameters for both sine waves were regression variables, the amplitudes, periods, and phases. A constant primary wave amplitude was not assumed. Only temperature data through 2013 were used. Table 5 shows the parameters for this fit where a primary period of 1,100 years was found.
Table 5
Dual Sine Cyclical Model Parameters 1850-2013
Two Sine Model
|
Sine 1
|
Sine 2
|
Period, years
|
1,100
|
68.4
|
Amplitude, deg C.
|
.875
|
.143
|
Phase, radians
|
102.
|
93.6
|
Data point weights
|
1
|
Next Max Temperature
|
14.9° C. in 2210
|
This dual sine model predicts that the Earth’s temperature will increase only about another 0.38° C. which will be achieved in about 200 years in the year 2210 with a maximum temperature then of about 14.9° C. Total temperature increases since 1850 would be about 1.34° C. Note that the amplitude of the first sine wave was found to be 0.875° C., not the 2° C. of the first regression. Also, the period of 1,100 years is shorter though not greatly different from the 1,126 years found by the first fit. Fig. 9 shows that Earth temperature oscillations agree much better with the Dual Sine Cyclical Model than with the Single Sine Model. The second sine wave oscillation has a period of 68 years according to this fit.
2.2.1 Dual Sine Fit 1850-2013 Data – Climatological Comparison
The primary sine period of 1,100 years correlates closely with the Bond Atlantic Drift Ice cyclicity where the initial period was reported at 1,470 years (Bond 1997) and was later reduced (Obrochta 2014) to 1,000 years. The 68 year secondary sine cycle found within the temperature data correlates closely with the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO). Fig. 10 shows this close correlation with the AMO Index (NOAA AMO) which had a 73 year oscillation between 1926 and 1999! Not only does the AMO have a similar period, but it is also in close sync with the increases and decreases of the temperature fluctuations. From Fig. 10 the expectation would be that there would be decreasing AMO index and cooling during the next 20 to 30 years. The PDO Index (NOAA PDO) also has a similar close correlation with the secondary temperature variations as shown in Fig. 11. The PDO period was about 62 years compared to the 68 years found in the secondary temperature oscillation. It also says that in the next 20 or 30 years cooling is expected.
When the Karl data in Fig. 3 from 1880-2016 (Karl 2016) are analyzed in this same way the results in Fig. 12 show that it too contains a temperature cycle of 71 years with further parameters in Table 6.
Table 6
Karl Data Dual Sine Cyclical Model Parameters 1880-2016
Two Sine Model
|
Sine 1
|
Sine 2
|
Period, years
|
1,076
|
71.3
|
Amplitude, deg C.
|
1.357
|
.115
|
Phase, radians
|
101.7
|
101.3
|
Data point weights
|
1
|
Next Max Temperature
|
15.37° C. in 2220
|
In summary, the 1,100 primary cycle period of the Dual Sine Cyclical Model is believed driven by solar activity and closely agrees with Bond Cycles, and the secondary temperature oscillations within the Earth temperature data agree with the AMO and PDO surface temperature oscillations. These conclusions were made possible by including both the five past historical temperature epochs and the measured Earth temperatures from 1850 to 2013. The final model fit next will include all data together from 1850-2020.
2.3 First Dual Sine Fit 1850-2020 Data
When the same analysis is done including the additional temperatures from 2014-2020, the nonlinear regression results are shown in Figs. 13 and 14. For this fit all parameters for both sine waves were again regression variables, the amplitudes, periods, and phases. Each temperature value was equally weighted. It is seen that the 2014-2020 data do not fit very well. There is a different characteristic in these data as already discussed about Fig. 2. Table 7 shows the resulting parameters for this fit.
Table 7
Dual Sine Cyclical Model Parameters 1850-2020 Equal weights
Two Sine Model
|
Sine 1
|
Sine 2
|
Period, years
|
1,050
|
69.6
|
Amplitude, deg C.
|
.936
|
.162
|
Phase, radians
|
101.5
|
96.53
|
Data point weights
|
1
|
Next Max Temperature
|
14.82° C. in 2200
|
2.3.1 Final Dual Sine Fit 1850-2020 Data
To force the 2014-2020 data into this model, the weights for those years were increased to five compared to one for the rest of the data points. Weights from 2 to 10 were evaluated with the 5 weights judged to be the best compromise with deviations for the 2014-2020 data greatly reduced while deviations for all other data not greatly increased. Figs. 15 and 16 and Table 8 show the final best fit for all temperature data. This fit is the best one can do to include the unusual increase in reported temperatures since 2013. If more consistent temperature values are ultimately reported, a final fit may not need to give the 2014-2020 data increased weight.
Table 8
Dual Sine Cyclical Model Parameters 1850-2020 weights=5 for 2014-2020
Two Sine Model
|
Sine 1
|
Sine 2
|
Period, years
|
1,060
|
76.1
|
Amplitude, deg C.
|
1.270
|
.243
|
Phase, radians
|
101.6
|
73.69
|
1850-2013 weights
|
1
|
2014-2020 weights
|
5
|
Next Max Temperature
|
15.4° C. in 2170
|
2.3.2 Final Dual Sine Fit 1850-2020 Data – Climatological Comparison
The correlation for this final model fit is virtually the same as that already seen in Figs. 10 and 11. The primary sine period of 1,060 years correlates closely with the Bond Atlantic Drift Ice cyclicity where the initial period was reported at 1,470 years (Bond 1997) and was later reduced (Obrochta 2014) to 1,000 years. The 76 year secondary sine cycle found within the temperature data for this final fit correlates closely with the Atlantic Multidecadal Oscillation (AMO) and the Pacific Decadal Oscillation (PDO). Fig. 17 shows this close correlation with the AMO (NOAA AMO) which had a 73 year oscillation between 1926 and 1999! Not only does the AMO have essentially the same period, but it is also in close sync with the increases and decreases of the temperature fluctuations. From Fig. 17 the expectation would be that there would be decreasing AMO index and cooling during the next 20 to 30 years. The PDO Index (NOAA PDO) also has a reasonable correlation with the secondary temperature variations as shown in Fig. 18. The PDO period was about 62 years compared to the 76 years found in the secondary temperature oscillation. It also says that in the next 20 or 30 years cooling is expected.
In summary, the 1,060 primary cycle period of the Final Dual Sine Cyclical Model is believed driven by solar activity and closely agrees with Bond Cycles, and the secondary temperature oscillations within the Earth temperature data agree with the AMO and PDO surface temperature oscillations. These conclusions were made possible by including both the five past historical temperature epochs and the measured Earth temperatures from 1850 to 2020.