**Guest post by Dan Pangburn**

**Introduction**

This monograph is a clarification and further refinement of Reference 10 (references are listed at the end of this paper) which also considers only average global temperature. It does not discuss weather, which is a complex study of energy moving about the planet. It does not even address local climate, which includes precipitation. It does, however, consider the issue of Global Warming and the mistaken perception that human activity has a significant influence on it.

The word ‘trend’ is used here for temperatures in two different contexts. To differentiate, Î±-trend applies to averaging-out the uncertainties in reported average global temperature measurements to produce the average global temperature oscillation resulting from the net ocean surface oscillation. The term Î²-trend applies to the slower average temperature change of the planet which is associated with change to the temperature of the bulk volume of the material (mostly water) involved.

The first paper to suggest the hypothesis that the sunspot number time-integral is a proxy for a substantial driver of average global temperature change was made public 6/1/2009. The discovery started with application of the first law of thermodynamics, conservation of energy, and the hypothesis that the energy acquired, above or below breakeven (appropriately accounting for energy radiated from the planet), is proportional to the time-integral of sunspot numbers. The derived equation revealed a rapid and sustained global energy rise starting in about 1941. The true average global temperature anomaly change Î²-trend is proportional to global energy change.

Measured temperature anomaly Î±-trends oscillate above and below the temperature anomaly Î²-trend calculated using only the sunspot number time-integral. The existence of ocean oscillations, especially the Pacific Decadal Oscillation, led to the perception that there must be an effective net surface temperature oscillation with all named and unnamed ocean oscillations as participants. Plots of measured average global temperatures indicate that the net surface temperature oscillation has a period of 64 years with the most recent maximum in 2005.

Combination of the effects results in the effect of the ocean surface temperature oscillation (Î±-trend) decline 1941-1973 being slightly stronger than the effect of the rapid rise from sunspots (Î²-trend) resulting in a slight decline of the trend of reported average global temperatures. The steep rise 1973-2005 occurred because the effects added. A high coefficient of determination, R^{2}, demonstrates that the hypothesis is true.

Over the years, several refinements to this work (often resulting from other's comments which may or may not have been agreeable) slightly improved the accuracy and led to the equations and figures in this paper.

**Prior work**

The law of conservation of energy is applied effectively the same as described in Reference 2 in the development of a very similar equation that calculates temperature anomalies. The difference is that the variation in energy ‘OUT’ has been found to be adequately accounted for by variation of the sunspot numbers. Thus the influence of the factor [T(i)/Tavg]^{4} is eliminated.

Change to the level of atmospheric carbon dioxide has no significant effect on average global temperature. This was demonstrated in 2008 at Reference 6 and is corroborated at Reference 2 and again here.

As determined in Reference 3, reported average global temperature anomaly measurements have a random uncertainty with equivalent standard deviation ≈ 0.09 K.

Global Warming ended more than a decade ago as shown here, and in Reference 4 and also Reference 2.

Average global temperature is very sensitive to cloud change as shown in Reference 5.

The parameter for average sunspot number was 43.97 (average 1850-1940) in Ref. 1, 42 (average 1895-1940) in Ref. 9, and 40 (average 1610-2012) in Ref. 10. It is set at 34 (average 1610-1940) in this paper. The procession of values for average sunspot number produces slight but steady improvement in R^{2} for the period of measured temperatures and progressively greater credibility of average global temperature estimates for the period prior to direct measurements becoming available.

**The sunspot number time-integral drives the temperature anomaly trend**

It is axiomatic that change to the average temperature trend of the planet is due to change to the net energy retained by the planet.

Table 1 in reference 2 shows the influence of atmospheric carbon dioxide (CO_{2}) to be insignificant (tiny change in R^{2} if considering CO_{2} or not) so it can be removed from the equation by setting coefficient ‘C’ to zero. With ‘C’ set to zero, Equation 1 in Reference 2 calculates average global temperature anomalies (AGT) since 1895 with 89.82% accuracy (R^{2}= 0.898220).

The current analysis determined that 34, the approximate average of sunspot numbers from 1610-1940, provides a slightly better fit to the measured temperature data than did 43.97 and other values^{ 9,10}. The influence, of Stephan-Boltzmann radiation change due to AGT change, on energy change is adequately accounted for by the sunspot number time-integral. With these refinements to Equation (1) in Reference 2 the coefficients become A = 0.3588, B = 0.003461 and D = ‑ 0.4485. R^{2} increases slightly to 0.904906 and the calculated anomaly in 2005 is 0.5045 K. Also with these refinements the equation calculates lower early anomalies and projects slightly higher (0.3175 vs. 0.269 in 2020) future anomalies. Measured anomalies are shown in Figure 2 of Reference 3. The excellent match of the up and down trends since before 1900 of calculated and measured anomalies, shown here in Figure 1, demonstrates the usefulness and validity of the calculations.

Projections until 2020 use the expected sunspot number trend for the remainder of solar cycle 24 as provided^{ 11} by NASA. After 2020 the limiting cases are either assuming sunspots like from 1925 to 1941 or for the case of no sunspots which is similar to the Maunder Minimum.

Some noteworthy volcanoes and the year they occurred are also shown on Figure 1. No consistent AGT response is observed to be associated with these. Any global temperature perturbation that might have been caused by volcanoes of this size is lost in the temperature measurement uncertainty. Much larger volcanoes can cause significant temporary global cooling from the added reflectivity of aerosols and airborne particulates. The Tambora eruption, which started on April 10, 1815 and continued to erupt for at least 6 months, was approximately ten times the magnitude of the next largest in recorded history and led to 1816 which has been referred to as ‘the year without a summer’. The cooling effect of that volcano exacerbated the already cool temperatures associated with the Dalton Minimum.

**Figure 1: Measured average global temperature anomalies with calculated prior and future trends using 34 as the average daily sunspot number.**

As discussed in Reference 2, ocean oscillations produce oscillations of the ocean surface temperature with no significant change to the average temperature of the bulk volume of water involved. The effect on AGT of the full range of surface temperature oscillation is given by the coefficient ‘A’.

The influence of ocean surface temperature oscillations can be removed from the equation by setting ‘A’ to zero. To use all regularly recorded sunspot numbers, the integration starts in 1610. The offset, ‘D’ must be changed to -0.1993 to account for the different integration start point and setting ‘A’ to zero. Setting ‘A’ to zero requires that the anomaly in 2005 be 0.5045 - 0.3588/2 = 0.3251 K. The result, Equation (1) here, then calculates the trend 1610-2012 resulting from just the sunspot number time-integral.

Trend3anom(y) = 0.003461/17 * Î£^{y}_{i = 1610} [s(i)-34] – 0.1993 (1)

Where:

Trend3anom(y) = calculated temperature anomaly Î²-trend in year y, K degrees.

0.003461 = the proxy factor, B, W yr m^{-2}.

17 = effective thermal capacitance of the planet, W Yr m^{-2} K^{-1}

s(i) = average daily Brussels International sunspot number in year i

34 ≈ average sunspot number for 1610-1940.

-0.1993 is merely an offset that shifts the calculated trajectory vertically on the graph, without changing its shape, so that the calculated temperature anomaly in 2005 is 0.3251 K which is the calculated anomaly for 2005 if the ocean oscillation is not included.

Sunspot numbers back to 1610 are shown in Figure 2 of Reference 1.

Applying Equation (1) to the sunspot numbers of Figure 2 of Reference 1 produces the trace shown in Figure 2 below.

**Figure 2: Anomaly trend from just the sunspot number time-integral using Equation (1).**

Average global temperatures were not directly measured in 1610 (thermometers had not been invented yet). Recent estimates, using proxies, are few. The anomaly trend that Equation (1) calculates for that time is roughly consistent with other estimates. The decline in the trace 1610-1700 on Figure 2 results from the low sunspot numbers for that period as shown on Figure 2 of Reference 1.

**How this phenomenon could take place**

Although the connection between AGT and the sunspot number time-integral is demonstrated, the mechanism by which this takes place remains somewhat theoretical.

Various papers have been written that indicate how the solar magnetic field associated with sunspots can influence climate on earth. These papers posit that decreased sunspots are associated with decreased solar magnetic field which decreases the deflection of and therefore increases the flow of galactic cosmic rays on earth.

Henrik Svensmark, a Danish physicist, found that decreased galactic cosmic rays caused decreased low level (< 3 km) clouds and planet warming. An abstract of his 2000 paper is at Reference 13. Marsden and Lingenfelter also report this in the summary of their 2003 paper^{ 14} where they make the statement “…solar activity increases…providing more shielding…less low-level cloud cover… increase surface air temperature.” These findings have been further corroborated by the cloud nucleation experiments^{ 15} at CERN.

These papers associated the increased low-level clouds with increased albedo leading to lower temperatures. Increased low clouds would also result in lower average cloud altitude and therefore higher average cloud temperature. Although clouds are commonly acknowledged to increase albedo, they also radiate energy to space so increasing their temperature increases radiation to space which would cause the planet to cool. Increased albedo reduces the energy received by the planet and increased radiation to space reduces the energy of the planet. Thus the two effects work together to change the AGT of the planet.

Simple analyses^{ 5} indicate that either an increase of approximately 186 meters in average cloud altitude or a decrease of average albedo from 0.3 to the very slightly reduced value of 0.2928 would account for all of the 20^{th} century increase in AGT of 0.74 °C. Because the cloud effects work together and part of the temperature change is due to ocean oscillation substantially less cloud change is needed.

**Combined Sunspot Effect and Ocean Oscillation Effect**

As a possibility, the period and amplitude of oscillations attributed to ocean cycles demonstrated to be valid after 1895 are assumed to maintain back to 1610. Equation (1) is modified as shown in Equation (2) to account for including the effects of ocean oscillations. Since the expression for the oscillations calculates values from zero to the full range but oscillations must be centered on zero, it must be reduced by half the oscillation range.

Trend4anom(y) = (0.3588,y) – 0.1794 + 0.003461/17 * Î£^{y}_{i = 1610} [s(i)-34] – 0.1993 (2)

The ocean oscillation factor, (0.3588,y) – 0.1794, is applied prior to the start of temperature measurements as a possibility. The effective sea surface temperature anomaly, (A,y), is defined in Reference 2.

Applying Equation (2) to the sunspot numbers from Figure 2 of Reference 1 produces the trend shown in Figure 3 next below. Available measured average global temperatures from Figure 2 in Reference 3 are superimposed on the calculated values.

**Figure 3: Calculated anomalies from the sunspot number time-integral plus ocean oscillation using Equation (2) with superimposed available measured data from Reference 3 and range estimates determined by Loehle.**

Figure 3 shows that temperature anomalies calculated using Equation (2) estimate possible trends since 1610 and actual trends of reported temperatures since they have been accurately measured world wide. The match from 1895 on has R^{2} = 0.9049 which means that 90.49% of average global temperature anomaly measurements are explained. All factors not explicitly considered must find room in that unexplained 9.51%. Note that a coefficient of determination, R^{2} = 0.9049 means a correlation coefficient of 0.95.

A survey^{ 12} of non-tree-ring global temperature estimates was conducted by Loehle including some for a period after 1610. A simplification of the 95% limits found by Loehle are also shown on Figure 3. The spread between the upper and lower 95% limits are fixed, but, since the anomaly reference temperatures might be different, the limits are adjusted vertically to approximately bracket the values calculated using the equations. The fit appears reasonable considering the uncertainty in all values.

Calculated anomalies look reasonable back to 1700 but indicate higher temps prior to that than most proxy estimates. They are, however, consistent with the low sunspot numbers in that period. They qualitatively agree with Vostok, Antarctica ice core data but decidedly differ from Sargasso Sea estimates during that time (see the graph for the last 1000 years in Reference 6). Credible worldwide assessments of average global temperature that far back are sparse. Ocean oscillations might also have been different from assumed.

**Possible lower values for average sunspot number**

Possible lower assumed values for average sunspot number, with coefficients adjusted to maximize R^{2}, result in noticeably lower estimates of early (prior to direct measurement) temperatures with only a tiny decrease in R^{2}. Calculated anomalies resulting from using an average sunspot number value of 26 are shown in Figure 4. The projected anomaly trend decline is slightly less steep (0.018 K warmer in 2020) than was shown in Figure 1.

**Figure 4: Calculated anomalies from the sunspot number time-integral plus ocean oscillation using 26 as the average sunspot number with superimposed available measured data from Reference 3 and range estimates determined by Loehle.**

**Carbon dioxide change has no significant influence**

The influence that CO_{2} has on AGT can be calculated by including ‘C’ in Equation (1) of Reference 2 as a coefficient to be determined. The tiny increase in R^{2} demonstrates that **consideration of change to the CO**_{2} level has no significant influence on AGT [average global temperature]. The coefficients and resulting R^{2} are given in Table

1.**Conclusions**

**Others that have looked at only amplitude or only duration factors for solar cycles got poor correlations with average global temperature. The good correlation comes by combining the two, which is what the time-integral of sunspot numbers does.** As shown in Figure 2, the anomaly trend determined using the sunspot number time-integral has experienced substantial change over the recorded period. Prediction of future sunspot numbers more than a decade or so into the future has not yet been confidently done although assessments using planetary synodic periods appear to be relevant^{ 7,8}.

**As displayed in Figure 2, the time-integral of sunspot numbers alone appears to show the estimated true average global temperature trend (the net average global energy trend) during the planet warm up from the depths of the Little Ice Age.**

**The net effect of ocean oscillations is to cause the surface temperature trend to oscillate above and below the trend calculated using only the sunspot number time-integral. Equation (2) accounts for both and also, because it matches measurements so well, shows that rational change to the level of atmospheric carbon dioxide can have no significant influence.**

**Long term prediction of average global temperatures depends primarily on long term prediction of sunspot numbers.**

**References:**