Sunday, December 15, 2013

The insignificant effect of CO2 on climate is quantified at  with a near-perfect explanation of annual average global temperatures since before 1900; R^2 = 0.97+

Calculated Mean Global Temperatures 1610-2012

This monograph considers only average global temperature (AGT). 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 tenaciously held but 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, α-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 β-trend applies to the slower average temperature change of the planet which is associated with change to the average 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 break-even (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 β-trend is proportional to global energy change.

Subsequent analysis revealed that the significant factor in calculating the β-trend is the sunspot number anomaly time-integral. The sunspot number anomaly is defined as the difference between the sunspot number in a specific year and an average sunspot number for many years.

Measured surface temperature anomaly α-trends oscillate above and below the temperature anomaly β-trend calculated using only the sunspot number anomaly time-integral.

The existence of multiple ocean oscillations led to the perception that there must be an effective net surface temperature oscillation for the planet 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 (α-trend) decline 1941-1973 being slightly stronger than the effect of the rapid rise from sunspots (β-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, R2, 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 corroborative) 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 which 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 number anomalies. 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. A substantial contributor to this variation appears to be the apparently random variation in magnitude and period of el Nino.

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. 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 20th century increase in AGT of 0.74 °C.

The value used for average sunspot number was 43.97 (average 1850-1940) in Ref. 1. It is set at 34 (average 1610-1940) in this paper. The procession of values for average sunspot number from 43.97 to 34 produces slight but steady improvement in R2 for the period of measured temperatures, and, progressively greater credibility of average global temperature estimates for the period prior to accurate, world wide, direct measurements becoming available.

Initial work is presented in several papers made public at

Ocean surface temperature oscillation
The ocean oscillation does not significantly add or remove planet energy. In the decades immediately prior to 1941 the amplitude range of the trends was not significantly influenced by any candidate internal forcing effect; so the observed amplitude of the effect on AGT of the net ocean surface temperature trend anomaly then, must be approximately the same as the amplitude of the part of the AGT trend anomaly due to ocean oscillations since then. This part is approximately 0.36 K total range with a period of approximately 64 years (verified below).

The AGT trajectory (Figure 1) suggests that the least-biased simple wave form of the ocean surface temperature oscillation is approximately saw-toothed. Ignoring the offset for the moment, the sea surface temperature anomaly oscillation can be described as varying linearly from 0.0 K in 1909 to approximately 0.36 K in 1941 and linearly back to the 1909 value in 1973. This cycle repeats before and after with a period of 64 years. This is consistent with the 50-70 year period previously observed by others.

Because the actual magnitude of the ocean oscillation in any year is needed, the expression to account for the contribution of the ocean oscillation to AGT is given by the following:

ΔTosc = (A,y)             K (degrees)                 (i)

where the contribution of the net of ocean oscillations to AGT change is the magnitude of the effect on AGT of the surface temperature anomaly trend of the oscillation in year y, and A is the maximum magnitude of the effect on AGT of the ocean surface temperature trend oscillation. 

Equation (i) is graphed in Figure 0.5.
Figure 0.5: Ocean surface temperature oscillations do not affect the bulk energy of the planet.

Although the peak-to-peak amplitude of the effect on AGT of the ocean oscillation is approximately 0.36 K, by definition, ocean oscillation is symmetrical with respect to zero. Therefore the contribution of Sea Surface Temperature (SST) trend oscillation to AGT since before 1900 has been approximately ±0.18 K.

The sunspot number anomaly time-integral drives the temperature anomaly β-trend
By definition, energy change divided by effective thermal capacitance is temperature change.

Table 1 in reference 2 shows the influence of atmospheric carbon dioxide (CO2) to be insignificant (tiny change in R2 if considering CO2 or not; also shown later in this paper to be insignificant) 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 (R2 = 0.898220).

The current analysis determined that 34, the approximate average of sunspot numbers from 1610-1940, provides a slightly better fit (in fact, the best fit) to the measured temperature data. The influence, of Stephan-Boltzmann radiation change due to AGT change, on energy change is adequately accounted for by the sunspot number anomaly time-integral. With these refinements to Equation (1) in Reference 2 the coefficients become A = 0.3588, B = 0.003461 and D = ‑ 0.4485.  R2 increases slightly to 0.904906 and the calculated anomaly in 2005 is 0.5045 K. Also with these refinements the equation calculates lower early temperature anomalies and projects a slightly higher (0.3175 K vs. 0.269 K in 2020) future anomaly trend. The resulting equation for calculating the AGT anomaly for any year, 1895 or later, is then:

Anom(y) = (0.3588,y) + 0.003461/17 Σyi=1895 (s(i) – 34) – 0.4485    (ii)

            Anom(y) = calculated temperature anomaly in year y, K
            (0.3588,y) = approximate contribution of ocean cycle effect to AGT in year y
            s(i) = average daily Brussels International sunspot number in year i
            17 = effective thermal capacitance, W yr m-2 K-1

Measured temperature anomalies are from Figure 2 of Reference 3. The excellent match of the up and down trends since before 1900 of calculated and measured temperature anomalies, shown here in Figure 1, and, for 5-year moving average temperature anomaly measurements, in Figure 1.1, demonstrate the usefulness and validity of the calculations. Figure 1.2 uses the same data as Figure 1 (which is prior to the corrupting ‘changes’ by the reporting agencies) with reported SSN and ‘best guess’ estimate of AGT anomalies for the years 2013-2015.

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. Note: The uncertainty is not in the method, or the measuring instruments themselves, but results from the effectively roiling (at this scale) of the object of the measurements.

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. (Last update 12/15/13)

Figure 1.1: Same as Figure 1 but with 5-year running average of measured temperatures. R2 = 0.973. (Added 5/26/15)

Figure 1.2: Same as Figure 1 but with 5-year running average of measured temperatures and extended through 2015. R2 = 0.973. (Added 1/6/16)

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 reported AGT of the full range of surface temperature oscillation is given by the coefficient ‘A’. (A, B, C, and D are the coefficients in Equation 1 of Reference 2)

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 anomaly time-integral.

Trend3anom(y) = 0.003461/17 * Σyi = 1610 [s(i)-34] – 0.1993                (1)

Trend3anom(y) = calculated temperature anomaly β-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 (β-trend) from just the sunspot number anomaly 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 temperature 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 anomaly time-integral is demonstrated, the mechanism by which this takes place remains somewhat speculative.

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 20th 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 would suffice.

Hind Cast Estimate of 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 * Σyi = 1610 [s(i)-34] – 0.1993   (2)

The ocean oscillation factor, (0.3588,y) – 0.1794, is applied to the period prior to the start of direct temperature measurements as a possibility. The effective sea surface temperature anomaly, (A,y), is defined above and 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 temperature anomalies from the sunspot number anomaly 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 1895-2012 has R2 = 0.9049 which means that 90.49% of average global temperature anomaly measurements are explained. All factors not explicitly considered (such as the 0.09 K s.d. random uncertainty in reported annual measured temperature anomalies, aerosols, CO2, other non-condensing ghg, volcanoes, ice change, etc.) must find room in that unexplained 9.51%. Note that a coefficient of determination, R2 = 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.

Smoothing of the measured temperatures using 5-year moving average, as shown in Figure 1.1, achieved R2 = 0.973. This accounts for most of the random uncertainty in reported annual measured temperature anomalies with only 2.7% left unexplained.

Calculated temperature anomalies look reasonable back to 1700 but indicate higher temperatures 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). Worldwide assessments of average global temperature that far back are sparse and speculative. 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 R2, result in noticeably lower estimates of early (prior to direct measurement) temperatures with only a tiny decrease in R2. Calculated temperature anomalies resulting from using an average sunspot number value of 26 are shown in Figure 4. The projected temperature anomaly trend decline is very slightly less steep (0.018 K warmer in 2020) than was shown in Figure 1.

Figure 4: Calculated temperature anomalies from the sunspot number anomaly 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 CO2 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 R2 demonstrates that consideration of change to the CO2 level has no significant influence on AGT. The coefficients and resulting R2 are given in Table 1.

Table 1: A, B, C, D, refer to coefficients in Equation 1 in Reference 2
Average daily SSN
ocean oscillation A
sunspots B
Coefficient of determination R2
% cause of 1909-2005 AGT change (0.9465 K)
Ocean oscillation
CO2 change
*Measured temperatures smoothed using 5-year moving average. (Added 5/26/15, rev 12/13/15)
**Equation calibrated through 1990
***Svalgaard 5 5 16 SSN, Average daily SSN set to 63

††Extended through 2015

The coefficient ‘B’ is effectively a combination of proxy factor and influence coefficient.

Assessment using Svalgaard SSN (rev 5/6/16)
Figure 5: SSN used in assessment results in Table 1 for Svalgaard 5 5 16. R2 = 0.968900 (1895-2012). (Pre 1700 SSN are Brussels International SSN/0.6).
Figure 6: Anomaly trend (β-trend) from just the sunspot number anomaly time-integral using SSN from Figure 5 (Svalgaard)

Possible explanation of why CO2 change has no significant effect on climate. (Added 10/2/14, revised 11/21/14, 11/29/14, 3/15/15)
1) Firmly acknowledge the established fact that gas molecules can absorb/emit photons only at specific discreet wavelengths (which might be broadened from pressure, etc.). This fact makes spectroscopy possible. Full spectrum (Plank’s law) Stephan-Boltzmann (S-B) radiation applies to liquids and solids, not to gases.
2) From gas kinetics, the time between atmospheric molecule collisions is extremely short (The Hyperphysics calculator calculates approximately 0.0001 microsecond at sea level pressure and temperature).
3) The elapsed time between absorption and emission of a photon by a CO2 gas molecule is perhaps shorter at higher temperature but must be greater than zero or there would be no evidence that absorption-emission had occurred.
4) At sea level conditions, some or all of the photon energy that is absorbed by a ghg molecule is immediately transferred to other molecules by collision. The process of absorbing a photon and transferring (thermal conduction in the gas) the added energy to other molecules is thermalization. A common observation of thermalization by way of water vapor is that cloudless nights cool faster when absolute water vapor content is lower.
5) The reduced radiation flux on both sides of the 15 micron CO2 absorption line, as observed in most Top Of Atmosphere (TOA) measurements 18 results because some of the EMR energy absorbed by CO2 has been thermalized.
6) Terrestrial radiation is nearly all in the wavelength range 6-100 microns. Thermalized energy carries no identity of the molecule that absorbed it.
7) Jostling between the molecules sometimes causes reverse-thermalization. At low to medium altitudes, EMR emission stimulated by reverse-thermalization is mostly by way of water vapor. The TOA spike at 15 microns results from reverse-thermalization to CO2  molecules at very high altitude.
8) The thermalized radiation warms the air, reducing its density, causing updrafts which are exploited by soaring birds, sailplanes, and occasionally hail. Updrafts are matched by downdrafts elsewhere, usually spread out but sometimes recognized by pilots and passengers as ‘air pockets’ and micro bursts.
9) The population gradient of ghg molecules, (especially water vapor above about 3 km, declining with increasing altitude) favors radiation to space. Ghg molecules that emit a photon are ‘recharged’ by reverse-thermalization. (or by absorbing a photon of appropriate wavelength)
10) Clouds (average emissivity about 0.5) consist of solid and/or liquid water particles (each particle containing millions of molecules) that radiate according to S-B law. Low amount of water vapor above clouds and widening molecule spacing allows substantial radiation directly to space.
11) The tiny increase in ghg from increased CO2 causes absorption/thermalization to occur at slightly lower altitude which very slightly increases the convection rate.

12) The increase in absorbing molecules near the surface is apparently compensated for by an equal increase in emitting molecules high in the atmosphere radiating energy from the planet.

Further discussion of ocean cycles (Added 6/23/14)
Ocean cycles are perceived to contribute to AGT in two ways: The first is the direct measurement of sea surface temperature (SST). The second is warmer SST increases atmospheric water vapor which acts as a forcing and therefore has a time-integral effect on temperature

The combined effect of temperature contribution to AGT of ocean cycles is approximated by a function that has a saw-tooth trajectory profile. It is represented earlier in this paper and in Equation (1) of Reference 2 by (A,y) where A is the total amplitude and y is the year. The uptrends and down trends are each determined to be 32 years long for a total period of 64 years. The total amplitude resulting from ocean oscillations was found here to be 0.3588 K for unsmoothed and 0.3482 for smoothed (cases highlighted in Table 1).

Thus, for an ocean cycle surface temperature uptrend, the contribution of ocean oscillations to AGT is approximated by adding (to the value calculated from the sunspot number anomaly time-integral) 0.3588 multiplied by the fraction of the 32 year period that has elapsed since a low. For an ocean cycle surface temperature down trend, the contribution is calculated by adding 0.3588 minus 0.3588 multiplied by the fraction of the 32 year period that has elapsed since a high. The calculated lows were found to be in 1909 and 1973 and the highs in 1941 and 2005. The resulting trajectory is shown in Figure 0.5, and, offset by half the amplitude, is shown as ‘approximation’ in Figure 5.

Temperature data is available for three named cycles: PDO index, ENSO 3.4 index and AMO index. Successful accounting for oscillations is achieved for PDO and ENSO when considering these as forcings (with appropriate proxy factors) instead of direct measurements. As forcings, their influence accumulates with time. The proxy factors must be determined separately for each forcing. The measurements are available since 1900 for PDO 16 and ENSO3.4 17. This PDO data set has the PDO temperature measurements reduced by the average SST measurements for the planet.

The contribution of PDO and ENSO3.4 to AGT is calculated by:
PDO_NINO = Σyi=1900 (0.017*PDO(i) + 0.009 * ENSO34(i))        (3)

            PDO(i) = PDO index 16 in year i
            ENSO34(i) = ENSO 3.4 index 17 in year i

How this calculation compares to the idealized approximation used in Equation (2) is shown in Figure 5. The high coefficient of determination in Table 1 and the comparison in Figure 5 corroborate the assumption that the saw-tooth profile provides an adequate approximation of the influence of all named and unnamed ocean cycles in the calculated AGT anomalies.
Figure 5: Comparison of idealized approximation of ocean cycle effect and the calculated effect from PDO and ENSO.

(added 9/21/2015)
The AMO index 22 is formed from area-weighted and detrended SST data. It is shown with two different amounts of smoothing in Figure 6 along with the saw-tooth approximation for the entire planet.
Figure 6: Comparison of idealized approximation of ocean cycle effect and the AMO index.

The high coefficient of determination in Table 1 and the comparisons in Figure 5 and 6 corroborate the assumption that the saw-tooth profile provides adequate approximation of the net effect of all named and unnamed ocean cycles in the calculated AGT anomalies.

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 number anomalies does. As shown in Figure 2, the temperature anomaly trend, determined using the sunspot number anomaly time-integral, exhibits 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 number anomalies 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 α-trend to oscillate above and below the β-trend calculated using only the sunspot number anomaly 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.

Figure 1.1 shows the near perfect match with calculated temperatures which occurs when random fluctuation in reported measured temperatures is smoothed out with 5-year moving average.

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

9. Deleted.
10. Deleted.
11. Graphical sunspot number prediction for the remainder of solar cycle 24
13. Svensmark paper, Phys. Rev. Lett. 85, 5004–5007 (2000)
14. Marsden & Lingenfelter 2003, Journal of the Atmospheric Sciences 60: 626-636
18. Barrett, ‘Greenhouse molecules, their spectra and function in the atmosphere’, Energy & Environment, Vol. 16, No. 6, 2005.
19. Deleted
20. Deleted
21 Deleted.