Recent posts have stressed the complexity of climates and their component variables. However, global warming was invented on the back of a single metric: rising global mean temperatures the last decades of last century. That was de-emphasized during the “pause” but re-emerged lately with the El-Nino-induced warming. So this post is focusing on that narrow aspect of climate change.
There are several papers on this blog referring to a quasi-60 year oscillation of surface temperatures due to oceanic circulations. I have also noted the attempts by many to make the link between solar activity (SA) and earth climate patterns.
Dan Pangburn is a professional engineer who has synthesized the solar and oceanic factors into a mathematical model that correlates with Average Global Temperature (AGT). On his blog is posted a monograph (here) Cause of Global Climate Change explaining clearly his thinking and the maths. I am providing some excerpts and graphs as a synopsis of his analysis, in hopes others will also access and appreciate his work on this issue.
The basis for assessment of AGT is the first law of thermodynamics, conservation of energy, applied to the entire planet as a single entity. Much of the available data are forcings or proxies for forcings which must be integrated (mathematically as in calculus, i.e. accumulated over time) to compute energy change. Energy change divided by effective thermal capacitance is temperature change. Temperature change is expressed as anomalies which are the differences between annual averages of measured temperatures and some baseline reference temperature; usually the average over a previous multiple year time period. (Monthly anomalies, which are not used here, are referenced to previous average for the same month to account for seasonal norms.)
At this point, it appears reasonable to consider two temperature anomaly data sets extending through 2015. These are co-plotted on Figure 8.Accuracy of the model is determined using the Coefficient of Determination, R 2, to compare calculated AGT with measured AGT.
Oceanic Climate Impacts
Approximation of the sea surface temperature anomaly oscillation can be described as varying linearly from –A/2 K in 1909 to approximately +A/2 K in 1941 and linearly back to the 1909 value in 1973. This cycle repeats before and after with a period of 64 years.
Comparison with PDO, ENSO and AMO
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 approximation, (A,y), accounts for both ways.
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 AMO index  is formed from area-weighted and de-trended SST data. It is shown with two different amounts of smoothing in Figure 3 along with the saw-tooth approximation for the entire planet per Equation (2) with A = 0.36.
The high coefficients of determination in Table 1 and the comparisons in Figures 2 and 3 corroborate the assumption that the saw-tooth profile with a period of 64 years provides adequate approximation of the net effect of all named and unnamed ocean cycles in the calculated AGT anomalies.
An assessment of this is that sunspots are somehow related to the net energy retained by the planet, as indicated by changes to average global temperature. Fewer sunspots are associated with cooling, and more sunspots are associated with warming. Thus the hypothesis is made that SSN are proxies for the rate at which the planet accumulates (or loses) radiant energy over time. Therefore the time-integral of the SSN anomalies is a proxy for the amount of energy retained by the planet above or below breakeven.
Also, a lower solar cycle over a longer period might result in the same increase in energy retained by the planet as a higher solar cycle over a shorter period. Both magnitude and time are accounted for by taking the time-integral of the SSN anomalies, which is simply the sum of annual mean SSN (each minus Savg) over the period of study.
The values for Savg are subject to two constraints. Initially they are determined as that which results in derived coefficients and maximum R2. However, calculated values must also result in rational values for calculated AGT at the depths of the Little Ice Age. The necessity to calculate a rational LIA AGT is a somewhat more sensitive constraint. The selected values for Savg result in calculated LIA AGT of approximately 1 K less than the recent trend which appears rational and is consistent with most LIA AGT assessments.
The sunspot number anomaly time-integral is a proxy for a primary driver of the temperature anomaly β-trend. By definition, energy change divided by effective thermal capacitance is temperature change.
Projections until 2020 use the expected sunspot number trend for the remainder of solar cycle 24 as provided  by NASA. After 2020 the ‘limiting cases’ are either assuming sunspots like from 1924 to 1940 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 9. 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 natural fluctuation of measured temperatures.
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.
These papers [14,15] associated the increased low-altitude clouds with increased albedo leading to lower temperatures. Increased low altitude 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 S-B 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  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 K. Because the cloud effects work together and part of the temperature change is due to ocean oscillation (low in 1901, 0.2114 higher in 2000), substantially less cloud change would suffice.
All of this leaves little warming left to attribute to rising CO2. Pangburn estimates CO2 forcing could be at most 18.6% or 0.23C added since 1895. Given uncertainties in proxies from the past, the estimate could be as low as 0.05C, and the correlation with natural factors would still be .97 R2.
However, all is not lost for CO2. It is still an important player in the atmosphere, despite its impotence as a warming agent.