Correlations of the first and second derivatives of atmospheric CO2 with   global surface temperature and the El Nino-Southern Oscillation respectively
  Understanding current global climate requires an understanding of trends both in Earth's atmospheric temperature and the El Nino-Southern Oscillation (ENSO), a characteristic large-scale distribution of warm water in the tropical Pacific Ocean and the dominant mode of year-to-year climate variability (Holbrook et al. 2009. However, despite much effort, the average projection of current climate models has become statistically significantly different from the observed 21st century global surface temperature trend (Fyfe 2013)and has failed to reflect the statistically significant evidence that annual-mean global temperature has not risen in the 21st century (Fyfe 2013, Kosaka 2013). Modelling also provides a wide range of predictions for future ENSO variability, some showing an increase, others a decrease and some no change (Guilyardi, et al. 2012; Bellenger, 2013). Here we present correlations which include the current era and do not have these drawbacks. The correlations arise as follows. First it has been shown (Kuo 1990, Wang W. et al. 2013) that the rate of change of the level of atmospheric CO2 (expressed as its first derivative) has a time-trend signature which is statistically significantly similar to that for global surface temperature. Second, we show here that the rate of this change - the second derivative of the level of atmospheric CO2 - is statistically significantly correlated with the separate signature displayed by the El Nino-Southern Oscillation. Third, we show that second-derivative atmospheric CO2 leads ENSO, first-derivative CO2 and temperature. Taken together the foregoing three points provide further lines of evidence for the key role of atmospheric CO2 as a driver of global climate. The results may also contribute to more accurate prediction of future global climate. 
L.M.W.Leggett and D.A. Ball
physics.ao-ph