Study of the interrelationship between global climate indices at different time scales

In this research, the possible interrelationship between several global climatic indices has been investigated by employing four statistical methods. At the first step, the spectral content of the data and their intensities were obtained at different harmonics with using the spectral analysis. Then the correlation analysis was used to calculate the correlation coefficients between each pair of climatic signals, in order to check any linear relationship between them at different time lags. Finally, by exploiting the sophisticated wavelet coherence method, the correlation of indices was analyzed at different frequencies and times, and in addition, the linkage between the indices was examined from the scatter plots. 
The selected indices include the number of sunspots (SN), Southern Oscillation Index (SOI), Quasi-Biennial Oscillation (QBO), North Atlantic Oscillation (NAO) and Mediterranean Oscillation (MO). The monthly values for these indices were used for the statistical period from 1979 to 2021.
Based on the analysis, evidences of the presence of 11-year cycle of sunspots in the SOI signal were found. The results show that if the SN signal is considered as the cause and the SOI signal as the effect, the minimum value of the 11-year component of the SOI signal occurs about 3 years after the maximum value of the same component in the SN signal.
 The correlation method shows a weak linear relationship between other signals such as NAO, QBO, SOI and MO in all time lags, and the highest correlation in this case is between NAO and MO signals. However, a closer look at the wavelet coherence plots shows that these signals are strongly correlated at decadal and to some extent at inter-annual time scales.
The lagged correlation method and the scatter plots also confirm an increase in the linear relationship between sunspot cycle with the SOI index for the time lag of 33 to 36 months, and no significant linear relationship was generally observed among other indices. The highest correlation (about 0.2) at the time lag of zero is revealed to belong to the NAO and MO signals, which according to the wavelet coherence analysis, it is due to the decadal common in-phase oscillation between these two signals. Furthermore, common oscillations with consistent phase were found sporadically between each pair of selected signals in the inter-annual scales for some periods of time. The scatter plots also showed more details about the possible relationship between the data.
Our results show the potential of the spectral and wavelet coherence analysis (WTC) for identifying common or dominant frequency components of the important climatic signals, and they can be exploited as a complementary to the traditional lagged correlation analysis and other preliminary statistical analysis such as the analyzing the scatter plots.