Influence of the Atlantic Multidecadal Oscillation and the Pacific Decadal Oscillation on Global Temperature by Wavelet-Based Multifractal Analysis

Oceanic–atmospheric patterns, Atlantic Multidecadal Oscillation (AMO) and Pacific Decadal Oscillation (PDO), and their respective influence on the global warming hiatus were the main interests of this study. In general, a fractal property is observed in the time series of dynamics of complex systems; hence, we investigated the relations among the AMO, PDO, and El Niño-Southern Oscillation (ENSO) from the point of view of multifractality, in which changes in fractality were detected with multifractal analysis using wavelet transform. For the periods 1950-1976 and 1998-2012, global temperature increased little, with positive AMO and negative PDO indices; subsequently, the rate of temperature increase weakened. Global temperature increased again in 1976, with the reversal of the AMO and PDO indices from negative to positive. More specifically, AMO, PDO, and Niño3.4 (ENSO index) exhibited fractality change from multifractality to monofractality, providing them stability. Generally, the PDO was influenced largely by the ENSO. But, around 1960 and around 2000, whose periods corresponded to hiatus periods in global warming, the influence of the ENSO on the PDO was weak. In 1998, the AMO increased and PDO decreased and global temperature increased little and the multifractality of PDO, and Niño3.4 was weak, which corresponded to the change from multifractality to monofractality in 1976. Wavelet analysis showed the leads of PDO and Niño3.4 indices with respect to global temperature. Consequently, the PDO and ENSO showed large influence on global temperature and, further, on the global warming hiatus.


Introduction
The Atlantic Multidecadal Oscillation (AMO) is a near-global scale mode of the observed multidecadal climate variability with alternating warm and cool phases over large parts of the Northern Hemisphere. Many prominent examples of regional multidecadal climate variability have been related to the AMO, including the North Eastern Brazilian and the African Sahel rainfall, the Atlantic hurricanes, and the North American and European summer climates (Knight et al., 2006). The AMO is a genuine quasiperiodic cycle of internal climate variability persisting for many centuries and is related to variability in the oceanic thermohaline circulation (THC) (Knight et al., 2005). The change in phase of the AMO in the 1960s may have caused a cooling of the US and European summer climates (Sutton & Hodson, 2005). When the phase of the AMO was positive, the Atlantic hurricane activity increased (Goldenberg et al., 2001). The importance of the AMO has been recognized by ecologists as a significant factor influencing ecosystems state (Nye et al., 2014). For instance, there have been studies focused on the impacts on soil moisture of AMO, along with another oceanic-atmospheric pattern, the Pacific Decadal Oscillation (PDO) (Tang et al., 2014). The rate of global mean surface temperature increase slowed between 1998 and 2012 and the change was often termed the "global warming hiatus" (Medhang et al., 2017).
Self-similarity, alternatively known as fractal property, exists in various objects in nature. Monofractality shows an approximately similar pattern at different scales and is characterized by a fractal dimension. Multifractality is a nonuniform, more complex fractal and is decomposed into many subsets characterized by different fractal dimensions. Moreover, fractal properties can be observed in the time series representing the dynamics of complex systems. A change in fractality accompanies a phase transition and changes of state. Multifractal properties of daily rainfall were investigated in two contrasting climates: an East Asian monsoon climate with extreme rainfall variability and a temperate climate with moderate rainfall variability (Svensson et al., 1996). In both climates, the frontal rainfall showed monofractality, whereas the convective-type rainfall showed multifractality.
On the above basis, climate change can be interpreted from the perspective of fractals. A change of fractality may be observed when a climate changes. We attempt to explain changes in climate, referred to as regime shifts, through fractality analysis.
For analyzing the multifractal behavior of the climate index, we apply the wavelet transform, as wavelet methods are useful in the analysis of complex non-stationary time series. The wavelet transform allows reliable multifractal analysis to be performed (Muzy et al., 1991). In terms of the multifractal analysis, we concluded in our previous paper (Maruyama & Morimoto, 2015) that a climatic regime shift corresponds to a change from multifractality to monofractality of the PDO index.
Thus, we present this study to investigate the relationship between the AMO

Data and Method of Analysis
We used a monthly time series provided by NOAA's Climate Prediction Center, USA (CPC), as detailed below, and applied the AMO, PDO, Niño3.4 indices, and global mean surface air temperature anomalies as inputs of the analysis.
We used the Daubechies wavelet as the analyzing wavelet because it is widely used in solving a broad range of problems, e.g., self-similarity properties of a signal or fractal problems and signal discontinuities. The data used consisted of a discrete signal that fitted the Daubechies mother wavelet with the capability of precise inverse transformation. Hence, precisely optimal value of τ(q) could be calculated as explained below. We then estimated the scaling of the partition function Z q (a), defined as the sum of the q-th powers of the modulus of the wavelet transform coefficients at scale a. In our study, the wavelet-transform coefficients did not become zero, and therefore, for a precise calculation, the summation was considered for the entire set. Muzy et al. (1991) defined Z q (a) as the sum of the q-th powers of the local maxima of the modulus to avoid division by zero. We obtained the partition function Z q (a): where is the wavelet coefficient of the function f, a is a scale parameter and b is a space parameter. The time window was set to 6 years for reasons outlined in the next statements. We calculated the wavelets using a time window of various periods: 10, 6, and 4 years. For a time window of 10 years, a slow change of fractality was observed. Such case was inappropriate for finding a rapid change of regime shift because when we integrated the wavelet coefficient over a wide range, small changes were canceled. In contrast, a fast change of fractality was observed for a time window of 4 years. Specifically, the first and subsequent data overlapped by 3 years, much shorter than the 9 years in the case of the 10-year calculation, thus leading to a large change of fractality. Moreover, for the 6-year time window, a moderate change of fractality was observed, and hence, we set the time window to this period. For small scales, we expect ( First, let us investigate the changes of Z q (a) in time series at a different scale a for each q, using a plot of the logarithm of Z q (a) against the logarithm of time scale a. Here τ(q) was the slope of the linear fitted line on the log-log plot for each q. Next, we plotted τ(q) vs q. The time window was then shifted forward 1 year and the process was repeated.
In this paper, we define monofractal and multifractal as follows: if τ(q) is linear with respect to q, then the time series is said to be monofractal; if τ(q) is convex upwards with respect to q, then the time series is classified as multifractal (Frish & Parisi, 1985). Further, we provided a definition for the value of R 2 , which is the coefficient of determination, for fitting the straight line as follows: if R 2 ≥ 0.98, then the time series is monofractal; otherwise, if 0.98 > R 2 , then the time series is multifractal.
Subsequently, we calculated τ(q) of different moments q for individual records for the Niño3.4 index. Figure 1 shows a plot of τ(q) between 1980 and 1994. The data were analyzed in 6-year sets, e.g., τ(q) of n80 was calculated for 1980-1985, and that of n81 was calculated for 1981-1986. For a study of the change of fractality, the time window was shifted forward to 1 year, and τ(q) was calculated from n80 up to n89. A monofractal signal would correspond to a straight line for τ(q), whereas τ(q) for a multifractal signal would be nonlinear. Most of the multifractality observed was due to the negative value of q, i.e., small fluctuations were more inhomogeneous than large fluctuations. From Figure 1, the data sets in the cases of was n80 -n82, n85 -n87, and n89 were monofractal, whereas those in the cases of n83, n84, and n88 were multifractal.
Accordingly, we plotted the value of the τ(−6) in each index. Here, a negative large value of τ(−6) showed large multifractality. More importantly for the τ(−6), q = −6 was the appropriate number showing the change of τ. Additionally, the value of τ(−6) did not always correspond to the fractality obtained from the value of R 2 .   phase (bottom) between the AMO and PDO index. The thick black contour encloses regions of greater than 95% confidence. The thin black contour encloses regions of greater than 90% confidence. The cone of influence, which indicates the region affected by edge effects, is shown with a black line. In the wavelet phase, the positive value shown by the blue and pink shading means that the AMO leads PDO index and the negative value shown by the green, yellow and red shading means that PDO index leads the AMO.

Relationship between the ENSO and Global Temperature
Figure 6 (top) shows a plot of the τ(−6) of the Niño3.4 and global temperature.
Changes in fractality were very similar for 1970s and 2000s, when the coherence was strong. Figure 6 shows the wavelet coherence (middle) and phase (bottom) between the Niño3.4 and global temperature using the Morlet wavelet. Specifically, coherence between the Niño3.4 and global temperature was strong for 1965-1975 and 1990-2005, and the lead of the Niño3.4 index was observed. The strong influence of the ENSO on the global temperature was shown.  But, around 1960 and around 2000, the Niño3.4 index laged the PDO, whose periods corresponded to hiatus periods in global warming. A relatively novel method for identifying running leading-aging LL-relations shows the same results (Seip & Wang, 2018). The sign of the PDO index changed at 1976, and both fractalities changed from multifractality to monofractality.

Discussion
There was a significant increase in global temperature for 1976-1998, during

Conclusion
In the present study, we investigated two oceanic-atmospheric patterns, namely, Niño3.4 (ENSO index) exhibited fractality change from multifractality to monofractality, providing them stability. Generally, the PDO was influenced largely by the ENSO. But, around 1960 and around 2000, whose periods corresponded to hiatus periods in global warming, the influence of the ENSO on the PDO was weak.
3) In 1998, the AMO increased and PDO decreased and global temperature increased little and the multifractality of PDO, and Niño3.4 was weak, which corresponded to the change from multifractality to monofractality in 1976.
Wavelet analysis showed the leads of PDO and Niño3.4 indices with respect to global temperature. Consequently, the PDO and ENSO showed large influence on global temperature and, further, on the global warming hiatus.
These findings will contribute to further studies on climate change.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.