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Wavelet-based multifractal analysis on a time series of solar activity and PDO climate index
Accepted Manuscript Wavelet-based multifractal analysis on a time series of solar activity and PDO climate index Fumio Maruyama, Kenji Kai, Hiroshi Morimoto PII: DOI: Reference:
S0273-1177(17)30411-8 http://dx.doi.org/10.1016/j.asr.2017.06.004 JASR 13258
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
20 August 2016 8 April 2017 2 June 2017
Please cite this article as: Maruyama, F., Kai, K., Morimoto, H., Wavelet-based multifractal analysis on a time series of solar activity and PDO climate index, Advances in Space Research (2017), doi: http://dx.doi.org/10.1016/j.asr. 2017.06.004
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Wavelet-based multifractal analysis on a time series of solar activity and PDO climate index
Fumio MARUYAMA*, Kenji KAI and Hiroshi MORIMOTO
Graduate School of Environmental Studies, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan
-------------------------------------* Corresponding author. E-mail: [email protected] (F. Maruyama) Keywords: sunspot number, solar flux, solar polar field strength, geomagnetic aa and Ap indices, PDO, climatic regime shift, wavelet, multifractal
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Abstract
There is increasing interest in finding the relation between solar activity and climate change. In general, fractal properties may be observed in the time series of the dynamics of complex systems, such as solar activity and climate. This study investigates the relations among solar activity, geomagnetic activity, and climatic regime shift by performing a multifractal analysis. To investigate the change in multifractality, we apply a wavelet transform to time series. The change in fractality of the sunspot number (SSN) correlates closely with that of the solar polar field strength. For the SSN and solar polar field strength, a weak multifractality or monofractality is present at the maximum SSN, minimum SSN, and maximum solar polar field strength. A strong multifractality is present two years before the maximum SSN.
The climatic regime shift
occurrs when the SSN increases and the disturbance of the geomagnetic activity is large. At the climatic regime shift, the changes in the fractality of the Pacific Decadal Oscillation (PDO) index and changes in that of the solar activity indices corresponded with each other. From the fractals point of view, we clarify the relations among solar activity, geomagnetic activity, and climatic regime shift. The formation of the magnetic field of the sunspots is correlated with the solar polar field strength. The solar activity seems to influence the climatic regime shift. These findings will contribute to investigating the relation between solar activity and climate change.
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1.
Introduction Various objects in nature exhibit the so-called self-similarity or fractal property.
Nature is full of fractals, for instance, trees, rivers, coastlines, mountains, clouds, and seashells. Monofractal signals are homogeneous in that they have the same scaling properties and are characterized by a fractal dimension.
On the other hand, multifractal
signals are nonuniform, more complex and can be decomposed into many subsets characterized by the different dimensions of the fractal. Fractal properties can also be observed in a time series representing the dynamics of complex systems. A change in fractality can appear with a phase transition or change of state. For example, the multifractal properties of daily rainfall were studied in an East Asian monsoon climate with extreme rainfall and in a temperate climate with moderate rainfall (Svensson et al., 1996). In both climates, the frontal rainfall and convective-type rainfall showed monofractality and multifractality, respectively. In another example, a healthy human heartbeat shows a multifractal character, while a diseased heart shows a monofractal character (Ivanov et al., 1999). In this study, we attempted to interpret climatic changes (that is, climatic regime shifts) by investigating fractality. We investigated fractality of the solar activity indices, F10.7 flux, SSN, aa and Ap indices for the first time. To examine the multifractal behavior of the climate index, we used a wavelet transform method because it is useful for the analysis of complex nonstationary time series. The wavelet transform can perform reliable multifractal analysis (Muzy et al., 1991); hence, we used the wavelet-based multifractal analysis to quantify the signals of higher complexity. We concluded that a climatic regime shift correlates to a change from multifractality to monofractality of the Pacific Decadal Oscillation (PDO) index (Maruyama et al., 2015). Minobe (1997) showed that a climatic regime shift is a sudden transition from one quasi-steady climatic state to another, and its transition period is much shorter than the individual epochs length of each climatic state. The climatic regime shift in the 1970s followed the warming of the tropical Pacific Ocean, the cooling of the central North Pacific, and the strengthening of the Aleutian Low. To identify the years when climatic
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regime shifts occurred in the Sea Surface Temperature (SST) field, the time series of the original gridded SST and those of the Empirical Orthogonal Function (EOF) modes were examined, and the 1925/26, 1945/46, 1957/58, 1970/71, 1976/77, and 1988/89 climatic regime shifts were detected from the 1910s to the 1990s (Yasunaka and Hanawa, 2002). The SSN has an 11-year cycle and the solar magnetic field has a 22-year cycle, exactly twice that of the sunspot cycle, because the polarity of the field returns to its original value every two sunspot cycles.
During the past ~120 years, Earth's surface
temperature has correlated with both the decadal averages and solar cycle minimum values of the geomagnetic aa index (Cliver et al., 1998a). The 11-year averages of the SSN and aa index have been highly correlated for the past 150 years (Cliver et al., 1998b). The maxima of the aa index occurred near the maximum SSN and (or) later in the declining phase of the SSN (Kane, 2002). High correlation coefficients were found among the geomagnetic activity, sea level atmospheric pressure and surface air temperature (Bucha et al., 1998). In this study, we investigated the relations among the SSN and solar polar field strength, the influence of the solar activity on the geomagnetic field, and the influence of the solar activity and geomagnetic activity on a climatic regime shift from the view point of fractals. To examine the changes in multifractality, we performed a multifractal analysis on the SSN, solar polar field strength, 10.7-cm solar radio flux (F10.7), geomagnetic aa and Ap indices, and PDO index by using the wavelet transform.
2.
Data and Method of Analysis We used several solar activity indices to see the solar activity in a multifaceted
manner. For the solar activity indices, we used F10.7 flux, SSN, aa and Ap indices and for the climate indices, PDO index and upper-air temperature. We made use of the monthly SSN provided by Solar Influences Data Analysis Center (sidc.oma.be), the solar radio flux at 10.7 cm provided by the National Oceanic and Atmospheric Administration’s (NOAA’s) Space Weather Prediction Center (www.swpc.noaa.gov). The F10.7 flux is an excellent indicator of the solar activity,
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because F10.7 indicates a good correlation with SSN. We used the solar polar field strength throughout the solar sunspot cycle provided by the Wilcox Solar Observatory (wso. stanford.edu) and the geomagnetic aa and Ap indices of the solar activity provided by NOAA. The aa index is a measure of disturbance in the Earth’s magnetic field based on magnetometer observations at two nearly antipodal stations in Australia and England. The Ap index provides a daily average level for the geomagnetic activity. We used the monthly PDO index provided by NOAA’s Climate Prediction Center (CPC). The PDO index is the leading principal component of the monthly SST anomalies in the North Pacific Ocean (Mantua et al., 1997).
We also used the global upper-air temperature
anomaly provided by the Met Office Hadley Centre. For the analysis, we used the Daubechies wavelet, which is widely used in solving a broad range of problems, e.g., self-similarity properties of a signal or fractal problems and signal discontinuities.
We used a discrete signal which was fitted the Daubechies
mother wavelet with the capacity of accurate inverse transformation. Thus, we can precisely calculate the following optimum τ(q), which can be regarded as a characteristic function of the fractal behavior.
We can define the τ(q) from the power-law behavior of
the partition function, as shown in equation (2). We then calculated the scaling of the partition function Zq(a), which is defined as the sum of the q-th powers of the modulus of the wavelet transform coefficients at scale a, where q is the q-th moment. In our calculation, the wavelet-transform coefficients did not become zero. Thus, for an accurate calculation, the summation was considered for the whole set. Muzy et al. (1991) defined Zq(a) as the sum of the q-th powers of the local maxima of the modulus to avoid dividing by zero. We obtained the following partition function Zq(a):
Zq a
W
f a, b
q
,
(1)
where Wφ[f](a, b), a, and b are the wavelet coefficient of function f, a scale parameter, and a space parameter, respectively.
W f a, b is defined as below.
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1
W f a, b
,where f t is data and
f t
a
*
t b dt a
(2)
is wavelet function. For small scales, we expect
Zq a ~ a
q
.
(3)
First, we investigated the changes in Zq(a) in the time series at a different scale a for each moment q. We plotted the logarithm of Zq(a) against that of time scale a. Here τ(q) is the slope of the fitted straight line for each q. Next, we plotted τ(q) versus q, which is the multifractal spectrum. The time window was advanced by one year, which was repeated.
Monofractal and multifractal signals were defined as follows: A
monofractal signal corresponds to a straight line for τ(q), while a multifractal signal τ(q) We calculated the R2 value, which is the
is nonlinear (Frish and Parisi, 1985).
coefficient of determination, for the fitted straight line. If R2 ≥ 0.98, the time series is monofractal; if 0.98 > R2, it is multifractal. A time window was fixed to 6 years for the following reasons. We calculated the wavelets with time windows of 10, 6, and 4 years. Initially, when a time window was 10 years, a fractality changed slowly.
By integrating the wavelet coefficient in a wide
range, small changes were canceled.
Thus, this case was inappropriate to find a fast
change of climatic regime shift. Next, when the time window was 4 years, the fractality changed quickly. The overlap of the first and following data was 3 years, which was shorter than the 9 years when the time window was 10 years, and the change of fractality was large. Thus, this case was also inappropriate. Finally, when the time window was 6 years, a moderate change in fractality was observed. Hence, the time window was fixed to 6 years. We calculated the multifractal spectrum τ(q) of the SSN between 1910 and 2010. The multifractal spectrum τ(q) between 1967 and 1979 is shown in Fig. 1. The data
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were analyzed in 6-year sets; as an example, the multifractal spectrum τ(q) of s67 was calculated between 1967 and 1972. To study the change in fractality, the time window was advanced by one year and the multifractal spectrum τ(q) was obtained from s67 to A monofractal signal corresponds to a straight line for τ(q), whereas for a
s76.
multifractal signal, τ(q) is nonlinear. In Fig. 1, the constantly changing curvature of the τ(q) curves for s67, s68, and s72 – s74 suggests multifractality. In contrast, τ(q) is linear for s69 – s71, which indicates monofractality. We plotted the τ(−6) of each index, where q = −6 is the appropriate number for showing a change in τ. The large negative values of τ(−6) show large multifractality. Thus, τ(−6) is not always equal to the fractality gained from the R2 value. We show the wavelet coherence and phase using the Morlet wavelet between the F10.7 flux and upper air temperature at 30 hPa. The Morlet wavelet is most often used and offers a good frequency resolution.
3. Results 3.1. The correlation between the SSN and the geomagnetic aa and Ap indices We investigate the relation between the SSN and the geomagnetic aa and Ap indices through multifractal analysis and the wavelet coherence and phase.
Figure 2 shows the
time series of the SSN and the aa and Ap indices as well as the solar cycle number of the SSN.
The sunspot cycles are typically asymmetric in shape with a rapid rise to
maximum and a slower decline to minimum.
Asymmetric solar polar field reversals are
simply a consequence of the asymmetry of solar activity (Svalgaard and Kamide, 2013). The magnetic cycle consists of two sunspot cycles.
Feynman (1982) noted that
geomagnetic activity has two different sources, one due to solar activity which follows the sunspot cycle, and another due to recurrent high-speed solar wind streams which peaks during the decline of each cycle. The changes in the aa and Ap indices are very similar. 11-year periodicity is observed in the aa and Ap indices, which indicates the influence of the Sun. The time series of the aa index has two peaks, which correspond to the maximum SSN and decrease in the SSN. When the SSN is at a minimum, the aa
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and Ap indices are also at a minimum and the geomagnetic disturbance is small.
Hence,
when solar activity is not strong, the disturbance of the geomagnetic field is weak. Ohl (1966) found that the minimum level of geomagnetic activity seen in the aa index near the time of the minimum sunspot cycle is a good predictor for the amplitude of the next cycle. We examine the correlation between the values of the aa and Ap indices before the maximum SSN and maximum SSN of the next cycle. correlation coefficients are r = 0.80 and r = 0.87, respectively.
The obtained
The stronger the
geomagnetic distarbance, the greater the maximum SSN of the next cycle. Hence, if the disturbance of the geomagnetic field is strong, the solar activity of the next cycle is also strong. We obtain the wavelet coherence and phase using the Morlet wavelet between the SSN and the aa index. The coherence between the SSN and aa index at around the 11-year scale is strong and the leads of the SSN are observed. We show the original versus 61-month running mean of the SSN in Fig. 3(a). The obtained high-frequency component, which is original minus the 61-month running mean of the SSN, is shown in Fig. 3(b). The high-frequency component of the SSN versus the aa index is shown in Fig. 3(c). The obtained high-frequency component is similar to the time series of the aa index. A change in the SSN appears to have influenced the geomagnetic activity.
3.2. The relation between the SSN and solar polar field strength We investigate the change in the SSN. To detect the changes in multifractality, we examine the multifractal analysis of the SSN. The τ(−6) of the SSN and SSN as well as the solar cycle number of the SSN are shown in Fig. 4. The SSN has an 11-year and a 22-year cycle for the reversal of the Sun's magnetic field.
τ(−6) is the value of τ for q =
−6 and indicates the fractality of the index. The red square and green circle represent monofractality and multifractality for the six years around the plotted year, respectively. For example, the red square for 1962 in the SSN shows monofractality between 1959 and 1964.
The data are excluded if we cannot distinguish between monofractality and
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multifractality. When the SSN is maximum (minimum), the τ(−6) of the SSN become maximum, as shown by 1 (3) in Fig. 4, when the SSN shows monofractality or weak multifractality. The τ(−6) of the SSN became minimum two years before the maximum SSN, shown by 2 in Fig. 4, when the change in the SSN is the largest. About three years before the minimum SSN, the τ(−6) of the SSN become maximum, shown by 4 in Fig. 4 and the multifractality is the weakest. The solar polar field strength (from Wilcox Solar Observatory data) is shown in Fig. 5. The changes in the solar polar field strength and SSN are large two years before the maximum SSN of the solar cycles 22, 23, and 24 as shown by the arrow in Fig. 5, which is determined from the slope of the change in the graph in Fig. 5. τ(−6) of the solar polar field strength in Fig. 6.
Next, we show the
At the maximum and minimum SSN,
the τ(−6) of the solar polar field strength become maximum, as shown by 1 and 3, respectively, in Fig. 6, when the solar polar field strength indicates monofractality or weak multifractality. The τ(−6) of the solar polar field strength become minimum two years before the maximum SSN shown by 2 in Fig. 6. Hence, two years before the maximum solar activity, the minimum τ(−6) of the SSN and the average solar polar field strength were observed, when the changes in the average solar polar field strength and SSN are large.
Before the maximum SSN, the multifractality of the SSN and solar polar
field strength become the strongest; at the maximum SSN they become the weakest, which indicates a change from multifractality to monofractality. A comparison of Fig. 5 with Fig. 6 indicates that when the change in the solar polar field strength is small, the multifractality is weak.
About three years before the
minimum SSN, the τ(−6) of the SSN and the solar polar field strength become maximum (shown by 4 in Fig. 6) and the multifractality is the weakest. We show the τ(−6) of the solar polar field strength and solar polar field strength in Fig. 7. When the solar polar field strength is maximum and minimum (approximately 0), the values of the τ(−6) become maximum and the multifractality is weak, as shown by 4 and 1, respectively, in Fig. 7.
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3.3. The relationship between solar activity and the geomagnetic field The τ(−6) of the geomagnetic aa index and the SSN are shown in Fig. 8. The result of the geomagnetic Ap index is not shown, while the tendency is smilar to that of aa index. When the SSN is minimum and maximum, the τ(−6) of the aa and Ap indices show, for most of the cases, weak multifractality, as shown by 3 and 1, respectively, in Fig. 8. The τ(−6) of the aa index at the minimum SSN is larger than that of the maximum SSN, because the disturbance of the aa index at the minimum SSN is small and the aa index indicates weak multifractality. Hence, when the disturbance of the geomagnetic field is small, the geomagnetic field indicates weak multifractality. Two years before the maximum SSN, the aa and Ap indices are small, the disturbance of the geomagnetic field is small, and the τ(−6) of the aa index becomes minimum, as shown by 2 in Fig. 8.
3.4. The influence of solar activity on climatic regime shift To investigate the influence of solar activity on the climate, we show the wavelet coherence and phase between the F10.7 flux and upper air temperature at 30 hPa in Fig. 9. We show the wavelet power spectra of the upper air temperature at 30 hPa in Fig. 10. A strong 10-year frequency is observed, which indicates the clear influence of the solar activity. The coherence is very strong in the 10-year cycle and the phase of F10.7 flux leaded; therefore, solar activity appears to have influenced the upper air temperature. The coherence becomes strong around the 1976/77 climatic regime shift; therefore, solar activity appears to have influenced the 1976/77 climatic regime shift. Solar activity influences the geomagnetic activity and North Atlantic Oscillation through solar wind (Thejll, 2003).
We examine the wavelet coherence and phase
between the geomagnetic aa index and the upper air temperature at 30 hPa.
The
coherence was very strong in 10-year cycle and the phase of the aa index leaded; therefore the geomagnetic activity appears to have influenced the upper air temperature. The coherence becomes strong around the 1976/77 climatic regime shift; therefore, the
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geomagnetic activity also appears to have influenced the 1976/77 climatic regime shift. We explain the influence of solar activity on climatic regime shift by means of fractals. We show the τ(−6) of the PDO index in Fig. 11. The changes at the climatic regime shifts of the SSN, aa, and Ap indices and the τ(−6) of the SSN, aa, Ap, and PDO indices at the Figs. 4, 8, and 11 are shown in Table 1. The climatic regime shifts occur when the SSN increases (except for the 1970/71 climatic regime shift) and the solar activity becomes strong.
4. Discussion 4.1. The correlation between the SSN and solar polar field strength When the SSN and solar polar field strength are maximum, the τ(−6) of the SSN and solar polar field strength become maximum.
The τ(−6) of the SSN becomes
minimum two years before the maximum SSN (as shown in Fig. 4), when the change of the SSN is large. The τ(−6) of the solar polar field strength and the SSN become minimum and two years before the maximum SSN of solar cycles 22, 23, and 24. The multifractalities of the SSN and solar polar field strength become strong two years before the maximum SSN, when the changes in the solar polar field strength and SSN are large. From the two years before the maximum SSN to the maximum SSN, the SSN and solar polar field strength indicate a change from multifractality to monofractality.
For 1980 to
1992, the changes in the SSN and solar polar field strength show strong similarity and correlated closely. When the change in the solar polar field strength is small, the multifractality is weak. There are three maximum τ(−6) of the SSN where the multifractality is the weakest: at the maximum SSN, at the minimum SSN, and at the SSN about three years before the minimum SSN. About three years before the minimum SSN, the τ(−6) of the SSN and solar polar field strength become maximum and the multifractality is the weakest. The solar polar field strength is the strongest during an interval of about three years before the sunspot minimum (Svalgaard and Kamide, 2005).
Hence, when the solar polar field
strength is maximum, the multifractality of the solar polar field strength and SSN become
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the weakest.
The solar polar field strength becomes large and the SSN becomes
minimum. The changes in the τ(−6) of the SSN and solar polar field strength correspond to each other, and therefore the SSN and solar polar field closely correlate. The peaks of τ(−6) for the SSN and solar polar field strength are classified into four peaks, the maximum peaks at the maximum SSN shown by 1 in Figs. 4 and 6, the minimum SSN shown by 3, and the maximum solar polar field strength shown by 4 and the minimum peak at two years before the maximum SSN shown by 2. The solar polar field strength makes the magnetic field of the sunspots when the solar cycle is the minimum (Upton and Hathaway, 2014). In this study, we compare the maximum SSN with the maximum solar polar field strength before the maximum SSN. When the maximum solar polar field strength before the maximum SSN is larger, the maximum SSN of the next cycle is larger.
Hence, the formation of the magnetic field of
the sunspots is correlated with the solar polar field strength. The fractal dimensions decrease with increasing mean magnetic field strength, implying that the magnetic field distribution is more regular in active regions (Ioshpa et al., 2008). The τ(−6) at the maximum solar polar field strength (shown by 4 in Fig. 7) is greater than that at the maximum SSN (shown by 1 in Fig. 7), when the solar polar field strength is nearly zero. Hence, when the solar polar field strength is large, the multifractality is weak, which coincides with the result of Ioshpa et al.(2008). However, when the change in the solar polar field strength is great as shown by the arrow in Fig. 7, τ(−6) become minimum and the multifractality is strong.
When the SSN and solar polar
field strength are maximum, the τ(−6) of the SSN and solar polar field strength become maximum.
4.2. The relation between solar activity and the geomagnetic field When the SSN is minimum, the geomagnetic aa and Ap indices are minimum (as shown in Fig. 2) and the values of the aa and Ap indices show weak multifractality (as shown by 3 in Fig. 8). Hence, when the solar activity is not strong, the disturbance of
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the geomagnetic field is small and the values of the aa and Ap indices show weak multifractality. Two years before the maximum SSN, the values of the aa and Ap indices are small, so the disturbance of the geomagnetic field is small. Hence, two years before the maximum solar activity, the τ(−6) of the SSN becomes minimum (as shown by 2 in Fig. 4), and the disturbance of the geomagnetic field is small, and the τ(−6) of the aa and Ap indices become minimum. The obtained correlation coefficients between the values of the aa and Ap indices before the maximum SSN and the maximum SSN of the next cycle are r = 0.80 and r = 0.87, respectively. The greater the geomagnetic disturbance, the larger the maximum SSN of the next cycle. We examine the correlation between the τ(−6) of the aa and Ap indices about two to three years before the maximum SSN and at the maximum SSN of the next cycle. The obtained correlation coefficient is r = 0.30 and r = 0.88 for the aa and Ap indices, respectively. When the τ(−6) of the aa and Ap indices are small and multifractality is strong, the SSN of the next cycle is large.
Hence, when the
disturbance of the geomagnetic field is large and the aa and Ap indices show strong multifractality, the solar activity of the next cycle is strong. Yoshida (2008) determined that when the minimum geomagnetic activity is large, then the maximum solar activity of the next cycle is strong, which coincides with our result.
4.3. The influence of solar activity on climatic regime shift Fractal behavior may be observed in many self-organization systems, representing the order or disorder included in the systems. There is a model that deals with an interaction between magnetism and fluid (Bartosz et al.,2000). It is reported that a phenomenon of self-organization occurs and a fractal is observed. Their model was performed in a micro level, but a similar mechanism might occur in an interaction between the Earth and Sun. Therefore, the study of fractals become an important issue to assess a relation between the solar activity and the Earth’s climate. As shown in 3.4, the climatic regime shift appears to be influenced by the solar activity. Additionally, at the climatic regime shift, the fractality of the PDO index changes from multifractality to
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monofractality and τ(−6) increases. For SSN and F10.7 flux, the change from multifractality to monofractality is not always seen at the climate regime shift. The τ(−6) of the SSN at three climatic regime shifts increases and those at two climatic regime shifts are minimum, just before the increase. All of the τ(−6) of the aa index are minimum and most of the τ(−6) of the Ap index are minimum. Hence, at the climatic regime shift, the changes in the τ(−6) of the PDO index and solar activity indices correspond to each other. The PDO appears to be influenced by the solar activity. The wavelet coherence between the SSN (F10.7) and PDO index is strong and the phase of SSN (F10.7) leaded (not shown). Hence, the PDO appears to be influenced by solar activity. A climatic regime shift is equal to a change from multifractality to monofractality of the PDO index (Maruyama et al., 2015). Hence a climatic regime shift is also influenced by the solar activity. These results show that the solar activity influences on the climate and climatic regime shift. Solar activity influences the climate through the ozone. A change in temperature in the stratosphere (due to a change in ultraviolet rays by the change of ozone) changes the stratosphere circulation. That spreads below through the dynamic effect and influences the climate of the troposphere (Kodera and Kuroda, 2002). A close correlation between cosmic ray flux, which increases when the solar activity is weak, and variations in Earth's cloud cover has been demonstrated (Svensmark, 2000). Because clouds are important for the Earth's energy balance, a solar influence on clouds can be the main cause for the observed correlations between the Sun and Earth's climate. A climatic regime shift occurs when the geomagnetic aa and Ap indices increase and the τ(−6) of the aa and Ap indices are minimum.
At the climatic regime shift, the
multifractality of the geomagnetic activity index, influenced by the Sun, became maximum and the disturbance of the geomagnetic activity is large.
5. Conclusions In this study, we investigated the relation between the SSN and the solar polar field strength, the influence of solar activity on the geomagnetic field, and the influence of
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solar activity on the climatic regime shift by performing a multifractal analysis. To detect the changes in multifractality, we performed a multifractal analysis on the SSN, F10.7 flux, solar polar field strength, geomagnetic aa and Ap indices, and PDO index using the wavelet transform.
Moreover we examined the wavelet coherence and phase
of these indices. (1) The relation between the SSN and solar polar field strength ・When the SSN and solar polar field strength are maximum, their τ(−6) become maximum and the multifractality becomes weak. The τ(−6) shows the fractality of the index. ・The minimum τ(−6) of the SSN and solar polar field strength are observed two years before the maximum SSN.
Before the maximum solar activity, the multifractality of the
SSN and solar polar field strength becomes strong, when the changes in the SSN and solar polar field strength are large. ・Solar polar field strength becomes maximum about three years before the minimum SSN, when the τ(−6) of the SSN and solar polar field strength become maximum and the multifractality becomes the weakest. The solar polar field strength becomes large and the SSN becomes minimum. ・The changes in the τ(−6) of the SSN correlate closely with those of the solar polar field strength. A weak multifractality or monofractality is shown at the maximum SSN, the minimum SSN, and the maximum solar polar field strength. A strong multifractality is shown at two years before the maximum SSN.
When the solar polar field strength
becomes stronger, the maximum SSN of the next cycle becomes large.
Consequently,
the formation of the magnetic field of the sunspots is correlated with the solar polar field strength. (2) The relation between solar activity and the geomagnetic field Disturbances in the geomagnetic field are small when solar activity is not active. When the SSN is maximum or minimum, the τ(−6) of the aa and Ap indices become maximum and the geomagnetic field shows weak multifractality. Two years before the maximum SSN, the τ(−6) of the aa and Ap indices become minimum. These results
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show a similar tendency of τ(−6) for the SSN and solar polar field strength, which indicates that solar activity influences geomagnetic activity.
The greater the
geomagnetic disturbance, the larger the maximum SSN of the next cycle.
If the
disturbance of the geomagnetic field is large and the aa and Ap indices show strong multifractality, the solar activity of the next cycle is strong. (3) The influence of solar activity on climatic regime shift The relation between the F10.7 flux and upper air temperature indicates that solar activity influences the upper air temperature. The climatic regime shift occurs when the SSN increases and the solar activity becomes active and the disturbance of the geomagnetic activity is large. At the climatic regime shift, the changes in the τ(−6) of the PDO and solar activity indices correspond to each other. By analyzing wavelet coherence and phase, we demonstrate that the solar activity relates to and influences the PDO. Hence, the PDO is influenced by the solar activity and a climatic regime shift seems to be influenced by solar activity.
16
References Bartosz, A. G., Howard, A. S., George, M. W., 2000. Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid-air interface. Nature, 405, 1033-1036. Bucha, V., Bucha, V. Jr., 1998. Geomagnetic forcing of changes in climate and in the atmospheric circulation. Journal of Atmospheric and Solar-Terrestrial Physics, 60, 145−169. Cliver, E. W., Boriakoff, V., Feynman, J., 1998a. Geomagnetic activity and the solar wind during the Maunder Minimum. Geophys. Res. Lett., 25, 897−900. Cliver, E. W., Boriakoff, V., Feynman, J., 1998b. Solar variability and climate change: Geomagnetic aa index and global surface temperature. Geophysical Res. Lett., 25, 1035−1038. Feynman, J., 1982. Geomagnetic and solar wind cycles, 1900−1975. J. Geophys. Res., 87, 6153−6162. Frish, U., Parisi, G., 1985. On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. edited by Ghil, M., R. Benzi, and G. Parisi, pp. 84−88, North-Holland, New York. Hathaway, D. H., 2009. Solar Cycle Forecasting. Space Sci. Rev., 144, 401−412. Ioshpa, B. A., Obridko, V. N., Rudenchik, E. A., 2008. Fractal Properties of Solar Magnetic Fields. Astronomy Letters, 34, 210−216. Ivanov, P. Ch., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. G., Struzikk, Z.R., Stanley, H. E., 1999. Multifractality in human heartbeat dynamics, Nature, 399, 461-465. Kane, R. P., 2002. Evolution of geomagnetic aa index near sunspot minimum. Annales Geophysicae, 20, 1519−1527. Kane, R. P., 2002. Prediction of solar activity: Role of long-term variations. J. Geophys. Res., 107, doi:10.1029/2001JA0700247. Kodera, K., Kuroda, Y., 2002. Dynamical response to the solar cycle. J. Geophys. Res.,
17
107, doi:10.1029/2002JD002224. Mantua, N. J., Hare, S. R., Zhang, Y., Wallace, J. M., Francis, R. C., 1997. A Pacific Interdecadal Climate Oscillation with Impacts on Salmon Production. Bulletin of the American Meteorological Society, 78, 1069−1079. Maruyama, F., Kai, K., Morimoto, H., 2015. Wavelet-based multifractal analysis on climatic regime shifts. J. Meteor. Soc. Japan, 93, 331−341. Minobe, S., 1997. A 50-70 year climatic oscillation over the North Pacific and North America. Geophys. Res. Lett., 24, 683−686. Miyahara, H., Yokoyama, Y., Masuda, K., 2008. Possible link between multi-decadal climate cycles and periodic reversals of solar magnetic field polarity. Earth and Planetary Science Letters, 272, 290−295. Muzy, J. F., Bacry, E., Arneodo, A., 1991. Wavelets and multifractal formalism for singular signals: Application to turbulence data. Phys. Rev. Lett., 67, 3515−3518. Ohl A. I., 1966. Forecast of sunspot maximum number of cycle 20. Solice Danie 9, 84. Russell, C. T., Luhmann, J. G., Jian, L. K., 2010. How unprecedented a solar minimum? Reviews of Geophysics, 48, RG2004. Svalgaard, L., Kamide, Y., 2005. Sunspot cycle 24: Smallest cycle in 100 years? Geophysical Research Letter 32, L01104. Svalgaard, L., Kamide, Y., 2013. Asymmetric solar polar field reversals. Astrophys. J., 763, doi: 10.1088/0004-637X/763/1/23. Svensmark, H., 2000. Cosmic Rays and Earth's Climate. Space Science Reviews, 93, 175-185. Svensson, C., Olsson, J., Berndtsson, R., 1996. Multifractal properties of daily rainfall in two different climates. Water Resources Research, 32, 2463−2472. Thejll, P., Christiansen, B., Gleisner, H., 2003. On correlations between the North Atlantic Oscillation, geopotential heights, and geomagnetic activity. Geophys. Res. Lett., 30, 1347, doi: 10.1029/2002GL016598. Trenberth, K. E., Hurrel, J. W., 1994. Decadal atmosphere-ocean variations in the Pacific. Clim. Dyn., 9, 303−319.
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Upton, L., Hathaway, D. H., 2014. Predicting the sun’s polar magnetic fields with a surface flux transport model. Astrophys. J., 780, doi: 10.1088/0004-637X/780/1/5. Yasunaka, S., Hanawa, K., 2002. Regime Shifts Found in the Northern Hemisphere SST Field. J. Meteor. Soc. Japan, 80, 119−135. Yoshida, A., 2008. A seasonal change and long-term change of geomagnetic activity. Thesis for a doctorate, The Graduate University for Advanced Studies.
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Figure Captions Fig. 1: Multifractal spectrum τ(q) for individual SSN between 1967 and 1979. Fig. 2: Time series of the SSN, aa, and Ap indices. Fig. 3 (a) Original versus 61-month running mean of the SSN.（b）High-Frequency component (original minus 61-month running mean) of the SSN. (c) High-Frequency component of the SSN versus the aa index. Fig. 4: Time series of the τ(−6) of the SSN and the SSN. Solar cycle number of the SSN is also shown. The τ (−6) is the value of τ for q = −6. The red squares show monofractality and the green circles show multifractality for a 6-year period centered on the year shown. Fig. 5: Time series of the average solar polar field strength and SSN. Fig. 6: Time series of the τ(−6) of the average solar polar field strength and SSN. Fig. 7: Time series of the τ(−6) of the average solar polar field strength and average solar polar field. Fig. 8: Time series of the τ(−6) of the aa index and SSN. Fig. 9: Wavelet coherence (top) and phase (middle) between the F10.7 flux and upper air temperature at 30 hPa. 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 F10.7 flux leads the upper air temperature at 30 hPa and the negative value shown by the green, yellow and red shading means that the upper air temperature at 30 hPa leads the F10.7 flux. Time series of the SSN is also shown (bottom). Fig. 10: Wavelet power spectra of the upper air temperature at 30 hPa. Fig. 11: Time series of the τ(−6) of the PDO index.
20
Fig. 1. Multifractal spectrum τ(q) for individual SSN between 1967 and 1979.
Fig. 2. Time series of the SSN, aa, and Ap indices.
21
(a)
(b)
(c) Fig. 3. (a) Original versus 61-month running mean of the SSN.（b）High-Frequency component (original minus 61-month running mean) of the SSN. (c) High-Frequency component of the SSN versus the aa index.
22
Fig. 4. Time series of the τ(−6) of the SSN and the SSN. Cycle number of the SSN is also shown. The τ(−6) is the value of τ for q = −6. The red squares show monofractality and the green circles show multifractality for a 6-year period centered on the year shown.
Fig. 5. Time series of the solar polar field strength and SSN.
23
Fig. 6. Time series of the τ(−6) of the solar polar field strength and the SSN.
Fig. 7. Time series of the τ(−6) of the solar polar field strength and the solar polar field strength.
Fig. 8. Time series of the τ(−6) of the aa index and SSN.
24
8 2
0
0.5
Scale [years]
1
Wavelet Coherence
1960
1970
1980
1990
2000
2010
2000
2010
Time [years]
0
8 2
-PI
0.5
Scale [years]
PI
Phase
1960
1970
1980
1990
Time [years]
Fig. 9. Wavelet coherence (top) and phase (middle) between the F10.7 flux and upper air temperature at 30 hPa. 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 F10.7 flux leads the upper air temperature at 30 hPa and the negative value shown by the green, yellow and red shading means that the upper air temperature at 30 hPa leads the F10.7 flux. Time series of the SSN is also shown (bottom).
25
1
8 16 4 2
0
0.5 1
Scale [years]
Wavelet Power Spectrum
1960
1970
1980
1990
2000
2010
Time [years]
Fig. 10. Wavelet power spectra of the upper air temperature at 30 hPa.
Fig. 11. The τ(−6) of the PDO index.
26
Table 1.
Change of indices at climatic regime shift
regime polarity
τ(−6) of
aa
τ(−6) of
Ap
τ(−6) of Ap
τ(−6) of
SSN
index
aa index
index
index
PDO index
SSN
shift
1925/26
−
increase
1945/46
−
1957/58
＋
1970/71
＋
decrease minimum constant
1976/77
＋
increase
1988/89
−
increase
increase
minimum increase increase
decrease
increase
increase minimum
increase
increase
constant
minimum
27
increase increase
increase maximum
polarity: the polarity of solar polar field strength
increase
increase
S0273-1177(17)30411-8 http://dx.doi.org/10.1016/j.asr.2017.06.004 JASR 13258
To appear in:
Advances in Space Research
Received Date: Revised Date: Accepted Date:
20 August 2016 8 April 2017 2 June 2017
Please cite this article as: Maruyama, F., Kai, K., Morimoto, H., Wavelet-based multifractal analysis on a time series of solar activity and PDO climate index, Advances in Space Research (2017), doi: http://dx.doi.org/10.1016/j.asr. 2017.06.004
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Wavelet-based multifractal analysis on a time series of solar activity and PDO climate index
Fumio MARUYAMA*, Kenji KAI and Hiroshi MORIMOTO
Graduate School of Environmental Studies, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan
-------------------------------------* Corresponding author. E-mail: [email protected] (F. Maruyama) Keywords: sunspot number, solar flux, solar polar field strength, geomagnetic aa and Ap indices, PDO, climatic regime shift, wavelet, multifractal
1
Abstract
There is increasing interest in finding the relation between solar activity and climate change. In general, fractal properties may be observed in the time series of the dynamics of complex systems, such as solar activity and climate. This study investigates the relations among solar activity, geomagnetic activity, and climatic regime shift by performing a multifractal analysis. To investigate the change in multifractality, we apply a wavelet transform to time series. The change in fractality of the sunspot number (SSN) correlates closely with that of the solar polar field strength. For the SSN and solar polar field strength, a weak multifractality or monofractality is present at the maximum SSN, minimum SSN, and maximum solar polar field strength. A strong multifractality is present two years before the maximum SSN.
The climatic regime shift
occurrs when the SSN increases and the disturbance of the geomagnetic activity is large. At the climatic regime shift, the changes in the fractality of the Pacific Decadal Oscillation (PDO) index and changes in that of the solar activity indices corresponded with each other. From the fractals point of view, we clarify the relations among solar activity, geomagnetic activity, and climatic regime shift. The formation of the magnetic field of the sunspots is correlated with the solar polar field strength. The solar activity seems to influence the climatic regime shift. These findings will contribute to investigating the relation between solar activity and climate change.
2
1.
Introduction Various objects in nature exhibit the so-called self-similarity or fractal property.
Nature is full of fractals, for instance, trees, rivers, coastlines, mountains, clouds, and seashells. Monofractal signals are homogeneous in that they have the same scaling properties and are characterized by a fractal dimension.
On the other hand, multifractal
signals are nonuniform, more complex and can be decomposed into many subsets characterized by the different dimensions of the fractal. Fractal properties can also be observed in a time series representing the dynamics of complex systems. A change in fractality can appear with a phase transition or change of state. For example, the multifractal properties of daily rainfall were studied in an East Asian monsoon climate with extreme rainfall and in a temperate climate with moderate rainfall (Svensson et al., 1996). In both climates, the frontal rainfall and convective-type rainfall showed monofractality and multifractality, respectively. In another example, a healthy human heartbeat shows a multifractal character, while a diseased heart shows a monofractal character (Ivanov et al., 1999). In this study, we attempted to interpret climatic changes (that is, climatic regime shifts) by investigating fractality. We investigated fractality of the solar activity indices, F10.7 flux, SSN, aa and Ap indices for the first time. To examine the multifractal behavior of the climate index, we used a wavelet transform method because it is useful for the analysis of complex nonstationary time series. The wavelet transform can perform reliable multifractal analysis (Muzy et al., 1991); hence, we used the wavelet-based multifractal analysis to quantify the signals of higher complexity. We concluded that a climatic regime shift correlates to a change from multifractality to monofractality of the Pacific Decadal Oscillation (PDO) index (Maruyama et al., 2015). Minobe (1997) showed that a climatic regime shift is a sudden transition from one quasi-steady climatic state to another, and its transition period is much shorter than the individual epochs length of each climatic state. The climatic regime shift in the 1970s followed the warming of the tropical Pacific Ocean, the cooling of the central North Pacific, and the strengthening of the Aleutian Low. To identify the years when climatic
3
regime shifts occurred in the Sea Surface Temperature (SST) field, the time series of the original gridded SST and those of the Empirical Orthogonal Function (EOF) modes were examined, and the 1925/26, 1945/46, 1957/58, 1970/71, 1976/77, and 1988/89 climatic regime shifts were detected from the 1910s to the 1990s (Yasunaka and Hanawa, 2002). The SSN has an 11-year cycle and the solar magnetic field has a 22-year cycle, exactly twice that of the sunspot cycle, because the polarity of the field returns to its original value every two sunspot cycles.
During the past ~120 years, Earth's surface
temperature has correlated with both the decadal averages and solar cycle minimum values of the geomagnetic aa index (Cliver et al., 1998a). The 11-year averages of the SSN and aa index have been highly correlated for the past 150 years (Cliver et al., 1998b). The maxima of the aa index occurred near the maximum SSN and (or) later in the declining phase of the SSN (Kane, 2002). High correlation coefficients were found among the geomagnetic activity, sea level atmospheric pressure and surface air temperature (Bucha et al., 1998). In this study, we investigated the relations among the SSN and solar polar field strength, the influence of the solar activity on the geomagnetic field, and the influence of the solar activity and geomagnetic activity on a climatic regime shift from the view point of fractals. To examine the changes in multifractality, we performed a multifractal analysis on the SSN, solar polar field strength, 10.7-cm solar radio flux (F10.7), geomagnetic aa and Ap indices, and PDO index by using the wavelet transform.
2.
Data and Method of Analysis We used several solar activity indices to see the solar activity in a multifaceted
manner. For the solar activity indices, we used F10.7 flux, SSN, aa and Ap indices and for the climate indices, PDO index and upper-air temperature. We made use of the monthly SSN provided by Solar Influences Data Analysis Center (sidc.oma.be), the solar radio flux at 10.7 cm provided by the National Oceanic and Atmospheric Administration’s (NOAA’s) Space Weather Prediction Center (www.swpc.noaa.gov). The F10.7 flux is an excellent indicator of the solar activity,
4
because F10.7 indicates a good correlation with SSN. We used the solar polar field strength throughout the solar sunspot cycle provided by the Wilcox Solar Observatory (wso. stanford.edu) and the geomagnetic aa and Ap indices of the solar activity provided by NOAA. The aa index is a measure of disturbance in the Earth’s magnetic field based on magnetometer observations at two nearly antipodal stations in Australia and England. The Ap index provides a daily average level for the geomagnetic activity. We used the monthly PDO index provided by NOAA’s Climate Prediction Center (CPC). The PDO index is the leading principal component of the monthly SST anomalies in the North Pacific Ocean (Mantua et al., 1997).
We also used the global upper-air temperature
anomaly provided by the Met Office Hadley Centre. For the analysis, we used the Daubechies wavelet, which is widely used in solving a broad range of problems, e.g., self-similarity properties of a signal or fractal problems and signal discontinuities.
We used a discrete signal which was fitted the Daubechies
mother wavelet with the capacity of accurate inverse transformation. Thus, we can precisely calculate the following optimum τ(q), which can be regarded as a characteristic function of the fractal behavior.
We can define the τ(q) from the power-law behavior of
the partition function, as shown in equation (2). We then calculated the scaling of the partition function Zq(a), which is defined as the sum of the q-th powers of the modulus of the wavelet transform coefficients at scale a, where q is the q-th moment. In our calculation, the wavelet-transform coefficients did not become zero. Thus, for an accurate calculation, the summation was considered for the whole set. Muzy et al. (1991) defined Zq(a) as the sum of the q-th powers of the local maxima of the modulus to avoid dividing by zero. We obtained the following partition function Zq(a):
Zq a
W
f a, b
q
,
(1)
where Wφ[f](a, b), a, and b are the wavelet coefficient of function f, a scale parameter, and a space parameter, respectively.
W f a, b is defined as below.
5
1
W f a, b
,where f t is data and
f t
a
*
t b dt a
(2)
is wavelet function. For small scales, we expect
Zq a ~ a
q
.
(3)
First, we investigated the changes in Zq(a) in the time series at a different scale a for each moment q. We plotted the logarithm of Zq(a) against that of time scale a. Here τ(q) is the slope of the fitted straight line for each q. Next, we plotted τ(q) versus q, which is the multifractal spectrum. The time window was advanced by one year, which was repeated.
Monofractal and multifractal signals were defined as follows: A
monofractal signal corresponds to a straight line for τ(q), while a multifractal signal τ(q) We calculated the R2 value, which is the
is nonlinear (Frish and Parisi, 1985).
coefficient of determination, for the fitted straight line. If R2 ≥ 0.98, the time series is monofractal; if 0.98 > R2, it is multifractal. A time window was fixed to 6 years for the following reasons. We calculated the wavelets with time windows of 10, 6, and 4 years. Initially, when a time window was 10 years, a fractality changed slowly.
By integrating the wavelet coefficient in a wide
range, small changes were canceled.
Thus, this case was inappropriate to find a fast
change of climatic regime shift. Next, when the time window was 4 years, the fractality changed quickly. The overlap of the first and following data was 3 years, which was shorter than the 9 years when the time window was 10 years, and the change of fractality was large. Thus, this case was also inappropriate. Finally, when the time window was 6 years, a moderate change in fractality was observed. Hence, the time window was fixed to 6 years. We calculated the multifractal spectrum τ(q) of the SSN between 1910 and 2010. The multifractal spectrum τ(q) between 1967 and 1979 is shown in Fig. 1. The data
6
were analyzed in 6-year sets; as an example, the multifractal spectrum τ(q) of s67 was calculated between 1967 and 1972. To study the change in fractality, the time window was advanced by one year and the multifractal spectrum τ(q) was obtained from s67 to A monofractal signal corresponds to a straight line for τ(q), whereas for a
s76.
multifractal signal, τ(q) is nonlinear. In Fig. 1, the constantly changing curvature of the τ(q) curves for s67, s68, and s72 – s74 suggests multifractality. In contrast, τ(q) is linear for s69 – s71, which indicates monofractality. We plotted the τ(−6) of each index, where q = −6 is the appropriate number for showing a change in τ. The large negative values of τ(−6) show large multifractality. Thus, τ(−6) is not always equal to the fractality gained from the R2 value. We show the wavelet coherence and phase using the Morlet wavelet between the F10.7 flux and upper air temperature at 30 hPa. The Morlet wavelet is most often used and offers a good frequency resolution.
3. Results 3.1. The correlation between the SSN and the geomagnetic aa and Ap indices We investigate the relation between the SSN and the geomagnetic aa and Ap indices through multifractal analysis and the wavelet coherence and phase.
Figure 2 shows the
time series of the SSN and the aa and Ap indices as well as the solar cycle number of the SSN.
The sunspot cycles are typically asymmetric in shape with a rapid rise to
maximum and a slower decline to minimum.
Asymmetric solar polar field reversals are
simply a consequence of the asymmetry of solar activity (Svalgaard and Kamide, 2013). The magnetic cycle consists of two sunspot cycles.
Feynman (1982) noted that
geomagnetic activity has two different sources, one due to solar activity which follows the sunspot cycle, and another due to recurrent high-speed solar wind streams which peaks during the decline of each cycle. The changes in the aa and Ap indices are very similar. 11-year periodicity is observed in the aa and Ap indices, which indicates the influence of the Sun. The time series of the aa index has two peaks, which correspond to the maximum SSN and decrease in the SSN. When the SSN is at a minimum, the aa
7
and Ap indices are also at a minimum and the geomagnetic disturbance is small.
Hence,
when solar activity is not strong, the disturbance of the geomagnetic field is weak. Ohl (1966) found that the minimum level of geomagnetic activity seen in the aa index near the time of the minimum sunspot cycle is a good predictor for the amplitude of the next cycle. We examine the correlation between the values of the aa and Ap indices before the maximum SSN and maximum SSN of the next cycle. correlation coefficients are r = 0.80 and r = 0.87, respectively.
The obtained
The stronger the
geomagnetic distarbance, the greater the maximum SSN of the next cycle. Hence, if the disturbance of the geomagnetic field is strong, the solar activity of the next cycle is also strong. We obtain the wavelet coherence and phase using the Morlet wavelet between the SSN and the aa index. The coherence between the SSN and aa index at around the 11-year scale is strong and the leads of the SSN are observed. We show the original versus 61-month running mean of the SSN in Fig. 3(a). The obtained high-frequency component, which is original minus the 61-month running mean of the SSN, is shown in Fig. 3(b). The high-frequency component of the SSN versus the aa index is shown in Fig. 3(c). The obtained high-frequency component is similar to the time series of the aa index. A change in the SSN appears to have influenced the geomagnetic activity.
3.2. The relation between the SSN and solar polar field strength We investigate the change in the SSN. To detect the changes in multifractality, we examine the multifractal analysis of the SSN. The τ(−6) of the SSN and SSN as well as the solar cycle number of the SSN are shown in Fig. 4. The SSN has an 11-year and a 22-year cycle for the reversal of the Sun's magnetic field.
τ(−6) is the value of τ for q =
−6 and indicates the fractality of the index. The red square and green circle represent monofractality and multifractality for the six years around the plotted year, respectively. For example, the red square for 1962 in the SSN shows monofractality between 1959 and 1964.
The data are excluded if we cannot distinguish between monofractality and
8
multifractality. When the SSN is maximum (minimum), the τ(−6) of the SSN become maximum, as shown by 1 (3) in Fig. 4, when the SSN shows monofractality or weak multifractality. The τ(−6) of the SSN became minimum two years before the maximum SSN, shown by 2 in Fig. 4, when the change in the SSN is the largest. About three years before the minimum SSN, the τ(−6) of the SSN become maximum, shown by 4 in Fig. 4 and the multifractality is the weakest. The solar polar field strength (from Wilcox Solar Observatory data) is shown in Fig. 5. The changes in the solar polar field strength and SSN are large two years before the maximum SSN of the solar cycles 22, 23, and 24 as shown by the arrow in Fig. 5, which is determined from the slope of the change in the graph in Fig. 5. τ(−6) of the solar polar field strength in Fig. 6.
Next, we show the
At the maximum and minimum SSN,
the τ(−6) of the solar polar field strength become maximum, as shown by 1 and 3, respectively, in Fig. 6, when the solar polar field strength indicates monofractality or weak multifractality. The τ(−6) of the solar polar field strength become minimum two years before the maximum SSN shown by 2 in Fig. 6. Hence, two years before the maximum solar activity, the minimum τ(−6) of the SSN and the average solar polar field strength were observed, when the changes in the average solar polar field strength and SSN are large.
Before the maximum SSN, the multifractality of the SSN and solar polar
field strength become the strongest; at the maximum SSN they become the weakest, which indicates a change from multifractality to monofractality. A comparison of Fig. 5 with Fig. 6 indicates that when the change in the solar polar field strength is small, the multifractality is weak.
About three years before the
minimum SSN, the τ(−6) of the SSN and the solar polar field strength become maximum (shown by 4 in Fig. 6) and the multifractality is the weakest. We show the τ(−6) of the solar polar field strength and solar polar field strength in Fig. 7. When the solar polar field strength is maximum and minimum (approximately 0), the values of the τ(−6) become maximum and the multifractality is weak, as shown by 4 and 1, respectively, in Fig. 7.
9
3.3. The relationship between solar activity and the geomagnetic field The τ(−6) of the geomagnetic aa index and the SSN are shown in Fig. 8. The result of the geomagnetic Ap index is not shown, while the tendency is smilar to that of aa index. When the SSN is minimum and maximum, the τ(−6) of the aa and Ap indices show, for most of the cases, weak multifractality, as shown by 3 and 1, respectively, in Fig. 8. The τ(−6) of the aa index at the minimum SSN is larger than that of the maximum SSN, because the disturbance of the aa index at the minimum SSN is small and the aa index indicates weak multifractality. Hence, when the disturbance of the geomagnetic field is small, the geomagnetic field indicates weak multifractality. Two years before the maximum SSN, the aa and Ap indices are small, the disturbance of the geomagnetic field is small, and the τ(−6) of the aa index becomes minimum, as shown by 2 in Fig. 8.
3.4. The influence of solar activity on climatic regime shift To investigate the influence of solar activity on the climate, we show the wavelet coherence and phase between the F10.7 flux and upper air temperature at 30 hPa in Fig. 9. We show the wavelet power spectra of the upper air temperature at 30 hPa in Fig. 10. A strong 10-year frequency is observed, which indicates the clear influence of the solar activity. The coherence is very strong in the 10-year cycle and the phase of F10.7 flux leaded; therefore, solar activity appears to have influenced the upper air temperature. The coherence becomes strong around the 1976/77 climatic regime shift; therefore, solar activity appears to have influenced the 1976/77 climatic regime shift. Solar activity influences the geomagnetic activity and North Atlantic Oscillation through solar wind (Thejll, 2003).
We examine the wavelet coherence and phase
between the geomagnetic aa index and the upper air temperature at 30 hPa.
The
coherence was very strong in 10-year cycle and the phase of the aa index leaded; therefore the geomagnetic activity appears to have influenced the upper air temperature. The coherence becomes strong around the 1976/77 climatic regime shift; therefore, the
10
geomagnetic activity also appears to have influenced the 1976/77 climatic regime shift. We explain the influence of solar activity on climatic regime shift by means of fractals. We show the τ(−6) of the PDO index in Fig. 11. The changes at the climatic regime shifts of the SSN, aa, and Ap indices and the τ(−6) of the SSN, aa, Ap, and PDO indices at the Figs. 4, 8, and 11 are shown in Table 1. The climatic regime shifts occur when the SSN increases (except for the 1970/71 climatic regime shift) and the solar activity becomes strong.
4. Discussion 4.1. The correlation between the SSN and solar polar field strength When the SSN and solar polar field strength are maximum, the τ(−6) of the SSN and solar polar field strength become maximum.
The τ(−6) of the SSN becomes
minimum two years before the maximum SSN (as shown in Fig. 4), when the change of the SSN is large. The τ(−6) of the solar polar field strength and the SSN become minimum and two years before the maximum SSN of solar cycles 22, 23, and 24. The multifractalities of the SSN and solar polar field strength become strong two years before the maximum SSN, when the changes in the solar polar field strength and SSN are large. From the two years before the maximum SSN to the maximum SSN, the SSN and solar polar field strength indicate a change from multifractality to monofractality.
For 1980 to
1992, the changes in the SSN and solar polar field strength show strong similarity and correlated closely. When the change in the solar polar field strength is small, the multifractality is weak. There are three maximum τ(−6) of the SSN where the multifractality is the weakest: at the maximum SSN, at the minimum SSN, and at the SSN about three years before the minimum SSN. About three years before the minimum SSN, the τ(−6) of the SSN and solar polar field strength become maximum and the multifractality is the weakest. The solar polar field strength is the strongest during an interval of about three years before the sunspot minimum (Svalgaard and Kamide, 2005).
Hence, when the solar polar field
strength is maximum, the multifractality of the solar polar field strength and SSN become
11
the weakest.
The solar polar field strength becomes large and the SSN becomes
minimum. The changes in the τ(−6) of the SSN and solar polar field strength correspond to each other, and therefore the SSN and solar polar field closely correlate. The peaks of τ(−6) for the SSN and solar polar field strength are classified into four peaks, the maximum peaks at the maximum SSN shown by 1 in Figs. 4 and 6, the minimum SSN shown by 3, and the maximum solar polar field strength shown by 4 and the minimum peak at two years before the maximum SSN shown by 2. The solar polar field strength makes the magnetic field of the sunspots when the solar cycle is the minimum (Upton and Hathaway, 2014). In this study, we compare the maximum SSN with the maximum solar polar field strength before the maximum SSN. When the maximum solar polar field strength before the maximum SSN is larger, the maximum SSN of the next cycle is larger.
Hence, the formation of the magnetic field of
the sunspots is correlated with the solar polar field strength. The fractal dimensions decrease with increasing mean magnetic field strength, implying that the magnetic field distribution is more regular in active regions (Ioshpa et al., 2008). The τ(−6) at the maximum solar polar field strength (shown by 4 in Fig. 7) is greater than that at the maximum SSN (shown by 1 in Fig. 7), when the solar polar field strength is nearly zero. Hence, when the solar polar field strength is large, the multifractality is weak, which coincides with the result of Ioshpa et al.(2008). However, when the change in the solar polar field strength is great as shown by the arrow in Fig. 7, τ(−6) become minimum and the multifractality is strong.
When the SSN and solar polar
field strength are maximum, the τ(−6) of the SSN and solar polar field strength become maximum.
4.2. The relation between solar activity and the geomagnetic field When the SSN is minimum, the geomagnetic aa and Ap indices are minimum (as shown in Fig. 2) and the values of the aa and Ap indices show weak multifractality (as shown by 3 in Fig. 8). Hence, when the solar activity is not strong, the disturbance of
12
the geomagnetic field is small and the values of the aa and Ap indices show weak multifractality. Two years before the maximum SSN, the values of the aa and Ap indices are small, so the disturbance of the geomagnetic field is small. Hence, two years before the maximum solar activity, the τ(−6) of the SSN becomes minimum (as shown by 2 in Fig. 4), and the disturbance of the geomagnetic field is small, and the τ(−6) of the aa and Ap indices become minimum. The obtained correlation coefficients between the values of the aa and Ap indices before the maximum SSN and the maximum SSN of the next cycle are r = 0.80 and r = 0.87, respectively. The greater the geomagnetic disturbance, the larger the maximum SSN of the next cycle. We examine the correlation between the τ(−6) of the aa and Ap indices about two to three years before the maximum SSN and at the maximum SSN of the next cycle. The obtained correlation coefficient is r = 0.30 and r = 0.88 for the aa and Ap indices, respectively. When the τ(−6) of the aa and Ap indices are small and multifractality is strong, the SSN of the next cycle is large.
Hence, when the
disturbance of the geomagnetic field is large and the aa and Ap indices show strong multifractality, the solar activity of the next cycle is strong. Yoshida (2008) determined that when the minimum geomagnetic activity is large, then the maximum solar activity of the next cycle is strong, which coincides with our result.
4.3. The influence of solar activity on climatic regime shift Fractal behavior may be observed in many self-organization systems, representing the order or disorder included in the systems. There is a model that deals with an interaction between magnetism and fluid (Bartosz et al.,2000). It is reported that a phenomenon of self-organization occurs and a fractal is observed. Their model was performed in a micro level, but a similar mechanism might occur in an interaction between the Earth and Sun. Therefore, the study of fractals become an important issue to assess a relation between the solar activity and the Earth’s climate. As shown in 3.4, the climatic regime shift appears to be influenced by the solar activity. Additionally, at the climatic regime shift, the fractality of the PDO index changes from multifractality to
13
monofractality and τ(−6) increases. For SSN and F10.7 flux, the change from multifractality to monofractality is not always seen at the climate regime shift. The τ(−6) of the SSN at three climatic regime shifts increases and those at two climatic regime shifts are minimum, just before the increase. All of the τ(−6) of the aa index are minimum and most of the τ(−6) of the Ap index are minimum. Hence, at the climatic regime shift, the changes in the τ(−6) of the PDO index and solar activity indices correspond to each other. The PDO appears to be influenced by the solar activity. The wavelet coherence between the SSN (F10.7) and PDO index is strong and the phase of SSN (F10.7) leaded (not shown). Hence, the PDO appears to be influenced by solar activity. A climatic regime shift is equal to a change from multifractality to monofractality of the PDO index (Maruyama et al., 2015). Hence a climatic regime shift is also influenced by the solar activity. These results show that the solar activity influences on the climate and climatic regime shift. Solar activity influences the climate through the ozone. A change in temperature in the stratosphere (due to a change in ultraviolet rays by the change of ozone) changes the stratosphere circulation. That spreads below through the dynamic effect and influences the climate of the troposphere (Kodera and Kuroda, 2002). A close correlation between cosmic ray flux, which increases when the solar activity is weak, and variations in Earth's cloud cover has been demonstrated (Svensmark, 2000). Because clouds are important for the Earth's energy balance, a solar influence on clouds can be the main cause for the observed correlations between the Sun and Earth's climate. A climatic regime shift occurs when the geomagnetic aa and Ap indices increase and the τ(−6) of the aa and Ap indices are minimum.
At the climatic regime shift, the
multifractality of the geomagnetic activity index, influenced by the Sun, became maximum and the disturbance of the geomagnetic activity is large.
5. Conclusions In this study, we investigated the relation between the SSN and the solar polar field strength, the influence of solar activity on the geomagnetic field, and the influence of
14
solar activity on the climatic regime shift by performing a multifractal analysis. To detect the changes in multifractality, we performed a multifractal analysis on the SSN, F10.7 flux, solar polar field strength, geomagnetic aa and Ap indices, and PDO index using the wavelet transform.
Moreover we examined the wavelet coherence and phase
of these indices. (1) The relation between the SSN and solar polar field strength ・When the SSN and solar polar field strength are maximum, their τ(−6) become maximum and the multifractality becomes weak. The τ(−6) shows the fractality of the index. ・The minimum τ(−6) of the SSN and solar polar field strength are observed two years before the maximum SSN.
Before the maximum solar activity, the multifractality of the
SSN and solar polar field strength becomes strong, when the changes in the SSN and solar polar field strength are large. ・Solar polar field strength becomes maximum about three years before the minimum SSN, when the τ(−6) of the SSN and solar polar field strength become maximum and the multifractality becomes the weakest. The solar polar field strength becomes large and the SSN becomes minimum. ・The changes in the τ(−6) of the SSN correlate closely with those of the solar polar field strength. A weak multifractality or monofractality is shown at the maximum SSN, the minimum SSN, and the maximum solar polar field strength. A strong multifractality is shown at two years before the maximum SSN.
When the solar polar field strength
becomes stronger, the maximum SSN of the next cycle becomes large.
Consequently,
the formation of the magnetic field of the sunspots is correlated with the solar polar field strength. (2) The relation between solar activity and the geomagnetic field Disturbances in the geomagnetic field are small when solar activity is not active. When the SSN is maximum or minimum, the τ(−6) of the aa and Ap indices become maximum and the geomagnetic field shows weak multifractality. Two years before the maximum SSN, the τ(−6) of the aa and Ap indices become minimum. These results
15
show a similar tendency of τ(−6) for the SSN and solar polar field strength, which indicates that solar activity influences geomagnetic activity.
The greater the
geomagnetic disturbance, the larger the maximum SSN of the next cycle.
If the
disturbance of the geomagnetic field is large and the aa and Ap indices show strong multifractality, the solar activity of the next cycle is strong. (3) The influence of solar activity on climatic regime shift The relation between the F10.7 flux and upper air temperature indicates that solar activity influences the upper air temperature. The climatic regime shift occurs when the SSN increases and the solar activity becomes active and the disturbance of the geomagnetic activity is large. At the climatic regime shift, the changes in the τ(−6) of the PDO and solar activity indices correspond to each other. By analyzing wavelet coherence and phase, we demonstrate that the solar activity relates to and influences the PDO. Hence, the PDO is influenced by the solar activity and a climatic regime shift seems to be influenced by solar activity.
16
References Bartosz, A. G., Howard, A. S., George, M. W., 2000. Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid-air interface. Nature, 405, 1033-1036. Bucha, V., Bucha, V. Jr., 1998. Geomagnetic forcing of changes in climate and in the atmospheric circulation. Journal of Atmospheric and Solar-Terrestrial Physics, 60, 145−169. Cliver, E. W., Boriakoff, V., Feynman, J., 1998a. Geomagnetic activity and the solar wind during the Maunder Minimum. Geophys. Res. Lett., 25, 897−900. Cliver, E. W., Boriakoff, V., Feynman, J., 1998b. Solar variability and climate change: Geomagnetic aa index and global surface temperature. Geophysical Res. Lett., 25, 1035−1038. Feynman, J., 1982. Geomagnetic and solar wind cycles, 1900−1975. J. Geophys. Res., 87, 6153−6162. Frish, U., Parisi, G., 1985. On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. edited by Ghil, M., R. Benzi, and G. Parisi, pp. 84−88, North-Holland, New York. Hathaway, D. H., 2009. Solar Cycle Forecasting. Space Sci. Rev., 144, 401−412. Ioshpa, B. A., Obridko, V. N., Rudenchik, E. A., 2008. Fractal Properties of Solar Magnetic Fields. Astronomy Letters, 34, 210−216. Ivanov, P. Ch., Amaral, L. A. N., Goldberger, A. L., Havlin, S., Rosenblum, M. G., Struzikk, Z.R., Stanley, H. E., 1999. Multifractality in human heartbeat dynamics, Nature, 399, 461-465. Kane, R. P., 2002. Evolution of geomagnetic aa index near sunspot minimum. Annales Geophysicae, 20, 1519−1527. Kane, R. P., 2002. Prediction of solar activity: Role of long-term variations. J. Geophys. Res., 107, doi:10.1029/2001JA0700247. Kodera, K., Kuroda, Y., 2002. Dynamical response to the solar cycle. J. Geophys. Res.,
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Upton, L., Hathaway, D. H., 2014. Predicting the sun’s polar magnetic fields with a surface flux transport model. Astrophys. J., 780, doi: 10.1088/0004-637X/780/1/5. Yasunaka, S., Hanawa, K., 2002. Regime Shifts Found in the Northern Hemisphere SST Field. J. Meteor. Soc. Japan, 80, 119−135. Yoshida, A., 2008. A seasonal change and long-term change of geomagnetic activity. Thesis for a doctorate, The Graduate University for Advanced Studies.
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Figure Captions Fig. 1: Multifractal spectrum τ(q) for individual SSN between 1967 and 1979. Fig. 2: Time series of the SSN, aa, and Ap indices. Fig. 3 (a) Original versus 61-month running mean of the SSN.（b）High-Frequency component (original minus 61-month running mean) of the SSN. (c) High-Frequency component of the SSN versus the aa index. Fig. 4: Time series of the τ(−6) of the SSN and the SSN. Solar cycle number of the SSN is also shown. The τ (−6) is the value of τ for q = −6. The red squares show monofractality and the green circles show multifractality for a 6-year period centered on the year shown. Fig. 5: Time series of the average solar polar field strength and SSN. Fig. 6: Time series of the τ(−6) of the average solar polar field strength and SSN. Fig. 7: Time series of the τ(−6) of the average solar polar field strength and average solar polar field. Fig. 8: Time series of the τ(−6) of the aa index and SSN. Fig. 9: Wavelet coherence (top) and phase (middle) between the F10.7 flux and upper air temperature at 30 hPa. 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 F10.7 flux leads the upper air temperature at 30 hPa and the negative value shown by the green, yellow and red shading means that the upper air temperature at 30 hPa leads the F10.7 flux. Time series of the SSN is also shown (bottom). Fig. 10: Wavelet power spectra of the upper air temperature at 30 hPa. Fig. 11: Time series of the τ(−6) of the PDO index.
20
Fig. 1. Multifractal spectrum τ(q) for individual SSN between 1967 and 1979.
Fig. 2. Time series of the SSN, aa, and Ap indices.
21
(a)
(b)
(c) Fig. 3. (a) Original versus 61-month running mean of the SSN.（b）High-Frequency component (original minus 61-month running mean) of the SSN. (c) High-Frequency component of the SSN versus the aa index.
22
Fig. 4. Time series of the τ(−6) of the SSN and the SSN. Cycle number of the SSN is also shown. The τ(−6) is the value of τ for q = −6. The red squares show monofractality and the green circles show multifractality for a 6-year period centered on the year shown.
Fig. 5. Time series of the solar polar field strength and SSN.
23
Fig. 6. Time series of the τ(−6) of the solar polar field strength and the SSN.
Fig. 7. Time series of the τ(−6) of the solar polar field strength and the solar polar field strength.
Fig. 8. Time series of the τ(−6) of the aa index and SSN.
24
8 2
0
0.5
Scale [years]
1
Wavelet Coherence
1960
1970
1980
1990
2000
2010
2000
2010
Time [years]
0
8 2
-PI
0.5
Scale [years]
PI
Phase
1960
1970
1980
1990
Time [years]
Fig. 9. Wavelet coherence (top) and phase (middle) between the F10.7 flux and upper air temperature at 30 hPa. 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 F10.7 flux leads the upper air temperature at 30 hPa and the negative value shown by the green, yellow and red shading means that the upper air temperature at 30 hPa leads the F10.7 flux. Time series of the SSN is also shown (bottom).
25
1
8 16 4 2
0
0.5 1
Scale [years]
Wavelet Power Spectrum
1960
1970
1980
1990
2000
2010
Time [years]
Fig. 10. Wavelet power spectra of the upper air temperature at 30 hPa.
Fig. 11. The τ(−6) of the PDO index.
26
Table 1.
Change of indices at climatic regime shift
regime polarity
τ(−6) of
aa
τ(−6) of
Ap
τ(−6) of Ap
τ(−6) of
SSN
index
aa index
index
index
PDO index
SSN
shift
1925/26
−
increase
1945/46
−
1957/58
＋
1970/71
＋
decrease minimum constant
1976/77
＋
increase
1988/89
−
increase
increase
minimum increase increase
decrease
increase
increase minimum
increase
increase
constant
minimum
27
increase increase
increase maximum
polarity: the polarity of solar polar field strength
increase
increase