are not periodic, fluctuations generally do not repeat at equal interval of time. The span between two specific consecutive maximum, or minimum, of SOI or SSTI is used to
Ž .
identify the corresponding interfluctuation period or interfluctuation span . The words signal or component refer to the contribution from a band along the length of data
record, or along another specified period.
3. Results
3.1. Fluctuations in the atmospheric mass oÕer the tropical pacific Fig. 2 shows five contributing bands to summer SOI from indicated wavelet periods
Ž .
the figure for winter SOI is not shown . The sub-decadal component is shown in the second panel from bottom. The remaining upper panels show the decadal and the other
interdecadal components. Ž .
The main features shown in Fig. 2 are: 1
an amplification of the bidecadal Ž .
component starting in mid-1930s, 2 an amplification of the decadal component starting Ž .
early 1960s. 3 A weak tridecadal component. We analyze next the total interdecadal component of SOI; the other components are analyzed in following sections.
The total interdecadal component of winter SOI is singled out in the upper panel of Ž
. Fig. 3 Summer components of SOI is similar, as seen in Fig. 4 . Main features are as
follows. Ž .
1 There is an amplitude modulation of the interdecadal component of the SOI. Apparently, a cycle of the amplitude modulation goes from around 1905 to mid-1970s.
Ž .
This suggests a 70-year span amplitude modulation for this realization sample of 100-year extent.
Ž Fig. 4. Contributions to winter SOI and to summer SOI, from total interdecadal band wavelet period 10
. years .
Table 1 Average span between consecutive extremes of same type for indicated components of SOI
Component Season
Average Standard
Ž .
Ž .
span year error year
Decadal Winter
13.1 1.8
Summer 12.8
1.6 Bidecadal
Winter 22
2.7 Summer
19.2 6.0
Total interdecadal Winter
15.6 4.7
Summer 12.8
2.3
Ž . 2 In 1982, year of a strong ENSO, the winter SOI reaches a large negative value
Ž .
see bottom panel of Fig. 3 that was attained with the strong negative contribution from Ž
. the interdecadal band
Figs. 3 and 4 . This results supports suggestions made by Ž
. Ž
. Brassington 1977 and Zhang et al. 1998 , based on indirect evidence obtained using
Fourier and principal component analyses respectively, on the existence of a construc- tive interaction between decadal and sub-decadal signals of the SOI.
A Kolmogorov–Smirnov test of goodness-of-fit indicates that inter-fluctuation peri- Ž
ods in the decadal, bidecadal and total interdecadal components of SOI winter and .
summer follow a normal distribution. The values of the parameters of the distributions are indicated in Table 1. The average interfluctuation period for these three components
does not vary significantly between seasons. Fig. 5 shows the fitting of a normal distribution to the interfluctuation period of the total interdecadal component of summer
SOI; we see that the interfluctuation period in this case is dominated by the decadal component. The decadal and bidecadal components of SOI have an average interfluctua-
Ž .
tion period of 13 years, and 21 years, respectively after averaging on both seasons .
Fig. 5. Straight line shows a normal distribution fitted to the span between minimum in total interdecadal component of winter SOI.
Fig. 6. Partial contributions to summer SSTI from bands of wavelet period as indicated in the y-axis. Year corresponds to November. The vertical scale is stretched for period greater than 10 years, to better depict
signal features.
3.2. Fluctuations of the sea surface temperature in the equatorial Pacific Fig. 6 shows partial contributions to the reconstruction of summer SSTI. Among
components with wavelet period 10-year, partial contributions from the decadal band are the largest. The magnitude of the bidecadal component is second in importance.
Partial contributions to SSTI from bands encompassing wavelet period larger than 27 year are small.
Seasonal differences in interdecadal components of winter and summer SSTI, Fig. 7, are not significant. There is an important maximum of SSTI centered in 1901. In
Fig. 7. Total interdecadal component of winter SSTI and summer SSTI.
Table 2 Average span between consecutive extremes of same type for indicated components of SSTI
Component Season
Average Standard
Ž .
Ž .
span year error year
Decadal Winter
13.8 1.2
Summer 11.6
2.9 Bidecadal
Winter 20.5
0.6 Summer
20.2 0.5
Total interdecadal Winter
13.3 1.6
Summer 11.4
2.6
mid-1940s, the quasi-regular pattern of pulsation changes into an almost linear ascend- Ž
. ing
warning trend. A new change occurs in early-1960s. Thereafter, the signal
amplifies. In another section, these features are related to changes in SOI. Kolmogorov–Smirnov goodness-of-fit tests indicate that interfluctuation period for
decadal, bidecadal and interdecadal components of SSTI follows a normal distribution; average values and standard errors are shown in Table 2. Fig. 8 shows the fitting of a
normal distribution to the interfluctuation periods observed in the interdecadal band of summer SSTI. Standard error values cast doubts on the potential of statistical models for
long-term — 10- to 20-year ahead — forecasting.
3.3. Similarity of interfluctuation period of total interdecadal component in SOI and in SSTI
A two-sample test statistics on the equality of average interdecadal oscillation period of SOI and SSTI indicate that interdecadal fluctuations in SSTI and SOI have a
Fig. 8. Fitting of a normal distribution to observed values of span between consecutive maxima of total interdecadal component in summer SSTI.
Fig. 9. Co-spectral density between total interdecadal component in summer SOI and summer SSTI. The 13-year wave period produces the largest contribution to cross-correlation between signals.
Ž dominant cadence with an average period equal to 13 years. Co-spectrum analysis Fig.
. 9 also indicates that SSTI and SOI signals are out of phase, with a negative peak value
at 13-year period. Negative sign of co-spectrum peak indicates that these components of SOI and SSTI are out of phase. The co-spectrum measures the contribution of oscilla-
tions of different frequencies to the total cross covariance at lag zero between the time
Ž .
series of SOI and SSTI Panofsky and Brier, 1968 . 3.4. Statistical relationship between amplitudes of fluctuations of SOI and SSTI
A linear regression between the amplitudes of associated extremes of opposite sign of SOI and SSTI were carried out. ‘Associated extremes of opposite sign’ means that each
Ž .
Ž .
positive negative extreme of SOI is related to the closest negative positive extreme of SSTI. Table 3 shows the results for three components. Amplitudes are negatively
Table 3 Liner regression between the value of an extreme in SOI and the closest extreme of opposite sign in SSTI, at
indicated components Null hypothesis is ‘‘slope of linear regression is zero’’, and it is rejected if p-level is smaller than 0.05. r is
the value of the linear correlation coefficient. Test statistics is Student’s t.
Component Season
Linear regression p-level
r Ž
. slope
Ž .
Total interdecadal Winter
SSTI s 0.049–0.70 SOI
q ´ 0, 0.157 0.0006
y0.84
ID ID
Ž .
Summer SSTI
s 0.048–0.54 SOI q ´ 0, 0.226
0.011 y0.68
ID ID
Ž .
Decadal Winter
SSTI s 0.006–0.639 SOI
q ´ 0, 0.070 0.00001
y0.91
De c Dec
Ž .
Summer SSTI
s 0.007–0.722 SOI q ´ 0, 0.143
0.001 y0.79
De c Dec
Ž .
Bidecadal Winter
SSTI s 0.004–0.894 SOI
q ´ 0, 0.061 0.001
y0.92
Bidc Bidc
Ž .
Summer SSTI
sy0.005–0.61 SOI q ´ 0, 0.050
0.001 y0.91
Bidc Bidc
Fig. 10. Regression between peak value of summer SSTI, on value of associated extreme of opposite sign of summer SOI, in the bidecadal band. Correlation coefficient is y0.91.
correlated at a significance level of 5. Fig. 10 is a graphical example of results for the summer bidecadal components of SOI and SSTI.
After a sudden decrease of amplitude, the bidecadal component of SSTI is amplifying Ž
. since 1915 see Fig. 11 . Amplitudes of bidecadal SOI are amplifying since start of
record.
3.5. RelatiÕe phase between components of SOI and SSTI In this section, an analysis of the relative phase between components of SOI and
Ž SSTI is carried out; results are expressed in terms of lag-BFOS see Section 2.4 for
. definition of lag-BFOS . Bidecadal, interdecadal, and decadal components of SOI and
Ž .
SSTI see Figs. 11–13
show a change in relative phase around mid-1930s. The lag-BFOS between components of SOI and SSTI was computed using cross-correlation
Ž .
Fig. 11. Bidecadal 17 to 27 years component of summer SSTI and summer SOI.
Fig. 12. Comparison of time series of total interdecadal component of winter SSTI and winter SOI.
analyses. Fig. 14 shows an example of cross-correlation analysis; this corresponds to the bidecadal winter components before 1935. Relative maximum of negative cross-correla-
tion closest to zero lag is found at a lag of 2 years; positive lag corresponds to SOI leading SSTI. Table 4 shows the results of cross-correlation analyses. Before mid-1930s,
the SOI component was leading the SSTI component at those three bands. After mid-1930s, the decadal and interdecadal components of SOI and SSTI are locked
Ž .
completely out of phase relative phase equal to 1808, or 0-year lag-BFOS8 ; while the bidecadal component of SSTI leads the SOI component.
Ž .
In the tridecadal band figure not shown , the SSTI leads SOI throughout the period under study.
Fig. 13. Partial contribution from decadal band to the reconstruction of winter SOI and winter SSTI.
Fig. 14. Cross-correlation function between bidecadal winter SOI and bidecadal winter SSTI. Vertical axis is Ž
. the lag year , horizontal axis is the cross-correlation value. Set of time series values before 1935. Positive lags
Ž correspond to SOI leading SSTI. Relative maximum of negative cross-correlation closest to zero lag this is the
. lag-BFOS is found at 2 years. Dashed lines enclose the 95 confidence interval.
3.6. Amplitude modulation of decadal SOI and SSTI A conspicuous joint amplitude modulation controls the amplitude of the decadal
Ž .
components of winter SOI and winter SSTI Fig. 13 . In 1912, the decadal component of SOI reaches its deepest negative value. Thereafter, the negative extreme of the SOI
Ž .
Ž climbs to its highest value less negative in 1951. Then, it decreases again becomes
. Ž
more negative . Opposite evolution is observed in the relative maxima positive — .
warmer — extremes of the decadal component of winter SSTI. The minimum
difference between an extreme of the decadal SOI and the corresponding extreme of opposite sign of the decadal SSTI is reached at 1957. After 1957, amplitudes amplify.
This realization of the amplitude modulation covers about 70 years. Both summer and winter SSTI show the amplitude modulation in the decadal component. On the other
hand, summer SOI do not show clearly the amplitude modulation of the decadal component.
Table 4 Ž
. Lag between fluctuations of opposite sign Lag-BFOS of SOI and SSTI
Results were obtained using cross-correlation functions. Component
Season Lag-BFOS
Conclusion Lag-BFOS
Conclusions Ž
. Ž
. years before
years after mid-1930s
mid-1930s Ž
. Decadal
Winter 2
SOI leads Out of phase 1808
Ž .
Summer 1
SOI leads Out of phase 1808
Bidecadal Winter
2 SOI leads
y2 SSTI leads
Summer 6
SOI leads y1
SSTI leads Ž
. Total interdecadal
Winter 3
SOI leads Out of phase 1808
Ž .
Summer 2
SOI leads Out of phase 1808
4. Discussion
