Long-term modulation of decadal and interdecadal fluctuations in annual rainfall amount has been found by other authors. For finding the explanation of that modulation, it is of
interest to know if there is a joint amplitude modulation of decadal and interdecadal fluctuations of SOI and of SSTI. Although indication of long-term modulation of time
series of SST is found in the literature, it has not been documented a joint amplitude modulation of the decadal and interdecadal components of the SOI and the SST in the
equatorial Pacific during this century. The existence and characteristics of a joint amplitude modulation is described in Section 3.6. Finally, Section 4 contains a discus-
sion of results, and Section 5 contains a summary of findings.
2. Data and methodology
2.1. Data Ž
. A monthly SSTI was obtained from Wright 1989 . This index is defined as the
w x w
monthly mean of SST anomaly over the region 6–28N, 170–908W ; 28N–68S, 180– x
w x
908W , and 6–108S, 150–1108W ; this region is in the central and eastern equatorial Pacific. SSTI can take on positive as well as negative values. The time series of monthly
values of SST index covers the 104-year interval 1882–1985. Values of SSTI are given in degree-Celsius.
Ž .
The operational SOI of the Climate Prediction Center CPC of NOAA is used in this research. SOI data covers the 101-year interval 1896–1996. The SOI is defined as:
SOI s T y D rM ,
Ž .
s s
s
where T is the standardized monthly mean of sea level pressure in Tahiti; standardized
s
Ž Darwin, D , is similarly defined. A variable is standardized if its mean value is zero
s
. and its standard deviation is 1 . M is given by:
s
1
2
M s Ý T y D
, where N s total number of summed months.
Ž .
s s
s
N Two sets of yearly data were constructed from the monthly SOI time series and from
Ž the monthly SSTI time series. One set is the average from November to April ‘summer
. season’ . Summer and winter months correspond to the Southern Hemisphere. The other
Ž .
set is the average from May to October ‘winter season’ . The time series of summer SOI is shown in the upper panel of Fig. 1. The time series of summer SSTI is shown in
Fig. 6. Hereafter, SOI and SSTI refer only to seasonal averages.
2.2. WaÕelet analysis Time-varying components at different frequencies, of both SSTI and SOI, are
individualized using the Morlet continuous wavelet transform. Wavelet analysis is increasingly being used in atmospheric sciences to analyze time series of several types
Ž .
of variables Hu et al. 1998; Meyers et al., 1993, among others .
Fig. 1. Top panel: Observed summer SOI. Bottom panel: Contributions to summer SOI, at indicated years Ž
. Ž
. from indicated wavelet period. Dotted solid contours indicate negative positive contributions.
Ž .
Following Torrence and Compo 1998 , let x be a time series of a variable X,
n
where n s 0 . . . N y 1; measurements are equally spaced at time interval d t equal to 1 year. In this research, the variable X is equal to the 6-month average of SOI or SSTI.
After wavelet decomposition of the time series made up of values x , the reconstruction
n
is given by:
J
R W s
Ž .
n j
x s A
Ý
n 1r2
s
j
js0
w Ž .x
where A is a constant, R W s is the real component of the complex wavelet
n j
transform, and s is the wavelet scale, J is the largest scale. To avoid unnecessary
j
Ž .
repetition, readers are referred to Torrence and Compo 1998 for details of the Morlet continuous wavelet transform.
Results of wavelet analysis are presented as a time series for contributions x for
n , j
each scale j as function of time n. For example, in Fig. 1, the upper panel shows the original time series of summer SOI. The bottom panel of this figure shows the time
4 Ž
. series x
, for all scales j j s 2, . . . ,64 . Ordinate unit is the wavelet period in years
n , j
Ž .
corresponding to each scale. In this figure, negative positive contours indicate negative Ž
. positive contributions to the value of the SOI. The sign of the contribution of each
Ž .
wavelet period indicates whether contribution increases positive sign or decreases Ž
. negative sign the SOI.
In this paper, the term wavelet period is used instead of wavelet scale. Wavelet scale is converted in such a way that if the analyzed wave is a periodic cosine or sine
function, the wavelet spectrum will be equal to Fourier spectrum. In the case of the Morlet wavelet, the conversion factor is l s 1.03 s, where s is the scale and l is the
equivalent period in Fourier representation. For the Morlet wavelet, the wavelet scale is
Ž .
almost equal to that Fourier period; see Torrence and Compo 1998 and Meyers et al. Ž
. 1993 for more details.
2.3. Partial contribution from bands of waÕelet period A convenient identification and description of the intensity of fluctuations is achieved
Ž .
summing the contributions from wavelets with scaling wavelet period within a given Ž
. Ž
. interval Lucero and Rodrıguez, 1999 . The contribution from an interval a band of
´
Ž .
wavelet period is called component of the SOI or of the SSTI in that band. Fig. 2 shows five components. Selection of intervals of wavelet period was carried out based
Ž on results of other authors on period of climatic fluctuations Trenberth and Hurrell,
. 1994; Knutson and Manabe, 1997, among others .
The SOI and SSTI time series were also separated out into two bands of contribution. One is the sub-decadal band, encompassing contributions from wavelet period smaller
than 10 years. The other is the total interdecadal band, encompassing contributions from wavelet period larger than 10 years. Fig. 3 is an example of this partitioning of the SOI.
In a study of North Pacific variability using empirical orthogonal functions, Zhang et al.
Fig. 2. Upper five panels: Partial contribution from bands of wavelet period identified in the y-axis, to summer SOI. Bottom panel shows the observed summer SOI. ‘‘Year’’ refers to the year of November. Vertical scale
stretches for wavelet period greater than 10 years.
Fig. 3. Partial contribution to winter SOI from bands of wavelet periods as indicated on top of upper and middle panel. Bottom panel is observed winter SOI.
Ž .
1997 also partitioned the global SST field into two components: ENSO-like timescale, and total interdecadal timescale.
2.4. Computation of lag In this research, the interest is in the lag between extremes of opposite signs of SOI
and SSTI because out-of-phase SOI and SSTI signals are commonly assumed to interact constructively. This type of lag is denominated Lag-BFOS for ‘lag between fluctuations
Ž .
of opposite sign’, and it is identified by the lag of the relative minimum negative value Ž
. of the cross-correlation function closest to lag zero see Fig. 14 . Positive lag-BFOS
indicates that the SOI signal leads the SSTI signal. Results are expressed in units of year. A lag-BFOS equal to zero year is equivalent to a conventional lag equal to 1808.
2.5. Definitions For ease of reference, definitions of some terms used in this paper are given in this
Ž section. The band including contributions from wavelet period within the interval 10 to
x Ž
. 17 years is denoted decadal band. The band 17 to 27 years is the bidecadal band.
The band including contributions from wavelet periods larger than 10 years is denoted Ž
. the total interdecadal band. An expression like decadal component of SOI or SSTI
Ž x
has the meaning of ‘partial contribution from the band 10 to 17 years to the
Ž .
reconstruction of SOI or SSTI ’. A fluctuation, or a pulsation, refers to a non-sta- Ž
tionary feature bounded in time by two consecutive extremes of the same type For .
example, bounded by two consecutive minima . Because components of SOI and SSTI
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
