Temporal variations in the Madden-Julian Oscillation (MJO) (Madden and Julian, 1971) and its relationship with El Niño-Southern Oscillation (ENSO) events has been previously investigated using classical spectral analysis; see, for example, Madden and Julian (1994) and references therein.Anderson et al. (1984) filtered two time series, atmospheric relative angular momentum (4 years) and the 850-200 mb shear of the zonal wind at Truk Island (25 years), with a filter designed to pass the frequency band corresponding to periods of 32-64 days. They noted that, with respect to the Truk Island series, a possible association with increased amplitude of the oscillation during the 1956-57, 1972-73, and 1976-77 ENSO warm events but noted that the duration of these increases were much longer than the ENSO events.
Madden (1986) performed a seasonally varying cross-spectral analysis on nearly twenty time series of rawinsonde data from tropical stations around the world. The MJO appears strongest during December-February and weakest during June-August, and that it is always stronger in the western Pacific and Indian oceans than elsewhere.
Gray (1988) performed a correlation analysis between daily station pressure data from Truk (7°N, 152°W), Balboa (9°N, 80°W), Darwin (12°S, 131°E) and Gan (1°S, 73°E), with seasonal sea surface temperature anomalies on a 5° grid. The data were partitioned into ENSO and non-ENSO years; in non-ENSO years a strong seasonal shift in frequency was found at all sites except Truk Island.
Kuhnel (1989) investigated the characteristics of a 40-50 day oscillation in cloudiness for the Australo-Indonesian region. Using data on a 10° by 5° grid, regions in the eastern Indian Ocean and western Pacific Ocean were found to have a pronounced 40-50 day peak with no obvious seasonal variation. Another region in the Indian Ocean (5-15°S, 95-100°E) showed a stronger oscillation in the March-June period. Regions around 5-15°S over northern Australia and in the Pacific Ocean showed a much stronger 40-50 day oscillation during the Australian monsoon season from December to March, than the rest of the year. The 40-50 day cloud amount oscillation did not appear to be affected by warm ENSO events.
Madden and Julian (1994) note the broadband nature of the oscillation by comparing the station pressure spectra for Truk Island (7.4°N, 151.8°W) during two time spans - 1967 to 1979 and 1980 to 1985. The MJO appears to have a 26-day period in the early 1980s.
The relationship between ENSO events and the MJO is a topic which could benefit markedly by using wavelet techniques. To investigate how these two atmospheric phenomena interact, Whitcher et al. (2000) analyzed two time series. The first one being the Southern Oscillation Index (SOI), which is an indicator of ENSO and usually defined to be the difference between monthly averages of the station pressure series from climate stations at Darwin, Australia (130.8°E, 12.4°S) and Tahiti, French Polynesia (149°W, 14°S); see Figure 1 for the locations of these climate stations. It was first introduced by Walker (1928) and came from the observation that pressure in the tropical Pacific Ocean is inversely related to pressure in the Indian Ocean.In Whitcher et al. (2000), they deviate from the usual definition of the SOI by introducing a daily version of it. Daily pressure readings were obatined from Darwin, Australia, and Tahiti, French Polynesia, starting in 1 June 1957 and continuing to 31 December 1992 (N = 12,998) and differenced; see Figure 2. The distance of the stations from the equator is apparent in the strong annual component in the time series. The measurements in the summer and winter of 1983 appear to be higher than those in adjacent years. This approximately corresponds to a large ENSO event in the early 1980s. Any missing values were filled in using one-step-ahead predictions from an ARIMA(3,1,0) model applied to the series.
Truk Island daily station pressure readings (7.4°N, 151.8°W) were also obtained as an indicator of the MJO. This series also exists from 1 June 1957 to 31 December 1992; see Figure 2. Unlike the SOI, there is no apparent annual trend since the station is quite close to the equator. Missing values were dealt with in the same manner as described for the SOI.
[ soi.txt and truk.txt ]
Daily measurements allow us to apply the maximal overlap discrete wavelet transform (MODWT) and analyze the sub-series which correspond to filtered series with approximate pass-band 1/2^{j+1} < |f| < 1/2^j. Due to the approximate bandpass nature of the MODWT, with the approximation improving as the length of the wavelet filter increases, it is unnecessary to remove any annual or semiannual components (a similar argument is made when bandpass filtering atmospheric time series in Anderson et al., 1984), which should be roughly captured in the level 7 and level 8 scales. The MJO is known to occur with periods of around 30-60 days. We therefore expect to see it in scale level 5, associated with changes of 16 days and an approximate pass-band of 1/64 < |f| < 1/32.A partial MODWT (J = 10) was applied to each series using the Daubechies extremal phase compactly supported wavelet filter of length L = 4 (Daubechies, 1992). Figures 3 and 4 give multiresolution analyses of the Truk Island station pressure series and SOI, respectively. For the Truk Island series, we observe only a slight annual trend in D8 and the obvious disruption in the early 1980s appears to primarily affect level 7 and level 8. The fifth scale appears to fade in and out in magnitude with no apparent pattern. The SOI multiresolution analysis exhibits a strong annual trend where the disturbance in the early 1980s affects level 8 and level 9. The scale associated with the MJO, level 5, exhibits numerous bursts across time.