The Fine Power Spectra of the Southern Oscillation Index and Its Components and their Implication

3.1 Time domain analysis

In Fig. 1, the time series of the SOI, Darwin, and Tahiti showed a feature of random noise. The feature somewhat contradicted the considerable magnitude of the one-point lag correlations of 0.63, 0.50, and 0.484 for the SOI, Darwin, and Tahiti, respectively. Their irregularity in time strongly contrasts with the SSN and QBO oscillations. The noisy feature was even clearer in the autocorrelation functions. Overall, the three autocorrelation functions show 2~3 years of oscillation, which primarily reflected the effects of the QBO.

At a lag of 28 months marked with the thick blue bars in the ACFs of Fig. 1, there is no positive correlation at all for Tahiti pressure (Fig. 1f). Such a deviation may originate from the weakening of the QBO frequency due to a modulation style interaction. The estimated modulation periods were 84 months (or 7.0 years) and approximately 6.5 months. The periods were obtained by calculating the modulation frequency of the following two components: the annual period of 12 months and the 14 months period that was half of the 28 months QBO period. The reason of having 14 months instead of 28 months will be discussed in Section 3.2. The resulted short period of 6.46 months from the modulation may be smeared with other high frequency components. We are more interested in the 84 months period from the modulation.

In Figs. 1d and 1f, the autocorrelation peak related to the SSN was more pronounced at Darwin and the QBO effects were stronger at Tahiti. The SOI comprised the both, which signified that it was illogical to say that the SOI reflected only one physical process, even though it was pertinent to represent the intensity of ENSO to a certain degree (Trenberth, 1984).

We expected a bilateral interaction of the QBO with the tropical convections resulted from the ENSO activity. However, we like to accept the former modulation of the latter from the fact that the QBO preceded two months the SOI in the cross-correlation function (not shown).

Fig. 1.

(a, c, e) Pressure time series and (b, d, f) autocorrelation functions. (a, b) are for the SOI, (c, d) for Darwin, and (e, f) for Tahiti. The red blobs in (a) marked the smoothed SOI values exceeding 1.5 in a unit of standard deviation. The smoothing was carried out by applying a 1-2-1 filter two times. The red shading in (c) shows the smoothed time series of SSN produced by applying the 1-2-1 filter five times and the autocorrelation of the raw SSN with dashed curve in (d). The blue shading in (e) shows the QBO time series and its autocorrelation with the blue dashed in (f). For comparison, we scaled down both of the autocorrelations by one-fifth. The dashed red bar marked a lag of 11 years and the thick blue a lag of 28 months. The thin blue bar locates a lag of 84 months or 7 years. The horizontal dashed lines denote a 5% significance level for the correlation.

The Darwin pressure responded to the forcing of the SSN for the short time span centered at a lag of 11 years. For the other lag times, it was difficult to designate similarity between the correlation functions of Darwin and the SSN (Fig. 1d). Such a localized correlation peak at a certain lag time appeared as power peaks at the integer multiples of the primary frequency in the corresponding spectrum function.

By considering the simultaneous effects of the sun and the QBO on the pressure change at the Darwin and Tahiti, we constructed scatter diagrams between the SOI and SSN and between “aSOI” and the QBO index for different values of SSN and QBO and displayed them in Fig. 2. We adopted a word “aSOI” to designate the orthogonal component to the SOI. The “aSOI” is the same as the quantity described as noise by Trenberth (1984). The crossed statistics for the above mentions parameters are given in Fig. 3. The both figures clearly show the contrasting features in the parallel and cross relationships between the SOI (aSOI) and the SSN (QBO).

During the most rapid decrease phase of SSN, SOI had a negative maximum value in Fig. 2a, even clearer in Fig. 2b. The SOI minima period accompanied high pressure at Darwin and low pressure at Tahiti. The level of correlation was high for the transition phases and with large values of SSN. Overall, the negative phase of SSN tendency favored the El-Niño condition (Fig. 2b). The relationship was consistent with the increase of convective motions by the evaporation of liquid water from the ocean surface rather than by direct warming of the sea surface due to the sun light (Lim et al., 2006).

From Figs. 2c and 2d, the relationship of “aSOI” to QBO did not show any dependency on the QBO tendency but had a clear dependency to the sign of QBO, in other words, to the direction of the wind. The above relationship implied the importance of convections than direct mechanical forcing of the QBO on the Walker circulation.

The dynamics inferred from Fig. 2 could not be ascribed to a chance originated from a stochastic process when considering the strong contrast between the features in Fig. 2 and a similar analysis but between SOI (aSOI) and QBO (SSN) shown in Fig. 3. The strong sun decreased the surface pressure at the both stations in general.

In Fig. 4, we were able to reconfirm the previous statistics inferred from Figs. 1, 2, and 3. The Fig. 4 showed the magnitude of SOI in the phase domain of SSN and SSN tendency. In this scatter diagrams, the distribution was noisy as consistent with the small correlation coefficients in the previous analysis. However, the features deviated from a uniform distribution in magnitude as well as position. The strong dissimilarity between Figs. 4a and 4b revealed the different mechanisms working on SOI and “aSOI”, which could be interpreted as Darwin and Tahiti affected by the different forcing.

In Fig. 4a, the spatial density of the circles revealed the strong positive skewness of SSN time series. The SSN had a long dormant period and a short peak time in general. With less clarity, we could point out that positive SSN tendency side was populated with blue bubbles and the negative side with red ones. In Fig. 4b, the sparse and dense phases with bubbles coincided with the rapid downward movement of westerly and the slow downward of easterly, respectively. The positive QBO phase had more blues than the negative phase did. In the positive phase of the QBO index, we had high pressure anomaly at Tahiti and Darwin simultaneously.

Fig. 2.

Scatter diagrams of the SOI versus the SSN tendency (a, b) and “aSOI” against the QBO index (c, d) for the different categories of parameters. (a) SOI versus the SSN tendency by the positive (686 red dots) and negative (970 blue dots) values of SSN. Color coded linear fits had correlation coefficients of 0.119, 0.003, and 0.081 for the entire pairs. (b) SOI versus the SSN tendency for the values of SSN from −0.005 through 3.0 with a linear fit in black. The obtained correlation coefficient 0.123 for 678 pairs was significant at least 5% based on the null hypothesis. (c) “aSOI” against QBO for the positive tendency (403 red dots) and negative tendency (389 blue dots) of the QBO. The nonlinear fit in black was for the entire 792 pairs. (d) As for (c) but for the different signs of the QBO. The magnitudes of SOI were marked as a function of QBO for the positive values (306 red dots) and the negative values (489 blue dots) of QBO. The corresponding linear fits had the correlation coefficients 0.309 and −0.072 in order. For analysis, the moving averages of SOI, “aSOI”, QBO, and the SSN tendency were produced by applying the 1-2-1 filter five times. The term “aSOI” means “not SOI” component, which is the averages of Darwin and Tahiti pressure anomalies.

Fig. 3.

As in Fig. 2, but for Scatter diagrams of the aSOI versus the SSN tendency (a, b) and “aSOI” against the SSN index (c, d) for the different categories of parameters.

3.2 Spectrum functions

For all of the spectrum functions in Fig. 5, the enhanced power appeared for a frequency band that ranged from the solar cycle at a low-frequency end to the QBO cycle at a high-frequency side. Over the frequency band, the most pronounced peak appeared at 3.5 years, which was close to the third harmonic of the sun with a 3.7 year period and coincided with the second harmonic of QBO-modulating annual cycle with a period of 3.5 years that was equal to 2/84 in a unit of cycle/month. The 84 month was the period of the primary frequency of the QBO-modulating annual cycle.

At high frequencies shorter than the QBO period of 28 months, a statistically significant peak did not exist, except for the SSN component with the 18-month period for the Darwin pressure only. Over the band with enhanced power, the overall coherency was 0.5 compared with the value of nearly zero for the high-frequency components. The shape of the spectrum functions deviated from the air temperature variation that was used for environmental influences and suitably described as red, brown, or black noise by Cuddington and Yodzis (1999) and Halley and Kunin (1999).

In the phase domain, the Darwin maximum pressure led the Tahiti minimum pressure by about 90 degrees rather than 180 degrees. The fact apparently contradicted the same 3-year lag between the Darwin and Tahiti pressure values with respect to the SSN and the almost simultaneous peaks of pressure with respect to the QBO by two-month delay only. The essential reason for such differences may arise from the combination of the results for the both periods simultaneously.

The Darwin pressure series had power peaks at 66 months (5.5 years) and 44 months (3.7 years). The periods were the harmonics of the primary 11-year period. In Fig. 5a, the power of the fundamental frequency was small compared with those of the second and the third harmonics. The harmonics might have originated from the strong nonlinear and indirect response of the surface pressure to the sun’s variation in intensity. The exact mechanism of amplification of a certain harmonic is beyond the object of this work.

The weak 11-year frequency peak in the spectra did not imply an absence of effects from the sun’s activity. For the nonlinear systems, the power of the forcing frequency could be transferred to its harmonics. The transfer was possible due to a spread spectrum style modulation (Barnston, 1982).

Maximum and minimum pressure values appeared at the period with the largest SSN tendency instead at the period of extreme values of SSN. When the SSN decreased rapidly, the Darwin pressure reached the highest value and the lowest pressure appeared at Tahiti. The Darwin pressure was proportional to the SSN value three years before. Such a relationship suggests that the tendency of SSN may be directly connected to the magnitude of the pressure anomalies at the both stations, which was discernable in Figs. 2a and 2b.

The Tahiti pressure showed a more wavelike response than the Darwin pressure. The shape was apt to produce the 7-year frequency peak by the QBO modulation of the annual cycle. The 7-year harmonics were possible when considering the surface pressure dependencies on the states of the QBO, together with the lowest pressures at the transition state of the QBO winds (Figs. 2c and 2d). More specifically, the annual cycle produced the 84-month period, while interacting with the QBO in the time scale of its half time instead of its full 28 months. The best fitting curve in Fig. 2c was based on the method by Cleveland (1979).

Fig. 4.

Scatter diagrams of (a) SOI anomaly in the phase domain of SSN and the SSN tendency and (b) the “aSOI” anomaly against the values and tendency of the QBO index. The position and size of a circle represent the phase domain coordinates and magnitude. Red and blue denote positive and negative anomaly, respectively. The legend digit 1, 2, 3 means the multiplier of the standard deviation of the corresponding time series. The tendency was computed by taking difference of pressures at before and after a given time. We normalized the resulted series with its standard deviation.

Fig. 5.

(a) Power and (b) cross spectra as a function of frequency. The SOI, Darwin, and Tahiti spectra are in grey, red, and blue curves, respectively. The spectrum functions were smoothed by applying a 1-2-1 filter five times. The smooth curves in grey, red, and blue mark a 5% significance against the red noises. All of the spectrum functions were normalized to have a unit magnitude at each frequency for the case of average. The SSN and QBO frequencies were marked with the downward red and blue arrows in (a) and with respective dashed bars in (b). The red and blue digits for the dotted lines were the harmonic numbers of the two basic periodicities of 132 months (11 years) related to the SSN and 84 months of the QBO-modulating annual cycle. Phase values were divided by π.

3.3 Spread spectrum modulation

To our knowledge, no one has investigated the outputs from the modulation of random time series with a sinusoidal function in time. It is worthwhile to examine the outputs from such a modulation or interaction. In atmospheric motions, the convections may be considered as stochastic processes when comparing with the large-scale motions. The latent heat is the second most important energy source after solar heating for the circulation of the atmosphere. We also have to remember that the latent heat also came from solar radiation.

For an illustration, let us assume a time series of random numbers multiplied by a cosine function of time. The latter could simulate an almost regular cycle such as the SSN. An ideal white noise process can be written by assuming a unit amplitude with an infinite number of frequencies. By considering the Fourier transform of a random process, we are able to express a modulated time series as follows:

The summation represents the assumed white noise process. A and B denote the amplitudes of the forcing and the forced processes, respectively. The amplitudes A and B did not significantly affect the final output of modulation. The modulation can produce a new frequency set as follows:

The modulated frequency is as follows:

No power of frequency zero (0.0) decreased by half the power of the forcing frequency in the spectrum of the modulated time series. The power of the forcing frequency moved by half to the adjacent frequencies separated by $$ f_0$$. At the same time, the displaced power of the frequency was compensated for by the powers from the adjacent frequencies. All of the frequencies, except for $$ f_0$$, released and regained equal amounts of power, so the modulation did not change the power of each frequency. The only exception was for the forcing frequency. The above simple and lucid model supported the disappearance of the power of the forcing frequency after modulation. Lim et al. (2014) detailed the characteristics of this type of modulation and successfully reproduced the previous theoretical prediction in their countless numerical experiments. On the other hand, the enhancement of the derived harmonics were yet unresolved and conceived as caused by a unique regime of the pressure variation at the two stations.

Fig. 6.

Wavelet power spectra (a, c, e) and the corresponding global spectra (b, d, f) of the SOI, Darwin, and Tahiti pressure anomalies. A significance level of 5% was marked by the thick contours in the left-hand side Figs. and the dashed curves in the right. The highly fluctuating grey curves in the right-hand side are the raw spectrum from the Fourier transform. The red-dashed lines mark the 11-year period.

3.4 Effects of the SSN and QBO

The high pressure at Darwin and low pressure at Tahiti appeared frequently for the period after three years from the maxima of the SSN. The time lag relationship resulted in the almost simultaneous linear relationship between the SOI index and the SSN tendency. Development of convective motions over the tropics might be affected by the time rate of change of SSN combined with the surface temperature changes of diverse time scales. The changes came from an annual cycle, the diurnal variation, and other various mechanism. The convective motions were mostly controlled by the available amount of water vapor and the stability of the atmosphere that had a strong connection with the heat capacity of the concerned region.

In a relative sense, Tahiti was in aqua world with large heat capacity. On the other hand, Darwin experienced dry climate with a small heat capacity due to the influences of the continents of Asia, Australia, and the Maritime Continent. During the phase of negative SSN tendency, Darwin could cool down much faster than Tahiti did. The fact suggested more stable atmospheric condition in the region of Darwin than of Tahiti. To the contrary, warm and water abundant Tahiti area would have stronger convective motions in the decreasing phase of SSN. During the period of a positive SSN tendency, we expected more frequent La-Niña situation.

The pressure increased at the both stations for the positive and negative extreme phases of the QBO. In Fig. 2c, the low pressure coincided with the null wind phase of the QBO. The relationship could be explained by considering the water vapor increase and the more active convection at this time of phase (Lim et al., 2008). The pressure changes were nearly independent of the time rate of change of QBO or its tendency, as evident from Fig. 2a.

The Coriolis force could be responsible for the asymmetrical response of the pressure to the westerly and easterly phases of the QBO. Due to the Coriolis deflection, the westerly converged toward the equator and the easterly diverged away from the equator meridionally. The convergence or divergence could suppress or enhance the convective motions over the tropics (Collimore et al., 1998; Lim et al., 2008). Over the tropics, the weak convective motions or less rainfall events resulted in high pressure at the surface.