(Received March 12, 2004; revised June 29, 2004)

Transcription

1 NO.1 HE Juanxiong, YU Zhihao and YANG Xiuqun 83 Temporal Characteristics of Pacific Decadal Oscillation (PDO) and ENSO and Their Relationship Analyzed with Method of Empirical Mode Decomposition (EMD) HE Juanxiong 1,2 (Ûò<), YU Zhihao 1 ( Í), and YANG Xiuqun 1 (fl?+) 1 Institute of Severe Weather and Climate, Nanjing University, Nanjing Guangzhou Institute of Tropical and Oceanic Meteorology of CMA, Guangzhou (Received March 12, 2004; revised June 29, 2004) ABSTRACT Pacific Decadal Oscillation (PDO) is a long-term ENSO-like variability of the North Pacific. It is the first principal component of EOF of the North Pacific SST. ENSO is the strongest signal of annular change of global climate system. Empirical Mode Decomposition (EMD) method is applied to two types of indices. One type of index is the Pacific Decadal Oscillation (PDO) index that represents a long-term ENSO-like variability of the North Pacific. The other type of indices such as Southern Oscillation (SO) index, Nino1+2 SST, Nino3 SST, Nino4 SST and Nino3.4 SST represents ENSO. The relationship between two types of indices shows the temporal characteristics and relationships between the two phenomena. It is found that, for PDO, the good correlations with ENSO only exist on the quasi-2-7-yr timescales and decadal and multidecadal timescales, suggesting that a close relationship between PDO and ENSO only exists among some special intrinsic mode functions at lower frequency band. Key words: EMD(Empirical Mode Decomposition), intrinsic mode function, PDO (Pacific Decadal Oscillation), ENSO 1. Introduction Pacific Decadal Oscillation (PDO) is a long-term ENSO-like variability of the North Pacific. It can be characterized by the first principal component of EOF of the North Pacific SST (Zhu and Yang, 2003; Trenberth, 1990; Yang and Zhang, 2003). ENSO is the strongest signal of annular change of global climate system (Trenberth, 1997). The spatial pattern of PDO is a wedge similar to El Niño. In the cool (warm) phases of PDO, the central and northwest Pacific is of warm (cold) SST and the coast of the North America is of cold (warm) SST. The temporal scale of PDO is longer than that of ENSO, and it persists years. The cool phases of PDO are from 1890 to 1924 and from 1947 to 1976; warm phases are from 1925 to 1946 and from 1977 to now. In recent years, the PDO has become a hot spot in the climate research for its important influence on the climate of the North America and the East Asia monsoon. ENSO is the strongest signal in the global climate system. It is well known that ENSO has a strong impact on global climate. ENSO occurs in the tropical Pacific region. It is worthwhile to make a research on the interrelationship between ENSO and PDO because of the resemblance on spatial patterns and the connection on the geography. The climate signal is nonlinear and nonstationary. The wavelet method and singular spectrum analysis (SSA) are often employed. Although the wavelet method has a capability of settling local time-frequency, it is not adaptive because the same wavelet is always employed to all time-frequency domains. SSA in essence is a kind of TEOF (triangular EOF), and it is adaptive and independent of the choice of the sine signal. It can amplify the sine signal and has been widely applied to the forecast and reconstruction of series. However, the principal components obtained must be steady. Otherwise, it makes the explanation difficultly because the length of the principal component is less than that of the series. The number of the principal components used in the analysis This work is jointly supported by the National Natural Science Foundation of China under Grant Nos ,

2 84 ACTA METEOROLOGICA SINICA VOL.19 is dependent on their contribution to the total variation. In order to rectify the above shortcomings of the two methods, a Huang-Hilbert method has been developed (Huang et al., 1998). As an efficient method in nonlinear and non-stationary time series analysis, the Huang-Hilbert method has been put into many practices in recent years. For example, Xie et al. (2002) used it to explore the interannual and decadal variability of landfalling tropical cyclones in southern coast of the United States. In the present paper, we use the Huang-Hilbert method to make a research on PDO, ENSO and their correlations on various time scales. 2. Data and methodology 2.1 Data PDO index (PDOI) data as the representation of PDO from 1900 to 2000 are the time series of the first principal component obtained by EOF of SST northward of 20 N in the North Pacific, which are available online at Southern Oscillation Indices (SOI) issued by Climate Prediction Center (CPC) are used as the representation of atmospheric component of ENSO. The time period is from 1950 to 2000 because some values are unavailable before The ocean component of ENSO is characterized by monthly SSTA (SST anomaly) over Nino1+2 (0-10 S, W), Nino3 (5 N-5 S, W), Nino4 (5 N-5 S, 160 E-150 W), and Nino3.4 (5 N-5 S, W) regions from 1950 to 2000 released by CPC. All of them are available online at fttp://ftpprd.ncep.noaa.gov/pub/cpc/wd52dg/ data/indices/. The SOI data from 1900 to 2000 available online at soi.htm put out by Climate Research Unit (CRU) are also used to make a test on the stability of our results (for further detail see Ropelewski and Jones, 1987; Allan et al., 1991). 2.2 Empirical Mode Decomposition (EMD) The essential part of Huang-Hilbert method is Empirical Mode Decomposition (EMD). The purpose of EMD is to figure out the IMF (intrinsic mode function). The latter is a time series which possesses two flavors: one is that the difference between the number of zero crossings and the number of local extremes is zero or one; the other is that the average over the envelopes of maximum and the envelopes of minimum is zero. IMF is not a monochromatic wave as sine or cosine functions used by Fourier transform, and it contains a great many waves. The process of EMD is as follows: (1) Identify the local maximum and minimum of the time series, and connect the local maximum and the local minimum with two cubic spline lines separately. (2) Denote m 1 as the average value of the two cubic spline lines, h 1 = X(t) m 1. (3) Identify the local extremum of h 1, and fit the local maximum and the local minimum with two cubic spline lines separately. (4) Denote the average value of the two cubic lines as m 11,h 11 = h 1 m 11. (5) Reiterate the above steps. In the k-th reiteration, set h 1(k 1) m 1(k 1) = h 1k,calculatingSD according to the following expression SD = [ T h1(k 1) (t) h 1k (t) 2 ] h 2 1(k 1) (t). (1) t=0 If SD is less than 0.2 or 0.3, then set c 1 = h 1k and c 1 is the first IMF. After test, it is found that SD limited to less than 0.2 or 0.3 is very strict and can guarantee the amplitude and frequency of the obtained IMF to have enough physical meaning. (6) Set r 1 = X(t) c 1, reiterating the above steps for r 1 until we can obtain the second IMF, denoting as c 2. (7) Set r 2 = h 2 c 2, reiterating the above steps for r 2 until we can obtain the third IMF, denoting as c 3. Reiterate the above steps until r n 1 c n = r n,where r n is a trend term or a residual, not an IMF. Finally, the original series can be decomposed into n-empirical modes and a trend (or a residual) in the form X(t) = n c i + r n. (2) i=1 For the cubic curve used by EMD, it is important for EMD to take the boundary condition into account. A mirror symmetric prolonging method is used

3 NO.1 HE Juanxiong, YU Zhihao and YANG Xiuqun 85 to solve this problem. For further detail, see Rilling et al. (2003). 2.3 Hilbert transform Setting that IMF is h(t), the Hilbert transform can be written as H(t) = 1 h(t) dτ. (3) π t τ Define the instantaneous amplitude a(t) of h(t) as a(t) = H 2 (t)+h 2 (t), (4) the instantaneous phase θ(t) as θ(t) = arctan H(t) h(t), (5) and the instantaneous frequency ω(t) as ω(t) = dθ dt. (6) The Hilbert spectrum diagram is that the amplitude a(t) isdisplayedonθ(t)-ω(t) plane. 3. EMD of PDOI and various indices of ENSO 3.1 EMD of PDOI Figure 1 represents seven modes and the trend obtained by the EMD applied to PDOI of CPC. The first three modes represent high frequency oscillations; the 4th and 5th modes represent quasi- 2-7-yr variability; and the 6th and 7th modes represent decadal and multi-decadal variations. The 5th and 6th modes are characterized by large amplitudes during the 1950s, 1960s and 1990s. The 7th mode has quasi-20-yr variability. The trend is at the warm phase when it is greater than zero after about 1977, and at the cold phase when it is less than zero before about Figure 2 depicts the Hilbert spectra of PDOI and SOI. In the figure the black shaded area is the area with large instantaneous amplitudes, the dotted area is the area with small instantaneous amplitudes and the blank area is zero. It can be seen from Fig. 2 that the oscillations of PDOI and SOI are concentrated at low frequencies. The large instantaneous amplitude of PDOI is closer to abscissa than SOI s and the significant frequency of PDOI is lower than SOI s. Note that in the paper, we settle the period of IMF according to the number of zero crossings. Set the number of zero crossings is N, the mode has about (N-1)/2 periods, then the oscillation periods are obtained by the time length dividing the number of periods. The low frequency timescales refer to quasi-2-7-yr variability and decadal and multi-decadal variability. Figure 3 is the SSA of PDOI with the embedding dimension of 100. The first 7 components account for 60% of the total variance, the first 10 account for 70% and the first 17 account for 80%. The first and second eigenvalue functions are decadal and multidecadal variations (Fig.3a), the 3rd and 4th eigenvalue functions are quasi-7-yr oscillation (Fig.3c). The highest lag coefficient between the first two principal components is small, being 0.27 at 70 months. The lag coefficient between the 3rd and 4th arrives the maximum (0.86) at 15 months. The length of the principal components is 513, which is less than the length (612) of the series (Figs.3b, 3d). It makes the explanation Fig.1. Trend and seven intrinsic mode functions of PDOI ( ).

4 86 ACTA METEOROLOGICA SINICA VOL.19 Fig.2. The Hilbert spectra of PDOI (a) and SOI (b). Fig.3. The first four principal components and eigenvalue functions of SSA of PDOI with an embedding dimension of 100. (a) The first (solid line) and second (dot-dashed) eigenvalue functions, (b) the first (solid) and second (dot-dashed) principal components, (c) the third (solid) and fourth (dot-dashed) eigenvalue functions, and (d) the third (solid) and fourth (dot-dashed) principal components. difficultly, thus the EMD is much terser than SSA. 3.2 EMD of SOI The EMD of SOI of CPC yields seven modes (Fig.4). These modes can be divided into three groups. Group one is the first three modes representing high frequency oscillations; group two is the 4th and 5th modes representing quasi-2-7-yr variability; and group three is the 6th and 7th modes representing decadal and multi-decadal variations. The amplitude of the 5th through 7th mode is smaller from the 1950s to the 1970s and becomes larger since the 1980s. Therefore, it can be found that PDOI and SOI have different temporal characteristics. 3.3 EMD of Nino1+2, Nino3.4, Nino3, and Nino4 The EMD of Nino1+2 (figure omitted) produces eight modes. The first three modes represent high frequency oscillations. The 4th and 5th modes have

5 NO.1 HE Juanxiong, YU Zhihao and YANG Xiuqun 87 quasi-2-7-yr variability. The 6th through 8th modes show decadal and multi-decadal variations. The 6th, 7th and 8th modes have a quasi-10-yr, quasi-15-yr, and quasi-25-yr variability separately. The EMD of Nino3, Nino4 and Nino3.4 generates seven modes (figure omitted). For each EMD, the first three modes characterize high frequency; the 4th and 5th modes characterize quasi-2-7-yr variability; and the 6th and 7th modes decadal and multi-decadal variations. The Hilbert spectrum graphs of these are similar to SOI (figure omitted). The variances are concentrated at low frequencies. Fig.4. Trend and seven intrinsic mode functions of SOI ( ). 3.4 Relationship between PDOI and the various indices of ENSO Because PDOI represents the variability of SST in the extratropical Pacific, and SOI, Nino1+2, Nino3, Nino4 and Nino3.4 represent the variability of the tropical Pacific atmosphere-ocean system, the correlation coefficients between them reflect, to some extent, links between the extratropical and the tropical zones. Although the correlation coefficients between PDOI and SOI, Nino1+2, Nino3, Nino4, Nino3.4 are , 0.437, 0.427, 0.416, and separately, we can make a further investigation on various timescales by the correlations of the modes. Table 1 lists the correlation coefficients between the modes of PDOI and SOI. The values are almost zero either between the high frequency modes of PDOI and each mode of SOI, or between the high frequency modes of SOI and each mode of PDOI. The correlation value is larger between the modes of PDOI and the modes of each index of ENSO at quasi-2-7-yr variability and at decadal and multi-decadal variability. For example, the (absolute) value between the two 6th modes is and the two 4th modes is Not any of modes at the low frequency may have a large value. For instance, the value between the 6th mode of PDOI and the 7th mode of SOI is almost zero (-0.096). The better relationship is not stochastic, and only exists at special timescale. Table 1. Correlation coefficients between the intrinsic mode functions of PDOI and SOI PDOI SOI

6 88 ACTA METEOROLOGICA SINICA VOL.19 We also calculated the correlation coefficients between the modes of PDOI and Nino1+2, Nino3, Nino4, and Nino3.4. Table 2 lists the situation of Nino3.4. The good relationship exists between their 4th modes (0.419), 5th modes (0.456), 6th modes (0.542), and 7th modes (0.478). All of them are significant at the 99.9% confidence level. Table 2. Correlation coefficients between the intrinsic mode functions of PDOI and Nino3.4 SST PDOI Nino , Table 3 lists correlation coefficients in the situation of Nino3. The high value exists between their 4th modes (0.471), 6th modes (0.618) and 7th modes (0.633). All of them are significant at the 99.9% confidence level. Table 3. Correlation coefficients between the intrinsic mode functions of PDOI and Nino3 SST PDOI Nino The large correlation coefficient only exists between the third mode of PDOI and the 4th mode of Nino4 (0.350, table omitted). The large correlation coefficients between the modes of PDOI and Nino1+2 (table omitted) are shown in that the 7th mode of PDOI correlates best with the 8th mode of SOI (0.906). Other good correlations exist in their 4th modes (0.382), 7th modes (0.373) and 6th modes (0.369). All of them are significant at the 99.9% confidence level. It can be found from the above analysis that the correlation is weaker between high frequency modes of PDOI and each mode of each index of ENSO, and between the high frequency modes of each index of ENSO and each mode of PDOI. The larger correlation coefficients exist at the quasi-2-7-yr variability and at the decadal and multi-decadal variability. The most remarkable characteristic is the concentration of the large correlation coefficients about the diagonal line of the tables. For example, the quasi-2-7-yr variability modes of PDO have a strong relationship with the quasi-2-7-yr variability modes of various indices of ENSO, but have a weak relationship with decadal and multi-decadal variability modes of various indices of ENSO. Otherwise, at low frequencies, the correlations between the modes of PDOI and SOI, Nino1+2, Nino3 and Nino3.4 are much better than the correlations between the modes of PDOI and Nino3.4. The correlations between the 6th IMF of PDOI and of SOI, Nino1+2, Nino3, and Nino3.4 are most stable. Lead/lag correlation coefficients about 10 years are also calculated further. The negative months in the abscissa represent that PDOI leads ahead of SOI, and

7 NO.1 HE Juanxiong, YU Zhihao and YANG Xiuqun 89 vice versa. It is found that the value between the modes at low frequency scale is also high. The value is the highest at about 2 years, and the second highest is from -60 to -90 months. For the modes at the quasi-2-7-yr scale the value become low out of -30 or 30 months. The value between the modes at high frequencies as well as between modes of high frequency and modes of low frequency is low. The good lag correlation does not arbitrarily exit even among the decadal or multi-decadal modes and quasi-2-7-yr oscillation modes. For example, the 4th IMF of PDOI has a bad lag correlation with the 6th IMF of SOI, Nino1+2, Nino3 and Nino3.4. In addition, the value between each mode of PDOI and Nino4 is always bad. In Fig.5a, the value has a cosine variation. SOI is out of phase with Nino3, and Nino3.4 leads Nino3 by about half a year. The value is best from -30 to 30 months, and better from -90 to -60 months. In Fig.5b, SOI is also out of phase with Nino3 and Nino3.4, with better correlation only from -24 to 24 months. In Fig.5c, the value is very small. In Fig.5d, a bad correlation exists between PDOI and Nino4. Fig.5. Lag correlation coefficients between the modes of PDOI and SOI, Nino1+2, Nino3, Nino4, Nino3.4. (a) Values between IMF6 of PDOI and IMF6 of SOI, Nino3, Nino3.4, (b) values between IMF4 of PDOI and IFM4 of SOI, Nino3, Nino3.4, (c) values between IMF4 of PDOI and the IMF6 of SOI, Nino3, Nino3.4, and (d) values between IMF6 of PDOI and IMF6 of Nino1+2, Nino Relationship between PDOI and SOI during To test whether the previous results are robust, we apply EMD to PDOI and SOI of CPC for the period. For PDOI (figure omitted), the first three of all eight IMF modes characterize high frequency. The 4th and 5th modes characterize quasi- 2-7-yr variability. The 6th through 8th modes characterize decadal and multi-decadal variations. The 8th mode is an oscillation throughout the 20th century. The elements of the 8th mode are larger than zero from 1925 to 1946 and from 1977 to the present, these two periods just correspond to the warm phases of PDO. The elements of the 8th mode are less than zero from 1900 to 1924 and from 1947 to 1976, these correspond to the cool phases of PDO. Comparison of Fig.2 with

8 90 ACTA METEOROLOGICA SINICA VOL.19 Fig.5 indicates that each mode of PDOI from 1950 to 2000 bears strong resemblance to the second half of the corresponding mode of PDOI from 1900 to In fact, is the spatial correlation coefficient between the first mode of PDOI from 1950 to 2000 and the second half of the first mode of PDOI from 1900 to Similarly, is for the second mode, is for the third, is for the 4th, is for the 5th, is for the 6th and is for the 7th mode is the correlation coefficient between the trend of PDOI from 1950 to 2000 and the second half of the 8th mode of PDOI. Therefore, the results by EMD are steady. Analogous to the situation of PDOI, EMD of SOI from CRU (Climate Research Unit) during obtains eight modes (figure omitted). The first three characterize high frequency variability. The 4th and 5th modes characterize quasi-2-7-yr variability. The 6th through 8th modes characterize decadal and multidecadal variations. Although the indices come from different institutes, each mode of SOI of CPC is similar to the second half of the corresponding mode of SOI of CRU is for the first mode, is for the second, is for the third, is for the 4th, is for the 5th, is for the 6th and is for the 7th is the correlation coefficient between the trend of SOI of CPC and the second half of the 8th mode of SOI of CRU. Although it is not better than the situation of PDOI, it also confirms that the EMD of SOI is stable. In Table 4, the correlation coefficients are almost zero between the high frequency modes of PDOI and each mode of SOI, and between the high frequency modes of SOI and each mode of PDOI. The correlation coefficients are large between the modes of SOI and the modes of PDOI at quasi-2-7-yr variability and at decadal and multi-decadal variability. Although different in the magnitude, it is consistent with Table 1. For example, the relationship between the 5th mode of PDOI and SOI (-0.386), as well as between the 7th mode of PDOI and SOI (-0.440) is fine, but it is bad between the 6th mode of PDOI and the 8th mode of SOI (-0.021). Table 4. Correlation coefficients between the intrinsic mode functions of PDOI and SOI PDOI SOI Discussion Atmosphere Bridge is the most commonly theory hired to explain the connection between the tropical and the extratropical zones. Atmosphere Bridge is an atmosphere channel linking the tropical with the extratropical zones. It is commonly thought that the variance of the wind field in the extratropics is partly due to the forcing of the tropics. The anomaly of the tropical SST triggers the anomaly of the tropical convection activity, which drives the wave train poleward (teleconnection pattern). When the wave train arrives at the extratropics, it produces the anomaly of atmosphere circulation. By Hasselmann mechanism (Hasselmann, 1976), wind stress field in the mid-latitude plays a role of white noise forcing, and the response of ocean has a red spectrum of decadal and multi-decadal variations. For example, Latif and Barnett (1994) suggested that more warm water could be transported from the tropical Pacific to the North Pacific because of subtropical gyre intensifying. Then, SST in the North Pacific increases and warm anomaly occurs. It

9 NO.1 HE Juanxiong, YU Zhihao and YANG Xiuqun 91 provokes two competitive mechanisms. The first is PNA (Pacific North America) Pattern that can reinforce the warm anomaly of SST. Thus PNA and the warm anomalous SST form a positive coupling system. The second is the anomalous wind stress curl that can spin down subtropical gyre transporting warm waters. PDO is the result of these two competitive mechanisms. On the other side, the variation of the atmosphere-ocean system in the extratropics can influence the tropics by the Atmosphere Bridge. Wang and An (2002) found that the decadal variability of tropical wind stress and the ocean surface dynamic process under the forcing of the tropical wind stress account for the decadal variability of ENSO. Besides, Atmosphere Bridge plays an important role. Extropical decadal variability signal is conveyed to tropics by Atmosphere Bridge. It makes a change in the tropical atmosphere circulation, and finally influences the tropical ocean dynamic process. Therefore, an explanation can be given by Atmosphere Bridge for the good synchronized correlations between PDOI, SOI, Nino3.4, Nino3 and Nino1+2 on the quasi-2-7-yr scale and decadal and multi-decadal scales. In spite of the connection between the tropical and the extratropical zones by Atmosphere Bridge, there exists the mode independent of each other. The Walker circulation of the tropical Indian Ocean may affect ENSO in addition to the North Pacific. Zhou et al. (2001) found that there is a North Pacific mode independent of ENSO linearly. Perhaps it is the reason for the weak correlations between PDOI and various indices at the high frequency. The different areas in the tropical Pacific have different relationships with the PDOI. Although the anomaly of the SST in the western Pacific may force the anomaly of the extratropical atmosphere, the anomaly is concentrated on East Asia not on the central of the North Pacific. The teleconnection pattern is the Pacific Japan pattern (Huang and Li, 1987). ENSO has a close link with the central and eastern tropical Pacific and it excites the PNA Pattern, which has a strong effect on PDOI. Perhaps, it is the cause for the weak correlation between PDOI and Nino4 and the strong correlation between PDOI and the other tropical indices. In the previous sections, it can be found that there exist high values, which surpass the 99.9% confidence level greatly, between PDOI, Nino3.4, Nino3 and Nino1+2 from -90 to -60 months. Because the lag time is so large, it is impossible the result of Atmosphere Bridge. The anomaly of mid-latitude SST may make an effect on tropical SST by the advection of ocean circulation. Gu and Philander (1996) assumed that there exists a shallow meridional circulation between tropical and the extratropical oceans. The anomaly of extropical SST travels along the isopycnal surface after subduction, changing the equatorial thermocline when it arrives the equator. By ocean water upwelling the anomaly is transported to the surface, making the sign of tropical Pacific SST reversed. Atmosphere Bridge has been employed to depict the connection between the tropical and the extratropical zones. It can not give a thorough explanation of the relationship between ENSO and PDO found in the previous sections. There are many puzzles. For example, it can possibly throw a light on the stable and better correlation at the timescale of quasi-2-7-yr and decade and intedecade between ENSO and PDO, but it can not give the cause why the decadal variation of ENSO has a poor relationship with quasi-2-7-yr variations of PDO and the cause for the weak correlation of the two systems at the high frequency. Is there an ocean pathway conveying the extratropical signal to the tropical? Is the signal exactly taken 60 to 90 months traveling to the tropical? It is dependent on the development of the concerned theories. 5. Conclusions Although PDO is ENSO-like in the spatial structure, PDO and ENSO have their respective temporal structure according to their respective IMFs obtained by the EMD. The stable and better correlation only exists on special modes. The weak relationship between PDO and ENSO at high frequencies suggests there exists the mode independent of each other. The strong correlations at the timescale of quasi-2-7-yr and decade and multi-decade suggest that PDO and ENSO

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