Oscillatory climatic modes over the North Atlantic, Ethiopian Plateau, and eastern Mediterranean were examined in instrumental and proxy records from these regions. Aside from the well-known North Atlantic Oscillation (NAO) index and the Nile River water-level records, the authors study for the first time an instrumental rainfall record from Jerusalem and a tree-ring record from the Golan Heights.
The teleconnections between the regions were studied in terms of synchronization of chaotic oscillators. Standard methods for studying synchronization among such oscillators are modified by combining them with advanced spectral methods, including singular spectrum analysis. The resulting cross-spectral analysis quantifies the strength of the coupling together with the degree of synchronization.
A prominent oscillatory mode with a 7–8-yr period is present in all the climatic indices studied here and is completely synchronized with the North Atlantic Oscillation. An energy analysis of the synchronization raises the possibility that this mode originates in the North Atlantic. Evidence is discussed for this mode being induced by the 7–8-yr oscillation in the position of the Gulf Stream front. A mechanism for the teleconnections between the North Atlantic, Ethiopian Plateau, and eastern Mediterranean is proposed, and implications for interannual-to-decadal climate prediction are discussed.
Interannual and interdecadal oscillations have been found in many variables and parts of the earth’s climate system; see Ghil and Vautard (1991), Dettinger et al. (1995), Plaut and Vautard (1994), Unal and Ghil (1995), Moron et al. (1998), and many others. These oscillations are usually correlated across large regions, on the order of thousands of kilometers, through “teleconnections” (Wallace and Gutzler 1981; Barnston and Livezey 1987) that have been attributed to several causes, including the propagation of atmospheric Kelvin and Rossby waves, changes in thermally direct circulations, intrinsic modes of atmospheric variability, and changes in the oceans’ wind-driven or thermohaline circulation. The terms “teleconnection” and “oscillation” have often been used synonymously in meteorology, ever since the discovery of the North Atlantic, North Pacific, and Southern Oscillations by G. Walker and associates (e.g., Walker and Bliss 1932, 1937); in these cases, the term “oscillation” was used merely to describe large-scale seesaw patterns in sea level pressures.
These pioneering studies did not imply the presence of distinct spectral peaks, although Walker (1931) did hint at their presence in the Southern Oscillation (SO). We now know that the SO is strongly coupled to El Niño (EN) sea surface temperature anomalies to form ENSO, which exhibits pronounced interannual periods (Philander 1990; Jiang et al. 1995a) and has near-global climate impacts through atmospheric teleconnections (Hoskins and Karoly 1981; Ropelewski and Halpert 1987; Chiang and Sobel 2002; Lintner and Chiang 2007). In the midlatitudes, the best-known oscillatory patterns are the North Atlantic Oscillation (NAO; Rogers 1990; Hurrell 1995) and the Pacific Decadal oscillation (Mantua et al. 1997; Chao et al. 2000), while the tropics are dominated by ENSO (Rasmusson and Carpenter 1982; Philander 1990).
We examine here interannual and interdecadal climate variability over the eastern Mediterranean during the last two centuries. Several instrumental and proxy records longer than 100 yr exist for different sites in the region. The records we analyze include precipitation records from Jerusalem, as well as tree-ring records from the Golan Heights (Feliks 1968), and the Nile River flow records previously analyzed by Kondrashov et al. (2005, hereafter KFG). We use advanced spectral methods, including singular-spectrum analysis (SSA), to find the most significant oscillatory modes in these climatic records (Ghil et al. 2002a) and relate them to similar modes over the North Atlantic.
We concentrate therewith on the eastern Mediterranean and on potential teleconnections with the North Atlantic Ocean and the Ethiopian Plateau. Previous teleconnection studies were based on correlations between climatic time series from the regions of interest (Wallace and Gutzler 1981); often the correlations were no larger than 0.6 and they did not clarify in which frequency band the teleconnection under investigation was most active.
The commonalities between the well-documented records of the NAO index and the records representing the focus area of our paper are considered here from the point of view of synchronization between chaotic oscillators (Rosenblum et al. 1996; Duane 1997; Boccaletti et al. 2002; Osipov et al. 2003; Duane and Tribbia 2004). From this point of view, each regional climatic index may be considered as a chaotic oscillator representing the regional dynamics of local climate; teleconnections then arise as a coupling in space between locations. Our synchronization analysis differs from that of previous studies inasmuch as we use multichannel SSA (M-SSA) to examine cross-spectral properties of the climate records from different regions and focus therewith on studying the commonalities between oscillatory components of the records. Based on this preliminary analysis, significant oscillatory modes are reconstructed and we apply the synchronization analysis to these modes. Such an approach allows us to identify which regions are teleconnected and, using a novel energy-ratio criterion, we can determine the strength of the coupling and the region in which each oscillatory mode arises.
In section 2, our modified synchronization analysis is described and it is illustrated by its application to a system of two coupled Rössler (1976) oscillators. In section 3, we provide a short analysis of each climate record separately, while in section 4 we examine the synchronization between the records from different regions and consider the mechanism of this synchronization. Concluding remarks appear in section 5.
2. Synchronization of chaotic time series
a. Basic methodology and its modification
We follow the overall approach of Rosenblum et al. (1996) and Osipov et al. (2003) in our analysis of the synchronization between two chaotic time series and will present some new concepts and tools after summarizing their approach. First, these authors, following Gabor (1946) and Panter (1965), define an analytic signal ψ(t) associated with a given time series x(t). This analytic signal is the complex-valued time series
where i = −1 is the imaginary unit and
is the Hilbert transform of x(t), y(t) = H[x(t)]. The instantaneous amplitude A(t) and phase ϕ(t) of the analytic signal ψ(t) are defined by the polar representation of Eq. (1a),
Two signals, x1(t) and x2(t), are said to be synchronized when the phase difference d(t) = ϕ1(t) − ϕ2(t) between the two does not grow with time. Note that the phase angle ϕ(t) is allowed to increase beyond 2π.
The integral in Eq. (1b) is ill posed, and thus the Hilbert transform y(t) = H[x(t)] of x(t) is often estimated by applying a discrete Fourier transform to x(t), setting all negative frequencies to zero, and transforming it back into the time domain. Marple (1999) discussed the mathematical properties of an analytic signal, as well as its numerical estimation by such a procedure. For each sine component x(t) = sin(t), the Hilbert transform returns its cosine counterpart H[x(t)] = −cos(t); Eq. (2b) thus yields the correct phase, but with the opposite sign, in this special case. Therefore, sometimes one uses, −H as a more convenient transform; since we will only need to consider phase differences here, we keep H as our transform, with the plus sign.
There is, however, a hitch: in general, the Hilbert transform is numerically ill posed; that is, H[x(t)] depends sensitively on small errors in x(t). Intuitively, for a broadband signal one might get a multitude of phases that are hard to separate from each other, rather than a single phase.
The climate records we consider in this paper are short, 100–200 yr or less, and there are only about 4–10 data points per period of interest. Hence the Marple (1999) method for the calculation of the Hilbert transform does not yield satisfactory results either. Moreover, it is well-nigh impossible to test over just a very few full periods whether the phase difference d(t) between two nearly equal periodicities in the time series x1(t) and x2(t) increases with time. We use therefore a different approach to find the phase and amplitude of an arbitrary signal; this approach has been tested extensively and found to be useful in studying oscillatory modes present in short and noisy climate records (Ghil et al. 2002a).
where x′(t) is the time derivative of x(t), while the instantaneous amplitude Ai(t) is given by the local extremum (maximum or minimum) of the signal that is nearest to the epoch t.
The key insight of these authors was precisely to only treat in this way the narrow-band signals obtained by SSA prefiltering, namely the reconstructed components (RCs) associated with an oscillatory mode. Since the RCs are such narrow-band, modulated signals, the Bedrosian (1963) theorem justifies the application of the Hilbert transform to them to obtain a well-defined, unique phase.
The synchronization between two signals is measured by the standard deviation σ of the instantaneous phase difference between the two time series,
and the correlation cor(A1, A2) between the instantaneous amplitudes:
where 〈·〉 denotes time averaging. Following Osipov et al. (2003), one can identify three types of synchronization in chaotic time series, which we label as follows:
Frequency locking (F) does not imply phase locking (FP), as can be easily seen from adding any perturbation with a broadband, continuous spectrum—whether random or deterministically chaotic, like any variable of the Lorenz (1963) model—to one of two harmonic oscillators having the same frequency. The threshold σ0 = 0.5 in FP is fairly arbitrary but clearly much smaller than π. When examining the coupling of two Rössler oscillators, Rosenblum et al. (1996) found that, as the strength of the coupling increases, the degree of synchronization increases as well, from F to FP and on to FPA.
The difficulty in estimating the quantities defined in Eqs. (1)–(4) and used in the definitions of synchronization F–FPA resides in the essential ill posedness of the differentiation of chaotic time series, especially in the presence of measurement and sampling noise (Maraun and Kurths 2004). This difficulty has given rise to several different techniques for estimating the phase and amplitude associated with a given time series. To obtain smooth and reliable estimates of these quantities, we apply two advanced, well-documented spectral methods to find the oscillatory modes shared by the two time series, namely SSA and, for verification purposes, the multitaper method (MTM) of Thomson (1982). These two methods, along with additional ones, are available in the freeware SSA-MTM toolkit at http://www.atmos.ucla.edu/tcd/ssa/.
M-SSA is an extension of SSA to several time series, with each “channel” corresponding to one of the scalar time series of interest (Broomhead and King 1986; Keppenne and Ghil 1993; Plaut and Vautard 1994; Ghil et al. 2002a). The method essentially diagonalizes the lag-covariance matrix associated with the vector time series whose components are the channels. The eigenvectors of this matrix are the space–time empirical orthogonal functions (ST-EOFs), and projection onto them yields the corresponding principal components.
The entire vector time series or parts thereof that correspond to trends, oscillatory modes or noise can be reconstructed by using linear combinations of these principal components and ST-EOFs, which provide the spatiotemporal RCs. A statistical significance test against red noise is performed by comparing the projection of the data onto a given ST-EOF for each channel with the corresponding projections of a large ensemble of red-noise surrogates (Allen and Robertson 1996; Ghil et al. 2002a).
We thus assess synchronization of a pair of records by applying a two-channel M-SSA, with each channel corresponding to one of the two records; each record is nondimensionalized by its standard deviation. Shared signals that are statistically significant at the 95% level are analyzed further to quantify their degree of synchronization. The standard deviation σ of the instantaneous phase difference between them is computed via (4a), using Eq. (3) to estimate each instantaneous phase.
b. Application to coupled Rössler oscillators
In this section, we illustrate the modified synchronization analysis method described in section 2a by applying it to a system of two coupled Rössler (1976) oscillators and examine the route to synchronization in the parameter range studied by Rosenblum et al. (1996). The Rössler oscillator provides a simple chaotic-system analog of regional climate. Its orbits spiral around a single pole, rather than the two poles of the Lorenz (1963) model’s “butterfly,” with its two wings; it is thus simpler to define instantaneous phase than in the “bipolar” Lorenz model.
The Rössler oscillator, sometimes called simply the “Rösselator” for short, is also very popular in studies of deterministic chaos. For instance, Janjarasjitt and Loparo (2008) studied recently a coupled system of two such oscillators with distinct parameters, while Brunnet et al. (1994) investigated the collective behavior of a two-dimensional square lattice of identical Rössler oscillators. The latter is a more sophisticated counterpart of the Lorenz (1991, 1996) models; in both cases, E. N. Lorenz studied a one-dimensional, regularly spaced lattice of oscillators. Either the Lorenz (1991, 1996) models or the Brunnet et al. (1994) model can serve for the conceptual investigation of teleconnections among local or regional climates.
The equations of the coupled Rössler system are, for i = 1, 2,
where ω1 = 1.015 and ω2 = 0.985 are the two angular frequencies of the uncoupled oscillators, and C is the coupling parameter between the two oscillators. We used the Adams–Bashforth scheme to integrate Eq. (5) with a time step of δt = 0.01 and stored the first components x1(t) and x2(t) of the two oscillators every Δt = 0.5 time units, that is, every 50 time steps. The synchronization analysis was carried out on the two model-generated time series x1(t) and x2(t) in the time interval t = 1000–4000.
Applying the synchronization criteria of Eqs. (1)–(4) to the two time series x1(t) and x2(t) we find that, for C < 0.027, the two time series are not synchronized at all: the standard deviation σ of the phase differences is large and the amplitude correlation is quite small; see Table 1 and Fig. 1a (C = 0.017). For C = 0.027, the standard deviation is σ = 7.9 over a time interval of 3000 time units, which is finite but already quite large (in particular larger than π) and actually tends to infinity as the time interval increases to infinity (not shown), while the amplitude correlation is small, cor(A1, A2) = 0.08, and does stay so as the interval increases.
When increasing the coupling strength to C = 0.035, the standard deviation of the phase difference drops to σ = 0.21 and changes but little as the time interval increases; thus both the frequencies and phases of the two oscillators are synchronized. On the other hand, the amplitude correlation only reaches 0.24 [cf. Fig. 1b (C = 0.035)]; so the synchronization type is only FP, and not FPA. From Fig. 1b, one sees in fact that—while the two time series keep a constant lead–lag relationship, unlike in Fig. 1b—the amplitude evolutions of the two series do not match. These results agree with those of Rosenblum et al. (1996).
Next we compute the cross-spectra of x1 and x2, by applying M-SSA with two channels, and using a window width of M = 60Δt = 30 time units (Ghil et al. 2002a). The RC of an oscillatory mode associated with a given pair (i, i+1) of ST-EOFs will be denoted by RC(i,i+1); the corresponding components will be called RC(i,i+1)x1 for channel x1 and RC(i,i+1)x2 for channel x2, and the energy (variance) in each of the two RC components will be identified as E(RC(i,i+1)x1) and E(RC(i,i+1)x2), respectively.
For C = 0.0, the oscillators are not coupled and we obtain five significant eigenvalues of the lag-covariance matrix: 2 eigenvalues equal to 0.168, 2 equal to 0.162, and one equal to 0.202; the significance is established with respect to the noise present in the two time series x1(t) and x2(t), due to the interaction between the numerical truncation and the nonlinearity (Henrici 1963; Isaacson and Keller 1966). The RC associated with the first ST-EOF pair, RC(1,2), captures 51% of the variance and has energy only in RC(1,2)x2. The second significant pair of RCs, RC(3,4), captures 35% of the variance and has energy only in RC(3,4)x1. There is no oscillatory mode that contains energy in both RCx1 and RCx2, as expected for the uncoupled system; the two pairs of eigenvalues simply correspond to the values of the frequencies ω1 and ω2. The RCs associated with the first four modes, RC(1–4), contain 51% + 35% = 86% of the variance. The amplitude correlation between RC(1–4)x1 and RC(1–4)x2 is 0.12, about the same as the amplitude correlation between the complete time series x1(t) and x2(t).
By increasing the coupling strength between the two oscillators, the system changes gradually as follows: (i) the five significant eigenvalues become closer in value and merge eventually; for C = 0.035, when the system experiences phase synchronization (see again Table 1), these five eigenvalues have completely merged (not shown). (ii) The variance of the RC(1,2) oscillatory mode increases and that of the RC(3,4) mode decreases as C increases; see columns 2 and 3 of Table 1. (iii) The energy ratio ɛ1,2 = E(RCx1)/E(RCx2) between the two components of the RC(1,2) oscillatory mode increases and tends to 1.0 as C increases (column 4 of Table 1); the energy ratio ɛ3,4 = E(RCx1)/E(RCx2) of the RC(3,4) oscillatory mode decreases (from infinity) and also tends to unity as C increases (not shown). The ratio ɛ34 for the oscillatory mode associated with the second pair is almost equal to that associated with the first pair. (iv) The standard deviation of the phase difference σ decreases as C increases (column 5 of Table 1) and we use it to measure the phase synchronization of the system.
The piling up of the variance in the RC pair (1, 2) is clearly of the essence in the synchronization process. The equalization of the energy ratios ɛ12 and ɛ34 between the two components of the RC(1,2) mode and the RC(3,4) mode, respectively, may be due to the symmetry of the coupled system, with two identical Rössler oscillators being involved.
Increasing C from 0.027 to 0.035, the energy in the RC(1,2) mode increases from 86% to 93% and that of the RC(3,4) mode decreases from 9% to 1%, while σ changes from 7.9 to 0.21 (see again Table 1 and Fig. 1), and thus the synchronization type of the system is FP for C = 0.035. Increasing C further to 0.12 increases the amplitude correlation from 0.24 to 0.96 (column 6 of Table 1, and Fig. 1c, with C = 0.12) and now the system’s synchronization is complete, that is, of FPA type; the two time series are now visually indistinguishable (Fig. 1c).
Thus our conclusions about the synchronization type in the coupled Rössler oscillator pair, as a function of the coupling strength, agree with those of Osipov et al. (2003) and Rosenblum et al. (1996). Their criteria, though, relied on the Lyapunov spectrum, while ours rely on M-SSA. The latter criteria seem more robust in the analysis of large systems and for short and noisy time series, like those available in climate dynamics and the geosciences in general: reliable calculation of Lyapunov exponents requires very long and relatively clean time series. To show that the results for the coupled Rössler oscillators in this section are not an artifact of the M-SSA analysis, we investigate in appendix A two even simpler synthetic-data examples.
Along the route to synchronization between the coupled Rössler oscillators, and for all values of C > 0, the synchronization between the components RCx1 and RCx2 that belong to the same oscillatory pair is of complete FPA type, since the corresponding σ is quite small, σ ≪ 1, and the amplitude correlation is about 1. A difference between the two components persists, though, for low coupling strength and is manifest in the energy ratio E(RCx1)/E(RCx2): this ratio tends to 1 for strong coupling and tends to 0 (respectively infinity) as C → 0. Therefore, in the analysis of the cross-spectra of several pairs of climatic records, we will examine the ratio ɛ = E(RCx1)/E(RCx2) to identify the coupling strength in significant frequency bands, in addition to the synchronization type.
Tentatively, the results for the coupled Rössler oscillators suggest that the direction of influence can be inferred from the energy ratio as follows: When the energy ratio is close to unity the direction of influence cannot be inferred via this criterion. For weak coupling, |E(RCx1)/E(RCx2)| ≪ 1, because of the much larger energy in RC(1,2)x2, the direction of influence in the first pair is from x2 to x1. The opposite direction of influence is obtained for the second pair. As the coupling increases, so does the degree of synchronization, while the energy ratio between the first and second component within each pair increases but is still less than unity; in this case, we can still identify the direction of influence, in the presence of strong coupling; that is, for synchronization of the FP or FPA type.
In the following we analyze the synchronization between different climatic records for the last one-to-two centuries; the records stem from the North Atlantic, the eastern Mediterranean, and North Africa. The methodology used is the one described so far. In this perspective, the climate system can be interpreted as being represented by a large number of coupled oscillators (Ghil and Childress 1987; Lorenz 1991, 1996). Significant modes that have similar periods and are found in distinct regions (Ghil et al. 2002a) may then be due to the coupling of such oscillators. We thus test not only (i) the degree of synchronization, but also (ii) the strength of the coupling, by evaluating the ratio ɛ = E(RCx1)/E(RCx2) for a given oscillatory mode of the cross-spectral analysis, and also, albeit tentatively, (iii) the direction of influence.
3. The climatic records
In this section we examine the spectral properties of the individual climatic records from the eastern Mediterranean, Ethiopian Plateau, and the North Atlantic. The climate records from the eastern Mediterranean are instrumental precipitation records from Jerusalem, and tree-ring records from the Golan Heights. The proxy records associated with the Ethiopian Plateau are the high-water records of the Nile River. For the North Atlantic we use the well-known NAO index.
Single-channel SSA is applied to each time series in turn. Statistical significance is assessed using Monte Carlo SSA (MC-SSA; Allen and Smith 1996) against a red-noise null hypothesis constructed from a surrogate data ensemble of 100 series, each with the same variance and lag-1 autocorrelation as the original record.
The surrogate time series (Allen and Smith 1996) were produced by projecting the first 10 principal components of the SSA analysis onto the basis vectors of a red-noise process. To check the robustness of the peaks identified via SSA, the Mann and Lees (1996) version of the MTM method (see also Ghil et al. 2002a, and the documentation available at http://www.atmos.ucla.edu/tcd/ssa/) was also applied, using 3 tapers and a spectral resolution of 0.03 cpy. In this MTM version, statistical significance is also assessed against a red-noise background and associated 50%, 90%, 95%, and 99% significance levels are computed. A summary of the modes identified as statistically significant at the 95% level using both MC-SSA and MTM are given in Table 2, along with the percentage of variance (of the detrended time series) captured by the corresponding SSA mode.
a. Jerusalem precipitation record
The annual mean precipitation records from Jerusalem cover the 133 yr from 1861 to 1994; see Fig. 2c. D. Ashbel compiled the records for the years 1861–1964 and the Israel Meteorological Service those for 1965–94. Dan Yakir from the Weizmann Institute of Science communicated these yet unpublished records to us.
To concentrate on the interannual periodicities in these precipitation records, as well as in the other records discussed below, we first removed the lowest-frequency components, including the nonlinear trend (Ghil et al. 2002a). In the rainfall records, the three components that fall in this category are (1, 2, 3), when carrying out the SSA analysis with a window width of M = 45 yr. Next we applied MC-SSA with a window of M = 40 yr to the detrended series. Very similar results were obtained with M = 50 yr (not shown).
The significant time EOFs (T-EOFs), ordered by decreasing variance captured, along with the associated periods and variances are the following: the pair (1, 2) has a period of 3.1 yr and a variance of 18%; the pair (3, 4) has a period of 2.2 yr and a variance of 14%; the pair (5, 6) has a period of 3.8 yr and a variance of 13%; the pair (7, 8) has a period of 14 yr and a variance of 10%; and the pair (9, 10) has a period of 7.8 yr and a variance of 7%. The above periods were also identified as significant at the 95% level using MTM applied to the detrended time series (Table 2), except the 14-year SSA pair (7, 8), which was significant in the MTM analysis at the 90% level only. In Fig. 3 we show the MC-SSA and MTM spectra, along with the statistical significance of the peaks.
b. The tree-ring record from the Golan Heights
The ring-width records of Quercus Infectoria (also called Aleppo oak or Cyprus oak) from the Golan Heights extend over 210 yr, 1757–1967; see Fig. 2b. This dataset is part of an extensive dendrochronological survey carried out in the Golan Heights in 1967 by Yehuda Feliks (the father of this paper’s lead author); see Feliks (1968).
SSA was applied to tree-ring records first by Cook et al. (1995). For our record, the nonlinear trend is captured by the first two eigenmodes of the SSA analysis with a 40-yr window. The detrended series was analyzed by MC-SSA with the same window width; very similar results were obtained with M = 50 yr. The significant pairs are the following: pair (1, 2) with a period of 25 yr and 37% of the variance; pair (3, 4) with a period of 14 yr and 20% of the variance; pair (6, 7) with a period of 10 yr and 12% of the variance; and pair (9, 10) with a period of 7.7 yr and 6% of the variance. We also applied MTM analysis, with 3 tapers and a spectral resolution of 0.03 cpy, to the detrended time series. The above frequencies were all significant at the 99% level.
The significant pairs with dominant periods that are equal to 7.7 yr or longer capture 75% of the variance of the detrended series. This raises the possibility that the trees integrate climatic effects over several years: the trees’ physiology seems to act like a low-pass filter on the climatic signals and thus enhance the low-frequency oscillations. The oscillatory modes of 14 and 7.7 yr are also prominent in the Jerusalem precipitation records.
c. The Nile River records
KFG studied in great detail the high- and low-water records for the Nile River over the 1300-yr time span a.d. 622–1922. They found periodicities of 2.2, 4.2, 7.2, 12, 19, and 64 yr; some of these periodicities had already been found and discussed in the abundant literature focusing on these records, as reviewed by KFG. For instance, the 2.2- and 4.2-yr signals had been attributed to the long-established connection between the Nile River discharge and the ENSO phenomenon in the Indo-Pacific Ocean (Ropelewski and Halpert 1987, 1989; Quinn 1992; Camberlin 1995; Wang and Eltahir 1999).
KFG’s novel and striking result was the clear presence of a 7.2-yr signal in the high-water record. A 7–8-yr peak has been well documented in several oceanic and atmospheric fields extending over the North Atlantic basin (Moron et al. 1998; Wunsch 1999; Da Costa and Colin de Verdière 2002; Dijkstra and Ghil 2005; Simonnet et al. 2005). Hence, KFG hypothesized that the 7.2-yr mode in the Nile River records demonstrates the extent of the North Atlantic influences, all the way into East Africa’s tropical regions. The research reported in the present paper was largely motivated by the need to examine more closely this hypothesis.
The high-water record corresponds to the maximum height of Nile floods, a height that is reached in August–September, when more than 85% of the flow in the Nile River is supplied by the Blue Nile and Atbara River, whose sources lie on the Ethiopian Plateau. This plateau reaches altitudes above 2200 m, and the summer monsoonal rainfall over it peaks between June and September (Wang and Eltahir 1999; Block and Rajagopalan 2007). Thus, although the Nile’s high-water stand was measured by the nilometer at Rhoda Island near Cairo, it is a proxy for rainfall over the Ethiopian Plateau. The measurements at Rhoda Island ceased in 1922 because of the lower, or older, Aswan Dam, built at the beginning of the twentieth century to regulate the Nile River flow.
To extend the records studied by KFG, we used existing measurements of the monthly volume flow for the century 1869–1969 at Aswan, provided in the hydrographic datasets of the National Center for Atmospheric Research (NCAR); see http://dss.ucar.edu/cgi-bin/rdabrowse?nb=true&c=topic&cv=Hydrosphere. There is a strong correlation between the summer (July–September) volume flow and the high level of the Nile River near Cairo, since 90% of the annual flow is during the summer and most of it is due to the Blue Nile. Thus we can extend the high-water records of the Nile River by using the volume flow for the years after 1922, and up to 1969, when the new, higher Aswan Dam was built.
The extension of the KFG high-water record to the more recent past is done by linear regression:
where h is the dependent variable, namely the high-water stand near Cairo, while the independent variable υ is the yearly mean volume flow near Aswan. The regression parameters, a = 1.66 and b = −0.83, were found from the overlapping time interval of h(t) and υ(t) during the years 1869–1922. During this interval, the correlation between the time series of h and υ is 0.93. In Fig. 4a, the observed high-water record and the one estimated according to Eq. (6) are shown for the interval of overlap. The estimated values are clearly quite accurate, and so the high-water record of the Nile at Cairo, extended until 1969, is shown in Fig. 2a. The spectral characteristics of this record (not shown) are quite similar to those in KFG, with the differences clearly attributable to this record’s relative shortness and to climatic changes between the earlier centuries and 1869–1969.
d. North Atlantic Oscillation
The NAO has been studied extensively over the last two decades (Hurrell et al. 2003), and its strength has been quantified using several indices. We use here the monthly index over the years 1823–2000, as compiled by Jones et al. (1997); see Fig. 2d. This index is the difference between the normalized sea level pressure at Gibraltar and the normalized sea level pressure over southwestern Iceland. Jones and colleagues used early instrumental data back to 1823. The dataset has been modified in November 2000 and the effect of this change is most evident in the summer; see http://www.cru.uea.ac.uk/~timo/projpages/nao_update.htm.
Here we revisit the index using a somewhat different approach; that is, we apply M-SSA with 12 channels, each channel being the NAO index in a given month of the year: channel 1 is the index for January, channel 2 for February, and so on, while the time step is 1 yr. The window width is again M = 40 yr.
The significant oscillatory modes for this M-SSA analysis have periods of 7.7, 5.5, 4, 3.2, and 2.2 yr. Each of these periods is dominant for a subset, or group, of calendar months. The most prominent oscillatory mode has the period of 7.7 yr that is familiar from KFG and it appears as the leading reconstructed component, RC(1,2). This mode is prominent during the winter months of December, January, and February (DJF), but also in August and October. In the other calendar months, the amplitude of this mode is smaller at least by a factor of 3. Rogers (1984), Robertson (2001), Gámiz-Fortis et al. (2002) and Paluš and Novotná (2004) also found an oscillation with a period of about 7.7 yr in the NAO index. A quasi-biennial component is present in the closely related Arctic Oscillation, constructed from hemispheric sea level pressures, as well (Trenberth and Paolino 1981; Robertson 2001). We applied also MTM analysis, with 3 tapers and a spectral resolution of 0.03 cpy, to the wintertime series [December–March (DJFM)] and in August. The above frequencies were all significant at the 95% level.
Below we concentrate mainly on two seasons. For boreal winter, that is, December–March, we use the average index value for these four months, while in summer we use the index value for August.
The main significant oscillatory modes of the above climatic time series are summarized in Table 2. The oscillatory modes appearing in the table are found to be significant at the 95% level against a null hypothesis of red noise, in both SSA and MTM.
4. Synchronized modes
Oscillatory modes with periods around 4, 7–8, and 14 yr are thus present in the records from the North Atlantic, eastern Mediterranean, and Ethiopian Plateau (Table 2). This clustering raises the possibility of teleconnections between the climate variations in these regions. We next examine separately the synchronization of each periodicity between regions to help determine the source region of the oscillatory mode and its role in the dynamics of each region.
The method of synchronization analysis was described in section 2. The analysis is applied to the overlapping portion of the two RC time series in question, with each series normalized by its variance in the overlapping part. In the following, we chose the window width M in the M-SSA to be about N′/3, where N′ is the length of the overlapping portion of the two time series in question.
a. Nile River floods and the NAO
The overlapping time interval of the extended high-water Nile records and the NAO index is 1825–1971. Monsoon rains over the Ethiopian Plateau occur during the boreal summer, along with the northward seasonal migration of the intertropical convergence zone (ITCZ), and hence we examine here the summer index of the NAO. The 7.7-yr oscillation is prominent in the NAO index during the boreal summer only in August.
We apply two-channel SSA where channel 1 consists of the high-water Nile records and channel 2 of the NAO index for August. The analyses carried out with a window width of M = 30 and 40 yr gave almost identical results. The significant oscillatory components have periods of 7.6, 4.3, 3.3, and 2 yr.
To concentrate on the interannual periodicities, we first removed the trend, as captured by RC(1,2) of the M-SSA analysis with a 35-yr window. We then applied M-SSA again to the detrended time series with M = 35 yr. The MC-SSA spectrum is shown in Fig. 4b, along with the statistical significance. The resulting leading oscillatory mode RC(1,2) has a period of 4.3 yr and accounts for 17% of the variance of the detrended series (see Fig. 5b).
The mean phase difference [cf. Eq. (2b)] is 3.11 rad = 2.1 yr, and the standard deviation [cf. Eq. (4a)] is σ = 0.11 rad = 0.96 month. The amplitude correlation [cf. Eq. (4b)] is 0.92, and so the synchronization is complete, that is, of FPA type, with a near phase opposition between the NAO index and Nile flow (see Fig. 5b). The energy ratio E(RCx1)/E(RCx2) is ɛ = 0.97, where x1 corresponds to the Nile record and x2 to the NAO index. This ratio is quite close to unity, suggesting that the coupling between the climate systems that trigger the 4.3-yr oscillation in the two regions is very strong. Since the energy ratio is about 1, we cannot identify the direction of influence, but physical plausibility clearly tilts toward the NAO being causal.
The second oscillatory mode is represented by RC(6,7), has a period of 7.6 yr, and captures 13% of the variance (see Fig. 5a). The mean phase difference is of 2.5 yr and is discussed below, while the standard deviation is σ = 0.23 rad = 0.28 yr. The amplitude correlation is 0.95, so the synchronization is complete, of FPA type. The energy ratio is ɛ = 0.28, suggesting that the coupling is moderate and that the NAO induces the 7.6-yr oscillation in the Nile River. (A potential teleconnection mechanism between the climate systems in the two regions that triggers this oscillation is discussed below, in connection with Fig. 7.)
The oscillatory mode RC(9,10) has a period of 3.3 yr and accounts for 10% of the variance (Fig. 5c). The mean phase difference is 1.37 rad = 0.72 yr (discussed below) and the standard deviation is σ = 0.42 rad = 0.22 yr. The amplitude correlation is 0.85, so the synchronization is almost complete. However, the ratio E(RCx1)/E(RCx2) = 0.11, suggesting that the coupling is quite weak.
To explore the teleconnection mechanism between the North Atlantic and the Nile River floods, Fig. 6a shows the mean low-level moisture fluxes during the month of August, computed from the National Centers for Environmental Prediction (NCEP)–NCAR atmospheric reanalyses (Kalnay et al. 1996); the pattern is qualitatively similar throughout the summer season (June–September; not shown). From the figure it follows that the rain over the Ethiopian Plateau is largely a result of moisture flux convergence from the south and north, associated with northward displacement of the ITCZ. Northward fluxes during boreal summer advect moisture from the equatorial Atlantic, and possibly the Indian Ocean—see also Rodwell and Hoskins (1996) and Raicich et al. (2003)—while the southward flux originates over the eastern Mediterranean.
The Somali jet transports humid air from the western part of the Indian Ocean northwestward, to the eastern slopes of the Ethiopian Plateau, which reach altitudes above 2000 m; this humidity flux is plotted as track 1 in Fig. 6b. As this humid air climbs above the plateau, it cools, condenses, and discharges its humidity as rain. Most of the rain from this source, however, falls over the eastern part of the Plateau and so its role in contributing to the flow of the Blue Nile is limited.
The second track of humid air to the Ethiopian Plateau starts off the equatorial Atlantic Ocean and is associated with the West African monsoon; this moisture flux is limited to the shallow layer beneath the African easterly jet at 700 hPa (Nicholson and Grist 2003) and is plotted as track 2 in Fig. 6b. A significant southward flux of moisture from the Mediterranean and Red Sea is visible in Fig. 6a, with some moisture flux divergence over the eastern Mediterranean; this humidity contribution is shown as track 3 in Fig. 6b. This third track is also found in maps of humidity flux drawn by James (1995, p. 250), Mariotti et al. (2002), and Mohamed et al. (2005). As these air masses reach the northern slopes of the Ethiopian Plateau, they ascend to levels above 3000 m, near the core of the African easterly jet. At these altitudes, the jet transports the ascending air over the Plateau, where it is cooled, condensed, and discharges its moisture as rain in the catchment area of the Blue Nile and the Atbara River.
Although most studies on the NAO have focused on the winter season, Folland et al. (2009) have recently studied its summertime expression during July and August. They demonstrate that the NAO’s positive phase is associated with anomalously wet conditions over the western Mediterranean and anomalously dry conditions across the Sahel; see also Hurrell and Folland (2002). Correlation maps between the NAO index and the low-level moisture flux vector (not shown) indicate a broad-scale influence of the NAO on the low-level flow over North Africa and the Mediterranean; in its positive phase, the summertime NAO tends to weaken the West African monsoon (described as track 2 above), while tending also to strengthen the flow across the Mediterranean and Egypt associated with track 3.
A mean phase difference of 2.5 yr in the 7.6-yr oscillation and 2.1 yr in the 4.2-yr oscillation is found in the coupled oscillatory modes of the NAO index and the Nile records. The almost exact phase opposition at the 4.2-yr period indicates that the negative phase of the NAO is associated with enhanced Nile River flow at this period, with the West African monsoon (track 2) most active. For the 7.6-yr period, the delay in the Nile River flow is closer to a quarter period than to a half period, and it is thus more likely that the eastern Mediterranean track 3 plays a greater role at this period.
b. The rain in Jerusalem and the NAO
The 134-yr time interval of overlap between the precipitation records in Jerusalem and the NAO index is 1861–1993. Since the rainy season in the eastern part of the eastern Mediterranean is during the winter months, we use the mean value of the NAO index during December–April in our analysis.
We apply M-SSA with channel 1 consisting of the NAO index and channel 2 being the precipitation record in Jerusalem. The analyses carried out with M = 30 and 40 yr gave almost the same results for both values of M. The significant oscillations have periods of 7.8, 3.8, 3.1, and 2 yr.
To better study the interannual periodicities we first removed the trend, captured by RC(1) of the M-SSA analysis with a 35-yr window. We then applied M-SSA to the detrended time series with M = 35 yr. The MC-SSA spectrum, along with the statistical significance, is shown in Fig. 7a.
The leading oscillatory mode, RC(1,2), has a period of 7.8 yr and accounts for 19% of the variance (Fig. 7b). The mean phase difference is −0.17 rad = −0.2 yr, with a standard deviation of σ = 0.05 rad = 0.06 yr. The amplitude correlation is 0.96, and so the synchronization is of FPA type. The energy ratio ɛ in this mode is E(RCx1)/E(RCx2) = 2.8, where x1 corresponds to the NAO index and x2 to the Jerusalem rainfall, suggesting that it is the NAO that induces the 7.8-yr oscillation in Jerusalem rainfall and that the coupling is moderate.
The in-phase relationship between the NAO index and Jerusalem rainfall is consistent with the results of Dünkeloh and Jacobeit (2003). In the positive phase of the NAO, higher pressures extend broadly over southern Europe and the Mediterranean, but over the far eastern Mediterranean surface pressures are lower, with an upper-level trough extending all the way from Iceland. This trough pushes very cold air over the relatively warm Mediterranean Sea and results in unstable air that leads to increased rainfall over the eastern and southeastern rim of the basin.
On the other hand, Eshel and Farrell (2000) claim the opposite for the eastern and southern parts of the eastern Mediterranean. They found that the correlation between the NAO index and the average precipitation in the whole eastern Mediterranean is negative. Their conclusion is based on averaging precipitation from the region, but in fact the stations in western Turkey, which are the densest, determine their average precipitation values. Thus the results of Eshel and Farrell (2000) are relevant only to the rainfall in western Turkey, and not to the eastern and southern part of the eastern Mediterranean, where the effect has the opposite sign.
The oscillatory mode RC(5,6) has a period of 3.1 yr, and accounts for 12% of the variance. The mean phase difference is 18.7 rad = 9.2 yr, with a standard deviation of σ = 10.6 rad = 5.2 yr. The amplitude correlation is 0.29, and so the eastern Mediterranean and the NAO are not synchronized at this period.
The oscillatory mode RC(7,8) has a period of 3.8 yr, and accounts for 11% of the variance (Fig. 7c). The mean phase difference is −0.27 rad = 0.13 yr, and the standard deviation, σ = 0.19 rad = 0.11 yr. The amplitude correlation is 0.84, and so the synchronization type is close to FPA. The energy ratio ɛ7,8 is E(RCx1)/E(RCx2) = 0.8, thus suggesting that the coupling is strong. Since the ratio is close to unity, it does not determine the direction of influence, but physical reasoning again leads us to suspect that it is the NAO that induces this oscillation in the eastern Mediterranean.
c. Tree rings and Jerusalem precipitation
The 107-yr time interval of overlap between the tree-ring record in the Golan Heights and Jerusalem precipitation extends over 1861–1968. We applied M-SSA to this time interval, with channel 1 consisting of the tree-ring widths and channel 2 of Jerusalem precipitation. The analysis was carried out with M = 30 and 40 yr, with very similar results in both cases. The significant oscillations have periods of 12.5, 7.8, and 3.8 yr.
To better study the interannual periodicities, we first removed the trend, captured by RC(1–4) with M = 30 yr. We then applied M-SSA to the detrended time series with the same window width.
The oscillatory mode RC(1,2) has a period of 3.8 yr, and accounts for 17% of the variance. The mean phase difference is 0.8 rad = 0.48 yr, with standard deviation σ = 1.23 rad = 0.7 yr. The amplitude correlation is 0.77, and so the synchronization type is FP. The energy ratio ɛ1,2 is E(RCx1)/E(RCx2) = 0.01, where x1 corresponds to the tree-ring records and x2 to the Jerusalem precipitation records. This ratio suggests that—while the coupling is quite weak—the rainfall induces the 3.8-yr oscillation in the tree growth, in agreement with intuition.
The oscillatory mode RC(3,4) has a period of 12.5 yr, and contains 16% of the variance; it is shown in Fig. 8. The mean phase difference is 1.57 rad = 3.3 yr and its standard deviation is σ = 1.64 rad = 3.26 yr, while the amplitude correlation is 0.98. It is surprising that the amplitude correlation is so high, although the phase is not synchronized; we thus appear to have a synchronization type that was not previously encountered but can be labeled FA. The energy ratio is E(RCx1)/E(RCx2) = 0.70; it shows that the coupling is strong and suggests it is the rainfall that induces the 12.5-yr oscillation in the tree growth, as expected.
The oscillatory mode RC(11,12) has a period of 7.8 yr, and captures 10% of the variance. The mean phase difference is 0.42 rad = 0.5 yr, and its standard deviation is σ = 2.47 rad = 3.07 yr; the amplitude correlation is −0.54, and so the synchronization type is F. The ratio ɛ34 is E(RCx1)/E(RCx2) = 0.41, suggesting that the coupling is moderate and that the rainfall induces the 7.8-yr oscillation in the tree-ring record as well.
It might appear somewhat surprising that the rainfall record in Jerusalem and the tree-ring record in the Golan Heights exhibit such a relatively weak degree of synchronization. Two main climatic factors determine tree growth: the amount of rain and the temperature. Usually more rain causes larger tree-ring growth, while cool winters result in less growth. Since the rainy winters are also cooler in the eastern Mediterranean, these two factors have an opposing effect on the tree growth, thus tending to weaken the rainfall signal in the tree-ring record.
5. Concluding remarks
Evidence for oscillatory modes of variability over the North Atlantic, Ethiopian Plateau, and eastern Mediterranean was examined using climate records from each region: the NAO index for the North Atlantic and Nile River high-water records for the Ethiopian Plateau, while the climate proxies for the eastern Mediterranean were precipitation in Jerusalem and a tree-ring record from the Golan Heights.
Statistically significant oscillatory modes—having periods of about 4 and 7–8 yr—were identified in all records. Such a clustering of periodicities raises the possibility of teleconnections between these regions (Ghil et al. 2002a). Other significant oscillatory modes were found in some of the records only; in particular, a 14-yr mode is present in Jerusalem rainfall and tree rings.
We have approached climatic teleconnections between regions in terms of synchronization of chaotic oscillators. Our approach is based, on the one hand, on the work of J. Kurths and colleagues (Rosenblum et al. 1996; Boccaletti et al. 2002; Pikovsky et al. 2003; Osipov et al. 2003) and, on the other, on prefiltering and spectral analysis of the time series of interest, via singular-spectrum analysis (SSA) and multichannel SSA (M-SSA). In addition, we examined the strength of the coupling and the direction of influence by examining the energy (or variance) ratio between various channels, in separate frequency bands identified by the M-SSA.
The methodology was presented in section 2, and illustrated by its application to the coupled Rössler oscillators previously used by Rosenblum et al. (1996). As the coupling between the oscillators increases, their time series synchronize, first in frequency (F), then in phase (FP), and finally in amplitude as well (FPA). In addition to following these steps in the synchronization, as defined by Osipov et al. (2003), we studied the route to synchronization by examining the joint spectra of the oscillatory modes, using two-channel M-SSA.
For the two coupled Rössler oscillators, the strength of the coupling can be measured, without explicit reference to the value of the coupling constant C in Eq. (5), by examining the ratio ɛk,k+1 of the energy (or variance) in the two channels of the reconstructed components (RCs) of the oscillatory mode (k, k + 1) under study. As the coupling between the two oscillators increases, the energy ratio tends to unity and the energy accumulates in a single oscillatory mode.
We applied the above method to examine the synchronization between our climatic time series, based on the broad theoretical concept that the climate system can be described as a large, or even infinite, number of coupled oscillators, as discussed at the end of section 2; see also Ghil and Childress (1987) and Lorenz (1991, 1996). Our hypothesis is that the commonality between several oscillatory modes found in the set of climate indices analyzed is due to the coupling between the oscillators. The significant oscillatory modes in the joint spectra of the climate records studied here, their synchronization type, and the energy ratio in each mode are summarized in Table 3.
The 7–8-yr oscillatory mode in all three regions and four types of records is completely synchronized, that is, of FPA type, and the energy-ratio analysis suggests that the NAO induces this mode in the other regions. We discuss the possible origin of this mode in the North Atlantic further below. The boreal summer teleconnection between the NAO and the Ethiopian Plateau’s rainfall, as captured by the Nile River’s high-water record, is likely to involve the NAO’s influence on the West African monsoon (Folland et al. 2009).
An FPA-type synchronization at 7.8 yr is also found between the North Atlantic and the eastern part of the eastern Mediterranean during winter, which is the rainy season in the latter region. This teleconnection can be explained using the results of Dünkeloh and Jacobeit (2003), who showed that, during the winter, a low pressure anomaly extends from Northern Europe to the southern and eastern part of the eastern Mediterranean. The NAO index determines the strength of this low pressure anomaly and, therewith, the amount of rain in the eastern and southern part of the eastern Mediterranean.
The 7–8-yr mode was found to be prominent in all our indices during December–February, August, and October. It might be induced by an oscillation of similar period in the position and strength of the Gulf Stream’s sea surface temperature (SST) front in the North Atlantic. This oscillation in SSTs and sea level pressures over the entire basin was identified in many previous studies (Deser and Blackman 1993; Moron et al. 1998; Joyce et al. 2000; Da Costa and Colin de Verdière 2002), while Wang et al. (2004) showed that it is the Gulf Stream region alone that affects the NAO. The 7–8-yr variability in the Gulf Stream, in turn, has been attributed to an oscillatory gyre mode of the North Atlantic’s wind-driven circulation (Jiang et al. 1995b; Ghil et al. 2002b; Dijkstra and Ghil 2005; Simonnet et al. 2005; Sushama et al. 2007).
Finally, Feliks et al. (2004, 2007) demonstrated in a simplified atmospheric the impact of a narrow and vigorous SST front, like the Gulf Stream front, on the atmosphere above it and downstream. Minobe et al. (2008) confirmed the predictions of Feliks et al. (2004, 2007) by carrying out calculations along these lines in a high-resolution, coupled general circulation model, with realistic SST forcing, and compared the model results with satellite observations.
An oscillatory mode with a period of about 4 yr is found in all our climate records as well. A teleconnection to the Ethiopian Plateau, related to the quasi-quadrennial (or low-frequency) ENSO mode (Rasmusson et al. 1990; Jiang et al. 1995a) has been the subject of several studies (e.g., Giannini et al. 2005; Chiang and Sobel 2002; Lintner and Chiang 2007); it appears to be associated with ENSO influence on changes in tropospheric static stability across the tropical atmosphere that affect convection over Ethiopia. There is no evidence, though, that this mode that is shared by the NAO and the eastern Mediterranean is induced by ENSO. Additional analyses not reported here show that the energy ratio in the joint M-SSA spectra of the NAO index and of the Southern Oscillation index is very small, ɛ = 0.1.
On the other hand, the energy ratio in the two-channel, 4-yr M-SSA mode for the NAO index and Jerusalem rainfall is ɛ = 0.8. We suggest that this mode may be induced in the North Atlantic climate by a 4-yr oscillation (possibly a harmonic of the 7–8-yr oscillation; see Moron et al. 1998) of the position of the Gulf Stream SST front. In the joint spectra of the Nile River record and the NAO index, a 4.2-yr oscillatory mode is also prominent. The synchronization type of this mode is FPA, and the coupling between the two regions is very strong, with an energy ratio of ɛ = 0.97. Thus the Ethiopian Plateau appears to exhibit independent teleconnections, at a period of about 4 yr, with both the North Atlantic and ENSO.
In the eastern Mediterranean, an oscillatory mode with a period of about 12–14 yr is prominent in the Jerusalem rainfall record and in tree rings from the Golan Heights. The oscillatory mode of the joint spectral analysis for the rainfall and tree rings has a period of 12.5 yr. Although the coupling between the rainfall record and the tree rings is relatively strong, the synchronization type is of FA type. This type differs from the three types introduced by Osipov et al. (2003)—which were also the ones we found useful in other two-channel M-SSA analyses here—since the amplitudes are well correlated, while the phases are not synchronized.
We attribute this unusual result to the physiology of tree-ring growth. Two principal climatic factors determine this growth, namely, rainfall and temperature. In the eastern Mediterranean, rainy winters are also cooler so these two factors have a tendency to cancel out. In the tree-ring record, the significant periods that are longer than 7 yr capture 75% of the variance of the detrended time series. This raises the possibility that the trees’ physiology acts like low-pass filter on the climatic signals.
This work grew out of KFG finding a 7–8-yr oscillation in a 1300-year-long record of Nile River water levels. Searching for the sources of the Nile has been a multimillennial quest that started with up-river military expeditions of the pharaohs, to find and protect the headwaters of the Nile. The quest continued with the one for the mythical Prester John in the Middle Ages, and only ended in the nineteenth century, with the European expeditions that finally reached the sources of the White and Blue Nile. Likewise, the search for the explanation of Joseph’s dream about the alternation of seven fat and seven lean years goes back several thousand years and might not be over yet.
We suppose that increasing the number of coupled oscillators in a system like that of section 2b here—as in the work of Brunnet et al. (1994) and of Lorenz (1991, 1996)—will make the synchronization problem much richer: the number of synchronized modes will increase, probably in rough proportion to the number of oscillators. The route to synchronization might still be similar, however: as the coupling between the oscillators increases, more energy will accumulate in the RCs of the oscillatory modes associated with lower frequencies and possibly broader spatial patterns. The whole system will synchronize when most of the energy accumulates in the RC mode obtained by projection onto the lowest pair of ST-EOFs, and the significant eigenvalues will all merge into one value. A model of this type might explain the prevalence of a small number of oscillatory patterns in the global climate system.
The evidence presented here for oscillatory climate modes with planetary-scale teleconnections may have implications for prediction, especially at lead times beyond seasonal ones, for which dynamical models are not yet useful. Robertson et al. (2001) demonstrated river-flow predictability up to 4 yr in advance for low-flow years on the Paraná River in southeastern South America. The predictions were based on an 8-yr periodicity in the river flow’s time series, as identified by SSA. Robertson and Mechoso (1998) found this periodicity to be related to SST anomalies in the tropical North Atlantic. Further work is required to relate these findings to those of the present paper.
It is a pleasure to thank D. Yakir for the precipitation records from Jerusalem, A. Zukerman for digitizing the tree-ring graphs, and many colleagues for useful discussions. Among the latter, G. Duane, A. Groth, J. Kurths, and J. J. Tribbia played a significant role in familiarizing us with the literature on synchronization of chaotic oscillators. We thank two anonymous reviewers for insightful and constructive comments. This work was supported by U.S. Department of Energy Grants DE-FG02-02ER63413, DE-FG02-07ER64429, and DE-FG02-07ER64439 from its Climate Modeling Programs, and by the European Commission’s No. 12975 (NEST) project “Extreme Events: Causes and Consequences (E2-C2).”
M-SSA and Synchronization
The purpose of this appendix is to show that the results for the coupled Rössler oscillators in section 2b are not an artifact of the M-SSA analysis. To do so, we illustrate the performance of our methods on two even simpler synthetic-data examples.
First, we apply our synchronization analysis to two phase- and amplitude-modulated, noise-perturbed sine curves:
Here A = 1, while ω1 and ω1 are white noises that are normally distributed, with mean zero and variance 1. In the first example (Figs. A1a,b) we set α = 1 so that f1 and f2 have the same underlying period but have different (noisy) phase modulations; the underlying periodicities will thus be synchronized, at least in frequency. The cross-spectra of f1 and f2 were computed by applying two-channel SSA, with a window width of M = 100Δt = 10 time units.
The RC associated with the first ST-EOF pair RC(1,2) is shown in Fig. A1b; it captures 16% of the variance. This pair is completely synchronized (σ = 0.05 radians) according to the phase-difference criteria of Osipov et al. (2003), with an energy ratio of E(RCx1)/E(RCx2) = 1.01. In the second example (Figs. A1c,d), we set α = 1.3 so that the periods are different and the underlying periodic signals are not synchronized. The two time series are shown in Fig. A1c; note that f1 is the same in both examples, and is visualized by the dashed curves in Figs. A1a,c. The RC associated with the first ST-EOF pair RC(1,2) is shown in Fig. A1d and captures 9% of the variance. This pair is not synchronized, since the phase-difference variance is σ = 17.5 radians, while the energy ratio E(RCx1)/E(RCx2) = 0.58.
These two examples illustrate the application of the analysis to noisy time series with known underlying periodicities and differing synchronization characteristics. It clearly shows that the method works when it should and does not when it should not.
* Additional affiliation: Department of Mathematics, Israel Institute for Biological Research, Ness Ziona, Israel
+ Additional affiliation: Geosciences Department and Laboratoire de Météorologie Dynamique, CNRS, IPSL, Ecole Normale Supérieure, Paris, France
Corresponding author address: Michael Ghil, Ecole Normale Supérieure, 24 rue Lhomond, Paris, 75005 France. Email: email@example.com