## 1. Introduction

The Arctic Oscillation (AO) proposed by Thompson and Wallace (1998) is defined as the empirical orthogonal function (EOF) first mode of the monthly mean sea level pressure (SLP) in the Northern Hemisphere. The spatial pattern is a seesaw between the Arctic region and the midlatitude. In this sense, the pattern is also called the annular mode (AM).^{1} Since the AO is related to many aspects in large-scale fields, it has been the subject of much attention and many research efforts.

On the other hand, the debate about the physical reality of the AO continues. A historical review of the debate is described in detail in appendix A. Its brief summary is as follows: in the debate, three other modes are involved. Two of them are well-known modes: the North Atlantic Oscillation (NAO) and the Pacific–North American Oscillation (PNA). The third is the EOF second mode, whose pattern shows a negative correlation between the Atlantic and Pacific regions (hereafter abbreviated as the NCM). This mode is indispensable in the debate because Wallace and Thompson (2002, hereafter WT02) ascribe uncorrelatedness between the Atlantic and Pacific to this mode. The standpoint of the apparent AO is that since the AO and NCM can be explained from the NAO and PNA, the AO and NCM are not real but apparent (e.g., Itoh 2002a, hereafter I02). The standpoint of the true AO is that the AO–NCM view (AO–NCM paradigm) is also possible as an alternative to the NAO–PNA view (WT02). Of course, there are many variations between these two.

The alternative view, however, may lead to an incorrect interpretation of atmospheric behavior. A typical example is the interpretation of the AM in the stratosphere. As is well known, when the geopotential height from the troposphere to the stratosphere is regressed upon the AO of the SLP, similar patterns emerge for all the layers. From the standpoint of the true AO, the interpretation may be that the AO (AM) is a unified mode penetrating from the troposphere to the stratosphere. This is, however, not the case. The regression of the geopotential height in the troposphere upon the AM in the stratosphere yields the NAO (e.g., Itoh and Harada 2004, hereafter IH04). Moreover, the regression on the NAO yields the AM in the stratosphere, which is the same as that on the AO. Therefore, the coupled mode should be interpreted as the NAO in the troposphere and the AM in the stratosphere. At least there should be a consensus that the combination of the AO in the troposphere and the AM in the stratosphere is not a simple unified mode.

Understanding the true nature of the AO has important consequences in applications such as prediction. If the AO is apparent, monitoring only the NAO and PNA is sufficient for predicting the AO. For instance, in the case of El Niño events, it is possible to predict that the positive phase of the AO might have difficulty emerging since the PNA excited by El Niño gives rise to a negative anomaly over the North Pacific. Although an effective long-lived forcing to the NAO is unknown, if the NAO can be predicted, similar things can be said. Furthermore, if the AO has a large amplitude, the NCM cannot have a large amplitude and vice versa, as will be shown later in this paper. On the other hand, if the AO is real, AO prediction must be undertaken in a completely different manner.

The present paper joins this debate as an extension of I02. Although it is sometimes claimed that the debate should be made from the dynamical view as well as the statistical view, this paper will treat the debate only from a statistical view. This is for two reasons. First, since the AO is statistically defined, it should be interesting to note the results obtained by further statistical analyses. Second, if the AO is apparent, dynamical studies may not yield fruitful results.

The present paper has three purposes. The first purpose is to examine the configuration of the four modes (AO, NCM, NAO, and PNA) in phase space. Although WT02 presented a schematic figure of the configuration of the four modes, which are located on the same plane, this feature has not been demonstrated in the real world. Quadrelli and Wallace (2004) show a figure representing the two-dimensional phase space spanned with EOFs 1 and 2 on which several patterns are projected. According to the figure, the PNA has an angle of 20° (calculated from a correlation of 0.94) from the EOF1–2 (i.e., AO–NCM) plane. It cannot be said that this angle is close to the plane. It is therefore worth examining the configuration of the four modes in phase space. If the AO–NCM and NAO–PNA systems are located on the same plane, it means that the two systems share the same variations on this common plane. This configuration makes the discussion of the AO–NCM very simple.

If the NAO–PNA and AO–NCM systems are on the same plane, WT02 and Christiansen (2002a, hereafter C02) believe the two cannot be distinguished. True and apparent oscillations, however, can be principally distinguished by using, for example, independent component analysis (ICA) and assuming independent oscillations to be true ones. Mori et al. (2006, hereafter M06) already applied ICA to this problem but their data selection was not suitable for ICA. Proper data selection might lead M06 to a different conclusion. The second purpose of this paper is to reexamine the reality of the AO–NCM by using simple ICA.

The third object of this work is to consider the AO–NCM system in light of evidence pertaining to the winter mean SLP field, where the leading EOFs extracted are not the AO or NCM. It is interesting to ask why, in contrast with the results from the monthly mean data, the AO–NCM does not appear. This absence must be related to the issue of the reality of the AO–NCM.

Concerning the terminology used in this paper, “true” or “real” modes are interchangeably used as physical modes, whereas “apparent” modes mean statistical artifacts. The term “oscillation,” as in the NAO or PNA, indicates both a spatial pattern and time variation of amplitude. When a spatial pattern is meant exclusively, the word “pattern” is generally attached, as in the “NAO pattern” or the “PNA pattern.” “Mode” and “component” (which appears only in section 4) have a similar meaning to “oscillation.” “System,” as in the “NAO–PNA system,” indicates a combination of two oscillations.

This paper is organized as follows. Section 2 will describe the data and an outline of the results from the SLP in I02. Issues are clarified by schematic figures for the three-point systems and are explained in detail in section 3. The relation of the four modes in phase space is also explored. ICA is performed under a proper data selection in section 4, and winter mean features are examined in section 5. After discussions in section 6, conclusions will be offered in section 7.

## 2. Data and a brief outline of previous results

The data used is the monthly mean SLP of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996). The data cover 53 yr from 1948 through 2000. This period is selected for consistency with I02. Extending the period gives almost the same results. They are available on 2.5° longitude by 2.5° latitude grids. Only winter (November through April) data are employed. The total number of months is then 318 (53 × 6). Anomalies are defined as deviations from the 53-yr mean of each calendar month.

An EOF analysis is performed for 10° longitude by 5° latitude grids north of 20°N. Sectorial as well as hemispheric EOF analyses are also carried out. The Euro-Atlantic and Pacific sectors are defined, respectively, as 80°W ∼ 80°E including 0° and 100°E ∼ 100°W including 180°. Hereafter, for simplicity, the Euro-Atlantic sector will be referred to as the Atlantic sector.

In the following, the spatial pattern of the *n*th mode in EOF analysis is called EOF*n*, while the time coefficient is notated as PC*n*. An “H” is sometimes used as a prefix to distinguish hemispheric EOF from sectorial EOF, that is, they are called HEOF1, HPC2, and so on.

For data with persistency, an equivalent number of independent samples is necessary to estimate significance. Since lag correlation with a lag of 1 month at each grid point is 0.4 at most, the independent sample size per year is estimated to be approximately 6(1 − 0.40)/(1 + 0.40) = 2.6. Based on this number, we can conservatively set the independent sample size at 2 per year, that is, 106 for all the samples. For correlation coefficients, 0.24 is 99% significant.

In I02, EOF analysis was first carried out for the SLP. The first mode (HEOF1) represents the AO, while the second mode (HEOF2) shows a negative correlation between the Atlantic and Pacific, which is called the NCM in this paper. See I02 for more on these patterns.

*i*and

*t*are space and time indices, respectively; SLP′ denotes the SLP anomaly; and

*i*∈ M.A. indicates that the summation is taken over the midlatitude (the northern boundary of 60°) Atlantic. The definition of HPC1(P) is similar. As a result, we find that HPC1(A) and HPC1(P) are almost uncorrelated. Although almost the same two oscillations can be obtained by the EOF analysis over the two individual sectors, this somewhat complicated procedure is employed because we want to show directly that the Atlantic and Pacific parts of the AO correspond to the NAO and PNA, respectively.

Regressions of SLP′ on HPC1(A) and HPC1(P) yield two seesaw patterns in the Atlantic and the Pacific, respectively, as shown in Fig. 1. The pattern of Fig. 1a is nothing but the NAO pattern. The pattern of Fig. 1b shows a variation of the Aleutian low, including a seesaw with the polar region. Performing the regression with the 500-hPa height on HPC1(P), we obtain a pattern which exhibits the PNA pattern with a wave train in the Atlantic [not shown here but see Fig. 3a of Ambaum et al. (2001)]. Together these patterns (the SLP and 500-hPa patterns) are therefore called the PNA pattern^{2} for simplicity.

## 3. Configuration of the AO, NCM, NAO, and PNA in phase space

### a. Consideration of the relationship between the “NAO–PNA” and “AO–NCM” systems in the three-point seesaw model

*r*

_{1}(

*t*) and

*r*

_{2}(

*t*) are two independent time series with zero mean (

*r̃*

_{1}and

*r̃*

_{2}also stand for two different time series (

*r̃*

_{1}and

*r̃*

_{2}will be called “AO” and “NCM,” respectively.

We should make three comments about the three-point models to avoid confusion. First, the word “point” does not literally mean point but actually refers to a region. For instance, the Arctic region covers not only the vicinity of the North Pole but also Iceland and the East Siberian Sea, which are two centers of action in the Arctic region in the PNA pattern (see Fig. 1b).

Second, the reader might think that the three-point models are unrealistic in that amplitudes in the three regions are very different from real values. These systems, however, are linear and arbitrary coefficients for *r _{n}* and

*r̃*

_{n}are permissible. Under these conditions, more realistic “patterns” for the four modes would be obtained (see Ambaum et al. 2001). Thus, the systems of (1) and (2) without any coefficients provide sufficient essential features.

Third, these two models with coefficients for *r _{n}* and

*r̃*

_{n}can describe various configurations of the “NAO–PNA” and the “AO–NCM” in phase space. When the “AO” and “NCM” patterns can be expressed as a linear combination of the “NAO” and “PNA” patterns, the four modes are located on the same plane in phase space (see Fig. 3). Otherwise, the “AO–NCM” system is placed on a different plane from that of the “NAO–PNA” system. An image of this configuration is depicted in Fig. 2.

For simplicity, in this subsection, *r _{n}* and

*r̃*

_{n}are assumed to be uniform random numbers with unit variance (i.e., maximum amplitude of

WT02 provide a schematic representation of the relation between the NAO–PNA and AO–NCM systems. Their figure, however, is incorrect (or at least confusing if angles among the four modes are to be ignored) because it depicts the “NAO” and “PNA” patterns as perpendicular to each other. This presentation means that the two patterns have no spatial correlation,^{3} but they do in the model and in the real atmosphere. A corrected figure is shown in Fig. 3. Since the spatial correlation between the “NAO” and “PNA” patterns in (1)—that is, (1, −1, 0) and (1, 0, −1)—is 0.5, they intersect with an angle of 60°. The “AO” pattern, which is expressed by (2, −1, −1), is the average of the “NAO” and “PNA” patterns, and the “NCM” pattern, which is expressed by (0, 1, −1), is perpendicular to the “AO” pattern. The AO pattern is therefore rotated by 30° from the “NAO” or “PNA” patterns.

The range (scatterplot area) of the “NAO” and “PNA” is represented by the shaded area in Fig. 3 denoted by the letter A. It can be easily understood from this figure why EOF1 and EOF2 of (1) are the “AO” and “NCM” patterns, respectively. Since the “NAO” and “PNA” patterns are not orthogonal, the axis with the largest variance is an average of the “NAO” and “PNA,” as is visualized in Fig. 3. This is the “AO.” Since the length of the “AO” axis is that of the “NCM” multiplied by

^{4}From this equation, we see that PC1 =

*r*

_{1}(

*t*) +

*r*

_{2}(

*t*) and PC2 =

*r*

_{2}(

*t*) −

*r*

_{1}(

*t*) (except for the normalization factor) for (1), and therefore they are uncorrelated but not independent; it is impossible for them to both have large (positive or negative) values independently.

Thus, (1) and (2) are not mathematically identical, contrary to C02’s conclusion. The difference in the PDFs of PC1 and PC2 makes it possible in principle to analyze differences between (1) and (2). The most general tool for this kind of discrimination is ICA.

Three comments associated with Fig. 3 should be given. First, if the angle between the “NAO” and “PNA” patterns is 90°, the apparent “AO” will never appear. Thus spatial correlation between the “NAO” and “PNA” patterns is essential for the existence of the apparent “AO.” Second, on the other hand, when there are two dominant variations with spatial correlation, they necessarily produce an apparent EOF mode. This is not restricted to the apparent AO but happens generally. We must be cautious about this kind of pitfall. Third, we have used uniform random numbers for *r*_{1}(*t*) and so on. This restriction is not, however, serious; almost all other time series yield the same conclusion: the PDFs of PC1 and PC2 are different between (1) and (2), therefore the discrimination between the two is possible. The only exception is the case where the amplitudes of these oscillations show a normal distribution, where it is impossible to discriminate between the two (for further details, see, e.g., Hyvärinen et al. 2001).

### b. The real relation between the NAO–PNA and the AO–NCM in phase space and the reality of the AO and NCM

The schematic figure for the simple three-point seesaw system was shown in Fig. 3, where the “AO” pattern is rotated by 30° from the “NAO” and “PNA” patterns. In the real relation, however, the AO pattern is located nearer to the NAO pattern, since the spatial pattern of the AO is much closer to that of the NAO. To display these relations, pattern correlations are calculated and converted into angles between two modes in phase space. For instance, when the NAO and PNA patterns (vectors) are expressed as **N** and **P**, respectively, the angle between them is cos^{−1}{(**N** · **P**)/**N**||**P**|**A** and **C**, respectively, the AO–NCM plane can be expressed as **H** ≡ **A** cos *α* + **C** sin *α* for an arbitrary angle *α* from the AO pattern. Then the angle from the AO to NAO patterns on the AO–NCM plane is calculated as *α* for which (**N** · **H**) (≡L) is maximum. Also, the angle between the NAO pattern and the AO–NCM plane is cos^{−1}{L/**N**||**H**|

The result is shown in Table 1, where each line representing the NAO or PNA pattern intersects the AO–NCM plane with an angle of about 13° in phase space. These four modes are therefore located on almost the same plane.

Figure 4 illustrates these relations (angles are approximate). The percent variances explained by the NAO and PNA are calculated as 19% and 14%, respectively. These variances share each “intrinsic variance” because the two patterns have some correlation. When “original variances” are denoted as *s*^{2}_{A} for the NAO and *s*^{2}_{P} for the PNA, the relations *s*^{2}_{A} + *s*^{2}_{P} cos^{2}*θ* = 0.19 and *s*^{2}_{P} + *s*^{2}_{A} cos^{2}*θ* = 0.14 are derived, where *θ* is the angle between the two patterns.

AO′ is drawn as the axis of the maximum variance (the angle from the NAO is about 21°). This angle *β* from the NAO pattern can be calculated as the angle for which the projection of the NAO and PNA onto the line with angle *β*, *s*^{2}_{A} cos^{2}*β* + *s*^{2}_{P} cos^{2}(*θ* − *β*), is maximum. If the NAO and PNA are completely independent, and if the AO is apparently produced only from the NAO and PNA, EOF1 obtained from the NAO and PNA should be located along AO′. The AO is remarkably close to AO′ and the amplitude of the AO is also reasonable; the percent variance is estimated as about 20% from this figure, while the actual percent variance is also 20%. On the other hand, the amplitude of the NCM estimated from this figure (about 9%) is considerably smaller than the real value (12%). However, since the NCM is the second mode, it can easily be contaminated by other variations located in different subspaces from this plane. This feature can readily be confirmed by the three-point models in which arbitrary perturbations are added at the three points (not shown). This difference is therefore not necessarily serious.

Considering the results in I02 and in previous parts of this paper, we can now say that the NAO–PNA and AO–NCM systems are located on almost the same plane in phase space. The evidence is summarized as follows: the AO and NCM were successfully produced from the NAO and PNA using real data in I02. The angles among the four modes did not contradict the hypothesis that the four modes are on the same plane. I02 and the previous discussion have made it clear that variances of the NAO–PNA and the AO–NCM were consistent.

Since the four modes are on the same plane, the AO–NCM and NAO–PNA systems share the same variations on the plane. Here we neglect the projection from other variations in different subspaces from the plane, since their contributions are considered to be relatively small. This inevitably leads to the relation AO + NCM = PNA + NAO in symbolic form. In a logical consideration of the reality of the AO–NCM, there are only four possibilities: 1) the NAO–PNA is real while the AO–NCM is apparent; 2) the AO–NCM is real while the NAO–PNA is apparent; 3) the AO–NCM and the NAO–PNA are both real; and 4) the AO–NCM and the NAO–PNA are both apparent.

Other than these four scenarios, it is possible that the NAO and PNA have high temporal correlations, thereby giving rise to the real AO in the sense that there is a coherent oscillation between the midlatitude Atlantic and Pacific. As shown in I02 and in previous analyses in this paper, however, this is not the case. Note also that such a situation would have substantially weakened the NCM. In practice, it can be shown that when the correlation of *r*_{1} and *r*_{2} is assumed to be *a* in (1), the two eigenvalues for the AO and NCM become 3(1 + *a*) and (1 − *a*), respectively, when the variances of *r*_{1} and *r*_{2} are unity. In other words, as long as the NCM, understood as a composite of the NAO and PNA, possesses sufficient amplitude to give rise to EOF2, the NAO and PNA must vary almost independently.

Among the four possibilities, the second and fourth ones are quite unexpected. Perhaps no one affirms these possibilities. Since many results have been and will be offered about the first possibility, let us then consider the third possibility. Generally speaking, it is hard to imagine that even three dominant physical modes are distributed on the same plane in phase space, because there are many degrees of freedom in that space. It is much harder to imagine the four modes on one plane. The configuration in which the AO is the average of the NAO and PNA, and the NCM is perpendicular to the AO is also surprising. We can say that the probability of such an occurrence, that is, the third possibility, is infinitesimally small.

## 4. Independent component analysis

In section 3, we stated that, even if the true and apparent oscillations are placed on the same plane in phase space, both are principally distinguishable by the difference of the PDFs. In this section, we try to achieve this by using ICA. Although M06 already applied ICA to this problem, we think that the data they used were not suitable for ICA.

“Independence” is an important keyword in this section and means “statistical independence.” Suppose we have two random variables *X*_{1} and *X*_{2} (usually time series in this paper), and any information on the value of *X*_{1} yields no information on the value of *X*_{2} and vice versa. Then *X*_{1} and *X*_{2} are deemed statistically independent. “Independence” has a stronger meaning than “uncorrelatedness” because uncorrelatedness is a necessary condition for independence but not a sufficient condition. For instance, the two time series data, sin(*t*) and cos(*t*), are uncorrelated but not independent because sin(*t*) uniquely determines cos(*t*), except for its sign. Another typical example of uncorrelated but dependent time series was already shown: *x*_{1}(*t*) = *r*_{1}(*t*) + *r*_{2}(*t*) and *x*_{2}(*t*) = *r*_{1}(*t*) − *r*_{2}(*t*), because both *x*_{1} and *x*_{2} cannot hold large (positive or negative) values at the same time.

From this meaning of independence, it is reasonable to assume that an independent oscillation is a true one. Therefore, ICA can find independent oscillations, that is, true oscillations.

Since the reader may not be familiar with ICA, it will be briefly described in the following. Details are provided in, for example, Hyvärinen et al. (2001). The principle is that PDFs of independent components show (global or local) maxima of non-Gaussianity, while PDFs of their combinations are relatively closer to the Gaussian distribution. This proof is shown in appendix B. M06 also proved that a necessary condition for the independence of a certain component is that it has maximum of non-Gaussianity. Thus, based on maximization of non-Gaussianity, we can distinguish independent components from their combinations.

Of the several methods of ICA, we adopt the simplest one, in which kurtosis is used to measure non-Gaussianity. Kurtosis is defined as *E*{*X*^{4}}/(*E*{*X*^{2}})^{2} − 3 in the normalized version, where *X* and *E*{} denote a random variable and expected value, respectively, and *E*{*X*} = 0 is assumed. Since the kurtosis is zero for the Gaussian distribution, independent components are extracted as those whose PDF kurtoses are far from zero. That is, independent components should comprise maxima (minima) for positive (negative) kurtosis.

We will explain how independent components are extracted more specifically with Fig. 5 illustrating the three-point seesaw system. Figure 5a is the 30° clockwise rotation of Fig. 3. We cannot, however, use this configuration directly for ICA, because, for instance, when the maximum amplitude of the “AO” is projected on the “NAO,” it is not the true amplitude of the “NAO” but, instead, a larger amplitude. Then whitening is necessary, where the direction of the “NCM” is increased as much as

We must be cautious about the data selection for ICA. M06 used whole-year data. These data, however, show positive kurtosis even for well-mixed components. This characteristic is easy to explain: let winter and summer data have normal distributions with large and small variances, respectively. In the case of M06, however, winter and summer data were used together (i.e., whole-year data), revealing typically a “spiky” distribution with heavy tails. Imagine that many samples with small amplitudes (summer samples) are added to a normal distribution with large variance (winter samples). Spiky PDFs with heavy tails mean positive kurtosis. Therefore, even if independent components in winter and summer have negative kurtosis, they may show near-zero kurtosis to whole-year data. As a whole, well-mixed components have a large positive kurtosis, while independent components have near-zero kurtoses. This is an apparent contradiction to ICA assumptions and results from mixing two types of samples with much different variances. This consideration indicates that we must use data such that well-mixed components have near-zero kurtoses. In this respect, whole-year data are not suitable for ICA. From preliminary calculations, we decide to use the 3-month (December, January, and February) data in the following analysis.

We now use the real SLP data, carrying out ICA for the two-dimensional space (i.e., plane) spanned by EOF1 and EOF2. Figure 6 shows kurtosis as a function of angle from EOF1. As a whole, it is clear that the kurtosis is negative. Patterns of 64° and 167° have locally minimum kurtoses, being independent components. These patterns are shown in Fig. 7 and can be identified as the NAO pattern and the PNA pattern with small amplitudes over the polar region. This NAO pattern is almost the same as Fig. 1a. On the other hand, this PNA pattern has a slightly larger amplitude over the Icelandic region compared with Fig. 1b, but the overall pattern is very similar.

Thus, we can say that the NAO and PNA are independent, that is, true components, while the AO and NCM are not. Statistical significance, however, cannot be obtained owing to the small sample size. We need more analyses, which appear in Itoh et al. (2007).

## 5. Winter mean field and an analysis using randomly selected samples

The first subject in this section is the winter mean field. Deser (2000) mentioned that EOF1 for the winter mean anomaly exhibits the NAO but not the AO. This seems curious, because similar longevity is expected from the nearly identical correlations at lag 1 month, which are calculated as 0.41 for both the AO and NAO. We now analyze how this “contradiction” occurs. The winter mean is defined as the winter (6 month) average of anomalies for individual months, which is the same as deviations of individual winter means from the winter mean climatology. We performed EOF analysis on this anomaly and illustrate the result in Figs. 8c,d. We can see that EOF1 evidently exhibits the NAO pattern, while EOF2 resembles the PNA pattern defined in this study, but without the significant amplitudes in the Arctic region. There is no AO or NCM.

From the standpoint that the AO and NCM apparently emerge when the NAO and PNA patterns are spatially correlated, it is conjectured that this correlation is greatly reduced in the winter mean anomaly field. To prove this hypothesis, we obtain the NAO and PNA patterns and calculate the spatial correlation between them.

*x*(

_{N}*t*) −

*r*

_{1}(

*t*) in (1a)—defined aswhere SLP′ is the SLP anomaly as already defined and NAO represents the pattern of Fig. 1a. As a result, the spatial correlation between AEOF1 and PEOF1 is only 0.05, corresponding to an angle of 87° in phase space. This indicates that the AO and NCM do not emerge without a spatial correlation between the NAO and PNA. The “contradiction” of the short longevity of the AO is thus resolved.

Where is the pattern of Fig. 8b [hereafter, referred to as the pure PNA (PPNA) pattern] located in phase space? It might be simple if it were located between the PNA (Fig. 1b) and NCM patterns because it intersects with the NAO pattern at a right angle, but the PPNA pattern, that is, a pattern without amplitudes over either the Atlantic or the Arctic region, cannot be expressed as a linear combination of the PNA and NCM patterns. Thus, it must be located on a different line from the PNA–NCM plane. The actual calculation indicates that the PPNA line intersects the AO–NCM plane (almost the same as the PNA–NCM plane) with an angle of 24°. This angle is consistent with the result of the PNA in Quadrelli and Wallace (2004; about 20°), although the PPNA is defined differently from their PNA. Figure 9 schematically represents the location of the PPNA. Patterns on the PNA–PPNA plane represent PNA patterns with various amplitudes in the Arctic region.

The above consideration leads to the possibility of a different analysis from the previous ones, an analysis using not all the 318 samples but a proper number of samples. Then we would obtain different EOFs, depending on the sample combination. The point here is whether HEOFs might not become the AO and NCM but, instead, the NAO and PPNA for certain combinations and, if so, what combinations would produce such HEOFs. This analysis also gives a clue to the nature of the AO. While small sample numbers are better for producing differences, large sample numbers are better for increasing significance. Following preliminary calculations, we decided to use 212 samples (two-thirds of the total), although similar results can be obtained for, say, half or three-quarters of the total. Random numbers were used for the selection of 212 out of 318 samples. We made 1000 combinations to determine general characteristics.

A typical combination demonstrating that HEOF1 and HEOF2 are the NAO and PPNA patterns is shown in Figs. 10c,d. AEOF1 and PEOF1 are also illustrated in Figs. 10a,b. It is clear that Figs. 10a,c are almost identical, as are Figs. 10b,d. Comparing Fig. 10b with Fig. 1b, we see that the amplitudes over the polar region in Fig. 10b are almost zero. In this case the NAO and PNA patterns have no spatial correlation. This example also supports the hypothesis that the AO results from a spatial correlation between the NAO and PNA.

Next, it will be shown that this typical example can be generalized. Figure 11 shows a scatterplot of 1000 combinations in which the abscissa represents the covariance over the polar region between AEOF1 and PEOF1, and the ordinate shows the variance in the Pacific region of HEOF1. The variance is calculated over grids of 5° by 5° south of 65°N in the Pacific sector where the sign of the amplitudes is different from that over the polar region. The smaller this variance, the closer the pattern of HEOF1 to the NAO pattern. The region where the covariance, defined as the average, is calculated is within the dash–dotted line in Fig. 10b. Figure 11 clearly shows that HEOF1 has smaller amplitudes in the Pacific, when the covariance between AEOF1 and PEOF1 is smaller. In other words, HEOF1 is not the AO but the NAO pattern, when the spatial correlation between AEOF1 and PEOF1 is small. It is only for cases of large covariance over the polar region that the AO appears. Figure 10 represents the case in which the covariance in the polar region is minimum in this figure.

Figure 12 illustrates a scatterplot of kurtosis as a function of the variance in the Pacific sector of HEOF1. Kurtosis here is defined as the average of that for HEOF1 and HEOF2. Because of reasons about the data selection described in section 4, 3-month data are used for Fig. 12. That is, kurtoses and variances are calculated for 1000 combinations of 106 randomly selected samples out of 159 (53 × 3) total samples. The 6-month data also show similar results, although the kurtoses are larger than those of Fig. 12 as a whole. This figure evidently shows that samples with small variances over the Pacific (close to the NAO pattern) generally indicate large negative kurtoses (independent), whereas samples with large variances (close to the AO pattern) exhibit near-zero kurtoses (dependent). Moreover, the bars of 95% significance, calculated by using the bootstrap method, are well separated between the top 3 and the bottom 3 of kurtosis. Thus, when HEOF1 and HEOF2 are the AO and NCM (NAO and PNA), respectively, PC1 and PC2 are mutually dependent (independent); that is, the AO and NCM (NAO and PNA) are apparent (true). Of course, more tests are needed to draw a final conclusion.

## 6. Discussion

In this section, we will discuss the NCM and PNA. For these oscillations more studies are needed in several areas.

First, we discuss the logic of WT02 concerning the reality of the AO and NCM. They carried out the regression of the 500-hPa height onto the NCM. Then, based on the fact that the PNA with a wave train over the Atlantic sector (which they call augmented PNA) appeared as the regressed pattern, they concluded that the NCM was real.

However, from the result in section 3b and Fig. 4, in which the NCM pattern is located near the PNA pattern in phase space, a pattern similar to the PNA pattern naturally appears. Furthermore, we should point out that the location and orientation of the wave train over the Atlantic in WT02’s Fig. 4 are different from those in the one-point correlations in the same figure. The one-point correlation maps are rather similar to those of Fig. 1b in this paper and the regressed pattern of the 500-hPa height on the SLP PNA (not shown). The implication is that the real PNA, which also has a wave train over the Atlantic but in another configuration, considerably differs from the NCM over the Atlantic region at sea level and at the 500-hPa level. Although there are certainly negative correlations between the Aleutian and Icelandic lows (van Loon and Rogers 1978; Honda et al. 2001), the center in the Atlantic sector differs from that of the NCM.

WT02 also calculated the temporal correlations of all grids with the Atlantic center of the AO for the field excluding the NCM. Consequently, the correlations between the Atlantic and the Pacific became high. It should be noted, however, that even the apparent AO model gives high correlations if the NCM is excluded. This is easily understood from (3).

It is fair to point out that further study of the PNA defined in this paper is also needed. There are two centers of action in the polar region in the PNA (Fig. 1b). One is located near Iceland. This reflects the Aleutian–Iceland seesaw. The second one, whose amplitude is larger than that at the Icelandic center, appears in the East Siberian Sea. To the author’s knowledge, this center, which seems different from the centers of action in the North Pacific Oscillation (Walker and Bliss 1932) and the western Pacific Oscillation (Wallace and Gutzler 1981), has not yet been investigated. In any event, both the Aleutian–Iceland seesaw and the Aleutian–East Siberian Sea seesaw are closely related to the AO. A preliminary analysis indicates that these two seesaws do not vary coherently. Thus, analyses of each variation and of their relative contributions to the AO remains as future work.

## 7. Conclusions

In an extension of I02, the reality of the Arctic Oscillation (AO, or Northern annular mode) is reconsidered. The existence of the AO is closely related to that of the negative correlation mode between the Atlantic and Pacific sectors (NCM).

First, we present a schematic figure to aid in considering the relationship between the North Atlantic Oscillation (NAO)–Pacific–North American Oscillation (PNA) and the AO–NCM in the simplest version of the three-point seesaw system. An examination of this figure enables us to visually understand why the AO–NCM is extracted from the NAO–PNA by EOF analysis. Although the NAO–PNA (apparent AO–NCM) system and the true AO–NCM system have the same EOF, the probability density functions for the time coefficients of the first and second EOF modes (PC1 and PC2) are different. Namely, PC1 and PC2 for the apparent AO–NCM system are dependent upon each other, whereas PC1 and PC2 for the true AO–NCM system are independent.

It is ascertained from several pieces of evidence that the NAO–PNA and AO–NCM systems are located on the same plane in phase space in the real world. This means that the AO–NCM and NAO–PNA systems share the same variations on the plane. Hence, when the NAO–PNA system is real, the AO–NCM is unlikely to be real.

Simple independent component analysis (ICA) is carried out to distinguish between the true and apparent AO–NCM systems. Here, true (apparent) oscillations are assumed to be independent (dependent). As a result, ICA indicates that the NAO and PNA are independent, while the AO and NCM are combinations of them.

The winter mean field is also examined. Although the NAO and AO have almost the same autocorrelation at lag 1 month, the EOF for the winter mean field does not show the AO–NCM but the NAO–PNA. This “contradiction” may come from the fact that the winter mean NAO and PNA patterns have little spatial correlation. Furthermore, from calculations using 212 samples (two-thirds of the total), it is found that when the NAO and PNA patterns have little spatial correlation, the AO never appears as EOF1, but the NAO and PNA themselves are extracted as the leading EOFs. When the NAO and PNA patterns are spatially correlated, the leading EOF is the AO.

We have shown that almost all the statistical characteristics of the AO and NCM can be explained from the combination of the NAO and PNA, which vary almost independently. Hence, we conclude again, with more significance than is found in I02, that the AO and NCM are almost apparent, where “almost” means that even if the true AO is present, its contribution is very small.

We have concluded by using the monthly mean data that the NAO and PNA vary almost independently. We do not, however, deny the interaction between the NAO and PNA. In fact, these relations have been elucidated upon by several studies, especially by those using daily data (e.g., IH04). It is important to examine the real interaction of the NAO and PNA in a future study.

The NCEP–NCAR reanalysis covers only about 60 yr. Such restricted data cannot achieve full statistical significance, especially in analyses with high sensitivity, like ICA. Furthermore, we should keep in mind also that the present climate could alter the stationarity of the modes, moving preferably from one system to the other. Studies using other long-term data are needed. Candidates are the Hadley Centre’s historical gridded mean sea level pressure dataset (HadSLP; Allan and Ansell 2006) and output from present climate experiments for the Intergovernmental Panel on Climate Change (IPCC) fourth Assessment Report. This study is now under way.

## Acknowledgments

I would like to express my appreciation to A. Mori for stimulating discussion about independent component analysis and the difference between independence and simple uncorrelatedness. Thanks are extended to three anonymous reviewers for giving many appropriate and constructive comments. I think the paper was greatly improved by these comments. NCEP–NCAR reanalysis data were provided by the CIRES Climate Diagnostics Center, Boulder, Colorado (available on their Web site at http://www.cdc.noaa.gov). This research is supported by a Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Science, Sports and Culture, and by the cooperative research project of the Center for Climate System Research, University of Tokyo. The GFD-DENNOU Library of the Japanese meteorological community was used to draft the figures.

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## APPENDIX A

### Review about the Debate around the Physical Reality of the AO

Since the AO was first introduced, it has been criticized as being identical to the NAO (e.g., Deser 2000). Another interesting issue is the identification of the center of action in the Pacific. Ambaum et al. (2001) consider a three-point seesaw system representing three centers of action, that is, the Arctic, Euro-Atlantic, and Pacific regions. They suspect that the PNA, together with the NAO, constitutes the AO. The Pacific center of the AO is produced from the PNA. In their analysis, there necessarily appears one more oscillation, defined as EOF2, with a negative correlation between the Atlantic and Pacific regions (NCM). Honda and Nakamura (2001) have also pointed out the participation of the PNA in the formation of the AO to the effect that a wave train emanating from the PNA reaches the vicinity of Iceland, resulting in a negative correlation between the Pacific and the Arctic region. Furthermore, Cohen and Saito (2002) conclude that the AO is nonannular.

WT02 refute the criticism of Ambaum et al. (2001) as follows: first, proposing a schematic figure explaining the three-point system, they point out that the NAO–PNA and the AO–NCM are located on the same plane in phase space and are rotated by 45° from each other. From the standpoint that the AO is real, the NCM must be real; although the AO view has been criticized in that there are no temporal correlations between the Atlantic and the Pacific, which should be seen if the AO really exists, the AO view can still be supported by the argument that coherent oscillations associated with the AO are masked by the NCM. WT02 then seek the physical reality of the NCM, finding it in the PNA pattern obtained by regression on the NCM, which features a wave train over the Atlantic at the 500-hPa geopotential height. By eliminating the NCM, the high correlation between the Atlantic and Pacific sectors in the AO is recovered. Thus, they state that the AO–NCM is a pair of dynamically significant modes and that the AO–NCM view is an alternative to the NAO–PNA view. From a different standpoint, Feldstein and Franzke (2006) state that the AO view is also important as well as the NAO view, because the AO and NAO are not distinguishable.

I02 questions the reality of the AO through a clearer analysis. The idea is to perform EOF analysis only in the midlatitude (midlatitude EOF), which is analogous to two-point EOF for the three-point seesaw system. Independent variations then appear in the Atlantic and Pacific regions, which are identified as the NAO and PNA, respectively; the annular mode in the midlatitude never emerges. By extending the area of analysis to the Arctic region, in-phase and out-of-phase patterns between the Atlantic and the Pacific simultaneously emerge, patterns which, in the hemispheric analysis, finally change to the AO and NCM patterns. This change occurs because the Arctic contains the common center of action of the two variations. In other words, the NAO and PNA patterns have a spatial correlation (i.e., pattern correlation). He further shows that the NAO and PNA can produce similar patterns to the AO and NCM, and then concludes that the observed AO is apparently extracted from the nearly independent NAO and PNA.

C02 criticizes I02 in that the two equations of the three-point seesaw system representing the true AO and the apparent AO are mathematically identical. The two are therefore not distinguishable by purely statistical analyses.

Another way of interpreting the AO is to examine it from the standpoint of nonlinear weather regimes. Monahan et al. (2001) state that the AO emerges as the difference between two regimes. On the other hand, using rotated EOF analysis, Christiansen (2002b) claims that the AO is a physical oscillation.

Very recently, M06 applied independent component analysis to extract independent components in the SLP data. This analysis can distinguish between independent and mixed patterns. They conclude that a seesaw-like pattern between the Aleutian and Icelandic lows and the cold ocean–warm land pattern are independent components. Neither the AO–NCM nor the NAO–PNA are independent components.

Thus, the discussion about the physical reality of the AO remains divided.

## APPENDIX B

### Non-Gaussianity of Independent Components and Their Combinations

Independent components can be extracted as maximization of non-Gaussianity. In this appendix, this will be explained by using kurtosis. Kurtosis is the simplest measure of non-Gaussianity; its value is zero for the Gaussian distribution, while the farther the PDF is from the Gaussian distribution, the larger its absolute value is. In the following, all random variables have zero mean.

*X*, kurt(

*X*), is defined aswhere

*E*{} denotes expected value. The following two formulas are useful:where

*X*

_{1}and

*X*

_{2}are two random variables and

*α*is a constant.

*X*

_{1}and

*X*

_{2}, whose distributions are not Gaussian (i.e., kurt(

*X*

_{1}) ≠ 0, kurt(

*X*

_{2}) ≠ 0). Their linear combination may be expressed as

*Y*≡

*a*

_{1}

*X*

_{1}+

*a*

_{2}

*X*

_{2}, where

*a*

_{1}and

*a*

_{2}are constants. Here, constraints,are added, which are equivalent to whitening. The second constraint means thatwhere

*α*denotes an angle from the

*X*

_{1}coordinate.

Then, we have the following theorem. The theorem is equivalent that independent components show the maximization of non-Gaussianity.

#### Theorem

The relationship *I* ≡ |kurt(*Y*)| shows global or local maxima at *α* = *kπ*/2 (*k* = 0, 1, 2, 3).

#### Proof

We will prove the theorem, by dividing into three cases.

- (i) When kurt(
*X*_{1}) > kurt(*X*_{2}) > 0 (positive case) or kurt(*X*_{1}) < kurt(*X*_{2}) < 0 (negative case),*I*shows 1) global maxima at*α*= 0 and*π*, and 2) local maxima at*α*=*π*/2 and 3*π*/2.In the negative case, only signs are different from the positive case. Since*I*is the same between the positive and negative cases, the positive case alone will be treated in the following.First, 1) will be proved. By using (B2), (B3), and (B4),an be easily derived under the condition of cos^{4}*α*+ sin^{4}*α*≤ 1. Equality holds if and only if cos^{4}*α*= 1, so that*I*takes global maxima at*α*= 0 and*π*.Next, we will prove 2). This is identical to the proof thatis satisfied for certain ranges of*α*straddling*α*=*π*/2 and 3*π*/2, and equality holds if and only if*α*=*π*/2 and 3*π*/2. By using cos^{4}*α*= (1 − sin^{2}This can be regarded as a quadratic inequality in the unknown sin^{2}*α*. Since two roots of the quadratic equation are sin^{2}*α*= 1 and(B7) is satisfied forEquality holds if and only if sin^{2}*α*= 1, so that*I*takes local maxima at*α*=*π*/2 and 3*π*/2. - (ii) When kurt(
*X*_{1}) = kurt(*X*_{2}) > 0 or kurt(*X*_{1}) = kurt(*X*_{2}) < 0,*I*shows global maxima at*α*=*kπ*/2. This proof is identical to 1) in (i). - (iii) When kurt(
*X*_{1}) > 0 > kurt(*X*_{2}),*I*shows global or local maxima at*α*=*kπ*/2. We can easily prove that kurt(*X*_{1}) ≥ kurt(*Y*) ≥ kurt(*X*_{2}), whose proof is identical to 1) in (i). This means that*I*takes global (local) maxima at*α*= 0 and*π*, and local (global) maxima at*α*=*π*/2 and 3*π*/2, when |kurt(*X*_{1})| > (<) |kurt(*X*_{2})|.

It is easy to extend the above result for 2 components to *N* components. (Further generalization leads to the central limit theorem.) Thus, we can extract independent components as (global or local) maximization of non-Gaussianity.

Angles among the four modes, AO, NCM, NAO, and PNA, angles from the AO on the AO–NCM plane (fourth row), and angles from the AO–NCM plane (fifth row) in phase space calculated from pattern correlations (in degrees).

^{1}

The AO is regarded as a surface (or at least lower tropospheric) pattern today. Instead, the Northern Hemisphere AM (NAM) is used for expressing the “deep coupled” mode. The AO, however, will be preferentially used even for the deep mode in the troposphere in this study, to avoid confusion with the AM in the stratosphere frequently used in this paper. In contrast with the AO, it is evident from various aspects that the AM in the stratosphere is a true mode.

^{2}

A PNA that has no amplitude over the polar region is discussed in section 5 and is called pure PNA.

^{3}

Strictly speaking, the term “normalized inner product” should be used instead of “spatial correlation,” because the spatial average must be subtracted in the latter. In this paper, however, “spatial correlation” will be used for simplicity.

^{4}

Although this transformation is shown by C02, the author has independently shown it in the original manuscript of I02 as well as in a Japanese article (Itoh 2002b).