The focus of this study was quadratic phase nonlinearities that give rise to time series skewness. Quadratic nonlinearities were associated with quadratic phase dependence among oscillators with periods P1, P2, and P3 and phases ϕ1, ϕ2, and ϕ3 satisfying 11P3=1P1+1P2 and 2ϕ3=ϕ1+ϕ2. The type of waveform resulting from quadratic nonlinearities can be measured using the biphase (King, 1996; Schulte, 2016; Maccarone, 2013), which is defined as 3ψ=ϕ3-ϕ1+ϕ2. A biphase of zero means that the quadratic nonlinearity results in positively skewed waveforms whose positive deviations from a horizonal axis are larger than the negative ones. On the other hand, waveforms associated with -180 biphase are characterized by larger negative deviations than positive ones. The biphase can also have values of -90 and 90, representing situations in which a time series rises faster than it falls (Schulte, 2016). This situation is not considered in this study because ENSO skewness is the focus of the paper.

The biphase is closely linked to the skewness of a distribution, which was computed using 4skewness=1N∑i=1N(xi-x‾)3S3, where S is the standard deviation of a time series x1,x2,…,xN with mean x‾. Positive (negative) skewness meant that the right (left) tail of the time series distribution was longer than the left (right) one, reflecting the tendency for positive (negative) time series events (i.e., anomalies) to be more intense than negative (positive) ones.

To measure the time evolution of skewness, we first partitioned a time series into overlapping segments and then computed the skewness for each individual segment. The segment length used in the calculations had to be chosen in advance, meaning that the results of the analysis depended on the chosen segment length. To see the segment length dependence, a sliding skewness analysis was applied to 5X1(t)=Acos⁡2πP1t, where A is amplitude and P1=8, 16, 32 is period (Fig. 1a).

In this situation, an appropriate measure of quadratic nonlinearity was constrained to be zero and constant because the cosine function is linear and stationary. Yet Fig. 2a shows that skewness is a function of time and P1 for a fixed segment length. In some cases, the skewness fluctuates between negative and positive values, which would suggest a change in the biphase even though the time series is stationary. It also appears that the cosine function with P1=16 tends to have greater skewness than the other cosine functions even though the cosine function is linear regardless of the period. These results imply that fluctuations in skewness could be erroneously deemed changes in time series nonlinearity in the presence of periodicities, especially for time series with low-frequency variability. In particular, we found that the impact of the periodicities is only strong when the chosen segment length is less than or close to the Fourier period so that computing sliding skewness of time series with low-frequency variability requires longer segment lengths to produce constant zero skewness for a stationary and linear time series. Thus, when comparing time series, it could be difficult to know whether one time series is truly more nonlinear than another.

It was also found that periodicities will also impact how one can interpret the skewness of truly nonlinear time series. For example, we considered the nonlinear time series given by 6X2(t)=cos⁡2πP1t+ϕ1+γ(t)cos⁡2πP3t+ϕ3, where 2P3=P1, ϕ1=0, and ϕ3=2ϕ1 so that sum rules (1) and (2) are satisfied. The quantity γ(t) was called a nonlinear coefficient because, as γ(t) approached unity, positive deviations from the mean became progressively larger than negative deviations (Schulte, 2016). In this example (Example 2), γ(t)=0.8, meaning that the degree of nonlinearity was constant. We considered the situations when P1=32, P1=16, and P1=8 to understand how skewness is impacted by periodicities of varying frequencies.

As shown in Fig. 1b, the quadratically phase-coupled oscillators composing X2(t) give rise to positively skewed waveforms whose positive excursions are larger than the adjacent negative ones. Although this time series is stationary and nonlinear, its skewness is a function of time and P1 (Fig. 2b), implying that skewness is not always a consistent measure of quadratic nonlinearity. At some time points, the skewness associated with the case P1=32 is close to zero, giving the false impression that X2(t) is linear during some time periods.

The deficiencies of traditional skewness motivated the construction of a new waveform skewness index that was more weakly influenced by periodicities. The construction of the waveform skewness index was also motivated by Fig. 1b that shows how positive deviations are larger than negative ones in the case of zero biphase. The waveform skewness index was constructed as follows. First standardized anomaly time series were decomposed into positive and negative events using an event decomposition approach (Schulte and Lee, 2019), where positive (negative) events are contiguous strings of positive (negative) anomalies. The peak intensity of a positive (negative) event was defined as the maximum (minimum) value obtained by the data points associated with the event. The waveform skewness of a positive time series event with peak intensity Ip was then defined as 7sp=Ip+Ina+Ins2, where Ina is the peak intensity of the antecedent negative event, and Ins is the peak intensity of the subsequent negative event. The waveform skewness index for negative events was calculated by multiplying a times series by -1, computing the waveform skewness indices of all fictitious positive events, and multiplying the resulting waveform skewness indices by -1.

Using the waveform skewness index, a waveform skewness time series was created by assigning to each time point the waveform skewness of the event to which the time point belongs. Performing this step for positive and negative events resulted in a waveform skewness time series whose length was nearly equal to that of the original time series, where the length inequality occurred because the waveform skewness index of events at the end of the time series could not be computed. By construction, the waveform skewness index measured time series asymmetry with respect to a horizonal axis at a moment in time.

The waveform time series is a transformed version of the original time series that need not be correlated with the original time series. Indeed, we found that on average a realization of a white noise process and its waveform skewness time series have a 0.4 correlation (not shown), meaning much of the information of the original time series is lost. For linear time series and nonlinear time series, waveform skewness can have no correlation with the original time series because waveform skewness could be constant even though the corresponding time series fluctuates (see below).

Although large negative or positive values of the waveform index were suggestive of time series nonlinearity, it is important to note that the waveform index could also be large for linear stochastic processes driven by non-Gaussian noise or even Gaussian white noise. For these reasons, we also evaluated the statistical significance of the waveform skewness (see below).

Unlike traditional skewness, the sliding waveform skewness time series for the linear cosine time series shown in Fig. 1a is zero and independent of time (Fig. 2a). The waveform skewness of zero is consistent with how the time series is stationary and linear so that the waveform skewness index is a more appropriate measure of quadratic nonlinearity in this situation. Similarly, the sliding waveform skewness time series corresponding to the nonlinear time series shown in Fig. 1b are nearly constant and always positive (Fig. 2b), reflecting the constant biphase of 0 and the constant degree of nonlinearity. The index appears to be slightly dependent on the period of the cosine function, though the dependence is not as strong as it is for traditional skewness. Thus, the waveform skewness index is a more theoretically consistent measure of quadratic nonlinearity given that these nonlinearities are less impacted by periodicities. Away from the edges of the time series, the sliding waveform skewness is not impacted by the chosen segment length (not shown), contrasting with sliding skewness time series whose depiction of nonlinearity depends on what segment length is chosen.

Despite these benefits of waveform skewness, the waveform skewness index at individual time points is highly influenced by noise because waveform skewness is only a function of three peak values. The sensitivity of waveform skewness to noise was confirmed by generating nonlinear time series like X2(t) and subjecting them to different levels of noise (see the Supplement). It was found that signal-to-noise ratios around 2.5 are needed to distinguish the waveform skewness of an underlying signal from background noise even for highly nonlinear time series (γ(t)=0.85). For this reason, it is recommended that one examines the waveform skewness at an individual time point in the context of surrounding waveform skewness. Time periods with stable waveform skewness are time periods where true nonlinearity may be present. Alternatively, one could filter time series using wavelet analysis and compute the waveform skewness of combined frequency components, as shown below.

Although the waveform skewness index measures local nonlinearity, it cannot determine the frequency components of the time series that are contributing to the time-domain waveform skewness. This frequency information is important because the frequency components determine how often positively or negatively skewed events will occur, as Fig. 1b suggests.

To determine the frequency components that are contributing to waveform skewness, we computed the waveform skewness of nonlinear modes embedded in time series, resulting in a waveform skewness spectrum. Following Schulte et al. (2020), a nonlinear mode was defined as the sum 8Xnonlinear=XP1+XP2+XP3 if all the periods are unequal and as the sum 9Xnonlinear=XP1+XP3 if P1=P2, where it was assumed that the sum rules in Eqs. (1) and (2) are satisfied. Each XPi was constructed using the three-step wavelet method implemented by Schulte et al. (2020). The first step of the wavelet method involved the wavelet transformation of X(t) that produced an array of wavelet coefficients containing time-frequency information about X(t). In the second step, all wavelet coefficients except those at P1, P2, and P3 were set to zero. Computing the waveform transformation of the resulting wave coefficients yielded Xnonlinear (Step 3). For X2(t) shown in Fig. 1b, the nonlinear mode is the sum of the two cosines.

After the computation of all possible nonlinear modes, the global waveform skewness of X(t) was computed as 10SP1,P2=∑SpP1,P2+∑SnP1,P2Np+Nn, where SpP1,P2 was a waveform skewness index of a positive Xnonlinear event, SnP1,P2 was a waveform skewness index of a negative Xnonlinear event, and the sum was computed across all Np positive events and Nn negative events associated with Xnonlinear. Repeating the calculations for all nonlinear modes resulted in the global waveform skewness spectrum.

The global waveform skewness represented the average waveform skewness index of a nonlinear mode. A positive value meant that a nonlinear mode was positively skewed, and a negative value indicated that a nonlinear mode was negatively skewed. In other words, positive (negative) values implied that the nonlinear mode was contributing to the positive (negative) waveform skewness of X(t).

It was found that even realizations of a red-noise process had large global waveform skewness even though they are linear. Thus, it was necessary to implement statistical significance tests. The statistical significance of global waveform skewness was assessed using a two-sided t test, where the null hypothesis was that the global waveform skewness is equal to zero. Because global waveform skewness is the average waveform skewness associated with individual time series events, auto-correlation did not pose a challenge for statistical significance testing.

The global waveform skewness spectrum is like the auto-bicoherence spectrum used by Schulte et al. (2020) but with a few notable differences. Unlike auto-bicoherence, high values of global waveform skewness will only occur for skewed waveforms defined with respect to a horizontal axis. On the other hand, auto-bicoherence can be high (close to 1) even if there is no skewness because the method detects waveforms that are asymmetric with respect to vertical and horizonal axes.

The second difference is that statistical significance of global waveform skewness can be assessed using a two-sided t test, whereas Monte Carlo methods are required for auto-bicoherence (Schulte, 2016), rendering statistical significance testing of auto-bicoherence slow and inefficient. Another notable difference is that time evolution of waveform skewness (i.e., local waveform skewness) corresponding to (P1, P2) is easier to compute than the time evolution of local auto-bicoherence. In local auto-bicoherence calculations, a smoothing operator is needed to create a time series that measures the degree of quadratic phase dependence (Schulte, 2016). The smoothing operation depends on wavelet scale, and there are three possible scales to choose from for a given nonlinear mode. Thus, the degree of nonlinearity will change based on the chosen scale. This problem is avoided in local waveform skewness calculations because the calculation of waveform skewness is done in the time domain. This problem is also avoided with traditional skewness, but traditional skewness calculations are influenced by periodicities, which is problematic because statistically significant nonlinear modes contain phase-locked periodic components that allow the biphase to be relatively stable (i.e., high auto-bicoherence).

To better understand what the global wavelet waveform spectrum measures, we considered the nonlinear and nonstationary time series given by 11X3(t)=X2(t)σ2+W(t)nσw, where σ2 is the standard deviation of X2(t), W(t) is a realization of a white noise process with standard deviation σw, and n is a real number representing the signal-to-noise ratio. Unlike in Example 2, we let 12γ(t)=t1002, so that the nonlinearity of X3(t) increased in time. In this case, n=0.8, ϕ1=0, P1=32, and P3=16 (Fig. 3a). The global waveform skewness spectrum for this time series was then compared to the corresponding auto-bicoherence spectrum, which was obtained using the Morlet wavelet with angular frequency equal to 6. The auto-bicoherence varied from 0 to 1, where a value of 1 indicated the strongest possible degree of phase dependence among oscillators satisfying the sum rules in Eqs. (1) and (2). The readers are referred to Schulte (2016) and Schulte et al. (2020) for more details.

As shown in Fig. 4a, the global waveform skewness is statistically significant around the point (32, 32), which indicates that there is quadratic phase dependence between oscillators with periods of P1=32 and P3=16. These statistically significant global waveform skewness values are positive, so that the quadratic phase dependence is contributing to the positive skewness of X3(t) seen in Fig. 3a. The auto-bicoherence spectrum shown in Fig. 4b also confirms that oscillators are contributing to the nonlinearity of X3(t). There are other points in the auto-bicoherence spectrum that are associated with statistical significance, but those points are associated with Type-1 errors. Furthermore, the corresponding points in the waveform spectrum are not associated with statistical significance, reducing confidence that the corresponding nonlinear modes are distinguishable from background noise.

The local waveform skewness time series corresponding to Xnonlinear32,32 indicates that the skewness of the nonlinear mode generally increases with time (Fig. 3b), consistent with how the nonlinear coefficient increases with time. Thus, the nonlinear mode is contributing to the positive waveform skewness of X3(t) to an increasing degree.

The above experiment shows that phase synchronization among frequency modes produces positive waveform skewness. In another example shown in the Supplement, we showed that large waveform skewness can also arise if there is covariance between amplitude and phase, a finding consistent with how nonlinearity can arise from such covariance (Pires and Hannachi, 2021).

As shown in Fig. 5, the ENSO time series comprise fluctuations of various magnitudes. For both the Niño 1+2 and Niño 3 indices, the most intense warm events are the 1982/1983 (Quinn et al., 1987) and 1997/1998 (McPhaden, 1999; Slingo and Annamalai, 2000) El Niño events. Surrounding many of the Niño 1+2 and Niño 3 warm events are cold events of lesser magnitude, reflecting the nonlinear character of ENSO (Timmermann, 2003). Other strong warm events are seen to have occurred around 1878 and 1889, but a relatively strong cold event appears after the 1889 event for the Niño 3 index, suggesting that this event is not as skewed as the 1982/1983, 1997/1998, and 2015/2016 events.

The waveform skewness time series associated with the ENSO time series better illustrate the temporal changes in nonlinearity. As shown in Fig. 5c, the two most skewed Niño 1+2 events are the 1982/1983 and 1997/1998 El Niño events. In contrast, the Niño 3 index time series comprises three strongly positive skewed events located around 1982/1983, 1997/1998, and 2015/2016 (Fig. 5d). The occurrence of these skewed events is consistent with how ENSO began to evolve more nonlinearly after the 1970s (Santoso et al., 2013). Despite the high intensity of the 1889 Niño 3 warm event (Fig. 5b), its waveform skewness is low, implying that there is no one-to-one relationship between event intensity and waveform skewness.

The transition of ENSO from a linear regime to a nonlinear one is evident from an inspection of the (standardized) sliding skewness and sliding waveform skewness time series. As shown in Figs. 6 and 7, the skewness and waveform skewness of the Niño 3 and Niño 1+2 indices are strongly positive around the 1980s and 1990s. The 10-year sliding skewness and waveform skewness time series also depict an abrupt decline in ENSO nonlinearity after the 1990s. For both ENSO indices, this decline is seen in the 20-year sliding waveform skewness time series (Figs. 7a and b) but not in the 20-year sliding skewness time series, suggesting more uncertainty in the extent of ENSO nonlinearity after the 1990s. However, Schulte et al. (2020) found the auto-bicoherence of the ENSO indices to decline after a peak in auto-bicoherence around the 1980s and 1990s so that the increase skewness could reflect the inability of skewness to always capture quadratic nonlinearities (Sect. 3).

The 1940s and 1950s appear to correspond to relatively low skewness and waveform skewness. In fact, the 10-year sliding skewness and waveform skewness time series associated with the Niño 1+2 index are negative around 1940 (Fig. 6a). The Niño 3 index is also associated with negative waveform skewness, but the negative waveform skewness occurs around the 1950s and 1960s, a time when strong warm Niño 3 events are absent (Fig. 5b). Prior to the 1940s, there is uncertainty regarding the nonlinearity of ENSO given the differences in the sliding skewness and waveform skewness time series. For example, the 10-year sliding skewness time series suggests that the Niño 1+2 index is nonlinear in the 1880s (Fig. 6a), whereas the 10-year sliding waveform skewness time series suggests that the Niño 1+2 index is linear during that time.

The waveform and auto-bicoherence spectra indicate that the nonlinearity of both ENSO indices is mainly the result of quadratic phase dependence among oscillators, with periods ranging from 24 to 64 months (Fig. 8). For example, a statistically significant peak is located at (60, 60) in both spectra for the Niño 3 and Niño 1+2 indices, indicating that phase dependence between modes with periods of 30 and 60 months is contributing to the skewness and waveform skewness seen in Figs. 6 and 7. The inferred cyclic behavior of this nonlinear mode implies a tendency for positively skewed ENSO events to occur every 60 months. For the Niño 1+2 index, there is also a statistically significant peak at (62, 44), which means that phase-dependent oscillators with periods of 26, 44, and 62 months are contributing to Niño 1+2 index skewness seen in Figs. 6a and 7a. It is worth noting that our findings are consistent with the Pires and Hannachi (2021) bispectral peaks in the 24- to 64-month period bands, supporting the idea the waveform skewness is a reliable estimator of nonlinearity.

As shown in Fig. 9, the relationship between seasonally averaged Niño 3 time series and AIR anomalies fluctuates on interdecadal timescales. The 20-year sliding intervals, which are nearly equal in length to the 21-year sliding intervals used in previous studies (Yun and Timmermann, 2018), especially highlight the interdecadal variability. The 10-year sliding analysis emphasizes the shorter timescale variations. Choosing other interval lengths produced curves with the general features of those depicted in Fig. 9.

From 1871 to 1970, the Niño 3–AIR relationship for the JJ and AS seasons is negative, consistent with the well-established idea that El Niño events are associated with Indian monsoon failures. However, after 1970, the AS Niño 3–AIR relationship weakens and becomes nearly positive. The weakening is not seen for the season JJ, which suggests that the processes influencing the AS Niño 3–AIR relationship are different from those influencing the relationship in the JJ season.

A comparison of Figs. 7b and 9 reveals that the weakest AS Niño 3–AIR correlation coincides with the greatest AS Niño 3 waveform skewness. Moreover, the AS relationship is seen to weaken when the Niño 3 waveform skewness increases after the 1970s but strengthens when the Niño 3 waveform skewness declines around the 1990s. These results support the idea that Niño 3 waveform skewness could be related to the weakening AS Niño 3–AIR relationship. Similar arguments hold for the Niño 1+2–AIR relationship.

To gain confidence that the sliding time series shown in Figs. 7 and 9 are related, the sliding Niño 3 and Niño 1+2 waveform skewness time series were correlated with the ENSO–AIR sliding correlation time series. The sliding seasonal waveform time series were obtained from the raw JJ and AS waveform time series associated with the Niño 3 and Niño 1+2 indices (Fig. S1 in the Supplement). In general, the seasonal waveform skewness time series (Fig. S2) were found to be like the ones associated with the full ENSO time series, but the seasonal waveform time series were easier to compare with the sliding correlation time series computed for the JJ and AS seasons. For comparison, the AS (JJ) sliding correlation time series were correlated with the seasonal skewness time series computed from the AS (JJ) Niño 3 and Niño 1+2 time series (not shown). The correlation between the sliding skewness and sliding correlation time series was calculated using different window lengths. Statistical significance was assessed using Monte Carlo methods (Appendix A).

As shown in Fig. 10a, there is a positive correlation between time series for AS Niño 1+2 skewness and waveform skewness and the AS Niño 3–AIR sliding correlation time series. The AS Niño 1+2 skewness relationship with the AS Niño 1+2–AIR sliding correlation time series is statistically significant at the 5 % level for most segment lengths, whereas the corresponding relationship with Niño 1+2 waveform skewness is only statistically at the 5 % level for segment lengths ranging from 20 to 25 years. On the other hand, the Niño 3 waveform skewness time series and the Niño 3–AIR sliding correlation time series are correlated with 5 % significance for most segment lengths (Fig. 10b). As shown in Fig. 10c, Niño 3 waveform skewness and skewness are more strongly related to changes in the AS Niño 1+2–AIR relationship than they are to changes in the AS Niño 3–AIR relationship. The positive correlations seen in Fig. 10 support the hypothesis that ENSO nonlinearity is related to changes in the ENSO–AIR relationship (Schulte et al., 2020).

Repeating the analysis for the JJ season revealed weaker relationships between ENSO nonlinearity and the AIR–ENSO relationship. Nevertheless, positive correlations were identified, which agrees with the idea that time periods of greater ENSO nonlinearity coincide with a weaker ENSO–AIR relationship, as originally suggested by Schulte et al. (2020). Furthermore, the stronger association for the AS season could explain why the Niño 3–AIR relationship weakens after the 1970s while the strength of the JJ relationship is more stable (Fig. 9).

A possible explanation for the relationships between ENSO nonlinearity and the strength of the ENSO–AIR relationship was determined by compositing AS ENSO indices and AIR anomalies based on AS ENSO waveform skewness bins (Appendix B). For example, we identified the years for which the AS waveform skewness was greater than the 95th percentile and computed the mean AS AIR anomaly for those years.

Figure 11a indicates that the intensity of Niño 3 anomalies is related to Niño 3 waveform skewness. For waveform skewness values less than 0.25, negative Niño 3 indices are preferred, whereas positive indices are preferred for waveform skewness values greater than 0.25. These results highlight the strong linear relationship between the Niño 3 index and Niño 3 waveform skewness. A similar analysis using the Niño 1+2 index also identified a linear relationship between the Niño 1+2 index and Niño 1+2 waveform skewness, further supporting a relationship between ENSO nonlinearity and ENSO intensity.

Consistent with a linear negative correlation between AIR and the Niño 3 index, AIR anomalies are preferentially positive for Niño 3 waveform skewness values less than 0 and negative for waveform skewness values greater than 0 (Fig. 11b). For the JJ season, the relationship appears to be generally linear, but the relationship between AS Niño 3 waveform skewness and AS AIR anomalies is slightly more complicated. For waveform skewness values ranging from -0.5 to 0.75, the composite mean AIR anomalies rapidly decrease in accordance with the tendency for greater Niño 3 indices to be associated with higher waveform skewness values. However, above a waveform skewness value of 0.75, AIR anomalies no longer decrease with increasing waveform skewness despite the increase in the composite mean Niño 3 indices. This nonlinear relationship also exists between Niño 1+2 waveform skewness and AIR anomalies (not shown) so that our results imply that the linear relationship between ENSO and AIR anomalies degrades for high ENSO waveform skewness, agreeing with the findings from the correlation analysis presented in Sect. 4.3.

To diagnose why the Niño 3–AIR relationship breaks down for high Niño 3 waveform skewness, we composited the magnitude of the TNI index based on Niño 3 waveform skewness. In this analysis, statistical significance of the composite means was assessed relative to the smallest composite mean TNI magnitude (Appendix B).

The results shown in Fig. 11c indicate that there is a propensity for the TNI magnitude to increase with increasing Niño 3 waveform skewness. For both seasons, the magnitude of the TNI appears to be preferentially small for Niño 3 waveform skewness, ranging from -0.75 to 0. The JJ TNI magnitude tends to be large (>0.7) for Niño 3 waveform skewness above 0.25, and the AS TNI magnitude tends to be greater than 0.7 for waveform skewness values greater than 0.5. Interestingly, we did not find such a relationship between the magnitude of Niño 3 anomalies and the TNI magnitude (not shown) in AS, suggesting that Niño 3 waveform skewness provides new information about the changing ENSO–Indian monsoon relationship not contained in the actual Niño 3 index.

A similar analysis conducted between Niño 1+2 waveform skewness and TNI magnitude did not reveal any relationship between them (not shown). However, the composite analysis assumes a simultaneous relationship between waveform skewness and the TNI magnitude that seems unlikely to exist given that most skewed Niño 1+2 events coincide with the strongest canonical El Niño events (Fig. 5a). Thus, we correlated sliding ENSO waveform skewness and skewness time series with sliding TNI magnitude time series to determine whether time periods of high ENSO nonlinearity coincide with time periods of high TNI magnitude.

As shown in Fig. 12, there is a statistically significant relationship between the waveform skewness of the Niño 1+2 and Niño 3 indices and the intensity of the TNI. The results indicate that time periods of enhanced ENSO nonlinearity occur with time periods of more intense TNI events. A similar relationship also exists between ENSO skewness and TNI magnitude (Fig. 13), though the relationship between Niño 3 skewness and TNI magnitude is not statistically significant at the 10 % level.