Idealized Frequency Experiments

To investigate the nonlinear atmospheric response to interannual ENSO SSTA, in the presence of the annual cycle, we use a similar experimental setup as in ref. 20. The eastern tropical Pacific SSTA pattern (Fig. 2A) is multiplied by a sinusoidal interannual time series (e.g., Fig. 2B for one experiment example) to derive the spatiotemporal evolution of the anomalous boundary forcing for a suite of atmospheric general circulation model (AGCM) experiments using the Community Earth System Model (CESM) Community Atmosphere Model version 4 (CAM4) (27) in a T42 horizontal resolution with 26 vertical levels (details in SI Appendix, SI Materials and Methods). The total boundary forcing comprises the observed SST annual cycle additional to the aforementioned ENSO anomalies. In the following, the warm pool annual cycle time evolution is given the variable A(t), and the ENSO time evolution is given the variable E(t).

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Fig. 2.

(A) ENSO SST anomaly forcing pattern for the AGCM experiments. The amplitude of the anomaly pattern is given by the colored shading (no units). The N3.4 (black) and NWP (cyan) regions are marked by two boxes. (B) The time evolution of the N3.4 SST anomaly index (black) and the anomalous ensemble mean circulation index NWP−AC(t)¯ (cyan) for the 2.5-y experiment.

The aforementioned nonlinear processes are an essential part of general circulation models such as CESM CAM4. For an SST boundary forcing that includes only ENSO (E) and the annual cycle (A), the nonlinear interaction during these processes can be expanded with a Taylor expansion, as a sum of terms ∑n=0kαn(k−n)EnAk−n. If we consider only the linear response (k=1) as well as quadratic (k=2) and cubic (k=3) interactions, the following frequencies are generated in the atmospheric response to the SST forcing: the ENSO forcing frequency (fE), its noninteraction overtones (2fE and 3fE), and the quadratic interaction terms (1±fE), as well as the cubic interaction terms (1±2fE and 2±fE). Here 1 denotes the annual (A) cycle frequency (f = 1 y^-1) and 2 denotes the semiannual (SA) cycle frequency (f = 2 y^-1).

For our experiments, we chose five different discrete idealized ENSO SSTA forcing frequencies (fE) spanning the observed interannual band (Fig. 1A): 3/7 y^-1, 2/5 y^-1, 3/10 y^-1, 3/13 y^-1, and 3/16 y^-1. These choices were motivated by an intended separation of all ENSO forcing frequencies and the resulting ENSO/annual cycle nonlinear interaction frequencies (SI Appendix, Table S1). We also added additional sinusoidal frequency experiments that have some overlapping forcing and interaction frequencies, fE = 1/2 y^-1, 1/3 y^-1, and 1/4 y^-1, to investigate the role of frequency overlapping in estimating the functional form of the atmospheric response. The expected response frequencies for these experiments, as well as the respective number of ensemble members for each experiment, are listed in SI Appendix, Table S1. The cubic ENSO noninteraction term (frequency 3fE) is omitted in the following analysis, as its amplitude is near zero.

When analyzing the nonlinear atmospheric response to the combined ENSO/annual cycle forcing, we focus here on the anomalous low-level NWP-AC because it acts as the crucial bridge between ENSO variability and the East Asian Monsoon system (20). To characterize this anomalous circulation, we define the anomalous surface wind stream function averaged over the NWP region (120°E−160°E, 5°N−20°N, Fig. 2) as our circulation index NWP-AC(t) for each experiment (as in ref. 20). To remove the noise (unforced internal atmospheric variability), the ensemble mean indices are calculated and reduplicated to give a time series of the same length as the original indices, which is labeled NWP−AC(t)¯. This time series can be considered as the deterministic atmospheric response in the region to the combined ENSO/annual cycle SST boundary forcing.

Higher-Order Combination Modes

The power spectra for the NWP−AC(t)¯ indices for a selected number of experiments (blue lines in SI Appendix, Figs. S2A, S3A, and S4A) show that most of the variance can be attributed to the quadratic combination tone peaks (1±fE). However, clear deterministic peaks are also evident for the ENSO noninteraction frequencies (fE, 2fE) as well as for the higher-order combination tones. The ensemble averaging has removed nearly all of the internally generated atmospheric noise from the circulation indices, leaving only the modes that originate from the ENSO/annual cycle interactions for our analysis.

Distinctively different anomalous low-level atmospheric circulation patterns (Fig. 3 G−K) are associated with the individual frequency components (Fig. 3 A−E), adding together to the full response pattern (Fig. 3L). Most of the meridionally antisymmetric response can be attributed to the ENSO interactions with either the annual or semiannual cycle. These distinct pattern and associated timescales demonstrate that ENSO’s impact extends far beyond its interannual timescale and canonical response pattern (Fig. 1 A and C).

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Fig. 3.

(A−E) Power spectral density for the individual reconstruction components for the 2.5-y Sine ENSO AGCM experiment using the averaged coefficients using the fast Fourier transform (FFT) method. (F) Power spectral density for the anomalous ensemble mean circulation index NWP−AC(t)¯ for this experiment (light blue lines) and the full reconstruction (red lines) using all of the terms depicted in A−E. (G−K) Regression coefficient of the normalized respective reconstruction component and the anomalous surface stream function of this experiment: (G) E, (H) E², (I) ExA, (J) E²xA, and (K) ExSA. (L) Regression coefficient of the normalized full reconstruction time series and the anomalous surface stream function of this experiment.

The very clearly identifiable peaks (SI Appendix, Figs. S2A, S3A, and S4A) at the theoretically expected frequencies (SI Appendix, Table S1) motivate us to use a least square optimization approach to determine the following amplitude (αn(k−n)) and phase (ϕm) coefficients for each experiment. The reconstruction Ψc(t) is based on the linear, quadratic, and cubic Taylor expansion (the curly brackets indicate the associated frequencies with each term),Ψc(t)=α10×E(t)︷fE+α20×E^(t)2︷2fE+α11×E(t)×cos(ωAt−2πϕ1)︷1±fE+α21×E^(t)2×cos(ωAt−2πϕ1)︸1±2fE+α12×E(t)×cos(ωSAt−2πϕ2)︸2±fE,

where ωA denotes the angular frequency of the annual cycle and ωSA denotes the angular frequency of the semiannual cycle. Because of the single sinusoidal ENSO frequency forcing (ωE) with a given phase (ϕE), we can write the ENSO time series E(t) with a given amplitude (αE) in the following form (the second equation has been adjusted to guarantee a zero mean):E(t)=αE×cos(ωEt−2πϕE),E^(t)2=E(t)2−0.5=αE2×0.5⁡cos[2(ωEt−2πϕE)].

We find that for the cases with nonoverlapping response frequencies, the estimated regression coefficients agree very well (SI Appendix, Table S2). We use these five experiments to calculate the averaged coefficients. The estimates clearly diverge for the experiments with 1/2 y^-1 and 1/3 y^-1 forcing frequencies, as the frequencies from different terms overlap (SI Appendix, Table S1). Using the averaged coefficients αn(k−n) and ϕm (last line in SI Appendix, Table S2), we now reconstruct our theoretical index time series for each experiment, which includes all of the terms (linear, quadratic, and cubic) listed in the equations above. We find very high correlations (R> 0.9), high spectral coherence, and phase relationships between our reconstructions Ψc (refer to Materials and Methods for the reconstruction nomenclature) and the ensemble mean indices (SI Appendix, Figs. S2, S3, and S4 and Table S3), with the largest contribution to the correlation coming from the quadratic combination tones. Furthermore, we see that the full reconstruction captures well the NWP−AC(t)¯ spectrum (Fig. 3F).

Application of the Concept

These results motivate us to test if we are also able to identify these higher-order combination tones in (i) an AGCM experiment with observed nonsinusoidal (broad spectral peak) ENSO forcing and (ii) the Japanese 55-year reanalysis (JRA-55) for the period 1958–2013 (28).

Consistent with the previous idealized frequency experiments, we find that the linear correlation coefficient (R) between the AGCM ensemble mean NWP−AC(t)¯ and the reconstruction using the linear ENSO term is only 0.27. Adding the quadratic and cubic terms increases the correlation significantly to 0.66 and 0.71, respectively. This confirms that our estimated coefficients also hold for a realistic ENSO temporal evolution with a broad spectral peak.

The JRA-55 index comprises, in addition to the fixed pattern ENSO response, attributions from different SSTA patterns as well as internal unforced variability that is not accounted for in our idealized AGCM experiments. Using a multiple linear regression approach, we calculate the amplitude regression coefficients αn(k−n) between the individual response components and the 4-mo low-pass filtered JRA-55 NWP-AC(t) index (SI Appendix, Table S4). The contribution of the quadratic C-mode term is larger in the reanalysis than in the model experiments, most likely because of the C-mode amplification due to thermodynamic air−sea coupling (20). Furthermore, the estimated coefficients agree very well with our previous estimates (from the individual frequency experiments) for both the linear ENSO term and the ENSO interaction term with the semiannual cycle. It is important to note that the JRA-55 reanalysis exhibits a different skewness (caused mostly by the quadratic noninteraction ENSO term) in the NWP circulation than estimated from the AGCM experiments. This is explained by model biases, specifically by a too-strong cyclonic response in the NWP region during the El Niño developing phase.

Using the linear ENSO term alone, we find a correlation coefficient of only 0.04 between the filtered JRA-55 NWP-AC(t) index and our reconstruction (Ψa*). This confirms the negligible direct impact of ENSO on the anomalous NWP circulation. In contrast, when adding the quadratic (quadratic + cubic) term(s), the correlation increases to 0.52 (0.53). Considering only ENSO conditions (when N3.4 is above 0.5°C or below −0.5°), R(JRA-55,Ψ*) values of 0.71 (0.72) can be obtained for El Niño conditions and 0.60 (0.60) for La Niña conditions. These correlations increase even more when considering only strong El Niño and La Niña events.

Our spectral analysis (SI Appendix, Fig. S5) shows a clear cross-spectral coherence between reanalysis, model experiment, and the theoretical reconstruction at the near-annual quadratic combination tone frequencies (1±fE). We find statistically significant coherence (above the 95% confidence level) for the frequency bands associated with the quadratic combination tones (1±fE) (SI Appendix, Fig. S5C). Furthermore, the phase estimation for the quadratic interaction term between model and theoretical reconstruction is well captured (red line in SI Appendix, Fig. S5B). Additionally, we find significant coherence and phase relationships when comparing the JRA-55 reanalysis with both our model experiment (green line) and the reconstruction Ψc* (light blue line). No clear coherence or phase signals can be found at higher frequencies associated with cubic interaction terms. Note that in our previous studies (19, 20), we discussed some ambiguity for the quadratic sum tone (1+fE) signal in the observations. Here we can confirm a clear coherence in this frequency range between the JRA-55 reanalysis and our reconstruction (SI Appendix, Fig. S5).

Predictability of the NWP Circulation During ENSO

To demonstrate the predictability of the NWP-AC, we use the National Centers for Environmental Prediction coupled forecast system model version 2 (CFSv2) (29) to hindcast the onset (SI Appendix, Fig. S6) and termination (Fig. 4) phases of major ENSO events. All selected termination phase hindcasts begin during the event peak phases in December. As expected from previous studies (17, 19), we see that the N3.4 prediction (thick magenta line for the ensemble mean and thin magenta lines for the 1 SD error) captures the observed evolution of the index (dashed black line) reasonably well. A hindcast of the NWP-AC is performed (thick solid orange line for the mean and thin orange lines for the 1 STD error) using the Ψc* reconstruction based on the hindcasted N3.4 index. Although large unforced variability is present in the NWP region (gray shading for a conservative estimate using a control AGCM experiment), it is clear that if the N3.4 hindcast (magenta lines) is successful, predictability for the anomalous NWP circulation exists up to 9 mo for both the onset and termination phases of ENSO events, which is a much longer lead time than previous estimates (24).

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Fig. 4.

Predictability of the observed NWP-AC circulation during major El Niño and La Niña events. Shown are the observed N3.4 index (dashed black line), the JRA-55 NWP-AC index (solid gray line), and the CFSv2 hindcast for the N3.4 index beginning in December (termination phase) of each event (thick magenta line). The error estimate for the NWP-AC index is 1 SD of the NWP-AC circulation in a 50-y integration of CESM CAM4 with only the annual cycle as forcing (gray shading). The error for the N3.4 hindcast is 1 SD in the ensemble spread (thin magenta lines). The reconstruction Ψc* is used for predicting the circulation (solid orange line for the prediction and thin orange lines for the 1 SD error) and on the observations (dashed orange line). A small offset between observations and the CFSv2 initialization is explained by both the CFSv2 climatology (1999−2010) and the observational uncertainty of the CFSv2 analysis (the SD of the initial conditions is shown by the range of the thin magenta lines during the first month).

For instance, for the hindcast starting in December 1982, we accurately predicted the NWP-AC in the following boreal summer. Persistence due to thermodynamic air−sea coupling, which is not included in our idealized AGCM experiments, plays a role in sustaining the C-mode response in boreal spring during some El Niño events (e.g., 1982–1983) (20). However, it is the atmospheric response (as captured by our AGCM experiments) that drives the crucial phase reversal of the anomalous NWP circulation and is responsible for the fast near-annual and subannual timescales in this region. For instance, these phase reversals include the rapid transition from cyclonic to anticyclonic circulation starting in October 1982 and the subsequent decay of the anticyclonic anomalies after the peak of the warm pool annual cycle in February 1983 (Fig. 4).