El Niño and the associated Southern Oscillation reorganize atmospheric circulation across the globe, affecting droughts, crops, ecosystems, and energy systems. The 2023–2024 event showed how devastating these impacts can be: drought and famine in southern Africa, heat stress on coral reefs. Yet the dynamical structure that organizes tropical Pacific variability and limits El Niño predictability remains incompletely understood. Observational records are getting longer and more detailed, but extracting physically meaningful and robust patterns from them is hampered by spectral density: hundreds of components sensitive to the choice of data and training period. We need a way to separate reliable climate modes from numerical artifacts without relying on heuristics like amplitude or explained variance fraction.
The work builds on Koopman operator theory, which describes nonlinear dynamics through a linear action on functions of the system state. To approximate this operator, they used a kernel variant of extended dynamic mode decomposition (k-EDMD) with a Gaussian kernel, applied to monthly sea surface temperature anomalies in the tropical Pacific from two independent products: ERA5 reanalysis and HadISST reconstruction. Data from 1940 to 2010 served as the training set, and the subsequent fifteen years as the validation set. To assess how well each computed eigenvalue–eigenfunction pair satisfies the Koopman equation, they computed residuals using the ResDMD method. Then, along the unit circle, they constructed a pseudospectral residual function, whose local minima indicate frequencies that are robust to perturbations. Spatial modes for these frequencies were reconstructed directly from the time series via spectral filtering, independent of the kernel dictionary. Finally, they searched the spectrum for integer powers and nonlinear products of fundamental components, checking period coincidence within a 10% tolerance and eigenfunction correlation above 0.7.
Pseudospectral analysis revealed nineteen stable frequencies in the period range from roughly one and a half years to a decade. These frequencies match between ERA5 and HadISST, even though the eigenvalue clouds themselves differ greatly between products and even when shifting the training window by one year. Among the found modes is a component with a period of about 3.6 years, whose spatial structure mimics canonical El Niño with anomalies in the central and eastern equatorial Pacific. Longer modes around 12.6 years have a broad basin-wide structure with noticeable meridional asymmetry, while short modes around 2.1–2.4 years are concentrated closer to the central equator. Reconstructing the Niño-3.4 index based on these nineteen modes explains 92% of the variance in the training period. For comparison, the largest-amplitude modes explain only 18% with standard k-EDMD eigenfunctions and 81% with pseudo-eigenfunctions. Forecasts based on the Koopman scaffold retain skill at lead times of 8–18 months, outperforming a linear inverse model and models using the full spectrum or only amplitude modes. The remaining hundreds of eigenpairs form a hierarchy: many coincide with integer powers of fundamental modes, with the residual growing with the power order, indicating increasingly less reliable components.
The main conclusion is that spectral redundancy is not chaos but reflects the algebraic structure of the Koopman operator. This changes how we interpret dense spectra: instead of selecting individual modes by heuristics, we should look for stable minima of the pseudospectral residual and treat the rest of the spectrum as generated by powers and products of these basic components. For climate science, this means that from long time series we can extract compact and predictive representations that are both physically interpretable and robust to data choice. Such an approach could become a general tool for physics-informed machine learning in Earth sciences, where scale interactions and spectral density often obscure the main features.
A natural next step is to test whether similar pseudospectral organization arises in other fields: subsurface temperature, wind stress, sea level, precipitation, or related ocean–atmosphere variables. Of particular interest is the physical meaning of the Koopman hierarchy: in a Fourier decomposition, harmonics are generated by fixed sinusoids, but here powers and products correspond to state-dependent eigenfunctions. A quantitative measure of this hierarchy could serve as a fingerprint of nonlinear dynamical organization and help compare climate regimes, attribute modes, and build predictive models. Longer climate model integrations will also be needed to determine whether the found scaffold is a property of the system itself or an artifact of the finite observational period.
The method directly affects seasonal and interannual El Niño forecasting, improving skill at lead times of 8–18 months. It is also useful for any Earth science where data are high-dimensional and contain interacting time scales, such as oceanography, paleoclimatology, or sea ice analysis.
The next step is to apply the same scheme to multivariate climate fields and test the reproducibility of the Koopman scaffold on long model simulations. In parallel, formal metrics should be developed to quantitatively assess the hierarchy of powers and nonlinear combinations.
In climatology, the question of which dynamical structures provide El Niño predictability beyond a few months remains unresolved. This work shows that stable Koopman modes, identified by a residual criterion, carry predictive memory, whereas amplitude-based or full-spectrum representations lose it. This links the problem of spectral interpretability to the fundamental challenge of identifying sources of long-term predictability in the nonlinear climate system.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The authors applied kernel EDMD with a Gaussian kernel to monthly sea surface temperature anomalies in the tropical Pacific from ERA5 and HadISST, using ResDMD to estimate residuals and pseudospectral analysis to select robust frequencies. The resulting Koopman framework with nineteen frequencies explains about 92% of the variability of the Niño-3.4 index and improves forecasts at lead times of 8–18 months, while the remaining spectrum is interpreted as nonlinear combinations of the fundamental modes.
Where its strength is
The methodological move of using pseudospectra and residuals to separate physically robust modes from numerical noise yields a compact and interpretable framework that is independent of the choice of kernel dictionary and robust to changes in the dataset.
Where it could go next
Worth trying
Why this may matter
If the proposed method proves transferable to other climate indices and regions, it could provide a general tool for extracting physically robust modes from noisy observational data, which is important for understanding climate predictability on seasonal to decadal timescales.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.52, against a usual 0.64. It looks like few have drilled nearby.
Out of about 800 eigenpairs that k-EDMD produces for each dataset, just 19 fundamental modes suffice to explain 92% of the variability of the main El Niño index. Meanwhile, the largest-amplitude modes, which are typically chosen in reduced models, explain only 18% of the same variability when using standard eigenfunctions.
The idea of a hidden low-dimensional scaffold behind a dense spectrum echoes the plot of Ted Chiang's 'Story of Your Life', where a complex nonlinear signal turns out to be generated by a small set of underlying patterns, and understanding them changes the perception of time and the prediction of the future.