An orchestra plays in a dark hall, and the listener hears only a rumble: hundreds of instruments merge into dense noise, where no melody can be discerned. That is roughly what spectra look like when extracted from climate data using methods such as the Koopman operator: hundreds of components, each with its own frequency, and it is unclear which are true musicians and which are the rattling of chairs. The Koopman operator is a mathematical conductor that describes nonlinear dynamics through linear action on functions of state, but its orchestra is too large to conduct.
The authors set out to find stable notes in this noise. They took monthly sea surface temperature anomalies from the tropical Pacific from two independent sources: the ERA5 reanalysis and the HadISST reconstruction. Data from 1940 to 2010 served as the training set, and the following fifteen years as the validation set. For each computed eigenvalue-eigenfunction pair, they estimated the residual using the ResDMD method: how well that pair satisfies the Koopman equation. Then, along the unit circle, they constructed a pseudospectral residual function. Local minima of this function indicate frequencies that are not knocked off by a slight perturbation, like a tuning fork that keeps ringing even if you clap your hands nearby.
These nineteen stable frequencies form a Koopman framework. Among them is a component with a period of about 3.6 years, whose spatial structure replicates the 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 to 2.4 years are concentrated closer to the central equator. Reconstructing the Niño-3.4 index from these nineteen modes explains 92% of the variance in the training period. For comparison, the largest-amplitude modes explain only 18% with standard eigenfunctions and 81% with pseudoeigenfunctions.
The remaining hundreds of eigenpairs are not independent musicians. Many of them coincide with integer powers of the fundamental modes, with the residual growing with the power order. This is as if, in the orchestra, in addition to the nineteen main instruments, their overtones and combinations were sounding: a flute and an oboe together produce a third tone that is not a separate instrument but merely their joint sound. Forecasts based on the Koopman framework maintain skill at lead times of 8 to 18 months, outperforming a linear inverse model and models that use the full spectrum or only amplitude modes.
In the dark hall, a melody is now audible. The nineteen notes that make up El Niño are not just a mathematical abstraction but real oscillations that can be tracked and used for prediction. The next time the Pacific Ocean decides to reorganize atmospheric circulation across the globe, we will have the score.
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.