Climate scientists have long sought an objective way to isolate persistent oscillations in the chaos of atmosphere and ocean. Alongside this work lie studies that predict El Niño a year and a half ahead, separate weather noise from climate signal, and analyze factors governing weather on a lone planet. These efforts converge on a common need: tools that extract slowly varying, coherent structures from high-dimensional data without imposing linearity or preselected frequencies.
The authors treat climate data as observations of an abstract dynamical system. They construct the Koopman operator or its generator from sea surface temperature time series in the Indo-Pacific region, using kernel methods from machine learning and delay embedding. Eigenfunctions of these operators with the largest eigenvalues in magnitude correspond to slowly decaying, nearly cyclic observables. For validation, they used 1300 years of CCSM4 model data and 50 years of ERSSTv4 reanalysis.
The generator spectrum for both datasets revealed branches corresponding to the annual cycle, El Niño with a period of about 4 years, and combination modes with the annual cycle. The El Niño eigenfunction provides a straightened representation of the cycle: in a two-dimensional phase space, the trajectory moves at nearly constant angular speed, in stark contrast to the chaotic behavior of the traditional Niño 3.4 index. Phase composites built from this eigenfunction show a more physically consistent life cycle, including asymmetry of transitions: La Niña occurs before mid-cycle, and anomalous westerly winds appear at the initiation stage of El Niño. Phases defined via eigenfunctions exhibit high equivariance: a six-month evolution maps one phase to the next with up to 100% accuracy for the reanalysis, whereas for the Niño 3.4 index this property is quickly lost.
This work shows that the operator approach extracts cycles from nonlinear dynamics without the linearity assumption inherent in EOF or POP analysis. Unlike neighboring studies that use statistical indices or filtering, here the Koopman operator eigenfunctions automatically straighten the cycle and separate fundamental modes from combination modes. This yields a more accurate description of El Niño diversity, including double-peaked events, and points to a stronger cyclic component than previously thought.
The method can be extended to other climate phenomena, such as the Madden-Julian oscillation or decadal modes. Reduced models based on straightened coordinates may be constructed, improving prediction. Application to nonstationary data with external forcings, such as volcanic eruptions, is also promising.
The approach will influence the diagnosis and prediction of climate variability, as well as the assessment of El Niño's impacts on ecosystems and economies.
Test the method on other datasets and for other modes, and integrate it into operational forecasting systems.
The work directly addresses the problem of objectively extracting coherent structures from climate data, which remains unsolved due to the system's nonlinearity and nonstationarity. Traditional methods mix time scales and require subjective filter choices, whereas the operator approach offers a systematic solution.
For the ERSSTv4 reanalysis, the El Niño eigenfunction remains almost perfectly cyclic even in the presence of a warming trend: the trend is captured by a separate eigenvalue with zero frequency, without distorting the cycle.
In Kim Stanley Robinson's novel 'Forty Signs of Rain', climatologists try to predict abrupt climate changes. The operator approach that straightens cycles echoes the attempt to find hidden order in chaos, which the characters seek in their data.