The Indo-Pacific region hosts several key climate oscillations: El Niño–Southern Oscillation (ENSO), the Indian Ocean Dipole (IOD), and the tropospheric biennial oscillation (TBO). Classical analysis methods, such as empirical orthogonal function (EOF) analysis, require prior filtering of data to isolate a desired time scale, which can distort or hide nonlinear interactions. The question of IOD independence from ENSO remains contentious: some studies consider IOD forced by ENSO, others as an intrinsic mode of the Indian Ocean. Similar debates surround the TBO. For decadal variability, the leading mode is thought to be the Pacific Decadal Oscillation (PDO) or the Interdecadal Pacific Oscillation (IPO), but short observational records limit reliable mode extraction. Against this backdrop, the study offers an alternative approach that requires no preprocessing and can extract modes of different scales simultaneously. Nearby in the archive are studies on gravitational waves, neutron stars, and quantum fluctuations, but this work stands apart, thematically adjacent to LIGO and baryon acoustic oscillations topics due to a shared interest in extracting weak signals from noise.
The authors apply nonlinear Laplacian spectral analysis (NLSA) to monthly mean sea surface temperature (SST) data in the Indo-Pacific region (28°E–70°W, 60°S–20°N). They use millennium-long control runs of CCSM4 (1300 years) and CM3 (800 years), as well as HadISST observations (140 years of the industrial era and 31 years of the satellite era). NLSA combines time-delay embedding ideas from dynamical systems theory with kernel methods from machine learning. A kernel is constructed that depends on the local direction of the data flow, allowing the system's intrinsic time scales to be identified. Unlike EOFs, which use a global covariance matrix, NLSA approximates the Laplace–Beltrami operator on the nonlinear manifold generated by the data. The eigenfunctions of this operator serve as nonlinear analogs of principal components. To enhance separation of time scales, an embedding with 240 monthly lags (20 years) is used, significantly more than in previous studies. Kernel parameters: ε = 1, ζ = 0.99. From 200 computed eigenfunctions, 18 are selected representing the main modes: the annual cycle and harmonics, ENSO, ENSO combination modes with the annual cycle, TBO and its combination mode, WPMM, and IPO. Spatiotemporal reconstructions are built by convolving eigenfunctions with the original data.
NLSA extracts a hierarchy of modes from seasonal to multi-decadal. The annual cycle and its first two harmonics explain up to 70% of raw variance in southern mid-latitudes. ENSO is represented by a pair of quadrature modes with a carrier frequency of about 4 years and a decadal envelope. ENSO combination modes with the annual cycle (ENSO-A1 and ENSO-A2) have peaks at frequencies 0.75 and 1.25 cycles/year and explain up to 35% of variance in the eastern Indian Ocean, the region used to define the Indian Ocean Dipole. This indicates that a significant portion of IOD variability may be deterministic modulation of the ENSO annual cycle. TBO emerges as a separate pair of modes with a period of about 2 years, but with amplitude an order of magnitude weaker than ENSO. The new western Pacific multi-decadal mode (WPMM) appears as a persistent SST anomaly in the western tropical Pacific with amplitude up to 0.25 K and explains about 1% of total variance, but up to 50% of non-periodic variance in some regions. IPO is also extracted, but with lower explained variance. In CM3 and HadISST data, similar modes are found, but decadal modes have lower quality due to short record lengths. Sensitivity analysis shows that records several centuries long are needed for reliable extraction of WPMM and TBO. Comparison with singular spectrum analysis (SSA) demonstrates NLSA's advantage in extracting combination modes and TBO.
The study shows that nonlinear analysis methods that do not require prior filtering can extract physically meaningful modes from raw data. This challenges the traditional view of Indian Ocean Dipole independence: a significant portion of its variability turns out to be deterministic modulation of the ENSO annual cycle. The discovery of WPMM as a distinct mode, separate from IPO, supports hypotheses about additional decadal modes in the Indo-Pacific region. The results are consistent with theoretical predictions of combination modes arising from quadratic nonlinearities in the coupled ocean–atmosphere system. Compared to neighboring works on gravitational waves, which also seek weak signals in noise, this study applies similar mathematical machinery to climate data, underscoring the universality of spectral analysis methods on manifolds.
The NLSA method can be extended to other climate variables, such as wind, precipitation, and thermocline depth, to study coupled patterns. Increasing observational record lengths or using paleoclimate reconstructions will allow more reliable extraction of multi-decadal modes. Further development of kernel methods and computational acceleration algorithms will make NLSA applicable to higher-dimensional data, including full atmospheric and oceanic fields. Investigating secondary ENSO modes and their role in El Niño and La Niña asymmetry could deepen understanding of nonlinear climate system dynamics.
The results may improve understanding of ENSO and Indian Ocean Dipole predictability, as well as their influence on global climate. The discovery of WPMM could be important for interpreting decadal sea level changes and regional precipitation, for example, in Australia.
In Part II, the authors plan to examine the connections of the extracted modes to wind, thermocline depth, and precipitation fields, as well as their climatic impacts. They also intend to investigate the role of secondary ENSO modes in phase asymmetry.
The work directly addresses the unresolved problem of the nature of the Indian Ocean Dipole: whether it is an intrinsic mode of the Indian Ocean or a regional manifestation of ENSO. The results show that a significant portion of IOD variability can be explained by ENSO combination modes, supporting the hypothesis of a forced IOD. It also touches on the problem of the existence and origin of the tropospheric biennial oscillation, which can only be isolated as a separate mode in long model runs.
The western Pacific multi-decadal mode (WPMM) is so slow that individual events can last longer than the entire satellite record of the ocean. For example, in the CCSM4 model, one positive WPMM event lasted about 65 years.
The idea of hidden periodicities governing climate echoes Isaac Asimov's novel 'The End of Eternity', where mathematicians compute cyclic changes in reality to manage human history.