  The Earth's climate system is a classical example of a multiscale,
multiphysics dynamical system with an extremely large number of active degrees
of freedom, exhibiting variability on scales ranging from micrometers and
seconds in cloud microphysics, to thousands of kilometers and centuries in
ocean dynamics. Yet, despite this dynamical complexity, climate dynamics is
known to exhibit coherent modes of variability. A primary example is the El
Ni\~no Southern Oscillation (ENSO), the dominant mode of interannual (3-5 yr)
variability in the climate system. The objective and robust characterization of
this and other important phenomena presents a long-standing challenge in Earth
system science, the resolution of which would lead to improved scientific
understanding and prediction of climate dynamics, as well as assessment of
their impacts on human and natural systems. Here, we show that the spectral
theory of dynamical systems, combined with techniques from data science,
provides an effective means for extracting coherent modes of climate
variability from high-dimensional model and observational data, requiring no
frequency prefiltering, while being capable of simultaneous recovery of
multiple timescales and their interactions. Lifecycle composites of ENSO are
shown to improve upon results from conventional indices in terms of dynamical
consistency and physical interpretability. In addition, the role of combination
modes between ENSO and the annual cycle in ENSO diversity is elucidated.
