We introduce new features of data-adaptive harmonic decomposition (DAHD) that are showcased to characterize spatiotemporal variability in high-dimensional datasets of complex and mutsicale oceanic flows, offering new perspectives and novel insights. First, we present a didactic example with synthetic data for identification of coherent oceanic waves embedded in high amplitude noise. Then, DAHD is applied to analyze turbulent oceanic flows simulated by the Regional Oceanic Modeling System and an eddy-resolving three-layer quasigeostrophic ocean model, where resulting spectra exhibit a thin line capturing nearly all the energy at a given temporal frequency and showing well-defined scaling behavior across frequencies. DAHD thus permits sparse representation of complex, multiscale, and chaotic dynamics by a relatively few data-inferred spatial patterns evolving with simple temporal dynamics, namely, oscillating harmonically in time at a given single frequency. The detection of this low-rank behavior is facilitated by an eigendecomposition of the Hermitian cross-spectral matrix and resulting eigenvectors that represent an orthonormal set of global spatiotemporal modes associated with a specific temporal frequency, which in turn allows to rank these modes by their captured energy and across frequencies, and allow accurate space-time reconstruction. Furthermore, by using a correlogram estimator of the Hermitian cross-spectral density matrix, DAHD is both closely related and distinctly different from the spectral proper orthogonal decomposition that relies on Welch’s periodogram as its estimator method.
The turbulent oceanic flows consist of ubiquitous complex motions—jets, vortices, and waves—that co-exist on very different spatiotemporal scales but also without a clear scale separation, and it brings natural challenge to characterize the whole complexity across the scales. In particular, the study of temporal scales has got less attention than of spatial scales. To that effect, we offer fresh perspectives and novel insights by introducing new features of data-adaptive harmonic decomposition (DAHD) that are applied to analyze complex high-dimensional spatiotemporal datasets of oceanic flows, including a synthetic example of identifying coherent oceanic waves embedded in high-amplitude noise and turbulent flows simulated by a hierarchy of oceanic models. DAHD results reveal striking low-rank behavior and a sparse representation of complex, multiscale, and chaotic flows by a relatively few data-inferred spatial patterns evolving with simple temporal dynamics, as well as well-defined scaling behavior across temporal frequencies, such as exponential-like shape and power law.