Multiscale Fractal Dimension Analysis of a Low-Order Coupled Ocean-Atmosphere Model based on Multivariate Empirical Mode Decomposition

Atmosphere and ocean dynamics display many complex features and are characterized by a wide variety of processes and couplings on different timescales. Here we use Multivariate Empirical Mode Decomposition (MEMD; Rehman and Mandic, 2010) to investigate the multivariate and multiscale properties of a conceptual model of the ocean-atmosphere coupled dynamics (Vannitsem, 2017). The MEMD allows to decompose the original data into a series of oscillating patterns with time-dependent amplitude and phase by exploiting the local features of data and without any a priori assumptions on the decomposition basis. Moreover, each oscillating pattern, usually named Multivariate Intrinsic Mode Function (MIMF), can be used as a source of local (in terms of scale) fluctuations and information allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales (Hentschel and Procaccia, 1983) that can be seen as a sort of multiscale/multivariate generalized fractal dimensions. With these two approaches we show that the ocean-atmosphere dynamics presents a rich variety of common features, although with a different nature of the fractal properties between the ocean and the atmosphere at different timescales. The MEMD results allow to capture the main dynamics of the phase-space trajectory that can be used for reconstructing the skeleton of the phase-space dynamics, while the evaluation of the fractal dimensions at different timescales allows to characterize the intrinsic complexity of oscillating patterns that can be related to the attractor properties. Our results could be helpful for the interpretation and the characterization of the coupled ocean-atmosphere dynamics as well as for the investigation of deterministic-chaotic dissipative dynamical systems in terms of invariant manifolds, bifurcations, as well as (strange) attractors in their phase-space, whose geometric and topological properties are a reflection of the dynamical regimes of the system at different scales.

References

Hentschel, H. G. E., Procaccia, I. (1983). Physica D, 8, 435-444.

Rehman, N., Mandic, D. P. (2010). Proceedings of the Royal Society A, 466, 1291-1302.

Vannitsem, S. (2017). Chaos, 27, 032101.