On the multiscale fractal features of a low-order coupled ocean-atmosphere model in comparison with reanalysis data
Authors: Alberti Tommaso , Donner Reik , Vannitsem Stéphane 
Published: March 2021

&lt;p&gt;Atmosphere and ocean dynamics display many complex features and are characterized by a wide variety of processes and couplings across different timescales. Here we use Multivariate Empirical Mode Decomposition (MEMD; Rehman and Mandic, 2010) to investigate the multivariate and multiscale properties of a low-order model of the ocean-atmosphere coupled dynamics (Vannitsem, 2017). The MEMD allows us 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. This information allows 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 multivariate and multiscale generalized fractal dimensions (Alberti et al., 2020). With these two approaches, we demonstrate that the coupled 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 us 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 characterize the intrinsic complexity of oscillating patterns that can be related to the attractor properties. Our results support the interpretation of the coupled ocean-atmosphere dynamics as well as the investigation of general 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. We compare the results obtained for the low-order dynamical model with those derived from the reanalysis data and demonstrate that a similar scale-dependent behavior is found, thus also confirming the suitability of the proposed system to model the ocean-atmosphere dynamics at different timescales and to describe topological and geometrical features of its phase-space.&lt;/p&gt;&lt;p&gt;&amp;#160;&lt;/p&gt;&lt;p&gt;&lt;strong&gt;References&lt;/strong&gt;&lt;/p&gt;&lt;p&gt;Alberti, T., Consolini, G., Ditlevsen, P. D., Donner, R. V., Quattrociocchi, V. (2020). Multiscale measures of phase-space trajectories. Chaos 30, 123116.&lt;/p&gt;&lt;p&gt;Alberti, T., Donner, R. V., and Vannitsem, S. (2021). Multiscale fractal dimension analysis of a reduced order model of coupled ocean-atmosphere dynamics. Earth Syst. Dynam. Discuss. [preprint], https://doi.org/10.5194/esd-2020-96, in review.&lt;/p&gt;&lt;p&gt;Hentschel, H. G. E., Procaccia, I. (1983). The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8, 435&amp;#8211;444.&lt;/p&gt;&lt;p&gt;Rehman, N., Mandic, D. P. (2010). Multivariate empirical mode decomposition. Proceedings of the Royal Society A, 466, 1291&amp;#8211;1302.&lt;/p&gt;&lt;p&gt;Vannitsem S., Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics, Chaos, 27, 032101, 2017.&amp;#160;&lt;/p&gt;