Multi-scale vortex structures constitute the intrinsic skeleton of turbulent flows and govern the energy cascade process in oceanic turbulence. Elucidating their evolutionary dynamics is crucial for understanding turbulent mixing and transport. In this study, we develop an innovative Energy Cascade Multilayer Directed Network (ECMDN) framework grounded in complex network theory to directly characterize nonlinear energy coupling pathways and directional transfers among multi-scale vortices in real marine environments. By integrating multi-parameter fusion node definitions, multi-scale interaction detection, and energy transfer direction identification, the ECMDN reconstructs the nonlinear turbulent system into a topologically interpretable structure. The emergent network properties enable quantitative characterization of intermittency and inhomogeneity in the energy cascade, offering new insights into vortex interactions and cross-scale energy transfer mechanisms. Compared with conventional cascade diagnostics including spectral flux, third-order velocity structure functions, multifractal analysis and shell models that require homogeneity and local equilibrium assumptions and only output global averaged energy flux, the proposed ECMDN multilayer network retains point-wise depth coordinates of each vortex interaction, separates directed forward/inverse energy edges, and quantifies intermittency via topological metrics. Analysis of the single Shenhu thermocline shear segment demonstrates these differentiated analytical capabilities of the proposed framework. Application to shear measurements from the Shenhu Sea reveals the simultaneous occurrence of forward and inverse energy cascades, manifesting a synchronous dual-energy-cascade pattern. This indicates that vortices at a given scale can concurrently transfer energy to larger- or smaller-scale structures and receive energy from larger- or smaller-scale counterparts during the cascade process. Our findings observe a typical synchronous dual-energy-cascade pattern in the strong thermocline of the Shenhu Sea, providing a novel theoretical and methodological framework for investigating the spatiotemporal evolution of stratified ocean turbulent mixing and advancing our understanding of geophysical fluid dynamics.
Zhang et al. (Tue,) studied this question.