Identification of the continuum field structure at multiple scale levels

For continuum fields such as turbulence, analyses of the field structure offer insights into their kinematic and dynamic properties. To ensure the analyses are quantitative rather than merely illustrative, two conditions are essential: space-filling and structure quantification. A pertinent example is the dissipation element (DE) structure, which is however susceptible to noisy interference, rendering it inefficient for extracting the large-scale features of the field. In this study, the multi-level DE structure is proposed based on the multi-level extremal point concept. At a given scale level, the entire field can be decomposed into the corresponding space-filling and non-overlapping DEs, each characterized by its length scale l and the scalar difference Δϕ between its two extremal points. We will first elaborate on the fundamental principles of this method. Results from an artificially constructed two-scale field indicate that the decomposed units adequately represent the geometry of the original field. In examining the fractal Brownian motion, a structure function equivalent ⟨Δϕ|l⟩ and an energy spectrum equivalent are introduced. The scaling relation derived from ⟨Δϕ|l⟩ corresponds with the Hurst number. Furthermore, the multi-level DE structure distinctly reveals the two different inertial ranges in two-dimensional turbulence. Overall, this novel structure identification approach holds significant potential for complex analyses concerning the field geometry.