Fig. 6: Causality in the turbulent energy cascade.
From: Decomposing causality into its synergistic, unique, and redundant components
a Visualization of the magnitude of the velocity field for four filter sizes Δi at the same instant. The parameter η is the Kolmogorov length-scale and represents the smallest scale in the flow. b An extract of the time history of Σ1, Σ2, Σ3, and Σ4. The time is non-dimensionalized by Tϵ, which is the integral time-scale of the flow. c Redundant (R), unique (U), and synergistic (S) causal contributions from SURD. Only the top 12 contributions, satisfying the condition \({\Delta I}_{(\cdot )\to j}/I({\Sigma }_{j}^{+};{{{\mathbf{\Sigma }}}})\ge 1{0}^{-3}\), are represented, where Σ = [Σ1, Σ2, Σ3, Σ4]. The gray bar is the causality leak. The results of CGC, CTE, CCM, and PCMCI are depicted on the right.
