Fig. 6. Causality in the turbulent energy cascade.

a Visualization of the magnitude of the velocity field for four filter sizes Δ_i 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, Σ_3, and Σ₄. The time is non-dimensionalized by T_ϵ, 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 ${\mathrm{\Delta }I}_{\left(\cdot \right)\to j}/I\left({\mathrm{\Sigma }}_j^{+};\mathbf{\Sigma }\right)\ge 10^{-3}$, 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.