Entropy-Based Uncertainty Ranking of Turbulent States

Chaotic dynamical systems are sensitive to perturbations in initial conditions, making long-term prediction equally challenging for all states, but in the short term some states are more predictable than others. Perturbation-based methods, such as local Lyapunov exponent analysis, have been used to assess state-dependent predictability. However, these approaches require knowledge of the governing equations, are computationally expensive, and are often impractical in data-driven or experimental settings where perturbations are infeasible. These limitations motivate the development of non-intrusive, data-driven methods that can assess predictability from partial observations without requiring explicit perturbations. In this work, we use conditional Shannon entropy to rank uncertainty growth in chaotic systems without perturbing the system. The approach is applicable to both fully and partially observed systems and is suitable for high-dimensional turbulent flows. We validate the method on the Lorenz system and the Rössler attractor by comparison with Lyapunov-based analysis. Then, we apply the method to the minimal turbulent channel flow at $Re_\tau=180$, analyzing the streamwise velocity field, to identify the most predictable and most unpredictable flow structures. Preliminary results show that wavy and broken low speed streaks are the most unpredictable states, while straight streaks are the most predictable states.