Pairwise interactions in turbulence time series: essential measures for validating synthetic data

Abstract

The synthesis of turbulence dynamics using machine learning, statistical, and dynamical system-based models has become an emerging research direction. Validation of synthetic turbulence time series is often based on marginal and second-order statistics, which may overlook whether the underlying interaction structure is preserved. In this work, we apply a set of linear and information-theoretic measures to evaluate interactions between pairs of turbulence time series obtained from a turbulent channel flow at various wall distances, as well as between their synthetic counterparts. To demonstrate the applicability of these measures, we assess synthetic time series generated from a high-order vector autoregressive (VAR) model. The results show that the VAR-generated time series preserve the linear interactions measured by auto- and cross-covariance functions, even for long-memory dynamics. However, when assessed by information-theoretic measures, such as time-delayed mutual information and transfer entropy, the agreement deteriorates. Despite under-predicting the magnitude of transfer entropy, the VAR-generated time series correctly capture the dominant direction of net information transfer in most of the pairs. Furthermore, when the time series embedding is optimally constructed, the interpretation of net transfer entropy is shown to be consistent with relevant physical interpretations of causal interactions reported in the turbulence literature. Future research should consider applying this framework to multivariate turbulence time series generated by other data-driven techniques to ensure more comprehensive validation.