Operator theoretic causality analysis of fluid flows using linearized dynamics

Abstract

This paper presents an operator-theoretic framework, Linear Operator Causality Analysis (LOCA), for analyzing causality in linearized dynamical systems, focusing here on fluid flows. Our proposed approach, which can be characterized as a special case of Dynamic Causal Effect (DCE) analysis, utilizes the matrix exponential of linearized differential equations to determine causal relationships between system modes at any future time. We further develop an upper bound that quantifies the presence and extent of global causality across all time horizons. This approach provides a physics-based alternative to data-driven statistical and information-theoretic causality measures such as Granger causality and transfer entropy. Unlike these data-driven techniques that infer causality from time-series data, LOCA leverages the linearized governing equations, yielding a physically-motivated and interpretable measure of causal interactions. We identify the conditions under which LOCA gives equivalent results to data-driven causality analysis methods, and further discuss connections to key system properties such as controllability, observability, and graph-theoretic transitive closure. To complement this operator-based approach, we introduce a data-driven methodology akin to Dynamic Mode Decomposition (DMD) that estimates causal connections directly from time series data by approximating the matrix exponential. We argue that LOCA also mitigates common issues in data-driven causality analyses, such as misleading inferences due to correlated variables or state truncation. We demonstrate our method on two fluid flow examples: linearized Couette flow, and a nonlinear wake flow featuring chaotic dynamics. In both cases, we demonstrate how our framework captures both direct and indirect causal interactions among flow structures.