Causal decomposition of flow variables using information theory

We present a method for decomposing a source variable into its causal and non-causal contributions relative to a target variable. The concept of causality is defined within the framework of information theory as the time flux of information between variables. The latter can be interpreted as how much the knowledge about the past of one variable aids the understanding of another variable in the future. The decomposition of the source variable into its causal and non-causal components is formulated as an optimization problem that seeks to maximize the information flux of the causal part to the target variable while simultaneously minimizing the information flux of the non-causal part to the target variable. We demonstrate the method in two cases. The first case consists of a problem with a known analytical solution. In the second case, we analyze the velocity regions in a turbulent channel flow that are causal to the wall shear stress. Our method offers a new approach to study interactions within complex flow phenomena by disentangling causal and non-causal contributions.