Tensor invariant approach to energy flux in magnetohydrodynamic turbulence

A scale-by-scale analysis of energy flux in the turbulent cascade can be performed using the spatially filtered magnetohydrodynamic (MHD) equations, while the gradient tensor invariants are widely used to characterize the structure of velocity and magnetic fields. Physical mechanisms responsible for energy flux require specific field configurations whose strength is quantified by these tensor invariants. We explore this requirement, showing that the tensor invariants act as proxies for mechanistic energy fluxes under quantifiable conditions. As a special case, the purely hydrodynamic contributions to energy flux can be expressed exactly in terms of the invariants of the velocity gradient tensor. We also show that the invariants bound the available energy flux for distinct physical mechanisms, formalizing the idea that each transfer mechanism requires field configurations with gradients of sufficient strength to support a given energy flux. Results are illustrated using three-dimensional simulations of freely decaying MHD turbulence. These findings advance our understanding of MHD turbulence by revealing key relationships between field topology and energy transfer across scales due to distinct physical mechanisms. They further provide a tool for probing the turbulent cascade using multispacecraft data where local tensor invariants are directly accessible.