On local and non-local energy transfers in Hall magnetohydrodynamic turbulence
A systematic study of inertial energy cascade in three-dimensional incompressible Hall magnetohydrodynamic turbulence is conducted to probe into the locality of energy conserving triads and the subsequent transfers. Based on the nature of triadic conservations, the energy transfer due to the Hall term is further decomposed into two channels BB and JB corresponding to the terms $d_i ({\bf j}\cdot\nabla){\bf b}$ and $ d_i ({\bf b}\cdot\nabla){\bf j}$, respectively (Halder et al., 2023; Banerjee and Halder,2024). Here ${\bf b}$ and ${\bf j}$ represent the magnetic field and the current in Alfvén units, whereas $d_i$ is the ion inertial scale. Using direct numerical simulations, we calculate the shell-to-shell energy transfer rates corresponding to both the channels, and convincingly show each of them to comprise a combination of local and non-local energy transfers. A local inverse transfer is consistently observed at all scales of the channel BB whereas for the channel JB, the local exchange of energy is associated with a gradual increase in strength as the scale is decreased, together with a transition from inverse to direct transfer across the Hall wavenumber, characterized by the ion inertial length $d_i$. Calculating mediator-specific transfer rates, we also conclude that a considerable amount of the local energy transfer is mediated by the non-local triads, especially at small scales of the channel JB. The observed results can be explained using the power-law behaviour of the modal fields. The present study captures the intricate dynamics of energy transfer due to the Hall term and hence can be used to develop more insightful analytical models (shell models, for example) for Hall magnetohydrodynamic cascade. The framework can be extended to segregate the local and the nonlocal heating in various turbulent flows including ferrofluids, binary fluids, etc.