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 investigate 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

and

script upper J script upper B J B $\mathcal{JB}$

corresponding to the terms

d Subscript i Baseline left parenthesis bold italic j bold dot bold nabla right parenthesis bold italic b d i ( j ⋅ ∇ ) b $d_i ({\boldsymbol{j}}\boldsymbol{\cdot }\boldsymbol{\nabla }){\boldsymbol{b}}$

and

minus d Subscript i Baseline left parenthesis bold italic b bold dot bold nabla right parenthesis bold italic j − d i ( b ⋅ ∇ ) j $-d_i ({\boldsymbol{b}}\boldsymbol{\cdot }\boldsymbol{\nabla }){\boldsymbol{j}}$

, respectively, where

bold italic b b $\boldsymbol{b}$

and

bold italic j j $\boldsymbol{j}$

represent the magnetic field and the current density (in Alfvén units) respectively, and

d Subscript i d i $d_i$

is the ion-inertial length. 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

script upper B script upper B B B $\mathcal{BB}$

whereas for the channel

script upper J script upper B J B $\mathcal{JB}$

, 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, characterised by the ion inertial length

d Subscript i d i $d_i$

. 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

script upper J script upper B J B $\mathcal{JB}$

. 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 non-local heating in various turbulent flows including ferrofluids, binary fluids, etc.