Anisotropic energy transfer rate quantified by LPDE and directional averaging methods in MHD turbulence

The energy cascade rate (ε) depicts the energy transfer in a turbulent system. In incompressible magneto-hydrodynamic (MHD)  turbulence, ε is linked to the third-order structure function (Yaglom vector) via the Yaglom/Politano–Pouquet law in the inertial range. In this study, we compare three estimators of ε in anisotropic MHD turbulence: (1) the lag polyhedral derivative ensemble (LPDE) technique that reconstructs the divergence of the Yaglom vector via tetrahedral linear gradients; (2) a directional-averaged third-order estimator that evaluates the Yaglom vector along a finite number of lag directions and averages over solid angle; and (3) the Yaglom vector on 60 degree with respect to the mean magnetic field direction.  To ensure a fair comparison in more realistic MHD turbulence, we emulate a multipoint virtual mission within anisotropic three-dimensional MHD simulations with a guide field B₀ along the z-axis. This work illuminates the reliable regime for LPDE and directional-averaging methods, and also tests whether 60 degree Yaglom vector is an accurate estimate of ε, providing practical guidance in both simulation and observational turbulence analysis.