Energy transfer and third-order law in forced anisotropic MHD turbulence with hyperviscosity*

The Kolmogorov-Yaglom (third-order) law links energy transfer rates in the inertial range of magneto-hydrodynamic (MHD) turbulence with third-order structure functions. Anisotropy, a typical property in the solar wind, largely challenges the applicability of the third-order law with isotropic assumption. To shed light on the energy transfer process in the presence of anisotropy, the present study conducted direct numerical simulations (DNSs) of forced MHD turbulence with normal and hyper-viscosity under various strengths of the external magnetic field (B₀), and calculated three forms of third-order structure function with or without averaging azimuthal or polar angles to B₀ direction. Correspondingly, three forms of estimated energy transfer rates were studied systematically with various B₀. The result shows that the peak of the estimated longitudinal transfer rate occurs at larger scales as closer to the B₀ direction, and its maximum shifts away from the B₀ direction at larger B₀. Compared with normal viscous cases, hyper-viscous cases can attain better separation of the inertial range from the dissipation range, thus facilitating the analyses of the inertial range properties and the estimation of the energy cascade rates. We find that the widespread use of the isotropic form of the third-order law in estimating the energy transfer rates is questionable in some cases, especially when the anisotropy arising from the mean magnetic field is inevitable. In contrast, the direction-averaged third-order structure function properly accounts for the effect of anisotropy and predicts the energy transfer rates and inertial range accurately, even at very high B₀. With limited statistics, the calculation of the third-order structure function shows a stronger dependence on averaging of azimuthal angles than the time, especially at high B₀ cases. These findings provide insights into the anisotropic effect on the estimation of energy transfer rates.