Angular dependence of third-order law in anisotropic MHD turbulence

In solar wind turbulence, the energy transfer/dissipation rate is typically estimated using MHD third-order structure functions calculated using spacecraft observations. However, the inherent anisotropy of solar wind turbulence leads to significant variations in structure functions along different observational directions, thereby affecting the accuracy of energy-dissipation rate estimation. An unresolved issue is how to optimise the selection of observation angles under limited directional sampling to improve estimation precision. We conduct a series of MHD turbulence simulations with different mean magnetic field strengths, $B_0$. Our analysis of the third-order structure functions reveals that the global energy dissipation rate estimated around a polar angle of $\theta = 60^\circ$ agrees reasonably with the exact one for $0 \le B_0/b_{rms} \le 5$, where $b_{rms}$ denotes the root-mean-square magnetic field fluctuation. The speciality of $60^\circ$ polar angle can be understood by the Mean Value Theorem of Integrals, since the spherical integral of the polar-angle component ($\widetilde{T_\theta}$) of the divergence of Yaglom flux is zero, and $\widetilde{T_\theta}$ changes sign around 60$^\circ$. Existing theory on the energy flux vector as a function of the polar angle is assessed, and supports the speciality of $60^\circ$ polar angle. The angular dependence of the third-order structure functions is further assessed with virtual spacecraft data analysis. The present results can be applied to measure the turbulent dissipation rates of energy in the solar wind, which are of potential importance to other areas in which turbulence takes place, such as laboratory plasmas and astrophysics.