Laws governing the behavior of statistical third-order moments in the inertial range are among the few rigorous results in the theory of statistically homogeneous incompressible magnetohydrodynamic turbulence. These fundamental laws apply to both isotropic and anisotropic turbulence. Assuming that the turbulence is stationary in time and statistically axisymmetric under proper rotations about the direction of the mean magnetic field, it is shown that the general mathematical form of the tensor quantities appearing in these laws is constrained by symmetry to have a particular form. Using these forms, the general solutions of the law for the vector and pseudovector third-order moments F and FC are obtained in the limit of large kinetic Reynolds number and large magnetic Reynolds number. The physical meaning of the different terms in F and FC are investigated and a method for obtaining the cascade rates of energy ϵ and cross-helicity ϵC from experimental data is described. The results show that the measurement of the cascade rates goes hand-in-hand with the measurement of the spatial anisotropy of the third-order moments F and FC. The theory developed here can be applied to measure the turbulent cascade rates of energy and cross-helicity in laboratory plasma experiments, numerical simulations, and the solar wind.
