Weak magnetohydrodynamic turbulence theory revisited

Two recent papers, P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021) and Yoon et al., Phys. Plasmas 29, 112303 (2022), utilized in the derivation of the kinetic equation for the intensity of turbulent fluctuations the assumption that the wave spectra are isotropic, that is, the ensemble-averaged magnetic field tensorial fluctuation intensity is given by the isotropic diagonal form, ⟨ δ B i δ B j ⟩ k = ⟨ δ B 2 ⟩ k δ i j. However, it is more appropriate to describe the incompressible magnetohydrodynamic turbulence involving shear Alfvénic waves by modeling the turbulence spectrum as being anisotropic. That is, the tensorial fluctuation intensity should be different in diagonal elements across and along the direction of the wave vector, ⟨ δ B i δ B j ⟩ k = 1 2 ⟨ δ B ⊥ 2 ⟩ k ( δ i j − k i k j / k 2 ) + ⟨ δ B ∥ 2 ⟩ k ( k i k j / k 2 ). In the present paper, we thus reformulate the weak magnetohydrodynamic turbulence theory under the assumption of anisotropy and work out the form of nonlinear wave kinetic equation.