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In a recent paper [P. H. Yoon and G. Choe, Phys. Plasmas 28, 082306 (2021)], the weak turbulence theory for incompressible magnetohydrodynamics is formulated by employing the method customarily applied in the context of kinetic weak plasma turbulence theory. Such an approach simplified certain mathematical procedures including achieving the closure relationship. The formulation in the above-cited paper starts from the equations of incompressible magnetohydrodynamic (MHD) theory expressed via Elsasser variables. The derivation of nonlinear wave kinetic equation therein is obtained via a truncated solution at the second-order of iteration following the standard practice. In the present paper, the weak MHD turbulence theory is alternatively formulated by employing the pristine form of incompressible MHD equation rather than that expressed in terms of Elsasser fields. The perturbative expansion of the nonlinear momentum equation is carried out up to the third-order iteration rather than imposing the truncation at the second order. It is found that while the resulting wave kinetic equation is identical to that obtained in the previous paper cited above, the third-order nonlinear correction plays an essential role for properly calculating derived quantities such as the total and residual energies.

 

A set of self-consistent equations of weak turbulence theory that describe the time evolution of the electron velocity distribution and of the spectra of Langmuir and ion sound waves is solved numerically, considering the presence of a core electron population and a ring-beam electron distribution. The results obtained show that the finite pitch angle of the beam relative to the direction of the ambient magnetic field leads to a spectrum of Langmuir waves which is more complex than the spectrum obtained in the case of beams with zero pitch angle, to an enlarged plateau in the beam region of the electron velocity distribution and to the generation of a prominent high-velocity population in the electron velocity distribution.