Interaction of shear-Alfvén wave packets: Implication for weak magnetohydrodynamic turbulence in astrophysical plasmas

The nonlinear interaction of shear-Alfvéan wave packets is a fundamental physical process underlying incompressible magnetohydrodynamics turbulence, as emphasized in the Iroshnikov-Kraichnan theory. In the weak turbulence limit, we give a detailed analytical and numerical treatment of the interaction between two colliding shear-Alfvén wave packets in the presence of a strong and uniform magnetic field B = B₀ ẑ. Using the ideal MHD equations, it is shown that three-wave interactions are generally nonzero if the k_z = 0 Fourier components of the wave packets are nonzero. From the reduced MHD equations, we calculate in closed form the three-wave and four-wave interaction terms, and show the latter to be generally asymptotically subdominant if the wave packets have no k_z = 0 component. Our results on the generic dominance of three-wave interactions contradict recent claims by Sridhar & Goldreich (1994) who have argued that three-wave interactions are empty and that the Iroshnikov-Kraichnan theory is incorrect because it describes weak three-wave turbulence. The principal implication of our results is that the Iroshnikov-Kraichnan theory is still a suitable point of departure for the study of Alfvénic turbulence in the interstellar medium.