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Turbulence in the magnetized plasma is well understood to be the consequence of wave interactions. The addition of the Hall term to classical magnetohydrodynamics (MHD) causes the shear Alfvén wave to become dispersive and circularly polarized. These effects are expected to cause Hall MHD turbulence to be more complex than its MHD counterpart. We develop and analyze a reduced model for wave–wave interactions for incompressible Hall MHD. Of all the nonlinear terms, we concentrate on a subclass of wave–wave interactions that will produce a response phase coherent with the mode under consideration. The principal idea is to calculate a “nonlinear dispersion relation” that modifies the linear dispersion relation due to the effect of the ensemble of waves in which the given mode is immersed. The amplitude dependent modification will, in general, result in growth, damping, and frequency shifts to the fiducial mode.
Hansen et al. (Wed,) studied this question.