Whistler and Cyclotron Wave Evolution and Interaction in Incompressible Hall MHD Turbulence

The incompressible Hall MHD system exhibits waves with whistler and cyclotron frequencies, and each branch can take the form of a positive or negative curl eigenstate. To study properties of wave interactions and turbulence, we decompose the magnetic and velocity fields of incompressible Hall MHD into the four waves of this system. The induction and velocity equations of Hall MHD are expanded in the basis of eigenmodes, in an extension of Mahajan 2021, to obtain evolution equations for the wave amplitudes. These equations are analyzed to predict the dominant interactions driving modes at various regimes in k-space. This decomposition is applied to results of a pseudospectral Hall MHD simulation to show how the total energy in each wave changes over the course of a simulation. An energy and helicity conserving integrator is applied to terms corresponding to the ideal Hall MHD system. Energy spectra, longitudinal and transverse structure functions, and the Goldreich-Sridhar nonlinearity parameter are obtained for each type of wave.