Properties of Hall magnetohydrodynamic waves modified by electron inertia and finite Larmor radius effects

The linear wave equation (sixth order in space and time) and the corresponding dispersion relation is derived for Hall magnetohydrodynamic (MHD) waves including electron inertial and finite Larmor radius effects together with several limiting cases for a homogeneous plasma. We contrast these limits with the solution of the full dispersion relation in terms of wave normal (k⊥,k∥) diagrams to clearly illustrate the range of applicability of the individual approximations. We analyze the solutions in terms of all three MHD wave modes (fast, slow, and Alfvén), with particular attention given to how the Alfvén branch (including the cold ideal field line resonance (FLR) [ D. J. Southwood, Planet. Space Sci. 22, 483 (1974) ]) is modified by the Hall term and electron inertial and finite Larmor radius effects. The inclusion of these terms breaks the degeneracy of the Alfvén branch in the cold plasma limit and displaces the asymptote position for the FLR to a line defined by the electron thermal speed rather than the Alfvén speed. For a driven system, the break in this degeneracy implies that a resonance would form at one field line for small k⊥ and then shift to another as k⊥→∞. However for very large ωk⊥/VA, Hall term effects lead to a coupling to the whistler mode, which would then transport energy away from the resonant layer. The inclusion of the Hall term also significantly effects the characteristics of the slow mode. This analysis reveals an interesting “swapping” of the perpendicular root behavior between the slow and Alfvén branches.