On waves in incompressible Hall magnetohydrodynamics

Hall magneto-hydrodynamics (HMHD) is a mono-fluid approximation extending the validity domain of the ordinary MHD system to spatial scales down to a fraction of the ion skin depth or frequencies comparable to the ion gyrofrequency. In the paper by Galtier (2006 J. Plasma Physics), an incompressible limit of the HMHD system is used for developing a wave turbulence theory. Nevertheless, the possibility and the consequences of such an approximation are different in HMHD and in MHD. Here, we analyse these differences by investigating the properties of the HMHD equations in the incompressible limit: the existence of linear modes, their dispersion relations and polarizations. We discuss the possibility of replacing the fluid closure equation of a complete HMHD system by an incompressibility hypothesis and determine the validity range.