On the Hall effect on vertical Alfvén waves in an isothermal atmosphere

Abstract

The Alfvén wave equation in an atmosphere in the presence of Ohmic and Hall currents is obtained. It is shown that, for vertical propagation and vertical external magnetic field, there is no coupling to gravity or compressibility, because the magnetic field and velocity perturbation are transverse, and hence horizontal. The wave equation is valid for Alfvén speed, external magnetic field, gyrofrequency, and Ohmic diffusivity arbitrary functions of altitude; under these conditions the transverse wave perturbation is shown to decouple into right- and left-polarized waves. The wave field in an isothermal atmosphere, under a uniform magnetic field, is affected by the Hall current only through a change of propagation speed. The neglect of Ohmic compared to Hall current is justified for low-density plasma, e.g., in the solar wind. It is shown that the Hall effect changes the eigenfrequencies and eigenfunctions of standing atmospheric waves. It does not change the scaling of asymptotic wave fields at high altitude, but it breaks equipartition of energy at low altitude, e.g., for propagating waves, even when the ray approximation holds. Possible extensions of this problem are discussed in connection with Alfvén waves in the solar wind.