Alfvén wave phase mixing in two-dimensional open magnetic configurations

The phase mixing of Alfvén waves in planar twodimensional open magnetic plasma configurations is considered. It is assumed that the characteristic vertical spatial scale of the configuration is much larger than the horizontal scale, and that the latter is of the order of a wavelength. The WKB method is used to derive the governing equation for the wave amplitude, which in appropriate coordinates is the diffusion equation with the diffusion coefficient being spatially dependent. The dependency of the energy flux on the vertical coordinate is obtained for monochromatic waves, and illustrated for three particular cases. In all three cases, at low heights phase-mixed Alfvén waves damp at the same rate as in a one-dimensional configuration. However, in the first and third cases phase mixing operates only at low and intermediate heights and practically stops at heights larger than a few characteristic vertical length scales. Only a part of the energy flux is damped due to phase mixing. In the second case the situation is reversed: the damping of the energy flux with height is much faster that in one-dimensional configurations. The rate of damping of the energy flux with height due to phase mixing in two-dimensional configurations thus depends strongly on the particular form of the configuration. The theory is applied to Alfvén wave damping in coronal holes.