Phase mixing of Alfven waves in two-dimensional magnetic plasma configurations with exponentially decreasing density

We study damping of phase-mixed Alfven waves propagating in axisymmetric magnetic plasma configurations. We use the linear magnetohydrodynamic (MHD) equations in the cold plasma approximation. The only dissipative process that we take into account is shear viscosity. We reduce the MHD equations describing the Alfven wave damping to a Klein-Gordon-type equation. We assume that the two terms in this equation, one describing the effect of inhomogeneity and the other the effect of viscosity, are small. Then we use the WKB method to derive the expression describing the wave energy flux attenuation with the height. We apply the general theory to particular equilibria with the exponentially divergent magnetic field lines with the characteristic scale H. The plasma density exponentially decreases with the height with the characteristic scale H-rho. We study the wave damping for typical parameters of coronal plumes and various values of the wave period, the characteristic scale of the magnetic field variation H, and kinematic shear viscosity H. We show that to have an appreciable wave damping at the height 6H we need to increase shear viscosity by at least six orders of magnitude in comparison with the value given by the classical plasma theory. Another important result is that the efficiency of wave damping strongly depends on the ratio H/H-rho. It increases fast when H/H-rho decreases. We present a physical explanation of this phenomenon.