  Phase mixing of Alfven waves is one of the most promising mechanisms for
heating of the solar atmosphere. The damping of waves in this case requires
small transversal scales, relative to the magnetic field direction. Here this
requirement is achieved by considering a transversal inhomogeneity in the
equilibrium plasma density profile. Using a single fluid approximation of a
partially ionized chromospheric plasma we study the effectiveness of the
damping of phase mixed shear Alfven waves and investigate the effect of varying
the ionization degree on the dissipation of waves. Our results show that the
dissipation length of shear Alfven waves strongly depends on the ionization
degree of the plasma, but more importantly, in a partially ionized plasma, the
damping length of shear Alfven waves is several orders of magnitude shorter
than in the case of a fully ionized plasma, providing evidence that phase
mixing could be a large contributor to heating the solar chromosphere. The
effectiveness of phase mixing is investigated for various ionization degrees,
ranging from very weakly to very strongly ionized plasmas. Our results show
that phase mixed propagating Alfven waves in a partially ionized plasma with
ionization degrees in the range 0.518 to 0.657, corresponding to heights of
1916 to 2150 km above the solar surface, can provide sufficient heating to
balance chromospheric radiative losses in the quiet Sun.
