  We study the propagation, reflection, and turbulent dissipation of Alfven
waves in coronal holes and the solar wind. We start with the Heinemann-Olbert
equations, which describe non-compressive magnetohydrodynamic fluctuations in
an inhomogeneous medium with a background flow parallel to the background
magnetic field. Following the approach of Dmitruk et al, we model the nonlinear
terms in these equations using a simple phenomenology for the cascade and
dissipation of wave energy, and assume that there is much more energy in waves
propagating away from the Sun than waves propagating towards the Sun. We then
solve the equations analytically for waves with periods of hours and longer to
obtain expressions for the wave amplitudes and turbulent heating rate as a
function of heliocentric distance. We also develop a second approximate model
that includes waves with periods of roughly one minute to one hour, which
undergo less reflection than the longer-period waves, and compare our models to
observations. Our models generalize the phenomenological model of Dmitruk et al
by accounting for the solar wind velocity, so that the turbulent heating rate
can be evaluated from the coronal base out past the Alfven critical point -
that is, throughout the region in which most of the heating and acceleration
occurs. The simple analytical expressions that we obtain can be used to
incorporate Alfven-wave reflection and turbulent heating into fluid models of
the solar wind.
