  Alfv\'{e}n waves are believed to play an important role in the heating and
acceleration of the fast solar wind emanating from coronal holes. Nonlinear
interactions between the dominant ${\bf z}_{+}$ waves and minority ${\bf
z}_{-}$ waves have the potential to transfer wave energy either to smaller
perpendicular scales ("direct cascade") or to larger scales ("inverse
cascade"). In this paper we use reduced magnetohydrodynamic (RMHD) simulations
to investigate how the cascade rates $\epsilon_{\pm}$ depend on perpendicular
wavenumber and radial distance from Sun center. For models with a smooth
background atmosphere we find that an inverse cascade ($\epsilon_{+} < 0$)
occurs for the dominant waves at radii between 1.4 and 2.5 $R_\odot$ and
dimensionless wavenumbers in the inertial range ($15 < a_\perp < 44$), and a
direct cascade ($\epsilon_{+} > 0$) occurs elsewhere. For a model with density
fluctuations there are multiple regions with inverse cascade. In both cases the
cascade rate $\epsilon_{+}$ varies significantly with perpendicular wavenumber,
indicating that the cacsade is a highly non-local process. As a result of the
inverse cascades, the enery dissipation rates are much lower than expected from
a phenomenological model, and are insufficient to maintain the temperature of
the background atmosphere. We conclude that RMHD models are unable to reproduce
the observed properties of the fast solar wind.
