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

Direct and Inverse Cascades in the Acceleration Region of the Fast Solar Wind