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

Abstract: 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.