### Heating and Acceleration of the Fast Solar Wind by Alfv\'{e}n Wave Turbulence

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We present numerical simulations of reduced magnetohydrodynamic (RMHD) turbulence in a magnetic flux tube at the center of a polar coronal hole. The model for the background atmosphere is a solution of the momentum equation, and includes the effects of wave pressure on the solar wind outflow. Alfv\'{e}n waves are launched at the coronal base, and reflect at various heights due to variations in Alfv\'{e}n speed and outflow velocity. The turbulence is driven by nonlinear interactions between the counter-propagating Alfv\'{e}n waves. Results are presented for two models of the background atmosphere. In the first model the plasma density and Alfv\'{e}n speed vary smoothly with height, resulting in minimal wave reflections and low energy dissipation rates. We find that the dissipation rate is insufficient to maintain the temperature of the background atmosphere. The standard phenomenological formula for the dissipation rate significantly overestimates the rate derived from our RMHD simulations, and a revised formula is proposed. In the second model we introduce additional density variations along the flux tube with a correlation length of 0.04 $R_\odot$ and with relative amplitude of $10 \%$. These density variations simulate the effects of compressive MHD waves on the Alfv\'{e}n waves. We find that such variations significantly enhance the wave reflection and thereby the turbulent dissipation rates, producing enough heat to maintain the background atmosphere. We conclude that interactions between Alfv\'{e}n- and compressive waves may play an important role in the turbulent heating of the fast solar wind.