Turbulence in the sub-Alfv\'enic solar wind driven by reflection of low-frequency Alfv\'en waves

We study the formation and evolution of a turbulent spectrum of Alfv\'en waves driven by reflection off the solar wind density gradients, starting from the coronal base up to 17 solar radii, well beyond the Alfv\'enic critical point. The background solar wind is assigned and 2D shell models are used to describe nonlinear interactions. We find that the turbulent spectra are influenced by the nature of reflected waves. Close to the base, these give rise to a flatter and steeper spectrum for the outgoing and reflected waves respectively. At higher heliocentric distance both spectra evolve toward an asymptotic Kolmogorov spectrum. The turbulent dissipation is found to account for at least half of the heating required to sustain the background imposed solar wind and its shape is found to be determined by the reflection-determined turbulent heating below 1.5 solar radii. Therefore reflection and reflection-driven turbulence are shown to play a key role in the accelerationof the fast solar wind and origin of the turbulent spectrum found at 0.3 AU in the heliosphere.