Spherically Polarized Alfv\'en Waves and the Gosling Boost

Alfv\'en waves are thought to play critical roles in solar wind acceleration and plasma heating in the solar corona and inner heliosphere. Parker Solar Probe (PSP) has highlighted the role of large amplitude Spherically Polarized Alfv\'en Waves (SPAWs), where the locally constant magnetic field magnitude $|\mathbf{B}|$ together with outward propagation explains the observed one sided radial velocity enhancement - the Gosling boost. Starting from the MHD equations, we derive the modified wave pressure and Poynting flux under the SPAW condition, and demonstrate both are governed solely by the transverse magnetic fluctuations. Using PSP data from Encounters 6--25, we define an unperturbed velocity baseline from the lower 10th-percentile running average and statistically characterize the radial evolution of Alfv\'enic fluctuations. The background solar wind velocity shows clear radial acceleration, while the velocity perturbation amplitude $\delta v$ decreases with heliocentric distance. This decay is anisotropic between the radial and perpendicular directions, which is a direct consequence of the growing magnetic deflection angle related to the spherical polarization. Our results demonstrate that radial velocity enhancements in the young solar wind arise naturally from SPAWs rather than from localized velocity jets, and provide direct observational evidence for the anisotropic radial evolution of SPAWs in the inner heliosphere.