Nonlinear reflection process of linearly-polarized, broadband Alfv\'en waves in the fast solar wind

Using one-dimensional numerical simulations, we study the elementary process of Alfv\'{e}n wave reflection in a uniform medium, including nonlinear effects. In the linear regime, Alfv\'{e}n wave reflection is triggered only by the inhomogeneity of the medium, whereas in the nonlinear regime, it can occur via nonlinear wave-wave interactions. Such nonlinear reflection (backscattering) is typified by decay instability. In most studies of decay instabilities, the initial condition has been a circularly polarized Alfv\'{e}n wave. In this study we consider a linearly polarized Alfv\'en wave, which drives density fluctuations by its magnetic pressure force. For generality, we also assume a broadband wave with a red-noise spectrum. In the data analysis, we decompose the fluctuations into characteristic variables using local eigenvectors, thus revealing the behaviors of the individual modes. Different from circular-polarization case, we find that the wave steepening produces a new energy channel from the parent Alfv\'en wave to the backscattered one. Such nonlinear reflection explains the observed increasing energy ratio of the sunward to the anti-sunward Alfv\'{e}nic fluctuations in the solar wind with distance against the dynamical alignment effect.