Numerical Study of Self-Modulation with Plasma Density Gradients

We study through numerical simulations the effect of a linear plasma density gradient on the self-modulation of a long proton bunch in plasma. % Numerical simulations give access to the details of the self-modulation process and wakefields along the bunch and along the plasma. % We obtain characteristics such as the bunch modulation frequency and wakefields frequency and phase evolution, as well as the charge of the modulated bunch. % These could then be compared to experimental results that can only be acquired at the end of the 10\,m plasma\footnote{F. Braunmueller and T. Nechaeva et al. (AWAKE Collaboration), to be submitted}. % In addition, they show the effect of the density gradient on the wakefields amplitude, a quantity so far not measured in experiments. % We use parameters that are similar to those of the AWAKE experiment\footnote{P. Muggli et al. (AWAKE Collaboration), Plasma Physics and Controlled Fusion, 60(1) 014046 (2017)}. % Detailed results will be presented. %