Cosmic-ray modified flows with effective wave damping

Non-linear Landau damping (NLLWD) is a fundamental mechanism that governs the confinement and transport of cosmic rays, influencing the thermal balance of the ISM and the dynamics of cosmic-ray-driven winds. This process involves two counter propagating self generated Alfvén waves which efficiently transfer energy to the thermal plasma. For this purpose, we adopt a hydrodynamic approach to study the propagation of cosmic rays in thermal plasma. A four-fluid model (containing cosmic rays, thermal plasma, forward and backward propagating Alfvén waves) accurately represents steady-state formation of continuous type and shock type profiles. We numerically simulate the coupled integral equations using the Runge-Kutta 4 (RK4) method. Our simulation elucidates the impact of cosmic-ray propagation by Alfvén waves having diffusion coefficients and energy transfer in thermal plasma. We represent an equal contribution of each species to the NLLWD and focus on the damping rate dependency on the magnetic fields and their astrophysical effects in ISM.