Time evolution from the simulations. Top panel: absolute value of the coefficient of the Fourier velocity (macroscopic u_x) inferred from the hybrid simulations corresponding to the excited wavelength in the system (m = 4 or equivalently m′ = 2). The dashed line represents the expected damping from the theoretical calculations based on the solution of the dispersion relation given by Eq. (19). In this plot β_e = 6 × 10⁻² and β_p = 10⁻². Bottom panel: same as the top panel, but in this case β_p = 2 × 10⁻², and therefore the attenuation is stronger.