Revisiting the role of cosmic-ray driven Alfv\'en waves in pre-existing magnetohydrodynamic turbulence. I. Turbulent damping rates and feedback on background fluctuations

Abstract

Alfv\'en waves (AWs) excited by the cosmic-ray (CR) streaming instability (CRSI) are a fundamental ingredient for CR confinement. The effectiveness of self-confinement relies on a balance between CRSI growth rate and damping mechanisms acting on quasi-parallel AWs excited by CRs. One relevant mechanism is the so-called turbulent damping, in which an AW packet injected in pre-existing turbulence undergoes a cascade process due to its nonlinear interaction with fluctuations of the background. The turbulent damping of an AW packet in pre-existing magnetohydrodynamic turbulence is re-examined, revised, and extended to include most-recent theories of MHD turbulence that account for dynamic alignment and reconnection-mediated regime. The case in which the role of feedback of CR-driven AWs on pre-existing turbulence is important will also be discussed. Particular attention is given to the nonlinearity parameter $\chi^w$ that estimates the strength of nonlinear interaction between CR-driven AWs and background fluctuations. We point out the difference between $\chi^w$ and $\chi^z$ that instead describes the strength of nonlinear interactions between pre-existing fluctuations. When $\chi^w$ is properly taken into account, one finds that (i) the turbulent damping rate of quasi-parallel AWs in anisotropic turbulence depends on the background-fluctuations' amplitude to the third power, hence is strongly suppressed, and (ii) the dependence on the AW's wavelength (and thus on the CR gyro-radius from which it is excited) is different from what has been previously obtained. Finally, (iii) when dynamic alignment of cascading fluctuations and the possibility of a reconnection-mediated range is included in the picture, the turbulent damping rate exhibits novel regimes and breaks. Finally, a criterion for CR-feedback is derived and simple phenomenological models of CR-modified turbulent scaling are provided.