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

TL;DR

This paper revisits the turbulent damping of Alfvén waves driven by cosmic rays in magnetohydrodynamic turbulence, incorporating recent theories of turbulence and feedback mechanisms to refine understanding of cosmic-ray confinement.

Contribution

It provides a revised analysis of turbulent damping rates of Alfvén waves, including effects of dynamic alignment, reconnection, and feedback on background turbulence, advancing the theoretical framework.

Findings

Damping rate depends on background fluctuations' amplitude cubed.

Dependence on wave wavelength differs from previous models.

Feedback mechanisms can modify turbulence scaling and damping regimes.

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.