Revisiting the role of cosmic-ray driven Alfvén waves in pre-existing magnetohydrodynamic turbulence

I. Turbulent damping rates and feedback on background fluctuations

Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France

Received: 5 February 2024
Accepted: 11 May 2024

Abstract

Context. Alfvén waves (AWs) excited by the cosmic-ray (CR) streaming instability (CRSI) are a fundamental ingredient for CR confinement. The effectiveness of such self-confinement relies on a balance between the CRSI growth rate and the damping mechanisms acting on quasi-parallel AWs excited by CRs. One relevant mechanism is 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.

Aims. The turbulent damping of an AW packet in pre-existing magnetohydrodynamic (MHD) turbulence is re-examined, revised, and extended to include the most recent theories of MHD turbulence that account for dynamic alignment and reconnection-mediated regimes. The case in which the role of feedback of CR-driven AWs on pre-existing turbulence is important is also discussed.

Methods. The Elsässer formalism is employed. Particular attention is given to the role of a nonlinearity parameter χ^w that estimates the strength of the nonlinear interaction between CR-driven AW packets and the background fluctuations. We point out the difference between χ^w and the parameter χ^z that instead describes the intrinsic strength of nonlinear interactions between pre-existing fluctuations. Turbulent damping rates of quasi-parallel AW packets and cosmic-ray feedback (CRF) are derived within this formalism.

Results. When the strength of the nonlinear interaction is properly taken into account, we find that (i) the turbulent damping rate of quasi-parallel AWs in sub-Alfvénic turbulence depends on the background-fluctuation amplitude to the third power, and hence is strongly suppressed; (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; and (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 CRF is derived and a simple phenomenological model of CR-modified scaling of background fluctuations is provided.