Cosmic-ray hydrodynamics: Alfvén-wave regulated transport of cosmic rays

Abstract

Star formation in galaxies appears to be self-regulated by energetic feedback processes. Among the most promising agents of feedback are cosmic rays (CRs), the relativistic ion population of interstellar and intergalactic plasmas. In these environments, energetic CRs are virtually collisionless and interact via collective phenomena mediated by kinetic-scale plasma waves and large-scale magnetic fields. The enormous separation of kinetic and global astrophysical scales requires a hydrodynamic description. Here, we develop a new macroscopic theory for CR transport in the self-confinement picture, which includes CR diffusion and streaming. The interaction between CRs and electromagnetic fields of Alfvénic turbulence provides the main source of CR scattering, and causes CRs to stream along the magnetic field with the Alfvén velocity if resonant waves are sufficiently energetic. However, numerical simulations struggle to capture this effect with current transport formalisms and adopt regularization schemes to ensure numerical stability. We extent the theory by deriving an equation for the CR momentum density along the mean magnetic field and include a transport equation for the Alfvén-wave energy. We account for energy exchange of CRs and Alfvén waves via the gyroresonant instability and include other wave damping mechanisms. Using numerical simulations we demonstrate that our new theory enables stable, self-regulated CR transport. The theory is coupled to magneto-hydrodynamics, conserves the total energy and momentum, and correctly recovers previous macroscopic CR transport formalisms in the steady-state flux limit. Because it is free of tunable parameters, it holds the promise to provide predictable simulations of CR feedback in galaxy formation.1 The Legendre polynomials are eigenfunctions of the pitch-angle Laplace operator ∂ t f | scatt = ∂ µ [ν(1 − µ 2 )/2 ∂ µ f ]. This operator describes pitch-angle diffusion and ν denotes the scattering frequency. Note that this simple Laplacian resembles the actual scattering operator as discussed in equation (51).

Cosmological simulations of galaxy formation

“…This leads to unphysical oscillations of the solution and small time steps especially near cosmic ray pressure maxima if not addressed in form of regularization techniques -such as replacing the sign function with the hyperbolic tangent function that ensures a smooth dependence of the streaming velocity on cosmic rays and gas properties 219,226,227 , albeit at the expense of a dependence of the solution on a numerical parameter. An elegant solution of this problem is to replace the equation for the cosmic ray energy by two equations for cosmic ray energy and flux that are coupled to the MHD system of equations 228,229 . This two-moment formulation can be derived from quasi-linear theory of cosmic ray transport and describes 15/34 cosmic ray streaming and diffusion self-consistently with a hyperbolic set of equations, which also contains the evolution equations for Alfvén waves that are self-generated by the streaming cosmic rays 229 .…”

Probing Cosmic-Ray Transport with Radio Synchrotron Harps in the Galactic Center

“…2. The streaming + diffusion model uses the more accurate description for CR transport of Thomas & Pfrommer (2019), which evolves the CR energy and momentum density. In addition, the energy contained in gyroresonant Alfvén waves is evolved and coupled to CRs using quasi-linear theory of CR transport.…”

“…In addition, the energy contained in gyroresonant Alfvén waves is evolved and coupled to CRs using quasi-linear theory of CR transport. We only consider non-linear Landau damping of Alfvén waves (see Thomas & Pfrommer 2019). The initial CR energy flux is chosen so that CRs stream with v a .…”

“…In comparison to the diffusion model, where the diffusion coefficient κ is constant, the diffusion coefficient in the streaming + diffusion model is calculated based on the local strength of Alfvén waves. We solve the equations of Thomas & Pfrommer (2019) using a finite volume method (Thomas et al in prep) in the moving mesh code AREPO (Springel 2010) for both models (in the diffusion model κ is constant). We use a one-dimensional grid with 4096 cells, a grid spacing of ∆x = 4 × 10 −3 pc, and outflowing boundary conditions.…”

Spectrally resolved cosmic ray hydrodynamics – I. Spectral scheme

“…A further simplification for numerical models is a grey approach, in which the distribution function times the kinetic energy per particle is integrated over momentum space and the resulting CR energy density is evolved in time and space. Recently, Jiang & Oh (2018) and Thomas & Pfrommer (2019) extended the traditional one-moment schemes in the scattering angle to two-moment descriptions that include a more self-consistent coupling of CRs to the plasma and captures streaming and diffusion relative to the gas rest frame.…”