Magnetohydrodynamic Particle-in-cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution

Abstract

The gyro-resonant cosmic-ray (CR) streaming instability is believed to play a crucial role in CR transport, leading to growth of Alfvén waves at small scales that scatter CRs, and impacts the interaction of CRs with the ISM on large scales. However, extreme scale separation (λ pc), low cosmic ray number density (n CR /n ISM ∼ 10 −9 ), and weak CR anisotropy (∼ v A /c) pose strong challenges for proper numerical studies of this instability on a microphysical level. Employing the recently developed magnetohydrodynamic-particle-in-cell (MHD-PIC) method, which has unique advantages to alleviate these issues, we conduct one-dimensional simulations that quantitatively demonstrate the growth and saturation of the instability in the parameter regime consistent with realistic CR streaming in the large-scale ISM. Our implementation of the δf method dramatically reduces Poisson noise and enables us to accurately capture wave growth over a broad spectrum, equally shared between left and right handed Alfvén modes. We are also able to accurately follow the quasi-linear diffusion of CRs subsequent to wave growth, which is achieved by employing phase randomization across periodic boundaries. Full isotropization of the CRs in the wave frame requires pitch angles of most CRs to efficiently cross 90 • , and can be captured in simulations with relatively high wave amplitude and/or high spatial resolution. We attribute this crossing to non-linear wave-particle interaction (rather than mirror reflection) by investigating individual CR trajectories. We anticipate our methodology will open up opportunities for future investigations that incorporate additional physics.

Testing physical models for cosmic ray transport coefficients on galactic scales: self-confinement and extrinsic turbulence at ∼GeV energies

“…Riquelme & Spitkovsky 2009;Ohira et al 2009) and the amplitudes predicted here are generally modest (for diffusivityκ29, the gyro-resonant |δB[rL]|/|B| ∼ 3 × 10 −4 (γL/BµGκ29) 1/2 ); however, two recent works studying the saturation of the gyro-resonant instability using the PIC method suggest possible ways that the effective ν might be lower than the QLT prediction. First, Bai et al (2019) find that the time required for the CR distribution to become fully isotropic in the Alfvén-wave frame is much longer than predicted by the QLT estimate. This behaviour arises because of particularly inefficient scattering across the zero pitch angle (µ = 0) barrier, which is both slow and requires scatterers of very short wavelength compared to rL (Völk 1973).…”

“…(2) Analytic and numerical work explicitly following transport and scattering of CRs on "micro-scales" (e.g. Bai et al 2015Bai et al , 2019Lazarian 2016;Holcomb & Spitkovsky 2019;van Marle et al 2019), coupled to improved intermediate-scale "effective fluid" theories (e.g. Zank 2014;Zweibel 2017;Thomas & Pfrommer 2018), has begun to yield more detailed prescriptions for the "effective" transport coefficients of CRs as a function of local plasma properties (appropriate on scales much larger than the CR gyro-radius, but much smaller than the scales of e.g.…”

“…In the WIM/CGM, we have β 1, ecr/eB 1, vst/vA ∼ 300 − 1000 1, regimes where the instability is not wellstudied and could potentially be strongly modified. 19 The results of Bai et al (2019) may again be of interest, if smaller-scale modes excited by low-µ and lower-energy particles are required to fully saturate the gyro-resonant instability. Since such particles are much less numerous, implying the growth rate of the resonant modes is lower, the damping-growth balance that is usually assumed to saturate the instability and determine κ (see §3.3) might occur at significantly lower Alfvén-wave amplitudes than usually assumed.…”

Impact of Low-Energy Cosmic Rays on Star Formation

“…If CR-produced ionisation exceeds thermal ionisation (inner < 1 au) and/or ionisation produced by radionuclides (disc midplane), CRs will regulate the cycle of MRI activation and accretion (Offner et al, 2019). However, the coupling between the mean magnetic field and the CRs is also mediated by the degree of turbulence, which in turn is dictated by the MRI activity (Rodgers- Lee et al, 2017;Bai et al, 2019) -circumscribing a thoroughly non-linear problem.…”

dHybridR: A Hybrid Particle-in-cell Code Including Relativistic Ion Dynamics

“…Recent works have tackled the study of the nonresonant instability with PIC and hybrid simulations (e.g., Ohira et al 2009;Riquelme & Spitkovsky 2009;Gargaté et al 2010) and of the resonant instability with PIC and PIC-MHD simulations (e.g. Bai et al 2019;Holcomb & Spitkovsky 2019;); these studies have generally found results consistent with theory for the fasting growing mode and corresponding growth rate for the linear phase. Nevertheless, the saturation of the CR streaming instability is a complex and non-linear physical phenomenon that is not yet completely understood.…”