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Register a new userMagnetohydrodynamic-Particle-in-Cell Simulations of the Cosmic-Ray Streaming Instability: Linear Growth and Quasi-linear Evolution
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The gyro-resonant cosmic-ray (CR) streaming instability is believed to play a crucial role in CR transport, leading to growth of Alfven waves at small scales that scatter CRs, and impacts the interaction of CRs with the ISM on large scales. However, extreme scale separation ($lambda ll rm pc$), low cosmic ray number density ($n_{rm CR}/n_{rm ISM} sim 10^{-9}$), and weak CR anisotropy ($sim v_A/c$) 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 $delta f$ method dramatically reduces Poisson noise and enables us to accurately capture wave growth over a broad spectrum, equally shared between left and right handed Alfven 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 $90^circ$, 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.
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We explore the physics of the gyro-resonant cosmic ray streaming instability (CRSI) including the effects of ion-neutral (IN) damping. This is the main damping mechanism in (partially-ionized) atomic and molecular gas, which are the primary components of the interstellar medium (ISM) by mass. Limitation of CRSI by IN damping is important in setting the amplitude of Alfven waves that scatter cosmic rays and control galactic-scale transport. Our study employs the MHD-PIC hybrid fluid-kinetic numerical technique to follow linear growth as well as post-linear and saturation phases. During the linear phase of the instability -- where simulations and analytical theory are in good agreement -- IN damping prevents wave growth at small and large wavelengths, with the unstable bandwidth lower for higher ion-neutral collision rate $ u_{rm in}$. Purely MHD effects during the post-linear phase extend the wave spectrum towards larger $k$. In the saturated state, the cosmic ray distribution evolves toward greater isotropy (lower streaming velocity) by scattering off of Alven waves excited by the instability. In the absence of low-$k$ waves, CRs with sufficiently high momentum are not isotropized. The maximum wave amplitude and rate of isotropization of the distribution function decreases at higher $ u_{rm in}$. When the IN damping rate approaches the maximum growth rate of CSRI, wave growth and isotropization is suppressed. Implications of our results for CR transport in partially ionized ISM phases are discussed.
The distribution of arrival directions of cosmic rays is remarkably isotropic, which is a consequence of their repeated scattering in magnetic fields. Yet, high-statistics observatories like IceCube and HAWC have revealed the presence of small-scale structures at levels of 1 part in 10,000 at hundreds of TeV, which are not expected in typical diffusion models of cosmic rays. We follow up on the suggestion that these small-scale anisotropies are a result of cosmic ray streaming in a particular realisation of the turbulent magnetic field within a few scattering lengths in our local Galactic neighbourhood. So far, this hypothesis has been investigated mostly numerically, by tracking test particles through turbulent magnetic fields. For the first time, we present an analytical computation that through a perturbative approach allows predicting the angular power spectrum of cosmic ray arrival directions for a given model of turbulence. We illustrate this method for a simple, isotropic turbulence model and we find remarkable agreement with the results of numerical studies.
Recent study suggests that the streaming instability, one of the leading mechanisms for driving the formation of planetesimals, may not be as efficient as previously thought. Under some disc conditions, the growth timescale of the instability can be longer than the disc lifetime when multiple dust species are considered. To further explore this finding, we use both linear analysis and direct numerical simulations with gas fluid and dust particles to mutually validate and study the unstable modes of the instability in more detail. We extend the previously studied parameter space by one order of magnitude in both the range of the dust-size distribution $[T_{s,min}, T_{s,max}]$ and the total solid-to-gas mass ratio $varepsilon$ and introduce a third dimension with the slope $q$ of the size distribution. We find that the fast-growth regime and the slow-growth regime are distinctly separated in the $varepsilon$-$T_{s,max}$ space, while this boundary is not appreciably sensitive to $q$ or $T_{s,min}$. With a wide range of dust sizes present in the disc (e.g. $T_{s,min}lesssim10^{-3}$), the growth rate in the slow-growth regime decreases as more dust species are considered. With a narrow range of dust sizes (e.g. $T_{s,max}/T_{s,min}=5$), on the other hand, the growth rate in most of the $varepsilon$-$T_{s,max}$ space is converged with increasing dust species, but the fast and the slow growth regimes remain clearly separated. Moreover, it is not necessary that the largest dust species dominate the growth of the unstable modes, and the smaller dust species can affect the growth rate in a complicated way. In any case, we find that the fast-growth regime is bounded by $varepsilongtrsim 1$ or $T_{s,max}gtrsim 1$, which may represent the favourable conditions for planetesimal formation.
Modelling of cosmic ray transport and interpretation of cosmic ray data ultimately rely on a solid understanding of the interactions of charged particles with turbulent magnetic fields. The paradigm over the last 50 years has been the so-called quasi-linear theory, despite some well-known issues. In the absence of a widely accepted extension of quasi-linear theory, wave-particle interactions must also be studied in numerical simulations where the equations of motion are directly solved in a realisation of the turbulent magnetic field. The applications of such test particle simulations of cosmic rays are manifold: testing transport theories, computing parameters like diffusion coefficients or making predictions for phenomena beyond standard diffusion theories, e.g. for cosmic ray small-scale anisotropies. In this review, we seek to give a low-level introduction to test particle simulations of cosmic rays, enabling readers to perform their own test particle simulations. We start with a review of quasi-linear theory, highlighting some of its issues and suggested extensions. Next, we summarise the state-of-the-art in test particle simulations and give concrete recipes for generating synthetic turbulence. We present a couple of examples for applications of such simulations and comment on an important conceptual detail in the backtracking of particles.
We use particle-in-magnetohydrodynamics-cells to model particle acceleration and magnetic field amplification in a high Mach, parallel shock in three dimensions and compare the result to 2-D models. This allows us to determine whether 2-D simulations can be relied upon to yield accurate results in terms of particle acceleration, magnetic field amplification and the growth rate of instabilities. Our simulations show that the behaviour of the gas and the evolution of the instabilities are qualitatively similar for both the 2-D and 3-D models, with only minor quantitative differences that relate primarily to the growth speed of the instabilities. The main difference between 2-D and 3-D models can be found in the spectral energy distributions (SEDs) of the non-thermal particles. The 2-D simulations prove to be more efficient, accelerating a larger fraction of the particles and achieving higher velocities. We conclude that, while 2-D models are sufficient to investigate the instabilities in the gas, their results have to be treated with some caution when predicting the expected SED of a given shock.
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