Cosmic-ray Streaming Instability Research Articles

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

Alfv\'en 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. 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. The Els\"asser formalism is employed. Particular attention is given to the role of a nonlinearity parameter $ that estimates the strength of the nonlinear interaction between CR-driven AW packets and the background fluctuations. We point out the difference between $ and the parameter $ 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. 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-Alfv\'enic 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.

Transonic plasma winds with cosmic-rays and waves

Transonic plasma winds are studied under the influence of gravitational potential well in the presence of cosmic rays and self excited Alfvén waves. We present interesting features of transonic winds originating from the gravitational potential well for a three-fluid system. The model is comprising of thermal plasma, cosmic rays and Alfvén waves. The analysis is carried out on the hydro-dynamical basis with a particular emphasis on cosmic-ray streaming instability and damping mechanism. Boundary conditions on the base of the gravitational potential well will help to explore steady-state transonic wind solutions. A critical analysis on various mass outflow rate is also presented which is applicable to know the behaviour of normal and star-burst galaxies. The dependence of the critical or sonic point is studied for different sets of physical parameters. For the graphical interpretation of the numerical results different contour plots presented to analyze impacts of different mass flow rates. We find that transonic solutions exist for a wide range of parameters. We examine the case very close to the base of the gravitational potential well and the effect of the cosmic ray diffusion is neglected in current study.

Outflows in the presence of cosmic rays and waves with cooling

Context. Plasma outflow from a gravitational potential well with cosmic rays and self-excited Alfvén waves with cooling and wave damping is studied in the hydrodynamics regime. Aims. We study outflows in the presence of cosmic ray and Alfvén waves including the effect of cooling and wave damping. We seek physically allowable steady-state subsonic-supersonic transonic solutions. Methods. We adopted a multi-fluid hydrodynamical model for the cosmic ray plasma system. Thermal plasma, cosmic rays, and self-excited Alfvén waves are treated as fluids. Interactions such as cosmic-ray streaming instability, cooling, and wave damping were fully taken into account. We considered one-dimensional geometry and explored steady-state solutions. The model is reduced to a set of ordinary differential equations, which we solved for subsonic-supersonic transonic solutions with given boundary conditions at the base of the gravitational potential well. Results. We find that physically allowable subsonic-supersonic transonic solutions exist for a wide range of parameters. We studied the three-fluid system (considering only forward-propagating Alfvén waves) in detail. We examined the cases with and without cosmic ray diffusion separately. Comparisons of solutions with and without cooling and with and without wave damping for the same set of boundary conditions (on density, pressures of thermal gas, cosmic rays and waves) are presented. We also present the interesting case of a four-fluid system (both forward- and backward-propagating Alfvén waves are included), highlighting the intriguing relation between different components.

Influence of Ion–Neutral Damping on the Cosmic-Ray Streaming Instability: Magnetohydrodynamic Particle-in-cell Simulations

Abstract We explore the physics of the gyroresonant 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 Alfvén waves that scatter cosmic rays (CRs) and control galactic-scale transport. Our study employs the magnetohydrodynamic (MHD)–particle-in-cell 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 IN collision rates ν in. Purely MHD effects during the post-linear phase extend the wave spectrum toward larger k. In the saturated state, the CR distribution evolves toward greater isotropy (lower streaming velocity) by scattering off of Alfvén 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 decrease at higher ν in. When the IN damping rate approaches the maximum growth rate of CRSI, wave growth and isotropization are suppressed. Implications of our results for CR transport in partially ionized ISM phases are discussed.

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 the growth of Alfvén waves at small scales that scatter CRs, and impacts the interaction of CRs with the interstellar medium (ISM) on large scales. However, extreme scale separation (λ ≪ pc), low CR 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 method, which has unique advantages to alleviate these issues, we conduct 1D 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 the pitch angles of most CRs to efficiently cross 90° and can be captured in simulations with relatively high wave amplitude and/or spatial resolution. We attribute this crossing to nonlinear wave–particle interaction (rather than mirror reflection) by investigating individual CR trajectories. We anticipate that our methodology will open up opportunities for future investigations that incorporate additional physics.

DAMPING OF ALFVÉN WAVES BY TURBULENCE AND ITS CONSEQUENCES: FROM COSMIC-RAY STREAMING TO LAUNCHING WINDS

ABSTRACT This paper considers turbulent damping of Alfvén waves in magnetized plasmas. We identify two cases of damping, one related to damping of cosmic-ray streaming instability, the other related to damping of Alfvén waves emitted by a macroscopic wave source, e.g., a stellar atmosphere. The physical difference between the two cases is that in the former case the generated waves are emitted with respect to the local direction of the magnetic field, and in the latter, waves are emitted with respect to the mean field. The scaling of damping is different in the two cases. We explore effects of turbulence in the regimes from sub-Alfvénic to super-Alfvénic to obtain analytical expressions for the damping rates and define the ranges of applicability of these expressions. In describing the damping of the streaming instability, we find that for sub-Alfvénic turbulence, the range of cosmic-ray energies influenced by weak turbulence is unproportionally large compared to the range of scales where weak turbulence is present. On the contrary, the range of cosmic-ray energies affected by strong Alfvénic turbulence is rather limited. A number of astrophysical applications of the process ranging from launching of stellar and galactic winds to propagation of cosmic rays in galaxies and clusters of galaxies is considered. In particular, we discuss how to reconcile the process of turbulent damping with the observed isotropy of the Milky Way cosmic rays.

Interpretation of Voyager 1 data on low energy cosmic rays in galactic wind model

The local interstellar energy spectra of galactic cosmic rays down to a few MeV/nucleon were directly measured in the experiment on the board of the Voyager 1 spacecraft. We suggest interpretation of these data based on our models of cosmic ray acceleration in supernova remnants and the diffusion in galactic wind where diffusion coefficient is determined by the cosmic ray streaming instability. The dependence of wind velocity on distance above the Galactic disk is determined.

INFLUENCE OF THE BACKREACTION OF STREAMING COSMIC RAYS ON MAGNETIC FIELD GENERATION AND THERMAL INSTABILITY

Using a multi-fluid approach, we investigate streaming and thermal instabilities of the electron-ion plasma with homogeneous cold cosmic rays propagating perpendicular to the background magnetic field. Perturbations are considered to be also across the magnetic field. The back-reaction of cosmic rays resulting in strong streaming instabilities is taken into account. It is shown that for sufficiently short wavelength perturbations, the growth rates can exceed the growth rate of cosmic-ray streaming instability along the magnetic field found by Nekrasov & Shadmehri (2012), which is in its turn considerably larger than the growth rate of the Bell instability (2004). The thermal instability is shown not to be subject to the action of cosmic rays in the model under consideration. The dispersion relation for the thermal instability has been derived which includes sound velocities of plasma and cosmic rays, Alfv\'{e}n and cosmic-ray streaming velocities. The relation between these parameters determines the kind of thermal instability ranging from the Parker (1953) to the Field (1965) instabilities. The results obtained can be useful for a more detailed investigation of electron-ion astrophysical objects such as supernova remnant shocks, galaxy clusters and others including the dynamics of streaming cosmic rays.

THE ORIGIN OF RADIALLY ALIGNED MAGNETIC FIELDS IN YOUNG SUPERNOVA REMNANTS

It has been suggested by radio observations of polarized synchrotron emissions that downstream magnetic field in some young supernova remnants are oriented radially. We study magnetic field distribution of turbulent supernova remnant driven by the Richtmyer-Meshkov instability -- in other words, the effect of rippled shock -- by using three-dimensional magnetohydrodynamics simulations. We find that the induced turbulence has radially biased anisotropic velocity dispersion that leads to a selective amplification of the radial component of the magnetic field. The Richtmyer-Meshkov instability is induced by the interaction between the shock and upstream density fluctuations. Future high-resolution polarization observations can distinguish the following candidates responsible for the upstream density fluctuations: (i) inhomogeneity caused by the cascade of large-scale turbulence in the ISM so-called the big-power-law-in-the-sky, (ii) structures generated by the Drury instability in the cosmic-ray modified shock, and (iii) fluctuations induced by the non-linear feedback of the cosmic-ray streaming instability.

Radio emission and nonlinear diffusive shock acceleration of cosmic rays in the supernova SN 1993J

The extensive observations of the supernova SN 1993J at radio wavelengths make this object a unique target for the study of particle acceleration in a supernova shock. To describe the radio synchrotron emission we use a model that couples a semianalytic description of nonlinear diffusive shock acceleration with self-similar solutions for the hydrodynamics of the supernova expansion. The synchrotron emission, which is assumed to be produced by relativistic electrons propagating in the postshock plasma, is worked out from radiative transfer calculations that include the process of synchrotron self-absorption. The model is applied to explain the morphology of the radio emission deduced from high-resolution VLBI imaging observations and the measured time evolution of the total flux density at six frequencies. Both the light curves and the morphology of the radio emission indicate that the magnetic field was strongly amplified in the blast wave region shortly after the explosion, possibly via the nonresonant regime of the cosmic-ray streaming instability operating in the shock precursor. The turbulent magnetic field was not damped behind the shock but carried along by the plasma flow in the downstream region. Cosmic-ray protons were efficiently produced by diffusive shock acceleration at the blast wave. We find that during the first ~8.5 years after the explosion, about 19% of the total energy processed by the forward shock was converted to cosmic-ray energy. However, the shock remained weakly modified by the cosmic-ray pressure. The high magnetic field amplification implies that protons were rapidly accelerated to energies well above 1 PeV. The results obtained for this supernova support the scenario that massive stars exploding into their former stellar wind are a major source of high-energy Galactic cosmic rays.

ENVIRONMENTAL LIMITS ON THE NONRESONANT COSMIC-RAY CURRENT-DRIVEN INSTABILITY

We investigate the so-called nonresonant cosmic-ray streaming instability, first discussed by Bell (2004). The extent to which thermal damping and ion-neutral collisions reduce the growth of this instability is calculated. Limits on the growth of the nonresonant mode in SN1006 and RX J1713.7-3946 are presented.

On the spectrum of high-energy cosmic rays produced by supernova remnants in the presence of strong cosmic-ray streaming instability and wave dissipation

The cosmic-ray streaming instability creates strong magnetohydrodynamic turbulence in the precursor of a SN shock. The level of turbulence determines the maximum energy of cosmic-ray particles accelerated by the diffusive shock accelera- tion mechanism. In this paper we present the continuation of previous work (Ptuskin & Zirakashvili 2003). We assume that Kolmogorov type nonlinear wave interactions together with ion-neutral collisions restrict the amplitude of the random mag- netic field. As a result, the maximum energy of the accelerated particles strongly depends on the age of a SNR. The average spectrum of cosmic rays injected in the interstellar medium in the course of the adiabatic SNR evolution (the Sedov stage) is approximately Q(p)p 2 ∝ p −2 at energies larger than 10−30 GeV/nucleon and with a maximum particle energy that is close to the position of the knee in the cosmic-ray spectrum observed at ∼4 × 10 15 eV. At an earlier stage of SNR evolution - the ejecta-dominated stage described by the Chevalier-Nadyozhin solution, the particles are accelerated to higher energies and have a rather steep power-law distribution. These results suggest that the knee may mark the transition from the ejecta-dominated to the adiabatic evolution of SNR shocks which accelerate cosmic rays.

Limits on diffusive shock acceleration in supernova remnants in the presence of cosmic-ray streaming instability and wave dissipation

The instability in the cosmic-ray precursor of a supernova shock moving in interstellar medium is studied. The level of magnetohydrodynamic turbulence in this region determines the maximum energy of particles accelerated by the diffusive shock acceleration mechanism. The high efficiency of cosmic ray acceleration is accepted, and the consideration is not limited by the case of weak turbulence. It is assumed that Kolmogorov type nonlinear wave interactions together with the ion-neutral collisions restrict the amplitude of random magnetic field. As a result, the maximum energy of accelerated particles strongly depends on the age of supernova remnant. The maximum energy can be as high as $\sim10^{17}Z$ eV in young supernova remnants and falls down to about $\sim10^{10}Z$ eV at the end of Sedov stage. Thus the standard estimate of maximum particle energy based on the Bohm limit calculated for the interstellar magnetic field strength is not justified in this case. This finding may explain why the supernova remnants with the age more than a few thousand years are not prominent sources of very high energy gamma-rays.

Hydrodynamic approach to cosmic ray acceleration

To study the structure and dynamics of a cosmic-ray-plasma system, hydrodynamic approach is a fairly good approximation. In this approach, there are three basic energy transfer mechanisms: work done by the plasma flow against pressure gradients, cosmic ray streaming instability and stochastic acceleration. The interplay between these mechanisms gives a range of structures. We present some results of different version of the hydrodynamic approach, e.g., flow structure, injection, instability, acceleration with and without shocks.

Continuous solutions of the hydrodynamic approach to cosmic-ray propagation

A hydrodynamic approach is employed to study a cosmic-ray–plasma system, which comprises thermal plasma, cosmic rays and two oppositely propagating Alfvén waves. The hydrodynamic approach is a good approximation in dealing with the structure or dynamics of the system. In this paper, we concentrate on the steady-state structures of the system, in particular, structures with continuous (or smooth) profiles. Three mechanisms are responsible for the energy exchange between different components. They are work done by plasma flow via pressure gradients, cosmic-ray streaming instability and stochastic acceleration. The interplay between these mechanisms generates several morphologically different structures. They may be divided into two categories: one looks like the test-particle picture and the other looks like a modified shock. Very often the profiles are non-monotonic, which is in sharp contrast to systems with only thermal plasma and cosmic rays, whose flow velocity (and cosmic-ray pressure) profiles are always monotonically decreasing (and increasing).

Burgers' equation and cosmic ray shocks

The time evolution of weak shocks in the classical theory of shock waves may be described by Burgers' equation, which is a small-amplitude, long-wavelength, but nonlinear wave equation. The present paper derives Burgers' equation for three different hydrodynamical models of cosmic ray shocks. The main development of the paper concerns the hydrodynamical model in which Alfvénic effects are neglected. For this model it is shown that the steady-state solution of Burgers' equation is equivalent to the weak, but smooth, transition solutions of the shock structure equation obtained previously. It is shown that Burgers' equation may be regarded as a wave energy equation, in which the interaction of the wave with the background medium is taken into account. Burgers' equation is also derived for Alfvénic models in which the Alfvén waves that scatter the cosmic rays are generated by the cosmic ray streaming instability, and propagate down the cosmic ray pressure gradient, at the Alfvén velocity relative to the background fluid.

Magnetohydrodynamic plasma instability driven by Alfvén waves excited by cosmic rays

Hydrodynamical equations describing the mutual interaction of cosmic rays, thermal plasma, magnetic field and Alfvén waves scattering the cosmic rays used in cosmic ray shock acceleration theory (e.g. McKenzie & Völk 1982; Drury 1983; Webb 1983) are analysed for long-wavelength linear compressive instabilities. The Alfvén wave field may contain a pre-existing component as well as a component excited by the cosmic ray streaming instability. In the case of no Alfvén wave damping, adiabatic wave growth and Alfvén wave generation by the cosmic ray streaming instability, it is found that the backward propagating slow magneto-acoustic mode is driven convectively unstable by the pressure of the self-excited Alfvén waves, provided the thermal plasmaβis sufficiently large. The equations are also analysed for the case where the Alfvén wave growth is balanced by some nonlinear damping mechanisms. In the latter case both the forward and backward propagating slow magneto-acoustic modes may be driven unstable if the plasmaβis sufficiently small. The conditions under which the instabilities occur are delineated, and sample calculations of growth rates given. Possible applications of the instabilities to astrophysical situations are briefly discussed.