Linear and non-linear evolution of the gyroresonance instability in Cosmic Rays

Transcription

1 Linear and non-linear evolution of the gyroresonance instability in Cosmic Rays DESY Summer Student Programme, 2016 Olga Lebiga Taras Shevchenko National University of Kyiv, Ukraine Supervisors Reinaldo Santos de Lima Prof. Huirong Yan September 5, 2016 Abstract In this work we study the evolution of the gyroresonance instability in astrophysical plasmas. This instability is driven by the presence of Cosmic Rays with anisotropic velocities distribution. Using 1d MHD+Particle-In-Cell numerical simulations we triggered this instability for different anisotropy degrees and investigate its main features. We followed the evolution of the magnetic field fluctuations and the CRs velocity distribution in linear and non-linear regimes of the instability.

2 CONTENTS 2 Contents 1 Introduction 1 2 Theoretical background 2 3 Methods 3 4 Results 5 5 Summary and Conclusions 12

3 1 1 INTRODUCTION 1 Introduction Cosmic rays (CRs) are relativistic particles present in most of diffusive astrophysical plasmas, such as the interstellar medium and the intracluster medium of galaxies. These high-energy particles are constituted by electrons, positrons, protons and heavier ions. CRs can be detected for example by the Cherenkov emission, which they produce when interacting with the Earth atmosphere. Their existence in diffuse media can be inferred by analysing the high-energy radiation produced by the CRs in their interaction with the medium: the synchrotron emission produced by cosmic-ray electrons in the presence of magnetic fields; the inverse Compton radiation produced by the scattering of background photons by the CRs; the gamma rays produced in the hadronic interactions of cosmic-ray protons with target protons, etc. CRs can originate close to their emission regions or they can be accelerated far and transported (by diffusion) to their interaction sites. Low-energy CRs ( 10 9 GeV) are of Galactic origin and they are mainly accelerated in supernova shockwaves through the so-called first-order Fermi mechanism. The CRs of highest energies are of extragalactic origin and their sources are a subject of important debate; potential candidates, among others, are the jets of Active Galactic Nuclei (AGNs). Also their acceleration mechanism is not completely determined; proposed mechanisms are shock acceleration, acceleration by magnetic reconnection, etc. A plasma is a fully (or partially) ionized gas which presents collective behaviour, such as electromagnetic waves. Astrophysical plasmas can be decomposed into two components: Thermal plasma Constituted by the particles (ions and electrons) with some degree of local thermodynamic equilibrium, which can be characterized by temperatures. We consider only plasmas with non-relativistic bulk velocities and temperatures, with ions and electrons each in thermodynamic equilibrium. This equilibrium is guaranted by the fast particle-particle collisions. The low frequency phenomena (frequencies much smaller than the particles Larmor frequency, and scales much larger than the Larmor radius) of the plasma can be described by the magnetohydrodynamic (MHD) approximation. Cosmic rays Composed by relativistic particles with velocities much higher than those of the thermal particles. As CRs have higher velocities than the thermal particles, their gyration radius around the magnetic field, the Larmor radius, are much larger. Kinetic effects like the resonances of the gyration of CRs with the electromagnetic waves of the plasma can not be described by a fluid approximation for the CRs. Because the rate of particle-particle collisions involving CRs is low, they can be described as "collisionless". Therefore, CRs constituted a non-thermal energy component of the plasma. They can be energetically relevant for the macroscopic dynamics, for example by providing an additional pressure component to the fluid. They can also modify the collective behaviour of the plasma, by interacting with the waves and absorbing their energy. The CRs themselves can be deflected and accelerated by the plasma waves, i.e. the electromagnetic fluctuations. Astrophysical plasmas are generally magnetized and turbulent. The large-scale compression/expansion of the gas by the compressible turbulent motions modify locally the magnetic field intensities. Due to the adiabatic conservation of the magnetic moment of charged particles p 2 /2B, where p the perpendicular momentum with respect to the local magnetic field of intensity B, the CR velocity distribution can become anisotropic, i.e. the energy in the perpendicular motions can become greater or smaller than the energy in the parallel ones. An anisotropic velocity distribution of CRs may produce unstable electromagnetic

4 2 THEORETICAL BACKGROUND 2 waves in the plasma, with wavelengths of the order of the cosmic-rays Larmor radius; this is the so-called Gyroresonance Instability ( [1], [2]). These electromagnetic fluctuations in turn scatter the resonant CRs, randomizing their velocity directions and then relaxing the velocity anisotropies. These wave-particle collisions reduce the mean-free-path of the CRs, and therefore changing their spatial diffusion rate. This project aims to analyse the linear and non-linear evolution of the Gyroresonance Instability. For this purpose we perform one-dimension hybrid (MHD + Particle-In-Cell) numerical simulations. 2 Theoretical background Each particle specie j in a plasma is described by a distribution function f j (x, p, t), which depends on coordinates and momentum of the particles. Their evolution is given by the Vlasov equation: f j t + v f j x + q [ j E(x, t) + m j ] v B(x, t) f j c p = 0, (1) where v, p are velocity and momentum of the particles, q j and m j is the charge and mass of specie j, c is the light speed, E and B are the electric and magnetic fields, described by the Maxwell s equations: B = 1 c E t + 4π c J = 1 E c t + 4π c n j q j j d 3 pvf j (x, p, t), (2) B = 0, (3) E = 1 c B t, (4) E = 4πρ = 4π j n j q j d 3 pf j (x, p, t), (5) where n j is the number dencity of particles in specie j.thermal particles dominate the mass of the system. We will consider only CRs proton. They are deflected by the background magnetic field and also modify the electromagnetic field. To investigate linear properties of the system we use perturbation theory. We take as background solution a zero-order distribution function f 0. Solution of the equation (1) with only uniform background magnetic field. Then we add plane wave perturbations to the background solution. In the linear order this set of equations gives a transverse electromagnetic wave solution propagating along the magnetic field, with circular polarization, with the following a dispersion relation 1 k2 c 2 ω 2 + j ɛ ± j (k, ω) = 0, (6)

5 3 3 METHODS where dielectric tensor ɛ ± j (k, ω) is given by ɛ ± j (k, ω) = 2πq2 ω 2 v d 3 p ω k v ± Ω ( f 0 kv + (ω k p v ) f ) 0, (7) p where ω is the wave frequency, and subscripts and means perpendicular and parallel to the direction of the background magnetic field [2]. Positive imaginary part of the frequency ω represents instabilities. It is demonstrated in [3] that for distribution functions with an isotropic velocity distribution the solutions are not growing(stable). But for CRs with anisotropic distribution function the solution can be unstable(the gyroresonance instability) with growth rate linearly proportional to the anisotropy accordingly to [2]. 3 Methods To check the theoretical prediction of the gyroresonance instability we use 1D space and 3D velocity simulation. Magnetohydrodynamics + Particle-In-Cell (MHD+PIC) equations [4] are solved for evolving the background thermal plasma and cosmic rays ρ t + (ρv g) = 0, (8) ρv g t ( ) + ρvg T v g BT B 4π + P g = (1 R)(n CR ε 0 + J CR B/c) = F CR, (9) [ E t + (E + Pg )v g (B v g)b + c ] 4π 4π (ε ε 0) B = (1 R)J CR ε 0 = u CR F CR, (10) B t = c ε, (11) where v g is the thermal plasma velocity, P g = P g + B2 8π and P g is thermal plasma pressure. Total energy density is defined as E = P g γ ρv2 g + B2 8π, (12) where γ is adiabatic index. We use γ = 5 3. n CR is the CRs charge density, J CR is the current density generated by CRs, u CR is the CRs bulk velocity, ε and ε 0 are the electric fields given by ε 0 = v g B, (13) c ε = v g c B n CR n e (u CR v g ) c B. (14) Finally R is the ratio between the CR charge density and the charge density of the thermal electrons: R = n CR n e. (15)

6 3 METHODS 4 # β 0 β 0 A 0 NX NP Table 1: In this table we present parameters which are different in the eleven simulations. β 0 and β 0 means E CR, E B and E CR, E B, A 0 is the initial anisotropy value, NX is a simulation resolution and NP is the number of particles per cell. The CRs particles are evolved by solving the motion equations dv ( j q ) dt = (cε + u j B), (16) j where ( q ) represents particle charge-to-mass ratio. Vector component of the four velocity j is defined as u j v j = γ j u j =, (17) 1 u2 j c 2 where γ j is Lorentz factor. The code used to evolve the MHD+PIC was implemented following closely the description in [4]. The MHD fluxes are calculated using the HLL solver with linear interpolation. Time integration was performed in identical way as [4]. The initial setup of all our simulations consist of MHD homogeneous field with zero gas bulk velocity and a uniform magnetic field aligned with the grid(the x-direction). An initial perturbation is applied to the B y and B z fields, with a flat spectrum δb k B in all the wave numbers allowed by the grid resolution. The CRs have an initial power law distribution in momentum: f 0 (p) ( p 2 + p 2 ) 1 α/2 (1 A) 2 (18) employed in [1]& [2]. This distribution function represents a power law in momentum, motivated by CRs observations and acceleration theory. We use the power law index α = 2.8. Parameter A 0 measures the order of the anisotropy in the CRs pressure. It is approximately A 0 = P CR P CR P CR, (19) where P CR and P CR are the CR parallel and perpendicular pressures.

7 5 4 RESULTS (a) A 0 = 1 (b) A 0 = 0 (c) A 0 = 0.5 Figure 1: Time evolution of β (blue) and β (red) with different values of initial velocity anisotropy A 0. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. 4 Results We performed 9 different simulations, in which we studied the influence of the initial anisotropy value A 0 on the instability evolution. We also performed 2 simulations changing the resolution grid and the number of particles per cell to check their influence on the results. The parameters of the simulations are presented in the table 1. The common parameters for the all simulations are: the Alfven speed of the gas in the initial conditions v a = 10 2 c, where c is light speed and the thermal beta parameter β g Pg E b density ρ CR = 10 2 ρ g, (γv) min c where Ω 0 = qb 0 are periodic. = 2, mass = 1.58 and (γv)max c = 158. The domain size is L = c is the non-relativistic Larmor frequency of the CRs. Boundary conditions Some parameters used in our simulations ( ρ CR ρ g, β CR, A 0 ) were made larger compared to the estimated values in realistic astrophysical media(see [2] for the values in the galactic ρ halo, warm interstellar medium and intercluster medium of galaxies: CR ρ g , β CR 1, A 0 1) n order to enhance the gyroresonance instability growth rate. It is justifiable due to the exploratory and quantitative nature of this study. Figure 1 presents the time evolution of β = E CR, E B and β = E CR, E B for different values of the initial anisotropy. These figures clearly indicate that the behavior of the β and β depends on the values the initial anisotropy: their order depends on the anisotropy sign. β and β tend to approach to each other because of the scattering of CRs by Ω 0,

8 4 RESULTS 6 (a) A 0 > 0 (b) A 0 < 0 Figure 2: Time evolution in the the energy of perturbations of the magnetic field δb2 B 2 for different initial values of anisotropy.green is A 0 = 0.1, blue is A 0 = ±0.25, cyan is A 0 = ±0.5, brown is A 0 = ±0.75 and violet is A 0 = 1. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. the electromagnetic instability, which randomize their pitch angles, until they reach some limiting values. These limiting values are not given by the condition of equality of the perpendicular and parallel pressures. Figure 2 shows the time evolution of the energy in the magnetic field perturbations for different initial anisotropy values. During a short time interval we see exponential energy growth that is consistent with instability in the linear regime. The maximum energy achieved grows with the absolute value of the initial anisotropy, in consistency with the instability being driven by the anisotropy in the momentum distribution. In later times non-linear effects become important. The instability saturates and decreases at the same time the anisotropy is relaxed. We see a decrease in the energy, with approaches some constant value. Figures 3 and 4 show the evolution of the momentum distribution for the simulations with A 0 = 0.5 and A 0 = 1. We see that initially anisotropic distributions becomes more isotropic as time evolves p 0 p 0 p p p p (a) t = 0 (b) t = 20 (c) t = 400 Figure 3: Momentum distribution at different times for the simulations with initial condition A 0 = 0.5. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. Momentum components p, are in units of In Figures 5 and 6 we present the time evolution of the CRs distribution in momentum 100.

9 7 4 RESULTS p 0 p 0 p p (a) t = p (b) t = p (c) t = 400 Figure 4: Momentum distribution at different times for the simulations with initial condition A 0 = 1. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. Momentum components p, are in units of 100. and pitch angle µ = cos θ, where θ is the angle between the CR velocity and the local magnetic field. It is clearly seen that initially anisotropic distribution becomes more isotropic with time. But it also is important to mention that particles with smaller momentum intensities become isotropic faster that particles with larger ones. It happens because of the magnetic fluctuations resonant with the lower energy particles grow faster than the fluctuations resonant with higher energy particles. The evolution of the total momentum for simulations with different anisotropy values is shown in figure 7. The main conclusion here is that the power law distribution in momentum remains unchanged, despite of the redistributing in pitch angles, of the particles. Figure 8a compares the evolution of the energy in the magnetic field perturbations for simulations with identical initial conditions, but different resolutions. We see that that the energy achieves higher values for the higher resolution simulation. At late times, the energy value in the higher resolution simulation becomes constant earlier when compared to the low resolution case. It can be expected because the higher resolution simulation can solve smaller fluctuations with wave lengths which contributes to the total magnetic energy perturbation. Figure 8b presents magnetic energy evolution for two simulations with different number of particles per cell, but identical initial conditions. We see that for these parameters the energy evolution behaves almost identically. Also it is interesting to observe that for this case with initially isotropic CRs pressure there is only small growth compared to the simulations with anisotropy(compare with figure 2). Figure 9 shows the time evolution of magnetic field power spectrum for simulations with different anisotropy values. In the cases where anisotropy is not equal zero, energy quickly grows near resonant wave mode k = 9 and then slightly fade away at the same time that energy for smaller ks grow. For the case A 0 = 0 there is no evident evolution in the magnetic field power spectrum, although we still observe some noise fluctuations, probably generated by small number of particles of simulation.

10 4 RESULTS Log(p) 2.8 Log(p) μ (a) t = μ (b) t = Log(p) μ (c) t = 400 Figure 5: Distribution of CRs in momentum and pitch angle (µ = cos θ, where θ is the CR pitch angle) at different times for the simulations with initial condition A 0 = 0.5. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. Momentum p is in units of 100.

11 9 4 RESULTS Log(p) Log(p) μ (a) t = μ (b) t = Log(p) μ (c) t = Figure 6: Distribution of CRs in momentum and pitch angle (µ = cos θ, where θ is the CR pitch angle) at different times for the simulations with initial condition A 0 = 1. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. Momentum p is in units of 100.

12 4 RESULTS frequency frequency log(p) (a) A 0 = log(p) (b) A 0 = 0 frequensy log(p) (c) A 0 = 0.5 Figure 7: Time evolution of the CRs momentum distribution for simulations with different values of initial A 0. Red is time=0, black is time=400. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs. Momentum p is in units of 100. (a) A 0 = 1 (b) A 0 = 0 Figure 8: Time evolution of the energy of perturbations of magnetic field δb2 B 2. (a)two setups with different resolutions are compared. The anisotropy value is A 0 = 1. Gray is NX=512 and brown is NX=256. (b)two setups with different number of particles per cell are compared. The anisotropy value is A 0 = 0. Gray is NP=512 and brown is NP=128. Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs.

13 11 4 RESULTS (a) A 0 = 1 (b) A 0 = (c) A 0 = 0.5 Figure 9: Time evolution of magnetic energy spectrum δbk 2 with different values of initial velocity anisotropy A 0.Time is in units 100 Ω 1 0, where Ω 0 = qb 0 is the non-relativistic Larmor frequency of the CRs.

14 REFERENCES 12 5 Summary and Conclusions In this project we have studied the evolution of gyroresonance instability driven by CRs velocities anisotropy in astrophysical plasmas. This effect means that under certain conditions, the energy of the fluctuations of magnetic field, with wavelengths resonant with the CRs grows exponentially. Study of this effect is important for understanding the CRs propagation. The linear phase of the instability remains while the fluctuations are small enough, but after some time non-linear effects become important. There is no theoretical framework to fully study the influence of non-linear effects, so we use simulation as a main tool for our investigation. We have compared the instability evolution for different values of initial anisotropy A 0, and we found that the energy in the magnetic field perturbations grows faster for larger absolute values of A 0, and the perturbations do not grow substantially in the case zero initial anisotropy, as shown in the figure 2. This is consistent with the analysis in [2]. Additionally we observed that initially anisotropic CR distribution becomes more isotropic with times. Moreover, results of our simulations (see figures 5, 6) indicate that particles with smaller values of p becomes isotropic faster. In all the simulations, the non-linear effects were important. After a short initial period of exponential growth of fluctuations, non-linear effects started to change dynamics of the system, saturating the instability growth. In future we plan to perform additional simulations in order to analyse quantitatively the instability growth rate for different parameters, specially the initial anisotropy values to access the dependence with the A 0. We need to check that the growth rate is proportional to A 0 and to measure the rate of isotropisation of the CRs. References [1] A. Lazarian and A. Beresnyak. Cosmic ray scattering in compressible turbulence. Monthly Notices of the Royal Astronomical Society, 373: , December [2] H. Yan and A. Lazarian. Cosmic Ray Transport Through Gyroresonance Instability in Compressible Turbulence. The Astrophysical Journal, 731:35, April [3] A. L. Brinca. On the electromagnetic stability of isotropic populations. Journal of Geophysical Research, 95: , January [4] X.-N. Bai, D. Caprioli, L. Sironi, and A. Spitkovsky. Magnetohydrodynamic-particlein-cell Method for Coupling Cosmic Rays with a Thermal Plasma: Application to Non-relativistic Shocks. The Astrophysical Journal, 809:55, August 2015.