collisionless damping of circularly polarized nonlinear ... - IOPscience

The Astrophysical Journal, 793:138 (7pp), 2014 October 1 C 2014.

doi:10.1088/0004-637X/793/2/138

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

´ WAVES COLLISIONLESS DAMPING OF CIRCULARLY POLARIZED NONLINEAR ALFVEN IN SOLAR WIND PLASMAS WITH AND WITHOUT BEAM PROTONS 1

Y. Nariyuki1 , T. Hada2 , and K. Tsubouchi3 Faculty of Human Development, University of Toyama, 3190, Toyama City, Toyama 930-8555, Japan; [email protected] 2 Department of Earth System Science and Technology, Kyushu University, 6-1, Kasuga City, Fukuoka 816-8580, Japan 3 Graduate School of Science, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan Received 2014 February 12; accepted 2014 August 4; published 2014 September 16

ABSTRACT The damping process of field-aligned, low-frequency right-handed polarized nonlinear Alfvén waves (NAWs) in solar wind plasmas with and without proton beams is studied by using a two-dimensional ion hybrid code. The numerical results show that the obliquely propagating kinetic Alfvén waves (KAWs) excited by beam protons affect the damping of the low-frequency NAW in low beta plasmas, while the nonlinear wave–wave interaction between parallel propagating waves and nonlinear Landau damping due to the envelope modulation are the dominant damping process in high beta plasmas. The nonlinear interaction between the NAWs and KAWs does not cause effective energy transfer to the perpendicular direction. Numerical results suggest that while the collisionless damping due to the compressibility of the envelope-modulated NAW plays an important role in the damping of the field-aligned NAW, the effect of the beam instabilities may not be negligible in low beta solar wind plasmas. Key words: instabilities – plasmas – solar wind – waves excite the ion acoustic waves (Derby 1978; Goldstein 1978; Matteini et al. 2010; Gao et al. 2013a). However, the high crosshelicity of solar wind plasmas does not support the occurrence of the simple decay instability observed in time evolution of monochromatic Alfvén waves, since the decay instability reduces the cross-helicity through the production of the sunward propagating daughter Alfvén waves. The growth of the decay instability is reduced by the broadband wave number spectra and the high beta ratio (Umeki & Terasawa 1992; Malara et al. 2000; Tenerani & Velli 2013). The nonlinear interaction among NAW and the obliquely and/or the quasi-perpendicularly propagating waves have also been studied by many authors (Mjølhus & Hada 1990; Vinas & Goldstein 1991; Laveder et al. 2002; Voitenko & Goossens 2005; Shukla & Stenflo 2005; Zhao et al. 2011; Gao et al. 2013a; Zhao et al. 2014). Notice that while the compressible waves are not negligible in these nonlinear interactions, most studies on the scaling law of the MHD turbulence assume incompressible plasmas (Goldreich & Sridhar 1995; Ng & Bhattacharlee 1996; Boldyrev 2006; Lithwick et al. 2007; Podesta 2011a; Boldyrev et al. 2011; Perez et al. 2012), in which the shear Alfvén mode is the only propagating MHD mode. It is also noteworthy that solar wind ions usually have nonequilibrium velocity distributions, which can excite the various instabilities (Marsch 2006; Maruca et al. 2011; Alexandrova et al. 2013) and may affect the damping range turbulence. Past studies pointed out that the dense and low speed ion beams such as those in solar wind plasmas can excite the obliquely propagating shear Alfvén waves (kinetic Alfvén waves (KAW); Daughton & Gary 1998; Yin et al. 2007; Hellinger & Travnicek 2011, 2013; Nariyuki et al. 2012; Verscharen & Chandran 2013) with which low-frequency NAWs can resonate (Voitenko & Goossens 2005; Shukla & Stenflo 2005; Zhao et al. 2014). On the other hand, it is also known that the presence of the NAW suppresses the ion instabilities (Gomberoff 2003; Araneda & Gomberoff 2004) and nonlinear evolution of parametric instabilities of parallel propagating Alfvén waves are affected by the presence of ion beams (Nariyuki et al. 2009; Gao et al. 2013b). Recently, numerical simulations with a two-dimensional ion hybrid code demonstrated that the monochromatic, left-handed

1. INTRODUCTION Low-frequency nonlinear Alfvén waves (NAWs), which have a clear correlation between magnetic field and plasma bulk velocities, are ubiquitous in solar wind plasmas (Ruiz et al. 2011; Gosling et al. 2011; Paschmann et al. 2013; Bruno & Carbone 2013). Since the NAW with the clear correlation is composed of the uni-directionally propagating Alfvén waves, the finite values of the cross-helicity are observed (Bavassano et al. 2000; Bruno & Carbone 2013). Such an NAW is the “imbalanced” magnetohydrodynamic (MHD) turbulence, which is the turbulence with the finite cross-helicity (Wicks et al. 2011; Boldyrev et al. 2011). To clarify, the scaling law of the imbalance MHD turbulence is one of the important issues in this field (Lithwick et al. 2007; Osman & Forbury 2009; Podesta 2011a, 2011b; Boldyrev et al. 2011; Wicks et al. 2011; Perez et al. 2012). It is also clarified that with the increasing heliocentric distance and the decreasing solar wind speed, slab Alfvénic turbulence, which may mainly be composed of the Alfvén waves with quasi-parallel wave number spectra (Ruiz et al. 2011), dissipates, while the fully developed turbulence (two-dimensional turbulence), which is mainly composed of the quasi-perpendicular wave number spectra, becomes dominant (Dasso et al. 2005; Ruiz et al. 2011). This suggests the damping of the NAW and/or the energy conversion from NAW into the fully developed turbulence in the interplanetary space (Dasso et al. 2005; Ruiz et al. 2011; Narita et al. 2011). However, the detailed processes of the radial evolution are still unknown. NAWs in solar wind plasmas may involve the envelope modulation, which corresponds to the broadband wave number/ frequency spectra (Bruno & Carbone 2013). Since the NAW in solar wind plasmas is usually a nonlinear fluctuation with a finite amplitude, the compressibility of the envelope-modulated NAW can play an important role in the damping of the NAW through the nonlinear Landau damping and/or the modulational instability (Mjølhus 1976; Mio et al. 1976; Mjølhus & Wyller 1988; Spangler 1989; Medvedev & Diamond 1996; Nariyuki et al. 2007, 2008a; Markovskii et al. 2009; Nariyuki et al. 2010). It is also well known that the parametric decay instability can dissipate the NAW with parallel wave number spectra and 1

The Astrophysical Journal, 793:138 (7pp), 2014 October 1

Nariyuki, Hada, & Tsubouchi Table 1 Parameters used in Simulation Runs

polarized Alfvén waves are rapidly dissipated due to the presence of the beam-induced KAWs (Nariyuki et al. 2012). A past study (Nariyuki et al. 2012) also suggested that the rapid damping due to the proton beams does not occur when the initial Alfvén waves are nonmonochromatic. However, the Alfvén waves discussed in the past study are dispersive and the nonlinearly driven modulational instability is strong. To the best of our knowledge, roles of beam protons in nonlinear evolution and damping of low-frequency Alfvén waves are still unclear. The purpose of the present study is to clarify the collisionless damping process of the field-aligned, low-frequency NAWs in solar wind plasmas with and without proton beams. The numerical setup is described in Section 2. The numerical results are presented in Section 3. In Section 4, the results are summarized and future issues are discussed.

l u exp(−ikx + φk ), ujp = Σkk=k 0 j0

(2)

ω0 − uj x0 k0 b0 , k0 (ω0 − uj x0 k0 + 1)

ω02 (ω0 − U k0 )2 1 η , + 1 + η 1 + ω0 1 + η 1 + (ω0 − U k0 )

1.8 1.8 0 1.8 1.8 0 1.8 0 1.8 0

3.1. Nonlinear Evolution of KAWs Without NAWs Before discussing the nonlinear evolution of the NAWs, nonlinear evolution of KAWs excited by the beam protons is discussed in this subsection. In the present study, the relatively dense (η = 0.15/0.85 = 0.176) and slow (U = 1.8) beam protons, which are typical values in solar wind plasmas (Marsch 2006), are initially given. As shown by the past study on linear analysis (Daughton & Gary 1998), in low beta plasmas, the beam protons excite the obliquely propagating KAWs through the Alfvén I instability. In contrast, the parallel beam instabilities (resonant and nonresonant instabilities), which excite the parallel propagating waves, are stable in low beta plasmas (Daughton & Gary 1998). Figure 1 (Run A0) shows time evolution of the wave number spectra of the complex transverse magnetic field (b = by + ibz ). The obliquely propagating KAWs are clearly observed at t = 135 (Figure 1(a)). Black lines in Figure 1(d) indicate the linear growth rates of Run A0. In this parameter, the maximum growth rate of the parallel instabilities (not shown) is about 1.4 × 10−6 , which is much smaller than the growth rates of the Alfvén I instability. The wave number mode with the maximum amplitude in Figure 1(a) (k = −0.7853, l = −1.325 and the propagation angle is about 59.◦ 4) agrees well with the maximum growing mode in Figure 1(d). As time elapses, the KAWs decay into the quasi-parallel parallel propagating waves with k = 0.7–1.2 (Figures 1(b) and (c)). The parallel propagating waves with k < 0 correspond to forwardly propagating left-handed polarized waves (ω/k > 0) or backwardly propagating right-handed polarized waves (ω/k < 0). The waves observed in Figures 1(b) and (c) are the former as shown in Section 3.3 (Figure 9).

(3)

given by the Walen relation (Hollweg et al. 1993), and uj x0 is the initial bulk velocity parallel to the x axis. In a similar way to the past studies (Araneda & Gomberoff 2004; Nariyuki et al. 2009, 2012), ω0 and k0 satisfy the dispersion relation of the parallel propagating electromagnetic waves in core proton– electron–beam proton plasmas with the fluid approximation k02 =

U

0 0.0625 0.0625 0 0.0625 0.0625 0.25 0.25 0.25 0.25

3. RESULTS

where bp = by + ibz and ujp = ujy + iuj z are the transverse magnetic field and bulk velocity parallel propagating transverse waves, k0 = 0.0123 and kl = 0.0982 (the number of wave number modes is 8), φk is the initial random phase, j = c (core protons) , b (beam protons), uj 0 = −

EW 0

0.1 0.1 0.1 1.0 1.0 1.0 0.1 0.1 1.0 1.0

where η = nb0 /nc0 is the density ratio between beam protons and core protons, and U = (ubx0 −ucx0 )/VA0 is the relative drift velocity between beam and core protons. We adopt the notation that the positive (negative) Re[ω0 ] corresponds to the righthanded (left-handed) polarized waves. Equation (4) is written in the frame of reference in which the core protons have no zeroth-order drift speed (ucx0 = 0). The initial current-free condition for the longitudinal components is satisfied from the charge neutrality. We here discuss the initial waves with Re[ω0 ]/k0 > 0, Re[ω0 ] > 0, and Im[ω0 ] = 0. The wave numbers in x and y direction are denoted as k and l, respectively. Parameters used in 16 simulation runs are tabulated in Table 1, where EW 0 = kk20 =k1 |bk0 |2 and βp = βc + βb .

We have carried out two-dimensional hybrid simulation runs (superparticle protons and an electron fluid; Terasawa et al. 1986; Tsubouchi 2011) with a setup similar to that in the past study (Nariyuki et al. 2012). Time and space are normalized to the reciprocal of the initial proton cyclotron frequency (Ωp0 ) and the proton inertial length (VA0 /Ωp0 ) defined by the background quantities, respectively. The ambient magnetic field bx0 is parallel to the x axis. The magnetic field is normalized to the initial background magnetic field (bx (t = 0) = bx0 = 1), the proton density ρ is normalized to ρe0 = mp ne0 , where mp is the mass of protons and ne0 is the initial electron number density, and the velocity is normalized to the initial background Alfvén velocity (VA0 ) defined by bx0 and ρe0 . We here assume that the electrons are isotropic, isothermal, and massless fluid. The grid size Δx = X/Lx = 0.5 and Δy = Y /Ly = 0.5, time step Δt = 0.04, the system size in the x direction X = 512 (the grid size Lx = 1024) and the y direction Y = 128 (the grid size Ly = 256). Two-hundred particles per cell are used in the runs. We have employed the rational Runge–Kutta scheme (Wambecq 1978) for time integration and a second-order finite difference scheme for evaluating spatial derivatives. The Alfvén waves propagating in the x direction are initially given as (1)

βp (= βe )

A0 A1 A2 B0 B1 B2 C1 C2 D1 D2

Note. The density ratio between beam protons and core protons η = 0.15/0.85 = 0.176 in all the runs.

2. NUMERICAL SETUP

l bp = Σkk=k b exp(−ikx + φk ), 0 0

Run

(4) 2


Nariyuki, Hada, & Tsubouchi

(a)

(b)

Figure 2. Wave number spectra of the complex transverse magnetic field (b = by + ibz ) in Run B0 at t = 135.

(c)

Figure 3. Wave number spectra of the complex transverse magnetic field (b = by + ibz ) in Run A2 at t = 750.

(d)

In contrast to low beta plasmas, the KAWs are not observed in the run with high beta plasma (Run B0). Figure 1(d) shows that the parallel instabilities have a finite growth rate in Run B0, while the Alfvén I instability is stable. This is because the Alfvén instability, which excites KAWs, is stabilized with increasing proton beta, while the one of the parallel resonant instability is destabilized (Daughton & Gary 1998). As shown in Figure 2, the quasi-parallel propagating waves are weakly excited at k = 0.5–0.8, since the growth rates are weaker and the unstable regime in k–l space are narrower than those of Run A0.

Figure 1. Wave number spectra of the complex transverse magnetic field (b = by + ibz ) in Run A0 at (a) t = 135, (b) t = 270, (c) t = 405, respectively. (d) The growth rate of the linear beam instabilities with the parameters of Run A0 (black lines) and B0 (gray lines). In order from left to right, black lines correspond to the propagation angle 70◦ , 60◦ , 50◦ , 40◦ , 30◦ , 20◦ , respectively. Two gray lines correspond to the propagation angle 0◦ and 10◦ .

3.2. Nonlinear Evolution of NAWs Without Beam Protons We next briefly discuss the nonlinear evolution of NAWs without beam protons (Runs A2, B2, C2, and D2). Figure 3 3



(a)

(a)

(b)

(b)

(c)

(c)

Figure 5. Wave number spectra of the complex transverse magnetic field in Run C1 at (a) t = 250, (b) t = 500, (c) t = 750, respectively.

Figure 4. Wave number spectra of the complex transverse magnetic field in Run A1 at (a) t = 250, (b) t = 500, (c) t = 750, respectively.

In the run with small amplitude and low beta plasmas (Run A2), the initial NAWs are almost stable within the simulation time scale (t < 1000). Time evolution of the magnetic energy density is discussed in the next section (Figures 7 and 8).

(Run A2) shows the wave number spectra of the complex transverse magnetic field at t = 750. In all the runs, time evolution of NAWs is almost one dimensional. This is in agreement with the past studies, in which two-dimensional simulations were carried out (Nariyuki et al. 2008b; Gao et al. 2013a). The damping of the wave energy is caused by the nonlinear Landau damping due to the envelope modulation in agreement with onedimensional simulation runs presented in past studies (Nariyuki et al. 2008a; Markovskii et al. 2009; Nariyuki et al. 2010). Notice that the decay instabilities are not important in the present runs, since the amplitude of each wave mode is not large and initial envelope modulation causes the nonlinear Landau damping and/or modulational instabilities (Nariyuki et al. 2007, 2008a).

3.3. Nonlinear Evolution of NAWs with Beam Protons Figures 4 and 5 show the wave number spectra at (a) t = 250, (b) t = 500, (c) t = 750 in Runs A1 and C1, respectively. When the initial amplitude of NAWs is relatively small (Run A1), time evolution of KAWs is similar to the one without NAWs (Run A0). On the other hand, as shown in Figure 5(a), the excitation of KAWs is weakened in the run with higher initial amplitude (Run C1). This is because the initial NAWs affect and 4



(a)

(a)

(b)

(b)

Figure 7. Time evolution of the transverse magnetic energy density with k > 0 and l = 0 (Erh ) in (a) Run A1 (solid line) and A2 (dashed line), and (b) Run B1 (solid line) and B2 (dashed line).

(a)

Figure 6. Wave number spectra of the complex transverse magnetic field at t = 750 in (a) Runs B1 and (b) D1, respectively.

change the growth rate and unstable wave numbers of the beam instabilities (Gomberoff 2003; Araneda & Gomberoff 2004; Nariyuki et al. 2012). In Run C1, KAWs inversely cascade instead of forming the quasi-parallel propagating waves. This inverse cascade is probably due to the nonlinear interaction with the large amplitude NAWs. In Run C1, power spectra of parallel propagating waves (l = 0) become broader than those in Run A1 due to the nonlinear interaction among the wave number modes in NAWs. This broadening in the wave number space may also make nonlinear interaction with KAWs easier. Figures 4(c) and 5(c) show that the resultant waves are affected by the initial amplitude of NAWs. In contrast to Run A0 and A1, the excitation of quasi-parallel propagating waves with k = 0.5–1.5 cannot be observed due to the suppression of the beam instability in Run C1 at t = 750. Figure 6 shows the wave number spectra of the complex transverse magnetic field at t = 750 in Runs (a) B1 and (b) D1, respectively. The wave number spectra in the runs with high beta plasmas is almost one dimensional (Runs B1 and D1), while the waves with various propagation angles can be observed in the wave number spectra of the runs with low beta plasmas (Runs A1 and C1). This is because the KAWs are not excited in the runs with high beta plasmas as shown in Section 3.1. It is important that the damping of NAWs is promoted by the excited KAWs. Figures 7 and 8 show the time evolution of the transverse magnetic energy density with k > 0 and l = 0 (Erh ) in each run. At t = 0, Erh = EW 0 . As is clearly

(b)

Figure 8. Time evolution of the transverse magnetic energy density with k > 0 and l = 0 (Erh ) in (a) Runs C1 (solid line) and C2 (dashed line), and (b) Runs D1 (solid line) and D2 (dashed line).

shown in Figures 7(a) and 8(a), the damping of NAWs in runs with beam protons is different from that in runs without beam protons in low beta plasmas. In the past study (Nariyuki et al. 2012), similar damping has been observed in nonlinear evolution of a monochromatic, left-handed polarized dispersive Alfvén wave. On the other hand, the effect of the beam protons on damping of NAWs is not large in runs with high beta plasmas, in which KAWs do not appear, as shown in Figures 7(b) and 8(b). In addition, the damping of initial NAWs due to the 5



the energy of Alfvénic (slab) turbulence into fully developed (two-dimensional) turbulence. It is also remarkable that the initial spectra of Alfvén waves may affect the nonlinear evolution (Nariyuki et al. 2007, 2012; Maneva et al. 2013). Actually, nonlinear evolution of Alfvén waves in solar atmosphere may be affected by the conditions of the solar surface and the configuration of the magnetic flux tubes (Suzuki 2012). The electron Landau damping of the KAWs, which is neglected in the present study, may also be important, especially in low beta plasmas (Yin et al. 2007). The perpendicular cascade of NAW ubiquitously occurs in incompressible, balanced Alfvénic turbulence (Ng & Bhattacharlee 1996; Howes et al. 2013). However, while how the perpendicular cascading occurs in a system with compressibility and kinetic effects is still unknown at this time, the limitation of the timescale and spatial dimension in the present study prevents the discussion of the spectral index in the inertial range. The relations among the energy cascade predicted in incompressible MHD systems and wave excitations due to the kinetic instabilities are also still unclear. It is important that the presence of the low-frequency NAW affects the wave dispersion and kinetic instabilities even at the linear stage (Stenflo 1976; Gomberoff 2003). The effect of the driving frequency on the excitation of magnetic turbulence was recently discussed using a two-dimensional hybrid simulation (Parashar et al. 2011). To discuss these issues comprehensively, larger scale computations should be performed in the future.

(a)

(b)

The computer simulation was performed on the KDK computer system at the Research Institute for Sustainable Humanosphere, Kyoto University.

Figure 9. ω − k spectra in (a) Run A1 and (b) C1 obtained for the periods 1 t 512.

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4. SUMMARY AND DISCUSSION In the present study, nonlinear evolution of nonmonochromatic, circularly polarized low-frequency NAWs is discussed. The numerical results indicate that obliquely propagating KAWs excited by beam protons affect the damping of the lowfrequency NAWs in low beta plasmas, while the parallel cascading (nonlinear steepening) and nonlinear Landau damping is the dominant damping process in high beta plasmas. This is because the beam instability is weakened (Daughton & Gary 1998) and the nonlinear Landau damping is promoted (Spangler 1989) in high beta plasmas. As shown in the past study (Nariyuki et al. 2012), the presence of a finite amplitude NAW suppresses the growth of the proton beam instability. Even when the KAWs affect the damping of NAWs, the quasi-parallel, unidirectional NAW is still dominant in our present simulation runs. This suggests that the presented damping processes may not cause effective perpendicular cascading, which may convert 6



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