The motions of relativistic particles in a magnetosonic shock wave propagatingobliquely to an external magnetic field are studied. In the zeroth-ordertheory, particles continue to move nearly parallel to the external magnetic field inthe shock transition region, when the shock speed is close to c cos θ, where c is thespeed of light and θ is the propagation angle. Perturbations to this zeroth-ordermotion are also analyzed for positrons and ions. The perturbation frequency ofpositrons is ω ? Ωp0γ?1 and that of ions is ω ? Ωi0γ?1/2, where Ωp0 and Ωi0 arethe non-relativistic gyrofrequencies of positrons and of ions, respectively, and γis the Lorentz factor. These theoretical predictions are confirmed with numericalsimulations

Syunsuke USAMI

Y. OHSAWA

National Institute for Fusion Science (NIFS-Repository)

Journal of Plasma Physicshttp://journals.cambridge.org/PLAAdditional services for Journal of Plasma Physics:Email alerts: Click hereSubscriptions: Click hereCommercial reprints: Click hereTerms of use : Click hereTheory and simulations of relativistic particle motions in a magnetosonic shock waveSHUNSUKE USAMI and Y. OHSAWAJournal of Plasma Physics / Volume 72 / Issue 06 / December 2006, pp 887 ­ 890DOI: 10.1017/S0022377806005058, Published online: 20 December 2006Link to this article: http://journals.cambridge.org/abstract_S0022377806005058How to cite this article:SHUNSUKE USAMI and Y. OHSAWA (2006). Theory and simulations of relativistic particle motions in a magnetosonic shock wave. Journal of Plasma Physics, 72, pp 887­890 doi:10.1017/S0022377806005058Request Permissions : Click hereDownloaded from http://journals.cambridge.org/PLA, IP address: 133.75.110.124 on 16 Nov 2012J. Plasma Physics (2006), vol. 72, part 6, pp. 887–890. c© 2006 Cambridge University Pressdoi:10.1017/S0022377806005058 Printed in the United Kingdom887Theory and simulations of relativistic particlemotions in a magnetosonic shock waveSHUNSUKE USAMI1 and Y. OHSAWA21Computer and Information Network Center, National Institute for Fusion Science,322-6 Oroshi-cho, Toki, Gifu 509-5292, Japan2Department of Physics, Nagoya University, Nagoya 464-8602, Japan(Received 15 August 2005 and accepted 20 December 2005)Abstract. The motions of relativistic particles in a magnetosonic shock wave propa-gating obliquely to an external magnetic ﬁeld are studied. In the zeroth-ordertheory, particles continue to move nearly parallel to the external magnetic ﬁeld inthe shock transition region, when the shock speed is close to c cos θ, where c is thespeed of light and θ is the propagation angle. Perturbations to this zeroth-ordermotion are also analyzed for positrons and ions. The perturbation frequency ofpositrons is ω ∼ Ωp0γ−1 and that of ions is ω ∼ Ωi0γ−1/2, where Ωp0 and Ωi0 arethe non-relativistic gyrofrequencies of positrons and of ions, respectively, and γis the Lorentz factor. These theoretical predictions are conﬁrmed with numericalsimulations.1. IntroductionMagnetosonic shock waves can accelerate thermal hydrogen ions, heavy ions, andelectrons with various non-stochastic mechanisms [1]. Recently, two accelerationmechanisms of non-thermal, relativistic particles in magnetosonic shock waves havebeen studied with theory and simulations. One mechanism is associated with large-radius gyromotions; particles absorb energy from the electric ﬁeld perpendicularto the magnetic ﬁeld B [2]. The other is the acceleration parallel to B [3, 4]. Here,we investigate particle motions in the latter. In Sec. 2, we describe the zeroth-orderand perturbation theories for relativistic particles. In Sec. 3, we verify them withsimulations. Section 4 gives a summary of our work.2. Theory2.1. Zeroth-order theoryWe analyze the motions of relativistic particles in a magnetosonic shock wavepropagating in the x direction with a speed vsh in an external magnetic ﬁeldB0 = B0(cos θ, 0, sin θ). If the particle speed v is very close to the speed of light c, aslight change in the particle speed can lead to a great change in the Lorentz factorγ. Therefore, ignoring γdv/dt compared with vdγ/dt, we obtain the zeroth-orderequation of motion for a particle with a massmj and a charge qj (j denotes particlespecies, j = p or i):mjdγ0dtv0 = qj(E+v0c× B), (2.1)where the subscript 0 refers to the zeroth-order quantities.888 S. Usami and Y. OhsawaWe consider particles moving with the wave, i.e. vx0 = vsh. When vsh ≈ c cos θ,we ﬁndvz0vsh≈ Bz0Bx0(2.2)from (2.1). It is also shown that |vy0| is much smaller than vx0 and vz0; thus, theparticles move nearly parallel to B0. Moreover, we obtain the time rate of changeof γ0 asdγ0dt=qjBx0mjvsh(E · B0)(B · B0) . (2.3)If the particle position in the wave does not change, γ0 continues to grow linearlywith time [3].2.2. Perturbation theoryThe zeroth-order theory is applicable to either positrons or ions. We do, how-ever, need to treat positron and ion perturbations separately [4]. We assume thatdv1/dt ∼ γ−10 Ωp0v1 for positrons and dv1/dt ∼ γ−1/20 Ωi0v1 for ions, where Ωp0and Ωi0 are the non-relativistic positron and ion gyrofrequencies, respectively, andthe subscript 1 refers to perturbed quantities. We then expand the exact equationof motion. After some algebra, the perturbation frequency ω of the positrons isobtained asω2 =(empc)2 (γ20/c2)(B0 · v0)2 + B2γ40, (2.4)which is obviously positive, while that of the ions is given asω2 = − qimiγ0γsh(dExdξ0+vy0cdBzdξ0− vz0cdBydξ0), (2.5)where γsh = [1 − (vsh/c)2]−1/2, ξ = x − vsht, and ξ0 is the center position of theperturbation; d/dξ0 designates the derivative at ξ = ξ0. The ion perturbation isstable when ω2 > 0.3. Numerical studiesWe numerically investigate the motions of relativistic particles. For positrons, weuse a one-dimensional, relativistic, electromagnetic particle simulation code [3]. Asin the theory, waves propagate in the x direction in an external magnetic ﬁeld B0.The ﬁeld strength is |Ωe0|/ωpe = 3, where |Ωe0| and ωpe are the electron gyro andplasma frequencies, respectively. The propagation angle is taken to be θ = 42◦. Thepropagation speed of a shock wave studied here is observed to be vsh = 2.4vA, wherevA is the Alfve´n speed. It has a typical shock proﬁle with the width of the transitionregion of the order of the ion inertial length as shown, for instance, in [3]. Figure 1shows the time variation of γ of a positron accelerated by a magnetosonic shockwave in an electron–positron–ion plasma with a positron-to-electron density ratioof 0.02 [3]. The energy increases up to γ ∼ 600. We did, however, ﬁnd oscillationsin γ. Figure 2 displays the oscillation frequency ω as a function of γ0 [4]. The datapoints represent simulation results, while the solid line shows the theoretical curvegiven by (2.4). The simulation results are explained by the theory.Theory and Simulations of Relativistic Particle Motions 88903006000 1000 1500γωpet500Figure 1. Time variation of γ of an accelerated positron. Here, ωpe is the electronplasma frequency.00.10.20.30.40 200 400 600ωγ0Ωp0Figure 2. Positron perturbation frequency ω versus γ0.0.6340.6350.6360 1000 2000100012001400γΩH0tvxcFigure 3. Time variations of γ and vx of an accelerated hydrogen ion. Here, ΩH0 is thenon-relativistic hydrogen gyrofrequency.For ions, we calculate test particle orbits; we ﬁrst obtain the electromagneticﬁelds in a shock wave from a particle simulation and then follow particle motionsin the ﬁelds, assuming stationary wave propagation. Here, plasma parameters are|Ωe0|/ωpe = 1.5, θ = 50◦, and vsh = 3.2vA. The initial velocities of the test particlesare given by the zeroth-order theory. In Fig. 3 we show the time variations of γ andvx of an ion with an initial energy γ = 1000. The energy increases from γ = 1000890 S. Usami and Y. Ohsawa00.20.40.60 10000 20000ωγ0ΩH0Figure 4. Ion perturbation frequency ω versus γ0.to approximately 1400. Also, we ﬁnd small-amplitude oscillations in vx. In Fig. 4,we display the perturbation frequency ω as a function of γ0. The simulation valuesﬁt well to the theoretical curve obtained from (2.5)4. SummaryThe motions of relativistic particles in a magnetosonic shock wave have been stud-ied. In the zeroth-order theory, where the relation γ|dv/dt| vdγ/dt is assumed,particles are accelerated almost parallel to the external magnetic ﬁeld, when vsh ≈c cos θ. This is applicable to either positrons or ions. Perturbation theories forpositrons and ions have been separately investigated. The perturbation frequencyof positrons is ω ∼ Ωp0γ−10 and that of ions is ω ∼ Ωi0γ−1/20 . The zeroth- andﬁrst-order theories have been veriﬁed with numerical simulations.References[1] Ohsawa, Y. 2004 Nonstochastic particle acceleration in collisionless shock waves. PhysicaScripta T107, 32–35.[2] Usami, S. and Ohsawa, Y. 2004 Evolution of relativistic ions incessantly accelerated byan oblique shock wave. Phys. Plasmas 11, 918–925.[3] Hasegawa, H., Usami, S. and Ohsawa, Y. 2003 Positron acceleration to ultrarelativisticenergies by a shock wave in a magnetized electron–positron–ion plasma. Phys. Plasmas10, 3455–3458.[4] Usami, S. and Ohsawa, Y. 2004 Motions of ultrarelativistic particles accelerated in anoblique plasma wave. Phys. Plasmas 11, 3203–3211.

Theory and simulations of relativistic particle motions in a magnetosonic shock wave

Abstract

Similar works

Full text

Available Versions

National Institute for Fusion Science (NIFS-Repository)