24P160-B
Detrapping of Energetic Electrons from an Oblique Shock Waveand its Dependence on External Magnetic Field
斜め衝撃波からの捕捉電子の解放と外部磁場に対する依存性
Junya Joho and Mieko Toida
城宝潤矢, 樋田美栄子
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
名古屋大学大学院理学研究科素粒子宇宙物理学専攻 〒 464-8602 名古屋市千種区不老町
With use of two-dimensional, relativistic, electromagnetic, particle simulations, the evolution of a large numberof electrons in an oblique shock wave is studied with attention to multi-dimensional eﬀects. Further, the motionsof the same number of test electrons are calcurated; the field data averaged along the direction of the shock front,
which are obtained from the two-dimensional simulation, are used in the equation of motion. Comparison of thetwo groups of electons verifies that multi-dimensional eﬀects can cause detrapping of electrons from the main
pulse region and subsequent acceleration by the shock wave. It is also confirmed that the detrapping of electrons
starts after two-dimensional fluctuations grow to large-amplitudes.
1. Introduction
Particle simulations have revealed [1] that prompt
electron acceleration to ultrarelativistic energies can occur
in a magnetosonic shock wave propagating obliquely to an> 1, where Ω and ω
external magnetic field with |Ωe |/ωpe ∼epe
are the electron gyro and plasma frequencies, respectively.
In such a wave, some electrons are reflected near the endof the main pulse of the shock wave, get trapped and areenergized in the main pulse. The acceleration is extremely
enhanced when the propagation speed of the shock wave
vsh is close to c cos θ, where c is the speed of light and θ isthe propagation angle of the shock wave. Once electrons
are traapped, they cannot readily escape from the wave [2].
In the above studies, the simulations were onedimensional (1D). Recently, two-dimensional (2D) particle simulations have shown [4] that after trapping and energization in the main pulse, some electrons can be detrapped from it and can suﬀer subsequent acceleration by
the mechanism studied in Refs. [3]. The detrapping canbe caused by magnetic fluctuations along the shock front.
In this study, we examine in detail the multi-dimensionaleﬀects. In addistion to a two-dimensional particle simulation, we calculate test particle orbits in the 1D filed averaged along the direction of the shock front; the field data
are obtained from the 2D particle simulation.
2. Physical Picture of Electron Detrapping
We here present a physical picture of detrapping. The
shock wave is assumed to propagate in the x direction witha speed vsh in the external magnetic field in the (x, z) plane,
B = B0 (cos θ, 0, sin θ). In the frame moving with the speed
vsh in the x direction, electric and magnetic fields may bewritten as
E j (x, t) = E jsh (x) + δE j1 (x, t) + δE j2 (x, y, z, t), (1)B j (x, t) = B jsh (x) + δB j1 (x, t) + δB j2 (x, y, z, t),(2)where the subscript j refers to x, y, or z, E jsh and B jsh are
the time-averaged values of 1D fields, and we assume that
Eysh = Ey0 = −vsh Bz0 /c, Ezsh = 0, which are equal to thevalues in the case of the 1D stationary shock wave.
Using the drift approximation, we can write the velocity of the electron asv = vk B/B + c(E × B)/B2 .

(3)

Substituting Eqs. (1) and (2) into Eq. (3) and neglecting
the second order temrs of the fluctuations, we obtainv x ' (c cos θ − vsh + δv x1 + δv x2 )Bo /Bsh ,δv xl 'c(c cos θ − vsh )(δEyl + δBxl ) −δBzl
B0Bzsh(4)(5)

where l =1 or 2. This indicates that some electrons canbe detrapped from the main pulse when they feel the bigchange in δv xl .
3. Simulation
We carry out 2D (two space coordinates and threevelocities), relativistic, electromagnetic particle simulatioswith full ion and electron dynamics. The total number ofthe simulation particles is N ' 1.1 × 109 . We follow themotions of 2.1×106 electrons in the 2D simulation and callthe electrons the 2Ds ones. We also compute the test particle orbits, integrating the relativistic equation of motion
dPe= e Ē(x, t) + v × B̄(x, t),dtc(6)where Ē(x, t) and B̄(x, t) are y-averaged data of the fielddata E(x, y, t) and B(x, y, t) obtained in the 2D simulation.
We call the test particles follwing Eq. (6) the 1Dt electrons.
The numbers of the 1Dt electrons is the same as that of
the 2Ds electrons. The initial positions and velocities of
the 1Dt electrons are exactly the same as those of the 2Dselectrons.

trapped in the main pulse by the time t (upper panel) andthat have been detrapped from it (lower panel). The detrapping of the 2Ds electrons starts at ωpe t ' 1000. Figure 4
displays time variations of the magnitudes of δv x1 and δv x2defined by Eq. (5). The values of |δv x2 | are much greaterthan those of |δv x1 |, which caused the diﬀerence between
the motions of 1Dt and 2Ds electrons. As predicted byEq. (5), the 2Ds electrons begins to be detrapped after theincrease of |δv x2 |.

Fig. 1 Phase space plots (x, γ) of the 1Dt and 2Ds electrons andmagnetic field profile at ωpe t = 3360.

Figure 1 shows the profile of Bz averaged over the ydirection in a shock wave with vsh ' 0.95c cos θ and theelectron number density in the (x, γ) plane at ωpe t = 3360,
where γ is the Lorentz factor. In the upper panel, someof the 1Dt electrons are trapped and energized in the main
pulse region, and there are no energetic electrons outside
the region. However, in the lower panel, many energetic
2Ds electrons exist in the wider region from the upstream
to downstream regions. The maximum energy of the 2Ds
electrons is higher than that of the 1Dt electrons.

Fig. 3 Time variations in the numbers of trapped and detrappedelectrons

4. Summary and Discussion
We have studied electron motion in an oblique shock
wave using the 2D particle simulations. Further, we havecalcurated test particle motion in the 1D field averaged
along the direction of the shock front. Comparison of
the simulation and test electrons confirmed that multidimensional eﬀects can cause detrapping from the main
pulse and subsequent acceleration by the shock wave. In
the presentation, we will discuss the dependence on external magnetic field, multi-dimensional eﬀects on reflection,and the evolution of two-dimensional flucutuations.
References

Fig. 2 Time variations in x and γ of 1Dt and 2Ds electrons withthe same initial velocities and positions

Figure 2 shows the time variations of γ and the position x of a 2Ds (black solid line) and a 1Dt (gray dashedline) electrons. The initial velocities and positions of thetwo electrons are exactly the same, and both of them arereflected near the end of the main pulse. However, theirorbits after the reflection are completely diﬀerent. The 1Dt
electron continues to be trapped in the main pulse, whilethe 2Ds electron has been detrapped from it to the upstream
region. The 2Ds electron then crosses the shock front several times due to the gyromotion and is further accelerated
by the the transverse electric field Ey in the shock wave.
Figure 3 shows the numbers of electrons that have got[1] N. Bessho and Y. Ohsawa, Phys. Plasmas 6 3076 (1999).
[2] A. Zindo, Y. Ohsawa, N. Bessho, and R. Sydora, Phys. Plasmas 12 052321 (2005) .
[3] S. Usami and Y. Ohsawa, Phys. Plasmas 9, 1069 (2002).
[4] K. Shikii and M. Toida, Phys. Plasmas 8 082316 (2010).

Fig. 4 Time variations in the magnitudes of fluctuations δv x1 andδv x2 near the center of the main pulse.