24P159-B
Dependence of Nonlinear Evolution of Magnetosonic Waves
on Ion Composition and Propagation Angle
非線形磁気音波のイオン組成と伝播角に対する依存性
Mieko Toida and Yuichi Kondo
樋田美栄子, 近藤憂一
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
The dependence of nonlinear evolution of magnetosonic waves on ion composition and propagation angle θis studied. First, the conditions necessary for KdV equation for the low-frequency mode in a two-ion-species
plasma is analytically obtained. The upper limit of the amplitude of the low-frequency-mode pulse is expressed
as a function of θ, density ratio, and cyclotron frequency ratio of two ion species. Next, with electromagnetic
particle simulations, the nonlinear evolution of a long-wavelength low-frequency-mode disturbance is examinedfor various θs in two plasmas with diﬀerent ion densities and cyclotron frequency ratios, and the theory for thelow-frequency-mode pulse is confirmed. It is also shown that if the pulse amplitude exceeds the theoretical valueof the upper limit of the amplitude, then shorter-wavelength low- and high-frequency-mode waves are generated.
1. Introduction
The presence of the multiple species ions influences
the properties of magnetosonic waves. For example, ina two-ion-species plasma, the magnetosonic wave propagating perpendicular to a magnetic field is split into twomodes; the low- and high-frequency modes. The frequencies of the low-frequency mode are in the range 0 < ω <ω−r , where ω−r is the ion-ion hybrid resonance frequency[1],
ω−r = [(ω2pa Ω2b + ω2pb Ω2a )/(ω2pa + ω2pb )]1/2 .

(1)Here, the subscripts a and b indicate ion species, and Ω jand ω p j ( j = a or b) represent their cyclotron and plasma
frequencies, respectively. The high-frequency mode has afinite cut-oﬀ frequency given by
ω+0 = (ω2pa /Ω2a + ω2pb /Ω2b )Ωa Ωb |Ωe |/ω2pe .

(2)Here, the subscript e refers to the electrons.
Although the dispersion curves of the high- and lowfrequency modes are quite diﬀerent in the long-wavelength
region, Korteweg-de Vries (KdV) equations have been derived for both the low- and high-frequency modes [2]. The
KdV equation for the high-frequency mode is valid forwave amplitudes (me /mi )1/2  ε  1. The KdV equation for the low-frequency mode is valid when ε < 2∆ω ,where ∆ω is defined as∆ω = (ω+0 − ω−r )/ω+0 ,(3)The value of ∆ω increases with increasing Ωa /Ωb , whereΩa > Ωb is assumed. For a fixed Ωa /Ωb , ∆ω has its maximum value when the ion charge densities are equal, na qa =nb qb . In the nonlinear evolution of magnetosonic waves,∆ω is an important parameter[3]. In fact, electromagnetic
particle simulations demonstrated that high-frequencymode pulses are generated from a long-wavelength lowfrequency-mode pulse if its amplitude exceeds 2∆ω . In

this study, we extend the above work to the case of obliquemagnetosonic waves.
2. Theory for Low-frequency mode
We derive the condition for the KdV equation for thelow-frequency mode to be valid, assuming that the waves
propagate in the x direction in a magnetic field in a (x, z)
plane, B0 = B0 (cos θ, 0, sin θ). We consider the cases ofθ < θcl , where θcl is defined as cos2 θcl = 2∆ω /r withr = ω2−r (Ω2a + Ω2b )/(Ω2a Ω2b ) − 1.

(4)

It is found that the linear dispersion relation of the lowfrequency mode can be written asω = vA k(1 + dl2 k2 /2).

(5)

(l)
(l)
for small wave numbers k  kinf, Here, kinf
and dl are
given by
[]1/2
(l)
kinf
3 sin2 θ(r cos2 θ − 2∆ω )=.
(6)kc20∆ω − 2(r cos2 θ − 2∆ω )2dl = |2∆ω − r cos2 θ|1/2 /(kc sin θ)(7)with kc = ω−r /vA .
As expected from Eq. (5), the nonlinear behaviorof the low-frequency mode can be described by the KdV
equation [4], and the characteristic wavenumber of thesolitary pulse can be estimated as k ∼ ε1/2 /dl . The dispersion form (5) is valid in the long wavelength region,(l)k  kinf. Then, we obtain a condition for the amplitude of(l−)
the rarefactive pulse as ε  ε(l−)
max , where εmax is defined asε(l−)
max =3(r cos2 θ − 2∆ω )2.
20∆ω − 2(r cos2 θ − 2∆ω )2(8)The value of ε(l−)
max increases with decreasing θ from θcl . As
∆ω increases, ε(l−)
max increases.

3. Simulation of Nonlinear Evolution
We study nonlinear evolution of oblique magnetosonic waves with numerical simulations, using a onedimensional (one space coordinate and three velocity components), electromagnetic particle code with full ion andelectron dynamics. We simulate H-T and H-He plasmas. The hydrogen-to-electron mass ratio is taken to bemH /me = 100. Initially, we have a sinusoidal disturbanceof the low-frequency mode with a wavelength L x =4096∆g ,where ∆g is the grid spacing, propagating in the positive xdirection. The amplitude of the magnetic field Bz is chosen
to be δBz /B0 = 0.1.

Fig. 2 Wave evolution and power spectrum of magnetic field forθ = 60◦ in the H-T plasma with nH = nT .

Fig. 1 Wave evolution of the low-frequency mode for θ = 55◦ inthe H-T plasma with nH = nT .

We firstly study the rarefactive pulse of the lowfrequency mode in the H-T plasma with nH /nT = 1, taking
the propagation angle to be θ = 55◦ . The value of ε(l−)
maxfor this case is 0.48. Figure 1 shows magnetic field profiles at various times. As a result of nonlinear evolution,some pulses are formed. These are the rarefactive pulsesof the low-frequency mode. High-frequency-mode pulses
are not found. At ΩH t = 320, the left pulse has the amplitude ε = (Bmax − Bmin )/B0 = 0.24. Even though themaximum pulse amplitude ε is greater than δBz /B0 of theinitial disturbance, it does not exceed ε(l−)
max .
We next consider a case for which the KdV equation is not valid. Figure 2 shows magnetic field profilesat ΩH t = 208, 256 and 320 and power spectrum for thecase of θ = 60◦ for which ε(l−)
max =0.18. As the result of
the wave steepening, rarefactive pulses are formed, and theamplitude of the left pulse exceeds ε(l−)
max ; for example, atΩH t = 256, the amplitude is ε = (Bmax − Bmin )/B0 = 0.23.
After ΩH t = 256, perturbations are generated behind theleft pulse. The power spectrum P(k, ω) in Fig. 2 showsthat the low-frequency-mode waves with the wavenum(l)bers kinf< k ≤ kc are created, in addition to the longer(l)wavelength waves with k < kinf.
We finally simulate an H-He plasma with nHe /nH =0.1. Figure 3 shows the result of the case with θ = 66◦ .
The value of ε(l−)
max for this case is 0.005, which is muchsmaller than those for the cases of the above H-T plasmas.
Power spectrum clearly shows that the high-frequencymode waves with ω > ω+0 and k > kc are generated.

4. Summary
We have studied the nonlinear evolution of obliquemagnetosonic waves. First, we analytically obtained thecondition for the KdV equation for the low-frequency
mode to be valid. The upper limit of the amplitudes ε(l−)
maxhas been given as a function of θ and ∆ω . Next, electromagnetic particle simulations demonstrated that when theamplitude of the low-frequency-mode pulse exceeds ε(l−)
max ,
shorter-wavelength low- and high-frequency-mode waves
are generated.
References[1] S. J. Buchsbaum, Phys. Fluids 3, 418 (1960).
[2] M. Toida and Y. Ohsawa, J. Phys. Soc. Jpn. 63, 573, (1994).
[3] M. Toida, H. Higashino, and Y. Ohsawa, J. Phys. Soc. Jpn.
76, 104502 (2007).
[4] S. Irie and Y. Ohsawa, Phys. Plasmas 10 1255 (2003).

Fig. 3 Wave evolution and power spectrum of magnetic field forθ = 66◦ in the H-He plasma with nH = 10nHe