Heavy-Ion Acceleration by Nonlinear Magnetosonic Waves in a Two-Ion-Species Plasma with a Large Difference in Cyclotron Frequencies

Abstract

In a two-ion-species plasma, a magnetosonic wave splits into two distinct modes: a low-frequency mode and a high-frequency mode. Nonlinear solitary waves corresponding to these modes can accelerate heavy ions in a plasma composed of hydrogen (H) and a heavier ion species (denoted by b). In this study, we theoretically examine the dependence of the heavy-ion acceleration on the heavy-ion mass (m_b) and the difference between cyclotron frequencies Ω_H and Ω_b. When the pulse amplitude is fixed, the heavy-ion acceleration weakens as m_b increases for both the low- and high-frequency-mode pulses. However, for the low-frequency-mode pulse, a maximum attainable amplitude exists, and this amplitude increases as the ratio Ω_H/Ω_b becomes larger. Consequently, in a plasma containing heavy ions with larger m_b, the low-frequency-mode pulse can have a larger amplitude, leading to the enhancement of the heavy-ion acceleration. These results indicate that the low-frequency-mode pulse can accelerate, for instance, oxygen (O) ions in an H-O plasma more effectively than helium (He) ions in an H-He plasma.

1.  Introduction

The presence of multiple ion species is common in both fusion and space plasmas. Therefore, understanding their effects on the propagation of magnetosonic waves and the associated energy transfer has been a long-standing issue [1–9]. It has been shown that nonlinear magnetosonic waves exhibit many properties unique to multi-ion-species plasmas [10–18].

In a plasma consisting of two ion species and electrons, the magnetosonic wave splits into two modes: a low-frequency mode and a high-frequency mode. The nonlinear behavior of these modes can be described by the Korteweg-de Vries (KdV) equations [10]. The characteristic soliton width of the high-frequency mode is on the order of the electron skin depth $c/{\omega }_{\mathrm{pe}}$, while that of the low-frequency mode is on the order of the ion inertial length $c/{\omega }_{\mathrm{p}i}$. The KdV equation for the high-frequency mode is valid for wave amplitudes ${(m_e/m_i)}^{1/2}\ll \epsilon \ll 1$, where $m_e/m_i$ is the electron-to-ion mass ratio. In contrast, the KdV equation for the low-frequency mode is valid when the amplitude is relatively small. The upper limit of the amplitudes, ${\epsilon }_{\max }$, depends on the ion composition [16]. Numerical simulations have demonstrated that when the amplitude of a low-frequency-mode pulse exceeds the threshold ${\epsilon }_{\max }$, high-frequency-mode pulses are generated from the low-frequency-mode pulse.

In a two-ion-species plasma where the density of hydrogen (H) ions is greater than that of heavy ions, the high-frequency-mode pulse accelerates heavy ions through the transverse electric field in the pulse [12]. This acceleration occurs in both a large-amplitude shock-like pulse and a small-amplitude soliton-like pulse, and it is more pronounced in the former case. The heavy-ion acceleration by the low-frequency-mode pulse was also investigated [15], but it was found to be significantly weaker than that caused by the high-frequency-mode pulse.

The previous studies have mainly considered hydrogen-helium (H-He) plasmas with a cyclotron frequency ratio of ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{\mathrm{He}}$ = 2, where the He charge number is 2. This choice reflects the average composition of space and astrophysical plasmas, where the abundance of He is approximately 10% of that of H, and other ions are far less abundant. Therefore, the wave structure is expected to be determined mainly by the H and He ions. In Ref. [13], small fractions of oxygen (O) and iron (Fe) ions, with ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{\mathrm{O}}$ = 16/7 and ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{\mathrm{Fe}}$ = 4 corresponding to the charge states at the average temperature of the solar corona, were introduced in the simulation of a large-amplitude shock wave. It was demonstrated that He, O, and Fe ions are accelerated to nearly the same speed by the high-frequency-mode shock wave. Furthermore, to examine effects of ion composition on ion acceleration, Ref. [17] considered deuterium-tritium (D-T) and hydrogen-tritium (H-T) plasmas. In these two-ion-species plasmas, the two ion cyclotron frequencies are of the same order, as in H-He plasmas. Such a condition is relevant to a wide range of fusion, space, and astrophysical plasma environments.

However, there also exist plasmas in which the ion cyclotron frequencies of the two main components are not of the same order of magnitude. For example, in the Earth’s magnetosphere, the primary ion components are H⁺ and O⁺, whose cyclotron frequencies differ significantly, with ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{\mathrm{O}}$ ≃ 16. Near comets, H⁺ and water-molecule ions H₂O⁺ can be dominant, where ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{{\mathrm{H}}_2\mathrm{0}}$ ≃ 18 (reviews on ion states in the Earth’s magnetosphere and near comets can be found in, for example, Refs. [19, 20], respectively). Such large differences in cyclotron frequencies can substantially affect the characteristics of magnetosonic waves. Furthermore, acceleration of heavy ions, such as O⁺ in the Earth’s magnetosphere and H₂O⁺ near comets, has been observed [19, 20], although the underlying mechanisms remain not fully understood. It is therefore important to examine whether magnetosonic waves can effectively accelerate heavy ions in plasmas where the cyclotron frequency difference between the dominant ion species is significant.

In this paper, we theoretically analyze heavy-ion motion in solitary waves for the low- and high-frequency modes in a two-ion-species plasma composed of H and a heavier ion species (denoted by $b$). Assuming that the pulse amplitude is fixed, we find that heavy-ion acceleration weakens as the heavy-ion mass ($m_b$) increases and the difference between the two ion cyclotron frequencies, ${\mathrm{\Omega }}_{\mathrm{H}}$ and ${\mathrm{\Omega }}_b$, becomes larger. This tendency holds for both high- and low-frequency-mode pulses. However, for the low-frequency mode, the maximum pulse amplitude strongly depends on the ratio ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_b$, increasing as this ratio becomes larger. As a result, the low-frequency-mode pulse can have a larger amplitude in a plasma containing heavy ions with larger $m_b$, leading to enhanced heavy-ion acceleration. Our findings suggest that the low-frequency-mode pulse can accelerate, for instance, O ions in an H-O plasma more effectively than He ions in an H-He plasma.

The rest of this paper is structured as follows. Section 2 reviews the theory of the low- and high-frequency-mode solitary waves based on the KdV equations and describes heavy-ion acceleration by these pulses, using an H-He plasma as an example. In Sec. 3, we investigate heavy-ion acceleration in plasmas with large differences in cyclotron frequencies, such as H-O plasmas. After presenting the wave properties, we show numerical results for heavy-ion motion in the high- and low-frequency-mode pulses. We then use theoretical formulas to examine how the heavy-ion velocity induced by the low-frequency-mode pulse depends on parameters, such as the heavy-ion mass, charge number, and density. Section 4 provides a summary of our findings.

2.  Theory for Magnetosonic Waves and Heavy-Ion Acceleration

We firstly review the theory for nonlinear magnetosonic waves in a two-ion-species plasma and associated heavy ion acceleration.

2.1  Low- and high-frequency modes

In a two-ion-species plasma, a magnetosonic wave splits into two modes; a high-frequency mode (${\omega }_{+}$) and a low-frequency mode (${\omega }_{-}$). Figure 1 shows the linear dispersion curves for these modes propagating perpendicular to the magnetic field in a hydrogen-helium (H-He) plasma with a density ratio of $n_{\mathrm{He}}/n_{\mathrm{H}}$ = 0.1 and a He charge number of $Z$ = 2. For simplicity, a cold-plasma model is adopted, and the displace current is neglected.

Fig. 1.  Linear dispersion curves of the low- and high-frequency modes (${\omega }_{\mathrm{-}}$ and ${\omega }_{\mathrm{+}}$) propagating perpendicular to the magnetic field in an H-He plasma with a density ratio of $n_{\mathrm{He}}/n_{\mathrm{H}}$ = 0.1 and a He charge number of $Z$ = 2.

The frequency of the low-frequency mode is within the range $0\le \omega \le {\omega }_{-\mathrm{r}}$, where ${\omega }_{-\mathrm{r}}$ is the ion-ion hybrid resonance frequency [1] defined as

  

$${\omega }_{-\mathrm{r}}={\left(\frac{{\omega }_{\mathrm{p}a}^2{\mathrm{\Omega }}_b^2+{\omega }_{\mathrm{p}b}^2{\mathrm{\Omega }}_a^2}{{\omega }_{\mathrm{p}a}^2+{\omega }_{\mathrm{p}b}^2}\right)}^{1/2}.$$(1)

Here, the subscripts $a$ and $b$ denote the two ion species and ${\mathrm{\Omega }}_i$ and ${\omega }_{\mathrm{p}i}$ ($i$ = $a$ or $b$) represent their cyclotron and plasma frequencies, respectively. We assume that $b$ is heavier than $a$ and ${\mathrm{\Omega }}_a>{\mathrm{\Omega }}_b$. The high-frequency mode has a finite cut-off frequency given by

  

$${\omega }_{+0}=\left(\frac{{\omega }_{\mathrm{p}a}^2}{{\mathrm{\Omega }}_a^2}+\frac{{\omega }_{\mathrm{p}b}^2}{{\mathrm{\Omega }}_b^2}\right)\frac{{\mathrm{\Omega }}_a{\mathrm{\Omega }}_b|{\mathrm{\Omega }}_{\mathrm{e}}|}{{\omega }_{\mathrm{pe}}^2},$$(2)

where the subscript e refers to electrons. Since ${\omega }_{+0}$ is slightly greater than ${\omega }_{-\mathrm{r}}$, the magnetosonic wave cannot propagate in the frequency gap ${\omega }_{-\mathrm{r}}<\omega <{\omega }_{+0}$. We define the normalized frequency gap ${\mathrm{\Delta }}_{\omega }$ as

  

$${\mathrm{\Delta }}_{\omega }=({\omega }_{+0}-{\omega }_{-\mathrm{r}})/{\omega }_{+0}.$$(3)

We also introduce the variable $r_{ab}$, defined as,

  

$$r_{ab}\equiv \frac{{\omega }_{+0}^2-{\omega }_{-\mathrm{r}}^2}{{\omega }_{+0}^2}=\frac{{\omega }_{\mathrm{p}a}^2{\omega }_{\mathrm{p}b}^2{({\mathrm{\Omega }}_a-{\mathrm{\Omega }}_b)}^2}{({\omega }_{\mathrm{p}a}^2+{\omega }_{\mathrm{p}b}^2)({\omega }_{\mathrm{p}a}^2{\mathrm{\Omega }}_b^2+{\omega }_{\mathrm{p}b}^2{\mathrm{\Omega }}_a^2)}.$$(4)

As ${\mathrm{\Delta }}_{\omega }$ increases, $r_{ab}$ also increases. When ${\omega }_{\mathrm{p}a}$ or ${\omega }_{\mathrm{p}b}$ is negligibly small or when ${\mathrm{\Omega }}_a$ and ${\mathrm{\Omega }}_b$ are close to each other, the difference between ${\omega }_{-\mathrm{r}}$ and ${\omega }_{+0}$ becomes small and $r_{ab}$ can be approximated as $2{\mathrm{\Delta }}_{\omega }$.

The dispersion curves for both low- and high-frequency modes have large curvatures near the wavenumber $k_c$ defined as

  

$$k_c={\omega }_{-\mathrm{r}}/v_{\mathrm{A}}.$$(5)

Here, $v_{\mathrm{A}}$ is the Alfvén speed,

  

$$v_{\mathrm{A}}=c/{({\omega }_{\mathrm{p}a}^2/{\mathrm{\Omega }}_a^2+{\omega }_{\mathrm{p}b}^2/{\mathrm{\Omega }}_b^2)}^{1/2},$$(6)

where the electron mass is neglected. In the long wave-wavelength region $k\ll k_c$, the linear dispersion relation for the low-frequency mode is given by

  

$$\omega =v_{\mathrm{A}}k(1-d_l^2{k}^2/2),$$(7)

where $d_l$ is defined as

  

$$d_l^2=\frac{v_A^6}{c^4}\left[\frac{{\omega }_{\mathrm{p}a}^2{\omega }_{\mathrm{p}b}^2}{{\mathrm{\Omega }}_a^2{\mathrm{\Omega }}_b^2}{\left(\frac{1}{{\mathrm{\Omega }}_a}-\frac{1}{{\mathrm{\Omega }}_b}\right)}^2\right].$$(8)

For the expression of $d_l$ including effects of electron mass, see Eq. (23) in Ref. [10]. When ${\omega }_{\mathrm{p}a}\sim {\omega }_{\mathrm{p}b}$, $d_l$ is on the order of ion inertial length. Using $r_{ab}$ and $k_c$, $d_l$ is rewritten as

  

$$d_l=r_{ab}^{1/2}/k_c.$$(9)

For the high-frequency mode, in the range of wavenumbers,

  

$${(m_e/m_i)}^{1/2}\ll ck/{\omega }_{\mathrm{pe}}\ll 1,$$(10)

the liner dispersion relation is approximated as

  

$$\omega =v_{\mathrm{h}}k[1-c^2{k}^2/(2{\omega }_{\mathrm{pe}}^2)],$$(11)

where $v_h$ is the characteristic phase velocity of this mode,

  

$$v_{\mathrm{h}}=\frac{{({\omega }_{\mathrm{p}a}^2+{\omega }_{\mathrm{p}b}^2)}^{1/2}|{\mathrm{\Omega }}_{\mathrm{e}}|c}{{\omega }_{\mathrm{pe}}^2}=v_{\mathrm{A}}\frac{{\omega }_{+0}}{{\omega }_{-\mathrm{r}}}.$$(12)

This equation indicates that $v_h$ is slithgly greater than $v_A$.

As expected from Eqs. (7) and (11), the nonlinear behavior of both the low- and high-frequency modes can be described by the KdV equation [10]. When the waves propagate in the x direction and the magnetic field is in the z direction, the solitary-wave solution is given by

  

$$\frac{(B_z-B_0)}{B_0}=b_n{\mathrm{sech}}^2\left[\frac{(x-M{v}_{\mathrm{p}0}t)}{D}\right].$$(13)

Here, $b_n$ is the normalized amplitude, $M$ is the Mach number, and $D$ is the characteristic width,

  

$$D={[2/(M-1)]}^{1/2}d=2d/{(\alpha b_n)}^{1/2},$$(14)

where we used the relation between $b_n$ and $M$,

  

$$b_n=2(M-1)/\alpha .$$(15)

The values of $v_{\mathrm{p}0}$, $\alpha$, and $d$ are

  

$$v_{\mathrm{p}0}=v_{\mathrm{A}},\alpha =1,d=d_l,$$(16)

for the low-frequency mode, and

  

$$v_{\mathrm{p}0}=v_{\mathrm{h}},\alpha =1+r_{ab}\frac{{\omega }_{-\mathrm{r}}^2}{{\mathrm{\Omega }}_a{\mathrm{\Omega }}_b},d=\frac{c}{{\omega }_{\mathrm{pe}}},$$(17)

for the high-frequency mode. Equations (16) and (17) indicate that the solitary pulse width for the low-frequency mode is much larger than that for the high-frequency mode when the two pulses have the same amplitude.

The longitudinal electric field $E_x$ in the pulse is written as

  

$$\frac{E_x(x,t)}{v_{\mathrm{p0}}{B}_0/c}=R{b}_{\mathrm{n}}^{3/2}{\mathrm{sech}}^2\left(\frac{x-M{v}_{\mathrm{p0}}t}{D}\right)\tanh \left(\frac{x-M{v}_{\mathrm{p0}}t}{D}\right),$$(18)

where the coefficient $R$ is given by

  

$$R=\frac{v_{\mathrm{A}}^3}{c^2{d}_l}\sum _j\frac{{\omega }_{pj}^2}{{\mathrm{\Omega }}_j^3},$$(19)

for the low-frequency mode, and

  

$$R={\alpha }^{1/2}{\omega }_{\mathrm{pe}}/{({\omega }_{\mathrm{p}a}^2+{\omega }_{\mathrm{p}b}^2)}^{1/2},$$(20)

for the high-frequency mode. The transverse electric field $E_y$ is given by

  

$$\frac{E_y(x,t)}{v_{\mathrm{p0}}{B}_0/c}=b_{\mathrm{n}}{\mathrm{sech}}^2\left(\frac{x-M{v}_{\mathrm{p0}}t}{D}\right).$$(21)

The above KdV theory is valid under the following conditions. For the high-frequency mode, the KdV equation is derived under the condition that ${(m_e/m_i)}^{1/2}\ll \epsilon \ll 1$, where $\epsilon$ is the pulse amplitude. For the low-frequency mode, the KdV equation is valid when

  

$$\epsilon <{\epsilon }_{\text{max}}=r_{ab}.$$(22)

The maximum amplitude, ${\epsilon }_{\max }$, significantly depends on the ion composition. In Ref. [16], numerical simulations demonstrated that when the amplitude of the low-frequency-mode pulse is smaller than ${\epsilon }_{\max }$, the pulse propagates steadily, but when it exceeds ${\epsilon }_{\max }$, high-frequency-mode pulses are generated from the low-frequency-mode pulse.

2.2  Heavy-ion acceleration

The equations of motion for particle $j$ in a nonlinear solitary pulse are written as

  

$$\frac{\mathrm{d}x}{\mathrm{d}t}=v_{jx},$$(23)

  

$$m_j\frac{\mathrm{d}{v}_{jx}}{\mathrm{d}t}=q_j\left[E_x(x,t)+\frac{v_{jy}}{c}{B}_z(x,t)\right],$$(24)

  

$$m_j\frac{\mathrm{d}{v}_{jy}}{\mathrm{d}t}=q_j\left[E_y(x,t)-\frac{v_{jx}}{c}{B}_z(x,t)\right].$$(25)

In a single-ion-species plasma, the relation $E_y-v_{jx}{B}_z/c$ ≃ 0 holds for both ions ($j=i$) and electrons ($j=e$) in a nonlinear magnetosonic wave [21–23]. This is because the drift approximation is sufficiently accurate for electrons ($v_{ex}\simeq c{E}_y/B_z$). Under the quasi-neutrality condition, the ion velocity $v_{ix}$ is nearly equal to the electron velocity $v_{ex}$. Therefore, the relation $E_y-v_{ix}B/c$ ≃ 0 is also satisfied for ions, even though the drift approximation itself is not applicable to them. This implies that the force acting on ions in the y-direction, $m_{i}d{v}_{iy}/dt$, is negligibly small in a nonlinear magnetosonic wave.

In a two-ion-species plasma containing hydrogen ions (H) and heavier ion species ($b$), the relation $v_{ex}\simeq v_{\mathrm{H}x}$ is satisfied when H is the dominant component. As a result, $m_{\mathrm{H}}d{v}_{\mathrm{H}y}/dt$ is also negligibly small. However, for heavy ions, the velocity $v_{bx}$ is smaller than $v_{\mathrm{H}x}$ due to their greater inertia. This means that the relation $E_y-v_{bx}{B}_z/c$ > 0 holds for heavy ions and $v_{by}$ can increase as a result of the acceleration by the transverse electric field $E_y$.

To explain the heavy-ion motion in more detail, we numerically solve Eqs. (23)–(25), substituting Eqs. (13), (18), and (21). Figure 2(a) shows the time evolution of $v_{by}$ in the high-frequency-mode pulse with an amplitude of $b_n$ = 0.2. Here, we consider the motion of a He ion in an H-He plasma with the density ratio $n_{\mathrm{He}}/n_{\mathrm{H}}$ = 0.1, where the He charge number is 2. The initial condition is that the He ion is located in the far upstream region, $x=x_0\gg D$, with its velocity $v_{bx}=v_{by}$ = 0. The gray line shows the magnetic field $B_z$ experienced by the He ion at its position $x(t)$ at time $t$. (Since the horizontal axis is $x-M{v}_{\mathrm{h}}t$, the He ion is observed to move from the right to the left as the time advances.) The transverse electric field $E_y$ has almost the same profile as $B_z$. It is seen that $v_{by}$ increases within the pulse due to the acceleration by $E_y$. The acceleration time $\mathrm{\Delta }t$, which is approximately the transit time of the He ion through the pulse, is on the order of $D/M{v}_{\mathrm{h}}$ and is much shorter than the cyclotron period $2\pi /{\mathrm{\Omega }}_b$. The value of $v_{by}$ reaches its maximum, $v_{by}$ = 8.4 × 10⁻³ $v_{\mathrm{h}}$, just behind the pulse and then gradually decreases due to gyromotion. In Ref. [12], the theoretical expression for the maximum speed was derived as

  

$$\frac{v_{bm}}{v_{\mathrm{h}}}=\frac{4}{{\alpha }^{1/2}}\frac{{\omega }_{\mathrm{pe}}{\omega }_{\mathrm{p}a}^2{\mathrm{\Omega }}_b}{{({\omega }_{\mathrm{p}a}^2+{\omega }_{\mathrm{p}b}^2)}^{3/2}{\mathrm{\Omega }}_{\mathrm{e}}}\left(1-\frac{{\mathrm{\Omega }}_b}{{\mathrm{\Omega }}_a}\right)b_n^{1/2}.$$(26)

The predicted value based on Eq. (26), $v_{bm}$ = 9.2 × 10⁻³ $v_{\mathrm{h}}$, agrees with the observed value in Fig. 2 with an error 10%.

Fig. 2.  Time variation of the He-ion velocity $v_{by}$ in the high-frequency-mode pulse with the amplitude $b_n$ = 0.2 (a), and in the low-frequency-mode pulse with $b_n$ = 0.02 (b). The gray lines in both panels represent the magnetic field experienced by the He ion.

Figure 2(b) shows the variation of $v_{by}$ in the low-frequency-mode pulse. As indicated by Eq. (22), the maximum amplitude ${\epsilon }_{\max }$ exists for this pulse, and it depends on the ion composition. For the H-He plasma with $n_{\mathrm{He}}/n_{\mathrm{H}}$ = 0.1, ${\epsilon }_{\max }$ ≃ 0.065. Therefore, we choose the amplitude of this pulse to be $b_n$ = 0.02, which is smaller than ${\epsilon }_{\text{max}}$. Equation (14) indicates that the width of this pulse is $D$ ≃ 2.5 × 10² $c/{\omega }_{\mathrm{pe}}$. (Correspondingly, the horizontal axis of Fig. 2 (b) is about 50 times wider than that of Fig. 2(a) for the high-frequency-mode pulse.) Because of this wide width, the transit time of the He ion through the low-frequency-mode pulse can be on the order of $2\pi /{\mathrm{\Omega }}_b$. As a result, the He ion can undergo gyromotion a few times while traversing the pulse region, which leads to the oscillation of $v_{by}$. After passing through the pulse, the He ion gyrates with a speed of $v_{bm}$ ≃ 2.7 × 10⁻³ $v_{\mathrm{A}}$, even though its initial speed was assumed to be zero. The theoretical expression for $v_{bm}$ caused by the low-frequency-mode pulse is approximately given by

  

$$\begin{array}{c}{v}_{bm}=4\pi v_{\mathrm{A}}\frac{n_{b0}{m}_b{n}_{a0}{m}_a}{M^3{\rho }_0^2}{\left(1-\frac{{\mathrm{\Omega }}_b}{{\mathrm{\Omega }}_a}\right)}^2\hfill \\ \times \left[\frac{n_a{m}_a}{{\rho }_0}\left(1-\frac{{\mathrm{\Omega }}_b}{{\mathrm{\Omega }}_a}\right)+\frac{b_n}{2}\right]\hfill \\ \times \mathrm{cosech}\left[\frac{\pi {(n_{a0}{m}_a{n}_{b0}{m}_b)}^{1/2}}{{\rho }_0{b}_n^{1/2}(1+b_n/2)}\left(1-\frac{{\mathrm{\Omega }}_b}{{\mathrm{\Omega }}_a}\right)\right].\end{array}$$(27)

In the limit of $b_n\to 0$ in the first and second lines of Eq. (27), this equation reduces to Eq. (21) in Ref. [15]. Equation (27) gives $v_{bm}$ ≃ 3.8 × 10⁻³ $v_{\mathrm{A}}$. Although this theoretical value overestimates the value observed in Fig. 2(b) by approximately 40%, Eq. (27) is sufficient for estimating the order of magnitude of $v_{bm}$.

3.  The Case for Large Difference in Ion Cyclotron Frequencies

We present the results for the case where the masses of the two-ion species, and consequently their cyclotron frequencies, are significantly different.

3.1  Wave properties

Figure 3 shows the linear dispersion curves of the high- and low-frequency modes in a hydrogen-oxygen (H-O) plasma with a density ratio of $n_{\mathrm{O}}/n_{\mathrm{H}}$ = 0.1 and an oxygen charge number of $Z$ = 1. The cyclotron frequency ratio, ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{\mathrm{O}}$ = 16, is much larger than the corresponding ratio in Fig. 1 for the H-He plasma (${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_{\mathrm{He}}$ = 2). It is evident from Figs. 1 and 3 that both ${\omega }_{-\mathrm{r}}$ and ${\omega }_{+0}$ are lower in the H-O than in the H-He plasma. The most notable difference is that the frequency gap, ${\omega }_{+0}-{\omega }_{-\mathrm{r}}$, is significantly wider for the H-O plasma compared to the H-He plasma. The normalized frequency gap ${\mathrm{\Delta }}_{\omega }=({\omega }_{+0}-{\omega }_{-\mathrm{r}})/{\omega }_{+0}$ is 0.27 for the H-O plasma and 0.032 for the H-He plasma.

Fig. 3.  Linear dispersion curves of the low- and high-frequency modes in an H-O plasma with a density ratio of $n_{\mathrm{O}}/n_{\mathrm{H}}$ = 0.1 and an O charge number of $Z$ = 1.

As described in Sec. 2, we introduced the important parameter $r_{ab}=({\omega }_{+0}^2-{\omega }_{-\mathrm{r}}^2)/{\omega }_{+0}^2$, which is defined by Eq. (4). The value of $r_{ab}$ increases as ${\mathrm{\Delta }}_{\omega }$ increases. As shown by Eqs. (9) and (22), $r_{ab}$ is a key factor determining the properties of the nonlinear solitary wave for the low-frequency mode. Specifically, the characteristic pulse width $d_l$ is proportional to $r_{ab}^{1/2}$, and the maximum amplitude for the KdV equation to be valid is given by ${\epsilon }_{\max }=r_{ab}$. The value of $r_{ab}$ depends on the ion composition.

Figure 4 presents $r_{ab}$ as a function of the normalized charge density for H-He plasmas (left panel) and for H-O plasmas (right panel). The curves correspond to various charge numbers $Z$ for He and O. The values of $Z$ and the cyclotron frequency ratio ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_b$ (where $a$ denotes H and $b$ denotes He or O) are listed in parentheses. It is found that when the ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_b$ ratio is the same for both the H-He and H-O plasmas, the $r_{ab}$ profiles as functions of $Z{n}_b/n_e$ are identical, regardless of the difference in $Z$. For a fixed ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_b$, $r_{ab}$ peaks at $Z{n}_b/n_e$ = 0.5. Conversely, for a fixed normalized density $Z{n}_b/n_e$, $r_{ab}$ increases with ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_b$. This indicates that a larger difference between ${\mathrm{\Omega }}_a$ and ${\mathrm{\Omega }}_b$ results in a higher value of $r_{ab}$. The ratio ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_b$ is at most 4 in the H-He plasma but reaches 16 in the H-O plasma. Consequently, the maximum value of $r_{ab}$ for the H-O plasma (${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_b$ = 16) is 0.75, which is much greater than any $r_{ab}$ for the H-He plasma. Because the maximum amplitude of the low-frequency-mode pulse is estimated as $r_{ab}$, this result clearly shows that the pulse amplitude in the H-O plasma can be significantly greater than that in the H-He plasma.

Fig. 4.  The value of $r_{ab}$ as a function of the normalized charge density for H-He plasmas (left panel) and for H-O plasmas (right panel). The curves correspond to various charge numbers $Z$ for He and O.

3.2  Oxygen ion acceleration

We present results of the numerical calculations based on Eqs. (23)–(25) for O-ion acceleration by nonlinear solitary pulses in an H-O plasma. The O charge number is $Z$ = 1, and the density ratio is $n_{\mathrm{O}}/n_{\mathrm{H}}$ = 0.1 (thus $Z{n}_{\mathrm{O}}/n_{\mathrm{e}}$ = 0.09). Figure 5(a) shows the time variation of $v_{by}$ for an O ion in the high-frequency-mode pulse with an amplitude of $b_n$ = 0.2. This amplitude is the same as that used in Fig. 2(a), where the He ion motion is shown. Similar to the He ion in the H-He plasma, the O ion is also accelerated in the y direction by the transverse electric field $E_y$ within the pulse. However, the maximum velocity $v_{bm}$ of the O ion, $v_{bm}$ ≃ 2.2 × 10⁻³ $v_{\mathrm{h}}$, is only one-fourth of that of the He ion in Fig. 2. This value of $v_{bm}$ is in good agreement with the theoretical value, $v_{bm}$ ≃ 2.4 × 10⁻³ $v_{\mathrm{h}}$, given by Eq. (26).

Fig. 5.  Time variation of the O-ion velocity $v_{by}$ in the high-frequency-mode pulse with the amplitude $b_n$ = 0.2 (a), in the low-frequency-mode pulse with $b_n$ = 0.02 (b), and in the low-frequency-mode pulse with a greater amplitude of $b_n$ = 0.2 (c).

Figure 5(b) shows the O-ion motion in the low-frequency-mode pulse with an amplitude of $b_n$ = 0.02. Although this amplitude is the same as that used in Fig. 2(b) for the H-He plasma, the pulse width in the H-O plasma is approximately ten times larger. This difference arises from the variation in $d_l$ given by Eq. (9): for the H-O plasma, $d_l$ ≃ 200 $c/{\omega }_{\mathrm{pe}}$, whereas for the H-He plasma, $d_l$ ≃ 18 $c/{\omega }_{\mathrm{pe}}$. A comparison between the He and O ions reveals a notable difference. The O-ion $v_{by}$ reaches 8 × 10⁻³ $v_{\mathrm{A}}$ within the pulse, which is greater than the maximum $v_{by}$ ≃ 2 × 10⁻³ $v_{\mathrm{A}}$ for the He ion. While the He ion continues to gyrate at nearly this speed after passing through the pulse, the O-ion speed decreases substantially to $v_{bm}$ ∼ 4 × 10⁻⁵ $v_{\mathrm{A}}$ behind the pulse. Consequently, the net acceleration of O ions is considerably weaker than that of He ions for the same pulse amplitude, $b_n$ = 0.02.

However, as discussed in the previous subsection, the low-frequency-mode pulse can have a much larger amplitude in the H-O plasma than in the H-He plasma. The maximum amplitude is ${\epsilon }_{\text{max}}$ ≃ 0.53 for the H-O plasma with $n_{\mathrm{O}}/n_{\mathrm{H}}$ = 0.1, whereas it is only 0.063 for the H-He plasma with $n_{\mathrm{He}}/n_{\mathrm{H}}$ = 0.1. Figure 5(c) shows the O-ion motion in the low-frequency-mode pulse with an amplitude of $b_n$ = 0.2. The pulse width, about one-third of that in Fig. 5(b), is given by Eq. (14), which predicts that $D\propto b_n^{-1/2}$. Comparing $v_{by}$ in Figs. 5(b) and (c), the maximum $v_{by}$ within the large-amplitude pulse, $v_{by}$ ≃ 3 × 10⁻² $v_{\mathrm{A}}$, is about three times higher than within the small-amplitude pulse. After passing through the large-amplitude pulse, the O ion continues to gyrate at nearly constant speed, $v_{bm}$ ≃ 4 × 10⁻² $v_{\mathrm{A}}$, which is three orders of magnitude greater than behind the small-amplitude pulse. Therefore, in the H-O plasma, the low-frequency-mode pulse can effectively accelerate O ions due to its larger attainable amplitude.

We discuss the physical origin of the difference between the acceleration of O and He ions. As briefly described below Eq. (25), the transverse electric field $E_y$ primarily causes the acceleration, which can be roughly estimated as

  

$$\mathrm{\Delta }{v}_{by}\sim \frac{q_b}{m_b}{E}_y\mathrm{\Delta }t.$$(28)

Here, $E_y\propto b_n$, based on Eq. (21), and the ion transit time through the pulse is approximated as $\mathrm{\Delta }t\sim D/v_{\mathrm{p}0}$, where the change in $v_{bx}$ is neglected. Under these assumptions, we obtain

  

$$\mathrm{\Delta }{v}_{by}\propto \frac{q_b}{m_b}\frac{D}{v_{\mathrm{p}0}}{b}_n\simeq \frac{q_b}{m_b}\frac{d}{v_{\mathrm{p}0}}{b}_n^{1/2},$$(29)

where $D\simeq d/b_n^{1/2}$ is used. Equation (29) shows that when the characteristic length $d$ and the amplitude $b_n$ are fixed, the acceleration weakens with decreasing $q_b/m_b$. This is valid for the high-frequency-mod pulse because $d$ (= $c/{\omega }_{\mathrm{pe}}$) is independent of $q_b/m_b$. In contrast, for the low-frequency mode, $d$ (= $r_{ab}^{1/2}/k_c$) increases as $q_b/m_b$ decreases, and we obtain

  

$$\mathrm{\Delta }{v}_{by}\propto \frac{q_b}{m_b}{r}_{ab}^{1/2}{b}_n^{1/2}.$$(30)

The values of $r_{ab}^{1/2}$ are 0.73 and 0.25 for the H-O plasma with $n_{\mathrm{O}}/n_{\mathrm{H}}$ = 0.1 (Fig. 5) and the H-He plasma with $n_{\mathrm{He}}/n_{\mathrm{H}}$ = 0.1 (Fig. 2), respectively. Although $r_{ab}^{1/2}$ is larger for the H-O plasma, the value of $(q_b/m_b)r_{ab}^{1/2}$ for O ions is smaller than that for He ions. This explains why, when the pulse amplitude $b_n$ is fixed, the O-ion acceleration is weaker than the He-ion acceleration even though the pulse width is wider in the H-O plasma. However, for the low-frequency mode, the pulse amplitude can become larger in the H-O plasma than in the H-He plasma, which enhances the O-ion acceleration. For the maximum amplitude $b_n=r_{ab}$, we obtain $\mathrm{\Delta }{v}_{by}\propto (q_b/m_b)r_{ab}$, indicating that $\mathrm{\Delta }{v}_{by}$ for O ions can exceed that for He ions. Therefore, the increasing amplitude due to $r_{ab}$ can result in enhanced O-ion acceleration.

We note that Eq. (29) provides only an extremely simplified estimate. More rigorous expressions for $v_{by}$, which include the variation of $v_{bx}$ in the pulse, are given by Eqs. (26) and (27). In the next section, we use these equations to systematically investigate the parameter dependence of heavy-ion acceleration.

3.3  Parameter dependence

In this subsection, we consider a plasma composed of H and a heavy-ion species $b$. Using the theoretical expressions for $v_{bm}$ given by Eqs. (26) and (27), we discuss how heavy-ion acceleration depends on parameters such as the heavy-ion mass $m_b$, charge number $Z$, and density $n_b$.

Figure 6(a) shows the variation of the heavy-ion speed $v_{bm}$, induced by the high-frequency-mode pulse with an amplitude of $b_n$ = 0.2, as a function of the mass ratio $m_b/m_{\mathrm{H}}$. The solid and gray lines correspond to charge numbers $Z$ = 1 and 2, respectively, while the density ratio is fixed at $n_b/n_{\mathrm{H}}$ = 0.1. As shown in the figure, $v_{bm}$ is larger for $Z$ = 2 than for 1, and it decreases with increasing $m_b$. These results confirm that heavy-ion acceleration by the high-frequency-mode pulse becomes inefficient when the difference between the ion cyclotron frequencies is large.

Fig. 6.  Heavy-ion mass dependence of $v_{bm}$ induced by the high-frequency-mode pulse with an amplitude of $b_n$ = 0.2 (a) and by the low-frequency-mode pulse with an amplitude of $b_n$ = 0.02 (b). The solid and gray lines represent charge numbers $Z$ = 1 and 2, respectively, while the density ratio is fixed at $n_b/n_{\mathrm{H}}$ = 0.1.

Figure 6(b) presents the dependence of $v_{bm}$ on the heavy-ion mass for the low-frequency-mode pulse with $b_n$ = 0.02. Consistent with the high-frequency-mode case, the $v_{bm}$ induced by the low-frequency-mode pulse is higher for $Z$ = 2 than for 1, and it rapidly decreases with $m_b$. Thus, for a fixed pulse amplitude, $v_{bm}$ decreases with increasing $m_b$ for both the high- and low-frequency-mode pulses.

We now discuss how the $v_{bm}$ induced by the low-frequency-mode pulse varies with the amplitude $b_n$. Figure 7 shows $v_{bm}$ for O ions as a function of $b_n$. The red, green, and gray lines correspond to charge numbers $Z$ = 1, 2, and 4, respectively. For comparison, the $v_{bm}$ for He ions with $Z$ = 2 is also plotted (blue line). The range of $b_n$ is determined by Eq. (22), which indicate that its upper limit is given by ${\epsilon }_{\max }=r_{ab}$.

Fig. 7.  Amplitude dependence of $v_{bm}$ induced by the low-frequency-mode pulse for O ions in plasmas with different density ratios ($n_b/n_{\mathrm{H}}$ = 0.05, 0.1, and 0.2). The lines for the O ions correspond to charge numbers $Z$ = 1 (red), 2 (green), and 4 (gray). For comparison, $v_{bm}$ for He ions with $Z$ = 2 is plotted (blue line).

The bottom panel of Fig. 7 shows the results for a density ratio of $n_b/n_{\mathrm{H}}$ = 0.2, where $b$ denotes O or He. In the small amplitude region ($b_n$ ≤ 0.1), $v_{bm}$ for the He ion exceeds that for the O ions. The dependence of $v_{bm}$ on $Z$ for the O ions is weak, although $v_{bm}$ is slightly greater for $Z$ = 4 than for 1 or 2. However, the low-frequency-mode pulse can attain much larger amplitudes in the H-O plasma than in the H-He plasma. As $b_n$ increases in the H-O plasma, the $Z$ dependence of $v_{bm}$ changes: in the large amplitude region ($b_n$ ≥ 0.2), $v_{bm}$ at a fixed $b_n$ increases slightly with decreasing $Z$. The most notable difference, however, is that the upper limit of $b_n$ increases as $Z$ decreases. Consequently, at the upper limit of $b_n$ for $Z$ = 1, $v_{bm}$ reaches its maximum value, $v_{bm}$ ≃ 0.32 $v_{\mathrm{A}}$.

Similar trends in $v_{bm}$ are observed in the top and middle panels of Fig. 7, which correspond to density ratios of $n_b/n_{\mathrm{H}}$ = 0.05 and 0.1, respectively. A comparison among the three panels reveals a clear dependence of $v_{bm}$ on the density ratio. As $n_b/n_{\mathrm{H}}$ decreases, the range of $b_n$ becomes narrower, but the rate of increase in $v_{bm}$ with respect to $b_n$ becomes significantly higher. As a result, the maximum $v_{bm}$ values in the three cases are comparable. Therefore, when $n_b/n_{\mathrm{H}}$ is small, the low-frequency-mode pulse can efficiently accelerate heavy ions even at small amplitudes. In contrast, when $n_b/n_{\mathrm{H}}$ ≳ 0.2, strong heavy-ion acceleration requires large pulse amplitudes. For such large amplitudes, the approximate formula, derived under the assumption that $b_n$ is small, may no longer be valid.

We next examine the mass dependence in more detail. Figure 8(a) shows the upper limit of the amplitude, ${\epsilon }_{\max }$, for the low-frequency mode pulse as a function of the mass ratio $m_b/m_{\mathrm{H}}$, with the density ratio fixed at $n_b/n_{\mathrm{H}}$ = 0.1. The black and gray lines correspond to charge numbers $Z$ = 1 and 2, respectively. This figure indicates that ${\epsilon }_{\text{max}}$ is greater for $Z$ = 1 than for 2, and that ${\epsilon }_{\max }$ increases with $m_b/m_{\mathrm{H}}$. Figure 8(b) shows the values of $v_{bm}$ at ${\epsilon }_{\text{max}}$, which therefore represent the upper limits of the heavy-ion speed induced by the low-frequency-mode pulse. As $m_b/m_{\mathrm{H}}$ increases, these upper limits of $v_{bm}$ also increase, although the rate of increase gradually decreases. Consequently, for example, the upper limit of $v_{bm}$ for $m_b/m_{\mathrm{H}}$ = 18 (H₂O⁺ ions) is nearly equal to that for $m_b/m_{\mathrm{H}}$ = 16 (O⁺ ions). Near comets, H⁺ and H₂O⁺ ions are major constituents. Therefore, our results suggest that O⁺ ions in the Earth’s magnetosphere and H₂O⁺ ions in the vicinity of comets can be effectively accelerated by the low-frequency-mode pulse.

Fig. 8.  (a) The upper limit of the amplitude, ${\epsilon }_{\max }$, for the low-frequency-mode pulse as a function of the mass ratio $m_b/m_{\mathrm{H}}$. (b) The corresponding heavy-ion speed achieved at ${\epsilon }_{\max }$.

4.  Summary and Discussion

We theoretically investigated heavy-ion dynamics in high- and low-frequency magnetosonic pulses in a two-ion-species plasma composed of H and a heavier ion species $b$. When the pulse amplitude is fixed, heavy-ion acceleration decreases with increasing heavy-ion mass ($m_b$) and cyclotron-frequency ratio ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_b$. This tendency appears in both the high- and low-frequency-mode pulses. However, for the low-frequency-mode pulse, the maximum attainable amplitude increases with ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_b$, resulting in enhanced acceleration for larger $m_b$. Consequently, it was found that the low-frequency-mode pulse can accelerate, for example, O ions in an H-O plasma more effectively than He ions in an H-He plasma, because ${\mathrm{\Omega }}_{\mathrm{H}}/{\mathrm{\Omega }}_b$ can be much larger in the H-O plasma.

The heavy-ion acceleration by the low-frequency-mode pulse is very efficient when the plasma beta is much smaller than unity. Such a very-low beta environment is observed in the polar region of the Earth’s magnetosphere. For example, the plasma beta is estimated to be on the order of 10⁻⁸ based on the values of $B_0$ = 0.1 G, $n_e$ = 1 cm^-3 [24], and $T_e$ = 5 eV [25] at 4,000 km altitude. In such a low-beta plasma, a low-frequency-mode pulse with the maximum amplitude can accelerate O ions to the speeds on the order of one-tenth of the Alfvén speed. Even when the amplitude is much smaller, the resulting acceleration speed can be much greater than the thermal speed before the ions enter the pulse. This acceleration occurs in the direction perpendicular to the magnetic field. If the perpendicular velocity is partially converted into the parallel speed by, for example, the mirror force, the parallel speed could reach values on the order of 10 km/s, which can exceed the typical O-ion outflow speeds [19]. Thus, if the low-frequency-mode pulse can be generated in the polar region, our study may offer a possible explanation for the O-ion outflow from the polar region, which has been unresolved for a long time.

The remaining physical issues in the present study are as follows. This study did not consider the feedback of heavy-ion acceleration on the pulses. The heavy-ion acceleration is expected to cause damping of the pulses [14], and the damping rate is likely to increase with the heavy-ion density. A quantitative evaluation of this damping and its effects on the energy gain of heavy ions is left for future work.

Ion kinetic effects associated with finite ion temperature are also important issues. When a magnetosonic solitary pulse has a sufficiently large amplitude, it can reflect ions with small velocities relative to the pulse, leading to the transition of the solitary pulse into a collisionless shock wave [23, 26]. Particle simulations have shown that high-frequency-mode shock waves can form in a two-ion-species plasma with ${\mathrm{\Omega }}_a\sim {\mathrm{\Omega }}_b$ [12, 17]. The effects of the low-frequency mode on collisionless shock formation in plasmas with substantially different values of ${\mathrm{\Omega }}_a$ and ${\mathrm{\Omega }}_b$ should be investigated by particle simulations.

Acknowledgments

This work was performed on the “Plasma Simulator” (NEC SX-Aurora TSUBASA) of NIFS with the support and under the auspices of the NIFS Collaboration research program (NIFS22KISS008, NIFS23KISS039, and NIFS24KISC005). It was supported by JSPS KAKENHI Grant Number 22J15207, 22K03570, 22KJ1887 and 24K17109; and by the NINS program of Promoting Research by Networking among Institutions (Grant Number 01422301).

Appendix

The derivation process for the maximum velocity of the heavy ion, given by Eq. (27), is presented here. We integrate Eqs. (24) and (25) over time, assuming that the initial particle velocity is zero, $v_{bx}(0)=v_{by}(0)=0$, and the initial particle position is located in the far upstream region, $x(0)\equiv x_0\gg D$. In these equations, the perturbation of $B_z$ is neglected, and the variable $x$ in $E_x$ and $E_y$ is approximated as $x=x_0$, under the assumption that the normalized pulse amplitude is much smaller than unity ($b_n$ ≪ 1). Taking the time derivative of Eq. (25) and eliminating $v_{bx}$ using Eq. (24), we obtain

  

$$\frac{d^2{v}_{by}}{d{t}^2}+{\mathrm{\Omega }}_b^2{v}_{by}=F(t),$$(31)

where $F(t)$ is defined as

  

$$\begin{array}{c}F(t)=\frac{{\mathrm{\Omega }}_b{v}_{\mathrm{p}0}^2}{d}\left(M-R\frac{{\mathrm{\Omega }}_{b}d}{v_{\mathrm{p}0}}\right)b_n^{3/2}\hfill \\ \times {\mathrm{sech}}^2\left(\frac{x_0-M{v}_{\mathrm{p0}}t}{D}\right)\tanh \left(\frac{x_0-M{v}_{\mathrm{p0}}t}{D}\right),\hfill \end{array}$$(32)

(in Ref. [15], $M$ in the large bracket in the first line is approximated as 1). Applying the Laplace transform to Eq. (31), the velocity $v_{by}$ for $t$ > 0 can be obtained, with the aid of the inversion formula, as

  

$$v_{by}(t)=\frac{1}{{\mathrm{\Omega }}_b}{\int }_0^t\sin [{\mathrm{\Omega }}_b(t-u)]F(u)du.$$(33)

Similarly, $v_{bx}$ can be expressed as

  

$$\begin{array}{ccc}\hfill v_{bx}(t)& \hfill =\hfill & -\frac{1}{{\mathrm{\Omega }}_b}{\int }_0^t\cos [{\mathrm{\Omega }}_b(t-u)]F(u)du\hfill \\ & & +v_{\mathrm{p0}}{b}_{\mathrm{n}}{\mathrm{sech}}^2\left(\frac{x_0-M{v}_{\mathrm{p0}}t}{D}\right).\hfill \end{array}$$(34)

We now derive $v_{by}$ in the downstream region at large $t$, assuming that $(x_0-M{v}_{\mathrm{p}0}t)/D\to -\mathrm{\infty }$. To this end, we introduce a variable $s$ defined as

  

$$s=(x_0-M{v}_{\mathrm{p}0}u)/D.$$(35)

Using $s$, ${\mathrm{\Omega }}_b(t-u)$ can be rewritten as

  

$${\mathrm{\Omega }}_b(t-u)={\mathrm{\Omega }}_b\left(t-\frac{x_0}{M{v}_{\mathrm{p}0}}\right)+ps,$$(36)

where $p$ is given by

  

$$p={\mathrm{\Omega }}_{b}D/M{v}_{\mathrm{p}0}.$$(37)

Then, Eq. (33) becomes

  

$$\begin{array}{ccc}\hfill v_{by}(t)& \hfill =\hfill & A\sin ({\mathrm{\Omega }}_b{t}^{\prime }){\int }_{s1}^{s2}ds\cos (ps){\mathrm{sech}}^2(s)\tanh (s)\hfill \\ & & +A\cos ({\mathrm{\Omega }}_b{t}^{\prime }){\int }_{s1}^{s2}ds\sin (ps){\mathrm{sech}}^2(s)\tanh (s),\hfill \end{array}$$(38)

where $A$ and $t^{\prime }$ are defined as

  

$$A=-\frac{v_{\mathrm{p}0}D}{Md}\left(M-R\frac{{\mathrm{\Omega }}_{b}d}{v_{\mathrm{p}0}}\right)b_n^{3/2},$$(39)

  

$$t^{\prime }=t-x_0/M{v}_{\mathrm{p}0}.$$(40)

The integration limits $s_1$ and $s_2$ are

  

$$s_1=\frac{x_0}{D},s_2=\frac{x_0-{\mathit{Mv}}_{\mathrm{p}0}t}{D}.$$(41)

Using the assumptions $x_0\gg D$ and $x_0-M{v}_{\mathrm{p}0}t\ll -D$, we can set the integral range as $s_1\to \mathrm{\infty }$ and $s_2\to -\mathrm{\infty }$. Under these conditions, the first integral in Eq. (38) vanishes, while the second can be evaluated as follows:

  

$$\begin{array}{c}{\int }_{+\mathrm{\infty }}^{-\mathrm{\infty }}ds\sin (ps){\mathrm{sech}}^2(s)\tanh (s)\hfill \\ =\frac{p}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}ds\sin (ps)d[{\mathrm{sech}}^2(s)]\hfill \\ =-\frac{p}{2}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}ds\cos (ps){\mathrm{sech}}^2(s)\hfill \\ =-\frac{\pi p^2}{2\sinh (\pi p/2)}.\hfill \end{array}$$(42)

Thus, the asymptotic form of $v_{by}$ at large $t$ is given by

  

$$v_{by}=v_{bm}\cos \left[{\mathrm{\Omega }}_b\left(t-\frac{x_0}{M{v}_{\mathrm{p}0}}\right)\right],$$(43)

where

  

$$\begin{array}{ccc}\hfill v_{bm}& \hfill =\hfill & -\frac{\pi p^{2}A}{2\sinh (\pi p/2)}\hfill \\ & \hfill =\hfill & \frac{4\pi {\mathrm{\Omega }}_b^2{d}^2}{M^3{v}_{\mathrm{p}0}{\alpha }^{3/2}}\left(M-R\frac{{\mathrm{\Omega }}_{b}d}{v_{\mathrm{p}0}}\right)\mathrm{cosech}\left({\displaystyle \frac{\pi p}{2}}\right).\hfill \end{array}$$(44)

For the low-frequency-mode pulse, neglecting electron inertia in $v_{\mathrm{p}0}=v_{\mathrm{A}}$ and $d=d_l$, we obtain $v_{bm}$ as given by Eq. (27) in Sec. 2. For the high-frequency-mode pulse, we can assume that $p$ is much smaller than unity, since the transit time of the heavy ion across the pulse, which is on the order of $D/M{v}_{\mathrm{p}0}$, is much shorter than $\pi /{\mathrm{\Omega }}_b$. Under this condition, $v_{bm}$ is reduced to Eq. (26) in Sec. 2, which was derived by a different method in Ref. [12]. Specifically, the substitution of the first-order approximation of $v_{bx}$, obtained from the KdV theory shown in Ref. [10], into Eq. (25) leads to Eq. (26).

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