Large-amplitude dust inertial Alfvén waves in an electron-depleted dusty plasma

Abstract

The existence of the large-amplitude dust inertial Alfvén waves (DIAWs) has been presented in an electron-depleted, two-fluid dust-ion plasma. Linear analysis shows that the DIAWs travel slower than the dust Alfvén waves. DIAWs are the obliquely (with respect to the external magnetic field) propagating oscillations of dust density, having the characteristics of a solitary wave. In order to observe the nonlinear behaviour of the DIAWs, the Sagdeev pseudopotential method has been used to derive the energy balance equation and from the expression of the Sagdeev pseudopotential, the existence conditions for the DIAWs have also been determined. It is observed that density rarefactions travelling at sub- and super-Alfvénic speeds are associated with DIAWs.

Appendix A. Linearisation and derivation of dispersion relation

The linearisation of eqs (10)–(17) is performed by assuming the following linearisation scheme: $n_{i}=n_{i0}+n_{i1}$, $n_{d}=n_{d0}+n_{d1}$, $v_{iz}=v_{iz1}$, $v_{dx}=v_{dx1}$, $v_{dz}=v_{dz1}$, $\phi _{\perp }=\phi _{\perp 1}$ and $\psi _{\parallel }=\psi _{\parallel 1}$. In this linearisation scheme, the quantities with the index ‘0’ represent the unperturbed value of that quantity, while the quantities with the index ‘1’ represent its first-order fluctuation. Thus, after the linearisation of eqs (10)–(14) and assuming all the first-order fluctuations to be of the form $A=A\mathrm {e}^{i(k_\perp x+k_\parallel z-\omega t)}$, we obtain the following set of equations respectively:

$$\begin{aligned}&\!\!\!\omega n_{i1}= k_\parallel n_{i0}v_{iz1}, \end{aligned}$$

(A.1)

$$\begin{aligned}&\!\!\!\omega v_{iz1}=\frac{Z_i e}{m_i} k_\parallel \psi _{\parallel 1}, \end{aligned}$$

(A.2)

$$\begin{aligned}&\!\!\!\omega n_{d1}=k_\perp n_{d0}v_{dx1}+k_\parallel n_{d0}v_{dz1}, \end{aligned}$$

(A.3)

$$\begin{aligned}&\!\!\!v_{dx1}=\frac{c}{B_0 \Omega _d} \omega k_\perp \phi _{\perp 1} \end{aligned}$$

(A.4)

and

$$\begin{aligned} \omega v_{dz1}=-\frac{Z_d e}{m_d} k_\parallel \psi _{\parallel 1}. \end{aligned}$$

(A.5)

Combining eqs (15) and (16), and linearising it, we obtain

$$\begin{aligned} k_{\perp }^2k_{\parallel }^2(\phi _{\perp 1}-\psi _{\parallel 1})=\frac{4\pi e}{c^2}[Z_i \omega ^2n_{i1}-Z_d \omega k_\parallel n_{d0}v_{dz1}].\nonumber \\ \end{aligned}$$

(A.6)

The linearisation of eq. (17) yields

$$\begin{aligned} Z_dn_{d1}=Z_in_{i1}. \end{aligned}$$

(A.7)

From eqs (A.1) and (A.2), we get

$$\begin{aligned} n_{i1}=\frac{k_{\parallel }^2}{\omega ^2}\frac{Z_i e n_{i0}}{m_i}\psi _{\parallel 1}. \end{aligned}$$

(A.8)

Similarly, from eqs (A.3)–(A.5), we get

$$\begin{aligned} n_{d1}=\frac{n_{d0}c}{B_0\Omega _d}k_{\perp }^2\phi _{\perp 1}-\frac{k_{\parallel }^2}{\omega ^2}\frac{Z_d e n_{d0}}{m_d}\phi _{\parallel 1}. \end{aligned}$$

(A.9)

Now, using eqs (A.8) and (A.9) in eq. (A.7), we get

$$\begin{aligned} \phi _{\perp 1}=\frac{1}{\lambda _{\mathrm{eff}}^2}\frac{V_{Ad}^2k_{\parallel }^2}{k_{\perp }^2\omega ^2}\psi _{\parallel 1}. \end{aligned}$$

(A.10)

Using eqs (A.5), (A.8) and (A.10) in eq. (A.6), we get the required dispersion relation of the dust inertial Alfvén waves

$$\begin{aligned} \omega =\frac{k_{\parallel } V_{Ad}}{\sqrt{1+k_{\perp }^2\lambda _{\mathrm{eff}}^2}}. \end{aligned}$$

(A.11)