Ion acoustic cnoidal waves in a magnetized plasma in the presence of ion pressure anisotropy

Abstract

An investigation of the ion acoustic nonlinear periodic (cnoidal) waves in a magnetized plasma with positive ions having anisotropic thermal pressure and Maxwellian electrons is carried out. The Korteweg-de Vries equation for the wave potential is derived via a reductive perturbation technique and its cnoidal wave solution is obtained. The effect of various relevant plasma parameters like ion pressure anisotropy and obliqueness of field on the characteristics of ion acoustic nonlinear periodic wave structures is investigated in detail. The present investigation could be useful in space and astrophysical plasma systems having ion pressure anisotropy, particularly, in the magnetosphere and near Earth magnetosheath.

Appendix: Derivation of Eqs. (9)–(11) from Eq. (2) using Eq. (3)

We consider the ion momentum equation (2), repeated below:

$$\begin{aligned}& \frac{\partial \vec{v}}{\partial t}+(\vec{v}\cdot \vec{\nabla }) \vec{v} =-\frac{Ze}{m} \vec{\nabla }\phi +\frac{Ze}{mc} ( \vec{v} \times B_{0}\hat{z} ) -\frac{1}{mn}\vec{\nabla }\cdot \tilde{P} \end{aligned}$$

(46)

We also know from Eq. (3) that the anisotropic pressure is

$$ \tilde{P}=p_{\perp }\hat{I}+ ( p_{\parallel }-p_{\perp } ) \hat{b}\hat{b} $$

(47)

where $\hat{I}$ and $\hat{b}\hat{b}$ in matrix form can be written as,

$$\hat{I}=\left ( \textstyle\begin{array}{c@{\quad }c@{\quad }c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right ) \quad \mbox{and}\quad \hat{b}\hat{b}=\left ( \textstyle\begin{array}{c@{\quad }c@{\quad }c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{array}\displaystyle \right ) . $$

The divergence of pressure tensor becomes

$$ \vec{\nabla }\cdot \tilde{P}=\vec{\nabla }p_{\perp }+\hat{b} \frac{ \partial }{\partial z} ( p_{\parallel }-p_{\perp } ) $$

(48)

Note that $\vec{\nabla }\cdot ( p_{\perp }\hat{I} ) = \vec{\nabla }p_{\perp }$. Further we may also write Eq. (48) in the following form

$$ \vec{\nabla }\cdot \tilde{P}=\hat{x}\frac{\partial p_{\perp }}{\partial x}+\hat{y} \frac{\partial p_{\perp }}{\partial y}+\hat{z}\frac{\partial p_{\parallel }}{\partial z} $$

(49)

From Eq. (4), we know that

$$ p_{\perp }=p_{\perp 0} \biggl( \frac{n}{n_{0}} \biggr) \quad \text{and}\quad p _{\parallel }=p_{\parallel 0} \biggl( \frac{n}{n_{0}} \biggr) ^{3} $$

(50)

Using Eq. (50) into Eq. (49), we get

$$ \vec{\nabla }\cdot \tilde{P}=\hat{x} \biggl( \frac{p_{\perp 0}}{n_{0}} \biggr) \frac{\partial n}{\partial x}+ \hat{y} \biggl( \frac{p_{\perp 0}}{n_{0}} \biggr) \frac{\partial n}{ \partial y}+\hat{z} \biggl( \frac{3p_{\parallel 0}}{n_{0}^{3}} \biggr) n^{2}\frac{\partial n}{\partial z} $$

(51)

We plug the above value of $\vec{\nabla }\cdot \tilde{P}$ into Eq. (46), and finally arrive at the set of Eqs. (9)–(11).