Comment on "Dispersion relation for MHD waves in homogeneous plasma"

TL;DR

This paper critiques Pandey & Dwivedi's claim that the dispersion relation for MHD waves is a sixth degree polynomial, clarifying that only five roots are relevant for wave propagation analysis.

Contribution

The authors demonstrate that Pandey & Dwivedi's derivation introduces an unnecessary root, emphasizing that only five roots are necessary for understanding wave modes.

Findings

Pandey & Dwivedi's sixth degree polynomial includes an extraneous root.

Five roots of the dispersion relation are sufficient for wave analysis.

The additional root is insignificant for the physical interpretation of wave modes.

Abstract

Pandey & Dwivedi (2007) again tried to claim that the dispersion relation for the given set of equations must be a sixth degree polynomial. Through a series of papers, they are unnecessarily creating confusion. In the present communication, we have shown how Pandey & Dwivedi (2007) are introducing an additional root, which is insignificant. Moreover, five roots of both the polynomials are common and they are sufficient for the discussion of propagation of slow-mode and fast-mode waves.

Equations

$$\frac{\partial \rho }{\partial t}+\nabla .(\rho \overrightarrow{v})=0$$

$$\rho \frac{D\overrightarrow{v}}{Dt}=-\nabla p+\frac{1}{4\pi }(\nabla \times \overrightarrow{B})\times \overrightarrow{B}-\nabla .\Pi$$

$$\frac{\partial \overrightarrow{B}}{\partial t}=\nabla \times (\overrightarrow{v}\times \overrightarrow{B})$$

$$\displaystyle\frac{Dp}{Dt}+\gamma p(\nabla.\stackrel{{\scriptstyle\rightarrow}}{{v}})=(\gamma-1)\Big{[}\nabla.\ \kappa\nabla T+Q_{vis}-Q_{rad}\Big{]}$$

$$p=\frac{2\rho k_{B}T}{m_p}$$

$$\displaystyle Q_{th}=\kappa_{\parallel}\Big{(}\frac{\partial T}{\partial z}\Big{)}^{2}T^{-1}\hskip 56.9055ptQ_{vis}=\frac{\eta_{0}}{3}(\nabla.\stackrel{{\scriptstyle\rightarrow}}{{v}})^{2}\hskip 56.9055ptQ_{rad}=n_{e}n_{H}Q(T)$$

$$\rho ={\rho }_0+{\rho }_1\overrightarrow{v}={\overrightarrow{v}}_1\overrightarrow{B}={\overrightarrow{B}}_0+{\overrightarrow{B}}_1$$

$$p=p_0+p_{1}T=T_0+T_1\Pi ={\Pi }_0+{\Pi }_1$$

$$\frac{\partial {\rho }_1}{\partial t}+{\rho }_0(\nabla .{\overrightarrow{v}}_1)=0$$

$${\rho }_0\frac{\partial {\overrightarrow{v}}_1}{\partial t}=-\nabla p_1+\frac{1}{4\pi }(\nabla \times \overrightarrow{B_1})\times \overrightarrow{B_0}-\nabla .{\Pi }_0$$

$$\frac{\partial \overrightarrow{B_1}}{\partial t}=\nabla \times (\overrightarrow{v_1}\times \overrightarrow{B_0})$$

$$\frac{\partial p_1}{\partial t}+\gamma p_0(\nabla .{\overrightarrow{v}}_1)+(\gamma -1){\kappa }_{\parallel }{k}_z^2{T}_1=0$$

$$\frac{p_1}{p_0}=\frac{{\rho }_1}{{\rho }_0}+\frac{T_1}{T_0}$$

$$\omega {\rho }_1-{\rho }_0(k_x{v}_{1x}+k_z{v}_{1z})=0$$

$$\omega {\rho }_0{v}_{1x}-k_x{p}_1-\frac{B_0}{4\pi }(k_x{B}_{1z}-k_z{B}_{1x})+\frac{i{\eta }_0}{3}(k_x^2{v}_{1x}-2k_x{k}_z{v}_{1z})=0$$

$$\omega {\rho }_0{v}_{1y}+\frac{B_0}{4\pi }(k_z{B}_{1y})=0$$

$$\omega {\rho }_0{v}_{1z}-k_z{p}_1+\frac{i{\eta }_0}{3}(4k_z^2{v}_{1z}-2k_x{k}_z{v}_{1x})=0$$

$$\omega B_{1x}+k_z{B}_0{v}_{1x}=0$$

$$\omega B_{1y}+k_z{B}_0{v}_{1y}=0$$

$$\omega B_{1z}-k_x{B}_0{v}_{1x}=0$$

$$i\omega p_1-i{\rho }_0{c}_s^2(k_x{v}_{1x}+k_z{v}_{1z})-(\gamma -1){\kappa }_{\parallel }{k}_z^2{T}_1=0$$

$$\frac{p_1}{p_0}-\frac{{\rho }_1}{{\rho }_0}-\frac{T_1}{T_0}=0$$

$$\displaystyle\Big{(}\omega^{2}\rho_{0}+\frac{i\omega\eta_{0}}{3}k_{x}^{2}-v_{A}^{2}\rho_{0}k^{2}\Big{)}v_{1x}-\frac{2i\omega\eta_{0}k_{x}k_{z}}{3}\ v_{1z}-k_{x}\omega p_{1}=0$$

$$\displaystyle\frac{2i\eta_{0}k_{x}k_{z}}{3}\ v_{1x}-\Big{(}\omega\rho_{0}+\frac{4i\eta_{0}}{3}k_{z}^{2}\Big{)}v_{1z}+k_{z}p_{1}=0\hskip 62.59596pt$$

$$(c_0{p}_0{k}_x-i{\rho }_0{c}_s^2{k}_x\omega )v_{1x}+(c_0{p}_0{k}_z-i{\rho }_0{c}_s^2{k}_z\omega )v_{1z}-(c_0\omega -i{\omega }^2)p_1=0$$

$$a_{11}{v}_{1x}+a_{12}{v}_{1z}+a_{13}{p}_1=0$$

$$a_{21}{v}_{1x}+a_{22}{v}_{1z}+a_{23}{p}_1=0$$

$$a_{31}{v}_{1x}+a_{32}{v}_{1z}+a_{33}{p}_1=0$$

$$\displaystyle a_{11}=\Big{(}\omega^{2}\rho_{0}+\frac{i\omega\eta_{0}}{3}k_{x}^{2}-v_{A}^{2}\rho_{0}k^{2}\Big{)};\hskip 28.45274pta_{12}=-\frac{2i\omega\eta_{0}}{3}k_{x}k_{z};\hskip 28.45274pta_{13}=-k_{x}\omega$$

$$a_{21}=\frac{2i{\eta }_0}{3}{k}_x{k}_z;a_{22}=-\omega {\rho }_0-\frac{4i{\eta }_0}{3}{k}_z^2;a_{23}=k_z$$

$$a_{31}=c_0{p}_0{k}_x-i{\rho }_0{c}_s^2{k}_x\omega ;a_{32}=c_0{p}_0{k}_z-i{\rho }_0{c}_s^2{k}_z\omega ;a_{33}=i{\omega }^2-c_0\omega$$

$$\displaystyle\begin{array}[]{ccc}\left|\begin{array}[]{lll}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{array}\right|&\begin{array}[]{c}=0\\ \\ \\ \end{array}\\ \end{array}$$

$${\omega }^5+iA{\omega }^4-B{\omega }^3-iC{\omega }^2+D\omega +iE=0$$

$$A=c_0+\frac{{\eta }_0}{3{\rho }_0}(k_x^2+4k_z^2)$$

$$B=\frac{c_0{\eta }_0}{3{\rho }_0}(k_x^2+4k_z^2)+(c_s^2+v_A^2)k^2$$

$$C=\frac{3{\eta }_0}{{\rho }_0}{c}_s^2{k}_x^2{k}_z^2+\frac{c_0{p}_0{k}^2}{{\rho }_0}+v_A^2{c}_0{k}^2+\frac{4{\eta }_0{v}_A^2{k}_z^2{k}^2}{3{\rho }_0}$$

$$D=\frac{3c_0{p}_0{\eta }_0{k}_x^2{k}_z^2}{{\rho }_0^2}+\frac{4{\eta }_0{c}_0{v}_A^2{k}_z^2{k}^2}{3{\rho }_0}+v_A^2{c}_s^2{k}_z^2{k}^2$$

$$E=v_A^2{c}_0{p}_0{k}_z^2{k}^2/{\rho }_0$$

$$(a_{11}{a}_{33}-a_{31}{a}_{13})v_{1x}+(a_{12}{a}_{33}-a_{32}{a}_{13})v_{1z}=0$$

$$(a_{21}{a}_{33}-a_{31}{a}_{23})v_{1x}+(a_{22}{a}_{33}-a_{32}{a}_{23})v_{1z}=0$$

$$(a_{11}{a}_{33}-a_{31}{a}_{13})(a_{22}{a}_{33}-a_{32}{a}_{23})=(a_{21}{a}_{33}-a_{31}{a}_{23})(a_{12}{a}_{33}-a_{32}{a}_{13})$$

$${\omega }^6+i{A}^{\prime }{\omega }^5-B^{\prime }{\omega }^4-i{C}^{\prime }{\omega }^3+D^{\prime }{\omega }^2+i{E}^{\prime }\omega -F^{\prime }=0$$

$$A^{\prime }=2c_0+c_1;B^{\prime }=(c_s^2+v_A^2)k^2+c_0(2c_1+c_0);$$

Full text

Comment on “Dispersion relation for MHD waves in homogeneous plasma”

Suresh Chandra, G.M. Dak, B.K. Kumthekar and Monika Sharma

School of Physics, Shri Mata Vaishno Devi University, Katra 182 320 (J&K), India

Email: suresh492000@yahoo.co.in

Abstract: Pandey & Dwivedi (2007) again tried to claim that the dispersion relation for the given set of equations must be a sixth degree polynomial. Through a series of papers, they are unnecessarily creating confusion. In the present communication, we have shown how Pandey & Dwivedi (2007) are introducing an additional root, which is insignificant. Moreover, five roots of both the polynomials are common and they are sufficient for the discussion of propagation of slow-mode and fast-mode waves.

Keywords: MHD - Sun: Corona - waves

1 Introduction

For application of magnetohydrodynamics (MHD) in solar physics as well as in plasma physics, dispersion relation plays key role. The basic equations under the present investigation can be expressed as (Pandey & Dwivedi, 2007, hereinafter referred to as PD)

[TABLE]

Here, symbols have their usual meaning. The quantities $Q_{th}$, $Q_{vis}$ and $Q_{rad}$ are

[TABLE]

where $\kappa_{\parallel}$ represents the conductivity along the magnetic field and is expressed by $\kappa_{\parallel}\approx 10^{-6}T^{5/2}$. For small perturbations from the equilibrium, we have

[TABLE]

where the equilibrium part is denoted by the subscript ‘0’ and the perturbation part by the subscript ‘1’. For the magnetic field taken along the $z$-axis, (i.e., $\stackrel{{\scriptstyle\rightarrow}}{{B}}_{0}=B_{0}\hat{z}$) and the propagation vector $\stackrel{{\scriptstyle\rightarrow}}{{k}}=k_{x}\hat{x}+k_{z}\hat{z}$, the equations (1) - (5) can be linearized in the following form (Chandra and Kumthekar, 2007):

[TABLE]

For the perturbations that are proportional to $\mbox{exp}[i(\stackrel{{\scriptstyle\rightarrow}}{{k}}.\stackrel{{\scriptstyle\rightarrow}}{{r}}-\omega t)]$, equations (6) - (10) reduce to the following equations:

[TABLE]

These equations (11) - (19) are the same as the equations (11) - (19) of PD. (In equation (11) of PD, $\rho$ must be $\rho_{1}$.) Equations (13) and (16) for the variables $v_{1y}$ and $B_{1y}$ are decoupled from the rest and describe the Alfvén waves. The rest of the equations for $p_{1}$, $\rho_{1}$, $T_{1}$, $B_{1x}$, $B_{1z}$, $v_{1x}$ and $v_{1z}$ describe damped magnetohydrostatic waves. Now, on substituting $B_{1x}$ and $B_{1z}$ from equations (15) and (17) in equations (12) and (14), respectively, we get

[TABLE]

and

[TABLE]

When we eliminate $\rho_{1}$ and $T_{1}$ from equations (11), (18) and (19), we get

[TABLE]

where $c_{0}=(\gamma-1)\kappa_{\parallel}k_{z}^{2}T_{0}/p_{0}$; $c^{2}_{s}=\gamma p_{0}/\rho_{0}$; and $v^{2}_{A}=B^{2}_{0}/4\pi\rho_{0}$. Thus, the equations (20), (21) and (22) are obtained from the equations which are the same as of PD.

2 Dispersion relation

For convenience, let us express equations (20) - (22) as

[TABLE]

where the coefficients $a$’s are:

[TABLE]

For a non-trivial solution of the set of equations (23), (24) and (25), we must have

[TABLE]

Expansion of this determinant and substitution of the values of $a$’s gives the fifth degree polynomial:

[TABLE]

where

[TABLE]

It obviously shows that the dispersion relation is a fifth degree polynomial. Now, question arises how PD are getting the sixth degree polynomial. It can be understood in the following manner.

3 How PD is getting sixth degree polynomial

The dispersion relation of PD can be derived in the following manner: On eliminating $p_{1}$ from equations (23) and (25), we get

[TABLE]

On eliminating $p_{1}$ from equations (24) and (25), we get

[TABLE]

From equations (35) and (36), we have

[TABLE]

Substitution of the values of $a$’s in equation (37), gives the dispersion relation

[TABLE]

where

[TABLE]

and

[TABLE]

This is the dispersion relation derived by PD. Now, here we can show that DP has introduced an addition root in this dispersion relation.

4 Discussion

It can be easily found that

[TABLE]

showing that five roots of both equations (34) and (38) are common. The additional root $\omega=-ic_{0}$ is introduced by PD. The five common roots, are of the form $i\alpha_{1}$, $-\beta_{2}+i\alpha_{2}$, $\beta_{2}+i\alpha_{2}$, $-\beta_{3}+i\alpha_{3}$ and $\beta_{3}+i\alpha_{3}$. The first root corresponds to the thermal motion whereas the rest four give the slow-mode and fast-mode waves. The sixth root $-ic_{0}$ also corresponds to the thermal motion. This has no significance, as we are interested only in the slow-mode and fast-mode waves. It is difficult to understand how science is affected when one considers the sixth degree polynomial derived by PD.

It can finally be concluded that the dispersion relation is of fifth degree polynomial. PD got the sixth degree polynomial, as they have introduced an additional root. Moreover, only five roots are sufficient for propagation of slow-mode and fast-mode waves.

Acknowledgments

We are thankful to the DST, New Delhi and ISRO, Bangalore for financial support in the form of research projects. A part of this work was done during the visit to the IUCAA, Pune. Financial support from the IUCAA, Pune is thankfully acknowledged.