Controversy on a dispersion relation for MHD waves

TL;DR

This paper investigates a controversy over the correct dispersion relation polynomial degree for MHD waves, analyzing conflicting results from previous studies and clarifying the mathematical discrepancies involved.

Contribution

It provides a detailed analysis of the conflicting derivations of the MHD wave dispersion relation, resolving the inconsistency regarding polynomial degree.

Findings

Identified the source of discrepancy in previous derivations.

Clarified that the equations are linear and should yield a consistent polynomial degree.

Highlighted the importance of correct algebraic elimination in deriving dispersion relations.

Abstract

Kumar et al. (2006) obtained a fifth order polynomial in $\omega$ for the dispersion relation and pointed out that the calculations preformed by Porter et al. (1994) and by Dwivedi & Pandey (2003) seem to be in error, as they obtained a sixth order polynomial. The energy equation of Dwivedi & Pandey (2003) was dimensionally wrong. Dwivedi & Pandey (2006) corrected the energy equation and still claimed that the dispersion relation must be a sixth order polynomial. The equations (11) $-$ (19) of Dwivedi & Pandey (2006) and the equations (24) $-$ (32) Kumar et al. (2006) are the same. This fact has been expressed by Kumar et al. (2006) themselves. Even then they tried to show this set of equations on one side gives the sixth order polynomial as they got; on the other side, the same set of equations gives the fifth order polynomial as Kumar et al. (2006) obtained. The situation appears to be non-scientific, as the system of equations is a linear one. These are simple algebraic equations where the variables are to be eliminated. However, it is a matter of surprise that by solving these equations, two scientific groups are getting polynomials of different degrees. In the present discussion, we have attempted to short out this discrepancy.