Fundamental Flaw in Kinetic Alfven Wave Dispersion

The well-known kinetic Alfven wave (KAW) dispersion is shown to be fundamentally incorrect because its derivation involves a mathematical error. The error results from the standard practice of setting to zero the dielectric tensor elements $K_{xy},$ $K_{yx}$ because these elements are of order $\omega/\omega_{ci}\ $ and $\omega/\omega_{ci}\ll1$. \ It is shown that in a warm electron, cold ion plasma the dielectric tensor elements $K_{yz},$ $K_{zy}$ are of order $\omega_{ci}/\omega$ so, when taking the full 3$\times $3 wave equation determinant, order of unity products of the sort $K_{xy}K_{yz}$ exist, are important, and are missing in the KAW derivation. This error was identified from the substantive discrepancy between the determinant of the matrix of a 3$\times$3 wave equation matrix reported by Hirose et al. [1] and the standard 2$\times$2 determinant used for deriving the KAW. The validity of the 3$\times$3 matrix elements and in particular the elements $K_{yz},K_{zy}$ was established by showing that 2- fluid theory gives the same matrix elements as the kinetic theory \ used in Ref. [1].\\[4pt] [1] A. Hirose et al. Phys. Letters A 330, 474 (2004).