Dispersion of electron Bernstein waves including weakly relativistic and electromagnetic effects. Part 2. Extraordinary modes

Extraordinary solutions of the weakly relativistic, electromagnetic dispersion relation are investigated for waves propagating perpendicular to a uniform magnetic field in a Maxwellian plasma. As in a companion paper, which treated ordinary modes, weakly relativistic effects are found to modify dramatically the dispersion predicted by strictly non-relativistic ‘classical’ theory in the neighbourhood of harmonics of the cyclotron frequency Ω_e. The infinite families of classical Gross-Bernstein and Dnestrovskii-Kostomarov modes are truncated to include only harmonics s satisfying s (ω²_p mc²/4k_B TΩ²_e)⅓ and s (ω_p/Ω_e)⅔/8 respectively where ω_p is the plasma frequency and T the temperature. All classical cut-offs and resonances are removed apart from the x- and z- mode cut-offs. The only coupling between large- and small-wave-vector modes is between the z mode and a Gross-Bernstein mode near the upper-hybrid frequency and between the x mode and the second Gross-Bernstein mode near 2Ω_e. Dispersion of the weakly relativistic counterpart of the x mode departs only slightly from that predicted by cold plasma theory except near Ω_e and 2Ω_e.