Dispersion relations for low-frequency waves in a warm, underdense, two-fluid plasma

Analytical forms of dispersion relations for three lower-frequency waves in a two-fluid plasma are presented. In underdense plasmas, ωp2≲Ωce2, that is, low density or strongly magnetized plasmas, where ωp and Ωce are respectively the plasma frequency and the electron cyclotron frequency, the displacement current can become important in higher-frequency (non-magnetohydrodynamic) region. Focusing on only three branches that become magnetohydrodynamic waves in the low-frequency limit, we derive dispersion relations applicable to both underdense and overdense plasmas by the full set of equations of two-fluid plasmas including the displacement current. It is shown that our analytical results well agree with the exact numerical ones in all frequency ranges when the following conditions are satisfied: normal (i.e., not pair) plasma; low plasma β, and not extremely small electron to ion temperature ratio, me/mi≪Te/Ti, while more exact dispersion relations are needed to perform polarization analysis.