Gyrokinetic Theory of Low-frequency Electromagnetic Waves in Finite-$\beta$ Anisotropic Plasmas

We present a gyrokinetic theory for the electromagnetic waves and instabilities with frequencies much lower than the ion cyclotron frequency in finite-$\beta$ anisotropic uniform plasmas. Here, $\beta$ is the ratio between plasma and magnetic pressures. Kinetic effects due to both the finite Larmor radii and wave-particle resonances are fully kept in the analysis. Corresponding linear dispersion relation and wave polarizations, valid for general $\beta$ value and perpendicular wavelength, are then specifically derived for a bi-Maxwellian plasma. Analytic expressions for the criteria of kinetic firehose and mirror instabilities are also given. The mode frequency, stability, and wave polarization of a broad spectrum of normal modes are then investigated numerically in a systematic study over a set of parameters. Our study clearly demonstrates that, due to the finite ion Larmor radius effect, the ion-sound wave, mirror mode, and shear Alfven wave are intrinsically coupled.