Low-frequency modes with high toroidal mode numbers: A general formulation

Low-frequency waves with high toroidal mode numbers in an axisymmetric toroidal configuration are studied. In particular, the relationship between the periodicity constraints imposed by the geometry, magnetic shear, and the spatial structure of eigenmodes is investigated. By exploiting the radial translational invariance and the poloidal periodicity of the gyro-kinetic and Maxwell equations, the two-dimensional problem can be converted into a one-dimensional one, and the mode structure can be expressed in terms of a single extended poloidal variable. This representation is used in the description of electromagnetic modes with phase velocities larger than the ion thermal velocity and with frequencies below the ion gyro-frequency. Trapped particle, curvature, and compressional effects are retained. The dispersion equations for drift modes and Alfvén-type modes are given in general geometry and simplified solutions are presented in the configuration of a double periodic plane slab.