Three-dimensional ray paths in cylindrical ducts: Upper branch cold plasma O and X modes

The guiding or ducting of O and X mode waves along magnetic-field-aligned density irregularities has been studied by integrating the Haselgrove equations in cylindrical coordinates r, ϕ z, where the z axis is parallel to the magnetic field (B). The density profile of the irregularity is a monotonic function of the radius r; linear and Gaussian functions of r are applied. The density rises from a minimum at r = 0 to a maximum at its outer limit, usually defined by the cutoff condition for the frequency and mode in question. Quasi-helical paths are obtained whose shape projected onto the r-ϕ plane perpendicular to B generally resembles a rotating ellipse. These shapes are similar to the orbital path of a particle in a central potential field. Indeed, for O mode propagation restricted to r and ϕ and a linear radial density profile, a closed-form solution is obtained in the form of an elliptical integral. Peculiarities are found, such as rays that are perfectly circular and others that have a fixed quasi-elliptical shape in the r-ϕ plane. Other parameters varied in the study include the relative depletion of the axial density below ambient density and the ray starting position and direction. The results suggest that three-dimensional effects may be observable in certain areas of high-frequency research in the ionosphere-magnetosphere.