Mixing of near-degenerate plasma modes on an elliptical plasma column.*

An unusual Trivelpiece-Gould (TG) mode frequency splitting pattern, due to interaction with an elliptical density perturbation caused by an $m_{\mathrm{\theta \thinspace }}=$ 2 diocotron mode, is observed for the first time in a magnetized pure electron plasma column. A single $m_{\mathrm{\theta \thinspace }}=$ 0,$_{\thinspace }m_{r}$ \textgreater 1 TG mode appears to branch out into frequency triplets as ellipticity increases. Here, $m_{\mathrm{\theta \thinspace }}$and $m_{r}$ are the azimuthal and radial wave numbers, respectively. For sufficiently small elliptical perturbations, the mode splitting $\Delta f_{\mathrm{\thinspace }}$/$f$ is linearly proportional to the plasma density quadrupole moment $q_{\mathrm{2}}$. An explanation of this effect involves mixing of the axisymmetric ($m_{\mathrm{\theta \thinspace }}=$ 0) mode with two non-axisymmetric ($m_{\mathrm{\theta }}\ne $0) nearly-degenerate plasma modes. For example, both the ($m_{\mathrm{\theta \thinspace }}=$ 0, $m_{r} \quad = \quad n)$ and the ($m_{\mathrm{\theta \thinspace }}=$ 2, $m_{r} \quad = \quad n$ -1) modes have frequencies determined by $k_{\mathrm{z}}r_{\mathrm{p}} \quad \approx \quad j_{\mathrm{1}}_{,n}$. We found that an elliptical density perturbation not only shifts the frequencies of the modes, but it also removes the orthogonality, with near-degeneracy allowing strong mixing of the eigenfunctions: the ($m_{\mathrm{\theta \thinspace }}=$ 2) modes pick up the ($m_{\mathrm{\theta \thinspace }}=$ 0) components resulting in the splitting pattern.