Plasma mode degeneracy and avoided crossings from plasma column curved ends

We observed parametric coupling between the axial center-of-mass mode (mr =1,mθ =1,mz =1) and the (2,0,3) mode as their frequencies become degenerate and they form an avoided crossing; the mode frequencies never overlap. In infinitely long (or periodic boundary) theory all the modes are independent (have orthogonal eigenfunctions) and therefore there would be no such coupling between the modes. However, in real experiments, the standing TG waves of the plasma column are supported by the (generally) curved end reflections, and hence some TG modes may become degenerate in the case of f_TG(mr,mθ,mz) ≈ f^*_TG(mr^*,mθ^*,mz^*) if they have the same mθ and nominal kz. The TG mode frequencies are set mostly by the (kz/k) ratio, with temperature T_e(t) as a fine tuning knob. There are two relatively simple but interesting cases: f_TG(1,0,1) ≈ f^*_TG(1,±1,2) and f_TG(1,0,1) ≈ f^*_TG(2,0,3). The former case shows no degeneracy and no modes coupling since those two modes have different θz-parity. In the latter case, the mode degeneracy does not allow for those two modes to have exactly same frequency, f_TG(1,0,1) ≠ f^*_TG(2,0,3). When these modes become strongly coupled during the temperature evolution, they form an avoided crossing behavior as their frequencies become heavily degenerate.