Phase tracking with Hilbert transform and nonlinear wave-wave coupling analysis on the HL-2A tokamak

A phase tracking method based on Hilbert transform algorithm is applied to the nonlinear wave-wave coupling analysis on HL-2A tokamak. Synthetic signal analysis is given to show the principle of phase analysis for the detection of nonlinear coupling. If the phase difference between two coherent modes is synchronized with the phase of a third mode, these three modes are nonlinearly coupled, vice versa. The time evolution of the phase of a coherent mode could be computed with Hilbert transform for experimental data, and a phase tracking flowchart is summarized. On HL-2A tokamak, the nonlinear coupling among two Alfvén modes (AMs) and a tearing mode (TM) has been confirmed with bicoherence analysis (Shi et al 2017 Phys. Plasmas 24 042509). With Hilbert transform, it is clearly observed that the phase difference between two AMs Δθ ₁₂ is synchronized with the phase of the TM θ _TM. The synchronization is confirmed with normalized cross-correlation. An alternative to check this synchronization is to observe the histogram of the phase difference Δθ ₁₂ − θ _TM. (paper)