Identification of mode numbers and wave numbers by multiple harmonics of MHD instabilities

Abstract Harmonic phenomena of plasma instabilities are frequently observed in tokamak plasmas. The harmonics of instabilities have higher toroidal mode numbers than their base modes, leading to proportionally larger Doppler shifts. This paper presents a method for determining the toroidal mode number based on the frequency differences between harmonics. Based on Doppler shift effect, the toroidal mode number can be expressed as n = ( f l − f 1 ) / ( ( l − 1 ) f v ϕ ) , where l is the harmonic order and f vφ is the toroidal rotation frequency. Compared to conventional techniques for mode number identification, the proposed approach offers advantages such as simplified data processing and lower requirement for the number of diagnostic channels. The harmonic components in the frequency spectrum can also be used to retrieve the instability frequency in the plasma frame. In this paper, it is demonstrated that the harmonics of plasma instabilities are not generated by intrinsic mode coupling, but are attributed to the nonlinear distortion of waveform. Diagnostic measurements, including soft x-ray and microwave interferometry, confirm that higher-order harmonics of plasma instabilities exist as real structural features inside the plasma and can interact with microwaves when the Bragg condition is satisfied. This property can be utilized to determine the perpendicular wave numbers of the instabilities. A simple physical model of the distribution of an instability and its harmonics is also proposed.