Trapped energetic electron driven modes with second stable regime in tokamak plasmas

Abstract

Features of fishbone-like trapped energetic electrons driven modes (TEEMs), which are a particular manifestation of the fishbone gap modes, are investigated taking into account model radial profile and the pitch angle distribution of the energetic electrons (EEs). The TEEMs are found unstable only when the beta values of EEs βh (=pressure of the energetic electrons/pressure of magnetic field) are higher than certain critical values which are proportional to perturbed energy δŴc of background plasma and much lower than that for traditional fishbone modes. In addition, TEEMs become stable again and enter a second stable regime when βh values are higher than second critical values. Furthermore, the modes can only be excited in plasmas which are stable for MHD instabilities. The real frequency and growth rate of the modes are approximately linear and parabolic functions of βh, respectively. The real frequency is very low but not zero in the vicinity of lower beta region, whereas it is comparable to the toroidal precession frequency of the EEs in higher critical beta region. The numerical results show that the second stable regime is easy to form when q = 1 flux surface is close to the magnetic axis. Suitable density gradient of EEs and magnetic shear are other two necessary conditions for excitation of the TEEMs.