Toroidicity-Induced Alfvén Eigenmodes (TAEs)

In the mid-1980s, magnetic fusion research reached a point at which large-scale thermonuclear plasmas with significant populations of energetic ions became possible. Several large tokamaks were built world-wide, and powerful auxiliary heating systems, neutral beam injection (NBI) and ion cyclotron resonance heating (ICRH), generating energetic ions were developed successfully. Two of the large tokamaks, TFTR (United States) and JET (United Kingdom), were designed to operate with DT plasmas, and hence, experimental studies of fusion-generated alpha particles in tokamak plasmas became possible.

Therefore, it is essential to consider in detail what novel physics features may arise from the presence of large populations of energetic ions in tokamak plasmas, especially if these populations were super-Alfvenic (i.e. speeds of the energetic ions exceeded the Alfven speed). Theoretically, it has been pointed out much earlier in Refs. [6.1,6.2] that coupling of super-Alfvenic ions to the Alfven waves might excite Alfven instabilities. Such instabilities, if the wave amplitudes would become sufficiently high, could result in a cross-field transport of the energetic ions much higher than the neoclassical transport due to Coulomb collisions. Consequently, due to the Alfven instabilities, plasma heating by the energetic ions could decrease significantly and change the power deposition profile. Moreover, enhanced losses of the energetic ions, if repeated discharge after discharge, could damage the first wall.

However, a principal uncertainty in calculating the critical threshold in energetic particle pressure for exciting Alfven instabilities was the strong damping of shear Alfven waves. As we saw in Chapter 5, in a sheared magnetic field in a slab geometry, these waves are highly localised at the surfaces a>= k^VA and experience strong continuum damping. An exception was found for global Alfven eigenmodes (GAEs), but such modes existing in cylindrical geometry with plasma current still experience a strong continuum damping in toroidal geometry [6.3].

Therefore, a dedicated search for weakly damped Alfven modes in toroidal geometry was performed. This resulted in the discovery of toroidal Alfven eigenmodes (TAE) [6.4]. TAE exists due to the periodic nature of the toroidal field, which gives rise to gaps in the Alfven continuum frequency spectrum, within which a discrete spectrum of TAEs may result from a finite magnetic shear. TAEs are discrete modes free of the continuum damping in the lowest order, and can be easily destabilised by alpha particles or other energetic ions [6.5,6.6].

The predicted weakly damped TAE modes residing inside the continuum gaps with a well- determined frequency were searched for in tokamak experiments. The instability of TAE was observed for the first time in TFTR and DIII-D experiments with NBI-produced energetic ions injected into plasmas with low magnetic field, B< 1 T, so that VA was comparable to the velocities of the beam ions [6.7,6.8]. Significant loss of the beam ions, up to 45% on DIII-D [6.8], was observed, indicating how crucial TAE instabilities could be in the presence of large population of energetic ions with velocities close to VA. These early experiments made the TAE issue one of the highest priorities for magnetic fusion studies world-wide. In addition, it was found in Refs. [6.7,6.8] that the experimentally observed thresholds for TAE instabilities were higher than those predicted theoretically. This attracted much attention to the theoretical studies of TAE damping and drive, but with more precision.

In present-day experiments, TAEs excited by several types of energetic ion populations are often observed in many tokamaks with auxiliary heating. In particular, TAEs driven by fusion-generated alpha-particles were detected in TFTR DT experiment [6.9]. Furthermore, several new types of weakly damped Alfven eigenmodes (AEs) were found, including the “gap” modes with frequencies within the gaps in the continua created by the ellipticity and triangularity of the plasma cross- section, and/or the geodesic curvature and plasma compressibility. Other types of present-day magnetic fusion machines, such as stellarators and reversed-field pinches, also often observe Alfven instabilities in discharges with auxiliary heating.

In this chapter and in Appendix C we present an analytical theory of TAE in the large aspect ratio tokamak with low magnetic shear, as well as qualitatively describe the main effects that determine TAE drive by energetic particles and TAE damping due to thermal plasma. We believe this would facilitate studies of other types of weakly damped AEs by using the TAE description here as an example.

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