Abstract
Effects of the energetic-ion-driven Geodesic Acoustic modes (GAM and E-GAM) on the toroidally passing energetic ions and the concomitant change of the neutron yield of beam-plasma fusion reactions in tokamaks are considered. It is shown that due to large perturbations of the plasma density, the resonant energetic ions driving the instability can be considerably slowed down for a few tens of the particle transit periods, which is much less than the collisional slowing down time. The time of the collisionless slowing down is actually determined by the period of the particle motion within the resonance island arising because of the GAM / E-GAM. Being trapped in the island, the resonant particles can not only lose their energy but also gain it. One more effect of GAMs is the flattening on the distribution function of the resonant particles. Due to conservation of the canonical angular momentum during a GAM / E-GAM instability, the change of the particle energy is accompanied by a radial displacement of the resonant particle for a distance up to the poloidal Larmor radius of energetic ions. The particles are displaced inwards or outwards, depending on the direction of their motion along the magnetic field. Expressions describing the change of the neutron yield due to GAM modes are derived. It is found that the distortion of the velocity distribution of the resonant particles can lead to a considerable drop of the neutron emission even when effects of the particle radial displacement are small. The developed theory is applied to an E-GAM experiment on the DIII-D tokamak. Relations for the period of the motion within the resonance island of passing (both well passing and marginally passing) particles and the width of the resonance of the energetic particles with GAM modes and low-frequency Alfvén modes are derived.
Highlights
An oscillation homogeneous in the toroidal direction, named Geodesic Acoustic Mode (GAM), was predicted to exist in toroidal plasmas by Winsor et al as long ago as in 1968 [1]
We conclude from here that the resonant interaction of the well-passing energetic ions and a global GAM / E-GAM mode can lead to the transfer of a large fraction of the energy of these ions to the mode, which is accompanied by radial displacements of the particles
Infer that in the framework of the model used the main mechanism affecting the change of the neutron emission is the velocity distortion of the energetic ion distribution function
Summary
The GAM modes can be considered as electrostatic, and harmonics of the scalar potential of the electric field with m ≠ 0 are negligible when β (the ratio of the plasma pressure to the magnetic field pressure) is sufficiently low, satisfying the condition βs ≡ c2s/v2A < (8q2)−1, where cs is the sound velocity, vA is the Alfvén velocity, and q is the tokamak safety factor [9, 10] This was the case in the mentioned DIII-D experiment where β ∼ 0.4% [6]. Appendix A contains the derivation of the width of the resonance of fast ions interacting with GAM modes or low-frequency Alfvén modes (whose frequency is small compared to the ion gyrofrequency) and the period of particle motion inside the resonance island, which uses a Hamiltonian formalism Both well-passing ions and marginally passing ions are considered. Effects of the plasma temperature and toroidal rotation on the rate of D–D fusion reaction in the beam-plasma system are considered in appendix C
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