Hamiltonian theory of the ion cyclotron minority heating dynamics in tokamak plasmas

Abstract

The question of heating a tokamak plasma by means of electromagnetic waves in the ion cyclotron range of frequencies (ICRF) is considered in the perspective of large rf powers and in the low collisionality regime. In such a case, the quasilinear theory (QLT) is validated by the Hamiltonian dynamics of the wave–particle interaction which exceeds the threshold of the intrinsic stochasticity. The Hamiltonian dynamics is represented by the evolution of a set of three canonical action angle variables well adapted to the tokamak magnetic configuration. This approach allows derivation of the rf diffusion coefficient with very few assumptions. The distribution function of the resonant ions is written as a Fokker–Planck equation but the emphasis is put on the QL diffusion instead of on the usual diffusion induced by collisions. The Fokker–Planck equation is then given a variational form from which a solution is derived in the form of a semianalytical trial function of three parameters: the percentage of resonant particles contained in the tail, an isotropic width ΔT, and an anisotropic width ΔP. This solution is successfully tested against real experimental observations. It is shown that in the case of the JET tokamak [Plasma Phys. Controlled Fusion 30, 1467 (1988)] the distribution function is influenced by adiabatic barriers which in turn limit the Hamiltonian stochasticity domain within energy values typically in the MeV range. Consequently and for a given ICRF power, the tail energy excursion is lower and its concentration higher than that from a bounce-averaged prediction. This may actually be an advantage for machines like JET [Plasma Phys. Controlled Fusion 30, 1467 (1988)] considering the energy range required to simulate the α-particle behavior in a relevant fusion reactor.

The behaviour of fast ions in tokamak experiments

“…Though the discrepancy was smaller, the helium spectra also suggested a broader heating profile than that suggested by calculations of wave propagation. Bkcoulet et al [234] suggest that these discrepancies may reflect a failure of the bounce averaged quasi-linear operator, but their simulations neglect the neutral density profile, which has an important effect upon the charge exchange spectra. Alternatively, fast ion transport may account for the discrepancy [232].…”

Monte Carlo operators for ions interacting with radio frequency waves

“…When the resonance functions are sufficiently broad and overlapping, one can perform the replacement, as shown in Ref. 10…”

An Equation of the Quasilinear Theory with Wide Resonance Region

“…The most known QL equation was proposed by Kennel and Engelmann (K&E) [1]. Later, quasilinear equations involving the inhomogeneity of the magnetic field of toroidal fusion devices were derived and analyzed (see, e.g., [2][3][4][5][6][7][8]). Nevertheless, many practical applications of the QL theory are associated with the K&E equation, because it often describes main features of the quasilinear distortion of the particle distribution function, being at the same time rather simple.…”