Nonlocal analyses of electrostatic and electromagnetic waves in hot, magnetized, nonuniform, bounded plasmas

Abstract

Heating of tokamak plasmas up to temperatures of the order of 10 keV (≈ 108 0K) is one of the main subjects in plasma physics research. Much experimental and theoretical effort has been devoted to the improvement of the heating efficiency and to the understanding of the beam-particle or wave-particle interactions. We have studied the latter subject. Many models describing the linear wave-particle interaction already exist, allowing one to analyze the absorption of the wave by the electrons and/or the ions. They can be separated into two main classes: ray-tracing models [Brambilla, 1986] and wave models [Willard et al., 1986]. The latter describe the global wave field (sum of incident, transmitted and reflected fields) taking into account the finiteness of the plasma, the boundary conditions, the plasma-vacuum interfaces and the antenna. With this wave approach, one is able to study scenarii where cut-off, reflected, resonant, mode converted and/or evanescent wave fields are present. Note also that this work is principally devoted to the ion cyclotron range of frequencies. The most advanced wave models are the models which use a second-order expansion in k⊥ρσ, where k⊥ is the perpendicular wavenumber and ρσ the Larmor radius of species σ [Appert et al., 1986a and 1987; Jaeger et al., 1988; Fukuyama et al., 1986; Edery and Picq, 1986; Brambilla and Krucken, 1988b]. In this way, one obtains a system of three second-order differential equations [Martin and Vaclavik, 1987]. However, the local models are limited to Larmor radii small compared with the wavelength or with the characteristic length of the inhomogeneous density and temperature profiles. Moreover, they are limited to frequencies lower than the third harmonic of the cyclotron frequency. In present day experiments, the temperature of the particles is very high, especially if tails of high energy particles are created. Also, the first experiments with D-T plasmas, generating alpha particles having a temperature of the order of 1 MeV, have been performed in JET. Moreover, increasing numbers of experiments use heating scenarii at high harmonic frequencies. Because these cases can no longer be studied using a local model, we have developed a nonlocal model which is not limited by the size of the Larmor radii nor by the harmonic considered. This model is based on the wave approach and therefore can treat the variety of problems mentioned above. Nevertheless, we have limited our work to uni-dimensional geometry, Maxwellian equilibrium distribution functions and slowly-varying equilibrium magnetic field. We have also neglected ky in the conductivity tensor, where y is the direction normal to the direction of the inhomogeneity and to the magnetostatic field. Starting from the linearized Vlasov-Maxwell equations, we have derived the equations in the Fourier and the configuration spaces (Chaps.3 and 4), which consist, in the latter case, of a system of three second-order integro-differential equations. We have also derived a formulation of the local power absorption allowing us to determine the profile of absorption of the wave by the particles. The equations are solved numerically using the finite element method (Chap.5). We have developed two codes, SEAL and SEMAL, which calculate the wave field in the electrostatic and electromagnetic cases, respectively. These codes have been tested, and SEAL has been used to simulate an experiment which studies the interaction of a Bernstein wave with a cylindrical plasma (Chap.6). SEMAL has mainly been used to study the effects of the alpha particles on ion cyclotron heating. We have shown that the local model was inadequate and have studied in more detail the effect of temperature and the strong influence of the alpha particle concentration (Chap.7). We have also studied the excitation of an ion Bernstein wave through mode conversion at the lower-hybrid frequency in the scrape-off layer (Chap.7).