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Nonlinear heating of ions by electron cyclotron frequency waves in tokamaks

The conditions for nonlinear heating of ions by two high frequency electron cyclotron (EC) beams are determined analytically. It is found that the extraordinary X mode is best suited for imparting wave energy to ions and modifying their distribution function. The eventual aim is to determine the possibility of affecting the transport of fusion alpha particles in tokamak plasmas by EC waves. Introduction It has been previously shown that high frequency electrostatic lower hybrid waves can interact with ions and nonlinearly energize them [1, 2]. This occurs either for a spectrum with a bandwidth larger than the ion cyclotron frequency or for two plane waves with their frequencies separated by a low integer multiple of the ion cyclotron frequency. We have developed a more general theory which considers the nonlinear interaction of electromagnetic waves with ions in a magnetized plasma. Of particular interest is the interaction with waves in the EC range of frequencies the X mode and the O mode. The electromagnetic waves propagate as a Gaussian beam into the plasma and the interaction with ions occurs in a region where two such beams overlap. The wave frequency of each beam is modulated around the the ion cyclotron frequency. The conditions for nonlinear interaction are studied using the Lie transform canonical perturbation theory for Hamiltonian systems [3]. Hamiltonian Perturbation Theory The motion of a charged particle in the presence of electromagnetic fields is given by the Hamiltonian h(q,p, t) = [ 1+ (p− eA(q, t))2 ]1/2 + eΦ(q, t). (1) where A and Φ are the vector and scalar potentials, respectively, for the fields, e is the charge of the particle, q is its position, and p is its momentum. The mass m and the speed of light c have been normalized out. A particle’s position and velocity are represented in phase space by the 6–dimensional vector z ≡ (q,p). We assume that A =A0+eA1. The first represents the background steady state magnetic field while the second term corresponds to the EM field. Φ is the scalar potential corresponding to 37 EPS Conference on Plasma Physics P5.157