Nonlinear Wave-Particle Interaction in a Toroidal Magnetic Field.

This thesis is a report of the research conducted to study the non-linear resonant interaction of highly energetic particles of a fusion plasma, confined in an axi-symmetric device, with externally imposed electromagnetic propagating waves. The relevance of this analysis to the Lagrangian particle transport and heating kinetic theories is discussed. Hamiltonian mechanics is used to be able to utilize modern theories on nonlinear dynamics. Since the conventional Hamiltonian perturbation theories are not adequate for electromagnetic waves, a general Hamiltonian perturbation theory is developed to include perturbations in phase space by a vector potential. Concepts in modern differential geometry are used to formulate a general Hamiltonian mechanics similar to Birkhoffian mechanics. The Hamilton's equations of motion are expressed in terms of new operators. A direct result is that explicit expressions for the perturbed equations of motion can be obtained. A general symplectic structure for the Hamiltonian phase flow is identified. The new perturbation method is based on the transformation theory of the general symplectic two-form of the phase space. Dynamics of a charged particle in resonant interaction with an electromagnetic field in a uniform magnetic field is examined. The criteria for the onset of global stochastic regions in phase space are derived. The dynamics of three dimensional motion of a charged particle in an axi-symmetrical magnetic field is examined. The effects of perturbations by an electromagnetic field to the phase space of such a particle is studied. It is demonstrated that a frequency -modulated wave can induce a global stochasticity in phase space which results in strong diffusion processes in action space of the particle. Since the required magnitude for the perturbing field to induce strong diffusion processes is very low, it may be well satisfied by the present experiments. The formulation is cast in a suitable coordinate such that the resulting Fokker-Planck kinematic and diffusion coefficients can be used in a Lagrangian kinetic theory.