Contained modes in inhomogeneous plasmas and their interaction with high energy particles

In experiments with fusing plasmas, enhanced emission at the harmonics of the ion cyclotron frequency of the fusion products has been observed. In this thesis a model is developed that explains the main characteristics of the emission, in particular the fact that the radiation peaks occur at frequencies corresponding to harmonics of the a (or D) cyclotron frequency, Q, at the outer edge of the plasma column. Our results indicate also a transition to a continuum spectrum at high frequencies that is consistent with the experiments. The modes corresponding to the discrete spectrum, are localized near the plasma edge. The radial containment is in fact a fundamental characteristic of our model, since only contained modes can generate sufficient growth of the mode to justify the detected emission power levels. The instability evolves on a time scale that is shorter than the slowing-down time of a-particles, and so the distribution function for the interacting particles, which are trapped particles near the outer edge of the plasma, can be described as strongly anisotropic in velocity space and having energies close to their value at birth from the fusion reaction. The contained mode is a solution to the ideal MHD equations, extended to include the Hall term, so the frozen-in law is replaced by the more general Ohm's equation E + i, x B = (1/en)! x B. We focus on quasi-flute modes with a high poloidal number, that are generalizations, to the case of an inhomogeneous plasma, of the magnetosonic-whistler modes. The wavevector has a small component parallel to the magnetic field, k11. The sensitivity of the localization of the mode to kll effects increases with frequency, and the interval over which modes can be excited extends toward the center of the plasma column. This suggest a mechaninsm for the transition to a continuum spectrum, since more a-particles can interact with the mode. The growth rate can be evaluated using a full toroidal calculation for the particle dymanics. The growth rate depends linearly on the a-particle density, and can be of the order of the bounce frequency of the interacting a-particles. Thesis Supervisor: Professor Bruno Coppi TABLE OF CONTENTS ABSTRACT . 2 TABLE OF CONTENTS . 4 ACKNOWLEDGMENTS . 5 0. Introduction . 8 1. The Magnetosonic-Whistler Mode . 13 2. The Mode Structure. A Contained-Interactive Mode . 23 3. The Whistler Contribution to the Contained Mode . 41 4. A Simple Picture of the Relevant a-particles Orbits . 54 5. The a-particles Distribution Function . 73 6. Quadratic Versus Linear Growth Rate . 79 7. The Growth Rate . 88 8. Conclusions . 99 Appendix A. A Brief Review of Anomalous Ion Cyclotron Emission (ICE) Observations in JET and TFTR . 103 Appendix B. The Coordinate System . 108 Appendix C. I as an Adiabatic Invariant . 110 Appendix D. The Distribution Function Obtained Through an Orbit Average of the Source. The Example of the Cyclotron Motion . 113 Appendix E. A Detailed Calculation of the Conductivity Tensor with O(A) Corrections . 118 Appendix F. The Curvature of the Poloidal Field. Example of an Explicit Calculation for an O(A/q) term . 126 Appendix G. The w, Contribution to the Perturbed Vlasov Operator 128 REFERENCES . 130