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PhD Thesis at MIT.

The nonlinear interaction of ions in a plasma with waves excited by RF power near the lower hybrid frequency is considered. The object of this study is to evaluate such an interaction as a method of heating the ions in a tokamak. The problem is simplified by assuming the magnetic field, \(B_0\), to be uniform and the lower hybrid wave (whose frequency, \(\omega\), is much greater than the ion cyclotron frequency, \(\Omega_i\)) to be propagating perpendicular to the magnetic field. It is shown that trapping by the wave significantly increases the energy of the ions on a time scale of a few cyclotron periods. For longer times the motion in a certain region of velocity space becomes stochastic for fields satisfying \(E/B_0 > \frac14(\Omega_i/\omega)^{1/3}\omega/k\) (\(k\) is the wavenumber of the wave). The stochastic region of velocity space extends from a lower limit given by \(\omega/k - \sqrt{qE/mk}\) (\(q\) and \(m\) are the ionic charge and mass, respectively) where the ion dynamics are determined by trapping to an upper limit which scales as \(E^{2/3}\). The stochasticity condition is independent of how close \(\omega\) is to a cyclotron harmonic. These results are extended to the case of propagation at an oblique angle to the magnetic field and to the case of inhomogeneous magnetic field. It is shown that stochastic ion heating can be an efficient method for supplementary heating of the ions in a tokamak.

Nonlinear excitation of ion waves by a pump near the lower hybrid frequency is examined. The ion waves considered are ion acoustic, electrostatic ion cyclotron and magnetosonic waves. The approach adopted is coupling of modes carried out to second order. We examine the effects of varying the angles of the excited waves, and apply the results to excitation of such waves in a bounded inhomogeneous plasma. The plasma is assumed uniform and is described by the two fluid equations. Considerable complexity is introduced by taking the angles of propagation to be arbitrary (subject to frequency and wave number matching conditions), and by taking the magnetic field to be finite. In order to treat this problem analytically, we have made extensive use of MACSYMA, a large symbolic computation system, that has been developed by the Mathlab group at M.I.T. Using this facility we have been able to obtain approximate analytic results for the growth rates of the excited waves. Using damping rates from kinetic theory, thresholds for these interactions have been calculated, for typical tokamak plasmas. With the interaction in a uniform plasma well understood, we have tried to examine some of the effects of exciting the pump from a waveguide at the wall of an inhomogeneous plasma. We find that in tokamak plasmas it is possible to excite electrostatic ion cyclotron waves, but since the pump ray is very narrow unstable pulses are unable to grow significantly before convecting out of the pump region. However if it is possible to set up an array of waveguides at the boundary of the plasma, a larger region of pump fields is attainable, and strong excitation of the electrostatic ion cyclotron wave should take place with reasonable external rf powers.