Stochastic Heating of Ions in a Tokamak by RF Power

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.