@PhdThesis{karney77a,
  author =       {Charles F. F. Karney},
  title =        {Stochastic Heating of Ions in a Tokamak by
                  Radio-Frequency Power},
  school =       {Massachusetts Institute of Technology},
  department =   {Department of Electrical Engineering and Computer
                  Science},
  reprint =      {thesis77},
  month =        may,
  year =         1977,
  abstract =     {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, <i>B</i><sub>0</sub>, to
                  be uniform and the lower hybrid wave (whose frequency,
                  &omega;, is much greater than the ion cyclotron
                  frequency, &Omega;<sub><i>i</i></sub>&nbsp;) 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 <i>E</i>/<i>B</i><sub>0</sub>
                  &gt;
                  &frac14;(&Omega;<sub><i>i</i></sub>&nbsp;/&omega;)<sup>1/3</sup>&omega;/<i>k</i>
                  (<i>k</i> is the wavenumber of the wave).  The
                  stochastic region of velocity space extends from a
                  lower limit given by &omega;/<i>k</i> &minus;
                  (<i>qE</i>/<i>mk</i>)<sup>1/2</sup> (<i>q</i> and
                  <i>m</i> are the ionic charge and mass, respectively)
                  where the ion dynamics are determined by trapping to
                  an upper limit which scales as <i>E</i><sup>2/3</sup>.
                  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.}
}