Fractional Resonances between Waves and Energetic Particles in Tokamak Plasmas

From numerical simulation and analytical modeling it is shown that fast ions can resonate with plasma waves at fractional values of the particle drift-orbit transit frequency when the plasma wave amplitude is sufficiently large. The fractional resonances, which are caused by a nonlinear interaction between the particle orbit and the wave, give rise to an increased density of resonances in phase space which reduces the threshold for stochastic transport. The effects of the fractional resonances on spatial and energy transport are illustrated for an energetic particle geodesic acoustic mode but they apply equally well to other types of MHD activity.

Figure 1

(a) E-GAM potential. (b) Energy excursion of a 50 keV deuteron launched at $R=2.02\mathrm{m}$ and $Z=0.0\mathrm{m}$, and at a pitch of 0.5, as a function of time in the presence of a 15 kHz E-GAM. The particle orbit during one energy cycle (0.33 ms) (c) without and (d) with the E-GAM present. Black (orange online) line: full orbit; light gray (blue online) line: guiding center.

Figure 2

Minimum and maximum particle energy excursion as a function of the initial transit frequency for various values of the electrical potential of the E-GAM. (a) At 0.05 kV only the linear resonances are present. (b) At 0.5 kV fractional resonances appear. (c) At 1.25 keV the threshold for mode overlap between linear and fractional resonances is passed, while (d) at 5 keV the region up to the fundamental resonance has become stochastic and subharmonic resonances are developing strongly. Integer and a number of fractional resonances are indicated.

Figure 3

Energy-pitch phase space with linear [solid (green online) line] and a small number of fractional resonances [dashed (red online) line] shown. The locations of the three ensembles that were used to investigate the effects of the fundamental resonance, the half resonance, and nonresonance on particle transport are indicated with the black squares.

Figure 4

Standard deviation of the (a) radial and (b) energy excursion of an ensemble of ions as a function of E-GAM mode amplitude for the fundamental resonance, the half resonance, and nonresonant particles.

Figure 5

Magnetic [dashed (black) line] and FILD [solid (red online) line] spectrum. Losses at half the E-GAM frequency are only observed with FILD.