Linear dispersion relation of geodesic acoustic modes driven by trapped and circulating energetic particles

Abstract

We derive the local dispersion relation of energetic-particle-induced geodesic acoustic modes (EGAMs) for both trapped and circulating ion beams with single pitch angle slowing-down and Maxwellian distributions, as well as a bump-on-tail distribution in tokamak plasmas. For slowing-down and Maxwellian particles, the solutions of the local dispersion relation give the spectrum, growth rate and thresholds of excitation as functions of the pitch angle, beam density and frequency of the energetic particles bounce motion. For circulating ions there is only one unstable branch with frequency below the GAM continuum and a threshold of excitation in the pitch angle, for both the slowing-down and single pitch Maxwellian distributions. Trapped particles cause no excitation of a mode for neither slowing-down nor Maxwellian ion beams, but they can excite a mode with a bump-on-tail distribution when the mean velocity of the beam is larger than the threshold and the energetic particle bounce frequency is high enough.

Gyrokinetic modeling of anisotropic energetic particle driven instabilities in tokamak plasmas

“…Whereas, positive gradients of the distribution functions in velocity space are needed to drive EGAMs, and as a consequence of which a certain portion of EPs will be redistributed to lower energies [28]. In previous work, analytical anisotropic distribution functions are bump-on-tail [10,11,28] and slowing down with pitch dependency [8,10,13,22], as well as single pitch Maxwellian [10]. Analytical theory and simulations adopting pitch dependent slowing down distribution function described it as function of Λ = μB/E, with μ being the magnetic moment of the particle, E total kinetic energy and B background magnetic field.…”

“…EP density is one of the main parameters affecting it. Simulations and analytical computations showed that an increase in EP concentration leads to an increase of growth rate of the mode, marking at a certain value of EP fraction with respect to electron density n EP /n e the threshold value for the transition from a damped to an excited mode [8][9][10][20][21][22][23]. On the other hand, the frequency is usually found to be decreasing because of the transition from an higher frequency GAM to a lower frequency EGAM for increasing EP fraction.…”

“…Particularly, the mode is found to be localized near the edge of the plasma column, since the damping rate is proportional to exp(−q 2 ) [7], where q is the tokamak safety factor. Energetic particles (EPs) can excite this mode, generating an energetic-particle driven GAM or energetic-particledriven geodesic acoustic mode (EGAM) [8][9][10][11][12][13]. The mode can be excited by inverse Landau damping with the resonant EPs, which causes a redistribution of the latter from higher to lower energies [14].…”

“…The mode can be excited by inverse Landau damping with the resonant EPs, which causes a redistribution of the latter from higher to lower energies [14]. Linear excitation of EGAMs with both circulating EPs [8,10], as well as trapped particles have been studied [10], using the deeply trapped particle model [15,16]. Additionally, theory of EGAMs excited by not fully sloweddown energetic ions was derived in reference [17] in reference to experimental observations in Large Helical Device [18].…”

“…For this purpose a new distribution function is formulated and implemented in the gyrokinetic code ORB5 [29] (section 2). Then, theoretical analysis of the newly implemented distribution function is presented, following the dispersion relation studies carried out in [8,10,21] (section 3). The numerical results and the effects of the shape of the distribution function in velocity space, in terms of its parameters, are reported and discussed (section 4).…”