Particle-in-Cell simulation of energetic particles driven instabilities

We present simulations of the evolution of energetic particles driven modes with the gyrokinetic turbulence code GEM\footnote{Y.~Chen and S.~E.~Parker, J. Comp. Phys. {\bf 220}, 839 (2007)}, except that kinetic electrons are replaced by a mass-less fluid model. PIC simulations of energetic particles use either the conventional full-f method or the $\delta\! f$ method. The latter is adequate for low-amplitude fluctuation amplitudes. The collisional $\delta \! f$-method\footnote{Y.~Chen and R.~White, Phys. Plasmas {\bf 4}, 3591 (1997)} is used to systematically account for collisions and particle source and sink. Steady state saturation amplitudes are benchmarked with predictions of analytic theory. We also employ full-f simulations\footnote{Y.~Todo {\it et.~al}, Phys. Plasmas 10, 2888 (2003)} to study bursty events in which the instabilities reach large amplitudes and cause macroscopic redistribution or loss of the particles. With full-f it is easy to retain all the nonlinear effects and treat accurately discontinuities in the distribution function at phase-space boundaries. Whereas the energetic particle current is neglegible in the Ampere's law in $\delta \!f$ simulations, it is important in full-f simulations. Thermal ion kinetic effects are observed to be important.