One and two dimensional electromagnetic gyrokinetic PIC simulation by $\delta f$ method

An electromagnetic gyrokinetic Particle-in-Cell simulation is studied aiming at long-wave-length magnetohydrodynamic instabilities. A fully nonlinear characteristic method ($\delta f$ method) of electrostatic gyrokinetic theory is employed.\footnote{S. E. Parker and W. W. Lee, Phys. Fluids B {\bf 5}, 77 (1993) } For a one dimensional geometry, ``0.5 dimension'' is taken in ``$y$-direction'' of the configuration space and another ``0.5 dimension'' is taken in the ``$v_z$-direction'' of the velocity space. Recent electromagnetic $\delta f$ simulation shows optimistic results.\footnote{A.Mishchenko and A. Zocco, Phys. Plasmas {\bf 19}, 122104, 2012; A. Bottino, T. Vernay, and B.Scott, Plasma Phys. Controlled Fusion {\bf 53}, 124027 (2011).} We revisit the importance of the conservation properties in the low dimensional geometries. This work is supported by National Science Council of Taiwan, NSC 100-2112-M-006-021-MY3 and NSC 103-2112-M-006-021-MY3.