Low dimensional gyrokinetic PIC simulation by $\delta f$ method

A step by step development of our low dimensional gyrokinetic Particle-in-Cell (PIC) simulation is reported. One dimensional PIC simulation of Langmuir wave dynamics is benchmarked.\footnote{C.Z.Cheng and G.Knorr, J. Comput. Phys. {\bf 22}, 330 (1976).} We then take temporal plasma echo as a test problem to incorporate the $\delta f$ method. Electrostatic driftwave simulation in one dimensional slab geometry\footnote{S~.E.~Parker and W.~W.~Lee, Phys. Fluids B {\bf 5}, 77 (1993).} is resumed in the presence of finite density gradients. By carefully diagnosing contour plots of the $\delta f$ values in the phase space, we discuss the saturation mechanism of the driftwave instabilities. A $v_\parallel$ formulation is employed in our new electromagnetic gyrokinetic method by solving Helmholtz equation for time derivative of the vector potential.\footnote{H.~Naitou (private communication, 2009); E.~A.~Starstev (private communication, 2014). Electron and ion momentum balance equations are employed in the time derivative of the Ampere's law.} This work is supported by Ministry of Science and Technology of Taiwan, MOST 103-2112-M-006-007 and MOST 104-2112-M-006-019.