Modeling full radial electric field and flow shears in gyrokinetic simulations

The radial electric field ($E_r$) is important in the turbulence of tokamak plasmas. It affects the growth rate of instabilities through the $E\times B$ shear and changes the real frequency of drift waves by adding a Doppler shift. The modeling of $E_r$ in simulations, however, was usually not complete. The full profiles of the main ion toroidal and poloidal flows were not implemented. In the gyrokientic electromagnetic particle code GEM, the poloidal flow was assumed to be zero by introducing a parallel flow. However, recent experiments show that the poloidal flow could be important.\footnote{B.A. Grierson et al., Nucl. Fusion 53 (2013) 063010.} In this study we add the full main ion rotation flows to GEM, following the comprehensive procedures of Sugama and Horton.\footnote{H. Sugama and W. Horton, Phys. Plasmas 5, 2560 (1998).} The major contribution to the $E_r$ from the ion toroidal flow is used as $E_{r0}$, and the result as $E_{r1}$. The effects to the growth rate and Doppler shift of all terms in the force balance equation are demonstrated using linear simulations of edge and core tokamak plasmas.