Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence

An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over {E}} \times \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over {B}} $ shear has been derived. In gyrokinetic Maxwell's equation, the particle charge density and current have been explicitly evaluated via pull-back transformation from the gyrocenter distribution function. Our generalized ordering takes$\rho _i <<\rho _{i\theta } \sim L_E \sim L_p $ as typically observed in the H-mode edge. We take $k_\bot \rho _i \sim 1$for generality, and keep the relative fluctuation amplitudes $e\delta \phi /T_i \sim \delta B/B<1$ up to the second order.