Nonlinear gyrokinetic equations for turbulence in core transport barriers

An energy-conserving set of the nonlinear electrostatic gyrokinetic Vlasov and Poisson equations is derived for the first time in the presence of equilibrium E×B velocity uE∼vTi, via phase-space Lagrangian Lie-perturbation theory. In this general formulation, only the basic small parameter ε with ω/Ω∼k∥/k⊥∼ε and δn/n0∼1/k⊥L∼ε, is used, while no device-specific expansion has been made. Here, L is the equilibrium scale length. For application to microturbulence in tokamak core transport barriers, an additional small ordering parameter δB≡Bθ/B≪1 is utilized. This leads to a useful form of the nonlinear gyrokinetic system which is applicable to a realistic situation in which the gradient lengths of the equilibrium radial electric field and pressure are of the same order as the ion poloidal gyroradius. The ordering for fluctuations is also modified to δn/n0∼εδB≪1/k⊥L∼δB for a better description of sub-mixing-length level fluctuations. uE/vTi∼δB and ρθi∼Lp put the pressure-gradient contribution to Er and the toroidal-flow contribution to Er at the same order. δB∼ε is shown to be a maximal ordering for studying the E×B flow shear suppression of turbulence.