Gyrokinetic equations for strong-gradient regions

Abstract

A gyrokinetic theory is developed under a set of orderings applicable to the edge region of tokamaks and other magnetic confinement devices, as well as to internal transport barriers. The result is a practical set equations that is valid for large perturbation amplitudes [qδψ/T=O(1), where δψ=δφ-ν∥δA∥/c], which is straightforward to implement numerically, and which has straightforward expressions for its conservation properties. Here, δφ and δA∥ are the perturbed electrostatic and parallel magnetic potentials, ν∥ is the particle velocity, c is the speed of light, and T is the temperature. The derivation is based on the quantity ɛ≡(ρ/λ⊥)qδψ/T≪1 as the small expansion parameter, where ρ is the gyroradius and λ⊥ is the perpendicular wavelength. Physically, this ordering requires that the E×B velocity and the component of the parallel velocity perpendicular to the equilibrium magnetic field are small compared to the thermal velocity. For nonlinear fluctuations saturated at “mixing-length” levels (i.e., at a level such that driving gradients in profile quantities are locally flattened), ɛ is of the order ρ/Lp, where Lp is the equilibrium profile scale length, for all scales λ⊥ ranging from ρ to Lp. This is true even though qδψ/T=O(1) for λ⊥∼Lp. Significant additional simplifications result from ordering Lp/LB=O(ɛ), where LB is the spatial scale of variation of the magnetic field. We argue that these orderings are well satisfied in strong-gradient regions, such as edge and scrapeoff layer regions and internal transport barriers in tokamaks, and anticipate that our equations will be useful as a basis for simulation models for these regions.