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 - v_par δA_par/c], which is straightforward to implement numerically, and which has straightforward expressions for its conservation properties. Here, q is the particle charge, δϕ and δA_par are the perturbed electrostatic and parallel magnetic potentials, v_par is the parallel velocity, c is the speed of light, and T is the temperature. The derivation is based on the quantity ε:=(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\times 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), ε is of order ρ/L, where L is the equilibrium profile scale length, for all scales λ ranging from ρ to L. This is true even though qδψ/T = O(1) for λ ~ L. Significant additional simplifications result from ordering L/R =O(ε), where R 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 screapeoff 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.