Bootstrap Current Coefficients in Stellarators

A method to derive analytical expressions for bootstrap current coefficients is studied. The drift kinetic equation is divided into two parts corresponding to the effects of local and global structures of magnetic fields. The divided trans- port coefficients can be approximated by connecting the results of only three types of asymptotic expansions of the divided equations. The current coefficients obtained by adding these two parts approximately reproduce results of a direct numerical calculation of the drift kinetic equation. The analytical theory of bootstrap currents in non-sym- metric toroidal plasmas (1-5) has been constructed based on the moment equation approach (6). In this approach, parallel (to the magnetic field) momentum balance equations satisfy- ing particle, momentum, and energy conservation are used together with neoclassical parallel viscosities given by solu- tions of the linearized drift kinetic equation (DKE). The vis- cosity forces include two parts; damping force against the parallel flows and driving force for the flows due to radial gradient forces. In stellarators, both of contributions due to ripple and banana trapped orbits are included in the driving force. These trapped orbits generate the driving forces in opposite directions to each other while their contributions to the damping force have same direction. Here we discuss a method to express analytically this driving force part of the viscosity under the co-existence of two types driving mecha- nisms. The treatments of the damping force part in general non-symmetric toroidal plasmas (7) are substantially identi- cal to those in symmetric plasmas such as tokamaks and thus details of them are out of scope of this paper. However, we should note that, even in the symmetric plasmas, the lin- earized DKE including the complete parts of the Vlasov and collision operators with the conservative properties could not be solved analytically. Therefore, for the damping and driv- ing forces in tokamaks (6), a connection formula was used to smoothly connect results of three types of asymptotic expan- sions of the DKE; (1) banana regime expansion giving the viscosity of ∝ν/υ, (2) plateau regime expansion giving that of ∝(ν/υ) 0 , and (3) Pfirsch-Schlueter regime expansion giv- ing that of ∝(ν/υ) −1 , where υ/ν is the collision mean free