Current Drive Calculations: Benchmarking Momentum Correction and Field-Line Integration Techniques

The adjoint approach is a standard tool for calculating the current drive (CD) efficiency in electron cyclotron ray-tracing codes. In the most common version of this approach, the first Legendre harmonic of the solution of the (first order) linearized drift-kinetic equation (DKE) with a parallel momentum conserving collision operator is used. This treatment is equivalent to a generalized Spitzer function for arbitrary collisionalities [1, 2] and would require the solution of the DKE in 4D-phase space for stellarators (3D for tokamaks). Momentum correction techniques are based on mono-energetic transport coefficients calculated from the solution of the equivalent DKE with the simple Lorentz form of the pitch-angle collision term without momentum conservation, where the radius, r, and the velocity, v, are only parameters. Only the flux-surface-averaged momentum-corrected parallel flows are then estimated without again solving the DKE. With precalculated databases of mono-energetic transport coefficients (from DKES [3] or NEO-MC [4] code), this technique is well suited for calculating the electric conductivity and the bootstrap current (also for NBCD) in arbitrary magnetic configurations [2]. The ECCD source function, i.e., the quasi-linear diffusion term with the Maxwellian in linear theory, however, is highly localized in 4D-phase space. In principle, this requires also the 4D-solution of the (adjoint) DKE which may only be obtained analytically in the collisional (classical Spitzer problem) and in the collisionless limits. The collisionless solution, g(x, ¸), given in Ref. [5] with the normalized magnetic moment, ¸, is constant on the flux-surface