Verification of continuum drift kinetic equation solvers in NIMROD

Abstract

Verification of continuum solutions to the electron and ion drift kinetic equations (DKEs) in NIMROD [C. R. Sovinec et al., J. Comp. Phys. 195, 355 (2004)] is demonstrated through comparison with several neoclassical transport codes, most notably NEO [E. A. Belli and J. Candy, Plasma Phys. Controlled Fusion 54, 015015 (2012)]. The DKE solutions use NIMROD's spatial representation, 2D finite-elements in the poloidal plane and a 1D Fourier expansion in toroidal angle. For 2D velocity space, a novel 1D expansion in finite elements is applied for the pitch angle dependence and a collocation grid is used for the normalized speed coordinate. The full, linearized Coulomb collision operator is kept and shown to be important for obtaining quantitative results. Bootstrap currents, parallel ion flows, and radial particle and heat fluxes show quantitative agreement between NIMROD and NEO for a variety of tokamak equilibria. In addition, velocity space distribution function contours for ions and electrons show nearly identical detailed structure and agree quantitatively. A H-centered, implicit time discretization and a block-preconditioned, iterative linear algebra solver provide efficient electron and ion DKE solutions that ultimately will be used to obtain closures for NIMROD's evolving fluid model. V C 2015 AIP Publishing LLC.

The impact of collisionality, FLR, and parallel closure effects on instabilities in the tokamak pedestal: Numerical studies with the NIMROD code

“…If the growth rates are modified beyond this small effect, it would be an indication that changes in the energy equation can impact the MHD response, and thus that closure effects are significant. Although the short-mean-free path Braginskii-like closure [34,35] is used in our NIMROD compuatations, we note that other closures, such as long-mean-free path [36] or the general approach of solving drift-kinetic equation [37] for a closure can be applied. However, for this case where the parallel-closure effects are negligible, different parallel closures are unlikely to significantly modify the results.…”

Effect of scrape-off-layer current on reconstructed tokamak equilibrium

“…The contributing integrals lie on top of each other for the low SOL-current case (160414). The improved equilibria provided by recomputation of the Grad-Shafranov solution are critical in NIM-ROD drift-kinetic computations [11]. For example, an accurate account of b · ∇ln|B| in latter integral of Eqn.…”

“…[6]). The effects encompassed by extended-MHD are variable: extended-MHD refers to models beyond resistive MHD that include some combination of anisotropic thermal conduction and stresses [7], two-fluid evolution [8], finite-Larmorradius closures [9,10], and/or advanced drift-kineticequation closures [11].…”

Coupling extended magnetohydrodynamic fluid codes with radiofrequency ray tracing codes for fusion modeling

“…We anticipate exploring this topic in greater depth in subsequent publications; investigation of the relationship of experimentally observed Fisch and Boozer [47] and Ohkawa [48] currents to RF-induced modifications of the closures is also a topic of interest. The recent development and implementation of continuum closure capabilities in NIMROD [49], via the solution of species drift-kinetic equations, is expected to enable significant progress in these efforts. In particular, the accurate inclusion of critical neoclassical effects such as bootstrap current, via the computation of parallel electron stresses and collisional friction in higher-β plasmas, will be crucial for modeling experimentally relevant NTM dynamics and stabilization in tandem with the methods we have proposed here.…”