# Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solver

## Abstract

The DK4D code has been written to solve a set of time-dependent, axisymmetric, finite-Larmor-radius drift-kinetic equations (DKEs) for the non-Maxwellian part of the electron and ion distribution functions using the full, linearized Fokker–Planck–Landau collision operator. The plasma is assumed to be in the low- to finite-collisionality regime, as is found in the cores of modern and future magnetic confinement fusion experiments. Each DKE is formulated such that the perturbed distribution function carries no net density, parallel momentum, or kinetic energy. Rather, these quantities are contained within the background Maxwellians and would be evolved by an appropriate set of extended magnetohydrodynamic (MHD) equations. This formulation allows for straight-forward coupling of DK4D to existing extended MHD time evolution codes. DK4D uses a mix of implicit and explicit temporal representations and finite element and spectral spatial representations. These, along with other computational methods used, are discussed extensively. Steady-state benchmarks are then presented comparing the results of DK4D to expected analytic results at low collisionality, qualitatively, and to the Sauter analytic fits for the neoclassical conductivity and bootstrap current, quantitatively. These benchmarks confirm that DK4D is capable of solving for the correct, gyroaveraged distribution function in stationary magnetic equilibria. Furthermore, the results presented demonstratemore »

- Authors:

- Princeton University, Princeton, New Jersey 08544 (United States)
- Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543-0451 (United States)
- Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 (United States)

- Publication Date:

- OSTI Identifier:
- 22410384

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 22; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1070-664X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; AXIAL SYMMETRY; BENCHMARKS; BOOTSTRAP CURRENT; COLLISIONS; COMPUTER CODES; DENSITY; DISTRIBUTION FUNCTIONS; ELECTRONS; FINITE ELEMENT METHOD; KINETIC ENERGY; KINETIC EQUATIONS; LARMOR RADIUS; MAGNETIC CONFINEMENT; MAGNETIC MOMENTS; MAGNETOHYDRODYNAMICS; PLASMA; STEADY-STATE CONDITIONS; TIME DEPENDENCE

### Citation Formats

```
Lyons, B. C., Jardin, S. C., and Ramos, J. J.
```*Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solver*. United States: N. p., 2015.
Web. doi:10.1063/1.4918349.

```
Lyons, B. C., Jardin, S. C., & Ramos, J. J.
```*Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solver*. United States. doi:10.1063/1.4918349.

```
Lyons, B. C., Jardin, S. C., and Ramos, J. J. Fri .
"Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solver". United States. doi:10.1063/1.4918349.
```

```
@article{osti_22410384,
```

title = {Steady-state benchmarks of DK4D: A time-dependent, axisymmetric drift-kinetic equation solver},

author = {Lyons, B. C. and Jardin, S. C. and Ramos, J. J.},

abstractNote = {The DK4D code has been written to solve a set of time-dependent, axisymmetric, finite-Larmor-radius drift-kinetic equations (DKEs) for the non-Maxwellian part of the electron and ion distribution functions using the full, linearized Fokker–Planck–Landau collision operator. The plasma is assumed to be in the low- to finite-collisionality regime, as is found in the cores of modern and future magnetic confinement fusion experiments. Each DKE is formulated such that the perturbed distribution function carries no net density, parallel momentum, or kinetic energy. Rather, these quantities are contained within the background Maxwellians and would be evolved by an appropriate set of extended magnetohydrodynamic (MHD) equations. This formulation allows for straight-forward coupling of DK4D to existing extended MHD time evolution codes. DK4D uses a mix of implicit and explicit temporal representations and finite element and spectral spatial representations. These, along with other computational methods used, are discussed extensively. Steady-state benchmarks are then presented comparing the results of DK4D to expected analytic results at low collisionality, qualitatively, and to the Sauter analytic fits for the neoclassical conductivity and bootstrap current, quantitatively. These benchmarks confirm that DK4D is capable of solving for the correct, gyroaveraged distribution function in stationary magnetic equilibria. Furthermore, the results presented demonstrate how the exact drift-kinetic solution varies with collisionality as a function of the magnetic moment and the poloidal angle.},

doi = {10.1063/1.4918349},

journal = {Physics of Plasmas},

issn = {1070-664X},

number = 5,

volume = 22,

place = {United States},

year = {2015},

month = {5}

}