# The Relationship between Flux Coordinates and Equilibrium-based Frames of Reference in Fusion Theory

## Abstract

The properties of two local reference frames based on the magnetic field and the current density are investigated for magnetized plasmas in toroidal geometry with symmetric angle. The magnetic field-based local frame of reference has been well-studied for example by Dewar and colleagues [Phys. Fluids 27, 1723 (1984)] An analogous frame based on the current density vector is possible because it is also divergence free and perpendicular to the gradient of the poloidal flux. The concept of straightness of a vector is introduced and used to elucidate the Boozer and Hamada coordinate systems. The relationship of these local frames to the more well-known Frenet frame of reference, which specifies a curve in terms of curvature and torsion, is given. As an example of the usefulness of the these formal relationships, we briefly review the ideal MHD theory and their use. We also present a new annihilation operator, useful for eliminating shorter time scales than the time scale of interest, for deriving the inner layer equations of Glasser, Greene, and Johnson [Phys. Fluids 18, 875 (1975)]. Compared to the original derivation that is based on the local frame of reference in terms of the magnetic field, the new annihilation operator thatmore »

- Authors:

- Tech-X Corp., Boulder, CO (United States)

- Publication Date:

- Research Org.:
- Tech-X Corp., Boulder, CO (United States); Tech-X Corporation, Boulder, CO (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Fusion Energy Sciences (FES) (SC-24)

- OSTI Identifier:
- 1557796

- Alternate Identifier(s):
- OSTI ID: 1543141

- Grant/Contract Number:
- SC0019067

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 26; Journal Issue: 8; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

### Citation Formats

```
Kruger, S. E. The Relationship between Flux Coordinates and Equilibrium-based Frames of Reference in Fusion Theory. United States: N. p., 2019.
Web. doi:10.1063/1.5098313.
```

```
Kruger, S. E. The Relationship between Flux Coordinates and Equilibrium-based Frames of Reference in Fusion Theory. United States. doi:10.1063/1.5098313.
```

```
Kruger, S. E. Mon .
"The Relationship between Flux Coordinates and Equilibrium-based Frames of Reference in Fusion Theory". United States. doi:10.1063/1.5098313. https://www.osti.gov/servlets/purl/1557796.
```

```
@article{osti_1557796,
```

title = {The Relationship between Flux Coordinates and Equilibrium-based Frames of Reference in Fusion Theory},

author = {Kruger, S. E.},

abstractNote = {The properties of two local reference frames based on the magnetic field and the current density are investigated for magnetized plasmas in toroidal geometry with symmetric angle. The magnetic field-based local frame of reference has been well-studied for example by Dewar and colleagues [Phys. Fluids 27, 1723 (1984)] An analogous frame based on the current density vector is possible because it is also divergence free and perpendicular to the gradient of the poloidal flux. The concept of straightness of a vector is introduced and used to elucidate the Boozer and Hamada coordinate systems. The relationship of these local frames to the more well-known Frenet frame of reference, which specifies a curve in terms of curvature and torsion, is given. As an example of the usefulness of the these formal relationships, we briefly review the ideal MHD theory and their use. We also present a new annihilation operator, useful for eliminating shorter time scales than the time scale of interest, for deriving the inner layer equations of Glasser, Greene, and Johnson [Phys. Fluids 18, 875 (1975)]. Compared to the original derivation that is based on the local frame of reference in terms of the magnetic field, the new annihilation operator that is based on the local frame of reference in terms of the current density simplifies the derivation.},

doi = {10.1063/1.5098313},

journal = {Physics of Plasmas},

number = 8,

volume = 26,

place = {United States},

year = {2019},

month = {7}

}

#### Figures / Tables:

Works referenced in this record:

##
Stability Criterion for Arbitrary Hydromagnetic Equilibria

journal, January 1962

- Greene, John M.; Johnson, John L.
- Physics of Fluids, Vol. 5, Issue 5

##
Resistive instabilities in a diffuse linear pinch

journal, June 1966

- Coppi, Bruno; Greene, John M.; Johnson, John L.
- Nuclear Fusion, Vol. 6, Issue 2

##
Magnetic coordinates for equilibria with a continuous symmetry

journal, January 1984

- Dewar, R. L.; Monticello, D. A.; Sy, W. N. -C.
- Physics of Fluids, Vol. 27, Issue 7

##
Resistive interchanges and the negative *V* " criterion

journal, January 1967

- Johnson, J. L.; Greene, J. M.
- Plasma Physics, Vol. 9, Issue 5

##
Resistive instabilities in general toroidal plasma configurations

journal, January 1975

- Glasser, A. H.; Greene, J. M.; Johnson, J. L.
- Physics of Fluids, Vol. 18, Issue 7

##
Plasma equilibrium with rational magnetic surfaces

journal, January 1981

- Boozer, Allen H.
- Physics of Fluids, Vol. 24, Issue 11

##
Generalized reduced magnetohydrodynamic equations

journal, December 1998

- Kruger, S. E.; Hegna, C. C.; Callen, J. D.
- Physics of Plasmas, Vol. 5, Issue 12

##
Time-dependent drift Hamiltonian

journal, January 1984

- Boozer, Allen H.
- Physics of Fluids, Vol. 27, Issue 10

##
Hydromagnetic equilibria and their proper coordinates

journal, January 1962

- Hamada, Shigeo
- Nuclear Fusion, Vol. 2, Issue 1-2

##
Ideal MHD stability calculations in axisymmetric toroidal coordinate systems

journal, January 1983

- Grimm, R. C.; Dewar, R. L.; Manickam, J.
- Journal of Computational Physics, Vol. 49, Issue 1

##
The direct criterion of Newcomb for the ideal MHD stability of an axisymmetric toroidal plasma

journal, July 2016

- Glasser, A. H.
- Physics of Plasmas, Vol. 23, Issue 7

##
Nonlinear magnetohydrodynamics simulation using high-order finite elements

journal, March 2004

- Sovinec, C. R.; Glasser, A. H.; Gianakon, T. A.
- Journal of Computational Physics, Vol. 195, Issue 1

##
Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria

journal, January 1983

- Hirshman, S. P.
- Physics of Fluids, Vol. 26, Issue 12

##
Local three-dimensional magnetostatic equilibria

journal, January 2000

- Hegna, C. C.
- Physics of Plasmas, Vol. 7, Issue 10

##
The second region of stability against ballooning modes

journal, April 1981

- Greene, J. M.; Chance, M. S.
- Nuclear Fusion, Vol. 21, Issue 4

##
A new form of the magnetohydrodynamic potential energy

journal, January 1996

- Greene, John M.
- Physics of Plasmas, Vol. 3, Issue 1

Figures / Tables found in this record:

*Figures/Tables have been extracted from DOE-funded journal article accepted manuscripts.*