# Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria

## Abstract

Using the recently developed multiregion, relaxed MHD (MRxMHD) theory, which bridges the gap between Taylor's relaxation theory and ideal MHD, we provide a thorough analytical and numerical proof of the formation of singular currents at rational surfaces in non-axisymmetric ideal MHD equilibria. These include the force-free singular current density represented by a Dirac δ-function, which presumably prevents the formation of islands, and the Pfirsch-Schlüter 1/x singular current, which arises as a result of finite pressure gradient. An analytical model based on linearized MRxMHD is derived that can accurately (1) describe the formation of magnetic islands at resonant rational surfaces, (2) retrieve the ideal MHD limit where magnetic islands are shielded, and (3) compute the subsequent formation of singular currents. The analytical results are benchmarked against numerical simulations carried out with a fully nonlinear implementation of MRxMHD.

- Authors:

- Max-Planck-Institut für Plasmaphysik, D-17491 Greifswald (Germany)
- (United States)
- Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton New Jersey 08543 (United States)

- Publication Date:

- OSTI Identifier:
- 22408096

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physics of Plasmas; Journal Volume: 22; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; BENCHMARKS; COMPUTERIZED SIMULATION; CURRENT DENSITY; DIRAC EQUATION; EQUILIBRIUM; MAGNETIC ISLANDS; MAGNETOHYDRODYNAMICS; MODE RATIONAL SURFACES; NONLINEAR PROBLEMS; PRESSURE GRADIENTS; RELAXATION; THREE-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Loizu, J., E-mail: joaquim.loizu@ipp.mpg.de, Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton New Jersey 08543, Hudson, S., Bhattacharjee, A., and Helander, P.
```*Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria*. United States: N. p., 2015.
Web. doi:10.1063/1.4906888.

```
Loizu, J., E-mail: joaquim.loizu@ipp.mpg.de, Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton New Jersey 08543, Hudson, S., Bhattacharjee, A., & Helander, P.
```*Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria*. United States. doi:10.1063/1.4906888.

```
Loizu, J., E-mail: joaquim.loizu@ipp.mpg.de, Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton New Jersey 08543, Hudson, S., Bhattacharjee, A., and Helander, P. Sun .
"Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria". United States.
doi:10.1063/1.4906888.
```

```
@article{osti_22408096,
```

title = {Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria},

author = {Loizu, J., E-mail: joaquim.loizu@ipp.mpg.de and Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton New Jersey 08543 and Hudson, S. and Bhattacharjee, A. and Helander, P.},

abstractNote = {Using the recently developed multiregion, relaxed MHD (MRxMHD) theory, which bridges the gap between Taylor's relaxation theory and ideal MHD, we provide a thorough analytical and numerical proof of the formation of singular currents at rational surfaces in non-axisymmetric ideal MHD equilibria. These include the force-free singular current density represented by a Dirac δ-function, which presumably prevents the formation of islands, and the Pfirsch-Schlüter 1/x singular current, which arises as a result of finite pressure gradient. An analytical model based on linearized MRxMHD is derived that can accurately (1) describe the formation of magnetic islands at resonant rational surfaces, (2) retrieve the ideal MHD limit where magnetic islands are shielded, and (3) compute the subsequent formation of singular currents. The analytical results are benchmarked against numerical simulations carried out with a fully nonlinear implementation of MRxMHD.},

doi = {10.1063/1.4906888},

journal = {Physics of Plasmas},

number = 2,

volume = 22,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}