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# Constructing stellarators with quasisymmetry to high order

## Abstract

Here, a method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren & Boozer’s expansion about the magnetic axis ( *Phys Fluids* B, vol. 3, 1991, p. 2805), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landreman *et al.* ( *J. Plasma Phys.*, vol. 85, 2019, 905850103), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect-ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer’s ideal scaling of quasisymmetry breaking with the cube of the inverse aspect ratio. We also demonstrate a strongly non-axisymmetric configuration (vacuum rotational transform $$\unicode[STIX]{x1D704} >0.4$$) in which symmetry-breaking mode amplitudes throughout a finite volume are $${<}2\times 10^{-7}$$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. In conclusion, the approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.

- Authors:

- Univ. of Maryland, College Park, MD (United States). Inst. for Research in Electronics and Applied Physics
- New York Univ. (NYU), NY (United States). Courant Inst. of Mathematical Sciences

- Publication Date:

- Research Org.:
- Univ. of Maryland, College Park, MD (United States)

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

- OSTI Identifier:
- 1597688

- Grant/Contract Number:
- [FG02-93ER54197; FG02-86ER53223; 560651]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Plasma Physics

- Additional Journal Information:
- [ Journal Volume: 85; Journal Issue: 6]; Journal ID: ISSN 0022-3778

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY

### Citation Formats

```
Landreman, Matt, and Sengupta, Wrick. Constructing stellarators with quasisymmetry to high order. United States: N. p., 2019.
Web. doi:10.1017/S0022377819000783.
```

```
Landreman, Matt, & Sengupta, Wrick. Constructing stellarators with quasisymmetry to high order. United States. doi:10.1017/S0022377819000783.
```

```
Landreman, Matt, and Sengupta, Wrick. Fri .
"Constructing stellarators with quasisymmetry to high order". United States. doi:10.1017/S0022377819000783.
```

```
@article{osti_1597688,
```

title = {Constructing stellarators with quasisymmetry to high order},

author = {Landreman, Matt and Sengupta, Wrick},

abstractNote = {Here, a method is given to rapidly compute quasisymmetric stellarator magnetic fields for plasma confinement, without the need to call a three-dimensional magnetohydrodynamic equilibrium code inside an optimization iteration. The method is based on direct solution of the equations of magnetohydrodynamic equilibrium and quasisymmetry using Garren & Boozer’s expansion about the magnetic axis (Phys Fluids B, vol. 3, 1991, p. 2805), and it is several orders of magnitude faster than the conventional optimization approach. The work here extends the method of Landremanet al. (J. Plasma Phys., vol. 85, 2019, 905850103), which was limited to flux surfaces with elliptical cross-section, to higher order in the aspect-ratio expansion. As a result, configurations can be generated with strong shaping that achieve quasisymmetry to high accuracy. Using this construction, we give the first numerical demonstrations of Garren and Boozer’s ideal scaling of quasisymmetry breaking with the cube of the inverse aspect ratio. We also demonstrate a strongly non-axisymmetric configuration (vacuum rotational transform $\unicode[STIX]{x1D704} >0.4$) in which symmetry-breaking mode amplitudes throughout a finite volume are ${<}2\times 10^{-7}$, the smallest ever reported. To generate boundary shapes of finite-minor-radius configurations, a careful analysis is given of the effect of substituting a finite minor radius into the near-axis expansion. In conclusion, the approach here can provide analytic insight into the space of possible quasisymmetric stellarator configurations, and it can be used to generate good initial conditions for conventional stellarator optimization.},

doi = {10.1017/S0022377819000783},

journal = {Journal of Plasma Physics},

number = [6],

volume = [85],

place = {United States},

year = {2019},

month = {11}

}

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