2022
2.
Perrella, David; Pfefferlé, David; Stoyanov, Luchezar
Rectifiability of Divergence-Free Fields along Invariant 2-Tori Journal Article
In: Partial Differential Equations and Applications, vol. 3, no. 4, pp. 50, 2022, ISSN: 2662-2971.
Links | BibTeX | Tags: cohomology, differential geometry, flux-surfaces, magnetic coordinates, MHD equilibrium, rotational transform
@article{perrellaRectifiabilityDivergencefreeFields2022,
title = {Rectifiability of Divergence-Free Fields along Invariant 2-Tori},
author = {David Perrella and David Pfefferl\'{e} and Luchezar Stoyanov},
doi = {10.1007/s42985-022-00182-3},
issn = {2662-2971},
year = {2022},
date = {2022-07-01},
urldate = {2022-07-01},
journal = {Partial Differential Equations and Applications},
volume = {3},
number = {4},
pages = {50},
keywords = {cohomology, differential geometry, flux-surfaces, magnetic coordinates, MHD equilibrium, rotational transform},
pubstate = {published},
tppubtype = {article}
}
2018
1.
Pfefferlé, D; Gunderson, L; Hudson, S R; Noakes, L
Non-planar elasticae as optimal curves for the magnetic axis of stellarators Journal Article
In: Physics of Plasmas, vol. 25, no. 9, pp. 092508, 2018.
Abstract | Links | BibTeX | Tags: elastica, euler-lagrange equations, linking number, magnetic axis, non-planar curves, rotational transform, stellarator, twist, variational problem, writhe
@article{pfefferle-elastica,
title = {Non-planar elasticae as optimal curves for the magnetic axis of stellarators},
author = {D Pfefferl\'{e} and L Gunderson and S R Hudson and L Noakes},
doi = {10.1063/1.5040894},
year = {2018},
date = {2018-09-26},
journal = {Physics of Plasmas},
volume = {25},
number = {9},
pages = {092508},
abstract = {The problem of finding an optimal curve for the target magnetic axis of a stellarator is addressed. Euler-Lagrange equations are derived for finite length three-dimensional curves that extremise their bending energy while yielding fixed integrated torsion. The obvious translational and rotational symmetries are exploited to express solutions in a preferred cylindrical coordinate system in terms of elliptic Jacobi functions. These solution curves, which, up to similarity transformations, depend on three dimensionless parameters, do not necessarily close. Two closure conditions are obtained for the vertical and toroidal displacement (the radial coordinate being trivially periodic) to yield a countably infinite set of one-parameter families of closed non-planar curves. The behaviour of the integrated torsion (Twist of the Frenet frame), the Linking of the Frenet frame, and the Writhe of the solution curves are studied in light of the C\u{a}lug\u{a}reanu theorem. A refreshed interpretation of Mercier's formula for the on-axis rotational transform of stellarator magnetic field-lines is proposed.},
keywords = {elastica, euler-lagrange equations, linking number, magnetic axis, non-planar curves, rotational transform, stellarator, twist, variational problem, writhe},
pubstate = {published},
tppubtype = {article}
}
The problem of finding an optimal curve for the target magnetic axis of a stellarator is addressed. Euler-Lagrange equations are derived for finite length three-dimensional curves that extremise their bending energy while yielding fixed integrated torsion. The obvious translational and rotational symmetries are exploited to express solutions in a preferred cylindrical coordinate system in terms of elliptic Jacobi functions. These solution curves, which, up to similarity transformations, depend on three dimensionless parameters, do not necessarily close. Two closure conditions are obtained for the vertical and toroidal displacement (the radial coordinate being trivially periodic) to yield a countably infinite set of one-parameter families of closed non-planar curves. The behaviour of the integrated torsion (Twist of the Frenet frame), the Linking of the Frenet frame, and the Writhe of the solution curves are studied in light of the Călugăreanu theorem. A refreshed interpretation of Mercier's formula for the on-axis rotational transform of stellarator magnetic field-lines is proposed.