Abstract
We consider the question raised by Enciso and Peralta-Salas (Arch Ration Mech Anal 220(1):243–260, 2016): what nonconstant functions f can occur as the proportionality factor for a Beltrami field \({{\mathbf {u}}}\) on an open subset \(U \subset \mathbb {R}^3\)? We also consider the related question: for any such f, how large is the space of associated Beltrami fields? By applying Cartan’s method of moving frames and the theory of exterior differential systems, we are able to improve upon the results given in Peralta-Salas (2016). In particular, the answer to the second question depends crucially upon the geometry of the level surfaces of f. We conclude by giving a complete classification of Beltrami fields that possess either a translation symmetry or a rotation symmetry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
While straightforward in principle, most of the computations in this algorithm are impractical to carry out by hand. We have used Maple for all computations, along with the Cartan package developed by the first author and available for free download at http://euclid.colorado.edu/~jnc/Maple.html.
References
Bishop, R.L., Goldberg, S.I.: Tensor Analysis on Manifolds, Dover Publications, Inc., New York, 1980, Corrected reprint of the 1968 original
Bryant, R.L., Chern, S.S., Gardner, R.B., Goldschmidt, H.L., Griffiths, P.A.: Exterior Differential Systems, Mathematical Sciences Research Institute Publications, vol. 18. Springer, New York 1991
Enciso, A., Peralta-Salas, D.: Knots and links in steady solutions of the Euler equation. Ann. Math. (2)175(1), 345–367, 2012
Enciso, A., Peralta-Salas, D.: Existence of knotted vortex tubes in steady Euler flows. Acta Math. 214(1), 61–134, 2015
Enciso, A., Peralta-Salas, D.: Beltrami fields with a nonconstant proportionality factor are rare. Arch. Ration. Mech. Anal. 220(1), 243–260, 2016
Enciso, A., Poyato, D., Soler, J.: Stability results, almost global generalized Beltrami fields and applications to vortex structures in the Euler equations. Commun. Math. Phys. 360(1), 197–269, 2018
Gascón, F.G., Peralta-Salas, D.: Ordered behavior in force-free magnetic fields. Phys. Lett. A292, 75–84, 2001
Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, vol. 61. American Mathematical Society, Providence, RI 2003
Marsh, G.E.: Force-Free Magnetic Fields: Solutions, Topology and Applications. World Scientific Publishing Co., River Edge, NJ 1996
Moawad, S.M.: Exact equilibria for nonlinear force-free magnetic fields with its applications to astrophysics and fusion plasmas. J. Plasma Phys. 80, 173–195, 2014
Sato, N., Yamada, M.: Local representation and construction of Beltrami fields. Phys. D391, 8–16, 2019
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Székelyhidi
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was supported in part by NSF grant DMS-1206272 and a Collaboration Grant for Mathematicians from the Simons Foundation (Grant 524130).
Rights and permissions
About this article
Cite this article
Clelland, J.N., Klotz, T. Beltrami Fields with Nonconstant Proportionality Factor. Arch Rational Mech Anal 236, 767–800 (2020). https://doi.org/10.1007/s00205-019-01481-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00205-019-01481-7