Finite Element Computation of Beltrami Fields

Abstract

Vector fields H satisfying curlH = λH, with λ being a scalar field, are called force-free fields. This name arises from magnetohydrodynamics, since a magnetic field of this kind induces a vanishing Lorentz force: F := J x B = curlH x (μH). In 1958 Woltjer [6] showed that the lowest state of magnetic energy density within a closed system is attained when λ is spatially constant. In such a case H is called a linear force-free field and its determination is naturally related with the spectral problem for the curl operator. The eigenfunctions of this problem are known as free-decay fields and play an important role, for instance, in the study of turbulence in plasma physics.
The spectral problem for the curl operator, curlH = λH, has a longstanding tradition in mathematical physics. A large measure of the credit goes to Beltrami [1], who seems to be the first who considered this problem in the context of uid dynamics and electromagnetism. This is the reason why the corresponding eigenfunctions are also called Beltrami fields. On bounded domains, the most natural boundary condition for this problem is H.η = 0, which corresponds to a field confined within the domain. Analytical solutions of this problem are only known under particular symmetry assumptions. The first one was obtained in 1957 by Chandrasekhar and Kendall [2] in the context of astrophysical plasmas arising in modeling of the solar crown.
A couple of numerical methods based on Nédélec finite elements have been introduced and analyzed in a recent paper [5] for the solution of the eigenvalue problem for the curl operator in simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane, as is shown in [3]. However, additional constraints can be added to the eigenvalue problem in order to recover a well posed problem with a discrete spectrum [3, 4]. We choose as additional constraints a zero-ux condition of the curl on all the cutting surfaces. We introduce two weak formulations of the corresponding problem, which are convenient variations of those studied in [5]; one of them is mixed and the other a Maxwell-like formulation. We prove that both are well posed and show how to modify the finite element discretization from [5] to take care of these additional constraints. We prove spectral convergence of both discretization as well as a priori error estimates. Finally, we report a numerical test which allows us to assess the performance of the proposed methods.
References
[1] E. Beltrami, Considerazioni idrodinamiche. Rend. Inst. Lombardo Acad. Sci. Let., vol. 22, pp. 122-131, (1889). (English translation: Considerations on hydrodynamics, Int. J. Fusion Energy, vol. 3, pp. 53-57, (1985)).
[2] S. Chandrasekhar, P.C. Kendall, On force-free magnetic fields. Astrophys. J., vol. 126, pp. 457-460, (1957).
[3] Z. Yoshida and Y. Giga, Remarks on spectra of operator rot. Math. Zeit., vol. 204, pp. 235-245, (1990).
[4] R. Hiptmair, P.R. Kotiuga and S. Tordeux, Self-adjoint curl operators. Ann. Mat. Pura Appl,, vol. 191, pp. 431-457, (2012).
[5] R. Rodríguez and P. Venegas, Numerical approximation of the spectrum of the curl operator. Math. Comp., vol. 83, pp. 553-577, (2014).
[6] L. Woltjer, A theorem on force-free magnetic fields. Prod. Natl. Acad. Sci. USA, vol. 44, pp. 489-491, (1958).