On the spectrum of the curl operator and the helicity of a bounded domain

Alberto Valli (Università degli Studi di Trento)

Thursday 19th November, 2020 14:00-15:00 ZOOM (ID: 999 9182 3581)


Recording: in progress

For linear isotropic media it is assumed that $\B=\mu \Hh$ ($\B$ magnetic induction, $\Hh$ magnetic field), where the scalar function $\mu$ is the magnetic permeability. For magnetostatic and eddy current problems it is assumed that $\J = \bcurl \, \Hh$ ($\J$ current density). In this situation a magnetic field satisfying $\bcurl \, {\Hh} = \eta {\Hh}$, with $\eta$ a scalar function, produces a vanishing magnetic force ${\J} \times {\B}$, and is called a {\sl force-free} field.


In fluid dynamics, force-free fields are named {\sl Beltrami} fields, and a tangential divergence-free Beltrami field $\uu$ is a steady solution of the Euler equations for incompressible inviscid flows (with pressure given by $p = - \frac{|\uu|^2}{2}$).


Eigenfunctions of the curl operator are therefore force-free fields and Beltrami fields, and are of relevant physical interest.


In this talk we are concerned with two topics: the formulation and analysis of the spectral problem for the curl operator in a multiply-connected domain, and the relation between the spectrum of the curl operator and the helicity of a domain.


Taking inspiration from the results in [1] (and extending the previous ones in [2], [3], [4]) by means of a saddle-point variational formulation we have proved in [5] that the curl operator is self-adjoint on suitable Hilbert spaces, each of them being contained in the space of vector fields $\vv$ for which $\bcurl \, \vv \cdot \n = 0$ on the boundary. It is important to note that additional conditions must be imposed when the physical domain is not topologically trivial: a viable choice is the vanishing of the line integrals of $\vv$ on suitable homological cycles lying on the boundary.


Concerning the helicity of a multiply-connected domain, in [7] we have shown how to find a gauge invariant definition of the helicity of a vector field, and how this formula is related to the Biot--Savart operator (which is a compact and self-adjoint operator). Following [6], through this relation it is possible to see that the helicity of a domain (namely, the largest absolute value of the helicity for vector fields defined in that domain and having a given energy) is the largest (in absolute value) eigenvalue of the Biot--Savart operator, or, in other words, the smallest (in absolute value) eigenvalue of the curl operator.

[1] R.Hiptmair, P.R. Kotiuga and S. Tordeux: Self-adjoint curl operators, Ann. Mat. Pura Appl. (4), 191 (2012), 431--457.

[2] R.Picard: Ein Randwertproblem in der Theorie kraftfreier Magnetfelder, Z. Angew. Math. Phys., 27 (1976), 169--180.

[3] R. Kress: On constant-alpha force-free fields in a torus, J. Engrg. Math., 20 (1986), 323--344.

[4] Z.Yoshida and Y. Giga: Remarks on spectra of operator rot, Math. Z., 204 (1990), 235--245.


[5] A. Alonso Rodrìguez, J. Camano, R. Rodrìguez, P. Venegas and A. Valli: Finite element approximation of the spectrum of the curl operator in a multiply-connected domain, Found. Comput. Math., 18 (2018), 1493--1533; 19 (2019), 243--244.

[6] A. Valli: A variational interpretation of the Biot--Savart operator and the helicity of a bounded domain, J. Math. Phys., 60 (2019), 021503.

[7] D. MacTaggart and A. Valli: Magnetic helicity in multiple connected domains, J. Plasma Phys., 85 (2019), 775850501.

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