Typical field lines of Beltrami flows and boundary field line behaviour of Beltrami flows on simply connected, compact, smooth manifolds with boundary

We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded 1-manifolds diffeomorphic to $${\mathbb {R}}$$ R , which approach the zero set as time goes to $$\pm \, \infty$$ ± ∞ . We then drop the assumptions of compactness and vanishing de Rham cohomology and prove that for almost every point on the given manifold, the field line passing through the point is either a non-constant, periodic orbit or a non-periodic orbit which comes arbitrarily close to the starting point as time goes to $$\pm \infty$$ ± ∞ . During the course of the proof, we in particular show that the set of points at which a Beltrami field vanishes in the interior of the manifold is countably 1-rectifiable in the sense of Federer and hence in particular has a Hausdorff dimension of at most 1. As a consequence, we conclude that for every eigenfield of the curl operator, corresponding to a non-zero eigenvalue, there always exists exactly one nodal domain.

Zero Set Structure of Real Analytic Beltrami Fields

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with (possibly empty) boundary and tame knots. Further, we consider the question of how complicated these tame knots can possibly be. To this end, we prove that on the standard (open) solid toroidal annulus in $${\mathbb {R}}^3$$ R 3 , there exist for any pair (p, q) of positive, coprime integers countable infinitely many distinct real analytic metrics such that for each such metric, there exists a real analytic Beltrami field, corresponding to the eigenvalue $$+1$$ + 1 of the curl operator, whose zero set is precisely given by a standard (p, q)-torus knot. The metrics and the corresponding Beltrami fields are constructed explicitly and can be written down in Cartesian coordinates by means of elementary functions alone.

Steady Euler flows and Beltrami fields in high dimensions

Using open books, we prove the existence of a non-vanishing steady solution to the Euler equations for some metric in every homotopy class of non-vanishing vector fields of any odd-dimensional manifold. As a corollary, any such field can be realized in an invariant submanifold of a contact Reeb field on a sphere of high dimension. The solutions constructed are geodesible and hence of Beltrami type, and can be modified to obtain chaotic fluids. We characterize Beltrami fields in odd dimensions and show that there always exist volume-preserving Beltrami fields which are neither geodesible nor Euler flows for any metric. This contrasts with the three-dimensional case, where every volume-preserving Beltrami field is a steady Euler flow for some metric. Finally, we construct a non-vanishing Beltrami field (which is not necessarily volume-preserving) without periodic orbits in every manifold of odd dimension greater than three.

On electromagnetics of an isotropic chiral medium moving at constant velocity

A medium which is an isotropic chiral medium from the perspective of a co-moving observer is a Faraday chiral medium (FCM) from the perspective of a non-co-moving observer. The Tellegen constitutive relations for this FCM are established. By an extension of the Beltrami field concept, these constitutive relations are exploited to show that plane wave propagation is characterized by four generally independent wavenumbers. This FCM can support negative phase velocity at certain translational velocities and with certain wavevectors, even though the corresponding isotropic chiral medium does not. The constitutive relations and Beltrami-like fields are also used to develop a convenient spectral representation of the dyadic Green functions for the FCM.

Scattering from a slit in a biisotropic medium

An analytical solution is developed for spherical wave scattering by a slit in an infinitely perfectly conducting sheet in a homogeneous biisotropic medium. Interestingly, the vector diffraction problem is reduced to the scattering of a single scalar field, this scalar field being the normal component of either a left-handed or a right-handed Beltrami field. The point source is assumed to be far from the slit so that the incident spherical wave is locally plane. The slit is wide and the sheet thin, both with respect to wavelength. By using the Fourier transform technique the boundary-value problem is transformed into Wiener–Hopf equation that is solved approximately. The diffracted wave field is studied in the far field of the slit. The diffracted field is the sum of the wave fields produced by the two edges of the slit and an interaction wave field.PACS Nos.: 41.20.q, 41.20.Jb, 52.35.Hr

WHICH ARE THE FUNDAMENTAL MAGNETIC FIELD CONFIGURATIONS IN NATURE?

The dipole field represents a magnetic paradigm. It occurs in a hierarchy of scale, from elementary particles, to ferromagnetic domains, to laboratory magnets, up to the earth itself. Nevertheless, the present magnetic paradigm for plasma physics and astrophysics is not the dipole, but the Beltrami field [Formula: see text], k a constant or scalar function. Such fields are regarded as the most stable, and are extensively used in models of cosmic and laboratory plasmas. The problem is that these two classes of supposedly paradigmatic vector fields are incompatible, as it is impossible to transform one into the other via a continuous limiting process. This paper attempts to resolve this basic contradiction by identifying the configurations corresponding to extrema of the field energy. It is found that, contrary to a long-held belief, Beltrami fields correspond to a point on an energy plane (the flat limiting case of a saddle) and not to a minimum. The classes of energy minima derived here include axisymmetric toroidal and axisymmetric poloidal fields. The latter naturally reduce to dipole fields in the limit of a microscopic current, in agreement with the magnetic paradigm. Some of the situations which have traditionally used Beltrami fields as a model can be explained with the minimum-energy fields derived here. The observed instability of axisymmetric magnetoplasmas follows from a separate energy equipartitioning mechanism.