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

Wadim Gerner

doi:10.1007/s10455-021-09768-3

Abstract

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}}, 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.

Highlights

We deal with the field line behaviour of so-called Beltrami fields
Beltrami fields are of interest in ideal magnetohydrodynamics, and in astrophysics in particular
We recall the usual form of the incompressible Euler equations tv + ∇vv = −∇p and div(v) = 0, where v is the fluid velocity and p is the pressure

Summary

Introduction

We deal with the field line behaviour of so-called Beltrami fields. Given an oriented, smooth Riemannian 3-manifold (M , g) with or without boundary, we call a smooth vector field X on Ma weak Beltrami field if there exists a locally bounded function λ ∶ M → R such that. If v and its curl are everywhere collinear, which includes the case of weak Beltrami fields, the topological properties of the flow can be much more complicated It is a consequence of Arnold’s structure theorem, see [3, Chapter II, proposition 6.2], that every no-where vanishing real analytic steady flow, which possesses a ’chaotic’ trajectory, i.e. a trajectory not contained in any two-dimensional subset, is necessarily a Beltrami field. A similar result regarding the Euclidean metric was obtained by Enciso and Peralta-Salas [11], who showed that there exists an eigenfield of the curl operator on R3 with the Euclidean metric, such that it admits all tame knots and (locally finite) links as field line configurations, see [12] These results have in common that one considers or constructs specific Beltrami fields with interesting topologies. We will show that the answer is simple, provided the corresponding eigenvalue is nonzero: It is always one component and this holds regardless of the boundary conditions we impose

Main results

X denote the