Local Representation and Construction of Beltrami Fields

TL;DR

This paper proves a local representation theorem for Beltrami fields using differential geometry, introduces a method to construct such fields with specified properties, and provides explicit examples with various proportionality factors.

Contribution

It establishes a local standard form for Beltrami fields and develops a construction method based on the eikonal equation, expanding the understanding of their structure and diversity.

Findings

Beltrami fields have a local standard form similar to Arnold-Beltrami-Childress flows.

Two local invariants characterize Beltrami flows: a physical plane coordinate and an angular momentum-like quantity.

A method to construct Beltrami fields with any given proportionality factor using solutions to the eikonal equation.

Abstract

A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie-Darboux theorem of differential geoemtry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold-Beltrami-Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors.