# Local representation and construction of Beltrami fields II. Solenoidal Beltrami fields and ideal MHD equilibria

## TL;DR

This paper develops a local geometric framework for constructing and analyzing solenoidal Beltrami fields and ideal MHD equilibria, revealing the prevalence of singular solutions and providing explicit examples.

## Contribution

It introduces a local representation theory for Beltrami fields using harmonic orthogonal coordinates and proves the existence of singular solutions in bounded domains.

## Key findings

- Explicit construction of solenoidal Beltrami fields with various topologies
- Proof of existence of singular solutions satisfying boundary conditions
- Application of theory to generate analytic examples

## Abstract

Object of the present paper is the local theory of solution for steady ideal Euler flows and ideal MHD equilibria. The present analysis relies on the Lie-Darboux theorem of differential geometry and the local theory of representation and construction of Beltrami fields developed in [1]. A theorem for the construction of harmonic orthogonal coordinates is proved. Using such coordinates families of solenoidal Beltrami fields with different topologies are obtained in analytic form. Existence of global solenoidal Beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered. It is shown that only singular solutions are admissible, an explicit example is given in a spherical domain, and a theorem on existence of singular solutions is proven. Local conditions for existence of solutions, and local representation theorems are derived for generalized Beltrami fields, ideal MHD equilibria, and general steady ideal Euler flows. The theory is applied to construct analytic examples.