Local representation and construction of Beltrami fields

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 nonsolenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors. arXiv:1809.03136v1 [math-ph]

Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

“…We are ready to construct nontrivial Beltrami field solutions of system (1) without continuous Euclidean symmetries. To achieve this goal, we apply the Clebsch-like parametrization of Beltrami fields derived in [10]. In particular, our aim is to find a coordinate system x 1 , x 2 , x 3 satisfying system (10).…”

“…with (Φ, Ψ, Θ) ∈ C ∞ (U ), and U a neighborhood of a point x ∈ Ω (see e.g. [10]). The vector field w is solenoidal provided that ∇ · w = ∆Φ + ∇Ψ · ∇Θ + Ψ∆Θ = 0 in U.…”

“…A result in this direction was obtained by [9], which showed that forĥ in an open and dense subset of C k (Ω), k ≥ 7, the only nontrivial Beltrami field solution of system (1) is w = 0. Nevertheless, this result does not prevent the existence of Beltrami fields with non-constantĥ; indeed, it has been shown in [10,11] that an infinite number of such solutions exist, and a systematic method to construct them was derived. This method relies on a local Clebsch-like parametrization of nontrivial Beltrami field solutions stemming from the Lie-Darboux theorem of differential geometry [12,13,14].…”

“…System (1) is thus converted into a set of geometric conditions for the metric tensor. A nontrivial Beltrami field solution can then be obtained by finding a coordinate system satisfying system (10). The treatment of the third class of solutions requires the notion of symmetry.…”

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

“…For a plasma system, w is the solenoidal magnetic field, κ = 0, and −P is the plasma pressure. Then, equations (1) and (2) describe ideal magnetohydrodynamic equilibria [3].…”