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

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.

Existence of ideal magnetofluid equilibria without continuous Euclidean symmetries

“…These conditions suggest that an intimate relationship exists between Beltrami fields and two-dimensional harmonic conjugate functions (this fact is discussed in [11]). Indeed, system (86) can be satisfied by setting x 3 = σ (z), x 1 = u (x, y), and x 2 = v (x, y), with σ a function of the Cartesian coordinate z, and u and v harmonic conjugate functions in the x-y plane such that…”

“…These conditions suggest that an intimate relationship exists between Beltrami fields and two-dimensional harmonic conjugate functions (this fact is discussed in [11]). Indeed, system (86) can be satisfied by setting…”

“…The purpose of the present section is to show that system (1) Furthermore, reflections are discrete transformations because the orbit of a point in the metric space under the isometry forms a discrete set. As mentioned in the introduction, if one postulates the existence of an ignorable coordinate x 3 such that the components w i , the pressure field χ, and the metric tensor g ij are all independent of x 3 , it is known that system (1) can be reduced to a single nonlinear elliptic second order partial differential equation, the Grad-Shafranov equation (11). For completeness, the derivation of the Grad-Shafranov equation is given in appendix A (we also refer the reader to [18,19] on this point).…”

“…If one can find a solution w satisfying (77), then such solution cannot be obtained as a solution of the Grad-Shafranov equation (11).…”

“…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].…”