Anosov branches of dynamo spectra in one dimensional plasmas

TL;DR

This paper analytically explores the spectral properties of dynamo operators in one-dimensional plasmas, revealing conditions for fast dynamo action and the influence of curvature, with connections to Anosov systems and magnetic field behavior.

Contribution

It provides new analytical solutions for turbulent dynamo filaments in resistive plasmas and investigates the effects of curvature on fast dynamo spectra in one-dimensional plasma flows.

Findings

Fast dynamo modes are obtained in resistive plasmas with zero kinetic helicity.

Weak curvature filaments support fast dynamo action in one-dimensional plasmas.

Non-Anosov oscillatory branches appear in the spectrum under certain conditions.

Abstract

Recently Guenther et al the globally diagonalized ${\alpha}^{2}$ dynamo operator spectrum [J Phys A 2007) in mean field media, and its Krein space related perturbation theory [J Phys A 2006). Earlier, an example of fast dynamos in stretch shear and fold Anosov maps have been given by Gilbert [PRSA [1993)). In this paper, analytical solutions representing general turbulent dynamo filaments are obtained in resistive plasmas. When turbulent diffusivity is present and kinetic helicity vanishes, a fast dynamo mode is obtained, and the Anosov eigenvalue obtained. The magnetic field lays down on a Frenet 2 plane along the filaments embedded in a 3D flow. Curvature effects on fast dynamo are also investigate. In case of weak curvature filaments the one dimensional manifolds in plasmas present a fast dynamo action. A parallel result has been obtained by Chicone et al [Comm Math Phys), in the case fast dynamo spectrum in two dimensional compact Riemannian manifolds of negative constant curvature, called Anosov spaces. While problems of embedding may appear in their case here no embedding problems appear since the one dimensional curved plasmas are embedded in three dimensional Euclidean spaces. In the examples considered here, equipartion between normal and binormal components of the magnetic field components is considered. In the opposite case, non Anosov oscillatory, purely imaginary, branches of the spectrum are found in dynamo manifold. Negative constant curvature non-compact $\textbf{H}^{2}$ manifold, has also been used in one-component electron 2D plasma by Fantoni and Tellez (Stat. Phys, (2008))