Dynamos and anti-dynamos as thin magnetic flux ropes in Riemannian spaces

TL;DR

This paper explores magnetic anti-dynamos in Riemannian spaces, demonstrating exponential decay of magnetic fields in certain geometries and analyzing the conditions under which dynamo action is suppressed or possible.

Contribution

It introduces specific Riemannian models of anti-dynamos, including conformally related metrics and thin flux ropes, and discusses their implications for magnetic field behavior.

Findings

Magnetic fields decay exponentially in the conformal Arnold metric model.

Thin magnetic flux ropes with near-zero torsion also exhibit decay, acting as anti-dynamos.

Planar configurations satisfy Zeldovich's theorem, preventing dynamo action.

Abstract

Two examples of magnetic anti-dynamos in magnetohydrodynamics (MHD) are given. The first is a 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-{\lambda}z}dp^{2}+e^{{\lambda}z}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The curvature decay as z-coordinates increases without bounds. Some of the Riemann curvature components such as $R_{pzpz}$ also undergoes dissipation while component $R_{qzqz}$ increases without bounds. The remaining curvature component $R_{pqpq}$ is constant on the torus surface. The other anti-dynamo which may be useful in plasma astrophysics is the thin magnetic flux rope or twisted magnetic thin flux tube which also behaves as anti-dynamo since it also decays with time. This model is based on the Riemannian metric of the magnetic twisted flux tube where the axis possesses Frenet curvature and torsion. Since in this last example the Frenet torsion of the axis of the rope is almost zero, or the possible dynamo is almost planar it satisfies Zeldovich theorem which states that planar dynamos do not exist. Changing in topology of this result may result on a real dynamo as discussed.