Ricci curvature and geodesic flows stability in Riemannian twisted flux tubes

TL;DR

This paper investigates the stability of geodesic flows in Riemannian twisted flux tubes by analyzing Ricci and sectional curvatures, revealing conditions under which flows become unstable due to negative sectional curvature.

Contribution

It provides a general analysis of flow stability in twisted flux tubes without assuming specific flow forms, using Riemannian geometry tools to identify instability conditions.

Findings

Thick planar tubes damp flow speed due to curvature effects.

Negative sectional curvature indicates flow instability.

Instability occurs with positive perturbations and angular flow speed.

Abstract

Ricci and sectional curvatures of twisted flux tubes in Riemannian manifold are computed to investigate the stability of the tubes. The geodesic equations are used to show that in the case of thick tubes, the curvature of planar (Frenet torsion-free) tubes have the effect ct of damping the flow speed along the tube. Stability of geodesic flows in the Riemannian twisted thin tubes (almost filaments), against constant radial perturbations is investigated by using the method of negative sectional curvature for unstable flows. No special form of the flow like Beltrami flows is admitted, and the proof is general for the case of thin tubes. It is found that for positive perturbations and angular speed of the flow, instability is achieved, since the sectional Ricci curvature of the twisted tube metric is negative.