Lagrangean stability of slow dynamos in compact 3D Riemannian manifolds

TL;DR

This paper investigates the stability of slow dynamos in compact 3D Riemannian manifolds, showing that they are Lagrangean stable when the sectional curvature vanishes, and explores conditions for stability and instability in various cases.

Contribution

It introduces modifications to a dynamo operator and analyzes Lagrangean stability in different geometric settings within 3D Riemannian manifolds.

Findings

Slow dynamos are Lagrangean stable with zero sectional curvature.

Stability depends on pressure derivatives along the filament.

Negative curvature dynamos exhibit different stability properties.

Abstract

Modifications on a recently introduced fast dynamo operator by Chiconne et al [Comm Math Phys 173, 379 (1995)] in compact 3D Riemannian manifolds allows us to shown that slow dynamos are Lagrangean stable, in the sense that the sectional curvature of the Riemann manifold vanishes. The stability of the holonomic filament in this manifold will depend upon the sign of the second derivative of the pressure along the filament and in the non-holonomic case, to the normal pressure of the filament. Lagrangean instability is also investigated in this case and again an dynamo operator can be defined in this case. Negative curvature (Anosov flows) dynamos are also discussed in their stability aspects.