Exponential stretching in filaments as fast dynamos in Euclidean and Riemannian 3D spaces

Abstract

A new antidynamo theorem for non-stretched twisted magnetic flux tube dynamo is obtained. Though Riemannian curvature cannot be neglected since one considers curved magnetic flux tube axis, the stretch can be neglect since one only considers the limit of thin magnetic flux tubes. The theorem states that marginal or slow dynamos along curved (folded), torsioned (twisted) and non-stretched flux tubes endowed with diffusionless plasma flows, if a constraint is imposed on the relation between poloidal and toroidal magnetic fields in the helical dynamo case. A formula for the stretch of flux tubes is derived. From this formula one shows that the Riemann flux tube is stretched by an interaction between the plasma flow vorticity and torsion, in accordance with our physical intuition. Marginal diffusionless dynamos are shown to exist obtained in the case of flux tube dynamos exponential stretching. Thus slow dynamos can be obtaining on the flux tube under stretching. Filamentary dynamos anti-dynamos are also considered. As flux tubes possess a magnetic axis torsioned filament, it can also be considered as thegerm of a fast dynamo in flux tubes Riemannian curved space. It is shown that for non-stretched filaments only untwist and unfold filaments can provide dynamo action in diffusive case. A condition for exponential stretching and fast dynamos in filaments is given. These results are actually in agreement with Vishik argument that fast dynamo cannot be obtained in non-stretched flows. Actually the flux tube result is the converse of Vishik's lemma.