Mathematics – Differential Geometry
Scientific paper
Oct 2007
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007phpl...14j2902g&link_type=abstract
Physics of Plasmas, Volume 14, Issue 10, pp. 102902-102902-6 (2007).
Mathematics
Differential Geometry
9
Classical Differential Geometry
Scientific paper
Two new analytical solutions of the self-induction equation in Riemannian manifolds are presented. The first represents a twisted magnetic flux tube or flux rope in plasma astrophysics, where the rotation of the flow implies that the poloidal field is amplified from toroidal field, in the spirit of dynamo theory. The value of the amplification depends on the Frenet torsion of the magnetic axis of the tube. Actually this result illustrates the Zeldovich stretch, twist, and fold method to generate dynamos from straight and untwisted ropes. Based on the fact that this problem was previously handled, using a Riemannian geometry of twisted magnetic flux ropes [Phys Plasmas 13, 022309 (2006)], investigation of a second dynamo solution, conformally related to the Arnold kinematic fast dynamo, is obtained. In this solution, it is shown that the conformal effect on the fast dynamo metric enhances the Zeldovich stretch, and therefore a new dynamo solution is obtained. When a conformal mapping is performed in an Arnold fast dynamo line element, a uniform stretch is obtained in the original line element.
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