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Title: The role of curvature and stretching on the existence of fast dynamo plasma in Riemannian space

Abstract

Vishik's anti-dynamo theorem is applied to a nonstretched twisted magnetic flux tube in Riemannian space. Marginal or slow dynamos along curved (folded), torsioned (twisted), and nonstretching flux tubes plasma flows are obtained. Riemannian curvature of the twisted magnetic flux tube is computed in terms of the Frenet curvature in the thin tube limit. It is shown that, for nonstretched filaments, fast dynamo action in the diffusive case cannot be obtained, in agreement with Vishik's argument that fast dynamos cannot be obtained in nonstretched flows. Instead of a fast dynamo, a nonuniform stretching slow dynamo is obtained. An example is given, which generalizes plasma dynamo laminar flows, recently presented by Wang et al. [Phys Plasmas 9, 1491 (2002)], in the case of low magnetic Reynolds number Re{sub m}{>=}210. Curved and twisting Riemannian heliotrons, where nondynamo modes are found even when stretching is present, shows that the simple presence of stretching is not enough for the existence of dynamo action. In this paper, folding plays the role of Riemannian curvature and can be used to cancel magnetic fields, not enhancing the dynamo action. Nondynamo modes are found for certain values of torsion, or Frenet curvature (folding) in the spirit of the anti-dynamomore » theorem. It is also shown that curvature and stretching are fundamental for the existence of fast dynamos in plasmas.« less

Authors:
 [1]
  1. Departamento de Fisica Teorica-IF-Universidade do Estado do Rio de Janeiro-UERJ, Rua Sao Francisco Xavier, 524 Cep 20550-003, Maracana, Rio de Janeiro, RJ (Brazil)
Publication Date:
OSTI Identifier:
21259700
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physics of Plasmas; Journal Volume: 15; Journal Issue: 12; Other Information: DOI: 10.1063/1.3041158; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
70 PLASMA PHYSICS AND FUSION TECHNOLOGY; 58 GEOSCIENCES; HELIOTRON; LAMINAR FLOW; MAGNETIC FLUX; PLASMA; RIEMANN SPACE

Citation Formats

Garcia de Andrade, L. C.. The role of curvature and stretching on the existence of fast dynamo plasma in Riemannian space. United States: N. p., 2008. Web. doi:10.1063/1.3041158.
Garcia de Andrade, L. C.. The role of curvature and stretching on the existence of fast dynamo plasma in Riemannian space. United States. doi:10.1063/1.3041158.
Garcia de Andrade, L. C.. Mon . "The role of curvature and stretching on the existence of fast dynamo plasma in Riemannian space". United States. doi:10.1063/1.3041158.
@article{osti_21259700,
title = {The role of curvature and stretching on the existence of fast dynamo plasma in Riemannian space},
author = {Garcia de Andrade, L. C.},
abstractNote = {Vishik's anti-dynamo theorem is applied to a nonstretched twisted magnetic flux tube in Riemannian space. Marginal or slow dynamos along curved (folded), torsioned (twisted), and nonstretching flux tubes plasma flows are obtained. Riemannian curvature of the twisted magnetic flux tube is computed in terms of the Frenet curvature in the thin tube limit. It is shown that, for nonstretched filaments, fast dynamo action in the diffusive case cannot be obtained, in agreement with Vishik's argument that fast dynamos cannot be obtained in nonstretched flows. Instead of a fast dynamo, a nonuniform stretching slow dynamo is obtained. An example is given, which generalizes plasma dynamo laminar flows, recently presented by Wang et al. [Phys Plasmas 9, 1491 (2002)], in the case of low magnetic Reynolds number Re{sub m}{>=}210. Curved and twisting Riemannian heliotrons, where nondynamo modes are found even when stretching is present, shows that the simple presence of stretching is not enough for the existence of dynamo action. In this paper, folding plays the role of Riemannian curvature and can be used to cancel magnetic fields, not enhancing the dynamo action. Nondynamo modes are found for certain values of torsion, or Frenet curvature (folding) in the spirit of the anti-dynamo theorem. It is also shown that curvature and stretching are fundamental for the existence of fast dynamos in plasmas.},
doi = {10.1063/1.3041158},
journal = {Physics of Plasmas},
number = 12,
volume = 15,
place = {United States},
year = {Mon Dec 15 00:00:00 EST 2008},
month = {Mon Dec 15 00:00:00 EST 2008}
}
  • The problem of the kinematic dynamo at large magnetic Reynolds numbers R/sub m/ is considered using as an example artificial flow with exponential particle stretching, simulating the stationary stochastic flow of a conducting fluid. The magnetic fields which have a periodic dependence on only one coordinate grow in time exponentially and without bound. Each Fourier harmonic of the deviation from this growing field increases initially rapidly with a rate independent of R/sub m/ during a time interval t/sub asterisk/roughly-equalt/sub 0/ln R/sub m/, and then decays very rapidly.
  • 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 ofmore » 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.« less
  • As is well-known, the Gauss theorem, according to which any 2-dimensional Riemannian metric can be mapped locally conformally into an euclidean space, does not hold in three dimensions. We define in this paper transformations of a new type, that we call principal. They map 3-dimensional spaces into spaces of constant curvature. We give a few explicit examples of principal transformations and we prove, at the linear approximation, that any metric deviating not too much from the euclidean metric can be mapped by a principal transformation into the euclidean metric.
  • We define the simplicial analogues of two concepts from differential topology: the concept of a point on the simplicial manifold and the concept of a tangent space on a simplicial manifold. We derive the simplicial analogues of parallel transport, the covariant derivative, connections, the Riemann curvature tensor, and the Einstein tensor. We construct the extrinsic curvature for a simplicial hypersurface using the simplicial covariant derivative. We discuss the importance of this simplicial extrinsic curvature to the 3+1 Regge-calculus program. It appears to us that the newly developed null-strut lattice is the most natural version of a 3+1 Regge lattice formore » the construction of extrinsic curvature. (A null-strut lattice is a 3+1 Regge spacetime lattice with TrK = const simplicial hypersurfaces, each connected to its two adjacent hypersurfaces entirely by simplicial light cones built of null struts.) Finally, we test the Regge-calculus version of the extrinsic curvature on a Bianchi type-IX simplicial hypersurface. The calculation agrees with the continuum expression to first order.« less