M ay 2 00 9 Spectrum of kinematic fast dynamo operator in Ricci flows May 24 , 2009

نویسنده

  • L. C. Garcia
چکیده

Spectrum of kinematic fast dynamo operators in Ricci compressible flows in Einstein 2-manifolds is investigated. A similar expression, to the one obtained by Chicone, Latushkin and Montgomery-Smith (Comm Math Phys (1995)) is given, for the fast dynamo operator. The operator eigenvalue is obtained in a highly conducting media, in terms of linear and nonlinear orders of Ricci scalar and diffusion-free limit. Spatial 3-Einstein manifold section of Friedmann-Robertson-Walker (FRW) cosmology is obtained in the limit of ideal plasma. Since only two dimensional Riemannian manifolds of negative curvature may support fast dynamo action, here only inflationary phase of the universe, of negative cosmological constant (Λ) in Ricci-Einstein flows may support dynamo action. When (Λ ≥ 0) magnetic field decays. As in Latushkin and Vishik (Comm Math Phys (2003)) Lyapunov exponents in kinematic dynamos are investigated. Since positive curvature scalars are preserved under Ricci flow, fast dynamos are also preserved. Bounds on cosmological model due to Vishik anti-fast dynamo theorem are also discussed. Investigations in the Riemannian geometry of magnetic dynamos ranged from the early investigations of Arnold, Zeldovich, Ruzmaikin and Sokoloff [1] to the more recently papers by Chiconne, Latushkin and their group [2] on the fast dynamo existence. Investigation of Riemannian geometry applications to plasma dynamos and anti-fast dynamo Vishik's theorem [3] has been performed by the Garcia de Andrade [4]. Chicone et al [2] have also shown that the fast dynamo operator spectrum for an ideally conducting fluid and the spectrum of the group acting on the associated compact Riemannian manifold. Yet more recently Latushkin and Vishik [5] investigated Lyapunov exponents in kinematic dynamos. On the more topological and dynamical system settings, Young and Klapper [6] has investigated the topological entropy systems and dynamo action. In this last paper Young and Klapper used the same concept of topological entropy used recently, by Fields medalist Grisha Perelman [7] to prove the long standing Poincare conjecture. In his important proof, Perelman has made used the concept of Ricci flows, proposed by Hamilton in 1982 [8]. In this paper a proof is given of the following theorem: Theorem: Let M be a two dimensional Riemannian manifold (M, g), not necessarily closed, endowed with a metric g(t), is given in the interval t ∈ [a, b] in the field of real numbers R of an Einstein 3-manifold and a Ricci flow given by the equation Definition: ∂g ∂t = −2Ric (I.1) where Ric represents …

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تاریخ انتشار 2009