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Physics  2008 

Lyapunov spectra in fast dynamo Ricci flows of negative sectional curvature

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Abstract:

Previously Chicone, Latushkin and Montgomery-Smith [\textbf{Comm. Math. Phys. \textbf{173},(1995)}] have investigated the spectrum of the dynamo operator for an ideally conducting fluid. More recently, Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated the anisotropies in magnetic field dynamo evolution, from finite-time, Lyapunov exponents, giving rise to a Riemann metric tensor, in the Alfven twist in magnetic flux tubes (MFTs). In this paper one investigate the role of Perelman Ricci flows constraints in twisted magnetic flux tubes, where the Lyapunov eigenvalue spectra for the Ricci tensor associated with the Ricci flow equation in MFTs leads to a finite-time Lyapunov exponential stretching along the toroidal direction of the tube and a contraction along the radial direction of the tube. It is shown that in the case of MFTs, the sectional Ricci curvature of the flow, is negative as happens in geodesic flows of Anosov type. Ricci flows constraints in MFTs substitute the Thiffeault and Boozer [\textbf{Chaos}(2001)] have vanishing of Riemann curvature constraint on the Lyapunov exponential stretching of chaotic flows. Gauss curvature of the twisted MFT is also computed and the contraints on a negative Gauss curvature are obtained.

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