Lyapunov spectra in fast dynamo Ricci flows of negative sectional curvature

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.Comment: Department of theoretical physics-UERJ-Brasi

A Riemannian geometrical method to classify tearing instabilities in plasmas

Riemannian geometrical tools, such as Ricci collineations and Killing symmetries, so often used in Einstein general theory of gravitation are here applied to plasma physics to build magnetic surfaces from Einstein plasma metrics used in tokamak devices. It is shown that the Killing symmetries are constrains the Einstein magnetic surfaces while the Killing vectors are built in terms of the displacement of the toroidal surface. The pressure is computed by applying these constraints to the pressure equations in tokamaks. A method, based on the sign of the only nontrivial constant Riemann curvature component, is suggested to classify tearing instability. Throughout the computations two approximations are considered: The first is the small toroidality and the other is the small displacement of the magnetic surfaces as Einstein spaces.Comment: Depto de Fisica de Teorica-if-uerj-rio-Brasi

Primordial magnetic fields of non-minimal photon-torsion axial coupling origin

Dynamo action is shown to be induced from homogeneous non-minimal photon-torsion axial coupling in the quantum electrodynamics (QED) framework in Riemann flat spacetime contortion decays. The geometrical optics in Riemann-Cartan spacetime is considering and a plane wave expansion of the electromagnetic vector potential is considered leading to a set of the equations for the ray congruence. Since we are interested mainly on the torsion effects in this first report we just consider the Riemann-flat case composed of the Minkowskian spacetime with torsion. It is also shown that in torsionic de Sitter background the vacuum polarisation does alter the propagation of individual photons, an effect which is absent in Riemannian spaces. It is shown that the cosmological torsion background inhomogeneities induce Lorentz violation and massive photon modes in this QED. Magnetic dynamos in this torsioned spacetime electrodynamics are simpler obtained in Fourier space than the cosmic ones, previously obtained by Bassett et al Phys Rev D, in Friedmann universe. By deriving plasma dispersion for linear electrodynamics in Riemann Cartan spacetime, dynamo action seems to be possible for plasma frequencies in some polarizations. The important cosmic magnetic field problem of breaking conformal flatness is naturally solved here since the photon torsion coupling breaks conformal flatness.Comment: depto de fisica teorica, if uerj, state rio de janeiro universit

Plasma torus dynamos versus laminar plasma dynamos

Earlier Wang et al [Phys Plasmas (2002)] have estimated a growth rate for the magnetic field of ${\gamma}=0.055$ and flow ionization velocity of $51{km}/{s}$ in a laminar plasma slow dynamo mode for aspect ratio of ${r_{0}}/L\approx{0.6}$, where $r_{0}$ is the internal straight cylinder radius, and L is the length scale of the plasma cylinder. In this paper, fast dynamo modes in curved Riemannian heliotron are shown to be excited on a plasma flow yielding a growth rate of ${\gamma}=0.318$ for an aspect ratio of ${r_{0}}/L\approx{0.16}$. It is interesting to note that the first growth rate was obtained in the Wang et al slow dynamo, where the magnetic Reynolds number of $Re_{m}=210$, while in the second one considered in this paper one uses the limit of $Re_{m}\to{\infty}$. These growth rates ${\gamma}$ are computed by applying the fast dynamo limit $lim_{{\eta}\to{0}}{\gamma}(\eta)>0$. This limit is used in the self-induced equation, without the need to solve these equations to investigate the fast dynamo action of the flow. In this sense the fast dynamo seems to be excited by the elongation of the plasma device as suggested by Wang group. The Frenet curvature of the tube is given by ${\kappa}_{0}\approx{0.5 m^{-1}}$. It is suggested that the small Perm torus could be twisted [Dobler et al, Phys Rev E (2003)] in order to enhance even more the fast dynamo effect. By considering the stability of the plasma torus one obtains a value for the fast dynamo growth rate as high as ${\gamma}=1.712$ from a general expression ${\gamma}=0.16{\omega}$ and a toroidal oscillation of a chaotic flow of ${\omega}=\frac{2{\pi}}{6}$.Comment: UER

Lagrangean stability of slow dynamos in compact 3D Riemannian manifolds

Modifications on a recently introduced fast dynamo operator by Chiconne et al [Comm Math Phys 173, 379 (1995)] in compact 3D Riemannian manifolds allows us to shown that slow dynamos are Lagrangean stable, in the sense that the sectional curvature of the Riemann manifold vanishes. The stability of the holonomic filament in this manifold will depend upon the sign of the second derivative of the pressure along the filament and in the non-holonomic case, to the normal pressure of the filament. Lagrangean instability is also investigated in this case and again an dynamo operator can be defined in this case. Negative curvature (Anosov flows) dynamos are also discussed in their stability aspects.Comment: Departamento de Fisica Teorica-IF-UERJ-Brasi

Geodesic plasma flows instabilities of Riemann twisted solar loops

Riemann and sectional curvatures of magnetic twisted flux tubes in Riemannian manifold are computed to investigate the stability of the plasma astrophysical tubes. The geodesic equations are used to show that in the case of thick magnetic tubes, the curvature of planar (Frenet torsion-free) tubes have the effect ct of damping the flow speed along the tube. Stability of geodesic flows in the Riemannian twisted thin tubes (almost filaments), against constant radial perturbations is investigated by using the method of negative sectional curvature for unstable flows. No special form of the flow like Beltrami flows is admitted, and the proof is general for the case of thin magnetic flux tubes. In the magnetic equilibrium state, the twist of the tube is shown to display also a damping effect on the toroidal velocity of the plasma flow. It is found that for positive perturbations and angular speed of the flow, instability is achieved, since the sectional Ricci curvature of the magnetic twisted tube metric is negative. Solar flare production may appear from these geometrical instabilities of the twisted solar loops.Comment: Departamento de Fisica Teorica-IF-UERJ-Rio-Brasi

Riemannian and filamentary geometries..

Riemannian and filamentary geometries are used to investigate the stretched filamentary dynamos against solar data in solar corona

Slow dynamos in Lorentz tori Anti-de Sitter spacetime embedded in Riemann 2D-space

Earlier Chicone, Latushkin and Montgomery-Smith [Comm Math Phys (1997)] have shown that a fast dynamo in compact two-dimensional manifold can be supported as long as its Riemannian curvature be negative. Recently Klebanov and Maldacena [Phys Today (2008)] showed that a similar flat spacetime embedding of a 2D negative Riemannian hyperbolic embedding in 2+1-D space-time, is achieved by a coordinate transformation. This embedding is used here to obtain a flat spacetime embedding of a slow dynamo in Riemannian 2D compact manifold of negative constant curvature. In is shown that a slow dynamo appears in anti-de Sitter space (AdS) Lorentz tori. This is in agreement with Bassett et al [Phys Rev D (2001)] cosmic dynamo where suppression of resonance by universe expansion slow dynamo action in comparison to preheating phases. Other example of flat embeddings, which keeps some resamblance with AdS slow dynamo, is given by the embedding of Moebius strip [Shukurov, Stepanov, Sokoloff, PRE (2008)] in the laboratory

Dynamo action at recombination epoch of open Friedmann universe spatial sections

Chicone et al [Comm Math Phys (1997)] investigated existence of fast dynamos by analyzing the spectrum kinematic magnetic dynamo. In real non-degenerate branch of the spectrum, the kinematic dynamo operator lies on a compact Riemannian 2D space of constant negative curvature. Here, generalization of Marklund and Clarkson [MNRAS (2005)], general relativistic GR-MHD dynamo equation to include mean-field dynamos is obtained. In the absence of kinetic helicity, adiabatic constant $\gamma={1/2}$ and gravitational colapse of negative Riemann curvature of spatial sections enhance dynamo effect $\frac{{\delta}B}{B}=2.6\times 10^{-1}$. Critical time where linear dynamo effects breaks down de to curvature. At recombination time, COBE temperature anisotropies, implies that magnetic field growth rate is ${\lambda}{\approx{10}^{-9}yr^{-1}}$. This places a bound on curvature till the recombination magnetic field is amplified to present value of $B_{0}=10^{-9}G$, by dynamo action. At present epoch, negative curvature becomes constant and the Chicone et al result is shown to be valid in cosmology. Since negative curvature is non-constant, Hilbert theorem which forbiddes negative constant curvature surfaces embeddeding in $\textbf{R}^{3}$ is bypassed.Comment: DFT uerj Brasi

Geodesic dynamo chaotic flows and non-Anosov maps in twisted magnetic flux tubes

Recently Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated the anisotropies in magnetic field dynamo evolution, from local Lyapunov exponents, giving rise to a metric tensor, in the Alfven twist in magnetic flux tubes (MFTs). Thiffeault and Boozer [\textbf{Chaos}(2001)] have investigated the how the vanishing of Riemann curvature constrained the Lyapunov exponential stretching of chaotic flows. In this paper, Tang-Boozer-Thiffeault differential geometric framework is used to investigate effects of twisted magnetic flux tube filled with helical chaotic flows on the Riemann curvature tensor. When Frenet torsion is positive, the Riemann curvature is unstable, while the negative torsion induces an stability when time $t\to{\infty}$. This enhances the dynamo action inside the MFTs. The Riemann metric, depends on the radial random flows along the poloidal and toroidal directions. The Anosov flows has been applied by Arnold, Zeldovich, Ruzmaikin and Sokoloff [\textbf{JETP (1982)}] to build a uniformly stretched dynamo flow solution, based on Arnold's Cat Map. It is easy to show that when the random radial flow vanishes, the magnetic field vanishes, since the exponential Lyapunov stretches vanishes. This is an example of the application of the Vishik's anti-fast dynamo theorem in the magnetic flux tubes. Geodesic flows of both Arnold and twisted MFT dynamos are investigated. It is shown that a constant random radial flow can be obtained from the geodesic equation. Throughout the paper one assumes, the reasonable plasma astrophysical hypothesis of the weak torsion. Pseudo-Anosov dynamo flows and maps have also been addressed by Gilbert [\textbf{Proc Roy Soc A London (1993)}Comment: Departamento de Fisica Teorica-IF-UERJ-Rio de Janeiro-Brasi