Riemannian and filamentary geometries..
Riemannian and filamentary geometries are used to investigate the stretched filamentary dynamos against solar data in solar corona.
astro-ph↗
arXiv subjects
Garcia de Andrade
Publications and source records attributed to Garcia de Andrade.
Testing a Riemannian twisted solar loop model from EUV data and magnetic topology
Compact Riemannian solar twisted magnetic flux tube surfaces model are tested against solar extreme ultraviolet (EUV) lines observations, allowing us to compute the diameter and height of solar plasma loops. The relation between magnetic and torsion energies is found for a nonplanar solar twisted (torsioned) loop to be $10^{9}$, which shows that the contribution of torsion energy to the solar loop is extremely weaker than the magnetic energy contribution. In this case solar loops of up $5000 km$ in diameter can be reached. The height of $220.000 km$ is used to obtain an estimate for torsion based on the Riemannian flux tube surface, which yields $τ_{0}=0.9{\times} 10^{-8} m^{-1}$ which coincides with one of the data of $(0.9{\pm}0.4){\times}10^{-8}m^{-1}$ obtained by Lopez-Fuentes et al (2003). This result tells us that the Riemannian flux tube model for plasma solar loops is consistent with experimental results in solar physics. These results are obtained for a homogeneous twisted solar loop. By making use of Moffatt-Ricca theorem for the bounds on torsional energy of unknotted vortex filaments, applied to magnetic topology, one places bounds on the lengths of EUV solar loops. New results as the vorticity of the plasma flow along the tube is also computed in terms of the flux tube twist.
astro-ph↗
Turbulent $α$-effect in twisted magnetic flux tubes dynamos in Riemannian space
Analytical solution of first order torsion $α$-effect in twisted magnetic flux tubes representing a flux tube dynamo in Riemannian space is presented. Toroidal and poloidal component of the magnetic field decays as $r^{-1}$, while grow exponentially in time. The rate of speed of the helical dynamo depends upon the value of Frenet curvature of the tube. The $α$ factor possesses a fundamental contribution from constant torsion tube approximation. It is also assumed that the curvature of the magnetic axis of the tube is constant. Though $α$-effect dynamo equations are rather more complex in Riemann flux tube coordinates, a simple solution assuming force-free magnetic fields is shown to be possible. Dynamo solutions are possible if the dynamo action is able to change the signs of torsion and curvature of the dynamo flux tube simultaneously.
astro-ph↗
Helical $α$-dynamos as twisted magnetic flux tubes in Riemannian space
Analytical solution of $α$-dynamo equation representing strongly torsioned helical dynamo is obtained in the thin twisted Riemannian flux tubes approximation. The $α$ factor possesses a fundamental contribution from torsion which is however weaken in the thin tubes approximation. It is shown that assuming that the poloidal component of the magnetic field is in principle time-independent, the toroidal magnetic field component grows very fast in time, actually it possesses a linear time dependence, while the poloidal component grows under the influence of torsion or twist of the flux tube. The toroidal component decays spatially with as $r^{-2}$ while vorticity may decay as $r^{-5}$ (poloidal component) where r represents the radial distance from the magnetic axis of flux tube. Toroidal component of vorticity decays as $r^{-1}$. In turbulent dynamos unbounded magnetic fields may decay at least as $r^{-3}$.
astro-ph↗
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.
astro-ph↗
Geometry of Liquid Crystals and the alignment of HIV-1 virus helical filaments
Geometry of Liquid crystals as detectors of HIV twisted helical filaments is presented.
cond-mat.mtrl-sci↗
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.
physics.plasm-ph↗
Lagrange Instability of Geodesics in Curved Double Twisted Liquid Crystals
It is shown that curved and flat helical double twisted liquid crystal (DTLC) in blue phase, can be unstable (stable) depending of the sign, negative (positive) of sectional curvature, depending on the pitch of the helix of the nematic crystal. In both cases Cartan torsion is presented. It is also shown that the instability or stability depends on the value of the pitch of the helix in nematic crystals. Frank energy stability A similar result using the method of Frank energy stability in the twist of cholesteric liquid crystal was given by Kiselev and Sluckin [PRE 71(2005)], where the free twist number determines the equilibrium value of the cholesteric liquid crystals (CLC) pitch of the helix. As a final example we solve the geodesic equations in twisted nematics with variable pitch helix and non-constant torsion. This non-Riemannian geometrical approach, also seems to unify two recent analysis of cylindrical columns given by Santangelo et al [PRL 99,(2007)] and the curved crystal endowed with torsion, given by Vitelli et al [Proc Nat Acad Sci (2006)]. Stability of toroidal curved surfaces were also previously considered by Bowick et al [PRE 69,(2004)], as an example of curvature-induced defect. Investigation of the Lagrangean instability may be useful in the investigation of HIV viruses and proteins.
cond-mat.soft↗
Non-Riemannian geometrical asymmetrical damping stresses on the Lagrange instability of shear flows
It is shown that the physical interpretation of Elie Cartan three-dimensional space torsion as couple asymmetric stress, has the effect of damping, previously Riemannian unstable Couette planar shear flow, leading to stability of the flow in the Lagrangean sense. Actually, since the flow speed is inversely proportional to torsion, it has the effect of causing a damping in the planar flow atenuating the instability effect. In this sense we may say that Cartan torsion induces shear viscous asymmetric stresses in the fluid, which are able to damp the instability of the flow. The stability of the flow is computed from the sectional curvature in non-Riemannian three-dimensional manifold. Marginal stability is asssumed by making the sectional non-Riemannian curvature zero, which allows us to determine the speeds of flows able to induce this stability. The ideas discussed here show that torsion plays the geometrical role of magnetic field in hydromagnetic instability of Couette flows recently investigated by Bonnano and Urpin (PRE, (2007,in press) can be extended and applied to plastic flows with microstructure defects. Recently Riemannian asymmetric stresses in magnetohydrodynamics (MHD) have been considered by Billig (2004).
physics.flu-dyn↗
Stable curved solitonic surfaces in nonholonomic frame
Assuming the stability of soliton surfaces of vanishing Ricci sectional curvature of soliton metric in the nonholonomic frame, we find a solution for the metric in the approximation of weak constant torsion curves with constant Frenet curvature. The computation of the Riemann tensor of the soliton metric shows that it does not vanish and therefore the solution is nontrivial. Heisenberg solitonic equation is also used to constrain the the soliton Riemann metric. The new feature here is that the coordinate curves on the soliton-like surface are composed of hydrodynamical filaments.
physics.flu-dyn↗
Ricci curvature and geodesic flows stability in Riemannian twisted flux tubes
Ricci and sectional curvatures of twisted flux tubes in Riemannian manifold are computed to investigate the stability of the tubes. The geodesic equations are used to show that in the case of thick 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 tubes. It is found that for positive perturbations and angular speed of the flow, instability is achieved, since the sectional Ricci curvature of the twisted tube metric is negative.
physics.flu-dyn↗
Stretch fast dynamo mechanism via conformal mapping in Riemannian manifolds
Two new analytical solutions of self-induction equation, in Riemannian manifolds are presented. The first represents a twisted magnetic flux tube or flux rope in plasma astrophysics, which shows that the depending on rotation of the flow the poloidal field is amplified from toroidal field which represents a dynamo. 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 (STF) method to generate dynamos from straight and untwisted ropes. Motivated by the fact that this problem was treated using a Riemannian geometry of twisted magnetic flux ropes recently developed (Phys Plasmas (2006)), we investigated a second dynamo solution which is conformally related to the Arnold kinematic fast dynamo. In this solution it is shown that the conformal effect on the fast dynamo metric only enhances the Zeldovich stretch, and therefore a new dynamo solution is obtained. When a conformal mapping is performed in Arnold fast dynamo line element a uniform stretch is obtained in the original line element.
math-ph↗
Chaotic flows and cosmic dynamos in anisotropic pseudo-Riemannian four-dimensional spacetime
It is shown that the existence of the cosmic kinematic fast dynamos in Bianchi type IX rotating cosmological models, faces severe difficulties, due to the fact that in these models rotation increases without bounds, which is strictly forbidden by CMB astronomical date which imposes strong bounds on its rotation with respect with its rapid expansion. The only way out of this difficulty is to assume that at least one of the expansion directions of this anisotropic universe decays as fast as the amplification rate of primordial magnetic fields. A solution is found where only one direction of the anisotropic universe expands while the other two remain constants. We compute an amplification of the seed magnetic field in the case where Bianchi IX degenerates into de Sitter metric, fields amplify from 10^{-6}G to 10^{-5}G in spiral galaxies for a cosmological constant of the order $|Λ|<10^{33}s^{-2}$ and considering that the age of universe of the order of 10^{10}yrs. Another example is given by the ABC chaotic flows in the pseudo-Riemannian spacetime representing the Kasner anisotropic nonsingular universe.
astro-ph↗
Differential rotation of stretched and twisted thick magnetic flux tube dynamos in Riemannian spaces
The topological mapping between a torus of big radius and a sphere is applied to the Riemannian geometry of a stretched and twisted very thick magnetic flux tube, to obtain spherical dynamos solving the magnetohydrodynamics (MHD) self-induction equation for the magnetic flux tubes undergoing differential (non-uniform) rotation along the tube magnetic axis. Constraints on the shear is also computed. It is shown that when the hypothesis of the convective cyclonic dynamo is used the rotation is constant and a solid rotational body is obtained. As usual toroidal fields are obtained from poloidal magnetic field and these fields may be expressed in terms of the differential rotation $Ω(r,θ(s))$. In the case of non-cyclonic dynamos the torsion in the Frenet frame is compute in terms of the dynamo constant. The flux tube shear $\frac{\partial}{{\partial}r}Ω$ is also computed. The untwisted tube case is shown to be trivial in the sense that does not support any dynamo action. This case is in agreement with Cowling antidynamo theorem, since in the untwisted case the tube becomes axially symmetric which the refereed theorem rules out. We also show that it is consistent with the Zeldovich antidynamo theorem which rules out planar dynamos. Knowledge of the differential rotation of the Earth, for example, allows one to place limits on the curvature and torsion of the flux tube axis and vice-versa, knowledge of the topology permit us to infer differential rotation and other physical parameters of the stars and planets.
astro-ph↗
An example of anti-dynamo conformal Arnold metric
A 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The Riemann-Christoffel connection and Riemann curvature tensor for the Arnold and its conformal counterpart are computed. The curvature decay as z-coordinates increases without bounds. Some of the Riemann curvature components such as $R_{pzpz}$ also undergoes dissipation while component $R_{qzqz}$ increases without bounds. The remaining curvature component $R_{pqpq}$ is constant on the torus surface. The Riemann curvature invariant $K^{2}=R_{ijkl}R^{ijkl}$ is found to be 0.155 for the $λ=0.75$. A simple solution of Killing equations for Arnold metric yields a stretch Killing vector along one direction and compressed along other direction in order that the modulus of the Killing vector is not constant along the flow. The flow is shown to be untwisted. The stability of the two metrics are found by examining the sign of their curvature tensor components.
physics.flu-dyn↗
Plasma planar filament instability and Alfven waves
Inhomogeneous plasmas filaments instabilities are investigated by using the techniques of classical differential geometry of curves where Frenet torsion and curvature describe completely the motion of curves. In our case the Frenet frame changes in time and also depends upon the other coordinates taking into account the inhomogeneity of the plasma. The exponential perturbation method so commonly used to describe cosmological perturbatons is applied to magnetohydrodynamic (MHD) plasma equations to find longitudinal modes describing Alfven waves propagation modes describing plasma waves in the medium. Stability is investigated in the imaginary axis of the spectra of complex frequencies $ω$ or $Im(ω)\ne{0}$.
math-ph↗
Plasma metric singularities in helical devices and tearing instabilities in tokamaks
Plasma toroidal metric singularities in helical devices and tokamaks, giving rise to magnetic surfaces inside the plasma devices are investigated in two cases. In the first we consider the case of a rotational plasma on an helical device with circular cross-section and dissipation. In this case singularities are shown to place a Ricci scalar curvature bound on the radius of the surface where the Ricci scalar is the contraction of the constant Riemannian curvature tensor of magnetic surfaces. An upper bound on the initial magnetic field in terms of the Ricci scalar is obtained. This last bound may be useful in the engineering construction of plasma devices in laboratories. The normal poloidal drift velocity is also computed. In the second case a toroidal metric is used to show that there is a relation between singularities and the type of tearing instabilities considered in the tokamak. Besides, in this case Ricci collineations and Killing symmetries are computed.The pressure is computed by applying these constraints to the pressure equations in tokamaks.
physics.plasm-ph↗
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.
physics.plasm-ph↗