Abstract:
Boozer addressed the role of magnetic helicity in dynamos [1]. He pointed out that the magnetic helicity conservation implies that the dynamo action is more easily attainable if the electric potential varies over the surface of the dynamo. This provided motivated us to investigate dynamos in Riemannian curved surfaces [2]. Thiffeault and Boozer [3] discussed the onset of dissipation in kinematic dynamos. In this paper, when curvature is constant and negative, a simple laminar dynamo solution is obtained on the flow topology of a Poincare disk, whose Gauss curvature is K = –1. By considering a laminar plasma dynamo [4] the electric current helicity λ ≈ 2.34 m–1 for a Reynolds magnetic number of Rm ≈ 210 and a growth rate of magnetic field |γ| ≈ 0.022 are obtained. Negative constant curvature non-compact H2 manifold, has also been used in onecomponent electron 2D plasma by Fantoni and Tellez [5]. Chicone et al. (CMP (1997)) showed fast dynamos can be supported in compact H2. PACS: 47.65.Md.

Abstract:
Since Kostelecky et al [Phys Rev Lett 100, 111102 (2008)], have shown that there is an intimate connection between spacetime with torsion and the possibility of constraining it to Lorentz violation, a renewed interest in torsion theories of gravity has arised. In this paper, minimal coupling between photons on a torsioned background is shown to allow us to obtain the galactic magnetic field strength μG without dynamo amplification. This agrees with recent results by Jimenez and Maroto (2011) for spiral galaxies, with galactic magnetic field constraints from Dark matter without dy- namo amplification. The approach discussed here allow us to get rid of the unpleasant photon mass by simply consider- ing the Lagrangean cut off for second order torsion terms. Therefore though the gauge and Lorentz symmetries are bro- ken here one does not have to deal with photon masses.

Abstract:
Recently
torsion fields were introduced in CP-violating cosmic axion a^{2}-dynamos
[Garcia de Andrade, Mod Phys Lett A, (2011)] in order to obtain Lorentz
violating bounds for torsion. Here instead, oscillating axion solutions of the
dynamo equation with torsion modes [Garcia de Andrade, Phys Lett B (2012)] are
obtained taking into account dissipative torsion fields. Magnetic helicity
torsion oscillatory contribution is also obtained. Note that the torsion presence
guarantees dynamo efficiency when axion dynamo length is much stronger than the
torsion length. Primordial axion oscillations due to torsion yield a magnetic
field of 10^{9} G at Nucleosynthesis epoch. This is obtained due to a
decay of BBN magnetic field of 10^{15} G induced by torsion. Since
torsion is taken as 10^{–20} s^{–1}, the dynamo efficiency is
granted over torsion damping. Of course dynamo efficiency is better in the
absence of torsion. In the particular case when the torsion is obtained from
anomalies it is given by the gradient of axion scalar [Duncan et al., Nuclear Phys B 87, 215] that a
simpler dynamo equation is obtained and dynamo mechanism seems to be efficient
when the torsion helicity, is negative while magnetic field decays when the
torsion is positive. In this case an extremely huge value for the magnetic
field of 10^{15} Gauss is obtained. This is one order of magnitude
greater than the primordial magnetic fields of the domain wall. Actually if one
uses t_{DW} ~ 10^{-}^{4} sone obtains B_{DW} ~ 10^{22} Gwhich is a
more stringent limit to the DW magnetic primordial field.

Abstract:
Magneto-curvature stresses could deform magnetic field lines and this would give rise to back reaction and restoring magnetic stresses [Tsagas, PRL (2001)]. Barrow et al [PRD (2008)] have shown in Friedman universe the expansion to be slow down in spatial section of negative Riemann curvatures. From Chicone et al [CMP (1997)] paper, proved that fast dynamos in compact 2D manifold implies negatively constant Riemannian curvature, here one applies the Barrow-Tsagas ideas to cosmic dynamos. Fast dynamo covariant stretching of Riemann slices of cosmic Lobachevsky plane is given. Inclusion of advection term on dynamo equations [Clarkson et al, MNRAS (2005)] is considered. In absence of advection a fast dynamo is also obtained. Viscous and restoring forces on stretching particles decrease, as magnetic rates increase. From COBE data ($\frac{{\delta}B}{B}\approx{10^{-5}}$), one computes stretching $\frac{{\delta}V^{y}}{V^{y}}=1.5\frac{{\delta}B}{B}\approx{1.5{\times}10^{-5}}$. Zeldovich et al has computed the maximum magnetic growth rate as ${\gamma}_{max}\approx{8.0{\times}10^{-1}t^{-1}}$. From COBE data one computes a lower growth rate for the magnetic field as ${\gamma}_{COBE}\approx{6.0{\times}10^{-6}t^{-1}}$, well-within Zeldovich et al estimate. Instead of the Harrison value $B\approx{t^{{4/3}}}$ one obtains the lower primordial field $B\approx{10^{-6}t}$ which yields the $B\approx{10^{-6}G}$ at the $1s$ Big Bang time.

Abstract:
Boozer addressed the role of magnetic helicity in dynamos [Phys Fluids \textbf{B},(1993)]. He pointed out that the magnetic helicity conservation implies that the dynamo action is more easily attainable if the electric potential varies over the surface of the dynamo. This provided us with motivation to investigate dynamos in Riemannian curved surfaces [Phys Plasmas \textbf{14}, (2007);\textbf{15} (2008)]. Thiffeault and Boozer [Phys Plasmas (2003)] discussed the onset of dissipation in kinematic dynamos. When curvature is constant and negative, a simple simple laminar dynamo solution is obtained on the flow topology of a Poincare disk, whose Gauss curvature is $K=-1$. By considering a laminar plasma dynamo [Wang et al, Phys Plasmas (2002)] the electric current helicity ${\lambda}\approx{2.34m^{-1}}$ for a Reynolds magnetic number of $Rm\approx{210}$ and a growth rate of magnetic field $|{\gamma}|\approx{0.022}$. Negative constant curvature non-compact $\textbf{H}^{2}$, has also been used in one-component electron 2D plasma by Fantoni and Tellez (Stat Phys, (2008)). Chicone et al (CMP (1997)) showed fast dynamos can be supported in compact $\textbf{H}^{2}$. PACS: 47.65.Md. Key-word: dynamo plasma.

Abstract:
Earlier, Chicone, Latushkin and Montgomery-Smith [Comm Math Phys (1997)] have proved the existence of a fast dynamo operator, in compact two-dimensional manifold, as long as its Riemannian curvature be constant and negative. More recently Gallet and Petrelis [Phys Rev \textbf{E}, 80 (2009)] have investigated saddle-node bifurcation, in turbulent dynamos as modelling for magnetic field reversals. Since saddle nodes are created in hyperbolic flows, this provides us with physical motivation to investigate these reversals in a simple kinematic dynamo model obtained from a force-free non-geodesic steady flow in Lobachevsky plane. Magnetic vector potential grows in one direction and decays in the other under diffusion. Magnetic field differential 2-form is orthogonal to the plane. A restoring forcing dynamo in hyperbolic space is also given. Magnetic field reversals are obtained from this model. Topological entropies [Klapper and Young, Comm Math Phys (1995)] are also computed.

Abstract:
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. Eigenvalue spectra shows that there is a relation between the Ricci scalar and expansion of the flow. Spatial 3-Einstein manifold section of Friedmann-Robertson-Walker (FRW) is obtained in the limit of ideal plasma. If the trace of the Ricci curvature tensor is negative, a contraction of the inflationary phase of the universe takes place, and the dynamo action takes place. When the universe expands a decaying magnetic field or non-dynamo is obtained. As in Latushkin and Vishik (Comm Math Phys (2003)) the Lyapunov exponents in kinematic dynamos is also investigated. Since positive curvature scalar are preserved under Ricci flow, it is shown that fast dynamos are preserved under this same flow.

Abstract:
Magnetic curvature effects, investigated by Barrow and Tsagas (BT) [Phys Rev D \textbf{77},(2008)],as a mechanism for magnetic field decay in open Friedmann universes (${\Lambda}<0$), are applied to dynamo geometric Ricci flows in 3D curved substrate in laboratory. By simple derivation, a covariant three-dimensional magnetic self-induced equation, presence of these curvature effects, indicates that de Sitter cosmological constant (${\Lambda}\ge{0}$), leads to enhancement in the fast kinematic dynamo action which adds to stretching of plasma flows. From the magnetic growth rate, the strong shear case, anti-de Sitter case (${\Lambda}<0$) BT magnetic decaying fields are possible while for weak shear, fast dynamos are possible. The self-induced equation in Ricci flows is similar to the equation derived by BT in $(3+1)$-spacetime continuum. Lyapunov-de Sitter metric is obtained from Ricci flow eigenvalue problem. In de Sitter analogue there is a decay rate of ${\gamma}\approx{-{\Lambda}}\approx{-10^{-35}s^{-2}}$ from corresponding cosmological constant ${\Lambda}$, showing that, even in the dynamo case, magnetic field growth is slower than de Sitter inflation, which strongly supports to BT result.

Abstract:
Geometrical tools, used in Einstein's general relativity (GR), are applied to dynamo theory, in order to obtain fast dynamo action bounds to magnetic energy, from Killing symmetries in Ricci flows. Magnetic field is shown to be the shear flow tensor eigendirection, in the case of marginal dynamos. Killing symmetries of the Riemann metric, bounded by Einstein space, allows us to reduce the computations. Techniques used are similar to those strain decomposition of the flow in Sobolev space, recently used by Nu\~nez [JMP \textbf{43} (2002)] to place bounds in the magnetic energy in the case of hydromagnetic dynamos with plasma resistivity. Contrary to Nu\~nez case, we assume that the dynamos are kinematic, and the velocity flow gradient is decomposed into expansion, shear and twist. The effective twist vanishes by considering that the frame vorticity coincides with Ricci rotation coefficients. Eigenvalues are here Lyapunov exponents. In analogy to GR, where curvature plays the role of gravity, here Ricci curvature seems to play the role of diffusion.

Abstract:
Recently Shukurov et al [Phys Rev E 72, 025302 (2008)], made use of non-orthogonal curvilinear coordinate system on a dynamo Moebius strip flow, to investigate the effect of stretching by a turbulent liquid sodium flow. In plasma physics, Chui and Moffatt [Proc Roy Soc A 451,609,(1995)] (CM), considered non-orthogonal coordinates to investigate knotted magnetic flux tube Riemann metric. Here it is shown that, in the unstretching knotted tubes, dynamo action cannot be supported. Turbulence there, is generated by suddenly braking of torus rotation. Here, use of CM metric, shows that stretching of magnetic knots, by ideal plasmas, may support dynamo action. Investigation on the stretching in plasma dynamos, showed that in diffusive media [Phys Plasma \textbf{15},122106,(2008)], unstretching unknotted tubes do not support fast dynamo action. Non-orthogonal coordinates in flux tubes of non-constant circular section, of positive growth rate, leads to tube shrinking to a constant value. As tube shrinks, curvature grows enhancing dynamo action.