Flow helicity in a simplified statistical model of a fast dynamo

The role of kinetic helicity in small-scale fast dynamo action is investigated by employing a simple statistical model for the underlying flow with statistics that are Gaussian distributed, temporally delta-correlated and spatially homogeneous and isotropic. In order to focus on small-scale dynamo action we restrict our attention to flows possessing no net kinetic helicity. With the help of a diagrammatic technique and a numerical calculation we show that the dynamo growth rate is independent of the kinetic helicity as the magnetic Reynolds number R_m → ∞. It is indicated that the latter enhances the growth of the magnetic energy only for finite R_m.     

Nonlinear current helicity fluxes in turbulent dynamos and alpha quenching

Large scale dynamos produce small scale current helicity as a waste product that quenches the large scale dynamo process (alpha effect). This quenching can be catastrophic (i.e., intensify with magnetic Reynolds number) unless one has fluxes of small scale magnetic (or current) helicity out of the system. We derive the form of helicity fluxes in turbulent dynamos, taking also into account the nonlinear effects of Lorentz forces due to fluctuating fields. We confirm the form of an earlier derived magnetic helicity flux term, and also show that it is not renormalized by the small scale magnetic field, just like turbulent diffusion. Additional nonlinear fluxes are identified, which are driven by the anisotropic and antisymmetric parts of the magnetic correlations. These could provide further ways for turbulent dynamos to transport out small scale magnetic helicity, so as to avoid catastrophic quenching.     

Investigation of the helicity of magnetic fields on the Sun within the Parker dynamo model

Construction of the butterfly diagrams for the magnetic helicity in several approximations of a Parker dynamo has been carried out. The diagrams are constructed both for the cases of efficient generation of the magnetic field (large dynamo numbers) and for the weak generation (a small dynamo number). The corresponding asymptotic solution to the solar dynamo is used in the first case. The butterfly diagrams for different values of the meridional circulation were studied due to this solution. The butterfly diagrams are constructed and based on the few-mode approximation, which is valid for moderate dynamo numbers. The issue of which butterfly diagram features are common in all these approximations and can be compared with observational data is discussed.     

Alpha effect in a family of chaotic flows

We perform numerical experiments to calculate the kinematic alpha effect for a family of maximally helical, chaotic flows with a range of correlation times. We find that the value of depends on the structure of the flow, on its correlation time and on the magnetic Reynolds number in a nontrivial way. Furthermore, it seems that there is no clear relation between alpha and the helicity of the flow, contrary to what is often assumed for the parametrization of mean-field dynamo models.     

Amplification of mean magnetic field and magnetic helicity production in hot lepton plasma of early universe

We argue that the magnetic helicity conservation is violated at the lepton stage in the evolution of early Universe owing to the parity violation in the Standard Model of electroweak interactions. As a result, a cosmological magnetic field which can be a seed for the galactic dynamo obtains from the beginning a substantial magnetic helicity which has to be taken into account in the magnetic helicity balance at the later stage of galactic dynamo. The particle physics mechanism suggested in our works depends neither on helicity of matter turbulence with plasma vortices resulting in the standard α effect in dynamo theory nor on general rotation. The mechanism can result in a self-exitation of an (almost) uniform cosmological magnetic field. The text was submitted by the authors in English.     

Magnetic dynamo action at low magnetic Prandtl numbers

Amplification of magnetic field due to kinematic turbulent dynamo action is studied in the regime of small magnetic Prandtl numbers. Such a regime is relevant for planets and stars interiors, as well as for liquid-metal laboratory experiments. A comprehensive analysis based on the Kazantsev-Kraichnan model is reported, which establishes the dynamo threshold and the dynamo growth rates for varying kinetic helicity of turbulent fluctuations. It is proposed that in contrast with the case of large magnetic Prandtl numbers, the kinematic dynamo action at small magnetic Prandtl numbers is significantly affected by kinetic helicity, and it can be made quite efficient with an appropriate choice of the helicity spectrum.     

Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium

We report the observation of dynamo action in the von Kármán sodium experiment, i.e., the generation of a magnetic field by a strongly turbulent swirling flow of liquid sodium. Both mean and fluctuating parts of the field are studied. The dynamo threshold corresponds to a magnetic Reynolds number R(m) approximately 30. A mean magnetic field of the order of 40 G is observed 30% above threshold at the flow lateral boundary. The rms fluctuations are larger than the corresponding mean value for two of the components. The scaling of the mean square magnetic field is compared to a prediction previously made for high Reynolds number flows.     

Magnetic helicity tensor for an anisotropic turbulence

The evolution of the magnetic helicity tensor for a nonzero mean magnetic field and for large magnetic Reynolds numbers in an anisotropic turbulence is studied. It is shown that the isotropic and anisotropic parts of the magnetic helicity tensor have different characteristic times of evolution. The time of variation of the isotropic part of the magnetic helicity tensor is much larger than the correlation time of the turbulent velocity field. The anisotropic part of the magnetic helicity tensor changes for the correlation time of the turbulent velocity field. The mean turbulent flux of the magnetic helicity is calculated as well. It is shown that even a small anisotropy of turbulence strongly modifies the flux of the magnetic helicity. It is demonstrated that the tensor of the magnetic part of the alpha effect for weakly inhomogeneous turbulence is determined only by the isotropic part of the magnetic helicity tensor.     

Fluid-magnetic helicity in axisymmetric stationary relativistic magnetohydrodynamics

The present work is intended to gain a fruitful insight into the understanding of the formations of magneto-vortex configurations and their role in the physical processes of mutual exchange of energies associated with fluid’s motion and the magnetic fields in an axisymmetric stationary hydromagnetic system subject to strong gravitational field (e.g., neutron star/magnetar). It is found that the vorticity flux vector field associated with vorticity 2-form is a linear combination of fluid’s vorticity vector and of magnetic vorticity vector. The vorticity flux vector obeys Helmholtz’s flux conservation. The energy equation associated with the vorticity flux vector field is deduced. It is shown that the mechanical rotation of vorticity flux surfaces contributes to the formation of vorticity flux vector field. The dynamo action for the generation of toroidal components of vorticity flux vector field is described in the presence of meridional circulations. It is shown that the stretching of twisting magnetic lines due to differential rotation leads to the breakdown of gravitational isorotation in the absence of meridional circulations. An explicit expression consists of rotation of vorticity flux surface, energy and angular momentum per baryon for the fluid-magnetic helicity current vector is obtained. The conservation of fluid-magnetic helicity is demonstrated. It is found that the fluid-magnetic helicity displays the energy spectrum arising due to the interaction between the mechanical rotation of vorticity flux surfaces and the fluid’s motion obeying Euler’s equations. The dissipation of a linear combination of modified fluid helicity and magnetic twist is shown to occur due to coupled effect of frame dragging and meridional circulation. It is found that the growing twist of magnetic lines causes the dissipation of modified fluid helicity in the absence of meridional circulations.     

Dynamo Effect in the Kraichnan Magnetohydrodynamic Turbulence

The existence of a dynamo effect in a simplified magnetohydrodynamic model of turbulence is considered when the magnetic Prandtl number approaches zero or infinity. The magnetic field is interacting with an incompressible Kraichnan-Kazantsev model velocity field which incorporates also a viscous cutoff scale. An approximate system of equations in the different scaling ranges can be formulated and solved, so that the solution tends to the exact one when the viscous and magnetic-diffusive cutoffs approach zero. In this approximation we are able to determine analytically the conditions for the existence of a dynamo effect and give an estimate of the dynamo growth rate. Among other things we show that in the large magnetic Prandtl number case the dynamo effect is always present. Our analytical estimates are in good agreement with previous numerical studies of the Kraichnan-Kazantsev dynamo by Vincenzi (J. Stat. Phys. 106:1073–1091, 2002).     

The Dynamo-Driven Magnetic Helicity Transport

In this paper, a new set of the evolution equations for the helicity of the mean magnetic field and the mean helicity of the fluctuating magnetic field is derived from the Maxwell equations and the generalized Ohm's law with the dynamo action. It is shown that there exist two kinds of the dynamo-driven magnetic helicity transport. One of them makes the mean magnetic field helicity transfer to the fluctuating magnetic field, yielding an anomalous loop voltage. The other makes the fluctuating magnetic field helicity transfer to the mean magnetic field, which provides a convincing evidence for the existence of the dynamo current. Therefore, the two kinds of the magnetic helicity transport describe the mutual conversion between the regular and irregular motions. The formulas of the loop voltage and the dynamo current are given. In particular, we give out the formula of the dynamo current-generated equilibrium magnetic field which provides a concrete mode of the magnetic field creation and maintenance in both astrophysical and laboratory (e.g., reversed-field pinch) plasmas.     

Dynamo in protostars

In this paper, we estimate the magnetic Reynolds number of a typical protostar before and after deuterium burning, and claim for the existence of dynamo process in both the phases, because the magnetic Reynolds number of the protostar far exceeds the critical magnetic Reynolds number for dynamo action. Using the equipartition of kinetic and magnetic energies, we estimate the steady-state magnetic field of the protostar to be of the order of kilogauss, which is in good agreement with observations.     

The Alpha Effect: The Connection between Cyclonic Events and Current Helicity

In a magnetized plasma, resistive diffusion of large-scale magnetic fields can be suppressed or even overcome by a turbulently generated electromotive force. For a plasma in which the turbulence is homogeneous and isotropic this EMF is characterized by the ensemble average = ?B0, where ?v and ?b represent the turbulent fields and B0 defines the large-scale field. Determination of the statistical properties of the turbulence that are required to generate a finite alpha effect, as it has become known, is one of the central subjects of dynamo theory. Parker has shown that helical velocity fluctuations possessing a net amount of kinetic helicity are capable of dynamo action. These "cyclonic events" produce electromagnetic fluctuations characterized by their own statistical properties. Within the context of "mean-field electrodynamics" we show that these fluctuations possess a net amount of current helicity, and find that a necessary condition for dynamo action is that the turbulent current helicity and the current helicity in the large-scale field be of opposite sign.     

Effect of electromagnetic boundary condition on dynamo actions

In this paper,based on the mean field dynamo theory,the influence of the electromagnetic boundary condition on the dynamo actions driven by the small scale turbulent flows in a cylindrical vessel is investigated by the integral equation approach.The numerical results show that the increase of the electrical conductivity or magnetic permeability of the walls of the cylindrical vessel can reduce the critical magnetic Reynolds number.Furthermore,the critical magnetic Reynolds number is more sensitive to the varying electrical conductivity of the end wall or magnetic permeability of the side wall.For the anisotropic dynamo which is the mean field model of the Karlsruhe experiment,when the relative electrical conductivity of the side wall or the relative magnetic permeability of the end wall is less than some critical value,the m=1(m is the azimuthal wave number)magnetic mode is the dominant mode,otherwise the m=0 mode predominates the excited magnetic field.Therefore,by changing the material of the walls of the cylindrical vessel,one can select the magnetic mode excited by the anisotropic dynamo.     

Numerical study of homogeneous dynamo based on experimental von Kármán type flows

A numerical study of the magnetic induction equation has been performed on von Kármán type flows. These flows are generated by two co-axial counter-rotating propellers in cylindrical containers. Such devices are currently used in the von Kármán sodium (VKS) experiment designed to study dynamo action in an unconstrained flow. The mean velocity fields have been measured for different configurations and are introduced in a periodic cylindrical kinematic dynamo code. Depending on the driving configuration, on the poloidal to toroidal flow ratio and on the conductivity of boundaries, some flows are observed to sustain growing magnetic fields for magnetic Reynolds numbers accessible to a sodium experiment. The response of the flow to an external magnetic field has also been studied: The results are in excellent agreement with experimental results in the single propeller case but can differ in the two propellers case.     

Critical magnetic Prandtl number for small-scale dynamo

We report a series of numerical simulations showing that the critical magnetic Reynolds number Rm(c) for the nonhelical small-scale dynamo depends on the Reynolds number Re. Namely, the dynamo is shut down if the magnetic Prandtl number Pr(m)=Rm/Re is less than some critical value Pr(m,c)< approximately 1 even for Rm for which dynamo exists at Pr(m)> or =1. We argue that, in the limit of Re-->infinity, a finite Pr(m,c) may exist. The second possibility is that Pr(m,c)-->0 as Re--> infinity, while Rm(c) tends to a very large constant value inaccessible at current resolutions. If there is a finite Pr(m,c), the dynamo is sustainable only if magnetic fields can exist at scales smaller than the flow scale, i.e., it is always effectively a large-Pr(m) dynamo. If there is a finite Rm(c), our results provide a lower bound: Rm(c) greater, similar 220 for Pr(m)< or =1/8. This is larger than Rm in many planets and in all liquid-metal experiments.     

Effect of hyperdiffusivity on turbulent dynamos with helicity

In numerical studies of turbulence, hyperviscosity is often used as a tool to extend the inertial subrange and to reduce the dissipative subrange. By analogy, hyperdiffusivity (or hyperresistivity) is sometimes used in magnetohydrodynamics. The underlying assumption is that only the small scales are affected by this manipulation. In the present paper, possible side effects on the evolution of the large-scale magnetic field are investigated. It is found that for turbulent flows with helicity, hyperdiffusivity causes the dynamo-generated magnetic field to saturate at a higher level than normal diffusivity. This result is successfully interpreted in terms of magnetic helicity conservation, which also predicts that full saturation is reached only after a time comparable to the large-scale magnetic (hyper)diffusion time.     

New dynamical mean-field dynamo theory and closure approach

We develop a new nonlinear mean field dynamo theory that couples field growth to the time evolution of the magnetic helicity and the turbulent electromotive force, E. We show that the difference between kinetic and current helicities emerges naturally as the growth driver when the time derivative of E is coupled into the theory. The solutions predict significant field growth in a kinematic phase and a saturation rate/strength that is magnetic Reynolds number dependent/independent in agreement with numerical simulations. The amplitude of early time oscillations provides a diagnostic for the closure.     

Impact of impellers on the axisymmetric magnetic mode in the VKS2 dynamo experiment

In the von Kármán Sodium 2 (VKS2) successful dynamo experiment of September 2006, the observed magnetic field showed a strong axisymmetric component, implying that nonaxisymmetric components of the flow field were acting. By modeling the induction effect of the spiraling flow between the blades of the impellers in a kinematic dynamo code, we find that the axisymmetric magnetic mode is excited. The control parameters are the magnetic Reynolds number of the mean flow, the coefficient measuring the induction effect alpha, and the type of boundary conditions. We show that using realistic values of alpha, the observed critical magnetic Reynolds number, Rm;{c} approximately 32, can be reached easily with ferromagnetic boundary conditions. We conjecture that the dynamo action achieved in this experiment may not be related to the turbulence in the bulk of the flow, but rather to the alpha effect induced by the impellers.     

Large-scale magnetic field generation by randomly forced shearing waves

A rigorous theory for the generation of a large-scale magnetic field by random nonhelically forced motions of a conducting fluid combined with a linear shear is presented in the analytically tractable limit of low magnetic Reynolds number (Rm) and weak shear. The dynamo is kinematic and due to fluctuations in the net (volume-averaged) electromotive force. This is a minimal proof-of-concept quasilinear calculation aiming to put the shear dynamo, a new effect recently found in numerical experiments, on a firm theoretical footing. Numerically observed scalings of the wave number and growth rate of the fastest-growing mode, previously not understood, are derived analytically. The simplicity of the model suggests that shear dynamo action may be a generic property of sheared magnetohydrodynamic turbulence.