Turbulent velocity waves and kinematic dynamo activity

It is demonstrated that a turbulent distribution of small amplitude velocity waves gives rise to kinematic dynamo activity on its own in an infinite medium. As the wave speed increases, the intensity of the waves has to increase in order to support dynamo regeneration. The theory given is statisticallyexact, but in view of the complexity of the resulting equations, the answers presented are for the long-wavelength (i.e. large-scale) magnetic field. In view of the fact that most astrophysical objects apparently contain turbulent wave motions, the present calculation is indicative of the extent to which turbulent dynamo activity may be physically importat in such objects.     

The Kinematic Theory of Solar Dynamo

Generation of the Sun‘s magnetic fields by self-inductive processes in the solar electrically conducting interior, the solar dynamo theory, is a fundamentally important subject in astrophysics. The kinematic dynamo theory concerns how the magnetic fields are produced by kinematically possible flows without being constrained by the dynamic equation. We review a number of basic aspects of the kinematic dynamo theory, including the magnetohydrodynamic approximation for the dynamo equation, the impossibility of dynamo action with the solar differential rotation, the Cowling‘s anti-dynamo theorem in the solar context, the turbulent alpha effect and recently constructed three-dimensional interface dynamos controlled by the solar tachocline at the base of the convection zone.     

Can the turbulent galactic dynamo generate large-scale magnetic fields?

Large-scale magnetic fields in galaxies are thought to be generated by a turbulent dynamo. However, the same turbulence also leads to a small-scale dynamo which generates magnetic noise at a more rapid rate. The efficiency of the large-scale dynamo depends on how this noise saturates. We examine this issue, taking into account ambipolar drift, which obtains in a galaxy with significant neutral gas. We argue as follows.
(i) The small-scale dynamo generated field does not fill the volume, but is concentrated into intermittent rope-like structures. The flux ropes are curved on the turbulent eddy scales. Their thickness is set by the diffusive scale determined by the effective ambipolar diffusion.
(ii) For a largely neutral galactic gas, the small-scale dynamo saturates, as a result of inefficient random stretching, when the peak field in a flux rope has grown to a few times the equipartition value.
(iii) The average energy density in the saturated small-scale field is subequipartition, since it does not fill the volume.
(iv) Such fields neither drain significant energy from the turbulence nor convert eddy motion of the turbulence on the outer scale into wave-like motion. The diffusive effects needed for the large-scale dynamo operation are then preserved until the large-scale field itself grows to near equipartition levels.     

P – ω Dynamo in Spherical Geometry

Based on the fundamental P – ω dynamo equation, using spherical polar coordinates, we carry out a study of turbulent plasma wave dynamo effect. For various rotation laws, different analytical solutions are derived. In the cases of no rotation and rigid rotation, the dynamo generates poloidal field only, while with differential rotation, regardless the differential rotation is radial or latitudinal, poloidal and toroidal fields are all generated. We may think that the solutions are the analytical forms of the magnetic field in a turbulent source region of celestial bodies. This revised version was published online in July 2006 with corrections to the Cover Date.     

The energy flux number and three types of planetary dynamo

It is proposed that the existence and nature of a planetary dynamo can be characterized by a dimensionless number Φ ≡ F_eR/ϱλ²ω, called the energy flux number, where F_e is the energy flux available for dynamo generation, R is the core radius (or thickness of the dynamo generating region), ϱ is the fluid density, λ is the magnetic diffusivity and ω is the angular velocity. For Φ ≲ 1, there is no dynamo. For 1 ≲ Φ ≲ 10^2.5 there is an “energy-limited dynamo”, in which F_e is insufficient to enable the dynamo to reach the dynamically desirable state A ≡ B²/8πϱλω ∼ 1, where B is a typical field amplitude (in Gauss). For 10^2.5 ≲ Φ ≲ 10⁵, there is a dynamically determined dynamo (Λ ∼ 1) in which the magnetic Reynolds number of turbulent eddies is small. For Φ ≳ 10⁵, there is a turbulent dynamo. Probable planetary examples of these three dynamo states are Mercury (Φ ∼ 10²-10³), Earth (Φ ∼ 10⁴) and Jupiter (Φ ∼ 10¹¹), respectively.     

List of forthcoming papers

This paper is largely a reply to Cowling's review of the present status of cosmic dynamo theory and its alternatives of primordial or fossil field models. Central is the question of turbulent diffusion, without which plasma dynamos will not work but primordial magnetic fields are retained. Turbulence does not shred or divide fields into small-scale elements as claimed; instead it creates these elements in addition to the large-scale field which remains after Ohmic diffusion has destroyed the small fields. The significance of the existence of a terrestrial dynamo is stressed and various objections to the existence of a solar-type dynamo are discussed, including the steady divergence of theory and observational evidence over a quarter century. Cowling's criticisms of the primordial field theory are discussed; these include turbulent diffusion, the timing of the solar magnetic cycle, and the importance attached to observations in active and quiet magnetic regions.Since this paper was communicated, a personal communication from Professor Cowling has partially resolved the difference of opinion about turbulent diffusion and its effects. This is discussed in a letter to the Editor, at the end of this volume, p. 477.     

Dynamo saturation in direct simulations of the multi‐phase turbulent interstellar medium

The ordered magnetic field observed via polarised synchrotron emission in nearby disc galaxies can be explained by a mean‐field dynamo operating in the diffuse interstellar medium (ISM). Additionally, vertical‐flux initial conditions are potentially able to influence this dynamo via the occurrence of the magnetorotational instability (MRI). We aim to study the influence of various initial field configurations on the saturated state of the mean‐field dynamo. This is motivated by the observation that different saturation behaviour was previously obtained for different supernova rates. We perform direct numerical simulations (DNS) of three‐dimensional local boxes of the vertically stratified, turbulent interstellar medium, employing shearing‐periodic boundary conditions horizontally. Unlike in our previous work, we also impose a vertical seed magnetic field. We run the simulations until the growth of the magnetic energy becomes negligible. We furthermore perform simulations of equivalent 1D dynamo models, with an algebraic quenching mechanism for the dynamo coefficients. We compare the saturation of the magnetic field in the DNS with the algebraic quenching of a mean‐field dynamo. The final magnetic field strength found in the direct simulation is in excellent agreement with a quenched α) dynamo. For supernova rates representative of the Milky Way, field losses via a Galactic wind are likely responsible for saturation. We conclude that the relative strength of the turbulent and regular magnetic fields in spiral galaxies may depend on the galaxy's star formation rate. We propose that a mean field approach with algebraic quenching may serve as a simple sub‐grid scale model for galaxy evolution simulations including a prescribed feedback from magnetic fields. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subcritical dynamos in shear flows

Identifying generic physical mechanisms responsible for the generation of magnetic fields and turbulence in differentially rotating flows is fundamental to understand the dynamics of astrophysical objects such as accretion disks and stars. In this paper, we discuss the concept of subcritical dynamo action and its hydrodynamic analogue exemplified by the process of nonlinear transition to turbulence in non‐rotating wall‐bounded shear flows. To illustrate this idea, we describe some recent results on nonlinear hydrodynamic transition to turbulence and nonlinear dynamo action in rotating shear flows pertaining to the problem of turbulent angular momentum transport in accretion disks. We argue that this concept is very generic and should be applicable to many astrophysical problems involving a shear flow and non‐axisymmetric instabilities of shearinduced axisymmetric toroidal velocity or magnetic fields, such as Kelvin‐Helmholtz, magnetorotational, Tayler or global magnetoshear instabilities. In the light of several recent numerical results, we finally suggest that, similarly to a standard linear instability, subcritical MHD dynamo processes in high‐Reynolds number shear flows could act as a large‐scale driving mechanism of turbulent flows that would in turn generate an independent small‐scale dynamo. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reconstructing the IRAS Point Source Catalog Redshift Survey with a generalized PIZA

We study the influence of different magnetic boundary conditions on the generation of magnetic fields by turbulent convection. It is found that the structure and strength of the generated field in the vicinity of the boundary is strongly dependent on the choice of boundary conditions. In the convective interior, however, the solutions remain largely insensitive to the boundary conditions. In all cases the overall efficiency of the dynamo process remains high with a steady state magnetic energy density between 12 and 25 per cent of the turbulent kinetic energy, and peak field values exceeding the equipartition level. These results support the idea that the solar granulation may constitute a dynamo source for magnetic fields in the quiet photosphere.     

Non‐holonomic filamentary dynamos as generalized Arnold's maps in Riemannian space

A filamentary non‐holonomic dynamo solution of self‐induction magnetic field equation is found by considering highly conducting filaments. It is shown that planar filaments cannot support dynamo action since the flow along the filament vanishes for torsion‐free filaments. This is a generalization of the Zeldovich theorem for linear magnetic dynamo filaments. The flow of filament is proportionally to the product between Frenet torsion and curvature. This shows that filamentary dynamos must possess Frenet torsion. A well‐known example of this result is the α ‐dynamo in solar physics. Magnetic helicity and magnetic energy for this filamentary dynamo are computed. Magnetic helicity vanishes by construction and the magnetic field decays with torsion energy in helicoidal dynamos. The approach considered here is useful for the investigation of anisotropic turbulent cascades. As a particular simple example it is shown that under certain constraints the solution can be reduced to the Arnold cat dynamo map solution where the non‐holonomic directional mixed derivative, would play the role of the Lyapunov exponent which appears on stretching the magnetic field in Riemannian space. The solution seems to describe marginal slow dynamos when the velocities involved in the dynamo flows are constants. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Configurations of dynamo waves in a two-layer medium and the 11-year solar cycle

The behavior of dynamo waves in a two-layer medium is investigated in terms of the Parker dynamo model. The solar cycle duration is shown to depend on the ratio of turbulent diffusivities in the layers. Meridional circulation has been incorporated into the Parker system. An increase in the intensity of meridional flows is shown to decelerate the propagation of dynamo waves. The minimum of solar magnetic activity can occur not only in the case of intense meridional circulation in both layers but also when a difference in physical characteristics arises between the layers and the meridional flows are moderate.     

Nonhelical mean‐field dynamos in a sheared turbulence

Mechanisms of nonhelical large‐scale dynamos (shear‐current dynamo and effect of homogeneous kinetic helicity fluctuations with zero mean) in a homogeneous turbulence with large‐scale shear are discussed. We have found that the shearcurrent dynamo can act even in random flows with small Reynolds numbers. However, in this case mean‐field dynamo requires small magnetic Prandtl numbers (i.e., when Pm < Pm^cr < 1). The threshold in the magnetic Prandtl number, Pm^cr = 0.24, is determined using second order correlation approximation (or first‐order smoothing approximation) for a background random flow with a scale‐dependent viscous correlation time τ_c = (νk 2)^–1 (where ν is the kinematic viscosity of the fluid and k is the wave number). For turbulent flows with large Reynolds numbers shear‐current dynamo occurs for arbitrary magnetic Prandtl numbers. This dynamo effect represents a very generic mechanism for generating large‐scale magnetic fields in a broad class of astrophysical turbulent systems with large‐scale shear. On the other hand, mean‐field dynamo due to homogeneous kinetic helicity fluctuations alone in a sheared turbulence is not realistic for a broad class of astrophysical systems because it requires a very specific random forcing of kinetic helicity fluctuations that contains, e.g., low‐frequency oscillations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamo mechanism for turbulent wave in celestial bodies

It is well known that under cosmic conditions the various modes of plasma turbulence waves (including MHD waves) are easily excited. In this paper we are trying to show that the turbulent wave also generates a source-term for the magnetic induced equations as does the turbulent fluid with nonzero helicity. By expanding the turbulent field in Fourier series, we have obtained dynamo equation for turbulent wave and a reasonable solution which indicates that the poloidal field may be built-up in the turbulent source region. Perhaps, we may think that the poloidal field of Equation (9) is the analytical form of the magnetic field in a turbulent source region of celestial bodies.     

Can turbulent plane‐shear flows support large‐scale dynamos?

Turbulent plane‐shear flow is found to show same basic effects of mean‐fieldMHD as rotating turbulence. In particular, the mean electromotive force (EMF) includes highly anisotropic turbulent diffusion and alpha‐effect. Only magnetic diffusion remains for spatially‐uniform turbulence. The question is addressed whether in this case a self‐excitation of a magnetic field by so‐called sher‐current dynamo is possible and the quasilinear theory provides a negative answer. The streamaligned component of the EMF has the sign opposite to that required for dynamo. If, however, the turbulence is not uniform across the flow direction then a dynamo‐active α ‐effect emerges. The critical magnetic Reynolds number for the alpha‐shear dynamo is estimated to be slightly above ten. Possibilities for cross‐checking theoretical predictions with MHD experiments are discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the generation of the large-scale and turbulent magnetic fields in solar-type stars

It is thought that the large-scale solar-cycle magnetic field is generated in a thin region at the interface of the radiative core (RC) and solar convection zone (SCZ). We show that the bulk of the SCZ virogoursly generates a small-scale turbulent magnetic field. Rotation, while not essential, increases the generation rate of this field.Thus, fully convective stars should have significant turbulent magnetic fields generated in their lower convection zones. In these stars the absence of a radiative core, i.e., the absence of a region of weak buoyancy, precludes the generation of a large-scale magnetic field, and as a consequence the angular momentum loss is reduced. This is, in our opinion, the explanation for the rapid rotation of the M-dwarfs in the Hyades cluster.Adopting the Utrecht's group terminology, we argue that the residual chromospheric emission should have three distinctive components: the basal emission, the emission due to the large-scale field, and the emission due to the turbulent field, with the last component being particularly strong for low mass stars.In the conventional dynamo equations, the dynamo frequencies and the propagation of the dynamo wave towards the equator are based on the highly questionable assumption of a constant . Furthermore, meridional motions, a necessary consequence of the interaction of rotation with convection, are ignored. In this context we discuss Stenflo's results about the global wave pattern decomposition of the solar magnetic field and conclude that it cannot be interpreted in the framework of the conventional dynamo equations.We discuss solar dynamo theories and argue that the surface layers could be essential for the generation of the poloidal field. If this is the case an -effect would not be needed at the RC-SCZ interface (where the toroidal field is generated). The two central problems facing solar dynamo theories may the transport of the surface poloidal field to the RC-SCZ interface and the uncertainty about the contributions to the global magnetic field by the small-scale magnetic features.Visitor, National Solar Observatory, National Optical Astronomy Observatories.The National Optical Astronomy Observatories are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.     

Predicting solar ‘climate’ by assimilating magnetic data into a flux-transport dynamo

Observational and theoretical knowledge about global-scale solar dynamo ingredients have reached the stage that it is possible to calibrate a flux-transport dynamo for the Sun by adjusting only a few tunable parameters. The important ingredients in this class of model are differential rotation (Omega-effect), helical turbulence (alpha-effect), meridional circulation and turbulent diffusion. The meridional circulation works as a conveyor belt and governs the dynamo cycle period. Meridional circulation and magnetic diffusivity together govern the memory of the Sun's past magnetic fields. After describing the physical processes involved in a flux-transport dynamo, we will show that a predictive tool can be built from it to predict mean solar cycle features by assimilating magnetic field data from previous cycles. We will discuss the theoretical and observational connections among various predictors, such as dynamo-generated toroidal flux integral, cross-equatorial flux, polar fields and geomagnetic indices. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The direction of propagation of the solar dynamo wave

We propose a solution to one of the oldest problems in the solar-dynamo theory: explaining the equatorward drift of magnetic activity in the solar cycle. The well-known suggestion that the dynamo waves propagate along the surfaces of constant angular velocity is shown to be restricted to an isotropic medium. Allowance for the rotation-induced anisotropy in turbulent diffusion leads to an equatorward deviation of the wave phase velocity from the isorotational surface. Estimates for the dynamo waves are illustrated with two-dimensional numerical models in a spherical geometry. The model with anisotropic diffusion also shows an equatorward drift of the toroidal magnetic field when the rotation is radially uniform.     

Mean-field dynamo in partially ionized plasmas – I

There are several astrophysical situations where one needs to study the dynamics of magnetic flux in partially ionized turbulent plasmas. In a partially ionized plasma, the magnetic induction is subjected to the ambipolar diffusion and the Hall effect in addition to the usual resistive dissipation. In this paper, we initiate the study of the kinematic dynamo in a partially ionized turbulent plasma. The Hall effect arises from the treatment of the electrons and the ions as two separate fluids and the ambipolar diffusion due to the inclusion of neutrals as the third fluid. It is shown that these non-ideal effects modify the so-called α effect and the turbulent diffusion coefficient β in a rather substantial way. The Hall effect may enhance or quench the dynamo action altogether. The ambipolar diffusion brings in an α which depends on the mean magnetic field. The new correlations embodying the coupling of the charged fluids and the neutral fluid appear in a decisive manner. The turbulence is necessarily magnetohydrodynamic with new spatial and time-scales. The nature of the new correlations is demonstrated by taking the Alfvénic turbulence as an example.     

On the Saturation of Astrophysical Dynamos: Numerical Experiments with the No-Cosines Flow

In the context of astrophysical dynamos we illustrate that the no-cosines flow, with zero mean helicity, can drive fast dynamo action and we study the dynamo’s mode of operation during both the linear and non-linear saturation regimes. It turns out that in addition to a high growth rate in the linear regime, the dynamo saturates at a level significantly higher than normal turbulent dynamos, namely at exact equipartition when the magnetic Prandtl number Pr_m∼ 1. Visualization of the magnetic and velocity fields at saturation will help us to understand some of the aspects of the non-linear dynamo problem.     

Isolated versus common envelope dynamos in planetary nebula progenitors

The origin, evolution and role of magnetic fields in the production and shaping of proto-planetary nebulae (PPNe) and planetary nebulae (PNe) are a subject of active research. Most PNe and PPNe are axisymmetric with many exhibiting highly collimated outflows; however, it is important to understand whether such structures can be generated by isolated stars or require the presence of a binary companion. Towards this end, we study a dynamical, large-scale α−Ω interface dynamo operating in a 3.0 M_⊙ Asymptotic Giant Branch (AGB) star in both an isolated setting and a setting in which a low-mass companion is embedded inside the envelope. The back reaction of the fields on the shear is included and differential rotation and rotation deplete via turbulent dissipation and Poynting flux. For the isolated star, the shear must be resupplied in order to sufficiently sustain the dynamo. Furthermore, we investigate the energy requirements that convection must satisfy to accomplish this by analogy to the Sun. For the common envelope case, a robust dynamo results, unbinding the envelope under a range of conditions. Two qualitatively different types of explosion may arise: (i) magnetically induced, possibly resulting in collimated bipolar outflows and (ii) thermally induced from turbulent dissipation, possibly resulting in quasi-spherical outflows. A range of models is presented for a variety of companion masses.