Magnetic helicity in stellar dynamos: new numerical experiments

The theory of large scale dynamos is reviewed with particular emphasis on the magnetic helicity constraint in the presence of closed and open boundaries. In the presence of closed or periodic boundaries, helical dynamos respond to the helicity constraint by developing small scale separation in the kinematic regime, and by showing long time scales in the nonlinear regime where the scale separation has grown to the maximum possible value. A resistively limited evolution towards saturation is also found at intermediate scales before the largest scale of the system is reached. Larger aspect ratios can give rise to different structures of the mean field which are obtained at early times, but the final saturation field strength is still decreasing with decreasing resistivity. In the presence of shear, cyclic magnetic fields are found whose period is increasing with decreasing resistivity, but the saturation energy of the mean field is in strong super‐equipartition with the turbulent energy. It is shown that artificially induced losses of small scale field of opposite sign of magnetic helicity as the large scale field can, at least in principle, accelerate the production of large scale (poloidal) field. Based on mean field models with an outer potential field boundary condition in spherical geometry, we verify that the sign of the magnetic helicity flux from the large scale field agrees with the sign of α. For solar parameters, typical magnetic helicity fluxes lie around 10⁴⁷ Mx² per cycle.     

The helicity constraint in spherical shell dynamos

The motivation for considering distributed large scale dynamos in the solar context is reviewed in connection with the magnetic helicity constraint. Preliminary accounts of 3-dimensional direct numerical simulations (in spherical shell segments) and simulations of 2-dimensional mean field models (in spherical shells) are presented. Interesting similarities as well as some differences are noted. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The helicity constraint in turbulent dynamos with shear

The evolution of magnetic fields is studied using simulations of forced helical turbulence with strong imposed shear. After some initial exponential growth, the magnetic field develops a large-scale travelling wave pattern. The resulting field structure possesses magnetic helicity, which is conserved in a periodic box by the ideal magnetohydrodynamics equations and can hence only change on a resistive time-scale. This strongly constrains the growth time of the large-scale magnetic field, but less strongly constrains the length of the cycle period. Comparing this with the case without shear, the time-scale for large-scale field amplification is shortened by a factor Q , which depends on the relative importance of shear and helical turbulence, and which also controls the ratio of toroidal to poloidal field. The results of the simulations can be reproduced qualitatively and quantitatively with a mean-field α Ω-dynamo model with alpha-effect and turbulent magnetic diffusivity coefficients that are less strongly quenched than in the corresponding α ²-dynamo.     

Analysis of the effect of a mean velocity field on a mean field dynamo

We study semi-analytically and in a consistent manner the generation of a mean velocity field     by helical magnetohydrodynamical (MHD) turbulence, and the effect that this field can have on a mean field dynamo. Assuming a prescribed, maximally helical small-scale velocity field, we show that large-scale flows can be generated in MHD turbulent flows via small-scale Lorentz force. These flows back-react on the mean electromotive force of a mean field dynamo through new terms, leaving the original α and β terms explicitly unmodified. Cross-helicity plays the key role in interconnecting all the effects. In the minimal τ closure that we chose to work with, the effects are stronger for large relaxation times.     

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)     

Laboratory plasma dynamos, astrophysical dynamos and magnetic helicity evolution

The term 'dynamo' means different things to the laboratory fusion plasma and astrophysical plasma communities. To alleviate the resulting confusion and to facilitate interdisciplinary progress, we pinpoint conceptual differences and similarities between laboratory plasma dynamos and astrophysical dynamos. We can divide dynamos into three types: 1. magnetically dominated helical dynamos which sustain a large-scale magnetic field against resistive decay and drive the magnetic geometry towards the lowest energy state, 2. flow-driven helical dynamos which amplify or sustain large-scale magnetic fields in an otherwise turbulent flow and 3. flow-driven non-helical dynamos which amplify fields on scales at or below the driving turbulence. We discuss how all three types occur in astrophysics whereas plasma confinement device dynamos are of the first type. Type 3 dynamos require no magnetic or kinetic helicity of any kind. Focusing on Types 1 and 2 dynamos, we show how different limits of a unified set of equations for magnetic helicity evolution reveal both types. We explicitly describe a steady-state example of a Type 1 dynamo, and three examples of Type 2 dynamos: (i) closed volume and time dependent; (ii) steady state with open boundaries; (iii) time dependent with open boundaries.     

The saturation of galactic dynamos

We summarize the current state of the long term discussion about the saturation mechanisms associated with rapid growth of small‐scale magnetic field, that operate in large‐scale galactic dynamos, and related problems with magnetic helicity conservation. Our general conclusion is that, taking into account magnetic helicity fluxes, large‐scale magnetic field can be amplified up to about the equipartition level. In contrast, models without helicity fluxes give an initial temporal magnetic field growth, but then decay. In our opinion, it is more appropriate to refer to the situation as a “potentially catastrophic scenario” rather than as “catastrophic α‐quenching” (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Small-scale magnetic helicity losses from a mean-field dynamo

Using mean-field models with a dynamical quenching formalism, we show that in finite domains magnetic helicity fluxes associated with small-scale magnetic fields are able to alleviate catastrophic quenching. We consider fluxes that result from advection by a mean flow, the turbulent mixing down the gradient of mean small-scale magnetic helicity density or the explicit removal which may be associated with the effects of coronal mass ejections in the Sun. In the absence of shear, all the small-scale magnetic helicity fluxes are found to be equally strong for both large- and small-scale fields. In the presence of shear, there is also an additional magnetic helicity flux associated with the mean field, but this flux does not alleviate catastrophic quenching. Outside the dynamo-active region, there are neither sources nor sinks of magnetic helicity, so in a steady state this flux must be constant. It is shown that unphysical behaviour emerges if the small-scale magnetic helicity flux is forced to vanish within the computational domain.     

Mean electromotive force proportional to mean flow in MILD turbulence

In mean‐field magnetohydrodynamics the mean electromotive force due to velocity and magnetic‐field fluctuations plays a crucial role. In general it consists of two parts, one independent of and another one proportional to the mean magnetic field. The first part may be nonzero only in the presence of mhd turbulence, maintained, e.g., by small‐scale dynamo action. It corresponds to a battery, which lets a mean magnetic field grow from zero to a finite value. The second part, which covers, e.g., the α effect, is important for large‐scale dynamos. Only a few examples of the aforementioned first part of the mean electromotive force have been discussed so far. It is shown that a mean electromotive force proportional to the mean fluid velocity, but independent of the mean magnetic field, may occur in an originally homogeneous isotropic mhd turbulence if there are nonzero correlations of velocity and electric current fluctuations or, what is equivalent, of vorticity and magnetic field fluctuations. This goes beyond the Yoshizawa effect, which consists in the occurrence of mean electromotive forces proportional to the mean vorticity or to the angular velocity defining the Coriolis force in a rotating frame and depends on the cross‐helicity defined by the velocity and magnetic field fluctuations. Contributions to the mean electromotive force due to inhomogeneity of the turbulence are also considered. Possible consequences of the above findings for the generation of magnetic fields in cosmic bodies are discussed (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Implications of mean field accretion disc theory for vorticity and magnetic field growth

In addition to the scalar Shakura–Sunyaev α _ss turbulent viscosity transport term used in simple analytic accretion disc modelling, a pseudo-scalar transport term also arises. The essence of this term can be captured even in simple models for which vertical averaging is interpreted as integration over a half-thickness and each hemisphere is separately studied. The additional term highlights a complementarity between mean field magnetic dynamo theory and accretion disc theory treated as a mean field theory. Such pseudo-scalar terms have been studied, and can lead to large-scale magnetic field and vorticity growth. Here it is shown that vorticity can grow even in the simplest azimuthal and half-height integrated disc model, for which mean quantities depend only on radius. The simplest vorticity growth solutions seem to have scales and vortex survival times consistent with those required for facilitating planet formation. In addition, it is shown that, when the magnetic back-reaction is included to lowest order, the pseudo-scalar driving the magnetic field growth and that driving the vorticity growth will behave differently with respect to shearing and non-shearing flows: the former pseudo‐scalar can more easily reverse sign in the two cases.     

Simple laminar dynamos

Although current interest in astrophysical dynamo theory is largely focussed on flows with both large‐ and small‐scale motions, historically the study of dynamos driven by laminar flows has been important. Some classical laminar flow dynamos are reviewed. These results were obtained in an asymptotic regime corresponding to small values of system parameters. Numerical simulations have since been used to extend these results outside of these asymptotic regimes; the asymptotic results remain useful approximations well outside of their formal regions of validity. By changing slightly the system geometries some interesting new results have recently been obtained The latter include the very simple “one‐roll” dynamo, with motions in a single meridional cell contained within a spherical volume of fluid, without differential rotation (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling spatio‐temporal nonlocality in mean‐field dynamos

When scale separation in space or time is poor, the mean‐field α effect and turbulent diffusivity have to be replaced by integral kernels by which the dependence of the mean electromotive force on the mean magnetic field becomes nonlocal. Earlier work in computing these kernels using the test‐field method is now generalized to the case in which both spatial and temporal scale separations are poor. The approximate form of the kernel for isotropic stationary turbulence is such that it can be treated in a straightforward manner by solving a partial differential equation for the mean electromotive force. The resulting mean‐field equations are solved for oscillatory α –shear dynamos as well as α² dynamos with α linearly depending on position, which makes this dynamo oscillatory, too. In both cases, the critical values of the dynamo number is lowered due to spatio‐temporal nonlocality.When scale separation in space or time is poor, the mean‐field α effect and turbulent diffusivity have to be replaced by integral kernels by which the dependence of the mean electromotive force on the mean magnetic field becomes nonlocal. Earlier work in computing these kernels using the test‐field method is now generalized to the case in which both spatial and temporal scale separations are poor. The approximate form of the kernel for isotropic stationary turbulence is such that it can be treated in a straightforward manner by solving a partial differential equation for the mean electromotive force. The resulting mean‐field equations are solved for oscillatory α –shear dynamos as well as α² dynamos     

Diffusion and dispersion in magnetohydrodynamic turbulence: The influence of mean magnetic fields

The properties of magnetohydrodynamic (MHD) turbulence under the influence of a strong mean magnetic field are investigated from the Lagrangian viewpoint by tracking fluid particles in direct numerical simulations. The particle trajectories show characteristic bends near vortex sheets. A strong mean magnetic field leads to preferential diffusion parallel to the mean magnetic field. The two‐particle relative dispersion process shows a dependence on the orientation of the initial separation vector. The relative dispersion is slowed down for initial separation vectors aligned with the mean magnetic field. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscosity alpha in rotating spherical shear flows with an external magnetic field

We study the turbulent behaviour induced by the magnetic shear instability for a magnetized, incompressible fluid in a spherical shell. A differential rotation that is decreasing outwards but hydrodynamically stable according to the Rayleigh criterion is prescribed, and an external, uniform magnetic field is imposed parallel to the rotation axis. Our main concern in this paper is the fully global treatment of this magnetohydrodynamical system, so we focus particular attention on the influence of the boundary conditions. Non-linear, steady solutions are presented for stress-free as well as for rigid boundary conditions for one specific model with a fixed strength of the external magnetic field and a fixed differential rotation rate. We calculate the eddy viscosity ν_T and the viscosity alpha α_SS resulting from the total stress tensor. These turbulence parameters turn out to differ drastically depending on the boundary conditions for the flow. An investigation of the radial structure of the viscosity alpha (whilst varying the differential rotation law) shows that the enhanced generation of turbulence takes place mainly in the boundary layers of the shell.     

The origin of our galactic magnetic field

A serious difficulty with the standard alpha‐omega theory of the origin of galactic magnetic fields involves the question of flux expulsion. This is intimately related to flux freezing. The alpha‐omega theory is shown in the context of the giant superbubble explosions that have a large impact on the physics of the interstellar medium. It is shown that superbubbles alone can duplicate the processes of the alpha‐omega dynamo and produce exponential growth of the galactic magnetic field. The possibility of the blow‐out of pieces of the magnetic field is discussed and it is shown that they have the potential to solve the flux‐expulsion problem. However, such an explanation must lead to apparent ‘gaps’ in the field in the galactic disc. These gaps are probably unavoidable in any dynamo theory and should have important observable consequences, one of which is an explanation for the escape of cosmic rays from the disc (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Turbulent viscosity,magnetic diffusivity,and heat conductivity under the influence of rotation and magnetic field

The old problem of turbulent diffusion is addressed to define the influence of rotation and magnetic field - the usual ingredients of astrophysical bodies - on the effective transport coefficients. Either rotation and magnetism produce the anisotropy and quenching. The tensorial structures of the diffusivities and their dependences on the angular velocity and the field strength are explicitly defined. An example of application of the theory to the global stellar circulation model is given and the implications for cosmic dynamos are briefly discussed.     

Magnetic field evolution in simulations with Euler potentials

Using two- and three-dimensional hydromagnetic simulations for a range of different flows, including laminar and turbulent ones, it is shown that solutions expressing the field in terms of Euler potentials (EP) are in general incorrect if the EP are evolved with an artificial diffusion term. In three dimensions, standard methods using the magnetic vector potential are found to permit dynamo action when the EP give decaying solutions. With an imposed field, the EP method yields excessive power at small scales. This effect is more exaggerated in the dynamic case, suggesting an unrealistically reduced feedback from the Lorentz force. The EP approach agrees with standard methods only at early times when magnetic diffusivity did not have time to act. It is demonstrated that the usage of EP with even a small artificial magnetic diffusivity does not converge to a proper solution of hydromagnetic turbulence. The source of this disagreement is not connected with magnetic helicity or the three-dimensionality of the magnetic field, but is simply due to the fact that the non-linear representation of the magnetic field in terms of EP that depend on the same coordinates is incompatible with the linear diffusion operator in the induction equation.