Second-order Correlation Approximation Research Articles

Interaction of Large- and Small-scale Dynamos in Isotropic Turbulent Flows from GPU-accelerated Simulations

Abstract Magnetohydrodynamical (MHD) dynamos emerge in many different astrophysical situations where turbulence is present, but the interaction between large-scale dynamos (LSDs) and small-scale dynamos (SSDs) is not fully understood. We performed a systematic study of turbulent dynamos driven by isotropic forcing in isothermal MHD with magnetic Prandtl number of unity, focusing on the exponential growth stage. Both helical and nonhelical forcing was employed to separate the effects of LSD and SSD in a periodic domain. Reynolds numbers ( ) up to ≈250 were examined and multiple resolutions used for convergence checks. We ran our simulations with the Astaroth code, designed to accelerate 3D stencil computations on graphics processing units (GPUs) and to employ multiple GPUs with peer-to-peer communication. We observed a speedup of ≈35 in single-node performance compared to the widely used multi-CPU MHD solver Pencil Code. We estimated the growth rates from both the averaged magnetic fields and their power spectra. At low LSD growth dominates, but at high SSD appears to dominate in both helically and nonhelically forced cases. Pure SSD growth rates follow a logarithmic scaling as a function of . Probability density functions of the magnetic field from the growth stage exhibit SSD behavior in helically forced cases even at intermediate . We estimated mean field turbulence transport coefficients using closures like the second-order correlation approximation (SOCA). They yield growth rates similar to the directly measured ones and provide evidence of α quenching. Our results are consistent with the SSD inhibiting the growth of the LSD at moderate , while the dynamo growth is enhanced at higher .

Electromotive force due to magnetohydrodynamic fluctuations in sheared rotating turbulence.

This article presents a calculation of the mean electromotive force arising from general small-scale magnetohydrodynamical turbulence, within the framework of the second-order correlation approximation. With the goal of improving understanding of the accretion disk dynamo, effects arising through small-scale magnetic fluctuations, velocity gradients, density and turbulence stratification, and rotation, are included. The primary result, which supplements numerical findings, is that an off-diagonal turbulent resistivity due to magnetic fluctuations can produce large-scale dynamo action-the magnetic analog of the "shear-current" effect. In addition, consideration of α effects in the stratified regions of disks gives the puzzling result that there is no strong prediction for a sign of α, since the effects due to kinetic and magnetic fluctuations, as well as those due to shear and rotation, are each of opposing signs and tend to cancel each other.

Homogeneous planar and two-dimensional mean-field antidynamo theorems with zero mean flow

In an electrically conducting fluid, two types of turbulence with a preferred direction are distinguished: planar turbulence, in which every velocity in the turbulent ensemble of flows has no component in the given direction; and two-dimensional turbulence, in which every velocity in the turbulent ensemble is invariant under translation in the preferred direction. Under the additional assumptions of two-scale and homogeneous turbulence with zero mean flow, the associated magnetohydrodynamic alpha- and beta-effects are derived in the second-order correlation approximation (SOCA) when the electrically conducting fluid occupies all space. Limitations of the SOCA are well known, but alpha- and beta-effects of a turbulent flow are useful in interpreting the dynamo effects of the turbulence. Two antidynamo theorems, which establish necessary conditions for dynamo action, are shown to follow from the special structures of these alpha- and beta-effects. The theorems, which are analogues of the laminar planar velocity and two-dimensional antidynamo theorems, apply to all turbulent ensembles with the prescribed alpha- and beta-effects, not just the planar and two-dimensional ensembles. The mean magnetic field is general in the planar theorem but only two-dimensional in the two-dimensional theorem. The two theorems relax the previous restriction to turbulence which is both two-dimensional and planar. The laminar theorems imply decay of the total magnetic field for any velocity of the associated turbulent ensemble. However, the mean-field theorems are not fully consistent with the laminar theorems because further conditions beyond those arising from the turbulence must be imposed on the beta-effect to establish decay of the mean magnetic field. In particular, negative turbulent magnetic diffusivities must be restricted. It is interesting that there is no inconsistency in the alpha-effects. The failure of the SOCA with the two-scale approximation to simply preserve the laminar antidynamo theorems at the beta-effect level is a further demonstration of the restricted validity of the theory and shows that negative diffusivity effects derived by approximation methods must be treated cautiously.

Mean-field transport in stratified and/or rotating turbulence (Corrigendum)

Fig. 1. The coefficients α⊥, β⊥, β‖, and μ, as well as β⊥ and β C ‖ for the Roberts flow, calculated in the second-order correlation approximation, as functions of k⊥/kf , where kf = √ 2k0 is the effective wavenumber of the flow. Results obtained with kx = ky and k‖/k⊥ = 1/ √ 2 ≈ 0.7 or k‖/k⊥ = 1/16 √ 2 ≈ 0.004 are represented by open squares and dotted lines or by open diamonds and dashed lines, respectively. Results with kx/ky = 0.75 [k⊥ = (3, 4, 0)k1] or kx/ky = 5 [k⊥ = (5, 1, 0)k1] and k‖/k⊥ = 0.2 are indicated by open or filled circles, respectively. Orange and black symbols correspond to the first and second expressions for β⊥ and μ in (21) or for β⊥ in (26).

Mean-field diffusivities in passive scalar and magnetic transport in irrotational flows

Certain aspects of the mean-field theory of turbulent passive scalar transport and of mean-field electrodynamics are considered with particular emphasis on aspects of compressible fluids. It is demonstrated that the total mean-field diffusivity for passive scalar transport in a compressible flow may well be smaller than the molecular diffusivity. This is in full analogy to an old finding regarding the magnetic mean-field diffusivity in an electrically conducting turbulently moving compressible fluid. These phenomena occur if the irrotational part of the motion dominates the vortical part, the Péclet or magnetic Reynolds number is not too large, and, in addition, the variation of the flow pattern is slow. For both the passive scalar and the magnetic cases several further analytical results on mean-field diffusivities and related quantities found within the second-order correlation approximation are presented, as well as numerical results obtained by the test-field method, which applies independently of this approximation. Particular attention is paid to nonlocal and noninstantaneous connections between the turbulence-caused terms and the mean fields. Two examples of irrotational flows, in which interesting phenomena in the above sense occur, are investigated in detail. In particular, it is demonstrated that the decay of a mean scalar in a compressible fluid under the influence of these flows can be much slower than without any flow, and can be strongly influenced by the so-called memory effect, that is, the fact that the relevant mean-field coefficients depend on the decay rates themselves.

Mean-field effects in the Galloway-Proctor flow

The coefficients defining the mean electromotive force in a Galloway-Proctor flow are determined. This flow shows a two-dimensional pattern and is helical. The pattern wobbles in its plane. Apart from one exception a circular motion of the flow pattern is assumed. This corresponds to one of the cases considered recently by Courvoisier, Hughes and Tobias (2006, Phys. Rev. Lett., 96, 034503). An analytic theory of the alpha effect and related effects in this flow is developed within the second-order correlation approximation and a corresponding fourth-order approximation. In the validity range of these approximations there is an alpha effect but no gamma effect, or pumping effect. Numerical results obtained with the test-field method, which are independent of these approximations, confirm the results for alpha and show that gamma is in general nonzero. Both alpha and gamma show a complex dependency on the magnetic Reynolds number and other parameters that define the flow, that is, amplitude and frequency of the wobbling motion. Some results for the magnetic diffusivity eta_t and a related quantity are given, too. Finally a result for alpha in the case of a randomly varying flow without the aforementioned circular motion is presented. This flow may be a more appropriate model for studying the alpha effect and related effects in flows that are statistical isotropic in a plane.

Mean-field dynamo in a turbulence with shear and kinetic helicity fluctuations

We study the effects of kinetic helicity fluctuations in a turbulence with large-scale shear using two different approaches: the spectral tau approximation and the second-order correlation approximation (or first-order smoothing approximation). These two approaches demonstrate that homogeneous kinetic helicity fluctuations alone with zero mean value in a sheared homogeneous turbulence cannot cause a large-scale dynamo. A mean-field dynamo is possible when the kinetic helicity fluctuations are inhomogeneous, which causes a nonzero mean alpha effect in a sheared turbulence. On the other hand, the shear-current effect can generate a large-scale magnetic field even in a homogeneous nonhelical turbulence with large-scale shear. This effect was investigated previously for large hydrodynamic and magnetic Reynolds numbers. In this study we examine the threshold required for the shear-current dynamo versus Reynolds number. We demonstrate that there is no need for a developed inertial range in order to maintain the shear-current dynamo (e.g., the threshold in the Reynolds number is of the order of 1).

α-effect dynamos with zero kinetic helicity

A simple explicit example of a Roberts-type dynamo is given in which the alpha effect of mean-field electrodynamics exists in spite of pointwise vanishing kinetic helicity of the fluid flow. In this way, it is shown that alpha-effect dynamos do not necessarily require nonzero kinetic helicity. A mean-field theory of Roberts-type dynamos is established within the framework of the second-order correlation approximation. In addition, numerical solutions of the original dynamo equations are given that are independent of any approximation of that kind. Both theory and numerical results demonstrate the possibility of dynamo action in the absence of kinetic helicity.

Closure tests for mean field magnetohydrodynamics using a self-consistent reduced model

The mean electromotive force and alpha effect are computed for a forced turbulent flow using a simple nonlinear dynamical model. The results are used to check the applicability of two basic analytic ansatze of mean-field magnetohydrodynamics - the second order correlation approximation (SOCA) and the tau approximation. In the numerical simulations the effective Reynolds number Re is 2-20, while the magnetic Prandtl number varies from 0.1 to $10^{7}$. We present evidence that the $\tau$ approximation may be appropriate in dynamical regimes where there is a small-scale dynamo. Catastrophic quenching of the $\alpha$ effect is found for high $P_{m}$. Our results indicate that for high $P_{m}$ SOCA gives a very large value of the $\alpha$ coefficient compared with the ``exact'' solution. The discrepancy depends on the properties of the random force that drives the flow, with a larger difference occuring for $\delta$-correlated force compared with that for a steady random force.

Mean-field electrodynamics: critical analysis of various analytical approaches to the mean electromotive force

There are various analytical approaches to the mean electromotive force crucial in mean-field electrodynamics, with u and b being velocity and magnetic field fluctuations. In most cases the traditional approach, restricted to the second-order correlation approximation, has been used. Its validity is only guaranteed for a range of conditions, which is narrow in view of many applications, e.g., in astrophysics. With the intention to have a wider range of applicability, other approaches have been proposed which make use of the so-called τ-approximation, reducing correlations of third order in u and b to such of second order. After explaining some basic features of the traditional approach a critical analysis of the approaches of that kind is given. It is shown that they lead in some cases to results which are in clear conflict with those of the traditional approach. It is argued that this indicates shortcomings of the τ-approaches and poses serious restrictions to their applicability. These shortcomings do not result from the basic assumption of the τ-approximation. Instead, they seem to originate in some simplifications made in order to derive without really solving the equations governing u and b. A starting point for a new approach is described which avoids the conflict.

Mean-field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo

Mean-field theory describes magnetohydrodynamic processes leading to large-scale magnetic fields in various cosmic objects. In this study magnetoconvection and dynamo processes in a rotating spherical shell are considered. Mean fields are defined by azimuthal averaging. In the framework of mean-field theory, the coefficients which determine the traditional representation of the mean electromotive force, including derivatives of the mean magnetic field up to the first order, are crucial for analyzing and simulating dynamo action. Two methods are developed to extract mean-field coefficients from direct numerical simulations of the mentioned processes. While the first method does not use intrinsic approximations, the second one is based on the second-order correlation approximation. There is satisfying agreement of the results of both methods for sufficiently slow fluid motions. Both methods are applied to simulations of rotating magnetoconvection and a quasi-stationary geodynamo. The mean-field induction effects described by these coefficients, e.g., the α-effect, are highly anisotropic in both examples. An α2-mechanism is suggested along with a strong γ-effect operating outside the inner core tangent cylinder. The turbulent diffusivity exceeds the molecular one by at least one order of magnitude in the geodynamo example. With the aim to compare mean-field simulations with corresponding direct numerical simulations, a two-dimensional mean-field model involving all previously determined mean-field coefficients was constructed. Various tests with different sets of mean-field coefficients reveal their action and significance. In the magnetoconvection and geodynamo examples considered here, the match between direct numerical simulations and mean-field simulations is only satisfying if a large number of mean-field coefficients are involved. In the magnetoconvection example, the azimuthally averaged magnetic field resulting from the numerical simulation is in good agreement with its counterpart in the mean-field model. However, this match is not completely satisfactory in the geodynamo case anymore. Here the traditional representation of the mean electromotive force ignoring higher than first-order spatial derivatives of the mean magnetic field is no longer a good approximation.

Mean electromotive force due to turbulence of a conducting fluid in the presence of mean flow

The mean electromotive force caused by turbulence of an electrically conducting fluid, which plays a central part in mean-field electrodynamics, is calculated for a rotating fluid. Going beyond most of the investigations on this topic, an additional mean motion in the rotating frame is taken into account. One motivation for our investigation originates from a planned laboratory experiment with a Ponomarenko-type dynamo. In view of this application the second-order correlation approximation is used. The investigation is of high interest in astrophysical context, too. Some contributions to the mean electromotive are revealed which have not been considered so far, in particular contributions to the effect and related effects due to the gradient of the mean velocity. Their relevance for dynamo processes is discussed. In a forthcoming paper the results reported here will be specified to the situation in the laboratory and partially compared with experimental findings.

Non-linear magnetic diffusivity in mean-field electrodynamics

ABSTRA C T We consider non-linear transport and drift processes caused by an inhomogeneous magnetic field in a turbulent fluid. The coefficients of magnetic diffusivity and drift velocity are calculated by making use of the second-order correlation approximation. Transport processes in the presence of a sufficiently strong magnetic field become anisotropic with larger diffusion rate and turbulent electrical resistivity across the field than along the field. Non-linear effects also lead to a drift of the magnetic field away from the regions with a higher magnetic energy.

Turbulent dynamo action in a shear flow

We consider the mean electromotive force and a dynamo-generated magnetic field, taking into account the stretching of turbulent magnetic field lines by a shear flow. Calculations are performed by making use of the second-order correlation approximation. In the presence of shear, the mirror symmetry of turbulence can be broken; thus turbulent motions become suitable for the generation of a large-scale magnetic field. Regardless of the shear law, turbulence can lead to a rapid amplification of the mean magnetic field. The growth rate of the mean magnetic field depends on the length-scale: it is faster for the fields with smaller length-scale. The mechanism considered is qualitatively different from the alpha dynamo, and can generate only a magnetic field that is inhomogeneous in the direction of flow. In contrast to the alpha dynamo, this mechanism also allows the generation of two-dimensional fields. The suggested mechanism may play an important role in the generation of magnetic fields in accretion discs, galaxies and jets.