Transport coefficients for the shear dynamo problem at small Reynolds numbers

We build on the formulation developed in Sridhar & Singh (JFM, 664, 265, 2010), and present a theory of the \emph{shear dynamo problem} for small magnetic and fluid Reynolds numbers, but for arbitrary values of the shear parameter. Specializing to the case of a mean magnetic field that is slowly varying in time, explicit expressions for the transport coefficients, $\alpha_{il}$ and $\eta_{iml}$, are derived. We prove that, when the velocity field is non helical, the transport coefficient $\alpha_{il}$ vanishes. We then consider forced, stochastic dynamics for the incompressible velocity field at low Reynolds number. An exact, explicit solution for the velocity field is derived, and the velocity spectrum tensor is calculated in terms of the Galilean--invariant forcing statistics. We consider forcing statistics that is non helical, isotropic and delta-correlated-in-time, and specialize to the case when the mean-field is a function only of the spatial coordinate $X_3$ and time $\tau\,$; this reduction is necessary for comparison with the numerical experiments of Brandenburg, R{\"a}dler, Rheinhardt & K\"apyl\"a (ApJ, 676, 740, 2008). Explicit expressions are derived for all four components of the magnetic diffusivity tensor, $\eta_{ij}(\tau)\,$. These are used to prove that the shear-current effect cannot be responsible for dynamo action at small \re and \rem, but for all values of the shear parameter.Comment: 27 pages, 5 figures, Published in Physical Review

Turbulent transport and dynamo in sheared MHD turbulence with a non-uniform magnetic field

We investigate three-dimensional magnetohydrodynamics turbulence in the presence of velocity and magnetic shear (i.e., with both a large-scale shear flow and a nonuniform magnetic field). By assuming a turbulence driven by an external forcing with both helical and nonhelical spectra, we investigate the combined effect of these two shears on turbulence intensity and turbulent transport represented by turbulent diffusivities (turbulent viscosity, α and β effect) in Reynolds-averaged equations. We show that turbulent transport (turbulent viscosity and diffusivity) is quenched by a strong flow shear and a strong magnetic field. For a weak flow shear, we further show that the magnetic shear increases the turbulence intensity while decreasing the turbulent transport. In the presence of a strong flow shear, the effect of the magnetic shear is found to oppose the effect of flow shear (which reduces turbulence due to shear stabilization) by enhancing turbulence and transport, thereby weakening the strong quenching by flow shear stabilization. In the case of a strong magnetic field (compared to flow shear), magnetic shear increases turbulence intensity and quenches turbulent transport

On the effects of turbulence on a screw dynamo

In an experiment in the Institute of Continuous Media Mechanics in Perm (Russia) an non--stationary screw dynamo is intended to be realized with a helical flow of liquid sodium in a torus. The flow is necessarily turbulent, that is, may be considered as a mean flow and a superimposed turbulence. In this paper the induction processes of the turbulence are investigated within the framework of mean--field electrodynamics. They imply of course a part which leads to an enhanced dissipation of the mean magnetic field. As a consequence of the helical mean flow there are also helical structures in the turbulence. They lead to some kind of $\alpha$--effect, which might basically support the screw dynamo. The peculiarity of this $\alpha$--effect explains measurements made at a smaller version of the device envisaged for the dynamo experiment. The helical structures of the turbulence lead also to other effects, which in combination with a rotational shear are potentially capable of dynamo action. A part of them can basically support the screw dynamo. Under the conditions of the experiment all induction effects of the turbulence prove to be rather weak in comparison to that of the main flow. Numerical solutions of the mean--field induction equation show that all the induction effects of the turbulence together let the screw dynamo threshold slightly, at most by one per cent, rise. The numerical results give also some insights into the action of the individual induction effects of the turbulence.Comment: 15 pages, 7 figures, in GAFD prin

The 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-like 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 $\alpha$--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.Comment: 16 pages, 2 figures, in PRE pres

Generation of coherent magnetic fields in sheared inhomogeneous turbulence: No need for rotation?

Coherent magnetic fields are often believed to be generated by the combination of stretching by differential rotation and turbulent amplification of magnetic field, via the so-called alpha effect. The latter is known to exist in helical turbulence, which is envisioned to arise due to both rotation and convection in solar-type stars. In this contribution, a turbulent flow driven by a nonhelical inhomogeneous forcing and its kinematic dynamo action are studied for a uniform magnetic field in the background of a linear shear flow. By using a quasilinear analysis and a nonperturbative method utilizing a time-dependent wave number, turbulence property and electromotive force are computed for arbitrary shear strength. Due to the large-scale shear flow, the turbulence is highly anisotropic, as a consequence, so is the electromotive force. The latter is found to exist even without rotation due to the combined effect of shear flow and inhomogeneous forcing, containing not only the alpha effect but also magnetic pumping (the gamma effect representing a transport of magnetic flux by turbulence). Specifically, without shear, only the magnetic pumping exists, aligned with the direction of inhomogeneity. For a weak but nonzero shear, the combined effects of shear and inhomogeneous forcing modify the structure of the magnetic pumping when the inhomogeneity is in the plane of the shear flow, the magnetic pumping becoming bidimensional in that plane. It also induces an alpha tensor which has nondiagonal components. When the inhomogeneity is perpendicular to the plane of the shear flow, the alpha effect has three nonzero diagonal components and one off-diagonal component. However, for a sufficiently strong shear, the gamma and alpha effects are suppressed due to shear stabilization which damps turbulence. A simplified dynamo model is then proposed where a large-scale dynamo arises due to the combined effect of shear flow and inhomogeneous forcing. In particular, the growth of a large-scale axisymmetric magnetic field is demonstrated in case of an inhomogeneity which is perpendicular to the plane of the shear flow. Interesting implications of these results for the structure of magnetic fields in star with slow rotation are discussed. (C) 2011 American Institute of Physics. [doi:10.1063/1.3551700

Nonlinear dynamo action in a precessing cylindrical container

It is numerically demonstrated by means of a magnetohydrodynamics (MHD) code that precession can trigger the dynamo effect in a cylindrical container. This result adds credit to the hypothesis that precession can be strong enough to be one of the sources of the dynamo action in some astrophysical bodies.Comment: 5 pages, 5 figures including subfigure

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 P\`eclet 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 non-local and non-instantaneous 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.Comment: 13 pages, 10 figures, published on PR

Dynamo quenching due to shear flow

We provide a theory of dynamo (α effect) and momentum transport in three-dimensional magnetohydrodynamics. For the first time, we show that the α effect is reduced by the shear even in the absence of magnetic field. The α effect is further suppressed by magnetic fields well below equipartition (with the large-scale flow) with different scalings depending on the relative strength of shear and magnetic field. The turbulent viscosity is also found to be significantly reduced by shear and magnetic fields, with positive value. These results suggest a crucial effect of shear and magnetic field on dynamo quenching and momentum transport reduction, with important implications for laboratory and astrophysical plasmas, in particular, for the dynamics of the Sun

A model of driven and decaying magnetic turbulence in a cylinder

Using mean-field theory, we compute the evolution of the magnetic field in a cylinder with outer perfectly conducting boundaries, an imposed axial magnetic and electric field. The thus injected magnetic helicity in the system can be redistributed by magnetic helicity fluxes down the gradient of the local current helicity of the small-scale magnetic field. A weak reversal of the axial magnetic field is found to be a consequence of the magnetic helicity flux in the system. Such fluxes are known to alleviate so-called catastrophic quenching of the {\alpha}-effect in astrophysical applications. Application to the reversed field pinch in plasma confinement devices is discussed.Comment: 7 pages, 4 figures, submitted to Phys. Rev.