Local and Nonlocal Magnetic Diffusion and Alpha-Effect Tensors in Shear Flow Turbulence

Abstract

Various approaches to estimate turbulent transport coefficients from numerical simulations of hydromagnetic turbulence are discussed. A quantitative comparison between the averaged magnetic field obtained from a specific three-dimensional simulation of a rotating turbulent shear flow in a slab and a simple one-dimensional alpha-omega dynamo model is given. A direct determination of transport coefficients is attempted by calculating the correlation matrix of different components of the field and its derivatives. This matrix relates the electromotive force to physically relevant parameters like the tensor components of the -effect and the turbulent diffusivity. The -effect operating on the toroidal field is found to be negative and of similar magnitude as the value obtained in previous work by correlating the electromotive force with the mean magnetic field. The turbulent diffusion of the toroidal field is comparable to the kinematic viscosity that was determined earlier by comparing the stress with the shear. However, the turbulent diffusion of the radial field component is smaller and can even be formally negative. The method is then modified to obtain the spectral dependence of the turbulent transport coefficients on the wavenumber. There is evidence for nonlocal behaviour in that most of the response comes from the smallest wavenumbers corresponding to the largest scale possible in the simulation. Again, the turbulent diffusion coefficient for the radial field component is small, or even negative, which is considered unphysical. However, when the diffusion tensor is assumed to be diagonal the radial component of the diffusion tensor is positive, supporting thus the relevance of a nonlocal approach. Finally, model calculations are presented using nonlocal prescriptions of the -effect and the turbulent diffusion. We emphasize that in all cases the electromotive force exhibits a strong stochastic component which make the -effect and the turbulent diffusion intrinsically noisy.

Sunspot cycles and Grand Minima

“…The analysis did not include variations of short time and latitudinal extent on the scales of turbulent vortices, which are obviously important for the short-term dynamics of solar activity. The presence of long-term variations in the alpha-coefficient has been reported from analysis of direct numerical simulations by Brandenburg & Sokoloff (2002) and Otmianowska-Mazur et al (2006).…”

Turbulence and its parameterization in accretion discs

“…This fact was also used in Brandenburg & Sokoloff (2002). The diagonal components ofη ij correspond to turbulent resistivity, while its off-diagonal components can be responsible for driving dynamo action [Ω×J and W ×J effects; see, Rädler (1969) and Rogachevskii & Kleeorin (2003, Rädler & Stepanov (2005)].…”

“…and the two components,η xx andη yy , are almost equally big. [We recall that in Brandenburg & Sokoloff (2002) it was found thatη yy (responsible for diffusion of B x ) was very small, but this result was perhaps not accurate. ]…”

“…A method that has been used for accretion disc turbulence is based on a fitting procedure of the measured mean field and the mean electromotive force to Eq. (22) (Brandenburg & Sokoloff 2002;Kowal et al 2005). A general procedure for determining the full α ij and η ijk tensors from a simulation is to calculate the electromotive force after applying test fields of different directions and with different gradients (Schrinner et al 2005).…”

Advances in Theory and Simulations of Large-Scale Dynamos

“…In that case one can not longer write the electromotive force in terms of products of α and the mean field or η t and the mean current density, but one has to write them as convolutions with corresponding integral kernels (e.g. Brandenburg & Sokoloff 2002). In Fourier space, a convolution corresponds to a multiplication.…”