Recent progress in astrophysical hydromagnetic turbulence is being reviewed. The physical ideas behind the now widely accepted Goldreich–Sridhar model and its extension to compressible magnetohydrodynamic turbulence are introduced. Implications for cosmic ray diffusion and acceleration is being discussed. Dynamo-generated magnetic fields with and without helicity are contrasted against each other. Certain turbulent transport processes are being modified and often suppressed by anisotropy and inhomogeneities of the turbulence, while others are being produced by such properties, which can lead to new large-scale instabilities of the turbulent medium. Applications of various such processes to astrophysical systems are being considered.

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Astrophysical Hydromagnetic Turbulence

Several recent advances in turbulent dynamo theory are reviewed. High resolution simulations of small-scale and large-scale dynamo action in periodic domains are compared with each other and contrasted with similar results at low magnetic Prandtl numbers. It is argued that all the different cases show similarities at intermediate length scales. On the other hand, in the presence of helicity of the turbulence, power develops on large scales, which is not present in non-helical small-scale turbulent dynamos. At small length scales, differences occur in connection with the dissipation cutoff scales associated with the respective value of the magnetic Prandtl number. These differences are found to be independent of whether or not there is large-scale dynamo action. However, large-scale dynamos in homogeneous systems are shown to suffer from resistive slow-down even at intermediate length scales. The results from simulations are connected to mean field theory and its applications. Recent work on magnetic helicity fluxes to alleviate large-scale dynamo quenching, shear dynamos, nonlocal effects and magnetic structures from strong density stratification are highlighted. Several insights which arise from analytic considerations of small-scale dynamos are discussed.

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Current Status of Turbulent Dynamo Theory. From Large-Scale to Small-Scale Dynamos

Aims. We study turbulent transport coefficients that describe the evolution of large-scale magnetic fields in turbulent convection. Methods. We use the test field method, together with three-dimensional numerical simulations of turbulent convection with shear and rotation, to compute turbulent transport coefficients describing the evolution of large-scale magnetic fields in mean-field theory in the kinematic regime. We employ one-dimensional mean-field models with the derived turbulent transport coefficients to examine whether they give results that are compatible with direct simulations. Results. The results for the α-effect as a function of rotation rate are consistent with earlier numerical studies, i.e. increasing magnitude as rotation increases and approximately cos θ latitude profile for moderate rotation. Turbulent diffusivity, ηt, is proportional to the square of the turbulent vertical velocity in all cases. Whereas ηt decreases approximately inversely proportional to the wavenumber of the field, the α-effect and turbulent pumping show a more complex behaviour with partial or full sign changes and the magnitude staying roughly constant. In the presence of shear and no rotation, a weak α-effect is induced which does not seem to show any consistent trend as a function of shear rate. Provided that the shear is large enough, this small α-effect is able to excite a dynamo in the mean-field model. The coefficient responsible for driving the shear-current effect shows several sign changes as a function of depth but is also able to contribute to dynamo action in the mean-field model. The growth rates in these cases are, however, well below those in direct simulations, suggesting that an incoherent α-shear dynamo may also act in the simulations. If both rotation and shear are present, the α-effect is more pronounced. At the same time, the combination of the shear-current and Ω × J-effects is also stronger than in the case of shear alone, but subdominant to the α-shear dynamo. The results of direct simulations are consistent with mean-field models where all of these effects are taken into account without the need to invoke incoherent effects.

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Alpha effect and turbulent diffusion from convection

The turbulent magnetic diffusivity tensor is determined in the presence of rotation or shear. The question is addressed whether dynamo action from the shear-current effect can explain large-scale magnetic field generation found in simulations with shear. For this purpose a set of evolution equations for the response to imposed test fields is solved with turbulent and mean motions calculated from the momentum and continuity equations. The corresponding results for the electromotive force are used to calculate turbulent transport coefficients. The diagonal components of the turbulent magnetic diffusivity tensor are found to be very close together, but their values increase slightly with increasing shear and decrease with increasing rotation rate. In the presence of shear, the sign of the two off-diagonal components of the turbulent magnetic diffusion tensor is the same and opposite to the sign of the shear. This implies that dynamo action from the shear--current effect is impossible, except perhaps for high magnetic Reynolds numbers. However, even though there is no alpha effect on the average, the components of the alpha tensor display Gaussian fluctuations around zero. These fluctuations are strong enough to drive an incoherent alpha--shear dynamo. The incoherent shear--current effect, on the other hand, is found to be subdominant.

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Magnetic diffusivity tensor and dynamo effects in rotating and shearing turbulence

The role of turbulence in various astrophysical settings is reviewed. Among the differences to laboratory and atmospheric turbulence we highlight the ubiquitous presence of magnetic fields that are generally produced and maintained by dynamo action. The extreme temperature and density contrasts and stratifications are emphasized in connection with turbulence in the interstellar medium and in stars with outer convection zones, respectively. In many cases turbulence plays an essential role in facilitating enhanced transport of mass, momentum, energy and magnetic fields in terms of the corresponding coarse-grained mean fields. Those transport properties are usually strongly modified by anisotropies and often completely new effects emerge in such a description that have no correspondence in terms of the original (non-coarse-grained) fields.

https://iopscience.iop.org/article/10.1088/0034-4885/74/4/046901/pdf

Astrophysical turbulence modeling

Aims. We determine the alpha effect and turbulent magnetic diffusivity for mean magnetic fields with profiles of different length scales from simulations of isotropic turbulence. We then relate these results to nonlocal formulations in which alpha and the turbulent magnetic diffusivity correspond to integral kernels. Methods. We solve evolution equations for magnetic fields that give the response to imposed test fields. These test fields correspond to mean fields with various wavenumbers. Both an imposed fully helical steady flow consisting of a pattern of screw-like motions (Roberts flow) and time-dependent, statistically steady isotropic turbulence are considered. In the latter case the evolution equations are solved simultaneously with the momentum and continuity equations. The corresponding results for the electromotive force are used to calculate alpha and magnetic diffusivity tensors. Results. For both, the Roberts flow under the second-order correlation approximation and the isotropic turbulence alpha and turbulent magnetic diffusivity are greatest on large scales and these values diminish toward smaller scales. In both cases, the alpha effect and turbulent diffusion kernels are approximated by exponentials, corresponding to Lorentzian profiles in Fourier space. For isotropic turbulence, the turbulent diffusion kernel is half as wide as the alpha effect kernel. For the Roberts flow beyond the second-order correlation approximation, the turbulent diffusion kernel becomes negative on large scales.

/pdf/scale-dependence-of-alpha-effect-and-turbulent-diffusivity-19wzpxyjc8.pdf

Scale dependence of alpha effect and turbulent diffusivity

Aims. We present an elegant method of determining the eigensolutions of the induction and dynamo equations in a fluid embedded in a vacuum. Methods. The magnetic field is expanded in a complete set of functions. The new method is based on the biorthogonality of the adjoint electric current and the vector potential with an inner product defined by a volume integral over the fluid domain. The advantage of this method is that the velocity and the dynamo coefficients of the induction and the dynamo equation do not have to be differentiated and thus even numerically determined tabulated values of the coefficients produce reasonable results. Results. We provide test calculations and compare with published results obtained by the classical treatment based on the biorthogonality of the magnetic field and its adjoint. We especially consider dynamos with mean-field coefficients determined from direct numerical simulations of the geodynamo and compare with initial value calculations and the full MHD simulations.

/pdf/an-efficient-method-for-computing-the-eigenfunctions-of-the-4t6t1ybds7.pdf

An efficient method for computing the eigenfunctions of the dynamo equation

Aims: To determine alpha effect and turbulent magnetic diffusivity for mean magnetic fields with profiles of different length scale from simulations of isotropic turbulence, and to relate these results to nonlocal formulations in which alpha and the turbulent magnetic diffusivity correspond to integral kernels. Methods: A set of evolution equations for magnetic fields is solved which gives the response to imposed test fields, that is, mean magnetic fields with various wavenumbers. Both an imposed fully helical steady flow consisting of a pattern of screw-like motions (Roberts flow) and time-dependent statistically steady isotropic turbulence are considered. In the latter case the aforementioned evolution equations are solved simultaneously with the momentum and continuity equations. The corresponding results for the electromotive force are used to calculate alpha and magnetic diffusivity tensors. Results: For both the Roberts flow under the second--order correlation approximation and isotropic turbulence alpha and turbulent magnetic diffusivity are largest at large scales and these values diminish toward smaller scales. In both cases the alpha effect and turbulent diffusion kernels are well approximated by exponentials, corresponding to Lorentzian profiles in Fourier space. For isotropic turbulence the turbulent diffusion kernel is half as wide as the alpha effect kernel. For the Roberts flow beyond the second--order correlation approximation the turbulent diffusion kernel becomes negative at large scales.

/pdf/scale-dependence-of-alpha-effect-and-turbulent-diffusivity-50zcydzhcx.pdf

We present an elegant method of determining the eigensolutions of the induction and the dynamo equation in a fluid embedded in a vacuum. The magnetic field is expanded in a complete set of functions. The new method is based on the biorthogonality of the adjoint electric current and the vector potential with an inner product defined by a volume integral over the fluid domain. The advantage of this method is that the velocity and the dynamo coefficients of the induction and the dynamo equation do not have to be differentiated and thus even numerically determined tabulated values of the coefficients produce reasonable results. We provide test calculations and compare with published results obtained by the classical treatment based on the biorthogonality of the magnetic field and its adjoint. We especially consider dynamos with mean-field coefficients determined from direct numerical simulations of the geodynamo and compare with initial value calculations and the full MHD simulations.

/pdf/an-efficient-method-for-computing-the-eigenfunctions-of-the-52hej7rlwn.pdf

M. Schrinner

Papers

Scale dependence of alpha effect and turbulent diffusivity

An efficient method for computing the eigenfunctions of the dynamo equation

Scale dependence of alpha effect and turbulent diffusivity

An efficient method for computing the eigenfunctions of the dynamo equation