We compute for the first time magnetic helicity and energy spectra of the solar active region NOAA 11158 during 11-15 February 2011 at 20^o southern heliographic latitude using observational photospheric vector magnetograms. We adopt the isotropic re
presentation of the Fourier-transformed two-point correlation tensor of the magnetic field. The sign of magnetic helicity turns out to be predominantly positive at all wavenumbers. This sign is consistent with what is theoretically expected for the southern hemisphere. The magnetic helicity normalized to its theoretical maximum value, here referred to as relative helicity, is around 4% and strongest at intermediate wavenumbers of k ~ 0.4 Mm^{-1}, corresponding to a scale of 2pi/k ~ 16 Mm. The same sign and a similar value are also found for the relative current helicity evaluated in real space based on the vertical components of magnetic field and current density. The modulus of the magnetic helicity spectrum shows a k^{-11/3} power law at large wavenumbers, which implies a k^{-5/3} spectrum for the modulus of the current helicity. A k^{-5/3} spectrum is also obtained for the magnetic energy. The energy spectra evaluated separately from the horizontal and vertical fields agree for wavenumbers below 3 Mm^{-1}, corresponding to scales above 2 Mm. This gives some justification to our assumption of isotropy and places limits resulting from possible instrumental artefacts at small scales.
Strongly stratified hydromagnetic turbulence has previously been found to produce magnetic flux concentrations if the domain is large enough compared with the size of turbulent eddies. Mean-field simulations (MFS) using parameterizations of the Reyno
lds and Maxwell stresses show a negative effective magnetic pressure instability and have been able to reproduce many aspects of direct numerical simulations (DNS) regarding the growth rate of this large-scale instability, shape of the resulting magnetic structures, and their height as a function of magnetic field strength. Unlike the case of an imposed horizontal field, for a vertical one, magnetic flux concentrations of equipartition strength with the turbulence can be reached. This results in magnetic spots that are reminiscent of sunspots. Here we want to find out under what conditions magnetic flux concentrations with vertical field occur and what their internal structure is. We use a combination of MFS, DNS, and implicit large-eddy simulations to characterize the resulting magnetic flux concentrations in forced isothermal turbulence with an imposed vertical magnetic field. We confirm earlier results that in the kinematic stage of the large-scale instability the horizontal wavelength of structures is about 10 times the density scale height. At later times, even larger structures are being produced in a fashion similar to inverse spectral transfer in helically driven turbulence. Using turbulence simulations, we find that magnetic flux concentrations occur for different values of the Mach number between 0.1 and 0.7. DNS and MFS show magnetic flux tubes with mean-field energies comparable to the turbulent kinetic energy. The resulting vertical magnetic flux tubes are being confined by downflows along the tubes and corresponding inflow from the sides, which keep the field concentrated.
An update is given on the current status of solar and stellar dynamos. At present, it is still unclear why stellar cycle frequencies increase with rotation frequency in such a way that their ratio increases with stellar activity. The small-scale dyna
mo is expected to operate in spite of a small value of the magnetic Prandtl number in stars. Whether or not the global magnetic activity in stars is a shallow or deeply rooted phenomenon is another open question. Progress in demonstrating the presence and importance of magnetic helicity fluxes in dynamos is briefly reviewed, and finally the role of nonlocality is emphasized in modeling stellar dynamos using the mean-field approach. On the other hand, direct numerical simulations have now come to the point where the models show solar-like equatorward migration that can be compared with observations and that need to be understood theoretically.
When scale separation in space and time is poor, the alpha effect and turbulent diffusivity have to be replaced by integral kernels. Earlier work in computing these kernels using the test-field method is now generalized to the case in which both spat
ial and temporal scale separations are poor. The approximate form of the kernel 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 alpha-shear dynamos as well as alpha^2 dynamos in which alpha is antisymmetric about the equator, making this dynamo also oscillatory. In both cases, the critical values of the dynamo number is lowered by the fact that the dynamo is oscillatory.
Some of the contributions of Chandrasekhar to the field of magnetohydrodynamics are highlighted. Particular emphasis is placed on the Chandrasekhar-Kendall functions that allow a decomposition of a vector field into right- and left-handed contributio
ns. Magnetic energy spectra of both contributions are shown for a new set of helically forced simulations at resolutions higher than what has been available so far. For a forcing function with positive helicity, these simulations show a forward cascade of the right-handed contributions to the magnetic field and nonlocal inverse transfer for the left-handed contributions. The speed of inverse transfer is shown to decrease with increasing value of the magnetic Reynolds number.
Numerical aspects of dynamos in periodic domains are discussed. Modifications of the solutions by numerically motivated alterations of the equations are being reviewed using the examples of magnetic hyperdiffusion and artificial diffusion when advanc
ing the magnetic field in its Euler potential representation. The importance of using integral kernel formulations in mean-field dynamo theory is emphasized in cases where the dynamo growth rate becomes comparable with the inverse turnover time. Finally, the significance of microscopic magnetic Prandtl number in controlling the conversion from kinetic to magnetic energy is highlighted.
The decay of kinetic helicity is studied in numerical models of forced turbulence using either an externally imposed forcing function as an inhomogeneous term in the equations or, alternatively, a term linear in the velocity giving rise to a linear i
nstability. The externally imposed forcing function injects energy at the largest scales, giving rise to a turbulent inertial range with nearly constant energy flux while for linearly forced turbulence the spectral energy is maximum near the dissipation wavenumber. Kinetic helicity is injected once a statistically steady state is reached, but it is found to decay on a turbulent time scale regardless of the nature of the forcing and the value of the Reynolds number.
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.
In the mean-field theory of magnetic fields, turbulent transport, i.e. the turbulent electromotive force, is described by a combination of the alpha effect and turbulent magnetic diffusion, which are usually assumed to be proportional respectively to
the mean field and its spatial derivatives. For a passive scalar there is just turbulent diffusion, where the mean flux of concentration depends on the gradient of the mean concentration. However, these proportionalities are approximations that are valid only if the mean field or the mean concentration vary slowly in time. Examples are presented where turbulent transport possesses memory, i.e. where it depends crucially on the past history of the mean field. Such effects are captured by replacing turbulent transport coefficients with time integral kernels, resulting in transport coefficients that depend effectively on the frequency or the growth rate of the mean field itself. In this paper we perform numerical experiments to find the characteristic timescale (or memory length) of this effect as well as simple analytical models of the integral kernels in the case of passive scalar concentrations and kinematic dynamos. The integral kernels can then be used to find self-consistent growth or decay rates of the mean fields. In mean-field dynamos the growth rates and cycle periods based on steady state values of alpha effect and turbulent diffusivity can be quite different from the actual values.
Using direct simulations, weakly nonlinear theory and nonlinear mean-field theory, it is shown that the quenched velocity field of a saturated nonlinear dynamo can itself act as a kinematic dynamo. The flow is driven by a forcing function that would
produce a Roberts flow in the absence of a magnetic field. This result confirms an analogous finding by F. Cattaneo & S. M. Tobias (arXiv:0809.1801) for the more complicated case of turbulent convection, suggesting that this may be a common property of nonlinear dynamos; see also the talk given also online at the Kavli Institute for Theoretical Physics (http://online.kitp.ucsb.edu/online/dynamo_c08/cattaneo). It is argued that this property can be used to test nonlinear mean-field dynamo theories.
