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.
We report on turbulent dynamo simulations in a spherical wedge with an outer coronal layer. We apply a two-layer model where the lower layer represents the convection zone and the upper layer the solar corona. This setup is used to study the coronal
influence on the dynamo action beneath the surface. Increasing the radial coronal extent gradually to three times the solar radius and changing the magnetic Reynolds number, we find that dynamo action benefits from the additional coronal extent in terms of higher magnetic energy in the saturated stage. The flux of magnetic helicity can play an important role in this context.
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.
We show how the 3DVAR data assimilation methodology can be used in the astrophysical context of a two-dimensional convection flow. We study the way this variational approach finds best estimates of the current state of the flow from a weighted averag
e of model states and observations. We use numerical simulations to generate synthetic observations of a vertical two-dimensional slice of the outer part of the solar convection zone for varying noise levels and implement 3DVAR when the covariance matrices are scalar. Our simulation results demonstrate the capability of 3DVAR to produce error estimates of system states between up to tree orders of magnitude below the original noise level present in the observations. This work exemplifies the importance of applying data assimilation techniques in simulations of the stratified convection.
Magnetic helicity fluxes in turbulently driven alpha^2 dynamos are studied to demonstrate their ability to alleviate catastrophic quenching. A one-dimensional mean-field formalism is used to achieve magnetic Reynolds numbers of the order of 10^5. We
study both diffusive magnetic helicity fluxes through the mid-plane as well as those resulting from the recently proposed alternate dynamic quenching formalism. By adding shear we make a parameter scan for the critical values of the shear and forcing parameters for which dynamo action occurs. For this $alphaOmega$ dynamo we find that the preferred mode is antisymmetric about the mid-plane. This is also verified in 3-D direct numerical simulations.
We consider mean-field dynamo models with fluctuating alpha effect, both with and without shear. The alpha effect is chosen to be Gaussian white noise with zero mean and given covariance. We show analytically that the mean magnetic field does not gro
w, but, in an infinitely large domain, the mean-squared magnetic field shows exponential growth of the fastest growing mode at a rate proportional to the shear rate, which agrees with earlier numerical results of Yousef et al (2008) and recent analytical treatment by Heinemann et al (2011) who use a method different from ours. In the absence of shear, an incoherent alpha^2 dynamo may also be possible. We further show by explicit calculation of the growth rate of third and fourth order moments of the magnetic field that the probability density function of the mean magnetic field generated by this dynamo is non-Gaussian.
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.
We present nonlinear mean-field alpha-Omega dynamo simulations in spherical geometry with simplified profiles of kinematic alpha effect and shear. We take magnetic helicity evolution into account by solving a dynamical equation for the magnetic alpha
effect. This gives a consistent description of the quenching mechanism in mean-field dynamo models. The main goal of this work is to explore the effects of this quenching mechanism in solar-like geometry, and in particular to investigate the role of magnetic helicity fluxes, specifically diffusive and Vishniac-Cho (VC) fluxes, at large magnetic Reynolds numbers (Rm). For models with negative radial shear or positive latitudinal shear, the magnetic alpha effect has predominantly negative (positive) sign in the northern (southern) hemisphere. In the absence of fluxes, we find that the magnetic energy follows an Rm^-1 dependence, as found in previous works. This catastrophic quenching is alleviated in models with diffusive magnetic helicity fluxes resulting in magnetic fields comparable to the equipartition value even for Rm=10^7. On the other hand, models with a shear-driven Vishniac-Cho flux show an increase of the amplitude of the magnetic field with respect to models without fluxes, but only for Rm<10^4. This is mainly a consequence of assuming a vacuum outside the Sun which cannot support a significant VC flux across the boundary. However, in contrast with the diffusive flux, the VC flux modifies the distribution of the magnetic field. In addition, if an ill-determined scaling factor in the expression for the VC flux is large enough, subcritical dynamo action is possible that is driven by the action of shear and the divergence of current helicity flux.
