Highwavenumber solar fmode strengthening prior to active region formation 

Axel Brandenburg Submitted: 20171021 07:56
We report a systematic strengthening of the local solar surface or fundamental fmode 12 days prior to the emergence of an active region (AR) in the same (corotating) location. Except for a possibly related increase in the kurtosis of the magnetic field, no indication can be seen in the magnetograms at that time. Our study is motivated by earlier numerical findings of Singh et al. (2014) which showed that, in the presence of a nonuniform magnetic field that is concentrated a few scale heights below the surface, the fmode fans out in the diagnostic kω diagram at high wavenumbers. Here we explore this possibility using data from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory and show for six isolated ARs, 11130, 11158, 11242, 11105, 11072, and 11768, that at large latitudinal wavenumbers (corresponding to horizontal scales of around 3000 km), the fmode displays strengthening about two days prior to AR formation and thus provides a new precursor for AR formation. Furthermore, we study two ARs, 12051 and 11678, apart from a magnetically quiet patch lying next to AR~12529, to demonstrate the challenges in extracting such a precursor signal when a newly forming AR emerges in a patch that lies in close proximity of one or several already existing ARs which are expected to pollute neighboring patches. We then discuss plausible procedures for extracting precursor signals from regions with crowded environments. The idea that the fmode is perturbed days before any visible magnetic activity occurs at the surface can be important in constraining dynamo models aimed at understanding the global magnetic activity of the Sun.
Authors: Singh, N. K., Raichur, H., & Brandenburg, A.
Projects: None

Publication Status: Astrophys. J. 832, 120 (2017)
Last Modified: 20171021 18:40



Stellar mixing length theory with entropy rain 

Axel Brandenburg Submitted: 20161224 07:54
The effects of a nongradient flux term originating from the motion of convective elements with entropy perturbations of either sign are investigated and incorporated into a modified version of stellar mixing length theory (MLT). Such a term, first studied by Deardorff in the meteorological context, might represent the effects of cold intense downdrafts caused by the rapid cooling in the granulation layer at the top of the convection zone of latetype stars. Such intense downdrafts were first seen in the strongly stratified simulations of Stein & Nordlund in the late 1980s. These downdrafts transport heat nonlocally, a phenomenon referred to as entropy rain. Moreover, the Deardorff term can cause upward enthalpy transport even in a weakly Schwarzschildstably stratified layer. In that case, no giant cell convection would be excited. This is interest in view of recent observations, which could be explained if the dominant flow structures were of small scale even at larger depths. To study this possibility, three distinct flow structures are examined: one in which convective structures have similar size and mutual separation at all depths, one in which the separation increases with depth, but their size is still unchanged, and one in which both size and separation increase with depth, which is the standard flow structure. It is concluded that the third possibility with fewer and thicker downdrafts in deeper layers remains most plausible, but it may be unable to explain the suspected absence of largescale flows with speeds and scales expected from MLT.
Authors: Axel Brandenburg
Projects: None

Publication Status: Astrophys. J. 832, 6 (2016)
Last Modified: 20161228 11:41



Bipolar region formation in stratified twolayer turbulence 

Axel Brandenburg Submitted: 20161031 05:48
This work presents an extensive study of the previously discovered formation of bipolar flux concentrations in a twolayer model. We interpret the formation process in terms of negative effective magnetic pressure instability (NEMPI), which is a possible mechanism to explain the origin of sunspots. In our simulations, we use a Cartesian domain of isothermal stratified gas that is divided into two layers. In the lower layer, turbulence is forced with transverse nonhelical random waves, whereas in the upper layer no flow is induced. A weak uniform magnetic field is imposed in the entire domain at all times. In this study we vary the stratification by changing the gravitational acceleration, magnetic Reynolds number, strength of the imposed magnetic field, and size of the domain to investigate their influence on the formation process. Bipolar magnetic structure formation takes place over a large range of parameters. The magnetic structures become more intense for higher stratification until the density contrast becomes around 100 across the turbulent layer. For the Reynolds numbers considered, magnetic flux concentrations are generated at magnetic Prandtl number between 0.1 and 1. The magnetic field in bipolar regions increases with higher imposed field strength until the field becomes comparable to the equipartition field strength of the turbulence. A larger horizontal extent enables the flux concentrations to become stronger and more coherent. The size of the bipolar structures turns out to be independent of the domain size. In the case of bipolar region formation, we find an exponential growth of the largescale magnetic field, which is indicative of a hydromagnetic instability. Additionally, the flux concentrations are correlated with strong largescale downward and converging flows. These findings imply that NEMPI is responsible for magnetic flux concentrations.
Authors: Jörn Warnecke, Illa R. Losada, Axel Brandenburg, Nathan Kleeorin, Igor Rogachevskii
Projects: None

Publication Status: Astron. Astrophys. 589, A125 (2016)
Last Modified: 20161031 14:12



Hydraulic effects in a radiative atmosphere with ionization 

Axel Brandenburg Submitted: 20160519 23:19
In a paper of 1978, Eugene Parker postulated the need for hydraulic downward motion to explain magnetic flux concentrations at the solar surface. A similar process has recently also been seen in simplified (e.g., isothermal) models of flux concentrations from the negative effective magnetic pressure instability. We study the effects of partial ionization near the radiative surface on the formation of such magnetic flux concentrations. We first obtain onedimensional (1D) equilibrium solutions using either a Kramerslike opacity or the H opacity. The resulting atmospheres are then used as initial conditions in twodimensional (2D) models where flows are driven by an imposed gradient force resembling a localized negative pressure in the form of a blob. To isolate the effects of partial ionization and radiation, we ignore turbulence and convection. In 1D models, due to partial ionization, an unstable stratification forms always near the surface. We show that the extrema in the specific entropy profiles correspond to the extrema in degree of ionization. In the 2D models without partial ionization, flux concentrations form close to the height where the blob is placed. In models with partial ionization, such flux concentrations form at the surface much above the blob. This is due to the corresponding unstable layer in specific entropy. With H opacity, flux concentrations are weaker due to the stably stratified deeper parts. We demonstrate that, together with density stratification, the imposed source of negative pressure drives the formation of flux concentrations. We find that the inclusion of partial ionization affects entropy profiles causing the strong flux concentrations to form closer to the surface. We speculate that turbulence is needed to limit the strength of flux concentrations and homogenize the specific entropy to a more nearly marginal stratification.
Authors: Pallavi Bhat, Axel Brandenburg
Projects: None

Publication Status: Astron. Astrophys. 587, A90 (2016)
Last Modified: 20160520 22:55



Magnetic helicity and energy spectra of a solar active region 

Axel Brandenburg Submitted: 20160313 06:49
We adopt an isotropic representation of the Fouriertransformed twopoint correlation tensor of the magnetic field to estimate the magnetic energy and helicity spectra as well as current helicity spectra of two individual active regions (NOAA 11158 and NOAA 11515) and the change of the spectral indices during their development as well as during the solar cycle. The departure of the spectral indices of magnetic energy and current helicity from 5/3 are analyzed, and it is found that it is lower than the spectral index of the magnetic energy spectrum. Furthermore, the fractional magnetic helicity tends to increase when the scale of the energycarrying magnetic structures increases. The magnetic helicity of NOAA 11515 violates the expected hemispheric sign rule, which is interpreted as an effect of enhanced field strengths at scales larger than 3060Mm with opposite signs of helicity. This is consistent with the general cycle dependence, which shows that around the solar maximum the magnetic energy and helicity spectra are steeper, emphasizing the largescale field.
Authors: Zhang, H., Brandenburg, A., & Sokoloff, D. D.
Projects: None

Publication Status: Astrophys. J. 819, 146 (2016)
Last Modified: 20160315 11:25



Intense bipolar structures from stratified helical dynamos 

Axel Brandenburg Submitted: 20160116 06:38
We perform direct numerical simulations of the equations of magnetohydrodynamics with external random forcing and in the presence of gravity. The domain is divided into two parts: a lower layer where the forcing is helical and an upper layer where the helicity of the forcing is zero with a smooth transition in between. At early times, a largescale helical dynamo develops in the bottom layer. At later times the dynamo saturates, but the vertical magnetic field continues to develop and rises to form dynamic bipolar structures at the top, which later disappear and reappear. Some of the structures look similar to δ spots observed in the Sun. This is the first example of magnetic flux concentrations, owing to strong density stratification, from selfconsistent dynamo simulations that generate bipolar, superequipartition strength, magnetic structures whose energy density can exceeds the turbulent kinetic energy by even a factor of ten.
Authors: Dhrubaditya Mitra, A. Brandenburg, N. Kleeorin, I. Rogachevskii
Projects: None

Publication Status: Mon. Not. Roy. Astron. Soc. 445, 761769 (2014)
Last Modified: 20160120 12:26



Magnetic flux concentrations from dynamogenerated fields 

Axel Brandenburg Submitted: 20150727 14:45
The meanfield theory of magnetized stellar convection gives rise to the two possibility of distinct instabilities: the largescale dynamo instability, operating in the bulk of the convection zone, and a negative effective magnetic pressure instability (NEMPI) operating in the strongly stratified surface layers. The latter might be important in connection with magnetic spot formation, but the growth rate of NEMPI is suppressed with increasing rotation rates, although recent direct numerical simulations (DNS) have shown a subsequent increase in the growth rate. We examine quantitatively whether this increase in the growth rate of NEMPI can be explained by an α squared meanfield dynamo, and whether both NEMPI and the dynamo instability can operate at the same time. We use both DNS and meanfield simulations (MFS) to solve the underlying equations numerically either with or without an imposed horizontal field. We use the testfield method to compute relevant dynamo coefficients. DNS show that magnetic flux concentrations are still possible up to rotation rates above which the largescale dynamo effect produces mean magnetic fields. The resulting DNS growth rates are quantitatively well reproduced with MFS. As expected, for weak or vanishing rotation, the growth rate of NEMPI increases with increasing gravity, but there is a correction term for strong gravity and large turbulent magnetic diffusivity. Magnetic flux concentrations are still possible for rotation rates above which dynamo action takes over. For the solar rotation rate, the corresponding turbulent turnover time is about 5 hours, with dynamo action commencing in the layers beneath.
Authors: Sarah Jabbari, Axel Brandenburg, Illa R. Losada, Nathan Kleeorin, Igor Rogachevskii
Projects: EUNIS

Publication Status: Astron. Astrophys. 568, A112 (2014)
Last Modified: 20150729 13:00



Nearpolytropic stellar simulations with a radiative surface 

Axel Brandenburg Submitted: 20141213 13:52
Studies of solar and stellar convection often employ simple polytropic setups using the diffusion approximation instead of solving the proper radiative transfer equation. This allows one to control separately the polytropic index of the hydrostatic reference solution, the temperature contrast between top and bottom, and the Rayleigh and Peclet numbers. We extend such studies by including radiative transfer in the gray approximation using a Kramerslike opacity with freely adjustable coefficients. We study the properties of such models and compare them with results from the diffusion approximation. We use the Pencil Code, which is a highorder finite difference code where radiation is treated using the method of long characteristics. The source function is given by the Planck function. The opacity is written as kappa=kappa_0 rho^a T^b, where b is varied from 3.5 to +5, and kappa_0 is varied by four orders of magnitude. We consider sets of one dimensional models and perform a comparison with the diffusion approximation. Except for the case where b=5, we find onedimensional hydrostatic equilibria with a nearly polytropic stratification and a polytropic index close to n=(3b)/(1+a), covering both convectively stable (n>3/2) and unstable (n<3/2) cases. For b=3 and a=1, the value of n is undefined a priori and the actual value of n depends then on the depth of the domain. For large values of \kappa_0, the thermal adjustment time becomes long, the Peclet and Rayleigh numbers become large, and the temperature contrast increases and is thus no longer an independent input parameter, unless the Stefan Boltzmann constant is considered adjustable. Proper radiative transfer with Kramerslike opacities provides a useful tool for studying stratified layers with a radiative surface in ways that are more physical than what is possible with polytropic models using the diffusion approximation.
Authors: Barekat, A., & Brandenburg, A.
Projects: None

Publication Status: Astron. Astrophys. 571, A68 (2014)
Last Modified: 20141215 14:00



Superflare occurrence and energies on G, K and Mtype dwarfs 

Axel Brandenburg Submitted: 20140925 21:24
Kepler data from G, K and Mtype stars are used to study conditions that lead to superflares with energies above 10^{34} { erg}. From the 117,661 stars included, 380 show superflares with a total of 1690 such events. We study whether parameters, like effective temperature or the rotation rate, have any effect on the superflare occurrence rate or energy. With increasing effective temperature we observe a decrease in the superflare rate, which is analogous to the previous findings of a decrease in dynamo activity with increasing effective temperature. For slowly rotating stars, we find a quadratic increase of the mean occurrence rate with the rotation rate up to a critical point, after which the rate decreases linearly. Motivated by standard dynamo theory, we study the behavior of the relative starspot coverage, approximated as the relative brightness variation. For faster rotating stars, an increased fraction of stars shows higher spot coverage, which leads to higher superflare rates. A turbulent dynamo is used to study the dependence of the Ohmic dissipation as a proxy of the flare energy on the differential rotation or shear rate. The resulting statistics of the dissipation energy as a function of dynamo number is similar to the observed flare statistics as a function of the inverse Rossby number and shows similarly strong fluctuations. This supports the idea that superflares might well be possible for solartype G stars.
Authors: Candelaresi, S., Hillier, A., Maehara, H., Brandenburg, A., & Shibata, K.
Projects: None

Publication Status: Astrophys. J. 792, 67 (2014)
Last Modified: 20140926 10:14



Magnetic Prandtl number dependence of the kinetictomagnetic dissipation ratio 

Axel Brandenburg Submitted: 20140730 00:01
Using direct numerical simulations of threedimensional hydromagnetic turbulence, either with helical or nonhelical forcing, we show that the ratio of kinetictomagnetic energy dissipation always increases with the magnetic Prandtl number, i.e., the ratio of kinematic viscosity to magnetic diffusivity. This dependence can always be approximated by a power law, but the exponent is not the same in all cases. For nonhelical turbulence, the exponent is around 1/3, while for helical turbulence it is between 0.6 and 2/3. In the statistically steady state, the rate of the energy conversion from kinetic into magnetic by the dynamo must be equal to the Joule dissipation rate. We emphasize that for both smallscale and largescale dynamos, the efficiency of energy conversion depends sensitively on the magnetic Prandtl number, and thus on the microphysical dissipation process. To understand this behavior, we also study shell models of turbulence and onedimensional passive and active scalar models. We conclude that the magnetic Prandtl number dependence is qualitatively best reproduced in the onedimensional model as a result of dissipation via localized Alfvén kinks.
Authors: Axel Brandenburg
Projects: None

Publication Status: Astrophys. J. 791, 12 (2014)
Last Modified: 20140730 15:22



Meanfield and direct numerical simulations of magnetic flux concentrations from vertical field 

Axel Brandenburg Submitted: 20140705 20:53
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. Meanfield simulations (MFS) using parameterizations of the Reynolds 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 largescale 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 largeeddy 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 largescale 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 meanfield 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.
Authors: A. Brandenburg, O. Gressel, S. Jabbari, N. Kleeorin, I. Rogachevskii
Projects: None

Publication Status: Astron. Astrophys. 562, A53 (2014)
Last Modified: 20140706 20:34



Meanfield dynamo action from delayed transport 

Axel Brandenburg Submitted: 20140529 01:11
We analyse the nature of dynamo action that enables growing horizontally averaged magnetic fields in two particular flows that were studied by Roberts in 1972, namely his flows II and III. They have zero kinetic helicity either pointwise (flow II), or on average (flow III). Using direct numerical simulations, we determine the onset conditions for dynamo action at moderate values of the magnetic Reynolds number. Using the testfield method, we show that the turbulent magnetic diffusivity is then positive for both flows. However, we demonstrate that for both flows largescale dynamo action occurs through delayed transport. Mathematically speaking, the magnetic field at earlier times contributes to the electromotive force through the offdiagonal components of the α tensor such that a zero mean magnetic field becomes unstable to dynamo action. This represents a qualitatively new meanfield dynamo mechanism not previously described.
Authors: Rheinhardt, M., Devlen, E., R?dler, K.H., & Brandenburg, A.
Projects: None

Publication Status: Mon. Not. Roy. Astron. Soc. 441, 116126 (2014)
Last Modified: 20140529 11:24



Magnetic flux concentrations in a polytropic atmosphere 

Axel Brandenburg Submitted: 20140503 00:28
Strongly stratified hydromagnetic turbulence has recently been identified as a candidate for explaining the spontaneous formation of magnetic flux concentrations by the negative effective magnetic pressure instability (NEMPI). Much of this work has been done for isothermal layers for which the density scale height is constant throughout. We now study the validity of earlier conclusions about the size and growth rate of magnetic structures in the case of polytropic layers, which scale height decreases sharply towards the surface. To allow for a continuous transition from isothermal to polytropic layers, we employ a generalization of the exponential function known as the qexponential. Now, the top of the polytropic layer shifts with the polytropic index such that the scale height at some reference height is always the same. We use both meanfield and direct numerical simulations of forced stratified turbulence to determine the resulting flux concentrations. Magnetic structures begin to form at a depth where magnetic field strength is about 34% the local equipartition field strength with respect to the turbulent kinetic energy. Unlike the isothermal case where stronger fields can give rise to magnetic flux concentrations at larger depths, in the polytropic one the growth rates decreases for structures deeper down. For vertical fields, magnetic structures of superequipartition strengths are formed because such fields survive downward advection, unlike NEMPI under horizontal magnetic fields. The horizontal crosssection of such structures is approximately circular. Results based on isothermal models can be applied locally to polytropic layers. For vertical fields, magnetic flux concentrations of superequipartition strengths form, which supports suggestions that sunspot formation might be a shallow phenomenon.
Authors: I. R. Losada, A. Brandenburg, N. Kleeorin, I. Rogachevskii
Projects: None

Publication Status: Astron. Astrophys. 564, A2 (2014)
Last Modified: 20140503 11:45



Effects of enhanced stratification on equatorward dynamo wave propagation 

Axel Brandenburg Submitted: 20140202 10:57
We present results from simulations of rotating magnetized turbulent convection in spherical wedge geometry representing parts of the latitudinal and longitudinal extents of a star. Here we consider a set of runs for which the density stratification is varied, keeping the Reynolds and Coriolis numbers at similar values. In the case of weak stratification, we find quasisteady dynamo solutions for moderate rotation and oscillatory ones with poleward migration of activity belts for more rapid rotation. For stronger stratification, the growth rate tends to become smaller. Furthermore, a transition from quasisteady to oscillatory dynamos is found as the Coriolis number is increased, but now there is an equatorward migrating branch near the equator. The breakpoint where this happens corresponds to a rotation rate that is about 37 times the solar value. The phase relation of the magnetic field is such that the toroidal field lags behind the radial field by about pi/2, which can be explained by an oscillatory α ^{2} dynamo caused by the sign change of the α effect about the equator. We test the domain size dependence of our results for a rapidly rotating run with equatorward migration by varying the longitudinal extent of our wedge. The energy of the axisymmetric mean magnetic field decreases as the domain size increases and we find that an m=1 mode is excited for a full 2pi azimuthal extent, reminiscent of the field configurations deduced from observations of rapidly rotating latetype stars.
Authors: Petri J. Käpylä, Maarit. J. Mantere, Elizabeth Cole, Jörn Warnecke, Axel Brandenburg
Projects: None

Publication Status: Astrophys. J. 778, 41 (2013)
Last Modified: 20140203 10:31



Selfassembly of shallow magnetic spots through strongly stratified turbulence 

Axel Brandenburg Submitted: 20131003 22:09
Recent studies have demonstrated that in fully developed turbulence, the effective magnetic pressure of a largescale field (nonturbulent plus turbulent contributions) can become negative. In the presence of strongly stratified turbulence, this was shown to lead to a largescale instability that produces spontaneous magnetic flux concentrations. Furthermore, using a horizontal magnetic field, elongated flux concentrations with a strength of a few per cent of the equipartition value were found. Here we show that a uniform vertical magnetic field leads to circular magnetic spots of equipartition field strengths. This could represent a minimalistic model of sunspot formation and highlights the importance of two critical ingredients: turbulence and strong stratification. Radiation, ionization, and supergranulation may be important for realistic simulations, but are not critical at the level of a minimalistic model of magnetic spot formation.
Authors: Axel Brandenburg, Nathan Kleeorin, Igor Rogachevskii
Projects: None

Publication Status: Astrophys. J. Lett. 776, L23 (2013)
Last Modified: 20131005 20:19



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