The effect of a dynamogenerated field on the Parker wind 

Axel Brandenburg Submitted: 20210303 20:24
Stellar winds are an integral part of the underlying dynamo, the motor of stellar activity. The wind controls the star's angular momentum loss, which depends on the magnetic field geometry which varies significantly in time and latitude. Here we study basic properties of a selfconsistent model that includes simple representations of both the global stellar dynamo in a spherical shell and the exterior in which the wind accelerates and becomes supersonic. We numerically solve an axisymmetric meanfield model for the induction, momentum, and continuity equations using an isothermal equation of state. The model allows for the simultaneous generation of a mean magnetic field and the development of a Parker wind. The resulting flow is transonic at the critical point, which we arrange to be between the inner and outer radii of the model. The boundary conditions are assumed to be such that the magnetic field is antisymmetric about the equator, i.e., dipolar. At the solar rotation rate, the dynamo is oscillatory and of α2 type. In most of the domain, the magnetic field corresponds to that of a split monopole. The magnetic energy flux is largest between the stellar surface and the critical point. The angular momentum flux is highly variable in time and can reach negative values, especially at midlatitudes. At rapid rotation of up to 50 times the solar value, most of the magnetic field is lost along the axis within the inner tangential cylinder of the model. The model reveals unexpected features that are not generally anticipated from models that are designed to reproduce the solar wind: highly variable angular momentum fluxes even from just an α2 dynamo in the star. A major caveat of our isothermal models with a magnetic field produced by a dynamo is the difficulty to reach small enough plasma betas without the dynamo itself becoming unrealistically strong inside the star.
Authors: Jakab, P., Brandenburg, A.
Projects: None

Publication Status: Astron. Astrophys. 647, A18 (2021)
Last Modified: 20210303 22:19



Turbulent viscosity and effective magnetic Prandtl number from simulations of isotropically forced turbulence 

Axel Brandenburg Submitted: 20201106 10:20
Context. Turbulent diffusion of largescale flows and magnetic fields plays a major role in many astrophysical systems, such as stellar convection zones and accretion discs. Aims. Our goal is to compute turbulent viscosity and magnetic diffusivity which are relevant for diffusing largescale flows and magnetic fields, respectively. We also aim to compute their ratio, which is the turbulent magnetic Prandtl number, Pmt , for isotropically forced homogeneous turbulence. Methods. We used simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed largescale sinusoidal shear flow. Turbulent viscosity was computed either from the resulting Reynolds stress or from the decay rate of the large scale flow. Turbulent magnetic diffusivity was computed using the testfield method for a microphysical magnetic Prandtl number of unity. The scale dependence of the coefficients was studied by varying the wavenumber of the imposed sinusoidal shear and test fields. Results. We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (Re) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. These results are similar to those obtained earlier for the turbulent magnetic diffusivity. The results for the turbulent transport coefficients appear to converge at sufficiently high values of Re and the scale separation ratio. However, a weak trend is found even at the largest values of Re, suggesting that the turbulence is not in the fully developed regime. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large Re. For small Re we find values between 0.5 and 0.6 but the data are insufficient to draw conclusions regarding asymptotics. We demonstrate that our results are independent of the correlation time of the forcing function. Conclusions. The turbulent magnetic diffusivity is, in general, consistently higher than the turbulent viscosity, which is in qualitative agreement with analytic theories. However, the actual value of Pmt found from the simulations (0.90.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.40.8).
Authors: Petri J. Käpylä, Matthias Rheinhardt, Axel Brandenburg, Maarit J. Käpylä
Projects: None

Publication Status: Astron. Astrophys. 636, A93 (2020)
Last Modified: 20201110 09:13



Helicity proxies from linear polarisation of solar active regions 

Axel Brandenburg Submitted: 20200912 08:56
Abstract: The α effect is believed to play a key role in the generation of the solar magnetic field. A fundamental test for its significance in the solar dynamo is to look for magnetic helicity of opposite signs in the two hemispheres, and at small and large scales. However, measuring magnetic helicity is compromised by the inability to fully infer the magnetic field vector from observations of solar spectra, caused by what is known as the "pi ambiguity" of spectropolarimetric observations. We decompose linear polarisation into parityeven and parityodd E and B polarisations, which are not affected by the "pi ambiguity". Furthermore, we study whether the correlations of spatial Fourier spectra of B and parityeven quantities such as E or temperature T are a robust proxy for magnetic helicity of solar magnetic fields. We analyse polarisation measurements of active regions observed by the Helioseismic and Magnetic Imager on board the Solar Dynamics observatory. Theory predicts the magnetic helicity of active regions to have, statistically, opposite signs in the two hemispheres. We then compute the parityodd E B and T B correlations, and test for systematic preference of their sign based on the hemisphere of the active regions. We find that: (i) E B and T B correlations are a reliable proxy for magnetic helicity, when computed from linear polarisation measurements away from spectral line cores, and (ii) E polarisation reverses its sign close to the line core. Our analysis reveals Faraday rotation to not have a significant influence on the computed parityodd correlations. The EB decomposition of linear polarisation appears to be a good proxy for magnetic helicity independent of the "pi ambiguity". This allows us to routinely infer magnetic helicity directly from polarisation measurements.
Authors: A. Prabhu, A. Brandenburg, M. J. Käpylä, A. Lagg
Projects: None

Publication Status: Astron. Astrophys. 641, A46 (2020)
Last Modified: 20200914 23:53



Sensitivity to luminosity, centrifugal force, and boundary conditions in spherical shell convection 

Axel Brandenburg Submitted: 20200804 12:33
We test the sensitivity of hydrodynamic and magnetohydrodynamic turbulent convection simulations with respect to Mach number, thermal and magnetic boundary conditions, and the centrifugal force. We find that varying the luminosity, which also controls the Mach number, has only a minor effect on the largescale dynamics. A similar conclusion can also be drawn from the comparison of two formulations of the lower magnetic boundary condition with either vanishing electric field or current density. The centrifugal force has an effect on the solutions, but only if its magnitude with respect to acceleration due to gravity is by two orders of magnitude greater than in the Sun. Finally, we find that the parameterisation of the photospheric physics, either by an explicit cooling term or enhanced radiative diffusion, is more important than the thermal boundary condition. In particular, runs with cooling tend to lead to more anisotropic convection and stronger deviations from the TaylorProudman state. In summary, the fully compressible approach taken here with the Pencil Code is found to be valid, while still allowing the disparate timescales to be taken into account.
Authors: Käpylä, P. J., Gent, F. A., Olspert, N., Käpylä, M. J., & Brandenburg, A.
Projects: None

Publication Status: Geophys. Astrophys. Fluid Dyn. 114, 834 (2020)
Last Modified: 20200805 14:12



fmode strengthening from a localized bipolar subsurface magnetic field 

Axel Brandenburg Submitted: 20200523 01:40
Recent numerical work in helioseismology has shown that a periodically varying subsurface magnetic field leads to a fanning of the fmode, which emerges from the density jump at the surface. In an attempt to model a more realistic situation, we now modulate this periodic variation with an envelope, giving thus more emphasis on localized bipolar magnetic structures in the middle of the domain. Some notable findings are: (i) compared to the purely hydrodynamic case, the strength of the fmode is significantly larger at high horizontal wavenumbers k, but the fanning is weaker for the localized subsurface magnetic field concentrations investigated here than the periodic ones studied earlier; (ii) when the strength of the magnetic field is enhanced at a fixed depth below the surface, the fanning of the fmode in the kω diagram increases proportionally in such a way that the normalized fmode strengths remain nearly the same in different such cases; (iii) the unstable Bloch modes reported previously in case of harmonically varying magnetic fields are now completely absent when more realistic localized magnetic field concentrations are imposed beneath the surface, thus suggesting that the Bloch modes are unlikely to be supported during most phases of the solar cycle; (iv) the fmode strength appears to depend also on the depth of magnetic field concentrations such that it shows a relative decrement when the maximum of the magnetic field is moved to a deeper layer. We argue that detections of fmode perturbations such as those being explored here could be effective tracers of solar magnetic fields below the photosphere before these are directly detectable as visible manifestations in terms of active regions or sunspots.
Authors: Authors:Nishant K. Singh, Harsha Raichur, Maarit J. Käpylä, Matthias Rheinhardt, Axel Brandenburg, Petri J. Käpylä
Projects: None

Publication Status: Geophys. Astro. Fluid 2020, 114, 196212 (2020)
Last Modified: 20200527 13:13



The nature of meanfield generation in three classes of optimal dynamos 

Axel Brandenburg Submitted: 20200417 13:13
Abstract: In recent years, several optimal dynamos have been discovered. They minimize the magnetic energy dissipation or, equivalently, maximize the growth rate at a fixed magnetic Reynolds number. In the optimal dynamo of Willis (2012, Phys. Rev. Lett. 109, 251101), we find meanfield dynamo action for planar averages. One component of the magnetic field grows exponentially while the other decays in an oscillatory fashion near onset. This behavior is different from that of an α ^{2} dynamo, where the two nonvanishing components of the planar averages are coupled and have the same growth rate. For the Willis dynamo, we find that the mean field is excited by a negative turbulent magnetic diffusivity, which has a nonuniform spatial profile near onset. The temporal oscillations in the decaying component are caused by the corresponding component of the diffusivity tensor being complex when the mean field is decaying and, in this way, timedependent. The growing mean field can be modeled by a negative magnetic diffusivity combined with a positive magnetic hyperdiffusivity. In two other classes of optimal dynamos of Chen et al. (2015, J. Fluid Mech. 783, 23), we find, to some extent, similar meanfield dynamo actions. When the magnetic boundary conditions are mixed, the two components of the planar averaged field grow at different rates when the dynamo is 15% supercritical. When the mean magnetic field satisfies homogeneous boundary conditions (where the magnetic field is tangential to the boundary), meanfield dynamo action is found for onedimensional averages, but not for planar averages. Despite having different spatial profiles, both dynamos show negative turbulent magnetic diffusivities. Our finding suggests negative turbulent magnetic diffusivities may support a broader class of dynamos than previously thought, including these three optimal dynamos.
Authors: Axel Brandenburg, Long Chen
Projects: None

Publication Status: J. Plasma Phys. 86 (2020) 905860110
Last Modified: 20200418 12:48



Spectral magnetic helicity of solar active regions between 2006 and 2017 

Axel Brandenburg Submitted: 20191207 00:53
We compute magnetic helicity and energy spectra from about 2485 patches of about 100 megameters (Mm) side length on the solar surface using data from Hinode during 20062017. An extensive database is assembled where we list magnetic energy and helicity, large and smallscale magnetic helicity, mean current helicity density, fractional magnetic helicity, and correlation length along with the Hinode map identification number (MapID), as well as Carrington latitude and longitude for each MapID. While there are departures from the hemispheric sign rule for magnetic and current helicities, the weak trend reported here is in agreement with the previous results. This is argued to be a physical effect associated with the dominance of individual active regions that contribute more strongly in the better resolved Hinode maps. In comparison with earlier work, the typical correlation length is found to be 68 Mm, while the length scale relating magnetic and current helicity to each other is found to be around 1.4 Mm.
Authors: Gosain, S., Brandenburg, A.
Projects: None

Publication Status: Astrophys. J. 882, 80 (2019)
Last Modified: 20191207 15:34



A global twoscale helicity proxy from piambiguous solar magnetic fields 

Axel Brandenburg Submitted: 20191015 22:42
If the α effect plays a role in the generation of the Sun's magnetic field, the field should show evidence of magnetic helicity of opposite signs at large and small length scales. Measuring this faces two challenges: (i) in weakfield regions, horizontal field measurements are unreliable because of the pi ambiguity, and (ii) one needs a truly global approach to computing helicity spectra in the case where one expects a sign reversal across the equator at all wavenumbers. Here we develop such a method using spin2 spherical harmonics to decompose the linear polarization in terms of the parityeven and parityodd E and B polarizations, respectively. Using simple one and twodimensional models, we show that the product of the spectral decompositions of E and B, taken at spherical harmonic degrees that are shifted by one, can act as a proxy of the global magnetic helicity with a sign that represents that in the northern hemisphere. We then apply this method to the analysis of solar synoptic vector magnetograms, from which we extract a pseudopolarization corresponding to a "piambiguated" magnetic field, i.e., a magnetic field vector that has no arrow. We find a negative sign of the global EB helicity proxy at spherical harmonic degrees of around 6. This could indicate a positive magnetic helicity at large length scales, but the spectrum fails to capture clear evidence of the wellknown negative magnetic helicity at smaller scales. This method might also be applicable to stellar and Galactic polarization data.
Authors: Axel Brandenburg
Projects: CORONASF/RESIK

Publication Status: Astrophys. J. 883, 119 (2019)
Last Modified: 20191016 12:42



Reversed dynamo at small scales and large magnetic Prandtl number 

Axel Brandenburg Submitted: 20190824 21:21
Abstract: We show that at large magnetic Prandtl numbers, the Lorentz force does work on the flow at small scales and drives fluid motions, whose energy is dissipated viscously. This situation is opposite to that in a normal dynamo, where the flow does work against the Lorentz force. We compute the spectral conversion rates between kinetic and magnetic energies for several magnetic Prandtl numbers and show that normal (forward) dynamo action occurs on large scales over a progressively narrower range of wavenumbers as the magnetic Prandtl number is increased. At higher wavenumbers, reversed dynamo action occurs, i.e., magnetic energy is converted back into kinetic energy at small scales. We demonstrate this in both direct numerical simulations forced by volume stirring and in large eddy simulations of solar convectively driven smallscale dynamos. Low density plasmas such as stellar coronae tend to have large magnetic Prandtl numbers, i.e., the viscosity is large compared with the magnetic diffusivity. The regime in which viscous dissipation dominates over resistive dissipation for large magnetic Prandtl numbers was also previously found in large eddy simulations of the solar corona, i.e., our findings are a more fundamental property of MHD that is not just restricted to dynamos. Viscous energy dissipation is a consequence of positive Lorentz force work, which may partly correspond to particle acceleration in closetocollisionless plasmas. This is, however, not modeled in the MHD approximation employed. By contrast, resistive energy dissipation on current sheets is expected to be unimportant in stellar coronae.
Authors: Axel Brandenburg, Matthias Rempel
Projects: None

Publication Status: Astrophys. J. 879, 57 (2019)
Last Modified: 20190825 16:48



Transition from axi to nonaxisymmetric dynamo modes in spherical convection models of solarlike stars 

Axel Brandenburg Submitted: 20181226 02:55
We seek to understand the transition from nearly axisymmetric configurations at solar rotation rates to nonaxisymmetric configurations for rapid rotation using 3D numerical simulations of turbulent convection and considering rotation rates between 1 and 30 times the solar value. We find a transition from axi to nonaxisymmetric solutions at around 1.8 times the solar rotation rate. This transition coincides with a change in the rotation profile from antisolar to solarlike differential rotation with a faster equator and slow poles. In the solarlike rotation regime, the field configuration consists of an axisymmetric oscillatory field accompanied by an m=1 azimuthal mode (two active longitudes), which also shows temporal variability. At slow (rapid) rotation, the axisymmetric (nonaxisymmetric) mode dominates. The axisymmetric mode produces latitudinal dynamo waves with polarity reversals, while the nonaxisymmetric mode often exhibits a drift in the rotating reference frame and the strength of the active longitudes changes cyclically over time between the different hemispheres. Most of the obtained dynamo solutions exhibit cyclic variability either caused by latitudinal or azimuthal dynamo waves. In an activityperiod diagram, the cycle lengths normalized by the rotation period form two different populations as a function of rotation rate or magnetic activity level. The slowly rotating axisymmetric population lies close to what is called the inactive branch in observations, while the rapidly rotating models are close to the superactive branch with a declining cycle to rotation frequency ratio with increasing rotation rate. We can successfully reproduce the transition from axi to nonaxisymmetric dynamo solutions for high rotation rates, but highresolution simulations are required to limit the effect of rotational quenching of convection at rotation rates above 20 times the solar value.
Authors: Viviani, M., Warnecke, J., Käpylä, M. J., Käpylä, P. J., Olspert, N., ColeKodikara, E. M., Lehtinen, J. J., & Brandenburg, A.
Projects: None

Publication Status: A&A 616, A160 (2018)
Last Modified: 20181226 12:06



Turbulent transport coefficients in spherical wedge dynamo simulations of solarlike stars 

Axel Brandenburg Submitted: 20180311 23:22
Aims: We investigate dynamo action in global compressible solarlike convective dynamos in the framework of meanfield theory.
Methods: We simulate a solartype star in a wedgeshaped spherical shell, where the interplay between convection and rotation selfconsistently drives a largescale dynamo. To analyze the dynamo mechanism we apply the testfield method for azimuthally (φ) averaged fields to determine the 27 turbulent transport coefficients of the electromotive force, of which six are related to the α tensor. This method has previously been used either in simulations in Cartesian coordinates or in the geodynamo context and is applied here for the first time to fully compressible simulations of solarlike dynamos.
Results: We find that the φφcomponent of the α tensor does not follow the profile expected from that of kinetic helicity. The turbulent pumping velocities significantly alter the effective mean flows acting on the magnetic field and therefore challenge the flux transport dynamo concept. All coefficients are significantly affected by dynamically important magnetic fields. Quenching as well as enhancement are being observed. This leads to a modulation of the coefficients with the activity cycle. The temporal variations are found to be comparable to the timeaveraged values and seem to be responsible for a nonlinear feedback on the magnetic field generation. Furthermore, we quantify the validity of the ParkerYoshimura rule for the equatorward propagation of the mean magnetic field in the present case.
Authors: Warnecke, J., Rheinhardt, M., Käpylä, P. J., Käpylä, M. J., & Brandenburg, A.
Projects: None

Publication Status: Astron. Astrophys. 609, A51 (2018)
Last Modified: 20180312 09:45



Spontaneous flux concentrations from the negative effective magnetic pressure instability beneath a radiative stellar surface 

Axel Brandenburg Submitted: 20180203 05:30
Abstract: The formation of sunspots requires the concentration of magnetic flux near the surface. The negative magnetic pressure instability (NEMPI) might be a possible mechanism for accomplishing this, but it has mainly been studied in simple systems using an isothermal equation of state without a natural free surface. We study NEMPI in a stratified Cartesian meanfield model where turbulence effects are parameterized. We use an ideal equation of state and include radiation transport, which establishes selfconsistently a free surface. We use a Kramerstype opacity with adjustable exponents chosen such that the deeper layers are approximately isentropic. No convection is therefore possible in this model, allowing us to study NEMPI with radiation in isolation. We restrict ourselves to twodimensional models. We use artificially enhanced meanfield coefficients to allow NEMPI to develop, making it therefore possible to study the reason why it is much harder to excite in the presence of radiation. NEMPI yields moderately strong magnetic flux concentrations a certain distance beneath the surface where optical depth is unity. The instability is oscillatory and in the form of upward travelling waves. This seems to be a new effect that has not been found in earlier models without radiative transport. The horizontal wavelength is about ten times smaller than what has been found previously in more idealized isothermal models. In our models, NEMPI saturates at field strengths too low to explain sunspots. Furthermore, the structures appear too narrow and too far beneath the surface to cause significant brightness variations at the radiative surface. We speculate that the failure to reproduce effects resembling sunspots may be related to the neglect of convection.
Authors: Perri, B., & Brandenburg, A.
Projects: None

Publication Status: A&A 609, A99 (2018)
Last Modified: 20180205 22:07



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