Author name code: kapyla ADS astronomy entries on 2022-09-14 author:"Kapyla, Petri J." ------------------------------------------------------------------------ Title: Transition from anti-solar to solar-like differential rotation: Dependence on Prandtl number Authors: Käpylä, Petri J. Bibcode: 2022arXiv220700302K Altcode: (abridged) Context: Late-type stars such as the Sun rotate differentially due to the interaction of turbulent convection and rotation. Aims: The aim of the study is to investigate the effects of the thermal Prandtl number on the transition from anti-solar (slow equator, fast poles) to solar-like (fast equator, slow poles) differential rotation. Methods: Three-dimensional hydrodynamic and magnetohydrodynamic simulations in semi-global spherical wedge geometry are used to model convection zones of solar-like stars. Results: The overall convective velocity amplitude increases as the Prandtl number decreases in accordance with earlier studies. The transition from anti-solar to solar-like differential rotation is insensitive to the Prandtl number for Prandtl numbers below unity but for Prandtl numbers greater than unity, solar-like differential rotation becomes significantly harder to excite. Magnetic fields and more turbulent regimes with higher fluid and magnetic Reynolds numbers help in achieving solar-like differential rotation in near-transition cases. Solar-like differential rotation occurs only in cases with radially outward angular momentum transport at the equator. The dominant contribution to such outward transport near the equator is due to prograde propagating thermal Rossby waves. Conclusions: The differential rotation is sensitive to the Prandtl number only for large Prandtl numbers in the parameter regime explored in the current study. Magnetic fields have a greater effect on the differential rotation, although the inferred presence of a small-scale dynamo does not lead to drastically different results in the present study. The dominance of the thermal Rossby waves in the simulations is puzzling given the non-detection in the Sun. The current simulations are shown to be incompatible with the currently prevailing mean-field theory of differential rotation. Title: Simulations of fully convective M dwarfs: dynamo action with varying magnetic Prandtl numbers Authors: Ortiz-Rodríguez, C. A.; Schleicher, D. R. G.; Käpylä, P. J.; Navarrete, F. H. Bibcode: 2022BAAA...63...62O Altcode: 2022arXiv220614123O M dwarfs are low-mass main-sequence stars, the most numerous type of stars in the solar neighbourhood, which are known to have significant magnetic activity. The aim of this work is to explore the dynamo solutions and magnetic fields of fully convective M dwarfs with varying magnetic Prandtl numbers PrM, and a rotation period (Prot) of 43 days. PrM is known to play an important role in the dynamo action; dynamos for low-PrM and large-PrM have very different properties. We performed three-dimensional magnetohydrodynamical (MHD) numerical simulations with the ``star-in-a- box'' model using stellar parameters for an M5 dwarf with 0.21 M. We found that the dynamo solutions are sensitive to Pr_. The simulations at this rotation period present periodic cycles of the large-scale magnetic field up to PrM ≤ 2; for higher values the cycles disappear and irregular solutions to arise. Our results are consistent with previous studies and suggest that the dynamos operating in fully convective stars behave similarly as those in partially convective stars. Title: Origin of eclipsing time variations: Contributions of different modes of the dynamo-generated magnetic field Authors: Navarrete, Felipe H.; Käpylä, Petri J.; Schleicher, Dominik R. G.; Ortiz, Carolina A.; Banerjee, Robi Bibcode: 2022A&A...663A..90N Altcode: 2021arXiv210211110N Context. The possibility to detect circumbinary planets and to study stellar magnetic fields through eclipsing time variations (ETVs) in binary stars has sparked an increase of interest in this area of research.
Aims: We revisit the connection between stellar magnetic fields and the gravitational quadrupole moment Qxx and compare different dynamo-generated ETV models with our simulations.
Methods: We present magnetohydrodynamical simulations of solar mass stars with rotation periods of 8.3, 1.2, and 0.8 days and perform a detailed analysis of the magnetic and quadrupole moment using spherical harmonic decomposition.
Results: The extrema of Qxx are associated with changes in the magnetic field structure. This is evident in the simulation with a rotation period of 1.2 days. Its magnetic field has a more complex behavior than in the other models, as the large-scale nonaxisymmetric field dominates throughout the simulation and the axisymmetric component is predominantly hemispheric. This triggers variations in the density field that follow the magnetic field asymmetry with respect to the equator, affecting the zz component of the inertia tensor, and thus modulating Qxx. The magnetic fields of the two other runs are less variable in time and more symmetric with respect to the equator, such that the variations in the density are weaker, and therefore only small variations in Qxx are seen.
Conclusions: If interpreted via the classical Applegate mechanism (tidal locking), the quadrupole moment variations obtained in the current simulations are about two orders of magnitude below those deduced from observations of post-common-envelope binaries. However, if no tidal locking is assumed, our results are compatible with the observed ETVs. Title: Turbulent Prandtl number from isotropically forced turbulence Authors: Käpylä, Petri J.; Singh, Nishant K. Bibcode: 2022arXiv220710335K Altcode: Turbulent motions enhance the diffusion of large-scale flows and temperature gradients. Such diffusion is often parameterized by coefficients of turbulent viscosity ($\nu_{\rm t}$) and turbulent thermal diffusivity ($\chi_{\rm t}$) that are analogous to their microscopic counterparts. We compute the turbulent diffusion coefficients by imposing large-scale velocity and temperature gradients on a turbulent flow and measuring the response of the system. We also confirm our results using experiments where the imposed gradients are allowed to decay. To achieve this, we use weakly compressible three-dimensional hydrodynamic simulations of isotropically forced homogeneous turbulence. We find that the turbulent viscosity and thermal diffusion, as well as their ratio the turbulent Prandtl number, ${\rm Pr}_{\rm t} = \nu_{\rm t}/\chi_{\rm t}$, approach asymptotic values at sufficiently high Reynolds and Peclét numbers. We also do not find a significant dependence of ${\rm Pr}_{\rm t}$ on the microscopic Prandtl number ${\rm Pr} = \nu/\chi$. These findings are in stark contrast to results from the $k-\epsilon$ model which suggests that ${\rm Pr}_{\rm t}$ increases monotonically with decreasing ${\rm Pr}$. The current results are relevant for the ongoing debate of, for example, the nature of the turbulent flows in the very low ${\rm Pr}$ regimes of stellar convection zones. Title: Magnetism, rotation, and nonthermal emission in cool stars. Average magnetic field measurements in 292 M dwarfs Authors: Reiners, A.; Shulyak, D.; Käpylä, P. J.; Ribas, I.; Nagel, E.; Zechmeister, M.; Caballero, J. A.; Shan, Y.; Fuhrmeister, B.; Quirrenbach, A.; Amado, P. J.; Montes, D.; Jeffers, S. V.; Azzaro, M.; Béjar, V. J. S.; Chaturvedi, P.; Henning, Th.; Kürster, M.; Pallé, E. Bibcode: 2022A&A...662A..41R Altcode: 2022arXiv220400342R Stellar dynamos generate magnetic fields that are of fundamental importance to the variability and evolution of Sun-like and low-mass stars, and for the development of their planetary systems. As a key to understanding stellar dynamos, empirical relations between stellar parameters and magnetic fields are required for comparison to ab initio predictions from dynamo models. We report measurements of surface-average magnetic fields in 292 M dwarfs from a comparison with radiative transfer calculations; for 260 of them, this is the first measurement of this kind. Our data were obtained from more than 15 000 high-resolution spectra taken during the CARMENES project. They reveal a relation between average field strength, ⟨B⟩, and Rossby number, Ro, resembling the well-studied rotation-activity relation. Among the slowly rotating stars, we find that magnetic flux, ΦB, is proportional to rotation period, P, and among the rapidly rotating stars that average surface fields do not grow significantly beyond the level set by the available kinetic energy. Furthermore, we find close relations between nonthermal coronal X-ray emission, chromospheric Hα and Ca H&K emission, and magnetic flux. Taken together, these relations demonstrate empirically that the rotation-activity relation can be traced back to a dependence of the magnetic dynamo on rotation. We advocate the picture that the magnetic dynamo generates magnetic flux on the stellar surface proportional to rotation rate with a saturation limit set by the available kinetic energy, and we provide relations for average field strengths and nonthermal emission that are independent of the choice of the convective turnover time. We also find that Ca H&K emission saturates at average field strengths of ⟨B⟩≈800 G while Hα and X-ray emission grow further with stronger fields in the more rapidly rotating stars. This is in conflict with the coronal stripping scenario predicting that in the most rapidly rotating stars coronal plasma would be cooled to chromospheric temperatures.

Table B.1 is only available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A/662/A41 Title: Solar-like Dynamos and Rotational Scaling of Cycles from Star-in-a-box Simulations Authors: Käpylä, Petri J. Bibcode: 2022ApJ...931L..17K Altcode: 2022arXiv220204329K Magnetohydrodynamic star-in-a-box simulations of convection and dynamos in a solar-like star with different rotation rates are presented. These simulations produce solar-like differential rotation with a fast equator and slow poles and magnetic activity that resembles that of the Sun with equatorward migrating activity at the surface. Furthermore, the ratio of rotation to cycle period is almost constant, as the rotation period decreases in the limited sample considered here. This is reminiscent of the suggested inactive branch of stars from observations and differs from most earlier simulation results from spherical shell models. While the exact excitation mechanism of the dynamos in the current simulations is not yet clear, it is shown that it is plausible that the greater freedom that the magnetic field has due to the inclusion of the radiative core and regions exterior to the star are important in shaping the dynamo. Title: Origin of eclipsing time variations in Post-Common-Envelope binaries: role of the centrifugal force Authors: Navarrete, Felipe H.; Schleicher, Dominik R. G.; Käpylä, Petri J.; Ortiz, Carolina; Banerjee, Robi Bibcode: 2022arXiv220503163N Altcode: The eclipsing time variations in post-common-envelope binaries were proposed to be due to the time-varying component of the stellar quadrupole moment. This is suggested to be produced by changes in the stellar structure due to an internal redistribution of the angular momentum and the effect of the centrifugal force. We examine this hypothesis presenting 3D simulations of compressible magneto-hydrodynamics (MHD) performed with the Pencil Code, modeling the stellar dynamo for a solar mass star with angular velocities of 20 and 30 times solar. We include and vary the strength of the centrifugal force, comparing with reference simulations without the centrifugal force and including a simulation where its effect is enhanced. The centrifugal force is causing perturbations in the evolution of the stars, so that the outcome in the details becomes different as a result of non-linear evolution. While the average density profile is unaffected by the centrifugal force, a relative change in density difference between high altitudes and the equator of order $\sim 10^{-4}$ is found. The power spectrum of the convective velocity is found to be more sensitive to the angular velocity than the strength of the centrifugal force. The quadrupole moment of the stars includes a fluctuating and a time-independent component which varies with the rotation rate. As very similar behavior is produced in the absence of the centrifugal force, we conclude that it is not the main ingredient for producing the time-averaged quadrupole moment of the star. In a real physical system, we thus expect contributions from both components, that is, due to the time-dependent gravitational force from the variation in the quadrupole term and due to spin-orbit coupling due to the persistent part of the quadrupole. Title: VizieR Online Data Catalog: Magnetic fields in 292 M dwarfs. (Reiners+, 2022) Authors: Reiners, A.; Shulyak, D.; Kaepylae, P. J.; Ribas, I.; Nagel, E.; Zechmeister, M.; Caballero, J. A.; Shan, Y.; Fuhrmeister, B.; Quirrenbach, A.; Amado, P. J.; Montes, D.; Jeffers, S. V.; Azzaro, M.; Bejar, V. J. S.; Chaturvedi, P.; Henning, T.; Kuerster, M.; Palle, E. Bibcode: 2022yCat..36620041R Altcode: The sample of stars used for our analysis, the number of spectra co-added for each star, and the approximate S/N around λ=8700Å are provided in Table B.1.

(1 data file). Title: On Strengthening of the Solar f-mode Prior to Active Region Emergence Using the Fourier-Hankel Analysis Authors: Waidele, Matthias; Roth, Markus; Singh, Nishant; Käpylä, Petri Bibcode: 2022arXiv220211236W Altcode: Recent results of Singh et al. (2016) show that the emergence of an active region (AR) can be seen in a strengthening of the f-mode power up to two days prior of the region's formation. In the original work, ring diagram analysis was used to estimate the power evolution. In this study, we make use of the Fourier-Hankel method, essentially testing the aforementioned results with an independent method. The data is acquired from SDO/HMI, studying the ARs 11158, 11072, 11105, 11130, 11242 and 11768. Investigating the total power as a function of time, we find a similar behavior to the original work, which is an enhancement of f-mode power about one to three days prior to AR emergence. Analysis of the absorption coefficient $\alpha$, yielded by a Fourier-Hankel analysis, shows neither absorption ($\alpha > 0$) nor emission ($\alpha < 0$) of power during the enhancement. Finding no changes of the absorption coefficient (i.e. $\alpha = 0$) is an important result, as it narrows down the possible physical interpretation of the original f-mode power enhancement, showing that no directional dependence (in the sense of inward and outward moving waves) is present. Title: Generation of mean flows in rotating anisotropic turbulence: The case of solar near-surface shear layer Authors: Barekat, A.; Käpylä, M. J.; Käpylä, P. J.; Gilson, E. P.; Ji, H. Bibcode: 2021A&A...655A..79B Altcode: 2020arXiv201206343B Context. Results from helioseismology indicate that the radial gradient of the rotation rate in the near-surface shear layer (NSSL) of the Sun is independent of latitude and radius. Theoretical models using the mean-field approach have been successful in explaining this property of the NSSL, while global direct or large-eddy magnetoconvection models have so far been unable to reproduce this.
Aims: We investigate the reason for this discrepancy by measuring the mean flows, Reynolds stress, and turbulent transport coefficients under conditions mimicking those in the solar NSSL.
Methods: Simulations with as few ingredients as possible to generate mean flows were studied. These ingredients are inhomogeneity due to boundaries, anisotropic turbulence, and rotation. The parameters of the simulations were chosen such that they matched the weakly rotationally constrained NSSL. The simulations probe locally Cartesian patches of the star at a given depth and latitude. The depth of the patch was varied by changing the rotation rate such that the resulting Coriolis numbers covered the same range as in the NSSL. We measured the turbulent transport coefficient relevant for the nondiffusive (Λ-effect) and diffusive (turbulent viscosity) parts of the Reynolds stress and compared them with predictions of current mean-field theories.
Results: A negative radial gradient of the mean flow is generated only at the equator where meridional flows are absent. At other latitudes, the meridional flow is comparable to the mean flow corresponding to differential rotation. We also find that the meridional components of the Reynolds stress cannot be ignored. Additionally, we find that the turbulent viscosity is quenched by rotation by about 50% from the surface to the bottom of the NSSL.
Conclusions: Our local simulations do not validate the explanation for the generation of the NSSL from mean-field theory where meridional flows and stresses are neglected. However, the rotational dependence of the turbulent viscosity in our simulations agrees well with theoretical predictions. Moreover, our results agree qualitatively with global convection simulations in that an NSSL can only be obtained near the equator. Title: Prandtl number dependence of stellar convection: Flow statistics and convective energy transport Authors: Käpylä, P. J. Bibcode: 2021A&A...655A..78K Altcode: 2021arXiv210508453K Context. The ratio of kinematic viscosity to thermal diffusivity, the Prandtl number, is much smaller than unity in stellar convection zones.
Aims: The main goal of this work is to study the statistics of convective flows and energy transport as functions of the Prandtl number.
Methods: Three-dimensional numerical simulations of compressible non-rotating hydrodynamic convection in Cartesian geometry are used. The convection zone (CZ) is embedded between two stably stratified layers. The dominant contribution to the diffusion of entropy fluctuations comes in most cases from a subgrid-scale diffusivity whereas the mean radiative energy flux is mediated by a diffusive flux employing Kramers opacity law. Here, we study the statistics and transport properties of up- and downflows separately.
Results: The volume-averaged rms velocity increases with decreasing Prandtl number. At the same time, the filling factor of downflows decreases and leads to, on average, stronger downflows at lower Prandtl numbers. This results in a strong dependence of convective overshooting on the Prandtl number. Velocity power spectra do not show marked changes as a function of Prandtl number except near the base of the convective layer where the dominance of vertical flows is more pronounced. At the highest Reynolds numbers, the velocity power spectra are more compatible with the Bolgiano-Obukhov k−11/5 than the Kolmogorov-Obukhov k−5/3 scaling. The horizontally averaged convected energy flux (F̅conv), which is the sum of the enthalpy (F̅enth) and kinetic energy fluxes (F̅kin), is independent of the Prandtl number within the CZ. However, the absolute values of F̅enth and F̅kin increase monotonically with decreasing Prandtl number. Furthermore, F̅enth and F̅kin have opposite signs for downflows and their sum F̅conv diminishes with Prandtl number. Thus, the upflows (downflows) are the dominant contribution to the convected flux at low (high) Prandtl numbers. These results are similar to those from Rayleigh-Benárd convection in the low Prandtl number regime where convection is vigorously turbulent but inefficient at transporting energy.
Conclusions: The current results indicate a strong dependence of convective overshooting and energy flux on the Prandtl number. Numerical simulations of astrophysical convection often use a Prandtl number of unity because it is numerically convenient. The current results suggest that this can lead to misleading results and that the astrophysically relevant low Prandtl number regime is qualitatively different from the parameter regimes explored in typical contemporary simulations. Title: Star-in-a-box simulations of fully convective stars Authors: Käpylä, P. J. Bibcode: 2021A&A...651A..66K Altcode: 2020arXiv201201259K Context. Main-sequence late-type stars with masses of less than 0.35 M are fully convective.
Aims: The goal is to study convection, differential rotation, and dynamos as functions of rotation in fully convective stars.
Methods: Three-dimensional hydrodynamic and magnetohydrodynamic numerical simulations with a star-in-a-box model, in which a spherical star is immersed inside of a Cartesian cube, are used. The model corresponds to a 0.2 M main-sequence M5 dwarf. A range of rotation periods (Prot) between 4.3 and 430 d is explored.
Results: The slowly rotating model with Prot = 430 days produces anti-solar differential rotation with a slow equator and fast poles, along with predominantly axisymmetric quasi-steady large-scale magnetic fields. For intermediate rotation (Prot = 144 and 43 days) the differential rotation is solar-like (fast equator, slow poles), and the large-scale magnetic fields are mostly axisymmetric and either quasi-stationary or cyclic. The latter occurs in a similar parameter regime as in other numerical studies in spherical shells, and the cycle period is similar to observed cycles in fully convective stars with rotation periods of roughly 100 days. In the rapid rotation regime the differential rotation is weak and the large-scale magnetic fields are increasingly non-axisymmetric with a dominating m = 1 mode. This large-scale non-axisymmetric field also exhibits azimuthal dynamo waves.
Conclusions: The results of the star-in-a-box models agree with simulations of partially convective late-type stars in spherical shells in that the transitions in differential rotation and dynamo regimes occur at similar rotational regimes in terms of the Coriolis (inverse Rossby) number. This similarity between partially and fully convective stars suggests that the processes generating differential rotation and large-scale magnetism are insensitive to the geometry of the star. Title: Erratum: Magnetohydrodynamical origin of eclipsing time variations in post-common-envelope binaries for solar mass secondaries Authors: Navarrete, Felipe H.; Schleicher, Dominik R. G.; Käpylä, Petri J.; Schober, Jennifer; Völschow, Marcel; Mennickent, Ronald E. Bibcode: 2021MNRAS.504.1676N Altcode: No abstract at ADS Title: Common dynamo scaling in late-type main sequence and evolved stars Authors: Lehtinen, Jyri; Spada, Federico; Käpylä, Maarit; Olspert, Nigul; Käpylä, Petri Bibcode: 2021csss.confE.193L Altcode: Both the magnetic activity and magnetic field strengths of late-type stars are observed to scale up with stellar rotation up to a certain saturation level. We show strong evidence from chromospheric Ca II H&K line emission observations that the same rotation-activity relation is shared by both main sequence stars and evolved giants. This indicates that the dynamos of both young and evolved stars follow the same underlying principles, irrespective of the stellar internal structure. The common scaling relation can only be found in relation to the Rossby number, i.e. the ratio of rotation period to the convective turnover time, meaning that this shared dynamo must be governed by turbulent convection. A more detailed analysis of the shape of the rotation-activity relation reveals a localised break in its slope at mid-activity levels. This knee point lies close to the threshold where large scale non-axisymmetric activity appears on fast rotating stars. It may thus be connected to a transition between axisymmetric and non-axisymmetric dynamo modes in slow and fast rotating stars. Title: Reynolds number dependence of Lyapunov exponents of turbulence and fluid particles Authors: Fouxon, Itzhak; Feinberg, Joshua; Käpylä, Petri; Mond, Michael Bibcode: 2021PhRvE.103c3110F Altcode: 2021arXiv210401235F The Navier-Stokes equations generate an infinite set of generalized Lyapunov exponents defined by different ways of measuring the distance between exponentially diverging perturbed and unperturbed solutions. This set is demonstrated to be similar, yet different, from the generalized Lyapunov exponent that provides moments of distance between two fluid particles below the Kolmogorov scale. We derive rigorous upper bounds on dimensionless Lyapunov exponent of the fluid particles that demonstrate the exponent's decay with Reynolds number Re in accord with previous studies. In contrast, terms of cumulant series for exponents of the moments have power-law growth with Re. We demonstrate as an application that the growth of small fluctuations of magnetic field in ideal conducting turbulence is hyperintermittent, being exponential in both time and Reynolds number. We resolve the existing contradiction between the theory, that predicts slow decrease of dimensionless Lyapunov exponent of turbulence with Re, and observations exhibiting quite fast growth. We demonstrate that it is highly plausible that a pointwise limit for the growth of small perturbations of the Navier-Stokes equations exists. Title: The Pencil Code, a modular MPI code for partial differential equations and particles: multipurpose and multiuser-maintained Authors: Pencil Code Collaboration; Brandenburg, Axel; Johansen, Anders; Bourdin, Philippe; Dobler, Wolfgang; Lyra, Wladimir; Rheinhardt, Matthias; Bingert, Sven; Haugen, Nils; Mee, Antony; Gent, Frederick; Babkovskaia, Natalia; Yang, Chao-Chin; Heinemann, Tobias; Dintrans, Boris; Mitra, Dhrubaditya; Candelaresi, Simon; Warnecke, Jörn; Käpylä, Petri; Schreiber, Andreas; Chatterjee, Piyali; Käpylä, Maarit; Li, Xiang-Yu; Krüger, Jonas; Aarnes, Jørgen; Sarson, Graeme; Oishi, Jeffrey; Schober, Jennifer; Plasson, Raphaël; Sandin, Christer; Karchniwy, Ewa; Rodrigues, Luiz; Hubbard, Alexander; Guerrero, Gustavo; Snodin, Andrew; Losada, Illa; Pekkilä, Johannes; Qian, Chengeng Bibcode: 2021JOSS....6.2807P Altcode: 2021JOSS....6.2807C; 2020arXiv200908231B The Pencil Code is a highly modular physics-oriented simulation code that can be adapted to a wide range of applications. It is primarily designed to solve partial differential equations (PDEs) of compressible hydrodynamics and has lots of add-ons ranging from astrophysical magnetohydrodynamics (MHD) to meteorological cloud microphysics and engineering applications in combustion. Nevertheless, the framework is general and can also be applied to situations not related to hydrodynamics or even PDEs, for example when just the message passing interface or input/output strategies of the code are to be used. The code can also evolve Lagrangian (inertial and noninertial) particles, their coagulation and condensation, as well as their interaction with the fluid. Title: Electrodynamics of turbulent fluids with fluctuating electric conductivity Authors: Rüdiger, G.; Küker, M.; Käpylä, P. J. Bibcode: 2020JPlPh..86c9018R Altcode: 2019arXiv191106611R Consequences of fluctuating microscopic conductivity in mean-field electrodynamics of turbulent fluids are formulated and discussed. If the conductivity fluctuations are assumed to be uncorrelated with the velocity fluctuations then only the turbulence-originated magnetic diffusivity of the fluid is reduced and the decay time of a large-scale magnetic field or the cycle times of oscillating turbulent dynamo models are increased. If, however, the fluctuations of conductivity and flow in a certain well-defined direction are correlated, an additional diamagnetic pumping effect results, transporting the magnetic field in the opposite direction to the diffusivity flux vector $\langle \unicode[STIX]{x1D702}^' }\boldsymbol{u}^' }\rangle$. In the presence of global rotation, even for homogeneous turbulence fields, an alpha effect appears. If the characteristic values of the outer core of the Earth or the solar convection zone are applied, the dynamo number of the new alpha effect does not reach supercritical values to operate as an $\unicode[STIX]{x1D6FC}2$-dynamo but oscillating $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}$-dynamos with differential rotation are not excluded. Title: Turbulent viscosity and magnetic Prandtl number from simulations of isotropically forced turbulence Authors: Käpylä, P. J.; Rheinhardt, M.; Brandenburg, A.; Käpylä, M. J. Bibcode: 2020A&A...636A..93K Altcode: 2019arXiv190100787K Context. Turbulent diffusion of large-scale 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 large-scale 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 large-scale 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 test-field 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 (≈0.9-0.95) at large Re and large scale separation ratio is higher than any of the analytic predictions (0.4-0.8). Title: Common dynamo scaling in slowly rotating young and evolved stars Authors: Lehtinen, Jyri J.; Spada, Federico; Käpylä, Maarit J.; Olspert, Nigul; Käpylä, Petri J. Bibcode: 2020NatAs...4..658L Altcode: 2020NatAs.tmp...46L; 2020arXiv200308997L One interpretation of the activity and magnetism of late-type stars is that these both intensify with decreasing Rossby number up to a saturation level1-3, suggesting that stellar dynamos depend on both rotation and convective turbulence4. Some studies have claimed, however, that rotation alone suffices to parametrize this scaling adequately5,6. Here, we tackle the question of the relevance of turbulence to stellar dynamos by including evolved, post-main-sequence stars in the analysis of the rotation-activity relation. These stars rotate very slowly compared with main-sequence stars, but exhibit similar activity levels7. We show that the two evolutionary stages fall together in the rotation-activity diagram and form a single sequence in the unsaturated regime in relation only to Rossby numbers derived from stellar models, confirming earlier preliminary results that relied on a more simplistic parametrization of the convective turn-over time8,9. This mirrors recent results of fully convective M dwarfs, which likewise fall on the same rotation-activity sequence as partially convective solar-type stars10,11. Our results demonstrate that turbulence plays a crucial role in driving stellar dynamos and suggest that there is a common turbulence-related dynamo mechanism explaining the magnetic activity of all late-type stars. Title: Sensitivity to luminosity, centrifugal force, and boundary conditions in spherical shell convection Authors: Käpylä, P. J.; Gent, F. A.; Olspert, N.; Käpylä, M. J.; Brandenburg, A. Bibcode: 2020GApFD.114....8K Altcode: 2018arXiv180709309K 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 large-scale 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 Taylor-Proudman 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. Title: f-mode strengthening from a localised bipolar subsurface magnetic field Authors: Singh, Nishant K.; Raichur, Harsha; Käpylä, Maarit J.; Rheinhardt, Matthias; Brandenburg, Axel; Käpylä, Petri J. Bibcode: 2020GApFD.114..196S Altcode: Recent numerical work in helioseismology has shown that a periodically varying subsurface magnetic field leads to a fanning of the f-mode, which emerges from a 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 localised bipolar magnetic structures in the middle of the domain. Some notable findings are: (i) compared to the purely hydrodynamic case, the strength of the f-mode is significantly larger at high horizontal wavenumbers k, but the fanning is weaker for the localised 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 f-mode in the ? diagram increases proportionally in such a way that the normalised f-mode 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 localised 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 f-mode 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 f-mode 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. Title: VizieR Online Data Catalog: Rotation and activity across HR diagram (Lehtinen+, 2020) Authors: Lehtinen, J. J.; Spada, F.; Kapyla, M. J.; Olspert, N.; Kapyla, P. J. Bibcode: 2020yCatp061000401L Altcode: Derived values for the chromospheric activity, rotational and convective characteristics, and the evolutionary stage, as well as literature data used for deriving these, are presented for 224 stars observed over four or more observing seasons in the Mount Wilson Observatory HK Project. Data for five additional stars with photometrically derived rotation periods (Lehtinen et al., 2016A&A...588A..38L) are presented in separate tables to supplement the short rotation period end of the sample. Moreover, for the stars with newly derived rotation periods, we present a comparison with period estimates found from the literature.

(5 data files). Title: Magnetohydrodynamical origin of eclipsing time variations in post-common-envelope binaries for solar mass secondaries Authors: Navarrete, Felipe H.; Schleicher, Dominik R. G.; Käpylä, Petri J.; Schober, Jennifer; Völschow, Marcel; Mennickent, Ronald E. Bibcode: 2020MNRAS.491.1043N Altcode: 2019MNRAS.tmp.2642N; 2019arXiv190606787N Eclipsing time variations have been observed for a wide range of binary systems, including post-common-envelope binaries. A frequently proposed explanation, apart from the possibility of having a third body, is the effect of magnetic activity, which may alter the internal structure of the secondary star, particularly its quadrupole moment, and thereby cause quasi-periodic oscillations. Here, we present two compressible non-ideal magnetohydrodynamical simulations of the magnetic dynamo in a solar mass star, one of them with three times the solar rotation rate ('slow rotator'), and the other one with 20 times the solar rotation rate ('rapid rotator'), to account for the high rotational velocities in close binary systems. For the slow rotator, we find that both the magnetic field and the stellar quadrupole moment change in a quasi-periodic manner, leading to O - C (observed minus corrected times of the eclipse) variations of ∼0.025 s. For the rapid rotator, the behaviour of the magnetic field as well as the quadrupole moment changes becomes considerably more complex, due to the less coherent dynamo solution. The resulting O - C variations are of the order of 0.13 s. The observed system V471 Tau shows two modes of eclipsing time variations, with amplitudes of 151 and 20 s, respectively. However, the current simulations may not capture all relevant effects due to the neglect of the centrifugal force and self-gravity. Considering the model limitations and that the rotation of V471 Tau is still a factor of 2.5 faster than our rapid rotator, it may be conceivable to reach the observed magnitudes. Title: Overshooting in simulations of compressible convection Authors: Käpylä, P. J. Bibcode: 2019A&A...631A.122K Altcode: 2018arXiv181207916K Context. Convective motions that overshoot into regions that are formally convectively stable cause extended mixing.
Aims: We aim to determine the scaling of the overshooting depth (dos) at the base of the convection zone as a function of imposed energy flux (ℱn) and to estimate the extent of overshooting at the base of the solar convection zone.
Methods: Three-dimensional Cartesian simulations of hydrodynamic compressible non-rotating convection with unstable and stable layers were used. The simulations used either a fixed heat conduction profile or a temperature- and density-dependent formulation based on Kramers opacity law. The simulations covered a range of almost four orders of magnitude in the imposed flux, and the sub-grid scale diffusivities were varied so as to maintain approximately constant supercriticality at each flux.
Results: A smooth heat conduction profile (either fixed or through Kramers opacity law) leads to a relatively shallow power law with dos ∝ ℱn0.08 for low ℱn. A fixed step-profile of the heat conductivity at the bottom of the convection zone leads to a somewhat steeper dependency on dos ∝ ℱn0.12 in the same regime. Experiments with and without subgrid-scale entropy diffusion revealed a strong dependence on the effective Prandtl number, which is likely to explain the steep power laws as a function of ℱn reported in the literature. Furthermore, changing the heat conductivity artificially in the radiative and overshoot layers to speed up thermal saturation is shown to lead to a substantial underestimation of the overshooting depth.
Conclusions: Extrapolating from the results obtained with smooth heat conductivity profiles, which are the most realistic set-up we considered, suggest that the overshooting depth for the solar energy flux is about 20% of the pressure scale height at the base of the convection zone. This is two to four times higher than the estimates from helioseismology. However, the current simulations do not include rotation or magnetic fields, which are known to reduce convective overshooting. Title: Stellar Dynamos in the Transition Regime: Multiple Dynamo Modes and Antisolar Differential Rotation Authors: Viviani, M.; Käpylä, M. J.; Warnecke, J.; Käpylä, P. J.; Rheinhardt, M. Bibcode: 2019ApJ...886...21V Altcode: 2019arXiv190204019V Global and semi-global convective dynamo simulations of solar-like stars are known to show a transition from an antisolar (fast poles, slow equator) to solar-like (fast equator, slow poles) differential rotation (DR) for increasing rotation rate. The dynamo solutions in the latter regime can exhibit regular cyclic modes, whereas in the former one, only stationary or temporally irregular solutions have been obtained so far. In this paper we present a semi-global dynamo simulation in the transition region, exhibiting two coexisting dynamo modes, a cyclic and a stationary one, both being dynamically significant. We seek to understand how such a dynamo is driven by analyzing the large-scale flow properties (DR and meridional circulation) together with the turbulent transport coefficients obtained with the test-field method. Neither an αΩ dynamo wave nor an advection-dominated dynamo are able to explain the cycle period and the propagation direction of the mean magnetic field. Furthermore, we find that the α effect is comparable or even larger than the Ω effect in generating the toroidal magnetic field, and therefore, the dynamo seems to be of α 2Ω or α 2 type. We further find that the effective large-scale flows are significantly altered by turbulent pumping. Title: Effects of small-scale dynamo and compressibility on the Λ effect Authors: Käpylä, Petri J. Bibcode: 2019AN....340..744K Altcode: 2019arXiv190304363K The Λ effect describes a rotation-induced nondiffusive contribution to the Reynolds stress. It is commonly held responsible for maintaining the observed differential rotation of the Sun and other late-type stars. Here, the sensitivity of the Λ effect to small-scale magnetic fields and compressibility is studied by means of forced turbulence simulations either with anisotropic forcing in fully periodic cubes or in density-stratified domains with isotropic forcing. Effects of small-scale magnetic fields are studied in cases where the magnetic fields are self-consistently generated by a small-scale dynamo. The results show that small-scale magnetic fields lead to a quenching of the Λ effect which is milder than in cases where also a large-scale field is present. The effect of compressibility on the Λ effect is negligible in the range of Mach numbers from 0.015 to 0.8. Density stratification induces a marked anisotropy in the turbulence and a vertical Λ effect if the forcing scale is roughly two times larger than the density scale height. Title: Antisolar differential rotation of slowly rotating cool stars Authors: Rüdiger, G.; Küker, M.; Käpylä, P. J.; Strassmeier, K. G. Bibcode: 2019A&A...630A.109R Altcode: 2019arXiv190204172R Rotating stellar convection transports angular momentum towards the equator, generating the characteristic equatorial acceleration of the solar rotation while the radial flux of angular momentum is always inwards. New numerical box simulations for the meridional cross-correlation ⟨uθuϕ⟩, however, reveal the angular momentum transport towards the poles for slow rotation and towards the equator for fast rotation. The explanation is that for slow rotation a negative radial gradient of the angular velocity always appears, which in combination with a so-far neglected rotation-induced off-diagonal eddy viscosity term ν provides "antisolar rotation" laws with a decelerated equator. Similarly, the simulations provided positive values for the rotation-induced correlation ⟨uruθ⟩, which is relevant for the resulting latitudinal temperature profiles (cool or warm poles) for slow rotation and negative values for fast rotation. Observations of the differential rotation of slowly rotating stars will therefore lead to a better understanding of the actual stress-strain relation, the heat transport, and the underlying model of the rotating convection. Title: Effects of a subadiabatic layer on convection and dynamos in spherical wedge simulations Authors: Käpylä, P. J.; Viviani, M.; Käpylä, M. J.; Brandenburg, A.; Spada, F. Bibcode: 2019GApFD.113..149K Altcode: 2018arXiv180305898K We consider the effect of a subadiabatic layer at the base of the convection zone on convection itself and the associated large-scale dynamos in spherical wedge geometry. We use a heat conduction prescription based on the Kramers opacity law which allows the depth of the convection zone to dynamically adapt to changes in the physical characteristics such as rotation rate and magnetic fields. We find that the convective heat transport is strongly concentrated towards the equatorial and polar regions in the cases without a substantial radiative layer below the convection zone. The presence of a stable layer below the convection zone significantly reduces the anisotropy of radial enthalpy transport. Furthermore, the dynamo solutions are sensitive to subtle changes in the convection zone structure. We find that the kinetic helicity changes sign in the deeper parts of the convection zone at high latitudes in all runs. This region expands progressively towards the equator in runs with a thicker stably stratified layer. Title: Magnetic and rotational quenching of the Λ effect Authors: Käpylä, P. J. Bibcode: 2019A&A...622A.195K Altcode: 2017arXiv171208045K Context. Differential rotation in stars is driven by the turbulent transport of angular momentum.
Aims: Our aim is to measure and parameterize the non-diffusive contribution to the total (Reynolds plus Maxwell) turbulent stress, known as the Λ effect, and its quenching as a function of rotation and magnetic field.
Methods: Simulations of homogeneous, anisotropically forced turbulence in fully periodic cubes are used to extract their associated turbulent Reynolds and Maxwell stresses. The forcing is set up such that the vertical velocity component dominates over the horizontal ones, as in turbulent stellar convection. This choice of the forcing defines the vertical direction. Additional preferred directions are introduced by the imposed rotation and magnetic field vectors. The angle between the rotation vector and the vertical direction is varied such that the latitude range from the north pole to the equator is covered. Magnetic fields are introduced by imposing a uniform large-scale field on the system. Turbulent transport coefficients pertaining to the Λ effect are obtained by fitting. The results are compared with analytic studies.
Results: The numerical and analytic results agree qualitatively at slow rotation and low Reynolds numbers. This means that vertical (horizontal) transport is downward (equatorward). At rapid rotation the latitude dependence of the stress is more complex than predicted by theory. The existence of a significant meridional Λ effect is confirmed. Large-scale vorticity generation is found at rapid rotation when the Reynolds number exceeds a threshold value. The Λ effect is severely quenched by large-scale magnetic fields due to the tendency of the Reynolds and Maxwell stresses to cancel each other. Rotational (magnetic) quenching of Λ occurs at more rapid rotation (at lower field strength) in the simulations than in the analytic studies.
Conclusions: The current results largely confirm the earlier theoretical results, and also offer new insights: the non-negligible meridional Λ effect possibly plays a role in the maintenance of meridional circulation in stars, and the appearance of large-scale vortices raises the question of their effect on the angular momentum transport in rapidly rotating stellar convective envelopes. The results regarding magnetic quenching are consistent with the strong decrease in differential rotation in recent semi-global simulations and highlight the importance of including magnetic effects in differential rotation models. Title: $f$-mode strengthening from a localized bipolar subsurface magnetic field Authors: Singh, Nishant K.; Raichur, Harsha; Käpylä, Maarit J.; Rheinhardt, Matthias; Brandenburg, Axel; Käpylä, Petri J. Bibcode: 2018arXiv180808904S Altcode: Recent numerical work in helioseismology has shown that a periodically varying subsurface magnetic field leads to a fanning of the $f$-mode, 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 $f$-mode 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 $f$-mode in the $k\omega$ diagram increases proportionally in such a way that the normalized $f$-mode 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 $f$-mode 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 $f$-mode 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. Title: Transition from axi- to nonaxisymmetric dynamo modes in spherical convection models of solar-like stars Authors: Viviani, M.; Warnecke, J.; Käpylä, M. J.; Käpylä, P. J.; Olspert, N.; Cole-Kodikara, E. M.; Lehtinen, J. J.; Brandenburg, A. Bibcode: 2018A&A...616A.160V Altcode: 2017arXiv171010222V Context. Both dynamo theory and observations of stellar large-scale magnetic fields suggest a change from nearly axisymmetric configurations at solar rotation rates to nonaxisymmetric configurations for rapid rotation.
Aims: We seek to understand this transition using numerical simulations.
Methods: We use three-dimensional simulations of turbulent magnetohydrodynamic convection in spherical shell wedges and considered rotation rates between 1 and 31 times the solar value.
Results: 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 solar-like differential rotation with a faster equator and slow poles. In the solar-like 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 slow drift in the rotating reference frame and the strength of the active longitudes changes cyclically over time between the different hemispheres. In the majority of cases we find retrograde waves, while prograde waves are more often found from observations. Most of the obtained dynamo solutions exhibit cyclic variability either caused by latitudinal or azimuthal dynamo waves. In an activity-period 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 in observations is called the inactive branch, where the stars are believed to have solar-like differential rotation, while the rapidly rotating models are close to the superactive branch with a declining cycle to rotation frequency ratio and an increasing rotation rate.
Conclusions: We can successfully reproduce the transition from axi- to nonaxisymmetric dynamo solutions for high rotation rates, but high-resolution simulations are required to limit the effect of rotational quenching of convection at rotation rates above 20 times the solar value. Title: Bihelical Spectrum of Solar Magnetic Helicity and Its Evolution Authors: Singh, Nishant K.; Käpylä, Maarit J.; Brandenburg, Axel; Käpylä, Petri J.; Lagg, Andreas; Virtanen, Ilpo Bibcode: 2018ApJ...863..182S Altcode: 2018arXiv180404994S Using a recently developed two-scale formalism to determine the magnetic helicity spectrum, we analyze synoptic vector magnetograms built with data from the Vector Spectromagnetograph instrument on the Synoptic Optical Long-term Investigations of the Sun telescope during 2010 January-2016 July. In contrast to an earlier study using only three Carrington rotations (CRs), our analysis includes 74 synoptic CR maps. We recover here bihelical spectra at different phases of solar cycle 24, where the net magnetic helicity in the majority of the data is consistent with a large-scale dynamo with helical turbulence operating in the Sun. More than 20% of the analyzed maps, however, show violations of the expected sign rule. Title: Large-scale dynamos in rapidly rotating plane layer convection Authors: Bushby, P. J.; Käpylä, P. J.; Masada, Y.; Brandenburg, A.; Favier, B.; Guervilly, C.; Käpylä, M. J. Bibcode: 2018A&A...612A..97B Altcode: 2017arXiv171003174B Context. Convectively driven flows play a crucial role in the dynamo processes that are responsible for producing magnetic activity in stars and planets. It is still not fully understood why many astrophysical magnetic fields have a significant large-scale component.
Aims: Our aim is to investigate the dynamo properties of compressible convection in a rapidly rotating Cartesian domain, focusing upon a parameter regime in which the underlying hydrodynamic flow is known to be unstable to a large-scale vortex instability.
Methods: The governing equations of three-dimensional non-linear magnetohydrodynamics (MHD) are solved numerically. Different numerical schemes are compared and we propose a possible benchmark case for other similar codes.
Results: In keeping with previous related studies, we find that convection in this parameter regime can drive a large-scale dynamo. The components of the mean horizontal magnetic field oscillate, leading to a continuous overall rotation of the mean field. Whilst the large-scale vortex instability dominates the early evolution of the system, the large-scale vortex is suppressed by the magnetic field and makes a negligible contribution to the mean electromotive force that is responsible for driving the large-scale dynamo. The cycle period of the dynamo is comparable to the ohmic decay time, with longer cycles for dynamos in convective systems that are closer to onset. In these particular simulations, large-scale dynamo action is found only when vertical magnetic field boundary conditions are adopted at the upper and lower boundaries. Strongly modulated large-scale dynamos are found at higher Rayleigh numbers, with periods of reduced activity (grand minima-like events) occurring during transient phases in which the large-scale vortex temporarily re-establishes itself, before being suppressed again by the magnetic field. Title: Small-scale dynamos in simulations of stratified turbulent convection Authors: Käpylä, P. J.; Käpylä, M. J.; Brandenburg, A. Bibcode: 2018AN....339..127K Altcode: 2018arXiv180209607K Small-scale dynamo action is often held responsible for the generation of quiet Sun magnetic fields. We aim to determine the excitation conditions and saturation level of small-scale dynamos in nonrotating turbulent convection at low magnetic Prandtl numbers. We use high-resolution direct numerical simulations of weakly stratified turbulent convection. We find that the critical magnetic Reynolds number for dynamo excitation increases as the magnetic Prandtl number is decreased, which might suggest that small-scale dynamo action is not automatically evident in bodies with small magnetic Prandtl numbers, such as the Sun. As a function of the magnetic Reynolds number (Rm), the growth rate of the dynamo is consistent with an Rm1/2 scaling. No evidence for a logarithmic increase of the growth rate with Rm is found. Title: Turbulent transport coefficients in spherical wedge dynamo simulations of solar-like stars Authors: Warnecke, J.; Rheinhardt, M.; Tuomisto, S.; Käpylä, P. J.; Käpylä, M. J.; Brandenburg, A. Bibcode: 2018A&A...609A..51W Altcode: 2016arXiv160103730W
Aims: We investigate dynamo action in global compressible solar-like convective dynamos in the framework of mean-field theory.
Methods: We simulate a solar-type star in a wedge-shaped spherical shell, where the interplay between convection and rotation self-consistently drives a large-scale dynamo. To analyze the dynamo mechanism we apply the test-field 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 solar-like dynamos.
Results: We find that the φφ-component 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 time-averaged values and seem to be responsible for a nonlinear feedback on the magnetic field generation. Furthermore, we quantify the validity of the Parker-Yoshimura rule for the equatorward propagation of the mean magnetic field in the present case. Title: Extended Subadiabatic Layer in Simulations of Overshooting Convection Authors: Käpylä, Petri J.; Rheinhardt, Matthias; Brandenburg, Axel; Arlt, Rainer; Käpylä, Maarit J.; Lagg, Andreas; Olspert, Nigul; Warnecke, Jörn Bibcode: 2017ApJ...845L..23K Altcode: 2017arXiv170306845K We present numerical simulations of hydrodynamic overshooting convection in local Cartesian domains. We find that a substantial fraction of the lower part of the convection zone (CZ) is stably stratified according to the Schwarzschild criterion while the enthalpy flux is outward directed. This occurs when the heat conduction profile at the bottom of the CZ is smoothly varying, based either on a Kramers-like opacity prescription as a function of temperature and density or a static profile of a similar shape. We show that the subadiabatic layer arises due to nonlocal energy transport by buoyantly driven downflows in the upper parts of the CZ. Analysis of the force balance of the upflows and downflows confirms that convection is driven by cooling at the surface. We find that the commonly used prescription for the convective enthalpy flux being proportional to the negative entropy gradient does not hold in the stably stratified layers where the flux is positive. We demonstrate the existence of a non-gradient contribution to the enthalpy flux, which is estimated to be important throughout the convective layer. A quantitative analysis of downflows indicates a transition from a tree-like structure where smaller downdrafts merge into larger ones in the upper parts to a structure in the deeper parts where a height-independent number of strong downdrafts persist. This change of flow topology occurs when a substantial subadiabatic layer is present in the lower part of the CZ. Title: Methods for compressible fluid simulation on GPUs using high-order finite differences Authors: Pekkilä, Johannes; Väisälä, Miikka S.; Käpylä, Maarit J.; Käpylä, Petri J.; Anjum, Omer Bibcode: 2017CoPhC.217...11P Altcode: 2017arXiv170708900P We focus on implementing and optimizing a sixth-order finite-difference solver for simulating compressible fluids on a GPU using third-order Runge-Kutta integration. Since graphics processing units perform well in data-parallel tasks, this makes them an attractive platform for fluid simulation. However, high-order stencil computation is memory-intensive with respect to both main memory and the caches of the GPU. We present two approaches for simulating compressible fluids using 55-point and 19-point stencils. We seek to reduce the requirements for memory bandwidth and cache size in our methods by using cache blocking and decomposing a latency-bound kernel into several bandwidth-bound kernels. Our fastest implementation is bandwidth-bound and integrates 343 million grid points per second on a Tesla K40t GPU, achieving a 3 . 6 × speedup over a comparable hydrodynamics solver benchmarked on two Intel Xeon E5-2690v3 processors. Our alternative GPU implementation is latency-bound and achieves the rate of 168 million updates per second. Title: Convection-driven spherical shell dynamos at varying Prandtl numbers Authors: Käpylä, P. J.; Käpylä, M. J.; Olspert, N.; Warnecke, J.; Brandenburg, A. Bibcode: 2017A&A...599A...4K Altcode: 2016arXiv160505885K Context. Stellar convection zones are characterized by vigorous high-Reynolds number turbulence at low Prandtl numbers.
Aims: We study the dynamo and differential rotation regimes at varying levels of viscous, thermal, and magnetic diffusion.
Methods: We perform three-dimensional simulations of stratified fully compressible magnetohydrodynamic convection in rotating spherical wedges at various thermal and magnetic Prandtl numbers (from 0.25 to 2 and from 0.25 to 5, respectively). Differential rotation and large-scale magnetic fields are produced self-consistently.
Results: We find that for high thermal diffusivity, the rotation profiles show a monotonically increasing angular velocity from the bottom of the convection zone to the top and from the poles toward the equator. For sufficiently rapid rotation, a region of negative radial shear develops at mid-latitudes as the thermal diffusivity is decreased, corresponding to an increase of the Prandtl number. This coincides with and results in a change of the dynamo mode from poleward propagating activity belts to equatorward propagating ones. Furthermore, the clearly cyclic solutions disappear at the highest magnetic Reynolds numbers and give way to irregular sign changes or quasi-stationary states. The total (mean and fluctuating) magnetic energy increases as a function of the magnetic Reynolds number in the range studied here (5-151), but the energies of the mean magnetic fields level off at high magnetic Reynolds numbers. The differential rotation is strongly affected by the magnetic fields and almost vanishes at the highest magnetic Reynolds numbers. In some of our most turbulent cases, however, we find that two regimes are possible, where either differential rotation is strong and mean magnetic fields are relatively weak, or vice versa.
Conclusions: Our simulations indicate a strong nonlinear feedback of magnetic fields on differential rotation, leading to qualitative changes in the behaviors of large-scale dynamos at high magnetic Reynolds numbers. Furthermore, we do not find indications of the simulations approaching an asymptotic regime where the results would be independent of diffusion coefficients in the parameter range studied here. Title: Influence of a coronal envelope as a free boundary to global convective dynamo simulations Authors: Warnecke, J.; Käpylä, P. J.; Käpylä, M. J.; Brandenburg, A. Bibcode: 2016A&A...596A.115W Altcode: 2015arXiv150305251W
Aims: We explore the effects of an outer stably stratified coronal envelope on rotating turbulent convection, differential rotation, and large-scale dynamo action in spherical wedge models of the Sun.
Methods: We solve the compressible magnetohydrodynamic equations in a two-layer model with unstable stratification below the surface, representing the convection zone, and a stably stratified coronal envelope above. The interface represents a free surface. We compare our model to models that have no coronal envelope.
Results: The presence of a coronal envelope is found to modify the Reynolds stress and the Λ effect resulting in a weaker and non-cylindrical differential rotation. This is related to the reduced latitudinal temperature variations that are caused by and dependent on the angular velocity. Some simulations develop a near-surface shear layer that we can relate to a sign change in the meridional Reynolds stress term in the thermal wind balance equation. Furthermore, the presence of a free surface changes the magnetic field evolution since the toroidal field is concentrated closer to the surface. In all simulations, however, the migration direction of the mean magnetic field can be explained by the Parker-Yoshimura rule, which is consistent with earlier findings.
Conclusions: A realistic treatment of the upper boundary in spherical dynamo simulations is crucial for the dynamics of the flow and magnetic field evolution. Title: Robustness of oscillatory α2 dynamos in spherical wedges Authors: Cole, E.; Brandenburg, A.; Käpylä, P. J.; Käpylä, M. J. Bibcode: 2016A&A...593A.134C Altcode: 2016arXiv160105246C Context. Large-scale dynamo simulations are sometimes confined to spherical wedge geometries by imposing artificial boundary conditions at high latitudes. This may lead to spatio-temporal behaviours that are not representative of those in full spherical shells.
Aims: We study the connection between spherical wedge and full spherical shell geometries using simple mean-field dynamos.
Methods: We solve the equations for one-dimensional time-dependent α2 and α2Ω mean-field dynamos with only latitudinal extent to examine the effects of varying the polar angle θ0 between the latitudinal boundaries and the poles in spherical coordinates.
Results: In the case of constant α and ηt profiles, we find oscillatory solutions only with the commonly used perfect conductor boundary condition in a wedge geometry, while for full spheres all boundary conditions produce stationary solutions, indicating that perfect conductor conditions lead to unphysical solutions in such a wedge setup. To search for configurations in which this problem can be alleviated we choose a profile of the turbulent magnetic diffusivity that decreases toward the poles, corresponding to high conductivity there. Oscillatory solutions are now achieved with models extending to the poles, but the magnetic field is strongly concentrated near the poles and the oscillation period is very long. By changing both the turbulent magnetic diffusivity and α profiles so that both effects are more concentrated toward the equator, we see oscillatory dynamos with equatorward drift, shorter cycles, and magnetic fields distributed over a wider range of latitudes. Those profiles thus remove the sensitive and unphysical dependence on θ0. When introducing radial shear, we again see oscillatory dynamos, and the direction of drift follows the Parker-Yoshimura rule.
Conclusions: A reduced α effect near the poles with a turbulent diffusivity concentrated toward the equator yields oscillatory dynamos with equatorward migration and reproduces best the solutions in spherical wedges. For weak shear, oscillatory solutions are obtained only for perfect conductor field conditions and negative shear. Oscillatory solutions become preferred at sufficiently strong shear. Recent three-dimensional dynamo simulations producing solar-like magnetic activity are expected to lie in this range. Title: Multiple dynamo modes as a mechanism for long-term solar activity variations Authors: Käpylä, M. J.; Käpylä, P. J.; Olspert, N.; Brandenburg, A.; Warnecke, J.; Karak, B. B.; Pelt, J. Bibcode: 2016A&A...589A..56K Altcode: 2015arXiv150705417K Context. Solar magnetic activity shows both smooth secular changes, such as the modern Grand Maximum, and quite abrupt drops that are denoted as grand minima, such as the Maunder Minimum. Direct numerical simulations (DNS) of convection-driven dynamos offer one way of examining the mechanisms behind these events.
Aims: In this work, we analyze a solution of a solar-like DNS that was evolved for roughly 80 magnetic cycles of 4.9 years and where epochs of irregular behavior are detected. The emphasis of our analysis is to find physical causes for such behavior.
Methods: The DNS employed is a semi-global (wedge-shaped) magnetoconvection model. For the data analysis tasks we use Ensemble Empirical Mode Decomposition and phase dispersion methods, as they are well suited for analyzing cyclic (non-periodic) signals.
Results: A special property of the DNS is the existence of multiple dynamo modes at different depths and latitudes. The dominant mode is solar-like (equatorward migration at low latitudes and poleward at high latitudes). This mode is accompanied by a higher frequency mode near the surface and at low latitudes, showing poleward migration, and a low-frequency mode at the bottom of the convection zone. The low-frequency mode is almost purely antisymmetric with respect to the equator, while the dominant mode has strongly fluctuating mixed parity. The overall behavior of the dynamo solution is extremely complex, exhibiting variable cycle lengths, epochs of disturbed and even ceased surface activity, and strong short-term hemispherical asymmetries. Surprisingly, the most prominent suppressed surface activity epoch is actually a global magnetic energy maximum; during this epoch the bottom toroidal magnetic field obtains a maximum, demonstrating that the interpretation of grand minima-type events is non-trivial. The hemispherical asymmetries are seen only in the magnetic field, while the velocity field exhibits considerably weaker asymmetry.
Conclusions: We interpret the overall irregular behavior as being due to the interplay of the different dynamo modes showing different equatorial symmetries, especially the smoother part of the irregular variations being related to the variations of the mode strengths, evolving with different and variable cycle lengths. The abrupt low-activity epoch in the dominant dynamo mode near the surface is related to a strong maximum of the bottom toroidal field strength, which causes abrupt disturbances especially in the differential rotation profile via the suppression of the Reynolds stresses. Title: Magnetic flux concentrations from turbulent stratified convection Authors: Käpylä, P. J.; Brandenburg, A.; Kleeorin, N.; Käpylä, M. J.; Rogachevskii, I. Bibcode: 2016A&A...588A.150K Altcode: 2015arXiv151103718K Context. The formation of magnetic flux concentrations within the solar convection zone leading to sunspot formation is unexplained.
Aims: We study the self-organization of initially uniform sub-equipartition magnetic fields by highly stratified turbulent convection.
Methods: We perform simulations of magnetoconvection in Cartesian domains representing the uppermost 8.5-24 Mm of the solar convection zone with the horizontal size of the domain varying between 34 and 96 Mm. The density contrast in the 24 Mm deep models is more than 3 × 103 or eight density scale heights, corresponding to a little over 12 pressure scale heights. We impose either a vertical or a horizontal uniform magnetic field in a convection-driven turbulent flow in set-ups where no small-scale dynamos are present. In the most highly stratified cases we employ the reduced sound speed method to relax the time step constraint arising from the high sound speed in the deep layers. We model radiation via the diffusion approximation and neglect detailed radiative transfer in order to concentrate on purely magnetohydrodynamic effects.
Results: We find that super-equipartition magnetic flux concentrations are formed near the surface in cases with moderate and high density stratification, corresponding to domain depths of 12.5 and 24 Mm. The size of the concentrations increases as the box size increases and the largest structures (20 Mm horizontally near the surface) are obtained in the models that are 24 Mm deep. The field strength in the concentrations is in the range of 3-5 kG, almost independent of the magnitude of the imposed field. The amplitude of the concentrations grows approximately linearly in time. The effective magnetic pressure measured in the simulations is positive near the surface and negative in the bulk of the convection zone. Its derivative with respect to the mean magnetic field, however, is positive in most of the domain, which is unfavourable for the operation of the negative effective magnetic pressure instability (NEMPI). Simulations in which a passive vector field is evolved do not show a noticeable difference from magnetohydrodynamic runs in terms of the growth of the structures. Furthermore, we find that magnetic flux is concentrated in regions of converging flow corresponding to large-scale supergranulation convection pattern.
Conclusions: The linear growth of large-scale flux concentrations implies that their dominant formation process is a tangling of the large-scale field rather than an instability. One plausible mechanism that can explain both the linear growth and the concentration of the flux in the regions of converging flow pattern is flux expulsion. A possible reason for the absence of NEMPI is that the derivative of the effective magnetic pressure with respect to the mean magnetic field has an unfavourable sign. Furthermore, there may not be sufficient scale separation, which is required for NEMPI to work.

Movies associated to Figs. 4 and 5 are available in electronic form at http://www.aanda.org Title: Magnetically controlled stellar differential rotation near the transition from solar to anti-solar profiles Authors: Karak, B. B.; Käpylä, P. J.; Käpylä, M. J.; Brandenburg, A.; Olspert, N.; Pelt, J. Bibcode: 2015A&A...576A..26K Altcode: 2014arXiv1407.0984K Context. Late-type stars rotate differentially owing to anisotropic turbulence in their outer convection zones. The rotation is called solar-like (SL) when the equator rotates fastest and anti-solar (AS) otherwise. Hydrodynamic simulations show a transition from SL to AS rotation as the influence of rotation on convection is reduced, but the opposite transition occurs at a different point in the parameter space. The system is bistable, i.e., SL and AS rotation profiles can both be stable.
Aims: We study the effect of a dynamo-generated magnetic field on the large-scale flows, particularly on the possibility of bistable behaviour of differential rotation.
Methods: We solve the hydromagnetic equations numerically in a rotating spherical shell that typically covers ± 75° latitude (wedge geometry) for a set of different radiative conductivities controlling the relative importance of convection. We analyse the resulting differential rotation, meridional circulation, and magnetic field and compare the corresponding modifications of the Reynolds and Maxwell stresses.
Results: In agreement with earlier findings, our models display SL rotation profiles when the rotational influence on convection is strong and a transition to AS when the rotational influence decreases. We find that dynamo-generated magnetic fields help to produce SL differential rotation compared to the hydrodynamic simulations. We do not observe any bistable states of differential rotation. In the AS cases we find coherent single-cell meridional circulation, whereas in SL cases we find multi-cellular patterns. In both cases, we obtain poleward circulation near the surface with a magnitude close to that observed in the Sun. In the slowly rotating cases, we find activity cycles, but no clear polarity reversals, whereas in the more rapidly rotating cases irregular variations are obtained. Moreover, both differential rotation and meridional circulation have significant temporal variations that are similar in strength to those of the Sun.
Conclusions: Purely hydrodynamic simulations of differential rotation and meridional circulation are shown to be of limited relevance as magnetic fields, self-consistently generated by dynamo action, significantly affect the flows. Title: Testing turbulent closure models with convection simulations Authors: Snellman, J. E.; Käpylä, P. J.; Käpylä, M. J.; Rheinhardt, M., Dintrans, B. Bibcode: 2015AN....336...32S Altcode: We compare simple analytical closure models of homogeneous turbulent Boussinesq convection for stellar applications with three-dimensional simulations. We use turbulent closure models to compute the Reynolds stresses and the turbulent heat flux as functions of rotation rate measured by the Taylor number. We also investigate cases with varying angles between the angular velocity and gravity vectors, corresponding to locating the computational domain at different latitudes ranging from the pole to the equator of the star. We perform three-dimensional numerical simulations in the same parameter regimes for comparison. The free parameters appearing in the closure models are calibrated by two fitting methods using simulation data. A unique determination of the closure parameters is possible only in the non-rotating case or when the system is placed at the pole. In the other cases the fit procedures yield somewhat differing results. The quality of the closure is tested by substituting the resulting coefficients back into the closure model and comparing with the simulation results. To eliminate the possibilities that the results obtained depend on the aspect ratio of the simulation domain or suffer from too small Rayleigh numbers we performed runs varying these parameters. The simulation data for the Reynolds stress and heat fluxes broadly agree with previous compressible simulations. The closure works fairly well with slow and fast rotation but its quality degrades for intermediate rotation rates. We find that the closure parameters depend not only on rotation rate but also on latitude. The weak dependence on Rayleigh number and on the aspect ratio of the domain indicates that our results are generally valid. Title: Quenching and Anisotropy of Hydromagnetic Turbulent Transport Authors: Karak, Bidya Binay; Rheinhardt, Matthias; Brandenburg, Axel; Käpylä, Petri J.; Käpylä, Maarit J. Bibcode: 2014ApJ...795...16K Altcode: 2014arXiv1406.4521K Hydromagnetic turbulence affects the evolution of large-scale magnetic fields through mean-field effects like turbulent diffusion and the α effect. For stronger fields, these effects are usually suppressed or quenched, and additional anisotropies are introduced. Using different variants of the test-field method, we determine the quenching of the turbulent transport coefficients for the forced Roberts flow, isotropically forced non-helical turbulence, and rotating thermal convection. We see significant quenching only when the mean magnetic field is larger than the equipartition value of the turbulence. Expressing the magnetic field in terms of the equipartition value of the quenched flows, we obtain for the quenching exponents of the turbulent magnetic diffusivity about 1.3, 1.1, and 1.3 for Roberts flow, forced turbulence, and convection, respectively. However, when the magnetic field is expressed in terms of the equipartition value of the unquenched flows, these quenching exponents become about 4, 1.5, and 2.3, respectively. For the α effect, the exponent is about 1.3 for the Roberts flow and 2 for convection in the first case, but 4 and 3, respectively, in the second. In convection, the quenching of turbulent pumping follows the same power law as turbulent diffusion, while for the coefficient describing the {\boldsymbolΩ} × \boldsymbol {{J}} effect nearly the same quenching exponent is obtained as for α. For forced turbulence, turbulent diffusion proportional to the second derivative along the mean magnetic field is quenched much less, especially for larger values of the magnetic Reynolds number. However, we find that in corresponding axisymmetric mean-field dynamos with dominant toroidal field the quenched diffusion coefficients are the same for the poloidal and toroidal field constituents. Title: On The Cause of Solar-like Equatorward Migration in Global Convective Dynamo Simulations Authors: Warnecke, Jörn; Käpylä, Petri J.; Käpylä, Maarit J.; Brandenburg, Axel Bibcode: 2014ApJ...796L..12W Altcode: 2014arXiv1409.3213W We present results from four convectively driven stellar dynamo simulations in spherical wedge geometry. All of these simulations produce cyclic and migrating mean magnetic fields. Through detailed comparisons, we show that the migration direction can be explained by an αΩ dynamo wave following the Parker-Yoshimura rule. We conclude that the equatorward migration in this and previous work is due to a positive (negative) α effect in the northern (southern) hemisphere and a negative radial gradient of Ω outside the inner tangent cylinder of these models. This idea is supported by a strong correlation between negative radial shear and toroidal field strength in the region of equatorward propagation. Title: Confirmation of bistable stellar differential rotation profiles Authors: Käpylä, P. J.; Käpylä, M. J.; Brandenburg, A. Bibcode: 2014A&A...570A..43K Altcode: 2014arXiv1401.2981K Context. Solar-like differential rotation is characterized by a rapidly rotating equator and slower poles. However, theoretical models and numerical simulations can also result in a slower equator and faster poles when the overall rotation is slow.
Aims: We study the critical rotational influence under which differential rotation flips from solar-like (fast equator, slow poles) to an anti-solar one (slow equator, fast poles). We also estimate the non-diffusive (Λ effect) and diffusive (turbulent viscosity) contributions to the Reynolds stress.
Methods: We present the results of three-dimensional numerical simulations of mildly turbulent convection in spherical wedge geometry. Here we apply a fully compressible setup which would suffer from a prohibitive time step constraint if the real solar luminosity was used. To avoid this problem while still representing the same rotational influence on the flow as in the Sun, we increase the luminosity by a factor of roughly 106 and the rotation rate by a factor of 102. We regulate the convective velocities by varying the amount of heat transported by thermal conduction, turbulent diffusion, and resolved convection.
Results: Increasing the efficiency of resolved convection leads to a reduction of the rotational influence on the flow and a sharp transition from solar-like to anti-solar differential rotation for Coriolis numbers around 1.3. We confirm the recent finding of a large-scale flow bistability: contrasted with running the models from an initial condition with unprescribed differential rotation, the initialization of the model with certain kind of rotation profile sustains the solution over a wider parameter range. The anti-solar profiles are found to be more stable against perturbations in the level of convective turbulent velocity than the solar-type solutions.
Conclusions: Our results may have implications for real stars that start their lives as rapid rotators implying solar-like rotation in the early main-sequence evolution. As they slow down, they might be able to retain solar-like rotation for lower Coriolis numbers, and thus longer in time, before switching to anti-solar rotation. This could partially explain the puzzling findings of anti-solar rotation profiles for models in the solar parameter regime. Title: Quantifying the effect of turbulent magnetic diffusion on the growth rate of the magneto-rotational instability Authors: Väisälä, M. S.; Brandenburg, A.; Mitra, D.; Käpylä, P. J.; Mantere, M. J. Bibcode: 2014A&A...567A.139V Altcode: 2013arXiv1310.3157V Context. In astrophysics, turbulent diffusion is often used in place of microphysical diffusion to avoid resolving the small scales. However, we expect this approach to break down when time and length scales of the turbulence become comparable with other relevant time and length scales in the system. Turbulent diffusion has previously been applied to the magneto-rotational instability (MRI), but no quantitative comparison of growth rates at different turbulent intensities has been performed.
Aims: We investigate to what extent turbulent diffusion can be used to model the effects of small-scale turbulence on the kinematic growth rates of the MRI, and how this depends on angular velocity and magnetic field strength.
Methods: We use direct numerical simulations in three-dimensional shearing boxes with periodic boundary conditions in the spanwise direction and additional random plane-wave volume forcing to drive a turbulent flow at a given length scale. We estimate the turbulent diffusivity using a mixing length formula and compare with results obtained with the test-field method.
Results: It turns out that the concept of turbulent diffusion is remarkably accurate in describing the effect of turbulence on the growth rate of the MRI. No noticeable breakdown of turbulent diffusion has been found, even when time and length scales of the turbulence become comparable with those imposed by the MRI itself. On the other hand, quenching of turbulent magnetic diffusivity by the magnetic field is found to be absent.
Conclusions: Turbulence reduces the growth rate of the MRI in the same way as microphysical magnetic diffusion does.

Appendix A is available in electronic form at http://www.aanda.org Title: An Azimuthal Dynamo Wave in Spherical Shell Convection Authors: Cole, Elizabeth; Käpylä, Petri J.; Mantere, Maarit J.; Brandenburg, Axel Bibcode: 2014ApJ...780L..22C Altcode: 2013arXiv1309.6802C We report the discovery of an azimuthal dynamo wave of a low-order (m = 1) mode in direct numerical simulations (DNS) of turbulent convection in spherical shells. Such waves are predicted by mean-field dynamo theory and have been obtained previously in mean-field models. An azimuthal dynamo wave has been proposed as a possible explanation for the persistent drifts of spots observed on several rapidly rotating stars, as revealed through photometry and Doppler imaging. However, this has been judged unlikely because evidence for such waves from DNS has been lacking. Here we present DNS of large-scale magnetic fields showing a retrograde m = 1 mode. Its pattern speed is nearly independent of latitude and does not reflect the speed of the differential rotation at any depth. The extrema of magnetic m = 1 structures coincide reasonably well with the maxima of m = 2 structures of the temperature. These results provide direct support for the observed drifts being due to an azimuthal dynamo wave. Title: Spoke-like Differential Rotation in a Convective Dynamo with a Coronal Envelope Authors: Warnecke, Jörn; Käpylä, Petri J.; Mantere, Maarit J.; Brandenburg, Axel Bibcode: 2013ApJ...778..141W Altcode: 2013arXiv1301.2248W We report on the results of four convective dynamo simulations with an outer coronal layer. The magnetic field is self-consistently generated by the convective motions beneath the surface. Above the convection zone, we include a polytropic layer that extends to 1.6 solar radii. The temperature increases in this region to ≈8 times the value at the surface, corresponding to ≈1.2 times the value at the bottom of the spherical shell. We associate this region with the solar corona. We find solar-like differential rotation with radial contours of constant rotation rate, together with a near-surface shear layer. This non-cylindrical rotation profile is caused by a non-zero latitudinal entropy gradient that offsets the Taylor-Proudman balance through the baroclinic term. The meridional circulation is multi-cellular with a solar-like poleward flow near the surface at low latitudes. In most of the cases, the mean magnetic field is oscillatory with equatorward migration in two cases. In other cases, the equatorward migration is overlaid by stationary or even poleward migrating mean fields. Title: Effects of Enhanced Stratification on Equatorward Dynamo Wave Propagation Authors: Käpylä, Petri J.; Mantere, Maarit J.; Cole, Elizabeth; Warnecke, Jörn; Brandenburg, Axel Bibcode: 2013ApJ...778...41K Altcode: 2013arXiv1301.2595K 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 quasi-steady 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 quasi-steady 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 three to seven times the solar value. The phase relation of the magnetic field is such that the toroidal field lags behind the radial field by about π/2, which can be explained by an oscillatory α2 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 2π azimuthal extent, reminiscent of the field configurations deduced from observations of rapidly rotating late-type stars. Title: Solar-like differential rotation and equatorward migration in a convective dynamo with a coronal envelope Authors: Warnecke, J.; Käpylä, P. J.; Mantere, M. J.; Brandenburg, A. Bibcode: 2013IAUS..294..307W Altcode: 2012arXiv1211.0452W We present results of convective turbulent dynamo simulations including a coronal layer in a spherical wedge. We find an equatorward migration of the radial and azimuthal fields similar to the behavior of sunspots during the solar cycle. The migration of the field coexist with a spoke-like differential rotation and anti-solar (clockwise) meridional circulation. Even though the migration extends over the whole convection zone, the mechanism causing this is not yet fully understood. Title: Role of longitudinal activity complexes for solar and stellar dynamos Authors: Mantere, Maarit J.; Käpylä, Petri J.; Pelt, Jaan Bibcode: 2013IAUS..294..175M Altcode: 2012arXiv1211.2990M In this paper we first discuss observational evidence of longitudinal concentrations of magnetic activity in the Sun and rapidly rotating late-type stars with outer convective envelopes. Scenarios arising from the idea of rotationally influenced anisotropic convective turbulence being the key physical process generating these structures are then presented and discussed - such effects include the turbulent dynamo mechanism, negative effective magnetic pressure instability (NEMPI) and hydrodynamical vortex instability. Finally, we discuss non-axisymmetric stellar mean-field dynamo models, the results obtained with them, and compare those with the observational information gathered up so far. We also present results from a pure α2 mean-field dynamo model, which show that time-dependent behavior of the dynamo solutions can occur both in the form of an azimuthal dynamo wave and/or oscillatory behavior related to the alternating energy levels of the active longitudes. Title: Flux concentrations in turbulent convection Authors: Käpylä, Petri J.; Brandenburg, Axel; Kleeorin, Nathan; Mantere, Maarit J.; Rogachevskii, Igor Bibcode: 2013IAUS..294..283K Altcode: 2012arXiv1211.2962K We present preliminary results from high resolution magneto-convection simulations where we find the formation of flux concentrations from an initially uniform magnetic field. The structures appear in roughly ten convective turnover times and live close to a turbulent diffusion time. The time scales are compatible with the negative effective magnetic pressure instability (NEMPI), although structure formation is not restricted to regions where the effective magnetic pressure is negative. Title: Oscillatory large-scale dynamos from Cartesian convection simulations Authors: Käpylä, P. J.; Mantere, M. J.; Brandenburg, A. Bibcode: 2013GApFD.107..244K Altcode: 2011arXiv1111.6894K We present results from compressible Cartesian convection simulations with and without imposed shear. In the former case the dynamo is expected to be of α2 Ω type, which is generally expected to be relevant for the Sun, whereas the latter case refers to α2 dynamos that are more likely to occur in more rapidly rotating stars whose differential rotation is small. We perform a parameter study where the shear flow and the rotational influence are varied to probe the relative importance of both types of dynamos. Oscillatory solutions are preferred both in the kinematic and saturated regimes when the negative ratio of shear to rotation rates, q ≡ -S/Ω, is between 1.5 and 2, i.e. when shear and rotation are of comparable strengths. Other regions of oscillatory solutions are found with small values of q, i.e. when shear is weak in comparison to rotation, and in the regime of large negative qs, when shear is very strong in comparison to rotation. However, exceptions to these rules also appear so that for a given ratio of shear to rotation, solutions are non-oscillatory for small and large shear, but oscillatory in the intermediate range. Changing the boundary conditions from vertical field to perfect conductor ones changes the dynamo mode from oscillatory to quasi-steady. Furthermore, in many cases an oscillatory solution exists only in the kinematic regime whereas in the nonlinear stage the mean fields are stationary. However, the cases with rotation and no shear are always oscillatory in the parameter range studied here and the dynamo mode does not depend on the magnetic boundary conditions. The strengths of total and large-scale components of the magnetic field in the saturated state, however, are sensitive to the chosen boundary conditions. Title: New Scaling for the Alpha Effect in Slowly Rotating Turbulence Authors: Brandenburg, A.; Gressel, O.; Käpylä, P. J.; Kleeorin, N.; Mantere, M. J.; Rogachevskii, I. Bibcode: 2013ApJ...762..127B Altcode: 2012arXiv1208.5004B Using simulations of slowly rotating stratified turbulence, we show that the α effect responsible for the generation of astrophysical magnetic fields is proportional to the logarithmic gradient of kinetic energy density rather than that of momentum, as was previously thought. This result is in agreement with a new analytic theory developed in this paper for large Reynolds numbers and slow rotation. Thus, the contribution of density stratification is less important than that of turbulent velocity. The α effect and other turbulent transport coefficients are determined by means of the test-field method. In addition to forced turbulence, we also investigate supernova-driven turbulence and stellar convection. In some cases (intermediate rotation rate for forced turbulence, convection with intermediate temperature stratification, and supernova-driven turbulence), we find that the contribution of density stratification might be even less important than suggested by the analytic theory. Title: Coronal ejections driven by convective dynamo action underneath the surface Authors: Warnecke, J.; Käpylä, P. J.; Mantere, M. J.; Brandeburg, A. Bibcode: 2012AGUFMSH51A2193W Altcode: Observations show that the Sun sheds mass through twisted magnetic flux configurations, like Coronal Mass Ejections (CMEs). Conventionally, CMEs are modeled by adopting a given distribution of magnetic flux at the solar surface and letting it evolve by shearing and twisting the magnetic field at its footpoints at the surface. Of course, ultimately such velocity and magnetic field patterns must come from a realistic simulation of the Sun's convection zone, where the field is generated by dynamo action. Therefore a unified treatment of convection zone and CMEs is needed. We combine a convectively driven dynamo in a spherical shell with a temperature and density stratification in the exterior that mimics a solar corona to study the emergence and ejections of magnetic field structures. This approach is an extension of earlier models where we use an isothermal corona (Warnecke et al. 2011) or even forced turbulence simulations to generate magnetic fields (Warnecke & Brandenburg 2010, Warnecke et al. 2011). A spherical wedge is used which consists of a convection zone and a corona up to 1.5 of the radius of the Sun. The wedge contains a quarter of the azimuthal extent of the sphere and 150o in latitude. The magnetic field is self-consistently generated by the turbulent motions due to convection underneath the surface. An oscillatory large-scale dynamo with equatorward migration in the lower layer can be found operating in the convection zone. In the Figure we show the equatorward migration of the radial magnetic field Br and the azimutal field Bφ as a function of time and latitude at the height of 0.97 solar radii. Magnetic fields are found to emerge at the surface and are ejected to the coronal part of the domain. These ejections occur in irregular intervals. We associate these events with coronal mass ejections on the Sun.; Variation of /line{Bφ } and /line{Br} in the convection zone at r=0.97R. Dark blue shades represent negative and light yellow positive values. The dotted horizontal lines show the location of the equator at θ =π /2. The magnetic field is normalized by the equipartition value. Title: Ejections of Magnetic Structures Above a Spherical Wedge Driven by a Convective Dynamo with Differential Rotation Authors: Warnecke, Jörn; Käpylä, Petri J.; Mantere, Maarit J.; Brandenburg, Axel Bibcode: 2012SoPh..280..299W Altcode: 2012SoPh..tmp..214W; 2011arXiv1112.0505W We combine a convectively driven dynamo in a spherical shell with a nearly isothermal density-stratified cooling layer that mimics some aspects of a stellar corona to study the emergence and ejections of magnetic field structures. This approach is an extension of earlier models, where forced turbulence simulations were employed to generate magnetic fields. A spherical wedge is used which consists of a convection zone and an extended coronal region to ≈ 1.5 times the radius of the sphere. The wedge contains a quarter of the azimuthal extent of the sphere and 150 in latitude. The magnetic field is self-consistently generated by the turbulent motions due to convection beneath the surface. Magnetic fields are found to emerge at the surface and are ejected to the coronal part of the domain. These ejections occur at irregular intervals and are weaker than in earlier work. We tentatively associate these events with coronal mass ejections on the Sun, even though our model of the solar atmosphere is rather simplistic. Title: Testing turbulent closure models with convection simulations Authors: Snellman, J. E.; Käpylä, P. J.; Käpylä, M. J.; Rheinhardt, M.; Dintrans, B. Bibcode: 2012arXiv1209.4923S Altcode: We compare simple analytical closure models of homogeneous turbulent Boussinesq convection for stellar applications with three-dimensional simulations. We use simple analytical closure models to compute the fluxes of angular momentum and heat as a function of rotation rate measured by the Taylor number. We also investigate cases with varying angles between the angular velocity and gravity vectors, corresponding to locating the computational domain at different latitudes ranging from the pole to the equator of the star. We perform three-dimensional numerical simulations in the same parameter regimes for comparison. The free parameters appearing in the closure models are calibrated by two fitting methods using simulation data. Unique determination of the closure parameters is possible only in the non-rotating case or when the system is placed at the pole. In the other cases the fit procedures yield somewhat differing results. The quality of the closure is tested by substituting the resulting coefficients back into the closure model and comparing with the simulation results. To eliminate the possibilities that the results obtained depend on the aspect ratio of the simulation domain or suffer from too small Rayleigh numbers we performed runs varying these parameters. The simulation data for the Reynolds stress and heat fluxes broadly agree with previous compressible simulations. The closure works fairly well with slow and fast rotation but its quality degrades for intermediate rotation rates. We find that the closure parameters depend not only on rotation rate but also on latitude. The weak dependence on Rayleigh number and the aspect ratio of the domain indicates that our results are generally valid Title: Cyclic Magnetic Activity due to Turbulent Convection in Spherical Wedge Geometry Authors: Käpylä, Petri J.; Mantere, Maarit J.; Brandenburg, Axel Bibcode: 2012ApJ...755L..22K Altcode: 2012arXiv1205.4719K We report on simulations of turbulent, rotating, stratified, magnetohydrodynamic convection in spherical wedge geometry. An initially small-scale, random, weak-amplitude magnetic field is amplified by several orders of magnitude in the course of the simulation to form oscillatory large-scale fields in the saturated state of the dynamo. The differential rotation is solar-like (fast equator), but neither coherent meridional poleward circulation nor near-surface shear layer develop in these runs. In addition to a poleward branch of magnetic activity beyond 50° latitude, we find for the first time a pronounced equatorward branch at around 20° latitude, reminiscent of the solar cycle. Title: Coronal ejections from convective spherical shell dynamos Authors: Warnecke, J.; Käpylä, P. J.; Mantere, M. J.; Brandenburg, A. Bibcode: 2012IAUS..286..154W Altcode: 2011arXiv1111.1763W We present a three-dimensional model of rotating convection combined with a simplified model of a corona in spherical coordinates. The motions in the convection zone generate a large-scale magnetic field which is sporadically ejected into the outer layers above. Our model corona is approximately isothermal, but it includes density stratification due to gravity. Title: Mean-field closure parameters for passive scalar turbulence Authors: Snellman, J. E.; Rheinhardt, M.; Käpylä, P. J.; Mantere, M. J.; Brandenburg, A. Bibcode: 2012PhyS...86a8406S Altcode: 2011arXiv1112.4777S Direct numerical simulations (DNSs) of isotropically forced homogeneous stationary turbulence with an imposed passive scalar concentration gradient are compared with an analytical closure model which provides evolution equations for the mean passive scalar flux and variance. Triple correlations of fluctuations appearing in these equations are described in terms of relaxation terms proportional to the quadratic correlations. Three methods are used to extract the relaxation timescales τi from DNSs. Firstly, we insert the closure ansatz into our equations, assume stationarity and solve for τi. Secondly, we use only the closure ansatz itself and obtain τi from the ratio of quadratic and triple correlations. Thirdly, we remove the imposed passive scalar gradient and fit an exponential law to the decaying solution. We vary the Reynolds (Re) and Péclet numbers (while fixing their ratio at unity) and the degree of scale separation and find for large Re a fair correspondence between the different methods. The ratio of the turbulent relaxation time of the passive scalar flux to the turnover time of the turbulent eddies is of the order of 3, which is in remarkable agreement with earlier work. Finally, we make an effort to extract the relaxation timescales relevant for the viscous and diffusive effects. We find two regimes that are valid for small and large Re, respectively, but the dependence of the parameters on scale separation suggests that they are not universal. Title: Negative effective magnetic pressure in turbulent convection Authors: Käpylä, P. J.; Brandenburg, A.; Kleeorin, N.; Mantere, M. J.; Rogachevskii, I. Bibcode: 2012MNRAS.422.2465K Altcode: 2011arXiv1104.4541K We investigate the effects of weakly and strongly stratified turbulent convection on the mean effective Lorentz force, and especially on the mean effective magnetic pressure. Earlier studies with isotropically forced non-stratified and stratified turbulence have shown that the contribution of the turbulence to the mean magnetic pressure is negative for mean horizontal magnetic fields that are smaller than the equipartition strength, so that the effective mean magnetic pressure that takes into account the turbulence effects can be negative. Compared with earlier cases of forced turbulence with an isothermal equation of state, we find that the turbulence effect is similar to or even stronger in the present case of turbulent convection. This is argued to be due to the anisotropy of turbulence in the vertical direction. Another important difference compared with earlier studies is the presence of an evolution equation for the specific entropy. Mean-field modelling with entropy evolution indicates that the negative effective magnetic pressure can still lead to a large-scale instability which forms local flux concentrations, even though the specific entropy evolution tends to have a stabilizing effect when applied to a stably stratified (e.g. isothermal) layer. It is argued that this large-scale instability could be important for the formation of solar large-scale magnetic structures such as active regions. Title: Verification of Reynolds stress parameterizations from simulations Authors: Snellman, J. E.; Brandenburg, A.; Käpylä, P. J.; Mantere, M. J. Bibcode: 2012AN....333...78S Altcode: 2011arXiv1109.4857S We determine the timescales associated with turbulent decay }and isotropization in closure models using anisotropically forced and freely decaying turbulence simulations and study the applicability of these models. We compare the results from anisotropically forced three-dimensional numerical simulations with the predictions of the closure models and obtain the turbulent timescales mentioned above as functions of the Reynolds number. In a second set of simulations, turning the forcing off enables us to study the validity of the closures in freely decaying turbulence. Both types of experiments suggest that the timescale of turbulent decay converges to a constant value at higher Reynolds numbers. Furthermore, the relative importance of isotropization is found to be about 2.5 times larger at higher Reynolds numbers than in the more viscous regime. Title: Dependence of the large-scale vortex instability on latitude, stratification, and domain size Authors: Mantere, M. J.; Käpylä, P. J.; Hackman, T. Bibcode: 2011AN....332..876M Altcode: 2011arXiv1109.4317M In an earlier study, we reported on the excitation of large-scale vortices in Cartesian hydrodynamical convection models subject to rapid enough rotation. In that study, the conditions for the onset of the instability were investigated in terms of the Reynolds (Re) and Coriolis (Co) numbers in models located at the stellar North pole. In this study, we extend our investigation to varying domain sizes, increasing stratification, and place the box at different latitudes. The effect of the increasing box size is to increase the sizes of the generated structures, so that the principal vortex always fills roughly half of the computational domain. The instability becomes stronger in the sense that the temperature anomaly and change in the radial velocity are observed to be enhanced. The model with the smallest box size is found to be stable against the instability, suggesting that a sufficient scale separation between the convective eddies and the scale of the domain is required for the instability to work. The instability can be seen upto the colatitude of 30 degrees, above which value the flow becomes dominated by other types of mean flows. The instability can also be seen in a model with larger stratification. Unlike the weakly stratified cases, the temperature anomaly caused by the vortex structures is seen to depend on depth. Title: Effects of stratification in spherical shell convection Authors: Käpylä, P. J.; Mantere, M. J.; Brandenburg, A. Bibcode: 2011AN....332..883K Altcode: 2011arXiv1109.4625K We report on simulations of mildly turbulent convection in spherical wedge geometry with varying density stratification. We vary the density contrast within the convection zone by a factor of 20 and study the influence of rotation on the solutions. We demonstrate that the size of convective cells decreases and the anisotropy of turbulence increases as the stratification is increased. Differential rotation is found to change from anti-solar (slow equator) to solar-like (fast equator) at roughly the same Coriolis number for all stratifications. The largest stratification runs, however, are sensitive to changes of the Reynolds number. Evidence for a near-surface shear layer is found in runs with strong stratification and large Reynolds numbers. Title: Pumping velocity in homogeneous helical turbulence with shear Authors: Rogachevskii, Igor; Kleeorin, Nathan; Käpylä, Petri J.; Brandenburg, Axel Bibcode: 2011PhRvE..84e6314R Altcode: 2011arXiv1105.5785R Using different analytical methods (the quasilinear approach, the path-integral technique, and the tau-relaxation approximation) we develop a comprehensive mean-field theory for a pumping effect of the mean magnetic field in homogeneous nonrotating helical turbulence with imposed large-scale shear. The effective pumping velocity is proportional to the product of α effect and large-scale vorticity associated with the shear, and causes a separation of the toroidal and poloidal components of the mean magnetic field along the direction of the mean vorticity. We also perform direct numerical simulations of sheared turbulence in different ranges of hydrodynamic and magnetic Reynolds numbers and use a kinematic test-field method to determine the effective pumping velocity. The results of the numerical simulations are in agreement with the theoretical predictions. Title: Starspots due to Large-scale Vortices in Rotating Turbulent Convection Authors: Käpylä, Petri J.; Mantere, Maarit J.; Hackman, Thomas Bibcode: 2011ApJ...742...34K Altcode: 2011arXiv1106.6029K We study the generation of large-scale vortices in rotating turbulent convection by means of Cartesian direct numerical simulations. We find that for sufficiently rapid rotation, cyclonic structures on a scale large in comparison to that of the convective eddies emerge, provided that the fluid Reynolds number exceeds a critical value. For slower rotation, cool cyclonic vortices are preferred, whereas for rapid rotation, warm anti-cyclonic vortices are favored. In some runs in the intermediate regime both types of cyclones coexist for thousands of convective turnover times. The temperature contrast between the vortices and the surrounding atmosphere is of the order of 5%. We relate the simulation results to observations of rapidly rotating late-type stars that are known to exhibit large high-latitude spots from Doppler imaging. In many cases, cool spots are accompanied with spotted regions with temperatures higher than the average. In this paper, we investigate a scenario according to which of the spots observed in the temperature maps could have a non-magnetic origin due to large-scale vortices in the convection zones of the stars. Title: Dynamo action and magnetic buoyancy in convection simulations with vertical shear Authors: Guerrero, G.; Käpylä, P. Bibcode: 2011sdmi.confE..62G Altcode: A hypothesis for sunspot formation is the buoyant emergence of magnetic flux tubes created by the strong radial shear at the tachocline. In this scenario, the magnetic field has to exceed a threshold value before it becomes buoyant and emerges through the whole convection zone. In this work we present the results of direct numerical simulations of compressible turbulent convection that include a vertical shear layer. Like the solar tachocline, the shear is located at the interface between convective and stable layers. We follow the evolution of a random seed magnetic field with the aim of study under what conditions it is possible to excite the dynamo instability and whether the dynamo generated magnetic field becomes buoyantly unstable and emerges to the surface as expected in the flux-tube context. We find that shear and convection are able to amplify the initial magnetic field and form large-scale elongated magnetic structures. The magnetic field strength depends on several parameters such as the shear amplitude, the thickness and location of the shear layer, and the magnetic Reynolds number (Rm). Models with deeper and thicker shear layers allow longer storage and are more favorable for generating a mean magnetic field. Models with higher Rm grow faster but saturate at slightly lower levels. Whenever the toroidal magnetic field reaches amplitudes greater a threshold value which is close to the equipartition value, it becomes buoyant and rises into the convection zone where it expands and forms mushroom shape structures. Some events of emergence, i.e., those with the largest amplitudes of the amplified field, are able to reach the very uppermost layers of the domain. These episodes are able to modify the convective pattern forming either broader convection cells or convective eddies elongated in the direction of the field. However, in none of these events the field preserves its initial structure. The back-reaction of the magnetic field on the fluid is also observed in lower values of the turbulent velocity and in perturbations of approximately three per cent on the shear profile. Title: Dynamo action and magnetic buoyancy in convection simulations with vertical shear Authors: Guerrero, G.; Käpylä, P. J. Bibcode: 2011A&A...533A..40G Altcode: 2011arXiv1102.3598G Context. A hypothesis for sunspot formation is the buoyant emergence of magnetic flux tubes created by the strong radial shear at the tachocline. In this scenario, the magnetic field has to exceed a threshold value before it becomes buoyant and emerges through the whole convection zone.
Aims: We follow the evolution of a random seed magnetic field with the aim of study under what conditions it is possible to excite the dynamo instability and whether the dynamo generated magnetic field becomes buoyantly unstable and emerges to the surface as expected in the flux-tube context.
Methods: We perform numerical simulations of compressible turbulent convection that include a vertical shear layer. Like the solar tachocline, the shear is located at the interface between convective and stable layers.
Results: We find that shear and convection are able to amplify the initial magnetic field and form large-scale elongated magnetic structures. The magnetic field strength depends on several parameters such as the shear amplitude, the thickness and location of the shear layer, and the magnetic Reynolds number (Rm). Models with deeper and thicker tachoclines allow longer storage and are more favorable for generating a mean magnetic field. Models with higher Rm grow faster but saturate at slightly lower levels. Whenever the toroidal magnetic field reaches amplitudes greater a threshold value which is close to the equipartition value, it becomes buoyant and rises into the convection zone where it expands and forms mushroom shape structures. Some events of emergence, i.e. those with the largest amplitudes of the initial field, are able to reach the very uppermost layers of the domain. These episodes are able to modify the convective pattern forming either broader convection cells or convective eddies elongated in the direction of the field. However, in none of these events the field preserves its initial structure. The back-reaction of the magnetic field on the fluid is also observed in lower values of the turbulent velocity and in perturbations of approximately three per cent on the shear profile.
Conclusions: The results indicate that buoyancy is a common phenomena when the magnetic field is amplified through dynamo action in a narrow layer. It is, however, very hard for the field to rise up to the surface without losing its initial coherence. Title: From convective to stellar dynamos Authors: Brandenburg, Axel; Käpylä, Petri J.; Korpi, Maarit J. Bibcode: 2011IAUS..271..279B Altcode: 2011arXiv1103.5475B Convectively driven dynamos with rotation generating magnetic fields on scales large compared with the scale of the turbulent eddies are being reviewed. It is argued that such fields can be understood as the result of an α effect. Simulations in Cartesian domains show that such large-scale magnetic fields saturate on a time scale compatible with the resistive one, suggesting that the magnitude of the α effect is here still constrained by approximate magnetic helicity conservation. It is argued that, in the absence of shear and/or any other known large-scale dynamo effects, these simulations prove the existence of turbulent α2-type dynamos. Finally, recent results are discussed in the context of solar and stellar dynamos. Title: Turbulence and magnetic spots at the surface of hot massive stars Authors: Cantiello, Matteo; Braithwaite, Jonathan; Brandenburg, Axel; Del Sordo, Fabio; Käpylä, Petri; Langer, Norbert Bibcode: 2011IAUS..273..200C Altcode: 2010arXiv1010.2498C Hot luminous stars show a variety of phenomena in their photospheres and in their winds which still lack clear physical explanations at this time. Among these phenomena are non-thermal line broadening, line profile variability (LPVs), discrete absorption components (DACs), wind clumping and stochastically excited pulsations. Cantiello et al. (2009) argued that a convection zone close to the surface of hot, massive stars, could be responsible for some of these phenomena. This convective zone is caused by a peak in the opacity due to iron recombination and for this reason is referred to as the ``iron convection zone'' (FeCZ). 3D MHD simulations are used to explore the possible effects of such subsurface convection on the surface properties of hot, massive stars. We argue that turbulence and localized magnetic spots at the surface are the likely consequence of subsurface convection in early type stars. Title: 3D MHD simulations of subsurface convection in OB stars Authors: Cantiello, Matteo; Braithwaite, Jonathan; Brandenburg, Axel; Del Sordo, Fabio; Käpylä, Petri; Langer, Norbert Bibcode: 2011IAUS..272...32C Altcode: 2010arXiv1009.4462C During their main sequence evolution, massive stars can develop convective regions very close to their surface. These regions are caused by an opacity peak associated with iron ionization. Cantiello et al. (2009) found a possible connection between the presence of sub-photospheric convective motions and small scale stochastic velocities in the photosphere of early-type stars. This supports a physical mechanism where microturbulence is caused by waves that are triggered by subsurface convection zones. They further suggest that clumping in the inner parts of the winds of OB stars could be related to subsurface convection, and that the convective layers may also be responsible for stochastic excitation of non-radial pulsations. Furthermore, magnetic fields produced in the iron convection zone could appear at the surface of such massive stars. Therefore subsurface convection could be responsible for the occurrence of observable phenomena such as line profile variability and discrete absorption components. These phenomena have been observed for decades, but still evade a clear theoretical explanation. Here we present preliminary results from 3D MHD simulations of such subsurface convection. Title: Reynolds stress and heat flux in spherical shell convection Authors: Käpylä, P. J.; Mantere, M. J.; Guerrero, G.; Brandenburg, A.; Chatterjee, P. Bibcode: 2011A&A...531A.162K Altcode: 2010arXiv1010.1250K Context. Turbulent fluxes of angular momentum and enthalpy or heat due to rotationally affected convection play a key role in determining differential rotation of stars. Their dependence on latitude and depth has been determined in the past from convection simulations in Cartesian or spherical simulations. Here we perform a systematic comparison between the two geometries as a function of the rotation rate.
Aims: Here we want to extend the earlier studies by using spherical wedges to obtain turbulent angular momentum and heat transport as functions of the rotation rate from stratified convection. We compare results from spherical and Cartesian models in the same parameter regime in order to study whether restricted geometry introduces artefacts into the results. In particular, we want to clarify whether the sharp equatorial profile of the horizontal Reynolds stress found in earlier Cartesian models is also reproduced in spherical geometry.
Methods: We employ direct numerical simulations of turbulent convection in spherical and Cartesian geometries. In order to alleviate the computational cost in the spherical runs, and to reach as high spatial resolution as possible, we model only parts of the latitude and longitude. The rotational influence, measured by the Coriolis number or inverse Rossby number, is varied from zero to roughly seven, which is the regime that is likely to be realised in the solar convection zone. Cartesian simulations are performed in overlapping parameter regimes.
Results: For slow rotation we find that the radial and latitudinal turbulent angular momentum fluxes are directed inward and equatorward, respectively. In the rapid rotation regime the radial flux changes sign in accordance with earlier numerical results, but in contradiction with theory. The latitudinal flux remains mostly equatorward and develops a maximum close to the equator. In Cartesian simulations this peak can be explained by the strong "banana cells". Their effect in the spherical case does not appear to be as large. The latitudinal heat flux is mostly equatorward for slow rotation but changes sign for rapid rotation. Longitudinal heat flux is always in the retrograde direction. The rotation profiles vary from anti-solar (slow equator) for slow and intermediate rotation to solar-like (fast equator) for rapid rotation. The solar-like profiles are dominated by the Taylor-Proudman balance.

Movies and Appendix A are available in electronic form at http://www.aanda.org Title: Magnetorotational instability driven dynamos at low magnetic Prandtl numbers Authors: Käpylä, P. J.; Korpi, M. J. Bibcode: 2011MNRAS.413..901K Altcode: 2010arXiv1004.2417K; 2011MNRAS.tmp..144K Numerical simulations of the magnetorotational instability (MRI) with zero initial net flux in a non-stratified isothermal cubic domain are used to demonstrate the importance of magnetic boundary conditions. In fully periodic systems, the level of turbulence generated by the MRI strongly decreases as the magnetic Prandtl number (Pm), which is the ratio of kinematic viscosity and magnetic diffusion, is decreased. No MRI or dynamo action below Pm = 1 is found, agreeing with earlier investigations. Using vertical field conditions, which allow the generation of a net toroidal flux and magnetic helicity fluxes out of the system, the MRI is found to be excited in the range 0.1 ≤ Pm ≤ 10, and the saturation level is independent of Pm. In the vertical field runs, strong mean-field dynamo develops and helps to sustain the MRI. Title: On global solar dynamo simulations Authors: Käpylä, P. J. Bibcode: 2011AN....332...43K Altcode: 2010arXiv1010.1249K Global dynamo simulations solving the equations of magnetohydrodynamics (MHD) have been a tool of astrophysicists who try to understand the magnetism of the Sun for several decades now. During recent years many fundamental issues in dynamo theory have been studied in detail by means of local numerical simulations that simplify the problem and allow the study of physical effects in isolation. Global simulations, however, continue to suffer from the age-old problem of too low spatial resolution, leading to much lower Reynolds numbers and scale separation than in the Sun. Reproducing the internal rotation of the Sun, which plays a crucial role in the dynamo process, has also turned out to be a very difficult problem. In the present paper the current status of global dynamo simulations of the Sun is reviewed. Emphasis is put on efforts to understand how the large-scale magnetic fields, i.e. whose length scale is greater than the scale of turbulence, are generated in the Sun. Some lessons from mean-field theory and local simulations are reviewed and their possible implications to the global models are discussed. Possible remedies to some current issues of solar simulations are put forward. Title: Turbulent transport in hydromagnetic flows Authors: Brandenburg, A.; Chatterjee, P.; Del Sordo, F.; Hubbard, A.; Käpylä, P. J.; Rheinhardt, M. Bibcode: 2010PhST..142a4028B Altcode: 2010arXiv1004.5380B The predictive power of mean-field theory is emphasized by comparing theory with simulations under controlled conditions. The recently developed test-field method is used to extract turbulent transport coefficients both in the kinematic and the nonlinear or quasi-kinematic cases. A striking example of the quasi-kinematic method is provided by magnetic buoyancy-driven flows that produce an α effect and turbulent diffusion. Title: Oscillatory Migrating Magnetic Fields in Helical Turbulence in Spherical Domains Authors: Mitra, Dhrubaditya; Tavakol, Reza; Käpylä, Petri J.; Brandenburg, Axel Bibcode: 2010ApJ...719L...1M Altcode: 2009arXiv0901.2364M We present direct numerical simulations of the equations of compressible magnetohydrodynamics in a wedge-shaped spherical shell, without shear, but with random helical forcing which has negative (positive) helicity in the northern (southern) hemisphere. We find a large-scale magnetic field that is nearly uniform in the azimuthal direction and approximately antisymmetric about the equator. Furthermore, the large-scale field in each hemisphere oscillates on nearly dynamical timescales with reversals of polarity and equatorward migration. Corresponding mean-field models also show similar migratory oscillations with a frequency that is nearly independent of the magnetic Reynolds number. This mechanism may be relevant for understanding equatorward migration seen in the solar dynamo. Title: Angular Momentum Transport in Convectively Unstable Shear Flows Authors: Käpylä, Petri J.; Brandenburg, Axel; Korpi, Maarit J.; Snellman, Jan E.; Narayan, Ramesh Bibcode: 2010ApJ...719...67K Altcode: 2010arXiv1003.0900K Angular momentum transport due to hydrodynamic turbulent convection is studied using local three-dimensional numerical simulations employing the shearing box approximation. We determine the turbulent viscosity from non-rotating runs over a range of values of the shear parameter and use a simple analytical model in order to extract the non-diffusive contribution (Λ-effect) to the stress in runs where rotation is included. Our results suggest that the turbulent viscosity is on the order of the mixing length estimate and weakly affected by rotation. The Λ-effect is non-zero and a factor of 2-4 smaller than the turbulent viscosity in the slow rotation regime. We demonstrate that for Keplerian shear, the angular momentum transport can change sign and be outward when the rotation period is greater than the turnover time, i.e., when the Coriolis number is below unity. This result seems to be relatively independent of the value of the Rayleigh number. Title: Open and closed boundaries in large-scale convective dynamos Authors: Käpylä, P. J.; Korpi, M. J.; Brandenburg, A. Bibcode: 2010A&A...518A..22K Altcode: 2009arXiv0911.4120K Context. Earlier work has suggested that large-scale dynamos can reach and maintain equipartition field strengths on a dynamical time scale only if magnetic helicity of the fluctuating field can be shed from the domain through open boundaries.
Aims: Our aim is to test this scenario in convection-driven dynamos by comparing results for open and closed boundary conditions.
Methods: Three-dimensional numerical simulations of turbulent compressible convection with shear and rotation are used to study the effects of boundary conditions on the excitation and saturation of large-scale dynamos. Open (vertical-field) and closed (perfect- conductor) boundary conditions are used for the magnetic field. The shear flow is such that the contours of shear are vertical, crossing the outer surface, and are thus ideally suited for driving a shear-induced magnetic helicity flux.
Results: We find that for given shear and rotation rate, the growth rate of the magnetic field is larger if open boundary conditions are used. The growth rate first increases for small magnetic Reynolds number, Rm, but then levels off at an approximately constant value for intermediate values of Rm. For large enough Rm, a small-scale dynamo is excited and the growth rate of the field in this regime increases as Rm1/2. Regarding the nonlinear regime, the saturation level of the energy of the total magnetic field is independent of Rm when open boundaries are used. In the case of perfect-conductor boundaries, the saturation level first increases as a function of Rm, but then decreases proportional to Rm-1 for Rm ⪆ 30, indicative of catastrophic quenching. These results suggest that the shear-induced magnetic helicity flux is efficient in alleviating catastrophic quenching when open boundaries are used. The horizontally averaged mean field is still weakly decreasing as a function of Rm even for open boundaries. Title: The α effect in rotating convection with sinusoidal shear Authors: Käpylä, P. J.; Korpi, M. J.; Brandenburg, A. Bibcode: 2010MNRAS.402.1458K Altcode: 2009arXiv0908.2423K; 2009MNRAS.tmp.1866K Using three-dimensional convection simulations, it is shown that a sinusoidal variation of horizontal shear leads to a kinematic α effect with a similar sinusoidal variation. The effect exists even for weak stratification and arises owing to the inhomogeneity of turbulence and the presence of impenetrable vertical boundaries. This system produces large-scale magnetic fields that also show a sinusoidal variation in the cross-stream direction. It is argued that earlier investigations overlooked these phenomena partly because of the use of horizontal averaging and also because measurements of α using an imposed field combined with long time averages give erroneous results. It is demonstrated that in such cases the actual horizontally averaged mean field becomes non-uniform. The turbulent magnetic diffusion term resulting from such non-uniform fields can then no longer be neglected and begins to balance the α effect. Title: Oscillatory migratory large-scale fields in mean-field and direct simulations Authors: Mitra, Dhrubaditya; Tavakol, Reza; Brandenburg, Axel; Käpylä, Petri J. Bibcode: 2010IAUS..264..197M Altcode: We summarise recent results form direct numerical simulations of both non-rotating helically forced and rotating convection driven MHD equations in spherical wedge-shape domains. In the former, using perfect-conductor boundary conditions along the latitudinal boundaries we observe oscillations, polarity reversals and equatorward migration of the large-scale magnetic fields. In the latter we obtain angular velocity with cylindrical contours and large-scale magnetic field which shows oscillations, polarity reversals but poleward migration. The occurrence of these behviours in direct numerical simulations is clearly of interest. However the present models as they stand are not directly applicable to the solar dynamo problem. Nevertheless, they provide general insights into the operation of turbulent dynamos. Title: Influence of Ohmic diffusion on the excitation and dynamics of MRI Authors: Korpi, M. J.; Käpylä, P. J.; Väisälä, M. S. Bibcode: 2010AN....331...34K Altcode: 2009arXiv0909.1724K In this paper we make an effort to understand the interaction of turbulence generated by the magnetorotational instability (MRI) with turbulence from other sources, such as supernova explosions (SNe) in galactic disks. First we perform a linear stability analysis (LSA) of non-ideal MRI to derive the limiting value of Ohmic diffusion that is needed to inhibit the growth of the instability for different types of rotation laws. With the help of a simple analytical expression derived under first-order smoothing approximation (FOSA), an estimate of the limiting turbulence level and hence the turbulent diffusion needed to damp the MRI is derived. Secondly, we perform numerical simulations in local cubes of isothermal nonstratified gas with external forcing of varying strength to see whether the linear result holds for more complex systems. Purely hydrodynamic calculations with forcing, rotation and shear are made for reference purposes, and as expected, non-zero Reynolds stresses are found. In the magnetohydrodynamic calculations, therefore, the total stresses generated are a sum of the forcing and MRI contributions. To separate these contributions, we perform reference runs with MRI-stable shear profiles (angular velocity increasing outwards), which suggest that the MRI-generated stresses indeed become strongly suppressed as function of the forcing. The Maxwell to Reynolds stress ratio is observed to decrease by an order of magnitude as the turbulence level due to external forcing exceeds the predicted limiting value, which we interpret as a sign of MRI suppression. Finally, we apply these results to estimate the limiting radius inside of which the SN activity can suppress the MRI, arriving at a value of 14 kpc. Title: Convective dynamos in spherical wedge geometry Authors: Käpylä, P. J.; Korpi, M. J.; Brandenburg, A.; Mitra, D.; Tavakol, R. Bibcode: 2010AN....331...73K Altcode: 2009arXiv0909.1330K Self-consistent convective dynamo simulations in wedge-shaped spherical shells are presented. Differential rotation is generated by the interaction of convection with rotation. Equatorward acceleration and dynamo action are obtained only for sufficiently rapid rotation. The angular velocity tends to be constant along cylinders. Oscillatory large-scale fields are found to migrate in the poleward direction. Comparison with earlier simulations in full spherical shells and Cartesian domains is made. Title: Stellar magnetic fields: observations and nonlinear modeling Authors: Lindborg, M.; Korpi, M.; Tuominen, I.; Hackman, T.; Ilyin, I.; Käpylä, P. Bibcode: 2009AGUFMSH13A1506L Altcode: Magnetic fields on a global scale are observed in a wide variety of astrophysical objects, spanning from planets, stars, and accretion disks to galaxies. These magnetic fields are anything but passive, taking part in the dynamics of their hosts, resulting for example in the familiar activity phenomena of the Sun, which also affect life here on the Earth. We have a long time series of temperature mapping and photometry of an active late-type star II Peg. II Peg is known as one of the most active RS Cvn stars representing the Sun at a younger age. We are trying to establish the link between the Sun and the more active rapidly rotating stars and compare the results to the dynamo models of the Sun. Preliminary results show some resemblance to the dynamo solutions. Title: The α effect with imposed and dynamo-generated magnetic fields Authors: Hubbard, A.; Del Sordo, F.; Käpylä, P. J.; Brandenburg, A. Bibcode: 2009MNRAS.398.1891H Altcode: 2009MNRAS.tmp.1219H; 2009arXiv0904.2773H Estimates for the non-linear α effect in helical turbulence with an applied magnetic field are presented using two different approaches: the imposed-field method where the electromotive force owing to the applied field is used, and the test-field method where separate evolution equations are solved for a set of different test fields. Both approaches agree for stronger fields, but there are apparent discrepancies for weaker fields that can be explained by the influence of dynamo-generated magnetic fields on the scale of the domain that are referred to as meso-scale magnetic fields. Examples are discussed where these meso-scale fields can lead to both drastically overestimated and underestimated values of α compared with the kinematic case. It is demonstrated that the kinematic value can be recovered by resetting the fluctuating magnetic field to zero in regular time intervals. It is concluded that this is the preferred technique both for the imposed-field and the test-field methods. Title: Reynolds stresses from hydrodynamic turbulence with shear and rotation Authors: Snellman, J. E.; Käpylä, P. J.; Korpi, M. J.; Liljeström, A. J. Bibcode: 2009A&A...505..955S Altcode: 2009arXiv0906.1200S Aims: We study the Reynolds stresses which describe turbulent momentum transport from turbulence affected by large-scale shear and rotation.
Methods: Three-dimensional numerical simulations are used to study turbulent transport under the influences of large-scale shear and rotation in homogeneous, isotropically forced turbulence. We study three cases: one with only shear, and two others where in addition to shear, rotation is present. These cases differ by the angle (0 or 90°) the rotation vector makes with respect to the z-direction. Two subsets of runs are performed with both values of θ where either rotation or shear is kept constant. When only shear is present, the off-diagonal stress can be described by turbulent viscosity whereas if the system also rotates, nondiffusive contributions (Λ-effect) to the stress can arise. Comparison of the direct simulations are made with analytical results from a simple closure model.
Results: We find that the turbulent viscosity is of the order of the first order smoothing result in the parameter regime studied and that for sufficiently large Reynolds numbers the Strouhal number, describing the ratio of correlation to turnover times, is roughly 1.5. This is consistent with the closure model based on the minimal tau-approximation which produces a reasonable fit to the simulation data for similar Strouhal numbers. In the cases where rotation is present, separating the diffusive and nondiffusive components of the stress turns out to be challenging but taking the results at face value, we can obtain nondiffusive contributions of the order of 0.1 times the turbulent viscosity. We also find that the simple closure model is able to reproduce most of the qualitative features of the numerical results provided that the Strouhal number is of the order of unity.

Tables B1-B3 are only available in electronic form at http://www.aanda.org Title: Turbulent Dynamos with Shear and Fractional Helicity Authors: Käpylä, Petri J.; Brandenburg, Axel Bibcode: 2009ApJ...699.1059K Altcode: 2008arXiv0810.2298K Dynamo action owing to helically forced turbulence and large-scale shear is studied using direct numerical simulations. The resulting magnetic field displays propagating wave-like behavior. This behavior can be modeled in terms of an αΩ dynamo. In most cases super-equipartition fields are generated. By varying the fraction of helicity of the turbulence the regeneration of poloidal fields via the helicity effect (corresponding to the α-effect) is regulated. The saturation level of the magnetic field in the numerical models is consistent with a linear dependence on the ratio of the fractional helicities of the small and large-scale fields, as predicted by a simple nonlinear mean-field model. As the magnetic Reynolds number (Re M ) based on the wavenumber of the energy-carrying eddies is increased from 1 to 180, the cycle frequency of the large-scale field is found to decrease by a factor of about 6 in cases where the turbulence is fully helical. This is interpreted in terms of the turbulent magnetic diffusivity, which is found to be only weakly dependent on the Re M . Title: Large-scale Dynamos in Rigidly Rotating Turbulent Convection Authors: Käpylä, Petri J.; Korpi, Maarit J.; Brandenburg, Axel Bibcode: 2009ApJ...697.1153K Altcode: 2008arXiv0812.3958K The existence of large-scale dynamos in rigidly rotating turbulent convection without shear is studied using three-dimensional numerical simulations of penetrative rotating compressible convection. We demonstrate that rotating convection in a Cartesian domain can drive a large-scale dynamo even in the absence of shear. The large-scale field contains a significant fraction of the total field in the saturated state. The simulation results are compared with one-dimensional mean-field dynamo models where turbulent transport coefficients, as determined using the test field method, are used. The reason for the absence of large-scale dynamo action in earlier studies is shown to be due to the rotation being too slow: whereas the α-effect can change sign, its magnitude stays approximately constant as a function of rotation, and the turbulent diffusivity decreases monotonically with increasing rotation. Only when rotation is rapid enough a large-scale dynamo can be excited. The one-dimensional mean-field model with dynamo coefficients from the test-field results predicts reasonably well the dynamo excitation in the direct simulations. This result further validates the test field procedure and reinforces the interpretation that the observed dynamo is driven by a turbulent α-effect. This result demonstrates the existence of an α-effect and an α2-dynamo with natural forcing. Title: Alpha effect and turbulent diffusion from convection Authors: Käpylä, P. J.; Korpi, M. J.; Brandenburg, A. Bibcode: 2009A&A...500..633K Altcode: 2008arXiv0812.1792K 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, η_t, 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. Title: Stellar nonlinear dynamos: observations and modelling Authors: Tuominen, Ilkka; Korpi, Maarit J.; Käpylä, Petri J.; Lindborg, Marjaana; Ilyin, Ilya Bibcode: 2009IAUS..259..417T Altcode: Recent numerical modelling of mean-field stellar dynamos with induction and Reynolds' equations show that, with increasing rotation, field symmetry changes from an axisymmetric solar type to a nonaxymmetric one, where the so-called active longitudes in the same stellar hemisphere are predicted to be of opposite polarities. It was originally named Active star Hale rule in Tuominen et al. (2002), being an analogue to the famous bipolar sunspot polarity rule but different in scale and being a global phenomenon. In addition to long timeseries of temperature mapping and photometry, during the last few years we have been able to measure accurately polarization spectra of an active late-type star II Peg with the upgraded spectropolarimetric option of the high-resolution spectrograph SOFIN at the Nordic Optical Telescope, La Palma. The magnetic inversions (Carroll et al., Kochukhov et al.) can be compared to the dynamo models and the preliminary results show some resemblance to the dynamo solutions. Title: Alpha effect and diffusivity in helical turbulence with shear Authors: Mitra, D.; Käpylä, P. J.; Tavakol, R.; Brandenburg, A. Bibcode: 2009A&A...495....1M Altcode: 2008arXiv0806.1608M Aims: We study the dependence of turbulent transport coefficients, such as the components of the α tensor (αij) and the turbulent magnetic diffusivity tensor (ηij), on shear and magnetic Reynolds number in the presence of helical forcing.
Methods: We use three-dimensional direct numerical simulations with periodic boundary conditions and measure the turbulent transport coefficients using the kinematic test field method. In all cases the magnetic Prandtl number is taken as unity.
Results: We find that with increasing shear the diagonal components of αij quench, whereas those of ηij increase. The antisymmetric parts of both tensors increase with increasing shear. We also propose a simple expression for the turbulent pumping velocity (or γ effect). This pumping velocity is proportional to the kinetic helicity of the turbulence and the vorticity of the mean flow. For negative helicity, i.e. for a positive trace of αij, it points in the direction of the mean vorticity, i.e. perpendicular to the plane of the shear flow. Our simulations support this expression for low shear and magnetic Reynolds number. The transport coefficients depend on the wavenumber of the mean flow in a Lorentzian fashion, just as for non-shearing turbulence. Title: Turbulent stresses as a function of shear rate in a local disk model Authors: Liljeström, A. J.; Korpi, M. J.; Käpylä, P. J.; Brandenburg, A.; Lyra, W. Bibcode: 2009AN....330...92L Altcode: 2008arXiv0811.2341L We present local numerical models of accretion disk turbulence driven by the magnetorotational instability with varying shear rate. The resulting turbulent stresses are compared with predictions of a closure model in which triple correlations are modelled in terms of quadratic correlations. This local model uses five nondimensional parameters to describe the properties of the flow. We attempt to determine these closure parameters for our simulations and find that the model does produce qualitatively correct behaviour. In addition, we present results concerning the shear rate dependency of the magnetic to kinetic energy ratio. We find both the turbulent stress ratio and the total stress to be strongly dependent on the shear rate. Title: Numerical study of large-scale vorticity generation in shear-flow turbulence Authors: Käpylä, Petri J.; Mitra, Dhrubaditya; Brandenburg, Axel Bibcode: 2009PhRvE..79a6302K Altcode: 2008arXiv0810.0833K Simulations of stochastically forced shear-flow turbulence in a shearing-periodic domain are used to study the spontaneous generation of large-scale flow patterns in the direction perpendicular to the plane of the shear. Based on an analysis of the resulting large-scale velocity correlations it is argued that the mechanism behind this phenomenon could be the mean-vorticity dynamo effect pioneered by Elperin, Kleeorin, and Rogachevskii [Phys. Rev. E 68, 016311 (2003)]. This effect is based on the anisotropy of the eddy viscosity tensor. One of its components may be able to replenish cross-stream mean flows by acting upon the streamwise component of the mean flow. Shear, in turn, closes the loop by acting upon the cross-stream mean flow to produce stronger streamwise mean flows. The diagonal component of the eddy viscosity is found to be of the order of the rms turbulent velocity divided by the wave number of the energy-carrying eddies. Title: Large-scale dynamos in turbulent convection with shear Authors: Käpylä, P. J.; Korpi, M. J.; Brandenburg, A. Bibcode: 2008A&A...491..353K Altcode: 2008arXiv0806.0375K Aims: To study the existence of large-scale convective dynamos under the influence of shear and rotation.
Methods: Three-dimensional numerical simulations of penetrative compressible convection with uniform horizontal shear are used to study dynamo action and the generation of large-scale magnetic fields. We consider cases where the magnetic Reynolds number is either marginal or moderately supercritical with respect to small-scale dynamo action in the absence of shear and rotation. Our magnetic Reynolds number is based on the wavenumber of the depth of the convectively unstable layer. The effects of magnetic helicity fluxes are studied by comparing results for the magnetic field with open and closed boundaries.
Results: Without shear no large-scale dynamos are found even if the ingredients necessary for the α-effect (rotation and stratification) are present in the system. When uniform horizontal shear is added, a large-scale magnetic field develops, provided the boundaries are open. In this case the mean magnetic field contains a significant fraction of the total field. For those runs where the magnetic Reynolds number is between 60 and 250, an additional small-scale dynamo is expected to be excited, but the field distribution is found to be similar to cases with smaller magnetic Reynolds number where the small-scale dynamo is not excited. In the case of closed (perfectly conducting) boundaries, magnetic helicity fluxes are suppressed and no large-scale fields are found. Similarly, poor large-scale field development is seen when vertical shear is used in combination with periodic boundary conditions in the horizontal directions. If, however, open (normal-field) boundary conditions are used in the x-direction, a large-scale field develops. These results support the notion that shear not only helps to generate the field, but it also plays a crucial role in driving magnetic helicity fluxes out of the system along the isocontours of shear, thereby allowing efficient dynamo action. Title: Lambda effect from forced turbulence simulations Authors: Käpylä, P. J.; Brandenburg, A. Bibcode: 2008A&A...488....9K Altcode: 2008arXiv0806.3751K Aims: We determine the components of the Λ-effect tensor that quantifies the contributions to the turbulent momentum transport even for uniform rotation.
Methods: Three-dimensional numerical simulations are used to study turbulent transport in triply periodic cubes under the influence of rotation and anisotropic forcing. Comparison is made with analytical results obtained via the so-called minimal tau-approximation.
Results: In the case where the turbulence intensity in the vertical direction dominates, the vertical stress is always negative. This situation is expected to occur in stellar convection zones. The horizontal component of the stress is weaker and exhibits a maximum at latitude 30° - regardless of how rapid the rotation is. The minimal tau-approximation captures many of the qualitative features of the numerical results, provided the relaxation time tau is close to the turnover time, i.e. the Strouhal number is of order unity.

Tables [see full textsee full textsee full text]-[see full textsee full textsee full text] are only available in electronic form at http.//www.aanda.org Title: A solar mean field dynamo benchmark Authors: Jouve, L.; Brun, A. S.; Arlt, R.; Brandenburg, A.; Dikpati, M.; Bonanno, A.; Käpylä, P. J.; Moss, D.; Rempel, M.; Gilman, P.; Korpi, M. J.; Kosovichev, A. G. Bibcode: 2008A&A...483..949J Altcode: Context: The solar magnetic activity and cycle are linked to an internal dynamo. Numerical simulations are an efficient and accurate tool to investigate such intricate dynamical processes.
Aims: We present the results of an international numerical benchmark study based on two-dimensional axisymmetric mean field solar dynamo models in spherical geometry. The purpose of this work is to provide reference cases that can be analyzed in detail and that can help in further development and validation of numerical codes that solve such kinematic problems.
Methods: The results of eight numerical codes solving the induction equation in the framework of mean field theory are compared for three increasingly computationally intensive models of the solar dynamo: an αΩ dynamo with constant magnetic diffusivity, an αΩ dynamo with magnetic diffusivity sharply varying with depth and an example of a flux-transport Babcock-Leighton dynamo which includes a non-local source term and one large single cell of meridional circulation per hemisphere. All cases include a realistic profile of differential rotation and thus a sharp tachocline.
Results: The most important finding of this study is that all codes agree quantitatively to within less than a percent for the αΩ dynamo cases and within a few percent for the flux-transport case. Both the critical dynamo numbers for the onset of dynamo action and the corresponding cycle periods are reasonably well recovered by all codes. Detailed comparisons of butterfly diagrams and specific cuts of both toroidal and poloidal fields at given latitude and radius confirm the good quantitative agreement.
Conclusions: We believe that such a benchmark study will be a very useful tool since it provides detailed standard cases for comparison and reference. Title: Magnetic Diffusivity Tensor and Dynamo Effects in Rotating and Shearing Turbulence Authors: Brandenburg, A.; Rädler, K. -H.; Rheinhardt, M.; Käpylä, P. J. Bibcode: 2008ApJ...676..740B Altcode: 2007arXiv0710.4059B 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 α 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. Title: The helicity constraint in spherical shell dynamos Authors: Brandenburg, A.; Käpylä, P. J.; Mitra, D.; Moss, D.; Tavakol, R. Bibcode: 2007AN....328.1118B Altcode: 2007arXiv0711.3616B The motivation for considering distributed large scale dynamos in the solar context is reviewed in connection with the magnetic helicity constraint. Preliminary accounts of 3-dimensional direct numerical simulations (in spherical shell segments) and simulations of 2-dimensional mean field models (in spherical shells) are presented. Interesting similarities as well as some differences are noted. Title: Turbulent viscosity and Λ-effect from numerical turbulence models Authors: Käpylä, P. J.; Brandenburg, A. Bibcode: 2007AN....328.1006K Altcode: 2007arXiv0710.5632K Homogeneous anisotropic turbulence simulations are used to determine off-diagonal components of the Reynolds stress tensor and its parameterization in terms of turbulent viscosity and Λ-effect. The turbulence is forced in an anisotropic fashion by enhancing the strength of the forcing in the vertical direction. The Coriolis force is included with a rotation axis inclined relative to the vertical direction. The system studied here is significantly simpler than that of turbulent stratified convection which has often been used to study Reynolds stresses. Certain puzzling features of the results for convection, such as sign changes or highly concentrated latitude distributions, are not present in the simpler system considered here. Title: Magnetic helicity effects in astrophysical and laboratory dynamos Authors: Brandenburg, A.; Käpylä, P. J. Bibcode: 2007NJPh....9..305B Altcode: 2007arXiv0705.3507B Magnetic helicity effects are discussed in laboratory and astro-physical settings. Firstly, dynamo action in Taylor Green flows is discussed for different boundary conditions. However, because of the lack of scale separation with respect to the container, no large-scale field is being produced and there is no resistively slow saturation phase as otherwise expected. Secondly, the build-up of a large-scale field is demonstrated in a simulation where a localized magnetic eddy produces field on a larger scale if the eddy possesses a swirl. Such a set-up might be realizable experimentally through coils. Finally, new emerging issues regarding the connection between magnetic helicity and the solar dynamo are discussed. It is demonstrated that dynamos with a nonlocal (Babcock Leighton type) α effect can also be catastrophically quenched, unless there are magnetic helicity fluxes. Title: Effects of rotation and input energy flux on convective overshooting Authors: Käpylä, Petri J.; Korpi, M. J.; Stix, M.; Tuominen, I. Bibcode: 2007IAUS..239..437K Altcode: 2006astro.ph..9350K We study convective overshooting by means of local 3D convection calculations. Using a mixing length model of the solar convection zone (CZ) as a guide, we determine the Coriolis number (Co), which is the inverse of the Rossby number, to be of the order of ten or larger at the base of the solar CZ. Therefore we perform convection calculations in the range Co = 0...10 and interpret the value of Co realised in the calculation to represent a depth in the solar CZ. In order to study the dependence on rotation, we compute the mixing length parameters alpha_T and alpha_u relating the temperature and velocity fluctuations, respectively, to the mean thermal stratification. We find that the mixing length parameters for the rapid rotation case, corresponding to the base of the solar CZ, are 3-5 times smaller than in the nonrotating case. Introducing such depth-dependent alpha into a solar structure model employing a non-local mixing length formalism results in overshooting which is approximately proportional to alpha at the base of the CZ. Although overshooting is reduced due to the reduced alpha, a discrepancy with helioseismology remains due to the steep transition to the radiative temperature gradient. In comparison to the mixing length models the transition at the base of the CZ is much gentler in the 3D models. It was suggested recently (Rempel 2004) that this discrepancy is due to the significantly larger (up to seven orders of magnitude) input energy flux in the 3D models in comparison to the Sun and solar models, and that the 3D calculations should be able to approach the mixing length regime if the input energy flux is decreased by a moderate amount. We present results from local convection calculations which support this conjecture. Title: Reynolds stresses and meridional circulation from rotating cylinder simulations Authors: Hupfer, C.; Käpylä, P. J.; Stix, M. Bibcode: 2006A&A...459..935H Altcode: Aims.The latitude and depth dependences of Reynolds stresses are obtained from numerical simulations of a solar-type convection zone where the star is assumed to rotate with a uniform angular velocity.
Methods: .A two-dimensional model, using a cylindrical annulus with axis perpendicular to the axis of rotation, is introduced as an approximation of a spherical section along the meridional plane. We solve the fully compressible Navier-Stokes equations numerically and use Cartesian and cylindrical geometries to simulate convection under the influence of rotation.
Results: .For moderate Coriolis numbers both models yield strong extrema of the Reynolds stress component Qθϕ at low latitudes near the equator, and a meridional cell pattern is found in the cylindrical model. For Coriolis numbers larger than about 10 the flow becomes aligned with the direction of the rotation axis.
Title: Solar dynamo models with α-effect and turbulent pumping from local 3D convection calculations Authors: Käpylä, P. J.; Korpi, M. J.; Tuominen, I. Bibcode: 2006AN....327..884K Altcode: 2006astro.ph..6089K Results from kinematic solar dynamo models employing α-effect and turbulent pumping from local convection calculations are presented. We estimate the magnitude of these effects to be around 2-3 m s-1, having scaled the local quantities with the convective velocity at the bottom of the convection zone from a solar mixing-length model. Rotation profile of the Sun as obtained from helioseismology is applied in the models; we also investigate the effects of the observed surface shear layer on the dynamo solutions. With these choices of the small- and large-scale velocity fields, we obtain estimate of the ratio of the two induction effects, C_α/C_Ω ≈ 10-3, which we keep fixed in all models. We also include a one-cell meridional circulation pattern having a magnitude of 10-20 m s-1 near the surface and 1-2 m s-1 at the bottom of the convection zone. The model essentially represents a distributed turbulent dynamo, as the α-effect is nonzero throughout the convection zone, although it concentrates near the bottom of the convection zone obtaining a maximum around 30° of latitude. Turbulent pumping of the mean fields is predominantly down- and equatorward. The anisotropies in the turbulent diffusivity are neglected apart from the fact that the diffusivity is significantly reduced in the overshoot region. We find that, when all these effects are included in the model, it is possible to correctly reproduce many features of the solar activity cycle, namely the correct equatorward migration at low latitudes and the polar branch at high latitudes, and the observed negative sign of B_r Bφ. Although the activity clearly shifts towards the equator in comparison to previous models due to the combined action of the α-effect peaking at midlatitudes, meridional circulation and latitudinal pumping, most of the activity still occurs at too high latitudes (between 5° ... 60°). Other problems include the relatively narrow parameter space within which the preferred solution is dipolar (A0), and the somewhat too short cycle lengths of the solar-type solutions. The role of the surface shear layer is found to be important only in the case where the α-effect has an appreciable magnitude near the surface. Title: Magnetoconvection and dynamo coefficients. III. α-effect and magnetic pumping in the rapid rotation regime Authors: Käpylä, P. J.; Korpi, M. J.; Ossendrijver, M.; Stix, M. Bibcode: 2006A&A...455..401K Altcode: 2006astro.ph..2111K Aims.The α- and γ-effects, which are responsible for the generation and turbulent pumping of large scale magnetic fields, respectively, due to passive advection by convection are determined in the rapid rotation regime corresponding to the deep layers of the solar convection zone.
Methods.A 3D rectangular local model is used for solving the full set of MHD equations in order to compute the electromotive force (emf), E = overline {u × b}, generated by the interaction of imposed weak gradient-free magnetic fields and turbulent convection with varying rotational influence and latitude. By expanding the emf in terms of the mean magnetic field, Ei = aij overline B_j, all nine components of aij are computed. The diagonal elements of aij describe the α-effect, whereas the off-diagonals represent magnetic pumping. The latter is essentially the advection of magnetic fields by means other than the underlying large-scale velocity field. Comparisons are made to analytical expressions of the coefficients derived under the first-order smoothing approximation (FOSA).
Results.In the rapid rotation regime the latitudinal dependence of the α-components responsible for the generation of the azimuthal and radial fields does not exhibit a peak at the poles, as is the case for slow rotation, but at a latitude of about 30°. The magnetic pumping is predominantly radially down- and latitudinally equatorward as in earlier studies. The numerical results compare surprisingly well with analytical expressions derived under first-order smoothing, although the present calculations are expected to lie near the limits of the validity range of FOSA.
Title: Alpha-Effect and Turbulent Pumping In The Rapid Rotation Regime - Implications For Solar Dynamo Models Authors: Käpylä, P. J.; Korpi, M. J.; Ossendrijver, M.; Stix, M.; Tuominen, I. Bibcode: 2006IAUJD...8E..46K Altcode: We use local 3D convection calculations to compute the alpha-effect and turbulent pumping of mean magnetic fields in the rapid rotation regime corresponding to the deep layers of the solar convection zone. We find that in this regime the alpha-effect responsible for generating the poloidal field out of the toroidal one peaks at around latitude 30 degrees, in contrast to the slow rotation case and the often adopted prescription in mean-field models of the solar dynamo, where the maximum values are found at the poles. Furthermore, the turbulent pumping of mean fields is predominantly down- and equatorward. We find that the downward pumping is decreased near the equator for rapid rotation and can be upward for the toroidal field component. In order to investigate the implications of the obtained local results for the problems in mean-field dynamo theory arising from the helioseismically determined solar rotation profile, namely the poleward migration of activity belts at low latitudes and the activity being concentrated at too high latitudes, we introduce the alpha-effect and turbulent pumping as they were found in the local calculations into a kinematic mean-field model of the solar dynamo. We also investigate the effect of a one-cell counter-clockwise meridional flow pattern on the dynamo solutions. We find that using the alpha-effect and turbulent pumping adapted from the results of the local calculations, the migration of the activity belts is equatorward also at low latitudes. When the meridional flow is added, the activity belts are shifted further closer to the equator, and a poleward migration belt appears at high latitudes. With all the effects included, the activity still appears at too high latitudes (5...60 degrees). Other remaining problems include the somewhat too short cycle periods for the solar-like dipole solutions. Title: Local models of stellar convection. III. The Strouhal number Authors: Käpylä, P. J.; Korpi, M. J.; Ossendrijver, M.; Tuominen, I. Bibcode: 2006A&A...448..433K Altcode: 2004astro.ph.10586K Aims.The Strouhal number (St), which is a nondimensional measure of the correlation time, is determined from numerical models of convection. The Strouhal number arises in the mean-field theories of angular momentum transport and magnetic field generation, where its value determines the validity of certain widely used approximations, such as the first order smoothing (hereafter FOSA). More specifically, the relevant transport coefficients can be calculated by means of a cumulative series expansion if St < Stcrit ≈ 1.Methods.We define the Strouhal number as the ratio of the correlation and turnover times, which we determine separately, the former from the autocorrelation of velocity, and the latter by following test particles embedded in the flow.Results.We find that the Strouhal numbers are, generally, of the order of 0.1 to 0.4 which is close to the critical value above which deviations from FOSA become significant. Increasing the rotational influence tends to shorten both timescales in such a manner that St decreases. However, we do not find a clear trend as a function of the Rayleigh number for the parameter range explored in the present study. Title: Local numerical modelling of magnetoconvection and turbulence - implications for mean-field theories Authors: Käpylä, Petri J. Bibcode: 2006PhDT.......275K Altcode: No abstract at ADS Title: Transport coefficients for solar and stellar dynamos Authors: Ossendrijver, M.; Käpylä, P. J. Bibcode: 2006IAUS..233....3O Altcode: The large-scale magnetic field of the Sun and solar-type stars is probably generated near the interface of the radiative core and the convection zone. One of the well-known difficulties of interface-type and other dynamo models is to explain the latitudinal distribution of magnetic fields as observed on the Sun. In this contribution new results of numerical MHD simulations of transport coefficients of magnetic fields in the rapid rotation regime, relevant for the base of the solar convection zone, are presented that may contribute to explaining the latitudinal distribution of magnetic fields observed on the Sun. A brief outlook on further numerical simulations of transport coefficients and their relevance for mean-field dynamo theory is given. Title: Connection between active longitudes and magnetic helicity Authors: Brandenburg, A.; Käpylä, P. J. Bibcode: 2005astro.ph.12639B Altcode: A two-dimensional mean field dynamo model is solved where magnetic helicity conservation is fully included. The model has a negative radial velocity gradient giving rise to equatorward migration of magnetic activity patterns. In addition the model develops longitudinal variability with activity patches travelling in longitude. These patches may be associated with active longitudes. Title: Local models of stellar convection. II. Rotation dependence of the mixing length relations Authors: Käpylä, P. J.; Korpi, M. J.; Stix, M.; Tuominen, I. Bibcode: 2005A&A...438..403K Altcode: 2004astro.ph.10584K We study the mixing length concept in comparison to three-dimensional numerical calculations of convection with rotation. In a limited range, the velocity and temperature fluctuations are linearly proportional to the superadiabaticity, as predicted by the mixing length concept and in accordance with published results. The effects of rotation are investigated by varying the Coriolis number, Co = 2 Ω τ, from zero to roughly ten, and by calculating models at different latitudes. We find that α decreases monotonically as a function of the Coriolis number. This can be explained by the decreased spatial scale of convection and the diminished efficiency of the convective energy transport, the latter of which leads to a large increase of the superadibaticity, δ = nabla - nabla_ad as function of Co. Applying a decreased mixing length parameter in a solar model yields very small differences in comparison to the standard model within the convection zone. The main difference is the reduction of the overshooting depth, and thus the depth of the convection zone, when a non-local version of the mixing length concept is used. Reduction of α by a factor of roughly 2.5 is sufficient to reconcile the difference between the model and helioseismic results. The numerical results indicate reduction of α by this order of magnitude. Title: Reynolds stresses - dependence on latitude Authors: Hupfer, C.; Käpylä, P.; Stix, M. Bibcode: 2005AN....326..223H Altcode: We present some results of three-dimensional hydrodynamic simulations of a compressible flow under the influence of rotation. The code implements a three-layer (stably-unstably-stably stratified) rectangular box placed at various latitudes in a convection zone of a star. We focus on the Reynolds stresses Qij = < u'i u'j > which in mean-field models have a crucial influence on angular momentum transport and differential rotation. Especially we examine the occurrence of a strong peak of Qθφ at low latitude very close to the equator, which may have implications to the theory of the angular velocity profile observed on the Sun. Title: Estimates of the Strouhal number from numerical models of convection Authors: Käpylä, P. J.; Korpi, M. J.; Ossendrijver, M.; Tuominen, I. Bibcode: 2005AN....326..186K Altcode: 2004astro.ph.10587K We determine the Strouhal number (hereafter St), which is essentially a nondimensional measure of the correlation time, from numerical calculations of convection. We use two independent methods to estimate St. Firstly, we apply the minimal tau-approximation (MTA) on the equation of the time derivative of the Reynolds stress. A relaxation time is obtained from which St can be estimated by normalising with a typical turnover time. Secondly, we calculate the correlation and turnover times separately, the former from the autocorrelation of velocity and the latter by following test particles embedded in the flow. We find that the Strouhal number is in general of the order of 0.1 to 1, i.e. rather large in comparison to the typical assumption in the mean-field theories that St ≪ 1. However, there is a clear decreasing trend as function of the Rayleigh number and increasing rotation. Furthermore, for the present range of parameters the decrease of St does not show signs of saturation, indicating that in stellar convection zones, where the Rayleigh numbers are much larger, the Strouhal number may indeed be significantly smaller. Title: The problem of small and large scale fields in the solar dynamo Authors: Brandenburg, A.; Haugen, N. E. L.; Käpylä, P. J.; Sandin, C. Bibcode: 2005AN....326..174B Altcode: 2004astro.ph.12364B Three closely related stumbling blocks of solar mean field dynamo theory are discussed: how dominant are the small scale fields, how is the alpha effect quenched, and whether magnetic and current helicity fluxes alleviate the quenching? It is shown that even at the largest currently available resolution there is no clear evidence of power law scaling of the magnetic and kinetic energy spectra in turbulence. However, using subgrid scale modeling, some indications of asymptotic equipartition can be found. The frequently used first order smoothing approach to calculate the alpha effect and other transport coefficients is contrasted with the superior minimal tau approximation. The possibility of catastrophic alpha quenching is discussed as a result of magnetic helicity conservation. Magnetic and current helicity fluxes are shown to alleviate catastrophic quenching in the presence of shear. Evidence for strong large scale dynamo action, even in the absence of helicity in the forcing, is presented. Title: Local models of stellar convection:. Reynolds stresses and turbulent heat transport Authors: Käpylä, P. J.; Korpi, M. J.; Tuominen, I. Bibcode: 2004A&A...422..793K Altcode: 2003astro.ph.12376K We study stellar convection using a local three-dimensional MHD model, with which we investigate the influence of rotation and large-scale magnetic fields on the turbulent momentum and heat transport and their role in generating large-scale flows in stellar convection zones. The former is studied by computing the turbulent velocity correlations, known as Reynolds stresses, the latter by calculating the correlation of velocity and temperature fluctuations, both as functions of rotation and latitude. We find that the horizontal correlation, Qθ φ, capable of generating horizontal differential rotation, attains significant values and is mostly negative in the southern hemisphere for Coriolis numbers exceeding unity, corresponding to equatorward flux of angular momentum. This result is also in accordance with solar observations. The radial component Qr φ is negative for slow and intermediate rotation indicating inward transport of angular momentum, while for rapid rotation, the transport occurs outwards. Parametrisation in terms of the mean-field Λ-effect shows qualitative agreement with the turbulence model of Kichatinov & Rüdiger (\cite{KiRu93}) for the horizontal part H ∝ (Qθ φ)/(cos θ), whereas for the vertical Λ-effect, V ∝ (Qr φ/(sin θ), agreement only for intermediate rotation exists. The Λ-coefficients become suppressed in the limit of rapid rotation, this rotational quenching being stronger and occurring with slower rotation for the V component than for H. We have also studied the behaviour of the Reynolds stresses under the influence of a large-scale azimuthal magnetic field of varying strength. We find that the stresses are enhanced by the presence of the magnetic field for field strengths up to and above the equipartition value, without significant quenching. Concerning the turbulent heat transport, our calculations show that the transport in the radial direction is most efficient at the equatorial regions, obtains a minimum at midlatitudes, and shows a slight increase towards the poles. The latitudinal heat transport does not show a systematic trend as a function of latitude or rotation. Title: Non-Fickian diffusion and tau approximation from numerical turbulence Authors: Brandenburg, Axel; Käpylä, Petri J.; Mohammed, Amjed Bibcode: 2004PhFl...16.1020B Altcode: 2003astro.ph..6521B Evidence for non-Fickian diffusion of a passive scalar is presented using direct simulations of homogeneous isotropic turbulence. The results compare favorably with an explicitly time-dependent closure model based on the tau approximation. In the numerical experiments three different cases are considered: (i) zero mean concentration with finite initial concentration flux, (ii) an initial top hat profile for the concentration, and (iii) an imposed background concentration gradient. All cases agree in the resulting relaxation time in the tau approximation relating the triple correlation to the concentration flux. The first order smoothing approximation is shown to be inapplicable. Title: Helical coronal ejections and their role in the solar cycle Authors: Brandenburg, Axel; Sandin, Christer; Käpylä, Petri J. Bibcode: 2004IAUS..223...57B Altcode: 2005IAUS..223...57B; 2004astro.ph..7598B The standard theory of the solar cycle in terms of an alpha-Omega dynamo hinges on a proper understanding of the nonlinear alpha effect. Boundary conditions play a surprisingly important role in determining the magnitude of alpha. For closed boundaries, the total magnetic helicity is conserved, and since the alpha effect produces magnetic helicity of one sign in the large scale field, it must simultaneously produce magnetic helicity of the opposite sign. It is this secondary magnetic helicity that suppresses the dynamo in a potentially catastrophic fashion. Open boundaries allow magnetic helicity to be lost. Simulations are presented that allow an estimate of alpha in the presence of open or closed boundaries, either with or without solar-like differential rotation. In all cases the sign of the magnetic helicity agrees with that observed at the solar surface (negative in the north, positive in the south), where significant amounts of magnetic helicity can be ejected via coronal mass ejections. It is shown that open boundaries tend to alleviate catastrophic alpha quenching. The importance of looking at current helicity instead of magnetic helicity is emphasized and the conceptual advantages are discussed. Title: What can we learn from Local Convection Simulations in the Context of mean Field Models of Stellar Rotation and Magnetism? Authors: Käpylä, Petri J.; Korpi, Maarit J.; Ossendrijver, Mathieu; Stix, Michael Bibcode: 2003ANS...324...63K Altcode: 2003ANS...324..I02K No abstract at ADS Title: Numerical study of the magnetorotational instability in weakly magnetised accretion disks: Resolution dependence of the Shakura-Sunyaev alpha Authors: Kapyla, P. J.; Korpi, M. J. Bibcode: 2001astro.ph.11554K Altcode: In this letter, we present numerical calculations made to investigate the possible resolution dependence of the Shakura & Sunyaev (1973) viscosity parameter alpha from local magnetohydrodynamic simulations of the magnetorotational instability (MRI). We find that the values of alpha do indeed depend significantly on the numerical resolution but also that when the highest resolutions attainable by the computational resources available are used, the growth of the alpha-parameter seems to saturate. The values of alpha are at most of the order of 10^(-3), which indicates that the sole presence of turbulence due to dynamo generated magnetic field in the disk is not enough to reproduce alphas of the order unity which could explain some observational results (e.g. Cannizzo 1993).