Dynamo, Dynamical Systems and Topology (25 July - 19 August 2011)

Final Report

Scientific goals and main achievements

The primary goal of the program was to discuss the current understanding and new approaches to solving the problem of
the origin of solar and stellar magnetic field, which is one of the central problems of physics and astrophysics, and a key to understanding the cosmic magnetism, in general. There are two main difficulties in studying this problem: 1) magnetic fields are generated by turbulent dynamos in convection zones below the visible surface, not accessible by direct observations; 2) solar and stellar magnetic activity is a multi-scale phenomenon, involving physical processes on very small scales, probably, below the current observational limit, and at the same time showing remarkable large-scale spatial and temporal organizations over a whole star. The prime target of our investigation is the Sun, which serves as the Rosetta Stone in this field and holds the key to unlocking the secrets of magnetic field generation in the Universe. The most detailed observational data and theoretical models have been obtained for the Sun, but also a very significant progress has been in observations and theories of magnetism on other stars.

The program was focused on the recent observational results from Solar Dynamics Observatory and large ground-based telescopes, Swedish Solar Telescope and New Solar Telescope, as well as on the results of turbulence theory and numerical simulations. The program brought together observers and theorists and provided a forum for broad discussions of the physical processes in the Sun and stars and related fundamental problems of theoretical physics, such as MHD turbulence, magnetic self-organization in plasma, radiative and turbulent energy transport, multi-scale dynamics, magnetic helicity and topological interpretation.

The main achievements of the program are in establishing links between the dynamo theory, numerical simulations and observational results, developing collaborations, approaches and ideas, and getting new science results. The Nordita program stimulated further development of the dynamo theory, numerical simulations and observational programs.

Overall structure of the program

The overall duration was 4 weeks. The program activity covered the following topics: Numerical simulations and mean-field theory, Recent results of numerical dynamo simulations, Effects of magnetic helicity and cross-helicity, Models of turbulent diffusivity and magnetic pumping, Progress in mean-field dynamo models.

As a rule, the morning hours were devoted to presentations and discussions, while the afternoons were left free for work, although most of us gathered for coffee each afternoon, during which some of the latest developments from participants were discussed.

The total number of program participants was 45 including 4 women, and 20 Nordic participant (Denmark 1, Finland 3, Sweden 16, of which 12 from Stockholm). A significant portion of the participants came from the US (5), Russia (4), Brazil (2), as well as Germany (6), Israel (2), and the UK (2). Of the 33 people from outside Stockholm, 24 stayed for 3 weeks or longer. The program marked a highlight in the ERC-supported Astrophysical Dynamo Project at Nordita, with two of the post-docs (Piyali Chatterjee and Gustavo Guerrero, right) just departing.

Scientific outcome

Solar dynamo models and effects of magnetic helicity and cross-helicity (A. Brandenburg, A. Bonanno, V. Pipin, D. Sokoloff, K. Kuzanyan, N. Yokoi, S. Candelaresi, D. Mitra, M. Reshetnyak, G. Guerrero)

An important quadratic invariant of MHD is cross helicity. Theory now predicts that at the solar surface is should be a negative multiple of the mean radial field. Results from new near-surface shear layer dynamos were presented, showing the importance of magnetic helicity. In this connection, new aspects of the Yoshizawa effect were also considered.

Flow-turbulence interaction in magnetic reconnection was investigated. Breakage of symmetry associated with mean-field configurations will lead to pseudoscalar generation in turbulence. Cross helicity (velocity--magnetic-field correlation) is a topological invariant of the magnetohydrodynamic equations. Cross-helicity effects in the momentum and magnetic-induction equations were analyzed. On the basis of this analysis, modulation of magnetic-reconnection rate due to the turbulent cross-helicity was estimated for the first time. Such work is expected to pave the way for understanding the turbulence effects in magnetic reconnection. Possibility and validity of the cross-helicity effect in magnetic reconnection were discussed with several researchers participating in this program. In relation to the cross-helicity dynamo, spatiotemporal distributions of the turbulent cross helicity were numerically investigated in the local and global convection domains in collaboration with some participants. In the context of solar activity cycle, incorporation of the cross-helicity effects into dynamical-system equations were examined.

Kinetic helicity is a topological invariant of the hydrodynamic equations. According to the turbulence theory of inhomogeneous hydrodynamic turbulence, inhomogeneity of turbulent helicity coupled with the rotational motion has been expected to play an important role in the large-scale vorticity generation. Such effects should be examined in numerical simulations. Validation of such effect using the pencil-code calculation has been preparing in collaboration with NORDITA researchers.

Due to the limited statistics some of the mean-field quantities considered in dynamo theory can fluctuate. These fluctuations can produce new features in the dynamo systems. On an example of the Parker's dynamo in the spherical shell we demonstrate how the small fluctuations (about 5 percents) of the mean level of the alpha-effect can cause significant equatorial asymmetry of the oscillatory magnetic field, as well as propagation of the dynamo wave through the equatorial plane.

Spherical convective rotating shells demonstrate existence of kinematic helicity even in the incompressible case. However, due to some theoretical background as well as to 3D simulations, spatial distribution of the kinetic helicity for incompressible and compressible systems can be quite different.

Solar dynamo with a positive alpha effect in the overshoot layer and a deep meridional flow was investigated. In this model the alpha effect is produced by a magnetic instability at the tachocline. The parity selection mechanism was investigated.

Turbulence models in dynamo theory (I. Rogachevskii, N. Kleeorin, K.-H. Raedler, N. Mansour, M. Rheinhardt, A. Brandenburg, Y. Snellman)

The tau approach and other closure hypotheses were discussed at length in a discussion session dedicated to this topic. Numerical evidence for its validity is was presented, but a lack of overlap with the second order correlation approximation was noted.

New work in connection with Reynolds stress modeling has been discussed and simulations have been presented to quantify the effect of the return to isotropy as a function of Reynolds number.

New work on two-dimensional turbulence has been performed using graphics card computations and the saturation behavior of the inversely cascading energy has been modeled with a 3-scale model.

Turbulent transport in stratified and/or rotating turbulence has been investigated. The test-field method is used where transport coefficients are determined by solving a set of equations with properly chosen mean magnetic fields or mean scalars. The method is adapted to mean fields which may depend on all thee space coordinates. Anisotropy of turbulent diffusion is found to be moderate in spite of rapid rotation or strong density stratification. Contributions to the mean electromotive force determined by the symmetric part of the gradient tensor of the magnetic field turn out to be important. In stratified rotating turbulence, the alpha effect is strongly anisotropic, suppressed along the rotation axis on large length scales, but strongly enhanced at intermediate length scales.

New ideas about the bottleneck effect in turbulence were developed. This work emerged from one of the extended coffee sessions. The closure in the one dimensional incompressible Boussinesq case was investigated, and tried to modify it to better match the numerical results from pencil-code. It would seem that the results from the Pencil-code are much more affected by the presence of vertical boundaries than the closure results. To shed light on this problem a variety of different closure approaches was investigated, none of which produced results similar to those from pencil-code without some serious side effects. A lot of new ideas and different approaches were discussed, and will most probably lead to important breakthroughs in this field in the near future.

Spontaneous Symmetry Breaking of the chiral symmetry in Tayler Instability was investigated. The basic idea is to test the recently proposed scenario for the SSB in MHD in the case of an instability produced in stably stratified medium. In addition, to study the Tayler Instability the pencil-code set-up was implemented for the study of the possible helical modes that can arise due to the instability. The aim is to see whether or not the instability can give birth to a breaking of a non-helical state, so causing the appearance of helicity. Several questions regarding this phenomena are still unanswered, like the difference between the linear and non-linear stages of the growth phase as well as the role that such a breaking can have in generating a dynamo.

Magnetic self-organization in turbulent plasma (I. Kitiashvili, N. Mansour, I. Rogachevskii, N. Kleeorin, K. Kemel, A. Brandenburg)

The origin of magnetic flux concentrations in active regions and sunspots was discussed. New work in connection with the negative effective magnetic pressure instability has been initiated. A related issue concerns the location of the solar dynamo, and more work has been done to examine the idea that the solar dynamo is a distributed one, shaped by near-surface shear.

Large-scale vortex instability and its applications to the starspots in rapidly rotating stars were discussed. The mechanisms of production of vorticity in the solar conditions were investigated.

The first numerical demonstration of the negative effective magnetic pressure instability was obtained in direct numerical simulations of stably-stratified, externally-forced, isothermal hydromagnetic turbulence. By the action of this instability, initially uniform horizontal magnetic field forms flux concentrations whose scale is large compared to the turbulent scale. It is shown that the magnetic energy of these large-scale structures is only weakly dependent on the magnetic Reynolds number. The results supports earlier mean-field calculations and analytic work which identified this instability. Applications to the formation of active regions in the Sun are discussed.

The magnetic self-organization effects were also investigated in realistic simulations of solar magnetoconvections, including spontaneous formation of magnetic structures and the filamentary structure of sunspots. The sunspot penumbra is a transition zone between the strong vertical magnetic field area (sunspot umbra) and the quiet Sun. The penumbra has a fine filamentary structure that is characterized by magnetic field lines inclined toward the surface. Numerical simulations of solar convection in inclined magnetic field regions have provided an explanation of the filamentary structure and the Evershed outflow in the penumbra. The radiative MHD simulations were used to investigate the influence of the magnetic field inclination on the power spectrum of vertical velocity oscillations. The results revealed a strong shift of the resonance mode peaks to higher frequencies in the case of a highly inclined magnetic field. The frequency shift for the inclined field is significantly greater than that in vertical-field regions of similar strength. This is consistent with the behavior of fast MHD waves.

Observational input and constraints for the dynamo problem and solar MHD (J. Stenflo, P. Goode, D. Sokoloff, V. Pipin, K. Kuzanyan, A. Kosovichev, G. Scharmer)

Regularity in tilt angles of bipolar regions has been discussed and violation of Hale's polarity law has been contrasted with naive expectations from the Babcock-Leighton dynamics. A lot of new developments connected with understanding the Waldmeier relations has been presented.

The possibility to explain the statistical relations of the sunspot activity cycle, like the so-called Waldmeier relations, by means of the mean field dynamo model with the fluctuating alpha-effect was investigated. The dynamo model includes the long-term fluctuations of the alpha-effect and two types of the nonlinear feedback of the mean-field on the alpha-effect including the algebraic quenching and the dynamic quenching due to the magnetic helicity generation. We found that the models are able to reproduce qualitatively the inclination and dispersion across the Waldmeier relations with the 20% fluctuations of the alpha-effect. The models with the dynamic quenching are in a better agreement with observations than the models with the algebraic alpha-quenching. Statistical distributions of the modeled parameters, like the amplitude, period, the rise and decay rates of the sunspot cycles, are compared with observations for 2D- and 1D-models.

A tendency toward superdiffusion in solar surface turbulence has been mentioned and discussed in connection with expectations from local surface dynamos.

Differences between the actual meridional flow speed, as obtained from Doppler measurements, and that obtained from magnetic feature tracking has been discussed.

The recent global analysis of the 15 year synoptic data set of SOHO/MDI full disk magnetograms was presented and discussed in depth, since the results may have profound implications for our understanding of how the Sun's dynamo operates. The tilt angles of the 160,000 measured bipolar magnetic regions are found to increase with the sine of the latitude, which implies that emerging regions of all sizes, from large to small, contribute to a general N-S dipole moment that will lead to the reversal of the old global dipole field and the regeneration of the new one. Violations of Hale's polarity law are rather common, at a few percent level. Examples have been identified, which show medium-sized bipolar regions of opposite orientations side by side in the same latitude zone of a given magnetogram. These examples are manifestly incompatible with the old Babcock-Leighton type paradigm that sunspots originate from a coherent subsurface toroidal flux system, but they are consistent with turbulent dynamo theories, according to which fluctuations at all scales play a fundamental role.

New observational results of helioseismology and magnetic field measurements from the Solar Dynamics Observatory, as well as high-resolution observations from the Swedish Solar Telescope and Big Bear Solar Observatory were discussed in relation to the theoretical dynamo and MHD turbulence studies. New data assimilations of solar convection have been developed.

Numerical MHD simulations
(F. Busse, R. Simitev, P. Käpylä, M. Mantere, A. Brandenburg, G. Guerrero, J. Warnecke, I. Kitiashvili, G. Di Bernardo)

Simulations of plasmoid ejections from convection in a spherical shell have been developed further.

The importance of fluctuating alpha effects have been discussed both in the geodynamo context, but also as a general mechanism to explain the phenomenon of what is known as shear dynamos.

The generation of large scale magnetic field by strongly-driven chaotic convection in rotating spherical fluid shells under the assumption of the Boussinesq approximation has been studied. This configuration is the simplest self-consistent model of the spherical dynamo problem and it can be applied, with some adjustments of the values of the basic governing parameters, both to planetary dynamos and the Solar dynamo. The model was studied by direct numerical simulation of the coupled Navier-Stokes, magnetic induction and energy equations, with the inclusion of all nonlinear effects and without ad-hoc assumptions. In the case of thick-shell dynamos, which is appropriate to the Geodynamo, the possibility of the existence of two essentially different dynamo attractors at identical parameter values was studied. In the case of thin-shell dynamos, which is appropriate to the solar dynamo, a large number of numerical simulations was performed in an attempt to model the profile of differential rotation in the solar interior and the dynamics of large-scale magnetic structures and active regions in the Sun. In addition, the self-consistent dynamo model was used to compute maps of various quantities which are of particular interest, such as magnetic, kinetic and cross helicity, and discussed the results with other participants.

In the Pencil code a numerical setup was implemented for the study of the possible events of magnetic reconnection, with or without turbulence. This work aims, among others, to test numerically a model of fast magnetic reconnection in the presence of weak turbulence proposed by Lazarian & Vishniac. First, 2-D Sweet-Parker reconnection was investigated, the stage before turbulence injection. A full understanding of this process is required in order to perform further analysis of reconnection in the presence of turbulence. Numerical simulation setup in order to successfully reproduce the Sweet-Parker scaling of reconnection was investigated. In particular the behaviour at low values for resistivity and the role of the Guide Field was investigated.
Links between solar, astrophysical and geophysical dynamos

Links between solar, astrophysical and geophysical dynamos (R. Arlt, F. Busse, R. Simitev, M. Reshetnyak, E. de Gouveia Dal Pino, V. Duez, S. Sur, A. Bobrick, Xing Wei, M. Mantere, F. del Sordo)

The occurrence of an alpha effect in the absence of any pseudo-scalars has been discussed in connection with spontaneous symmetry breaking. New work on the Tayler instability has been initiated and the possible relevance to magnetars has been mentioned.

Turbulent fast reconnection and its role on the diffusive transport of magnetic flux in astrophysical environments were discussed. Potential applications of this transport to dynamo problems is foreseen.

Outflow ejection by supernovae in galaxy disks was studied. Numerical simulations of supernova triggered galactic outflow formation were performed including realistic galactic disk/coronal setups for the Milky Way and starburst galaxies, the possibility of using these setups in magnetic helicity transport studies related to galactic dynamos has been discussed. The role of kinectic effects on turbulent dynamo models related to the IGM using a Kinectic-MHD description was also discussed.

Progress on research on the generation of large scale magnetic field by strongly-driven chaotic convection in rotating spherical fluid shells under the assumption of the Boussinesq approximation has been reported. In addition, self-consistent dynamo models have been developed to compute maps of various quantities which are of particular interest, such as magnetic, kinetic and cross helicity, and discussed the results with other participants.

Thin shell dynamos were investigated with the goal of understanding as many solar magnetic phenomena as is possible with numerical simulations of a most simple physically consistent convection driven dynamo model. With a given ratio of 0.65 between inner and outer radii of the rotating spherical there are only 4 parameters of the coupled Navier-Stokes equations, heat equation, and equation of magnetic induction that can be varied: Rayleigh number R, rotation parameter tau, Prandtl number and magnetic Prandtl number.

Although the geostrophic differential rotation found in the simulations clearly differs from the solar differential rotation, the oscillatory dynamo shares many features with the observed solar magnetic cycle. The most important new result is the strong non-axisymmetric (m=1) component of the oscillating magnetic field, the energy of which far exceeds the energy of the axisymmetric component. The latter is traditionally assumed to be the dominant component of the solar magnetic field. The new result is in good agreement with the phenomenon of active longitudes on the sun.

In geodynamo, it's believed that Busse rolls generate the alpha effect, namely the fluid helical motions in cyclonic and anti-cyclonic convection rolls twist the toroidal field lines to create the poloidal field lines. An alternative is the precession which can generate inertial waves arising from the destabilization of the internal shear layers (the so-called triad resonance) to induce the alpha effect. The topography or ellipticity can help the generation of these waves. However, with the spectral code, it's impossible to calculate for the topography and large ellipticity. Geodynamo calculations with the pencil code have been discussed.


As a result of the program, several Nordita preprints have already appeared or are expected to appear:

Excerpts from selected responses from the participants

Other items

During the meeting, Koen Kemel defended his licentiate thesis on "Mean-field theory in magneto-hydrodynamics and the generation of magnetic flux concentrations" with Sasha Kosovichev as opponent.

  • Photo gallery (from CK)