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ERC Adv. Grant Project 227952 AstroDyn

Fourth reporting period: 1 August 2013 - 31 January 2014 (months 55-60)
Project's full title: Astrophysical dynamos
Name of the PI: Axel Brandenburg
Name of PI's host institution for project: Nordita
Project website:

Management of relation between PI and HI:

The budget of the AstroDyn project grant has been managed by Ms Marianne Persson Söderlind since August 2011, as planned.

Expenses incurred during reporting period 4:

The fourth of the PhD students on the AstroDyn grant finished his work in August. Assistant professor Mitra and visiting professor Rheinhardt finished their work during the last 6 months of the grant.

Personnel paid during reporting period 4:

J. Warnecke      1 months   PhD student, tasks 9 and 10
M. Rheinhardt    6 months   visiting professor, task 2
D. Mitra         6 months   assistant professor, tasks 2, 5, and 14

1/2 page overview of achievements for the reporting period

The fourth of the PhD students on the AstroDyn grant finished his work in August (Warnecke) with an important paper on the formation of bipolar regions from the negative effective magnetic pressure instability (task 8). He now went to the Max Planck Institute for Solar Systems Research in Göttingen on a Marie Curie Grant. Assistant professor Dhrubaditya Mitra is now working the interaction between the negative effective magnetic pressure instability and and underlying turbulent dynamo (task 8). Together with visiting professor Matthias Rheinhardt, he is also working on the implementation of the test-field method for spherical shell simulations. Matthias Rheinhardt also finished work on a qualitatively new mean-field dynamo mechanism that works solely from time-delayed transport. Full details are available on The full list of 105 referred papers (published or in press) and 8 submitted ones, all acknowledging ERC is given under No problems or delays have occurred.

Detailed objectives for the reporting period and corresponding achievements

The following 14 items in italics are excerpts from the original proposal of 2008. The present status of the achievements is described for all items. All the papers that are quoted acknowledge the ERC grant. The results of Periods I and II are summarized in small letters and documented by 59 peer-reviewed papers plus 18 additional papers that acknowledge the ERC, but whose scope falls outside the originally anticipated goals. The results of Period III are summarized in normal letters and documented by 21 peer-reviewed papers plus 6 additional papers addressing other goals. Several papers mentioned under Period II were published only in Period III, but they remain listed under Period II, which explains the relatively large number of papers in that period.

  1. Code validation.  Continue testing the spherical extension of the PENCIL CODE by comparing with other codes. Much of this has already been completed successfully, but there are some issues connected with the treatment of boundary conditions in large scale $\alpha$ effect dynamos where the comparison is not yet satisfactory. (Phase 1)

    Results from reporting periods I-III.  The implementation of spherical geometry in the PENCIL CODE has been developed and a number of additional tests and code enhancements have been carried out by Drs Mitra and Plasson as well as Mr. Svedin. Further tests have been performed by Dr. Babkovskaia in connection with applications to turbulent combustion. During the PENCIL CODE User Meeting 2010 in New York, an important development was undertaken to restructure the mean-field modelling as part of a substructure underneath the magnetic field modules. The main scientific outcomes of this are reported below in connection with the formation of magnetic structures in stratified turbulence. In addition, Drs Chatterjee and Mitra have continued working on the implementation of the anelastic solver. The following papers where mentioned in reporting period I:

    Current status.  The PENCIL CODE is currently at revision number r21458. Among many other developments, it allows the use of complex test fields and its use for two-dimensional test fields has been improved.

  2. Nonlinear test-field method.  Determination of the quenching of the nonlinear $\alpha$ effect and the turbulent diffusivity by large scale magnetic fields using the test-field method. The importance of the small scale current helicity for the $\alpha$ effect is still not entirely settled, so it is important to extend work along those lines. The idea is to calculate not only the response of each test field on the small scale velocity, but also on the small scale magnetic field. For this work the Cartesian configuration of the PENCIL CODE will be used. (Phase 1)

    Results from reporting periods I-III.  A new test-field method has been developed by Rheinhardt & Brandenburg that is able to account for the effects of MHD background turbulence. This method has been tested for the Roberts flow and the paper has now appeared. In addition, we have identified pitfalls in determining the correct $\alpha$ effect using the more traditional imposed-field method. This work has led to new possibilities for determining magnetic helicity fluxes both in mean-field and in direct simulations, which in turn has led to the realization that diffusive magnetic helicity fluxes can be more important than previously thought. We have extended the determination of turbulent transport coefficients to contributions that do not depend on the magnetic field, but depend just on rotation (the Yoshizawa effect). In this connection we have for the first time determined numerically its quenching due to self-consistently generated magnetic fields. We have also extended the test-field method to irrotational flows, to helical shear flows, and to passive scalar transport. Using the linear test-field method we now have the first solid confirmation of a negative turbulent diffusivity dynamo.

    New achievements.  A qualitatively new type of mean-field dynamo mechanism has been discovered. It works solely from time-delayed transport and makes use of earlier developments concerning the memory effect in turbulent transport coefficients.

    The work on the nonlinear test-field method is still to be extended to determine the turbulent viscosity tensor as well as other contributions such as the AKA and $\Lambda$ effects.

  3. Catastrophic quenching in a spherical shell.  Reproduce the catastrophic quenching behavior in a closed sphere or spherical shell sector using perfectly conducting boundary conditions and forced turbulence. Some work in this direction has already been done, but the results are not yet well understood nor entirely conclusive. (Phase 1)

    Results from reporting periods I-III.  In pursuit of this problem, Dr. Mitra has come across a new type of solution that yields equatorward migration even without shear. This result is quite surprising and has now been published. Additional work is in progress and has been combined with dynamos in spherical shells (item 7 in this list). To understand catastrophic quenching, we need to understand the magnetically generated $\alpha$ effect, whose value is characterized by magnetic helicity, which is a conserved quantity at large magnetic Reynolds numbers. There is the possibility that there might be higher order invariants that could also matter. This possibility has been tested in a recent paper where some evidence for the need of higher order invariants has been found.

    New achievements.  The ideas that lead to catastrophic quenching and the understanding how to get rid of this also predict the existence of bi-helical magnetic field, for which there is now evidence from solar wind measurements. We have now developed a new method that allows us to determine potentially bi-helical magnetic fields in galaxies

  4. Dynamo effect from the MRI.  Calculate the nonlinear $\alpha$ effect and the turbulent diffusivity for turbulence driven by the magneto-rotational instability (MRI). Some work in this direction has already been done, but only a few representative test cases at relatively low resolution were done. This work is primarily relevant to accretion discs. However, understanding this case may also teach us general aspects of magnetically driven dynamos that may in some form also work in the Sun. (Phase 1)

    Results from reporting periods I-III.  This work has been started with the help of a student from the ENS in Paris, Emeric Bron, who visited us for 1/2 year on an internship. Preparatory work on this topic has already been published with another student who also came on an internship. Both works have led to new issues concerning the importance of using open boundary conditions for the magnetic field. This has also led to new work that helped resolving the question of the dependence of the onset of MRI on the value of the magnetic Prandtl number. In now turns out that with open boundary conditions the onset is independent of the magnetic Prandtl number. The possibility of explaining MRI dynamo action as the result of an incoherent $\alpha$-shear dynamo has now been explored further and has been shown to obey the observed linear scaling of growth rate with shear rate. The MRI has now been explored further in systems where the Hall effect is important. This following work appeared in the previous reporting period.

    New achievements.  We have shown that turbulence lowers the growth rate of the MRI in exactly the way expected from turbulent diffusion.

    This work showed to our surprise that turbulent diffusion is not strongly affected by the spatio-temporal nonlocality nor nonlinearity.

    Main objectives originally scheduled for the second reporting period:

  5. Test-field method in spherical geometry.  Adapt the test-field method to spherical coordinates. Originally the test-field method was developed in connection with full spheres, and then the test-fields consisted of field components of constant value or constant slope. However, only afterwards it became clear that the scale (or wavenumber) of the field components must be the same for one set of all tensor components, and so it is necessary to work with spherical harmonic functions as test-fields. In other words, constant and linearly varying field components are insufficient. (Phase 2)

    Results from reporting periods I-III.  In preparation of this task, Dr. Mitra has implemented a helical flow in spherical geometry that will allow us to validate the test-field method in spherical geometry. Drs Käpylä, Mitra, and Rheinhardt have now modified the test-field_xz module to work in spherical geometry.

    Current status.  Dr Rheinhardt has now derived analytical expressions for the spatial dependence of the $\alpha$ effect that depends on two spatial coordinates.

  6. Alpha effect from convection.  The calculation of the $\alpha$ effect in convective turbulence is at the moment rather unclear. There are some results suggesting that $\alpha$ goes to zero in the limit of large magnetic Reynolds numbers even for kinematically weak magnetic fields. There remain however several open questions regarding the amount of stratification ($\alpha$ should be proportional to the local stratification gradient and should hence be absent in Boussinesq convection) and regarding the degree of scale separation. (Phase 2)

    Results from reporting periods I-III.  In the mean time the situation has changed dramatically. Significant progress in this direction has been made by Dr. Käpylä and collaborators in a series of papers: We have now found a surprising occurrence of oscillatory large-scale dynamo action from Cartesian convection simulations. It turned out that the alpha effect is proportional to $\nabla\ln(\rho u_{\rm rms}^2)$, and not, as previously thought, proportional to $\nabla\ln(\rho u_{\rm rms})$.

    Current status.  We are still pursuing this research with another post-doc beyond the AstroDyn project.

  7. Dynamo in open shells with and without shear.  Calculate the saturation of the magnetic field and the underlying dynamo effects with open boundary conditions in a spherical shell sector with and without shear. One expects low saturation amplitude with magnetic energy of the mean field being inversely proportional to the magnetic Reynolds number in the absence of shear, but of order unity in the presence of shear. The shear is here critical, because it is responsible for the local driving of small scale magnetic helicity fluxes. (Phase 2)

    Results from reporting periods I-III.  Significant progress has also been made in understanding the nature of magnetic helicity and its fluxes. Additional items connected with understanding magnetic helicity fluxes. While the progress has been significant, our work also showed that the magnetic helicity fluxes are still small compared with microscopic diffusion unless the magnetic Reynolds number exceeds values around $10^3$ to $10^4$. Most surprisingly, there is now evidence that the Vishniac-Cho flux may not exist. We have also established that in a stationary but open system the divergence of magnetic helicity fluxes (even separately for those of the small-scale field) is gauge-invariant if the magnetic helicity density is found to be statistically steady. Furthermore, magnetic helicity fluxes can begin to alleviate catastrophic quenching at magnetic Reynolds numbers just a little above 1000, which is significantly lower than previously thought (which was closer to 30,000).

    New achievements.  In an attempt to find observational evidence for magnetic helicity in the Sun and in Galaxies, we have devised a new method for determining magnetic helicity at the solar surface.

    Achievement of goals scheduled for grant period III:

  8. Magnetic flux concentrations near the surface.  Test the scenario that the emergence of active regions and sunspots can be explained as the result of flux concentrations from local dynamo action via negative turbulent magnetic pressure effects or turbulent flux collapse. (Phase 2)

    Results from reporting periods I-III.  This work constitutes one of the corner stones of our project in that we must explore scenarios for being able to explain the formation of magnetic flux concentrations in the absence of deep-rooted hypothetical flux loops at the bottom of the convection zone. This work has been started with Professors Kleeorin and Rogachevskii, as well as Mr. Kemel (one of our PhD students). The field has now seen a major transformation with the detection of the negative effective magnetic pressure instability in turbulence simulations. This has been a major milestone for this project that contributed to putting this effect on the map. Furthermore, Dr. Käpylä has started look at the possibility of producing magnetic flux concentration in stratified convection. Major new developments include the detection of strong magnetic spots and even bipolar regions. As advertised, one of the important new developments includes the treatment of radiative transport. A Master's Thesis on this topic has been completed (Ms. Barekat) and this work has been submitted. This and other new papers are listed below.

    New achievements.  We now understand that the negative effective magnetic pressure instability can affect the flow in regions well above the optimal range of its operation by producing a downflow along vertical magnetic field lines that leads to cooling of the upper layers. We also made progress in the combined modeling of dynamo and negative effective magnetic pressure instabilities.

    One of the next urgent items remains the inclusion of hydrogen ionization.

  9. CME-like features above the surface.  Analyze the nature of the expelled magnetic field in simulations that couple to a simplified version of the lower solar wind. It is possible that the magnetic field above the surface might resemble coronal mass ejections (CMEs), in which case more detailed comparisons with actual coronal mass ejections would be beneficial. (Phase 3)

    Results from reporting periods I-III.  This project has been started with Mr. Jörn Warnecke, one of our PhD students who arrived in August 2009. Our first steps in this direction include a simple Cartesian model with a force-free outer layer above the turbulence zone. This work has been extended to include spherical geometry. We have also now included convection. Using observations from the Ulysses space craft we have for the first time determined the magnetic helicity spectrum in the solar wind. Similar results have now also been verified using the simulations of Warnecke et al. We have found a new explanation for the unusual sign of the magnetic helicity in the solar wind and published it in J. Spa. Weather Spa. Clim. We have also now extended work on convection-driven dynamos with an outer coronal envelope.

    Current status.  The last paper on this topic is now published.

  10. Convective dynamo in spherical shell.  Set up convection in the spherical shell. If the resulting scale of the flow is small enough and there is scale separation it would be useful to simulate the resulting magnetic field, compare with forced turbulence simulations in spherical shells and see whether contact can be made both with the Sun and with improved mean field models. (Phase 3)

    Results from reporting period I.  This work has been started under the initiative of Dr. Käpylä and first results have been published. We have now obtained convection-driven dynamo in spherical wedges. Self-consistently driven differential rotation has been studied for different degree of stratification and rotation rates. A major development includes the discovery of equatorward migration from spherical shell convective dynamos. The likely cause of this is an oscillatory $\alpha^2$ dynamo.

    New achievements.  At rapid rotation, nonaxisymmetric dynamo modes develop. They take the form of an azimuthal dynamo wave. We have also now suggested that the solar-like differential rotation of the Sun might be a result of initial conditions and would not be obtained if one were to model the Sun from scratch.

  11. Buoyancy-driven dynamo.  The turbulence in accretion discs is believed to be driven by the magnetorotational instability. It was one of the first examples showing cyclic dynamo action somewhat reminiscent of the solar dynamo. It was believed to be a prototype of magnetically driven dynamos. In the mean time, another example of a magnetically driven dynamo has emerged, where magnetic buoyancy works in the presence of shear and stratification alone. This phenomenon is superficially similar to a magnetically dominated version of the shear-current effect. We are now in a good position to identify the governing mechanism by using the recently developed test-field method. (Phase 3)

    Results from reporting periods I-III.  Dr. Chatterjee has started with this project. Surprisingly, we have found that in a completely mirror-symmetric system, an $\alpha$ effect can still emerge as a result of spontaneous symmetry breaking. Furthermore, Drs Guerrero and Käpylä have modeled convection with a strong shear layer (tachocline) at the bottom. They find the emergence of flux tubes at the top of the domain, but the field has become rather weak by the time it reaches the surface. We have now provided a more thorough understanding helicity-generating instabilities by studying the Tayler instability. This work is now published.

    Current status.  This project has reached a successful conclusion.

  12. Deep convection dynamo.  The deeper layers of the Sun are characterized by rather low values of the energy flux relative to the natural units given by $\rho c_{\rm s}^3$, where $\rho$ is the density and $c_{\rm s}$ is the speed of sound. In the Sun this is accomplished by nearly perfectly adiabatic conditions, which implies low Mach numbers on the order of $10^{-4}$. Such conditions cannot be economically simulated with compressible codes, so it is necessary to turn to an anelastic configuration of the PENCIL CODE. This should not be so hard to do because a Poisson solver has already been implemented in connection with solving for self-gravitating flows. Another possibility would be multigrid solvers. One such multigrid solver is also already present in the PENCIL CODE, but this subroutine still need to be parallelized. In discussions with Professor J Toomre from Boulder concerning near-future peta-flop computing it became clear that there is great interest in mesh-based codes that are able to solve anelastic flows in spherical shells. (Phase 4)

    Results from reporting periods I-III.  Dr. Chatterjee has started implementing an anelastic solver into the PENCIL CODE. This work has been presented at the last PENCIL CODE User Meeting in New York ( We did make some progress in understanding the effect of strong stratification in the stably stratified tachocline and the layers beneath.

    Current status.  This project has been essentially completed, but it has now led to the award of a PRACE computing grant that allows us to model deep spherical shell convective dynamo action.

  13. Solar dynamo models and solar cycle forecast.  Among the popular applications of solar dynamo theory and solar magnetohydrodynamics are solar cycle predictions, solar subsurface weather, and space weather. Also of interest are predictions of solar activity during its first 500 thousand years. This has great relevance for predicting the loss of volatile elements from the Earth's atmosphere, for example, and for understanding the conditions on Earth during the time when life began colonizing the planet. In this connection it is also of interest to calculate the deflection of cosmic ray particles by the Sun's magnetic field and on the scale of the galaxy which is relevant for galactic cosmic rays. (Phase 4)

    Results from reporting periods I-III.  In a preparatory step of this work, Mr. Svedin has started developing a data assimilation package for the PENCIL CODE. The first steps of this work are currently being written up. For future models of the solar dynamo, the effects of magnetic helicity fluxes have now been studied in more detail both in Cartesian as well as on spherical mean-field models. While the effects of the near-surface shear layer have still not been taken into account into much of this modeling, new insights have been gained by modelling the meridional advection of magnetic structures on the solar surface. A paper on data assimilation for stratified convection has recently been published.

    Current status.  Dr. Svedin continues to work part time at Nordita to continue on new developments in data assimilation.

  14. Applications to laboratory liquid sodium dynamos.  Unexpected beneficial insights have come from recent laboratory dynamo experiments. Unlike numerical dynamos, experimental liquid metal dynamos are able to address the regime of rather low values of the magnetic Prandtl number of the order of $10^{-5}$, which is of interest for solar and stellar conditions. At the same time the magnetic Reynolds number can be large enough (above 100) to allow for dynamo action. It is hoped that such work can teach us important aspects about small-scale dynamos at low magnetic Prandtl number, which is relevant to the Sun, but hard to address numerically. (Phase 4)

    Results from reporting periods I-III.  There is significant hope to be able to determine for the first time the alpha effect in a turbulent liquid-metal plane Couette flow. Preparations have been performed through analytical and numerical calculations. We have also pursued turbulent dynamo calculations at low magnetic Prandtl numbers that are relevant to liquid metal dynamos. There has now also been an unexpected development with applications to planetesimal formation by what is called collisional fusion.

    Current status.  The work with Rüdiger has now been resubmitted and would provide a test bed for measuring $\alpha$ in a turbulent shear flow.

The ultimate goal of the project is of course to establish the cause of the equatorward migration of magnetic activity belts at low solar latitudes. Is it the rather feeble meridional circulation, as assumed in the now rather popular flux transport models, even though one has to assume unrealistic values of the turbulent magnetic Prandtl number, or is it perhaps the near-surface shear layer, which would have indeed the right sign?

Several reviews have been published that outline our current thinking:

The success of our project is further evidenced by a number of publications on other timely aspects of dynamo theory.
Papers from reporting periods I-III

In all these papers, support from the ERC is acknowledged.

3. Explanation of the use of resources

A detailed working plan is given in the extended synopsis of Section 2. We summarize here the intermediate goals as described in detail in that section, where the goals were ordered by the phase within the grant period.

Table: Status of completion. Columns 2-5 sum to unity, so all entries summed together give 14 for the 14 objectives. The sum of each of the 4 columns is therefore 14/4.
objective I II III IV task
1 0.8 0.2 0.0 0.0 code validation
2 0.3 0.3 0.3 0.1 (nonlinear) test-field
3 0.3 0.3 0.2 0.2 catastr. quenching
4 0.1 0.3 0.4 0.2 MRI dynamo
5 0.1 0.2 0.4 0.3 spherical test-field
6 0.7 0.2 0.1 0.0 alpha in convection
7 0.2 0.1 0.4 0.3 open shell dynamos
8 0.3 0.6 0.1 0.0 magn flux concentrations
9 0.2 0.2 0.3 0.3 CME-like features
10 0.2 0.2 0.2 0.4 conv shell dynamos
11 0.1 0.2 0.3 0.4 buoyancy-driven dynamos
12 0.0 0.2 0.3 0.5 deep convection
13 0.2 0.2 0.2 0.4 solar dynamos/forecast
14 0.0 0.3 0.3 0.4 laboratory dynamos
vertical sum 14/4 14/4 14/4 14/4  
expenses 574 716 467 72 Sum=1,829 k EUR

  1. Code validation, nonlinear test-field method, catastrophic quenching in a spherical shell, dynamo effect from the MRI (items 1-4 in Sect. 2).
    Task completed: resources consumed in Period IV: xx,000.00 EUR (246,000.00, 225,000.00, and 120,000.00 in Periods I-III). Completion date from Annex I: July 2010. Comments: this follows the revised plan of September 2012.

  2. Test-field method in spherical geometry, alpha effect from convection, dynamo in open shells with and without shear (items 5-7 in Sect. 2).
    Task completed: resources consumed in Period IV: xx,000.00 EUR (164,000.00, 102,000.00, and 120,000.00 in Periods I-III). Completion date from Annex I: February 2012. Comments: this follows the revised plan of September 2012.

  3. Magnetic flux concentrations near the surface, CME-like features above the surface, convective dynamo in spherical shell, buoyancy-driven dynamo (items 8-11 in Sect. 2).
    Task completed: resources consumed in Period IV: xx,000.00 EUR (131,000.00, 245,000.00 and 120,000.00 in Periods I-III). Completion date from Annex I: July 2013. Comments: this follows the revised plan of September 2012.

  4. Deep convection dynamo, solar cycle forecast, applications to laboratory liquid sodium dynamos (items 12-14 in Sect. 2).
    Task completed: resources consumed in Period IV: xx,000.00 EUR (33,000.00, 143,000.00, and 106,000.00 in Periods I-III). Completion date from Annex I: February 2014. Comments: this follows the revised plan of September 2012.

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Axel Brandenburg 2018-08-12