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ERC Adv. Grant Project 227952 AstroDyn

Reporting period: 1 February 2009 - 31 July 2010 (months 1-18)
Project's full title: Astrophysical dynamos
Name of the PI: Axel Brandenburg
Name of PI's host institution for project: Nordita
Project website: http://www.nordita.org/~brandenb/AstroDyn/


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.

  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)

    Status of the achievements.  The implementation of spherical geometry in the PENCIL CODE has now been explained in the following paper:


    A number of additional tests and code enhancements have been carried out by Drs Mitra and Plasson as well as Mr. Svedin. Additional tests have been performed by Dr. Babkovskaia in connection with applications to turbulent combustion:


    In addition, Drs Chatterjee and Mitra have worked on the implementation of the anelastic solver (described below in more detail).

  2. Nonlinear testfield method.  Determination of the quenching of the nonlinear $\alpha$ effect and the turbulent diffusivity by large scale magnetic fields using the testfield 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)

    Status of the achievements.  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 is now in press:


    This work is currently being extended to determine the turbulent viscosity tensor as well as other contributions such as the AKA and $\Lambda$ effects. In addition, we have identified pitfalls in determining the correct $\alpha$ effect using the more traditional imposed-field method, on which we have produced the following publications:


    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. (This is described further below in more detail.)

  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)

    Status of the achievements.  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).

  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)

    Status of the achievements.  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. This work is still to be written up. 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. This work has now been submitted:


    More work is planned to clarify the role of boundary conditions further.

    Objectives for the remainder of the grant period and status report, with emphasis on already obtained results:

  5. Testfield method in spherical geometry.  Adapt the testfield method to spherical coordinates. Originally the testfield method was developed in connection with full spheres, and then the testfields 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 testfields. In other words, constant and linearly varying field components are insufficient. (Phase 2)

    Status of the achievements. In preparation of this task, Dr. Mitra has implemented a helical flow in spherical geometry that will allow us to validate the testfield method in spherical geometry. We plan to implement this method soon.

  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)

    Status of the achievements.  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:


    Additional work in this direction has been mentioned under item 2 in this list. More work is planned, depending on how our recently published work is being perceived by the community.

  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)

    Status of the achievements.  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 $10^4$.

  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)

    Status of the achievements. 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). One paper has now been published and one has been submitted.


    In addition, Dr. Käpylä has started look at the possibility of producing magnetic flux concentration in stratified convection. In future work we plan to investigate the scale dependence of this effect.

  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)

    Status of the achievements.  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. A revised version of the paper


    Our next step is to include spherical geometry. This work has already been started.

  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)

    Status of the achievements.  this work has been started under the initiative of Dr. Käpylä and first results have been published.


    We are currently extending this work to larger domains at increased resolution. For this reason we are currently applying for major European computing resources.

  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 testfield method. (Phase 3)

    Status of the achievements.  Dr. Chatterjee has started with this project. First results have been reported in the following review paper:


    A more dedicated publication is in preparation. Another related approach is currently being pursued by Drs Guerrero and Käpylä who model 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.

  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)

    Status of the achievements.  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 (http://www.nordita.org/software/pencil-code/UserMeetings/2010/).

  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)

    Status of the achievements. 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:


    In future work we plan to take into account the effects of the near-surface shear layer and to combine these efforts with data assimilation techniques using both direct and mean-field models.

  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. (This magnetic Prandtl number is not to be confused with the turbulent magnetic Prandtl number that was mentioned before in connection with flux transport dynamos!) The Cadarache experiment is particularly interesting to us. Simulations of this flow have been attempted by various groups using the Taylor-Green flow as a model. Again, the nature of the resulting dynamo effect has not yet been elucidated. It would be useful to analyze the resulting flows using the testfield method. 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. Another relevant application is precession-driven dynamos. Preparatory simulations in Cartesian simulations have been carried out in collaboration with Agris Gailitis from Riga/Latvia. (Phase 4)

    Status of the achievements.  this work has not yet started.

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?

Status of the achievements. Two 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:

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 0.5 0.2 0.0 nonlinear testfield
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 0.4 0.3 0.2 spherical testfield
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 0.4 0.3 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 0.1 0.3 0.6 laboratory dynamos
  14/4 14/4 14/4 14/4

  1. Code validation, nonlinear testfield method, catastrophic quenching in a spherical shell, dynamo effect from the MRI (items 1-4 in Sect. 2).
    Estimated completion: 43%, resources consumed: 202,000.00 EUR. Completion date from Annex I: July 2010. Comments: other topics such as near-surface magnetic flux concentrations and CME-like features have become extremely timely, so some of the resources have been diverted to those topics.

  2. Testfield method in spherical geometry, alpha effect from convection, dynamo in open shells with and without shear (items 5-7 in Sect. 2).
    Estimated completion: 37%, resources consumed: 174,000.00 EUR. Completion date from Annex I: February 2012. Comments: Studies of the alpha effect from convection have been nearly completed.

  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).
    Estimated completion: 14%, resources consumed: 66,000.00 EUR. Completion date from Annex I: July 2013. Comments: studies of near-surface magnetic flux concentrations and CME-like features have already been 40% completed.

  4. Deep convection dynamo, solar cycle forecast, applications to laboratory liquid sodium dynamos (items 12-14 in Sect. 2).
    Estimated completion: 6%, resources consumed: 28,000.00 EUR. Completion date from Annex I: February 2012. Comments: a number of preparatory steps toward solar cycle forecast have been completed.


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Axel Brandenburg 2010-09-18