The Test-field method - a powerful tool for extracting turbulent transport coefficients
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Gressel, Oliver
The interstellar medium of the Milky Way and nearby disk galaxies harbours large-scale coherent magnetic fields of Microgauss strength, that can be explained via the action of a mean-field dynamo. As in our previous work, we aim to quantify dynamo effects that are self-consistently emerging in realistic direct magnetohydrodynamic simulations, but we generalise our approach to the case of a non-local (non-instantaneous) closure relation, described by a convolution integral in space (time). To this end, we leverage our comprehensive simulation framework for the supernova-regulated turbulent multi-phase interstellar medium. By introducing spatially (temporally) modulated mean fields, we extend the previously used test-field method to the spectral realm - providing the Fourier representation of the convolution kernels. The resulting spectra of the dynamo mean-field coefficients that we obtain broadly match expectations and allow to rigorously constrain the degree of scale separation in the Galactic dynamo.
 So we immediately move to the next talk. It's the talk of Oliver Gressel. So you have a little bit time because I just want to mention, okay, also a little introduction. I see, oh yeah, we have to start from the bottom. So you wear it in tubing, make some masters in tubing, and then went to Potsdam to do the PhD. We're also some other positions in between University of London, I see. And so in the Nordic countries, Stockholm, Copenhagen. That's right. So and now you are at the Leibniz Institute for Astrophysics or your local people, so to say, and we are looking forward to see for a method to extract turbulent transport coefficients for what reason. However, I don't know, maybe also for astrophysics, maybe for others. Yeah, okay, please. Can you hear me okay? You should speak to the microphone. Others can hear, that's what I learn now. Okay, let me pay on. Alright, thanks for the introduction. My name is Oliver Gressel. I'm the head of the MHD Intervalence Section at the Leibniz Institute for Astrophysics here in Potsdam. So I returned after 10 years abroad and this is work explaining magnetic field in nearby spiral galaxies that we believe are caused by the turbulent motions in the interstellar medium, in fact, and by a dynamo process. And we describe this by means of an effective theory for unresolved scales. So this is work done with my colleague Detlef Elster. And I would like to actually make this in the memoriam of Karlheinz-Gridler, the founding head of Institute of the Leibniz, then called Astrophysics Institute Potsdam. Karlheinz-Gridler who passed away two years ago and who worked actively in this research area and he actually was one of the three people who invented this framework, invented the theory. This has proven extremely powerful since the 60s when this was first conceived. Okay, here we have a picture of a whirlpool galaxy. This is a nearby spiral galaxy. It's just below the big dipper in the sky. So you can't see it with the eye. This is a Hubble HST image. You can see spiral arms. Let's see if I can use a pointer here. It's an awkward projection here to use actually this pointer because I got a lot of reflections on the screen. Let's see if I can tweak this a little. Maybe I'll use this screen. Sorry for that. You can see galaxies, they rotate differentially. So the center, there's more mass. Everything has to rotate faster to actually not fall onto the center. So everything is constantly wound up and with it the plasma is stretched and wound up and in this process magnetic fields actually get amplified. You can look at the mathematical structure of the induction equation and you can see if you have a velocity field that has a shear in it or compression or then you can actually squeeze the magnetic field lines. You can stretch them out and with that by the motion of the plasma you actually amplified a magnetic field. So in the plasma there's a current. The current has a magnetic field. The magnetic field exerts a Lorentz force on the plasma so this is a very intricate mathematical dynamics and quite amazingly if you look at this in the synchrotron polarization you can see that the light that we receive in these millimeter radio bands that Luciano already showed for the M87 case is highly polarized and this means that there are magnetic fields, large-scale ordered magnetic fields, but the electrons, the relativistic electron spiral round and hence emit polarized emission that we collect with these radio telescopes. So this is a polarization image that's been done with a 100 meter Eiffel's Berg telescope in Bonn and if you look at this on first glance it might appear okay the magnetic field lines that just follow the spiral arms and so this just means the magnetic field is wound up by the motion of the plasma around the collected center but that's actually not correct if you if you look you will see that the pitch angle of the polarized the magnetic field direction will actually have an inclination towards the earthen-methyl direction. So there is actually something producing by an induction effect of radial magnetic field in scales of kilopassets and scales of the galaxy as a whole. So this galaxy is a plasma that is coherent enough to actually produce a large-scale magnetic field and this at first glance might appear surprising because the star formation, the supernova we're going off driving shocks through the supernova remnant shocks into the interstellar medium create a lot of turbulence. So the plasma is insanely high Reynolds numbers so there's essentially no dissipation whatsoever in terms of the microscopic dissipation. So you get turbulent cascade you get large eddies breaking down into small eddies dissipating at the same time producing induction effects in the magnetic field and if you do an order of magnitude calculation you will actually arrive at the conclusion that a turbulent plasma will not sustain any large-scale field because the field will decay away as the action of the turbulence so you will diffuse your coherent magnetic fields but just in timescales that are comparable to the rotation timescales. So you need something that actually overcomes this dissipative effect of the turbulence something that creates an order out of the chaos and it turns out what you actually need is two ingredients the first one is rotation making the plasma motions as whirl because you get Coriolis forces Coriolis accelerations that actually can sort of twist motions around additionally and the second thing is that this galaxy has a vertical structure so it's actually a stratified disc like the atmosphere in the earth so it's high density in the middle and then the density falls off towards the halo of the galaxy and that means that you actually have a preferred direction. So if you have a supernova remnant expanding it will swirl up but at the same time it will inflate a little more on the top side than it will inflate on the bottom side and this characteristic twist actually means that there's a net effect on the scale of the galaxy as a whole. So in principle you can put this galaxy on a computer and people have done this I would argue that it's very challenging to actually arrive at high enough resolution even the biggest supercomputer on this planet to really model the turbulence as a what's called a direct simulation where you resolve both the outer scale where the turbulence is driven and the dissipation scales or a proxy for the dissipation scales at least. So what we would like to do or what we've done is to actually focus on a small box that we put inside this entire galactic disc and then do highly resolved local simulations and then in some way measure infer probe the effect of the turbulence see how isotropic is a turbulence how homogeneous is a turbulence are there systematic deviations that actually lead to an induction effect as a whole and then encapsulate that into a effective theory a sub grid scale model and then run global simulations where we actually have a sub grid model for these rotational effects and so the framework for that is been invented by Steindegg, Krause-Riedler and the 60s so Karl-Heinz-Riedler was one of the minds that came up with this and you look at the induction equation so the induction equation this is a very beautiful mathematically topologically so you have a curl of a velocity field crossed with a magnetic field so you can separate this out by a vector identity one of the four terms vanishes because we know magnetic fields are divergence free first term means if you have a divergence of the velocity you can compress or decompress magnetic field displays a role by when you have interacting galaxies like you actually have with this M51 there's a nearby galaxy that's in the outskirts interacting you can have a B dot grad V that just means that you can as soon as you have a gradient of the velocity that you protect into the direction of the beefy you can stretch out the field and create amplify the field and the last term is a V dot grad so this is just an affection term so you actually the magnetic field lines are frozen into the plasma they are moving around now this mean field approach basically says we split our velocity field and our magnetic field into mean with an over bar and a fluctuation so that's just the rest when you subtract the mean from the total field and you can actually write down an induction equation for the mean magnetic field and you can see this is the curl of the electromotive force and the electromotive force is essentially this curl V cross B plus a dissipative term that I won't talk too much about it this is important as far as topologies are concerned but since we're in the limit of basically infinite magnetic Reynolds number typically this term gets ignored in the realm of the simplified approaches so the really exciting thing is that the mean field induction equation is exactly the same as induction equation but it has an additional term this epsilon with the over bar which is the turbulent electromotive force so this bodies all the correlations between the small scale velocities the fluctuating velocities in the small scale magnetic field so this is really the heart of the dynamo mechanism as soon as this is correlated with the mean magnetic field you will have an exponential amplification so on the left hand side you have the time change of the field and you can see this is related to the curl of this mean electromotive force so if this mean electromotive force is proportional to a magnetic field this gets an exponential amplification and that's what you actually need to sustain a large scale order in the scales of this galaxy and the name of the game is to encapsulate the properties of the turbulence in these tensors alpha and eta so eta is related to the dissipation effects the turbulent diffusion eddy diffusion the alpha effect is related to actually the broken symmetries in the system so as soon as you have something that's helical so as a result of a rotation and in how much you need to a stratification you actually get helical motions that directly connect the mean electric field to the mean magnetic field and hence enable the dynamo mechanism so a little bit of the derivation how do you get this mean field induction equation you actually just write down your induction equation it's top line and then you average it you see all the bees and the bees are split you get some mixed terms there in the curl the cross beat part and you need some properties for your averaging procedure so ideally these would be ensemble averages in practice practical terms these are can be time averages or spatial averages and in these boxes we typically have a horizontal average and there's only a vertical dependence remaining what you need is that the mixed products actually vanish and that's can be seen because you can pull out then one of the averages and then you average with the average of a fluctuation and that's by default zero so when you do this on the entire equation the mixed terms fall out and and you get the curl V over bar cross B over bar that's just the induction of the mean field with the mean flow plus you get the curl of the correlations in the fluctuating velocities in the flag like fluctuating magnetic field so as soon as your magnetic fluctuations actually know about the velocity fluctuations or vice versa then this effect is actually there and it's important can you estimate actually where this effect might come from so to do this you want to express your electric field the turbulent electromotive force as a time integral so you try to predict how your magnetic fluctuations evolve in time this is sort of very hand-wavyly written by this partial time derivative under the integral and once you know this time evolution of your fluctuations you can integrate in time again and you can predict your electromotive force as a function of time how do you find an equation to predict the evolution of the fluctuations this is actually also very straightforward you take the total equation this is a top line here but now note that there's no over bar so this is not the average equation this is just a very bulky way of writing the actual induction equation split into means and fluctuations and below that equation and I'm subtracting it so if I'm subtracting the mean equation from the total equation I get the equation for the fluctuations and you can see there's a lot of terms now because the cancellation effect doesn't show up now because we haven't averaged the equation so they are actually these five terms in the first four ones they can tell you over bar v cross B prime this is advection stretching and compression of magnetic fluctuations by large-scale flow the next one is related to what's called the small scale dynamo or the fluctuation dynamo so there's fluctuations in the magnetic field being amplified by fluctuations in the velocity this is interesting in its own right because you can see the B prime on the left hand side the time derivative is also proportional to the B prime there so this can actually also exponentially amplify the field but this will not happen on the scale of the galaxy as a whole but only on the scales much smaller than the the eddy scales so this is something that's often invoked by cosmologists today who like to do global simulations of discs forming but there are arguments that this will never actually produce any any large-scale coherent fields with significant radial fields you might just wind up the field and end up with with some residual field but that would be of insignificant pitch angle the next two terms it's the turbulent electromotive force again you can see that this actually drains energy out of the large on the small-scale fields and puts it into the large-scale fields again but the really central term is the next one the v prime cross B over bar so this tells you that as soon as there is a turbulent motion of fluctuating motion v prime and a pre-existing large-scale field then this will actually produce correlations in the B prime on the left hand side that are by definition correlated with the v prime so you can see by tangling off small-scale magnetic fluctuations from a pre-existing large-scale field you produce magnetic fluctuations that are correlated with a velocity and that lead to this mean electromotive force as a consequence so this is what motivated Karl-Heinz-Reyedler 50 years ago to parametrize the electromotive force really as a linear function in the mean magnetic field and when you postulate that the mean magnetic field is large scale slowly varying you can actually pull it out of this integral on the top and that means that you actually then you have a v prime and the B prime you see there's another v prime you can see mathematically that this actually makes the kinetic helicity the twirliness of the the helicalness of the velocity flow show up in the mean field equation this is a famous alpha effect that Potsdam is is so famous for. Alright so how do we actually measure this in the simulations and there's been a method coming about I quickly skip this over when I did my PhD and that was sort of a Eureka moment that I had that this is really the way to go and so what you actually do is you solve induction equations if you just take the magnetic field that you have in the simulations can be highly degenerate the gradients can be very poorly defined so it's very difficult to actually extract the alpha and the eta tensors simultaneously from the simulations and so an ingenious idea was to actually prescribe a set of magnetic fields that I evolved passively with the simulation but that are not talking back to the simulation they're just fields that have various you can see here these are just more harmonic functions in space and time with components pointing in orthogonal ways this is true in a way to construct a basis where you have a linear system of equations you see in the bottom where you when you compute the electric fields the electromotive forces you can directly convert these into your turbulence coefficients so the properties of the turbulence are probed by imposing magnetic fields that are varying in space pointing in directions that are non degenerate basis to actually do the linear equation so this comes at the price of having to solve for additional induction equations in the code and these induction equations they take the velocity field from the simulation but they actually they produce magnetic fluctuations that are the result of this imposed test field not the actual field the turbulence itself will be affected by the presence of the magnetic field so if the fields get strong you get what's called quenching so the turbulence effects will sort of die away if the field gets strong enough to actually suppress the turbulence or suppress the anisotropy in the turbulence so this can actually be measured we did this so in a way this is very powerful we have these additional induction equations and we we evolve them taking the velocity field of the simulations as an input and then producing the correlations again of this v prime the v prime of these test fields and you can see here I've written this down as an example for harmonic mode so we typically a fields that were right along the vertical direction if the box is sort of sitting in the disc and there's a vertical coordinate above and below the disc and so far we've assumed that this is a local relation so that the electric field as a result of imposing a homogeneous field is an instantaneous and local point-by-point effect it turns out turbulence is not a local and instantaneous phenomenon turbulent eddies have a certain characteristic length scale they have a characteristic time scale so it means that the the information that there is a magnetic field in the background that gets twirled up and producing fluctuation this is a finite time effect and the time scale also depends on the length scale because of the riches and cascade and so on so the actual mathematical way to formulate this is that the relation between the mean electric field the mean electromagnetic force on the left hand side as a result of a pre existing mean magnetic field is not an instantaneous and local function but it can be described as a as a non instantaneous effect so this is splits the time and the space dependence here you can actually do both at the same time for clarity I'm showing that here you think of in the center think of the mean electromagnetic force as an integral with a convolution kernel that is your your turbulent coefficient your mean field closure and then the magnetic field in sort of a history backward in time so this would be sometimes called a memory effect so the turbulence is aware of what happened at this location in the past has there been a magnetic field in the past at this location or rather the blob that's been affected from that location so there's a certain domain of dependence and the time of dependence and the same you can do in the spatial dependence so the electric field at a certain position is not an instantaneous consequence of imposing a mean field at that location or a mean field being created by the turbulence at that location but it's actually a history of all the turbulent eddies in the vicinity that sort of collect and bring together the magnetic fluctuations from the turbulence in the surroundings and so this can be described as a convolution in space with the turbulence coefficient as a convolution kernel and the magnetic field being probed in an environment and you can see that in Fourier space these have these nice Lorentzian shapes and we assume this as a model and this is actually really nicely met by the data that we have looked at alright and so the tool that we use is this test-filled method where we have the imposed test fields and so this is a simple test case we actually only solving the hydrodynamic equations you can see them mass conservation and momentum conservation with some artificial forcing function this is a helical forcing so we basically put in by hand the effect you want to measure just to see is a method sensitive and you can see here as an example the amplitude of the alpha effect as a function of wave number and you can see it has this Lorentzian shape so this means that it has a finite domain of dependence if you translate this actually to real space is Lorentzian with quadratic in the set in the k means you get an exponential decay so a one over one over distance decay so if you if you go away from a point the influence decays exponentially with distance on producing the electromotive force as a consequence of turbulence same is true actually for the turbulent diffusion that's shown in lower panel and the the coefficients actually quite comparable they're not exactly the same but they're very comparable and so the same thing you can do for the dependence in time so if you have a magnetic field a mean magnetic field that pulsates for example that's what we did with the amonic approach or exponentially grows or exponentially decays this will have depending on the time scale of this temporal change a different sensitivity on the turbulence effects and this is plotted in the upper panel you can see the real and imaginary part so this means you can actually have a face leg of the electromotive force as a result of the mean field so you have a mean field of a dynamo mode you know the large-scale field in the Sun for example or this large-scale field in the galaxies on the planet interior so this is very agnostic of where you apply it to and maybe you have an oscillatory dynamo mode the solar cycle 22 years obviously is a little longer than the turbulence turnover time but in some extreme cases you might actually get these two closer together and you can see that at intermediate time lags there is actually a significant imaginary part so that means that you actually get a get a face leg so there might be a mean field coming up and then because of the turbulent at is sometime later there will be a phase shift that responds in the electric field and the electric field itself will create the magnetic field right so the changes the magnetic field is a curl of the electric field by Maxwell's equations so there's a dynamical coupling and you see if you have time delay effects it actually changes the dynamics significantly so this can actually lead to a change of dynamo cycle period also on so the curves for the turbulent diffusion aspects of the turbulence is similar you can see a slight difference that there's a bit of an overshoot for the for the alpha then the media time so this is more in a way less simplistic in a way so the turbulent diffusion appears to be somewhat monotonic while there's really seems to be an intrinsic time scale for the eddies to actually twirl up the fields and make them helical and so on okay so this has been for a test case but we are actually then also renders on two really real world cases so the one was simulations of the turbulent interstellar medium where we really put this is actually a lot of fun I did this for my pg already we put 10 to 51 erg into a single grid cell and it produces a supernova remnant that the travels through the interstellar medium and produces a shock and then we have several thousands of them that interact and everything gets clumpy because you have a thermal instability in the cooling and their clumps raining down you get the galactic fountain you actually have hot material being blown out of the galaxy leading to galactic winds you have clumps raining down and it turns out that we were actually successful in measuring that this galactic fountain played a decisive role in enabling the galactic dynamo that was actually not appreciated before so the turbulence in the galactic disk is so vigorous that it actually drives a galactic wind so constantly plasma is flowing away from the galactic disk if you amplify your magnetic field in the disk and you blow it away then you actually extinguish this effect you dilute it so the induction equation works the way if you have a positive divergence you dilute your magnetic energy density we've seen that the clumps from the thermal instability raining down they actually drag the field lines down again so there's a pumping effect so this is a topological effect this is the off diagonal elements of the dynamo tensor and this pumping actually counteracts the effect of the wind so you blow away everything but you pump topologically your mean magnetic field down and this is actually what enables the past timescale of the mean field dynamo in the galaxy and makes it work another thing that we looked at was magneto rotational turbulence so there's a magnetic instability in accretion discs this is relevant for the picol accretion discs that Luciano showed and this creates magnetized turbulence and we were wondering does this turbulence actually also have mean field effects such as this producer dynamo and these simulations actually all they all of them show that there's a butterfly diagram just like the one in the Sun and so we could actually quantitatively show that the oscillation periods seen in the simulations while it might not reflect the real conditions in the disks this can actually be explained by this alpha omega type of dynamo and recently we've even also measured the the non-incentanous and the non-local effects in this and so in a way we're building a quantitative theory of accretion disc turbulence and we hope that we can then actually incorporate this also in global simulations and the mean fields dynamo might actually be relevant for producing these large-scale coherent fields that we've then seen drive the jets and outflows in the systems and so this is a theory developed 50 years ago and and we've really systematically confronted with the simulations and mathematical modeling to actually test these ideas that have been developed before computers were actually available for this type of thing and we find that they actually really beautifully agree with this theory so with that I'm happy to take questions thank you yeah for this talk and other questions from the audience or from the what will the people in the zoom the magnetic field should be divergence free how do you manage this on the numerical level I'm very puristic in that regard so there are two approaches the ones where you actually by means of your discredit station have it correct by machine precision so this is called constrained transport so you have a discretized version of stokes theorem on your discredit station element your grid cell so you have a surface integral of the magnetic flux so your magnetic flux is a surface quantity and the electric field is an edge quantity and so you're solving a stokes integral along the edges and that's exactly then if you have been able to tell to two edges always cancel and so by by definition that the curl of the entire volume is then preserved and this actually is from the 90s by evanson holly and if you start with a magnetic field on your grid that's discretized with zero divergence it actually maintains its divergence to the other approaches you you don't care so much about it but you use some cleaning methods so there's some hyperbolic cleaning parabolic cleaning so you allow for a divergence B and you sort of try to dissipate it away and there's quite some arguing in the community obviously the one with the cleaning is more affordable it's easier to do the bigger simulations but especially in the realm of dynamo theory it's been found that it's actually mandatory to use constrained transport because otherwise you might end up with very spurious effects that having said they're actually the these topological things they're related to magnetic illicit and that is actually not very well conserved at the grid scale by any existing code so it's not clear actually how well we we can model these aspects of the magnetic field that the topology cannot be changed if you have two magnetic lines that are interlinked like this you cannot actually just separate them and these things are very difficult to model on very difficult to assess in a numerical way how well you actually conserve these topological quantities the magnetic illicit as a prime one so there's some interesting avenues in the future to actually develop discretization that might have a built-in property of maintaining this topological aspect okay thank you any other question yes the first one is if in your modeling of these passive tracers film I saw just two dimensions away are you really three dimensional or those are three dimensional fields typically we have boxes where the fields point either in the X or Y so that they're always quadruplets as one pointing X one pointing Y and then one of each is a sinusoidal and a cosine so that just to make always safe okay make them non degenerate in terms of the linear equation relating in F with the D fields by terms of these tensors okay and the second question is about do you know anyone who has extended this to a relativistic regime I'm not aware so the I think the most pressing extension of this framework to make it usable for a creation this is really actually there are two contributions in EMF it's a V cross B right a small-scale V or fluctuating V cross a fluctuating B and correlated but at the moment we only model the B prime and we take the V prime as given from the simulation it's called the kinematic or the quasi kinematic but if you have a magnetic instability like buoyancy or the MRI you create magnetic fluctuations that are pre-existing in a way in the turbulence and that way you would actually like to have a mean field approach also in the momentum equation where you actually have as more symmetric approach and there's a framework that's been developed ten years ago by Matthias Reinhardt and Axel Brandenburg they call it the non kinematic test-field method and there you you have you do a trick I mean the momentum equation is non-linear intrinsically right and you can write it down in a linearized fashion so it's in the way it's a trick but you can define a test-field approach to the ponder ponderow motive force as well unfortunately the approach is riddled with mathematical instability so there's some eigenvalues so we're interested in the inhomogeneous part of the equations but there's always exponential growth in the homogeneous part of the equation as well and how do you beat that down how do you still make it sensitive to what we actually want to measure from turbulence and that's been hard to overcome so they just take a mistake and beat down the fluctuations every any turn over time and they claim that they've arrived at a sound solution but it's not obvious but that would be what is really needed to make it a non kinematic and really make it applicable to black hole abrasion discs fundamentally consistent level okay thank you okay so I think you once again on let's say both of the speakers of the first session
 So we immediately move to the next talk. It's the talk of Oliver Gressel. So you  have a little bit time because I just want to mention, okay, also a little  introduction. I see, oh yeah, we have to start from the bottom. So you wear it in  tubing, make some masters in tubing, and then went to Potsdam to do the  PhD. We're also some other positions in between University of London, I see. And  so in the Nordic countries, Stockholm, Copenhagen. That's right. So and now you  are at the Leibniz Institute for Astrophysics or your local people, so to  say, and we are looking forward to see for a method to extract turbulent  transport coefficients for what reason. However, I don't know, maybe also for  astrophysics, maybe for others. Yeah, okay, please.  Can you hear me okay? You should speak to the microphone. Others can hear, that's what I learn now.  Okay, let me pay on. Alright, thanks for the introduction. My name is Oliver Gressel. I'm the head of the  MHD Intervalence Section at the Leibniz Institute for Astrophysics here in  Potsdam. So I returned after 10 years abroad and this is work explaining  magnetic field in nearby spiral galaxies that we believe are caused by the  turbulent motions in the interstellar medium, in fact, and by a dynamo process.  And we describe this by means of an effective theory for unresolved scales.  So this is work done with my colleague Detlef Elster. And I would like to  actually make this in the memoriam of Karlheinz-Gridler, the founding head of  Institute of the Leibniz, then called Astrophysics Institute Potsdam.  Karlheinz-Gridler who passed away two years ago and who worked actively in this  research area and he actually was one of the three people who invented this  framework, invented the theory. This has proven extremely powerful since the 60s  when this was first conceived. Okay, here we have a picture of a whirlpool galaxy.  This is a nearby spiral galaxy. It's just below the big dipper in the sky. So you  can't see it with the eye. This is a Hubble HST image. You can see spiral  arms. Let's see if I can use a pointer here. It's an awkward projection here to  use actually this pointer because I got a lot of reflections on the screen.  Let's see if I can tweak this a little. Maybe I'll use this screen. Sorry for that.  You can see galaxies, they rotate differentially. So the center, there's  more mass. Everything has to rotate faster to actually not fall onto the  center. So everything is constantly wound up and with it the plasma is stretched  and wound up and in this process magnetic fields actually get amplified. You can  look at the mathematical structure of the induction equation and you can see if  you have a velocity field that has a shear in it or compression or then you  can actually squeeze the magnetic field lines. You can stretch them out and with  that by the motion of the plasma you actually amplified a magnetic field. So  in the plasma there's a current. The current has a magnetic field. The magnetic  field exerts a Lorentz force on the plasma so this is a very intricate  mathematical dynamics and quite amazingly if you look at this in the  synchrotron polarization you can see that the light that we receive in these  millimeter radio bands that Luciano already showed for the M87 case is  highly polarized and this means that there are magnetic fields, large-scale  ordered magnetic fields, but the electrons, the relativistic electron  spiral round and hence emit polarized emission that we collect with these  radio telescopes. So this is a polarization image that's been done with a  100 meter Eiffel's Berg telescope in Bonn and if you look at this on first  glance it might appear okay the magnetic field lines that just follow the spiral  arms and so this just means the magnetic field is wound up by the motion of the  plasma around the collected center but that's actually not correct if you if you  look you will see that the pitch angle of the polarized the magnetic field  direction will actually have an inclination towards the earthen-methyl  direction. So there is actually something producing by an induction effect of  radial magnetic field in scales of kilopassets and scales of the galaxy as a  whole. So this galaxy is a plasma that is coherent enough to actually  produce a large-scale magnetic field and this at first glance might appear  surprising because the star formation, the supernova we're going off driving  shocks through the supernova remnant shocks into the interstellar medium  create a lot of turbulence. So the plasma is insanely high Reynolds numbers so  there's essentially no dissipation whatsoever in terms of the microscopic  dissipation. So you get turbulent cascade you get large eddies breaking down into  small eddies dissipating at the same time producing induction effects in the  magnetic field and if you do an order of magnitude calculation you will actually  arrive at the conclusion that a turbulent plasma will not sustain any large-scale  field because the field will decay away as the action of the turbulence so you  will diffuse your coherent magnetic fields but just in timescales that are  comparable to the rotation timescales. So you need something that actually  overcomes this dissipative effect of the turbulence something that creates an  order out of the chaos and it turns out what you actually need is two ingredients  the first one is rotation making the plasma motions as whirl because you get  Coriolis forces Coriolis accelerations that actually can sort of twist motions  around additionally and the second thing is that this galaxy has a vertical  structure so it's actually a stratified disc like the atmosphere in the earth so  it's high density in the middle and then the density falls off towards the halo  of the galaxy and that means that you actually have a preferred direction. So  if you have a supernova remnant expanding it will swirl up but at the  same time it will inflate a little more on the top side than it will inflate on  the bottom side and this characteristic twist actually means that there's a net  effect on the scale of the galaxy as a whole. So in principle you can put this  galaxy on a computer and people have done this I would argue that it's very  challenging to actually arrive at high enough resolution even the biggest  supercomputer on this planet to really model the turbulence as a what's called  a direct simulation where you resolve both the outer scale where the turbulence  is driven and the dissipation scales or a proxy for the dissipation scales at  least. So what we would like to do or what we've done is to actually focus on a  small box that we put inside this entire galactic disc and then do highly  resolved local simulations and then in some way measure infer probe the effect  of the turbulence see how isotropic is a turbulence how homogeneous is a  turbulence are there systematic deviations that actually lead to an  induction effect as a whole and then encapsulate that into a effective  theory a sub grid scale model and then run global simulations where we actually  have a sub grid model for these rotational effects and so the framework  for that is been invented by Steindegg, Krause-Riedler and the 60s so Karl-Heinz-Riedler  was one of the minds that came up with this and you look at the induction  equation so the induction equation this is a very beautiful mathematically  topologically so you have a curl of a velocity field crossed with a magnetic  field so you can separate this out by a vector identity one of the four terms  vanishes because we know magnetic fields are divergence free first term means if  you have a divergence of the velocity you can compress or decompress magnetic  field displays a role by when you have interacting galaxies like you actually  have with this M51 there's a nearby galaxy that's in the outskirts  interacting you can have a B dot grad V that just means that you can as soon as  you have a gradient of the velocity that you protect into the direction of the  beefy you can stretch out the field and create amplify the field and the last  term is a V dot grad so this is just an affection term so you actually the  magnetic field lines are frozen into the plasma they are moving around now this  mean field approach basically says we split our velocity field and our  magnetic field into mean with an over bar and a fluctuation so that's just the  rest when you subtract the mean from the total field and you can actually write  down an induction equation for the mean magnetic field and you can see this is  the curl of the electromotive force and the electromotive force is essentially  this curl V cross B plus a dissipative term that I won't talk too much about  it this is important as far as topologies are concerned but since we're in  the limit of basically infinite magnetic Reynolds number typically this term gets  ignored in the realm of the simplified approaches so the really exciting thing  is that the mean field induction equation is exactly the same as induction  equation but it has an additional term this epsilon with the over bar which is  the turbulent electromotive force so this bodies all the correlations between  the small scale velocities the fluctuating velocities in the small scale magnetic field  so this is really the heart of the dynamo mechanism as soon as this is  correlated with the mean magnetic field you will have an exponential  amplification so on the left hand side you have the time change of the field and  you can see this is related to the curl of this mean electromotive force so if  this mean electromotive force is proportional to a magnetic field this  gets an exponential amplification and that's what you actually need to  sustain a large scale order in the scales of this galaxy and the name of the  game is to encapsulate the properties of the turbulence in these tensors alpha  and eta so eta is related to the dissipation effects the turbulent  diffusion eddy diffusion the alpha effect is related to actually the broken  symmetries in the system so as soon as you have something that's helical so as  a result of a rotation and in how much you need to a stratification you  actually get helical motions that directly connect the mean electric field  to the mean magnetic field and hence enable the dynamo mechanism so a little  bit of the derivation how do you get this mean field induction equation you  actually just write down your induction equation it's top line and then you  average it you see all the bees and the bees are split you get some mixed terms  there in the curl the cross beat part and you need some properties for your  averaging procedure so ideally these would be ensemble averages in  practice practical terms these are can be time averages or spatial averages and  in these boxes we typically have a horizontal average and there's only a  vertical dependence remaining what you need is that the mixed products actually  vanish and that's can be seen because you can pull out then one of the  averages and then you average with the average of a fluctuation and that's by  default zero so when you do this on the entire equation the mixed terms fall out  and and you get the curl V over bar cross B over bar that's just the induction of the  mean field with the mean flow plus you get the curl of the correlations in the  fluctuating velocities in the flag like fluctuating magnetic field so as soon as  your magnetic fluctuations actually know about the velocity fluctuations or  vice versa then this effect is actually there and it's important can you estimate  actually where this effect might come from so to do this you want to express  your electric field the turbulent electromotive force as a time integral so  you try to predict how your magnetic fluctuations evolve in time this is sort  of very hand-wavyly written by this partial time derivative under the  integral and once you know this time evolution of your fluctuations you can  integrate in time again and you can predict your electromotive force as a  function of time how do you find an equation to predict the evolution of  the fluctuations this is actually also very straightforward you take the total  equation this is a top line here but now note that there's no over bar so this  is not the average equation this is just a very bulky way of writing the  actual induction equation split into means and fluctuations and below that  equation and I'm subtracting it so if I'm subtracting the mean equation from  the total equation I get the equation for the fluctuations and you can see  there's a lot of terms now because the cancellation effect doesn't show up now  because we haven't averaged the equation so they are actually these five terms in  the first four ones they can tell you over bar v cross B prime this is advection  stretching and compression of magnetic fluctuations by large-scale flow the  next one is related to what's called the small scale dynamo or the fluctuation  dynamo so there's fluctuations in the magnetic field being amplified by  fluctuations in the velocity this is interesting in its own right because you  can see the B prime on the left hand side the time derivative is also  proportional to the B prime there so this can actually also exponentially  amplify the field but this will not happen on the scale of the galaxy as a  whole but only on the scales much smaller than the the eddy scales so this is  something that's often invoked by cosmologists today who like to do global  simulations of discs forming but there are arguments that this will never  actually produce any any large-scale coherent fields with significant radial  fields you might just wind up the field and end up with with some residual field  but that would be of insignificant pitch angle the next two terms it's the  turbulent electromotive force again you can see that this actually drains  energy out of the large on the small-scale fields and puts it into the  large-scale fields again but the really central term is the next one the v  prime cross B over bar so this tells you that as soon as there is a turbulent  motion of fluctuating motion v prime and a pre-existing large-scale field then  this will actually produce correlations in the B prime on the left hand side  that are by definition correlated with the v prime so you can see by tangling  off small-scale magnetic fluctuations from a pre-existing large-scale field you  produce magnetic fluctuations that are correlated with a velocity and that  lead to this mean electromotive force as a consequence so this is what  motivated Karl-Heinz-Reyedler 50 years ago to parametrize the electromotive  force really as a linear function in the mean magnetic field and when you  postulate that the mean magnetic field is large scale slowly varying you can  actually pull it out of this integral on the top and that means that you actually  then you have a v prime and the B prime you see there's another v prime you can  see mathematically that this actually makes the kinetic helicity the twirliness  of the the helicalness of the velocity flow show up in the mean field equation  this is a famous alpha effect that Potsdam is is so famous for. Alright so  how do we actually measure this in the simulations and there's been a method  coming about I quickly skip this over when I did my PhD and that was sort of a  Eureka moment that I had that this is really the way to go and so what you  actually do is you solve induction equations if you just take the magnetic  field that you have in the simulations can be highly degenerate the gradients  can be very poorly defined so it's very difficult to actually extract the alpha  and the eta tensors simultaneously from the simulations and so an ingenious idea  was to actually prescribe a set of magnetic fields that I evolved passively  with the simulation but that are not talking back to the simulation they're  just fields that have various you can see here these are just more harmonic  functions in space and time with components pointing in orthogonal ways  this is true in a way to construct a basis where you have a linear system of  equations you see in the bottom where you when you compute the electric fields  the electromotive forces you can directly convert these into your  turbulence coefficients so the properties of the turbulence are probed by  imposing magnetic fields that are varying in space pointing in directions  that are non degenerate basis to actually do the linear equation so this  comes at the price of having to solve for additional induction equations in the  code and these induction equations they take the velocity field from the  simulation but they actually they produce magnetic fluctuations that are the  result of this imposed test field not the actual field the turbulence itself  will be affected by the presence of the magnetic field so if the fields get  strong you get what's called quenching so the turbulence effects will sort of  die away if the field gets strong enough to actually suppress the turbulence or  suppress the anisotropy in the turbulence so this can actually be measured  we did this so in a way this is very powerful we have these additional  induction equations and we we evolve them taking the velocity field of the  simulations as an input and then producing the correlations again of this  v prime the v prime of these test fields and you can see here I've written this  down as an example for harmonic mode so we typically a fields that were right  along the vertical direction if the box is sort of sitting in the disc and  there's a vertical coordinate above and below the disc and so far we've assumed  that this is a local relation so that the electric field as a result of  imposing a homogeneous field is an instantaneous and local point-by-point  effect it turns out turbulence is not a local and instantaneous phenomenon  turbulent eddies have a certain characteristic length scale they have a  characteristic time scale so it means that the the information that there is a  magnetic field in the background that gets twirled up and producing fluctuation  this is a finite time effect and the time scale also depends on the length  scale because of the riches and cascade and so on so the actual mathematical  way to formulate this is that the relation between the mean electric field  the mean electromagnetic force on the left hand side as a result of a pre  existing mean magnetic field is not an instantaneous and local function but it  can be described as a as a non instantaneous effect so this is splits  the time and the space dependence here you can actually do both at the same  time for clarity I'm showing that here you think of in the center think of the  mean electromagnetic force as an integral with a convolution kernel that is  your your turbulent coefficient your mean field closure and then the magnetic  field in sort of a history backward in time so this would be sometimes called a  memory effect so the turbulence is aware of what happened at this location in  the past has there been a magnetic field in the past at this location or rather  the blob that's been affected from that location so there's a certain domain of  dependence and the time of dependence and the same you can do in the spatial  dependence so the electric field at a certain position is not an instantaneous  consequence of imposing a mean field at that location or a mean field being  created by the turbulence at that location but it's actually a history of  all the turbulent eddies in the vicinity that sort of collect and bring  together the magnetic fluctuations from the turbulence in the surroundings and  so this can be described as a convolution in space with the turbulence  coefficient as a convolution kernel and the magnetic field being probed in an  environment and you can see that in Fourier space these have these nice  Lorentzian shapes and we assume this as a model and this is actually really  nicely met by the data that we have looked at  alright and so the tool that we use is this test-filled method where we have  the imposed test fields and so this is a simple test case we actually only  solving the hydrodynamic equations you can see them mass conservation and  momentum conservation with some artificial forcing function this is a  helical forcing so we basically put in by hand the effect you want to measure  just to see is a method sensitive and you can see here as an example the  amplitude of the alpha effect as a function of wave number and you can see  it has this Lorentzian shape so this means that it has a finite domain of  dependence if you translate this actually to real space is Lorentzian with  quadratic in the set in the k means you get an exponential decay so a one over  one over distance decay so if you if you go away from a point the influence  decays exponentially with distance on producing the electromotive force as a  consequence of turbulence same is true actually for the turbulent diffusion  that's shown in lower panel and the the coefficients actually quite comparable  they're not exactly the same but they're very comparable and so the same thing  you can do for the dependence in time so if you have a magnetic field a mean  magnetic field that pulsates for example that's what we did with the amonic  approach or exponentially grows or exponentially decays this will have  depending on the time scale of this temporal change a different sensitivity  on the turbulence effects and this is plotted in the upper panel you can see  the real and imaginary part so this means you can actually have a face leg of  the electromotive force as a result of the mean field so you have a mean field  of a dynamo mode you know the large-scale field in the Sun for example or  this large-scale field in the galaxies on the planet interior so this is very  agnostic of where you apply it to and maybe you have an oscillatory dynamo mode  the solar cycle 22 years obviously is a little longer than the turbulence  turnover time but in some extreme cases you might actually get these two closer  together and you can see that at intermediate time lags there is actually  a significant imaginary part so that means that you actually get a get a face  leg so there might be a mean field coming up and then because of the  turbulent at is sometime later there will be a phase shift that responds in the  electric field and the electric field itself will create the magnetic field  right so the changes the magnetic field is a curl of the electric field by Maxwell's  equations so there's a dynamical coupling and you see if you have time delay  effects it actually changes the dynamics significantly so this can actually  lead to a change of dynamo cycle period also on so the curves for the turbulent  diffusion aspects of the turbulence is similar you can see a slight difference  that there's a bit of an overshoot for the for the alpha then the media time so  this is more in a way less simplistic in a way so the turbulent diffusion appears  to be somewhat monotonic while there's really seems to be an intrinsic time  scale for the eddies to actually twirl up the fields and make them helical and  so on okay so this has been for a test case but we are actually then also  renders on two really real world cases so the one was simulations of the  turbulent interstellar medium where we really put this is actually a lot of  fun I did this for my pg already we put 10 to 51 erg into a single grid cell and  it produces a supernova remnant that the travels through the interstellar  medium and produces a shock and then we have several thousands of them that  interact and everything gets clumpy because you have a thermal instability  in the cooling and their clumps raining down you get the galactic fountain you  actually have hot material being blown out of the galaxy leading to galactic  winds you have clumps raining down and it turns out that we were actually  successful in measuring that this galactic fountain played a decisive  role in enabling the galactic dynamo that was actually not appreciated before  so the turbulence in the galactic disk is so vigorous that it actually drives a  galactic wind so constantly plasma is flowing away from the galactic disk if  you amplify your magnetic field in the disk and you blow it away then you  actually extinguish this effect you dilute it so the induction equation  works the way if you have a positive divergence you dilute your magnetic  energy density we've seen that the clumps from the thermal instability  raining down they actually drag the field lines down again so there's a pumping  effect so this is a topological effect this is the off diagonal elements of  the dynamo tensor and this pumping actually counteracts the effect of the  wind so you blow away everything but you pump topologically your mean magnetic  field down and this is actually what enables the past timescale of the mean  field dynamo in the galaxy and makes it work another thing that we looked at  was magneto rotational turbulence so there's a magnetic instability in  accretion discs this is relevant for the picol accretion discs that Luciano  showed and this creates magnetized turbulence and we were wondering does  this turbulence actually also have mean field effects such as this producer  dynamo and these simulations actually all they all of them show that there's a  butterfly diagram just like the one in the Sun and so we could actually  quantitatively show that the oscillation periods seen in the simulations  while it might not reflect the real conditions in the disks this can actually  be explained by this alpha omega type of dynamo and recently we've even also  measured the the non-incentanous and the non-local effects in this and so in a  way we're building a quantitative theory of accretion disc turbulence and we  hope that we can then actually incorporate this also in global  simulations and the mean fields dynamo might actually be relevant for producing  these large-scale coherent fields that we've then seen drive the jets and  outflows in the systems and so this is a theory developed 50 years ago and and  we've really systematically confronted with the simulations and mathematical  modeling to actually test these ideas that have been developed before  computers were actually available for this type of thing and we find that they  actually really beautifully agree with this theory so with that I'm happy to  take questions thank you yeah for this talk and other questions from the  audience or from the what will the people in the zoom the magnetic field  should be divergence free how do you manage this on the numerical level I'm  very puristic in that regard so there are two approaches the ones where you  actually by means of your discredit station have it correct by machine  precision so this is called constrained transport so you have a  discretized version of stokes theorem on your discredit station element your  grid cell so you have a surface integral of the magnetic flux so your  magnetic flux is a surface quantity and the electric field is an edge quantity  and so you're solving a stokes integral along the edges and that's exactly then  if you have been able to tell to two edges always cancel and so by by definition  that the curl of the entire volume is then preserved and this actually is  from the 90s by evanson holly and if you start with a magnetic field on your  grid that's discretized with zero divergence it actually maintains its  divergence to the other approaches you you don't care so much about it but you  use some cleaning methods so there's some hyperbolic cleaning parabolic  cleaning so you allow for a divergence B and you sort of try to dissipate it away  and there's quite some arguing in the community obviously the one with the  cleaning is more affordable it's easier to do the bigger simulations but  especially in the realm of dynamo theory it's been found that it's actually  mandatory to use constrained transport because otherwise you might end up with  very spurious effects that having said they're actually the these topological  things they're related to magnetic illicit and that is actually not very  well conserved at the grid scale by any existing code so it's not clear  actually how well we we can model these aspects of the magnetic field that the  topology cannot be changed if you have two magnetic lines that are interlinked  like this you cannot actually just separate them and these things are very  difficult to model on very difficult to assess in a numerical way how well you  actually conserve these topological quantities the magnetic illicit as a  prime one so there's some interesting avenues in the future to actually  develop discretization that might have a built-in property of maintaining this  topological aspect okay thank you any other question yes  the first one is if in your modeling of these passive tracers film I saw just  two dimensions away are you really three dimensional or those are three  dimensional fields typically we have boxes where the fields point either in  the X or Y so that they're always quadruplets as one pointing X one pointing  Y and then one of each is a sinusoidal and a cosine so that just to make always  safe okay make them non degenerate in terms of the linear equation relating  in F with the D fields by terms of these tensors okay and the second question is  about do you know anyone who has extended this to a relativistic regime I'm not  aware so the I think the most pressing extension of this framework to make it  usable for a creation this is really actually there are two contributions in  EMF it's a V cross B right a small-scale V or fluctuating V cross a fluctuating B  and correlated but at the moment we only model the B prime and we take the V  prime as given from the simulation it's called the kinematic or the quasi  kinematic but if you have a magnetic instability like buoyancy or the MRI you  create magnetic fluctuations that are pre-existing in a way in the turbulence  and that way you would actually like to have a mean field approach also in the  momentum equation where you actually have as more symmetric approach and  there's a framework that's been developed ten years ago by Matthias Reinhardt  and Axel Brandenburg they call it the non kinematic test-field method and  there you you have you do a trick I mean the momentum equation is non-linear  intrinsically right and you can write it down in a linearized fashion so it's  in the way it's a trick but you can define a test-field approach to the  ponder ponderow motive force as well unfortunately the approach is riddled  with mathematical instability so there's some eigenvalues so we're interested  in the inhomogeneous part of the equations but there's always exponential  growth in the homogeneous part of the equation as well and how do you beat  that down how do you still make it sensitive to what we actually want to  measure from turbulence and that's been hard to overcome so they just take a  mistake and beat down the fluctuations every any turn over time and they claim  that they've arrived at a sound solution but it's not obvious but that would be  what is really needed to make it a non kinematic and really make it applicable  to black hole abrasion discs fundamentally consistent level okay thank  you okay so I think you once again on let's say both of the speakers of the  first session
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