Telling you today about experiments we're carrying out at the University of Wisconsin, using plasmas to look at the Dynamo problem.
And this is intended to be a little bit of a provocative title.
I'll try and explain to you what fast dynamos are, where we haven't even gotten to slow dynamos yet.
So many dynamos. So this is this is an experimentalist in his lab and these are dynamos dynamos are objects that sorry about that
dynamos are objects that create magnetic energy from kinetic energy and you might just know them as generators.
These are electrical generators. And this phrase, if you haven't seen it before, is the same spelled backwards.
It's a palindrome. So, so many dynamos this. If you haven't seen that, you should see it.
I'll be talking about astrophysical dynamos and and so these are a bunch of those.
Everywhere we look in the universe, we see that it's magnetised mostly we see plasma and and and it's usually turbulent and magnetic fields
exist on a huge range of scales from things like the planetary magnetic fields of our own planets.
Stars have magnetic fields, pulsars have magnetic fields, galaxies have self-generated magnetic fields, probably.
And deep in the middle of galaxies,
sometimes there are black holes that have dynamos that can create magnetic energy that spews out into the intergalactic medium.
This is an example showing magnetic lobes which ultimately have some source of a dynamo here in the core,
and maybe even clusters of galaxies are dynamos. So Dynamo is what are they?
Again, these are systems which continuously convert kinetic energy of flowing plasma into magnetic energy, and that's the subject of the talk.
So how can those objects, such as the sun, become magnets?
The first person to talk about this was Joseph Lorimer.
And he he put forward this hypothesis in the early twenties or late night in 1919,
said it's possible for the internal cyclic motion of the sun to act in the manner of a self
exciting dynamo and create a magnetic field at the expense of some of the internal energy.
So this term self exciting dynamo already existed even at this point.
And the idea of connecting that term to astrophysical objects was what he introduced.
Before that time, people were working with mechanical dynamo generators, if you will.
And let me just try and illustrate for you what a simple dynamo might be.
And this is known as a disk dynamo. The idea is maybe we have some highly conducting metal like copper that's spinning, so it has some kinetic energy.
And from Faraday's law of induction, we know that if there is some motion and there's a little tiny bit of seed field,
present currents can be generated through the emotional EMF.
So now let's imagine I have a seed field which is like this field here that penetrates through the disk.
There is an EMF that's generated from the centre to the outside and that EMF can be hooked up to an external coil to drive the current in the coil.
And if you hook it up in the right way so that it creates a magnetic field that reinforces the original seed field,
you get a possibility of having a positive feedback cycle in which the EMF that's generated creates a little bit more magnetic field.
And this process can run away and one gets an instability, if you will, which which depends upon a few things.
So if you can write down the induction equation, you actually in this case it's a the circuit equations for this circuit.
And you can show that the time rate of change of the current sorry, the time rate of change of the current obeys a linear equation for the current,
where there's a pre factor out front that depends upon the speed that the disk is rotating,
the resistance of the wire that's used, and sort of the mutual inductance between these two things.
And if this factor is positive, this equation has exponentially growing solutions.
Okay? If it's negative, you have decaying solutions.
And we're looking for systems which have exponentially growing solutions.
Those are dynamos. Now that combination of parameters, you can work it out.
It turns out that this combination has to be dimensionless and in I'll refer to this combination of the conductivity.
Times the size, times the speed as a quantity referred to as the magnetic Reynolds number.
And this number needs to be big enough for this system to self excite.
So this is the first lesson. There's a quantity, a dimensionless number that characterises dynamos, which is the magnetic Reynolds.
Number. And it's a product of size times speed, times conductivity, and that needs to be big.
Of course, there's more to it.
There's also an equation of motion and some sort of back reaction of the magnetic field on, say, the torque which is applied.
That slows things down. But I won't get into that right now. Okay.
So this process, basically what I just showed you was used by Siemens to make the first self exciting electrical generator.
And his idea was to take. So before this time, permanent magnets were used and spinning disks and currents were generated.
And the maximum power that was possible depended upon the strength of the permanent magnets which were used.
Siemens idea was to use a little bit of the current to create the magnetic field that was there to begin with.
And this allowed much, much stronger generators to become possible.
And it really brought in the electric age and his. And he called these self exciting generators.
Okay. Okay. So let's before I move on, imagine taking this object now,
which has ferromagnetic components that direct the magnetic energy, you know, magnetic flux in certain places.
I have electrical conductors which are solid, covered by insulators that direct the current to flow in a particular path.
And I've made it all just the right geometry.
Imagine melting this, turning it into a liquid metal or a plasma and then ask, can it still behave in this way?
And that's the that's the problem I'm addressing today.
What happens if you make this simple? The system very simply connected,
a homogeneous body of conducting material and then somehow stir it in such a way that it creates its own magnetic field.
That's the dynamo problem. Okay.
So what I'm going to do in the rest of the talk is I'm going to begin with sort of an introduction, if you will, a theory to the theory of Dynamos.
But I'm going to approach it in a in a very particular way.
I'm going to talk about dynamos, which I called buildable, things that we might actually put in the lab.
And the philosophy here that I'm following is that what I can't create?
I don't understand. If I understand something as an experimentalist, I should be able to build it.
I should be able to conceptualise what the device would look like constructed in the lab and sort of test to some extent,
my understanding, and this is the philosophy of doing laboratory plasma astrophysics,
I'll try to connect these theories to astrophysical dynamos to show you that
we're working on problems that are important for understanding astrophysics.
And then I'll talk about experiments. We do experiments using liquid metals, which are a good model for understanding things like geophysical dynamos.
And now we're embarking on a new direction which is using plasmas to study dynamos.
And I'll show you how we're doing this. Okay. So.
So if I strip away all those insulators, I make the metal a fluid.
We're stuck dealing with some more sophisticated equations that that describe the magnetic and velocity fields.
And since this is a physics colloquium, I should be able to use these equations.
I think if you combine an Ampere as law, Faraday's law and a and a a model for resistivity,
you can remove eliminate several variables to come up with a single equation which describes how the magnetic field evolves.
And it depends upon the conductivity of the fluid, the velocity, whatever it is.
And you can see this is an equation which is linear in the magnetic field and it depends upon the velocity through this induction term.
And this term over here represents the resistive decay or diffusion of magnetic field, which always eliminates magnetic field.
So the ratio of those two terms you can show just comparing the magnitude of this to the magnitude of this.
This gives the magnetic Reynolds number that I already showed you.
So the product of size times, speed times, there is the conductivity.
And when this term is big magnetic induction, that is the creation of magnetic field from flow dominates over the resistive diffusion.
Okay. Now, I want to talk about one more thing, which is there's also an equation of motion.
So something governs what the velocity is doing. And this is just the Navy stokes equation governing the fluid velocity,
but with an additional term, which is the magnetic forces which might act on the flow.
And there are two limits we can think about for plasma physics, if you will.
One is a regime in which this force dominates.
So I'll call those magnetically dominated plasmas, and those would be things like fusion experiments or the solar corona,
where the magnetic field is very strong and completely dominates the dynamics.
And when this is big, when the magnetic field is strong, you tend to get waves,
often waves which propagate at a characteristic alpha speed that depends upon the density and the magnetic field strength.
I'll be thinking about a different regime, which is the flow dominated regime,
and this is a regime in which at least initially will ignore the effects of magnetic field and
we'll assume that the velocity is given so that I can think about the induction growth of magnetic
field independently from this this Lorenz force and this regime in which the inertial forces
dominate over the magnetic forces requires that the flow speed be much larger than the alpha speed.
We call this the the the flow dominated regime.
So in the flow dominated regime, I have plasmas which are flowing no magnetic fields or very weak magnetic fields.
And I ask, what do those flows do to manipulate magnetic fields?
And in this regime, when the plasma is flow dominated and the magnetic Reynolds number is big,
we can understand how the flow acts on magnetic fields by essentially what I would call the fundamental tenet of plasma astrophysics,
which is the idea that magnetic fields are frozen into moving fluids when the plasma is very conducting.
And this is just nothing more than lenses law.
The idea that if I have a highly conducting fluid elements, the magnetic field through it doesn't change with time, even as it moves.
And so you can think about field lines, if you will, as being an vectored along by some flow.
So now let's imagine I start with a seed field which is vertical here, and I push it to the right.
What will happen is that this field will become stretched out,
carried along by the fluid elements, and until it becomes strong enough to act back on the flow.
And this only happens when the plasma is flow dominated, the inertial forces are strong enough to push the magnetic field lines around,
if you will, so you can see that this magnetic this creates a magnetic field which points to the right here, points to the left here.
So if you follow the lines and the density of the field lines, which represents the strength of the field,
can become really strong if you let this shear act on the flow for a long period of time.
And this is producing magnetic field,
but it's not a dynamo because there isn't a feedback mechanism to take this shear field and reinforce the original seed field.
There's no positive feedback mechanism by which this system acts like the disk dynamo that I showed you.
Okay, so how does this happen in so you can you can see that this might be relatively easy to do in things like differentially rotating stars.
So imagine I have a star or an earth that has some inner core which is rotating at one speed and the surface is rotating at a.
And speed and see this system with a magnetic field line, which might be something like North-South,
and ask what happens to the field line as the rotation happens.
And so you can picture this in the following way.
Imagine these field lines getting their frozen into the moving fluid because the middle is moving, spinning faster than the outside.
This wraps up the north south field into the east west direction, making a very strong field in the east west direction.
I call this a toroidal field and this can be much stronger in astrophysical objects than perhaps the péladeau field,
which seeds that the initial seed field that we started with.
Now, is this a dynamo? No, it's not a dynamo, because this this field doesn't reinforce the initial seed field that you started with.
There's no closing of the feedback loop that allows this system to regenerate, if you will.
The initial North-South field that we started with. Okay. So something more is needed.
Okay. So the Dynamo regime, just just to summarise then, is a regime in which the magnetic Reynolds number is big.
This this gives us frozen in flux and flow dominated so the plasma inertial can stretch and amplify magnetic field lines.
And this is unexplored by plasma experiments.
It turns out that in order to get the high conductivity that's required to make the magnetic Reynolds number big one needs a very hot plasma,
and that is difficult to do unless some confinement is present.
And so in Tokamaks, we use strong magnetic fields to provide the thermal insulation that allows the
temperature to become high and the plasma therefore become very conducting.
And magnetic fields are usually needed for something like that.
It's much harder to do this without a magnetic field, which is the second condition that it be flow dominated.
In addition, you need some technique for stirring the plasma, and that's not so easy, it turns out.
And I'll show you how we're doing this as things go further.
Okay. So now let's think about buildable dynamo.
So I showed you how magnetic field lines are stretched.
This is an example of a geometry that could be built in the lab, which is a very, very nice dynamo.
And I'll show you how it works. So the idea here is I have two counter-rotating vortices,
and the vortices might be produced by two propellers which are rotating in the
opposite directions and providing thrust out along the poles of this system.
So the idea is the fluid elements would trace out lines like I'm showing you here.
It's two vortices packed inside a sphere. Very simple.
And this is the magnetic field, which is spontaneously created from that flow.
This is a dipolar magnetic field.
It sticks out at the North Pole and at the South Pole here, where the poles are actually orthogonal to the axis on which the flow is being stirred.
Okay. And this dynamo works because of the following idea.
So imagine you start with two seed fields, which are that dipole field.
And now I have two propellers which are rotating in opposite directions and ask what happens to those field lines as the flow carries them around?
The field lines are frozen in. What happens is they initially they get pulled out because of the rotation.
They get twisted.
So the field line, which was coming down through the centre, is now back up and around on the outside and then it gets laid back down in the centre.
So each of these field lines are following their fluid elements and being stretched and amplified and, and manipulated in the following way.
And what you end up with is a feedback mechanism by which the stretched,
amplified magnetic field reinforces the original seed field that you started with in the centre.
Okay. And so it has the stretching which is magnetic fields and the magnetic energy gain from flow,
and it has this geometry that twists the field into just the right shape to work as a dynamo.
Okay. Now you'll note here, these field lines are a huge mess in the middle.
This dynamo is quite particular in that it needs just a little bit of diffusion to cancel oppositely directed magnetic fields in this central region.
It requires some diffusion to work. Okay. And I'll show you what that means in practice in a second.
One can look at the growth rate for that dynamo as a function of the magnetic Reynolds number.
And what you find is when the magnetic Reynolds number is zero, when the system's not moving at all,
the magnetic field is just it decays away resistivity and that's a negative growth rate.
As you increase the flow speed, sorry, as you increase the flow speed, this flow, this dynamo becomes less and less damped.
And there's a critical magnetic Reynolds number above which the growth rate becomes possible.
That's the same type of linear growth that I was mentioning before.
And for this dynamo that corresponds to a magnetic Reynolds number of about 50, which is very small, I'll tell you.
And and it becomes tractable to do this with something like liquid sodium.
So liquid sodium for a device, which is a half a metre across, has a magnetic Reynolds number,
which is sort of six times the flow speed in metres per second. So for ten metres per second flow speeds in liquid sodium, you can actually.
We achieve this. You work very hard to do this, but you can achieve it in the lab.
Okay. Now what would happen if you increase the magnetic Reynolds number further?
This dynamo has a particular property, which is. It turns out as you make the system more and more conducting.
The growth rate actually turns over and it ceases to be a dynamo at very high magnetic Reynolds number.
So if the plasma or liquid metal is too conducting and the magnetic fields are too frozen in that cancellation,
that that resisted, the fusion that I mentioned earlier doesn't take place in this dynamo simply turns off.
And this is something. So some resistivity is required.
And this is a characteristic of something called a slow dynamo.
Okay. I think. Okay, so I'm just going to tell you that for a second.
I'll come back to it in a bit. Now, now we're going to talk about buildable dynamos.
And the one I just showed you is buildable.
I could use some propellers which stirred the plasma in the way that I showed you.
Or if I stir liquid metal, it turns out that we've invented a way in plasma to confine a hot plasma on the boundary using permanent magnets.
I'll show you how this works. And control the speed, but only at the boundary.
Okay. So this is the an analogue of having a bucket of fluid that's very low viscosity and
spinning the bucket and having the momentum penetrate into the fluid through viscosity.
We've developed a technique for controlling this, the rotation speed on the surface of a simply connected plasma.
And so take it from me for now and I'll come back and show you how this works later.
We can control the rotation as a function of latitude in this spherical domain.
So if I make the plasma spin one way in this hemisphere and the opposite direction in the lower hemisphere,
I have controlled the flow on the boundary.
If you look at what this does, this creates a centrifugal force which throws the plasma out so that it circulates.
So this is the political circulation which develops. So the self-consistent solution to the Na'vi stokes equation,
given those boundary conditions on the flow, looks something like this and we can solve that.
That's just a simple solution to known equations of motion.
So you can do this and we've done this. And this looks for steady state solutions to the Nabi Stokes equation, given the boundary conditions,
and it depends upon the boundary conditions, but also the plasma viscosity.
The viscosity matters because it determines what the flow looks like.
So I'm introducing another parameter, not just conductivity, but now viscosity also controls what the how the magnetic field is going to behave.
Okay. So this is the solution.
I can then take that solution and this is for a fluid Reynolds number of 300 and I can solve the kinematic conduction equation,
which is the thing that governs the magnetic eigen modes and it depends upon the magnetic Reynolds number, this parameter here.
And it turns out that this flow also is a slow dynamo.
It's stamped for zero magnetic Reynolds number. There's a critical value of about 300.
It turns over and it becomes and we have a Dynamo regime about here.
Okay, so I'm sure I'm showing you some math,
but I'm I'm trying to show you that the viscosity matters and the conductivity matters and the speed matters through these parameters.
Okay. So the viscosity determines the type of flow.
If I take the very, very viscous plasma and it looks like this is what the flow looks like,
as I increase the Reynolds number or decrease the viscosity, the flow becomes a better and better dynamo.
And you can compute what the critical magnetic Reynolds number as a function of the fluid Reynolds number in this sort of space.
And another spoiler is that this is where we're now operating our plasma is, or even better than this,
we're able to make plasmas that have just the right set of parameters to put us up in this regime.
Okay. So now let me come back to this slow versus fast team I showed you.
Slow dynamos are things which turn on at some critical magnetic Reynolds number, but then eventually turn off.
And so you might get a critical you might get a growth rate that looks something like this.
And you'll note that as the magnetic Reynolds number becomes very big, these go away.
There are other types of dynamos I'll call them fast for now that don't go away
at very large magnetic Reynolds numbers and they're fundamentally different.
Okay. And I'm going to try and show you a buildable version of a fast dynamo now
using the same idea of controlling the rotation on the surface of the sphere.
And the idea is this this is a. Solution to the Navy stokes equation time dependent as I'll show you,
in which I use the same stirring that I just showed you to create this two vortex flow.
Okay, so I know how to do this. I claim what I'm going to do then is I'm going to slosh this plasma at the equator.
I'm going to go back and forth. And what's going to happen is you're going to get a flow which does this.
The vortices go up and down. And I'm driving all of this from the boundary where I'm sloshing this back and forth like a washing machine.
Okay. This is still an axis symmetric flow. That is it.
It's the same around the. The vertical axis. But you can see that something is fun.
Adding time dependence fundamentally changes things.
And it turns out that if you look at the growth rate for magnetic field in this system, let me one more point.
If you follow fluid trajectories in this system, you'll find that when there is no sloshing, the particles do just what I showed you before.
They sit on these tauri in each hemisphere.
As you increase the sloshing the particle direction, factories start to chaotically fill the entire volume.
And when you really make it slosh a lot, any two points inside the fluid are exponentially diverging.
There's a finite spin off exponent for this system for the velocity field,
which is which gives completely chaotic streamlines for the flows in this system.
Okay.
If you can do this, if you can make a flow chaotic and then you ask what happens to a little magnetic field line which is penetrating through this,
it turns out that it grows exponentially just by virtue of the fact that the flow has chaos.
And this is what's known as a fast dynamo.
The really interesting thing about this is the magnetic field which develops looks nothing at all like the one I showed you.
It ends up being stretched and twisted like spaghetti on very small scales.
You get long filaments of magnetic field stretched by each of these little diverging fluid elements.
And what's most amazing here, and this is why it's important for astrophysics,
is that this dynamo doesn't go away as you go to very large magnetic Reynolds numbers.
This is a fast dynamo that remains even at very large magnetic Reynolds numbers.
Which, as you'll see, is what characterises most astrophysical systems.
Okay. Okay. So you don't what I just showed you was a very smooth laminar flow that had chaotic streamlines.
The easiest way to get chaos in fluids is to just make some turbulent flow.
And and it turns out that this is well understood.
People like Alex have worked on this problem. If you make a turbulent velocity field,
it almost always just spontaneously creates a magnetic field that's of this small scale and can be made sufficiently fast.
And the details of this are important, but I don't think I'll get into it.
The details depend upon whether or not the Reynolds number is bigger or smaller than the magnetic Reynolds number.
Okay, so I've introduced a lot of things here.
I've talked about small and large scale. So what scale does the magnetic field show up at?
Is there some average magnetic field or just magnetic turbulence?
Astrophysical systems have large scale magnetic fields that we observe.
There are slow versus fast. Slow only works at modest arms.
As you become very highly conducting or big large arm, it needs to be a fast dynamo of astrophysical dynamos.
Are all they? Can't they? The ones that are interesting have large scale fields.
They're fast. And there's still a big question, which is how does nature do it?
And it's really not answered yet.
So what I'd like to do now is give you an idea of what people think, that basically the standard model for how dynamos might work in astrophysics.
And the idea is as follows.
So I'm going to show you the standard model for Dynamos and then talk about how astrophysics has to astrophysical objects might do this.
The starting point is this differential rotation that I showed you before.
That is the idea that if I have differential rotation in an astrophysical object, I can stretch magnetic fields out into the east west direction.
This is called the omega effect because omega is rotation and usually there's some shear in most astrophysical objects.
This isn't a dynamo. As I mentioned, something is needed to make it a dynamo.
And the second step in the standard model of Dynamos took a long time to figure out.
It turns out there was a period of time where people. We're very good at proving things like anti dynamo theorems.
There there's a dark ages, if you will, for dynamo theory. And.
And so a tip of one of the most famous is known as Cowlings Theorem.
And he proved that with magnetic fields and fluid motions are symmetric, no dynamo exists.
And the answer to this dilemma wasn't really discovered until the mid fifties.
And the first person to really look at this was Eugene Parker.
And his idea was that if you take this strong east west field that I'm creating by differential rotation and I have some convection,
if you will, in a rotating medium that can twist those field lines in the right way.
And then you ask, how does that field behave as it's acted upon by some turbulence, which is behaving in just the right way?
What happens is that magnetic field can be twisted as it moves upward.
So the idea is I have some upward moving plumes of plasma acted out by the Coriolis force that takes the field and twists it in just the right way.
This takes the magnetic field and it creates a small component that reinforces the initial seed field.
Now, this gets very complex because suddenly we're going to try and describe a turbulent system in which the eddies are all
acting in just the right way to take a little bit of the east west field and convert it back into the little field.
And that becomes a problem of turbulence. And all of the issues that go with turbulence are come into this.
And and I want to show you a couple of things about this theory of turbulence,
which are important, because it turns out we can measure some of these things later.
So the first thing is, he said the action of this is equivalent to some current being generated in the east west direction
due to the pre-existing east west field and some parameter that characterises the turbulence.
I call this the alpha effect and people went on to develop a theory for how
fluctuations in magnetic fields and velocity fields could act to drive currents.
So this is what's known as a mean field theory for driving current.
And the idea is we have correlations between the velocity fluctuations and the magnetic
field fluctuations that on average can drive a current in some other direction.
And the theory is the simplest version of these theories, and it gets very complex quickly, is that this EMF, this turbulent EMF,
which is like the viscosity in a river that transports momentum around on average or something along those lines.
The idea was there's this term that Parker suggested was important,
but there's another term which is effectively an anomalous resistivity which enters into the into the system.
And this alpha effect, if you will, always comes with an enhancement to the resistivity.
And in liquid metal experiments, it'll turn out that what happens every time we have turbulence is we make the effect of resistivity much higher.
I'll show you why this this is very important in these theories and I'll show you measurements of this in in a bit.
Okay. But nonetheless, this is what we think is important.
So now let me talk about how that theory folds into astrophysical dynamos and what what
characterises the astrophysical dynamo is that we might want to that we might want to study.
So planetary dynamos are almost certainly so dynamo is the magnetic Reynolds number is relatively small the fluid Reynolds number is very big.
That's a property of liquid metals. This means that in order to make a magnetic field grow,
you're almost always dealing with very turbulent convective flows in something like the core of the earth,
we can simulate the magnetic field generation. This is a classic example of a simulation of the Earth's magnetic field.
You see the strong east west field deep inside the north south field that pokes out and so forth.
Oh, this is from differential rotation, as I mentioned, the stretching, if you will.
If you look over time and this is an example of a movie that was reconstructed from measurements of the Earth's magnetic fields,
from captain's logbooks over the last 400 years or so,
you can actually work out the time history of the variation of the of the Earth's magnetic field.
And so you see in this movie of the Earth's magnetic field that it's turbulent.
There are small scale fluctuations that exist on top of the large scale field that we observe on the surface of the earth.
So small scale turbulence is also present. It's probably not a small scale dynamo.
But just the large scale field being advocated about. But it's not clear.
The sun, our stars, dynamo is clearly magnetic.
There are strong magnetic fields. We know it has very interesting time dynamics.
Does it have a small scale dynamo or a large scale dynamo?
Is it fast or slow? Well, the magnetic Reynolds number is enormous.
So it's probably almost certainly has to be something like a fast dynamo.
If you look at the magnetic field on the surface of the sun and this is really well observed now by satellites,
this is the strength of the magnetic field sticking out. You get the sunspots show up as blue sticking out and yellow sticking in.
And this is a movie for how the sun's magnetic field evolves with time.
Over the last 20 years, you can see that the magnetic field, it comes and goes and there is strong differential rotation as this goes forward.
So this is the early eighties. You can see the equator is moving one speed, the poles are moving another speed.
The magnetic field goes away. Every 11 years it comes back.
The sunspots during the active cycle are present. The polarity changes, so the yellow leads blue for 11 years, then it flips.
So a lot is going on. If you average that, we find that there is a large scale magnetic field which is especially strong in between active times.
It sticks out at the North Pole and in at the South Pole, but it's very, very weak.
Only ten gauss or so this is two orders of magnitude weaker than the strong fluctuations of the magnetic field,
which are observed on the surface of the sun.
The sunspots themselves are probably representative of a strong field from deep inside, but in any case, is there a small scale magnetic field?
We think that there is a small scale dynamo.
Also looking in detail at the granular convection that takes place, there's a much stronger magnetic field,
which is always present even during those periods of solar minimum when the dipole is present.
So we're stuck in something like the sun dealing simultaneously with a fast,
a fast, large scale dynamo and a small scale dynamo simultaneously acting okay.
And this field is usually much bigger. All the building blocks are there.
There's differential rotation, it's measured. We have something like magnetic buoyancy, which could give us something like the alpha effect.
But once you combine this with the small scale dynamo and the magnetic turbulence,
so far it's not been possible to reconcile these two these two two themes.
How about the galaxy? The galaxy has a really physically big magnetic Reynolds number enormous, if you will.
The fluid Reynolds number is also big. So it's turbulent. We can see from Faraday rotation that it has large scale magnetic fields and we
can measure the strength of those fields and even our own galaxy has fields.
It's really remarkable to me to some extent that we know more about the magnetic
field structure of galaxies far away than we do even about our own galaxy.
And we can actually measure the details,
the three dimensional details of the of the of these galactic magnetic fields better than things like the core of the earth,
which is right right here. In any case, an alpha omega dynamo of the sort I showed you is consistent with the galactic dynamo.
So the idea is you have some quadrupole or magnetic field, there's differential rotation in the disk.
This creates a strong toroidal field. And then the idea is you have something like supernova driven turbulence that
gives you the alpha effect or the helical twist that I mention now in the galaxy.
Is it is there a small scale dynamo acting?
Well, it turns out you can measure the total field from something like synchrotron emission and compare it to the order field,
which is measured through Faraday rotation. The average field is at least a factor of four or more larger than the average field.
And so almost certainly in something like the galaxy, there's a small scale dynamo also acting, okay, so how are we?
What are we going to do with all this? Okay, I think I'm a skip this and move on to experiments.
Sorry. Really bad.
Okay. So with that interlude experiments,
Fermi said in MHC one shouldn't believe the product of a long and complicated piece of mathematics if unsupported by observation.
So let's see what we can do in the lab. So Dynamo experiments require a big magnetic Reynolds number.
They need to be flow dominated. As I mentioned, this is a new research.
For plasma experiments. Usually plasma experiments are magnetically dominated, especially when you have a highly conducting plasma.
We've also learned that the Fluid Reynolds number is critical for governing the behaviour of these systems.
Plasmas are hard. They're difficult to stir. Some confinement is needed and so until now, people have been using liquid metal experiments.
The confinement is free. I build a bucket, I pour the liquid metal in, I stir it, and I look for magnetic fields to grow.
There's a very unfortunate power scaling for these experiments.
The magnet, the power required to reach a certain magnetic Reynolds number goes is the cube of the magnetic Reynolds number.
And so building these experiments costs money.
You need a certain size lab. It gets more dangerous as you get bigger.
We worked very, very hard to accumulate this much liquid sodium and stir it in such a way that you have a magnetic Reynolds number,
which is just barely above the critical magnetic Reynolds number for observing a dynamo.
This takes about 100 kilowatts of power, and this is just barely above threshold.
And this is because these systems are very turbulent. Now, let me show you why that turbulence is so important.
This is our experiment at Wisconsin. It's a liquid metal experiment. It's one metre diameter spherical vessel.
We have two 100 horsepower motors that drive flows of about ten metres per second.
And this would be fine. This is the geometry we these two propellers do just what I showed you earlier.
The problem is. This is what the flow actually looks like on the scale that the magnetic field is trying to be generated.
This dipolar smooth dipole that I showed you. The turbulence in the fluid makes the velocity field appear extremely rough,
and this turbulent velocity field x to transport to take the magnetic field which
is trying to build up and move it around because the magnetic field is frozen
into the moving fluid it transports it just like turbulence mixes anything else
it would turbulent mix the magnetic field which is being created a simulation.
This is a simulation at a magnetic REYNOLDS Number of a very fluid.
Reynolds Number of about 2000, just barely beyond, you know, just barely enough to get turbulence.
It's much less than the ten to the seven that the experiments happen at.
So what we discovered in our experiments is that this turbulent EMF plays an absolutely
essential role in these experiments for governing the magnetic field evolution.
And what we did was we went into the fluid and measured the turbulent fluctuations of velocity and the turbulent fluctuations of the magnetic field.
And what we found was we had a probe, we just stuck it inside here, and we did experiments where we applied fields to study it.
We measured for the first time in the lab a turbulent EMF.
On average, the fluctuations created an EMF.
And what we then also discovered was that the alpha effect that we had expected wasn't present.
It was all an anomalous resistivity. That is when we tried to create current in one direction.
The fluctuations would act to oppose that current, and that's in essence diffusing the magnetic field away more rapidly.
This is so strong that the effective resistivity, it increases as you go up in speed.
And it also quenched any dynamo action that we were hoping to observe.
And this is because our flow was so open and unconstrained that the turbulence dominated
and so nonetheless self excited dynamos are observed in liquid metal experiments.
So they've about 15 years ago Rega used a set of pipes to constrain flows in a cylindrical geometry to make a dynamo.
The Karlsruhe experiment packed about 30 of these into another tank to do the same thing.
And then most recently, the Von Karman experiment in Catterall, France did this.
Each of these experiments, to some extent, I would say, used to cheat.
Now I use that term in the most generous way.
They are either very heavily constraining the flow to get rid of the turbulence or in the cataract experiment you're using,
ferromagnetic iron impeller is to help amplify the flow.
And so none of these experiments are the simple experiment I described to you where I melted away,
the insulators, got rid of the ferromagnetic and allowed the flow to be homogeneous.
Okay, so this is what liquid metal experiments have done for us.
We've done, we've learned a lot, but now we're ready to move on.
And I think the next step is to move to plasma experiments.
And so we're doing this and that plasma does offer a bunch of things.
They offer the possibility of very high magnetic Reynolds numbers, the ability to independently vary the fluid.
Reynolds Number and magnetic Reynolds Number, which allows us to go from laminar to turbulent, which I think is very important.
And then a host of effects that are beyond MHC that might be interesting that distinguish plasma dynamos from more simple MHC dynamos.
And I'll tell you about this now. I'll tell you how we hold the plasma, how we stir it, and how we do this with all the magnetic field of plasma.
The fluid Reynolds number depends upon the plasma density, the temperature of the plasma, the flow speed and so forth.
The magnetic Reynolds number also depends upon the temperature. These things we can compute them for typical laboratory plasma.
And what we find is that it's relatively straightforward to get interesting fluid numbers and magnetic Reynolds
numbers and and I'll show you that we create plasmas that have a temperature of about 20 V densities here.
It's a very diffuse plasma, and we can make them flow very fast. This gives us magnetic.
Reynolds numbers and fluid. REYNOLDS Numbers which are big.
So this requires two things that requires a dense plasma to make the discuss of the viscosity low
enough that the fluid Reynolds number is big and it requires a hot a high electron temperature.
And the way we do this is in the following device. So this is this is our plasma dynamo experiment.
It's a three metre diameter spherical vacuum vessel that confines plasma.
Using an array of permanent magnets. I'll show you in a second.
Also shown, here are some coils for applying fields to test it.
You can look through a window and see the plasma inside. We have probes all over to probe and measure the properties of the plasma.
If you look through actually if you look through this hole here at the opposite side inside,
this is what you would see when there's no plasma present.
And this is with a plasma helium plasma. So these are rings of very strong permanent magnets which have been bolted to the surface
of the inside of the sphere to create a magnetic field which is localised to the edge.
Okay, so this is the strength of the magnetic field plotted in cross-section and each of these magnet rings mostly cancel their nearby magnet rings.
And so in the centre of this plasma is a perfectly on magnetised volume.
And we use this very high order multiple to create a magnetic field that holds the plasma on the surface.
It's a bucket. It's a magnetic bucket for holding plasma, if you will.
And we use cathodes that emit electrons biased relative to anodes to inject current into the plasma that then ionise and heats it.
And so you can see these glowing cathodes are hot cathodes that are injecting power into the plasma.
These permanent magnets have the effect of only allowing the plasma to sneak out along the field lines which intersect the permanent magnets.
And so the plasma can't go across this part. It can only be lost in a little tiny region.
And you can see that black stripe down the middle of the magnet ring. That's where the plasma is lost.
It's not a millimetre across. And so we effectively limit the loss area to this tiny, tiny set of rings, which are which has some big advantages.
This is what the magnets look like inside.
It turns out we were able to we were able to put in every other magnet by hand and arm, and then the one in between is tricky.
The magnets have a very strong force between them. About a £200 pole force.
I'll illustrate that with a movie in a second.
And we had to use sort of a robotic technique to go down along the neutral point where this magnetic field and this magnetic field nearly cancel.
And we were able to push the magnet down before bolting it in.
So this was a fun construction technique. This is what happens.
Magnets don't play nicely if if you lose control.
So these are two magnets coming together. They destroy each other.
This is a very strong pull force.
And so this this system has an enormous amount of magnetic energy which is stored up in the edge that holds the plasma in place.
So we inject power in. We put a couple of hundred kilowatts of power in.
This is that's the green trace here. These plasmas can last for 20 seconds or so.
The plasma heats up. This is 15 EV, which I think is about 300,000 degrees centigrade.
So it's quite hot. The densities are high enough for our purposes, it turns out, and they're all almost nearly fully ionised.
At least half of the particles are ionised. There are still some neutrals present in helium, which is hard to ionise.
Okay, so now I'm going to show you how we stir plasma.
And I told you, this is the magnetic field.
It's there's no magnetic field in the centre. There's a magnetic field at the edge.
These are the cathodes. The red. The red dots are the electron emitting cathodes.
And the grey dots are on anodes, which will be collecting the electrons.
And and so they're emitting current, if you will.
Our idea for stirring plasma is the following.
If I pull these cathodes back into the edge region, in the magnet, in the Magnetised Edge region, you get a picture that looks like this.
So here are the magnet. Here are the magnetic field lines going from the South Pole to the North Pole.
Here's the cathode, which is collecting currents.
The red arrow is the current because it's emitting electrons.
And here's the magnetic field.
Now you can see that the current, which is flowing across the field lines radially, creates a j cross B force, which is in the as neutral direction.
It's a torque on the plasma, which makes it spin.
And so the cathodes, when pulled back into the Magnetised Edge region, provide a talk to control the flow in the Magnetised Edge.
And now I can control the rotation as a function of latitude by adjusting the potential
between each of these probes to drive as much current or as little current as I want.
And by controlling where I do it so it becomes possible to drive flow in one direction in one hemisphere,
and the opposite direction in the other hemisphere.
And so this is consistent with the picture I showed you earlier where I have a simply connected plasma domain,
where I can control the rotation as a function of latitude on the surface of the plasma volume.
And this is exactly the control we need to chase slow dynamos at a minimum.
Okay. So now I'm going to show you that this stirs plasma,
and I'll show you an example of an experiment that we've done where we pulled just two of these cathodes back into the edge.
And then we stick a probe in to measure the flow along a cord which goes there.
I'll show you that in a second. So we we can build probes.
We call them mock probes. They're a variant on the Langmuir probe measures flow to two different phases,
and by measuring the difference in current to the two different phases, we can infer what the velocity is.
This is easy, relatively straightforward to do, and we take this probe and we scan and measure the velocity along this cord.
And what we measured we can plotted here is a function of radius from the centre of the machine.
This is the measured East-West rotation of the plasma.
Here is where our cathode is. The magnetic field is gone by about this spot.
So here is our very fast up here. This is our very fast ten kilometres per second flow speeds are measured here where
we're stirring and it viciously penetrates in to the on magnetised centre of the plasma.
And this is this is how that falls off. This is consistent with viscosity, transporting the momentum inward.
At the same time, I'm spinning this, I'm making it spin, toroidal or East-West.
And so there's a centrifugal force that actually throws the plasma out in this region generating this polygonal circulation.
Okay, so this is flow in the north south direction and we actually detect about two kilometres per second of palatal flow being thrown out.
And this this is a very good thing because this is the effect that we wanted to control the rotation with.
So I'm demonstrating to you here that we have the knobs to stir the plasma and control it on the on the surface of the plasma.
Look, in this case, it was a four sided mock probe which measured the two components.
Okay. So this is a big step for us.
We are now in the process of optimising the flow and putting in magnetic sensors to look for magnetic fields.
So we haven't yet put a magnetic probe in the experiment that could even observe this.
We're just starting this and we're also adding the control.
This is an example where we used eight cathodes to spin one hemisphere, one way, the opposite hemisphere, the other direction.
And and these are the global fluid, Reynolds numbers.
In this case, the fluid Reynolds number was about 600 and the magnetic Reynolds number was about 400.
But it's not a fully optimised plasma. We have to balance the flows and and add a little bit more control, we think, to get a dynamo.
So let me show you what we're thinking will happen and this will sort of finish things up.
This is the flow we would like. This is the one I showed you at the beginning.
We control the rotation as a function of latitude and this is the resulting palatal flow.
To do this, we need to add more cathodes and in fact, we need a little bit of counter rotation, the opposite direction from the poles at the equator.
And we'll do that with at least 1212 cathodes that are gone in, go, go, going in.
At the same time, we need to measure the flow so we know what it is that we're what we're creating.
And we're already we're already making the plasma parameters,
the turning of plasmas and the densities which are required to get the viscosity and the resistivity is just right.
So this is underway and I would stay tuned.
Okay. And that's and that will create this magnetic field.
That's where that's a slow dynamo and that's the baseline that we've been working towards.
But we think we also have more that can be done.
And I'd like to leave you with the following simulation.
So this is now a time some are a time dependent calculation in which I'm going to vary the fluid Reynolds number
and the magnetic Reynolds numbers within what we think are the types of parameters that the experiment can access.
And so the fluid Reynolds number is we'll start out at 100 with a magnetic Reynolds number of 200.
This corresponds to three kilometres per second, a 10th of plasma of this density and so forth.
I can increase the fluid Reynolds number.
The flow will change a little bit and then I if I go just a little bit further, the flow actually becomes turbulent.
Okay? There's a there's a critical hydrodynamic instability which sets in.
The flow becomes turbulent and I get chaotic flows.
Okay. If at the same time I increase the electron temperature so that it becomes very.
Conducting I can make the magnetic Reynolds number big and we think we could probably get up
to temperatures as high as 40 V It's about twice what we're doing now by adding more power.
And this gives the possibility of a very big magnetic Reynolds number.
And this is what happens in the simulations in that case. So here is the velocity here and we're stirring on the outside.
Here's the polite flow. As I increase the fluid Reynolds number, it starts to fluctuate a little bit.
A hydrodynamic instability is setting in and then eventually it just becomes turbulent.
And this is what turbulence looks like driving it with the flow drive that I'm using.
Okay, now I increase the electron temperature also.
So it's a very highly conducting plasma. And what happens then is we start to see magnetic field energy develop.
And this simulation shows that we have a small scale fast dynamo occurring and now we're starting to look at Dynamo.
Is that happen at very large magnetic Reynolds numbers?
What we might hope is that we could use this object to then look for large
scale magnetic fields also being generated and attack the fast dynamo problem,
which is what is most important in astrophysics. So that's my talk.
In summary, fast, large scale dynamos are the most important ones in nature.
This is still a theoretical challenge.
Liquid metal experiments see self excite under some conditions, but they also allow us to measure things which we're thought to be important.
Like the turbulent EMF.
I've shown you that very turbulent liquid metal experiments are much more resistive on average then than they are if they're laminar.
And I've also tried to show you that plasma dynamos are now underway.
They have the possibility of operating at much larger magnetic Reynolds number.
We can vary the fluid. Reynolds number, which is the ratio between the magnetic Reynolds number and the fluid.
Reynolds Number and we're in the process of optimising the flow that requires good
measurements and many control tools to make the flow just right for a dynamo,
but that's what we're doing. So with that, let me acknowledge all these people and thank you for your attention.