1 00:00:07,450 --> 00:00:16,420 Telling you today about experiments we're carrying out at the University of Wisconsin, using plasmas to look at the Dynamo problem. 2 00:00:16,450 --> 00:00:20,919 And this is intended to be a little bit of a provocative title. 3 00:00:20,920 --> 00:00:27,220 I'll try and explain to you what fast dynamos are, where we haven't even gotten to slow dynamos yet. 4 00:00:29,470 --> 00:00:40,720 So many dynamos. So this is this is an experimentalist in his lab and these are dynamos dynamos are objects that sorry about that 5 00:00:40,720 --> 00:00:49,990 dynamos are objects that create magnetic energy from kinetic energy and you might just know them as generators. 6 00:00:50,230 --> 00:00:55,930 These are electrical generators. And this phrase, if you haven't seen it before, is the same spelled backwards. 7 00:00:56,260 --> 00:01:00,190 It's a palindrome. So, so many dynamos this. If you haven't seen that, you should see it. 8 00:01:01,330 --> 00:01:08,260 I'll be talking about astrophysical dynamos and and so these are a bunch of those. 9 00:01:08,260 --> 00:01:19,510 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 10 00:01:19,510 --> 00:01:25,000 exist on a huge range of scales from things like the planetary magnetic fields of our own planets. 11 00:01:25,420 --> 00:01:31,959 Stars have magnetic fields, pulsars have magnetic fields, galaxies have self-generated magnetic fields, probably. 12 00:01:31,960 --> 00:01:34,360 And deep in the middle of galaxies, 13 00:01:34,360 --> 00:01:40,980 sometimes there are black holes that have dynamos that can create magnetic energy that spews out into the intergalactic medium. 14 00:01:41,000 --> 00:01:46,630 This is an example showing magnetic lobes which ultimately have some source of a dynamo here in the core, 15 00:01:47,140 --> 00:01:51,730 and maybe even clusters of galaxies are dynamos. So Dynamo is what are they? 16 00:01:52,060 --> 00:02:01,330 Again, these are systems which continuously convert kinetic energy of flowing plasma into magnetic energy, and that's the subject of the talk. 17 00:02:02,260 --> 00:02:07,420 So how can those objects, such as the sun, become magnets? 18 00:02:08,350 --> 00:02:11,530 The first person to talk about this was Joseph Lorimer. 19 00:02:11,800 --> 00:02:18,550 And he he put forward this hypothesis in the early twenties or late night in 1919, 20 00:02:18,670 --> 00:02:23,680 said it's possible for the internal cyclic motion of the sun to act in the manner of a self 21 00:02:23,680 --> 00:02:29,680 exciting dynamo and create a magnetic field at the expense of some of the internal energy. 22 00:02:29,680 --> 00:02:34,459 So this term self exciting dynamo already existed even at this point. 23 00:02:34,460 --> 00:02:39,730 And the idea of connecting that term to astrophysical objects was what he introduced. 24 00:02:40,270 --> 00:02:44,829 Before that time, people were working with mechanical dynamo generators, if you will. 25 00:02:44,830 --> 00:02:49,719 And let me just try and illustrate for you what a simple dynamo might be. 26 00:02:49,720 --> 00:02:59,140 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. 27 00:03:00,640 --> 00:03:06,280 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, 28 00:03:06,280 --> 00:03:10,060 present currents can be generated through the emotional EMF. 29 00:03:10,330 --> 00:03:16,150 So now let's imagine I have a seed field which is like this field here that penetrates through the disk. 30 00:03:16,480 --> 00:03:24,760 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. 31 00:03:24,940 --> 00:03:32,110 And if you hook it up in the right way so that it creates a magnetic field that reinforces the original seed field, 32 00:03:32,740 --> 00:03:40,570 you get a possibility of having a positive feedback cycle in which the EMF that's generated creates a little bit more magnetic field. 33 00:03:40,750 --> 00:03:49,629 And this process can run away and one gets an instability, if you will, which which depends upon a few things. 34 00:03:49,630 --> 00:03:55,720 So if you can write down the induction equation, you actually in this case it's a the circuit equations for this circuit. 35 00:03:56,080 --> 00:04:06,610 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, 36 00:04:07,060 --> 00:04:12,640 where there's a pre factor out front that depends upon the speed that the disk is rotating, 37 00:04:13,930 --> 00:04:19,510 the resistance of the wire that's used, and sort of the mutual inductance between these two things. 38 00:04:19,870 --> 00:04:25,510 And if this factor is positive, this equation has exponentially growing solutions. 39 00:04:25,750 --> 00:04:28,840 Okay? If it's negative, you have decaying solutions. 40 00:04:28,840 --> 00:04:33,489 And we're looking for systems which have exponentially growing solutions. 41 00:04:33,490 --> 00:04:39,459 Those are dynamos. Now that combination of parameters, you can work it out. 42 00:04:39,460 --> 00:04:47,500 It turns out that this combination has to be dimensionless and in I'll refer to this combination of the conductivity. 43 00:04:47,980 --> 00:04:54,730 Times the size, times the speed as a quantity referred to as the magnetic Reynolds number. 44 00:04:54,730 --> 00:04:59,560 And this number needs to be big enough for this system to self excite. 45 00:04:59,950 --> 00:05:06,970 So this is the first lesson. There's a quantity, a dimensionless number that characterises dynamos, which is the magnetic Reynolds. 46 00:05:07,050 --> 00:05:12,030 Number. And it's a product of size times speed, times conductivity, and that needs to be big. 47 00:05:14,010 --> 00:05:15,089 Of course, there's more to it. 48 00:05:15,090 --> 00:05:22,499 There's also an equation of motion and some sort of back reaction of the magnetic field on, say, the torque which is applied. 49 00:05:22,500 --> 00:05:25,890 That slows things down. But I won't get into that right now. Okay. 50 00:05:27,030 --> 00:05:36,839 So this process, basically what I just showed you was used by Siemens to make the first self exciting electrical generator. 51 00:05:36,840 --> 00:05:45,060 And his idea was to take. So before this time, permanent magnets were used and spinning disks and currents were generated. 52 00:05:45,390 --> 00:05:50,340 And the maximum power that was possible depended upon the strength of the permanent magnets which were used. 53 00:05:53,010 --> 00:05:57,930 Siemens idea was to use a little bit of the current to create the magnetic field that was there to begin with. 54 00:05:58,260 --> 00:06:02,100 And this allowed much, much stronger generators to become possible. 55 00:06:02,100 --> 00:06:06,930 And it really brought in the electric age and his. And he called these self exciting generators. 56 00:06:07,130 --> 00:06:13,380 Okay. Okay. So let's before I move on, imagine taking this object now, 57 00:06:13,380 --> 00:06:20,459 which has ferromagnetic components that direct the magnetic energy, you know, magnetic flux in certain places. 58 00:06:20,460 --> 00:06:27,480 I have electrical conductors which are solid, covered by insulators that direct the current to flow in a particular path. 59 00:06:27,750 --> 00:06:29,760 And I've made it all just the right geometry. 60 00:06:29,760 --> 00:06:37,060 Imagine melting this, turning it into a liquid metal or a plasma and then ask, can it still behave in this way? 61 00:06:37,080 --> 00:06:41,100 And that's the that's the problem I'm addressing today. 62 00:06:41,100 --> 00:06:44,640 What happens if you make this simple? The system very simply connected, 63 00:06:44,970 --> 00:06:52,200 a homogeneous body of conducting material and then somehow stir it in such a way that it creates its own magnetic field. 64 00:06:52,200 --> 00:06:54,240 That's the dynamo problem. Okay. 65 00:06:56,040 --> 00:07:05,910 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. 66 00:07:06,210 --> 00:07:09,420 But I'm going to approach it in a in a very particular way. 67 00:07:09,420 --> 00:07:15,329 I'm going to talk about dynamos, which I called buildable, things that we might actually put in the lab. 68 00:07:15,330 --> 00:07:21,059 And the philosophy here that I'm following is that what I can't create? 69 00:07:21,060 --> 00:07:25,500 I don't understand. If I understand something as an experimentalist, I should be able to build it. 70 00:07:25,500 --> 00:07:31,200 I should be able to conceptualise what the device would look like constructed in the lab and sort of test to some extent, 71 00:07:31,200 --> 00:07:35,880 my understanding, and this is the philosophy of doing laboratory plasma astrophysics, 72 00:07:36,990 --> 00:07:41,040 I'll try to connect these theories to astrophysical dynamos to show you that 73 00:07:41,040 --> 00:07:44,820 we're working on problems that are important for understanding astrophysics. 74 00:07:45,300 --> 00:07:55,470 And then I'll talk about experiments. We do experiments using liquid metals, which are a good model for understanding things like geophysical dynamos. 75 00:07:56,190 --> 00:08:01,200 And now we're embarking on a new direction which is using plasmas to study dynamos. 76 00:08:01,200 --> 00:08:09,230 And I'll show you how we're doing this. Okay. So. 77 00:08:10,720 --> 00:08:14,930 So if I strip away all those insulators, I make the metal a fluid. 78 00:08:15,530 --> 00:08:21,930 We're stuck dealing with some more sophisticated equations that that describe the magnetic and velocity fields. 79 00:08:21,950 --> 00:08:26,450 And since this is a physics colloquium, I should be able to use these equations. 80 00:08:26,450 --> 00:08:36,830 I think if you combine an Ampere as law, Faraday's law and a and a a model for resistivity, 81 00:08:37,100 --> 00:08:45,010 you can remove eliminate several variables to come up with a single equation which describes how the magnetic field evolves. 82 00:08:45,020 --> 00:08:50,599 And it depends upon the conductivity of the fluid, the velocity, whatever it is. 83 00:08:50,600 --> 00:08:57,980 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. 84 00:08:58,160 --> 00:09:04,490 And this term over here represents the resistive decay or diffusion of magnetic field, which always eliminates magnetic field. 85 00:09:05,780 --> 00:09:12,210 So the ratio of those two terms you can show just comparing the magnitude of this to the magnitude of this. 86 00:09:12,710 --> 00:09:15,770 This gives the magnetic Reynolds number that I already showed you. 87 00:09:15,770 --> 00:09:21,080 So the product of size times, speed times, there is the conductivity. 88 00:09:21,500 --> 00:09:30,560 And when this term is big magnetic induction, that is the creation of magnetic field from flow dominates over the resistive diffusion. 89 00:09:30,740 --> 00:09:37,100 Okay. Now, I want to talk about one more thing, which is there's also an equation of motion. 90 00:09:37,100 --> 00:09:44,179 So something governs what the velocity is doing. And this is just the Navy stokes equation governing the fluid velocity, 91 00:09:44,180 --> 00:09:49,400 but with an additional term, which is the magnetic forces which might act on the flow. 92 00:09:50,000 --> 00:09:55,520 And there are two limits we can think about for plasma physics, if you will. 93 00:09:56,210 --> 00:09:59,760 One is a regime in which this force dominates. 94 00:10:00,230 --> 00:10:06,230 So I'll call those magnetically dominated plasmas, and those would be things like fusion experiments or the solar corona, 95 00:10:06,230 --> 00:10:10,640 where the magnetic field is very strong and completely dominates the dynamics. 96 00:10:12,440 --> 00:10:17,329 And when this is big, when the magnetic field is strong, you tend to get waves, 97 00:10:17,330 --> 00:10:23,090 often waves which propagate at a characteristic alpha speed that depends upon the density and the magnetic field strength. 98 00:10:24,380 --> 00:10:28,130 I'll be thinking about a different regime, which is the flow dominated regime, 99 00:10:28,130 --> 00:10:34,160 and this is a regime in which at least initially will ignore the effects of magnetic field and 100 00:10:34,160 --> 00:10:38,899 we'll assume that the velocity is given so that I can think about the induction growth of magnetic 101 00:10:38,900 --> 00:10:46,340 field independently from this this Lorenz force and this regime in which the inertial forces 102 00:10:46,340 --> 00:10:51,950 dominate over the magnetic forces requires that the flow speed be much larger than the alpha speed. 103 00:10:51,950 --> 00:10:55,340 We call this the the the flow dominated regime. 104 00:10:55,700 --> 00:11:01,549 So in the flow dominated regime, I have plasmas which are flowing no magnetic fields or very weak magnetic fields. 105 00:11:01,550 --> 00:11:06,800 And I ask, what do those flows do to manipulate magnetic fields? 106 00:11:07,490 --> 00:11:15,380 And in this regime, when the plasma is flow dominated and the magnetic Reynolds number is big, 107 00:11:16,970 --> 00:11:26,390 we can understand how the flow acts on magnetic fields by essentially what I would call the fundamental tenet of plasma astrophysics, 108 00:11:26,390 --> 00:11:33,469 which is the idea that magnetic fields are frozen into moving fluids when the plasma is very conducting. 109 00:11:33,470 --> 00:11:35,330 And this is just nothing more than lenses law. 110 00:11:35,360 --> 00:11:42,950 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. 111 00:11:42,950 --> 00:11:49,040 And so you can think about field lines, if you will, as being an vectored along by some flow. 112 00:11:49,040 --> 00:11:53,210 So now let's imagine I start with a seed field which is vertical here, and I push it to the right. 113 00:11:53,510 --> 00:11:56,120 What will happen is that this field will become stretched out, 114 00:11:56,840 --> 00:12:02,270 carried along by the fluid elements, and until it becomes strong enough to act back on the flow. 115 00:12:02,630 --> 00:12:09,860 And this only happens when the plasma is flow dominated, the inertial forces are strong enough to push the magnetic field lines around, 116 00:12:09,860 --> 00:12:16,849 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. 117 00:12:16,850 --> 00:12:21,800 So if you follow the lines and the density of the field lines, which represents the strength of the field, 118 00:12:22,100 --> 00:12:27,770 can become really strong if you let this shear act on the flow for a long period of time. 119 00:12:27,770 --> 00:12:29,840 And this is producing magnetic field, 120 00:12:30,170 --> 00:12:37,819 but it's not a dynamo because there isn't a feedback mechanism to take this shear field and reinforce the original seed field. 121 00:12:37,820 --> 00:12:44,750 There's no positive feedback mechanism by which this system acts like the disk dynamo that I showed you. 122 00:12:44,870 --> 00:12:58,249 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. 123 00:12:58,250 --> 00:13:07,490 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. 124 00:13:07,730 --> 00:13:13,040 And speed and see this system with a magnetic field line, which might be something like North-South, 125 00:13:13,040 --> 00:13:16,939 and ask what happens to the field line as the rotation happens. 126 00:13:16,940 --> 00:13:18,979 And so you can picture this in the following way. 127 00:13:18,980 --> 00:13:26,540 Imagine these field lines getting their frozen into the moving fluid because the middle is moving, spinning faster than the outside. 128 00:13:26,870 --> 00:13:34,760 This wraps up the north south field into the east west direction, making a very strong field in the east west direction. 129 00:13:35,030 --> 00:13:42,379 I call this a toroidal field and this can be much stronger in astrophysical objects than perhaps the péladeau field, 130 00:13:42,380 --> 00:13:45,500 which seeds that the initial seed field that we started with. 131 00:13:46,100 --> 00:13:58,510 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. 132 00:13:58,520 --> 00:14:04,520 There's no closing of the feedback loop that allows this system to regenerate, if you will. 133 00:14:04,760 --> 00:14:09,110 The initial North-South field that we started with. Okay. So something more is needed. 134 00:14:11,420 --> 00:14:18,500 Okay. So the Dynamo regime, just just to summarise then, is a regime in which the magnetic Reynolds number is big. 135 00:14:18,830 --> 00:14:26,570 This this gives us frozen in flux and flow dominated so the plasma inertial can stretch and amplify magnetic field lines. 136 00:14:27,020 --> 00:14:30,230 And this is unexplored by plasma experiments. 137 00:14:30,590 --> 00:14:40,880 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, 138 00:14:41,240 --> 00:14:46,430 and that is difficult to do unless some confinement is present. 139 00:14:46,520 --> 00:14:52,309 And so in Tokamaks, we use strong magnetic fields to provide the thermal insulation that allows the 140 00:14:52,310 --> 00:14:56,480 temperature to become high and the plasma therefore become very conducting. 141 00:14:57,230 --> 00:15:01,100 And magnetic fields are usually needed for something like that. 142 00:15:01,100 --> 00:15:06,320 It's much harder to do this without a magnetic field, which is the second condition that it be flow dominated. 143 00:15:07,400 --> 00:15:13,180 In addition, you need some technique for stirring the plasma, and that's not so easy, it turns out. 144 00:15:13,190 --> 00:15:18,020 And I'll show you how we're doing this as things go further. 145 00:15:18,230 --> 00:15:22,380 Okay. So now let's think about buildable dynamo. 146 00:15:22,400 --> 00:15:25,040 So I showed you how magnetic field lines are stretched. 147 00:15:25,070 --> 00:15:34,969 This is an example of a geometry that could be built in the lab, which is a very, very nice dynamo. 148 00:15:34,970 --> 00:15:40,220 And I'll show you how it works. So the idea here is I have two counter-rotating vortices, 149 00:15:40,220 --> 00:15:43,640 and the vortices might be produced by two propellers which are rotating in the 150 00:15:43,640 --> 00:15:49,280 opposite directions and providing thrust out along the poles of this system. 151 00:15:50,660 --> 00:15:55,580 So the idea is the fluid elements would trace out lines like I'm showing you here. 152 00:15:56,060 --> 00:15:59,090 It's two vortices packed inside a sphere. Very simple. 153 00:16:00,590 --> 00:16:04,220 And this is the magnetic field, which is spontaneously created from that flow. 154 00:16:04,250 --> 00:16:06,500 This is a dipolar magnetic field. 155 00:16:06,500 --> 00:16:15,380 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. 156 00:16:15,770 --> 00:16:24,330 Okay. And this dynamo works because of the following idea. 157 00:16:24,350 --> 00:16:29,600 So imagine you start with two seed fields, which are that dipole field. 158 00:16:29,600 --> 00:16:37,669 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? 159 00:16:37,670 --> 00:16:44,840 The field lines are frozen in. What happens is they initially they get pulled out because of the rotation. 160 00:16:45,290 --> 00:16:46,280 They get twisted. 161 00:16:47,390 --> 00:16:55,879 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. 162 00:16:55,880 --> 00:17:04,610 So each of these field lines are following their fluid elements and being stretched and amplified and, and manipulated in the following way. 163 00:17:04,610 --> 00:17:08,300 And what you end up with is a feedback mechanism by which the stretched, 164 00:17:08,510 --> 00:17:14,150 amplified magnetic field reinforces the original seed field that you started with in the centre. 165 00:17:14,360 --> 00:17:22,430 Okay. And so it has the stretching which is magnetic fields and the magnetic energy gain from flow, 166 00:17:22,730 --> 00:17:28,610 and it has this geometry that twists the field into just the right shape to work as a dynamo. 167 00:17:29,780 --> 00:17:33,860 Okay. Now you'll note here, these field lines are a huge mess in the middle. 168 00:17:34,280 --> 00:17:45,319 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. 169 00:17:45,320 --> 00:17:51,290 It requires some diffusion to work. Okay. And I'll show you what that means in practice in a second. 170 00:17:52,970 --> 00:17:58,970 One can look at the growth rate for that dynamo as a function of the magnetic Reynolds number. 171 00:17:59,420 --> 00:18:03,530 And what you find is when the magnetic Reynolds number is zero, when the system's not moving at all, 172 00:18:03,860 --> 00:18:08,749 the magnetic field is just it decays away resistivity and that's a negative growth rate. 173 00:18:08,750 --> 00:18:16,729 As you increase the flow speed, sorry, as you increase the flow speed, this flow, this dynamo becomes less and less damped. 174 00:18:16,730 --> 00:18:21,350 And there's a critical magnetic Reynolds number above which the growth rate becomes possible. 175 00:18:21,350 --> 00:18:25,640 That's the same type of linear growth that I was mentioning before. 176 00:18:26,690 --> 00:18:32,809 And for this dynamo that corresponds to a magnetic Reynolds number of about 50, which is very small, I'll tell you. 177 00:18:32,810 --> 00:18:37,520 And and it becomes tractable to do this with something like liquid sodium. 178 00:18:37,520 --> 00:18:42,889 So liquid sodium for a device, which is a half a metre across, has a magnetic Reynolds number, 179 00:18:42,890 --> 00:18:49,910 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. 180 00:18:50,000 --> 00:18:53,810 We achieve this. You work very hard to do this, but you can achieve it in the lab. 181 00:18:53,970 --> 00:18:59,270 Okay. Now what would happen if you increase the magnetic Reynolds number further? 182 00:18:59,690 --> 00:19:07,640 This dynamo has a particular property, which is. It turns out as you make the system more and more conducting. 183 00:19:07,940 --> 00:19:14,300 The growth rate actually turns over and it ceases to be a dynamo at very high magnetic Reynolds number. 184 00:19:14,540 --> 00:19:22,609 So if the plasma or liquid metal is too conducting and the magnetic fields are too frozen in that cancellation, 185 00:19:22,610 --> 00:19:27,920 that that resisted, the fusion that I mentioned earlier doesn't take place in this dynamo simply turns off. 186 00:19:29,030 --> 00:19:32,720 And this is something. So some resistivity is required. 187 00:19:33,440 --> 00:19:38,240 And this is a characteristic of something called a slow dynamo. 188 00:19:38,650 --> 00:19:46,129 Okay. I think. Okay, so I'm just going to tell you that for a second. 189 00:19:46,130 --> 00:19:52,190 I'll come back to it in a bit. Now, now we're going to talk about buildable dynamos. 190 00:19:52,190 --> 00:19:55,339 And the one I just showed you is buildable. 191 00:19:55,340 --> 00:19:59,180 I could use some propellers which stirred the plasma in the way that I showed you. 192 00:19:59,180 --> 00:20:11,320 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. 193 00:20:11,330 --> 00:20:16,520 I'll show you how this works. And control the speed, but only at the boundary. 194 00:20:16,560 --> 00:20:21,530 Okay. So this is the an analogue of having a bucket of fluid that's very low viscosity and 195 00:20:21,530 --> 00:20:26,090 spinning the bucket and having the momentum penetrate into the fluid through viscosity. 196 00:20:26,570 --> 00:20:34,520 We've developed a technique for controlling this, the rotation speed on the surface of a simply connected plasma. 197 00:20:35,000 --> 00:20:39,770 And so take it from me for now and I'll come back and show you how this works later. 198 00:20:40,280 --> 00:20:45,769 We can control the rotation as a function of latitude in this spherical domain. 199 00:20:45,770 --> 00:20:51,650 So if I make the plasma spin one way in this hemisphere and the opposite direction in the lower hemisphere, 200 00:20:51,830 --> 00:20:54,740 I have controlled the flow on the boundary. 201 00:20:56,210 --> 00:21:01,340 If you look at what this does, this creates a centrifugal force which throws the plasma out so that it circulates. 202 00:21:01,340 --> 00:21:07,370 So this is the political circulation which develops. So the self-consistent solution to the Na'vi stokes equation, 203 00:21:07,370 --> 00:21:12,019 given those boundary conditions on the flow, looks something like this and we can solve that. 204 00:21:12,020 --> 00:21:16,820 That's just a simple solution to known equations of motion. 205 00:21:17,690 --> 00:21:25,610 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, 206 00:21:26,090 --> 00:21:29,930 and it depends upon the boundary conditions, but also the plasma viscosity. 207 00:21:29,930 --> 00:21:33,079 The viscosity matters because it determines what the flow looks like. 208 00:21:33,080 --> 00:21:41,390 So I'm introducing another parameter, not just conductivity, but now viscosity also controls what the how the magnetic field is going to behave. 209 00:21:42,440 --> 00:21:44,030 Okay. So this is the solution. 210 00:21:44,690 --> 00:21:52,819 I can then take that solution and this is for a fluid Reynolds number of 300 and I can solve the kinematic conduction equation, 211 00:21:52,820 --> 00:21:58,430 which is the thing that governs the magnetic eigen modes and it depends upon the magnetic Reynolds number, this parameter here. 212 00:21:59,450 --> 00:22:05,329 And it turns out that this flow also is a slow dynamo. 213 00:22:05,330 --> 00:22:09,650 It's stamped for zero magnetic Reynolds number. There's a critical value of about 300. 214 00:22:09,890 --> 00:22:15,500 It turns over and it becomes and we have a Dynamo regime about here. 215 00:22:15,650 --> 00:22:18,889 Okay, so I'm sure I'm showing you some math, 216 00:22:18,890 --> 00:22:26,150 but I'm I'm trying to show you that the viscosity matters and the conductivity matters and the speed matters through these parameters. 217 00:22:26,930 --> 00:22:31,489 Okay. So the viscosity determines the type of flow. 218 00:22:31,490 --> 00:22:36,400 If I take the very, very viscous plasma and it looks like this is what the flow looks like, 219 00:22:36,410 --> 00:22:41,870 as I increase the Reynolds number or decrease the viscosity, the flow becomes a better and better dynamo. 220 00:22:42,170 --> 00:22:48,800 And you can compute what the critical magnetic Reynolds number as a function of the fluid Reynolds number in this sort of space. 221 00:22:49,310 --> 00:22:56,300 And another spoiler is that this is where we're now operating our plasma is, or even better than this, 222 00:22:57,080 --> 00:23:01,820 we're able to make plasmas that have just the right set of parameters to put us up in this regime. 223 00:23:01,940 --> 00:23:08,509 Okay. So now let me come back to this slow versus fast team I showed you. 224 00:23:08,510 --> 00:23:15,229 Slow dynamos are things which turn on at some critical magnetic Reynolds number, but then eventually turn off. 225 00:23:15,230 --> 00:23:18,740 And so you might get a critical you might get a growth rate that looks something like this. 226 00:23:19,280 --> 00:23:23,930 And you'll note that as the magnetic Reynolds number becomes very big, these go away. 227 00:23:26,240 --> 00:23:32,120 There are other types of dynamos I'll call them fast for now that don't go away 228 00:23:32,150 --> 00:23:35,390 at very large magnetic Reynolds numbers and they're fundamentally different. 229 00:23:35,930 --> 00:23:41,060 Okay. And I'm going to try and show you a buildable version of a fast dynamo now 230 00:23:41,480 --> 00:23:46,640 using the same idea of controlling the rotation on the surface of the sphere. 231 00:23:47,360 --> 00:23:53,300 And the idea is this this is a. Solution to the Navy stokes equation time dependent as I'll show you, 232 00:23:53,570 --> 00:23:58,160 in which I use the same stirring that I just showed you to create this two vortex flow. 233 00:23:59,020 --> 00:24:07,300 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. 234 00:24:07,310 --> 00:24:13,850 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. 235 00:24:14,660 --> 00:24:21,710 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. 236 00:24:22,820 --> 00:24:28,070 Okay. This is still an axis symmetric flow. That is it. 237 00:24:28,670 --> 00:24:35,540 It's the same around the. The vertical axis. But you can see that something is fun. 238 00:24:35,750 --> 00:24:38,540 Adding time dependence fundamentally changes things. 239 00:24:39,980 --> 00:24:49,640 And it turns out that if you look at the growth rate for magnetic field in this system, let me one more point. 240 00:24:50,600 --> 00:25:00,300 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. 241 00:25:00,320 --> 00:25:02,510 They sit on these tauri in each hemisphere. 242 00:25:03,080 --> 00:25:09,560 As you increase the sloshing the particle direction, factories start to chaotically fill the entire volume. 243 00:25:09,980 --> 00:25:17,010 And when you really make it slosh a lot, any two points inside the fluid are exponentially diverging. 244 00:25:17,010 --> 00:25:21,200 There's a finite spin off exponent for this system for the velocity field, 245 00:25:21,500 --> 00:25:27,380 which is which gives completely chaotic streamlines for the flows in this system. 246 00:25:27,410 --> 00:25:27,700 Okay. 247 00:25:28,610 --> 00:25:37,490 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, 248 00:25:38,690 --> 00:25:44,540 it turns out that it grows exponentially just by virtue of the fact that the flow has chaos. 249 00:25:45,020 --> 00:25:47,150 And this is what's known as a fast dynamo. 250 00:25:47,390 --> 00:25:52,880 The really interesting thing about this is the magnetic field which develops looks nothing at all like the one I showed you. 251 00:25:53,120 --> 00:25:57,769 It ends up being stretched and twisted like spaghetti on very small scales. 252 00:25:57,770 --> 00:26:03,620 You get long filaments of magnetic field stretched by each of these little diverging fluid elements. 253 00:26:04,760 --> 00:26:08,330 And what's most amazing here, and this is why it's important for astrophysics, 254 00:26:08,690 --> 00:26:14,100 is that this dynamo doesn't go away as you go to very large magnetic Reynolds numbers. 255 00:26:14,120 --> 00:26:18,590 This is a fast dynamo that remains even at very large magnetic Reynolds numbers. 256 00:26:18,920 --> 00:26:22,790 Which, as you'll see, is what characterises most astrophysical systems. 257 00:26:23,340 --> 00:26:32,450 Okay. Okay. So you don't what I just showed you was a very smooth laminar flow that had chaotic streamlines. 258 00:26:32,450 --> 00:26:36,220 The easiest way to get chaos in fluids is to just make some turbulent flow. 259 00:26:36,890 --> 00:26:40,219 And and it turns out that this is well understood. 260 00:26:40,220 --> 00:26:45,470 People like Alex have worked on this problem. If you make a turbulent velocity field, 261 00:26:45,740 --> 00:26:54,920 it almost always just spontaneously creates a magnetic field that's of this small scale and can be made sufficiently fast. 262 00:26:54,920 --> 00:26:59,510 And the details of this are important, but I don't think I'll get into it. 263 00:27:00,020 --> 00:27:05,720 The details depend upon whether or not the Reynolds number is bigger or smaller than the magnetic Reynolds number. 264 00:27:06,710 --> 00:27:10,850 Okay, so I've introduced a lot of things here. 265 00:27:10,850 --> 00:27:15,350 I've talked about small and large scale. So what scale does the magnetic field show up at? 266 00:27:15,620 --> 00:27:18,710 Is there some average magnetic field or just magnetic turbulence? 267 00:27:19,460 --> 00:27:22,640 Astrophysical systems have large scale magnetic fields that we observe. 268 00:27:23,390 --> 00:27:27,320 There are slow versus fast. Slow only works at modest arms. 269 00:27:30,800 --> 00:27:37,580 As you become very highly conducting or big large arm, it needs to be a fast dynamo of astrophysical dynamos. 270 00:27:37,580 --> 00:27:42,229 Are all they? Can't they? The ones that are interesting have large scale fields. 271 00:27:42,230 --> 00:27:47,120 They're fast. And there's still a big question, which is how does nature do it? 272 00:27:48,110 --> 00:27:49,460 And it's really not answered yet. 273 00:27:50,240 --> 00:28:00,740 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. 274 00:28:02,210 --> 00:28:06,590 And the idea is as follows. 275 00:28:06,590 --> 00:28:13,380 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. 276 00:28:14,330 --> 00:28:19,730 The starting point is this differential rotation that I showed you before. 277 00:28:19,760 --> 00:28:27,260 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. 278 00:28:27,270 --> 00:28:33,470 This is called the omega effect because omega is rotation and usually there's some shear in most astrophysical objects. 279 00:28:35,300 --> 00:28:40,850 This isn't a dynamo. As I mentioned, something is needed to make it a dynamo. 280 00:28:40,910 --> 00:28:47,389 And the second step in the standard model of Dynamos took a long time to figure out. 281 00:28:47,390 --> 00:28:52,810 It turns out there was a period of time where people. We're very good at proving things like anti dynamo theorems. 282 00:28:53,650 --> 00:28:56,770 There there's a dark ages, if you will, for dynamo theory. And. 283 00:28:57,300 --> 00:29:01,720 And so a tip of one of the most famous is known as Cowlings Theorem. 284 00:29:01,720 --> 00:29:07,260 And he proved that with magnetic fields and fluid motions are symmetric, no dynamo exists. 285 00:29:07,270 --> 00:29:14,950 And the answer to this dilemma wasn't really discovered until the mid fifties. 286 00:29:15,070 --> 00:29:19,990 And the first person to really look at this was Eugene Parker. 287 00:29:20,200 --> 00:29:30,969 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, 288 00:29:30,970 --> 00:29:35,590 if you will, in a rotating medium that can twist those field lines in the right way. 289 00:29:35,590 --> 00:29:43,030 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? 290 00:29:43,450 --> 00:29:48,429 What happens is that magnetic field can be twisted as it moves upward. 291 00:29:48,430 --> 00:29:57,880 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. 292 00:29:58,420 --> 00:30:04,810 This takes the magnetic field and it creates a small component that reinforces the initial seed field. 293 00:30:04,840 --> 00:30:14,229 Now, this gets very complex because suddenly we're going to try and describe a turbulent system in which the eddies are all 294 00:30:14,230 --> 00:30:20,799 acting in just the right way to take a little bit of the east west field and convert it back into the little field. 295 00:30:20,800 --> 00:30:28,040 And that becomes a problem of turbulence. And all of the issues that go with turbulence are come into this. 296 00:30:29,530 --> 00:30:34,540 And and I want to show you a couple of things about this theory of turbulence, 297 00:30:34,540 --> 00:30:38,440 which are important, because it turns out we can measure some of these things later. 298 00:30:39,550 --> 00:30:46,360 So the first thing is, he said the action of this is equivalent to some current being generated in the east west direction 299 00:30:46,690 --> 00:30:52,899 due to the pre-existing east west field and some parameter that characterises the turbulence. 300 00:30:52,900 --> 00:30:58,840 I call this the alpha effect and people went on to develop a theory for how 301 00:30:58,840 --> 00:31:04,959 fluctuations in magnetic fields and velocity fields could act to drive currents. 302 00:31:04,960 --> 00:31:08,080 So this is what's known as a mean field theory for driving current. 303 00:31:08,530 --> 00:31:13,989 And the idea is we have correlations between the velocity fluctuations and the magnetic 304 00:31:13,990 --> 00:31:18,400 field fluctuations that on average can drive a current in some other direction. 305 00:31:19,900 --> 00:31:28,030 And the theory is the simplest version of these theories, and it gets very complex quickly, is that this EMF, this turbulent EMF, 306 00:31:28,030 --> 00:31:36,160 which is like the viscosity in a river that transports momentum around on average or something along those lines. 307 00:31:36,490 --> 00:31:40,360 The idea was there's this term that Parker suggested was important, 308 00:31:40,360 --> 00:31:48,099 but there's another term which is effectively an anomalous resistivity which enters into the into the system. 309 00:31:48,100 --> 00:31:54,430 And this alpha effect, if you will, always comes with an enhancement to the resistivity. 310 00:31:54,730 --> 00:32:02,120 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. 311 00:32:02,140 --> 00:32:11,650 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. 312 00:32:12,130 --> 00:32:15,520 Okay. But nonetheless, this is what we think is important. 313 00:32:16,690 --> 00:32:24,549 So now let me talk about how that theory folds into astrophysical dynamos and what what 314 00:32:24,550 --> 00:32:29,680 characterises the astrophysical dynamo is that we might want to that we might want to study. 315 00:32:29,680 --> 00:32:41,530 So planetary dynamos are almost certainly so dynamo is the magnetic Reynolds number is relatively small the fluid Reynolds number is very big. 316 00:32:42,190 --> 00:32:47,409 That's a property of liquid metals. This means that in order to make a magnetic field grow, 317 00:32:47,410 --> 00:32:52,299 you're almost always dealing with very turbulent convective flows in something like the core of the earth, 318 00:32:52,300 --> 00:32:58,900 we can simulate the magnetic field generation. This is a classic example of a simulation of the Earth's magnetic field. 319 00:32:58,900 --> 00:33:04,090 You see the strong east west field deep inside the north south field that pokes out and so forth. 320 00:33:05,170 --> 00:33:11,080 Oh, this is from differential rotation, as I mentioned, the stretching, if you will. 321 00:33:14,770 --> 00:33:21,610 If you look over time and this is an example of a movie that was reconstructed from measurements of the Earth's magnetic fields, 322 00:33:21,610 --> 00:33:25,270 from captain's logbooks over the last 400 years or so, 323 00:33:25,540 --> 00:33:31,239 you can actually work out the time history of the variation of the of the Earth's magnetic field. 324 00:33:31,240 --> 00:33:36,250 And so you see in this movie of the Earth's magnetic field that it's turbulent. 325 00:33:36,280 --> 00:33:43,990 There are small scale fluctuations that exist on top of the large scale field that we observe on the surface of the earth. 326 00:33:44,290 --> 00:33:49,360 So small scale turbulence is also present. It's probably not a small scale dynamo. 327 00:33:49,440 --> 00:33:53,040 But just the large scale field being advocated about. But it's not clear. 328 00:33:54,630 --> 00:33:59,660 The sun, our stars, dynamo is clearly magnetic. 329 00:33:59,670 --> 00:34:05,070 There are strong magnetic fields. We know it has very interesting time dynamics. 330 00:34:06,000 --> 00:34:09,030 Does it have a small scale dynamo or a large scale dynamo? 331 00:34:09,990 --> 00:34:15,360 Is it fast or slow? Well, the magnetic Reynolds number is enormous. 332 00:34:15,990 --> 00:34:19,410 So it's probably almost certainly has to be something like a fast dynamo. 333 00:34:20,010 --> 00:34:26,940 If you look at the magnetic field on the surface of the sun and this is really well observed now by satellites, 334 00:34:26,940 --> 00:34:33,600 this is the strength of the magnetic field sticking out. You get the sunspots show up as blue sticking out and yellow sticking in. 335 00:34:33,900 --> 00:34:38,040 And this is a movie for how the sun's magnetic field evolves with time. 336 00:34:38,970 --> 00:34:49,890 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. 337 00:34:49,920 --> 00:34:55,950 So this is the early eighties. You can see the equator is moving one speed, the poles are moving another speed. 338 00:34:55,950 --> 00:34:59,489 The magnetic field goes away. Every 11 years it comes back. 339 00:34:59,490 --> 00:35:07,290 The sunspots during the active cycle are present. The polarity changes, so the yellow leads blue for 11 years, then it flips. 340 00:35:07,620 --> 00:35:19,979 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. 341 00:35:19,980 --> 00:35:23,760 It sticks out at the North Pole and in at the South Pole, but it's very, very weak. 342 00:35:23,760 --> 00:35:30,930 Only ten gauss or so this is two orders of magnitude weaker than the strong fluctuations of the magnetic field, 343 00:35:30,930 --> 00:35:32,760 which are observed on the surface of the sun. 344 00:35:32,970 --> 00:35:40,650 The sunspots themselves are probably representative of a strong field from deep inside, but in any case, is there a small scale magnetic field? 345 00:35:40,660 --> 00:35:42,719 We think that there is a small scale dynamo. 346 00:35:42,720 --> 00:35:49,890 Also looking in detail at the granular convection that takes place, there's a much stronger magnetic field, 347 00:35:49,890 --> 00:35:56,340 which is always present even during those periods of solar minimum when the dipole is present. 348 00:35:56,790 --> 00:36:01,470 So we're stuck in something like the sun dealing simultaneously with a fast, 349 00:36:02,340 --> 00:36:08,440 a fast, large scale dynamo and a small scale dynamo simultaneously acting okay. 350 00:36:09,120 --> 00:36:14,909 And this field is usually much bigger. All the building blocks are there. 351 00:36:14,910 --> 00:36:21,540 There's differential rotation, it's measured. We have something like magnetic buoyancy, which could give us something like the alpha effect. 352 00:36:21,540 --> 00:36:25,860 But once you combine this with the small scale dynamo and the magnetic turbulence, 353 00:36:26,220 --> 00:36:31,230 so far it's not been possible to reconcile these two these two two themes. 354 00:36:33,720 --> 00:36:41,220 How about the galaxy? The galaxy has a really physically big magnetic Reynolds number enormous, if you will. 355 00:36:41,790 --> 00:36:48,569 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 356 00:36:48,570 --> 00:36:54,090 can measure the strength of those fields and even our own galaxy has fields. 357 00:36:54,090 --> 00:36:57,989 It's really remarkable to me to some extent that we know more about the magnetic 358 00:36:57,990 --> 00:37:02,370 field structure of galaxies far away than we do even about our own galaxy. 359 00:37:02,670 --> 00:37:05,100 And we can actually measure the details, 360 00:37:05,100 --> 00:37:11,190 the three dimensional details of the of the of these galactic magnetic fields better than things like the core of the earth, 361 00:37:11,190 --> 00:37:21,540 which is right right here. In any case, an alpha omega dynamo of the sort I showed you is consistent with the galactic dynamo. 362 00:37:21,540 --> 00:37:26,370 So the idea is you have some quadrupole or magnetic field, there's differential rotation in the disk. 363 00:37:26,700 --> 00:37:33,059 This creates a strong toroidal field. And then the idea is you have something like supernova driven turbulence that 364 00:37:33,060 --> 00:37:40,410 gives you the alpha effect or the helical twist that I mention now in the galaxy. 365 00:37:40,410 --> 00:37:42,899 Is it is there a small scale dynamo acting? 366 00:37:42,900 --> 00:37:48,930 Well, it turns out you can measure the total field from something like synchrotron emission and compare it to the order field, 367 00:37:48,930 --> 00:37:57,780 which is measured through Faraday rotation. The average field is at least a factor of four or more larger than the average field. 368 00:37:57,780 --> 00:38:05,880 And so almost certainly in something like the galaxy, there's a small scale dynamo also acting, okay, so how are we? 369 00:38:05,920 --> 00:38:13,020 What are we going to do with all this? Okay, I think I'm a skip this and move on to experiments. 370 00:38:17,040 --> 00:38:22,740 Sorry. Really bad. 371 00:38:26,520 --> 00:38:30,330 Okay. So with that interlude experiments, 372 00:38:30,900 --> 00:38:38,700 Fermi said in MHC one shouldn't believe the product of a long and complicated piece of mathematics if unsupported by observation. 373 00:38:38,700 --> 00:38:45,779 So let's see what we can do in the lab. So Dynamo experiments require a big magnetic Reynolds number. 374 00:38:45,780 --> 00:38:49,380 They need to be flow dominated. As I mentioned, this is a new research. 375 00:38:49,590 --> 00:38:55,890 For plasma experiments. Usually plasma experiments are magnetically dominated, especially when you have a highly conducting plasma. 376 00:38:55,920 --> 00:39:01,140 We've also learned that the Fluid Reynolds number is critical for governing the behaviour of these systems. 377 00:39:03,000 --> 00:39:12,540 Plasmas are hard. They're difficult to stir. Some confinement is needed and so until now, people have been using liquid metal experiments. 378 00:39:12,720 --> 00:39:19,140 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. 379 00:39:20,040 --> 00:39:22,920 There's a very unfortunate power scaling for these experiments. 380 00:39:23,910 --> 00:39:29,879 The magnet, the power required to reach a certain magnetic Reynolds number goes is the cube of the magnetic Reynolds number. 381 00:39:29,880 --> 00:39:33,690 And so building these experiments costs money. 382 00:39:34,620 --> 00:39:39,390 You need a certain size lab. It gets more dangerous as you get bigger. 383 00:39:39,810 --> 00:39:48,389 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, 384 00:39:48,390 --> 00:39:52,860 which is just barely above the critical magnetic Reynolds number for observing a dynamo. 385 00:39:53,190 --> 00:39:57,930 This takes about 100 kilowatts of power, and this is just barely above threshold. 386 00:39:58,710 --> 00:40:04,680 And this is because these systems are very turbulent. Now, let me show you why that turbulence is so important. 387 00:40:06,060 --> 00:40:11,129 This is our experiment at Wisconsin. It's a liquid metal experiment. It's one metre diameter spherical vessel. 388 00:40:11,130 --> 00:40:16,380 We have two 100 horsepower motors that drive flows of about ten metres per second. 389 00:40:17,070 --> 00:40:22,020 And this would be fine. This is the geometry we these two propellers do just what I showed you earlier. 390 00:40:22,920 --> 00:40:34,139 The problem is. This is what the flow actually looks like on the scale that the magnetic field is trying to be generated. 391 00:40:34,140 --> 00:40:42,660 This dipolar smooth dipole that I showed you. The turbulence in the fluid makes the velocity field appear extremely rough, 392 00:40:43,440 --> 00:40:52,049 and this turbulent velocity field x to transport to take the magnetic field which 393 00:40:52,050 --> 00:40:56,220 is trying to build up and move it around because the magnetic field is frozen 394 00:40:56,220 --> 00:41:00,390 into the moving fluid it transports it just like turbulence mixes anything else 395 00:41:00,390 --> 00:41:04,830 it would turbulent mix the magnetic field which is being created a simulation. 396 00:41:05,640 --> 00:41:09,240 This is a simulation at a magnetic REYNOLDS Number of a very fluid. 397 00:41:09,240 --> 00:41:14,160 Reynolds Number of about 2000, just barely beyond, you know, just barely enough to get turbulence. 398 00:41:14,160 --> 00:41:19,020 It's much less than the ten to the seven that the experiments happen at. 399 00:41:22,500 --> 00:41:29,129 So what we discovered in our experiments is that this turbulent EMF plays an absolutely 400 00:41:29,130 --> 00:41:33,060 essential role in these experiments for governing the magnetic field evolution. 401 00:41:33,060 --> 00:41:41,970 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. 402 00:41:42,300 --> 00:41:48,840 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. 403 00:41:49,320 --> 00:41:53,250 We measured for the first time in the lab a turbulent EMF. 404 00:41:53,250 --> 00:41:56,580 On average, the fluctuations created an EMF. 405 00:41:57,810 --> 00:42:02,730 And what we then also discovered was that the alpha effect that we had expected wasn't present. 406 00:42:03,270 --> 00:42:08,340 It was all an anomalous resistivity. That is when we tried to create current in one direction. 407 00:42:08,700 --> 00:42:16,200 The fluctuations would act to oppose that current, and that's in essence diffusing the magnetic field away more rapidly. 408 00:42:17,280 --> 00:42:23,670 This is so strong that the effective resistivity, it increases as you go up in speed. 409 00:42:23,880 --> 00:42:28,650 And it also quenched any dynamo action that we were hoping to observe. 410 00:42:28,650 --> 00:42:34,560 And this is because our flow was so open and unconstrained that the turbulence dominated 411 00:42:35,580 --> 00:42:43,409 and so nonetheless self excited dynamos are observed in liquid metal experiments. 412 00:42:43,410 --> 00:42:52,080 So they've about 15 years ago Rega used a set of pipes to constrain flows in a cylindrical geometry to make a dynamo. 413 00:42:52,500 --> 00:42:58,020 The Karlsruhe experiment packed about 30 of these into another tank to do the same thing. 414 00:42:58,470 --> 00:43:04,320 And then most recently, the Von Karman experiment in Catterall, France did this. 415 00:43:04,620 --> 00:43:07,739 Each of these experiments, to some extent, I would say, used to cheat. 416 00:43:07,740 --> 00:43:10,140 Now I use that term in the most generous way. 417 00:43:10,440 --> 00:43:17,639 They are either very heavily constraining the flow to get rid of the turbulence or in the cataract experiment you're using, 418 00:43:17,640 --> 00:43:21,120 ferromagnetic iron impeller is to help amplify the flow. 419 00:43:21,420 --> 00:43:25,950 And so none of these experiments are the simple experiment I described to you where I melted away, 420 00:43:25,950 --> 00:43:31,020 the insulators, got rid of the ferromagnetic and allowed the flow to be homogeneous. 421 00:43:31,170 --> 00:43:35,790 Okay, so this is what liquid metal experiments have done for us. 422 00:43:36,360 --> 00:43:39,659 We've done, we've learned a lot, but now we're ready to move on. 423 00:43:39,660 --> 00:43:43,260 And I think the next step is to move to plasma experiments. 424 00:43:44,010 --> 00:43:47,969 And so we're doing this and that plasma does offer a bunch of things. 425 00:43:47,970 --> 00:43:54,270 They offer the possibility of very high magnetic Reynolds numbers, the ability to independently vary the fluid. 426 00:43:54,270 --> 00:43:59,550 Reynolds Number and magnetic Reynolds Number, which allows us to go from laminar to turbulent, which I think is very important. 427 00:44:00,060 --> 00:44:09,000 And then a host of effects that are beyond MHC that might be interesting that distinguish plasma dynamos from more simple MHC dynamos. 428 00:44:09,450 --> 00:44:18,330 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. 429 00:44:19,710 --> 00:44:27,330 The fluid Reynolds number depends upon the plasma density, the temperature of the plasma, the flow speed and so forth. 430 00:44:27,660 --> 00:44:37,560 The magnetic Reynolds number also depends upon the temperature. These things we can compute them for typical laboratory plasma. 431 00:44:37,680 --> 00:44:43,409 And what we find is that it's relatively straightforward to get interesting fluid numbers and magnetic Reynolds 432 00:44:43,410 --> 00:44:52,830 numbers and and I'll show you that we create plasmas that have a temperature of about 20 V densities here. 433 00:44:52,830 --> 00:44:57,479 It's a very diffuse plasma, and we can make them flow very fast. This gives us magnetic. 434 00:44:57,480 --> 00:45:01,170 Reynolds numbers and fluid. REYNOLDS Numbers which are big. 435 00:45:01,180 --> 00:45:06,899 So this requires two things that requires a dense plasma to make the discuss of the viscosity low 436 00:45:06,900 --> 00:45:12,300 enough that the fluid Reynolds number is big and it requires a hot a high electron temperature. 437 00:45:12,450 --> 00:45:18,540 And the way we do this is in the following device. So this is this is our plasma dynamo experiment. 438 00:45:19,290 --> 00:45:25,860 It's a three metre diameter spherical vacuum vessel that confines plasma. 439 00:45:26,130 --> 00:45:29,400 Using an array of permanent magnets. I'll show you in a second. 440 00:45:30,690 --> 00:45:34,500 Also shown, here are some coils for applying fields to test it. 441 00:45:34,530 --> 00:45:41,730 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. 442 00:45:44,220 --> 00:45:51,150 If you look through actually if you look through this hole here at the opposite side inside, 443 00:45:51,450 --> 00:45:55,439 this is what you would see when there's no plasma present. 444 00:45:55,440 --> 00:46:06,239 And this is with a plasma helium plasma. So these are rings of very strong permanent magnets which have been bolted to the surface 445 00:46:06,240 --> 00:46:11,730 of the inside of the sphere to create a magnetic field which is localised to the edge. 446 00:46:11,850 --> 00:46:22,310 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. 447 00:46:22,320 --> 00:46:26,520 And so in the centre of this plasma is a perfectly on magnetised volume. 448 00:46:26,880 --> 00:46:33,850 And we use this very high order multiple to create a magnetic field that holds the plasma on the surface. 449 00:46:33,870 --> 00:46:37,170 It's a bucket. It's a magnetic bucket for holding plasma, if you will. 450 00:46:39,150 --> 00:46:50,620 And we use cathodes that emit electrons biased relative to anodes to inject current into the plasma that then ionise and heats it. 451 00:46:50,640 --> 00:46:56,400 And so you can see these glowing cathodes are hot cathodes that are injecting power into the plasma. 452 00:46:57,090 --> 00:47:06,120 These permanent magnets have the effect of only allowing the plasma to sneak out along the field lines which intersect the permanent magnets. 453 00:47:06,120 --> 00:47:11,370 And so the plasma can't go across this part. It can only be lost in a little tiny region. 454 00:47:11,370 --> 00:47:15,419 And you can see that black stripe down the middle of the magnet ring. That's where the plasma is lost. 455 00:47:15,420 --> 00:47:25,890 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. 456 00:47:27,630 --> 00:47:29,730 This is what the magnets look like inside. 457 00:47:31,080 --> 00:47:40,310 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. 458 00:47:40,320 --> 00:47:45,360 The magnets have a very strong force between them. About a £200 pole force. 459 00:47:45,630 --> 00:47:47,610 I'll illustrate that with a movie in a second. 460 00:47:47,850 --> 00:47:54,750 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. 461 00:47:55,020 --> 00:47:58,020 And we were able to push the magnet down before bolting it in. 462 00:47:58,440 --> 00:48:02,640 So this was a fun construction technique. This is what happens. 463 00:48:02,880 --> 00:48:07,860 Magnets don't play nicely if if you lose control. 464 00:48:08,550 --> 00:48:12,720 So these are two magnets coming together. They destroy each other. 465 00:48:12,720 --> 00:48:13,950 This is a very strong pull force. 466 00:48:13,950 --> 00:48:21,810 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. 467 00:48:24,500 --> 00:48:28,250 So we inject power in. We put a couple of hundred kilowatts of power in. 468 00:48:28,610 --> 00:48:33,290 This is that's the green trace here. These plasmas can last for 20 seconds or so. 469 00:48:33,950 --> 00:48:40,399 The plasma heats up. This is 15 EV, which I think is about 300,000 degrees centigrade. 470 00:48:40,400 --> 00:48:47,720 So it's quite hot. The densities are high enough for our purposes, it turns out, and they're all almost nearly fully ionised. 471 00:48:48,350 --> 00:48:53,660 At least half of the particles are ionised. There are still some neutrals present in helium, which is hard to ionise. 472 00:48:55,820 --> 00:49:00,140 Okay, so now I'm going to show you how we stir plasma. 473 00:49:02,210 --> 00:49:06,920 And I told you, this is the magnetic field. 474 00:49:07,620 --> 00:49:11,000 It's there's no magnetic field in the centre. There's a magnetic field at the edge. 475 00:49:11,390 --> 00:49:17,690 These are the cathodes. The red. The red dots are the electron emitting cathodes. 476 00:49:18,110 --> 00:49:23,899 And the grey dots are on anodes, which will be collecting the electrons. 477 00:49:23,900 --> 00:49:26,990 And and so they're emitting current, if you will. 478 00:49:28,640 --> 00:49:31,340 Our idea for stirring plasma is the following. 479 00:49:31,640 --> 00:49:40,790 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. 480 00:49:40,800 --> 00:49:45,890 So here are the magnet. Here are the magnetic field lines going from the South Pole to the North Pole. 481 00:49:46,550 --> 00:49:49,910 Here's the cathode, which is collecting currents. 482 00:49:49,940 --> 00:49:53,240 The red arrow is the current because it's emitting electrons. 483 00:49:53,720 --> 00:49:55,459 And here's the magnetic field. 484 00:49:55,460 --> 00:50:07,670 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. 485 00:50:07,700 --> 00:50:10,460 It's a torque on the plasma, which makes it spin. 486 00:50:12,440 --> 00:50:22,670 And so the cathodes, when pulled back into the Magnetised Edge region, provide a talk to control the flow in the Magnetised Edge. 487 00:50:24,890 --> 00:50:36,200 And now I can control the rotation as a function of latitude by adjusting the potential 488 00:50:36,590 --> 00:50:40,730 between each of these probes to drive as much current or as little current as I want. 489 00:50:41,030 --> 00:50:47,330 And by controlling where I do it so it becomes possible to drive flow in one direction in one hemisphere, 490 00:50:47,720 --> 00:50:50,570 and the opposite direction in the other hemisphere. 491 00:50:51,480 --> 00:50:58,730 And so this is consistent with the picture I showed you earlier where I have a simply connected plasma domain, 492 00:50:59,030 --> 00:51:05,090 where I can control the rotation as a function of latitude on the surface of the plasma volume. 493 00:51:05,540 --> 00:51:10,490 And this is exactly the control we need to chase slow dynamos at a minimum. 494 00:51:10,970 --> 00:51:16,610 Okay. So now I'm going to show you that this stirs plasma, 495 00:51:16,620 --> 00:51:22,950 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. 496 00:51:23,580 --> 00:51:30,380 And then we stick a probe in to measure the flow along a cord which goes there. 497 00:51:30,390 --> 00:51:33,390 I'll show you that in a second. So we we can build probes. 498 00:51:33,660 --> 00:51:39,390 We call them mock probes. They're a variant on the Langmuir probe measures flow to two different phases, 499 00:51:39,660 --> 00:51:45,060 and by measuring the difference in current to the two different phases, we can infer what the velocity is. 500 00:51:45,360 --> 00:51:54,450 This is easy, relatively straightforward to do, and we take this probe and we scan and measure the velocity along this cord. 501 00:51:55,650 --> 00:52:01,410 And what we measured we can plotted here is a function of radius from the centre of the machine. 502 00:52:01,830 --> 00:52:09,360 This is the measured East-West rotation of the plasma. 503 00:52:09,990 --> 00:52:14,610 Here is where our cathode is. The magnetic field is gone by about this spot. 504 00:52:15,000 --> 00:52:22,860 So here is our very fast up here. This is our very fast ten kilometres per second flow speeds are measured here where 505 00:52:22,860 --> 00:52:29,729 we're stirring and it viciously penetrates in to the on magnetised centre of the plasma. 506 00:52:29,730 --> 00:52:36,720 And this is this is how that falls off. This is consistent with viscosity, transporting the momentum inward. 507 00:52:37,830 --> 00:52:43,110 At the same time, I'm spinning this, I'm making it spin, toroidal or East-West. 508 00:52:43,440 --> 00:52:51,809 And so there's a centrifugal force that actually throws the plasma out in this region generating this polygonal circulation. 509 00:52:51,810 --> 00:52:59,940 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. 510 00:52:59,940 --> 00:53:06,000 And this this is a very good thing because this is the effect that we wanted to control the rotation with. 511 00:53:06,780 --> 00:53:15,330 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. 512 00:53:15,750 --> 00:53:21,840 Look, in this case, it was a four sided mock probe which measured the two components. 513 00:53:23,430 --> 00:53:26,520 Okay. So this is a big step for us. 514 00:53:28,650 --> 00:53:36,510 We are now in the process of optimising the flow and putting in magnetic sensors to look for magnetic fields. 515 00:53:36,510 --> 00:53:40,110 So we haven't yet put a magnetic probe in the experiment that could even observe this. 516 00:53:40,110 --> 00:53:44,130 We're just starting this and we're also adding the control. 517 00:53:44,910 --> 00:53:52,500 This is an example where we used eight cathodes to spin one hemisphere, one way, the opposite hemisphere, the other direction. 518 00:53:52,830 --> 00:53:56,790 And and these are the global fluid, Reynolds numbers. 519 00:53:56,790 --> 00:54:02,880 In this case, the fluid Reynolds number was about 600 and the magnetic Reynolds number was about 400. 520 00:54:03,270 --> 00:54:11,220 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. 521 00:54:12,720 --> 00:54:17,460 So let me show you what we're thinking will happen and this will sort of finish things up. 522 00:54:20,460 --> 00:54:24,000 This is the flow we would like. This is the one I showed you at the beginning. 523 00:54:24,120 --> 00:54:29,550 We control the rotation as a function of latitude and this is the resulting palatal flow. 524 00:54:29,880 --> 00:54:37,980 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. 525 00:54:38,310 --> 00:54:43,170 And we'll do that with at least 1212 cathodes that are gone in, go, go, going in. 526 00:54:43,500 --> 00:54:48,030 At the same time, we need to measure the flow so we know what it is that we're what we're creating. 527 00:54:48,780 --> 00:54:52,920 And we're already we're already making the plasma parameters, 528 00:54:53,160 --> 00:54:59,160 the turning of plasmas and the densities which are required to get the viscosity and the resistivity is just right. 529 00:54:59,430 --> 00:55:03,120 So this is underway and I would stay tuned. 530 00:55:04,320 --> 00:55:07,440 Okay. And that's and that will create this magnetic field. 531 00:55:07,440 --> 00:55:12,690 That's where that's a slow dynamo and that's the baseline that we've been working towards. 532 00:55:14,670 --> 00:55:18,150 But we think we also have more that can be done. 533 00:55:19,860 --> 00:55:22,530 And I'd like to leave you with the following simulation. 534 00:55:22,530 --> 00:55:30,599 So this is now a time some are a time dependent calculation in which I'm going to vary the fluid Reynolds number 535 00:55:30,600 --> 00:55:37,140 and the magnetic Reynolds numbers within what we think are the types of parameters that the experiment can access. 536 00:55:37,770 --> 00:55:43,770 And so the fluid Reynolds number is we'll start out at 100 with a magnetic Reynolds number of 200. 537 00:55:44,970 --> 00:55:49,380 This corresponds to three kilometres per second, a 10th of plasma of this density and so forth. 538 00:55:49,920 --> 00:55:51,629 I can increase the fluid Reynolds number. 539 00:55:51,630 --> 00:55:57,570 The flow will change a little bit and then I if I go just a little bit further, the flow actually becomes turbulent. 540 00:55:57,720 --> 00:56:02,400 Okay? There's a there's a critical hydrodynamic instability which sets in. 541 00:56:02,430 --> 00:56:05,790 The flow becomes turbulent and I get chaotic flows. 542 00:56:06,000 --> 00:56:12,480 Okay. If at the same time I increase the electron temperature so that it becomes very. 543 00:56:12,820 --> 00:56:18,910 Conducting I can make the magnetic Reynolds number big and we think we could probably get up 544 00:56:18,910 --> 00:56:24,550 to temperatures as high as 40 V It's about twice what we're doing now by adding more power. 545 00:56:24,790 --> 00:56:28,480 And this gives the possibility of a very big magnetic Reynolds number. 546 00:56:28,750 --> 00:56:37,690 And this is what happens in the simulations in that case. So here is the velocity here and we're stirring on the outside. 547 00:56:37,900 --> 00:56:44,010 Here's the polite flow. As I increase the fluid Reynolds number, it starts to fluctuate a little bit. 548 00:56:44,020 --> 00:56:48,370 A hydrodynamic instability is setting in and then eventually it just becomes turbulent. 549 00:56:48,370 --> 00:56:52,960 And this is what turbulence looks like driving it with the flow drive that I'm using. 550 00:56:53,050 --> 00:56:57,250 Okay, now I increase the electron temperature also. 551 00:56:57,250 --> 00:57:05,260 So it's a very highly conducting plasma. And what happens then is we start to see magnetic field energy develop. 552 00:57:05,650 --> 00:57:15,160 And this simulation shows that we have a small scale fast dynamo occurring and now we're starting to look at Dynamo. 553 00:57:15,160 --> 00:57:18,250 Is that happen at very large magnetic Reynolds numbers? 554 00:57:18,580 --> 00:57:22,719 What we might hope is that we could use this object to then look for large 555 00:57:22,720 --> 00:57:27,129 scale magnetic fields also being generated and attack the fast dynamo problem, 556 00:57:27,130 --> 00:57:33,760 which is what is most important in astrophysics. So that's my talk. 557 00:57:34,660 --> 00:57:40,630 In summary, fast, large scale dynamos are the most important ones in nature. 558 00:57:41,110 --> 00:57:43,390 This is still a theoretical challenge. 559 00:57:44,380 --> 00:57:53,560 Liquid metal experiments see self excite under some conditions, but they also allow us to measure things which we're thought to be important. 560 00:57:53,560 --> 00:57:54,850 Like the turbulent EMF. 561 00:57:54,850 --> 00:58:04,180 I've shown you that very turbulent liquid metal experiments are much more resistive on average then than they are if they're laminar. 562 00:58:05,020 --> 00:58:11,170 And I've also tried to show you that plasma dynamos are now underway. 563 00:58:12,610 --> 00:58:16,330 They have the possibility of operating at much larger magnetic Reynolds number. 564 00:58:16,600 --> 00:58:21,309 We can vary the fluid. Reynolds number, which is the ratio between the magnetic Reynolds number and the fluid. 565 00:58:21,310 --> 00:58:26,320 Reynolds Number and we're in the process of optimising the flow that requires good 566 00:58:26,320 --> 00:58:30,820 measurements and many control tools to make the flow just right for a dynamo, 567 00:58:31,090 --> 00:58:39,790 but that's what we're doing. So with that, let me acknowledge all these people and thank you for your attention.