 Hello everybody and welcome to video number seven of the free online version of the future research lecture You might remember that in the last video we introduced the major Properties the main geometry of the magnetic feed configuration the rotational transform and In this video, you're going to talk about flux surfaces something which we already mentioned in the last video. So flux surfaces The topic of this video So before talking about flux surfaces We have to be sure that we all are on the same boat when we talk about magnetic field lines. So what is the magnetic field line? magnetic field line Now a magnetic field line is a curve that is everywhere tangential to the direction of the magnetic field. So that is a curve That is tangential to Well everywhere to the direction of the magnetic field Of the magnetic field Okay Now we also or you might also remember that the magnetic twisted the twisted magnetic field line is something is essential for system where we aim on Confining a plasma meaning that if the field lines go around toroidally, they are shifted poloid by a certain angle so if our field line goes around toroidally the field line Going around Toroidally then Going around toroidally, then it is shifted poloidally by a certain angle and this angle we called Yota and more often Yota bar is used, which is just Yota over 2 pi In addition to Yota we talked about Q we introduced Q which is The inverse of Yota bar and it is 2 pi over Yota and if Yota is a rational It means that the field line closes itself. So if Yota is rational the field line closes after M Sorry, usually it's in this case. It's first N after N toroidal turns And M poloidal turns Meaning that Q then equal to N over M is a rational number and if we draw an example This could look for example This is a poloidal cross section and if we start in this poloidal cross section by following a field line here Which goes around the torus and then after one toroidal Circumference that ends up here after a second toroidal circumference that ends up here When then after a third toroidal circumference it exactly ends up where it started three toroidal circumference as it go We have a Q number equals to N equals 3 and equals 1 thus Q equals 3 Would be here in this example our safety factor now if Q on the other hand is a rational This means that the field line will trace out an entire surface. So it means that the field line will trace out an Entire Surface if we follow it around the torus Meaning there is a surface Which is covered ergodically by field line meaning there's a surface covered ergodically by one field line and Ergodically here means basically that if a line or that it means is that one field line will pass Arbitrarily close to any point on a given surface if we just follow it long enough This means if we again go to our poloidal cross section example if we start at the same position here Then go around toroidal once then we might end up here then toroidal around another Turn another toroidal circumference another toroidal circumference another one Sorry another one another one another one and so on and so on and eventually all our Intersection points would fill out the entire circle here. That's the entire surface and this Surface filled out by the field line is called flux surface This is basically the flux surface Sometimes it's also called magnetic surface You might read that also once in a while magnetic surface okay, so Let's give you an example. So here I have plotted one field line So this is all the one magnetic field line which goes around the torus it goes around the torus It goes around the torus and so on and so on Meaning if you would follow one field line Let's look at this for example. It goes around the torus Goes around the torus and then it's hidden underneath this plot Then it might end up somewhere here and so on it goes around further toroid Lee Spending up an entire surface and of this surface I have drawn this surface in blue, but only half of the surface for visibilia reasons the reasons So here I have drawn one flux surface as an example Okay You might remember as the next step from your plasma physics one lecture the equilibrium condition the equilibrium condition was introduced when you talked about or you learned about the single fluid model a single fluid model and Within the single fluid model you might remember an equation of motion was derived equation of motion was derived and Then assuming stationary conditions Meaning that the temporal derivative vanishes Corresponding to equilibrium condition to an equilibrium condition then actually the so-called equilibrium condition was derived and it says that the pressure gradient so the plasma pressure gradient is balanced by J cross B the diamagnetic current times the magnetic field Now the equilibrium condition has a few implications implications one Since we have a cross product over there one application is that the magnetic field is perpendicular to the pressure gradient then B the same is true for the current This is also perpendicular to the pressure gradient and If we look at a then we can also say that B times the pressure gradient times the pressure gradient is zero or Or that the pressure gradient parallel to the magnetic field is zero and that means that the magnetic field lines B magnetic field lines a the magnetic field lies on surfaces of Constant plasma Pressure and These surfaces are the flux surfaces So on one flux surface we have the constant plasma pressure Okay from the beginning of the flux surface I have given in the sorry from the definition of the flux surface I had given in the beginning it is clear that flux surfaces are never crossed by magnetic field lines, so flux surfaces are never crossed By field lines This also means that on one flux surface. We have a fixed value of the magnetic flux psi. So there is a fixed Value fixed value of the magnetic flux psi on the flux surface and It is does not surprising that psi is used as a label for Oops, sorry as a label for flux surfaces now, let's again draw our polar cross section to Illustrate a few things about the magnetic flux. So here we have our typical cross section Extending in this direction then We have magnetic field lines pointing into this direction here since we have a purely toroid example here just for illustration purposes So this is B And now the magnetic flux is basically a measure on how many magnetic field lines are going through this particular surface Through this particular surface. So these are sticking here through the surface Meaning that if we bring that into a more formal way, we see we say that psi is the surface integral On the surface S. So this is over the surface S. This is the surface S then B dot product DS Where DS is the surface normal Meaning it is a vector perpendicular to the surface. This is a surface normal And this basically implies that if there are more field lines sticking through that surface the magnetic flux is higher and We can rearrange that using Stokes theorem Into a different form Saying that The line integral around a curve around the boundary of the surface is the same as the expression above So the line integral and then using the vector potential a which I will define in a minute times the tangential unit vector DL. So this is the the tangential unit vector of a curve Surrounding our surface S Meaning that in our example it could be for example simply Curves surrounding our surface S could look in the simplest case like this Then DL would be here the tangential unit vector But the curve could also look like this It would also surround the surface having some other shape and then the tangential unit vector would maybe look like this now a what is a a is the magnetic Vector potential something which you might remember from your electrodynamics lecture magnetic vector potential A and a is defined such that the rotation of a Gives us the or defines the magnetic field so a is a very Useful quantity as we will see and I just write vector potential Into the box here because often you just write a C vector potential although it's probably better to call it magnetic vector potential and Since the rotation of a gives you B. This also defines that the divergence of B is equal to 0 Okay, this is the magnetic flux psi. Let's look an example We just set that on one flux surface We have Psi equal to Psi as a certain number a constant number and that should not to be zero Now if we look at the example at the example which I have drawn there We have three flux surfaces there. So first of all the innermost flux surface here. This is oops, this is Psi one then use another one. This is psi two then use another one This is Psi three and you might remember The magnetic flux size and measure for how many field lines basically are enclosed inside of that flux surface And this implies of course that Psi three is larger than Psi two larger than Psi one so the magnetic flux increases for Outer If we go further outwards increases for outer flux surfaces You should probably write Psi increases for outer flux surfaces and since They are not cross-time magnetic field lines You can also write that the gradient of Psi In a function of r then the dot product with B And this is equal to zero Important equation Okay, now how to define Psi an actual example if we assume a twisted magnetic field, which is our standard our default case if we have a twisted magnetic field and If we separate the poloidal and the toroid magnetic field direction, then there are basically two ways To define Psi in a such a twisted magnetic field system Now let's try to explain so Let's again draw our typical cross-section like this and Then another flux surface may be like this And you're on the other side like this and then maybe like this More or less good. Okay Then let's draw these lines connecting these two to get half a half torus half of a torus Also the inner flux of faces of course goes around toroidally like this Then the coordinate system is as usual that Z is pointing upwards and As I said, there are now two ways to define Psi Let's first just look at the Toroil magnetic field component Now if you just look at the Toroil magnetic field component, then the flux through this surface Gives us the first way to define Magnetic flux and this is the Toroil magnetic flux the Toroil flux Psi Psi for indicating in the Toroil direction and this is then using or taking the Toroil magnetic field direction component the Toroil magnetic field component times the surface normal ds Psi and this would point if we Here perpendicular. This is supposed to be perpendicular to the surface Which I have drawn there. So this would be ds Psi Now, you know, however, that there is also the at all. Sorry a polar magnetic field component So we can also draw some surface which Encloses the polar magnetic field vectors sticking through it. So this would be the surface and This would be a surface of Z equal constant direction. This is here, of course the r direction, right? Oops in the r direction and that surface would lie at a z equal constant value and That surface defines the pole loyal flux. So that is the pole loyal flux Psi theta That is given by the pole loyal magnetic field component the pole loyal magnetic field line sticking through it times the surface normal ds theta Which if we look at one point here for example, it's it's pointing upwards since this is a surface normal ds theta And of course, yeah, yes, I have not written it explicitly here But you remember that if we go around in this direction, this is the angle phi Okay, those are the two methods to define the magnetic flux Psi in a twisted magnetic field system The total magnetic flux and the polar magnetic flux Now why do we talk so much about flux surfaces because flux surfaces are crucial for the confinement. So flux surfaces are crucial for the confinement now First of all, you remember I said The first part of the lecture one of the first videos that we need the energy from the alpha particles in the fusion process To further heat up the plasma This however takes some time. So in a typical fusion device In a typical reactor type device in the alpha particles They need to make Ten to the five toroid returns before they have delivered the energy to the plasma so the alpha particles from the fusion process They take ten to the five turns to Deliver their energy to the plasma to deliver the energy to the plasma So we need a good Magnetic confinement. We need good flux surfaces such that these alpha particles are confined and stay on the flux surfaces So they can deliver the energy to the plasma and The flux surfaces Then also important due to the fact that we have a very general a very fast movement of charged particles and our plasma only consists of charged particles or mostly So we have a fast parallel movement Whenever you read parallel and magnetic confinement Configuration context always refers to the almost always refers to the direction of the magnetic field So we have a fast parallel movement of particles meaning that the temperature as Basically is basically a function of psi and the same is true for the density being also a function of psi and As we just said two slides ago This can be in most cases translated to a radial coordinate since the magnetic flux psi increases if you go further outwards or radially outwards Okay, that's it for this video. We talked about flux surfaces We talked about how they are defined So they are basically defined by a field line going around and around and around a torus to royally Filling out an entire surface thus or let's say tracing out an entire surface We said that on one flux surface The magnetic flux is constant and thus psi the magnetic flux psi is also used as a label for the flux surface And there are two ways to define the magnetic flux in a twisted magnetic field system the toroidal and the polo magnetic flux We also introduced the magnetic vector potential which we will make use of in the next video and Finally I said on this slide that flux surfaces are crucial for our confinement And that the shape of the flux surfaces basically gives us an indication on the shape of our temperature and density profile Okay, that's it hope to see you on the next video