 Hello everybody and welcome to video number 13 of the free online version of the fusion research lecture My name is Alp and this is a YouTube channel called Der Plasma You might remember in the last video we were still talking about magnetic field configurations in particular about this Delerator and In this video we will start to talk about what limits in the performance infusion plasmas meaning we will start with chapter 3 so today's topic is Chapter 3 and this deals with parameter Limits for Fusion plasmas So you might remember from video number. I think it was video number five that One way to describe the goal and fusion research is the following formula is to express it in the triple product Where we have the density times the temperature Times the energy confinement time and that this triple product should be larger than a certain number to achieve a burning plasma That was 0.5 mega joule second over meter cubed That was for a self Sustained so for a burning fusion plasma plasma a different way to write that inequality or to express it is by writing it as follows b squared over four mu naught Times beta Times the energy confinement time that this should be larger than this number Why writing it this way because here you can basically first Distinguish between what? Well, you can kind of separate the the factors in what? Which parameters need to be increased the first parameter here is the magnetic field and this is kind of fixed This is fixed due to technological constraints because we cannot get the magnetic field arbitrary large Then you might remember from the discussion with the energy confinement time that this is something we are working on trying to increase that and In addition here the parameter beta and by the way Hope you remember that but in case you don't remember. Let's quickly write that again beta was the Kinetic pressure so the plasma pressure Over the magnetic field pressure so over b squared over to you not And beta is actually also limited so beta is limited by on one hand side the so-called equilibrium limit equilibrium limit We will talk about that equilibrium beta limit in this video and then there is also the stability beta limit a stability limit something we will discuss in the next videos and The stability or generous stability limits can be further separated into a current Driven whoops, I think current is usually written with a T current Into current and pressure driven instabilities current and pressure driven Instabilities Now of these two limits of the equilibrium beta limit and the stability beta limit the letter one is usually the most severe limit the stronger limit Okay, to start with this topic I have a nice quote from Edward Teller which describes the problem. I think very well Edward Teller might be known to you as the father of the hydrogen bomb, but he was also a very prominent and active part of the peaceful use of fusion energy especially in the beginning and in 1954 when everything was still classified he made a conference the statement about magnetic confinement fusion This is like trying to confine plasma with magnetic field lines It's like trying to hold a blob of jelly with rubber bands and I think that describes the problem really well So let's now start to talk about the equilibrium beta limit the equilibrium Beta limit So in a simple magnetized torus that is a torus with no pole oil feed But a purely toroid field if we would have a simple Magnetized torus As I said meaning no pole oil field, but a purely toroid field or to write a chart having no twist in the magnetic field line No twist Then as you might remember from the one of the very first videos The diamagnetic current the divergence of the diamagnetic current is not zero Which means there is basically no confinement. So we had immediate loss of the plasma So we need twisted magnetic field lines I'm just to tell you that simple magnetized torus are experiments which are still Operated usually at universities to investigate losses and transport and also confinement properties But now let's have a look at the twisted magnetic field situation Thus to do this. Let's first draw again one of our beloved cross sections So let's try first to draw a circular cross section Bung Circular good Okay, now let's try to duplicate that maybe something like this and Now let's Connect these two So we are looking at the cross section which extends into this direction This is supposed to be some kind of plasma cross section Maybe for maybe it's easier to Understand this if we Get rid of this part here Make it maybe a bit clearer Okay, now in this cross section we first have the magnetic field pointing into Into the cylinder This is of course a curved cylinder. So well, let's say this group hinge to start with So twist the magnetic field line magnetic field pointing into it then We also have a magnetic field gradient So well, we have we now assume a toroid experiment as well We have a magnetic field gradient and that magnetic field gradient Points into this direction and Having a magnetic field gradient you remember that this means the diamagnetic Current is different on both sides of the plasma So we have on the outer board side where we have a lower magnetic field We have a larger diamagnetic current going for example like this This was explained in one of the first videos To indicate that this is a larger current. I try to draw a thick Arrow here and on the other side on the high field side on the inboard side We have a lower diamagnetic current because magnetic field is higher Trying to indicate this by a thinner arrow So and this green line is the diamagnetic current likewise this one of course and having now this Current being larger on the output side as one of the input side results in charge separation You might remember this plus minus and we would also have the same such separation in each polar cross section But now we have however as you remember hopefully twisted magnetic field line Connecting these areas With these areas here at the bottom and also if you were to draw them on the other side not visible here like this So having magnetic field lines blue magnetic field line going around the Taurus and Along this along these magnetic field lines charges can move freely Thus the charge accumulation cancels out so we have the fish Schlüter current remember This was called fish Schlüter current which cancels out the charge separation and here flowing In the other direction the inboard side Jps for fish Schlüter and this is what you for sure remember, right that we having a helical Magnetic field results in a additional current called the fish Schlüter current and the divergence now of the sum of the diamagnetic current and the fish Schlüter current is Again zero Now this fish Schlüter current however is of course also accompanied by magnetic fields It also creates the magnetic field and it creates a vertical magnetic field. So this creates the Vertical magnetic field now, let's try to visualize that so looking again at the pool oil cross section Oops, nope. This was not what I wanted to do looking a second trial looking again at the pool oil cross section So supposed to be concentric circuits more or less So now, please assume With a bit of fantasy Imagination sorry with a bit of imagination that these have all the same center And now this is supposed to be the pool oil magnetic field component only so we're looking only at the pool oil magnetic field component which goes like this right it goes around it and If we look at the cross section on the left hand side, we have the fish Schlüter current We have it on the left hand side flowing into the board. So here the fish Schlüter current would be like this Whereas on the other side the fish Schlüter current Sorry here points into the board sorry on the other end it points out of the board For Schlüter current if you have a look on the drawing on the left hand side and this first Schlüter current creates magnetic field Surrounding it and if you use your right hand you can get the direction into which the magnetic field points and Now the superposition of the fish Schlüter current on the low-field side and the high-field side Results in a magnetic field which points approximately upwards like this So this is the magnetic field created by the fish Schlüter currents on the low and on the high-field side oops by the Like this and this is a magnetic field. This is a vertical magnetic field due to the fish Schlüter current yes, so Due to the fish Schlüter current now the Interaction of this magnetic field with a background magnetic field in the plasma Results in a shift in a modification of the shape of the plasma because now The magnetic axis is shifted to the left in this example meaning it would look like this Flux surfaces would look like this for example So it would result in a shift So here This would be the shift of The flux surfaces of the center of the flux surface and this shift is called data Okay, so a vertigo magnetic field results in a shift and this shift is called the Shafranoff shift data Since it is so important Give it oops a yellow box and Now this Shafranoff shift increases with beta now the Shafranoff shift increases with beta because Well beta is proportional to the plasma pressure And the plasma pressure higher plasma pressure means having a higher diamagnetic current and Having a higher diamagnetic current means we have a higher we require a higher fish Schlüter current And this means the Shafranoff shift gets larger okay Now an interesting or Good an interesting question is How to get the final shape of the flux surfaces because apparently this fish Schlüter current is something which appears in every Toroidal magnetic confinement experiment where we have a twist and that is a very good question and I will only Outline the Quantitative description of how to describe this here Roughly and not to derive it in detail So I will how to give a quantitative Description and I will only outline this Now what you need to do first what we do of course is to make life simple we assume We have a system which has not a troll Royal but Tor Royal symmetry Symmetry in cylinder coordinates in cylinder Coordinates and the symmetry is with respect to the Torah direction with respect to Phi so like a talker mug and Then we can Calculate or get the magnetic field components from a few Assumptions and quantities so first of all we look at the poor loyal flux When we talked about the poor loyal flux in video 7 and video 10 and this can be Described by I was given by the following formula to pi and then the integration from 0 to capital R Then we have our prime B Z Being a function of our prime and to set in the integration with respect to our prime and this gives us the Z component Remember we use the vector potential to calculate these quantities then the Condition that the divergence of the magnetic field is zero This gives us the radial magnetic field component and then The equation that the rotation of the magnetic field is equal to mu not J this gives us the J Z component and This in turn allows us to calculate B Phi Meaning here we have the magnetic field components that we need and Now what one has to do is one has to insert Those components into the Radial component of The equilibrium condition of the equilibrium condition which read The gradient of the plasma pressure as to be balanced by J cross B and We make use of the following Quantity sorry the following equation that the radial derivative of the plasma pressure being a function of Psi Po-loyde, there's a function of R is equal to the derivative with respect to Psi of The pressure times the derivative with respect to R of Psi And what we have done here you have just Well made use of the following expansion. You have just written D the derivative of the pressure with respect to Psi as the derivative of the pressure this Sorry, there of course should be an R As you might have guessed already Um So that this derivative is the same as the derivative of the pressure with respect to Psi and then here we have Psi with respect to the Radius and this is just the exponent that we have used here And this is typically or often written as P prime so the derivative with respect to Psi is often abbreviated with P prime then times the radial derivative of Psi and into this equation we have to insert J and B the J and B components and if we are doing this We finally get And I just make this arrow to indicate that this is a bit of work but to make it short we get the famous grad Sha run off equation This is the equation which you found if you go to the max plug is a bit of plasma physics they are selling coffee max there and You have the grudge of our equation on it. So Now how does is a grudge of our equation reads let's write it down one at least one time here are and then the derivative with respect to R One over R derivative of R with respect to of Psi plus the second derivative Into the z direction of Psi This is the same as minus you not to pi R squared P prime minus you not squared I Z prime Z And the expression and you might recognize that this when we discussed about the current loop for example a few years ago and The whole expression on the right-hand side. This is what is sometimes also referred to as hoop force and This is what pushes the plasma outwards So this is pushing the plasma outwards Which is the Shafranov shift now? This equation describes the equilibrium Now this equation so this one here describes the Equilibrium of Toroil Xe Symmetric system So basically of a tokamak and What you usually do is you solve this numerically you solve this numerically and then Get functions for Psi So which allows you then to reconstruct your equilibrium In a stellar radar due to the lack of symmetry we do not have such Let's call it simple equation. So in this stellar radar stellar radar due to the lack of Symmetry as I explained in the last video There is no simple differential equation like the one here on the board and Thus we need numerical tools numerical tools to get the equilibrium to get the equilibrium and by the way very famous tool for equilibrium Full equilibrium reconstruction is in the v-mech code which has been around for quite a while and it's Yeah Used for quite a while and this is a very robust and famous code for that now here is shown or Illustrated the shaffer on a shift in a stellarator. So what do you see here? You see poloil cross sections the solid lines are The flux surfaces as you would Expect them from the equilibrium reconstruction and the color-coded contours are The intensity of soft x-ray emission. So the color-coded here is the intensity of Soft x ray Emission which is a function of Density and temperature thus of plasma pressure and with an array of line of sites You get a 2d profile and this was by the way the w7 as Stellarator and you can see two plots on the left-hand side. We have oops. Sorry on the left-hand side We have a plasma beta of 1.35 on the right-hand side of 4.41 And you can see a see how the center of the magnetic axis is shifted to the right corresponding to the shaffer on a shift and you can also see that the Predicted flux surfaces the shape of the flux surfaces corresponds very well with the measurements. So this is a very nice illustration of how the shaffer on a shift is proportional to The plasma beta Okay, if you were more a few more words, sorry about the shaffer on a shift so the Shaffer on a shift As you might have imagined already leads to steep Gradients on the low-feed side Where the flux surfaces are pushed together and this results in locally increased transport Because the gradients are stronger. So locally increased Transport something to be aware of and Now finally the equilibrium beta limit is achieved if the shaffer on a shift is equal to the minor plasma radius So the equilibrium the equilibrium beta Limit Achieved if the shaffer on a shift is equal to the minor plasma radius because then the Shift can no further increase it is the maximum value because otherwise you would have you would get plasma Well, you reconnection of the flux surfaces and huge islands and thus large loss of Particles and energy You go back one slide. So if on the right-hand side is this Magnetic axis would shift further here to the separate checks This would correspond to then to the equilibrium beta limit being reached now One can try to estimate the equilibrium beta limit. So there's an estimation estimation for an axi symmetric Toroidal plasma and It reads that the beta equilibrium limit Is approximately given by the minor plasma radius divided by 2 are not and the safety factor squared oops Which is it could also be written in a slightly different form that we have a minor plasma radius over two times are not times Utah bar squared Or to again write it in a different way one half What times of the inverse aspect ratio times Utah bar squared and this Estimation can be understood from simple considerations so This can be Understood from Simple Considerations Which are if we have a larger value of Yota bar This means we have steeper field lines. So this means we get steeper Field lines and Having steeper field lines meaning a means that the first Schluter current is less to Reutel So it has less influence on the overall shape of the flux surfaces. So steeper field lines means less influence of the First Schluter current on the shape of the flux Surfaces due to being less to Reutel Thus having a reduced vertical magnetic field and the same is true if you think about If you look into the equation of having a larger minor radius and a smaller major radius Then you also have steeper field lines thus less influence of the first Schluter current on the shape of the flux surfaces So this as I said this estimation can be understood from these simple physics considerations now The estimation also tells us that the high beta value can be achieved with either having a low aspect ratio Which is for example realized in a spherical Tokamax if you know about these Spherical Tokamax which are small and compact devices which have a low aspect ratio There's for example nstx upgrade in the US or mass upgrade in the UK Quest in Japan, and there's also a company working on spherical Tokamax Tokamak energy also based in the UK and in addition well or having High values of Yota bar Now this is in general more crucial for Stellarators not crucial for stellarators. So the equilibrium beta condition due to Their high aspect ratio due to the high aspect Ratio remember in video 11 we talked about that that stellarators usually need to have high aspect ratios And this is why it's important in stellarators to Have a high Yota bar. So this is why it is important in stellarators to have a high Yota bar value and Just to finish here with an example if We have a stellarator for example w7x There we have the Yota bar value of approximately one we have inverse aspect ratio of one over ten and This results in an equilibrium Beta limit of approximately 5% so usually beta is given in percent If we think of the Tokamak Of the Tokamak which has a safety factor of approximately two and In aspect ratio an inverse aspect ratio of approximately one over three So this is already a much smaller device That's also a cheaper device just because the aspect ratio is more compact This would result if you calculate that in equilibrium beta limit of four percent but Due to the safety factor Decreasing Towards the center remember the safety factor was a function of the radius in the Tokamak towards center Usually beta is higher so with a Cheaper Tokamak device we can achieve an equal amount of beta here This is just to illustrate that the equilibrium beta condition is more crucial in stellarators Okay, that's it for this video where we talked about parameter limits for fusion plasmas and we started to talk about the equilibrium beta limit Where we needed to introduce the shafranoff shift, which is the simple simply an effect of the fish Schluter currents, which we have in every twisted magnetic field device and We talked briefly about the grad shrafanoff equation and that the equilibrium beta Limit is achieved if the shafranoff shift is the same as the minor plasma radius and Then we talked a bit about how to get a larger beta equilibrium and There are two ways basically for that having either a large Yota bar value or having a large aspect ratio and a small larger, sorry or having a large minor radius and a small large plasma radius thus a Small aspect ratio so a more compact device Okay, that's it hope to see you the next video