 Hello everybody and welcome to video number 11 of the free online version of the Fusion Research Lecture. My name is Alf and we are in chapter 2 magnetic field configuration. And you might remember from last video where we talked about what is needed to build a tokamak and in this video, well in terms of the magnetic field I should probably add. And in this video we will talk about the Stellarator. So today's video is about the Stellarator and you remember probably or hopefully also from one of the very first videos that a Toroidal device for magnetic confinement needs a twisted magnetic field. So Toroidal devices need a twisted magnetic field. They require twisted magnetic field lines. Otherwise we will have no confinement and the plasma is lost immediately more or less. And a quantity to measure that was the rotational transform yota. So the rotational transform yota that was the quantity we introduced to describe to quantify the rotation of the twisted magnetic field lines. There are basically three methods to get the yota to get such a twist. There are three methods to get yota which was nicely summarized in an old paper by Mercier in 1964. The first method is a Toroidal current. The Toroidal current as in the tokamak where the Toroidal current IP then resides in a poloidal magnetic field which can be calculated as mu not the plasma current over 2 pi and then if we are at the outboard side 2 pi a the minor radius. In addition to the poloidal magnetic field you just added the Toroidal magnetic field from the strong field coils and together these make the twisted magnetic field line. Now method number two is a helical rotation so like in a screw for example a helical rotation of the magnetic axis of a device of the magnetic axis which corresponds basically to winding the torus. So you can say this is something like winding the torus around the magnetic axis as we will see and then a third method this is the rotation of the poloidal cross section along the toroidal coordinate along the toroidal coordinate and this is for example when you wind the coils this is achieved winding the coils. So three methods a Toroidal plasma current which is basically the tokamak and then two methods applying to the stellar configuration one is the helical rotation of the magnetic axis and the other one is the rotation of the poloidal cross section along the toroidal coordinate which is also achieved when you look at method two but here we talk then about winding the coils in method three. So let's first just look at method two so this is method two which is for obvious reasons called figure which an example for that is sorry the figure 8 stellerator and I say for obvious reasons because have a look at the picture on the right hand side where we have the top view of the stellerator this is a design this is from a paper from Lyman Spitzer from 1958 the father of the stellerator for example from this top view start here and then going around here this is a straight section and then here we have a curved section then here we have another straight section and then a curved section and on the right hand side this is a view from the site and you remember in one of the early videos lecture videos I showed you the photography of such a stellerator and that could fit on a table so these were like table top experiments. Now how do these experiments achieve a twisted magnetic field line now to understand that let's have a look at the magnetic field line so let's draw a cross section so this is supposed to be a polar cross section and let's duplicate that by some magic and let's do a bit more magic and duplicate that as well so easy to do that on a computer harder to do it on a classical blackboard trust me okay and now we have what they're supposed to be on the same high side maybe more like this yeah good okay so the first one on the left hand side this corresponds to cross section k so this is cross section k see the letters on the picture on the top right hand side and if we start mark the center at each of the cross sections and then at k we go to cross section L and we have a straight line so L this would be cross section L and this corresponds to one of the straight components or sections and then from L we go to cross section M and as you can see on the drawing this corresponds to an curved section so if you go down here to cross section M A to M this corresponds to a curved section a curved section then going from M to N you have another straight section so this is N and then finally going from N back to K we have another curved section something like this okay now how is the twist achieved so let's have a look at a magnetic field line starting for example here so this is just a continuation of this line so this is our first position now if we follow that line to the next cross section it will still be here since it is a straight cross section but if we would follow this further along then we would see that here we have the angle alpha so this corresponds to an angle which we call alpha and if we now follow this along further along and then since this is the curved cross section so basically going from L to M so if we would be here on the inner side with the field line the field line would be still on the inner side here meaning that it will end up somewhere like roughly here I guess so this will be number three and here we have still the angle alpha but if we would follow further along here we would also have the angle alpha meaning that in total we would have the angle to alpha now following the line further along the field line we end up here number four and then if we continue to measure the angle here this would correspond to the angle to alpha like on the cross section M and if we would follow the field line further along you would have to add another angle alpha so we would have three alpha and if we now going along the next curved section the intersection with of the magnetic field line in that polo cross section would probably be roughly here I guess this would be number five and then if we measure the angle this one would be three alpha and following it further along here we would have to add another angle alpha meaning that in total this would be four alpha so we would have an angle of a twist of four alpha after a full circumference meaning Yota would correspond to four alpha or if we write it in Yota bar which is just Yota over two pi we have four alpha over two pi and in writing it in terms of the safety factor this would then mean Qs just to be complete here is two pi over four alpha and the point is that with a twisted magnetic field line here we get a formation of flux surfaces so such a stellarator such a design results in the formation of closed flux surfaces with this simple design however it is a bit let's say rather large if you want to have large plasma cross sections and you need a lot of coils and these are hard to line properly so this design concept was abandoned relatively early now another method to achieve a helical rotation sorry to achieve a twisted magnetic field line by a helical rotation is this so-called heliac so this is also for method two which was the helical rotation of the magnetic axis so the heliac which is short for helical axis stellarator and in that stellarator we have a central conductor we have a central conductor or well or like a current loop on the picture on the right hand side taken from a paper from elin boozer from 1998 you can see the central conductor indicated by the green color then we have two royal field coils then we have two royal field coils like in a classical stellarator you know you need them two royal field coils and those follow the helical path they follow a helical path the two royal field coils are indicated by the blue color in the drawing and you can see how they sort of rotate around the central conductor following a helical path and then the resulting flux surfaces you can indicate it let me try to draw that indicated here these are flux surfaces you can see how they're basically rotate around the central conductor and an example for this experiment is the TJ2 stellarator being operated in and built in Madrid whoops the TJ2 stellarator Europe's second largest stellarator okay let's now talk about method 3 so let's talk about method 3 which are to use helical coils now you might remember that in a classical stellarator the classical stellarator oops stellarator we have the current in the helical coils flowing in alternating directions so in a classical stellarator in there we have helical coils with the current flowing in alternating directions right meaning that if we would look at the drawing on the bottom there are two coils indicated here so here the there's one coil with the current flowing more into the bottom direction and in the other one here we have another coil more flowing into the top direction and what we've drawn here is a linear or what the figure shows is a linear stellarator that equals one to one period linear stellarator and this or these coils they create a helical magnetic field component they create a helical magnetic field component bh and we use a small h for that and if you look at the drawing at the sketch you can see here we have the helical magnetic field component from one of the coils and here from the other coil and then in addition we have the strong toroid magnetic field from the toroid field coils capital BZ not so indicated in the drawing at the top we have here the strong toroid magnetic field by the toroid magnetic field coils ok now how to estimate the poloidal displacement after one period because in the end we want to estimate we want to get an estimation for Yoda for the twist so let's try to get an estimation on the poloidal displacement after one period so estimate the poloidal displacement after one period and to do that we will introduce the so-called field line equation first in 2d the field line equation just actually something very simple so if we draw the following coordinate system we have here y and then here x then we have a magnetic field line like this this is a magnetic field line and then going down here this is the component dx and here we have dy and the since we have a linear magnetic field here assumed the slope is given simply by dy over dx being equal to the magnetic field component into y direction and into x direction so this is by not be five sorry by and then in 3d you can write equivalently and then doing a small rearrangement but which would then read dx over bx is equal to dy over by is equal to dz over bz so this is based on the same drawing but just in 3d which I dare not to draw here for the better of this video I guess now this can be transformed or written in cylinder coordinates so in cylinder coordinates the field line equation basically reads R and then d theta over b theta being equal to dz over over bz this means we can now give or have an expression for the poloidal displacement after one period and that corresponds to R d theta over dz so this is the poloidal displacement after one period so v theta over vz okay now how to further quantify that what to do with that equation again we have the same situation as previously the figure so meaning here is equals one a linear stellarator this is the current and the coils flowing in this direction and in the other anti-parallel to that then we have a residing magnetic field line like this usually in blue going down like this okay now if we start again with the equation which we just had which was our d theta over dz which is b theta over bz and then then b theta can be written capital b theta as the sum of the left of the contribution from the left coil and from the right coil meaning this corresponds then to minus b theta so small b theta over bz not plus bz so this is from the left coil or from the left part of the drawing so this one here this part where the bz component the small bz points into the same direction as the capital bz but b theta has a minus sign there because it points upwards and then from the right part of the coil so from this one from the right coil we get plus b theta over bz not minus bz because this points now in the in a different direction and then this can be brought in one nominate one denominator such that we have two b let me write that a bit more below and also indicate just here this was from the right coil such that we can write two b theta bz over bz not squared minus bz squared and then looking closely at the angle alpha which is defined alpha c here you can see it is the angle between small b theta and small b helical and if we make use of that definition and make in addition use of the trigonometric expression that two sine alpha times cosine alpha is the same as sine of two alpha then we can approximately write bh squared over bz not squared times sine of two times alpha c so alpha c roughly corresponds to the truth of the coils and why have you done this or what can we learn from that let's say so first of all looking at the equation we see one important point and that is that the twist of the magnetic field line follows the coil twist so the twist hardly readable the twist follows the coil twist because the very theta is positive right and very importantly there is only a moderate contribution per period to the twist there is only a moderate contribution a moderate contribution per period and obviously it is a maximum for alpha c equal to pi half which would be a bit hard to realize because then it means we have basically coils which not really wind around um um helically right there's only a moderate contribution per period because the term in front of the sign is small so we have the behelical squared over bz squared and you know that the two royal magnetic field component is the dominant magnetic field component so we have only a moderate contribution per period to the twist and that requires uh implies that we need a large number of periods to have a large yota so we need a large number of periods for a large yota bar and now you might wonder why do we want to have a large yota bar well for performance reason if you remember when we talked about the scaling laws which was a scaling law for the energy confinement time the energy confinement time was dependent on yota bar and larger yota bar meant we have or means that we have larger energy confinement time because as we will see in a few videos having a larger yota increases the stability of our device okay so having a large yota helps to get a better performance in general and this means stellarators need to have large or usually large aspect ratios large aspect ratio which unfortunately also makes them a bit expensive right now nowadays we do not have these this strict separation between the type of stellarators in an optimized stellarator optimized stellarator like for example w seven x we always have a helical magnetic axis and also coils like the classical stellarator so we have a combination of these two effects plus coils plus coils like classical stellarator okay so that's it in this video we talked about the stellarator and what is needed in the stellarator to achieve a rotational transform you might remember just as a reminder a rotational transform so a twisted magnetic field is mandatory to get a confined plasma at all there are in general three methods to get a yota so to get a twisted magnetic field line to royal current this is the tokamak and then two ways to get it in a stellarator a helical rotation of the magnetic axis and or a rotation of the pole loyal cross section along the toroidal coordinate i explained or tried to explain the figure eight stellarator and how the twist is achieved there then the heliac where tj2 was an example as shown as an example for that and then we talked about the classical stellarator and how to estimate the pole loyal displacement after one period and we might have learned that there is only a moderate only a small contribution of twist per period which means to have a large yota bar which we need for stability reasons we need a large number of periods and that means we need a large aspect ratio and in an optimized stellarator we always have a helical axis plus coils one around the torus so a combination of the two methods okay that's it hope to see you in the next video