 Hello everybody and welcome to video number 18 of the free online version of the future research lecture as You might remember in the last video. We talked about or started to talk about pressure driven instabilities So we are still in chapter 3 parameter limits for fusion plasmas and in today's video We will dive deeper into that topic and to be more precise the topic of today's video is our ballooning instabilities Ballooning Instabilities Now what are ballooning instabilities? So Let's now say that we allow Poloidally asymmetric modes. This is something new which we haven't allowed before so we allow now Poloidally Asymmetric modes Meaning instead of having XI Being a function well with the mode numbers MN being just a function of R Psy now Psy now I really have to practice this writing. I guess as this letter MN is a function of R and theta and The residing modes or the corresponding modes are called ballooning modes so it's ballooning modes and On the left hand side, you can see an example for this. So here we have Geometry, this is a polar cross section for the magnetic field gradient to the left and Here we have modes which have larger amplitude on the low Magnetic field side on the outboard side So this is as you remember from last time the bed curvature region. So this is the bed Curvature region Region and on the high field side. It's the good curvature region Good curvature region where we have Modes with reduced amplitude and so these are ballooning modes Now how to handle them To describe them or their the residing stability limits, we need to introduce a few quantities. So to get a proper description of the Stability limit we need to introduce two quantities and the first one is the normalized pressure gradient So the normalized pressure gradient and this is alpha Defined as minus and then to mu knot R not Over b phi squared times qs squared times d P over dr. So the pressure gradient So the left term so so this This expression here basically Includes the machine Parameters, so we have the radius and the magnetic field and the safety factor and this one basically is the destabilizing pressure gradient In addition to the normalized pressure gradient We need another quantity expression that you already know which is the shear the shear was defined as s equal to R over Qs and then D Qs Over d r. So these are the two Important or the two Quantities we need and now if we apply the energy principle Oops so Applying the Energy Principle yields Oops, sorry yields s Alpha diagrams yields as alpha diagrams which characterize Correction rising for each flux surface stability against Those ballooning modes so to say that again The energy principle gives us an s-alpha diagram or s-alpha diagrams Which characterizes or which allows us to characterize for each flux surface the stability against the ballooning modes Now let's have a look at that first to start with a circular plasma Let's first start with a circular plasma and When I say circular plasma, I of course refer to the polar cross-section being circular So this is the diagram which you see on the right-hand side you see s So the normal as pressure as sorry the shear then as a function of the pressure gradient and You see two or you see basically two type of regions the non shaded area This is a region of stability, which is usually referred to as first stable Region then we have a region of instability which here is color-coded I'm not sure if this is some kind of magenta. I guess. I'm sorry. I'm slightly colorblind, but I guess it's magenta then we have the here below the second stable Region and as I said in between is a region of instability so the first stable region Which has oh Which is characterized by small alpha small alpha means also small pressure gradient So it's not surprising that we have a stable region there because the destabilizing forces are small them so the first stable region is limited by a linear function by s equal to 1.67 7 alpha so if you have a look on the Drawing the diagram. So here we can roughly draw a linear function. This corresponds to the Parameter where to the function which I've written there on the left-hand side Now there's also a second stable region the second stable region and that has Instead large pressure gradient So large normalized pressure gradient large alpha, but small shear but small shear Now what does that mean? So let's Briefly discuss that having a large pressure gradient means we have a large shavranoff shift Shavranoff shift data Having a large shavranoff shift now Means that we have steep field lines on the low-field side remember that discussion from a few videos ago. So this means we have steep field line steep magnetic field lines on the low-field side this means that the so to say the length in the good Curvature region is longer or higher the length in good curvature region is higher or longer of the magnetic field line and This means that there's an overall increase Increased influence increase influence of the good curvature region of Good curvature region Thus we're getting an overall second region of stability there Okay, that's it for a circular plasma for plasma having a circular polo cross-section now How can we change that we can change that or extend that? Regions of stability by plasma shaping so if we perform some plasma shaping which basically means having a non Circular poloidal cross-section This has positive effects has positive Effects As you can see on the right-hand side So now we can see that there is a dark purple area. This is the region of instability And this is much smaller for a non-circular cross-section plasma than for the circle and remember all this region here Which we had previously has been previously Instable or unstable for a circular plasma and now is stable. So we have increased now our parameter space where we can operate our plasma and what happens is basically that we reduce All the what what we do basically by plasma shaping is that we reduce the field line length in the bed curvature region and the bed Curvature region For example by triangular cross-sections by bean shaped cross-section. So basically it's elongated along the vertical direction With the overall goal to reduce the field line length in the bed curvature region So to get a higher influence of the good curvature region thus increasing our accessible parameter space now This leads us to the stability beta limit the stability beta limit the Maximum achievable beta as in each experiment is subject of active research as you can imagine because beta higher beta means a more efficient plasma device a more efficient fusion device for this reason the maximum achievable beta in each Experiment is a subject of active research so To remember just as a reminder the beta beta maximum beta is defined or was defined by the Pressure the normalized plasma pressure as sorry the average plasma pressure So yeah, since we want to have the maximized beta is also the maximized plasma pressure then over the magnetic field pressure Then we use the normalized parameters we introduced a few slides ago so we use the normalized parameters Introduce a few slides ago for sorry for comparison Or for comparing Between experiments that means we express the averaged pressure in terms of S and Alpha How does that look like so let's write this down so we have we start with p the normalized Sorry the average pressure plasma pressure being two Over a squared with a factor of two is due to the fact that we have electrons and ions then a squared in the denominator is the Normalization then we have the integral from Plasma boundary to the plasma center from zero to a then PR R times dr and what we then perform is integration by parts integration By parts Making use of the fact that the plasma pressure at the boundary is Defined as zero And instead of writing every step down adjust right now the result on of the integration by parts Whereas I'm only missing here one step, so it's easy to do that on your own so that I'm hiding something here Mine is one over a squared zero to a then p prime so the radiate derivative of p R squared dr and Now we finally insert the normalized values then we get in front of the integral we have b phi squared Over to you not our A squared Are not a squared Then the integral From zero to a and it's alpha r squared Over qs squared dr Okay, this is just another expression not too useful, so let's we have to do one more step and oops, I Wanted to insert a page here So We have to do one more step and this is to insert the Maximum pressure gradient in a circular cross section plasma against ballooning instability, which we just did use from our s alpha diagram So what we now do is to use the knowledge about the maximum pressure gradient in Circular to come back and again by circular to come back. I mean a plasma having a circular pull loyal cross section Maximum pressure gradient in a circular to come back, which is stable Against ballooning instability that was alpha being equal to s over 1.67 or To third times s approximately inserting that yields for the maximum averaged pressure Approximately b phi squared over three times mu not are not a squared Then the integral from zero to a Then s r squared Over qs squared dr and this now allows us to write for the Maximum beta value an expression that reads approximately two times Over three a squared are not squared The integral from zero to a Then qs prime so the derivative with respect to the radius Qs to the power of three dr Now we have an expression for the Maximum beta value against ballooning instabilities and If you have a close look you realize something important that writing it in this way the Maximum beta value depends only on Qs and this is why we also Why this factor is also called a set safety factor one of the reasons Because this defines basically or sets if we have a stable plasma if you get this higher than yeah Well, it it tells us the maximum beta value we can achieve Writing it in this way Now, okay, this is still just a formula now. What does this actually mean for actual values of the beta? So let's see what we can do trying to maximize beta So we are now trying to get as large beta as possible. Oops trying to Maximize beta and we do this with the Radial profiles as given in this plot where we have the plasma pressure the safety factor and the corresponding Plasma current necessary To get the maximum beta and what we said is a boundary condition is to the we set the safety factor as a plasma Boundary to be three which is then stable against external king instability plus a safety margin Then we can calculate KQ KQ late the resulting integral Which we had in the previous slide which gives us a beta maximum of approximately 5.6 times IP the plasma current Over a the minor plasma radius and B5 Yes, we now have a Formula to actually calculate the beta maximum the units here are such that IP The plasma current is given in mega ampere The magnetic field is given in Tesla The minor plasma radius is given in meter and beta is given in percentage Okay, now these are ideal profiles with these step functions. It's of course unrealistic Now what happens if we go from? ideal Profiles to more realistic profiles Profiles then We get a beta Maximum of 2.8 Then again times to the plasma current Over the minor radius and the magnetic field B5 this supposed to be five So this is an important limitation This is the oops. Sorry the so-called toy on limit which we have written down here This is the toy on limit Important enough to give it a yellow box the toy on limit and This is for circular for a circular polo cross section for a circular polo cross-section plasma and This value 2.8 which we have here in front This is also abbreviated with beta n and What is often used for comparing the beta values between different experiments is actually this value so the Normalized beta the normalized beta something which we use often which is often used for comparison and Beta n is as you can see they're defined by the actual achieved the beta maximum in an experiment over I P over a B Phi at the plasma center oops So this is the normalized beta Now let's have a look at the actually achieved values here You can see a diagram of the beta as a function of this normalized parameter So I over the minor radius and the magnetic field strength and you can see the achieved values you can see there is a Linear function along which the values Order so to say so here this one is Linear function where beta max is given as beta n times I over a B Phi and the Beta oops, sorry the beta n value which we have here so you can see it's 3.5 meaning It is higher Than the value for which we got for circular cross-section plasmas Circular cross sections Again, it shows us that it pays off to have plasmas with non-circular cross-section Let's have a look again at the the plot you can see there are Discharges from a few experiments where we actually exceed the this 3.5 for example here or here or here These are Strongly shaped plasmas where we have achieved very high beta values kind of record values You also see that here at the bottom so different colored regions correspond basically to different experiments And in the beginning the beta was very low but then it started to increase and as I said Shaping the plasma pays off because you get an overall higher beta This is 3.5 instead of the 2.8, which was a trial limit There is however Another limitation for beta and this is something we should definitely mention here apart from the ballooning instability. There is often a Practical limitation hitting earlier a practical Limitation limitation which you have to be aware of and you can tackle it But you have to be aware of we can deal with it and this is due to the NTM occurrence Remember Neoclassical tiering modes Something you would like to avoid at a large scale experiment because it can lead to disruptions and we talked about NTM's and how to deal with them Just as a small remark here There's often a practical limitation in terms of the maximum achievable beta Due to the occurrence the onset of NTM's at very high plasma pressures Now to finish this video Small reminder why beta is important. So before we come to the plot If you will do introducing remarks a reminder so the fusion power fusion power increases With plasma pressure, so it's good to have a high plasma pressure Then the magnetic field times the volume is what sets the price of a fusion device so this is what calls money and For this region we need to Maximize beta for a commercial fusion power plant For a commercial Fusion plant And this is indicated in the plot on the left-hand side which shows the coast of Electricity of a fusion power plant as a function of the beta and as you can see the higher the beta is the Cheaper the electricity is the lower the costs are and this is a nice illustration of why pays off to have As high beta as possible Okay, that's it for today's video where we talked about ballooning instabilities which are poloidally asymmetric modes and to apply the energy Analysis we introduced the normalized pressure gradient and the shear then we got these as alpha diagrams telling us stable regions for each flux surface and From that we derived the stability beta limit. We came up with a Trojan limit for a circular cross-section plasma However, the all modern experiments achieved higher limits in terms of the normalized beta due to plasma shaping and Here on the last slide. We had a reminder on why It is worth to get the beta as high as possible because it increases the cost of electricity That's it hope to see you the next video you