 Hello and welcome to video number 33 of the online version of the Fusion Research Lecture. We are in chapter 6 Turbulent Transport and in the last video we talked about the differences between 2D and 3D turbulence. And I told you that or you might remember that magnetized plasma has more features of 2D turbulence due to the magnetic field which imposes a preferential direction of movement onto the plasma which are as you know charged particles. Okay and in this video we will have a closer look at electrostatic turbulence, how transport arises from that and how these turbulence structures are actually created. So let's start with electrostatic turbulence. Since it is electrostatic turbulence the transport here is only due to the resulting E cross B drift. So transport from the electrostatic turbulence is only due to the E cross B drift. Now let's try to illustrate that with two drawings. So first of all let's try to draw a vortice maybe something like this, yes good. This is supposed to be a perturbation in the potential with the peak in the center and then it falls off to the edges and for now we just assume that this is somehow created. Let's talk about later how these are created. Let's assume they are created. So the local perturbation in phi creates of course an electric field. An electric field if we have a center and the maximum of phi then the electric field will look like this. So it will point outwards like this, like this, like this, so radially outwards I guess you get it by now and here also into this direction. Okay and this is supposed to be the electric field generated by the perturbation okay so E twiddle E tilde and I have not said that here. Our background system our coordinate system is such that the magnetic there's a magnetic field pointing into the board and we have a density gradient pointing to the left. Then if we start with a local perturbation in phi that creates E twiddle so perturbed electric field and as you're well aware of perturbed electric field creates the E cross B drift I just mentioned which will here be if you make the cross product E cross B you will see that it points that it goes circular like this so it goes around here like this and here around like this so red here corresponds to the E cross B drift so the perturbed electric field creates a circular E cross B drift and there is no transport yet when we start with this situation so this is just an initial perturbation and phi is being created creating an electric field and a resulting E cross B drift is created and by the way looking at this picture with the perturbation in phi here having a typical size so this would correspond to so if we continue this down here this corresponds to oh this is also called L perp this is the structure size the characteristic structure size perpendicular to the magnetic to the background magnetic field L perp okay let's now make an assumption that this vortex is not moving so now we make an assumption vortex is not moving and how does the situation then it looks like okay so let's instead of drawing everything again we make use of the fact that we are using a computer here so let's copy this paste and let's get rid of the things we don't need and these are this lines here these white lines so okay this is of course be here now if the vortex is not moving we get you remember that there is this density gradient pointing into this direction so we have higher density on the left hand side due to the E cross B drift as it is indicated here if the vortex is not moving more and more density is transported outwards meaning that in this area we will have higher density and in this area lower density and this means if we were to draw a contour of the density along going from the bottom to the top the amplitude of the density maybe we use yellow for that that would then look maybe something like this that we have something like this where we have the maximum density here then the density decreases and it is minimum here and then it decreases again so this would correspond to a line in density going from the bottom to the top due to the vortex not moving but pushing due to the cross B drift density as it is depicted by this line here so if we have a not moving vortex then there is a perturbation in the density is created due to the density gradient being fixed here and this now can lead of course to transport because if the vortex is still not moving the density being transported in the bottom half of this vortex from the left to the right is getting more and more so there is transport going on and what is now the difference between the left and the right case on the right hand case we have an initial density and initial sorry perturbation in phi plus the density perturbation and that is required for getting transport so transport requires an initial density and potential perturbation we need both for that okay and now let's have a look if the vortex is moving what is going on then because as we will see this is again a slightly different case so let's copy this again copy and paste paste it here so density gradient pointing into this direction and then paste it again this then as we said oops density gradient pointing to the left if the vortex is moving now moving to the bottom then it might look like this this is again a line of the density going from the bottom to the top so a cut along the profile and now the assumption is that the vortex is moving for this scenario the assumption is that the vortex is moving and we see that in such a case there is no transport because now the e cross b drift at the bottom of the vortex is creating again pushing a plasma a little bit outwards and in the top a little bit inwards such that here instead of getting a net transport we just get something like a wave moving further downwards of the density perturbation so if we have a moving vortex we have in this in this example no transport and obviously there is another important factor responsible here which we should know it's not only that we need density and potential perturbation but the phase between these two perturbations is important so the phase is important and let's move this a little bit up so we get a little bit more space here whoop this is so useful um so the phase is actually important and we only get transport transport only if or for a phase phi between the density and potential perturbation being different from zero only then we will get a finite transport and this is something important to realize we only so requirement for the transport is to have density and potential perturbation and that the phase between these two perturbations is different from zero okay let's now have a look at the fundamental linear electrostatic instabilities creating such vortices or turbulence in general so let's have a look at fundamental linear oops sorry linear electrostatic instabilities instabilities being responsible for creating such vortices and thus in the end turbulence so there are two instabilities we will look at here one is the interchange instability one is the interchange instability which you might remember this is from non favorable curvature and this is more into the core direction so if you're going more into the core direction of fusion plasmas going from the outside inside it's more into core direction of fusion plasmas and if we were to make a quick drawing of that so let's draw a box maybe like this and then like this like this and this and then we have magnetic field pointing into the board so this is the direction of the magnetic field into the board then pressure plasma pressure gradient pointing to the left then we assume we have an initial perturbation for example a sinusoidal perturbation maybe like this and due to the pressure sorry pressure gradient pointing to the left we have somewhat higher plasma density plasma pressure on the left hand side and we start with such a perturbation and we assume the perturbation to have a k parallel of zero meaning it is only a two-dimensional perturbation it has no variation into the parallel direction where parallel always refers to the direction of the background magnetic field which we hear for simplicity just said to be the toroidal direction then we have the curvature drift curvature loop curvature drift which is proportional to the curvature radius times the magnetic field direction over the charge and the curvature radius and the magnetic field amplitude and if we look again at the drawing on the left hand side we assume the magnetic field gradient pointing to the left and the curvature radius pointing to the right if we look at these quantities the pressure gradient to the left curvature radius to the right then we are here in the situation of the bad curvature so this is a bad curvature case corresponding to the low field side LFS low field side now this curvature drift it points upwards for ions if you look at the proportionalities on the right hand side you see we have a charge at the denominator for the electrons it points downwards and since we have higher pressure on the left hand side more density of the left hand side then we have thus oops just looking for an appropriate color um then we have uh stick with white that means we have more ions in this area here so more ions can move upwards thus we get here a positive space charge charge accumulation here a negative one for the same reasons and here again a positive one meaning that we get an electric field like this so pointing downwards hope you can see that this is an electric field pointing downwards here we get the same electric field pointing upwards let's make a bit thicker that you can see that and this is the perturbed electric field so the curvature drift creates charge separation and creates charge separation and thus also the perturbed electric field as I have indicated in the drawing um with a phase here between density and now it's potential which is pi half shifted which is the electric field the perturbed electric field shifted by pi half so the phase between density and potential perturbation is pi half for the interchange stability and now the important point is that the resulting e cross b drift um enhances the initial perturbation so if you would uh calculate the e cross b drift here then we get here this points here into this direction and here into this direction right e cross b going to the left end to the right so um sorry this is the e cross b drift so this enhances the initial perturbation so the resulting e cross b drift enhances the initial perturbation um yes that is for the interchange instability and how this can lead on the bad curvature side as you know um to instabilities so the curvature drift also depends on the particles energy I have not talked about that here but uh that means so I left that out in the proportionality in the formula um well in this formula here but since it also depends on the particles energy this means that a gradient in the temperature profile can be destabilizing because if you have like a large temperature gradient that this can further enhance this of um amplify this instability these are then the so-called etg electron trap nodes or itgs okay this was the interchange instability let's now look at the second one which we discuss here b and this is the drift wave instability this is the drift wave instability um and this is happening more at the edge region of fusion plasmas more the edge region of fusion plasmas and let's try to indicate that so first of all again we make a drawing so um maybe like this like this this and like this and then what we do now is we copy that shift it to here um now what we do is we start again with an initial perturbation but that perturbation now is supposed to have a k parallel different from zero although k perpendicular is still supposed to be much larger than k parallel meaning this is a very long scale length perturbation along the magnetic field let's go back to the drawing on the left hand side so these are two poloial cross sections somewhere um or parts of the poloial cross section and each of these cross sections the pressure gradient points again to the left pressure gradient points to the left the magnetic field goes um from one to the other cross section part of the cross section so this is supposed to be the direction of the magnetic field then then we assume the initial perturbation for example like this oops sorry this is a should look more sinusoidal well well roughly again uh more plasma on the left hand side due to the pressure gradient as indicated on the bottom here we assume a similar perturbation but since we now have allow for a variation along the magnetic field line meaning having a k parallel not being zero it might look like this um having such a perturbation what then happens next is that the electrons they react very fast along the magnetic field because you know that the electrons are much lighter so they have a higher mobility and they can move basically freely along the magnetic field that means since here we have more plasma so to say the electrons move from this area immediately to this area and this will create a charge separation so here we will have a plus here we'll have minus just because the electrons moved so quickly along the magnetic field line the same is true here we will have a positive charge and here we will have more negative charge so the electrons reacting fast this is also what we are calling usually adiabatic electrons adiabatic electrons that basically means if we say that an instantaneous reaction on a parallel perturbation so this is an instantaneous reaction instantaneous reaction parallel to the magnetic field line um this will create a charge separation a charge separation and of course this charge separation will result in an electric field so it will result in electric field which goes like this and here from plus to minus like this so this is the perturbed electric field created by our initial perturbation so each twiddle and of course that will create an e cross b drift and the e cross b drift will be such that if you look at the quantities here since we have an initial field going down here from plus to minus this will also be a perturbed electric field then if you create a calculate the resulting e cross b drift you will find that exact this position we will have an e cross b drift pointing to the right so this will be here the e cross b drift and same on this on the upper cross section so if we look at this position we will see that here the e cross b drift will point to the left being the largest so this is here the e cross b drift meaning we get basically a propagation of a downwards propagating wave because this one increases or moves basically the peak to the bottom and this one at the other one moves to the hole to the bottom corresponding to wave and maybe just to complete that here the phase between density and potential perturbation here is zero so they are in phase and this finally results in a propagation in a propagation of the perturbation into the downwards direction so propagation downwards and this is why this is called a drift wave instability now you might wonder why I called it instability since here we have just a prop downwards propagating wave there's nothing unstable true a few more comments on that first why we call it instability the first common a comment is that the cross phase so the phase between the quantities density and potential is an indicator and that is important realize is an indicator for the type of instability for type of instability remember for the for the interchange instability it was basically pi half whereas for the drift wave it was basically zero now the drift wave can become unstable so the drift wave becomes unstable for assuming non adiabatic electrons for non adiabatic electrons as soon as the reaction parallel to the magnetic field is no longer instantaneously then there is a lack a phase lack between density and potential and thus the phase difference is no longer zero that will result in an unstable situation thus creating turbulence this can be non adiabaticity can be due to collisionality for example collision collisionality or induction landau damping landau damping or trapped particles i'm pretty sure there are more reasons for that which i don't have in my head at the moment but just a few examples important point is that this leads to a phase delay between density and potential perturbation and this again or furthermore tells us that the cross phase is an important quantity for the transport so the cross phase between density and potential is an important quantity for the transport okay that's it with this video where we talked about electrostatic turbulence how this can lead to transport that we need density and potential perturbation and we need a finite phase between these two quantities then we looked at fundamental linear electrostatic instabilities creating actually the turbulence there were two type of instabilities two major instabilities the interchange instability and the drift wave instability the interchange instability has a phase between density and potential on the order of pi half for the drift wave it is close to zero but we only get an unstable drift wave if it is not exactly zero but if it's close to zero and this one was given by the non adiabacity adiabaticity sorry of the electrons okay that's it for this video see you in the next video