 Hello everybody and welcome to video number 34 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 underlying electrostatic instabilities the interchange instability and the drift of instability leading to turbulence This video we will have a look at an Approximate or estimating expression for the transport So we would actually talk about the turbulent transport Turbulent transport and In on this slide. I have shown you two time traces From measurements of a plasma on the left hand side. This is the plasma potential and on the right hand side This is the plasma density both measured by means of longer probes and You can see how both quantities are varying over time. So this is measured at a fixed position So they are the quantities are fluctuating and these are basically the quantities we want to measure. So, sorry the quantities to measure in plasmas Because they are mostly relatively easy to measure and they allow us To get quite a decent amount of information out of it. Let me rewrite that again Quantities to measure in plasmas This is like on the left hand side of the potential being a function of both both position and time Where we have like a background potential phi not Which is independent of the time just a function of the position and then plus the Varying part to the twiddle Which is a function of both position and time and the same is true for the density being a function of position and time where we have the unperturbed background density and not plus the perturbed Part which is then again a function of position and time Okay, now what to do with with such time traces One important thing to do is calculate the auto or cross correlation function Not to understand the a cross correlation function. It's easiest to look at the auto correlation function so calculation of the auto and cross Correlation function As illustrated here So let's first focus on the auto correlation function now What is the auto correlation function when you calculate that what you need to do you first need to duplicate? Duplicate a time series after You have duplicated the time series you need to shift one of the time series by a certain interval tau so you need to shift one of the time series by an interval by a time interval tau and then you need to multiply and and Integrate in the two time series you need to multiply the two time series with each other and the product need to be integrated and Integrate and that basically corresponds to the auto correlation function and if you do that then you Apply an additional normalization such that The auto correlation function the indice the index here is nn because I Correlated the density with the density so with itself That the auto correlation of zero so of a time like of zero is one So that is per definition If you look at the time trace shown on the left-hand side or the auto correlation function You can see that here At position zero so at time like zero we have a correlation of one again This is achieved by duplicating a time series shifting one of the time series by an interval tau multiply and integrate it then and We get this one over e value here This is something like this is called the correlation time and this tells us something about the lifetime of the structure at that position So this is how long the structure is correlated at the position where it was measured so the auto correlation function tells us allows us to deduce the correlation time the correlation time tau Core and the same is true. We for a spatial Correlation function, so we would make a spatial auto correlation function So shift it in space if we would have a measurement in space and not in time here We are in time then we would get from that the correlation length. So this is equivalent We can also obtain a correlation length Elcor from an equivalent measurement in space And now the cross correlation function is just doing the same with density and potential instead of density density so the cross correlation the cross correlation is the same But using density and potential and remember in the last video we talked about the cross correlation that the Cross-phase we get from that which is a quantity which you can get from the cross correlation is an important quantity for the transport now Let's try to make an estimation for the turbulent diffusion coefficient so a turbulent diffusion Coefficient Based on the random walk approach for the step size We take the correlation length and for the step time we take the correlation time For the step time we take the correlation time Meaning that the diffusion coefficient Based on random walk approach can then be approximated by a correlation the correlation length squared over tau correlation and the characteristic length the correlation length is given basically By the E cross B drift. So the characteristic characteristic tick length L car is given by The E cross B drift due to the varying electric field remember we have electrostatic Instabilities electrostatic turbulence here it is given defined by the product of the Drift velocity the radio drift velocity times the correlation time so the distance such Vortice travels within the correlation time and then This allows us to estimate a turbulent diffusion coefficient inserting in the above equation to be on the order of R the Velocity due to the E cross B drift squared times the correlation time As an estimation for the diffusion coefficient Now let's look at the transport so the transport is as we already said a few times given by a Diffusion coefficient now here for the turbulent scenario times density gradient grad n Then for the diffusion coefficient, we can now insert what we just had on the previous slide so the E cross B velocity times the correlation time and then the density gradient and then inserting the definition of the or inserting the Expression the expression for the correlation length which we had on the bar on the previous slide and we get minus l car Times grad n times the Drift velocity and Now we apply the so-called mixing length model the mixing length Model which basically tells us that the that the amplitude of the Fluctuation is given by the correlation length by the structure size the size of the vortices so the fluctuation amplitude can be roughly estimated by the Size of the vortices which is the correlation length and then times the density gradient So the larger the gradient the larger the fluctuation amplitude something easy to imagine if you have a Higher gradient. You have basically more free energy which can lead to larger fluctuation amplitude. This is the mixing length model and then We can write the The transport as and Twittles so the density variation since a core times grad n is the density variation times the drift velocity and Now we should have in mind that the drift velocity Is proportional to or can be estimated at least its quantity its amplitude by the radial electric field over the magnetic field and If we now were to look at the transport then we need Since you've seen that it varies it varies over time and we need to make a temporal average So an average over time indicated by these brackets and the t in the index is that is an average over time then we assume by We want to just know in order of magnitude expression setting the magnetic field to one here that this is the temporal average over the Density variation and then times the electric field variation Just inserting the expression for the drift velocity and What we now do is we take these time traces as Fourier series meaning if we have Some quantity f t then we've Take its take it into Fourier space or using its Fourier transformation such that f t Can be expressed by Normalization 1 over 2 pi times the integral of the amplitude components f omega Sorry, this is supposed to read f omega. Let's probably write that again F head With the amplitude components or omega So every frequency component has its own amplitude times e To the power of i omega t d omega Inserting that for the time series of density and electric field then we can finally write it as the integral over n head of omega times e head of omega and Now we can with it both with the exponential functions there. We can rewrite it finally as cosine of So this is like a bit of algebra involved here cosine of Phi e Minus Phi n d omega and This is the cross face. I talked about earlier This is the cross face. I talked about earlier So here we have this expression Which was basically Kind of mentioned in the last video Namely that the transport it depends on the cross face between density and electric field or density and potential and Here you can see this expression and by the way just as a reminder note that the face between Electric field and density is the face between Sorry this supposed to be a Phi the face between Potential and density plus or minus Pi half. So the electric field shifted with respect to the potential to its face by Pi half Okay, that was very briefly just in video to get an Approximate expression for the turbulent transport. I briefly showed you in the beginning of this video the quantities we interested in that is the Potential and the density and how these quantities depend to both on time and position I briefly talked about the auto correlation function and the cross correlation function Then we used the random walk model to describe Diffusion coefficient where we used the correlation length as the step size and the correlation time as the step time and Then the E the cross B drift resulting from the varying electric field Which defines to define a correlation length and interpreting it as a characteristic length and then here we have the in estimation for the turbulent transport most importantly we have an estimation which shows us if we do not regard the absolute numbers, but which shows us that the Turbulent transport depends on the cross phase between density and potential fluctuation in the face of these two quantities That's it for this video. Hope to see you in the next video