 Hello everybody and welcome to video number 32 of the online version of the future research lecture We aren't chapter 6 turbulent transport and in the last video We basically talked about turbulent transport in neutral fluids and showed some examples and introduced the Navier-Stokes equation and the Borgers equation being similar to the 1d Navier-Stokes equation without a pressure gradient term Now in this video, we will have a look at the energy transfer between different scales different size scales So we will have To be more specific an overview over eddies and the energy cascade eddies being flow bodies as you remember So let's have a look at eddies and The energy cascade the energy cascade Sorry Okay, so Let's assume we start with some large eddy by either stirring a tank or having a large scale instability So let's try to draw one large eddy Maybe something like this Oops And just to indicate This is some something which We steer we draw it like this. So this is now supposed to be an eddy and now when Well, let's probably write down that we start here. So this is starting by either Stirring a large tank the fluid in a large tank or we have some large scale instability and instability And what then happen is what you have probably seen already in the pictures I've shown the photography so on simulations and you know that that this large scale eddy um oops Decays into smaller scale eddies for example like this So let's try to indicate that by putting them down here And then two eddies you have another one here And then that these eddies again Decay into smaller sized eddies For example like this like this like this Then let's make one more um Cascade like this even smaller smaller Then each of the eddies Decays into these in the such two eddies So this is something which we have basically seen in the photographies we looked at or in the simulations and What now finally happens is whoops, I didn't meant to change the Color of that So what then eventually happens, of course, they cannot be infinitesimally small Eventually they will be the energy will be dissipated So dissipation will happen here dissipation By viscosity by inner friction of the system by viscosity And so in this direction basically so going downwards This corresponds to how the energy flows. So this is the Direction of the energy flow meaning we start with some large scale eddy um this one Here which then decays into smaller scale eddies like this one and into smaller scale eddies Smaller scale eddies and so on until the energy is dissipated by viscosity So now to describe that in a bit more detail what we want to look at is the energy transfer between different spatial scales So we are interested in the energy transfer between Different spatial Scales and that is basically a turbulence Energy spectrum what we are looking what we want to look at a turbulence energy spectrum And a very famous one for turbulence in the two fluids is the so-called k 41 theory k 41 theory and it is named according to Kolmogorov according oops According to Kolmogorov Who published this in 1941 and what did he do is so he looked at the spectrum Meaning we have here one axis which is the in logarithmic units The energy density log e so this corresponds to the energy density Which is proportional to the square of the the velocity And then in this direction We have the the spatial dimension so um oops Also logarithmic units the wave number which is the inverse of the wavelength So usually we use wave number space here And then we start somewhere with injecting energy into the system. So here we have somewhere is the injection range Where we basically inject energy into the system For example by instabilities Or usually by some type of instabilities For example Rayleigh-Taylor instability or Kevin Hamel's instability And then the energy Um decays so the the the um the structures decay. So here we have In this representation a linear slope Something like this And this corresponds to how the energy flows So this is the flow of the energy and This is called the Inertial range It's called the inertial range where we have this direct or such a direct cascade A direct cascade meaning that the energy is transferred from large scale eddies to smaller ones This is why it is called and generally a direct cascade. So direct refers to the energy being transferred from large Scale eddies to smaller ones to smaller ones And now the interesting point is that this slope This slope Follows The equation k to the power of minus Five over three and that is a universal universal law So there's something very interesting and finally when this if the structures are small enough We have the dissipation range. So this is the dissipation range Where the energy is transferred into heat. So here we have transformation By usually by inner friction into heat This is the k 41 theory and let's try to write that down in And in some some some words and some sentences. So what we have just seen is that in 3d neutral Fluid turbulence in 3d neutral fluid Turbulence Which is supposed to be homogeneous So the turbulence is supposed to be homogeneous An isotropic so spread out over the whole system isotropic And in such a system the energy is Transferred Between self similar Eddies So flow what he says A long. Oops. Sorry typo here Along a so-called direct Cascade from large To small scales from large to small scales Following The law Where the energy density Being proportional to k to the power of minus five over three And this is observed Since we are talking as I said about 3d neutral fluid turbulence observed in gas and fluid systems gas and fluid flows and So let's just highlight it. We have a 3d system 3d neutral fluid turbulence We have a direct energy cascade here a direct cascade Where the energy Transfer is described by this law And this is basically one of the most celebrated results in turbulence theory due to its universal Universal character. Yeah, so this is a very important result here Now what about 2d turbulence? So let's talk about 2d turbulence 2d turbulence For example thinking of a magnetized plasma Because in a magnetized plasma The turbulence is not isotropic in all three dimensions The turbulence Is not We'd rather underline that here is not isotropic in all Three dimensions due to the magnetic field which imposes Certain preferential direction So parallel to the magnetic field We have the thermal movement Thermal movement Whereas pep and dickler to the magnetic field we have the drifts And so this there's a difference Now how does the corresponding spectrum looks like? So if you can plot the energy density on this axis So in logarithmic units the energy density And here the the size Again in logarithmic units. So look at Rhythmic units And then if we have some injection range here So this is supposed to be the injection. This is where we inject our energy injection Then we have now two Um processes going on we have in transfer to larger scales And one to smaller scales And This is the Energy cascade and indirect energy cascade going upwards and This direction this is the endstrophy here The endstrophy Which we will define in a minute The energy cascade the indirect energy cascade Sorry the inverse energy cascade I wanted to say Has a slope of k to the power of minus five over three as we have seen it before and the endstrophy Cascade now has a slope of k to the power of minus three Now what is the endstrophy? So to define the endstrophy we first have to define the vorticity The vorticity Is Corresponds basically to the strength of the spinning of a vortex Is defined by capital omega Then the rotation of the Velocity so it corresponds to the strength of the spinning of a vortex then the endstrophy is basically Corresponding to the energy so it is The average over the squared of the vorticity so this roughly corresponds to the energy and these two quantities So these two quantities are conserved Sorry conserved quantities in 2d turbulence like the energy usually Okay, so let's write down what we have in for 2d turbulence in 2d turbulence. We have a dual cascade In 2d turbulence a dual cascade is observed in 2d turbulence So we have on one hand side the inverse energy cascade We have on one hand side the inverse energy Cascade And if you might wonder the infrared catastrophe is avoided due to the interaction with the vessel wall So finally if the structure become too large they just interact with the vessel wall and dissipate the energy in that way So the Infrared Catastrophe Similar or not too different to the ultraviolet catastrophe in the Rayleigh Jean's law from the black body emission, which was avoided by By making the by introducing discrete energy particles, right? So the infrared catastrophe Here is avoided due to The eventual interaction with the vessel at too Large Structure sizes corresponding to a small case at too large Structure sizes corresponding of course, as you know to two small k's And then in addition to the inverse energy cascade we have the direct endstrophy Cascade And this is what is observed in 2d turbulence And as I said on the previous slide, this is what we expect For plasma turbulence. So this is expected For Plasma turbulence Since The Um Drift Cascade Drift Velocity Drift Of the perturbed Quantities so due to an initial perturbation of the density for example Um particular to the magnetic field since this has The 2d nature Okay, that is important to be aware of that the 3d spectrum looks different from the 2d spectrum That in a plasma in a magnetized plasma we expect the more the 2d case Now if you want to simulate turbulence Turbulence in magnetized plasmas In magnetized plasmas There are multiple codes to do that multiple ways to do that relatively Well, the oldest one probably is to use a two fluid model To use a two fluid model Um, well the two fluid model to describe Plasma turbulence And the fluid is assumed to be in local thermo dynamical equilibrium And the input parameters for that are a A continuity equation Then B Momentum equation Momentum equation And then C The energy equation So making use of the conservation laws here of continuity Um of mass basically of a momentum and of energy And this is what I think you should have talked about in plasma will talk about in plasma physics 2 lecture Um and the source of free energy So the plasma is described by this using the two fluid model And then what you use as a source of free energy is a pressure gradient. So this is the source of free energy Which creates instabilities as you might know instabilities And these instabilities will finally lead to Turbulence As we will discuss in the next video Okay, so this video was about eddies and the energy cascade We looked at two different cases a three-dimensional neutral fluid case where we have the k 41 theory applying Where we have an energy a direct energy cascade going from larger scales to smaller scales until The energy is finally dissipated due to viscosity enough friction And then the 2d turbulent case the 2d turbulence case Which more corresponds to magnetized plasma because the magnetic field Imposes a certain preferential direction There we have a Inverse energy cascade and a direct enstrophy cascade And then here I've just given an example Probably the oldest certainly the oldest model how to describe plasma turbulence by applying the two fluid model using the The conservation laws as I've written down it here and then Taking the pressure gradient as the source of the free energy which creates instabilities which then lead eventually to turbulence Okay, that's it for this video. Hope to see you in the next video