 And I'm going to talk about the problem, really, that is about as old and about as open. And I'll tell you where the difficulty is, what pregis has been made, some of the history. But it's an issue. Now, what I want to talk about is about can you write equations of motion, microscopic equations of motion, like Navistokes for classical fluids, for quantum fluids. When I say quantum fluids, I think about helium-4, helium-3, below the phase transition, the lambda point, when you have already at least some proportion of superfluid in the thing. And what is the major difference between superfluids and classical fluids is that in superfluids, vorticity is quantized. And you have real objects, Vodex lines. So these Vodex lines are really objects that are there, cannot be destroyed. They go either in a loop or from one side of the box to the other. And they can reconnect. They can collide, reconnect, and have all kinds of things. First question is, do the quantized vortices matter in tablets? Do you need to know that you have these quantized vortices? Do they have a different or important effect on the statistics of the table flow, et cetera? And things here had a history. For example, experimentally and numerically, the answer is, in fact, no. Oops. When you talk about mechanically excited flows, so there is a very famous experiment that was already 1998 by a good friend, Patrick Tabeling and his group, in which they had a mechanically excited turbulence, what you call the French washing machine, helium, below the lambda point. And they have concluded that the energy spectrum of the super flow is compatible with chromogoron scaling. That is to say, they couldn't see in the statistics of the flow, in the statistical characteristics, much different from the classical fluid. And the two-fluid model, the classical two-fluid model of Landau and Tissa, in which you have a super fluid and a normal fluid. The normal fluid just has the Davies-Toch equation. The super fluid has the Euler equation. And with coupling through a mutual friction, works and seems to give you chromogoron spectrum. However, when you do other flows, for example, counterflow, counterflow is the turbulence you get if you have a channel of say, liquid helium 4 with temperature T1 here, temperature T2 there. And then you have a counterflow in the sense that the normal fluid goes one way and the super fluid goes the other way. Suddenly, it has been discovered that already intermediate scales, intermediate scales meaning between the macro scale of the size of the system and the scale, which is the important scale, which is the intervortix distance, the dynamics of vortex lines is crucial and strongly coupled to velocities of the normal and the super flow. So the two-fluid model is not sufficient. You need at least one more order parameter. Now, which one is this order parameter that you need to add? Well, we go back 60 years, as I promised. Before that, let me show you some visualization of these flows. These are from the laboratory of Den Lathrop in Maryland. I think it's very beautiful because what he does, he sees the vortex lines with hydrogen. And you see now the motion of these vortex lines. They collide. They reconnect. And you also see that they have Kelvin waves when they, after their interaction. So these are all of interesting dynamics. These are very beautiful experiments, rare that you can see the events and the actual action of the vortex lines. And the question is, as I said, how this affects the observed statistics and observed macro dynamics of the system. So in helium-4, it depends on which helium. Helium-4, the vortex core, is one angstrom. Helium-3, which is not boson, is fermion. You need cooper pairing. And the vortex core is 800 angstrom. But besides that, the dynamics you see is not entirely different on temperatures that are not too low. Temperature becomes too low. Helium-3, very complex status, has been discovered already by Volovic and others. There's a lot of interesting stuff. I'm not going to talk about Helium-3. Joe Vynen, already in the 1950s, is this competing with you? Not entirely, right? But it's almost there. Has already offered the concept that you need at least one more order parameter, which is the density of the vortex lines. And Vynen wrote an equation. L is going to be the density of the vortex lines. So it's one over area because it's length over volume. So it has a dimension of area 2 minus 1. Has, of course, a dL dt, has a production term, and a decay term. Everybody can write this equation. It's still empty. But then he has predicted on dimensional, analyzed on dimensional considerations a form of the production line. He felt in his stomach that the production should be proportional to the difference between the normal and the super velocity. So this VNS is V normal minus V super. And this is the only combination you can make with linearity in VNS that has the right dimensions. And the form of the decay, he predict, is going to be proportional to L squared. Of course, it cannot depend on VNS because you have a decay also where there is no difference in VN and VS. Now, as I say, L is a vortex line density. VNS is VN minus VS. Alpha is the so-called parameter known as a mutual friction parameter. C1 and C2 are fitting parameters. And kappa, which appears here, is the quantum of circulation that for helium 4, we know this number rather accurately. Is this true? Well, in fact, if you do dimensional considerations carefully, you realize that both the production and the decay can be written in terms of the circulation and the density as kappa L squared, and then any function of a dimensionless parameter. So the units are already taken here by the kappa L squared. And our question is, what is x? And x, if we follow Vinan, is going to be VNS squared over kappa squared L. This is dimensionless. Now, question is, how f of x and g of x depend on x? The possibilities are myriad. If you say that they go like square root x, you get the choice of Joe Vinan. That is proportional to VNS. And if you say square root x and L appears like L to the half, and therefore the L squared becomes L to the 3 halves. If you do a London-kind approach saying things are analytic, you'd say f of x starts like x. And then you're going to get this proportional to L VNS squared, because VNS squared is here. You'll have L. You get the L squared becomes an L. But why not x to the 3 halves? If this f of x is going to be like x to the 3 halves, then you get something that is proportional to VNS cubed. How can you determine? How can you say what is correct? Well, people have tried for a long time. People have tried with experiments on homogeneous flows to make the distinction. They didn't think about this crazy idea of x to the 3 halves. But these two forms were already clear in the 90s. And Nemirovsky, for example, in 1993, tried to fit experiments. So these data here are experiments. And he tried to compare the prediction of the equation with the first form and the second form. And you see these are the two predictions. And you cannot tell which one is right, because the error bars are sufficiently large that both forms agree. So we had the idea, let's not do it in a homogeneous case. Let's do it in a channel. So about three years ago, we started to propose, let's consider, inhomogeneous channel flows. Maybe there, the difference is going to be more clear. And indeed, it's much clearer. And when you do it in a channel that looks like this, so you have a channel. You make here these numerical simulations. You choose a profile for the difference between the normal and the super velocity. And then you solve for the vortex lines. You do vortex methods. I'm not going to go through the numerics much. It's careful numerics. It's done well. And you can now ask questions. It turns out that tentative result that we published really two or three years ago is that, in fact, it's the third form that fits the data best. That is to say, in a channel, you could write an equation like this. Now, of course, if the system is inhomogeneous, there's also a flux of the density of vortex lines that we look like this. Alpha of a kappa, VNS, DVS, DY. This is a result of the closure calculation we did. And we said, look, p3, not p1 or vinyl, not p2, but p3 is the right thing. Let me show you. Why? I mean, this comes really from a microscopic theory. And the microscopic theory, in order to get what you see, you have to start with the equation for the vortex line segment. This is really following the work of Schwartz already from the 70s and 80s. And you write an equation of motion for xi. Is an element of a vortex line, the excited T. S is going to be the vector position along this vortex line. Prime and double prime is the derivative and double derivative. We don't need to go to details. This equation contains two unknown parameters. Now they're actually measured alpha and alpha prime. They're known as the mutual friction parameter. And the point is that when you have this equation, you need to integrate this equation over the volume, the whole volume. And then you have an equation for the vortex line density, because xi is just the element. So the way that this increases or decreases average of the whole volume is the vortex line density. And then when you do this, you get all kinds of terms. And you need to estimate them as a function of the velocities and line density. You need to do closure. I'm not going to get to the technical details too much. I'm going to show you what is the difficulties. So when you're very happy, you ask, can we express these integrals in terms of velocities and vortex line density only? And you do this. But you have to remember that, in principle, closure can be sensitive to details. It may require further characteristics of the vortex tangles, not just density. If you are inhomogeneous, it can be an anisotropy parameter, curvature of the tangles, et cetera. But we'll come back to this. So there are additional candidates or the parameters such as the tangled curvature and the tangled anisotropy. But three years ago, we were still very ambitious and optimistic. So we looked at this production term. We did the analysis. And this is how it looks. So it has V and S. It has the S derivative, an S double derivative. You argue that the velocities are changing on a large scale. Whereas these vortex characteristics are changing quickly on a small scale. So you can take out the V and S. And then you get a prediction that the production term is going to be the normal super contrast velocity times S prime as double prime, where this means an average over the whole volume. Now, see, this is an object that is neither the velocities nor the vortex line density. So you need to close it. And by the way, just as a comment for people, I'm not sure anybody has looked at this problem before. But there was in the literature an attempt to estimate this with absolute magnitude. Here is a channel calculation of S prime, S double prime on the average. And the absolute magnitude is very different. So you're not allowed to do this. You have to do this. And we did analysis. And we came up with an understanding that the only dimension variables that you may want to consider, at least for a first go, as I said, is the one that was related to the vinyl dimensionless objects. Zeta, you can make others. Zeta 2, for example, is also dimensionless. And there are characteristics of the vortex tangle. And we call the one IL. And it's related to total this S prime, S double prime, and the other one is non-dimensionalized by L to the half, the other one by S double prime, et cetera. So these are dimensionless objects that you need to consider. And then when you consider this, we have to ask how the non-dimensional integrants depend on the dimensionless variables. And we came up with the following prediction that said, let's jump on this, that indeed we can compare in a numerical simulation what happens when you have a parabolic profile to the VNS, so the difference between the normal and the super. And you see there are three lines here. One is the prediction P1 of vinyl. The other one is the prediction of P2. This is the experimental or the numerical calculation of P3. This is the prediction of P3. So you see that neither P1 nor P2 agree. In fact, the non-intuitive result that those dimensionless functions have to go like the dimensionless variable to the three halves fit the data better. The decay term agrees very well with vinyl's approximation. And we have here the new term that nobody looked at, which is the flux of the vortex lines. And it fits the data very well. So we have a good closure and this will fit. But maybe we're just lucky. And why we're just lucky? Because we looked at parabolic profiles of these VNS. And indeed, when you look at various parabolic profiles and you look at this object that they call the anisotropy parameter as a function of the dimensionless variable zeta of vinyl, you see it's always almost a parabola. And this is what gives the result that I just showed you. This parabolic dependence is not a great function. It's not a single function, but it's not so bad. So this works pretty well. However, now, is this the end of the story? Absolutely not. Let's now do different profiles of VNS. OK, so now I'm looking at different forcing by the normal and super velocity. And I want to know what is the effect on the vortex characteristics. So you do the parabolic, but you do also sine wave, you do all kinds of things, and disaster. You see this doesn't fall on the function at all. It's not only a function of zeta. If this were only a function of zeta, as you would require in order to get the nice closure that I just showed you, then this will be a function. OK, it'll be just one line. There's no data collapse. It's not true. If you do now, this fellow is a function of two of these guys, the zeta and the zeta two that I showed you before. There are two dimensionless variables. Wow, suddenly the thing falls on the surface. So again, it's a function. So you see, if you have two closing variables, it looks much better. But now you see, this needs more. So we need to get back to the microscopic theory and improve the theory, because you now have more than just the density of the vortex lines. You have to continue. So you have to write now a new equation for the density dL dt. You do the microscopics. You now get that this is proportional to the n-isotropy term that before we've closed only in terms of the single zeta. There is the object kappa squared, which is the curvature of the vortex lines. Again, I don't want to get into much details. And when everything is said and done, the sky is the s double prime. And you have this, as I said, the dimensionless object that is made from s prime, s double prime. Now they appear explicitly. And you have a closure problem. You need to know how to close in terms of the dimensionless object that you have. So to the equations of the density of vortex lines, we need to couple an equation of the n-isotropy parameter at least. And you can do this. You can write an equation for the n-isotropy parameter. Again, you go to the microscopics. You do the theory. You do the integrals. You do the approximations. You get a form. You get a form. And are you confused? Well, you won't be after the next episode where you're going to put the time dependence. What I showed you is that the problem is not closed. Let's say you can have other objects that appear when you put in more physics. This is not as nice as classical turbulence, in which you have one important closure assumption. You have the velocity field. You have the flux of momentum. You assume that the flux of momentum is proportional to gradient of the velocity. You get the Navier-Stokes term, and it works. Here, things are much more dodgy. So you do have important physics of the vortex lines. The vortex lines are there. They are important. And now when you try to couple the velocity fields with the vortex lines, life is not so simple. The problem is it has many snakes under the grass. You have to be very careful. And the answer can depend on the conditions. Are you homogeneous? Inhomogeneous? Are you in a channel? Are you in a co-flow like the French washing machine? It's an interesting subject, and I just wanted to show you a little bit of the difficulties and the fact that the story is not yet finished. And I thank you very much for your attention.