 Proste, Prof. Ishida. OK. Čakaj. Zdaj, da imamo to unifaj, imamo seori krepiču prezentaciju. Zdaj, možemo izgledati, kaj je tudi topologi. Zdaj, če je tudi izgleda, in in tudi, da je inčen, je vse zelo zelo zelo, kako nekaj je izgleda. Seveda, if the flow is initially 2D incompressible, it will continue to be 2D incompressible, and also, if the initial flow is helicity free, it will be maintained. Zato mu je tako pravda, da je totopoložice, pravda prijeljega. En vse prijeljamo govori, da te pravda prijeljem je totopoložice, to je, da je totopoložice imeljno in vsega ideje revolučijo. In što prijeljamo, da pa se počušimo, ki je totopoložice kod vzvečilje in vzvečje vzvečje, tudi v zelo v zelo načinu. Tako tukaj teznih teznih teznih teznih teznih will give some influence to the turbulence that I am going to discuss. So the background of this talk is the first, there was a suggestion by Fukumoto that these teznih teznih teznih teznih can be unified as cross-heresities. This is the one essential idea. And also I will draw heavily on this work, collaboration with Phil Morison. And that means the hierarchy of flow field parameterization. And that hierarchy means that we have these kind of special type of flows as hierarchical structures. The outline of my talk is, I am going to say that Krebš parameters are fundamental field underlying fluid observables. So I will explain why I am going to consider this. And fundamental means we can consider the communical Hamiltonian mechanics in Krebš parameters, phase space. And that is not directly visible, but we consider some canonical Hamiltonian mechanics of Krebš parameters. And we consider that the fluid observables like densities, the entropy or the velocity field, they are reduction from these Krebš parameters. So that I will call reduction to fluid variables. And then on this visible fluid mechanical parameters, we can construct fluid mechanics equation just from the canonical Hamiltonian mechanics of Krebš parameters. So that is the basic idea. And this reduction means the degree of freedom is reduced. And so therefore this reduction is in some sense redundant. And that redundancy allows us to consider some gauge symmetry. And this gauge symmetry will give you the so-called cashmere as data charges. And it is also well known that the helicities and these kind of things are the data charges. So these cashmere can be regarded as the data charges of this gauge symmetry of the reduction. And so that means the cashmere basically describes that the fluid mechanics is constrained. And that constraint gives some constrained mechanics as the fluid mechanics. And so that is the geometrical picture of topological picture of the fluid mechanics. And we can formulate generalized entropy, which is conserved in 3D ideal flow. You know, the generalized entropy is usually assumed that it's a constant motion of 2D flow. But we can formulate further generalized entropy that is conserved in 3D ideal flow. And this generalized entropy basically reflects the circulation law. But it is more convenient to delineate the topological constraints in the 3D ideal flow. And so this generalized entropy may put obstacle in the way of cascade, because it's the conserved, topological conserved. So in the inertia range, the fluid motion is topologically constrained by this generalized entropy. And this generalized entropy is degenerating to the conventional generalized entropy in the 2D fluid. So therefore we know that in a two-dimensional situation, the inverse cascade occurs. So then the question is what happens in the 3D? Because in the 3D, usually we never pay attention to the entropy. However, we do have something similar to the entropy in the 3D vortex dynamics. So then in the 3D, we show that the vortex tube strength gives the pass for the cascade. So that is the explanation of the difference between 2D and 3D. So I will start from reviewing the Hamiltonian mechanical formulation of fluid mechanics. That is already well known. So to avoid the complexity induced by the boundary condition, we just consider three-dimensional torres. I mean neglected boundary. And consider the ideal flow obejing these. And by ideal, I mean inviscid and barotropic. So this is a barotropic pressure. And so this is, I think, the very well-known equation for the fluid mechanics community. Most simple. And so then we can put this in the Hamiltonian form. Hamiltonian, but it's the infinite dimension Hamiltonian form. And to do that, we first define the phase space. And naturally, we consider this rho and v as these are the fluid observables. This span the phase space. So we consider the phase space of rho and v. Rho is the density and v is the velocity. And then we can put these two equations into the Hamiltonian form, which I avoid to write down here. It is very well known. And we can formulate non-canonical Poisson bracket. And the energy is just simple fluid energy Hamiltonian. And then we can write it in Hamiltonian form. And because then this Poisson bracket of these Hamiltonian mechanics turns out to be non-canonical. Non-canonical or the degenerate. And the non-canonicality is reflected as the creation of the cashmere. And the cashmere are the constants of motion. There are two. One is the total mass. And the other is the fluid helicity. And these two are known to be the cashmere of this Poisson algebra. And these are the constants of motion in the context of physics. But in the context of mathematics, these are the center of the Poisson algebra or the cashmere. And cashmere are the constant of motion. But it is very interesting in the sense that that is independent of the choice of the Hamiltonian. And usually in physics we believe that the constant of motion is some attribute of some symmetry of the system. So usually the constant of motion comes from the symmetry in the Hamiltonian. However, these cashmere are independent of the any choice of the Hamiltonian. So the origin of these cashmere are not in the Hamiltonian, but it is in the symmetry or the sum of gauge symmetry beneath this, the fundamental, beneath this fluid variables. So the idea is we may relate these cashmere as the gauge symmetry of the Poisson algebra. So then the question is what is the more fundamental field beneath the fluid variables. So there are any kind of mathematical possibilities, but I am proposing here that the Krebjš field is the fundamental field beneath the fluid mechanical variables. So we can prove that by formulating canonical Hamiltonian mechanics on the Krebjš parameters, and then by reduction we can reproduce fluid mechanics. So that is the scenario. So here we introduce the Krebjš parameters as the underlying basic quantities. And that I write like this. Here we invoke six functions like this, rho, phi, p, q, r, s. And so these six component functions, among them rho, p, r, are three forms, and phi, q, s are the scalars, zero forms. And three forms means something like the density. And scalars, just scalars, like you know that. So these three of them are three forms, and three of them are the scalars. And we define the canonical Poisson bracket, just canonical like this. And this Jc is a canonical symplectic matrix. So we formulate canonical Poisson bracket on this phase space. And then fluid system, the previous fluid system is obtained by relating these Krebjš parameters to fluid variables like this. Rho density is this curly rho. I put star to become a rigorous mathematically, but star means Hojcidual. Hojcidual means basically you divide by the volume form. And then velocity V is written like this. And this is the well-known Krebjš parameterization in some part mathematical representation. But we write that these fundamental Krebjš parameters appear as this, and rho and velocity in this very specific form. And then we may write the Hamiltonian, that is the fluid energy. And the fluid energy is, of course, at the sum of the kinetic energy and the thermal energy, multiplied by the density and integrate over the volume. So because this part is velocity V, so this represents the kinetic energy, and by this epsilon, I'm representing the valotropic thermal energy. And just plotting in this Hamiltonian to the previous canonical system, we obtain this Hamiltonian's equation of motion. And that is like this. And you will find that only the equation for phi is a little bit complicated, but four others are very similar form. And you will notice that, for instance, this is nothing but the mass conservation law. Right? And that is this, rho, as I said, is a three-form. So this means that rho is redrug as a three-form. So it is the maintained if you integrate over the moving volume. And others, P, R, these are the scalars. No, no, no. These Qs are the scalars. That is just also redrug the scalars. And P, R are the three-forms. So they are just, you know, the constant of motion along the flow motion. So therefore, the Hamiltonian's equation of motion is very simple. And it says that basically these, the gravity potentials are accepting phi, just, you know, the transported by the flow. And using this, then we can easily verify that, you know, the relating rho and V in this way, we can easily check that this equation of motion is equivalent to the fluid mechanics equation. So therefore, we can say that these Krebš parameters, Hamiltonian mechanics of Krebš parameters is the fundamental representation of the fluid mechanics. Yes. Yeah, phi is the Bernoulli constant. Yeah, from here. It's basically the energy density. So for the Krebš parameterization, we have this kind of lemma. That says in the n-dimension-based space, we can represent every one form, it's something like a vector in this form. So this is the Krebš one form, which we call Krebš one form. That is special because this first component is exact. It's just the gradient of something. And others, you know, the general way. So we can say that in the n-dimension, every field is written like this. So therefore, in the two-dimension space, every velocity field can be written like this. And in the three-dimension space, every field can be written like this. So you notice that we need another R-spare in the 3D. In 2D, this is enough. And even in the 3D geometry, we can consider what we named AP two-dimensional flow. That has only this component. So we can consider this kind of special class of flow in the 3D space. And we will show that this is the helicity-free flow. And in the two-dimensional geometry, as I said, every velocity field can be written in this way. So therefore, the phase space is a little bit smaller. We need only p cubed, and we do not need rs. And then we have in the two-dimensional fluid mechanics, we have the so-called generalized entropy as the famous constant of motion. And that is written like this. And f is an arbitrary function. And we basically divide, this is the so-called potential velocity, and any function of potential velocity multiplied by the density 2,4 gives you the constant of motion. And you can easily verify this is the constant of motion. But here, we notice that the basic point is in the 2D, the velocity is always 2,4. It is the curl of the velocity. And velocity, if you consider, velocity is one form, curl v is omega, the voltage is 2,4. So if you divide 2 form by 2 form, you get scalar. So the idea of formulating this is you first determine the scalar field, and then you integrate over multiply density, and you get this constant of motion. So we can do the same thing in the 3D. So in the 3D, we have this generalized entropy conservation. So how we can formulate the generalized entropy is for the flow by taking a curl, we have two components contributing to the velocity. First one comes from the first component which I denote by omega plus. And the second one comes from the second component of the curvature that I denote omega minus. And for each omega plus and omega minus, we can formulate this as the generalized three-dimensional entropy. And to define that, we need another moving redrugged scalar, sigma, and by help of this redrugged sigma, we can formulate this. And we can show that this generalized entropy is a constant of motion in the 3D. And this sigma to define this may be any moving scalar field. And that may be either phantom or axial. By phantom means any scalar field which does not enter the Hamiltonian. And axial field I mean some field entering in the Hamiltonian. And in a two-dimensional case, this sigma may be simply the coordinate, perpendicular coordinate z. And then you can easily show that this collapse to the conventional entropy. But for the 3D, these are the generalizations. And also it is interesting that you can create the 3D helicity by this, just combining these two generalized entropies. So we will call this generalized entropy as charges, topological charges. And so therefore these are the more fundamental topological charges than the 3D helicity. So the 3D helicity is just a combination of these two generalized helicities. Entropies. And topological constraints given by this. So we now notice that even in the 3D we have topological constraints given by these topological charges. So any ideal flow must conserve these topological charges. So in 2D, as I said, this is just the conventional entropy. So therefore it is known that this and also if the flow is almost incompressible, the conservation of this charge is the conservation of omega. So therefore the omega cannot change in that 2D. So therefore the slow change, because this is the constraint, slow change of this omega in the Fourier space by the action of this convection blocks the cascade of the energy. So the energy cascade is blocked by the topological constraint imposed by this. So therefore the energy must go to the other directions of the inverse cascade of gas. So this is the well-known scenario. So what happens in the 3D? In the 3D fortunately the conservation of these two charges does not directly influence the magnitude of omega plus and omega minus because that is the multiple of the gradient sigma and this and this gradient sigma is basically the perpendicular direction. So therefore if this gradient sigma is reduced this omega may increase. So this is the well-known vortex stretching because the degrees of the variation of sigma means the elongation in the direction of sigma. So that means the stretching can help to omega to be cascaded much much faster than the usual situation. I'm finishing. So therefore the vortex stretching paves are the path for the normal cascade. And this is how this stretching will modify the omega. So let me conclude. So invokings are the Kurevich parameters. We can formulate canonize freed Hamiltonian system and we have topological charges that is the generalized enstorofis whose conservation causes topological consultant on the 3D fluid motion. And the well-known heresities are the derivative of these charges. And in 2D the topological charges generate the enstorofi conservation gives the inverse cascade. In a 3D the vortex tube stretching enables rapid cascade of omega enabling normal cascade. This is, of course, very long. But the point is this omega plus minus invariance places morphological consultant on the cascading process because these are topological consultants. So by use of these things we can make more detailed analysis of how cascade will create the internal change of the structures in the fluid. So, thank you very much.