 Does the clicker work? Yes, this one? No, it does not. Oh, yes, here. OK. I got one. OK, so you have a pointer? A pointer here. Thank you. Good afternoon. Thank you to the organizer for this invitation. I will speak about LECT. These LECT everywhere are nowhere. And it's a preliminary study, which will be continued with my colleague Fabien Godfair from my own lab in Lyon, France, and with Aziz Salih from Tunis. First, my main topic is the spontaneous generation of LECTs, LECTs with S, because, for example, in MHD, we have a free kind of LECTs. And without initialization, no forcing. I will present briefly, first, the case of purely rotating turbulence. After, I will move to the NSF context. It's not the National Science Foundation. It's N for stratification, like the Brunweizela frequency N. S for shear, and F for the Coriolis parameter. And it's an approach in which Casio-Homogenius or Homogenius to Brunweiz is subjected to constant uniform rate of buoyancy gradient, velocity gradient, and Coriolis force. I will move to MHD using an analogy between a true cross-elicity in MHD and a subrogate proposed by Gibbon and Holmes and illustrated by, I will show you on the next slide. Of course, local LECTs everywhere in snapshot, but it's local. Often, it is very difficult to generate bulk LECTs with statistically relevant. For example, in Homogenius to Brunweiz, John Nomé suggested that if the symmetry in a flow, the center of symmetry, you cannot generate LECTs. On the other hand, there are a lot of DNS reinforcing by the group of Minini-Pouquet. There are also surgery, for example, helical modes. You can expand any velocity field in helical modes which concentrate LECTs. But often, if you add helical mode with different polarities, you cancel LECTs. And only if you perform some surgery in killing some helical mode with a given polarization, you have a lot of LECTs as illustrated by resistance today by Luca Biferalé and co-worker. Nevertheless, LECTs spectrum, the statistically relevant LECTs, is very important for the alpha effect, the dynamo effect, and so on. And I will mention some work by Marino in the same team of Minini and Pouquet after. By definition, LECTs are kinematic LECTs because I will have other kind of LECTs. Is the correlation of a scalar product of a vorticity, vorticity fluctuation, and velocity fluctuation? The first question is, what kind of averaging we can use? In a statistically homogeneous turbulence, two-point correlation like that correspond exactly by Fourier transform to an LECTs spectrum. Often, presence of LECTs means beltramization, so that the ratio of the long vector, the modulus of a long vector over this scalar product quantifies the non-linearity. And it is expected that when u is more aligned with omega, non-linearity is dramatically reduced. So is there a correlation between a kind of joint probability of alignment with velocity spectrum? It's an open question, not completely solved. Now looking at the symmetries, there are all LECTs, I will define the other LECTs after, are always product, scalar or inner product, of a vector by a pseudo vector. Omega is a pseudo vector. It's change of sign by inversion, so that there is a breaking of mirror symmetry. So if you have mirror symmetry, you have no statistically relevant LECTs. And I will follow my color schedule in the rest of the talk. Red will mean breaking of mirror symmetry. So just a bit of statistics, I will look at the two-point second-order correlation like that with our separation vector. This involves the second-order structure function, for example, but a bit more. And we have a complete equivalence in at least in homogeneous turbulence, in homogeneous even in strontian dystopic, a complete equivalence between this tensor and its Fourier transform with respect to the separation vector, the wave vector correspond to the separation vector r. And it only in Fourier space, because we can take explicitly a count of incompressibility, divergence-free contribution, this guy can be represented by only four pseudo-scalar, energy e, polarization z, and LECT. z is complex value, so that this corresponds to four independent scalar. Calligraphic e, of course, if you spherically average, give classical energy spectrum. And we can distinguish directional and polarization anisotropy by some expansion in terms of spherical harmonics. The first harmonic gives, for example, the directional tensor for e and polarization tensor for z. Roughly speaking, the two kinds of anisotropy are not completely formal. Directional anisotropy means the departure of the distribution of calligraphic e from a spherical distribution. And it could be expanded in terms of classical spherical harmonics. And z is a bit more complicated. It corresponds to the tensorial structure of the spectral tensor at a given wave vector. And it could be generated also by spherical harmonics, but more complicated. In fact, z represents a tensor, a deviatory tensor. I will stop here, of course, over the possibility of simplified statistics by averaging. But h is lost if you take the spherical average of h by symmetry, it is 0. So some odd angular harmonics could survive in h, but it's an overstory. No time to discuss more of that. I will concentrate on the first level. And I will begin first by the case of rotating turbulence, purely rotating turbulence. I will say that in homogeneous turbulence, even arbitrary anisotropic, there is no spontaneous emergence of act in rotating turbulence. Even if imaginary part of z, I will comment briefly on the next slide, and h both break the mirror symmetry, but there is no coupling. Even in turbulence from a cloud, inertial waves propagates from the cloud in a rotating frame. But by pair, so that you have always inertial waves with positive polarization and inertial waves with negative polarization. If you add both, you have zero ECT. This is not discussed in the paper of the Davidson team, but you can check that you have exact consolation. It's a very nice illustration that two modes that you have exact consolation of electricity when you add two helical modes, which individually concentrate electricity. I have an example. It was not really discussed in the paper also, but with a boundary near a wall. This corresponds to a Ekman layer. In this case, if you consider up and down eddies near a wall, one eddies moving towards the wall, one eddies moving in the other direction, their sense of rotation are correlated with their actual velocity, so that you add velocity. In this case, and only in this case, to my knowledge, rotating turbulence generates net electricity, but you need a wall. It's not homogeneous turbulence at all. Let me continue with a kind of formalism. I mentioned that the spectral tensor, which is the covariance matrix in Fourier space of velocity correlation, can be expressed in terms of four scalar. And what is the idea? The idea is that in Fourier space, you take into account the incompressibility condition very simply, so that the velocity fluctuation has only two components, one toroidal of type toroidal, one of type toroidal, and the divergent is 0, because the divergence u equal to 0 means Fourier transform of u perpendicular to the wave vector. So if you correlate, if you have a cross-correlation, you have this kinematic block in which you have toroidal energy, toroidal energy, and a red term, correct the mirror symmetry, you find, of course, electricity, but you find also the imaginary part of the polarization term. In terms of helical mode, you recombine now the two components in Fourier space, they are two components normal to the wave vector. You combine them in terms of imaginary part. You have helical mode with plus, helical mode with minus, and 0. And in this case, electricity appear on the diagonal because the helical modes diagonalize the curl. And z appear naturally, the polarization term appear naturally as a cross-correlation between helical modes of opposite polarity. I am surprised that almost nobody outside our team use this polarization term. So no problem. I continue. There are, of course, a question, a bit complicated. I will not discuss all these questions. Craya introduced a very nice concept of homogeneous turbulence. Homogeneity restricted to fluctuation in the presence of uniform mean velocity gradient. Mean velocity gradient can have a symmetric and anti-searmacy symmetric part. And you can write a question for all the guy which generates second-order correlation, like E vector angle dependent, polarization z, and electricity. There is a very important term which break the mirror symmetry due to the rotational effect, due to the rotational effect of the shear, and also to the system rotation if you are in a rotating frame. But you have no linear coupling. These linear parts are exact. And you have no linear coupling between EZ and H. It is expected because this equation is very close to the Karmann-Warth equation you will see over in physical space in over torque, in which you have a purely dissipative term. You have the action of a symmetric part of a gradient, the linear vortex stretching, in fact. And you have nonlinear cubic term because the nonlinearity of Navier-Stock's quadratic, you have right-hand side in terms of cubic correlation. And it could be possible to couple electricity with energy through a nonlinear term, but it's not a complete coupling. If we have a theoretical study, like inertial wave turbulence, in which we can show that if electricity is zero, it remains zero. So it's not definite because wave turbulence or statistical model for this guy are not completely the truth. But we have a long experience with both nonlinear model and DNS. In DNS, it's possible to introduce through three-periodic DNS in terms of triple-periodic boxes. It is possible to take into account the advection by the mean with K time dependent. I will not insist on that. It's not important in my talk. I continue with now I introduce the stratification. So according to Bosnian assumption, the velocity field remains divergent-free, so that I can keep the solenoidal or divergent-free part of the second-order correlation. And I can consider the buoyancy. Of course, the velocity is divergent-free, but the buoyancy can fluctuate. And it's possible to add a buoyancy part in this compact decomposition. And the buoyancy is more considered as a vector because it corresponds roughly to the gradient of a scalar. And a gradient in Fourier space is along K as the divergence in this. So instead of zero here, I have buoyancy contribution. Very importantly, I have given Brunweisela frequency. So stratification, stable stratification is answered by constant uniform Brunweisela frequency and the buoyancy fluctuation, fluctuate, of course. So in this line, I have a toroidal buoyancy flux, a polyoidal buoyancy flux, and BB, scaled as an energy, give a potential energy. Without rotation, with only stratification, the red term are irrelevant, and gravity waves affect all these blocks. Now, if I add rotation, I have this new term, F-tor, the toroidal flux, which generates, which break the mirror symmetry. And as a recent result, Marino et al. have shown that elicity following I'd have shown that elicity is no longer conserved quantities. There is a linear coupling with a correlation of B buoyancy with a vertical vorticity component. And this correlation is completely linked to what I call the toroidal energy flux. So the elicity equation in one point of physical space is like that. And its spectral counterpart includes a new linear term involving the toroidal energy flux. And of course, there is a non-linear term, but with zero integral, which means that when you integrate over k this equation, you recover that, of course. I will not comment this technical point. Just one of the main results of Marino et al. is this interesting relationship, which involves not only the full elicity, but only the elicity linked to the horizontal component of the velocity. But it is only to simplify the equation. And I think they could find a similar result with a complete elicity, because elicity is really a scholar. It's not linked to a particular component, of course. Now, let me continue and finish with M-I-H-D. If now we consider a combination of N and F, we have a vertical stabilizing stratification characterized by N. You have Coriolis force characterized by F. And we have a coupling of H with the toroidal flux of energy. And the following I, the Marino et al, found a nice equivalent of elicity in terms of ratio N over F. But I think the result depends on a bit of non-zero initial either after or either high. If they are strictly zero initially, the linear solution gives no generation. So the generation is probably due to weak initial noise or some weak nonlinearity. If you cancel completely nonlinearity or others, strictly zero elicity and zero toroidal flux, you have nothing. So it was also found by Pierietta, my colleague, who found you control perhaps a bit better the initial data. And then they have no kinematicity. But in Coutre part, they find the illustrate a nice proposal by Gibbon and Holmes to define a strange correlation like that, a new B vector, which is completely equivalent without the same equation as cross elicity in M-H-D. This will be my transition to M-H-D. I will not have no time to discuss that. You will have to look at the original reference because it's complicated. It's the gradient of the potential vorticity cross the gradient of buoyancy. And this is the only PLL of my long career in which you have some discussion of that. I move to M-H-D, and I will finish soon, I think. No, when you have true M-H-D, you have a free kind of elicity, of course. Kinematic elicity, the same for the magnetic elicity. With caveat is that it's not B cross omega. Magnetic B cross G, G in English is the current field, people use more the magnetic vector A, so they construct more A cross B, a scalar product B than G cross B. But there is only a K2 factor between them. And the elicity B rotation, curl of B, can be called supermagnetic elicity. Often people call it super when the spectrum is multiplied by K2. And of course, cross elicity is only the scalar product of U with B. And we can define the spectra and cospectra for all these quantities. Now, just I continue my little game to identify the most important correlator with a minimum number of components taking into account incompressibility, divergence free for both U and B. You have two components of U in Fourier space, two components of B in Fourier space using, for example, the helical modes. So you recover exactly the kinematic block here as before. You have exactly a similar magnetic block here in which M, all these quantities are affected by M as magnetic. And you have a more complicated cross block, cross velocity magnetic field block, which contain cross elicity and also the spectrum of the electromotive force. Over Z term are not completely identified, interpreted just now. But we have an interesting thing to look at. First calculation, DNS with my colleague, also, Faviet Al, with true MHD, but with additional rotation and additional mean magnetic field give very important new result in terms of the LZR number, LZR number, which is close to a Rosby number with omega in the denominator. So that small LZR number means strong rotation. And I will finish with a few result, not new, unfortunately. Here we can see that the recuperation between magnetic and kinetic energy is broken by rotation. So that at strong rotation, the magnetic energy is much smaller than the kinetic energy. About alignment, alignment is something to do with cross elicity, I remember. You can find with PDF that without rotation, U and B are almost aligned. And with increasing rotation, U and B becomes more and more perpendicular. And unfortunately, my own new result with respect to Faviet Al, I will give more, you can read them. If you have not a good ice, you can read this. It's an angular spectrum of cross elicity, purely linear. Now we will have a very rapidly, completely nonlinear result from DNS. And this scale, this is modulated by, in term of VEA, VEA is the alpha velocity, so the mean magnetic field is scaled as a velocity over F, the Coriolis parameter. And it's possible to show that, of course, if you average on the angle between the wave vector and the direction of a mean magnetic field, you have 0. But you have non-zero angle-dependent spectra. I move to my perspective. So sorry for giving you so few new results. It's a study in progress. But very formally, we will derive all the lean equation, equation in spectral space for the angle-dependent spectra for all the stuff I have shown. We have done that for magnetic and kinetic energy. We will do for free energy to look at all the coupling and a possible scenario in the presence of B and twice omega. So in line with the previous DNS, I have not seated Lennert. I didn't introduce also rotating the ones. I keep more thought also, look at rotating MHD. And a scenario with emergence, a successive emergence of a fully city from the one of cross-elicity in addition to previous suggestions because they are not complete calculation by Heide and by Keith Moffat. Our application, it's a toy model because if we are in homogeneous turbulence without boundary, it's a toy model. But we hope to have possible application to geodynamo with additional replication taking into account all the ingredients B, omega, B, F and also N. And perhaps an approximation like quasi-static MHD could be useful for liquid metal in the core of the Earth. Thank you for your attention.